mitrofan choban - romaiprof. mitrofan choban is known in romania as well as in the wide world with...

Academician of the Academy of Sciences of MOLDOVA

MITROFAN CHOBAN

at 70th year anniversary January 5, 2012

A TRIBUTE TO

Mitrofan M. CHOBAN at his 70-th anniversary

The Romanian Society of Applied and Industrial Mathematics - ROMAI isvery honored to celebrate one of its most distinguished members, AcademicianMitrofan M. CHOBAN, at his 70-th anniversary.

Professor Choban was born in the village Copciac, Tighina county, Moldova,on January 5, 1942. He attended the courses of the primary school in his nativevillage. Then, in 1959, he finished his high school studies at the high-school fromvillage Volontiri. In 1960 he enrolled at the Tiraspol State Pedagogical Institute(Moldova). Since 1963 he attended the courses of the Faculty of Mathematicsand Mechanics of the State University ”M. Lomonosov” of Moscow. He brilliantlygraduated this Faculty in 1967. Here he attracted the attention of the famousand exigent teaching staff. He was recommended to Professor A.V. Arhangel’skij,who marked his whole evolution as a mathematician, so that at the beginning ofyear 1967, being yet a student, he published a valuable paper in the prestigiousjournal Sovietic Mathematiceskij Doklady (see Sov. Math. Dokl. 8, 1967, p.603-605). He was privileged to study and work in the prestigious collective ofresearchers in topology, brilliantly dominated by the great personality of A.N.Kolmogorov. A year after his graduation of the faculty he has begun preparinghis Doctor’s Degree - the chosen specialty being Topology - at the State University”M. Lomonosov” of Moscow. He graduated in 1970 the Ph.D. in Mathematicswith the thesis ”Relations between classes of topological spaces”, his adviser beingProfessor A.V. Arhangel’skij. In 1980, he became Dr. Sc. in Mathematics withthe thesis ”Set-valued mappings and their applications” (scientific consultantagain being A.V. Arhangel’skij). In 1995, he was elected corresponding memberof the Academy of Sciences of Moldova. Then, in 2000 he was elected Memberof the Academy of Moldova, the highest forum of Moldavian spirituality and thehighest recognition which a scholar may receive.

Professor Mitrofan Choban started his didactic career at the Tiraspol StateUniversity (Moldova), and continuously worked in this university till now. Hewas in succession senior lecturer at the Department of Geometry and Didacticsof Mathematics (1970-1974), associate professor at the Department of Geome-try and Didactics of Mathematics (1975-1976), the head of the Department ofGeometry and Didactics of Mathematics (1976-1983), scientific secretary of theuniversity (1983-2002), president of the University (2002-2009) and the head of

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the Department of Algebra, Geometry and Topology (2009-present). Since 1981,Professor Mitrofan Choban was adviser for PhD Thesis as well as for Dr. Sc.Thesis. He advised 13 doctors of sciences and 4 Dr. Sc.

His teaching activity concerned the courses of ”Geometry” as well as of ”SetTheory and Topology”. He also taught several special courses: Functional spaces,Topological groups, Algebraic theory of automata, Topological universal algebras.

Due his prestige in the world of Mathematics he became: 1. Member of theEditorial Boards of: - Buletinul Academiei de Stiinte a Moldovei, Matematica,ROMAI Journal, Scientific Annals of Oradea University, Qusigroups and relatedsystems; 2. President of the Mathematical Society of Republic Moldova (1999-present); 3. Vice-President of the Romanian Society of Applied and IndustrialMathematics (ROMAI) (1995-present); 4. Member of the Moscow MathematicalSociety; 5. Member of the Romanian Mathematical Society.

The special appreciation of his scientific work brought him several prizes,titles and orders, namely: prize of the All-Union Presidium of the Scientific-Technical Societies (1968), prize ”Boris Glavan” of the Komsomol of Moldova, inMathematics (1974), title Excellent of the High Education of the USSR (1980),order ”Gloria Muncii” (Glory of Labor) of the Republic of Moldova (2000), StatePrize of the Republic of Moldova (2002), Honorary citizen of the Stefan Vodacounty, Republic of Moldova (2005), prize ”Academician Constantin Sibirschi”(2006), Doctor Honorius Causa of the Oradea University (2006), order ”Honor”of the Republic of Moldova (2010).

He attended more then 70 scientific forums: 1) International mathematicalcongresses (Moscow, Zurich, Berlin), conferences (Moscow, New York, Baku,Sofia, Pitesti, Oradea, Sozopol, Bucuresti, Timisoara, Brasov, Chisinau, Novosi-birsk, Tbilisi, Lecce, Iasi, Constanta, Sicilia, Livov, Varna, Borovets, Ohrid),2) symposiums (Prague, Eger, Burgas, Genova, Marseille), 3) All-Union mathe-matical conferences and symposiums (Minsk, Moscow, Tiraspol, Chisinau, Livov,Sankt-Petersburg, Novosibirsk, Tobolsk, Tartu), 4) several national conferences.

Having a great prestige in the world of Mathematics, Professor MitrofanChoban has been invited to lecture by the well-known institutions: Institute ofMathematics and Informatics of the Academy of Science of Bulgaria, the Universi-ties of Oradea, Tartu, Tbilisi, Tashkent, Tsukuba, Bishkek, North Bay (Canada).

Moreover, he was Invited speaker of the forums: V-th Prague Topol. Symp.(1981), Topological Colloq., Eger, Ungary (1983), International Moscow Topo-logical Conference (1979), Soviet-Japan Topological Symposium, Niigata (1991),Workshop on General Topology and Geometric Topology, Tsukuba (1991), Work-shop ”Well-Posedness in Optimization, Margarita di Liguri, Italy (1991), In-ternational Conference on group theory, Timisoara, Romania (1991), Workshop”Well-Posedness in Stability and Optimization, Sozopol, Bulgaria (1993), Con-ferences on Applied and Industrial Mathematics, Romania (1994-2010), Inter-national Congress of Mathematical Society of South Europe, Borovets, Bulgaria(2003), International Conference ”Geometric Topology, Discrete Geometry and

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Set Theory” in celebration of the centennial of Ljumila V.Keldysh, Moscow(2004), International Conference ”Quality in Formal and non Formal Education”,Iasi, Romania (2010), Centennial Conference ”Alexandru Myller” MathematicalSeminar”, Iasi, Romania (2010), ICTA Islamabad, Pakistan (2011), CAIM - Con-ferences in Applied and Industrial Mathematics (1993-2011), 8-th InternationalConference on Applied Mathematics, Baia Mare, Romania (2011), etc..

Professor Choban authored more than 200 papers and 20 books in manybranches of Mathematics. He brought important contributions in: Hausdorff’sproblem on Borelian classes of sets; Alexandroff’s problem about the struc-ture of compact subsets of countable pseudocharacter in topological groups;Arhangel’skii’s problem on the zero-dimensional representation of topologicaluniversal algebras; two Maltsev’s problems on free topological universal alge-bras; two Michael’s problems about G -sections of open mappings of compactspaces and of the k-coverings of open compact mappings of paracompact spaces;Phelps’ problem about the structure of the set of points of Gateaux differentiabil-ity of convex functionals (with P.Kenderov and J.Revalski); Tichonoff’s problemabout well-possedness of optimization problems in the Banach spaces of continu-ous functions (with P.Kenderov and J.Revalski); Confort’s problem about Baireisomorphism of compact groups; Pasynkov’s problem about Raikov completionof topological groups; Arhangel’skii’s problem on metrizability of o-metrizabletopological groups (with S.Nedev); Pelczinski’s and Semadeni’s problems aboutstructure of Banach spaces of continuous functions on special compact subsets ofquotient spaces of topological groups.

To all these we must add the contribution brought to the activity of ROMAI,Prof. Mitrofan Choban being involved, as Vice-President, in all the aspects ofthis activity, from the organization of conferences CAIM, to the editorial activityfor ROMAI Journal. If ROMAI succeeded in existing for 20 years, in better orworse conditions from the Romanian institutional framework, this is due in agood measure to Prof. Mitrofan Choban.

He shared Professor‘s Adelina Georgescu ideas concerning the necessity offinding ways of collaboration between mathematicians from both sides of Prut,and continuously activated in this sense. The friendship and appreciation thatProfessors Adelina Georgescu and Mitrofan Chobanu had one for the other werethe germs for friendship relations between many Romanian and Moldavian math-ematicians. We hope that the tree of these relations will grow with vigor in theyears to come.

Prof. Mitrofan Choban is known in Romania as well as in the wide worldwith his results and his monographs in topology, topological algebraic systemsand topological algebras.

We wish to the important and the distinguished mathematician Mitrofan M.CHOBAN many happy years in good health and keep up the good work.

ROMAI DIRECTORY COMMITTEE

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SOME SELECTED PUBLICATIONS OF

Mitrofan CHOBAN

1. M.M. Choban, Open finite-to-one mappings, Soviet. Math. Dokl. 8 (1967),603-605.2. M.M. Choban, Continuous images of complete spaces, Trans. Moscow Math.Soc. (1974), 25 - 63.3. M.M. Choban, On Baire isomorphisms and Baire topologies. Solution of aproblem of Comfort, Soviet. Math. Dokl. 30:3 (1984), 780 - 784.4. M.M. Choban, On the theory of topological algebraic systems, Transl. Amer.Math. Soc. 48 (1986), 115-159.5. M.M. Choban, The theory of stable metrics, Math. Balkanika, 2:4 (1988),357-373.6. A.V. Arhangel’skii and M.M. Choban, Extenders of Kuratovski-van Douwenand classes of spaces, Comptes Rendus Acad. Bulgare Sci. 45 (1992), 5-7.7. M.M. Choban, The structure of locally compact algebras, Serdica. 18 (1992),129-137.8. M.M. Choban, Descriptive Set Theory and Topology, Enciclopedia of Math.Sciences, v. 51, Springer, 1995, 157 - 219.9. M.M. Choban, P. Kenderov and J. Revalski, Generic well-posedness of opti-mization problems in topological spaces, Mathematika, 36 (1989), 301 - 324.10. M.M. Choban, The open mappings and spaces - Suplim. Rendicanti CircoloMatem. di Palermo, 29 (1992), 51 - 104.11. M.M. Choban, Some topics in topological algebra - Topology and its Appli-cations, 54 (1993), 183 - 202.12. M.M. Choban, P. Kenderov and J.Revalski, Densely defined selections ofmultivalued mappings, Trans. Amer. Math. Soc., 344 (1994), 533 - 552.13. M.M. Choban, P. Kenderov and J. Revalski, Caracterizations of topologi-cal spaces with almost completeness properties, Seminare d’initiation a analise1993/94 (G. Choquef, G. Godefroy, M. Rogalski, eds), 1995, 3-9.14. M.M. Choban, Well possedness of optimization problems and measurablefunctions, Math. Balkanica, 10 (1996), 211 - 236.15. M.M. Choban, S.I. Nedev, Paracompact extentions and orderability of spaces,Buletinul Acad. de Stiinte a RM, Matematica, 3 (1996), 43-51.

16. A.V. Arhangel’skii and M.M. Choban, On continuous mappings of Cp-spaces and extenders, Proceed. Steclov. Institute of Math. 211 (1996), 28-31.17. M. M. Choban, Well possedness of optimization problems and measurablefunctions, Math. Balkanika, 10, 2-3, 211-236.18. M.M. Choban, P. Kenderov and J.Revalski, Densely defined selections ofopen mappings, Comp.Rend. Acad. Sc. Bulgar., 50, no 9 - 10 (1997), 5 - 8.19. M.M. Choban, Isomorphism problems for the Baire function spaces of topo-logical spaces, Serdica Math. J. 24 (1998), 5-20.

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20. M.M. Choban, General theorems on functional equivalence of topologicalspaces, Topology and its Appl. 89 (1998), 223-239.21. M.M. Choban, Isomorphism of function spaces, Mathematica Balkanica, 12(1998), 59 - 91.22. M.M. Choban, S.I. Nedev, Continuons selections for mappings with general-ized ordered domain , Math. Balcanica, 11, 1-2 (1997), 87-95.23. M.M. Choban, On topologically complementable spaces, Buletinul Stiintific alUniv. din Pitesti, Matematica si Informatica, 2 (1998), 45 - 50.24. M.M. Choban and L. Calmutschi, On universal propierties, Analele Univer-sitatii din Oradea, Fascicola Matematica tom VII, 1999-2000, 73-78.25. M. M. Cioban, Functional equivalence of topological spaces, Topology and itsapplications, 111 (2001), 105-134.26. M. M. Cioban, Mappings and Prohorov spaces, Topology and its applications,153 (2006), 2320-2350.27. M. M. Cioban, D. Pavel, Almost periodic functions on quasigroups, AnaleleUniversitatii din Oradea. Fascicola Matematica, XIII (2006), 99-124.28. M. Choban and L. Kiriyak, The homomorphisms of fuzzy algebras, AnaleleUniversitatii din Oradea, Fascicola Matematica 8 (2001), 131 - 138.29.M. Cioban, Spaces, mappings and compact subsets, Buletinul A.S.R.M. Matem-atica, 2 (2001), 3-52.30. M. Cioban, Algebrical equivalence of topological spaces, Buletinul A.S.R.M.Matematica, Nr.1 (43) 2001,31. M. Cioban, L. Kiriyak, Decomposition of some groupoids with topologies andits resolvability, Buletinul A.S.R.M. Matematica, Nr.3 (43) 2001, p. 27 - 37.32. M. Cioban, L. Kiriyak, The topological quasigroups with multiple identities,Quasigroups and Related Systems, 8 (2002), 1 - 15.33. M. Choban, L. Kiriyak, Compact subsets of free algebras with topologies andequivalence of spaces, Hadronic Journal 25 (2002), 609 - 631.34. M. Cioban, N. Dodon, Bijections onto scattered hereditarily paracompactcompact spaces, Buletinul A.S.R.M. Matematica, 3 (2002), 37 - 45.35. M. Cioban, N. Dodon, Some factorization theorems for scattered spaces,Buletinul A.S.R.M. Matematica, 2, (2002), 88 - 99.36. M.M. Choban, On some problems of descriptive set theory in topologicalspaces, Russian Math. Surveys 60:4 (2005), 699-719.37. M. Choban, P. Kenderov and S. Nedev, Professor Alexander Arhangel’skiiand his Topological Galaxies, Matematica Balkanica, 22:1-2(2008), 1-9.38. M.M. Choban, Reduction principles in the theory of selections, Topology andits applications 155(2008), 787-796.

39. M.M. Choban, E.P. Mihaylova and S.I. Nedev, On selections and classesof spaces, Topology and its applications 155 (2008), 797-804.40. M. Cioban and A.-V. Ion, Homage to Professor Adelina Georgescu at the ageof 65, ROMAI Journal, 3:1 (2007), 1-8.

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41. M. Choban and I. Ciobanu, Compactness and free topological algebras, RO-MAI Journal, 3:2 (2007), 55- 85.42. L. Calmutchi, M. Cioban, Some problems of the theory of compactificationsof topological spaces, ROMAI Journal, 2: 1 (2006), 25-51.43. A.V. Arhangel’skii and M.M. Choban, Semitopological Groups and the The-orems of Motgomery and Ellis, Comptes Rendus de l’Academie Bulgare des Sci-ences, 62:8, 2009,917-922.44. M. Choban and L. Calmutchi, Algebras of continuous functions and compact-ifications of spaces, ROMAI Journal, 4:1, 2008, 61-67.45. M. Cioban , Academician Vladimir Arnautov, Buletinul Academiei de Stiintea Moldovei, Matematica 3 (2009), 130 - 136.46. A.V. Arhangel’skii and M.M. Choban, Completeness type properties of semi-topological groups and the theorems of Montgomery and Ellis, Topology Proceed-ings. 2011, 37, (E-published on April 29, 2010).47. A.V. Arhangel’skii, M.M. Choban and P.S. Kenderov, Topological games andcontinuity of group operations, Topol. Appl. 157 (2010), 2542-2552.48. A.V. Arhangel’ski i and M.M. Choban, Remainders of rectifiable spaces,Topology and its Applications, 157 (2010), 789-799.49. M.M. Choban and M.I. Ursul , Applications of the Stone Duality in the theoryof precompact Boolean rings, In: D. V.Huzanh and S.R.Lopez-Permouth (Eds),Advances in Ring Theory, Birkhauser, New York, 2010, 85-111.50. A.V. Arhangel’skii and M.M. Choban, Some addition theorems for rectifiablespaces, Bulletin of Academy of Sciences of the Republic of Moldova. Mathematics,2011, 67 (2).51. A.V. Arhangel’ski i and M.M. Choban, Some generalizations of the conceptof the p-space, Topology and i ts Applications, 2011, 158, 1381-1389.52. M.M. Choban and L. Chiriac, Automatons and Topological Algebras, Proceed-ings IIS, International Workshop on Intelligent Information Systems, September13-14, 2011, Chisinau, Proceedings IIS, Institute of Mathematics and ComputerScience, Chisinau, 2011, 147-150.53. A.V. Arhangel’ski i and M.M. Choban, Some generalizations of the conceptof the p-space, Topology and i ts Applications, 2011, 158, 1381-1389

FUZZY INCLUSION AND DESIGN OFMEASURE OF FUZZY INCLUSION

ROMAI J., v.8, no.1(2012), 1–15

Ismat BegDepartment of Mathematics and Statistics, Faculty of Management Studies,University of Central Punjab, Lahore, [email protected]

Abstract Fuzzy inclusion between fuzzy subsets of a crisp universe is defined as a binary opera-tion on the set of all fuzzy subsets of a universe of discourse X. The fuzzy set definedas a fuzzy set of inclusion is then converted into a degree of inclusion with the help ofa suitable measure. It is shown that the pointwise character of fuzzy inclusion allowsmany interesting properties to hold. Furthermore, the technique of applying a fuzzymeasure to a fuzzy set of inclusion is used to construct mappings which provide degreesof inclusion.

Keywords: fuzzy inclusion; measure of fuzzy inclusion.2010 MSC: 03E72, 46S40, 68T37.

1. INTRODUCTIONFuzzy subsethood or fuzzy inclusion is an important concept in the field of fuzzy

set theory and it provides a basis for fuzzy similarity and measures of similarity (see[18] and [25]). First attempt to define fuzzy subsethood was made by Zadeh [26]; thegraphical interpretation of his definition is: he defines the fuzzy set A as a subset ofthe fuzzy set B, if the graph of A never goes above the graph of B. Later on it wasrealized that defining fuzzy subsethood in this way is though highly appreciable anduseful but is still against the spirit of fuzzy set theory, in the sense that it represents acrisp decision about being a subset or not (for details see [1] and [14]). Researchersworking in the area of fuzzy inclusion remained interested in assigning a degree ofinclusion of one fuzzy set into another (for details see [2], [4], [5], [6], [7], [8],[10], [13], [14], [19] and [24]). Many measures of inclusion have been proposed inliterature; most of these take values in [0, 1] or sometimes in the Boolean lattices L.Many researchers formulated axioms of inclusion i.e., they provide a list of propertieswhich a reasonable inclusion measure must satisfy. A significant contribution inthis direction is by Sinha and Dougherty [22]; they list properties that a reasonableinclusion measure should possess. Cornelis [10] later proved that the scalar definedby Bandler and Kohout [1] as inf

x∈XI(A(x), B(x) is a fuzzy inclusion satisfying all the

axioms of fuzzy inclusion constructed by Sinha and Dougherty.

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2 Ismat Beg

In this paper, a different approach towards fuzzy inclusion is adopted. Basic intuitionis that the inclusion defined should capture both the features of amount of overlap andorientation of one fuzzy set with respect to the other fuzzy set. This is in accordancewith the spirit of fuzzy set theory. The properties possessed by such an inclusionare observed to be similar to the ones stated as axioms of inclusion by Sinha andDougherty [10]. In our work, we use the Lukasiewicz’s implicator IW along with itscorresponding t-norm and t-conorm. The only exception to this setting is Proposition3.7 where max and min are used along with Lukasiewicz’s implicator as per require-ment of Sinha and Dougerty axioms of inclusion. The implicator IW selects betweentwo values depending upon the region of its domain. If a fuzzy set A is subset of an-other fuzzy set B in Zadeh’s sense, the Lukasiewicz’s implicator assigns to it a degreeof inclusion equal to 1 at each point of its domain. When measure is applied to sucha fuzzy set of inclusion it assigns a degree of inclusion of the fuzzy set A into B to beequal to 1. Intuitively speaking, it partially agrees with the idea of Zadeh. It differsfrom Zadeh’s inclusion in the fact that Zadeh allocates a degree zero to mInc(A, B)(see Definition 4.2) if A takes a value greater than B even at a single point of the do-main, while mInc(A, B) calculated through Lukasiewicz’s implicator would take intoaccount all the other 1s and would not assign a value equal to zero abruptly at thepoint where A(x) ≤ B(x) is violated. Measure mInc is much closer to reality than allthe previously defined measures of inclusion. The important part of the theory is thatthe fuzzy set of inclusion provides with a permanent basis for inclusion theories andmeasures of inclusion. Every time one selects an appropriate fuzzy measure by it’sapplication to the fuzzy set of inclusion, one gets a new example of measure of fuzzyinclusion.

2. PRELIMINARIESDefinition 2.1. [26, Sec II, p.339] Let F(X) be the set of all fuzzy subsets of a universeX. For all A, B ∈ F(X), A is said to be a subset of B if for all x ∈ X, A(x) ≤ B(x),where A(x) and B(x) represent the membership grades of x in A and B respectively.In this case, we write A ⊆ B and call it the Zadeh’s inclusion. Two fuzzy sets A andB are said to be equal if and only if A(x) = B(x) for all x ∈ X.

Fuzzy sets X and ∅ will be denoted by 1 and 0 respectively. This notation has beenborrowed from Boolean lattices in order to remain specific about the maximal andminimal elements of the range of the newly introduced mappings in this paper. Thet-norms and t-conorms will be used for the pointwise conjunction and disjunction offuzzy sets respectively.

Definition 2.2. [20,Definition 1.13 p. 11] The triangular norm (t-norm) T and trian-gular conorm (t-conorm) δ are increasing, associative, commutative and [0, 1]2 →[0, 1] mappings satisfying: T (1, a) = a and δ(a, 0) = a for all a ∈ [0, 1].

Fuzzy inclusion and design of measure of fuzzy inclusion 3

In the following there is a list [17, Sec 3 p 219] of some choices for t-norms and theirdual t-conorms:1. The min and max operators M(x, y) = min(x, y), and M∗(x, y) = max(x, y).2. The product and probabilistic sum P(x, y) = xy, and P∗(x, y) = x + y − xy.3. The Lukasiewicz’s pair

W(x, y) = max(x + y − 1, 0), and W∗(x, y) = min(x + y, 1).In this paper the notation T (A, B) shall represent a fuzzy set defined as: T (A, B)(x) =

T (A(x), B(x)) for any t-norm T and for all x ∈ X.

Definition 2.3. [14, Sec2.2, Definition 1 p. 3] A negator N is an order-reversing[0, 1] → [0, 1] mapping such that N(0) = 1 and N(1) = 0. A negator is called strictif it is continuous and strictly decreasing. A strict negator is said to be strong if it isinvolutive too i.e., N(N(x)) = x for all x ∈ [0, 1].

The negators are used to model pointwise complements in the literature of fuzzysets. Throughout this paper Ac, the complement of a fuzzy set A will be calculatedby the standard negator i.e., Ac(x) = N(x) = 1 − x.

Next we give an introduction to the fuzzy implicator which is the back bone of thetheory on fuzzy set of similarity.

Definition 2.4. [11,Sec III (C) (2), p.277] A fuzzy implicator is a binary operation on[0, 1] with order reversing first partial mappings and order preserving second partialmappings such that:

I(0, 1) = I(0, 0) = I(1, 1) = 1, I(1, 0) = 0.

Remark 2.1. [9, Section2, p.213, Definition 1] From an axiomatic point of view thefollowing properties are very important for fuzzy implicators I:

A1. Contraposition: (∀x, y ∈ [0, 1]), I(x, y) = I(N(y),N(x));A2. Exchange Principle: (∀x, y, z ∈ [0, 1]), I(x, I(y, z)) = I(y, I(x, z));A3. Hybrid Monotonicity: stated in definition 2.4;A4. Ordering Principle: x ≤ y⇐⇒ I(x, y) = 1 for all x, y ∈ [0, 1];A5. Neutrality Principle: I(1, x) = x for all x ∈ [0, 1];A6. Continuity: I is continuous.Where N is a strong negator.

Definition 2.5. [14, Sec 2.4.1, Definition 7] For a t-norm T, the R-implicator (orresiduated implicator) IT is defined as:

IT (a, b) = supx ∈ [0, 1] | T (a, x) ≤ b ∀a, b ∈ [0, 1].

Definition 2.6. [1,De f inition 2.1 (5)] The Lukasiewicz implicator (IW) is defined as:

IW(x, y) = min(1 − x + y, 1) for all x, y ∈ [0, 1].

4 Ismat Beg

The Lukasiewicz implicator possesses all the properties from A1 to A6, Moreoverit is an R-implicator with respect to the t-norm W and satisfies following property ofR-implicators [21, Sec 2, p 3]:

W(IW(x, y), IW(y, z)) ≤ IW(x, z). (1)

Definition 2.7. [12, Sec 1.3.2 ] The scalar cardinality of a fuzzy subset A of a finiteuniverse X, is defined as:

|A| =∑

x∈X

A(x).

Definition 2.8. [16, Definition 2.2] Given a t-norm T, a T-equivalence relation on aset X is a fuzzy relation E on X that satisfies:

(i) E(x, x) = 1 for all x ∈ X. (Reflexivity)(ii) E(x, y) = E(y, x) for all x, y ∈ X. (Symmetry)(iii) T (E(x, y), E(y, z)) ≤ E(x, z) for all x, y, z ∈ X. (T−transitivity)

Definition 2.9. [10, Definition 2.1]Let Inc be a F(X) × F(X)→ [0, 1] mapping, andA, B,C ∈ F(X) be fuzzy sets in a given universe X. Sinha-Dougherty axioms imposedon Inc for it to be an inclusion operator are as follows:Axiom 1. Inc(A, B) = 1 if and only if A ⊆ B (in Zadeh’s sense).Axiom 2. Inc(A, B) = 0 if and only if Ker(A) ∩ (supp(B)c) , ∅, where ker(A = x ∈X | A(x) = 1, supp(B) = x ∈ X | B(x) > 0.Axiom 3. B ⊆ C implies Inc(A, B) ≤ Inc(A,C), i.e., Inc has increasing second partialmappings.Axiom 4. B ⊆ C implies Inc(B, A) ≥ Inc(C, A), i.e., Inc has decreasing first partialmappings.Axiom 5. Inc(A, B) = Inc(S (A), S (B)) where S is a F(X) → F(X) mapping definedby, for every x ∈ X, S (A)(x) = A(s(x)), s denoting an X → X mapping.Axiom 6. Inc(A, B) = Inc(Bc, Ac).Axiom 7. Inc(B ∪C, A) = min(Inc(B, A), Inc(C, A)).Axiom 8. Inc(A, B ∩C) = min(Inc(A, B), (A,C)).

Definition 2.10. [3] Let R be a fuzzy relation on X, I any fuzzy implicator and T at-norm, then the fuzzy transitivity relation trI,T

R is a fuzzy relation on X defined as:

trI,TR (x, z) = inf

y∈XI(T (R(x, y),R(y, z)),R(x, z)).

The transitivity function so defined, assigns a degree of transitivity to the givenfuzzy relation at each point of X × X.

Definition 2.11. [3] For a given fuzzy relation R, the measure of transitivity of R isgiven by:

mTr(R) = m(trI,TR );


where, m is Sugeno’s measure (see [23]). In this paper plinth of a fuzzy set will betaken as the measure which is defined as: Plinth(A) = inf

x∈XA(x), while only IW and W

will be used in place of I and T respectively, consequently

mTr(R) = infx,y,z∈X

(IW(W(R(x, y),R(y, z)),R(x, z))).

For this specific choice we shall write tr instead of trI,TR and Tr instead of mTr. If

Tr(R) = ε, then R is called an ε-fuzzy transitive relation. A fuzzy relation R is calledfuzzy transitive if Tr(R) > 0, strong fuzzy transitive if ε ≥ 0.5 and R is nontransitiveif Tr(R) = 0.

Definition 2.12. [3] A binary operation O on F(X) is an ε−local fuzzy order on F(X),if for all x ∈ X and A, B,C ∈ F(X), following conditions hold :

E1. fuzzy Reflexivity at x: if and only if O(A, A)(x) = 1.E2. antisymmetry at x: O(A, B)(x) = O(B, A)(x) > 0⇒ A(x) = B(x).E3. ε−fuzzy transitivity at x:

IM(min(O(A, B)(x),O(B,C)(x)),O(A,C)(x)) = ε.If ε > 0, then O is called fuzzy transitive at x. If ε ∈ [0.5, 1], then O will be calledstrong fuzzy transitive at x and it is called weak fuzzy transitive at x, otherwise.

Definition 2.13. [3] A fuzzy relation R on F(X) is called a fuzzy order relation onF(X) if for all A, B,C ∈ F(X) :

(i) R(A, A) = 1; (fuzzy reflexivity)(ii) R(A, B) = R(B, A)⇒ A = B; (fuzzy antisymmetry)(iii) IW(W(R(A, B),R(B,C)),R(A,C)) = ε > 0. ( fuzzy transitivity)

If ε ∈ [0.5, 1], then R will be called Strong fuzzy transitive order relation. A fuzzyrelation satisfying (i) and (iii) only is called a fuzzy quasi order.

3. FUZZY INCLUSION UNDER IW

Definition 3.1. The fuzzy inclusion is a mapping Inc : F(X) × F(X) → F(X), whichassigns to every A, B ∈ F(X) a fuzzy set Inc(A, B) ∈ F(X) defined as:

Inc(A, B)(x) = I(A(x), B(x)) for all x ∈ X. (2)

We will only use IW in this paper for the study of properties of this inclusion.

Proposition 3.1. For any A, B ∈ F(X), for any x ∈ X,

Inc(A, B)(x) = Inc(B, A)(x) if and only if A(x) = B(x).

Proof. For any A, B ∈ F(X) and x ∈ X,Inc(A, B)(x) = Inc(B, A)(x).⇐⇒ IW(A(x), B(x)) = IW(B(x), A(x)).⇐⇒ 1 = 1 − B(x) + A(x) (without loss of generality, let A(x) ≤ B(x)).

6 Ismat Beg

⇐⇒ A(x) = B(x).

Theorem 3.1. The fuzzy inclusion defined in Definition 3.1 is a local fuzzy order onF(X) which is 1−fuzzy transitive.

Proof. . For all A, B,C ∈ F(X).E1. local fuzzy reflexivity: For any x ∈ X, Inc(A, A)(x) = IW(A(x), A(x)) = 1 bydefinition of IW . So, Inc(A, A) = X.E2. local fuzzy antisymmetry: Follows from Proposition 3.1.E3. Strong local fuzzy transitivity: Let x ∈ X,

IW(W(Inc(A, B)(x), Inc(B,C)(x)), Inc(A,C)(x)))= IW(W(IW(A(x), B(x)), IW(B(x),C(x)), IW(A(x),C(x))). (3)

By the property stated in (1)

W(IW(A(x), B(x)), IW(B(x),C(x))) ≤ IW(A(x),C(x)). (4)

using (4) in (3), we get for all x ∈ X,

IW(W(Inc(A, B)(x), Inc(B,C)(x)), Inc(A,C)(x)) = 1.

Hence Inc is 1−fuzzy transitive fuzzy local order.

Proposition 3.2. For all A, B ∈ F(X) following hold:1. Inc(A, B) = X ⇐⇒ A(x) ≤ B(x) for all x ∈ X.2. Inc(A, B) = ∅⇐⇒ A = X and B = ∅.3. Inc(A, Ac) = ∅⇐⇒ A = X.

Proof. For all A, B ∈ F(X),1. Let Inc(A, B) = X ⇐⇒ for all x ∈ X, IW(A(x), B(x)) = 1 ⇐⇒ A(x) ≤ B(x) for allx ∈ X by definition of IW .2. Let Inc(A, B) = ∅⇐⇒for all x ∈ X, IW(A(x), B(x)) = 0⇐⇒for all x ∈ X, 1 − A(x) + B(x) = 0⇐⇒for all x ∈ X, A(x) = 1 and B(x) = 0.3. 2 implies 3.

Proposition 3.3. For all A, B ∈ F(X), we have:

Inc(A, B) = Inc(Bc, Ac).

Proof. For any x ∈ X, there arise two cases:1. A(x) ≤ B(x) : It implies that Bc(x) ≤ Ac(x). Therefore

Inc(Bc, Ac)(x) = IW(Bc(x), Ac(x)) = 1= IW(A(x), B(x)) = Inc(A, B)(x).


2. A(x) > B(x) : It implies that Bc(x) > Ac(x). Thus

Inc(Bc, Ac)(x) = IW(Bc(x), Ac(x)) = min(1, 1 − Bc(x) + Ac(x))= B(x) + 1 − A(x) = 1 − A(x) + B(x)= IW(A(x), B(x)) = Inc(A, B)(x).

Proposition 3.4. For all A, B ∈ F(X),

W∗(Inc(A, B), Inc(B, A)) = X.

Proof. If A, B ∈ F(X), then we have for any x ∈ X, without loss of generality assumethat A(x) ≤ B(x),

W∗(Inc(A, B), Inc(B, A))(x) = W∗(Inc(A, B)(x), Inc(B, A)(x))= W∗(IW(A(x), B(x)), IW(B(x), A(x)))= W∗((1), (1 − B(x) + A(x))= 1 by definition of W∗.

HenceW∗(Inc(A, B), Inc(B, A)) = X.

Proposition 3.5. For all A, B,C ∈ F(X),1. B ⊆ C =⇒ Inc(A, B) ⊆ Inc(A,C) i.e., the fuzzy inclusion is increasing in secondvariable.2. B ⊆ C =⇒ Inc(C, A) ⊆ Inc(B, A) i.e., the fuzzy inclusion is decreasing in firstvariable.

Proof. For all A, B,C ∈ F(X), there arise following cases:1. If B ⊆ C, then B(x) ≤ C(x) for all x ∈ X. Using definition of IW for any x ∈ X,so, Inc(A, B)(x) = IW(A(x), B(x)) ≤ IW(A(x),C(x)) due to the fact that IW orderpreserving second partial mappings.2. Again B ⊆ C implies that B(x) ≤ C(x) for all x ∈ X. Using definition of IW forany x ∈ X, Inc(C, A)(x) = IW(C(x), A(x)) ≤ IW(B(x), A(x)) due to the fact that IW hasorder reversing first partial mappings.

Proposition 3.6. For all A, B,C,D ∈ F(X), we have

W(Inc(A, B), Inc(C,D))⊆ W[Inc(W(A,C),W(B,D)), Inc(W∗(A,C),W∗(B,D))] (5)

⊆ W∗[Inc(W(A,C),W(B,D)), Inc(W∗(A,C),W∗(B,D))] (6)

8 Ismat Beg

⊆ W∗[Inc(A, B), Inc(C,D)]. (7)

Proof. . For any x ∈ X, there arise following possibilities:Case I. A(x) ≤ B(x) and C(x) ≤ D(x),

W(Inc(A, B)(x), Inc(C,D)(x)) = W(IW(A(x), B(x)), IW(C(x),D(x)))= 1 ∧ 1 = 1 by assumption. (8)

Similarly, the assumption also implies that for this particular x,W(A(x),C(x)) ≤ W(B(x),D(x)) and W∗(A(x),C(x)) ≤ W∗(B(x),D(x)).

W(Inc(W(A,C),W(B,D)), Inc(W∗(A,C),W∗(B,D)))(x)= W(IW(W(A(x),C(x)),W(B(x),D(x))),

IW(W∗(A(x),C(x)),W∗(B(x),D(x))))= W(1, 1) = 1 by assumption and definition of IW .

(9)

Similar argument implies that for the x selected in case 1,

W∗(Inc(W(A,C)(x),W(B,D)(x)), Inc(W∗(A,C),W∗(B,D)(x)) = 1, (10)

andW∗(Inc(A, B), Inc(C,D))(x) = 1. (11)

From equations (8), (9), (10) and (11), we obtain the result.Case II. x ∈ X, is such that A(x) > B(x) and C(x) > D(x)

W(Inc(A, B)(x), Inc(C,D)(x)) = W(IW(A(x), B(x)), IW(C(x),D(x)))= W(1 − A(x) + B(x), 1 −C(x) + D(x))= max(0, 1 − (A(x) + C(x)) + B(x) + D(x)),

(12)and

W(Inc(W(A,C),W(B,D)), Inc(W∗(A,C),W∗(B,D)))(x)= W(IW(W(A(x),C(x)),W(B(x),D(x))),

IW(W∗(A(x),C(x)),W∗(B(x),D(x))))= W(1 −W(A(x),C(x)) + W(B(x),D(x)), 1 −W∗(A(x),C(x))

+W∗(B(x),D(x)))= max(0, 1 − (W(A(x),C(x)) + W∗(A(x),C(x)))+

W(B(x),D(x)) + W∗(B(x),D(x)))= max(0, 1 − (A(x) + C(x)) + B(x) + D(x)).

(13)

We used the fact W(x, y) + W∗(x, y) = x + y in reaching at the last expression in (13).Lastly, Comparing (12), (13) the equality is again obtained.Case III. For any x ∈ X, A(x) ≤ B(x) and C(x) > D(x). So,

W(Inc(A, B), Inc(C,D))(x) = W(IW(A(x), B(x)), IW(C(x),D(x)))


= W(1, 1 −C(x) + D(x))

= 1 −C(x) + D(x) (14)

There arise many different cases for the right side of (5), without loss of generalitywe state the proof only for the situation when W(B(x),D(x)) < W(A(x),C(x)) andW∗(B(x),D(x)) < W∗(A(x),C(x)), all the other situations can be handled similarly.

W(Inc(W(A,C),W(B,D)), Inc(W∗(A,C),W∗(B,D)))(x)= W(IW(W(A(x),C(x)),W(B(x),D(x))), IW(W∗(A(x),C(x)),W∗(B(x),D(x))))= 1 − (A(x) + C(x)) + B(x) + D(x)= 1 −C(x) + D(x) + (B(x) − A(x))≥ 1 −C(x) + D(x).

(15)The proof of (5) is complete.Case IV can be proved on more or less the same lines as Case III so we omit theproof.The inequality (6) is straightforward outcome of results about conjunction and dis-junction respectively, the last inequality (7) can be proved on very same lines as(5).

Proposition 3.7. For all A, B,C ∈ F(X) we have:1. Inc(max(A, B),C) = min(Inc(A,C), Inc(B,C)).2. Inc(min(A, B),C) = max(Inc(A,C), Inc(B,C)).3. Inc(A,min(B,C)) = min(Inc(A, B), Inc(A,C)).4. Inc(A,max(B,C)) = max(Inc(A, B), Inc(A,C)).

Proof. 1. For all A, B,C ∈ F(X) and x ∈ X,

min(Inc(A,C)(x), Inc(B,C)(x))

= min(IW(A(x),C(x)), IW(B(x),C(x)))

= min(min(1 − A(x) + C(x), 1), min(1 − B(x) + C(x), 1)). (16)

Similarly,

Inc(max(A, B),C)(x) = IW(max(A(x), B(x)),C(x))

= min(1 −max(A(x), B(x)) + C(x), 1)

= min(min(1 − A(x), 1 − B(x)) + C(x), 1)

= min(min(1 − A(x) + C(x), 1 − B(x) + C(x), 1) (17)

Comparing equations (16) and (17) we get the result.2. It can proved on the similar lines as 1.3. For all A, B,C ∈ F(X) and for any x ∈ X, without loss of

10 Ismat Beg

generality, suppose B(x) ≤ C(x),

min(Inc(A, B), Inc(A,C)(x)) = min(IW(A(x), B(x)), IW(A(x),C(x)))

= min(1 − A(x) + B(x), 1 − A(x) + C(x))

= 1 − A(x) + B(x). (18)

Next we consider the other side

Inc(A,min(B,C))(x) = IW(A(x) + min(B(x),C(x)))= min(1 − A(x) + min(B(x),C(x), 1)= min(1 − A(x) + B(x), 1)

according to the selection made in eq (18).

4. Proof is similar to part 3.

Proposition 3.8. For all A, B,C ∈ F(X) we have

Inc(A, B) ≤ W(Inc(W(A,C),W(B,C)), Inc(W∗(A,C),W∗(B,C))).

Proof. This inequality can be proved only by putting D = C in the first part ofinequalities of Proposition 3.6.

4. MEASURE OF FUZZY INCLUSIONFuzzy inclusion as defined in Section 3 is a fuzzy set and defines inclusion in a verygeneral setting. The advantage of this approach is that it provides a basis for all scalarvalued inclusions. Several researchers have tried to give a degree of inclusion of onefuzzy set into another. In this section we aim at refining this approach with the helpof fuzzy measures to the fuzzy set of inclusions. We observe that all the previouslydefined measures of inclusion are examples of this newly defined concept. Two typesof observations are highly important in dealing with measures of inclusion insteadof fuzzy inclusion. The measure of fuzzy inclusion of a set loses some propertiessatisfied by the fuzzy set of inclusion itself particularly antisymmetry. If we define thefuzzy set of inclusion and local fuzzy order in the above fashion then we successfullyorder the given fuzzy sets by fuzzy inclusion. But if we attempt to order the givenfuzzy sets with the help of measure of fuzzy inclusion then we can at the most obtaina quasi order. Moreover the scalar associated with fuzzy inclusion has advantages. Ithas been proved about certain measures of inclusion that they satisfy all the axiomsof Sinha and Dougherty [22].

Definition 4.1. [15,Definition 2.7] Let (X, ρ) be a measurable space. A functionm : ρ→ [0,∞[ is a fuzzy measure if it satisfies the following properties:m1: m(∅) = 0, and m(X) = 1; m2: A ⊆ B implies that m(A) ≤ m(B).


The concept of measure considers that ρ ⊆ 0, 1X , but this consideration can beextended to a set of fuzzy subsets = of X, = ⊆ F(X), satisfying the properties ofmeasurable space (F(X),=).Remark 4.1. Some times it is beneficial to require following property from the mea-sure m: For any A ∈ F(X), m(A) = 1 implies that A = X and m(A) = 0 implies thatA = ∅.

Definition 4.2. A fuzzy measure of inclusion is a mapping mInc:F(X) × F(X) −→[0, 1], which allocates to all A,B ∈ F(X) a value in the interval [0, 1] defined as:

mInc(A, B) = m(Inc(A, B)),

where Inc(A, B) is the fuzzy set of inclusion of A into B defined in (2), and m is a fuzzymeasure defined in Definition 4.1.

Examples 4.1. Here are some examples of the measure m: for all A ∈ F(X),1a. m1(A) = Plinth(A) = inf

x∈XA(x);

2a. m2(A) = 12 [Plinth(A) + Height(A)] = 1

2 [infx∈X

A(x) + supx∈X

A(x)];

3a. In case of finite universes,

m3(A) =|A||X| ;

4a. In case of bounded universes equipped with a measure m,

m4(A) =

∫A(x)dm

m(X).

Applying these measures on the fuzzy set of inclusion defined in (2), we get:1b. m1Inc(A, B) = inf

x∈XIW(A(x), B(x)).

2b.m2Inc(A, B) =

12

[infx∈X

IW(A(x), B(x)) + supx∈X

IW(A(x), B(x))].

3b. In case of finite universes:

m3Inc(A, B) =|Inc(A, B)||X × X| .

4b. In case of bounded universes equipped with a measure m the measure m4Inc(A, B)is defined as:

m4Inc(A, B) =

∫Inc(A, B)dm

m(X × X).

Fuzzy inclusion measure defined in 1b is the one defined in [1]. Next we give anexample to demonstrate the role of Inc(A, B) and comparison of different measures.For simplicity we take crisp sets A and B.

12 Ismat Beg

Example 4.2. Let X = 1, 2, 3, ....10, A = 1, 2, 3, 8, 9 and B = 2, 3, 4, 5, 6, 7 then:

Inc(A, B) = (1, 0), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (7, 1), (8, 0), (9, 0), (10, 1),Inc(B, A) = (1, 1), (2, 1), (3, 1), (4, 0), (5, 0), (6, 0), (7, 0), (8, 1), (9, 1), (10, 1)

where, The first coordinate of every ordered pair represents the element and the sec-ond coordinate represents its membership in the fuzzy set of inclusion.1. m1Inc(A, B) = 0 and m1Inc(B, A) = 0;2. m2Inc(A, B) = 0.5 and m2Inc(B, A) = 0.5;3. m3Inc(A, B) = 0.7 and m3Inc(B, A) = 0.6.

Theorem 4.1. If the fuzzy measure m defined in Definition 4.1 also possesses theadditional property stated in Remark 4.1, then it is a fuzzy quasi order on F(X).

Proof. We will prove the two conditions one by oneO1. Reflexivity: For all A ∈ F(X),Using property m1 and Proposition 3.2 we get mInc(A, A) = m(X) = 1.O3. Fuzzy transitivity: Let on contrary suppose that mInc is not fuzzy transitive so,

IW(W( mInc(A, B),mInc(B,C)),mInc(A,C)) = 0.

It implies that

mInc(A, B) = 1, mInc(B,C) = 1 and mInc(A,C) = 0.

It further implies that

Inc(A, B) = X and Inc(B,C) = X and mInc(A,C) = 0.

ThereforeA ⊆ B and B ⊆ C and mInc(A,C) = 0.

a contradiction. So mInc is fuzzy transitive.

Proposition 4.1. For all A, B ∈ F(X) and the measure m satisfying the additionalproperty stated in Remark 4.1, then we have:

1. mInc(A, B) = 1 if and only if A(x) ≤ B(x) for all x ∈ X;2. mInc(A, B) = 0 if and only if A = X and B = ∅;3. Inc(A, Ac) = 0 if and only if A = X.

Proof. For all A, B ∈ F(X),1. Let A(x) ≤ B(x). It implies that Inc(A, B) = X, applying measure to both sides

it further implies that mInc(A, B) = 1.Conversely let mInc(A, B) = 1 which implies that Inc(A, B) = X,which further impliesthat A(x) ≤ B(x).

2. A = X and B = ∅ implies that Inc(A, B) = ∅ by Proposition 3.2, applyingmeasure to both sides we get mInc(A, B) = 0.


Conversely let mInc(A, B) = 0 which implies that Inc(A, B) = ∅ due to property ofthe measure stated in Remark 4.1, and which in turn implies that A = X and B = ∅due to Proposition 3.2.

3. Using the Proposition 4.1 (2) with Ac = B.

Proposition 4.2. For all A, B ∈ F(X) we have:1. mInc(A, B) = mInc(Bc, Ac).2. m(W∗(Inc(A, B), Inc(B, A))) = 1.

Proof. These identities can be obtained by applying fuzzy measure to both sides ofthe statements proved in Propositions 3.5 and 3.6.

Proposition 4.3. For all A, B,C ∈ F(X),1. B ⊆ C =⇒ mInc(A, B) ≤ mInc(A,C).2. B ⊆ C =⇒ mInc(C, A) ≤ mInc(B, A).

Proof. 1. Since B ⊆ C =⇒ Inc(A, B) ⊆ Inc(A,C) by Proposition 3.3. Using proper-ties of fuzzy measure we get

mInc(A, B) ≤ mInc(A,C).

2. Proof is on the same lines as of part 1.

Proposition 4.4. For all A, B,C ∈ F(X),1. mInc(max(A, B),C) = m(min(Inc(A,C), Inc(B,C)));2. mInc(min(A, B),C) = m(max(Inc(A,C), Inc(B,C)));3. mInc(A,min(B,C)) = m(min(Inc(A,C), Inc(B,C)));4. mInc(A,max(B,C)) = m(max(Inc(A,C), Inc(B,C))).

Proof. All these equalities can be obtained by applying measures to both sides ofequalities in Proposition 3.7.

Theorem 4.2. Fuzzy inclusion measure defined in :

mInc(A, B) = infx∈X

IW(A(x), B(x))

is a 1-transitive quasi order.

Proof. For all A, B,C ∈ X, we have1. mInc is reflexive since mInc(A, A) = inf

x∈XIW(A(x), A(x)) = 1.

2. Strong Fuzzy transitivity: The use of Theorem 3.1 yields

W(mInc(A, B),mInc(B,C)) = W(infx∈X

(Inc(A, B)(x)), infx∈X

(Inc(B,C)(x)))

≤ infx∈X

(Inc(A,C)(x)) = mInc(A,C).

14 Ismat Beg

It implies that mInc is a W-transitive fuzzy equivalence relation on F(X) and hence1-fuzzy transitive.

Acknowledgements. Research partially supported by Higher Education Commis-sion of Pakistan.

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DIRECT GLOBAL OPTIMIZATION BYCELLULAR EXCLUSION IN ABIOCONCENTRATION PROBLEM

ROMAI J., v.8, no.1(2012), 17–26

Stefan-Gicu Cruceanu, Dorin Marinescu”Gh. Mihoc - C. Iacob” Institute of Mathematical Statistics and Applied Mathematics of Ro-manian Academy, Bucharest, [email protected], [email protected]

Abstract Models of practical interest used to determine a set of unknown parameters lead o nu-merical global optimization problems. Since most of the known computational methodsconduct to local instead of global extrema or require an expensive numerical effort, con-structing efficient algorithms for global optimization remains of current interest. It iswell known [1] that cellular exclusion is a powerful tool for locating the solutions ofstrongly nonlinear algebraic systems of equations. In this work, we show how cellularexclusion can be successfully adapted for direct solving global optimization problems.Here, we use this technique to determine unknown parameters associated to a bioac-cumulation model governed by a system of ODEs. Finally, we analyze and comparethe results, advantages and disadvantages provided by the direct and indirect cellularexclusion methods in our parameter estimating problem.

Keywords: global optimization, parameter estimation, bioaccumulation, exclusion test.2010 MSC: primary: 65H10, secondary: 65G40, 92B05, 37N25.

1. INTRODUCTIONThere is a large variety of potentially toxic chemicals resulting from technological

processing or chemicals used in agriculture either to prevent pests or to fertilize thesoil. Unfortunately, these substances enter the food chain and they can affect thehealth of the organisms. Therefore it is important to be able to understand and todescribe the bioaccumulation processes.

One assumes a contaminated environment by some toxin of uniform concentra-tion cw and N species S 1, S 2, . . . , S N that are interacting each other as well as withthis environment. Each species S r can feed from the environment and only from thespecies S r−1, S r−2, . . . , S 1. Therefore the toxin passes directly form the environmentinto the species, as well as from one species to another one. The concentration ofthe contaminant inside the species S r is denoted by Cr. This concentration can growdue to the uptake fluxes form the environment or form the species S r−1, S r−2, . . . ,S 1 and it can decrease due to the release flux from the species S r. Supposing thateach of these fluxes is proportional to its corresponding concentration, a bioconcen-tration model (e.g. [2, 3]) can now be described by the following linear system of N

17

18 Stefan-Gicu Cruceanu, Dorin Marinescu

differential equations

dCr

dt= ku

r cw − kerCr +

r−1∑

p=1

kurpCp, for r = 1, . . . ,N. (1)

Here, kur cw and ku

rpCp represent the uptake fluxes into the species S r from the envi-ronment and from the species S p (p = 1, . . . , r − 1), respectively, while ke

rCr is therelease flux for the species S r.

From a phenomenological point of view, a natural assumption is that all kur are

pairwise different each other. Under this assumption, the general analytic solutionof the Cauchy problem associated to (1) for the initial conditions Cr(0) = Cr,0 (r =

1, . . . ,N) has the following form:

Cr(t) =

r∑

m=1

arm e−kemt + cr, r = 1, . . . ,N, (2)

where the coefficients arm and cr are some quantities depending on kurp, ke

p, cw andCp,0 (with p = 1, . . . , r) (see [4] for more details).

Note that, it is very difficult to determinate the coefficients kurp, ke

p by direct mea-surements. Here, our purpose is to present a method that allows the evaluation ofthese coefficients by indirect measurements.

We now fix our attention on the rth component of (2). Let y1, . . . , yn be a setof measurements taken for the contaminant concentration Cr at the moments of timet1, . . . , tn. Suppose also that the concentration cw and the values of the parameterske

m are known for all m = 1, . . . , r − 1 (the parameters ke1, . . . , k

er−1 can be determined

following the same procedure described below for experimental measurements onC1, . . . ,Cr−1, respectively). Using the information obtained from the above men-tioned experimental measurements, our purpose is to determine the unknown param-eters arm (∀ m = 1, . . . , r), ke

r , and cr appearing in the expression of the concentrationCr given in (2).

Since r is fixed, let us denote

−→a := (a1, . . . , ar+2),

where am := arm, ∀ m = 1, . . . , r, ar+1 := −ker , ar+2 := cr.

Define now h : R × Rr+2 → R by

h(t,−→a ) := ar+2etar+1 +

r∑

m=1

am ϕm(t), (3)

where ϕm(t) = e−tkem , ∀ m = 1, . . . , r − 1 and ϕr(t) = 1.

Some of the most used methods to find the unknown parameters for which a math-ematical model ”best fits” the measured data consist in l2 or l1 minimization. In order

Direct global optimization by cellular exclusion in a bioconcentration problem 19

to determine −→a , we look for the global minimum of the function H or G in a domainΩ ⊂ Rr+2:

H : Ω→ R+, H(−→a ) :=n∑

s=1

(h(ts,−→a ) − ys)2 (l2−minimization), (4)

G : Ω→ R+, G(−→a ) :=n∑

s=1

|h(ts,−→a ) − ys| (l1−minimization). (5)

Note that, the indirect method presented in [5] and used in [4] with an improveddominant function determines the global minimum of the objective function only if itis a critical point inside Ω, and cannot be applied to non-differentiable functions (asG). The direct method we present here is also capable to find the global minimumeven if it is not among the critical points in Ω.

2. GLOBAL MINIMIZATION BY CELLULAREXCLUSION

Cellular Exclusion Algorithm is an efficient computational tool developed and im-proved by K. Georg and E. Allgower in order to find all the roots of a nonlinearsystem

F(x) = 0 (6)

inside a bounded domain Ω, where F : Ω ⊂ Rd → Rd is a smooth nondegeneratedfunction. This method will essentially provide a number of ”small”, disjoint subsetsΩl of Ω such that (6) has no solutions inside Ω \ (∪Ωl). The reader can refer to[6, 7, 8, 9] for more information.

In order to adapt this method for global optimization problems, let us first intro-duce some notations. Let d ∈ N∗ and consider the component-wise ”≤” as a partialordering in Rd. A box-interval (rectangular cell) σ ⊂ Rd is a set of the form

σ = [mσ − ρσ,mσ + ρσ] := x ∈ Rd∣∣∣∣ [mσ − ρσ ≤ x ≤ mσ + ρσ,

where mσ, ρσ ∈ Rd, with ρσ a vector of positive components. mσ and ρσ are calledthe center and the radius of σ, respectively.

In what it follows, for the sake of simplicity, we consider the domain Ω to be sucha box-interval and f : Ω ⊂ Rd → R be an objective function for which we want tofind the global minimum. Assuming that an inferior bound bσ of f on any cell σ ⊂ Ω

can be evaluated, we can build the following algorithm:

Global Minimization Algorithm by Cellular Exclusion

1 Initiate Γ := Ω; split Ω into smaller box-intervals σ and let P(Γ) be the setconsisting of these cells; let µΩ be the value of f at an arbitrary point in Ω.


2 Calculate νP(Γ) := minσ∈P(Γ)

f (mσ) and update µΩ by µΩ := minµΩ, νP(Γ)

.

3 Exclude all the box-intervals σ from P(Γ) for which bσ > µΩ and replace Γ

with the set σ ∈ P(Γ) | bσ ≤ µΩ.4 If a stopping criteria is met, stop the algorithm. Otherwise, split each σ ∈ Γ

into smaller box-intervals and take P(σ) to be the set consisting of these cells.Now, let P(Γ) :=

⋃

σ∈ΓP(σ). Go back to step 2.

The aim of this algorithm is to evaluate the set of minimal points

M f (Ω) :=x ∈ Ω

∣∣∣∣ f (x) = miny∈Ω

f (y)

(7)

for a given accuracy ε > 0. For this, one can choose as a stopping criteria

‖ρσ‖∞ ≤ ε, ∀σ ∈ Γ,

and M f (Ω) will therefore be ”approximated” by Γ. The efficiency of the algorithmis dependent on the choice of the inferior bounds bσ of f inside the box-intervals σ.A good choice of lower bound estimates bσ was introduced by means of dominantfunctions in [6].

For any multi-index α := (α1, . . . , αd) ∈ Nd and a variable x := (x1, . . . , xd) ∈ Rd,let us introduce the following notations:

|α| :=∑

i

αi, α! :=∏

i

αi, xα :=∏

i

xαii , ∂α :=

∏

i

∂αi .

Assuming f permits a Taylor expansion, we can write

f (mσ + x) = f (mσ) +∑

0<|α|<q

∂α f (mσ)xα

α!+

∑

|β|=q

∫ 1

0∂α f (mσ + tx)ωq(dt)

xβ

β!, (8)

where ωq(dt) := q(1 − t)q−1dt. By the triangle inequality, for |x| ≤ ρσ, we obtain

| f (mσ + x) − f (mσ)| ≤∑

0<|α|<q

|∂α f (mσ)|ρα

α!+

∑

|β|=q

Mβρβσ

β!, (9)

where Mβ are some constants such that Mβ ≥ |∂β f | on σ. Consequently, one canimmediately see that

bσ( f ) := f (mσ) −∑

0<|α|<q

|∂α f (mσ)|ρασ

α!−

∑

|β|=q

Mβρβσ

β!. (10)


is an inferior bound of f on σ.Sometimes, the sums in the r.h.s. of (10) may have a lot of terms. In such situations

it is difficult to implement the cellular exclusion method using the expression givenabove and it is preferably to use another method which provides a simpler expressionfor the inferior bound. By example, one can use the technique presented in [6] basedon combinations of dominant functions. Though, the formula (10) can provide abetter value of the inferior bound leading to a more efficient algorithm.

In the case when we choose f = H, with H given by (4), where d = r + 2, a lot ofderivatives vanish in (10). One has only κ non-null derivatives of order α, where

κ =

r + 2, |α| = 1r(r + 1)

2+ 4, |α| = 2

r + 4, |α| ≥ 3

.

Therefore, for the l2-minimization case, we can use (easy to computationally con-struct) bσ(H) from (10) as inferior bound of H in σ.

For the l1-minimization, the function G given by (5) is not differentiable, and con-sequently, (10) cannot applied to find an inferior bound of G. Since only Lipschitzestimations do not provide a quite good inferior bound for G, we can still take advan-tage of the smoothness of h defined in (3).

Let σ be a box-interval. Define

sign(σ, t, y) :=

1, if h(t,−→a ) > y, ∀ −→a ∈ σ0, if ∃ −→a ∈ σ, h(t,−→a ) = y−1, if h(t,−→a ) < y, ∀ −→a ∈ σ

. (11)

Let Gσ be the restriction of G on σ. We can write Gσ = Gσ + Gσ, where

Gσ(−→a ) =

n∑

s=1

sign(σ, ts, ys)(h(ts,−→a ) − ys).

Gσ(−→a ) =

n∑

s=1

(1 − |sign(σ, ts, ys)|)|h(ts,−→a ) − ys|.

Obviously, Gσ is smooth. Therefore, bσ(Gσ) calculated with (10) for d = r + 2is an inferior bound for Gσ and since Gσ is obviously positive, we can easily seethat bσ(Gσ) is also an inferior bound for Gσ. Thus, to apply the algorithm, one maychoose bσ(Gσ) := bσ(Gσ).

3. NUMERICAL RESULTSConsider r = 1 and denote a = a3, b = a2, c = a1.As in [4], starting with a set of n = 250 non-perturbed data ysns=1 generated for

(a, b, c) = (3,−1, 0.5) and corresponding to some arbitrary moments of time tsns=1 ⊂


[0,T ], with T = 10, we have applied the direct method described in this article (forq = 3) and, as expected, we recovered the values of these parameters with very goodaccuracy in both types (l2 and l1) of minimization.

We then decided to perturb each exact datum ys with a random measurement errorbetween [−p%, p%] of itself. The estimated parameters (am, bm, cm) are found usingthe algorithm described in this paper, starting with the initial box-interval

σ : mσ = (4,−2, 3.5), ρσ = (3.5, 1.5, 3).

Letting the algorithm to run up to the resolution level l = 30, (where the radius ofany kept box-interval is ρ = ρσ · 2−30), we obtained a set of cells; all these cells areincluded in a box-interval of the form

J = [am − δa, am + δa] × [bm − δb, bm + δb] × [cm − δc, cm + δc].

The numerical results obtained through the direct (l2 and l1) minimization algorithmas described in this article for p = 2 and p = 8 are presented in Table 1, where weused the following distances between the ”measured” data and the one provided bythe model:

d(a, b, c) :=

√√1n

n∑

s=1

(aebts + c − ys

)2,

for the case of l2- minimization, and

d(a, b, c) :=1n

n∑

s=1

∣∣∣aebts + c − ys∣∣∣ ,

for the case of l1- minimization. d and dm stand for d(a, b, c) and d(am, bm, cm),respectively.

Note that, for the same input data, the direct l2-minimization method presentedhere and the indirect minimization method presented in [4], practically provide thesame values of the estimated parameters.

Another type of test was carried out for non-uniform perturbed data. Starting againfrom (a, b, c) = (3,−1, 0.5), we generate n = 250 data ysns=1 corresponding to somearbitrary moments of time tsns=1 ⊂ [0, T ], with T = 10. n1 = 240 out of them areperturbed with random measurement errors between [−p1%, p1%] (of their values)and the other n2 = 10 data with random measurement errors between [−p2%, p2%].Beginning with the same initial box-interval as in the previous case, and running thealgorithm up to the level l = 30, we have obtained the results presented in Table 2.


Numerical results for uniform perturbed data.

p 2 8

Min. l2- min. l1- min. l2- min. l1- min.

am 2.9803021 2.9630732 2.92152865 2.8550699

bm −0.99320240 −0.98613883 −0.97265576 −0.94536232

cm 0.500071955 0.499067625 0.500203925 0.4959229

δa 5 · 10−8 5 · 10−8 10 · 10−8 5 · 10−8

δb 2.5 · 10−8 5.5 · 10−8 2.5 · 10−8 10.5 · 10−8

δc 10−8 2 · 10−8 10−8 5.5 · 10−8

d 0.011475 0.008114 0.045900 0.032457

dm 0.010993 0.007710 0.043971 0.030841

Table 2: Numerical results for non-uniform perturbed data.

Min. l2- min. l1- min.

am 3.62848455 2.99164335

bm −1.1597097 −0.994863625

cm 0.51421461 0.499817865

δa 10 · 10−8 10 · 10−8

δb 0.5 · 10−8 2 · 10−8

δc 1.5 · 10−8 2 · 10−8

d 0.316948 0.051731

dm 0.293963 0.051604

The perturbed data and the fitting functions associated to the estimated parameters(am, bm, cm) are illustrated in Figure 1. It is known [10, 11] that, as compared to theleast square methods, the l1-optimization is less affected by a small number of mea-surements with large errors. The effect of such gross measurement errors (outliers)can also be seen in this nonlinear case (Figure 2).


Figure 1: The perturbed data (p1 = 2, n1 = 240, p2 = 90, n2 = 10,T = 0) and the functions f associated to the parameters determined byl1 and l2 minimization.

0

1

2

3

4

5

6

0 2 4 6 8 10

f(t)

t

Input dataf(t) for l1 - minimizationf(t) for l2 - minimization

Figure 2: A zoom in Figure 1.

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

2 3 4 5 6 7 8 9

f(t)

t

Input dataf(t) for l1 - minimizationf(t) for l2 - minimization

ObviouslyM f (Ω) ⊂ Γ ⊂ J. (12)


where f is any of the functions H or G. As the centers of the remaining box-intervals are concentrated around (am, bm, cm) with a tolerance (δa, δb, δc), where0 ≤ δa, δb, δc ≤ ε ≈ 10−7, one can consider that (am, bm, cm) is the solution ofthe minimization problem.

A quick comparison between the values of d and dm in both tables tells us thatthe estimated parameters fit the perturbed data set in a better way than the initialparameters. We have expected this since the data obtained from the initial values ofthe parameters a, b, and c were perturbed and therefore, the global minima of theobjective functions G and H from (4) and (5), respectively, will now be attained forother values of these parameters.

4. FINAL REMARKS AND CONCLUSIONSA first remark that we should emphasize is that the direct optimization method

presented in this article works for both Least Square and Least Absolute Values min-imization problems.

In comparison with the indirect method from [4], this new algorithm comes withsome advantages which are not negligible. First of all, the laborious work requiredfor the construction of the exclusion inequalities is simplified. As the reader canobserve from step 3 of the Global Minimization Algorithm, we now only have tobuild 1 instead of r + 2 exclusion inequalities.

Also, from a computing point of view, the indirect method is more costly sinceit first determines all the critical points of a nonlinear system of equations and thenselect the one (ones) that minimizes the cost function. The new method proposed inthis paper focuses directly on the global minima (the set M f (Ω) from (7)) and candetermine them even for the case of a non-critical points (reached on the boundaryfor example).

Another advantage of the direct over the indirect method relates to the fact that itcan also be applied to non-differentiable functions.

As expected, the older method does not have only disadvantages when comparedto the new one. A higher number of box-intervals were eliminated in a neighborhoodof the global minimum (there were more inequalities to be satisfied) and we haveobserved slightly more accurate results for the case of global minimum attained on acritical point.

References

[1] K. Georg, Improving the Efficiency of Exclusion Algorithms, Advances in Geometry, 1, (2001),193-210.

[2] S. Galassi, M. Gatto, B. Zanetti, BIOCON: A program for the parameter estimation and thesimulation of a simple bioconcentration model, Environmental Software, 4, (1989), 157-161.

[3] S. N. Luoma , P. S. Rainbow Why is metal bioaccumulation so variable? Biodynamics as aunifying concept, Environ. Sci. Technol., 39(7), (2005), 1921-1931.


[4] S. G. Cruceanu, D. Marinescu Parameter evaluation for bioconcentration model, using cellularexclusion method, ROMAI J., 7, 1(2011), 63-77.

[5] S. G. Cruceanu, D. Marinescu, Cellular exclusion algorithm for parameter estimation with appli-cations in biological models, Topics in mathematical modelling of life science problems, (2011),37-50.

[6] E. L. Allgower, M. Erdmann, K. Georg, On the complexity of exclusion algorithms for optimiza-tion, J. Complexity, 18, (2002), 573-588.

[7] E. L. Allgower, S. G. Cruceanu, S. Tavener, Application of numerical continuation to detect allthe solutions of semilinear elliptic equations, Advances in Geometry, 9, (2009), 371-400.

[8] B. L. Meng, Z. D. Xing, A cell exclusion algorithm for finding all solutions to systems of nonlin-ear equations. Pure Appl. Math. (Xian), 15, (1999), 72-76.

[9] E. Allgower, M. Erdmann, K. Georg, On the complexity of exclusion algorithms for optimization,J. Complex., 18, (2002), 573-588.

[10] D. Birkes, Y. Dodge, Alternative Methods of Regression, Wiley Series in Probability and Statis-tics, (1993), p.240.

[11] S. Bektas, Y. Sisman, The comparison of L1 and L2-norm minimization methods, Int. J. Phys.Sci., 5(11), (2010), 1721-1727.

A SEVEN-EQUATION MODEL FOR THERELATIVISTIC DYNAMICS OF ATWO-PHASE FLOWS

ROMAI J., v.8, no.1(2012), 27–42

Sebastiano Giambo Serena Giambo, Veronica La RosaDepartment of Mathematics, University of Messina, [email protected], [email protected], [email protected]

Abstract An interface-capturing method is used to describe the relativistic dynamics of a two-phase flow. Following Allaire, Clerc and Kokh, we are able to build a seven-equationmodel of conservation equations for particle number density of each fluid component,for the energy-momentum tensor of the mixture and an advection equation for the vol-ume fraction of one of the two fluids. The relativistic model of such a mixture needs anadditional law in order to get a closure. We study two different closure laws: isobaricand isothermal closure. The weak discontinuities propagating in the mixture for bothmodels are examined.

Keywords: general relativity, relativistic fluid dynamics, two-phase flow, nonlinear waves.2010 MSC: 83C99, 80A10, 80A17, 76T99, 74J30.

1. INTRODUCTIONTwo-phase fluid flow problems appear in many topics of classical [1, 20] and rela-

tivistic context [21, 47]. In these problems, the medium consist of two compressiblefluids, which do not mix. Moreover, a sharp interface separates the two fluids.

Such situations appear frequently in astrophysics, physics of explosives, nuclearphysics, power engineering and many other applications. Following difficulties areconcerned with interaction, which includes the transfer of mass, momentum and en-ergy across the interfaces. The manner of treatment of the interfaces is the keypointof each model.

A class of two-fluid models in known as interface-capturing techniques (see Wackers-Koren paper [13]). These methods do not use an explicit interface model. Instead,the fluid is modeled as a mixture of two pure fluids everywhere.

In this paper, a capturing method is presented, which is a relativistic extensionof the method introduced by Allaire, Clerc and Kokh [6] for classical compressibletwo-fluid flow.

On the physical ground, the model describes the relativistic two-fluid flow, inwhich the entire flow domain is filled with a mixture of two fluids.

However, in this underlying two-phase model, the fluids are supposed not to bemixed on the molecular level. So, the fluid is a mixture in the mascroscopic sense.

27

28 Sebastiano Giambo Serena Giambo, Veronica La Rosa

Then, each phase still has its own particle number density, specific internal energy,pressure and temperature, but a single four-velocity is definded for the whole mixture.

In our relativistic two-phase model, seven unknown variables appear in our de-scription of two-phase flow: the four-velocity and the pressure of the whole mixture,the particle number density of each fluid component and another quantity for therelative concentrations of the components.

This paper is organized as follows: Section 2 is devoted to a description of the rel-ativistic mixture model and to the derivation of the flow equations. Afterwards, someclosure relations must be introduced and they are based on the previous thermody-namical considerations. In particular two algebraic closure for our model: isobaricand isothermal closure, respectively, are investigated. In both cases the completesystem of governing differential equations is deduced. In Section 3, the weak discon-tinuities propagating in the mixture are examined and the expressions for their speedsof propagation are obtained. Finally, a special case in which each fluid is supposedto satisfy the equation of state of perfect gases is considered.

Space-time is a four-dimensional manifold V4, whose normal hyperbolic metricds2, with signature +,−,−,−, is expressed in local coordinates in the usual formds2 = gµνdxµdxν, the metric tensor is assumed to be of class C1 and piecewise C2;the 4-velocity is defined as uν = dxν/ds, which implies its unitary character uνuν = 1;∇ν is the operator of covariant differentiation with respect to the given metric; in whatfollows the units are such that the velocity of light is unitary, i.e. c = 1.

2. THE SEVEN-EQUATION MODELIn this section we describe a model for a mixture of two relativistic fluid. Each

fluid k = 1, 2 is assumed to be compressible with particle number density rk, specificinternal energy εk, pressure pk, and temperature Tk. We also define the energy densityρk, the “classical” specific enthalpy hk and “relativistic” specific enthalpy fk of eachfluid component k, respectively, by

ρk = rk (1 + εk) ,hk = εk +

pkrk,

fk = 1 + hk.(1)

Throughout this paper, both fluids in the mixture are supposed to share the same4-velocity uν. Moreover, we introduce the volume fractions of the two componentsX1 = X, X2 = 1 − X. The bulk particle number density is then defined as

r = X1r1 + X2r2 . (2)

The conservation laws that hold in this context are: conservation equation of particlenumber of each fluid component

∇α (X1r1uα

)= 0,

A seven-equation model for the relativistic dynamics of a two-phase flows 29

∇α (X2r2uα

)= 0 (3)

and conservation equation of the bulk energy-momentum tensor, Tαβ, of the mixture

∇αTαβ = 0 . (4)

Moreover, from equations (3) we can achieve the following conservation law

∇α(ruα) = 0 . (5)

Now, we introduce the mass fractions Yk of the fluid k given by

Yk =Xkrk

r, with Y1 + Y2 = 1 . (6)

After that, taking into account (3) and (5) we obtain

DY = 0 , (7)

where we set D = uα∇α.The energy-momentum tensor Tαβ is given by

Tαβ = r f uαuβ − pgαβ , (8)

being f the “relativistic” specific enthalpy of the mixture

f = 1 + h = 1 + ε +pr

=ρ + p

r, (9)

with h = ε +pr

classical specific enthalpy, ε specific internal energy, p pressure andρ = r(1 + ε) energy density of the fluid mixture. We observe that ε can be expressedin terms of the specific internal energy of each fluid component as

ε = Y1ε1 + Y2ε2 . (10)

The spatial projection and the projection along uµ of equation (4), give respectively

r f uν∇νuµ − γµν∂νp = 0, (11)

Dρ + (ρ + p)ϑ = 0 , (12)

where we set ϑ = ∇νuν.Moreover, following Allaire, Clerck, Kokh [6], we assume that the following trans-

port equation for the volume fraction X holds

DX = 0 . (13)

This seven-equation model described by (3)1, (3)2, (5), (12) and (13) is not closed.Indeed, there are seven equations in eight independent variables, which are X, r1,


r2, uα, ε1, ε2. Therefore, another relation is needed in order to close system. Againfollowing Allaire, Clerck, Kokh [6], we analyze two different possible closure laws:the isobaric closure and the isothermal one.

A general closure relationship for the seven-equation system is the definition of anequation of state for the pressure p of the mixture of the type

p = p (X, Xr1, (1 − X)r2, ρ) . (14)

Clearly, (3)1 and (3)2 are formally equivalent to

D (X1r1) = −X1r1ϑ, (15)

D (X2r2) = −X2r2ϑ. (16)

Then, we can write an evolution equation for pressure. Since total pressure p is afunction of the variables X, X1r1, X2r2 and ρ, its differential is

dp =∂p∂ρ

dρ +∂p

∂X1r1d (X1r1) +

∂p∂X2r2

d (X2r2) +∂p∂X

dX . (17)

Since DX = 0, the evolution of pressure is governed by

Dp =∂p∂ρ

Dρ +∂p

∂X1r1D (X1r1) +

∂p∂X2r2

D (X2r2) , (18)

which, using (12), (15) and (16), can be re-written as

Dp = −(r f∂p∂ρ

+ X1r1∂p

∂X1r1+ X2r2

∂p∂X2r2

)ϑ . (19)

At this point, we are able to deduce an evolution equation for the entropies of eachcomponent.

Let us suppose that each constituent k admits a temperature Tk and an entropydensity S k, which satisfy Gibb’s relation

TkdS k = dεk + pkd1rk, k = 1, 2 . (20)

For the global fluid, entropy is given by

S = Y1S 1 + Y2S 2 . (21)

According to Lagoutiere [4], we suppose that there exists a priori a function, T , ofall the thermodynamic variables of the problem, such that

T DS = Dε + pD1r. (22)


This hypothesis is called “central hypothesis”.Using (8), (10) and (20), we can write

Y1T1DS 1 + Y2T2DS 2 = Dε + Y1 p1D1r1

+ Y2 p2D1r2

(23)

and, for the central hypothesis (22), equation (19) gives the compatibility condition

Y1T1DS 1 + Y2T2DS 2 − T DS = Y1 p1D1r1

+ Y2 p2D1r2− pD

1r. (24)

By virtue of (2), (8) and (13) and taking into account that

1r

=Y1

r1+

Y2

r2, (25)

equation (24) can be written as

Y1T1DS 1 + Y2T2DS 2 − T DS = (X1 p1 + X2 p2 − p)D1r. (26)

Now, we make the further assumption that the closure relationship must verify thatthe two expression on the right and left side of (26) vanish. It must be

Y1T1DS 1 + Y2T2DS 2 = T DS , (27)

(X1 p1 + X2 p2 − p)D1r

= 0. (28)

It is noted that (27) is verified where we have the following closure in temperatureand entropy:

i) T1DS 1 = TaDS 2 , with1T

=Y1

T1+

Y2

T2, (29)

ii) DS 1 = DS 2 , with T = Y1T1 + Y2T2, (30)

iii) T1 = T2 = T (isothermal closure), (31)

whereas (28) requires closure in pressure:

i) p = X1 p1 + X2 p2, (32)

ii) p1 = p2 = p (isobaric closure). (33)

Now, we assume that by virtue of (5), (12) and (22)

DS = 0. (34)

Moreover, if (29) and (30) are valid, we have, respectively

T1DS 1 = T2DS 2 = T DS , (35)


orDS 1 = DS 2 = DS , (36)

and, hence for regular solutions of system (5),(6), (9) and (9) it is

DS 1 = DS 2 = 0 ⇔ DS = 0 . (37)

Let us now discuss separately the two cases of isobaric and isothermal closure.

2.1. ISOBARIC CLOSUREIn order to get a closure for system (5), (6), (9) and (9), we add the algebraic

relation (33) requiring that the pressures of the two components are equal.The equations of state for both fluids are supposed to be of the form

ρk = ρk (rk, p) , k = 1, 2 . (38)

At this point, we want to characterize the evolution equation for pressure (19) with theisobaric assumption. In particular, we want to prove that the differential of pressure

p = p (X, Xr1, Xr2, ρ) (39)

has the following form

Dp =1ξ

[Dρ − ζ1D (X1r1) − ζ2D (X2r2)

], (40)

whereξ = X1

∂ρ1

∂p+ X2

∂ρ2

∂p, (41)

ζk =∂ρk

∂rk, k = 1, 2 . (42)

In fact, sinceρ = X1ρ1 + X2ρ2 , (43)

we obtainDρ = D

(X1ρ1 + X2ρ2

)= X1Dρ1 + X2Dρ2

= X1

(∂ρ1

∂r1Dr1 +

∂ρ1

∂pDp

)+ X2

(∂ρ2

∂r2Dr2 +

∂ρ2

∂pDp

)

= X1ζ1Dr1 + X2ζ2Dr2 +

(X1∂ρ1

∂p+ X2

∂ρ2

∂p

)Dp

= X1ζ1Dr1 + X2ζ2Dr2 + ξDp

= ζ1D (X1r1) + ζ2D (X2r2) + ξDp , (44)

which is exactly equation (40).


Therefore, the evolution of the pressure p is governed by equation (40), which canbe written also in the equivalent form

ξDp = − [r f − ζ1X1r1 − ζ2X2r2

]ϑ . (45)

The complete set describing the evolution of the concerned model is

D (X1r1) = −X1r1ϑ,

D (X2r2) = −X2r2ϑ,

r f uα∇αuβ = γαβ∂αp (46)

ξDp = − [r f − (

X1r1ζ1 + X2r2ζ2)]ϑ,

DX = 0.

2.2. ISOTHERMAL CLOSUREThe isothermal closure is based on the assumption that the temperatures of the two

fluid components are equal, i.e. relation (29) is valid.In this section we assume that each fluid k is equipped with an equation of state

defined in terms of its particle number density rk and temperature T , which reads as

εk = εk (rk,T ) ,

ρk = ρk (rk,T ) , (47)

pk = pk (rk, T ) .

As first, we want to prove that the equation describing the evolution of the totalpressure, p, of the mixture in terms of the variables ρ, X1r1, X2r2 is

ξDp = Dρ − ζ1D (X1r1) − ζ2D (X2r2) , (48)

with1ξ

=1

rCV

X1

(∂p1

∂T

)

r1

+ X2

(∂p2

∂T

)

r2

, (49)

rCV = X1r1

(∂ε1

∂T

)

r1

+ X2r2

(∂ε2

∂T

)

r2

, (50)

ζk = −ξ(∂pk

∂rk

)

T+ fk − T

rk

(∂pk

∂T

)

rk

, k = 1, 2. (51)


In fact, by virtue of (32), we get

Dp = X1Dp1 + X2Dp2

= X1

(∂p1

∂r1

)

TDr1 +

(∂p1

∂T

)

r1

DT

+X2

(∂p2

∂r2

)

TDr2 +

(∂p2

∂T

)

r2

DT

=

(∂p1

∂r1

)

TD (X1r1) +

(∂p2

∂r2

)

TD (X2r2)

+

X1

(∂p1

∂T

)

r1

+ X2

(∂p2

∂T

)

r2

DT

=

(∂p1

∂r1

)

TD (X1r1) +

(∂p2

∂r2

)

TD (X2r2) +

rCV

ξDT.

(52)From the definition of ρ

ρ = X1ρ1 + X2ρ2 = X1 (r1 + r1ε1) + X2 (r2 + r2ε2) = X1r1 (1 + ε1) + X2r2 (1 + ε2) ;(53)

we get that

Dρ = (1 + ε1) D (X1r1) + X1r1Dε1 + (1 + ε2) D (X2r2) + X2r2Dε2= (1 + ε1) D (X1r1) + (1 + ε2) D (X2r2)

+X1r1

(∂ε1

∂r1Dr1 +

∂ε1

∂TDT

)+ X2r2

(∂ε2

∂r2Dr2 +

∂ε2

∂TDT

)

= (1 + ε1) D (X1r1) + (1 + ε2) D (X2r2) + r1∂ε1

∂r1D (X1r1)

+r2∂ε2

∂r2D (X2r2) +

(X1r1

∂ε1

∂T+ X2r2

∂ε2

∂T

)DT

= (1 + ε1) D (X1r1) + (1 + ε2) D (X2r2)

+r1∂ε1

∂r1D (X1r1) + r2

∂ε2

∂r2D (X2r2) + rCV DT ,

so the expression of DT is deduced.At this point, equation (52) gives

Dp =∂p1

∂r1D (X1r1) +

∂p2

∂r2D (X2r2) +

1ξ

Dρ − 1ξ

(1 + ε1) D (X1r1)

−1ξ

(1 + ε2) D (X2r2) − r1

ξ

∂ε1

∂r1D (X1r1) − r2

ξ

∂ε2

∂r2D (X2r2)


=1ξ

Dρ +1ξ

(ξ∂p1

∂r1− (1 + ε1) − r1

∂ε1

∂r1

)D (X1r1)

+1ξ

(ξ∂p2

∂r2− (1 + ε2) − r2

∂ε2

∂r2

)D (X2r2)

=1ξ

Dρ +1ξ

(ξ∂p1

∂r1−

(1 + ε1 +

p1

r1

)+

Tr1

∂p1

∂T

)D (X1r1)

+1ξ

(ξ∂p2

∂r2−

(1 + ε2 +

p2

r2

)+

Tr2

∂p2

∂T

)D (X2r2)

=1ξ

Dρ − 1ξζ1D (X1r1) − 1

ξζ2D (X2r2) ,

which is exactly what we wanted to prove, i.e. the evolution equation for total pres-sure (48). Using (15) and (16), equation (48) writes also in the form

ξDp = − [r f − (

ζ1X1r1 − ζ2X2r2)]ϑ . (54)

We can conclude that complete system governing the evolution of the consideredmixture is

D (X1r1) = −X1r1ϑ,

D (X2r2) = −X2r2ϑ,

r f uν∇νuµ = γµν∂νp (55)

ξDp = − [r f − (

X1r1ζ1 + X2r2ζ2)]ϑ,

DX = 0.

Observe that the coefficients ξ and ζk, defined by (49) and (51), appearing in thecontext of the isothermal closure, are different from the corresponding ones definedby (41) and (42) in Sect. 2.1, in the context of the isobaric closure.

3. THE CHARACTERISTIC MANIFOLDS ONWAVE FRONTS

In a domain Ω of space-time manifold V4, let Σ be a regular hypersurface, notgenerated by flow lines, and ϕ (xµ) = 0 be its local equation. We set Lµ = ∂µϕ. Aswe will see below, the hypersurface Σ is a space-like type, i.e. LµLµ < 0. Moreover,we introduce the normalized vector Nµ

Nµ =Lµ√−LνLν

, NµNµ = −1 . (56)

We are interested in a particular class of solutions of systems (46) and (55) namely,weak discontinuity waves Σ, across which field variables uµ, r1, r2, p, X are continu-ous, but they exhibit jump discontinuities in their normal derivatives. In this case, if


Z denotes any of these variables, then there exists [48, 49] the distribution δZ, withsupport over Σ, such that

δ[∇µZ

]= NµδZ , (57)

where δ is the Dirac measure defined by ϕ with Σ as support, square brackets denotethe jump across Σ, δ being an operator of infinitesimal discontinuity. Using equation(57), from the systems (46) and (55), we obtain the following linear homogeneoussystem in the distributions δX, δr1, δr2, Nνδuν and δp

Lr1δX + LXδr1 + Xr1Nνδuν = 0,Lr2δX + LX2δr2 − X2r2Nνδuν = 0,

r f Lδuν − γνµNµδp = 0,ξLδp +

r f − (

X1r1ζ1 + X2r2ζ2)

Nνδuν = 0,LδX = 0,

(58)

where L = uνLν.Moreover, from the unitary character of uν, we have

uνδuν = 0 . (59)

Now, we investigate the normal speeds of propagation of the various waves withrespect to an observer moving with the velocity uν of the mixture. The normal speedof propagation of the wave front Σ, denoted by λΣ, with respect to an observer, de-scribed by a time-like world-line of tangent vector field uν (that is with respect to thetime direction uν), is given by [48, 49]

λ2Σ =

L2

`2 , `2 = 1 + L2 . (60)

The local causality condition, i.e. the requirement that the characteristic hypersur-face Σ is time-like or null (or, equivalently, that the normal Nµ is space-like or null,that is NµNµ ≤ 0) is equivalent to the condition

0 ≤ λ2Σ ≤ 1. (61)

From the above equations, we obtain as first the solution L = 0, which represent awave moving with the mixture. For the corresponding discontinuities, we find

Nνδuν = 0,

δp = 0. (62)

From equations (58) and (59), the coefficients characterizing the discontinuities havefive degrees of freedom, and so the system admits five independent eigenvectors cor-responding to L = 0 in the space of the field variables.


At this point, we can suppose L , 0. Equation (58)3, multiplied by Nν, gives us

r f LNµδuµ + `2δp = 0 . (63)

Consequently, (58)4 and (63) represent a linear homogeneous system in the two scalardistributions Nµδuµ and δp, which admits non trivial solutions only if the determinantof the coefficients vanishes. Therefore we find the equation

H ≡ r f ξL2 − ω`2 = 0 , (64)

whereω = r f − (

X1r1ζ1 + X2r2ζ2). (65)

This equation corresponds to the hydrodynamical waves propagating in such a two-fluid system. Their speed of propagation is given by

ξλ2Σ =

ω

r f(66)

and the condition 0 <ω

r f ξ≤ 1 ensures their spatial orientation.

The associated discontinuities can be written in terms of ψ = Nµδuµ as follows

δuν = −1`

nνψ,

δr1 = −r1

Lψ,

δr2 = −r2

Lψ,

δp = − ωξLψ,

δX = 0, (67)

where nν is the unitary space-like 4-vector defined by

nµ =1`

(Nµ − Luµ

). (68)

If the condition, given above, ensuring that the surfaces have space-like orientationsis fulfilled, then system of governing equations is hyperbolic. In fact, all velocities(eigenvalues) are real and there is a complete set of eigenvectors in the space of fieldvariables: seven independent eigenvectors (five from L = 0 and two from H = 0) forthe seven independent field variables uν, r1, r2, p and X.

Now, let us express velocity of propagation of the hydrodynamical waves givenby (66), in both isobaric and isothermal cases, considering the special case in whicheach fluid satisfies the equation of state of an ideal gas.


– In the isobaric case we have the following expressions for the coefficients ω,given by (65), and ξ, given by (41). In fact, using (42)

ω = r f − (X1r1ζ1 + X2r2ζ2)

= r f −(X1r1

∂ρ1

∂r1+ X2r2

∂ρ2

∂r2

)

= X1r1 f1 + X2r2 f2 − X1r1∂ρ1

∂r1− X2r2

∂ρ2

∂r2

= X1r1

(f1 − ∂ρ1

∂r1

)+ X2r2

(f2 − ∂ρ2

∂r2

). (69)

Therefore, from (66) the following expression for the velocity of propagation ofthe dydrodynamical waves is obtained

λ2Σ =

X1r1

(f1 − ∂ρ1

∂r1

)+ X2r2

(f2 − ∂ρ2

∂r2

)

r f(X1∂ρ1

∂p+ X2

∂ρ2

∂p

) . (70)

Furthermore, each fluid k = 1, 2 is assumed to be compressible with a particlenumber density rk, a specific internal energy εk, a pressure pk and a temperature Tksatisfying perfect gas equation of state

pk = (γk − 1)rkεk , εk = cVk Tk . (71)

In this case, for the corresponding global equation of state we have

p = (γ − 1)rε , (72)

with1

γ − 1=

X1

γ1 − 1+

X2

γ2 − 1. (73)

Having, from (71) that

ρk = rk(1 + εk) = rk +pk

γk − 1, (74)

we obtain that∂ρk

∂rk= 1 ,

∂ρk

∂pk=

1γk − 1

(75)

and equation (70) can be rewritten as

λ2Σ =

X1r1( f1 − 1) + X2r2( f2 − 1)

r f(

X1

γ1 − 1+

X2

γ2 − 1

) , (76)


or also, using (1)2 and (73),

λ2Σ =

X1r1h1 + X2r2h2

r f(γ − 1) =

rhr f

(γ − 1) . (77)

Finally, eq. (70) reduces toλ2

Σ =γpr f

. (78)

Moreover, from (73) we can write

1γ − 1

r fλ2Σ = X1r1 f1

h1

f1

(γ1 − 1)γ1 − 1

+ X2r2 f2h2

f2

(γ2 − 1)γ2 − 1

, (79)

and ultimately1

γ − 1r fλ2

Σ =X1r1 f1γ1 − 1

λ21 +

X2r2 f2γ2 − 1

λ22 , (80)

having denoted byλ2

k = γkp

rk fk(81)

the hydrodynamical speed of propagation of fluid k.Equation (80), since

γ

γ − 1=

X1γ1

γ1 − 1+

X2γ2

γ2 − 1, (82)

can be write also as1γ

r fλ2Σ =

X1

γ1r1 f1λ2

1 +X2

γ2r2 f2λ2

2 . (83)

– In the isothermal case we have the following expressions for the coefficients ω,

given by (65), and ξ, given by (49). In fact, using (51)

ω = r f −[X1r1

(−ξ∂p1

∂r1+ f1 − T

r1

∂p1

∂T

)]+

[X2r2

(−ξ∂p2

∂r2+ f2 − T

r2

∂p2

∂T

)](84)

= ξ

(X1r1

∂p1

∂r1+ X2r2

∂p2

∂r2

)+ T

(X1∂p1

∂T+ X2

∂p2

∂T

).

Therefore, from (66), the following expression for the velocity of propagation ofthe hydrodynamical waves is obtained

r fλ2Σ =

(X1r1

∂p1

∂r1+ X2r2

∂p2

∂r2

)+

TrcV

(X1∂p1

∂T+ X2

∂p2

∂T

)2

. (85)

In the particular case in which (71) and (72) are valid, with

γ =cp

cV, γ1 =

cp1

cV1

, γ2 =cp2

cV2

(86)


andcp = Y1cp1 + Y2cp2 , cV = Y1cV1 + Y2cV2 , (87)

so that we can rewrite as

pk = RkrkT with Rk = cpk − cVk , (88)

p = RrT with R0Y1R1 + Y2R2, (89)

equation (85) reduces toλ2

Σ =γpr f

. (90)

Moreover, by virtue of (32), equation (90) can be written as

r fλ2Σ = γ(X1 p1 + X2 p2) , (91)

or1γ

r fλ2Σ = X1

r1 f1γ1

γ1 p1

r1 f1+ X2

r2 f2γ2

γ2 p2

r2 f2(92)

and, ultimately,1γ

r fλ2Σ =

X1r1 f1γ1

λ21 +

X2r2 f2γ2

λ22 , (93)

with λ2k given by (81). Equation (93) is analogous to (83) and, by virtue of (86) and

(87), it can be write in the following form

1γ − 1

r fλ2Σ =

X1

γ1 − 1r1 f1λ2

1 +X2

γ2 − 1r2 f2λ2

2 , (94)

havingγRγ − 1

= Y1γ1R1

γ1 − 1+ Y2

γ2R2

γ2 − 1. (95)

Note that equation (94) is analogous to (80) of the isobaric case.Work supported by G.N.F.M of I.N.d.A.M., by Tirrenoambiente s.p.a. of Messina and by research

grants of the University of Messina.

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ABOUT FINITENESS CONDITIONSFOR COMMUTATIVE MOUFANG LOOPS

ROMAI J., v.8, no.1(2012), 43–48

Aliona GurdishTechnical University of Molddova, Chisinau, Republic of [email protected]

Abstract It is proved that a commutative Moufang loop satisfies one of the following properties:is finite; is finitely generated; has a finite (special) rank; maximum condition for itssubloops; minimum condition for its subloops if and only if this property is satisfied bythe centralizer of one of its finitely generated subloops.

Keywords: commutative Moufang loop, centralizer, finitely generated, special rank, minimum condi-tion for subloops, maximum condition for subloops.2010 MSC: 20N05.

1. INTRODUCTIONIt is known that, in some classes of groups or loops, the different finiteness con-

ditions on their centers are transferred to these groups, loops, respectively. For ex-ample, in [1] it was proved that if the centre of finitely generated nilpotent group isfinite then the group itself is finite. Further, let Ω denote one of the following classesof loops: the class of finite loops; the class of finitely generated loops; the class ofloops of finite rank; the class of loops with maximum conditions for its subloops; theclass of loops with minimum conditions for its subloops. In [2 – 4] it is proved that acommutative Moufang ZA-loop belongs to a class Ω if and only if its centre belongsto the same class Ω.

There exists a commutative Moufang loop (CML) with trivial centre [5]. Thenfor the described CML with finiteness conditions it is reasonable to use the notionof centralizer, more general that the notion of centre. This paper generalizes theaforementioned result for ZA-loops. It is proved that a CML belongs to a class Ω ifand only if the centralizer of one of its finitely generated subloops belongs to Ω.

2. THEORETICAL BACKGROUNDLet us recall some notions and results of the theory of the commutative Moufang

loops (abbreviated CMLs) from [5], which are the commutative loops characterizedby the identity x2 · yz = xy · xz.

43

44 Aliona Gurdish

The associator (a, b, c) of the elements a, b, c of the CML L is defined by theequality ab · c = (a · bc)(a, b, c). The identities:

(xy, u, v) = (x, u, v)((x, u, v), x, y)(y, u, v)((y, u, v), y, x), (1)

(x, y, z) = (y−1, x, z) = (y, x, z)−1 = (y, z, x) (2)

hold in the CML L.The centre Z(L) of the CML L is a normal subloop

Z(L) = x ∈ L|(x, y, z) = 1, ∀y, z ∈ L. The upper central series of the CML L is the series

1 = Z0 ⊆ Z1 ⊆ Z2 ⊆ . . . ⊆ Zα ⊆ . . .of the normal subloops of the CML L, satisfying the conditions:

1) Zα =∑β<α Zβ for the limit ordinal;

2) Zα+1/Zα = Z(L/Zα) for any α.If the CML possesses a central series, then this loop is called ZA-loop.If the upper central series of the ZA-loop have a finite length, then the loop is called

centrally nilpotent.The least of such a length is called the class of the central nilpotentcy.

Lemma 2.1. (Bruck-Slaby Theorem). Let n be a positive integer, n ≥ 3. Then everycommutative Moufang loop L, which can be generated by n elements, is centrallynilpotent of class at most n − 1.

Lemma 2.2. For any CML, L, with centre Z(L), the quotient loop L/Z(L) is locallyfinite 3-loop of exponent 3 and it is finite if L is finitely generated [5].

3. MAIN RESULTSThe following concept is the natural generalization of the concept of centre.Let M be a subset and H be a subloop of the CML L. The set ZH(M) = x ∈

H|x · yz = xy · z, ∀y, z ∈ M is called centralizer of the subset M into subloop H.ZH(M) is a subloop of L [6].

The (special) rank of loop L is called the least positive number rL with the follow-ing feature: any finitely generated subloop of loop L can be generated by rL elements;if there are not such numbers, then we suppose that rL = ∞.

Lemma 3.1. A centrally nilpotent CML L belongs to class Ω if and only if the cen-tralizer of some of its finitely generated subloop H also belongs to class Ω.

Proof. The necessity of lemma is obvious. To prove the sufficiency, it is enough toproceed by induction on the class of central nilpotence of CML L. The sufficiency

About finiteness conditions for commutative Moufang loops 45

may be proved by using the well known facts from [2-4]: if the center of commutativeMoufang ZA-loop belongs to class Ω, then the CML itself belongs to Ω. In [2-4] thisresult is proved in different way for various classes. We prove the sufficiency only forthe case of rank finiteness as the proof of other cases of class Ω are identical to thiscase.

Let the subloop H be generated by the set A = a1, a2, . . . , an and let the central-izer ZL(H) have a finite rank. We will suppose that the CML L is non-associative, asfor abelian groups (centrally nilpotent CML of class k = 1) the statement holds.

Let Z be the centre of the CML L and let k be the class of the central nilpotence.Obviously, Z ⊆ ZL(H), hence the rank of Z is finite. As the quotient loop L/Z iscentrally nilpotent of class k − 1, then by inductive supposition, the rank of L/Z willbe finite if the centralizer D/Z of image HZ/Z of subloop H into the quotient loopL/Z has a finite rank. We will prove this below.

Indeed, let x, y ∈ D, and let Ai = ai1 , ai2, i = 1, 2, . . . , t, be an arbitrary fixed pairof elements ai1 , ai2 ∈ A. We have (x, ai1 , ai2) ∈ Z, then by (1), it follows (xy, ai1 , ai2) =

(x, ai1 , ai2)(y, ai1 , ai2). This equality shows that the mapping x → (x, ai1 , ai2) is ahomomorphism of D into Z.

For each Ai we consider the homomorphisms ϕi(x) = (x, ai1 , ai2), x ∈ D. Obvi-ously, kerϕi = ZD(Ai). We mentioned above that the centre Z has a finite rank. Hencethe quotient loops D/ZD(Ai) are abelian groups of finite ranks. In particular they arefinitely generated. Since t is a finite integer, then the direct product

∏ti=1 D/ZD(Ai)

is a finitely generated abelian group. It is known that: if an arbitrary Abelian grouphas m generators then any of its subgroup have at most m generators. Then fromdefinition of special rank it follows that the direct product

∏ti=1 D/ZD(Ai) has a finite

rank. Further, by (1) and (2), it is easy to see that⋂t

i=1 ZD(Ai) = ZD(H).Analogously to Remak Theorem for groups [7], it may be proved that the quotient

loop D/⋂t

i=1 ZD(Ai) = D/ZD(H) is isomorphic to a subloop of the direct product∏ti=1 D/ZD(Ai). Then D/ZD(H) has a finite rank r(D/ZD(H). As ZD(H) ⊆ ZL(H),

then ZD(H) also has a finite rank r(ZD(H). Thus, from the definition of special rankit follows that and CML D have a finite rank ≤ r(D/ZD(H))r(ZD(H)).

Consequently, L/Z is a CML of finite rank r(L/Z). The centre Z has a finite rankr(Z). Then the CML L also has a finite rank r(L/Z)r(Z).

Lemma 3.2. Let H be a finitely generated subloop of the CML L. If the centralizerZL(H) belongs to class Ω then the centralizer ZL/Z(L)(Z(L)H/Z(L)) belongs to classΩ.

Proof. In [9, 10] it is proved that for a CML the condition of finite generation andmaximum condition for subloops are equivalent, and, in [3], it was proved that thisconditions are equivalent with the maximum condition for associative subloops. Fur-ther, in [2], it was proved that, for a CML, the minimum condition for subloops andthe minimum condition for associative subloops are equivalent, and for p-loops, theseconditions are equivalent with the condition of finiteness of rank [4].

46 Aliona Gurdish

Now, let Z(L) be the centre of the CML L, L = L/Z(L) and H =

Z(L)H/Z(L). Let us suppose that the centralizer ZL(H) does not belong to class S .By Lemma 2.2, the quotient loop L/Z(L) satisfies the identity x3 = 1. Then bythe aforementioned, ZL(H) contains an infinite elementary abelian 3-group B, whichdecomposes into a direct product of cyclic groups of order 3. Let A/Z(L) = A =

A1 × A2 × . . . × Ai × . . . be the maximal subgroup of B regarding to property Ai * H,and let Ai =< ai >. We denote by M(ai) a maximal subloop of the CML R =< A,H >such that ai < M(ai). As the element ai has order 3 then Ai ∩ M(ai) = 1. Every max-imal subloop of a CML is normal in this CML [9]. Let I(R) be the inner mappinggroup of the CML R. Every inner mapping of the CML is an automorphism of thisCML [5]. Then I(R)Ai = Ai. Hence the subloop Ai is normal in R. In [10] it wasproved that, if an element of order 3 of CML generates a normal subloop then thiselement belongs to the centre of this CML. Hence Ai ⊆ Z(R). A ⊆ Z(R). From hereit follows that R = AH. But A ∩ H = 1. Then from I(R)A = A it follows thatI(R)H = H, i.e. the subloop H is normal in R. Consequently, R = A × H.

The subloop H is finitely generated. Then, by Lemma 2.1, it is centrally nilpo-tent. The subloop H is also centrally nilpotent. Then the subloop R is also centrallynilpotent. As B ⊆ R and B do not belong to the class S then R does not belong toclass S , too. The inverse image of R, under the homomorphism L → L/Z(L), is AH.This CML is centrally nilpotent and does not belong to class S . Then, by Lemma3.1, the centralizer ZAH(H) does not belong to class S . We get a contradiction, asZAH(H) ⊆ ZL(H) and ZL(H) belong to class S . Consequently, the centralizer ZL(H)belongs to class S .

Lemma 3.3. Let H be a finitely generated subloop of a CML L. If the centrali-zer ZL(H) belongs to a class Ω, then the centre Z(L) belongs to Ω if the CML L isdifferent from unity loop.

Proof. If a ∈ L is an element of infinite order, then by Lemma 2.2, 1 , a3 ∈ Z(L).Let us suppose that L is a periodic CML. In these cases, L decomposes into a directproduct of its maximal p-subloops Lp, and, in addition, Lp belongs to the centre Z(L)under p , 3. Hence, in order to prove Lemma 3.1, it is sufficient to suppose that Lis a 3-loop. By Lemma 2.1, every finitely generated CML is centrally nilpotent, andconsequently we will suppose that CML L is infinite.

By analogy with the group theory [7], we say that the system Gα (α ∈ I) ofsubloops of loop G is a local system if the union

⋃α∈I Gα coincides with G and every

two members of this system are contained in a certain third member of this system.Using the definition of the local system, it is easy to prove the statement: if Gα(α ∈ I) is some local system of loop G and I = I1

⋃I2

⋃. . .

⋃Ik is a certain partition

of the set of indices I into a finite number of subsets, I j, j = 1, 2, . . . , k, then at leastone subset I j corresponds to the set of subloops Gβ, β ∈ I j, which will also be alocal system for loop G.

About finiteness conditions for commutative Moufang loops 47

Let now Lα, α ∈ I, be the local system of all finitely generated subloops of CMLL, which contain the subloop H. By Lemma 2.1, Z(Lα) , 1. For each α ∈ I, wefixed an arbitrary non-unitary element aα in the centre Z(Lα) and let K be the subloopof L generated by all chosen aα, α ∈ I. From K ⊆ ZL(H) it follows that K belongs toclass Ω. We suppose that the 3-subloop K has a finite rank. Then by [4], K satisfiesthe minimum condition for its subloops and, by [2], K = R × T , where R ⊆ Z(L) andT is a finite subloop. Further, any periodic CML is locally finite [5]. Hence, to proveLemma 3.1, it is sufficient to consider that K is a finite subloop.

Further, let us decompose the set of indices I into a finite number of subsetsI = I1

⋃I2

⋃. . .

⋃Ik by rule: β, γ ∈ I j if and only if aβ = aγ. According to the

aforementioned statement we have received that at least one of the subsets I j (e.g. I1)corresponds to the subset of subloops Lα, α ∈ I1, which will be a local system forthe CML L. Next, let us fix index α ∈ I1, and consider the set of indices S ⊆ I1,such that Lα ⊆ Lβ, β ∈ S . We notice that the set S corresponds to the set of subloopsLβ, β ∈ S , which will give a local system for CML L. Let us denote the value of thecorresponding members by b, b = aα = aβ = . . .. Then b ∈ Z(Lβ) for all β ∈ S and,consequently, b ∈ Z(L).

Theorem 3.1. A CML L belongs to class Ω if and only if the centralizer of some ofits finitely generated subloop H also belongs to class Ω.

Proof. The “necessity” is obvious. Conversely, let the centralizer ZL(H) belong toclass Ω. We denote L = L/Z(L), H = HZ(L)/Z(L). Any periodic CML is locallyfinite [5]. Then from Lemma 2.2 it follows that the subloop H is finite. From Lemmas3.1, 3.2, it follows that the upper central series of CML L has the form 1 ⊂ Z1(L) ⊂. . . ⊂ Zk(L) ⊂ . . ., and, for a natural number n, Zn(L) , Zn+1(L) if Zn(L) , L. AsH is finite, then, for some k, H * Zk−1(L) but H ⊆ Zk(L). Hence ZL(H) = L, whereL = L/Zk, H = HZk/Zk.

By Lemma 2.2, L is a 3-loop and, by Lemma 3.2, L belongs to the class Ω. In [4] itwas proved that the minimum condition for subloops and the condition of finitenessrank are equivalent for the CML L. In this case L = R × T , where R ⊆ Z(L),and T is a finite CML which, by Lemma 2.1, is centrally nilpotent. Then CMLL = L/Zk = (L/Z)/(Zk/Z) L/Zk is also centrally nilpotent. From central nilpotenceof L/Zk it follows the central nilpotence of L, and by Lemma 3.1, L belongs to classΩ.

References.

[1] P. Hall, On the finiteness of certain soluble groups, Proc. London Math. Soc., 1959, 9, 36, 595 –622.

[2] N. I. Sandu, Commutative Moufang loops with minimum condition for subloops I, BuletinulAcademiei de Stiinte a Republicii Moldova, Matematica, 2003, 3(43), 25 - 40.

[3] A. Babiy, N. Sandu, The commutative Moufang loops with maximum conditions for subloops,Buletinul Academiei de Stiinte a Republicii Moldova, Matematica, 2006, 2(51), 53 – 61.

48 Aliona Gurdish

[4] A. Babiy, N. Sandu, About commutative Moufang loops of finite special rank, BuletinulAcademiei de Stiinte a Republicii Moldova, Matematica, 2006, 1(50), 92 – 100.

[5] Bruck R. H., A survey of binary systems. Springer Verlag, Berlin-Heidelberg, 1958.

[6] Sandu N. I.: Commutative Moufang loops with finite classes of conjugate elements, Mat. Zametki,2003, 73, 2, 269 – 280 (In Russian).

[7] Kurosh A. G., Group Theory, Moscow, Nauka, 1967 (In Russian).

[8] Evans T., Identities and Relations in Commutative Moufang Loops. J. Algebra, 31, 1974, 508 –513.

[9] Sandu N. I., About Centrally Nilpotent Commutative Moufang Loops. Quasigroups and Loops(Matem. issled., 51), Kishinev, 1979, 145 – 155 (In Russian).

[10] Sandu N. I., Commutative Moufang Loops with Minimum Condition for Subloops II. BuletinulAcademiei de Stiinte a Republicii Moldova, Matematica, 2004, 2(45), p. 33–48.

MONOTONE ITERATIVE TECHNIQUE FORINTEGRAL BOUNDARY VALUE PROBLEMSOF SINGULAR DIFFERENTIAL EQUATIONSON THE WHOLE LINE

ROMAI J., v.8, no.1(2012), 49–72

Yuji Liu1, Xingyuan Liu2

1Department of Mathematics, Guangdong University of Business Studies, Guangzhou,P. R. China2Department of Mathematics, Sahoyang University, Hunan Shaoyang, P. R. [email protected], [email protected]

Abstract This paper deals with the existence of positive solutions for some boundary value prob-lems of the singular differential equations on the whole line. Our approach is based onthe fixed point theorem and the monotone iterative technique. Without the assumption ofthe existence of lower and upper solutions, we obtain not only the existence of nonneg-ative solutions for the problems, but also establish iterative schemes for approximatingthe solutions.

Keywords: second order differential equation with p−Laplacian on the whole line, integral type bound-ary value problem, solution, fixed point theorem.2010 MSC: 34B10, 34B15, 35B10.

1. INTRODUCTIONDifferential equations governed by nonlinear differential operators have been widely

studied. In this setting the most investigated operator is the classical p−Laplacian,that is Φp(x) = |x|p−2x with p > 1, which, in recent years, has been general-ized to other types of differential operators, that preserve the monotonicity of thep−Laplacian, but are not homogeneous. These more general operators, which areusually referred to as Φ−Laplacian (or quasi-Laplacian), are involved in some mod-els, e.g. in non-Newtonian fluid theory, diffusion of flows in porous media, nonlinearelasticity and theory of capillary surfaces. The related nonlinear differential equationhas the form

[Φ(x′)]′ = f (t, x, x′), t ∈ (−∞,+∞),

where Φ : R → R is an increasing homeomorphism such that Φ(0) = 0. Morerecently, equations involving other types of differential operators have been studiedfrom a different point of view arising from other types of models, e.g. reaction diffu-sion equations with non-constant diffusivity and porous media equations. This leadsto consider nonlinear differential operators of the type [a(t, x, x′)Φ(x′)]′, where a is a

49

50 Yuji Liu, Xingyuan Liu

positive continuous function. For a comprehensive bibliography on this subject, seee.g. [1-10].

Philos and Purnaras [1] study a class of the boundary value problem (BVP forshort) for the second order nonlinear ordinary differential equations on the wholeline. Two existence results are established in [1]. The first theorem is established bythe use of the Schauder theorem and concerns the existence of solutions, while thesecond theorem is concerned with the existence and uniqueness of solutions and isderived by the Banach contraction principle.

In [2], Bianconi and Papalini investigate the existence of solutions of the followingboundary value problem

[Φ(x′(t))]′ + a(t, x(t))b(x(t), x′(t)) = 0, t ∈ R,limt→−∞ x(t) =: x(−∞) = 0,limt→+∞ x(t) =: x(+∞) = 1,

(1)

where Π is a monotone function which generalizes the one-dimensional p−Laplacanoperator. The criteria for the existence and non-existence of solutions of BVP(1.1) isestablished.

In [3,4], Avramescu and Vladimirescu study the following boundary value prob-lem

x′′(t) + 2 f (t)x′(t) + x(t) + g(t, x(t)) = 0, t ∈ R,limt→±∞ x(t) =: x(±∞) = 0,limt→±∞ x′(t) =: x(±∞) = 0,

(2)

where f and g are given functions. The existence of solutions of BVP(1.2) is ob-tained.

In [5], Avramescu and Vladimirescu study the following boundary value problem

x′′(t) + f (t, x(t), x′(t)) = 0, t ∈ R,limt→−∞ x(t) = limt→+∞ x(t),limt→−∞ x′(t) = limt→+∞ x′(t),

(3)

under adequate hypothesis and using the Bohnenblust-Karlin fixed point theorem, theexistence of solutions of BVP(1.3) is established.

Cabada and Cid [6] prove the solvability of the boundary value problem on thewhole line

[Φ(x′(t)]′ + f (t, x(t), x′(t)) = 0, t ∈ R,limt→−∞ x(t) = −1,limt→+∞ x(t) = 1,

(4)

where f is a continuous function, Φ : (−a, a) → R is a homeomorphism with a ∈(0,+∞), i.e., Φ is singular.

Monotone iterative technique for integral boundary value problems... 51

Calamai [7] and Cristina Marcelli and Papalini [8] discuss the solvability of thefollowing strongly nonlinear BVP:

[a(x(t))Φ(x′(t)]′ + f (t, x(t), x′(t)) = 0, t ∈ R,limt→−∞ x(t) = α,limt→+∞ x(t) = β,

(5)

where α < β, Φ is a general increasing homeomorphism with bounded domain (sin-gular Φ-Laplacian), a is a positive, continuous function and f is a Caratheodorynonlinear function. conditions for the existence and non-existence of heteroclinic so-lutions in terms of the behavior of y → f (t, x, y) and y → Φ(y) as y → 0, and oft → f (t, x, y) as |t| → +∞. The approach is based on fixed point techniques suitablycombined to the method of upper and lower solutions.

Most of the known papers only consider the existence and uniqueness of positivesolutions of various boundary value problems. A natural question which arises is”How can we find the solutions when they are known to exist?”

To fill this gap, we consider the following more generalized boundary value prob-lem for the second order differential equation on the whole line with p−Laplacian

[Φ(ρ(t)a(t, x(t), x′(t))x′(t))]′ + f (t, x(t), x′(t)) = 0, t ∈ R,limt→−∞ ρ(t)x′(t) = −

∫ +∞−∞ g(s, x(s), x′(s))ds,

limt→+∞ x(t) =∫ +∞−∞ h(s, x(s), x′(s))ds,

(6)

where• f , g, h defined on R3 are nonnegative Caratheodory functions, f (t, 0, 0) . 0 on

each subinterval of R.• ρ ∈ C(R, (0,∞)) satisfies

∫ +∞

0

1ρ(s)

ds < +∞,∫ +∞

−∞

1ρ(s)

ds = +∞.

• a : R × R × R → (0,+∞) is continuous and satisfies that there exist constantsm,M > 0 such that

M ≥ a(t,(1 +

∫ +∞

t

1ρ(s)

ds)

x,yρ(t)

)≥ m, t ∈ R, x ∈ R, y ∈ R

and for each r > 0, |x|, |y| ≤ r imply that a(t,(1 +

∫ +∞t

1ρ(s) ds

)x, y

ρ(t)

)→ a±∞ uni-

formly as t → ±∞ and

(x, y)→ a(t,(1 +

∫ +∞

t

1ρ(s)

ds)

x,yρ(t)

)is uniformly continuous on [−r, r] × [−r, r].

• Φ : R → R is a sup-multiplicative-like function and its inverse function isdenoted by Φ−1.


The purpose of this paper is to establish the existence of positive solutions forBVP(1.6). Our approach is based on the fixed point theorem and the monotone itera-tive technique. Without the assumption of the existence of lower and upper solutions,we obtain not only the existence of positive solutions for the problems, but also es-tablish iterative schemes for approximating the solutions.

The remainder of this paper is organized as follows: the preliminary results aregiven in Section 2, the main results are presented in Section 3.

2. PRELIMINARY RESULTSIn this section, we first present some background definitions in Banach spaces and

state an important three fixed point theorem. Then the main results are given andproved.

Definition 2.1. F defined on R3 is called a quasi-Carathedory function if(i) t → f

(t,(1 +

∫ +∞t

1ρ(s) ds

)x, 1

ρ(t) y)

is measurable for any (x, y) ∈ R2,

(ii) (x, y)→ f(t,(1 +

∫ +∞t

1ρ(s) ds

)x, 1

ρ(t) y)

is continuous for a.e. t ∈ R,

(iii) for each r > 0, there exists nonnegative function φr ∈ L1(R) such that|x|, |y| ≤ r implies

∣∣∣∣∣∣ f(t,(1 +

∫ +∞

t

1ρ(s)

ds)

x,1ρ(t)

y)∣∣∣∣∣∣ ≤ φr(t), a.e.t ∈ R.

Definition 2.2. [11]. An odd homeomorphism Φ of the real line R onto itself is calleda sup-multiplicative-like function if there exists a homeomorphism ω of [0,+∞) ontoitself which supports Φ in the sense that for all v1, v2 ≥ 0 it holds

Φ(v1v2) ≥ ω(v1)Φ(v2). (7)

ω is called the supporting function of Φ.

Remark 2.1. Note that any sup-multiplicative function is sup-multiplicative-likefunction. Also any function of the form

Φ(u) :=k∑

j=0

c j|u| ju, u ∈ R

is sup-multiplicative-like, provided that c j ≥ 0. Here a supporting function is definedby ω(u) := minuk+1, u, u ≥ 0.

Remark 2.2. It is clear that a sup-multiplicative-like function Φ and any corre-sponding supporting function ω are increasing functions vanishing at zero and more-over their inverses Φ−1 and ν respectively are increasing and such that

Φ−1(w1w2) ≤ ν(w1)Φ−1(w2), (8)


for all w1,w2 ≥ 0. ν s also called the supporting function of Φ−1.

In the sequel, we suppose that Φ is a sup-multiplicative-like function with its sup-porting function ω. The inverse function of Φ is denoted by Φ−1, whose supportingfunction is denoted by ν.

Choose

X =

x ∈ C1(R) :

x(t)1+

∫ +∞t

1ρ(s) ds

is bounded on R and the limits

limt→+∞ x(t), limt→−∞ ρ(t)x′(t),and limt→+∞ ρ(t)x′(t) exist

.

For x ∈ X1, define the norm of x by

||x|| = max

supt∈R

|x(t)|1 +

∫ +∞t

1ρ(s) ds

, supt∈R

ρ(t)|x′(t)| .

One can prove that X is a Banach space with the norm ||x|| for x ∈ X.

For x, y ∈ X, we call x ≤ y if x(t) ≤ y(t) and |x′(t)| ≤ |y′(t)| hold for all t ∈ R. Then(X, || · ||,≤, ) is a partially ordered Banach space.

Denote τ(t) =∫ +∞

tdsρ(s) . Define the cone P in X by

P = x ∈ X : x(t) ≥ 0, t ∈ R.For x ∈ X, define T x by

(T x)(t) =

∫ +∞

−∞h(u, x(u), x′(u))du

+

∫ +∞

t

Φ−1(Φ

(a−

∫ +∞−∞ g(u, x(u), x′(u))ds

)+

∫ s−∞ f (u, x(u), x′(u))du

)

a(s, x(s), x′(s))ρ(s)ds,

t ∈ (−∞,∞).

Lemma 2.1. Suppose that f , g, h are Caratheodory functions. Then(i) T : X → X is well defined and T x satisfies

[Φ(ρ(t)a(t, x(t), x′(t))(T x)′(t))]′ + f (t, x(t), x′(t)) = 0, t ∈ (−∞,+∞),limt→−∞ ρ(t)(T x)′(t) = −

∫ +∞−∞ g(s, x(s), x′(s))ds,

limt→+∞(T x)(t) =∫ +∞−∞ h(s, x(s), x′(s))ds,

(9)

(ii) T x ∈ P for each x ∈ P;(iii) x ∈ X is a positive solution of BVP(1.6) if and only if x ∈ P is a solution of

the operator equation x = T x.


Proof. (i) For x ∈ X, we know that

r = max

supt∈R

|x(t)|1 +

∫ +∞t

1ρ(s) ds

, supt∈R

ρ(t)|x′(t)| < +∞.

Since f , g, h are Caratheodory functions, there exist φr ∈ L1(R) such that

| f (t, x(t), x′(t))| =∣∣∣∣∣∣∣∣f

t,(1 +

∫ +∞

t

dsρ(s)

)x(s)

1 +∫ +∞

tdsρ(s)

,1ρ(t)

ρ(s)x′(s)

∣∣∣∣∣∣∣∣≤ φr(t)

and| f (t, x(t), x′(t))| ≤ φr(t), |g(t, x(t), x′(t))| ≤ φr(t), t ∈ R.

Then∫ +∞

−∞| f (u, x(u), x′(u))|du,

∫ +∞

−∞|g(u, x(u), x′(u))|du,

∫ +∞

−∞|h(u, x(u), x′(u))|du

are convergent. So T x ∈ C0(R) and

(T x)′(t) = −Φ−1

(Φ

(a−

∫ +∞−∞ g(u, x(u), x′(u))ds

)+

∫ t−∞ f (u, x(u), x′(u))du

)

a(t, x(t, ), x′(t))ρ(t).

It is easy to see that (T x)′ ∈ C0(R) and

(T x)(t)

1 +∫ +∞

t1ρ(s) ds

is bounded on R and there exist the limits

limt→+∞(T x)(t) and lim

t→−∞ ρ(t)(T x)′(t), limt→+∞ ρ(t)(T x)′(t).

Hence T x ∈ X. So T : X → X is well defined. It is easy to show that (2.9) holds.

(ii) Since f , g, h are nonnegative Caratheodory functions, we get that T : P→ P.

(iii) It is easy to show that x ∈ X is a positive solution of BVP(1.6) if and only ifx ∈ P is a solution of the operator equation x = T x.

Lemma 2.2. T : X → X is completely continuous.

Proof. From Lemma 2.1, T : X → X is well defined. Now we prove that T iscontinuous and maps bounded subsets into relatively compact sets.


First, we show that T is continuous. Let xn → x0 as n→ +∞ in X, then we get

r = supn=0,1,2,···

max

supt∈R

|xn(t)|1 +

∫ +∞t

1ρ(s) ds

, supt∈R

ρ(t)|x′n(t)| < +∞.

Because f , g, h are Caratheodory functions, then there exists φr ∈ L1(−∞,+∞) suchthat

∣∣∣ f (t, xn(t), x′n(t)

)∣∣∣ =

∣∣∣∣∣∣∣∣f

t,(1 +

∫ +∞

t

dsρ(s)

)xn(s)

1 +∫ +∞

tdsρ(s)

,1ρ(t)

ρ(s)x′n(s)

∣∣∣∣∣∣∣∣≤ φr(t), t ∈ R,∣∣∣g (

t, xn(t), x′n(t))∣∣∣ ≤ φr(t), t ∈ R,∣∣∣h (

t, xn(t), x′n(t))∣∣∣ ≤ φr(t), t ∈ R.

Then

∫ +∞

−∞f (s, x(s), x′(s))ds,

∫ +∞

−∞g(s, x(s), x′(s))ds and

∫ +∞

−∞h(s, x(s), x′(s))ds

are convergent.We need to prove that T xn → T x0 as n→ ∞.From the definition of T , we find that

(T xn)(t) =

∫ +∞

−∞h(u, xn(u), x′n(u))du

+

∫ +∞

t

Φ−1(Φ

(a−

∫ +∞−∞ g(u, xn(u), x′n(u))ds

)+

∫ s−∞ f (u, xn(u), x′n(u))du

)

a(s, xn(s, ), x′n(s))ρ(s)ds,

and

ρ(t)(T xn)′(t)

= −Φ−1

(Φ

(a−

∫ +∞−∞ g(u, xn(u), x′n(u))ds

)+

∫ t−∞ f (u, xn(u), x′n(u))du

)

a(t, xn(t, ), x′n(t))


for n = 0, 1, 2, · · · . By the definition of G, we get|(T xn)(t) − (T x0)(t)|

1 +∫ +∞

tdsρ(s)

≤ 1

1 +∫ +∞

tdsρ(s)

∫ +∞

−∞|h(u, xn(u), x′n(u)) − h(u, x0(u), x′0(u))|du

+1

1 +∫ +∞

tdsρ(s)

×

∫ +∞

t

∣∣∣∣∣∣∣∣Φ−1

(Φ

(a−

∫ +∞−∞ g(u, xn(u), x′n(u))du

)+

∫ s−∞ f (u, xn(u), x′n(u))du

)

a(s, xn(s), x′n(s))ρ(s)

−Φ−1

(Φ

(a−

∫ +∞−∞ g(u, x0(u), x′0(u))du

)+

∫ s−∞ f (u, x0(u), x′0(u))du

)

a(s, x0(s), x′0(s))ρ(s)

∣∣∣∣∣∣∣∣ds

≤∫ +∞


+1

1 +∫ +∞

tdsρ(s)

∫ +∞

t

1ρ(s)a(s, xn(s, ), x′n(s))

×∣∣∣∣∣∣Φ−1

(Φ

(a−

∫ +∞

−∞g(u, xn(u), x′n(u))du

)+

∫ s

−∞f (u, xn(u), x′n(u))du

)

−Φ−1(Φ

(a−

∫ +∞

−∞g(u, x0(u), x′0(u))du

)+

∫ s

−∞f (u, x0(u), x′0(u))du

)∣∣∣∣∣∣ ds

+1

1 +∫ +∞

tdsρ(s)

×

∫ +∞

t

Φ−1(Φ

(a−

∫ +∞−∞ g(u, x0(u), x′0(u))du

)+

∫ s−∞ f (u, x0(u), x′0(u))du

)

ρ(s)

×|a(s, xn(s, ), x′n(s)) − a(s, x0(s, ), x′0(s))|a(s, xn(s), x′n(s))a(s, x0(s, ), x′0(s))

ds

≤∫ +∞


+1m

1

1 +∫ +∞

tdsρ(s)

∫ +∞

t

1ρ(s)

×∣∣∣∣∣∣Φ−1

(Φ

(a−

∫ +∞


)+

∫ s


)

−Φ−1(Φ

(a−

∫ +∞

−∞g(u, x0(u), x′0(u))ds

)+

∫ s

−∞f (u, x0(u), x′0(u))du

)∣∣∣∣∣∣ ds


+1

1 +∫ +∞

tdsρ(s)

∫ +∞

t

Φ−1(Φ

(a−

∫ +∞−∞ φr(u)du

)+

∫ +∞−∞ φr(u)du

)

ρ(s)

×|a(s, xn(s, ), x′n(s)) − a(s, x0(s), x′0(s))|m2 ds

≤∫ +∞


+1m

1

1 +∫ +∞

tdsρ(s)

∫ +∞

t

1ρ(s)

×∣∣∣∣∣∣Φ−1

(Φ

(a−

∫ +∞


)+

∫ s


)

−Φ−1(Φ

(a−

∫ +∞

−∞g(u, x0(u), x′0(u))ds

)+

∫ s

−∞f (u, x0(u), x′0(u))du

)∣∣∣∣∣∣ ds

+Φ−1(Φ

(a−

∫ +∞

−∞φr(u)du

)+

∫ +∞

−∞φr(u)du

)×

|a(s, xn(s), x′n(s)) − a(s, x0(s, ), x′0(s))|m2 ds.

For any ε > 0, since∫ +∞

−∞|h(u, xn(u), x′n(u)) − h(u, x0(u), x′0(u))|du ≤ 2

∫ +∞

−∞φr(s)ds,

by the Lebesgue dominated convergence theorem, there exists N1 > 0 such that∫ +∞

−∞|h(u, xn(u), x′n(u)) − h(u, x0(u), x′0(u))|du < ε, n > N1. (10)

Similarly, we get

a−∫ +∞

−∞g(u, xn(u), x′n(u))du→ a−

∫ +∞

−∞g(u, x0(u), x′0(u))du as n→ ∞,

and∫ s

−∞f (u, xn(u), x′n(u))du→

∫ s

−∞f (u, x0(u), x′0(u))du uniformly as n→ ∞.

Since ∣∣∣∣∣∣a−∫ +∞


∣∣∣∣∣∣ ≤ |a−|∫ +∞

−∞φr(s)ds =: r1,


and ∣∣∣∣∣∫ s


∣∣∣∣∣ ≤∫ +∞

−∞φr(s)ds =: r2,

and Φ is uniformly continuous on [−r1, r1], Φ−1 is uniformly continuous on [−Φ(r1)−r2,Φ(r1) + r2], we can easily know that there exists N2 > 0 such that

∣∣∣∣Φ−1(Φ

(a−

∫ +∞−∞ g(u, xn(u), x′n(u))du

)+

∫ s−∞ f (u, xn(u), x′n(u))du

)

−Φ−1(Φ

(a−

∫ +∞−∞ g(u, x0(u), x′0(u))ds

)+

∫ s−∞ f (u, x0(u), x′0(u))du

)∣∣∣∣

< mε, n > N2.

(11)

By|a(s, xn(s, ), x′n(s)) − a(s, x0(s, ), x′0(s))|

=

∣∣∣∣∣∣a(s,

(1 +

∫ +∞t

dsρ(s)

)xn(s)

1+∫ +∞

tdsρ(s), 1ρ(t)ρ(t)x′n(s)

)

−a(t,(1 +

∫ +∞t

dsρ(s)

)x0(t)

1+∫ +∞

tdsρ(s), 1ρ(t)ρ(t)x′0(t)

)∣∣∣∣∣∣ ,and

a(t,(1 +

∫ +∞

t

dsρ(s)

)x,

1ρ(t)

y)

is uniformly continuous on [−r, r] × [−r, r]

we get the there exists δ > 0 such that∣∣∣∣a

(t,(1 +

∫ +∞t

dsρ(s)

)x1,

1ρ(t) y1

)− a

(t,(1 +

∫ +∞t

dsρ(s)

)x2,

1ρ(t) y2

)∣∣∣∣

< m2ε

Φ−1(Φ(a−

∫ +∞−∞ φr(u)du

)+∫ +∞−∞ φr(u)du

)

holds for |x1 − x2| < δ, |y1 − y2| < δ. From xn → x0, we get that there exists N3 > 0such that ∣∣∣∣∣∣∣∣

xn(s)

1 +∫ +∞

tdsρ(s)

− x0(s)

1 +∫ +∞

tdsρ(s)

∣∣∣∣∣∣∣∣< δ,

and|ρ(t)x′n(s) − ρ(t)x′0(s)| < δ

hold for n > N3 and all t ∈ R. Hence n > N3 implies that∣∣∣∣a (

t, xn(t), x′n(t)) − a

(t, x0(t), x′0(t)

)∣∣∣∣

< m2ε

Φ−1(Φ(a−

∫ +∞−∞ φr(u)du

)+∫ +∞−∞ φr(u)du

) .(12)


It follows from (2.10)-(2.12) that n > maxN1,N2,N3 implies that

|(T xn)(t) − (T x0)(t)|1 +

∫ +∞t

dsρ(s)

< ε + ε + ε = 3ε. (13)

Furthermore, we have

ρ(t)|(T xn)′(t) − (T x0)′(t)|

≤∣∣∣∣∣∣∣∣Φ−1

(Φ

(a−

∫ +∞−∞ g(u, xn(u), x′n(u))ds

)+

∫ t−∞ f (u, xn(u), x′n(u))du

)

a(t, xn(t, ), x′n(t))

−Φ−1

(Φ

(a−

∫ +∞−∞ g(u, x0(u), x′0(u))ds

)+

∫ t−∞ f (u, x0(u), x′0(u))du

)

a(t, x0(t, ), x′0(t))

∣∣∣∣∣∣∣∣2∫ +∞−∞ Mr(s)ds∫ +∞−∞

drρ(r)

+

∫ +∞

−∞Mr(s)ds.

By the same methods used above, we get

|ρ(t)(T xn)′(t) − ρ(t)(T x0)′(t)| → 0 uniformly as n→ +∞, (14)

So, (2.13) and (2.14) imply that T xn → T x0 as n→ ∞. So T is continuous.

Second, we show that T maps bounded sets into relatively compact sets.Recall W ⊂ X is relatively compact if(i) it is bounded,(ii) both 1

1+τ(t) W and ρ(t)W are equi-continuous on any closed subinterval [e, f ]of (−∞,+∞),

(iii) both 11+τ(t) W and ρ(t)W are equi-convergent at t = −∞,

(iv) both 11+τ(t) W and ρ(t)W are equi-convergent at t = +∞.

Let Ω be any bounded subset of X. Then there exists r > 0 such that ||x|| ≤ r for allx ∈ Ω. Because f , g, h are Caratheodory functions, then there exists φr ∈ L1(−∞,+∞)such that

∣∣∣ f (t, x(t), x′(t)

)∣∣∣ ≤ φr(t), t ∈ R,∣∣∣g (t, x(t), x′(t)

)∣∣∣ ≤ φr(t), t ∈ R,∣∣∣h (t, x(t), x′(t)

)∣∣∣ ≤ φr(t), t ∈ R,a(t, x(t), x′(t)) ≥ m, t ∈ R.

(i) We show that TΩ is bounded.


Obviously, for x ∈ Ω, one sees from the definition of G that

|(T x)(t)|1+τ(t) = 1

1+τ(t)

∣∣∣∣∫ +∞−∞ h(u, x(u), x′(u))du

+∫ +∞

t

Φ−1(Φ(a−

∫ +∞−∞ g(u,x(u),x′(u))du

)+∫ s−∞ f (u,x(u),x′(u))du

)

a(s,x(s,),x′(s))ρ(s) ds∣∣∣∣∣

≤∫ +∞−∞ φr(s)ds + 1

1+τ(t)

∫ +∞t

Φ−1(Φ(a−

∫ +∞−∞ φr(s)ds

)+∫ +∞−∞ φr(s)

)

mρ(s) ds

≤∫ +∞−∞ φr(s)ds + 1

mΦ−1(Φ

(a−

∫ +∞−∞ φr(s)ds

)+

∫ +∞−∞ φr(s)

).


ρ(t)|(T x)′(t)| = Φ−1(Φ(a−

∫ +∞−∞ g(u,x(u),x′(u))ds

)+∫ s−∞ f (u,x(u),x′(u))du

)

a(t,x(t,),x′(t))

≤ Φ−1(Φ(a−

∫ +∞−∞ φr(s)ds

)+∫ +∞−∞ φr(u)du

)

m .

We get that TΩ is bounded.(ii) We show that both 1

1+τ(t) TΩ and ρ(t)(T x)′ : x ∈ Ω are equi-continuous onany closed subinterval [a, b] of (−∞,+∞),

For t1, t2 ∈ [a, b] with t1 < t2, one has

∣∣∣∣ (T x)(t1)1+τ(t1) − (T x)(t2)

1+τ(t2)

∣∣∣∣

≤∣∣∣∣ 11+τ(t1) − 1

1+τ(t2)

∣∣∣∣∫ +∞−∞ |h(s, x(s), x′(s))|ds

+

∣∣∣∣∣ 11+τ(t1)

∫ +∞t1

Φ−1(Φ(a−

∫ +∞−∞ g(u,x(u),x′(u))du

)+∫ s−∞ f (u,x(u),x′(u))du

)

a(s,x(s,),x′(s))ρ(s) ds

− 11+τ(t2)

∫ +∞t2

Φ−1(Φ(a−

∫ +∞−∞ g(u,x(u),x′(u))du

)+∫ s−∞ f (u,x(u),x′(u))du

)

a(s,x(s,),x′(s))ρ(s) ds∣∣∣∣∣

≤ |τ(t1) − τ(t2)|∫ +∞−∞ φr(s)ds

+ 11+τ(t1)

∫ t2t1

Φ−1(Φ(a−

∫ +∞−∞ g(u,x(u),x′(u))du

)+∫ s−∞ f (u,x(u),x′(u))du

)


∣∣∣∣ 11+τ(t1) − 1

1+τ(t2)

∣∣∣∣∫ +∞

t2

Φ−1(Φ(a−

∫ +∞−∞ g(u,x(u),x′(u))du

)+∫ s−∞ f (u,x(u),x′(u))du

)



≤ |τ(t1) − τ(t2)|∫ +∞−∞ φr(s)ds

+∫ t2

t11

mρ(s) dsΦ−1(Φ

(a−

∫ +∞−∞ φr(s)ds

)+

∫ +∞−∞ φr(s)ds

)

|τ(t1)−τ(t2)|1+τ(t2)

∫ +∞t2

1mρ(s) dsΦ−1

(Φ

(a−

∫ +∞−∞ φr(s)ds

)+

∫ +∞−∞ φr(s)ds

)

≤ |τ(t1) − τ(t2)|∫ +∞−∞ φr(s)ds

+∫ t2

t11ρ(s) ds 1

mΦ−1(Φ

(a−

∫ +∞−∞ φr(s)ds

)+

∫ +∞−∞ φr(s)ds

)

|τ(t1) − τ(t2)| 1mΦ−1

(Φ

(a−

∫ +∞−∞ φr(s)ds

)+

∫ +∞−∞ φr(s)ds

)

→ 0 uniformly as t1 → t2

and

|ρ(t1)(T x)′(t1) − ρ(t2)|(T x)′(t2)|

=

∣∣∣∣∣∣∣∣Φ−1

(Φ

(a−

∫ +∞−∞ g(u, x(u), x′(u))ds

)+

∫ t1−∞ f (u, x(u), x′(u))du

)

a(t1, x(t1), x′(t1))

−Φ−1

(Φ

(a−

∫ +∞−∞ g(u, x(u), x′(u))ds

)+

∫ t2−∞ f (u, x(u), x′(u))du

)

a(t2, x(t2), x′(t2))

∣∣∣∣∣∣∣∣

≤∣∣∣∣∣∣∣∣Φ−1

(Φ

(a−

∫ +∞−∞ g(u, x(u), x′(u))ds

)+

∫ t1−∞ f (u, x(u), x′(u))du

)

a(t1, x(t1), x′(t1))

−Φ−1

(Φ

(a−

∫ +∞−∞ g(u, x(u), x′(u))ds

)+

∫ t2−∞ f (u, x(u), x′(u))du

)

a(t1, x(t1), x′(t1))

∣∣∣∣∣∣∣∣

+

∣∣∣∣∣1

a(t1, x(t1), x′(t1))− 1

a(t2, x(t2), x′(t2))

∣∣∣∣∣

×Φ−1(Φ

(a−

∫ +∞

−∞g(u, x(u), x′(u))ds

)+

∫ t2

−∞f (u, x(u), x′(u))du

)

≤∣∣∣∣∣∣∣∣Φ−1

(Φ

(a−

∫ +∞−∞ g(u, x(u), x′(u))ds

)+

∫ t1−∞ f (u, x(u), x′(u))du

)

m

−Φ−1

(Φ

(a−

∫ +∞−∞ g(u, x(u), x′(u))ds

)+

∫ t2−∞ f (u, x(u), x′(u))du

)

m

∣∣∣∣∣∣∣∣


+1

m2

∣∣∣a(t1, x(t1), x′(t1)) − a(t2, x(t2), x′(t2))∣∣∣

×Φ−1(Φ

(a−

∫ +∞

−∞φr(s)ds

)+

∫ +∞

−∞φr(s)ds

)

=

∫ t2t1φr(s)ds

m+

1m2

∣∣∣a(t1, x(t1), x′(t1)) − a(t2, x(t2), x′(t2))∣∣∣

×Φ−1(Φ

(a−

∫ +∞

−∞φr(s)ds

)+

∫ +∞

−∞φr(s)ds

)

→ 0 uniformly as t1 → t2.

It follows that both TΩ1+τ(t) and ρ(t)(T x)′ : x ∈ Ωare equi-continuous any subinterval

[a, b].

(iii) We show that both 11+τ(t) TΩ and ρ(t)(T x)′ : x ∈ Ω are equi-convergent at

t = −∞,One sees that

∣∣∣∣∣∣∣(T x)(t)1 + τ(t)

−a−

∫ +∞−∞ g(s, x(s), x′(s))ds

a(t, x(t), x′(t))

∣∣∣∣∣∣∣

≤∫ +∞−∞ h(s, x(s), x′(s))ds

1 + τ(t)

+

∣∣∣∣∣∣∣∣∣∣

∫ +∞t

Φ−1(Φ(a−

∫ +∞−∞ g(u,x(u),x′(u))ds

)+∫ s−∞ f (u,x(u),x′(u))du

)


1 + τ(t)

−a−

∫ +∞−∞ g(s, x(s), x′(s))ds

a(t, x(t), x′(t))

∣∣∣∣∣∣∣

≤∫ +∞−∞ φr(s)ds

1 + τ(t)+

11 + τ(t)

+

∣∣∣∣∣∣∣∣∣∣

∫ +∞t

Φ−1(Φ(a−

∫ +∞−∞ g(u,x(u),x′(u))ds

)+∫ s−∞ f (u,x(u),x′(u))du

)


1 + τ(t)

−∫ +∞

t

Φ−1(Φ(a−

∫ +∞−∞ g(s,x(s),x′(s))ds

))

a(t,x(t),x′(t))ρ(s) ds

1 + τ(t)

∣∣∣∣∣∣∣∣∣∣


=

∫ +∞−∞ φr(s)ds

1 + τ(t)+

11 + τ(t)

+1m

11 + τ(t)

×∫ +∞

t

∣∣∣∣∣∣∣∣Φ−1

(Φ

(a−

∫ +∞−∞ g(u, x(u), x′(u))ds

)+

∫ s−∞ f (u, x(u), x′(u))du

)

ρ(s)

−Φ−1

(Φ

(a−

∫ +∞−∞ g(s, x(s), x′(s))ds

))

ρ(s)

∣∣∣∣∣∣∣∣ds.

For any ε > 0, since Φ−1 is uniformly continuous on [−Φ(r1) − r2,Φ(r1) + r2], thereexists δ > 0 such that

|Φ−1(x) − Φ−1(y)| < ε, x, y ∈ [−Φ(r1) − r2,Φ(r1) + r2], |x − y| < δ.By ∣∣∣∣Φ

(a−

∫ +∞−∞ g(u, x(u), x′(u))ds

)+

∫ t−∞ f (u, x(u), x′(u))du

−Φ(a−

∫ +∞−∞ g(s, x(s), x′(s))ds

)∣∣∣∣

≤∫ t−∞ φr(s)ds→ 0 uniformly as t → −∞,

So there exists T1 > 0 such that t < −T1 implies that∣∣∣∣Φ

(a−

∫ +∞−∞ g(u, x(u), x′(u))ds

)+

∫ t−∞ f (u, x(u), x′(u))du

−Φ(a−

∫ +∞−∞ g(s, x(s), x′(s))ds

)∣∣∣∣ < δ

holds uniformly for x ∈ Ω. Then t < −T1 implies that∣∣∣∣Φ−1

(Φ

(a−

∫ +∞−∞ g(u, x(u), x′(u))ds

)+

∫ s−∞ f (u, x(u), x′(u))du

)

−Φ−1(Φ

(a−

∫ +∞−∞ g(s, x(s), x′(s))ds

))∣∣∣∣ < ε

holds uniformly for x ∈ Ω. It is easy to see that there exists T2 > 0 such that∫ +∞−∞ φr(s)ds

1 + τ(t)+

11 + τ(t)

< ε, t < −T1.

Hence t < −min−T1,−T2 implies that∣∣∣∣∣ (T x)(t)

1+τ(t) −a−

∫ +∞−∞ g(s,x(s),x′(s))ds

a(t,x(t),x′(t))

∣∣∣∣∣ < ε + 1m

11+τ(t)

∫ +∞t

ερ(s) ds <

(1 + 1

m

)ε. (15)


Furthermore, we have∣∣∣∣∣∣ρ(t)(T x)′(t) +

∫ +∞

−∞g(u, x(u), x′(u))ds

∣∣∣∣∣∣

=

∣∣∣∣∣∣∣∣Φ−1

(Φ

(a−

∫ +∞−∞ g(u, x(u), x′(u))ds

)+

∫ t−∞ f (u, x(u), x′(u))du

)

a(t, x(t, ), x′(t))

−Φ−1

(Φ

(a(t, x(t, ), x′(t))

∫ +∞−∞ g(s, x(s), x(s), x′(s))ds

))

a(t, x(t, ), x′(t))

∣∣∣∣∣∣∣∣

≤ 1m

∣∣∣∣∣∣Φ−1

(Φ

(a−

∫ +∞

−∞g(u, x(u), x′(u))ds

)+

∫ t

−∞f (u, x(u), x′(u))du

)

−Φ−1(Φ

(a(t, x(t, ), x′(t))

∫ +∞

−∞g(s, x(s), x(s), x′(s))ds

))∣∣∣∣∣∣ .

It is easy to see that∣∣∣∣Φ

(a−

∫ +∞−∞ g(u, x(u), x′(u))ds

)+

∫ t−∞ f (u, x(u), x′(u))du

∣∣∣∣

≤ Φ(M

∫ +∞−∞ φr(s)ds

)+

∫ +∞−∞ φr(s)ds =: r4,

and ∣∣∣∣∣∣Φ(a(t, x(t), x′(t))

∫ +∞

−∞g(u, x(u), x′(u))ds

)∣∣∣∣∣∣ ≤ Φ

(M

∫ +∞

−∞φr(s)ds

)≤ r4.

Since Φ−1 is uniformly continuous on [−r4, r4], for any ε > 0, there exists δ1 > 0such that

|Φ−1(x) − Φ−1(y)| < ε, x, y ∈ [−r4, r4], |x − y| < δ1.

One sees that ∣∣∣∣∣∣Φ(a−

∫ +∞

−∞g(u, x(u), x′(u))ds

)+

∫ t

−∞f (u, x(u), x′(u))du

−Φ

(a(t, x(t, ), x′(t))

∫ +∞

−∞g(s, x(s), x(s), x′(s))ds

)∣∣∣∣∣∣

≤∫ t

−∞φr(s)ds +

∣∣∣∣∣∣Φ(a−

∫ +∞

−∞g(u, x(u), x′(u))ds

)

−Φ

(a(t, x(t, ), x′(t))

∫ +∞

−∞g(s, x(s), x(s), x′(s))ds

)∣∣∣∣∣∣ .

Since ∣∣∣∣∣∣a(t, x(t, ), x′(t))∫ +∞

−∞g(s, x(s), x(s), x′(s))ds

∣∣∣∣∣∣ ≤ M∫ +∞

−∞φr(s)ds =: r3,


and Φ is uniformly continuous on [−r3, r3], we get that there exists δ2 > 0 such that

|Φ(u) − Φ(v)| < δ1

2, v, u ∈ [−r3, r3], |u − v| < δ2.

It is easy to see that there exists T1 > 0 such that∫ t

−∞φr(s)ds <

δ1

2, t < −T1.

Since a(t, x(t), x′(t))→ a− uniformly as t → −∞, we get that there exists T2 > 0 suchthat ∣∣∣a− − a(t, x(t), x′(t))

∣∣∣ < δ2∫ +∞−∞ φr(s)ds

holds uniformly for t < −T2. Then t < −T2 implies that∣∣∣∣a−

∫ +∞−∞ g(u, x(u), x′(u))ds − a(t, x(t), x′(t))

∫ +∞−∞ g(u, x(u), x′(u))ds

∣∣∣∣

≤ |a(t, x(t), x′(t)) − a−|∫ +∞−∞ φr(s)ds < δ2.

Hence ∣∣∣∣∣∣Φ(a−

∫ +∞

−∞g(u, x(u), x′(u))ds

)+

∫ t

−∞f (u, x(u), x′(u))du

−Φ

(a(t, x(t, ), x′(t))

∫ +∞

−∞g(s, x(s), x(s), x′(s))ds

)∣∣∣∣∣∣

<δ1

2+δ1

2= δ1, t < −maxT1,T2.

Then ∣∣∣∣∣∣ρ(t)(T x)′(t) +

∫ +∞

−∞g(u, x(u), x′(u))ds

∣∣∣∣∣∣ <1mε, t < maxT1,T2. (16)

It follows from (2.15) and (2.16) that TΩ1+τ(t) and ρ(t)(T x)′ : x ∈ Ω are equiconvergent

at −∞.(iv) We show that both 1

1+τ(t) TΩ and ρ(t)T x)′ : x ∈ Ω are equi-convergent att = +∞.

Similarly we can show that

(T x)(t)1 + τ(t)

→∫ +∞

−∞h(u, x(u), x′(u))ds uniformly as t → +∞

andρ(t)(T x)′(t)

→ −Φ−1(Φ(a−

∫ +∞−∞ g(u,xn(u),x′n(u))ds

)+∫ +∞−∞ f (u,xn(u),x′n(u))du

)

a+


uniformly as t → +∞. So 11+τ(t) TΩ and ρ(t)T x)′ : x ∈ Ω are equi-convergent at

t = +∞.Above all discussion implies that T : X → X is completely continuous. The proof

is completed.

3. MAIN RESULTSIn this section, we present the main theorem and its proof. Let Φ be a sup-

multiplicative-like function with its supporting function ω. The inverse function ofΦ is denoted by Φ−1, whose supporting function is denoted by ν.

For nonnegative functions ϕ, ψ1, ψ2 ∈ L1(−∞,+∞), and a > 0, define

n =

∫ +∞

−∞ψ2(s)ds +

1mν

1

ω

(1

a−∫ +∞−∞ ψ1(s)ds

) +

∫ +∞

−∞ϕ(s)ds

.

Theorem 3.1. Suppose that there exists a constant a > 0 such that(H1) the inequalities

f(t, (1 + τ(t))x1,

1ρ(t)

y1

)≤ f

(t, (1 + τ(t))x2,

1ρ(t)

y1

),

g(t, (1 + τ(t))x1,

1ρ(t)

y1

)≤ g

(t, (1 + τ(t))x2,

1ρ(t)

y1

),

h(t, (1 + τ(t))x1,

1ρ(t)

y1

)≤ h

(t, (1 + τ(t))x2,

1ρ(t)

y1

)

hold for allt ∈ (−∞,+∞), 0 ≤ x1 ≤ x2 ≤ a, |y1| ≤ |y2| ≤ a.

(H2) there exist nonnegative functions ϕ, ψ1, ψ2 such that ϕ, ψ1, ψ2 ∈ L1(0,∞) and

f(t, (1 + τ(t))a,

1ρ(t)

a)≤ Φ

(an

)ϕ(t),

g(t, (1 + τ(t))a,

1ρ(t)

a)≤ a

nψ1(t),

h(t, (1 + τ(t))a,

1ρ(t)

a)≤ a

nψ2(t)

for all t ∈ (−∞,+∞).Then BVP(1.6) has two positive solutions w∗ and v∗ such that 0 < ||w∗|| ≤ a and

limn→∞ wn = limn→∞ T nw0 = w∗ with

w0(t) = a(1 + τ(t)), t ∈ (0,∞),


and 0 < ||v∗|| ≤ a with limn→∞ vn = limn→∞ T nv0 = w∗ and v0(t) ≡ 0 on (0,∞).

Proof. By Lemmas 2.1 and 2.2, we know that T : P→ P is completely continuous.Denote

Pa = x ∈ P : ||x|| ≤ a.

Step 1. We prove that T : Pa → Pa. If x ∈ Pa, then ||x|| ≤ a. We have

r = max sup

t∈(−∞,+∞)

|x(t)|1 + τ(t)

, supt∈(−∞,+∞)

ρ(t)|x′(t)| < +∞.

From (H2) and (H1), we have

f (t, x(t), x′(t)) = f(t, (1 + τ(t)) x(t)

1+τ(t) ,1ρ(t)ρ(t)x′(t)

)

≤ f(t, (1 + τ(t))a, 1

ρ(t) a)

≤ Φ(

an

)ϕ(t), t ∈ R.

Similarly we have

g(t, x(t), x′(t)

) ≤ anψ1(t), h

(t, x(t), x′(t)

) ≤ anψ2(t), t ∈ R.

Hence, we get from (2.7) and (2.8) that

|(T x)(t)|1 + τ(t)

=1

1 + τ(t)

∣∣∣∣∣∣∫ +∞

−∞h(u, x(u), x′(u))du

+

∫ +∞

t

Φ−1(Φ

(a−

∫ +∞−∞ g(u, x(u), x′(u))ds

)+

∫ s−∞ f (u, x(u), x′(u))du

)

a(s, x(s, ), x′(s))ρ(s)ds

∣∣∣∣∣∣∣∣

≤ 11 + τ(t)

∫ +∞

−∞

anψ2(s)ds

+1

1 + τ(t)

∫ +∞

t

Φ−1(Φ

(a−

∫ +∞−∞

anψ1(s)ds

)+

∫ s−∞Φ

(an

)ϕ(s)ds

)

mρ(s)ds


≤ an

∫ +∞−∞ ψ2(s)ds

+ 11+τ(t)

∫ +∞t

1mρ(s) dsΦ−1

(Φ

(a−

∫ +∞−∞

anψ1(s)ds

)+

∫ +∞−∞ Φ

(an

)ϕ(s)ds

)

≤ an

∫ +∞−∞ ψ2(s)ds + 1

mΦ−1(Φ

(a−

∫ +∞−∞

anψ1(s)ds

)+

∫ +∞−∞ Φ

(an

)ϕ(s)ds

)

≤ an

∫ +∞−∞ ψ2(s)ds + 1

mΦ−1

Φ( a

n )ω

(1

a−∫ +∞−∞ ψ1(s)ds

) +∫ +∞−∞ Φ

(an

)ϕ(s)ds

= an

∫ +∞−∞ ψ2(s)ds + 1

mΦ−1

Φ(

an

)

1

ω

(1

a−∫ +∞−∞ ψ1(s)ds

) +∫ +∞−∞ ϕ(s)ds

≤ an

∫ +∞−∞ ψ2(s)ds + 1

manν

1

ω

(1

a−∫ +∞−∞ ψ1(s)ds

) +∫ +∞−∞ ϕ(s)ds

= a.


ρ(t)|(T x)′(t)|

=

∣∣∣∣∣∣∣∣−

Φ−1(Φ

(a−

∫ +∞−∞ g(u, x(u), x′(u))ds

)+

∫ t−∞ f (u, x(u), x′(u))du

)

a(t, x(t), x′(t))

∣∣∣∣∣∣∣∣

≤Φ−1

(Φ

(a−

∫ +∞−∞

anψ1(s)ds

)+

∫ +∞−∞ Φ

(an

)ϕ(s)ds

)

m

≤∫ +∞−∞

anψ2(s)ds +

∫ +∞−∞

anψ1(s)ds

∫ +∞−∞

drρ(r)

+

∫ +∞t

∫ +∞s

drρ(r)

anϕ(s)ds +

∫ t−∞

∫ s−∞

drρ(r)

anϕ(s)ds

∫ +∞−∞

drρ(r)

≤∫ +∞−∞

anψ2(s)ds +

∫ +∞−∞

anψ1(s)ds

∫ +∞−∞

drρ(r)

+

∫ +∞

−∞

anϕ(s)ds

≤ 1m

Φ−1

Φ(

an

)

ω

(1

a−∫ +∞−∞ ψ1(s)ds

) +

∫ +∞

−∞Φ

(an

)ϕ(s)ds


=1m

Φ−1

Φ

(an

)

1

ω

(1

a−∫ +∞−∞ ψ1(s)ds

) +

∫ +∞

−∞ϕ(s)ds

≤ 1m

anν

1

ω

(1

a−∫ +∞−∞ ψ1(s)ds

) +

∫ +∞

−∞ϕ(s)ds

≤ a.

Hence we have show that T : Pa → Pa.Step 2. We establish iterative schemes for approximating the solutions.Choose

w0(t) = a(1 + τ(t)).

(i) one sees easily that ||w0|| ≤ a. Then w0 ∈ Pa.(ii) Let w1 = Tw0 and w2 = Tw1. Then above discussion implies that w1,w2 ∈ Pa.

We denote wn+1 = Twn = T nw0 for n = 1, 2, · · · . Since T : Pa → Pa, we havewn ∈ Pa for all n = 1, 2, 3, · · · . It follows from the complete continuity of T thatwn : n = 0, 1, 2, 3, · · · is a sequentially compact set.

(iii) We prove that

wn+1 ≤ wn, n = 1, 2, 3, · · · .

By (H1) and (H2), we get

0 ≤ (Tw0)(t)1+τ(t) = 1

1+τ(t)

∫ +∞−∞ h(u, ω0(u), ω′0(u))du

+ 11+τ(t)

∫ +∞t

Φ−1(Φ(a−

∫ +∞−∞ g(u,ω0(u),ω′0(u))ds

)+∫ s−∞ f (u,ω0(u),ω′0(u))du

)

a(s,ω0(s,),ω′0(s))ρ(s) ds

≤ 11+τ(t)

∫ +∞−∞

anψ2(s)ds

+ 11+τ(t)

∫ +∞t

Φ−1(Φ(a−

∫ +∞−∞

anψ1(s)ds

)+∫ s−∞ Φ( a

n )ϕ(s)ds)

mρ(s) ds

≤ an

∫ +∞−∞ ψ2(s)ds

+ 11+τ(t)

∫ +∞t

1mρ(s) dsΦ−1

(Φ

(a−

∫ +∞−∞

anψ1(s)ds

)+

∫ +∞−∞ Φ

(an

)ϕ(s)ds

)

≤ an

∫ +∞−∞ ψ2(s)ds + 1

mΦ−1(Φ

(a−

∫ +∞−∞

anψ1(s)ds

)+

∫ +∞−∞ Φ

(an

)ϕ(s)ds

)


≤ an

∫ +∞−∞ ψ2(s)ds + 1

mΦ−1

Φ( a

n )ω

(1

a−∫ +∞−∞ ψ1(s)ds

) +∫ +∞−∞ Φ

(an

)ϕ(s)ds

= an

∫ +∞−∞ ψ2(s)ds + 1

mΦ−1

Φ(

an

)

1

ω

(1

a−∫ +∞−∞ ψ1(s)ds

) +∫ +∞−∞ ϕ(s)ds

≤ an

∫ +∞−∞ ψ2(s)ds + 1

manν

1

ω

(1

a−∫ +∞−∞ ψ1(s)ds

) +∫ +∞−∞ ϕ(s)ds

= a.

Thenω1(t) = (Tω0)(t) ≤ (1 + τ(t))a = ω0(t).


ρ(t)|(Tw0)′(t)| ≤Φ−1

(Φ

(a−

∫ +∞−∞

anψ1(s)ds

)+

∫ +∞−∞ Φ

(an

)ϕ(s)ds

)

m

≤∫ +∞−∞

anψ2(s)ds +

∫ +∞−∞

anψ1(s)ds

∫ +∞−∞

drρ(r)

+

∫ +∞t

∫ +∞s

drρ(r)

anϕ(s)ds +

∫ t−∞

∫ s−∞

drρ(r)

anϕ(s)ds

∫ +∞−∞

drρ(r)

≤∫ +∞−∞

anψ2(s)ds +

∫ +∞−∞

anψ1(s)ds

∫ +∞−∞

drρ(r)

+

∫ +∞

−∞

anϕ(s)ds

≤ 1m

Φ−1

Φ(

an

)

ω

(1

a−∫ +∞−∞ ψ1(s)ds

) +

∫ +∞

−∞Φ

(an

)ϕ(s)ds

=1m

Φ−1

Φ

(an

)

1

ω

(1

a−∫ +∞−∞ ψ1(s)ds

) +

∫ +∞

−∞ϕ(s)ds

≤ 1m

anν

1

ω

(1

a−∫ +∞−∞ ψ1(s)ds

) +

∫ +∞

−∞ϕ(s)ds

≤ a.


Thenρ(t)|(Tw0)′(t)| ≤ a = ρ(t)|ω′0(t)|, |ω′1(t)| = |(Tw0)′(t)| ≤ |ω′0(t)|.

Henceω1 ≤ ω0.

By the monotonic property of f , g, and h and the definition of T , we get

w2(t) = (Tw1)((t) ≤ (Tw0)(t) = w1(t), |ω′2(t)| = |(Tw1)′(t)| ≤ |ω′1(t)| t ∈ (0,∞).

So ω2 ≤ ω1. By induction, we get

wn+1 ≤ wn, n = 1, 2, 3, · · · .Thus there exists w∗ ∈ Pa such that wn → w∗ as n → ∞. Applying the continuity ofT and wn+1 = Twn, we get that w∗ = Tw∗.

Choosev0(t) = 0 for all t ∈ (−∞,∞).

Then v0 ∈ Pa. Let v1 = Tv0 and v2 = Tv1. One can prove that v1 ∈ Pa and v2 ∈ Pa.We denote vn+1 = Tvn = T nv0 for n = 1, 2, 3, · · · . Since T : Pa → Pa, we havevn ∈ Pa. for all n = 1, 2, 3, · · · . It follows from the complete continuity of T that vnis a sequentially compact set.

Since v1 = Tv0 ∈ Pa, we have

v1(t) = (Tv0)(t) ≥ 0 ≡ v0, t ∈ (−∞,∞).

On the other hand, we have

ρ(t)|v′1(t)| = ρ(t)|(Tv0)′(t)|

=Φ−1

(Φ

(a−

∫ +∞−∞ g(u, v0(u), v′0(u))ds

)+

∫ t−∞ f (u, v0(u), v′0(u))du

)

a(t, v0(t), v′0(t))≥ 0 = ρ(t)|v′0(t)|, t ∈ (−∞,∞).

It follows that |v′1(t)| ≥ |v′0(t)|, t ∈ (−∞,∞). Then v1 ≥ v0. By induction, we get

vn+1 ≥ vn, n = 1, 2, 3, · · · .Thus there exists v∗ ∈ Pa such that vn → v∗ as n → ∞. Applying the continuity of Tand vn+1 = Tvn, we get that v∗ = Tv∗.

Since f (t, 0, 0) . 0, we see that the zero function is not the solution of BVP(1.6).Thus both w∗ and v∗ are positive solution of the operator equation x = T x in Pa.

It is well known that each fixed point of T in P is a solution of BVP(1.6). Hencew∗ and v∗ are two positive solutions of BVP(1.6). The proof is complete.


Remark 3.1. The iterative schemes in Theorem 3.1 are w0(t) = a(1 + τ(t)) andwn+1(t) = (Twn)(t) for n = 0, 1, 2, 3, · · · and v0(t) ≡ 0, vn+1(t) = (Tvn)(t) forn = 0, 1, 2, 3, · · · . They start off with a known simple function and the zero functionrespectively. This is convenient for application.

Remark 3.2. One sees that the ideas of the proof of the main theorem are alsoobtainable from the metric version of Theorem 3 in [12] (rephrased as Theorem 8 in[13]).

Acknowledgement. The first author is supported by the Natural Science Foundation of Guangdongprovince (No: S2011010001900) and the Guangdong Higher Education Foundation for High-leveltalents and the second author supported by the Natural Science Foundation of Hunan province (No:12JJ6006) and the Science Foundation of Department of Science and Technology of Hunan province(No. 2012FJ3107).

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[6] A. Cabada, J.A. Cid, Heteroclinic solutions for non-autonomous boundary value problems withsingular Φ-Laplacian operators, Discrete Contin. Dyn. Syst. 2009, Dynamical Systems, Differ-ential Equations and Applications. 7th AIMS Conference, suppl., 118–122.

[7] Alessandro Calamai, Heteroclinic solutions of boundary value problems on the real line in-volving singular Φ-Laplacian operators, Journal of Mathematical Analysis and Applications,378(2011)667-679.

[8] Cristina Marcelli, Francesca Papalini, Heteroclinic connections for fully non-linearnon-autonomous second-order differential equations, Journal of Differential Equations,241(2007)160-183.

[9] G. Cupini, C. Marcelli, F. Papalini, On the solvability of a boundary value problem on the realline, Boundary Value Problem, 2011 (2011): 26, 1-17.

[10] G. Cupini, C. Marcelli, F. Papalini, Heteroclinic solutions of boundary value problems on thereal line involving general nonlinear differential operators, Differential and Integral Equations,24 (2011) 619-644.

[11] G. L. Karakostas, Positive solutions for the Φ−Laplacian when Φ is a sup-multiplicative-likefunction, Electron. J. Diff. Eqns., Vol. 68(2004)1-12.

[12] M. Turinici, Abstract comparison principles and multivariable Gronwal-Bellman inequalities, J.Math. Anal. Appl. 117(1986)100-127.

[13] M. Turinici, Nieto-Lopet theorems in ordered metric spaces, Archiv, 105-2401-v2, 20 May, 2011.

CUBIC SPLINE METHOD AND FRACTIONALSTEPS SCHEME TO APPROXIMATE THEPHASE-FIELD SYSTEM WITHNON-HOMOGENEOUS CAUCHY-NEUMANNBOUNDARY CONDITIONS

ROMAI J., v.8, no.1(2012), 73–91

Costica Morosanu“Al. I. Cuza” University of Iasi, [email protected]

Abstract A scheme of fractional steps type, associated to the nonlinear phase-field transition sys-tem with non-homogeneous Cauchy-Neumann boundary conditions, is considered in thepresent paper. To approximate the solution of the linear parabolic system introduced bysuch approximating scheme, a cubic spline method have been used. A stability resultfor this new approach is proved and some numerical experiments, like simulation ofseparation zone between the phases of the material that is involved in the solidificationprocess, are performed too.

Keywords: boundary value problems, fractional steps method, cubic spline method, phase changes,solidification process.2010 MSC: 65Nxx, 65N12, 65N30, 80A22.

1. INTRODUCTIONConsider the nonlinear parabolic boundary value problem

ρc ∂

∂t u + `2∂∂tϕ = k∆u

τ ∂∂tϕ = ξ2∆ϕ + 12a (ϕ − ϕ3) + 2u

in Q := (0,T ] ×Ω, (1.1)

subject to the non-homogeneous Cauchy-Neumann boundary conditions:∂∂νu + hu = w(t, x)∂∂νϕ = 0

on Σ := (0,T ] × ∂Ω, (1.2)

and the initial conditions:

u(0, x) = u0(x), ϕ(0, x) = ϕ0(x) on Ω, (1.3)

where:

Ω is a bounded domain in Rn with smooth boundary ∂Ω,

T > 0 is a given positive number,

73

74 Costica Morosanu

the unknown functions u and ϕ represent the reduced temperature distributionand the phase function (used to distinguish between the phases of Ω), respec-tively,

u0, ϕ0 : Ω→ Rn are given functions,

w : [0,T ] × ∂Ω → R also is a given function - the temperature surrounding at∂Ω,

the positive parameters ρ, c, τ, ξ, `, k, h, a, have the following physical mean-ing: ρ - is the density, c - is the heat capacity, τ - is the relaxation time, ξ- is the length scale of the interface, ` - denotes the latent heat, k - the heatconductivity, h - the heat transfer coefficient and a is an probabilistic measureon the individual atoms (a depends on ξ).

The mathematical model (1.1), introduced by Caginalp [2], has beenestablished in literature as an alternative of the classic two-phase Stefan problem tocapture, among others, the effects of surface tension, supercooling, and superheating.

As regards the existence, it is known that under appropiate conditions on u0, ϕ0and w, the system (1.1)-(1.3) has a unique solution u, ϕ ∈ W2,1

p (Q) ∩ L∞(Q), p > 32

(see Morosanu [5]).Numerical approximation of the phase-field system (1.1) subject to the homoge-

neous Neumann boundary conditions: ∂∂νu + hu = 0 on Σ, has been analyzed in

Morosanu [4]. For other numerical investigation of the phase-field model (subject tovarious other boundary conditions), see Benincasa & Morosanu [1], Morosanu [3, 5]and references there in.

To formulate the scheme of fractional steps type in order to approximate the abovenonlinear problem, we denote

Qε = (t, x) ∈ Q, ε ≤ t ≤ T and Σε = (t, x) ∈ Σ, ε ≤ t ≤ T ,for every ε > 0. Thus, we associate to problem (1.1)-(1.3) the following approximat-ing scheme (also known in literature as the Lie-Trotter product formula):

ρc ∂

∂t uε + `

2∂∂tϕ

ε = k∆uε

τ ∂∂tϕ

ε = ξ2∆ϕε + 12aϕ

ε + 2uεin Qε, (1.4)

∂∂νuε + huε = w(t, x)∂∂νϕ

ε = 0on Σε, (1.5)

uε(ε, x) = u0(x)ϕε(ε, x) = z(ε, ϕ0(x)) on Ω, (1.6)

where z(ε, ϕ0(x)) is the solution of Cauchy problem:

z′(s) + 12a z3(s) = 0, s ∈ (0, ε),z(0) = ϕ0(x). (1.7)

Cubic Spline Method and Fractional Steps Scheme to approximate... 75

The following result (proved in Benincasa & Morosanu [1], Morosanu [5]) estab-lishes the link between the solutions (u, ϕ) and (uε , ϕε):

Theorem 1.1. Assume that u0, ϕ0 ∈ W1∞(Ω) satisfying ∂∂νu0 +hu0 = w(0, x), ∂

∂νϕ0 = 0and w ∈ W1([0,T ], L2(∂Ω)). Furthermore, Ω ⊂ Rn (n = 1, 2, 3) is a bounded domainwith a smooth boundary. Let (uε, ϕε) be the solution of the approximating scheme(1.4)-(1.7). Then, for ε→ 0, one has

(uε(t), ϕε(t))→ (u(t), ϕ(t)) stronglyinL2(Ω) foranyt ∈ (0,T ], (1.8)

where u, ϕ ∈ W2,1p ([0,T ]; L2(Ω)) ∩ L2([0, T ]; H2(Ω)) is the solution of the nonlinear

system (1.1)-(1.3).

Based on the result of convergence given by Theorem 1, we will concerned in thiswork with the numerical approximation of the solution (uε, ϕε) of the linear system(1.4)-(1.7).

The rest of paper is organized as follows: in Section 2 we have introduced, viacubic spline method, the discrete equations (2.22), corresponding to the linear system(1.4)-(1.6); consequently, a conceptual algorithm has been formulated: algspline1D.A stability result for this new approach is stated and proved, too. Some numericalexperiments are reported in the last Section.

2. NUMERICAL METHODIn this Section we are concerned with the cubic spline approximation of the un-

knowns uε, ϕε in the linear system (1.4)-(1.6), obtained via a fractional steps methodin order to approximate the nonlinear phase-field transition system (1.1)-(1.3). Next,we will work in one dimension (1D) and then: ∆uε = uεxx, ∆ϕε = ϕεxx. To fix theideas, let Ω = [0, b] ⊂ IR+ and introduce over it the grid with N equidistant nodes

x j = ( j − 1)dx j = 1, 2, . . . ,N, dx = b/(N − 1).

Given positive values ε,T (T >> ε) and considering M as the number of equidis-tant nodes in which in divided [ε,T ], we set

ti = ε + (i − 1)dt i = 1, 2, . . . , M, dt = (T − ε)/(M − 1).

Now we denote byui

j, ϕij

the approximate values in the point (ti, x j) of the un-

known functions uε, ϕε in (1.4)-(1.6); more precisely:

uij = uε(ti, x j)

ϕij = ϕε(ti, x j)

i = 1, 2, . . . ,M, j = 1, 2, . . . ,N, (2.1)

or, for later use

ui not=

(ui

1, ui2, . . . , u

iN

), ϕi not

=(ϕi

1, ϕi2, . . . , ϕ

iN

), i = 1, 2, . . . ,M.

76 Costica Morosanu

Corresponding to the initial conditions (1.6) we put

u1j = uε(t1, x j) = u0(x j) j = 1, 2, . . . ,N, (2.2)

and, involving the separation of variables method to solve the Cauchy problem (1.7)(see Morosanu [3]), we get

ϕ1j = ϕε(t1, x j) = z(ε, ϕ0(x j)) = ϕ0(x j)

√a

a + ε · ϕ20(x j)

, (2.3)

for j = 1, 2, . . . ,N.For approximating the partial derivative with respect to time, we employed a first-

order scheme, namely:

∂

∂tuε(ti, x j) ≈

uij − ui−1

j

dt,

∂

∂tϕε(ti, x j) ≈

ϕij − ϕi−1

j

dt(2.4)

while, to approximate 12aϕ

ε(ti, x j) + 2uε(ti, x j) in (1.4) (the reaction term), we willinvolve an implicit formula, i.e.:

12aϕε(ti, x j) + 2uε(ti, x j) ≈ 1

2aϕi

j + 2uij, (2.5)

i = 1, 2, . . . , M, j = 1, 2, . . . ,N.Let’s denote by S i(x) and Vi(x) the cubic splines that interpolates uε(t, x) and

ϕε(t, x), respectively, in nodes x j, j = 1,N at the time level i, i = 1,M. Then,making use of (2.4) and (2.5), we associate to (1.4) at the point (ti, x j), i = 1,M,j = 1,N, the following general approximating scheme:

ρc

ui+1j −ui

jdt + `

2ϕi+1

j −ϕij

dt = k[θMi+1, j + (1 − θ)Mi, j

],

τϕi+1

j −ϕij

dt = ξ2[θNi+1, j + (1 − θ)Ni, j

]+ 1

2aϕij + 2ui

j,(2.6)

where 0 ≤ θ ≤ 1 and

Mi, j = S ′′i (x j) Ni, j = V ′′i (x j), i = 1,M, j = 1,N.

It is known that at the time level t = ti the cubic splines S i(x), Vi(x) have thefollowing expression (see Prenter [6])

S i(x) =

(ui

j−1 −dx2

6Mi, j−1

)x j − x

dx+

(ui

j −dx2

6Mi, j

)x − x j−1

dx

+Mi, j−1 (x j − x)3

6dx+ Mi, j (x − x j−1)3

6dx, x ∈ [x j−1, x j],


Vi(x) =

(ϕi

j−1 −dx2

6Ni, j−1

)x j − x

dx+

(ϕi

j −dx2

6Ni, j

)x − x j−1

dx

+Ni, j−1 (x j − x)3

6dx+ Ni, j (x − x j−1)3

6dx, x ∈ [x j−1, x j],

for i = 1,M, j = 2,N.We now turn our attention only on the relation (2.6)1. In other words, we will deal

with S i(x). So, let’s consider the intervals [x j−1, x j], [x j, x j+1]. Corresponding, wehave

S ′i (x) =

(ui

j−1 −dx2

6Mi, j−1

)· −1

dx+

(ui

j −dx2

6Mi, j

)· 1

dx

−Mi, j−1 (x j − x)2

2dx+ Mi, j (x − x j−1)2

2dx,

x ∈ [x j−1, x j], j = 2,N,

S ′i (x) =

(ui

j −dx2

6Mi, j

)· −1

dx+

(ui

j+1 −dx2

6Mi, j+1

)· 1

dx

−Mi, j (x j+1 − x)2

2dx+ Mi, j+1 (x − x j)2

2dx,

x ∈ [x j, x j+1], j = 1,N − 1.

From the continuity of S ′i (x) at the interior nodes x j, j = 2,N − 1, expressedbelow:

S ′i (x j − 0) =1

dx

[−ui

j−1 +dx2

6Mi, j−1 + ui

j −dx2

6Mi, j +

dx2

2Mi, j

]

=1

dx

[−ui

j +dx2

6Mi, j + ui

j+1 −dx2

6Mi, j+1 − dx2

2Mi, j

]= S ′i (x j + 0),

we get ( j = 2,N − 1)

Mi, j−1 + 4Mi, j + Mi, j+1 =6

dx2 [uij−1 − 2ui

j + uij+1]. (2.7)

Doing the same computations relative to the time level t = ti+1, we derive ( j =

2,N − 1)

Mi+1, j−1 + 4Mi+1, j + Mi+1, j+1 =6

dx2 [ui+1j−1 − 2ui+1

j + ui+1j+1]. (2.8)

Now, multiplying (2.7) by k(1− θ), (2.8) by kθ and summing the results, we obtain( j = 2,N − 1)

k[θMi+1, j−1 + (1 − θ)Mi, j−1] + 4k[θMi+1, j + (1 − θ)Mi, j

]

78 Costica Morosanu

+k[θMi+1, j+1 + (1 − θ)Mi, j+1

]

= 6k

dx2

[θ(ui+1

j−1 − 2ui+1j + ui+1

j+1

)+ (1 − θ)

(ui

j−1 − 2uij + ui

j+1

)].

Let’s observe that the left term in the expression above represents the right side ofthe equality (2.6)1, expressed in nodes x j−1, x j, x j+1, at the level ti of time. Thus, thelatter can be written equivalent as ( j = 2,N − 1)

ρc[(ui+1

j−1 − uij−1) + 4(ui+1

j − uij) + (ui+1

j+1 − uij+1)

](2.9)

+`

2

[(ϕi+1

j−1 − ϕij−1) + 4(ϕi+1

j − ϕij) + (ϕi+1

j+1 − ϕij+1)

]

= 6dt

dx2 k[θ(ui+1

j−1 − 2ui+1j + ui+1

j+1) + (1 − θ)(uij−1 − 2ui

j + uij+1)

].

Setting now r = dtdx2 and performing some straightforward computation, the rela-

tion (2.9) leads to ( j = 2,N − 1)

(ρc − 6rkθ)ui+1j−1 + (4ρc + 12rkθ)ui+1

j + (ρc − 6rkθ)ui+1j+1 (2.10)

+`

2ϕi+1

j−1 + 2`ϕi+1j +

`

2ϕi+1

j+1

=[ρc + 6rk(1−θ)] ui

j−1 +[4ρc − 12rk(1−θ)] ui

j +[ρc + 6rk(1−θ)] ui

j+1

+`

2ϕi

j−1 + 2`ϕij +

`

2ϕi

j+1.

Following the same sequence of computations as in (2.6)1, from the second equa-tion in relation (2.6) one obtains ( j = 2,N − 1)

Ni, j−1 + 4Ni, j + Ni, j+1 =6

dx2 [ϕij−1 − 2ϕi

j + ϕij+1], (2.11)

(τ − 6rξ2θ

)ϕi+1

j−1 +(4τ + 12rξ2θ

)ϕi+1

j +(τ − 6rξ2θ

)ϕi+1

j+1 (2.12)

=

(τ + 6rξ2(1 − θ) +

dt2a

)ϕi

j−1 +

(4τ − 12rξ2(1 − θ) +

dta

)ϕi

j

+

(τ + 6rξ2(1 − θ) +

dt2a

)ϕi

j+1 + 2dt(uij−1 + ui

j + uij+1).

To treat the boundary conditions (1.5), we have to consider the nodes x1 ( j = 1)and xN ( j = N). In the sequel, we will present a detalied computation only for (1.5)1and j = 1; for the remaining cases ((1.5)1 and j = N, (1.5)2 and j = 1, (1.5)2 andj = N) we will give only the final result.


The boundary ∂Ω corresponding to Ω already chosen in 1D is reduced to the set0, b. So the boundary condition (1.5)1 becomes

−ux(0) + h u(0) = w(t, 0)ux(b) + h u(b) = w(t, b), (2.13)

where the sign in front of ∂∂νu = ux is ∓ because the normal to [0, b] at 0 (b) points in

the negative (positive) direction.Invoking relation (2.13), we have (let’s remember that in the grid associated to

[0, b], we have noted: 0 = x1, b = xN) −S ′i (0) + hui

1 = wi(0)S ′i (b) + hui

N = wi(b), (2.14)

where wi(0) = w(ti, 0), wi(b) = w(ti, b), i = 1, 2, ...,M.

If we limit our study in relation (2.14) to the case j = 1 (so, on [x1, x2]) at the levelti of time, we know from before thatS ′i (0) = 1

dx

[−ui

1 + dx2

6 Mi,1 + ui2 − dx2

6 Mi,2 − dx2

2 Mi,1]

= − 1dx

[dx2

6

(2Mi,1 + Mi,2

)+ ui

1 − ui2

],

while, at the level ti+1 of timeS ′i+1(0) = 1

dx

[−ui+1

1 + dx2

6 Mi+1,1 + ui+12 − dx2

6 Mi+1,2 − dx2

2 Mi+1,1]

= − 1dx

[dx2

6

(2Mi+1,1 + Mi+1,2

)+ ui+1

1 − ui+12

].

Substituting the expressions S ′i (0) and S ′i+1(0) in (2.14)1, one obtains

2Mi,1 + Mi,2 =(− 6

dx2 − 6dx h

)ui

1 + 6dx2 ui

2 + 6dx wi(0),

2Mi+1,1 + Mi+1,2 =(− 6

dx2 − 6dx h

)ui+1

1 + 6dx2 ui+1

2 + 6dx wi+1(0).

(2.15)

Multiplying (2.15)1 by k(1 − θ), (2.15)2 by kθ and summing, we deduce the equality

2k[θMi+1,1 + (1 − θ)Mi,1] + k[θMi+1,2 + (1 − θ)Mi,2

]

+

(6

dx2 +6

dxh)

kθui+11 − 6

dx2 kθui+12

= −(

6dx2 +

6dx

h)

k(1 − θ)ui1 +

6dx2 k(1 − θ)ui

2

+6

dx

[k(1 − θ)wi(0) + kθwi+1(0)

].

Combining the last relation with (2.6)1, we conclude that (2.14)1 takes the discretizedform (i = 1, 2, ...,M − 1, r = dt

dx2 )(2ρc + 6rkθ + 6

dtdx

khθ)

ui+11 + (ρc − 6rkθ) ui+1

2 + `ϕi+11 +

`

2ϕi+1

2 (2.16)

80 Costica Morosanu

=

[2ρc − 6rk(1 − θ) − 6

dtdx

kh(1 − θ)]

ui1 +

[ρc + 6rk(1 − θ)] ui

2

+`ϕi1 +

`

2ϕi

2 + 6dtdx

[k(1 − θ)wi(0) + kθwi+1(0)

].

In the case j = N (i.e. on [xN−1, xN]), we obtain from (2.14)2 that (i = 1, 2, ...,M −1, r = dt

dx2 )

Mi,N−1 + 2Mi,N =6

dx2 uiN−1 −

(6

dx2 +6

dxh)

uiN +

6dx

wi(b), (2.17)

(ρc − 6rkθ) ui+1N−1 +

(2ρc + 6rkθ + 6

dtdx

khθ)

ui+1N +

`

2ϕi+1

N−1 + `ϕi+1N (2.18)

=[ρc + 6rk(1 − θ)] ui

N−1 +

[2ρc − 6rk(1 − θ) − 6

dtdx

kh(1 − θ)]

uiN

+`

2ϕi

N−1 + `ϕiN + 6

dtdx

[k(1 − θ)wi(b) + kθwi+1(b)

].

Treating the boundary conditions (1.5)2 in the same manner, we arrive at

2Ni,1 + Ni,2 = − 6dx2

(ϕi

1 − ϕi2

),

Ni,N−1 + 2Ni,N = 6dx2

(ϕi

N−1 − ϕiN

),

(2.19)

(2τ + 6rξ2θ

)ϕi+1

1 +(τ − 6rξ2θ

)ϕi+1

2 =

[2τ−6rξ2(1−θ)+ dt

2a

]ϕi

1 (2.20)

+

[τ+6rξ2(1−θ)+ dt

2a

]ϕi

2 +2dt(ui1 + ui

2),

for j = 1, i = 1, 2, ..., M − 1, and

(τ − 6rξ2θ

)ϕi+1

N−1 +(2τ + 6rξ2θ

)ϕi+1

N =

[τ+6rξ2(1−θ)+ dt

2a

]ϕi

N−1 (2.21)

+

[2τ−6rξ2(1−θ)+ dt

2a

]ϕi

N + 2dt(uiN−1 + ui

N),

for j = N, i = 1, 2, ..., M − 1.Setting

e1 = 6rkθ e2 = 6rξ2θ e3 = 6rk(1 − θ) e4 = 6rξ2(1 − θ),then the discrete system (2.10)-(2.11), coupled with (2.16)-(2.21), can be written inmatrix form as:

E(ui+1ϕi+1

)= F

(uiϕi

)+

(di

1di2

)i = 1, 2, ...,M − 1, (2.22)


where

E =E11 E120 E22

2N×2N F =F11 E12F21 F22

2N×2N

with E11, E12, E22, F11, F21, F22 having the same size N × N, and

E11 =

ea ρc − e1 0 · · · 0 0 0ρc − e1 eb ρc − e1 · · · 0 0 0

......

.... . .

......

...0 0 0 · · · ρc − e1 eb ρc − e10 0 0 · · · 0 ρc − e1 ea

ea = 2ρc + e1 + 6 dtdx khθ > 0,

eb = 4ρc + 2e1 > 0,

E22 =

2τ + e2 τ − e2 0 · · · 0 0 0τ − e2 ec τ − e2 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · τ − e2 ec τ − e20 0 0 · · · 0 τ − e2 2τ + e2

ec = 4τ + 2e2 > 0,

E12 =

2c3 c3 0 · · · 0 0 0c3 4c3 c3 · · · 0 0 0...

......

. . ....

......

0 0 0 · · · c3 4c3 c30 0 0 · · · 0 c3 2c3

c3 =`

2,

F11 =

f a ρc + e3 0 · · · 0 0 0ρc + e3 f b ρc + e3 · · · 0 0 0

......

.... . .

......

...0 0 0 · · · ρc + e3 f b ρc + e30 0 0 · · · 0 ρc + e3 f a

f a = 2ρc − e3 − 6 dtdx kh(1 − θ),

f b = 4ρc − 2e3,

F22 =

f c f d 0 · · · 0 0 0f d f e f d · · · 0 0 0...

......

. . ....

......

0 0 0 · · · f d f e f d0 0 0 · · · 0 f d f c

82 Costica Morosanu

f c = 2τ − e4 + dt2a ,

f d = τ + e4 + dt2a ,

f e = 4τ − 2e4 + dta ,

F21 =

dt dt 0 · · · 0 0 0dt dt dt · · · 0 0 0...

......

. . ....

......

0 0 0 · · · dt dt dt0 0 0 · · · 0 dt dt

di1 =

6 dtdx

[k(1 − θ)wi(0) + kθwi+1(0)

]

0...0

6 dtdx

[k(1 − θ)wi(b) + kθwi+1(b)

]

N×1

di2 =

0...0

N×1

.

The following sequence (again, i denotes the time level) can be used to calculatethe approximate solution (ui+1, ϕi+1):

Begin algspline1DChoose ε > 0, T > ε, b > 0;Choose M > 0, N > 0 and compute dt, dx;Choose u0, ϕ0,w;Compute z (ε, ·) from (2.3);

i := 1→ u1, ϕ1 from initial conditins (2.2)-(2.3);

For j = 1,N:

Compute M1, j, solving (2.7)+(2.15)1+(2.17) (i = 1);Compute N1, j, solving (2.11)+(2.19) (i = 1);

End-for;

For i := 1 to M − 1 do

Compute ui+1, ϕi+1 solving the linear system (2.22);

For j = 1,N, compute (see (2.6)):

Mi+1, j = θ−1θ Mi, j +

ρckθdt (u

i+1j − ui

j) + `2

1kθdt (ϕ

i+1j − ϕi

j);

Ni+1, j = θ−1θ Ni, j + τ

ξ2θdt(ϕi+1

j − ϕij) − 1

2aξ2θϕi

j − 2ξ2θ

uij;

End-for;

End-for;

End.

3. STABILITY CONDITIONSTo establish conditions of stability for the discrete equation (2.22), we will use the

Lax-Richtmyer definition of stability, expressed in terms of norm ‖ · ‖∞ (see Smith


[7], pp. 48). Let’s rewrite the linear system (2.22) in a more convenient form, namely(ui+1

ϕi+1

)= E-1F

(ui

ϕi

)+ E-1

(di

1di

2

)i = 1, 2, ...,M − 1, (3.1)

(the existence of E-1 will be proved in Proposition 3.1 below). Also, the matrix Ecan be written in the form

E = D2(I + D-12 G2) (3.2)

where D2 = diag(ea, eb, · · · , eb, ea, 2τ + e2, ec, · · · , ec, 2τ + e2) and G2 = A − D2.Thus we conclude that the matrix D-1

2 G2 is expressed as

0 g1ea 0 .. 0 0 0 2c3

eac3ea 0 .. 0 0 0

g1eb 0 g1

eb .. 0 0 0 c3eb

4c3eb

c3eb .. 0 0 0

......

......

......

......

......

. . ....

......

0 0 0 ..g1eb 0 g1

eb 0 0 0 .. c3eb

4c3eb

c3eb

0 0 0 .. 0 g1ea 0 0 0 0 .. 0 c3

ea2c3ea

0 0 0 .. 0 0 0 0 g2g3

0 .. 0 0 00 0 0 .. 0 0 0 g2

ec 0 g2ec .. 0 0 0

......

......

......

......

......

. . ....

......

0 0 0 .. 0 0 0 0 0 0 ..g2ec 0 g2

ec0 0 0 .. 0 0 0 0 0 0 .. 0 g2

g30

with g1 = ρc−e1, g2 = τ − e2, g3 = 2τ+e2.

The sum of each distinct line in array above is written in vector v below

v =

[ρc−e1 + 3c3

ea, 2

ρc−e1 + 3c3

eb,τ − e2

2τ + e2, 2

τ − e2

ec

]. (3.3)

Let’s denote by0 < vmax = max|ρc−e1 + 3c3|, |τ − e2|,

and0 < vmin = minea, eb, 2τ + e2, ec.Now we are able to prove the following result with respect to the stability in matrix

equation (3.1).

Proposition 3.1. Suppose that vmin − vmax > 0.I. If 4ρc + 3 `

2 ≤ 6 dtdx kh(θ − 1) and one of the following conditions is true:

I1. 2ρc + 3 `2 − 6 dt

dx kh(1 − θ) > 6τ + 3dt + 2 dta &

2ρc + 3 `2 − 6 dt

dx kh(1 − θ)vmin − vmax

< 1,

84 Costica Morosanu

orI2. 2ρc + 3 `

2 − 6 dtdx kh(1 − θ) ≤ 6τ + 3dt + 2 dt

a &

6τ + 3dt + 2 dta

vmin − vmax< 1,

then the equation (3.1) is stable.

II. If 4ρc+3 `2 > 6 dt

dx kh(θ−1) and one of the following conditions is true: II1. 6ρc+

3` > 6τ + 3dt + 2 dta &

6ρc + 3`vmin − vmax

< 1,

or

II2. 6ρc + 3` ≤ 6τ + 3dt + 2 dta &

6τ + 3dt + 2 dta

vmin − vmax< 1,

then the equation (3.1) is stable.Otherwise, it is unstable.

Proof. As usual, the proof is reduced to checking the condition of stability which,based on the Lax-Richtmyer definition mentioned above and, taking into account therelation (3.1), it reduces to check the inequality

‖E−1F‖∞ < 1.

First of all let’s determine an estimate for ‖D−12 G2‖∞. Based on relation (3.3), this is

equivalent with: ‖D−12 G2‖∞ = max |v|, wherefrom we easily derive the estimate

‖D−12 G2‖∞ < vmax

vmin. (3.4)

The estimate (3.4) allows us to prove the existence of E-1. Indeed, since by hy-pothesis we have assumed that vmax < vmin, making use of (3.4) we get ‖D−1

2 G2‖∞ < 1which guarantees that there exist (I + D-1

2 G2)−1. Moreover, there exist E-1 andE-1 = (I + D-1

2 G2)-1D-12 . Using the well known inequality: ‖(I + D-1

2 G2)-1‖∞ ≤1

1 − ‖D−12 G2‖∞

and making use of (3.2), it follows that

‖E−1‖∞ ≤ ‖(I + D-12 G2)-1‖∞‖D−1

2 ‖∞ (3.5)

≤ 11 − ‖D−1

2 G2‖∞‖D−1

2 ‖∞.

How ‖D−12 G2‖∞ ≤ 1 imply that 1−‖D−1

2 G2‖∞ ≥ 1− vmax

vmin> 0, we easily deduce from

this that0 <

11 − ‖D−1

2 G2‖∞≤ vmin

vmin − vmax.


Since ‖D−12 ‖∞ ≤ 1

vminand involving the above estimate, from (3.5) we obtain

‖E−1‖∞ < 1vmin − vmax

which allows us to deduce

‖E−1F‖∞ ≤ ‖E−1‖∞‖F‖∞ < 1vmin − vmax

‖F‖∞, (3.6)

where‖F‖∞ = max f a + ρc + e3 + 3c3, f b + 2ρc + 2e3 + 6c + 3,

2dt + f d + f c, 3dt + 2 f d + f e

= max

2ρc+3`

2−6

dtdx

kh(1−θ), 6ρc+3`, 6τ+3dt+2dta

(3.7)

=

2ρc + 3 `2 if4ρc + 3 `

2 ≤ 6 dtdx kh(θ − 1)

−6 dtdx kh(1 − θ) 2ρc + 3 `

2 − 6 dtdx kh(1 − θ) > 6τ + 3dt + 2 dt

a ,

6τ + 3dt + 2 dta if4ρc + 3 `

2 ≤ 6 dtdx kh(θ − 1)

2ρc + 3 `2 − 6 dt

dx kh(1 − θ) ≤ 6τ + 3dt + 2 dta ,

6ρc + 3` if4ρc + 3 `2 > 6 dt

dx kh(θ − 1)6ρc + 3` > 6τ + 3dt + 2 dt

a ,

6τ + 3dt + 2 dta if4ρc + 3 `

2 > 6 dtdx kh(θ − 1)

6ρc + 3` ≤ 6τ + 3dt + 2 dta .

The last equality, which gives us ‖F‖∞, was established after comparing the firsttwo elements in the set of (3.7). Substituting now in (3.6) the corresponding value of‖F‖∞, we constat that in either cases I or II we get the estimate ‖E−1F‖∞ < 1 as weclaimed at beginning of proof.

4. NUMERICAL EXPERIMENTSThe aim of this Section is to present numerical experiments implementing the con-

ceptual algorithm algspline1D. In what follows, all computations were performedtaking ε = .01. Corresponding to input data T , b, M, N, we have used several dif-ferent values while, for the model’s parameters we have considered industrial values,which are:

- the casting speed (c = 12.5 mm/s),

- physical parameters:• the density (ρ = 7.85 kg/m3),• the latent heat (` = 65.28 kcal/kg),• the thermal conductivity (k = 7.8e − 2),

86 Costica Morosanu

Fig. 1. The initial conditions ϕ0 and u0.

• the length of separating zone (ξ = .3),• the relaxation time (τ = 1.0e + 3 ∗ ξ2),• the coefficients of heat transfer (h = 32.012),• a =

√ξ ;

The initial values ϕ0(x j), j = 1, 2, ...,N, plotted in Figure 4.1 - left side, werecomputed via Matlab function csapi(fi0) - cubic spline interpolant to the givendata:

fi0=[-1.4 -1.4 -1.44 -1.42 -1.42 -1.44 -1.43 -1.43 -1.42 -1.42 -1.4 -1.4 -1.25 -1.2 -1.17 -1.15 ...

-1.1 -1.08 -1.0 -.95 -.9 -.85 -.88 -.6 .0 .5 -.92 -.25 .8 -.7 .58 .75 .58 -.63 -.59 .69 -.72 .7 -.59 -.5 ...

.7 -.79 -.87 -.88 .0 .72 -.8 .81 .0 -.89 .0 .7 .55 .68 -.49 .79 .0 -.1 -.8 -.78 -.83 .69 -.8 .68 .5 .7 ...

.59 1. 1.08 1.1 1.15 1.17 1.2 1.25 1.3 1.3 1.25 1.24 1.3 1.31 1.3 1.32 1.3 1.3];

The initial values u0(x j), j = 1, 2, ...,N, plotted in Figure 4.1 - right side, werecomputed as solution of the discrete form to the stationary equation (2a)−1[ϕ0(x) −ϕ3

0(x)] + 2u0(x) = 0 (see Caginalp [2]), i.e.:

(2a)−1[ϕ0(x j) − (ϕ0(x j)3)] + 2u0(x j) = 0 j = 1, 2, ...,N.

Now we are able to calculate the vectors: z(ε, ·) =(z(ε, ϕ0(x j)

))j=1,N

(see Figure 4.2), ϕ1 =(ϕ1

j

)j=1,N

, and u1 =(u1

j

)j=1,N

(see (2.2)-(2.3)).We will continue with the presentation of numerical experiments regarding the

stability of equation (3.1) (see Proposition 3.1). The shape of the graphs plotted


Fig. 2. The approximate solution z(ε, ·) of Cauchy problem (1.7)

in Figures 4.3A and 4.3B shows the stability and accuracy of the numerical resultsobtained by algorithm algspline1D. For this test we have used T = 1, b = 1, M = 50,N = 100 and the temperature surrounding at ∂Ω=0, b given by: w(ti, 0) = −15,w(ti, b) = 7.5, i = 1, 2, ..., M.

Taking now ξ=.35, we can verify that vmin − vmax=112.1467 which means that thefirst hypothesis in Proposition 3.1 in verified, but II1 is not verified:

6ρc + 3` = 784.59, 6τ + 3dt + 2 dta = 735.1289 & 6ρc+3`

vmin−vmax= 6.9961.

Consequently the numerical scheme is unstable. Figure 4.4A shows that it reallyis. Furthermore, if we keep ξ=.35 and take M = 60 (in place of M = 50), we getalso vmin − vmax > 0 and 6ρc+3`

vmin−vmax= 592.017. So, again we are in a unstable case.

Moreover, analyzing the graph in Figure 4.4B we found a more pronounced instabil-ity. Let’s remark that the instability of the solution occurred following a slight change(modification) of only two physical parameters (ξ and dt in this case). This highlightsthe strong dependence of approximation scheme regarding physical parameters.

We turn to numerical stability conditions and we change the temperature surround-ing at 0 ∈ ∂Ω by setting w(ti, 0) = −60 and w(ti, 0) = −120, i = 1, 2, ...,M. Thenumerical results were plotted in Figures 4.5A and Figure 4.5B, respectively. Ana-lyzing the approximations near to zero, we observe a instability just for u, due to thenature of boundary conditions that we have considered (1.2)1.

88 Costica Morosanu

Fig. 3. Example of numerical stability: ui at different levels of time

Fig. 4. Example of numerical stability: ϕi at different levels of time


Fig. 5. An example of slight numerical instability

Fig. 6. An example of strong numerical instability

90 Costica Morosanu

Fig. 7. ui corresponding to wi(0) = −60.

Fig. 8. ui corresponding to wi(0) = −120.


5. CONCLUSIONSAs the novelty of this work we notice the use of a cubic spline method in order to

approximate the linear system given by a scheme of fractional steps type. Notice firstthat conditions of stability are sustained by both theory and numerical experimentand that are significantly dependent on the physical parameters (see figures 4.4A,4.4B). Moreover, analyzing the numerical results in terms of physical phenomena,we constat that the temperature distribution tends to become parabolic and the phasefunction distribution say that the instability of the portion of material will disappear.More precisely, analyzed together the figures 4.3A and 4.3B, highlight theoreticalmeaning assigned to functions u and ϕ as well as the zone of separation betweenmaterial phases.

The numerical solution obtained by this way can be considered as an admissibleone for the corresponding boundary optimal control problem. Generally, the numer-ical method considered here can be used to approximate the solution of a nonlinearparabolic phase-field system containing a general nonlinear part.

References[1] T. Benincasa and C. Morosanu, Fractional steps scheme to approximate the phase-field transition

system with nonhomogeneous Cauchy-Neumann boundary conditions, Numer. Funct. Anal. andOptimiz., Vol. 30 (3-4)(2009), 199-213.

[2] G. Caginalp, An analysis of a phase field model of a free boundary, in ”Arch. Rat. Mech. Anal.”,92(1986), 205-245.

[3] C. Morosanu, Approximation and numerical results for phase field system by a fractional stepscheme, Revue d’analyse numerique et de theorie de l’approximation, Tome 25, 1-2(1996), 137-151.

[4] C. Morosanu, Approximation of the phase-field transition system via fractional steps method,Numer. Funct. Anal. Optimiz., 18, 5& 6(1997), 623-648.

[5] C. Morosanu, Analysis and optimal control of phase-field transition system, Nonlinear Funct.Anal. & Appl., Vol. 8, 3(2003), 433-460.

[6] P.M. Prenter, Splines and variational methods, J. Wiley, New York, London, 1979.

[7] G.D. Smith, Numerical Solution of Partial Differential Equations: Finite Difference Methods,Third Edition, Clarendon Press, Oxford, 1985.

ON CONVEXITY OF HELE- SHAW CELLS

ROMAI J., v.8, no.1(2012), 93–101

Ibrahim W. RabhaInstitute of Mathematical Sciences, University Malaya, Kuala Lumpur, [email protected]

Abstract The main aim of this paper is to apply methods of the theory of univalent functions andgeometric functions to some problems of fluid mechanics. We study the time evolutionof the free boundary of a viscous fluid for planar flows in Hele- Shaw cells under injec-tion; we prove the invariance in time of convexity of complex order for two basic cases:the inner problem and the outer problem.

Keywords: free boundary problem, conformal map, complex analysis, univalent functions, unit disk,Hele-Shaw cells, Hadamard product, convex function, convex of complex order.2010 MSC: 30C45,76S05, 76D99.

1. INTRODUCTIONOne of the basic models is a fluid flow in a Hele-Shaw cell. Hele-Shaw [1] first

described his “cell”, which was an experimental device for studying fluid flow bypumping a viscous liquid into the gap between two closely-separated glass plates.Using dye-lines, he was able to observe the flow patterns generated when the flow wasimpeded by various kinds of obstacles, such as aerofoil sections, placed between theplates. If the displacing fluid has lower viscosity than the displaced fluid the interfacewill develop hydrodynamical instability which results in highly ramified patterns [2-4]. This phenomenon is known as viscous fingering, and it may also occur whenelasticity of the fluids acts as another driving mechanism [5-7]. Pattern formationof similar type has been observed in a variety of non equilibrium systems besidesviscous fingering, such as crystal growth [8], electrodeposition [9] and solidification[10].

The canonical mathematical model for Hele-Shaw flows is Darcy’s law, where theflow velocity is proportional to the pressure gradient:

Darcy Law : vn(z) ∼ H(z); z ∈ γ(t),

where vn is the velocity of the curve, γ(t) is a simple planar curve - boundary of asimply-connected domain and H(z) is the normal gradient of a solution of the Dirich-let problem in a fluid domain D with a source (a sink) at a distant location (oftenat infinity). Under the assumption that the flow is incompressible, the pressure fieldsatisfies a Laplace’s equation; therefore, such an evolution of the free interface isalso called Laplacian growth process. The method of conformal mapping resides its

93

94 Ibrahim W. Rabha

strength of transforming the generally difficult task of solving a moving-free bound-ary problem into finding solutions to a single differential equation of an analyticfunction on a fixed domain, usually the half plane or the interior of the unit disk.Indeed, the simply connected domains are preserved in rotating cell [11]. Recently,the stability of this method was considered in [12].

The time evolution of the free boundary of a viscous fluid for planar flows in Hele-Shaw cells under injection was studied by many authors. By using methods of uni-valent functions theory, they proved that certain geometric properties (such as star-likeness, directional convexity Φ−like and E−family) are preserved in time [13-20].In fact, as any zero-surface tension problem is timer-eversible, blow-up solutions canbe generated by injecting with a non-smooth free boundary and then reversing thesequence of solutions so obtained. This procedure can reveal unexpected features,an example being the “waiting time” that can occur when injection takes place intoan initial domain with a corner [21]. The analysis of this situation again involves theBaiocchi transform of the pressure, and this device is also instrumental in the analysisof allowable cusps in injection problems [22].

In this paper, we continue their study by proving the invariance in time of anothergeometric property. We study the time evolution of the free boundary of a viscousfluid for planar flows in Hele- Shaw cells under injection. Applying methods fromthe theories of univalent functions and geometric functions; we prove the invariancein time of convexity of complex order property for two basic cases: the inner problemand the outer problem.

2. HELE-SHAW CELLS PROBLEMWe study the flow of a viscous fluid in a planar Hele-Shaw cell under injection

through a source (of constant strength Q,Q < 0 in case of injection) which is situatedat the origin. Suppose that for the initial time, the phase domain Ω0 occupied bythe fluid is simply connected and bounded by a smooth curve Γ0. The evolution ofthe phase domains Ω(t),Ω(0) = Ω0 is described by an auxiliary conformal mappingf (ζ, t), f (ζ, 0) = f0(ζ) on the unit disk U := z ∈ C : |z| < 1 onto Ω(t),Γ(t) = ∂Ω(t)normalized by f (0, t) = 0, f ′(0, t) > 0. We denote the derivatives by f ′(z, t) =

∂ f∂z

and f (z, t) =∂ f∂t , where t is the time parameter. The function f (ζ, 0) = f0(ζ) poses

a parametrization of Γ0 = f0(eiθ), θ ∈ [0, 2π), while the moving boundary isparameterized by Γ(t) = f (eiθ, t), θ ∈ [0, 2π). This mapping satisfies the equation(see [23-27])

<[ f (ζ, t)ζ f ′(ζ, t)] = − Q2π, ζ = eiθ. (1)

In the case of the problem of injection Q < 0 of the fluid into a bounded simplyconnected with small surface tension γ > 0, the Polubarinova-Galin equation [18] isof the form:

On convexity of Hele- Shaw cells 95

<[ f (ζ, t)ζ f ′(ζ, t)] = − Q2π

+ γH[i∂k∂θ

(eiθ, t)](θ), ζ = eiθ, (2)

where

k(eiθ, t) =1

| f ′(eiθ, t)|<(1 +

eiθ f ′′(eiθ, t)f ′(eiθ, t)

), θ ∈ [0, 2π),

∂k∂θ

(eiθ, t) = −=(e2iθS f (eiθ, t)| f ′(eiθ, t)| ),

S f = (f ′′

f ′)′ − 1

2(

f ′′

f ′)2

is the Schwarzian derivative, and H[Φ](θ) is the Hilbert transform given by

H[Φ](θ) =1π

P.V.θ

∫ 2π

0

Φ(eiθ′)1 − ei(θ−θ′) dθ′

(PV denotes the Cauchy principal value) satisfying

∂

∂θ[i∂k∂θ

(eiθ, t)](θ) = H[iA](θ),

where

A(ζ) =( 1| f ′(ζ)|

)[<

(2ζ2S f (ζ)+ζ(

f ′′(ζ)f ′(ζ)

)′−ζ(f ′′(ζ)f ′(ζ)

)(f ′′(ζ)f ′(ζ)

)′)+=

(ζ f ′′(ζ)f ′(ζ)

)=ζ2S f (ζ)

].

The case of unbounded domain with bounded complement virtues as the dynamicsof a contracting bubble in a Hele-Shaw cell since the fluid occupies a neighborhoodof infinity and injection (of constant strength Q < 0) is supposed to take place atinfinity. Again, we denote by Ω(t) the domain occupied by the fluid at the momentt, Γ(t) = ∂Ω(t). By using the Riemann mapping theorem, the domain Ω(t) can bedescribed by a univalent function F(ζ, t) from the exterior of the unit disk U = ζ :|ζ | > 1 onto Ω(t). The equation satisfied by the free boundary is [16,18]

<[F(ζ, t)ζF′(ζ, t)] =Q2π, ζ = eiθ, (3)

for the zero tension surface model and

<[F(ζ, t)ζF′(ζ, t)] =Q2π− γH[i

∂k∂θ

(eiθ, t)](θ), ζ = eiθ, (4)

for the small surface tension model.

96 Ibrahim W. Rabha

3. THE INNER PROBLEMLet us define the classes of convex functions which will parameterize our phase

domains. If a function f (z) maps U onto a convex domain and f (0) = 0, then we saythat f (z) is a convex function. A necessary and sufficient condition for a function fto be convex is the inequality<(1 +

z f ′′(z)f ′(z) ) > 0.

In this section, we impose the invariance in time of convex of complex order for theinner problem. Starting with an initial bounded domain Ω(0) which is convex ofcomplex order, we prove that at each moment t ∈ [0,T ) the domain Ω(t) is convex ofcomplex order.

Definition 3.1. Let f be a holomorphic function on U such that f (0) = 0, f ′(0) , 0.A function f is called convex of complex order b (b ∈ C\0) if it satisfies the followinginequality

<1 +1b

(z f ′′(z)f ′(z)

− 1) > 0, (z ∈ U).

We denoted this class C(b).

Definition 3.2. A simply connected domain Ω , C is convex in the direction of thereal axis R if each line parallel to R either misses Ω or the intersection with Ω is aconnected set. If a function f (ζ) maps U(the unit disk) onto a domain which is convexin the direction of the real axis, f (0) = 0, then we say that f (z) is a convex functionin the direction of the real axis and denote the class of such function by CR and byCR(b) for convex of complex order in the direction of the real axis.

Theorem 3.1. Let Q < 0 and f0 ∈ CR(b) on U and univalent in U. Let f (ζ, t) be theclassical solution of the Polubarinova-Galin equation (1) with the initial conditionf (ζ, 0) = f0(ζ). Moreover, let Ω :=

⋃t∈[0,T ) Ω(t) =

⋃t∈[0,T ) f (U, t), where T is the

blow-up time. If<b = 2 and 1 < =b < 2 then f (ζ, t) ∈ CR(b) in the direction of thereal axis R for |ζ | ∈ [ε, 1), 0 < ε < 1.

Proof. Assume that there exists a complex number such that

ζ0 := (eiθ0)b−1

b , |ζ0| ≤ 1

which satisfies the equality

arg ζ0

(b f ′(ζ0, t0)

) bb−1

=π

2(or − π

2), b , 0

and for ε > 0 there is t > t0 and θ ∈ (θ0 − ε, θ0 + ε) such that

arg ζ(b f ′(ζ, t)

) bb−1 ≥ π

2(or − π

2).


Then∂

∂θarg(eiθ)

b−1b(b f ′(ei( b−1

b )θ, t)) b

b−1 ≥ 0,

for ζ = (eiθ)b−1

b , which yields

<1 +

1b

[ζ f ′′(ζ, t0)

f ′(ζ, t0)− 1]

> 0.

Since f ′(ei( b−1b )θ, t) , 0 we obtain

=ζ0(b f ′(ζ0, t0))b

b−1 > 0.

Since ζ0 is a critical point then we deduce that

∂

∂θarg ei( b−1

b )θ(b f ′(ei( b−1

b )θ, t)) b

b−1∣∣∣∣θ=θ0,t=t0

= 0

and∂

∂rarg ei( b−1

b )θ(b f ′(rei( b−1

b )θ, t)) b

b−1∣∣∣∣θ=θ0,t=t0,r=1

≥ 0.

We put

<(1 +

1b

[ζ0 f ′′(ζ0, t0)

f ′(ζ0, t0)− 1]

)= 0,

=(1 +

1b

[ζ0 f ′′(ζ0, t0)

f ′(ζ0, t0)− 1]

)> 0.

Equality does not hold for the last assertion because the level lines preserve the prop-erty of being convex in the direction of R. By the assumptions<b = 2 and =b < 2,we obtain

<(1 +

ζ0 f ′′(ζ0, t0)f ′(ζ0, t0)

)= 0, (5)

=(1 +

ζ0 f ′′(ζ0, t0)f ′(ζ0, t0)

)> 0. (6)

Then we derive∂

∂targ ζ(b f ′(ζ, t))

bb−1 = = 1

b − 1

∂∂t f ′(ζ, t)f ′(ζ, t)

. (7)

By differentiating the Polubarinova-Galin equation with respect to θ for |ζ | = 1, weobtain

=(

f ′(ζ, t)∂

∂tf ′(ζ, t) − ζ f ′(ζ, t) f ·(ζ, t) − ζ2 f ′′(ζ, t) f ·(ζ, t)

)= 0

or

| f ′(ζ, t)|2=( ∂∂t f ′(ζ, t)f ′(ζ, t)

− ζ f ′(ζ, t) f ·(ζ, t) − ζ2 f ′′(ζ, t) f ·(ζ, t))

= 0

98 Ibrahim W. Rabha

and consequently, we have

| f ′(ζ, t)|2= 1b − 1

∂∂t f ′(ζ, t)f ′(ζ, t)

= = 1b − 1

ζ f ′(ζ, t) f ·(ζ, t)(ζ f ′′(ζ, t)

f ′(ζ, t)− 1

)

Substituting (1) and (5) in last equation, we have

∂

∂targ ζ

(b f ′(ζ, t)

) bb−1

∣∣∣∣ζ=ζ0,t=t0

=Q

2π| f ′(ζ0, t0)|2=1

b − 1

(ζ f ′′(ζ0, t0)f ′(ζ0, t0)

+ 1).

The right-hand side of this equality is strictly negative because of (6) and =b > 1.Hence


) bb−1 <

π

2for t close to t0 and in a neighborhood of θ0 and this is a contradiction.

4. THE OUTER PROBLEMIn this section, we obtain the invariance in time of the same geometric property

(denoted by C(b)) for the outer problem (injection at infinity). Here we prove that ourproblem preserves geometric properties of domains with C(b) complements. Let ussuppose that the complement of the fluid domain contains the origin and is in the classC(b) with respect to the origin at the initial instant, we have the following definition:

Definition 4.1. Let F be a holomorphic function on U such that

F(ζ) = aζ + a0 +a−1

ζ+ ..., a , 0.

The C(b) is defined so that

<1 +1b

(zF′′(z)F′(z)

− 1) > 0, ζ ∈ U. (8)

Straightforward computations give the following facts:

Remark 4.1. (a) If the function F belongs to the C(b) in U then the function f :

U −→ C given by

f (z) =1

F( 1z ), z , 0, f (0) = 0

is a member in C(b) on U.


(b) If the function f belongs to the C(b) in U then the function F : U −→ C givenby

F(ζ) =1

f ( 1ζ ), ζ , 0,

is a member in C(b) on U.

(c) If F ∈ C(b), then it is univalent on U.

Theorem 4.1. Let Q < 0 and F0 be a function which is in the class C(b) on U and

univalent in U. Let F(ζ, t) be the classical solution of the Polubarinova-Galin equa-tion (3) with the initial condition F(ζ, 0) = F0(ζ). Moreover, let Ω :=

⋃t∈[0,T ) Ω(t) =⋃

t∈[0,T ) F(U, t), where T is the blow-up time. If If <b = 2 and 1 < =b < 2 thenF(ζ, t) is in C(b).

Proof. By assuming the function f (ζ, t) = 1/F( 1ζ ), the Polubarinova-Galin equation

(3) can be rewritten in terms of f as follows:

<[ f (ζ, t)ζ f ′(ζ, t)] = −Q| f (ζ, t)|42π

, |ζ | = 1. (9)

According to the previous remark, the function F(ζ, t), ζ ∈ U is in C(b) if and only iff (ζ, t), ζ ∈ U is in C(b). Thus, it suffices to prove that the functions f (ζ, t), ζ ∈ U, t ∈[0,T ), is in C(b). Suppose by contrary that the previous statement is not true. Thenthere exist t0 ≥ 0 and ζ0 = eiθ0 such that equations (5-7) are held. Calculation gives

∂

∂targ ζ

(b f ′(ζ, t)

) bb−1

∣∣∣∣ζ=ζ0,t=t0

=Q| f ′(ζ0, t0)|2

2π= 1

b − 1

(ζ f ′′(ζ0, t0)f ′(ζ0, t0)

+ 1).

The right-hand side of this equality is strictly negative because of (6) and =b > 1.Hence


) bb−1 <

π

2

for t close to t0 and in a neighborhood of θ0 or

arg[1 +1b

(eiθ f ′′(eiθ, t)

f ′(eiθ, t)− 1)] <

π

2

and this is a contradiction. Hence f (ζ, t), ζ ∈ U is in the C(b) and consequentlyF(ζ, t), ζ ∈ U is in C(b).

100 Ibrahim W. Rabha

5. CONCLUSIONWe imposed sufficient conditions for convexity of Hele- Shaw Cells in complex

domain. This problem was studied by Gustafsson, Friedman, Hohlov, Markina,Prokhorov, Vasil’ev and others. We considered a modification of a logarithmic an-tiderivative of the convex of complex order. The functional that assumed here taking

the form ζ(b f ′(ζ, t)

) bb−1 such that<b = 2 and 1 < =b < 2. This functional satisfied

arg ζ0

(b f ′(ζ0, t0)

) bb−1

=π

2(or − π

2), b , 0

whereζ0 := (eiθ0)

b−1b , |ζ0| ≤ 1.

References[1] H.S. Hele-Shaw, The flow of water, Nature, 58(1898), 33-36.

[2] P. Saffman, G. I. Taylor, The penetration of a fluid into a porous medium or Hele-Shaw cellcontaining a more viscous fluid, Proc. R. Soc. London, Ser. A, 245(1958), 312-329.

[3] L. Paterson, Radial fingering in a Hele-Shaw cell, J. Fluid Mech. 113(1981), 513-529.

[4] D. Bensimon, Stability of viscous fingering, Phys. Rev., A 33(1986), 1302-1308.

[5] E. Lemaire, P. Levitz, G. Daccord , H. Damme, From viscous fingering to viscoelastic frac-turing, Phys. Rev. Lett., 67(1991), 2009-2012.

[6] T. Podgorski, M. C. Sostarecz, S. Zorman, A. Belmonte, Fingering instabilities of a reactivemicellar interface, Physical Review E, 76( 016202)(2007), 1-6.

[7] S. Mora, M. Manna, Saffman-Taylor instability of viscoelastic uids: From viscous fingering toelastic fractures, Physical Review E, 81(026305) (2010), 1-10.

[8] J. Langer, Instabilities and pattern formation in crystal growth, Rev. Mod. Phys., 52(1980), 1-28.

[9] M. Matsushita, M. Sano, Y. Hayakawa, H. Honjo, Y. Sawada, Fractal structures of zinc metalleaves grown by electrodeposition, Phys. Rev. Lett., 53(1984), 286-289.

[10] J. Hunt, Pattern formation in solidification, Materials Science & Technology, 15(1999), 9-15.

[11] V.M. Entov, P.I. Etingof, D. Ya, Kleinbock, On nonlinear interface dynamics in Hele-Shaw flows,Eur. J. Appl. Math., 6(1995), 399-420.

[12] A. He, A. Belmonte, Inertial effects on viscous fingering in the complex plane, J. Fluid Mech.,668(2011), 436-445.

[13] Y. E. Hohlov, D. V. Prokhorov, A. J. Vasil’ev, On geometrical properties of free boundaries inthe Hele-Shaw flows moving boundary problem, Lobachevskii Journal of Mathematics, 1(1998),3-12.

[14] K. Kornev, A. Vasil’ev, Geometric properties of the solutions of a Hele-Shaw type equation,Proceedings of the American Mathematical Society, 128, 9(2000), 2683-2685.

[15] D. Prokhorov , A. Vasil’ev, Convex dynamics in Hele-Shaw cells, International Journal of Math-ematics and Mathematical Sciences, 31, 11(2002), 639-650.

[16] A. Vasil’ev, Univalent functions in the dynamics of viscous flows, Computational Methods andFunction Theory, 1, 2(2001), 311-337.


[17] A. Vasil’ev, Univalent functions in two-dimensional free boundary problems, Acta ApplicandaeMathematicae, 79, 3(2003), 249-280.

[18] A. Vasil’ev, I. Markina, On the geometry of Hele-Shaw flows with small surface tension, Inter-faces and Free Boundaries, 5, 2(2003), 183-192.

[19] P. Curt, D. Fericean, A Special class of univalent functions in Hele-Shaw flow problems, Abstractand Applied Analysis Volume 2011, Article ID 948236, 1-10.

[20] R.W. Ibrahim, On Geometric Properties for Hele- Shaw Cells, Complex Variables and EllipticEquations. To appear.

[21] J.R. King, , A.A. Lacey, J.L. Vazquez, Persistence of corners in free boundaries in Hele-Shawflow, Europ. J. Appl. Math., 6(1995), 455-490.

[22] S.D Howison, Cusp development in Hele-Shaw flow with a free surface, SIAM J. Appl. Math.,46(1986), 20-26.

[23] L. A. Galin, Unsteady filtration with a free surface, Doklady Akademii Nauk USSR, 47(1945),246- 249, (Russian).

[24] P. Y. Polubarinova-Kochina, On a problem of the motion of the contour of a petroleum shell,Doklady Akademii Nauk USSR, 47(1945), 254-257, (Russian).

[25] P. J. Poloubarinova-Kochina, Concerning unsteady motions in the theory of filtration, 9, 1(1945),79-90, (in Russian).

[26] L. Fejer, Mechanische Quadraturen mit positiven Cotesschen Zahlen, Math. Zeitschrift, 37, 2(1933), 287-309.

[27] B. Gustafsson, A. Vasil’ev, Conformal and Potential Analysis in Hele-Shaw Cells, Advances inMathematical Fluid Mechanics, Birkhauser, Basel, Switzerland, 2006.

CONVERGENCE THEOREMS OF THREE-STEPITERATIVE ALGORITHM WITH ERRORS FORASYMPTOTICALLY QUASI-NONEXPANSIVEMAPPINGS IN THE INTERMEDIATE SENSE

ROMAI J., v.8, no.1(2012), 103–114

Gurucharan S. SalujaDepartment of Mathematics and Information Technology, Govt. Nagarjuna P.G. College ofScience, Raipur, [email protected]

Abstract The purpose of this paper is to establish weak and strong convergence theorems ofthree-step iterative algorithm with errors to approximate a common fixed point for non-Lipschitzian asymptotically quasi-nonexpansive mappings in the intermediate sense inthe setup of Banach spaces. The results presented in this paper extend and improve thecorresponding results of [1, 7, 8, 14, 15, 16, 17, 18, 21] and many others.

Keywords: asymptotically quasi-nonexpansive mapping in the intermediate sense, common fixed point,three-step iterative algorithm with errors, Banach space, strong convergence, weak convergence.2010 MSC: 47H09, 47H10, 47J25.

1. INTRODUCTION AND PRELIMINARIESLet C be a nonempty subset of a real Banach space E. Let T : C → C be a map-

ping. We use F(T ) to denote the set of fixed points of T , that is, F(T ) = x ∈ C :T x = x. Recall the following concepts.

(1) T is nonexpansive if

‖T x − Ty‖ ≤ ‖x − y‖ , (1)

for all x, y ∈ C.

(2) T is quasi-nonexpansive if F(T ) , ∅ and

‖T x − p‖ ≤ ‖x − p‖ , (2)

for all x ∈ C and p ∈ F(T ).

(3) T is asymptotically nonexpansive [3] if there exists a sequence an in [1,∞)with an → 1 as n→ ∞ such that

103

104 Gurucharan S. Saluja

∥∥∥T nx − T ny∥∥∥ ≤ an ‖x − y‖ , (3)

for all x, y ∈ C and n ≥ 1.

(4) T is uniformly L-Lipschitzian if there exists a constant L > 0 such that

∥∥∥T nx − T ny∥∥∥ ≤ L ‖x − y‖ , (4)

for all x, y ∈ C and n ≥ 1.

(5) T is asymptotically quasi-nonexpansive [7] if F(T ) , ∅ and there exists a se-quence an in [1,∞) with an → 1 as n→ ∞ such that

∥∥∥T nx − p∥∥∥ ≤ an ‖x − p‖ , (5)

for all x ∈ C, p ∈ F(T ) and n ≥ 1.

It is clear that every nonexpansive mapping is asymptotically nonexpansive andevery asymptotically nonexpansive is uniformly Lipschitzian. Also, if F(T ) , ∅,then a nonexpansive mapping is a quasi-nonexpansive mapping and an asymptoti-cally nonexpansive mapping is an asymptotically quasi-nonexpansive mapping butthe converse is not true in general.

The class of asymptotically nonexpansive mappings was introduced by Goebeland Kirk [3] as a generalization of the class of nonexpansive mappings. Recall alsothat a mapping T : C → C is said to be asymptotically quasi-nonexpansive in theintermediate sense [22] provided that T is uniformly continuous and

lim supn→∞

supx∈C, p∈F(T )

( ∥∥∥T nx − p∥∥∥ − ‖x − p‖

)≤ 0. (6)

From the above definitions, it follows that asymptotically nonexpansive mappingmust be asymptotically quasi-nonexpansive and asymptotically quasi-nonexpansivemapping in the intermediate sense. But the converse does not hold as the followingexample:

Example 1. (see [6]) Let X = R, K = [− 1π ,

1π ] and |λ| < 1. For each x ∈ K, define

T (x) =

λ x sin( 1

x ), if x , 0,0, if x = 0.

Then T is an asymptotically nonexpansive mapping in the intermediate sense butit is not asymptotically nonexpansive mapping.

Convergence theorems of three-step iterative algorithm with errors... 105

It is well known that the concept of asymptotically nonexpansive mappings wasintroduced by Goebel and Kirk [3] who proved that every asymptotically nonexpan-sive self-mapping of nonempty closed bounded and convex subset of a uniformlyconvex Banach space has a fixed point. In 1973, Petryshyn and Williamson [13]gave necessary and sufficient conditions for Mann [9] iterative sequence to convergeto fixed points of quasi-nonexpansive mappings. In 1997, Ghosh and Debnath [1]extended the results of Petryshyn and Williamson [13] and gave necessary and suffi-cient conditions for Ishikawa iterative sequence to converge to fixed points for quasi-nonexpansive mappings.

Liu [8] extended the results of [1, 13] and gave necessary and sufficient conditionsfor Ishikawa iterative sequence with errors to converge to fixed point of asymptoti-cally quasi-nonexpansive mappings. In 2001, Noor [10, 11] introduced the three-stepiterative sequences and studied the approximate solutions of variational inclusions inHilbert spaces. The three-step iterative approximation problems were studied exten-sively by Noor [10, 11], Glowinsky and Le Tallec [2], and Haubruge et al [4]. It hasbeen shown [2] that three-step iterative scheme gives better numerical results thanthe two step and one step approximate iterations. Thus we conclude that three stepscheme plays an important and significant role in solving various problems, whicharise in pure and applied sciences.

In 2002, Xu and Noor [21] introduced and studied a three-step iteration scheme toapproximate fixed points of asymptotically nonexpansive mappings in the frameworkof real uniformly convex Banach spaces.

In 2006, Quan [14] studied some necessary and sufficient conditions for three-stepIshikawa iterative sequences with error terms for uniformly quasi-Lipschitzian map-pings to converge to fixed points. The results presented in [14] extend and improvethe corresponding results of Liu [7, 8], Xu and Noor [21] and many others.

Inspired and motivated by [14, 21] and others, we study the following three-stepiterative algorithm with errors:

Definition 1.1. Let E be a normed linear space, C be a nonempty convex subset of E,and T1, T2,T3 : C → C be given mappings. Then for arbitrary x1 ∈ C, the iterativesequences xn, yn, zn defined by

zn = (1 − γn − νn)xn + γnT n3 xn + νnun, n ≥ 1,

yn = (1 − βn − µn)xn + βnT n2 zn + µnvn, n ≥ 1,

xn+1 = (1 − αn − λn)xn + αnT n1 yn + λnwn, n ≥ 1, (7)


where un, vn, wn are bounded sequences in C and αn, βn, γn, λn, µn, νnare appropriate sequences in [0, 1], is called the three-step iterative algorithm witherror terms of Ti for all i = 1, 2, 3.

We note that the usual modified Ishikawa and Mann iterations are special cases ofthe above three-step iterative scheme. If γn = νn = 0 and T1 = T2 = T , then (7)reduces to the usual modified Ishikawa iterative scheme with errors,

yn = (1 − βn − µn)xn + βnT nxn + µnvn, n ≥ 1,

xn+1 = (1 − αn − λn)xn + αnT nyn + λnwn, n ≥ 1, (8)

where vn, wn are bounded sequences in C and αn, βn, λn, µn are appropriatesequences in [0, 1].

If βn = µn = 0, then (8) reduces to the usual modified Mann iterative scheme witherrors,

x1 ∈ C,

xn+1 = (1 − αn − λn)xn + αnT nxn + λnwn, n ≥ 1, (9)

where wn is a bounded sequence in C and αn, λn are appropriate sequences in[0, 1].

We say that a Banach space E satisfies the Opial’s condition [12] if for each se-quence xn in E weakly convergent to a point x and for all y , x

lim infn→∞ ‖xn − x‖ < lim inf

n→∞ ‖xn − y‖ .

The examples of Banach spaces which satisfy the Opial’s condition are Hilbertspaces and all Lp[0, 2π] with 1 < p , 2 fail to satisfy Opial’s condition [12].

Let K be a nonempty closed convex subset of a Banach space E. Then I − T isdemiclosed at zero if, for any sequence xn in K, condition xn → x weakly andlimn→∞ ‖xn − T xn‖ = 0 implies (I − T )x = 0.

The purpose of this paper is to investigate necessary and sufficient condition forthree-step iterative algorithm with errors for asymptotically quasi-nonexpansive map-pings in the intermediate sense to converge to common fixed points in Banach spaces.The results obtained in this paper extend and improve the corresponding results of[1, 7, 8, 14, 15, 16, 17, 18, 21] and many others.

In the sequel, we shall need the following lemma:


Lemma 1.1. (see [20]) Let an and bn be sequences of nonnegative real numberssatisfying the inequality

an+1 ≤ an + bn, n ≥ 1.

If∑∞

n=1 bn < ∞, then limn→∞ an exists. In particular, if an has a subsequence con-verging to zero, then limn→∞ an = 0.

2. MAIN RESULTSWe are now in a position to prove our main results of this paper.

Theorem 2.1. Let E be a real Banach space, C be a nonempty closed convex subset ofE. Let Ti : C → C, (i = 1, 2, 3) be uniformly L-Lipschitzian and asymptotically quasi-nonexpansive mappings in the intermediate sense with F = ∩3

i=1F(Ti) , ∅. Let xnbe the sequence defined by (7) with the restrictions

∑∞n=1 αn < ∞ and

∑∞n=1 λn < ∞.

Put

cn = max0, sup

x∈C, p∈F

( ∥∥∥T ni x − T n

i p∥∥∥ − ‖x − p‖

): i = 1, 2, 3

, (10)

such that∑∞

n=1 αncn < ∞. Then xn converges to a common fixed point of the map-pings T1, T2 and T3 if and only if

lim infn→∞ d(xn, F) = 0,

where d(x, F) = infp∈F d(x, p).

Proof. The necessity is obvious. Thus we only prove the sufficiency. Let p ∈ F. Itfollows from (7) and (10) that

‖zn − p‖ =∥∥∥(1 − γn − νn)xn + γnT n

3 xn + νnun − p∥∥∥

≤ (1 − γn − νn) ‖xn − p‖ + γn

∥∥∥T n3 xn − p

∥∥∥ + νn ‖un − p‖

≤ (1 − γn − νn) ‖xn − p‖ + γn[‖xn − p‖ + cn] + νn ‖un − p‖

≤ ‖xn − p‖ + γncn + νn ‖un − p‖ ,

(11)


using (7) and (11), we have

‖yn − p‖ =∥∥∥(1 − βn − µn)xn + βnT n

2 zn + µnvn − p∥∥∥

≤ (1 − βn − µn) ‖xn − p‖ + βn

∥∥∥T n2 zn − p

∥∥∥ + µn ‖vn − p‖

≤ (1 − βn − µn) ‖xn − p‖ + βn[‖zn − p‖ + cn] + µn ‖vn − p‖

≤ (1 − βn − µn) ‖xn − p‖ + βn ‖zn − p‖ + βncn + µn ‖vn − p‖

≤ (1 − βn − µn) ‖xn − p‖ + βn

[‖xn − p‖ + γncn + νn ‖un − p‖

]

+βncn + µn ‖vn − p‖

≤ ‖xn − p‖ + βncn(1 + γn) + βnνn ‖un − p‖ + µn ‖vn − p‖

≤ ‖xn − p‖ + 2βncn + νn ‖un − p‖ + µn ‖vn − p‖ ,

(12)

again using (7) and (12), we have

‖xn+1 − p‖ =∥∥∥(1 − αn − λn)xn + αnT n

1 yn + λnwn − p∥∥∥

≤ (1 − αn − λn) ‖xn − p‖ + αn∥∥∥T n

1 yn − p∥∥∥ + λn ‖wn − p‖

≤ (1 − αn − λn) ‖xn − p‖ + αn[‖yn − p‖ + cn] + λn ‖wn − p‖

≤ (1 − αn − λn) ‖xn − p‖ + αn[‖xn − p‖ + 2βncn + νn ‖un − p‖

+µn ‖vn − p‖]

+ αncn + λn ‖wn − p‖

≤ ‖xn − p‖ + αncn(1 + 2βn) + αnνn ‖un − p‖ + αnµn ‖vn − p‖ + λn ‖wn − p‖

≤ ‖xn − p‖ + 3αncn + αn ‖un − p‖ + αn ‖vn − p‖ + λn ‖wn − p‖

= ‖xn − p‖ + tn,(13)

where

tn = 3αncn + αn ‖un − p‖ + αn ‖vn − p‖ + λn ‖wn − p‖ .

Since by hypothesis∑∞

n=1 αncn < ∞,∑∞

n=1 αn < ∞,∑∞

n=1 λn < ∞ and un, vn, wnare bounded in C, it follows that

∑∞n=1 tn < ∞. From (13) and Lemma 1.1, we have


limn→∞ ‖xn − p‖ exists. Also from (13), we obtain

d(xn+1, F) ≤ d(xn, F) + tn, (14)

for all n ≥ 1. From Lemma 1.1 and (14), we know that limn→∞ d(xn, F) exists. Sincelim infn→∞ d(xn, F) = 0, we have that limn→∞ d(xn, F) = 0.

Now, we shall prove that xn is a Cauchy sequence. Let limn→∞ ‖xn − p‖ = r. Forany given ε > 0, since un, vn, wn are bounded in C, there exists a constant M > 0,such that for all n ≥ 1, ‖xn − p‖ ≤ M, ‖un − p‖ ≤ M, ‖vn − p‖ ≤ M, ‖wn − p‖ ≤ Mhold. Because

∑∞n=1 αncn < ∞,

∑∞n=1 αn < ∞,

∑∞n=1 λn < ∞, there exists a positive

n1, such that for all n ≥ n1, we have

∞∑

i=n

αi <ε

12M,

∞∑

i=n

λi <ε

6M, (15)

and∞∑

i=n

αici <ε

9. (16)

From (7) and (12), it can be obtained that

‖xn+1 − xn‖ =∥∥∥αn(T n

1 yn − xn) + λn(wn − xn)∥∥∥

≤ αn∥∥∥T n

1 yn − xn∥∥∥ + λn ‖wn − xn‖

≤ αn∥∥∥T n

1 yn − p∥∥∥ + αn ‖xn − p‖ + λn ‖wn − p‖ + λn ‖xn − p‖

≤ αn[‖yn − p‖ + cn] + αn ‖xn − p‖ + λn ‖wn − p‖ + λn ‖xn − p‖

≤ αn[‖xn − p‖ + 2βncn + νn ‖un − p‖ + µn ‖vn − p‖

]+ αncn

+αn ‖xn − p‖ + λn ‖wn − p‖ + λn ‖xn − p‖

≤ (2αn + λn) ‖xn − p‖ + αncn(1 + 2βn) + αnνn ‖un − p‖

+αnµn ‖vn − p‖ + λn ‖wn − p‖

≤ 2αn ‖xn − p‖ + 3αncn + λn ‖xn − p‖ + αn ‖un − p‖

+αn ‖vn − p‖ + λn ‖wn − p‖

≤ 4αnM + 3αncn + 2λnM.(17)


Thus for all n ≥ n1 and m ≥ 1, we have

‖xn+m − xn‖ ≤m∑

i=1

‖xn+i − xn+i−1‖

≤ 4Mm∑

i=1

αn+i−1 + 3m∑

i=1

αn+i−1cn+i−1 + 2Mm∑

i=1

λn+i−1

< 4Mε

12M+ 3

ε

9+ 2M

ε

6M< ε.

(18)

This implies that xn is a Cauchy sequence. Thus limn→∞ xn exists. Let limn→∞ xn =

p. We shall prove that p is a common fixed point, that is, p ∈ F.

Since limn→∞ xn = p, for all ε1 > 0, there exists a positive integer n2 such thatwhen n ≥ n2, we have

‖xn − p‖ < ε1

2(L + 1). (19)

Moreover, limn→∞ d(xn, F) = 0 implies that there exists a positive integer n3 ≥ n2,such that when n ≥ n3, we have

d(xn, F) <ε1

2(L + 1), d(xn3 , F) <

ε1

2(L + 1). (20)

Thus there exists a p∗ ∈ F, such that∥∥∥xn3 − p∗

∥∥∥ = d(xn3 , p∗) <ε1

2(L + 1). (21)

It follows from (19), (21) and for i = 1, 2, 3 that

‖Ti p − p‖ =∥∥∥Ti p − p∗ + p∗ − xn3 + xn3 − p

∥∥∥≤ ‖Ti p − p∗‖ +

∥∥∥xn3 − p∗∥∥∥ +

∥∥∥xn3 − p∥∥∥

≤ L ‖p − p∗‖ +∥∥∥xn3 − p∗

∥∥∥ +∥∥∥xn3 − p

∥∥∥≤ L

∥∥∥xn3 − p∥∥∥ + L

∥∥∥xn3 − p∗∥∥∥ +

∥∥∥xn3 − p∗∥∥∥ +

∥∥∥xn3 − p∥∥∥

≤ (L + 1)∥∥∥xn3 − p

∥∥∥ + (L + 1)∥∥∥xn3 − p∗

∥∥∥< (L + 1). ε1

2(L+1) + (L + 1). ε12(L+1) = ε1.

(22)

By the arbitrariness of ε1 > 0, we have Ti p = p for i = 1, 2, 3, that is, p is a commonfixed point of the mappings T1, T2 and T3. This completes the proof.



∑∞n=1 αn < ∞ and

∑∞n=1 λn < ∞.


Put

cn = max0, sup

x∈C, p∈F

( ∥∥∥T ni x − T n

i p∥∥∥ − ‖x − p‖

): i = 1, 2, 3

,

such that∑∞

n=1 αncn < ∞. Then xn converges to a common fixed point p of themappings T1, T2 and T3 if and only if there exists some infinite subsequence of xnwhich converges to p.

Proof. The proof of Theorem 2.2 follows from Lemma 1.1 and Theorem 2.1.



∑∞n=1 αn < ∞ and

∑∞n=1 λn < ∞.

Put

cn = max0, sup

x∈C, p∈F

( ∥∥∥T ni x − T n

i p∥∥∥ − ‖x − p‖

): i = 1, 2, 3

,

such that∑∞

n=1 αncn < ∞. Suppose that the mappings T1, T2 and T3 satisfy the fol-lowing conditions:

(C1)

limn→∞ ‖xn − T1xn‖ = 0, lim

n→∞ ‖xn − T2xn‖ = 0, limn→∞ ‖xn − T3xn‖ = 0;

(C2) there exists a constant A > 0 such that‖xn − T1xn‖ + ‖xn − T2xn‖ + ‖xn − T3xn‖

≥ Ad(xn, F), ∀n ≥ 1.

Then xn converges strongly to a common fixed point of the mappings T1, T2 andT3.

Proof. From conditions (C1) and (C2), we have limn→∞ d(xn, F) = 0, it follows as inthe proof of Theorem 2.1, that xn must converges strongly to a common fixed pointof the mappings T1, T2 and T3. This completes the proof.

Theorem 2.4. Let E be a real Banach space satisfying Opial’s condition and Cbe a weakly compact subset of E. Let Ti : C → C, (i = 1, 2, 3) be uniformly L-Lipschitzian and asymptotically quasi-nonexpansive mappings in the intermediatesense with F = ∩3

i=1F(Ti) , ∅. Let xn be the sequence defined by (7) with the re-strictions

∑∞n=1 αn < ∞ and

∑∞n=1 λn < ∞. Put

cn = max0, sup

x∈C, p∈F

( ∥∥∥T ni x − T n

i p∥∥∥ − ‖x − p‖

): i = 1, 2, 3

,


such that∑∞

n=1 αncn < ∞. Suppose that T1, T2 and T3 have a common fixed point,I − Ti for i = 1, 2, 3 is demiclosed at zero and xn is an approximating common fixedpoint sequence for Ti for i = 1, 2, 3, that is, limn→∞ ‖xn − Tixn‖ = 0, for i = 1, 2, 3.Then xn converges weakly to a common fixed point of the mappings T1, T2 and T3.

Proof. First, we show that ωw(xn) ⊂ F = ∩3i=1F(Ti). Let xnk → x weakly. By

assumption, we have limn→∞ ‖xn − Tixn‖ = 0 for i = 1, 2, 3. Since I−Ti for i = 1, 2, 3is demiclosed at zero, x ∈ F = ∩3

i=1F(Ti). By Opial’s condition, xn possesses onlyone weak limit point, that is, xn converges weakly to a common fixed point of themappings T1, T2 and T3. This completes the proof.

Remark 2.1. The main result of this paper can be extended to a finite family ofasymptotically quasi-nonexpansive mappings in the intermediate sense Ti : 1 ≤ i ≤N by introducing the following iteration scheme:

Let T1,T2 . . . ,TN : C → C be N asymptotically quasi-nonexpansive mappings inthe intermediate sense. Let x1 ∈ C be a given point. The sequence xn defined by

xn+1 = (1 − an1 − bn1)xn + an1T n1 yn1 + bn1un1 ,

yn1 = (1 − an2 − bn2)xn + an2T n2 yn2 + bn2un2 ,

...yn(N−2) = (1 − an(N−1) − bn(N−1))xn + an(N−1)T

nN−1yn(N−1) + bn(N−1)un(N−1) ,

yn(N−1) = (1 − anN − bnN )xn + anN T nN xn + bnN unN , n ≥ 1,

(23)

is called N-step iterative sequence with errors of T1,T2, . . . ,TN , where uni∞n=1, i =

1, 2, . . . ,N, are N bounded sequences in C, and ani∞n=1, bni∞n=1, i = 1, 2, . . . ,N, areN appropriate sequences in [0, 1].

Remark 2.2. Theorem 2.1 extends, improves and unifies the corresponding re-sults of [1, 7, 8, 13, 15, 19]. Especially Theorem 2.1 extends, improves and unifiesTheorem 1 and 2 in [8], Theorem 1 in [7] and Theorem 3.2 in [19] in the followingways:

(1) The asymptotically quasi-nonexpansive mapping in [7], [8] and [19] is ex-tended to more general asymptotically quasi-nonexpansive mapping in the interme-diate sense.

(2) The usual Ishikawa [5] iteration scheme in [7], the usual modified Ishikawaiteration scheme with errors in [8] and the usual modified Ishikawa iteration schemewith errors for two mappings are extended to the three-step iterative algorithm witherrors for three mappings because three-step iteration scheme gives better numericalresults than two-step and one-step approximate iterations for more details see [2].

Remark 2.3. Theorem 2.2 extends, improves and unifies Theorem 3 in [8] andTheorem 2.3 extends, improves and unifies Theorem 3 in [7] in the following aspects:


(1) The asymptotically quasi-nonexpansive mapping in [7] and [8] is extended tomore general asymptotically quasi-nonexpansive mapping in the intermediate sense.

(2) The usual Ishikawa iteration scheme in [7] and the usual modified Ishikawaiteration scheme with errors in [8] are extended to the three-step iterative algorithmwith errors for three mappings the explanation is as per remark (2.2)(2).

Remark 2.4. Our results also extend the corresponding results of Quan [14] to thecase of more general class of uniformly quasi-Lipschitzian mapping considered inthis paper.

Remark 2.5. Our results also extend the corresponding results of Xu and Noor[21] to the case of more general class of asymptotically nonexpansive mapping con-sidered in this paper.

Remark 2.6. Theorem 2.4 extends and improves Theorem 2.6 and 2.7 of Sahuand Jung [15] to the case of more general class of asymptotically quasi-nonexpansivetype mapping and modified three-step iterative algorithm with errors considered inthis paper.

Remark 2.7. Theorem 2.1 also extends the corresponding result of Yao and Liou[22] to the case of three-step iterative algorithm with errors considered in this paper.

Remark 2.8. (i) Our result also extends the corresponding result of Saluja [16] tothe case of more general class of asymptotically quasi-nonexpansive mapping andthree-step iteration process with bounded errors considered in this paper.

(ii) Our results also extend and improve the corresponding results of Saluja [17, 18]to the case of more general class of asymptotically quasi-nonexpansive and asymp-totically quasi-nonexpansive type mapping considered in this paper.

Acknowledgement. The author thanks the referee for his careful reading and valuable suggestionson the manuscript.

References[1] M.K. Ghosh and L. Debnath, Convergence of Ishikawa iterates of quasi-nonexpansive mappings,

J. Math. Anal. Appl. 207(1997), 96-103.[2] R. Glowinski and P. Le Tallec, ”Augemented Lagrangian and Operator-Splitting Methods in Non-

linear Mechanics” Siam, Philadelphia, (1989).[3] K. Goebel and W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc.

Amer. Math. Soc. 35(1972), 171-174.[4] S. Haubruge, V.H. Nguyen and J.J. Strodiot, Convergence analysis and applications of the

Glowinski Le Tallec splitting method for finding a zero of the sum of two maximal monotoneoperators, J. Optim. Theory Appl. 97(1998), 645-673.


[5] S. Ishikawa, Fixed point by a new iteration method, Proc. Amer. Math. Soc. 44(1974), 147-150.[6] G.E. Kim and T.H. Kim, Mann and Ishikawa iterations with errors for non-Lipschitzian mappings

in Banach spaces, Comput. Math. Appl. 42(2001), 1565-1570.[7] Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings, J. Math. Anal.

Appl. 259(2001), 1-7.[8] Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error mem-

ber, J. Math. Anal. Appl. 259(2001), 18-24.[9] W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4(1953), 506-510.

[10] M.A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl.251(2000), 217-229.

[11] M.A. Noor, Three-step iterative algorithms for multivalued quasi variational inclusions, J. Math.Anal. Appl. 255(2001).

[12] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive map-pings, Bull. Amer. Math. Soc. 73 (1967), 591-597.

[13] W.V. Petryshyn and T.E. Williamson, Strong and weak convergence of the sequence of successiveapproximations for quasi-nonexpansive mappings, J. Math. Anal. Appl. 43(1973), 459-497.

[14] J. Quan, Three-step iterative sequences with errors for uniformly quasi-Lipschitzian mappings,Numer. Math. J. Chinese Univ. (English Ser.) 15(4) (2006), 306-311.

[15] D.R. Sahu and J.S. Jung, Fixed point iteration processes for non-Lipschitzian mappings of asymp-totically quasi-nonexpansive type, Int. J. Math. Math. Sci. 33(2003), 2075-2081.

[16] G.S. Saluja, Strong convergence theorem for two asymptotically quasi-nonexpansive mappingswith errors in Banach spaces, Tamkang J. Math. 38(1) (2007), 85-92.

[17] G.S. Saluja, Approximating fixed points of Noor iteration with errors for asymptotically quasi-nonexpansive mappings, Funct. Anal. Appro. Comput. 1(1) (2009), 31-48.

[18] G.S. Saluja, Weak and strong convergence of common fixed points for asymptotically quasi-nonexpansive type mappings in Banach spaces, Krag. J. Math. 35(3) (2011), 451-462.

[19] N. Shahzad, A. Udomene, Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications, Vol. 2006, Arti-cle ID 18909, Pages 1-10.

[20] K.K. Tan and H.K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawaiteration process, J. Math. Anal. Appl. 178(1993), 301-308.

[21] B.L. Xu and M.A. Noor, Fixed point iterations for asymptotically nonexpansive mappings in Ba-nach spaces, J. Math. Anal. Appl. 267(2002), 444-453.

[22] Y. Yao and Y.C. Liou, New iterative schemes for asymptotically quasi-nonexpansive map-pings, Journal of Inequalities and Applications, Volume 2010, Article ID 934692, 9 pages,doi:10.1155/2010/934692.

RADICALS AND EMBEDDINGS OF MOUFANGLOOPS IN ALTERNATIVE LOOP ALGEBRAS

ROMAI J., v.8, no.1(2012), 115–154

Nicolae I. SanduTiraspol State University of Moldova, Chisinau, Republic of [email protected]

Abstract The paper defines the notion of alternative loop algebra F[Q] for any nonassociativeMoufang loop Q as being any non-zero homomorphic image of the loop algebra FQ ofa loop Q over a field F. For the class M of all nonassociative alternative loop algebrasF[Q] and for the class L of all nonassociative Moufang loops Q are defined the radicalsR and S, respectively. Moreover, for classes M, L is proved an analogue of WedderburnTheorem for finite dimensional associative algebras. It is also proved that any Moufangloop Q from the radical class R can be embedded into the loop of invertible elementsU(F[Q]) of alternative loop algebra F[Q]. The remaining loops in the class of all nonas-sociative Moufang loops L cannot be embedded into loops of invertible elements of anyunital alternative algebras.

Keywords: Moufang loop, alternative loop algebra, circle loop, loop of invertible elements, radical,analog of Wedderburn Theorem, embedding.2010 MSC: 20N05.

1. INTRODUCTIONEmbedding of a Moufang loop into a loop of invertible elements U(A) of an alter-

native algebra with unit A (see, for example, [11], [34] is one of the major questionsin the Moufang loop theory. In general, the answer to this question is negative [34],[29]. Nevertheless, many authors look into such Moufang loops assuming that theycan be embedded into a loop of type U(A) (see, Section 6). The question on embed-ding a Moufang loop into a loop of type U(A) is fully solved in this paper.

To solve this question (section 6) the notion of alternative loop algebra F[Q] forany Moufang loop Q is introduced. The algebra F[Q] is alternative and it is a non-zero homomorphic image of the loop algebra FQ for the loop Q over a field F (Sec-tion 2). Moreover, the radicals R and S are introduced for the class M of all alter-native loop algebras F[Q] and for the class L of all Moufang loops (section 4). It isalso introduced the class RA into the alternative loop algebras. Let A ∈ M and letA = R] = R ⊕ Fe, i.e A is obtained by adjoining the exterior unit element e to R.Then A ∈ RA when and only when R ∈ R. Algebras of type A = R] are considered inSection 3.

The semisimple classes P and S corresponding to radicals R and S respectively areconsidered in section 5.

115

116 Nicolae I. Sandu

Proposition 7.3 and Theorem 7.4 are the crucial structure results for the examina-tion of Moufang loops. These statements are similar with Wedderburn Theorem forassociative algebras, which is regarded as the beginning of radical theory. In 1908 heproved that every finite dimensional associative algebra is an extension of the directsum of full matrix algebras over corps with the help of nilpotent algebra.

Proposition 7.3. Let F[Q] be an alternative loop algebra from the class M andlet R(F[Q]) be its radical. Then algebra (R(F[Q]))] = F[G], G ⊆ Q, F[G] ∈ RA,is nonassociative antisimple with respect to nonassociativity or, equivalently, it doesnot contain subalgebras that are nonassociative simple algebras and the quotient-algebra F[Q]/R(F[Q]) is a direct sum of Cayley-Dickson algebras over their centre.

Theorem 7.4. Let ∆ be a prime field, let P be its algebraic closure, and let Fbe a Galois extension over ∆ in P. Then the radical S(Q) of a Moufang loop Q isnonassociative antisimple with respect to nonassociativity or, equivalently, it doesnot contain subloops that are nonassociative simple loops and quotient-loop Q/S(Q)is isomorphic to a direct product of matrix Paige loops M(F).

For the basic properties of Moufang loops see [2], [5], and of alternative algebrassee [36].

The Cayley-Dickson algebras (simple alternative algebras) and the Paige loops(simple Moufang loops) are quite well explored, see [36], [33].

Let Q ∈ L. By definition Q ∈ S if and only if F[Q] ∈ RA. Then from Proposition7.3 and Theorem 7.4 it follows that the construction of nonassociative Moufang loopsQ by module of nonassociative simple Moufang loops is limited to the examinationof alternative loop algebras from radical class RA. Such algebras are described inPropositions 4.3, 4.4, Corollary 4.5.

According to Lemma 7.1 and Proposition 7.2 the following statements are equiv-alent for a nonassociative Moufang loop Q ∈ L:

r1) Q ∈ S;r2) Q is antisimple with respect to nonassociativity of loop;r3) the loop Q does not have subloops that are simple loops. Then the contrary

statements hold:nr1) G < S, i.e. G ∈ L\S;nr2) G are not antisimple with respect to nonassociativity loop;nr3) the loop G contains subloops that are simple loops hold for any nonassociative

Moufang loop G ∈ L\S.From the definition of class of alternative loop algebras RA, the definition of class

of loops S and Theorem 4.2 it follows that if a nonassociative Moufang loop Q sat-isfies the condition r1) then the loop Q can be embedded into the loop of invertibleelements U(F[Q]) of alternative loop algebra F[Q]. On the other hand, in [29], it wasproved that if a nonassociative Moufang loop G satisfies the condition nr2) then theloop Q is not imbedded into the loop of invertible elements U(A) for a suitable unitalalternative F-algebra A, where F is an associative commutative ring with unit. As

Radicals and embeddings of Moufang loops in alternative loop algebras 117

S⋂

(L\S) = ∅ then the main result of this paper follows from the above-mentionedstatements.

Theorem 7.5. Any nonassociative Moufang loop Q that satisfies one of the equiv-alent conditions r1) - r3) can be embedded into a loop of invertible elements U(F[Q])of alternative loop algebra F[Q]. The remaining loops in the class of all nonasso-ciative Moufang loops L, i.e. the loops G ∈ L that satisfy one of the equivalentconditions nr1 - nr3 cannot be embedded into loops of invertible elements of anyunital alternative algebras.

From Corollary 6.10 and Theorem 7.5 followsCorollary 7.6. Any commutative Moufang loop Q can be embedded into a loop of

invertible elements U(F[Q]) of alternative loop algebra F[Q].Recently a series of papers have been published, which look into the Moufang

loops with the help of the powerful instrument of group theory, in particular finitegroup theory (see, for example, [7], [10], [9], [13], [16]). For this purpose the corre-spondence between Moufang loops and groups with triality [7] is used. The proofsbased on the correspondence are complex and cumbersome.

This paper offers another simpler approach method: to use the Theorem 7.5, Corol-lary 7.6 instead of the the correspondence between Moufang loops and groups withtriality. Such examples are presented into the end of section 6 and section 7. In sec-tion 7 this is proved on the basis of Theorem 7.5 of the known results from [9]: everyfinite Moufang p-loop is centrally nilpotent. Paper [9] introduces the notion of groupwith triality. We note that Theorem 7.5 was also used in [32] for proving the nextstatements.

If three elements a, b, c of Moufang loop Q are tied by the associative law ab · c =

a · bc, then they generate an associative subloop (Moufang Theorem).The intersection of the terms of the lower central series of a free Moufang loop

LX(M) is the unit loop.Any finitely generated free Moufang loop is Hopfian.We will examine only nonassociative Moufang loops and nonassociative alterna-

tive algebras over a fixed field F. Particularly, the alternative loop algebra F[Q]corresponding to nonassociative loop Q is nonassociative. If the loop Q is commuta-tive then algebra F[Q] is also commutative. Then, in the commutative case, we willconsider that char F = 0 or 3 as there are no nonassociative commutative alternativealgebras over fields of characteristic , 0; 3 [36].

Any algebra A with unit e is always considered nontrivial by definition, thereforeall such algebras A contain one dimensional central subalgebra Fe = αa|α ∈ F(with the same unit e , 0, which allows to identify Fe and F).

If J is an ideal of algebra A and the quotient algebra is an algebra with unit e, then Jis a proper ideal of A (J , A) and e < J. Besides, by definition, the homomorphismsof algebras with unit e is always unital, i.e. keep the unit. Hence, if ϕ : A → B


is a homomorphism of algebras with unit e, then kerϕ is a proper ideal of A, asϕe = e , 0.

Let A be an associative algebra. By I(A) we denote the set of such elements u ∈ Athat

u + v + uv = 0, u + v + vu = 0 (1)

for some v ∈ A and by J(A) we denote the set of all quasiregular elements of A, i.e.the set of such elements a ∈ A that

a + b − ab = 0, a + b − ba = 0 (2)

for some b ∈ A. In the past, almost simultaneously, with various goals on the elementsof the set I(A) different authors (see., for example, [8], [18], [24]) have introducedthe group operation (⊗):

u ⊗ v = uv + u + v. (3)

However, at present the so-called circle operation ():

a b = a + b − ab (4)

on the elements of the set J(Q), as in such a case the strong instrument of the theoryof quasiregular associative algebras (see., for example, [15], [25]) can be used toconsider operation () (we mentioned that this paper is influenced by [25]).

The operations (⊗) and () on the sets I(A) and J(A) respectively are groups. Thus,this paper established by analogy a link between alternative algebras and Moufangloops. In [12] this link is established with the help of the operation, defined by (3),but we believe it is not successful. In such a form it is impossible to use the developedtheory of quasiregular alternative algebras, though in [12] the elements defined by (1)are wrongly called quasiregular. According to [36] a quasiregular alternative algebracan be characterized (defined) as alternative algebra A satisfying the property thatthe set A form a loop with respect to circle operation (), defined by (4). Then theset J(A) ⊆ A define the Zhevlakov (quasiregular) radical of alternative algebra A,the analog of Jacobson radical of associative algebra theory. In [12] the notion ofZhevlakov radical is defined with respect to operation (⊗) that not correspond [36].However, the loops (I(A),⊗) and (J(A), ) are isomorphic by Corollary 2.7.

Let A be an alternative algebra. Unlike [12], this paper establishes a link betweenalternative algebras and Moufang loops with the help of relation (4). In such a casethe developed theory of quasiregular alternative algebras can be used. For example,in [12, Theorem 1], it is quite cumbersomely proved that groupoid (I(A),⊗) is aMoufang loop, but this paper proves such a result for (J(A), ) (Proposition 2.4) quiteeasily. We will call loop (J(A), ) circle loop of algebra J(A). The paper also givesa full answer to the modified question from [12] about embedding a Moufang loopinto circle loop of a suitable alternative algebra (Corollaries 7.8, 7.9).


2. CIRCLE MOUFANG LOOPSLet A be an algebra over a field F. Let’s consider that the field F is a module over

itself. The unit e of F is the generating element of the F-module Fe. We consider thedirect sum A] = A ⊕ Fe of the modules A and Fe and define on it the multiplication:

(a + α · e)(b + β · e) = (ab + αb + βa) + αβ · ewhere a, b ∈ A, α, β ∈ F. It is easy to see that e is the unit of algebra A] and A is anideal of A]. A] is called the algebra obtained by adjoining the exterior unit element eto A.

Consequently, it is always possible to pass, from any algebra R to algebra R] =

R⊕ F · e with externally attached unit e and R be an ideal of algebra R]. In general, itis not always possible to restore algebra R from the algebra R]: it is possible that R]1 =

A = R]2, though the algebras R1 and R2 are not isomorphic. However, if the algebrasare given R and A = R] then for any algebra B with unit ε every homomorphismϕ : R → B unequivocally proceeds up to homomorphism ϕ : A → B by rule:ϕ(αe + r) = αε + ϕr. Particularly, the homomorphism π : A → F, defined byπ(αe + r) = α, will be the only unital homomorphism of algebra A = R] in algebraF ≡ Fe, continuing the null homomorphism of algebra R. Moreover, hold.

Lemma 2.1. An algebra A with unit e will be an algebra with externally adjoinedunit (i.e. A = R] for some algebra R) when and only when there exists a homomor-phism π : A → πA = F of algebra A. In such a case A = R] = R ⊕ Fe, whereR = ker π.

Proof. For the homomorphism π with ker π = R we have A/R F. Besides, π isidentical on Fe ≡ F and according to decomposition a = πa + (a − πa) = αe + rof elements a ∈ A and equality πA = Fe we have A = R]. On the other hand,R

⋂Fe = 0 for any proper ideal R and hence if A = R] = R ⊕ Fe then A→ A/R F

will be the only unital homomorphism R] → F, continuing the null homomorphismof algebra R. This completes the proof of Lemma 2.1.

An alternative algebra is an algebra in which x · xy = x2y and yx · x = yx2 areidentities. Any alternative algebra satisfies the Moufang identity

(x · yx)z = x(y · xz). (5)

The loop, satisfying the identity (5), is called Moufang loop.Let A be an alternative algebra with unit e. The element a ∈ A is said to have an

inverse, if there exists an element a−1 ∈ A such that aa−1 = a−1a = e. It is wellknown that for an alternative algebra A with the unit e the set U(A) of all invertibleelements of A forms a Moufang loop with respect to multiplication [21].

Lemma 2.2 [36]. Let A be an alternative algebra. Then, the following statementsare equivalent:


a) the elements a and b are invertible;b) the elements ab and ba are invertible.The element a of alternative algebra A is called quasiregular if it satisfies the

relation (2). The element b of (2) is called quasiinverse of a. An alternative algebrais called quasiregular if any of its elements is quasiregular.

Lemma 2.3 [36]. The following statements are equivalent:a) the element a of the alternative algebra A is quasiregular with quasiinverse b;b) the element e − a of the algebra A] is inverse with the inverse element e − b.Quasiregularity is a fundamental concept in algebra theory because it allows to

define one of the most important radicals. An ideal is called quasiregular if it con-sists entirely of quasiregular elements. Every alternative algebra A has the largestquasiregular ideal J(A) such that A/J(A) has no non-zero quasiregular ideals. Thisideal J(A) is called the Zhevlakov radical and it is, of course, like the Jacobson radicalof associative algebra theory [36].

Proposition 2.4. Let A be an alternative algebra and let J(A) be its Zhevlakovradical. Then the set J(A) forms a Moufang loop with respect to operation x y =

x + y − xy.

Proof. We suppose that x and y are quasiregular elements with quasiinverses a andb respectively. We denote u = e − x, v = e − y where e is the unit of algebra A].From Lemma 2.3 it follows that u−1 = e − a, v−1 = e − b, where uu−1 = e, vv−1 =

e, and from Lemma 2.2 it follows that (uv)(v−1u−1) = e. From here we get that((e−x)(e−y))((e−b)(e−a)) = e, (e−x−y+xy)(e−b−a+ba) = e, (e−xy)(e−ba) = e.Hence the element e − x y is inverse with the element e − b a and from Lemma2.3 it follows that x y is quasiregular with quasiinverse b a. Consequently, the setJ(A) is closed under the operation ().

It is easy to see that the 0 element of A is an unit for (). To prove that the setJ(A) forms a loop under (), it sufficient to show that (x y)b = x and similarly thata (xy) = y. Indeed, by Lemma 10.5 from [6]?? xy ·b = x ·yb and as −yb+y+b = 0then (xy)b = (−xy+ x+y)b = xy ·b− xb−yb− xy+ x+y+b = xy ·b− xb− xy+ x =

x · yb − xb − xy + x = x(yb − b − y) + x = x · 0 + x = x. In this manner it is provedthat a (x y) = y. Hence (J(A), ) is a loop.

Finally, in order to prove the validity of Moufang identity (5) in the loop (J(A), )it is sufficient to evaluate the difference ((x y) x) z) − x (y (x z)) by (4)and by using the identity (5) for the algebra A, diassociativity of A and the identityxy · z + yx · z = x · yz + y · xz obtained through linearization of algebra identityxx · z = x · xz. As a result we have obtained that this difference is 0. This completesthe proof of Proposition 2.4.

Corollary 2.5. Let J(A) be the Zhevlakov radical of the alternative algebra A.Then the Moufang loop (J(A), ) is isomorphic to (I(A),⊗).


Proof. Let x ∈ J(A). J(A) is a module, then −x ∈ J(A). Let −y be a quasiinverse for−x. By (2) we have (−x) + (−y) − (−x)(−y) = 0,−x − y − xy = 0,−(x + y + xy) =

0, x + y + xy = 0. Hence by (1) x ∈ I(A), i.e. J(A) ⊆ I(A). Inversely, let x ∈ I(A) andlet x + y + xy = 0. Then (−x) + (−y) − (−x)(−y) = 0, and by (1) −x ∈ J(A), x ∈ J(A),i.e. I(A) ⊆ J(A). Hence J(A) = I(A).

Now for x ∈ J(A) we define ϕ(x) = −x. ϕ is a one-to-one map of J(A) onto I(A).Moreover,

ϕ(x y) = −(x + y − xy) = −x + (−y) + (−x)(−y) = ϕ(x) ⊗ ϕ(y),

so ϕ is an isomorphism of (J(A), ) onto (I(A),⊗), as required.

Further the Moufang loop (J(A), ), considered in Proposition 2.4 will be calledthe circle loop of algebra A and denoted by U∗(A). If A contains the unit e then thecorrespondence e − a → a maps the multiplicative loop of simple inverse elementsof A upon U∗(A) and, in this case, the circle operation does not offer anything new.Therefore, we will assume further that algebra A is without unit element.

Let now A be an arbitrary alternative algebra with externally adjoined unit e, i.e.by Lemma 2.1 with clearly distinguished one-dimensional subalgebra Fe ≡ F withsame unit e. We define the mapping η : A → A by the rule ηa = e − a ∈ A forany a ∈ A (particularly, ηe = 0 , e = η0). Then, from the definition of circle loop(Proposition 2.4), we get the equality

(e − a)(e − b) = e − a b, (6)

which, by replacing of type x→ ηx, it is rewritten as follows:

(e − a) (e − b) = e − ab. (7)

Obviously, η = η−1. By replacing of type c → e − c in Lemma 2.3, we get that anelement a ∈ A is invertible if and only if the element e − a ∈ A is quasiregular. Thenfrom (6), (7) and Proposition 2.4 it follows that η is an isomorphism, which connectsthe group of invertible elements U(A) of algebra A with circle group of quasiregularelements U∗(A) by rule

U∗(A) = η(U(A)) = a ∈ A|e − a ∈ U(A). (8)

Hence the rule

a ∈ U∗(A)→ e − (e − a)−1 = −a(e − a)−1 = a∗ ∈ U∗(A) (9)

defines on A, by isomorphism η, the operation a → a∗ of taking the quasiinverse,defined on U∗(A) and corresponding precisely to the operation of taking the inverse,defined on U(A), according to equality a a∗ = 0 = a∗ a and the isomorphism

U∗(A) U(A). (10)


By definition, an alternative algebra is quasiregular if any of its elements is quasireg-ular. Consequently, from definitions of Zhevlakov radical J(A) and circle loop U∗(A)it follows that for any alternative algebra A the Zhevlakov radical J(A) is a loop withrespect to circle operation () and

J(A) = U∗(A). (11)

In particular, an alternative algebra A is quasiregular if and only if the algebra Acoincides with its circle loop U∗(A). According to (9), it means that, on algebra A,there also exists the unique operation x x∗, of taking the quasiinverse, related withthe basic operations of identity x + x∗ = xx∗ = x∗x (i.e. r r∗ = 0 = r∗ r for allr ∈ A by the construction of loop U∗(A)).

Hence the class of all quasiregular algebras K∗ form a variety, if considered withan additional unitary operation x x∗ of taking the quasiinverse. Then, by Birkhoff

Theorem, the class K∗ is closed with respect to the taking of quasiregular subal-gebras, of direct product of quasiregular subalgebras and of homomorphic imagesof homomorphisms of quasiregular subalgebras. But the class K∗ is also closed inrespect to usual homomorphisms, i.e. to homomorphisms of algebras. Indeed, thefollowing result holds.

Lemma 2.6. Let ϕ be a homomorphism of algebra A ∈ K∗. Then ϕA ∈ K∗.

Proof. Let a ∈ A. From definition of quasiregular elements it follows that ϕa is aquasiregular element. Let a∗, (ϕa)∗ be their quasiinverse elements. The homomor-phism ϕ is unital. Hence from (9) it follows that (ϕa)∗ = ϕa∗. Then the homomor-phism αϕ saves the identity x + x∗ = xx∗ = x∗x, distinguishing quasiregular algebras.Consequently, ϕA ∈ K∗, as required.

From equalities ab + (ab) = a + b it follows easily that a subspace R of algebra Ais it subalgebra then and only then R is a subgroupoid of the groupoid (A, ). Hencethe circle subgroupoid (R, ) of circle groupoid (A, ), isomorphic to multiplicativesubgroupoid e−R of algebra A is linked with any subalgebra R of alternative algebraA by rule

e − R = e − r|r ∈ R = ηR. (12)

Then, according with (6) - (11), with

U∗(R) = η(U(e − R)) (13)

and with the definition of multiplicative loop of algebra, we get the circle loop U∗(R)of algebra R, which is a subloop of loop U∗(A) and it is isomorphic to multiplicativeloop U(e − R) of loop U(A) U∗(A) by (10). In particular, as F ≡ Fe then themultiplicative group U(F) = F\0 of field F and circle group U∗(F) = F\0 arecentral subloops of loops U(A) and U∗(A), respectively.


From equalities a + x + ax = x + a + xa = 0 it follows that the set of all elementsof some subalgebra R of algebra A, that have quasiinverses in R, is a subloop ofloop U∗(A), U∗(R) ⊆ U∗(A) . By (11) an alternative algebra A is quasiregular if Acoincides with circle loop U∗(A), J(A) = U∗(A). The Zhevlakov radical J(A) of anyalternative algebra A is hereditary, i.e. J(R) = R

⋂J(A) for any ideal R of A. Hence

U∗(R) = R⋂

U∗(A) = r ∈ R|e − r ∈ U(A). (14)

The following result holds, too.Proposition 2.7. Let A be an alternative algebra and R be an ideal of A. Then

U∗(R) is a normal subloop of U∗(A) and U∗(A)/U∗(R) U∗(A/R).

Proof. Let x, y ∈ U∗(A), u ∈ U∗(R) and let a, b will be the quasiinverses of x, yrespectively. In the proof of Proposition 2.4 it is shown that the element x y isquasiregular with a quasiinverse b a. By definition the subloop U∗(R) is normal inU∗(A) if x U∗(R) = U∗(R) x, x (y U∗(R)) = (x y) U∗(R), (U∗(R) x) y =

U∗(R) (x y) for any x, y ∈ U∗(A). Any Moufang loop is an IP-loop, i.e. it satisfiesthe identities x−1 · xy = y, yx · x−1 = y. Then to show that U∗(R) is normal in U∗(A) itis sufficient to show that t1 = (xu)a ∈ U∗(R) and t2 = ((u x)y) (ba) ∈ U∗(R).From the aforementioned we have that t1, t2 ∈ U∗(A). Further, u ∈ R, then by (4)t1 = xu · a− xa− ua− xu + x + u + a = xu · a− ua− xu + u ∈ R since −xa + x + a = 0.We similarly have t2 = r − xy · ba + xy · b + xy · a + x · ba− xb + y · ba− ya− xy− ba,where r ∈ R. We denote x = e− x, y = e− y, a = e− a, b = e− b, where e is the unitof algebra A]. Let us express t2 in terms of x, y, a, b over x, y, a, b respectively. Weget that t2 = r − xy · ba + xa + yb − e. By Lemmas 2.2 and 2.3 xy · ba = xa = yb = e.Hence t2 = r. Then t1, t2 ∈ U∗(R) and, consequently, the subloop U∗(R) is normal inU∗(A). We proved that the ideal R of algebra A induced the normal subloop U∗(R) ofloop U∗(A).

As noted above, the homomorphic image of circle loop U∗(A) under homomor-phism A → A/R is a circle loop. Hence the quotient loopU∗(A)/U∗(R) is a circle loop. By (10) U∗(A/R) = J(A/R). The Zhevlakov radi-cal J(A/R) of a maximal ideal of A/R. Hence U∗(A/R) is a maximal subloop ofmultiplicative groupoid of algebra A/R.

We will show that the quotient loop U∗(A)/U∗(R) is isomorphic to the correspond-ing subloop of circle loop U∗(A/R). Indeed, if x1 and x2 belong to the same cosetof U∗(A) modulo U∗(R), then x1 = x2 r where r is a quasiregular element of R.But x2 r = −x2r + x2 + r, consequently, x1 − x2 = r − x2r ∈ R. Conversely, ifx1 − x2 ∈ R and a is a quasiinverse for x2, i.e. a = x−1

2 , then x1 = x2 + r (r ∈ R),x2 + a − ax2 = 0, x−1

2 x1 = a + x1 − ax1 = a + x2 + r − ax1 = ax2 + r − ax1 =

ax2 + r − ax1 = r − a(x1 − x2) = r − ar ∈ R. Hence x−12 x1 ∈ U∗(R).

We proved that U∗(A)/U∗(R) ⊆ U∗(A/R) or, by (10), J(A)/J(R) ⊆ J(A/R). IfB/J(R) = J(A/R), then by [36, Lemma 13, cap. 10] it follows that B ⊆ J(A). Con-


sequently, J(A)/J(R) J(A/R) or, by (19), U∗(A)/U∗(R) U∗(A/R), as it wasrequired.

Corollary 2.8. Let R be an arbitrary non-zero alternative algebra and let A =

R] = Fe ⊕ R. Then the following results hold:(i) the circle loop U∗(A) is a direct product of the central subloop U∗(F) and the

normal subloop U∗(R)(ii) the loop of invertible elements U(A) is a direct product of central subloop U(F)

and normal subloop U(e − R).

Proof. By Proposition 2.7 and (10) and the above-mentioned result as well, U∗(R),U∗(F) are normal subloops of U∗(A) and U(e − R), U(F) are normal subloops ofU(A). As R is a proper ideal of algebra A then

U∗(R)⋂

U∗(F) = 0, U(e − R)⋂

U(F) = e. (15)

Besides, as e < R then R⋂

U(F) = ∅ = R⋂

U(A). Hence

U(A) ⊆ A\R = αe + r | 0 , α ∈ F, r ∈ R, (16)

U(A) = α(e − u) | α ∈ U(F), u ∈ U∗(R). (17)

Now, Corollary 2.8 follows from (14), (15) and (17).

According to Lemma 2.1 the peculiarity of algebras with externally adjoined unitA = R] is linked with the homomorphisms π : A → F of the considered algebrason the one-dimensional algebra F ≡ Fe. An algebra A with unit e will be algebrawith externally adjoined unit (i.e. A = R] for some algebra R) when and only whenA ,

⋂J C A|A/J F or, equally, the set S F(A) =⋂J C A|A/J F is non-empty.

As indicated in the beginning of the section, it could be the case that R1,R2 ∈ S F(A),i.e. that R]1 = A = R]2, though algebras R1 and R2 are quite different. But for circleloops U∗ U(e − R) from Corollary 2.8 the following result holds.

Corollary 2.9. If R1,R2 ∈ S F(A) then U∗(R1) U∗(R2).

3. ALTERNATIVE LOOP ALGEBRASLet F be a field (with unit 1) and Q be a Moufang loop with unit e. We remind

that, by its definition, the loop algebra FQ ≡ F(Q) is a free F-module with thebasis g|g ∈ Q and the product of the elements of this basis is just their product inthe loop Q. Any element g ∈ Q is identified with the element 1g, and any elementλ ∈ F is identified with the element λe. In particular, the unit of algebra FQ maybe considered both as unit of field F and as unit of loop Q. In this case, everyhomomorphism ϕ of algebra FQ must be unital, i.e. it has to maintain the unit,ϕe = e. Since ϕe = e , 0 then kerϕ is a proper ideal of FQ.


Let H be a normal subloop of the loop Q and let ωH ≡ ω(H) be the ideal of theloop algebra FQ, generated by the elements e−h (h ∈ H). If H = Q, then ωQ will becalled the augmentation ideal of loop algebra FQ. In [1, Lemma 1] it is proved that

F(Q/H) FQ/ωH. (18)

By definition the Moufang loop Q satisfies the Moufang identity (xy·x)z = x(y·xz).It is easy to see that the loop algebra FQ does not always satisfy the Moufang identityif the loop Q is nonassociative. This is an equivalent to the fact that the equalities

(a, b, c) + (b, a, c) = 0, (a, b, c) + (a, c, b) = 0 ∀a, b, c ∈ Q, (19)

where the notation (a, b, c) = ab · c − a · bc means that the associator in algebra, doesnot always hold in loop algebra FQ. This means that algebra FQ is not alternative.We remind that algebra A is called alternative if the identities (x, x, y) = (y, x, x) = 0hold in it.

Let I(Q) denote the ideal of algebra FQ, generated by all elements of the left partof equalities (19). It follows from the definition of loop algebra and diassociativity ofMoufang loops that FQ/I(Q) will be an alternative algebra. Further for the alternativealgebra FQ/I(Q) we use the notation F[Q] and we call them alternative loop algebra.

In [31], [32] it is proved that a free Moufang loop L is isomorphically embeddedunder homomorphism η : FL → F[L] into loop of invertible elements of algebraF[L]. If the image L is identified with L then the following holds.

Lemma 3.1. Any free Moufang loop L is a subloop of the loop of invertible ele-ments U(F[L]) of the alternative loop algebra F[L].

From the definition of loop algebra FL and Lemma 3.1 it follows.Corollary 3.2. Any element u of the alternative loop algebra F[L] of any free

Moufang loop L is a finite sum u =∑k

i=1 αigi, where αi ∈ F, gi ∈ L.Further we will use the following statement proved in [31], [32].Lemma 3.3. Let A be an alternative algebra and let Q be a subloop of the loop

of invertible elements U(A). Then the restriction of any homomorphism of algebraA upon Q will be a loop homomorphism. Consequently, any ideal J of A induces anormal subloop Q

⋂(e + J) of loop Q.

Let H be a normal subloop of free Moufang loop L with unit e. We denote theideal of algebra F[L], generated by the elements e − h (h ∈ H) by ω[H]. If H = L,then ω[L] will be called the augmentation ideal of the alternative loop algebra F[L].

For a Moufang loop Q, let L be a free Moufang loop such that the loop Q hasa presentation Q = L/H. We consider the mapping µ : FL → FQ induced byhomomorphism µ : L→ L/H = Q by

µ((FL)∑

g∈L

αgg) =

(FL)∑

g∈L

αgµ(g) =

(FQ)∑

a∈Q

αaa, (20)


where a = µ(g), αg, αa ∈ F and∑(FL) means sum in F-module FL,

∑(FQ) means sumin F-module FQ. The mapping µ is defined correctly because FL is an F-modulewith basis g | ∀g ∈ L, FQ is an F-module with basis a|∀a ∈ Q and µ is an epimor-phism. Moreover, as FL is a free module, then µ is an epimorphism of F-modules.Further, let x =

∑(FL)g∈L αgg, y =

∑(FL)h∈L βhh. Then µ(xy) = µ(

∑(FL)g,h∈L αgβh(gh)) =

∑(FL)g,h∈L αgβhµ(gh) =

∑(FL)g,h∈L αgβhµ(g)µ(h) =

∑(FL)g∈L αgµ(g) · ∑(FL)

h∈L βhµ(h) = µ(x)µ(y).Consequently, µ : FL→ FQ is a homomorphism of algebras, and by (18)

ker µ = ωH, FQ = FL/ωH. (21)

Let µQ be the homomorphism of the alternative loop algebra F[L] induced byhomomorphism µ of loop algebra FL, µQ(F[L]) = µ(FL)/µ(I(L)) = FQ/µ(I(L)). Toexclude the null homomorphisms we will consider that the induced homomorphismµQ is unital.

The algebra F[L] is alternative and the algebra FQ/µ(I(L)) is alternative, as well.In this case I(Q) ⊆ µ(I(L)). Further, by definition, the ideal I(L) of the loop algebraFL is generated by the set (u, v,w) + (v, u,w), (u, v,w) + (u,w, v)|∀u, v,w ∈ L. Sinceµ((u, v,w) + (v, u,w)) = (µ(u), µ(v), µ(w)) + (µ(v), µ(u), µ(w)), µ((u, v,w) + (u,w, v)) =

(µ(u), µ(v), µ(w)) + (µ(u), µ(w), µ(v)) and µ(u), µ(v), µ(w) ∈ Q, thenµ(I(L) ⊆ I(Q) and µ(I(L)) = I(Q). Consequently,

µQ(F[L]) = FQ/I(Q) = F[Q]. (22)

Further, according to (21) and homomorphism theorems it follows I(Q) =

µQ(I(L)) = (I(L) + ωH)/ωH ωH/(ωH⋂

I(L)), i.e.

I(Q) ωH/(ωH⋂

I(L)).

We denote by ” - ” the difference in loop algebra FL, by ” ” we denote thedifference in alternative loop algebra F[L] and by θ - the restriction on ideal ωHof natural homomorphism η : FL → FL/I(L) = F[L]. It is obvious that ker θ =

ωH⋂

I(L) and θ(ωH) = ωH/(ωH⋂

I(L)).By definition, the ideal ωH is generated by set e−h|∀h ∈ H. From Lemma 3.1, it

follows that η(H) = H. Then the algebra θ(ωH) is generated by the set eh|∀h ∈ H.We have θ(ωH) = η(ωH). Recall that we have above proved the equality θ(ωH) =

ωH/(ωH⋂

I(L)). By the homomorphisms theorem it results ωH/(ωH⋂

I(L)) (ωH + I(L))/I(L). Hence the ideal (ωH + I(L))/I(L) of algebra FL/I(L) is generatedby the set e h|∀h ∈ H. Consequently,

(ωH + I(L))/I(L) = ω[H]. (23)

Now, by (21), homomorphism theorems and (23) it follows

µQ(F[L]) = µQ(FL/I(L)) = µ(FL)/µ(I(L)) =


((FL + ωH)/ωH)/((I(L) + ωH)/ωH) (FL + ωH)/(I(L) + ωH) =

FL/(I(L) + ωH) (FL/I(L))/((I(L) + ωH)/I(L)) = F[L]/ω[H].

According to (22), it results µQ(F[L]) = F[Q], i.e.

ker µQ = ω[H]. (24)

The homomorphism of alternative loop algebras µQ : F[X] → F[Q] is inducedby homomorphism of loop algebras µ : FX → FQ which is induced, in its turn,by the homomorphism of loops µ : X → Q. Then, from (20), it follows that anyhomomorphism of loops µ : X → Q induces a homomorphism of alternative loopalgebras µQ : F[X]→ F[Q], defined by

µQ((F[L])∑

g∈L

αgg) =

(F[Q])∑

a∈Q

αaµ(g) =

(F[Q])∑

a∈Q

αaa, (25)

where a = µ(g), αg, αa ∈ F and∑(F[L]) means the sum in the F-module F[L],

∑(F[Q])

means the sum in the F-module F[Q].We remind that in order to exclude the case kerµQ = F[Q] we assume that the

homomorphism µQ is unital. Then from (24) the following results.Proposition 3.4. Let L be a free Moufang loop, let Q be a Moufang loop which

has the presentation Q = L/H such that the homomorphism µQ, induced by (25) byhomomorphism µ : L→ Q, is unital. Then the alternative loop algebra F[Q] has thepresentation F[Q] = F[L]/ω[H].

Corollary 3.5. The alternative loop algebra F[Q] of a Moufang loop Q is gener-ated as an F-module by the set q|q ∈ Q.

The statement follows from Proposition 3.4 and (25).Now we consider a homomorphism ρ of the alternative loop algebra F[L]. From

Lemma 3.1, it follows that F[L] is generated as F-module by set g | g ∈ L. Then theF-module ρ(F[L]) is generated by set ρ(g)|g ∈ L. Hence any element x ∈ ρ(F[L])has a form x =

∑g∈L αgρ(g).

By Lemma 3.3 ρ induces a normal subloop H of loop L. From (25) it follows thatthe homomorphism µ : L → L/H = Q induces a homomorphism of alternative loopalgebras µQ : F[L]→ F[Q], defined by

µQ((F[L])∑

g∈L

αgg) =

(F[Q])∑

a∈Q

αaµ(g).

Since µ(g) = gH = ρ(g), it follows η(F[X]) = µQ(F[X]) = F[Q]. Hence we provedthe next result.


Proposition 3.6. Let L be a free Moufang loop. The homomorphic images of theform µQ(F[L]) = F[Q] are the only alternative loop algebras. The homomorphismsµQ are unital and are induced by homomorphisms of loops µ : L→ Q by rules (25).

Corollary 3.7. Let ϕ be an unital homomorphism of alternative loop algebraF[Q]. Then the homomorphic image ϕ(F[Q]) is a non-zero alternative loop algebra.

Proof. We consider the homomorphism µQ : F[L]→ F[Q] from Proposition 3.6 andlet ϕ be a homomorphism of alternative loop algebra F[Q]. The product ϕµQ is ahomomorphism of alternative loop algebra F[L] on algebra ϕ(F[Q]). By Proposition3.6, ϕ(F[Q]) is an alternative loop algebra, as it was required.

Let L be a free Moufang loop with unit e. By Corollary 3.2 any element a ∈ F[L]has a form a =

∑ki=1 αiui, where αi ∈ F, ui ∈ L. Let H be a normal subloop of loop L

and let ϕ : L→ L/H be the natural homomorphism. It is easy to see that the mappingϕ : F[L]→ F[L/H], defined by rule

ϕ(∑

g∈L

αgg) =∑

g∈L

αgϕ(g) =∑

g∈L

αggH (26)

is a homomorphism. Then it necessarily follows

F[L/H] F[L]/ kerϕ. (27)

We assume that F[L]/ kerϕ is an algebra with externally adjoined unit. Then ϕ is anunital homomorphism, i.e. ϕ(e) = e , 0. In such a case e < kerϕ.

Lemma 3.8. Let ϕ be a homomorphism defined in (26) and we assume thatF[L]/ kerϕ is an algebra with externally adjoined unit. Then

1) h ∈ H if and only if e − h ∈ ω[H],2) F[L/H] F[L]/ω[H],3) ω[H] = kerϕ.

Proof. 1). As the mapping ϕ is F-linear, then for u ∈ F[L] and h ∈ H

ϕ((e − h)u) = (ϕe − ϕh)ϕu = (e − H)(uH) = uH − uH = 0

andω[H] ⊆ kerϕ. (28)

If g < H then gH , H and ϕ(e − g) = H − gH , (0). Hence e − g < kerϕ ⊇ ω[H]by (28), i.e. e − g < ω[H].

2). Let the ideal ω[H] of algebra F[L] induces, by Lemma 3.3, the normal subloopK = L

⋂(e − ω[H]) of loop L and, hence, F[L/K] ≡ F[L]/ω[H]. From the first

relation, we get 1 − K ⊆ ω[L]. By item 1) K = H, hence F[L/H] ≡ F[L]/ω[H].3). The isomorphism ξ : F[L]/ω[H]→ F[L]/ kerϕ follows from (27) and item 2).

For any element u ∈ F[L] we denote by u the image of u into F[L]/ω[H] and by u we


denote the image of u into F[L]/ kerϕ. Let 0 , u ∈ kerϕ\ω[H]. As a < ω[H] then

0 , u. Hence ξ(0) , ξ(u), 0 , u. But as u ∈ kerϕ then 0 = u what is a contradiction.Hence ω[H] = kerϕ. This completes the proof of Lemma 3.8.

4. ALTERNATIVE LOOP ALGEBRAS WITHEXTERNALLYADJOINED UNIT

Let now ω[L] be the augmentation ideal of the alternative loop algebra F[L] of thefree Moufang loop L , e. According to Corollary 3.2 any element a ∈ F[L] has theform a =

∑ki=1 αiui, where αi ∈ F, ui ∈ L. We denote R = ∑u∈L λuu|∑u∈L λu = 0.

Obviously, ω[L] ⊆ R. Conversely, if r ∈ R and r =∑

u∈L λuu, then −r = −∑u∈L λuu =

(∑

u∈L λu)e −∑u∈L λqq =

∑u∈L λu(e − u) ∈ ωL, i.e. R ⊆ ω[L]. Hence

ω[L] = ∑

u∈L

λuu|∑

u∈L

λu = 0. (29)

From (29) it follows that ω[L]⋂

L = ∅. Then the algebra ω[L] will be non-zerowhen and only when L , e, i.e. when F[L] , Fe = 0]. In such case for any t ∈ Lthe equalities Ft

⋂ω[L] = 0, Ft +ω[L] = F[L] hold and, by (29), t− s ∈ ω[L]. Then,

the set Bt(ω[L]) = t − s | s ∈ L, t , s generates the F-module ω[L] for any t ∈ L asL , t and the set t⋃ Bt(ω[L]) generates the F-module F[L] by Corollary 3.2. Inparticular, the set e⋃ Be(ω[L]) generates the F-module F[L] and the set Be(ω[L])generates the F-module ω[L]. Then F[L]/ω[L] Fe and, by Lemma 2.1,

F[L] = Fe ⊕ ω[L] = (ω[L])]. (30)

If u ∈ L then by Lemma 2.3 e − u is a quasiregular element. The set Be(ω[L])generates the F-module ω[L]. By [36, Lemma 10.4.12], in an alternative algebra,the sum of quasiregular elements is a quasiregular element. Hence the augmentationideal ω[L] is a quasiregular algebra and from (30) it follows that ω[L] coincides withZhevlakov radical J(F[L]), ω[L] = J(F[L]).

Hence we have proved the next result.Lemma 4.1. Let L be a free Moufang loop and let ω[L] be the augmentation ideal

of the alternative loop algebra F[L]. Then1) ω[L] is generated as an ideal of the algebra F[L], as well as an F-module, by

set L0 = e − u | for allu ∈ L,2) ω[L] is a quasiregular algebra, i.e. ω[L] = J(F[L]), where J(F[L]) is the

Zhevlakov radical of the algebra F[L].

Theorem 4.2. Let Q be a Moufang loop with unit e such that the alternative loopalgebra F[Q] is an algebra with externally adjoined unit e. Then the loop Q canbe embedded into the loop of invertible elements U(F[Q]) of the alternative loopalgebra F[Q].


Proof. Let Q = L/H, where L is a free Moufang loop, let ϕ : L→ L/H be the naturalhomomorphism and let ϕ : F[L] → F[L/H] = F[L]/ kerϕ be the homomorphismdefined in (20) by

ϕ(∑

g∈L

αgg) =∑

g∈L

αgϕ(g) =∑

g∈L

αggH.

By item 3) of Lemma 3.8, F[L]/ kerϕ F[L]/ω[L] and, by Proposition 3.4,F[L]/ω[L] F[Q]. Hence F[Q] F[L]/ kerϕ.

According to (30) F[L] = Fe ⊕ ω[L]. Fe is a field, then from F[L]/ω[L] Fe itfollows that ω[L] is a maximal proper ideal of F[L]. Further, ω[H] ⊆ ω[L]. Then it iseasy to see that ϕ(ω[L]) = ∆ is a maximal proper ideal of F[Q] F[L]/ω[H]. Sincethe homomorphism ϕ is unital, then ϕ(Fe) = Fe and F[Q]/∆ Fe. By Lemma 2.1F[Q] = ∆] = Fe ⊕ ∆.

We denote B(∆) = e − q|e , q ∈ Q, Q0 = B(∆)⋃0 = e − q | q ∈ Q. By

item 1) of Lemma 4.1 the augmentation ideal ω[L] of algebra F[L] is generated asideal of F[L] and as F-module by set B(ω[L]) = e − g | g ∈ L, e , g. Further,ϕB(ω[L]) = ϕ(e − g) | e , ϕg ∈ ϕL = e − q | e , q ∈ Q = B(∆) = Q0\0. Hencethe ideal ∆ is generated, as an ideal of the algebra F[Q] and as well as an F-module,by the set B(∆) = Q0\0. From F[Q] = Fe ⊕ ∆ it follows that the algebra F[Q] isgenerated by set Q. According to Corollary 3.5, any element a ∈ F[Q] have the form

k∑

i=1

αiqi, (31),

where αi ∈ F, qi ∈ Q.By item 2) of Lemma 4.1 the ideal ω[L] is a quasiregular and ω[L] = J(F[L]).

From ω[H] ⊆ ω[L] it follows that ω[H] is a quasiregular ideal. Then ω[H] =

J(ω[H]). According to (11) and Propositions 2.7, 3.4, J(F[Q]) = J(F[L]/ω[H]) =

J(F[L])/J(ω[H]) = ω[L]/ω[H] = ϕ(ω[L]) = ∆. Hence J(F[Q]) = ∆ and, by (11), ∆

coincide with circle loop U∗(F[Q]), i.e.

∆ = U∗(F[Q]). (32)

We consider the mapping η : F[Q] → F[Q] defined by the rule ηu = e − u,∀u ∈F[Q]. From (11) and (32) it follows that the rule ηQ : ηQb = e − b, ∀b ∈ ∆, definesan isomorphism between the circle loop (∆, ) and the loop of invertible elementsU(F[Q]) of algebra F[Q] because η−1 = η. Particularly, the restriction ηQ of ηQon Q0 is an isomorphism of subloop (Q0, ) ⊆ (∆, ) and loop Q defined by rule:ηQb = e − b0, ∀b0 ∈ Q0. Consequently, the given Moufang loop Q is a subloop ofmultiplicative loop of invertible elements U(F[Q]) of algebra F[Q]. This completesthe proof of Theorem 4.2.

Now we define the class RA of alternative F-algebras. Any alternative loop algebrawith externally adjoined unit F[Q] belong to class RA. Remind that if F[Q] ∈ RA


then the loop Q, the field F and the algebra F[Q] have the same unit e and ϕ(e) = efor any homomorphism ϕ of F[Q]. Let F[Q] ∈ RA. Then, from Theorem 4.2, itfollows that Q ⊆ U(F[Q]). This fact suggest us to give the following definition. LetF[Q] be an alternative loop algebra and let H be a normal subloop of loop Q suchthat H ⊆ U(F[Q]). In such a case, we denote by ω[H] the ideal of F[Q] generatedby the set e− h|h ∈ H. If H = Q, then ω[Q] will be called an augmentation ideal ofthe alternative loop algebra F[Q].

Proposition 4.3. Let ω[Q] be the augmentation ideal of an alternative loop alge-bra F[Q] with externally adjoined unit e, i.e. let F[Q] ∈ RA. Then:

1) any element a ∈ F[Q] has the form∑k

i=1 αiqi, where αi ∈ F, qi ∈ Q;2) F[Q] = (ω[Q])] = ω[Q] ⊕ Fe;3) if F[Q] = R], then R = ω[Q];4) ω[Q] is generated as F-module by the set B(ω[Q]) = e − q|e , q ∈ Q;5) any isomorphism ϕ of algebra F[Q] induces the identical isomorphism on loop

Q and on ideal ω[Q], as well;6) ω[Q] = ∑q∈Q αqq|∑q∈Q αq = 0;7) the algebra ω[Q] is quasiregular and coincides with the Zhevlakov radical

J(F[Q]), ω[Q] = J(F[Q]);8) ω[Q] coincides with the circle loop U∗(F[Q]) (ω[Q] = U∗(F[Q])), i.e., by

(9), on the algebra ω[Q] there exists and it is unique, the unary operation x x∗of taking the quasiinverse, that is connected with the basic operations by identityx + x∗ = xx∗ = x∗x (i.e. r r∗ = 0 = r∗ r for all r ∈ ω[Q] from the constructionof loop U∗(ω[Q])). The circle loop U∗(F[Q]) is isomorphic to the loop U(F[Q]) ofinvertible elements under the isomorphism η : u → e − u, ∀u ∈ ω[Q]. The subloopQ0 = B(ω[Q])

⋃0 = e−q | q ∈ Q of the loop U∗(F[Q]) is isomorphic to the givenloop Q, i.e. η(Q0) = Q;

9) F[Q]\ω[Q] = U(F[Q]), i.e. the algebra ω[Q] coincides with set of all non-invertible elements of algebra F[Q];

10) J(I) = I⋂

J(F[Q]), U∗(I) = I⋂

U∗(F[Q]) for all ideals I of F[Q].

Proof. The item 1) was already proved (see the proof of Theorem 4.2 for the equality(31)).

Theorem 4.2 is proved by showing that the ideal ∆ coincides with augmentationideal ω[Q]. Then the statements 2), 4), 7), 8) are contained in proof of Theorem 4.2.

3) Let F[Q] = R] = R ⊕ Fe. By items 2), 7), 8) F[Q] = (ω[Q])] = ω[Q] ⊕ Fe,ω[Q] = J(Q), (ω[Q], ) = U∗(ω[Q]) and by Corollary 2.9 U∗(ω[Q]) U∗(R).An alternative algebra is quasiregular if it coincides with its circle loop. Hence theideal R is quasiregular. Fe is a field. Then from relation F[Q]/R Fe it followsthat R is a maximal ideal of F[Q]. Hence R = J(R). As J(R) = J(ω[Q]) thenR = J(R) = J(ω[Q]) = ω[Q], as required.

5) As ϕ is an isomorphism, then kerϕ = 0. By Lemma 3.3 it follows that ϕinduces the normal subloop Q

⋂(e + kerϕ) = e of loop Q. Hence ϕ induces on Q the


identical isomorphism ε. By item 4) the ideal ω[Q] is generated as F-module by setB(ω[Q]) = e − q|e , q ∈ Q. But ϕ(e − q) = e − εq = e − q. Consequently, again byitem 4), ϕ(ω[Q]) = ω[Q].

Using the item 1) the statement 6) is proved similarly as equality (29).By item 7) U∗(ω[Q]) = ω[Q]. Then the item 9) follows from the description of

the loop U(A) by equalities (16), (17).According to (11), the item 10) is just the equality (14). This completes the proof

of Proposition 4.3.

Let H be a normal subloop of the Moufang loop Q and let F[Q] ∈ RA. By item 2)of Proposition 4.3, e < ω[Q]. Then e < ω[H] and hence F[H] ∈ RA.

Let us determine the homomorphism of F-algebras ϕ: F[Q] → F[Q/H] by therule ϕ(

∑αqq) =

∑αqHq. The following result holds.

Proposition 4.4. Let F[Q] ∈ RA and let H,H1,H2 be normal subloops of the loopQ. Then:

1) kerϕ = ω[H];2) e − h ∈ ω[H] if and only if h ∈ H;3) if the family of elements hi generates the subloop H, then the family of elements

e − hi generates the ideal ω[H];4) if H1 , H2, then ω[H1] , ω[H2]; if H1 ⊂ H2, then ω[H1] ⊂ ω[H2]; if

H = H1,H2, then ω[H] = ω[H1] + ω[H2];5) F[Q]/ω[H] F[Q/H], ω[Q]/ω[H] ω[Q/H].

Proof. The statement 1) is proved similarly with item 3 of Lemma 3.8. To prove itis necessary only to use Theorem 4.2 instead of Lemma 3.1 and, in particular, to usethe item 1) of Proposition 2 instead of Corollary 3.2.

2). If q < H, then Hq , H. Consequently, ϕ(e − q) = H − Hq , 0, i.e., by 1),e − q < kerϕ = ω[H].

3), 4). Let elements hi generate subloop H and let I be the ideal, generated bythe elements e − hi. Obviously I ⊆ ω[H]. Conversely, let g ∈ H and let g = g1g2,where g1, g2 are words from hi. We suppose that e − g1, e − g2 ∈ I. Then e − g =

(e − g1)g2 + e − g2 ∈ I, i.e. ω[H] ⊆ I and I = ω[H]. Let H1 , H2 (respect. H1 ⊂ H2)and g ∈ H1, g < H2. Then, by item 2), e − g ∈ ω[H1], but e − g < ω[H2]. HenceωH1 , ωH2 (respect. ωH1 ⊂ ωH2). If H = H1,H2, then by the first statement of3), ωH = ωH1 + ωH2.

5). Mapping ϕ : F[Q] → F[Q/H] is the homomorphism of alternative loopalgebras and, as by item 1), kerϕ = ω[H], then F[Q/ω[H] F[Q/H]. The mappingω[Q] → ω[Q]/ω[H] save the sum of coefficients then from item 5) of Proposition4.3 it follows that ωL/ωH ω(L/H).

Corollary 4.5. For a normal subloop H of a Moufang loop Q with unit e thefollowing statements are equivalent:


1) F[H] ∈ RA;2) F[H] = ω[H] ⊕ F1;3) e < ω[H];4) ω[H] is a proper ideal of algebra F[Q];5) ω[H] = ∑q∈H αqq|∑q∈H αq = 0.

Proof. The equivalence of items 1), 2) follows from Lemma 2.1 and item 3) of Propo-sition 4.3. The implications 2)⇔ 3), 3)⇔ 4) are obvious. The implication 1)⇔ 5)follows from items 2), 6) of Proposition 4.3.

5. RADICALS IN ALTERNATIVE LOOPALGEBRAS AND MOUFANG LOOPS

Let M denote the class of all alternative loop algebras and its augmentation ideals.If ϕ is a non-zero homomorphism of algebra F[Q] ∈ M then, by Corollary 3.7, thehomomorphic image ϕ(F[Q]) will be an alternative loop algebra. Hence ϕ(F[Q]) ∈M.

Let now ω[Q] be the ideal of the alternative loop algebra F[Q] ∈ M with unit e,defined above, and let ψ be a homomorphism ofω[Q]. If e ∈ ω[Q] then ω[Q] = F[Q]and ψ is the zero homomorphism. We suppose that e < ω[Q]. Then F[Q] ∈ R and,by Corollary 4.5, F[Q] = ω[Q] ⊗ Fe. We extend ψ to the homomorphism ϕ of F[Q]considering that ϕ(e) = e. Let ϕ(F[Q]) = F[G]. Then F[G] = ψ(ω[Q]) ⊗ ϕ(Fe) =

ψ(ω[Q]) ⊗ Fe. By Lemma 2.1 it follows F[G] ∈ R. Then by item 3) of Proposition4.3 ψ(ω[Q] = ω[G]. Consequently, we proved that the class M is closed with respectto homomorphic images.

Let J be an ideal of the alternative loop algebra F[Q]. By Lemma 3.3 J induces thenormal subloop H = Q

⋂(e + J) of loop Q. In its turn H induces the homomorphism

ϕ : F[Q] → F[Q]/H defined by ϕ(∑αqq) =

∑αqqH, αq ∈ F, q ∈ Q. We have

kerϕ = J. According to item 1) of Proposition 4.4 J = kerϕ = ω[H]. If J1 is anotherideal of F[Q], J1 , J, then ω[H] = J , J1 = ω[H1], where H1 = Q

⋂(e + J1). By

item 3) of Proposition 4.4, it follows that H1 , H. Consequently, various ideals ofalgebra F[Q] induce various normal subloops of the loop Q.

Conversely, let H , H1 be normal subloops of loop Q. The subloops H,H1 inducethe homomorphisms ϕ, ϕ1 of algebra F[Q] and, by items 1), 3) of Proposition 4.4,kerϕ = ω[H] , ω[H1] = kerϕ1. Hence various normal subloops of the loop Q in-duce various ideals of algebra F[Q]. The proper ideals of algebra F[Q] have the formω[H], where H is a normal subloop of the loop Q and ω[H] is the augmentation idealwithout unit of the alternative loop algebra F[H]. Consequently, the correspondenceω[H] → H is an one-to-one mapping between all normal subloops H of the loop Qand all ideals of the algebra F[Q]. Further let’s consider, that all considered algebrasbelong to class M, i.e. they have the form F[Q], any of its ideals J , F[Q] has theform ω[H], where ω[H] is the augmentation ideal of some alternative loop algebraF[H] where H is a normal subloop of loop Q.


Now we consider the class of algebras RA ⊆ M, defined in section 3. This classwas analysed in Propositions 4.3, 4.4 and Corollary 4.5. The class RA is characterizedby the property that any algebra from RA is an alternative loop algebra F[Q] such thatits ideals J , F[Q] are augmentation ideals ω[H] without unit of some alternativeloop algebras F[H] from RA, where H ⊂ Q is a normal subloop of the loop Q.

We denote by R the class of such augmentation ideals ω[Q] of its alternative loopalgebra F[Q] ∈ RA.

An ideal I of algebra A ∈ M will be called R-ideal if it belongs to class R. Analternative loop algebra F[Q] ∈ M, containing non-zero ideals will be called RA −algebra if its augmentation ideal ω[Q] belongs to class R, i.e. the ideal ω[Q] iswithout unit and (ω[Q])] = F[Q].

Lemma 5.1. Any algebra A of class M contains a unique maximal R-ideal R(A).The ideal R(A) coincides with augmentation ideal ω[K] of some alternative loopalgebra F[K] ∈ RA.

Proof. Let Σ denote the set of all R-ideals of algebra A. The set Σ is non-empty, sincethe ideal (0) ∈ Σ. By Zorn Lemma the set Σ contains a maximal R-ideal R(A).

Let’s show that R(A) is an unique maximal R-ideal. Let R(B) also be a maximalR-ideal and let x ∈ R(A), y ∈ R(B). As R(A),R(B) ∈ R, then, by item 5) of Corollary4.5, x =

∑g∈Q αgg with

∑g∈Q αg = 0 and y =

∑g∈Q βgg with

∑g∈Q βg = 0. Then

x + y =∑

g∈Q γgg with∑

g∈Q γg = 0. Hence R(A) + R(B) ∈ R. If R(A) , R(B) thenR(A) + R(B) strictly contain R(A). Contradiction. Consequently, R(A) is the uniquemaximal R-ideal.

The second statement of lemma follows from relation R(A) ∈ R and the construc-tion of ideals of class R. This completes the proof of Lemma 5.1.

Theorem 5.2. The class R of all augmentation ideals without unit is radical inclass M of all alternative loop F-algebras and its ideals.

Proof. According to the definition of radical [36] should prove the statements:(a) any homomorphic image of any R-ideal is an R-ideal;(b) each algebra A from M contains an R-ideal R(A), containing all R-ideals of

algebra A;(c) the quotient-algebra A/R(A) does not contain any non-null R-ideals.Really, let ω[H] ∈ R and let x =

∑h∈H αhh ∈ ω[H]. By item 5) of Corollary 4.5,∑

h∈H αh = 0. Any homomorphism ϕ of the ideal ω[H] does not change the sum ofcoefficients,

∑αh. From here, it follows that ϕ(ω[H]) ∈ R and the statement (a) is

proved.The Lemma 5.1 is just the statement (b).Let A ∈ M. The homomorphism ϕ : A → A/R(A) maintains the sum of coeffi-

cients. Hence if J , (0) is an R-ideal of A/R(A) then the inverse image ϕ−1J will bean R-ideal and R(A) ⊂ ϕ−1J. But this contradicts the maximality of R-ideal R(A).


Consequently, J = (0) and the statement (c) is proved. This completes the proof ofTheorem 5.2.

Let A ∈ M. The mapping A → R(A) is called radical, defined in class of algebraM; denote it by R.

Let now introduce some notions, derived from the general concepts of the theory ofradicals [36]. The ideal R(A) of algebra A ∈M is called its R-radical. An alternativeloop algebra F[Q] ∈M will be called R-radical if F[Q] ∈ RA and ω[Q] = R(F[Q]).Non-zero algebras F[Q] ∈ M, whose radical is null, will be called R-semisimples.The class P of all R-semisimple algebras of class M is called semisimple class ofradical R.

Let A ∈ M. According to Lemma 5.1 the radical R(A) coincides with the aug-mentation ideal ω[K] of some alternative loop algebra F[K] ∈ RA. Then, by item7) of Proposition 4.3, R(A) coincides with the Zhevlakov radical J(F[K]) which, byitem 10) of Proposition 4.3, is hereditary. Remind that the radical R in the class ofalgebras A is called hereditary if R(J) = J

⋂R(A) for any algebra A ∈ A and any it

ideal J. Then the following holds.

Corollary 5.3. The radical R is hereditary in class M.Corollary 5.4. Let A be an algebra of class M and let J be an ideal of A. The

following results hold:(1) if A ∈ R then J ∈ R;(2) if A ∈ P then J ∈ P.It follows from Corollary 1 and [36, Theorem 3, cap. 8].

Let F[Q] be the alternative loop algebra of a Moufang loop Q with unit e. Ac-cording to Corollary 3.5 any element a ∈ F[Q] is a finite sum a =

∑q∈Q αqq, where

αq ∈ F. Then we may define the ideal of F[Q] generated by set e−q|q ∈ Q. We de-note it by ω[Q]. Note that in a similar way we have above defined the augmentationideal ω[Q] of algebra F[Q] ∈ RA.

Corollary 5.5. For any algebra A of class P the following statements hold:1) if A = ω[Q] then ω[Q] = F[Q];2) if x ∈ A and x =

∑g∈Q αgg then

∑g∈Q αg , 0;

3) any ideal J of algebra A has the form J = F[H] and, if J , 0, then J isnonassociative.

Proof. If A ∈ M, then the from definition of class M, it follows that A = F[Q] forsome alternative loop algebra F[Q]. By Lemma 5.1, R(A) = ω[H]. If A ∈ P thenR(A) = 0. From here it follows that the algebra A does not have non-zero properideals. Then ω[Q] = F[Q].

The item 2) follows from item 6) of Proposition 4.3.


Further, A ∈ P implies J ∈ P for any ideal J of algebra A, by Corollary 5.4. Wehave above proved that any ideal I of A have the form I = ω[H]. Then, by item 1),it follows J = F[H]. If F[H] is associative then H is a group and F[H] is a groupalgebra. From the definition of group algebra it follows that F[H] is a free F-modulewith bases h ∈ H. Then e < ω[H] and F[H] , ω[H]. Contradiction. Consequently,the ideal J , 0 cannot be associative. This completes the proof of Corollary 5.5.

In the beginning of the section we showed that for any alternative loop algebraF[Q] the mapping ω[H]→ H is an one-to-one mapping between all normal subloopsH of loop Q and all ideals of algebra F[Q]. Moreover, the following statement holds.

Lemma 5.6. Let F[Q] be an alternative loop algebra and let H,H1,H2 be normalsubloops of loop Q. Then:

1) e − h ∈ ω[H] if and only if h ∈ H;2) if the elements hi generate the subloop H, then the elements 1 − hi generate

the ideal ω[H]; if H1 , H2, then ω[H1] , ω[H2]; if H1 ⊂ H2, then ω[H1] ⊂ ω[H2];if H = H1,H2, then ω[H] = ω[H1] + ω[H2].

Proof. If e < ω[H] then F[H] ∈ RA and the statement 1) is the statement 2) ofProposition 4.4. If e ∈ ω[H] then ω[H] = F[H] and statement 1) follows, fromproperty (31), that F[H] is generated as an F-module by the set H.

The statement 2) is proved similarly as item 3) of Proposition 4.4. We have to usethe item 1), only.

By L denote the class of all Moufang loops and by S denote the class of Moufangloops G such that F[G] ∈ RA or, equivalently, e < ω[G] (by Proposition 4.3). Anyloop from the class L (respect. S) will be called L-loop (respect. S-loop). Now, letQ ∈ L be a Moufang loop, F[Q] ∈ M be it alternative loop algebra and, accordingto Theorem 5.2, let R(F[Q]) be the R-radical of F[Q]. By Lemma 5.1, R(F[Q]) =

ω[S(Q)], where ω[S(Q)] is the augmentation ideal of some alternative loop algebraF[S(Q)] ∈ RA. By Theorem 5.2 the mapping R : F[Q] → R(F[Q]) = ω[S(Q)] is aradical of class M. Obviously, R induces the mapping S : Q→ S(Q).

Note that, with the help of Lemma 3.3, it is easy to see that from R(F[Q])= 0 it follows S(Q) = e. Further, we will show that the mapping S is a radicalof the class L of loops. For this, the class of loops S should satisfy the followingconditions:• the homomorphic image of any S-loop is a S-loop;• each L-loop Q contains a normal S-subloop S(Q), containing all normal S-

subloops of the loop Q;• the quotient loop Q/S(Q) does not contain non-unitary normal

S-subloops.

Theorem 5.7. The class S is radical in the class L of all Moufang loops.


Proof. Let G ∈ S. Then F[G] ∈ RA. Any homomorphism ϕ of loop G inducesa homomorphism ϕ : F[G] → F[ϕG] defined by rules ϕ(

∑g∈G αgg) =

∑g∈G αgϕg,

ϕ(F[G]) = F[ϕG]. By Theorem 5.2 ϕ(F[G]) ∈ RA. Then F[ϕG] ∈ R. Hence ϕG ∈ S.Consequently, the class S is closed under homomorphisms.

Let now Q be a Moufang loop and let, by Theorem 5.2, R(Q) be the R-radical ofF[Q]. By Lemma 3.3 the ideal R(Q) of F[Q] induces the normal subloop S(Q) =

Q⋂

(e + R(Q)) of the loop Q. Let ψ : F[Q] → F[Q/S(Q)] be the homomorphismdefined by:

∑g∈Q αgg → ∑

g∈Q αg(gS(Q)). Then, F[Q]/ kerψ F[Q/S(Q)] andR(Q) ⊆ kerψ.

If e ∈ kerψ, then kerψ = F[Q] and F[Q/S(Q)] F[Q]/ kerψ =

F[Q]/F[Q] = 0. We get a contradiction because e ∈ F[Q/S(Q)]. Hence kerψis a proper ideal of F[Q]. Then, as it was shown in the beginning of section, kerψis an augmentation ideal. The radical R(Q) is a maximal augmentation ideal of F[Q].Hence kerψ = R(Q) and F[Q/S(Q)] F[Q]/R(Q), R(F[Q/S(Q)]) R(F[Q]/R(Q)). By Theorem 5.2, R(Q)) is a maximal ideal of F[Q] such that theR-radical of the quotient-algebra F[Q]/R(Q) is zero. HenceR(F[Q/S(Q)]) = 0 and S(Q/S(Q)) = 0. Consequently, the normal subloop S(Q)of loop Q is maximal and such that S(Q/S(Q)) = 0. This completes the proof ofTheorem 5.7.

Let Q ∈ L. By Theorem 5.7, the mapping Q → S(Q) is a radical defined in theclass of loops L; denote it by S. The normal subloop S(Q) of loop Q will be calledits S-radical. A loop coinciding with its S-radical will be called S-radical, and thenon-unitary loops, whose radical is equal to unit, will be called S-semisimples. Theclass T of all S-semisimples algebras in class L will be called semisimple class ofradical S.

Proposition 5.8. The radical S is hereditary in the class L of all Moufang loops,i.e. for any loop Q ∈ L and its normal subloop H, S(H) = H

⋂S(Q).

Proof. By Theorem 5.7, the radical S(G) is a maximal normal subloop H of the loopG with respect to property ω[H] ∈ R. From item 3) of Proposition 4.4 it follows thatω[S(G)] is a maximal ideal ω[H] of algebra ω[G] with respect to property ω[H] ∈ R.Then by Theorem 5.2 ω[S(H)] = R(ω[H]).

Let now H be a normal subloop of loop Q. Then ω[H] will be a normal subloopof loop ω[Q]. By Corollary 5.3 R(ω[H]) = ω[H]

⋂R(ω[Q]) and by the precious

equality ω[S(H)] = ω[H]⋂ω[S(Q)]. Then from item 3) of Proposition 5.3 it follows

from here that ω[S(H)] = ω[H⋂

S(Q)], S(H) = H⋂

S(Q). This completes the proofof Proposition 5.8.

Corollary 5.9. Let Q be a Moufang loop and let K be a normal subloop of Q.Then the following statements hold:

(i) if Q ∈ S then K ∈ S;


(ii) if Q ∈ T then K ∈ T.These follow from Proposition 3 and [36, Theorem 3, cap. 8].

6. SEMISIMPLE ALTERNATIVE LOOPALGEBRAS ANDSEMISIMPLE MOUFANG LOOPS

Let A be an algebra. The sum of ideals Is|s ∈ S of algebra A is called the idealI of A generated by reunion

⋃s∈S Is. The ideal I consists of elements x, presented in

the form x = x1 + . . . + xk, where x j ∈ Is j for some s j ∈ S and denote I =∑

s∈S Is.The sum is called direct if Is

⋂∑s,t∈S It = 0. Denote I =

∑⊕ Is and I = I1 ⊕ . . . ⊕ Ikfor finite sum of ideals.

By analogy, the product N of normal subloops Ns|s ∈ S of the loop Q consistsof elements x, presented in the form x = x1 · . . . · xk, where x j ∈ Ns j for some s j ∈ Sand denote N =

∏s∈S Ns. The product is called direct if Ns

⋂∏s,t∈S Nt = 1. Denote

N =∏⊗ Ns and N = N1 ⊗ . . . ⊗ Nk for a finite factor product.

An ideal J of the algebra A is called simple if J does not have other ideals of Abesides the null and ideal J itself and it is called principal if it is generated by oneelement. A normal subloop N of loop Q will be called simple if N does not have othernormal subloops of Q besides the unitary subloop and loop N itself.

Lemma 6.1. The following statements are equivalent for the simple normal subloopsNs|s ∈ S of a loop Q:

1) Q =∏

s∈S Ns;2) Q =

∏⊗s∈S Ns.

Lemma 6.2. The following statements are equivalent for the family of simpleideals Is|s ∈ S of algebra A:

1) A =∑

s∈S Is;2) A =

∑⊕s∈S Is.

Proof. Lemmas 6.1, 6.2 have similar proofs. Let us prove Lemma 6.2.Let T be a maximal subset of S such that the sum

∑t∈T It is direct. The sum

∑t∈T It

is an ideal of A. Let us show that this sum coincides with A. For this it is enough toshow that each ideal I j is contained in this sum. The intersection of our sum with I jis an ideal in A and, consequently, equals 0 or I j. If it equals 0, then subset T is notmaximal, as we can add j to it. Consequently, I j is contained in the sum

∑t∈T It. This

completes the proof of Lemmas 6.2.

Let A be an F-algebra and let I(M) be the ideal of A generated by set M ⊆ A. Theideal I(M) consists of all possibly types of finite sums of elements of form

ϕ(x1, . . . , x j, a, x j+1, . . . , xn)α, (33)

where ϕ ∈ F, a ∈ M, xi ∈ A, α is a certain distribution of parenthesis.


Let a, b ∈ A. Then from (33) it follows that

I(a + b) ⊆ I(a) + I(b). (34)

Now, we consider an ideal ω[Q] of the alternative loop algebra F[Q] of a Moufangloop Q. If a, b ∈ Q then e − ab = (e − a) + (e − b) − (e − a)(e − b). Denote e − u = u.By (34), I(ab) = I(a + b − ab), I(ab) ⊆ I(a) + I(b) − I(ab). From (33), it follows thatI(ab) ⊆ I(a). Then

I(ab) ⊆ I(a) + I(b). (35)

Moreover, the following result holds.Lemma 6.3. Let consider a principal ideal I(a), for a ∈ Q, of ideal ω[Q] which is

not simple. Then, there exists an element b ∈ Q such that I(a) = I(b) + I(c), wherea = bc, and I(b) is a proper ideal of algebra I(a).

Proof. Let J be a proper ideal of I(a). By Lemma 3.3 the normal subloops B andA of the loop Q correspond to the ideals J, I(a) and, by item 2) of Lemma 5.6,B ⊂ A. Let b ∈ B and let a = bc. Then b, c ∈ A and, by item 1) of Lemma 5.6,e− b, e− c ∈ ω[A] ⊆ I(a). Hence I(b) ⊆ I(a), I(c) ⊆ I(a). By (35), I(a) ⊆ I(b) + I(c).Then I(a) = I(b) + I(c), as required.

Let F[Q] ∈ P and let I(a), where a = e − a, for a ∈ Q, be a principal ideal ofF[Q]. As F[Q] ∈ P then, by item 2) of Corollary 5.4, I(a) ∈ P. Let A be the normalsubloop of the loop Q induced, via Lemma 3.3, by the ideal I(a). Then, by item 3) ofCorollary 5.5, I(a) = F[A] and any element of I(a) has the form

n∑

i=1

αiui,

n∑

i=1

αi , 0, (36)

where αi ∈ F, ui ∈ Q.Further, for principal ideals I(ai), I(b j), . . . we use the notations F[Ai] = I(ai),

F[B j] = I(b j), . . . The symbols FY F[A] will denote the F-modules FY , F[A].Lemma 6.4. Let F[Q] ∈ P and let I(a), where a = e − a, a ∈ Q, be a principal

ideal of F[Q]. Then, there exists an element b ∈ Q such that I(a) = I(b) + I(c), wherea = bc, I(b) is a proper ideal of algebra I(a) and F[A] = F[B] ⊕ M[K], where M[K]denotes the F-submodule of F[C] generated by set K = A\B.

Proof. Let Q = L/H, where L is a free Moufang loop. We consider the homomor-phisms ϕ : LX → LX/I = F[X], ψ : F[X] → F[L]/ω[H] = F[Q] (see Proposition3.4). From item 5) of Corollary 5.5, it follows that any element in ω[H] has the form

m∑

j=1

β jh j,

m∑

j=1

β j = 0, (37)


where β j ∈ F, h j ∈ H. By Lemma 6.3, I(a) = I(b) + I(c) and I(b) is a proper ideal ofI(a).

If we denote ψ−1(A) = XA, ψ−1(B) = XB, ψ−1(C) = XC , thenψ−1(I(a)) = ψ−1(F[A]) = F[XA], ψ−1(I(b)) = F[XB], ψ−1(I(c)) = F[XC]. By (37),the homomorphism F[X] → F[X]/ω[H] maintains the sum of coefficients, thus anyelement in F[XA], F[XB], F[XC] has the form

k∑

i=1

γixi,

k∑

i=1

γi , 0, (38)

where γi ∈ F, xi ∈ X. Then from (37), (38) it follows that F[XA]⋂ω[H] = 0,

F[XB]⋂ω[H] = 0, F[XC]

⋂ω[H] = 0. Consequently, F[A] = ψ(F[XA]) =

(F[XA] + ω[H])/ω[H] F[XA]/(F[XA]⋂ω[H]) = F[XA]/0

= F[XA], i.e. F[XA] F[A]. Similarly, F[XB] F[B], F[XC] F[C].According to Lemma 3.1 ϕ−1(XA) = XA, ϕ−1(XB) = XB, ϕ−1(XC) = XC . Hence

ϕ−1(F[XA]) = FXA, ϕ−1(F[XB]) = FXB, ϕ−1(F[XC]) = FXC .From the definition of an ideal I of the loop algebra FX it follows that any element

of I has the form∑n

j=1 β jx j with∑

j=1 β j = 0, where β j ∈ F, x j ∈ X. Then from(38) it follows that F[XA]

⋂I = 0, F[XB]

⋂I = 0, F[XC]

⋂I = 0 and FX[A] =

ϕ(FXA) = (FXA + I)/I FXA/(FXA⋂

I)= F[XA]/0 = F[XA]. Hence FXA F[XA]. Before we have proved that F[XA] F[A] = I(a). Consequently, FXA I(a). Similarly, FXB F[B] = I(b), FXC F[C] = I(c).

The inverse image of equality I(a) = I(b) + I(c) regarding homomorphism ϕψ, isthe equality FXA = FXB + FXC of the loop algebra FX. The loop algebra FX is afree F-module with basis x|x ∈ X. Then FXA = FXB ⊕ M(XA\XB), M(XA\XB) ⊆FXC . Hence F[A] = F[B]

⋂M[A\B], M[A\B] ⊆ F[C]. This completes the proof of

Lemma 6.4.

Proposition 6.5. Let F[Q] ∈ P and let I(a), where a = e − a, for a ∈ Q, be aprincipal ideal of F[Q]. Then I(a) decomposes into a direct sum of finite number ofsimple nonassociative principal ideals I(a) = I(b1) ⊕ . . . ⊕ I(bn).

Proof. Inductively we construct two series

I(b1) ⊃ I(b2) ⊃ . . . ⊃ I(bn) ⊃ . . . ,I(d1) ⊆ I(d2) ⊆ . . . ⊆ I(dn) ⊆ . . . (39)

of proper non-zero ideals of the algebra I(a) such that I(a) = I(bn)+ I(dn) and a series

M[K1] ⊂ M[K2] ⊂ . . .M[Kn] ⊂ . . . (40)

of F-submodules of the F-module F[A] such that M[Ki] ⊆ M[Di] and F[A] = F[Bi]⊕M[Ki], Ki = A\Bi. The inductive process stops if an ideal I(bn) is simple for someinteger n.


Let the ideal I(a) be non-simple. Then by Lemma 6.4 I(a) = I(b1) + I(c1), whereI(b1) is a proper ideal of I(a) and F[A] = F[B1] ⊕ M[D1], D1 = A\B1, M[D1] ⊆F[C1]. If at least one of the ideals I(b1), I(c1) is simple then the inductive processends. Conversely, let us consider that the ideal I(b1) is non-simple. By Lemma 6.4,let I(b1) = I(b2) + I(c2), where I(b2) is an ideal of F[A] and is a proper ideal of I(b1).Again by Lemma 6.4 I(a) = I(b2) + I(d2) and F[A] = F[B2] ⊕ M[K2], K2 = A\B2,M[K1] ⊆ F[D2].

Let us continue the inductive process. Let I(a) = I(bn) + I(dn), F[A] = F[Bn] ⊕M[Kn], Kn = A\Bn, M[Kn] ⊆ F[Cn] and let the ideal I(bn) be non-simple. By Lemma6.4 I(bn) = I(bn+1) + I(cn+1) and I(a) = I(bn+1) + I(dn+1) and F[A] = F[Bn+1] ⊕M[Kn+1], Kn+1 = A\Bn+1. From I(bn) ⊃ I(bn+1) it follows that I(dn) ⊂ I(dn+1) andKn ⊂ Kn+1, M[Kn] ⊂ M[Kn+1]. Consequently, the series (39), (40) with propertyF[A] = F[Bn] ⊕ M[Kn], n = 1, 2, . . . are defined.

The modules M[Kn] in the ascending series (40) satisfy the property b1 < M[Kn].Then, by Zorn Lemma, this series have a maximal proper ideal J in I(a) such thatb1 < J. Let b ∈ I(a) \ J. As J is a maximal ideal of I(a) then I(a) = I(b) + J.

Let the ideal I(b) be non-simple. Then, by Lemma 6.4, I(b) = I(b1) + I(b2), whereI(b1) is a proper non-zero ideal of I(a), I(a) = I(b1) + I(b2) + J and the ideal I(b2) + Jstrictly contain the maximal ideal J. Contradiction. Hence the ideal I(b) is simple.

By Lemma 6.4 I(a) = I(b) + I(d) for some proper ideal I(d) of I(a). The idealI(b) is simple. Let the ideal I(d) is non-simple. Then I(a) = I(b1) + I(d1), whereI(d1) = I(d) and I(b1) = I(b) is a simple ideal. Further, I(a) = I(b) + I(b2) +

. . . + I(bk) + . . ., where I(bi), i = 1, 2, . . ., is a simple ideal. Then, by Lemma 6.2,I(a) = I(b) ⊕ I(b2) ⊕ . . . ⊕ I(bk) ⊕ . . ., where each I(bi), i = 1, 2, . . ., is a simple ideal.

It is known that any element of the direct sum is written unequivocally as thesum of a finite number of non-zero elements, taken one from some ideals I(bi j). Leta = b1 + . . . + bn, where b j ∈ I(bi j). Then by (34) I(a) ⊆ I(bi1) ⊕ . . . ⊕ I(bin) and,consequently, I(a) = I(bi1) ⊕ . . . ⊕ I(bin). As F[Q] ∈ P then by [36, Theorem 3, cap.8] I(bi j) ∈ P and by item 3) of Corollary 5.5 the ideals I(bi j) are nonassociative. Thiscompletes the proof of Proposition 6.5.

Lemma 6.6. Let Q ∈ T be a nonassociative semisimple Moufang loop, let F[Q] ∈P be it corresponding alternative loop algebra and let A, B be normal subloops ofthe loop Q. Then A ⊂ B (respect. A = B) when and only when ω[A] ⊂ ω[B] (respect.ω[A] = ω[B]).

Proof. Let Q = L/H, where L is a free Moufang loop and let ϕ : LX → LX/I =

F[X], ψ : F[X] → F[L]/ω[H] = F[Q] be the homomorphisms considered in proofof Lemma 6.4. Let ψ−1(A) = XA, ψ−1(B) = XB, XA, XB ⊆ X. It is proved thatFXA F[XA] F[A] = ω[A], FXB F[XB] F[B] = ω[B]. From (36) –(38), it follows easily that the restrictions of homomorphism ψ on XA and on XB areisomorphisms of loops XA, A and XB, B, respectively.


From the mentioned isomorphisms, it follows that the inclusions F[A] ⊂ F[B]in the alternative loop algebra F[Q] and FXA ⊂ FXB in the loop algebra FX areequivalent. The loop algebra FX is a free F-module with basis x|x ∈ X. Then theinclusion FXA ⊂ FXB is equivalent to inclusion XA ⊂ XB of subloops in the loopX. Further, from isomorphisms of loops XA, A and XB, B it follows that the inclusionXA ⊂ XB is equivalent to inclusion A ⊂ B. Consequently, the inclusion F[A] ⊂ F[B]in the alternative loop algebra F[Q] is equivalent to inclusion A ⊂ B of subloops inthe loop Q. The facts that the equalities F[A] = F[B] and A = B are equivalent areanalogously proved.

Proposition 6.7. Let Q ∈ T be nonassociative semisimple Moufang loop andlet F[Q] ∈ P be its corresponding alternative loop algebra. Then for any elementa ∈ Q the normal subloop N(a) of Q generated by element a decompose into a directproduct of finite number of nonassociative simple loops.

Proof. By Proposition 6.5 I(a) = I(a1)⊕ . . .⊕ I(ak), where each I(ai) is a simple idealof F[Q] generated by element ai = e − ai, for ai ∈ Q, i = 1, . . . , k. By Lemma 3.3,the ideal I(ai) induces in Q the normal subloop Hi = Q

⋂(e+ I(ai)). Let N(ai) denote

the normal subloop of Q generated by the element ai ∈ Q. It is clear that ai ∈ Hi.Then N(ai) ⊆ Hi. If N(ai) ⊂ Hi (strictly) then by Lemma 6.6 ω[N(ai)] ⊂ ω[Hi](strictly). Butω[Hi] = I(a). Henceω[N(ai)] ⊂ I(a) (strictly), i.e. ω[N(ai)] is a properideal of I(ai). We get a contradiction because I(ai) is a simple ideal. Consequently,ω[N(ai)] = I(ai).

If K is a proper normal subloop of N(ai) then, by Lemma 6.6, ω[K] is a properideal of ω[N(ai)] = I(ai). Again we get a contradiction. Hence the normal subloopsN(ai), i = 1, . . . , k, are simple. We have

I(a) = I(a1) ⊕ . . . ⊕ I(ak) (41)

or ω[N(a)] = ω[N(a1)] + . . . +ω[N(ak)]. Then, by item 2) of Lemma 6.6, ω[N(a)] =

ω[N(a1)] · . . . · N(ak)] and, by Lemma 6.6, N(a) = N(a1)] · . . . · N(ak). The subloopsN(ai), i = 1, . . . , k, are simple. Then, by Lemma 6.1,

N(a) = N(a1) ⊗ . . . ⊗ N(ak). (42)

This completes the proof of Proposition 6.7.

Corollary 6.8. Let Q ∈ T be a nonassociative semisimple Moufang loop and letF[Q] ∈ P be it corresponding alternative loop algebra. Then:

1) any nonassociative simple subloop of the loop Q has the form H = I(a), whereI(a) is a normal subloop of F[Q], with a = e − a for some a ∈ Q;

2) any nonassociative simple subalgebra of algebra F[Q] has the form F[H] =

ω[H], where H = I(a), with a = e − a, for some a ∈ Q;


3) a nonassociative subalgebra F[H] of algebra F[Q] is simple when and onlywhen the nonassociative normal subloop H of loop Q is simple.

The corollary follows from (41), (42) and Lemma 6.6.Lemma 6.9. Any algebra A of semisimple class P of radical R decomposes into a

direct sum of nonassociative simple algebras.

Proof. By item 3) of Corollary 5.5, any algebra A of semisimple class P has the formA = F[Q] and F[Q] = ω[Q]. The idealω[Q] is generated as ideal by set e−g|g ∈ Q.Let g1 ∈ Q. As (e − g)g1 = (e − gg1) − (e − g1), g1(e − g) = (e − g1g) − (e − g1)then ω[Q] is generated as F-module by elements of form e− g, where g ∈ Q. Denotee − g = g and let I(g) be the (principal) ideal generated by element g ∈ F[Q]. Then

F[Q] =∑

g∈Q

I(g). (43)

As F[Q] ∈ P then by item 2) of Corollary 5.4 I(g) ∈ P. By Proposition 6.5,I(g) = I(b1) ⊕ . . . ⊕ I(bk), where I(bi) is a simple ideal of F[Q]. Then, from (43), itfollows that F[Q] =

∑I(bi) and, by Lemma 6.2, F[Q] =

∑⊕ I(bi), where I(bi) aresimple ideals of algebra F[Q]. This completes the proof of Lemma 6.9.

Corollary 6.10. Let charF = 0 or charF = 3. Then any nonassociative com-mutative Moufang loop Q is S-radical and any it alternative loop algebra F[Q] isR-radical.

Proof. According to Theorem 5.2 F[Q]/R(F[Q]) = P(F[Q]), where P(F[Q]) ∈ P.We assume that P(F[Q]) , Fe. As the loop Q is commutative then by (31) the algebraP(F[Q]) also is commutative. From Lemma 6.9 it follows that P(F[Q]) decomposesinto a direct sum of nonassociative simple algebras. But any commutative simplealternative algebra is a field [36, pag. 172]. We get a contradiction. Hence P(F[Q]) =

Fe. Then, by using the definitions, F[Q] ∈ R and Q ∈ S, as required.

Lemma 6.11. Any nonassociative semisimple Moufang loop Q ∈ T decomposesinto a direct product of nonassociative simple loops.

The statement follows from (41 - 43) and Lemmas 6.1, 6.6.

7. MAIN RESULTSLet us consider the following notions. In the beginning of the paper, we have

mentioned that, in the literature, an algebra is called antisimple, if none of its two-sided ideals allows homomorphism on a simple algebra.

An alternative loop algebra F[Q] ∈ M will be called antisimple with respect tononassociativity if for any its ideal ω[H] the algebra (ω[H])] = F[H] does not allowhomomorphism on a simple nonassociative algebra.


Analogously, a loop Q ∈ L will be called antisimple with respect to nonassociativ-ity if none of its normal subloops allows homomorphism on a simple nonassociativeloop.

Let A be a class of algebras, let B be its radical class and let C be it semisimpleclass. By the definition of radical B any homomorphic image of B-algebra is a B-algebra. In [36, Proposition 1, pag. 184] it is proved that the radical class B of A

is the totality of algebras from A, not reflected homomorphically on the algebras ofclass C. Then from Lemmas 6.9, 6.11 it follows the next result.

Lemma 7.1. The class of all antisimple with respect to nonassociativity alterna-tive loop algebras F[Q] coincides with the radical class RA of all alternative loopalgebras of type F[Q] = (ω[Q])]. The class of all antisimple with respect to nonas-sociativity Moufang loops coincides with the radical class S of Moufang loops.

Proposition 7.2 An alternative loop algebra F[Q] is antisimple with respect tononassociativity when and only when F[Q] does not have subalgebras that are nonas-sociative simple algebras. A Moufang loop Q is antisimple with respect to nonas-sociativity when and only when Q does not have subloops that are nonassociativesimple loops.

Proof. Let F[Q] ∈ M and, according to Theorem 5.2, let F[Q]/R(F[Q])= P(F[Q]), where P(F[Q]) ∈ P. By Lemma 7.1, the algebra F[Q] is antisimplewith respect to nonassociativity when and only when P(F[Q]) = Fe.

We assume that P(F[Q]) , Fe. By Lemma 3.3, the ideal R(F[Q]) of algebra F[Q]induces the normal subloop R = Q

⋂(e+R(F[Q])) of loop Q. As P(F[Q]) , Fe then

R(F[Q]) is a proper ideal of F[Q] and, according to the one-to-one mapping amongall ideals of F[Q] and all normal subloops of Q the normal subloop R is proper andF[Q/R] F[Q]/R(F[Q]) = P(F[Q]). By Lemma 6.9, P(F[Q]) , Fe decomposesinto a direct sum of nonassociative simple algebras. According to Corollary 6.8, letF[H] = ω[H], where H is the normal subloop of loop Q/R generated by one elementhR = h + R(F[Q]), be one of such nonassociative simple algebras. We denote by Hthe normal subloop of the loop Q generated by the element h ∈ Q.

Clearly, the inverse image of subalgebra F[H] under the natural homomorphismF[Q] → F[Q]/R(F[Q]) = P(F[Q]) is F[H] + R(F[Q]. For a ∈ R(F[Q]), wehave a =

∑q∈Q αqq with

∑q∈Q αq = 0, by item 5) of Corollary 4.5, and for b ∈

F[H], we have b =∑

g∈H βgg with∑

g∈H βg , 0, by item 2) of Corollary 5.5.The extension F[Q]/R(F[Q]) does not change the sum of coefficients. Hence ifc ∈ F[H] then c =

∑g∈H γgg with

∑g∈H γg , 0. Then F[H]

⋂R(F[Q]) = (0)

and, by homomorphism theorems, it follows F[H] (F[H] + R(F[Q]))/R(F[Q]) F[H]/(F[H]

⋂R(F[Q]) = F[H]. Hence, the subalgebra F[H] of the algebra F[Q] is

a nonassociative simple algebra. Then, by item 3) Corollary 6.8, H is a nonassocia-tive simple loop. Consequently, if P(F[Q]) , (0) then:

(i) the algebra F[Q] contains a nonassociative simple algebra F[H];


(ii) the loop Q contains a nonassociative simple loop H.If the algebra F[Q] does not contain a nonassociative simple loop then, from item

2) of Corollary 5.5, it follows that∑

q∈Q αq = 0 for any element a =∑

q∈Q αqq ∈ω[Q]. In such a case, F[Q] = (ω[Q])] and, from Lemma 5.1, it follows that P(F[Q]) =

Fe. This completes the proof of the first statement.Now, let Q ∈ L be a Moufang loop and, according to Theorem 5.7, let Q/S(Q) =

T (Q), where T (Q) ∈ T. By Lemma 7.1, the loop Q is antisimple with respect tononassociativity when and only when T (Q) = e. We prove that the equality T (Q) =

e is equivalent to the property that the loop Q does not contain a subloop isomorphicto simple nonassociative loop.

Indeed, we assume that T (Q) , 1. Then, from the relation Q/S(Q) = T (Q),it follows that Q , S(Q) and, from the definition of the class S, it follows thatR(F[Q]) , F[Q] and P(F[Q]) , Fe, in accordance with the relationF[Q]/R(F[Q]) = P(F[Q]). In such a case, the loop Q contains a nonassociativesimple loop H, by statement (ii).

Now let the loop Q not contain any nonassociative simple loop as subloop. Then,from item 3) of Corollary 6.8, it follows that the alternative loop algebra F[Q] doesnot contains nonassociative simple algebra as subalgebra. Thus, by the first case,P(F[Q]) = Fe or R(F[Q]) = F[Q], S(Q) = Q, T (Q) = e. This completes the proofof Proposition 7.2.

Let us consider the analogue for alternative loop algebras of the Wedderburn The-orem for finite dimensional associative algebras.

By Kleinfeld Theorem [36] any nonassociative simple alternative algebra is aCayly-Dickson algebra over its centre. Then, from Theorem 5.2 and Lemmas 6.9,7.1, it follows the next result.

Proposition 7.3. Let F[Q] be an alternative loop algebra from class M and letR(F[Q]) be its radical. Then algebra (R(F[Q]))] = F[G], G ⊆ Q, F[G] ∈ RA, isa nonassociative antisimple with respect to nonassociativity or, equivalently, it doesnot contain subalgebras that are nonassociative simple algebras and the quotient-algebra F[Q]/R(F[Q]) is a direct sum of Cayley-Dickson algebras over their centre.

As it was above mentioned, the nonassociative antisimple with respect to nonas-sociativity algebras are considered in Propositions 4.3, 4.4, 7.2 and Corollary 4.5.

Let now Q ∈ L be a nonassociative Moufang loop. According to Theorem 5.7,Q/S(Q) = T (Q), where T (Q) ∈ T, S is the radical class, T is the semisimple class forclass loop L. Further, as a rule in the theory of algebraic systems, in order to studythe loops of class L we will consider the loops of classes S and T separately.

To describe class T, we remind the description of nonassociative simple Moufangloops from [33]. Let M(F) denote the matrix Paige loop constructed, over the fieldF, as in [22]. That is, M(F) consists of vector matrices


M∗(F) =

(α1 α12α21 α2

), where α1, α2 ∈ F, α12, α21 ∈ F3,

det M∗ = α1α2 − (α12, α21) = 1, and where M∗ is identified with −M∗.The multiplication in M(F) coincides with the Zorn matrix multiplication

(α1 α12α21 α2

) (β1 β12β21 β2

)=

(α1β1 + (α12, β21) α1β12 + β2α12 − α21 × β21β1α21 + α2β21 + α12 × β12 α2β2 + (α21, β12)

),

where, for vectors γ = (γ1, γ2, γ3), δ = (δ1, δ2, δ3) ∈ A3, (γ, δ) = γ1δ1 + γ2δ2 + γ3δ3denotes their scalar product and γ×δ = (γ2δ3−γ3δ2, γ3δ1−γ1δ3, γ1δ2−γ2δ1) denotesthe cross vector product.

Let ∆ be a prime field and let P be its algebraic closure. In [33], it was proved thatonly and only the Paige loops M(F), where F is a Galois extension over ∆ in P are,up to an isomorphism, nonassociative simple Moufang loops. [33] also describes thefinite nonassociative simple Moufang loops and the set of generators, the group ofautomorphisms of nonassociative simple Moufang loops.

An analogue for loops of the Wedderburn Theorem (for finite-dimensional asso-ciative algebras), follows from Theorem 5.7, Lemmas 6.10, 7.1 and Proposition 7.2.

Theorem 7.4. Let ∆ be a prime field, P be its algebraic closure, and F be a Galoisextension over ∆ in P. Then the radical S(Q) of a Moufang loop Q is nonassociativeantisimple with respect to nonassociativity or, equivalently, it does not contain anysubloops that are nonassociative simple loops and quotient-loop Q/S(Q) is isomor-phic to the direct product of matrix Paige loops M(F).

Let G be a finite Moufang loop. Obviously, from the finiteness of G it followsthat for any subloop H of G there exists a normal subloop K of H such that thecomposition factor H/K is a simple loop. According to [14], a finite Moufang loop Git said to be a loop of group type if all composition factors of G are groups. It is clearthat the homomorphic image of loop of group type is a loop of group type and theproduct of two normal subloops of group type is again a loop of group type. Henceevery finite Moufang loop has a unique maximal normal subloop of group type ([14,Proposition 1]). We denote this maximal normal subloop of group type by Gr(G). Itis obvious that Gr(G/Gr(G)) = e. Hence Gr(G) is a radical of G. By [14] Gr(G) iscalled the group-type radical of G. Obviously, for any finite Moufang loop G, thedefinition of normal subloop Gr(G) is equivalent with the condition: Gr(G) does notcontain any subloops that are nonassociative simple loops, by Theorem 7.4. Hence,for finite loops, the group-type radical Gr coincides with radical S.

In the proof of the main result from [14] about the existence of quasi-p-Sylowsubloops in every finite Moufang loop, the following structural Theorem B is used es-sentially: every finite Moufang loop M contains uniquely determined normal subloops


Gr(M) and M0 such that Gr(M) ≤ M0, M/M0 is an elementary abelian 2-group,M0/Gr(M) is the direct product of simple Paige loops M(q) (where q may vary), thecomposition factors of Gr(M) are groups and Gr(M/Gr(M)) = 1.

The proof of Theorem B is based on the correspondence between Moufang loopsand groups with triality [7]. The proof of Theorem B is quite cumbersome and usesdeep results from finite groups. Moreover, the Theorem B in such a version does nothold true. For example, the case when M is a simple loop leads to a contradictionwith the condition that M/M0 is an elementary abelian 2-group. In reality, M/M0is the unitary group, i.e. M = M0. In such a case Theorem B is a particular caseof Theorem 7.4. Hence if the corresponding results from this paper are used in theproofs of the main results from [14], then these proofs become as it is shown below.

From Theorem 7.4, it follows that the loops from semisimple class T are welldescribed. However, unlike the class T, much less is known about the qualities andconstruction of loops from the radical class S. A new approach is suggested for thestudy of the loops in the class S (though some authors made some attempts earlier):

a) by using the one-to-one mapping between loops Q ∈ S and alternative loopalgebras F[Q] ∈ R, below indicated in Theorem 7.5;

b) by using the developed theory of alternative algebras, in particular, of the alge-bras with externally adjoined unit, of Zhevlakov radicals, of circle loops and others.

According to Lemma 7.1 and Proposition 7.2 the following statements are equiv-alent for a nonassociative Moufang loop Q ∈ L:

r1) Q ∈ S;r2) Q is an loop antisimple with respect to nonassociativity;r3) the loop Q does not have any subloops that are simple loops.

Then the opposite statementsnr1) G < S, i.e. G ∈ L\S;nr2) G is not antisimple with respect to nonassociativity loop;nr3) the loop G has subloops that are simple loops

hold for any nonassociative Moufang loop G ∈ L\S.From the definition of class of alternative loop algebras RA, definition of class of

loops S and Theorem 4.2, it follows that if a nonassociative Moufang loop Q satisfiesthe condition r1) then the loop Q can be embedded into the loop of invertible elementsU(F[Q]) of the alternative loop algebra F[Q]. On the other hand, [29] proves that ifa nonassociative Moufang loop G satisfies the condition nr2) then the loop Q is notimbedded into the loop of invertible elements U(A) for a suitable unital alternative F-algebra A, where F is an associative commutative ring with unit. As S

⋂(L\S) = ∅

then the main result of this paper follows from the above statements.Theorem 7.5. Any nonassociative Moufang loop Q that satisfies one of the equiv-

alent conditions r1) - r3) can be embedded into a loop of invertible elements U(F[Q])of alternative loop algebra F[Q]. The remaining loops of class of all nonassociativeMoufang loops L, i.e.the loops G ∈ L that satisfy one of the equivalent conditions


nr1) - nr3) cannot be embedded into a loop of invertible elements of any unital alter-native algebras.

From Corollary 6.10 and Theorem 7.5 the following Corollary follows.Corollary 7.6. Any commutative Moufang loop Q can be embedded into a loop of

invertible elements U(F[Q]) of the commutative alternative loop algebra F[Q].Corollary 7.7. Any finite Moufang p-loop Q can be embedded into a loop of

invertible elements U(F[Q]) of alternative loop algebra F[Q].Proof. According to [33], with up to an isomorphism, only Paige loops M(q) overa finite field Fq are finite simple Moufang loops. By [22] the order of M(Q) is(1/d)q3(q4 − 1), where d = gcd(2, q − 1), and is the product of two coprime numbersq3(q2 − 1) and (1/d)q3(q2 + 1). Q is a finite p-loop if the order of elements from Q isa power of p (for Moufang loops, this is equivalent to the condition that the order ofQ be a power of p). From here it follows that the finite p-loop satisfies the conditionr3). By Theorem 7.5 the Corollary 7.7 is proved.

From Theorem 7.5, Corollary 7.6 and (10) the following Corollary follows.Corollary 7.8. Any nonassociative Moufang loop Q that satisfy one of the equiva-

lent conditions r1) - r3) can be embedded into a circle loop (U?(F[Q]), ) of alterna-tive loop algebra F[Q]. The remaining loops of class of all nonassociative Moufangloops L, i.e.the loops G ∈ L that satisfy one of the equivalent conditions nr1) - nr3)cannot be embedded into circle loops of any unital alternative algebras.

Corollary 7.9. Any commutative Moufang loop Q can be embedded into a circleloop (U?(F[Q]), ) of the alternative loop algebra F[Q].

In [12], the circle loops are examined under condition that the underlying setsdefining the alternative algebra and the loop are identical. Under this supposition itis proved that for any prime p, there are no Moufang circle loops of order dividingp4 which are not associative, though non associative Moufang loops of order p4 exist[4]. It is also proved that no commutative Moufang which is not associative is thecircle loop of an alternative nil ring of index 2. But from Corollaries 7.6, 7.7 thefollowing Corollary follows.

Corollary 7.10. Any commutative Moufang loop Q can be embedded into a circleloop (U(F[Q], , 0) of the alternative loop algebra F[Q].

Corollary 7.11. Any finite Moufang p-loop Q can be embedded into a circle loopof invertible elements (U(F[Q], , 0) of the alternative loop algebra F[Q].

Now let us present some examples that were proved on the basis on the corre-spondence between commutative Moufang loops and loops of invertible elements ofcommutative alternative algebras (Corollary 6.6).

10. Bruck’s Theorem. This theorem is one of the profound results in the theory ofcommutative Moufang loops: a commutative Moufang loop with n (n ≥ 2) generators


is centrally nilpotent of class at most n−1 [2], Chap. VIII]. The proof of this assertionis very cumbersome; it is based on a complicated inductive process and uses severalhundred nonassociative identities. In [19, Chap. 1] Manin used group methods toprove a weaker assertion, namely that any finite commutative Moufang loop of period3 is centrally nilpotent. Although less calculative, his proof is by no means simple;it uses deep facts from finite group theory. The supremum of the central nilpotenceclass of a commutative Moufang loop with n generators is equal to n − 1 [17].

In [32], the relationship between commutative Moufang loops and alternative com-mutative algebras, i.e. the Corollary 7.6, (in [32] the proof is not very convincing) isused to prove (rather simply) that any finitely generated commutative Moufang loopis centrally nilpotent. In the proof, we use the fact that any alternative commutativenil-algebra of index 3 is locally nilpotent, only.

In [23], a Moufang loop E is called special, if it can be embedded in the loop U(A)of invertible elements of an alternative algebra A with unit. The Bruck Theorem isproved in a quite transparent manner (and an accurate appraisal is made) for specialcommutative Moufang loops. In the proof the assertion that the commutator idealof the multiplication algebra of a free commutative alternative algebra with n freegenerators is nilpotent of index n − 1 is transferred on such loops. Consequently,according to Corollary 7.5, in [23] the Bruck Theorem (and the accurate appraisal) isproved for any commutative Moufang loops.

20. Infinite independent system of identities. In [6] Slin’ko has formulatedthe question: if any variety of solvable alternative algebras would be finitely based.Umirbaev has got an affirmative answer to this question for alternative algebras over afield of characteristic , 2, 3 (see [35]), while Medvedev [20] gave a negative answerfor characteristic 2. The topic of work [28] is the transfer of infinite independentsystems of a commutative Moufang loop, constructed in [26] on solvable alternativecommutative algebra over a field of characteristic 3 (another example was constructedby Badeev, see [1]), provided that holds the Corollary 7.6. Consequently, the lastresult together with the former results, completes the statement of Slin’ko problemfor solvable alternative algebras.

30. The order of free commutative Moufang loops of exponent 3. Let Ln bethe free commutative Moufang loop on n generators of exponent 3 with unit e andlet | Ln | = 3δ(n). The Manin problem asks to calculate δ(n) [19]. One of the mainresults of paper [13] is that δ(3) = 4, δ(4) = 12, δ(5) = 49, δ(6) = 220, δ(7) = 1014or 1035 and δ(7) = 1014 if and only if Ln can be embedded into a loop of invertibleelements of a unital alternative commutative algebra. It is also proved that the freeloop Ln on n < 7 generators is embedded into a loop of invertible elements U(A) fora unital alternative commutative algebra A. Moreover, L7 may be embedded in U(A)if and only if the following identity is true for commutative Moufang loops

((((a, x, y), z, b), t, c), b, c)((((a, x, z), y, b), t, c), b, c)

((((a, x, t), y, b), z, c), b, c)−1((((a, x, b), y, z), t, c), b, c) (44)


((((a, x, c), y, z), t, b), b, c)((((a, x, b), y, c), z, t), b, c) = e.

Here (x, y, z) = (xy)z · (x(yz))−1. According to Corollary 7.6, from here it follows thatδ(7) = 1014 and the identity (44) is true for any commutative Moufang loop.

8. FINITE MOUFANG P-LOOPSLet Q be a loop with unit e. The set z ∈ Q | zx = xz, zx · y = z · xy, xz · y =

x · zy, xy · z = x ·yz ∀x, y ∈ Q is a subloop Z(Q) of Q, the centre. Z(Q) is an abeliangroup, and every subgroup of Z(Q) is a normal subloop of Q. If Z1(Q) = Z(Q),then the normal subloops Zi+1(Q) : Zi+1(Q)/Zi(Q) = Z(Q/Zi(Q)) are inductivelydetermined. A loop Q is called centrally nilpotent of class n, if its upper centralseries have the form

e ⊂ Z1(Q) ⊂ . . . ⊂ Zn−1 ⊂ Zn(Q) = Q.

If N is a normal subloop of Q, there is a unique smallest normal subloop M of Qsuch that N/M is part of the centre of Q/M, and we write M = [N,Q]. The lowercentral series of Q is defined by Q1 = Q,Qi+1 = [Qi,Q] (i ≥ 1). The loop Qis centrally nilpotent of class n if and only if its lower central series have the formQ ⊃ Q1 ⊃ . . . ⊃ Qn−1 ⊃ Qn = e [2].

The associator (x, y, z) and commutator (x, y) of elements x, y, z ∈ Q are definedby the equalities xy · z = (x · yz)(x, y, z) and xy = (yx)(x, y) for an arbitrary loop Q.The commutator-associator of weight n is defined inductively:

1) any associator (x, y, z) and any commutator (x, y), where x, y, z ∈ Q, are commutator-associator of the weight 1;

2) if a is a commutator-associator of weight i, then (a, x, y) or (a, x), where x, y ∈ Q,is a commutator-associator of the weight i + 1.

Lemma 8.1 [3]. The subloops Qi (i = 1, 2, . . .) of the lower central series of aMoufang loop Q are generated by all commutator-associators of weight i of Q.

If A is an F-algebra, then its n degree An is an F-module with a basis, consistingof products from any of its n elements with any bracket distribution. Algebra A iscalled nilpotent if An = (0) for a certain n.

Lemma 8.2. Let Q be a finite Moufang p-loop and F be a field of characteristicp. Then, the ideal ω[Q] of the alternative loop algebra F[Q] is nilpotent.

Proof. In accordance with Corollary 7.7 we assume that Q ⊆ F[Q]. By Theorem2.2 from [2, pag. 92] in a finite Moufang loop Q the order of any of its elementdivides the order of Q. Hence gk = e, where k = pn, for g ∈ Q. We have (e − g)k =

e −C1k g + . . . + (−1)iCi

kgi + ... + (−1)kgk. All binomial coefficients Cik can be divided

by p, therefore (e−g)k = e + (−1)kgk. If p = 2, then (e−g)k = e + gk = e + e = 2 = 0,because F is a field of characteristic 2. But if p > 2, then (e−g)k = e−gk = e−e = 0.Then we can apply the following statement to algebra ωQ: any alternative F-algebra,


generated as an F-module by a finite set of nilpotent elements, is nilpotent [36, pages144, 408]. Consequently the augmentation ideal ω[Q] is nilpotent, as required.

Let now A be an alternative F-algebra with unit e and B be a subalgebra from A,satisfying the law

xm = 0. (45)

Then e − B = e − b|b ∈ B will be a loop and (e − b)−1 = e + b + . . . + bm−1. Weremind that inscription (a, b, c) = ab · c− a · bc, (a, b) = ab− ba means the associatorand commutator in algebra, but [a, b, c] = (a · bc)−1 · (ab · c), [a, b] = a−1b−1 · ab areassociator and commutator in IP-loops.

Lemma 8.3. Let A be an alternative algebra with unit e and B its subalgebra,satisfying the law (45). Then for u, v,w ∈ B [e − u, e − v, e − w] = e − ((e + w + . . . +wm−1)(e + v + . . .+ vm−1) · (e + u + . . .+ um−1))(u, v,w), [u, v] = (e + u + . . .+ um−1)(e +

v + . . . + vm−1)(u, v).

Proof. We denote e−u = a, e− v = b, e−w = c. Then we have [e−u, e− v, e−w] =

(a · bc)−1(ab · c) = (a · bc)−1(ab · c) − (a · bc)−1(a · bc) + e = e + (a · bc)−1(a, b, c) =

e + (((e−w)−1 · (e− v)−1)(e− u)−1)(e− u, e− v, e−w) = e− ((e−w)−1(e− v)−1 · (e−u)−1)(u, v,w) = e− ((e+w+ . . .+wm−1)(e+v+ . . .+vm−1) · (e+u+ . . .+um−1))(u, v,w).The second equality is analogously proved.

Lemma 8.4. Let Q be a Moufang loop and let the ideal ω[Q] of the alternativeloop algebra F[Q] be nilpotent. Then the loop Q is centrally nilpotent.

Proof. It follows, from the definition of ideal ω[Q], that Q = Q0 ⊆ e − ω[Q] =

e − ω[Q]2·0+1. We suppose that Qi−1 ⊆ e − (ω[Q])2·i+1. Then it follows from Lemma8.4 that Qi ⊆ e − (ω[Q])2·(i+1)+1. The algebra ω[Q] is nilpotent; we suppose that(ω[Q])2·k+3 = (0). Then Qk = e. Hence the loop Q is centrally nilpotent, as re-quired.

The following Proposition follows from Lemmas 8.2 and 8.5.

Proposition 8.5. Any finite Moufang p-loop is centrally nilpotent.Theorem 8.6. Let a Moufang loop Q belong to radical class S. Then the following

statements are equivalent:1) the augmentation ideal ω[Q] of alternative loop algebra F[Q] is nilpotent;2) Q is a p-loop and the field F has a characteristic p;3) ωQ is artinian.

Proof. Let the algebra ω[Q] be nilpotent, for example, of index r and let 0 , x ∈(ω[Q])r−1. By (31) the element x will be written in form x = α1g1 + · · ·+αkgk, where


αi ∈ F, gi ∈ Q. We suppose that gi , g j if i , j. As Q ∈ S then from definition of theclass S it follows that F[Q] ∈ R and by item 4) of Proposition 4.3, we have x(e−u) = 0for any u ∈ Q. Hence α1g1 + · · · + αkgk = α1g1u + · · · + αkgku in the alternative loopalgebra F[Q]. But F[Q] = FQ/I. Thus α1g1 + · · ·+αkgk = α1g1u+ · · ·+αkgku(modI)in the loop algebra FQ. We suppose that the loop Q is infinite. Then, there exist u ∈ Qsuch that α1g1u < α1g1, . . . , αkgk. By the definition the loop algebra FQ is a free F-module with the basis g|g ∈ Q. Then α1g1u ∈ I. From here it follows that I = FQ.But this contradicts Theorem 7.5. Hence the loop Q is finite.

By [2, pag. 92], in the finite Moufang loop Q the order of any of its elementdivides the order of Q. If e , g ∈ Q is an element of simple order p then, byitem 6) of Proposition 4.3, a = p − (e + g + g2 + . . . + gp−1) ∈ ω[Q]. We have(e + g + g2 + . . . + gp−1)(e + g + g2 + . . . + gp−1) = p(e + g + g2 + . . . + gp−1). Thena2 = p2 − p(e + g + g2 + . . . + gp−1) and, by induction of n, it is easy to show thata2n

= p2n − p2n−1(e + g + g2 + . . . + gp−1). We choose an n such that (ω[Q])2n= (0).

Then a2n= 0. We suppose that F does not have the characteristic p. It follows, from

the equalities 0 = p2n− p2n−1(e+g+g2 + . . .+gp−1) = p24−1(p−(e+g+g2 + . . .+gp−1),that 0 = p − (e + g + g2 + . . . + gp−1), p = e + g + g2 + . . . + gp−1, i.e. pg = p, g = e.We have obtained a contradiction as g , e. Consequently, F has the characteristic pand Q is p-loop. Consequently, 1)⇒ 2).

Conversely, let the field F have a characteristic p and let Q be a finite p-loop. ByProposition 8.6 it will be centrally nilpotent loop. Let H =< a > be a cyclic group oforder p from the center Z(Q) of Q. We will prove that the product

(e − ai1)(e − ai2) . . . (e − aim)

equals zero, if m ≥ p. Indeed, if we use the identity e − xy = (e − x) + (e − y) − (e −x)(e − y), then the last product is the sum of the factors of type (e − a)k, k ≥ p. Then

(e − a)p = e −C1pa + C2

pa2 − . . . ± ap.

As all binomial coefficients Cip divide by p, then they are zero in the field F. Conse-

quently, (ω[H])p = (0), where ω[H] means the augmentation ideal of the alternativeloop algebra F[H]. Let µH means the ideal of the alternative loop algebra F[Q],generated by the set e− h|h ∈ H. As the subloop H belongs to the center Z(Q), thenthe equality (ω[H])p = (0) entails the equality (µH)p = (0).

We will prove the nilpotency of augmentation idealω[Q] via induction on the orderof the loop Q. As H ⊆ Z(Q), then H is normal in Q and H induces the homomorphismF[Q] → F[Q/H]. By item 4) of Proposition 4.4, we have ω[Q]/µH ω[Q/H]. Byinductive hypotheses, the augmentation ideal ω[Q/H] is nilpotent, for example, ofindex k. Then (ω[Q])k ⊆ µH and (ω[Q])kp ⊆ (µH)p = (0). Consequently, the idealω[Q] is nilpotent. Hence 2)⇒ 1). Conversely, let us prove 1)⇔ 2).

The equivalence of 1) and 2) follows from Proposition 4.3. Now we suppose that2) holds. Let g1, g2, . . . , gk be all elements of Q. By item 5) of Lemma 6 ??? ωQ is a


finite sum of modules Fui, where ui = e − gi. The field F has a characteristic p andgpn

= e for some n. Then upn

i = (e − gi)pn= 0. Hence Fui satisfies the minimum

condition for submodules. It easily follows from here that ωQ is Artinian, i.e theitem 3) holds. Furthermore, it is known [6] that the Zhevlakov radical of an Artinianalternative algebra is nilpotent. By item 7) of Proposition 4.3, J(F[Q]) = ω[Q]. Thusfrom 3) it follows 1). This completes the proof of Theorem 8.7.

References.

[1] Badeev A. V. On the Specht property of varieties of commutative alternative alebras over a fieldof characteristic 3 and of commutative Moufang loops, Sibirsk. Mat. Zb., 41(2000), 1252 – 1268(Russian).

[2] Bruck R. H. A survey of binary systems, Berlin-Gottingen-Heidelberg, Springer-Verlag, 1958.

[3] Covalschi A. V., Sandu N. I. On the generalized nilpotent and generalized solvable loops I,ROMAI Jurnal, 7, 1(2011), 39 – 63.

[4] Chein O. Moufang loops of small order, Trans. Amer. Math. Soc., 188(1974), 31 – 51.

[5] Chein O., Pflugfelder H. O., Smith J. D. H. Quasigroups and Loops: Theory and applications,Berlin, Helderman Verlag, 1990.

[6] The Dniester Notebook: Unsolved problems in the theory of and modules, Third edition; Akad.Nauk SSSR Sibirsk Otdel., Inst. Mat., Novosibirsk 1982 (Russian).

[7] Doro S. Simple Moufang loops, Math. Proc. Camb. Phil. Soc., 83(1978), 377 – 392.

[8] Foster A. The indempotent elements of a commutative ring form a Boolean algebra, Duke Math.J., 12(1945), 143 – 152.

[9] Glauberman G. On loops of odd order II, J. Algebra, 8(1968), 393 – 414.

[10] Glauberman G., Wright C. R. R. Nilpotence of finite Moufang 2-loops, J. Algebra, 8(1968), 415– 417.

[11] Goodaire E. C. A brief history of loop rings. 15th Brasilian School of Algebra (Canela, 1998,)Mat.Contemp., 16(1999), 93–109.

[12] Goodaire E. C. Circle loops of radical alternative rings, Algebras, groups and geometries,4(1987), 461 – 474.

[13] Grishkov A. N, Shestakov I. P. Commutative Moufang loops and alternative algebras, J. Algebra,333, 1(2011), 1 – 13; arXiv:0811.3787v1.

[14] Grishkov A. N, Zavarnitsine A. V. Sylow’s theorem for Moufang loops, J. Algebra, 321, 7,1(2009), 1813 – 1825; arXiv:0709.2696v1.

[15] Jacobson N. Sructure of rings, Amer. Math. Soc. Colloq. Publ., XXXVII, 1956.

[16] Liebeck M. W. The classification of finite Moufang loops, Math. Proc. Camb. Phil. Soc.,102(1987), 33 – 47.

[17] Malbos J.-P. Sur la classe de nilpotence des boucles commutative de Moufang et des espacesmediaus, C.R. Acad. Sci. Paris Ser. A, 287(1980), 691 – 693.

[18] Mal’cev A. I. About the decomposition of algebra into a direct sum of radical and semisimplesubalgebra, Dokl. AN SSSR, 36(1942), 46 – 50 (Russian).

[19] Manin Yu. I. Cubic forms, Amsterdam: North-Holland, 1979.

[20] Medvedev Yu. A. Example of a variety of alternative algebras over a field of characteristic two,Algebra i Logika, 19(1980), 300 – 313 (Russian).


[21] Moufang R. Zur Structur von Alternativekorpern, Math. Ann., 110(1935), 416 – 430.

[22] Paige L. J. A class of simple Moufang loops, Proc. Amer. Math. Soc., 7(1956), 471 – 482.

[23] Pchelintsev S. V. Structure of finitely generated commutative alternative algebras and specialMoufang loops, Matem. zametki, 80(2006), 413 – 420 (Russian).

[24] Perlis S. A characterization of the radical of an algebra, Bull. Amer. Math. Soc., 48(1942), 128– 132.

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[26] Sandu N., Infinite irreducible systems of identities of commutative Moufang loops and distributiveSteiner’s quasigroups, Izv. Ak. Nauk (USSR), ser. mat., 51, 6(1987), 171 – 189 (Russian).

[27] Sandu N. I. On the Bruck-Slaby Theorem for Commutative Moufang Loops, Matem. zametki, 66,2(1999), 275 – 281(Russian).

[28] Sandu N. I. Infinite independent systems of identities of alternative commutative algebra overa field of characteristic three, Discussiones Mathematicae. General Algebra and Applications,24(2004), 5 – 30.

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ON CLASSES OF GENERALIZEDQUASICONFORMAL MAPPINGS

ROMAI J., v.8, no.1(2012), 155–163

Victoria StanciuUniversity “Politehnica” of Bucharest, [email protected]

Abstract We present our recent results which generalize the normality and compactness proper-ties of families of K-quasiconformal (K − QC) and K-quasiregular (K − QR) mappingson Riemann surfaces to the cases: a) of mappings whose distortion is dominated by aBMOloc - or by a FMOloc - function, b) when instead of QC (or QR) mappings we workwith ring FMOloc - homeomorphisms (or with ring FMOloc - QR mappings), c) whenthe mappings are of finite length distortion (FLD) homeomorphisms.

Keywords: quasiconformal (QC), quasiregular (QR), BMOloc - QC, BMOloc - QR, FMOloc, FMD,FLD - homeomorphism, ring Q(p) - homeomorphism, Riemann surface.

1. INTRODUCTIONConformal mappings and analytic functions of one complex variable were first

generalized in the work of H. Grotzsch, M.A. Lavrentiev, L.V. Ahlfors, O. Teichmullerand others and then extended to the spaces Rn, n ≥ 2 in the theory of quasiconformal(QC) and quasiregular (QR) mappings. The subject is treated in many monographsby O. Lehto and K.I. Virtanen [13], L.V. Ahlfors [1], C. Andreian Cazacu [2] forn = 2, J. Vaisala [35], P. Caraman [9], Yu.G. Reshetnyak [17], M. Vuorinen [36], S.Rickman [18] and others for n ≥ 3. Later on, the dilatation of the QC mappings wasdominated by a given measurable function Q, which can be BMO, BMOloc (boundedmean oscillation ), FMO, FMOloc (finite mean oscillation), L1

loc etc. A list of theworks with this subject can be founded in [15] and [26]. We present in a generalframe the results obtained with C. Andreian Cazacu [6,7,8] and alone [27-32] aboutconvergence in the 2 - dimensional case.

In the following, D and D′ will designate domains in C and g : D → D′ a sense-preserving ACL (absolutely continous on lines) mapping.

Let µ : D → C the complex dilatation, i.e. a measurable function with |µ(z)| ≤ 1a.e., g solution of the Beltrami equation

∂g∂z

= µ(z)∂g∂z

(1)

155

156 Victoria Stanciu

andK(z, g) =

1 + |µ(z)|1 − |µ(z)| , (2)

is the (maximal) dilatation of function g.

The classical geometric definition of K - QC mappings due to H. Grotzsch [1]for n = 2, where g is a homeomorphism, is

M(Γ)/K ≤ M(gΓ) ≤ KM(Γ), (3)

for every path family Γ in D. The (conformal) modulus of Γ is

M(Γ) = infρ∈admΓ

"

Cρ2(z)dm(z), (4)

with dm(z) the Lebesgue measure in C and ρ : C → [0,∞] a Borel function calledadmissible for Γ (ρ ∈ admΓ) with the property∫

γρds ≥ 1 (5)

for each locally rectifiable γ ∈ Γ. M(Γ) being interpreted as +∞ if admΓ = ∅. HereK(z, g) ≤ K a.e. and K ≥ 1 is a constant.

By Theorem 1.1, p. 24 in [13], or Theorem 34.3 in [35], a homeomorphism g isQC in the geometric sense if and only if

M(gΓ) ≤ KM(Γ), (6)

for some K ∈ [1,∞) and for every path family Γ in D, that means that it is sufficientthat

supM(gΓ)M(Γ)

< ∞, (7)

where the supremum is taken over all path families Γ in D for which M(Γ) and M(gΓ)are not simultaneously 0 or∞.

Taking into account the relation between (3) and (6), O. Martio gives a naturalextension of the definition of QC mappings. (see [15], chap. 4, p. 81, [13], p. 221,or [26], p. 551 ):

Let Q : D → [1,∞] be a measurable function. We say that a homeomorphismg : D→ C = C ∪ ∞ is a Q(z) - homeomorphism, if

M(gΓ) ≤"

DQ(z)ρ2(z)dm(z) (8)

for Γ family of paths in D and ρ ∈ admΓ. This concept is related in a natural wayto the theory of moduli with weights, see [3,4]. We deal with some subclasses ofQ(z) - homeomorphisms, which are introduced in [12,20 - 25], K(z, g) ≤ Q(z), Q inBMOloc or FMOloc, and we extend some of convergence results obtained in the planeto Riemann surfaces.

On classes of generalized quasiconformal mappings 157

2. DEFINITIONSA function Q : D→ R is called of bounded mean oscillation in D, Q ∈ BMO(D),

if Q ∈ L1loc(D) and

‖Q‖∗ = supB

1|B|

"

B|Q(z) − QB|dxdy < ∞,

where the supremum is taken over all disks B in D,

QB =1|B|

"

BQ(z)dxdy,

and |A| denotes the Lebesgue measure of A ⊂ C.

A function Q : D → R is called of finite mean oscillation at a point z0 ∈ D, if Qis integrable in a neighborhood of the point z0 and

dQ(z0) = limε→0

1|D(z0, ε)|

"

D(z0,ε)|Q(z) − Qε(z0)|dxdy < ∞,

whereQε =

1|D(z0, ε)|

"

D(z0,ε)Q(z)dxdy,

is the mean value of the function Q(z) over

D(z0, ε) = z ∈ C : |z − z0| < ε.We call dQ(z0) the dispersion of the function Q at z0. We say that an L1

loc(D)function Q : D → R is of finite mean oscillation in the domain D, abbr. Q ∈FMO(D) or simply Q ∈ FMO, if Q has a finite dispersion at every point z ∈ D. Thefunction Q ∈ FMOloc(D) if Q |U∈ FMO(U) for every relatively compact subdomainU of D.

Given a domain D and two sets E and F in C, Γ(E, F,D) denotes the family of allpaths γ : [a, b] → C which join E and F in D, i.e. γ(a) ∈ E, γ(b) ∈ F and γ(t) ∈ Dfor a < t < b. Let R = R(C1,C2) be a ring domain, i.e. a doubly connected domainin C with the connected components C1 and C2 of C \ R. The capacity of R can bedefined by

capR(C1,C2) = M(Γ(C1,C2,R)).

Given a domain D ⊂ C, z0 ∈ D, ε0 > 0 with D(z0, ε0) ⊂ D, and Q : D(z0, ε0) →[1,∞], we say that a homeomorphism f : D → f (D) ⊂ C is a ring Q - homeomor-phism at a point z0 ∈ D if

M(Γ( fC1, fC2, f R)) ≤"

RQ(z)η2(|z − z0|)dxdy


for all circles C1 = z ∈ C; |z − z0| = ε1 and C2 = z ∈ C; |z − z0| = ε2 in D with 0 <ε1 < ε2 < ε0 = dist(z0, ∂D) and for every measurable function η : (ε1, ε2) → [0,∞]such that ∫ ε2

ε1

η(r)dr = 1.

For x ∈ E ⊂ C and a mapping ϕ : E → C, we set

L(x, ϕ) = limy→x

supy∈E

|ϕ(y) − ϕ(x)||y − x| ,

andl(x, ϕ) = lim

y→xinfy∈E

|ϕ(y) − ϕ(x)||y − x| .

We say that a continuous mapping g : D → C is said to be of finite metricdistortion, abbr. g ∈ FMD, if g has the Lusin(N) - property and

0 < l(x, g) ≤ L(x, g) < ∞ a.e.

Recall that a mapping g : X → Y between measurable spaces (X,Σ, µ) and(X′,Σ′, µ′) is said to have the (N) - property if µ′(g(S )) = 0 whenever µ(S ) = 0.Similarly, g has the (N−1) - property if µ(S ) = 0 whenever µ′(g(S )) = 0.

We say that a property ℘ holds for almost every path (a.e. path) γ in a family Γ ifthe subfamily of all path in Γ for which ℘ fails has modulus zero.

If γ : ∆ → C, where ∆ is an interval in R, is a locally rectifiable path, then thereis the unique increasing length function lγ of ∆ onto a length interval ∆γ ⊂ R with aprescribed normalization lγ(t0) = 0 ∈ ∆γ, t0 ∈ ∆, such that lγ(t) is equal to the lengthof the subpath γ | [t0, t] of γ if t > t0, t ∈ ∆, and lγ(t) is equal to −l(γ | [t0, t]) if t < t0,t ∈ ∆. Let g : γ(∆)→ C be a continuous mapping, and suppose that the path γ = gγis also locally rectifiable. Then there is a unique increasing function Lγ,g : ∆γ → ∆γ,such that

Lγ,g(lγ(t)) = lγ f or all t ∈ ∆.

We say that a mapping g : D → C has the (L) - property if the following twoconditions hold:

(L1) for a.e. path γ in D, γ = g γ is locally rectifiable and the function Lγ,g hasthe (N) - property;

(L2) for a.e. path γ in g(D), each lifting γ of γ is locally rectifiable and the functionLγ,g has the (N−1) - property.

A path γ in D is a lifting of a path γ in C under g : D → C if γ = g γ. Note thatthe condition (L2) applies only to path γ which have a (maximal) lifting.

We say that a mapping g : D → C is of finite length distortion abbr. g ∈ FLD, ifg is FMD and has the (L) - property.


3. CONVERGENCEStarting with the classical theorems on normality and compactness of meromor-

phic functions, which have been extended to QC or QR mappings for instance by O.Lehto and K. I. Virtanen [13], II, 5, p. 71, many authors generalized this topic onother classes of mappings, e.g. G. David [11], P. Tukia [34], V. Ryazanov [19], V.Ryazanov, U. Srebro and E. Yakubov [20-25], [14] or [10].

Let Y and Y ′ be homeomorphic Riemann surfaces, (Y ,Π,Y) and(Y ′,Π′, Y ′) their universal coverings, where Π : Y → Y and Π′ : Y ′ → Y ′. Here Yand Y ′ are either C, ∆ - the unit disk or C with the corresponding metric: Euclidean,hyperbolic or spherical. The metrics on Y and Y ′ induce by Π and Π′ the metricsof Y and Y ′. The convergence is always taken with respect to these metrics. LetQ : Y → R be a function defined on Y such that the composition Q = QΠ : Y → R,be a BMOloc or FMOloc function. Then, we say that a mapping g : Y → Y ′ is a Q(p)- QC (or QR) mapping if the lifting g : Y → Y ′ is a Q(z) - QC (or QR) mapping(cf. [16], p. 9). In this paper, homeomorphism (embedding) means homeomorphismonto (into, respectively) and QC - mapping is a general name for both cases.

K(p, g) ≤ Q(p) (9)

holds if and only if K(z, g) ≤ Q(z), Π(z) = p.

Let be p0 ∈ Y and p′0 ∈ Y ′ two arbitrary but fixed points of Y and Y ′ respectively.We consider the family G of mappings g : Y → Y ′ normalized by the condition:

g(p0) = p′0. (10)

Then G is normal if every sequence of mappings in G contains a locally uniformly(l.u.) convergent subsequence. G is compact if it contains the limit function of everyl.u. convergent sequence in G.

Theorem 3.1. (i) If Y ′ is non conformally equivalent to either C or C, then G is nor-mal if1. Q ∈ BMOloc(Y) and g from G is a Q(p) - QC mapping;2. Q ∈ FMOloc(Y) and g from G is a ring Q(p) - homeomorphism;3. g ∈ G is a FLD - homeomorphism, Q(p) ∈ FMO(Y), with (9).

(ii) if Y is non conformally equivalent to C or ∆, and gn is a sequence in G, whichl. uniformly converges to g0, then g0 is a:1. Q(p) - QC embedding, if gn ∈ G is a Q(p) - QC mapping Q ∈ BMOloc(Y);2. ring Q(p) - embedding, if gn ∈ G is a ring Q(p) - homeomorphism Q ∈ FMOloc(Y);3. FLD - embedding, if gn ∈ G is a FLD - homeomorphism Q ∈ FMO(Y), with (9).


(iii) G is closed in the following cases:1. if Y ′ is compact, even for C, g ∈ G is a Q(p) - QC mapping Q ∈ BMOloc(Y) (orring Q(p) - homeomorphism Q ∈ FMOloc(Y));2. if Y ′ is non conformally equivalent to C, g ∈ G is a Q(p) - QC mapping Q ∈BMO(Y) [6];3. if Y , C or ∆ and there exists a function C ∈ BMOloc(Y ′) (FMOloc(Y ′) respectively)such that g−1 is a C(p′) - QC mapping (ring C(p′) - homeomorphism) for every g ∈ G.

These normality results have been extended [28, 30, 32] to the families G′ andG′′ of BMOloc(Y) Q(p) - QC mappings (FMOloc(Y) ring Q(p) - homeomorphisms,FLD homeomorphisms) g : Y → Y ′ which map a given compact subset M ⊂ Y into,respectively onto another given compact subset M′ ⊂ Y ′.

Remark 1. The following example shows that the conclusion of Theorem 3.1, doesnot hold in the case excluded by its hypotheses.

Example 1. Take D =

z ∈ C/|z| < 1

ne−

α4

, n > 2, fn(z) =

1n

exp−

√α

2log

1|z|

z|z| . Using a formula from [13], p. 220 for the complex di-

latation, we deduce K(z, fn) =

√8α

log1|z| . Choosing Q(z) =

√8α

log1|z| ∈ BMO(C)

[16], Corollary 2 to Lemma 3, we can infer that fn is a Q(z) - QC embedding. The

sequence gn(z) =

fn(z), z ∈ D1n

z|z|, z ∈ C \ Dis of Q(z) - QC mappings, where Q(z) =

maxQ(z), 2 ∈ BMO(C) by [16], p.2. This shows that gn converges l.u. in C to aconstant g0 ≡ 0, hence G is not closed.

We present now normality criteria for mappings of QR type.

By using [22], Corollary 5.6, p.15, a normality criterion was proved for BMOloc -QR (or for ring FMOloc - QR) mappings in [29] and [31].

The BMO - QR solutions of (1) are obtained by Stoilow’s factorization theorem,which says that every open and discrete mapping g : D→ C admits a representation

g = h ϕ (11)

where ϕ is an embedding and h a meromorphic function in ϕ(D) ([33], V, 5, p.120).

Theorem 3.1 does not hold for the class of Q(p) - QR mappings, as follows fromthe


Example 2. Consider the sequence gn(z) = n(

1log |z|e

i arg z +1α

)−1, α > 2 of BMO

- QR mappings defined on the set D = z ∈ C/|z| < e−1,−(π/3) < arg z < (2π/3).The preimage of −1 is e−α and the limit mapping is g0(z) =

−1, z = e−α∞, z , e−α .

Theorem 3.2. Let Y and Y ′ be two homeomorphic Riemann surfaces, Y ′ is not con-formally equivalent to either C or C, z j ∈ Y, ζ j ∈ Y ′, j = 0, 1, z0 , z1, ζ0 , ζ1 andQ ∈ BMOloc(Y) (Q ∈ FMOloc(Y)). If Φ is a family of Q(z) - QR (ring Q(z) - QR)mappings f : Y → Y ′ such that f −1(ζ j) = z j, j = 0, 1, then Φ is normal.

Remark 2. Theorem 3.2 is not true if either (i) Y ′ = C or (ii) Y ′ = C, as followsfrom

Example 3. (i) If fn : C → C, fn(z) = nz then fn are Q(z) - QR maps such that

f −1n (0) = 0, f −1

n (∞) = ∞, but f0(z) =

0, z = 0∞, z , 0 hence f0(z) being discontinuous,

the family of Q(z) - QR maps from C to C which having 0 and ∞ as fixed points, isnot normal.(ii) Take fn : C → C, fn(z) = zen(1−z), then f −1

n (0) = 0, f −1n (1) = 1 yielding f0(z) =

1, z = 1∞, z , 1 .

The proof method is to lift the problem to the universal coverings, obtain the resultsthere, and then factorize, using [5], Proposition 1 and 2, p.173.

4. CONCLUSIONSTheorems 3.1 and 3.2 are original contributions and opening prospects of general-

izations to wider classes, e.g. to strong ring Q(z) - homeomorphisms or super Q(z) -homeomorphisms.


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STRONG CONVERGENCE FORASYMPTOTICALLY GENERALIZEDΦ−HEMICONTRACTIVE MAPPINGS

ROMAI J., v.8, no.1(2012), 165–171

Balwant Singh ThakurSchool of Studies in Mathematics, Pt.Ravishankar Shukla University, Raipur, 492010, [email protected]

Abstract The purpose of this work is to prove a strong convergence theorem for pair of asymptoti-cally Φ−hemicontractive, nearly uniformly L−Lipschitzian mappings in Banach spaces.The main result of this paper is improvement and generalization of well-known corre-sponding results.

Keywords: asymptotically generalized Φ−hemicontractive mapping, generalized uniformly Lipschi-tizian mapping, nearly uniformly Lipschitzian mapping.2010 MSC: 47H09, 47H10.

1. INTRODUCTIONLet E be a real Banach space, E∗ be its dual space, K a nonempty closed convex

subset of E and J : E → 2E∗ the normalized duality mapping defined by

Jx = f ∈ E∗ : 〈x, f 〉 = ‖x‖2 = ‖ f 2‖, ‖ f ‖ = ‖x‖, for all x ∈ E,

where 〈·, ·〉 denote the duality pairing between E and E∗. The single-valued normal-ized duality mapping is denoted by j.Recall that a mapping T : K → K is said to be

(1) Lipschitzian if there exists a constant L > 0 such that

‖T x − Ty‖ ≤ L ‖x − y‖ , for all x, y ∈ K ,

(2) uniformly L−Lipschitzian if for each n ∈ N there exists a constant L > 0 suchthat

‖T nx − T ny‖ ≤ L ‖x − y‖ , for all x, y ∈ K ,

(3) nearly L−Lipschitzian [10] with respect to an if for each n ∈ N there exists aconstant L > 0 such that

‖T nx − T ny‖ ≤ L (‖x − y‖ + an)

for all x, y ∈ K, where an is a fixed sequence in [0,∞) with an → 0 as n→ ∞,

165

166 Balwant Singh Thakur

(4) asymptotically generalized Φ− hemicontractive if F(T ) = x ∈ K : x = T x ,∅ and for each n ∈ N, x ∈ K and x∗ ∈ F(T ), there exists constant kn ≥ 1with limn→∞ kn = 1 and strictly increasing function Φ : [0,∞) → [0,∞) withΦ(0) = 0 and j(x − x∗) ∈ J(x − x∗) satisfying

〈T nx − x∗, j(x − x∗)〉 ≤ kn‖x − x∗‖2 − Φ(‖x − x∗‖) .

It is important to note that Lischitzian mappings are always continuous but nearlyLipschtizian mapping need not be continuous.Example 1[7]. Let E = R and T : K → K be defined by

T x =

x2 , if x ∈ [0, 1);0, if x = 1 .

Then, T is a discontinuous mapping which is not Lipschitzian, but it is nearly uni-formly 1

2 Lipschitizian with sequence 12n .

Recently, Chang et al. [3] proved the following result for strong convergence ofcommon fixed point of pair of asymptotically generalized Φ−hemicontractive map-pings which are uniformly L−Lipschitzian :

Theorem 1.1. Let E be a real Banach space, K be a nonempty closed convex subset ofE, Ti : K → K, i = 1, 2 be two uniformly Li−Lipschitzian asymptotically generalizedΦ− hemicontractive mapping with F(T1) ∩ F(T2) , ∅, where F(Ti) is the set offixed points of Ti in K and x∗ is a point in F(T1) ∩ F(T2). Let αn and βn betwo sequences in [0, 1] satisfying the conditions :

∑∞n=1 αn = ∞,

∑∞n=1 α

2n < ∞,∑∞

n=1 βn < ∞,∑∞

n=1 αn(kn − 1) < ∞. Then for any x1 ∈ K, the sequence xn definedby

xn+1 = (1 − αn)xn + αnT n1 yn

yn = (1 − βn)xn + βnT n2 xn

converges strongly to x∗.

More recently, Kim et al. [7] has improved the results of Chang [1], Chidume etal. [4, 5], Gu [6] and proved the following result for convergence of asymptoticallygeneralized Φ−hemicontractive mapping, which is nearly uniformly L− Lipschitzian:

Theorem 1.2. Let E be a real Banach space, K be a nonempty closed convex subsetof E, T : K → K a nearly uniformly L−Lipschitzian mapping with sequence an andasymptotically generalized Φ− hemicontractive mapping with F(T ) , ∅. Let αnbe a sequences in [0, 1] satisfying the conditions :

anαn

is bounded,

∑∞n=1 αn = ∞,∑∞

n=1 α2n < ∞ and

∑∞n=1 αn(kn − 1) < ∞. Then the sequence xn in K generated from

arbitrary x1 ∈ K by

xn+1 = (1 − αn)xn + αnT nxn, n ∈ N

Strong convergence for asymptotically generalized Φ−hemicontractive mappings 167

converges strongly to a unique fixed point of T .

From the above theorems, the following questions arise:

(1) Is it possible to weaken the necessity of uniformly Lipschitz continuity of T inTheorem 1.1?

(2) Is it possible to generalize the Theorem 1.2 for the pair of mappings ?

(3) Is it possible to improve the Theorem 1.2 by taking less restrictive setting i.e.is it possible to drop some control condition of Theorem 1.2 ?

The purpose of this paper is to provide affirmative answers for the above questions.

2. MAIN RESULTS:Following lemmas will be needed for the proof of our main result:

Lemma 1. [2]. Let E be a real Banach space and J : E → 2E be the normalizedduality mapping. Then, for any x, y ∈ E

‖x + y‖2 ≤ ‖x‖2 + 2〈y, j(x + y)〉,for all j(x + y) ∈ J(x + y).

Lemma 2. [8] Let θn be a sequence of nonnegative real numbers and λn be a realsequence satisfying the following conditions :

0 ≤ λn ≤ 1 ,∞∑

n=1

λn = ∞ .

If there exists a strictly increasing function φ : [0,∞)→ [0,∞) such that

θ2n+1 ≤ θ2

n − λnφ(θn+1) + σn ,

for all n ≥ n0, where n0 is some nonnegative integer and σn is a sequence ofnonnegative numbers such that σn = o(λn). Then θn → 0 as n→ ∞.

Lemma 3. [7] Let δn, βn, γn and σn be four sequences of nonnegative realnumbers such that

δ2n+1 ≤ (1 + βn)δ2

n + γn(δn + σn)2

for all n ∈ N. If∑∞

n=1 βn < ∞,∑∞

n=1 γn < ∞ and σn is bounded, then limn→∞ δnexists.

Now we state and prove the main result of this paper :

Theorem 2.1. Let E be a real Banach space, K be a nonempty closed convex subsetof E and Ti : K → K, i = 1, 2 be two asymptotically generalized Φ−hemicontractive,


nearly uniformly Li−Lipschitzian mappings with sequence an and F(T1) ∩ F(T2) ,∅, where F(Ti) is the set of fixed points of Ti in K. Let αn and βn be two sequencesin [0, 1] satisfying the following conditions :

(i)∑∞

n=1 αn = ∞;

(ii)∑∞

n=1 α2n < ∞;

(iii)∑∞

n=1 βn < ∞;

(iv)∑∞

n=1 αn(kn − 1) < ∞ .

Let xn be a sequence in K generated from arbitrary x1 ∈ K by

xn+1 = (1 − αn)xn + αnT n1 yn

yn = (1 − βn)xn + βnT n2 xn , n ∈ N . (1)

Then xn converges strongly to x∗ ∈ F(T1) ∩ F(T2).

Proof. Let x∗ ∈ F(T1) ∩ F(T2) and set L = maxL1, L2, then

‖xn+1 − x∗‖2 = ‖(1 − αn)(xn − x∗) + αn(T n1 yn − x∗)‖2

≤ (1 − αn)2‖xn − x∗‖2 + 2αn〈T n1 yn − x∗, j(xn+1 − x∗)〉

≤ (1 − αn)2‖xn − x∗‖2 + 2αn〈T n1 xn+1 − x∗, j(xn+1 − x∗)〉

+ 2αn〈T n1 yn − T n

1 xn+1, j(xn+1 − x∗)〉≤ (1 − αn)2‖xn − x∗‖2 + 2αn

kn‖xn+1 − x∗‖2 − Φ(‖xn+1 − x∗‖)

+ 2αn L ‖yn − xn+1‖ + an ‖xn+1 − x∗‖ (2)

Note that,

‖xn+1 − yn‖ = ‖(1 − αn)(xn − yn) + αn(T n1 yn − yn)‖

≤ (1 − αn)‖xn − yn‖ + αn‖T n

1 yn − x∗‖ + ‖yn − x∗‖

≤ (1 − αn)‖xn − yn‖ + αnL(‖yn − x∗‖ + an

)+ ‖yn − x∗‖

≤ (1 − αn)‖xn − yn‖ + αn(1 + L)‖yn − x∗‖ + an

≤ (1 − αn)‖xn − yn‖ + αn(1 + L)(‖xn − yn‖ + ‖xn − x∗‖) + an

≤ (1 + αnL)‖xn − yn‖ + αn(1 + L)‖xn − x∗‖ + an

≤ (1 + αnL)βn‖xn − T n2 xn‖ + αn(1 + L)

‖xn − x∗‖ + an

≤ (1 + αnL)βn

‖xn − x∗‖ + ‖x∗ − T n

2 xn‖

+ αn(1 + L)‖xn − x∗‖ + an

≤ (1 + αnL)βn‖xn − x∗‖L (‖xn − x∗‖ + an

)+ αn(1 + L)

‖xn − x∗‖ + an

≤ (1 + αnL)(1 + L)βn‖xn − x∗‖ + an

+ αn(1 + L)

‖xn − x∗‖ + an

= (1 + L)(1 + αnL)βn + αn

(‖xn − x∗‖ + an)

= dn‖xn − x∗‖ + an

(3)


where dn = (1 + L)(1 + αnL)βn + αn

. Conditions (i)-(iv) imply

∑∞n=1 αndn < ∞ .

Substituting (3) into (2), we have

‖xn+1 − x∗‖2 ≤ (1 − αn)2‖xn − x∗‖2 + 2αnkn‖xn+1 − x∗‖2 − Φ(‖xn+1 − x∗‖)

+ 2αn Ldn‖xn − x∗‖ + (1 + dn)an

‖xn+1 − x∗‖≤ (1 − αn)2‖xn − x∗‖2 + 2αn

kn‖xn+1 − x∗‖2 − Φ(‖xn+1 − x∗‖)

+ 2αn L(1 + dn)‖xn − x∗‖ + an

‖xn+1 − x∗‖≤ (1 − αn)2‖xn − x∗‖2 + 2αn

kn‖xn+1 − x∗‖2 − Φ(‖xn+1 − x∗‖)

+ αn L(1 + dn)(‖xn − x∗‖ + an

)2+ ‖xn+1 − x∗‖2

which on simplification gives

‖xn+1 − x∗‖2 ≤(1 +

An

Bn

)‖xn − x∗‖2 +

αn L(1 + dn)Bn

(‖xn − x∗‖ + an)2

− 2αn

BnΦ(‖xn+1 − x∗‖) (4)

where An = 2αn(kn − 1) + α2 and Bn = 1 − (2αnkn + αn(1 + dn)L).Now Bn → 1 since αn → 0 as n → ∞, so there exists a positive integer n0 such that12 < Bn ≤ 1 for all n ≥ n0, then from (4), we have

‖xn+1 − x∗‖2 ≤ (1 + 2An) ‖xn − x∗‖2 + 2αn L(1 + dn)(‖xn − x∗‖ + an

)2

− 4αn Φ(‖xn+1 − x∗‖) (5)

≤ (1 + 2An) ‖xn − x∗‖2 + 2αn L(1 + dn)(‖xn − x∗‖ + an

)2 (6)

By condition∑∞

n=1 αn(kn − 1) < ∞ and∑∞

n=1 α2n < ∞, we have 2

∑∞n=1 An < ∞,

also since an is convergent, it is bounded, we obtain from above inequality (6) andLemma3, that limn→∞ ‖xn − x∗‖ exists.Hence ‖xn − x∗‖ ≤ M1 and |an| ≤ M2 for some positive constant M1, M2, then from(5), we have

‖xn+1 − x∗‖2 ≤ ‖xn − x∗‖2 − 4αn Φ(‖xn+1 − x∗‖)+ 2AnM1 + 4αn L(1 + dn)(M2

1 + M22) (7)

Taking θn = ‖xn − x∗‖, λn = 4αn and σn = 2[AnM1 + 2αn L(1 + dn)(M21 + M2

2)],inequality (7) can be written as

θ2n+1 ≤ θ2

n − λnΦ(θn+1) + σn , for all n ≥ n0.

By conditions (i) − (iv), we see that all the conditions of the Lemma 2 satisfied.Therefore it follows that ‖xn − x∗‖ → 0, i.e. xn → x∗.This completes the proof.


Remark 2.1.

1. Theorem 2.1 extends and improve Theorem 2.1 of Chang et al. [3] from uni-formly L−Lipschtizian mappings to nearly uniformly L−Lipschtizian mappings.

2. Under the suitable conditions the sequence xn considered in Theorem 2.1 canbe generalized to sequence with errors.

3. Theorem 2.1 generalizes the Theorem 2.5 of Kim et al. [7] for pair of mappings.Also the “control condition an

αn bounded” is not needed in our result.

From Theorem 2.1, we immediately obtain the following theorem

Theorem 2.2. Let E be a real Banach space, K be a nonempty closed convex subsetof E, and T : K → K be asymptotically generalized Φ−hemicontractive, nearlyuniformly L−Lipschitzian mapping with sequence an and F(T ) , ∅, where F(T )is the set of fixed points of T in K. Let αn be a sequences in [0, 1] satisfying thefollowing conditions :

(i)∑∞

n=1 αn = ∞;

(ii)∑∞

n=1 α2n < ∞;

(iii)∑∞

n=1 αn(kn − 1) < ∞ .

Let xn be a sequence in K generated from arbitrary x1 ∈ K by

xn+1 = (1 − αn)xn + αnT nxn , n ∈ N .

Then xn converges strongly to x∗ ∈ F(T ).

Remark 2.2.

1. Theorem 2.2 extend and improve Theorem 2.2 of Chang et al. [3] from uni-formly L−Lipschtizian mappings to nearly uniformly L−Lipschtizian mappings.

2. Theorem 2.2 is also a generalization of Theorem 3.2 of Ofoedu [9].

3. Theorem 2.2 improves Theorem 2.5 of Kim et al. [7] with less restrictive set-ting, i.e. it is without the “control condition an

αn bounded” that was used in

[7].

References[1] S.S.Chang, Some results for asymptotically pseudocontractive mappings and asymptotically non-

expansive mappings, Proc. Amer. Math. Soc., 129 (2001), 845–853.

[2] S.S.Chang, On Chidume’s open questions and approximation solutions of multi-valued stronglyaccretive mapping equation in Banach spaces, J. Math. Anal. Appl., 216 (1997) 94–111.


[3] S.S.Chang, Y.J.Cho, J.K.Kim, Some results for uniformly L-Lipschitzian mappings in Banachspaces, Appl. Math. Letters, 22 (2009) 121–125.

[4] C.E.Chidume, C.O.Chidume, Convergence theorems for fixed points of uniformly continuousgeneralized Ψ−hemi-contractive mappings, J. Math. Anal. Appl., 303 (2005), 545–554.

[5] C.E.Chidume, C.O.Chidume, Convergence theorem for zeros of generalized Lipschitz general-ized phi-quasi-accretive operators, Proc. Amer. Math. Soc., 134 (2006), 243–251.

[6] F.Gu, Convergence theorems form φ−pseudocontractive type mappings in normed linear spaces,Northeast Math. J., 17(3) (2001), 340–346.

[7] J.K.Kim, D.R.Sahu, Y.M.Nam, Convergence theorems for fixed points of nearly uniformly L-Lipschitzian asymptotically generalized Φ− hemicontractive mappings, Nonlinear Anal., 71(2009) e2833–e2838.

[8] C.Moore, B.V.Nnoli, Iterative solution of nonlinear equations involving set-valued uniformlyaccretive operators, Comput. Math. Appl., 42 (2001) 131–140.

[9] E.U.Ofoedu, Strong convergence theorem for uniformly L-Lipschitzian asymptotically pseudo-contractive mapping in a real Banach space, J. Math. Anal. Appl., 321 (2006) 722–728.

[10] D.R.Sahu, Fixed point of demicontinuous nearly Lipschitzian mappings in Banach spaces, Com-ment. Math. Univ. Carolin., 46 (2005) 653–666.

NUMERICAL INVESTIGATIONS ON THESTABILITY OF THE PILOT-AIRPLANE SYSTEMUSING FREQUENCY DOMAIN ANALYSIS

ROMAI J., v.8, no.1(2012), 173–188

Adrian Toader1, Ionel Iorga2

1National Institute for Aerospace Research ”Elie Carafoli”, Bucharest, Romania2University of Craiova, Craiova, [email protected], [email protected]

Abstract In the paper the absolute stability of a specific pilot-airplane system is analysed by usingthe Popov Criterion. The absolute stability Lurie problem defined by the presence in thesystem mathematical model of the actuator rate saturation is used. The case study is thatof a longitudinal control chain of ADMIRE airplane model and the human operator isrepresented by a simple gain, without a specific delay.

Keywords: asymptotic absolute stability, Lurie problem, Popov Criterion, ADMIRE mathematicalmodel, rate saturation.2010 MSC: 93D10.

1. INTRODUCTIONThe pilot induced oscillations phenomenon (shortly, PIO) is described in the litera-

ture as ”sustained or uncontrollable oscillations resulting from the efforts of the pilotto control the airplane” [1], or, for example, as a ”misadaptation between the pilotand the aircraft during some task in which tight closed loop control of the aircraft isrequired from the pilot” [12].

1.1. TYPES OF PIOThe following types of PIO are known [12]:

1. category I: linear oscillations of the pilot-vehicle system resulting from exces-sive lags introduced by filters, actuators, feel system and digital system timedelays;

2. category II: quasi-linear oscillations which are mainly due to actuator rate lim-iting;

3. category III: severe life-threatening PIO, which are caused by nonlinearitiesand transitions in pilot or effective airplane dynamics.

If the reader wants to consult more about the PIO clasiffication a good reference is[12].

173

174 Adrian Toader, Ionel Iorga

Fig. 1. YF-22A Pilot-Induced Oscillation [3]

Fig. 2. YF-22A PIO time history [4]

1.2. AN EXAMPLE OF PIO: THE YF-22A PIOACCIDENT

In [2] is described an example of longitudinal PIO which has started after theretraction of the landing gear and, also, happened nearly the same time as the after-burner was initiated in order to proceed another overflight above the runway. Theoscillations which led to crash took about eleven seconds (their shape can be seen inFigure 1).The accident happened in April 1992 at Edwards Air Force Base, in California andthe test pilot survived. After four to five oscillations the aircraft impacted the runway.In Figure 2 the pitch rate and the pitch angle vs. time are shown.

In the YF-22A accident, the rate limit was involved, therefore it can be concludedthat the PIO II phenomenon was present. See details in [11].

2. CLOSED LOOP CONTROL OF THE PILOTEDSYSTEM

In Figure 3, the block diagram of the pilot-aircraft system, with rate limiter, isshown. The airplane dynamics is represented by the transfer function

G(s) = cT (sI − A)−1b (1)

Numerical investigations on the stability of the pilot-airplane system ... 175

Fig. 3. Closed loop pilot-aircraft system with rate saturation of the flight control surface actuator

where χ is the output of the linear system, cT , A, b are common notations from thestate-space representation and s is the complex variable from the Laplace transform.

The actuator dynamics is modeled by a first order lag system with transfer function

δe(s)δc(s)

=1

τs + 1=

1τ

s + 1τ

=ω0

s + ω0(2)

where τ > 0 is the actuator time constant, δc represents the input due to the stick andδe is the deflection of the longitudinal flight control surface. In other words,

δe + ω0δe = ω0δc

andδe = ω0(δc − δe) (3)

whereδc(t) = κPθ(t) (4)

The rate saturation of the deflection of the longitudinal flight control surface, δe,is defined as:

δe =

|δe|, if |δe| < VLVL, if δe ≥ VL−VL, if δe ≤ −VL

(4)

where VL is the rate limit value, herein VL = 1.57 rads .

3. ABSOLUTE STABILITY ANALYSISAbsolute stability refers to the global asymptotic stability of the zero equilibrium

point of the general nonlinear system

x(t) = Aαx(t) − bαψ(cTαx(t)) (6)

having sector restricted nonlinearities [13]:


Fig. 4. Absolute stability feedback structure (Lurie problem)

0 ≤ ψ ≤ ψ(y)y≤ ψ ≤ ∞, ψ(0) = 0 (7)

3.1. LURIE PROBLEMLurie problem (see [10], for details) refers to finding necesarry and sufficient con-

ditions for system (6)-(7) to be absolutely stable. The absolute stability feedbackstructure is shown in Figure 4, where L is a linear block (obtained by linearization ofthe original nonlinear system - in this paper the system (26) - around an equilibriumpoint) and N represents a non-linear block - modeled by the continuous function ψ(herein, ψ(y) represents a saturation, like in the left part of Figure 5) which fulfills thegeneral sector condition (7).The continous functions ψ fulfilling the condition (7) belong to the class describedby the right part of Figure 5.

An equivalence between rate limiter blocks (Figures 3 and 6) proof is made in thefollowing. The starting point is the following relation

χ(s) = G(s)δe(s) (8)

From the form of the Lurie problem the following output is considered

yα(t) = cT x(t) + δe(t) (9)

where

δe(t) = −ω0ψ(yα(t))u(t) = −ψ(yα(t)) (9)

The Laplace transform is applied


Fig. 5. Standard saturation nonlinearity. Sector restricted nonlinearity

sx(s) = Ax(s) + bδe(s)sδe(s) = ω0u(s) (10)

yα(s) = cT x(s) + δe(s) (12)

From (1) it follows that:

x(s) = (sI − A)−1bδe(s)δe(s) =

ω0

su(s) (13)

Therefore

x(s) =ω0

s(sI − A)−1bu(s) (14)

From (12), (13) and (14) follows

yα(s) = cT(sI − A)−1bω0

su(s) +

ω0

su(s) (15)

and, further on

G(s) = cT (sI − A)−1 b =(

0 0 κP)

(sI − A)−1 b (16)

from (15) and (16) the following relation is determined

yα(s) = G(s)ω0

su(s) +

ω0

su(s) (17)


Fig. 6. Pilot-aircraft system with rate limiter written in terms of the Lurie problem

which is equivalent to

yα(s) = L(s)u(s) = ω0

(G(s)

1s

+1s

)u(s)

= ω0

[(0 0 κP

)(sI − A)−1 b

1s

+1s

]u(s)

(18)

The relation (18) shows that the transfer function is in the critical case of one sim-ple pole in origin (the transfer function denominator - the characteristic polynomial -has all the roots in the open-left half plane, exception one which is zero).

When the coupled pilot-aircraft system represented in Figure 3 isrewritten as a Lurie system, cT in (34) has the form cT = (0 0 κP).

3.2. THE POPOV CRITERION - ASIMTOPTICSTABILITY WITH ONE POLE IN THEORIGIN

The study of the absolute stability will be made by using the Popov criterion.First, the statement of the Popov Criterion [13] is given and then the criterion willbe applied to a low-order ADMIRE aircraft mathematical model. Consider systemswith one nonlinearity

xα(t) = Aαxα(t) − bαψ(yα(t))yα(t) = cT

α xα(t) (18)

where Aα, bα, cTα , xα, yα are the usual notations for the time-domain realisation of the

L(s) transfer function


Aα =

(A b

0(1, n) 0

); bα =

(0(n, 1)

1

); cT = (cT 1) (20)

(n ∈ N \ 0, 1) and ψ is a continous function having the property (7).From [15] the following theorem is adapted (with ψ = 0):

Theorem 3.1. The trivial solution of the system (19) is asymptotic absolutely stable- under conditions (7)- if the following conditions are satisfied

det(sIn − A) = 0 has all the roots in the open left half planeand

1 − cT A−1b > 0 (21)

There exists a number ξ ≥ 0, ξ < ∞ such that

1ψ

+ Re[(1 + ıωξ)L(ıω)] > 0 (22)

for every real number ω.

Relation (22) can be written as:

1ψ

+ Re[(1 + ıωξ)(U(ω) + ıV(ω))] > 0, ∀ω ≥ 0 (23)

where

U(ω) = Re[L(ıω)]V(ω) = Im[L(ıω)] (23)

which is equivalent with

1ψ

+ U(ω) − ωξV(ω) > 0, ∀ω ≥ 0 (25)

Remark 3.1. In the previous relations ω is restricted to positive values because,from the general theory of functions of a complex variable with real coefficients, it isknown that, for every function from the class mentioned, the real part is even and theimaginary part is odd (in conclusion, the above expression is always even).

4. THE ADMIRE MODELThe Generic Aerodata Model (GAM) is a theoretical model of a small fighter air-

craft. This model has been developed by the Saab AB (the Swedish company thatproduces the JAS-39) and was used by the Swedish Defence Research Agency (FOI),


in order to develop a mathematical model for a single-seated fighter with a delta ca-nard configuration (ADMIRE). A detailed and explicit mathematical description ofthe two models is not given but informations can be obtained by consulting datatables. Suggestions regarding the distribution of the mass concerning the GAM the-oretical model can be found in [5], also, a good reference for the ADMIRE is [6].The original ADMIRE system has twelve states [6], but for the simplicity of the anal-ysis a low order non-linear explicit differential model was used. This model has beendescribed in [7], [8], [9].

From [7] the following nonlinear system is considered:

α = zαα + q +g

V0cos(θ) + zδeδe

q = mαα + mqq − 1aαq +

gV0

(mα cos θ − a sin θ) + mδeδe

θ = q(25)

The state system is X = (α, q, θ), where α is the incidence angle, θ represents thepitch angle and q is the pitch rate. The input vector u one dimensional, u = (δe), δerepresents the angle of the elevon.

Let consider equilibrium points of the above system. Using only the first twoequations (from the third one results q = 0), it follows that

α = − 1zα

(g

V0cos θ1 + zδe δe) (27)

which, introduced in the second equation and with γ = tan( θ2 ) substitution, gives

γ2(δe%3 − %1) + γ(2%2) + (%1 + δe%3) = 0 (28)

where

%1 =g

V0(mα − mα

zα)

%2 = −a gV0

%3 = mδe − zδezα

mα

(28)

The positivity condition on the discriminant associated to the above equation equa-tion gives

|δe| ≤

√%2

1 + %22

|%| (30)

With a fixed value of δe, the roots of the equation (28) are:

γ1, 2 =−%2 ±

√%2

2 − (δe%3 − %1)(%1 + δe%3)

δe%3 − %1(31)


Therefore:

θ = 2tan−1(γ) (32)

4.0.1 The simplified linearized ADMIRE system. The linearizedsystem associated to (26) has the form:

∆x = A∆x + b∆δe∆y = cT ∆x (32)

where:

A =

zα 1 − g

V0sin θ

mα κ2 − gV0κ1

0 1 0

; ∆x =

∆αq

∆θ

; b =

zδe

mδe

0

; cT = (0 0 1) (34)

andκ1 = mα sin θ + a cos θκ2 = mq − 1

a α(34)

Let consider the notation:∆ν = ν − ν (36)

where ν is the equilibrium point (trim value) of the state ν, ν ∈ α, q, θ.Therefore, the low-order transfer function of the ADMIRE aircraft is:

G(s) =mδe s + ε

s3 − s2(κ2 + zα) + µ1s + µ2g

V0

(37)

where

ε = mαzδe − mδezαµ1 = zακ2 +

gV0κ1 − mα

µ2 = mα sin θ − κ1zα(37)

The open-loop transfer function in the case of rate saturation is denoted by L(s)and from (18) we obtain

L(s) = ω0sκPmδe + κPε

P(s)+ω0

s(39)

where P(s) is the characteristic polynomial


P(s) = s[s3 − s2(κ2 + zα) + µ1s + µ2g

V0] (40)

Using the Routh-Hurwitz criterion, the answer to the question if the transfer func-tion (39) is stable (and, consequently, if a ’sector rotation’, as described in [14], isneeded) can be provided.

Consider the polynomial

D(s) = s3 − s2(κ2 + zα) + µ1s + µ2g

V0(41)

The Routh-Hurwitz condition (from [16]) is

−(κ2 + zα)µ1 > µ2g

V0(42)

The condition (21) from Theorem (3.1) is

1 − κPV0

g(κ1zα − mα)(mαzδe − zαmδe) (43)

4.1. STABILITY ANALYSIS IN THE CASE OFRATE LIMITER

If the following substition is made s→ ıω in (39), then

L(ıω) = Re[L(ıω)] + ıIm[L(ıω)] (44)

where

Re[L(ıω)] = ω0κPω2[mδe(κ2 + zα) + ε] + mδe

gV0µ2 − εµ1

[ω2(κ2 + zα) +g

V0µ2]2 + ω2(µ1 − ω2)2

Im[L(ıω)] = −ω0ω[κP−ω4mδe + ω2[ε(κ2 + zα) + mδeµ1] + ε

gV0µ2

ω2[ω2(κ2 + zα) +g

V0µ2]2 + ω2(µ1 − ω2)2 −

1ω2 ]

(44)

Based on (24) and (25), the following frequency domain inequality holds

−ω0κPmδeξω4 + ω2[ω0κP(τ1ξ + τ2)] + β2εω0κPξ + τ3ω0κP

(ω2β1 + β2)2 + ω2(µ1 − ω2)2+ ξω0 + 1 > 0 (46)

where the following notations were introduced:

β1 = κ2 + zα; β2 =g

V0µ2; τ1 = εβ1 + mδeµ1; τ2 = mδe β1 + ε; τ3 = mδe β2 − εµ1 (46)


With the notations:

f (ξ, ω) = −mδeξω

4 + ω2(τ1ξ + τ2) + β2εξ + τ3g(ξ, ω) = (ω2β1 + β2)2 + ω2(µ1 − ω2)2 (47)

we get

h(ξ, ω) = ω0κPf (ω)g(ω)

+ ξω0 + 1

= ω0κP−mδeξω

4 + ω2(τ1ξ + τ2) + β2εξ + τ3

ω6 + ω4β4 + ω2β2 + β0+ ξω0 + 1 (49)

where

β4 = β21 − 2µ1; β2 = µ2

1 + 2β1β2; β0 = β22 (49)

Thus, the relation (46) can be written as

h(ξ, ω) > 0, ∀ω ≥ 0 (51)

Let now study the monotony of h(ξ, ω). First, the function behaviour is analyzedin the frontier points 0 and∞:

limω→∞ h(ξ, ω) = ξω0 + 1

limω→0 h(ξ, ω) = ω0κPβ2εξ + τ3

β22

+ ξω0 + 1 (51)

Because we have ω0 > 0, ξ ≥ 0 (from (2) and Theorem 3.1) then the first relation ofthe above system is strictly positive.In these conditions the sign of the second relation of the above system is given by thesign of

β2εξ + τ3 (53)

and, if the above relation is strictly positive, then

κP > − (ξω0 + 1)β22

ω0(β2εξ + τ3)

Finally, from positivity of κP, we denote that


κP ∈ (0,∞) (54)

On the other hand, if relation (53) is strictly negative, then

κP < − (ξω0 + 1)β22

ω0(β2εξ + τ3)

and, one obtains

κP ∈ (0, − (ξω0 + 1)β22

ω0(β2εξ + τ3)) (55)

We notice that (54) and (55) are conditions obtained at limit (when ω → 0), thuswe cannot say that they are valid for every ω but, however, if we want a maximuminterval for κP we can assert the fact that the strict positive sign of (53) is in ouradvantage.In what follows a discussion about restrictions imposed for ξ, in conditions in whichκP > 0 is fixed, is made. First, if we denote by υ = ω2, we observe that the sign of(49) is given in relation with the sign of quantity

−mδeξυ2 + (τ1ξ + τ2)υ + β2εξ + τ3 (56)

For the above relation the following discriminant is computed

∆ = ξ2τ21 + ξ(2τ1τ2 + 4mδe β2ε) + τ2

2 + 4mδeτ3 (57)

We observe the fact that, for the positivity of (49), is sufficient that relation (56) tobe strictly positive. The strict positivity of (56) is true if the above discriminant isstrictly negative and if −mδe is strictly positive (the latter condition is obtained fromthe sign of (56) when υ→ ∞), that is:

ξ2τ2

1 + ξ(2τ1τ2 + 4mδe β2ε) + τ22 + 4mδeτ3 < 0

−mδe > 0 (57)

In order to make the evaluation of the above expression the following discriminant(relative to ξ) must be positive (because, otherwise, only complex values for ξ areobtained):

∆ξ = 42mδe(τ1τ2β2ε + mδe β22ε

2 − τ21τ3) (59)

Using the above result ξ1,2 is determined:

ξ1,2 = −τ1τ2 + 2mδe β2ε ± 2

√mδe(τ1τ2β2ε + mδe β

22ε

2 − τ21τ3)

τ21

(60)


In order for (51) to be true the following must hold

ξ ∈ (ξ1, ξ2) (61)

because τ22 + 4mδe < 0 (this is obtained from the fact that (59) is considered strictly

positive and, obviously, from the strict positivity of τ21).

The roots of the first derivative of h(ξ, ω) - with respect to ω (ξ is supposed fixed)- denoted by h

′(ω), are studied in order to evaluate the sign and magnitude of the

extremum points associated to h(ξ, ω).When computing h

′(ω) the intermediary relation is simplified with 1

2ω0κPω:

h′(ω) =

f′(ω)g(ω) − f (ω)g

′(ω)

g2(ω)

=ω8(mδeξ) + ω6$6 + ω4$4 + ω2$2 +$0

(ω6 + ω4β4 + ω2β2 + β0)2 (62)

where

$6 = −2(τ1ξ + τ2)$4 = −[(mdeβ2 + τ1β4 + 3β2ε)ξ + (τ2β4 + 3τ3)]$2 = −2[(β4β2ε + mdeβ0)ξ + β4τ3]$0 = (τ1 − β2β2ε)ξ + (τ2 − β2τ3)

(62)

From (64), the following data are used:

g = 9.81 m

s2 ; V0 = 84.5 ms ; a = −.2424; a = 1.424

zα = −.7986; zδe = −.2603; mδe = −8.2668mα = −6.5315; mq = −.6957; mα = −.162

(63)

These get (see (29))

%1 = −0.9683; %2 = −0.1653; %3 = −6.1379 (64)

and, further on, the relations (65), (30), (32) and (27), yield:

θ = 0.055 rad; α = 0.197 rad; q = 0 rad/s; with δe = −0.159 rad (65)


The following values are obtained (from (35) and (38)) using the above equilibriumpoint:

κ1 = 1.413; κ2 = 0.117; µ1 = 6.6; µ2 = 0.77; ε = −4.9 (66)

β1 = −0.6817; β2 = 0.0894; τ1 = −51.2374; τ2 = 0.7336; τ3 = 31.6231 (67)

From (67) and (68) we obtain that (53) is strictly positive and (54) is true (themaximum possible interval for κP is achieved).Also, we are in the case in which τ2

2 + 4mδeτ3 < 0 and, from (60) and (61), takinginto account that ξ ≥ 0, we obtain:

ξ ∈ [0, 0.642625) (69)

Using the above values it follows that Routh-Hurwitz condition (42) is true (there isno need for a ’sector rotation’) and, also, the condition (43) holds.

The coefficients of h(ξ, ω) are computed:

β4 = −12.7396; β2 = 43.4667; β0 = 0.008 (69)

In the numeric simulations, performed in the MatLab environment, the followingvalues were employed:

κP = 12 ; ξ = 0.64; ω0 = 20 rad

s (70)

With the usage of the previous values, the reader can see that the second relationof the system (52) is strictly positive.The coefficients of (62) are given bellow

$6 = 70.265; $4 = −290; $2 = 797.74; $0 = −1395.5 (71)

Taking into account the Remark 3.1, for table (1), only the strictly positive roots of(62) are used:In the following figure, for the ξ = 0.64, h(ξ, ω) has the following extremum valuesare determined numerically


Table 1 h(ξ, ω) extremum points

0 + 1.8795 + 2.6815 + ∞

h′(ξ, ω) −1.3935 · 103 - 0 + 0 - 0

h(ξ, ω) 3.923 · 104 + 9.66088 + 42.77232 + 13.8

Fig. 7. Graphical representation of h(ξ, ω) function, for ξ = 0.64

max[h(ξ, ω)] = 3.923 · 104; min[h(ξ, ω)] = 9.66088 (72)

5. CONCLUSIONSFrom Table 1 or, equivalently, from relation (73), it becomes clear that h(ξ, ω) is

strictly positive, thus implying that (25) is true. In conclusion, in the case of the pilotmodel acting like a simple gain, the simplified ADMIRE system, with one saturationnonlinearity given by the rate limiter, the asymptotic absolutely stability property isfulfilled as follows from the Popov Criterion, in the mentioned conditions.

References[1] ***Flying Qualities of Piloted Airplanes, (U.S. Military Specification) MIL-F-8785C, Nov.

1980.


[2] M.A. Dornheim, Report Pinpoints Factors Leading to YF-22 Crash, Aviation Week & SpaceTechnology, Nov. 9, 1992, pp. 53-54.

[3] M.R. Anderson, A.B. Page, Unified Pilot Induced Oscillation Theory. Vol. III: Pio Analysis usingmultivariable Methods, Technical Report, Wright Laboratory, Wright-Patterson Air Force Base,U.S.A, December 1995

[4] *** Flight Control Design – Best practices, NATO Technical Report 29, 2000.

[5] H. Backstrom,Report on the usage of the Generic Aerodata Model, SAAB Aircraft AB, SAABMilitary Aircraft, S-58199 Linkping Sweden, May 1997.

[6] L. Forssel, U. Nillson, ADMIRE The Aero-Data Model In a Research Envinronment, Version 4.0,Swedish Defence Re Description, Report FOI-R–1624–SE ISSN-1650-1942, December 2005.

[7] A. Ionita, A.M. Balint, St. Balint, Limit Cycle Behaviour in Aircraft Longitudinal TerminalPhase, Seventh International Conference on Mathematical Problems in Engineering, Aerospaceand Sciences 2008, Genova.

[8] A.M. Balint,St. Balint,A. Ionita, Oscillation Susceptibility Analysis of the ADMIRE Aircraftalong the Path of Longitudinal Flight Equilibriums in Two Different Mathematical Models, Hin-dawi Publishing, Corporation Differential Equations and Nonlinear Mechanics. Volume 2009,Article ID 842656, 26 pages doi:10.1155/2009/842656

[9] St. Balint et al., Oscillation susceptibility analysis along the path of longitudinal flight equilibri-ums,Nonlinear Analysis (2008), Nonlinear Analysis: Theory, Methods & Applications, In Press,2008, available on-line doi:1010.16/j.na.2008.10.054

[10] H. K. Khalil, Nonlinear systems, 3rd edition, ISBN 0-13-067389-7, Prentice Hall, 2002.

[11] F. Amato, R. Iervolino, S. Scala, L. Verde,Actuator design for aircraft robustness versus categoryII PIO, Proc. of the 7th Mediterranean Conference on Control and Automation, Haifa, Israel, June28-30, 1999.

[12] D. T. McRuer, D. H. Klyde, T.T. Myers, Development of a Comprehensive PIO Theory, AIAApaper 96-3433 (1996), pp. 581-597.

[13] V. M. Popov, Hiperstability of Control Systems, ISBN 0387063730, 1973, Springer Verlag,Berlin-Heidelberg-New York,1st edition.

[14] . Vl. Rasvan, D. Danciu, D. Popescu, On absolute (robust) stability: slope restrictions and sta-bility multipliers, Int. J. Robust. Nonlinear Control (2011), Published online in Wiley OnlineLibrary (wileyonlinelibrary.com). DOI: 10.1002/rnc.1821.

[15] Vl. Rasvan Absolute stability of control systems whith delay (in Romanian), Editura Academiei,Bucharest, 1975.

[16] M. Voicu, Stability Analysis Techniques of the Automatic Control Systems (in Romanian), Edi-tura Tehnica, Bucuresti, 1986.

FUNCTION CONTRACTIVE MAPS IN PARTIALMETRIC SPACES

ROMAI J., v.8, no.1(2012), 189–207

Mihai Turinici”A. Myller” Mathematical Seminar; ”A. I. Cuza” University; Iasi, [email protected]

Abstract Some fixed point results are given for a class of functional contractions over partialmetric spaces. These extend some contributions in the area due to Ilic et al [Math.Comput. Modelling, 55 (2012), 801-809].

Keywords: partial metric space, functional contraction, fixed point..2010 MSC: 47H10 (Primary), 54H25 (Secondary).

1. INTRODUCTIONLet (X, d) be a metric space; and T : X 7→ X be a selfmap of X. Denote the class

of its fixed points in X as Fix = z ∈ X; z = Tz. Concerning the existence anduniqueness of such elements, a basic result is the 1922 one due to Banach [4]; it saysthat, if (X, d) is complete and

(a01) d(T x,Ty) ≤ λd(x, y), ∀x, y ∈ X,

for some λ in [0, 1[ then, the following conclusions hold:i) Fix is a singleton, z; ii) T nx→ z as n→ ∞, for each x ∈ X.This result found a multitude of applications in operator equations theory; so, it

was the subject of many extensions. For example, a natural way of doing this is byconsidering ”functional” contractive conditions of the form

(a02) d(T x,Ty) ≤ F(d(x, y), d(x,T x), d(y,Ty), d(x, Ty), d(y,T x)), ∀x, y ∈ X;

where F : R5+ → R+ is an appropriate function. For more details about the possible

choices of F we refer to Collaco and E Silva [7], Kincses and Totik [12]. Park[16] and Rhoades [17]; see also Turinici [20]. Another way of extension is that ofconditions imposed upon d being modified. Some results of this type were obtainedin the last decade over the realm of partial metric spaces, introduced as in Matthews[13].

However, for the subclass of zero-complete partial metric spaces, most of thesestatements based on contractive conditions like in (a02) are nothing but clones oftheir corresponding (standard) metrical counterparts; see, for instance, Abdeljawad[1], Altun et al [2], Altun and Sadarangani [3], Dukic et al [8], Karapinar and Erhan[11], Oltra and Valero [14], Paesano and Vetro [15], Romaguera [18], Valero [21],

189

190 Mihai Turinici

and the references therein. Concerning the non-reducible to (a02) contractive mapsin complete partial metric spaces, the first specific result in the area was obtained byIlic et al [9]. It is our aim in the following to show that all these are again reducibleto a line of argument used in the standard metrical case. Further aspects will bedelineated elsewhere.

2. PARTIAL METRICSLet X be a nonempty set. By a symmetric on X we mean any map d : X × X → R+

with

(b01) d(x, y) = d(y, x), ∀x, y ∈ X.

Fix such an object; and put (for x, y ∈ X)

b(x, y) = (1/2)(d(x, x) + d(y, y)), c(x, y) = maxd(x, x), d(y, y).Call the (symmetric) d, reflexive-triangular, if

(b02) d(x, z) ≤ d(x, y) + d(y, z) − d(y, y), ∀x, y, z ∈ X.

According to Matthews [13], we say that the symmetric d is a partial metric when itis reflexive-triangular and

(b03) (b(x, y) ≤)c(x, y) ≤ d(x, y), ∀x, y ∈ X (Matthews property)

(b04) d(x, y) = d(x, x) = d(y, y) =⇒ x = y (weak sufficiency).

In this case, (X, d) is called a partial metric space. Denote in what follows

(b05) e(x, y) = 2(d(x, y) − b(x, y)), x, y ∈ X.

Lemma 2.1. The mapping e(., .) is a (standard) metric on X; in addition,

|d(x, x) − d(y, y)| ≤ e(x, y), ∀x, y ∈ X. (1)

Proof. The first part is clear; so, we do not give details. For the second one, letx, y ∈ X be arbitrary fixed. Without loss, one may assume that d(x, x) ≥ d(y, y); for,otherwise, we simply interchange x and y. In this case, (1) becomes

d(x, x) − d(y, y) + d(x, x) + d(y, y) ≤ 2d(x, y).

This is evident, from the Matthews property; hence the conclusion.

Technically speaking, all constructions and results below are ultimately based onthe (standard) metric e; but, for an appropriate handling, these are expressed in termsof the partial metric d.

(A) Denote, for x ∈ X, ε > 0,

Function contractive maps in partial metric spaces 191

Xd(x, ε) = y ∈ X; d(x, y) < d(x, x) + ε

(the open d-sphere with center x and radius ε). Let T(d) stand for the topologyhaving as basis the family Xd(x, ε); x ∈ X, ε > 0; it is T0-separated, as established inBukatin et al [5]. The sequential convergence structure attached to this topology maybe depicted as follows. Let N := 0, 1, ... stand for the set of all natural numbers;and, for each k ∈ N, put N(k,≤) = n ∈ N; k ≤ n. By a sequence over (the nonemptyset) A, we shall mean any map n 7→ x(n) := xn from N(n0,≤) to A, where n0 dependson this map only; it will be also written as (xn; n ≥ n0), or (xn), when n0 is genericallyunderstood. Likewise, by a doubly indexed sequence in A we shall mean any map(m, n) 7→ y(m, n) := ym,n from N(p0,≤) × N(p0,≤) to A, where p0 depends on thismap only; it will be also written as (ym,n; m, n ≥ p0), or (ym,n), where p0 is genericallyunderstood. Now, for the sequence (xn; n ≥ n0) in X and the point x ∈ X, define

(xn; n ≥ n0)→ x (modulo T(d)) iff∀ε > 0, ∃n(ε) ≥ n0: n ≥ n(ε) =⇒ xn ∈ Xd(x, ε).

When n0 is generic, it will be also written as xn → x (modulo T(d)). Note that, bythe very definition of our topology,

xn → x (modulo T(d)) iff limn d(xn, x) = d(x, x). (2)

Further, let T(e) stand for the Hausdorff topology attached to the metric e. We shallwrite (xn; n ≥ n0) → x (modulo T(e)) as (xn; n ≥ n0)

e−→ x; or, simply, xne−→ x. By

definition, this means e(xn, x)→ 0 as n→ ∞; i.e.:

∀α > 0,∃n(α) ≥ n0 : n ≥ n(α) =⇒ e(xn, x) < α.

A useful interpretation of this in terms of d may be given as follows.

Lemma 2.2. Under the above conventions,

xne−→ x iff limn d(xn, x) = limm,n d(xm, xn) = d(x, x). (3)

Proof. If (3) holds, then

limn

d(xn, x) = d(x, x), limn

d(xn, xn) = d(x, x);

so that,lim

ne(xn, x) = lim

n[2d(xn, x) − d(xn, xn) − d(x, x)] = 0.

Conversely, assume that e(xn, x)→ 0 as n→ ∞. By definition, this means

[d(xn, x) − d(xn, xn] + [d(xn, x) − d(x, x)]→ 0.

192 Mihai Turinici

Since the quantities in the square brackets are positive, this yields limn d(xn, x) =

limn d(xn, xn) = d(x, x). Finally, let m, n be arbitrary fixed. By the reflexive-triangularproperty,

d(xm, xn) ≤ d(xm, x) + d(x, xn) − d(x, x),

d(xm, xn) ≥ d(xm, x) − d(x, xn) + d(xn, xn).

Since the limit in the right hand side is d(x, x) when m, n→ ∞, we are done.

Note that, as a consequence of this, the convergence structures attached to T() andT() are quite distinct; we do not give details.

(B) Remember that, a sequence (xn; n ≥ n0) in X is e-Cauchy when e(xm, xn)→ 0as m, n→ ∞; that is,

∀α > 0,∃p(α) ≥ n0 : m, n ≥ p(α) =⇒ e(xm, xn) < α.

As before, we are interested to characterize this property in terms of d.

Lemma 2.3. The generic property below is true:

(xn; n ≥ n0) is e-Cauchy⇐⇒ limm,n d(xm, xn) exists (in R+). (4)

Proof. i) Assume that (xn; n ≥ n0) is taken as in the right hand of (4); i.e.,

limm,n d(xm, xn) = γ, for some γ ∈ R+.

In particular, (for m = n), we must have limn d(xn, xn) = γ. But then,

limm,n

e(xm, xn) = limm,n

[2d(xm, xn) − d(xm, xm) − d(xn, xn)] = 0;

so that, (xn) is e-Cauchy.ii) Assume that (xn; n ≥ n0) is e-Cauchy; that is: limm,n e(xm, xn) = 0. By the very

definition of our metric, this means

limm,n

[d(xm, xn) − d(xm, xm) + d(xm, xn) − d(xn, xn)] = 0. (5)

Let α > 0 be arbitrary fixed. By the imposed hypotheses and Lemma 2.1, there existsk := k(ε) ≥ n0 such that

|d(xm, xm) − d(xn, xn)| ≤ e(xm, xn) < α, ∀m, n ≥ k.

This tells us that (d(xn, xn); n ≥ n0) is a Cauchy sequence in R+; wherefrom limn d(xn, xn) =

γ, for some γ ∈ R+. Combining with (5), limm,n d(xm, xn) = γ; hence, (xn) fulfills theproperty of the right hand side in (4).

(C) Denote, for each sequence (xn; n ≥ n0) in X,


e − limn(xn) = x ∈ X; xne−→ x.

As e(., .) is a metric, e − limn(xn) is either empty or a singleton; in this last case, wesay that (xn) is e-convergent. Again by the metric property of e, we have that eache-convergent sequence is e-Cauchy too; when the reciprocal holds, (X, e) is referredto as complete.

In the following, a useful result is given about the e-semi-Cauchy sequences thatare not endowed with the e-Cauchy property. Some conventions are needed. Given asequence (τn; n ≥ n0) in R and a point τ ∈ R, we have the equivalence

τn → τ iff (τn; n ≥ k)→ τ, for all k ≥ n0.

Also, for the same initial data, put

τn ↓ τ if [τn ≥ τ,∀n ≥ n0] and τn → τ

τn τ if (τn; n ≥ k) ↓ τ, for some k ≥ n0.

The implication below is evident:

τn ↓ τ =⇒ (τn; n ≥ k) ↓ τ, ∀k ≥ n0.

On the other hand, if (τn; n ≥ n0) τ then τn ↓ τ is not in general true. Likewise,given the double indexed sequence (σm,n; m, n ≥ p0) in R and the point σ ∈ R,

σm,n → σ iff (σm,n; m, n ≥ k)→ σ, for all k ≥ p0.

Also, for the same initial data, put

σm,n ↓ σ if [σm,n ≥ σ,∀m, n ≥ p0] and σm,n → σ

σm,n σ if (σm,n; m, n ≥ k) ↓ σ, for some k ≥ 0.

As before, the implication below is evident:

σm,n ↓ σ =⇒ (σm,n; m, n ≥ k) ↓ σ, ∀k ≥ p0.

On the other hand, if σm,n σ then σm,n ↓ σ is not in general true.Now, given a sequence (xn; n ≥ n0) in X, call it e-semi-Cauchy provided

(αn := d(xn, xn)) γ, (ρn := d(xn, xn+1)) γ, for some γ ∈ R+.

Note that, in such a case, the e-Cauchy property of (xn) means d(xm, xn) γ.

Proposition 2.1. Suppose that (xn; n ≥ n0) is e-semi-Cauchy but not e-Cauchy. Thereexist then k ≥ n0, ε > 0, j(ε) ≥ k, and a couple of rank-sequences (m( j); j ≥ k),(n( j); j ≥ k) with

194 Mihai Turinici

j ≤ m( j) ≤ n( j), d(xm( j), xn( j)) ≥ γ + ε, ∀ j ≥ k (6)

n( j) − m( j) ≥ 2, d(xm( j), xn( j)−1) < γ + ε, ∀ j ≥ j(ε) (7)

(d(xm( j), xn( j)); j ≥ k) ↓ γ + ε (8)

(d(xm( j)+p, xn( j)+q); j ≥ k)→ γ + ε, ∀p, q ∈ 0, 1. (9)

Proof. By the very definition of e-semi-Cauchy property, there must be some k ≥ n0in such a way that (αn; n ≥ k) ↓ γ, (ρn; n ≥ k) ↓ γ. In this case, the conclusion to bederived writes (d(xm, xn); m, n ≥ k) ↓ γ; i.e.,

∀ε > 0, ∃ j = j(ε) ≥ k : j ≤ m ≤ n =⇒ d(xm, xn) < γ + ε.

The negation of this property means that, there exists ε > 0, such that: for each j ≥ k,there may be found a couple of ranks m( j), n( j) with the property (6). Given thisε > 0, there exists, by the e-semi-Cauchy property, some j(ε) ≥ k with

γ ≤ αi ≤ ρi < γ + ε, ∀i ≥ j(ε). (10)

We claim that this is our desired rank for the remaining conclusions in the statement.In fact, for each j ≥ j(ε), let n( j) be the minimal rank fulfilling (6) (with n in placeof n( j)). For the moment, (7) is clear, via (10). Further, by this relation and thereflexive-triangular inequality,

γ + ε ≤ d(xm( j), xn( j)) ≤ d(xm( j), xn( j)−1) + ρn( j)−1 − αn( j)−1 ≤γ + ε + ρn( j)−1 − αn( j)−1, ∀ j ≥ j(ε).

So, passing to limit as j → ∞ yields (8). Finally, again by the reflexive-triangularinequality, one has, for all j ≥ j(ε),

d(xm( j), xn( j)+1) ≤ d(xm( j), xn( j)) + ρn( j) − αn( j),

d(xm( j), xn( j)+1) ≥ d(xm( j), xn( j)) − ρn( j) + αn( j)+1.

By a limit process upon j one gets the case (p = 0, q = 1) of (9). The remaining onesmay be obtained in a similar way.

3. NORMAL FUNCTIONSLet F(A) stand for the class of all functions from A to itself.(A) Call ϕ ∈ F(R+), normal when

(c01) ϕ(0) = 0; ϕ(t) < t, for all t in R0+ :=]0,∞[.

For any such function ϕ, and any s in R0+, denote

(c02) lim supt→s+ ϕ(t) = infε>0 Φ[s+](ε), whereΦ[s+](ε) = supϕ(t); s ≤ t < s + ε, ε > 0


(c03) lim supt→s ϕ(t) = infε>0 Φ[s](ε), whereΦ[s](ε) = supϕ(t); s − ε < t < s + ε, ε > 0.

Note that, again by the choice of ϕ, one has

ϕ(s) ≤ lim supt→s+

ϕ(t) ≤ lim supt→s

ϕ(t) ≤ s, ∀s ∈ R0+. (11)

Lemma 3.1. Let ϕ ∈ F(R+) be normal; and s ∈ R0+ be arbitrary fixed. Then,

i) lim supn ϕ(tn) ≤ lim supt→s+ϕ(t), for each sequence (tn; n ≥ n0) with tn s

ii) lim supn ϕ(tn) ≤ lim supt→s ϕ(t), for each sequence (tn; n ≥ n0) with tn → s.

Proof. By definition, there exists k ≥ n0 with (tn; n ≥ k) ↓ s. Given ε > 0, thereexists a rank p(ε) ≥ k such that s ≤ tn < s + ε, ∀n ≥ p(ε); whence

lim supn

ϕ(tn) ≤ supϕ(tn); n ≥ p(ε) ≤ Φ[s+](ε).

It suffices taking the infimum (=limit) as ε → 0+ in this relation to get the desiredfact. The second part is proved in a similar way.

We say that the normal function ϕ ∈ F(R+) is right limit normal (respectively:limit normal), if

(c04) lim supt→s+ϕ(t) < s (respectively: lim supt→s ϕ(t) < s), ∀s ∈ R0

+.

In particular, for the normal function ϕ ∈ F(R+), this holds whenever ϕ is rightusc (respectively: usc) on R0

+. Note that this property is fulfilled when ϕ is rightcontinuous (respectively: continuous) on R0

+.(B) Let ψ ∈ F(R+) be a function. Denote

(c05) lim inft→∞ ψ(t) = supα≥0 Ψ(α), where Ψ(α) = infψ(t); t ≥ α, α ≥ 0.

Call ψ, semi-coercive, provided lim inft→∞ ψ(t) > 0. For example, this is valid when-ever ψ is coercive: limt→∞ ψ(t) = ∞.

Lemma 3.2. Let ψ ∈ F(R+) be some function; and (tn; n ≥ n0) be a sequence with(tn; n ≥ n0)→ ∞. Then,

iii) lim infn ψ(tn) ≥ lim inft→∞ ψ(t),iv) lim infn ψ(tn) > 0, whenever ψ is semi-coercive.

The proof is very similar to the one of Lemma 3.1; so, we omit it.(C) Let (an; n ≥ n0) be a bounded in R+ sequence. From this choice, An :=

supah; h ≥ n exists in R+, for all n ≥ n0; moreover, (An; n ≥ n0) is descending.Now, by definition, lim supn(an) = infn≥n0(An); this, by the descending property ofthis last sequence, writes: lim supn(an) = limn(An).

196 Mihai Turinici

Lemma 3.3. Let F : R3+ → R+ be a function with

(c06) F is increasing in all variables,

(c07) F is continuous at the right over R3+;

and let (an; n ≥ n0), (bn; n ≥ n0), (cn; n ≥ n0) be bounded sequences. Then,

lim supn

F(an, bn, cn) ≤ F(a, b, c),

where a = lim supn(an), b = lim supn(bn), c = lim supn(cn).

Proof. Denote (for each n ≥ n0) An := supah; h ≥ n, Bn := supbh; h ≥ n, Cn :=supch; h ≥ n. Let ε > 0 be arbitrary fixed. By the remark above, there existsh(ε) ≥ n0 in such a way that An < a + ε, Bn < b + ε, Cn < c + ε, for all n ≥ h(ε). This(by the properties of F), yields for all n ≥ h(ε),

lim supn

F(an, bn, cn) ≤ supF(an, bn, cn); n ≥ h(ε) ≤ F(a + ε, b + ε, c + ε).

Passing to limit as ε→ 0 gives the desired fact.

As we shall see, the usual particular case to be considered is that of F(a, b, c) =

a + maxb, c, a, b, c ∈ R+. Clearly, (c06) and (c07) are fulfilled here.

4. MAIN RESULTLet X be a nonempty set. Take a partial metric d(., .) over it; and let e(., .) stand for

its associated (standard) metric. Assume in the following that

(d01) (X, e) is complete.

Let T : X → X be a selfmap of X. We say that z ∈ X is a d-fixed point of T , whend(z,Tz) = d(z, z). The set of all these points will be denoted as Fix(T ; d).

Loosely speaking, such points are to be determined as e–limits of iterative pro-cesses. Precisely, let us say that x ∈ X is a (d; T )-Picard point, when

(d02) (T nx; n ≥ 0) is e-convergent and x∗ := e − limn(T nx) is in (T ; d).

If this holds for each x ∈ X, then T will be referred to as a d-Picard operator; cf. Rus[19, Ch 2, Sect 2.2].

Now, sufficient conditions for getting this property are of functional contractivetype. Denote, for x, y ∈ X,

M1(x, y) = maxd(x, y), d(x,T x), d(y,Ty), M2(x, y) =

(1/2)[d(x,Ty) + d(T x, y)], M(x, y) = maxM1(x, y),M2(x, y).Note that, if y = x, then (by the Matthews property)

M1(x, x) = maxd(x, x), d(x,T x) = d(x,T x), M2(x, x) = d(x,T x);


hence

M(x, x) = d(x, T x), x ∈ X. (12)

Likewise, if y = T x, then

M1(x,T x) = maxd(x,T x), d(T x,T 2x),M2(x,T x) = (1/2)[d(x,T 2x) + d(T x, T x)] ≤(1/2)[d(x, T x) + d(T x,T 2x)] ≤ maxd(x,T x), d(T x,T 2x);

henceM(x,T x) = maxd(x,T x), d(T x,T 2x), x ∈ X. (13)

Given the normal function ϕ ∈ F(R+) and g ∈ b, c, let us say that T is (M; g;ϕ)-contractive (modulo d) when

(d03) d(T x,Ty) ≤ maxϕ(M(x, y)), g(x, y), ∀x, y ∈ X.

We are now in position to give the first main result of this exposition.

Theorem 4.1. Suppose that T is (M, c;ϕ)-contractive (modulo d) for some right limitnormal function ϕ ∈ F(R+). Then, each x ∈ X is a (d; T )-Picard point, in the sense

i) (ρn(x) := d(T nx,T n+1x); n ≥ 0) is descending; hence, ρn(x) ↓ γ(x), for someγ(x) ∈ R+

ii) (αn(x) := d(T nx, T nx); n ≥ 0) γ(x); so that, (T nx; n ≥ 0) is e-semi-Cauchyiii) (δm,n(x) := d(T mx, T nx); m, n ≥ 0) γ(x); whence, (T nx; n ≥ 0) is e-Cauchyiv) x∗ := e − limn(T nx) is an element of Fix(T; d).Hence, in particular, T is a d-Picard operator.

Proof. Fix in the following some point x0 ∈ X; and put xn = T nx0, n ≥ 0. Denote,for simplicity reasons, ρn = ρn(x0), αn = αn(x0), n ≥ 0; that is: ρn = d(xn, xn+1),αn = d(xn, xn), n ≥ 0. By the contractive condition written at (x = xn, y = xn), wehave (taking (12) into account)

αn+1 ≤ maxϕ(ρn), αn, ∀n ≥ 0. (14)

On the other hand, by the contractive condition written at (x = xn, y = xn+1), we have(along with (13) above)

ρn+1 ≤ maxϕ(maxρn, ρn+1), αn, αn+1, ∀n ≥ 0. (15)

Now (for some n ≥ 0), the alternative ρn+1 ≤ ϕ(maxρn, ρn+1) gives (as ϕ is normal),ρn+1 ≤ ϕ(ρn); so that, (15) becomes

ρn+1 ≤ maxϕ(ρn), αn, αn+1), ∀n ≥ 0.

This, finally combined with (14), gives

ρn+1 ≤ maxϕ(ρn), αn, ∀n ≥ 0. (16)

198 Mihai Turinici

Now, by the Matthews property of the partial metric d,

αn ≤ ρn, ∀n ≥ 0. (17)

This, combined with the above relation, gives

ρn+1 ≤ maxϕ(ρn), ρn ≤ ρn, ∀n ≥ 0.

The sequence (ρn; n ≥ 0) is therefore descending; so that, ρn ↓ γ, for some γ ∈ R+.(I) If γ = 0 then, by (17), αn ↓ 0; so that, (xn; n ≥ 0) is e-semi-Cauchy. Assume

now that γ > 0. By (16),

γ ≤ maxϕ(ρn), αn), ∀n ≥ 0. (18)

As ρn ↓ γ and ϕ is right limit normal, there must be a sufficiently large k = k(γ)in such a way that ϕ(ρn) < γ, ∀n ≥ k. This tells us that, for n ≥ k, the alternativeγ ≤ ϕ(ρn) cannot hold in (18); so, necessarily,

γ ≤ αn, ∀n ≥ k. (19)

Combining with (17) yields (αn; n ≥ k) ↓ γ; or, equivalently: (αn : n ≥ 0) γ;hence the assertion.

(II) Further, we claim that (xn; n ≥ 0) is e-Cauchy. This, by the above ob-tained facts, writes (d(xm, xn); m, n ≥ 0) γ; or, equivalently (by (19) above)(d(xm, xn); m, n ≥ k) ↓ γ. Assume that such a property does not hold. By Propo-sition 2.1, there exist ε > 0, j(ε) ≥ k and a couple of rank-sequences (m( j); j ≥ k),(n( j); j ≥ k), with the properties (6)-(9). Denote, for j ≥ k,

t j := M(xm( j), xn( j)), s j := d(xm( j)+1, xn( j)+1),β j := c(xm( j), xn( j))(= maxαm( j), αn( j)).

By the quoted relations (and the e-semi-Cauchy property)

(t j; j ≥ k) ↓ γ + ε, (s j; j ≥ k)→ γ + ε, (β j; j ≥ k) ↓ γ.

On the other hand, by the contractivity condition at (x = xm( j), y = xn( j)),

s j ≤ maxϕ(t j), β j, ∀ j ≥ k.

Passing to lim sup as j→ ∞, one gets (via Lemma 3.3 and Lemma 3.1)

γ + ε ≤ max lim supt→(γ+ε)+

ϕ(t), γ,

in contradiction with the choice of ϕ. Hence, the working assumption about (xn; n ≥0) cannot be true; so, this sequence is e-Cauchy.


(III) As (X, e) is complete, there exists a uniquely determined x∗ ∈ X with xne−→

x∗; or, equivalently (cf. Lemma 2.2)

limn

d(xn, x∗) = d(x∗, x∗) = γ(= limm,n

d(xm, xn)).

Denote for simplicity δ := d(x∗, T x∗). Clearly, γ ≤ δ (by the Matthews property).Assume by contradiction that γ < δ; and let η > 0 be such that γ + 2η < δ. Asd(xn, x∗) ↓ γ, d(xn, xn+1) ↓ γ, there exists k = k(η) with

d(xn, x∗), d(xn, xn+1) < γ + η < δ, ∀n ≥ k; (20)

and this, by definition, yields M1(xn, x∗) = δ, ∀n ≥ k. Further, by the reflexive-triangular inequality,

d(xn, T x∗) ≤ d(xn, x∗) + δ − γ < δ + η;

wherefrom, combining with (20),

M2(xn, x∗) ≤ (1/2)[δ + γ + 2η[< δ, ∀n ≥ k.

Hence, summing up, M(xn, x∗) = δ, ∀n ≥ k. This, along with the contractivitycondition written for (x = xn, y = x∗), gives

d(xn+1,T x∗) ≤ maxϕ(δ), αn, γ = maxϕ(δ), αn, ∀n ≥ k.

As a direct consequence, one gets (again by the reflexive-triangular inequality)

δ = d(x∗, T x∗) ≤ d(x∗, xn+1) + d(xn+1, T x∗) − αn+1≤ d(x∗, xn+1) + maxϕ(δ), αn − αn+1, ∀n ≥ k.

Passing to limit as n→ ∞ one gets (via Lemma 3.3) δ ≤ maxϕ(δ), γ, contradiction.Hence, γ = δ; i.e., x∗ is an element of Fix(T; d). The proof is complete.

5. FIXED POINT STATEMENTLet again X be a nonempty set. Take a partial metric d over it; and let e stand for

its associated metric. As before, assume that (d01) holds.Let T : X → X be a selfmap of X. Remember that, under the conditions of

Theorem 4.1, each x ∈ X is a (d; T )-Picard point, in the sense precise there. Inparticular, this tells us that Fix(T; d) is a nonempty subset of X. Note that, when d isa (standard) metric, Fix(T; d) = Fix(T); whence, Fix(T) is nonempty; but, in general,this is not possible. It is our aim in the following to identify – in the ”partial” setting– sufficient conditions under which the non-emptiness of Fix(T) be still retainable.Note that, by the weak sufficiency of d,

z ∈ X, d(z,Tz) = d(z, z) = d(Tz,Tz) =⇒ z = Tz. (21)

200 Mihai Turinici

As a consequence of this,

z ∈ Fix(T; d), d(z, z) = d(Tz,Tz) =⇒ z ∈ Fix(T). (22)

Hence – assuming that Theorem 4.1 holds – it will suffice getting points z ∈ Fix(T; d)with d(z, z) = d(Tz,Tz) to conclude that Fix(T) is nonempty.

Technically speaking, such a conclusion is deductible under a stronger contractivecondition upon T than the one in Theorem 4.1. This firstly refers to the mappingc(., .) appearing there being substituted by b(., .); i.e.,

(e01) T is (M; b;ϕ)-contractive (modulo d).

Secondly, the function ϕ ∈ F(R+) in this condition must be taken so as

(e02) ϕ is limit normal and ψ := ι − ϕ is semi-coercive.

Here, ι ∈ F(R+) is the identity function (ι(t) = t, t ∈ R+). It is our aim in the followingto show that, under these requirements, a positive answer to the posed question isavailable.

(A) Note that, under such a setting, the working conditions of Theorem 4.1 hold.Hence, in particular, Fix(T; d) is nonempty. For an easy reference, we shall collectsome basic facts about this set. These are valid even in the absence of (e01)+(e02)above.

Proposition 5.1. Let the selfmap T of X be such that Fix(T; d) , ∅. Then,

d(Tz,Tz) ≤ d(z, z), ∀z ∈ Fix(T; d) (23)

d(y, Tz) ≤ d(y, z), ∀y ∈ X, ∀z ∈ Fix(T; d) (24)

e(z,Tz) = d(z, z) − d(Tz,Tz), ∀z ∈ Fix(T; d) (25)

M(z,w) = d(z,w), ∀z,w ∈ Fix(T; d). (26)

Proof. j) Let z ∈ Fix(T; d) be arbitrary fixed. By the Matthews property of d, one hasd(Tz,Tz) ≤ d(z, Tz) = d(z, z); and this proves (23).

jj) For each y ∈ X, z ∈ Fix(T; d), we have, by the reflexive-triangular inequality,

d(y, Tz) ≤ d(y, z) + d(z, Tz) − d(z, z) = d(y, z);

hence, (24) follows.jjj) For each z ∈ Fix(T; d), one has, by definition,

e(z,Tz) = 2d(z,Tz) − d(z, z) − d(Tz,Tz) = d(z, z) − d(Tz,Tz);

and, from this, (25) is proved.


jv) Take some couple z,w ∈ Fix(T; d). We have

M1(z,w) = maxd(z,w), d(z, Tz), d(w,Tw) =

maxd(z,w), d(z, z), d(w,w) = d(z,w);

as well as, by (24),

M2(z,w) = (1/2)[d(z, Tw) + d(w,Tz)] ≤ d(z,w).

Hence, by simply combining these, (26) follows.

(B) We may now pass to the effective part of our developments. Assume that (e01)and (e02) hold; and put

(e03) θ = infd(z, z); z ∈ Fix(T; d), X(T ; d) = z ∈ Fix(T; d); d(z, z) = θ.Proposition 5.2. Under the admitted conditions, we have

X(T ; d) ⊆ Fix(T); i.e.: z ∈ Fix(T; d), d(z, z) = θ =⇒ z = Tz (27)

Fix(T) is at most singleton; hence, so is X(T ; d). (28)

Proof. h) If θ = 0, we are done. For, in such a case, d(z, z) = d(z,Tz) = 0 implies(by Proposition 5.1) d(Tz, Tz) = 0; whence z = Tz. Assume in the following θ > 0.Taking z as a starting point in Theorem 4.1, one gets (by its conclusions) that

(ρn := d(T nz,T n+1z); n ≥ 0) is descending; hence γ := limn(ρn) exists (29)

z∗ := e − limn(T nz) exists, as an element of Fix(T; d),with d(z∗, z∗) = γ; hence θ ≤ d(z∗, z∗) ≤ d(Tz, T 2z). (30)

On the other hand, by the initial choice of z, we must have θ = d(T nz,T n+1x), ∀n ≥ 0;whence θ = γ. From the contractive condition written at (x = z, y = Tz),

θ = d(Tz,T 2z) ≤ maxϕ(M(z,Tz)), b(z,Tz).

In addition, by (13) and (29),

M(z,Tz) = maxd(z,Tz), d(Tz, T 2z) = d(z,Tz) = d(z, z) = θ.

Hence, by simply combining these relations,

θ ≤ maxϕ(θ), (1/2)[θ + d(Tz,Tz)].

The alternative θ ≤ ϕ(θ) is impossible. Hence, the alternative θ ≤ (1/2)[θ+d(Tz,Tz)]must be true; and then, θ ≤ d(Tz,Tz) ≤ d(z, Tz) = θ; wherefrom d(Tz, Tz) = θ =

d(z, z). This, along with (22), yields z = Tz.

202 Mihai Turinici

hh) Assume that u, v ∈ X are such that u = Tu, v = Tv. Then, by the Matthewsproperty of d,

M1(u, v) = d(u, v), M2(u, v) = d(u, v);

hence, M(u, v) = d(u, v). By the contractive condition,

d(u, v) = d(Tu,Tv) ≤ maxϕ(d(u, v)), b(u, v).If d(u, v) ≤ ϕ(d(u, v)) then d(u, v) = 0 (wherefrom b(u, v) = 0); since this givese(u, v) = 0, we must have u = v. If d(u, v) ≤ b(u, v) then (combining with (b03)), onehas d(u, v) = b(u, v). This, by definition, yields e(u, v) = 0; wherefrom u = v. Theproof is thereby complete.

(C) It therefore follows that, a positive answer to the posed question is obtainable,if we can establish that X(T ; d) is nonempty. The following intermediate statementwill be useful for this purpose.

(C) It therefore follows that, a positive answer to the posed question is obtainable,if we can establish that X(T ; d) is nonempty. The following intermediate statementwill be useful for this purpose.

Proposition 5.3. Let the same conditions hold. Then,

∀ε > 0,∃z ∈ Fix(T; d) : d(z, z) < θ + ε, d(z, z) − d(Tz,Tz) < 2ε. (31)

Proof. The conclusion is clear when θ = 0. In fact, let z ∈ Fix(T; d) be such thatd(z, z) < ε; then (by Proposition 5.1), 0 ≤ d(Tz,Tz) ≤ d(z, z) < ε < 2ε. It remainsnow to discuss the case of θ > 0. The conclusion is again clear when θ ≤ ε; because,taking z ∈ Fix(T; d) according to d(z, z) < θ + ε, we have [by (23)]

0 ≤ d(Tz, Tz) ≤ d(z, z) < θ + ε ≤ 2ε.

So, we have to see what happens when 0 < ε < θ. Assume by contradiction that theconclusion in the statement is not true; i.e., for some ε in ]0, θ[,

(e04) ∀z ∈ Fix(T; d): d(z, z) < θ + ε =⇒ d(z, z) − d(Tz, Tz) ≥ 2ε.

Note that, in such a case, one derives

∀z ∈ Fix(T; d) : d(z, z) < θ + ε =⇒ d(Tz,Tz) ≤ d(z, z) − 2ε < θ − ε. (32)

As (in particular) ϕ is right limit normal, there exists η in ]0, ε[, such that

ϕ(t) < θ, whenever θ ≤ t < θ + η. (33)

Given this η, there exists some z ∈ Fix(T; d) with θ ≤ d(z, z) < θ + η. Taking z asa starting point in Theorem 4.1, the relations (29) and (30) are still retainable here.From the contractive condition written at (x = z, y = Tz),

θ ≤ d(Tz,T 2z) ≤ maxϕ(M(z,Tz)), b(z,Tz).


By (13), the choice of z, and (29),

M(z, Tz) = maxd(z,Tz), d(Tz, T 2z) = d(z,Tz) = d(z, z).

Replacing in the previous relation gives

θ ≤ d(Tz, T 2z) ≤ maxϕ(d(z, z)), b(z,Tz).The first alternative of this, [θ ≤ ϕ(d(z, z))], is unacceptable, by the choice of z ∈Fix(T; d) and (33). Moreover, the second alternative of the same, [θ ≤ b(z,Tz)],yields, by the choice of z ∈ Fix(T; d) and (32) (modulo η)

θ ≤ (1/2)[d(z, z) + d(Tz,Tz)] < (1/2)[θ + η + θ − η] = θ;

again a contradiction. So, (e04) cannot be true; and conclusion follows.

Having these precise, we may now answer the posed question.

Proposition 5.4. Let (e01) and (e02) be in use. Then, X(T ; d) is nonempty.

Proof. Let (µn; n ≥ 0) be a strictly descending sequence in R0+, with µn → 0. (For

example, one may take µn = 2−n, n ≥ 0). By Proposition 5.3, there exists, for eachn ≥ 0, some zn ∈ Fix(T; d) with

d(zn, zn) < θ + µn, d(zn, zn) − d(Tzn, Tzn) < 2µn. (34)

Let m, n ≥ 0 be arbitrary for the moment. We have (by the reflexive-triangular prop-erty and (zn; n ≥ 0) ⊆ Fix(T; d))

d(zm, zn) ≤ d(zm,Tzm) + d(Tzm, zn) − d(Tzm,Tzm) =

d(zm, zm) − d(Tzm, Tzm) + d(Tzm, zn) < 2µm + d(Tzm, zn).

The distance in the right hand side may be evaluated as

d(Tzm, zn) ≤ d(Tzm,Tzn) + d(Tzn, zn) − d(Tzn, Tzn) =

d(Tzm,Tzn) + d(zn, zn) − d(Tzn,Tzn) < 2µn + d(Tzm,Tzn).

So, replacing in the previous one, we get

d(zm, zn) ≤ 2[µm + µn] + d(Tzm, Tzn), ∀m, n ≥ 0;

and this, combined with the contractive condition, yields (via Proposition 5.1)

d(zm, zn) ≤ 2(µm + µn) + maxϕ(d(zm, zn)), b(zm, zn), ∀m, n ≥ 0. (35)

Part 1. Denote, for simplicity

αn := d(zn, zn), ρn := d(zn, zn+1), βn := b(zn, zn+1), n ≥ 0.

204 Mihai Turinici

By (34), (αn; n ≥ 0) ↓ θ; and this, by definition, yields (βn; n ≥ 0) ↓ θ. On the otherhand, by the Matthews property of d,

θ ≤ αn ≤ ρn, ∀n ≥ 0; hence θ ≤ σ := lim supn(ρn).

We claim that σ = θ; or, equivalently (see above) ρn ↓ θ. Suppose by contradictionthat σ > θ; note that σ ≤ ∞. As a direct consequence of (35),

ρn ≤ 2[µn + µn+1] + maxϕ(ρn), βn, ∀n ≥ 0. (36)

On the other hand, by the definition of σ, there must be a rank-sequence (n( j); j ≥ 0)with n( j) → ∞ and ρn( j) → σ. Let ε > 0 be such that θ + ε < θ + 2

e−→< σ. Asβn ↓ θ, µn ↓ 0, there must be some j(ε), in such a way that

ρn( j) > θ + 2ε, βn( j) < θ + ε, µn( j) < ε/4, ∀ j ≥ j(ε);

whence, it is the case that

ρn( j) > 2[µn( j) + µn( j)+1] + βn( j), ∀ j ≥ j(σ).

This, along with (36), shows us that we must have (as alternative)

ρn( j) ≤ 2[µn( j) + µn( j)+1] + ϕ(ρn( j)), ∀ j ≥ j(ε). (37)

Sub-case 1a. Suppose that σ = ∞. By the introduced in (e02) notation,

ψ(ρn( j)) ≤ 2[µn( j) + µn( j)+1], ∀ j ≥ 0.

Then, passing to limit as j→ ∞, one gets lim j ψ(ρn( j)) = 0; in contradiction with theproperty imposed upon ψ and Lemma 3.2.

Sub-case 1b. Suppose that σ < ∞. Passing to limit as j → ∞ in (37), one getsσ ≤ lim supt→σ ϕ(t); again a contradiction.

Summing up, (ρn; n ≥ 0) ↓ θ; wherefrom, (zn; n ≥ 0) is e-semi-Cauchy.Part 2. We show that (zn; n ≥ 0) is e-Cauchy. Suppose that this is not true. By

Proposition 2.1, there exist then ε > 0, j(ε) ≥ 0, and a couple of rank-sequences(m( j); j ≥ 0), (n( j); j ≥ 0) with the properties (6)-(9). Further, by (35),

d(zm( j), zn( j)) ≤ 2[µm( j) + µn( j)]+maxϕ(d(zm( j), zn( j))), b(zm( j), zn( j)), ∀ j ≥ 0. (38)

Passing to limit as j→ ∞, one gets, via (8), Lemma 3.3 and Lemma 3.1,

θ + ε ≤ maxlim supt→θ+ε

ϕ(t), θ;

impossible by the choice of ϕ. Hence the conclusion.


Part 3. As (X, e) is complete, there exists z ∈ X with zne−→ z; i.e.,

d(z, z) = limn

d(zn, z) = limm,n

d(zm, zn) = θ.

We claim that z ∈ Fix(T; d); this, by the relation above, amounts to d(z,Tz) = θ. Forthe moment, we have (cf. Proposition 5.1)

d(Tzn, Tzn) ≤ d(zn, zn) ≤ d(z, zn), ∀n ≥ 0;

henceδn := d(z, zn) − d(Tzn, Tzn) ≥ 0, ∀n ≥ 0.

In addition, by the choice of (zn; n ≥ 0),

δn = d(z, zn) − d(zn, zn) + d(zn, zn) − d(Tzn,Tzn) ≤ d(z, zn) − θ + 2µn, ∀n ≥ 0;

and this gives δn → 0 as n → ∞. On the other hand, by the reflexive-triangularinequality and Proposition 5.1, one has, for all n,

d(z,Tz) ≤ d(z, zn) + d(zn, Tz) − d(zn, zn),

d(zn,Tz) ≤ d(zn,Tzn) + d(Tzn, Tz) − d(Tzn,Tzn) ≤d(zn, zn) + d(Tzn,Tz) − d(Tzn, Tzn);

so, by combining these,

d(z, Tz) ≤ δn + d(Tzn,Tz), ∀n ≥ 0. (39)

Now, d(z,Tz) ≥ d(z, z) = θ. Assume by contradiction that d(z,Tz) > θ; and let ε > 0be such that θ + 2ε < d(z,Tz). There exists k = k(ε) such that, for all n ≥ k,

d(zn, z) < θ + ε < d(z,Tz),

By the contractive condition,

d(Tzn,Tz) ≤ maxϕ(M(zn, z)), b(zn, z), ∀n ≥ 0.

To evaluate the right member, we have (from Proposition 5.1 and the preceding rela-tion), for all n ≥ k,

M1(zn, z) = maxd(zn, z), d(zn,Tzn), d(z, Tz) =

maxd(zn, z), d(zn, zn), d(z,Tz) = maxd(zn, z), d(z,Tz) = d(z,Tz),

M2(zn, z) = (1/2)[d(zn,Tz) + d(Tzn, z)] ≤(1/2)[d(zn, z) + d(z,Tz) − d(z, z) + d(zn, z)] =

(1/2)[2d(zn, z) − d(z, z) + d(z, Tz)] <(1/2)[θ + 2ε + d(z,Tz)] < d(z,Tz).

206 Mihai Turinici

This yields M(zn, z) = d(z,Tz), ∀n ≥ k; so that, the contractive condition becomes

d(Tzn, Tz) ≤ maxϕ(d(z, Tz)), b(zn, z), ∀n ≥ k;

wherefrom, replacing in (39),

d(z, Tz) ≤ δn + maxϕ(d(z,Tz)), b(zn, z), ∀n ≥ k. (40)

Note thatb(zn, z) = (1/2)[d(zn, zn) + d(z, z)]→ θ, asn→ ∞.

So, passing to limit as n→ ∞ in (40), gives, via Lemma 3.3,

d(z,Tz) ≤ maxϕ(d(z,Tz)), θ;contradiction. Hence, the working assumption about d(z,Tz) cannot be accepted;wherefrom d(z,Tz) = θ. This completes the argument.

(D) Now, as a direct consequence of all these, we have the second main result ofthis exposition.

Theorem 5.1. Suppose that (e01) and (e02) hold. Then, Fix(T) is a nonempty sin-gleton; i.e.: T has a unique fixed point in X.

In particular, assume ϕ is linear; i.e., ϕ(t) = αt, t ∈ R+, for some α in [0, 1[. Then,Theorem 5.1 is just the main result in Ilic et al [10]. But, we must say that the way ofproving it (via Theorem 4.1) is different from the one proposed in that paper. Furtheraspects may be found in Chi et al [6].

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ADSORPTION MODELS FOR TREATMENTOF EXPERIMENTAL DATA ON REMOVALFLUORINE FROM WATER BYOXIHYDROXIDESOF ALUMINUM

ROMAI J., v.8, no.1(2012), 209–215

Veacheslav Zelentsov, Tatiana Datsko, Elena DvornikovaInstitute of Applied Physics of the Academy of Sciences of Moldova, Chisinau,Republic of [email protected]

Abstract The adsorption of fluorine ions from aqueous solutions by oxihydroxides of aluminumwas studied. The equilibrium sorption was explained by using the Lengmuire, Fre-undlich, Bet and Redlich-Peterson models of isotherms. The results obtained allow oneto conclude that the mathematical model of adsorption Freundlich is the best for describ-ing the measured experimental data, which testifies to the heterogeneity of the surface.The parameters of all equations of adsorption were calculated.

Keywords: adsorption models; equilibrium; aluminum oxihydroxides; fluorine adsorption isotherms;constants of the adsorption equations.2010 MSC: 80A50.

1. INTRODUCTIONExperimental data on adsorption from solutions are, as a rule, presented in the

form of dependence of the quantity of substance adsorbed on solid surface vs itsequilibrium containing in solution - in form of adsorption isotherm [1 -5]. Adsorptionisotherms are used for understanding the mechanism and quantifying the distributionof the adsorbate between the liquid phase and solid adsorbent phase at equilibriumduring the adsorption process. For adsorption isotherms interpretation there are sev-eral empirical models, among them the most commonly used in sorption studies arethe Langmuir [6], Freundlich [7], Redlich-Peterson [8] and BET (Brunauer - Emmett- Teller) [9] models. Langmuir model - a model based on the fact that on the surfaceof the sorbent is formed monomolecular layer of adsorbate, and all active sites are ofequal energy and enthalpy of adsorption. Langmuir equation is as follows:

a =amKLCe

1 + KLCe. (1)

Freundlich model is used to describe the adsorption on heterogeneous surfaces.Since the adsorption centers of the surface have different values of adsorption en-ergy, the most active sorption centers with maximum energy are filled the first. Here

209

210 Veacheslav Zelentsov, Tatiana Datsko, Elena Dvornikova

Freundlich equation is:

a = KFC1ne . (2)

Redlich-Peterson model combines models of Freundlich and Langmuir and in-tends to describe in addition of the heterogeneity of the sorbent surface and a cer-tain number of adsorption sites with the same adsorption potential. The equation byRedlich-Peterson is the following:

a =KRCe

1 + αCβe

. (3)

BET model provides coverage of the surface with adsorbate multilayers, and asactive sites have different energies, multilayers can be formed in different parts ofthe surface and may arise as to the completion of the monolayer, and after it. Theequilibrium constant KBET characterizes the interaction energy of the adsorbate withthe surface of the adsorbent. The sign of this constant (positive or negative) indicatesthe applicability or inapplicability of this model to describe the adsorption in thissystem:

a =amKBETCeCs

(Cs −Ce)[Cs + (KBET − 1)Ce]. (4)

In these equations:

a and am - the amount of adsorbed substance at 1g of sorbent at equilibriumand the adsorption capacity of the sorbent at saturation;

Ce and Cs - equilibrium and limiting concentration (its solubility at a giventemperature) of substance in solution;

KL, KF , KR and KBET -equilibrium constants of the equations of Langmuire,Freundlich, Redlich-Peterson and BET;

n, α and β - parameters of the equations of Freundlich and Redlich-Peterson.

To the adsorption equilibrium data interpret all the equations are used in linearform:

Langmuir modelCe

a=

1KLam

+Ce

am; (5)

Adsorption models for treatment of experimental data on removal fluorine... 211

Freundlich modelln a = ln KF +

1n

ln Ce; (6)

BET modelCe

a(Cs −Ce)=

1amKBET

+(KBET − 1)Ce

amKBETCs; (7)

Redlich-Peterson model

ln(KRCe

a− 1

)= lnα + β ln(Ce). (8)

The equilibrium constants of the equations are calculated from the slope and theintersection of the line graph in the corresponding coordinates of linear equations(5)-(8): Ce

a vs Ce, ln a vs ln Ce, Cea(Cs−Ce) vs Ce

Csand ln

(KRCe

a − 1)

vs ln(Ce).

2. MAIN RESULTSThis paper studies the equilibrium adsorption of fluoride ions in the Al2O3 - NaF

system and clarifies the applicability of different models for interpreting experimen-tal data on adsorption of fluoride - ion on aluminum oxihydroxides. All the abovemodels have been applied for calculation of the isotherms of fluorine adsorption onaluminum oxihydroxides and the calculated isotherms have been compared with theexperimental ones. For the absorption of fluorine from aqueous solutions were used2 samples of aluminum oxyhydroxide A100 and A600. The first sample presentsaluminum trihydrate containing 2.85 moles of water per 1 mole of Al2O3 ; A600 -consisting of a crystalline aluminum oxide γ-Al2O3 with composition of 0.09 moleH2O/mol Al2O3 . Adsorption of fluorine from solutions with initial concentration of0.05 - 0.25 mol F / L was carried out in acetate buffer at optimum pH of 5.9. [10].The amount of adsorbed fluoride was calculated by using the formula:

a =(C0 −Ce)V

m, mmol/g (9)

where C0 and Ce - initial and equilibrium concentration of fluoride in solution, mmol/ L; V - volume of initial solution, l; m - mass of the sample sorbent, g. The fluorideadsorption isotherms are shown in Fig. 1. The figure shows that for small (2-5mmol/ L) concentrations of fluorine (the initial parts of the isotherms), the adsorption isalmost directly proportional to the concentration of fluoride and there is a sharp riseof the curves. This indicates a strong interaction of the adsorbate with the surface ofthe adsorbent. With increasing concentration of fluoride in the solution the degree


Fig. 1. Adsorption isotherms of fluorine on aluminum oxihydroxides

of its extraction increases. With further growth of the concentration of fluoride inthe solution curves almost reach saturation. Comparison of sorption isotherms forsamples A100 and A600 shows that the isotherm of the first sample is located higherthan the second, indicating a greater affinity of fluoride ions to hydroxide than alu-minum oxide. Plots designed on the base of linear equations (5)-(8) are shown inFig. 2 It should be noted that in contrast to equations of models (1), (2), (4), whichcontains two parameters, equation of model (3) - Redlich-Peterson has 3 constants,and therefore it is impossible to use its linear form to determine the constants. In thiscase, we used a procedure to maximize [11]. Linear forms of the isotherms of (5) -(8) express the minimum deviation between the experimental equilibrium adsorptiondata and theoretical calculations.

It is important to note that the data for A100 and A600 fit well on straight lines(Fig. 2a, b, c, d), which serves as proof that the models can be used to describe thesorption of fluoride by these samples. The calculated values of the constants of equa-tions and correlation coefficients R2 are presented in the table. The data in the tableshow that Freundlich model (highest R2 = 0.9935 and 0.9961 for A100 and A600,respectively) fits best for description of fluoride sorption on the studied samples. Ac-cording to this model, adsorption occurs on heterogeneous surfaces and active siteshave different values of adsorption energy. Langmuir model, which assumes homo-geneity, including energy, the surface gives the minimum value of R2, which mayserve as confirmation of the applicability of the Freundlich model.

On the adequacy of the description of the adsorption of fluoride on the samplesA100 and A600 the models are arranged in the following order:


Fig. 2. Adsorption isotherms of fluorine on A100 and A600 in coordinates of linearized equations:(a) - Langmuir; (b) - Freundlich; (c) - BET; (d) - Redlich-Peterson

Table 1 The parameters of Lengmuire, Freundlich, BET and Redlich-Peterson equations at the ad-sorption of fluorine on aluminum oxihydroxides

sampleLengmuire model Freundlich modelKL,L/mmol

am,mmol/g

R2 KF ,(mmol/g)·(L/mmol)1/n

1/n R2

A100 0.230 36.23 0.8981 20.90 0.116 0.9935

A600 0.065 19.60 0.9607 6.65 0.201 0.9961

sampleBET model Redlich-Peterson modelKBET ,g/mmol

am,mmol/g

R2 KR,L/g

α β ,L/mmol

R2

A100 9.239 30.80 0.9702 15.40 0.527 0.945 0.9876

A600 0.151 15.60 0.9886 4.31 0.520 0.854 0.9948


Adsorption model Freundlich > Redlich-Peterson > BET > LangmuireA100, R2 0.9935 0.9876 0.9702 0.8981A600, R2 0.9961 0.9948 0.9886 0.9607

Fig. 3. Experimental adsorption isotherm -of fluorine on A100 (dots) and calculated according themodels: 1- Freundlich; 2 - Lengmuire; 3 - BET; 4 - Redlich-Peterson

Fig. 4. Experimental adsorption isotherm -of fluorine on A600 (dots) and calculated according themodels: 1- Freundlich; 2 - Langmuir; 3 - BET; 4 - Redlich-Peterson.

Fig. 3 and 4 show the experimental adsorption isotherms of fluoride on the sampleA100 and A600 and the isotherms, calculated by the models of Langmuir, Freundlich,Redlich - Peterson and BET for comparison which are plotted in the same graphs.

From the figures it is clear that the Freundlich model best describes the experimen-tal results.

3. CONCLUSIONSA comparative study of the applicability of adsorption models of Langmuir, Fre-

undlich, BET, and Redlich-Peterson to describe the experimental adsorption isothermsof fluoride on 2 samples of aluminum oxihydroxides, (A100) -hydroxide and (A600)


-alumina has been carried out. The constants and parameters of these equations havebeen defined. Comparing the regression coefficients R2 shows that the Freundlichmodel best describes the experimental data on the adsorption of fluoride with studiedsamples. This indicates that the adsorption of fluoride occurs on a heterogeneoussurface and that the majority of active sites have different quantities of energy.

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[10] Zelentsov V. I., Datsko T. Ya., Dvornikova E. E., Fluorine Adsorption by Aluminum Oxihy-drates Subjected to Thermal Treatment, Surface Engineering and Applied Electrochemistry , 44,1(2008), 64-68.

[11] Allen S.J., Gan Q., Matthews R., Johnson P.A., Comparison of optimized isotherm models forbasic dye adsorption by kudzu, Bioresour. Technol., 88(2003), 143-152.

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