research article nearly contraction mapping principle...

Hindawi Publishing CorporationISRN Applied MathematicsVolume 2013 Article ID 379498 4 pageshttpdxdoiorg1011552013379498

Research ArticleNearly Contraction Mapping Principle for Fixed Points ofHemicontinuous Mappings

Xavier Udo-utun M Y Balla and Z U Siddiqui

Department of Mathematics and Statistics University of Maiduguri Maiduguri Nigeria

Correspondence should be addressed to M Y Balla myballayahoocom

Received 20 May 2013 Accepted 13 August 2013

Academic Editors M Hermann and K Karamanos

Copyright copy 2013 Xavier Udo-utun et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We extend the application of nearly contraction mapping principle introduced by Sahu (2005) for existence of fixed pointsof demicontinuous mappings to certain hemicontinuous nearly Lipschitzian nonlinear mappings in Banach spaces We haveapplied certain results due to Sahu (2005) to obtain conditions for existencemdashand to introduce an asymptotic iterative processfor constructionmdashof fixed points of these hemicontractions with respect to a new auxiliary operator

1 Introduction

In this paper we have applied certain results due to Sahu[1] on nearly contraction mapping principle to obtain condi-tions for existence of fixed points of certain hemicontinuousmappings and introduced an asymptotic iterative process forconstruction of fixed points of these hemicontinuous map-pings with respect to a new auxiliary operator Our resultsare important generalizations and an extension of importantand fundamental aspect of a branch of asymptotic theory offixed points of non-Lipschitzian nonlinear mappings in realBanach spaces

Let119864 and119884 be real Banach spaces and 119870 sube 119864 a nonemptysubset of 119864 A mapping 119879 119870 rarr 119884 is said to be (see eg[2])

(i) demicontinuous if whenever a sequence 119909119899 sub 119883 con-

verges strongly to 119909 isin 119883 it implies that the sequence119879119909119899 converges weakly to 119879119909 isin 119884

(ii) hemicontinuous if whenever a sequence 119909119899 sub

119883 converges stronly on a line to 119909 isin 119883 it implies thatthe sequence 119879119909

119899 converges weakly to 119879119909 isin 119884 that

is 119879(1199090+ 119905119899119909) 119879119909

0as 119905119899rarr 0

Asymptotic fixed point theory which has been studied byso many authors [1 3ndash6] has a fundamental role in nonlinearfunctional analysis concerning existence and construction

of fixed points of Lipschitzian mappings 119871-uniformly Lip-schitzian mappings and non-Lipschitzian mappings amongother classes of operators (see eg [5 7ndash9]) A very impor-tant branch of the theory of asymptotic fixed point relates tothe important class of asymptotically nonexpansivemappingswhich have been studied by various authors in specific typesof Banach spaces

Motivated by the need to relax continuity conditioninherent in asymptotic nonexpansiveness of asymptoticallynonexpansive mappings in certain applications Sahu [1]considered and introduced the nearly contraction mappingprinciple into the study of asymptotic fixed point theoryconcerning nearly Lipschitzian mappings and obtained thefollowing results among others

Lemma 1 Let 119862 be a nonempty subset of a Banach space andlet 119879 119862 rarr 119862 be hemicontinuous Suppose that 119879119899119906 =

119901 as 119899 rarr infin for some 119906 119901 isin 119862 Then 119901 is an elementof Fix(119879) the set of fixed point of 119879

Theorem 2 Let 119862 be a nonempty closed subset of a Banachspace 119883 and 119879 119862 rarr 119862 a demicontinuous nearly Lipschit-zian mapping with sequence (119886

119899 120578(119879119899

)) Suppose 120578infin(119879) =

lim119899rarrinfin

[120578(119879119899

)]1119899

lt 1 Then we have the following

(a) 119879 has unique fixed point 119909lowast isin 119862

2 ISRN Applied Mathematics

(b) for each 1199090isin 119862 the sequence 119879119899119909

0 converges strongly

to 119909lowast

(c) 1198791198991199090minus119909lowast

le (1199090minus1198791199090119872)sum

infin

119894=119899120578(119879119894

) for all 119899 isin N

The aim of this work is appling Lemma 1 to obtain condi-tions for existence and uniqueness of asymptotic fixed pointof a new auxiliary operator and appling Theorem 2 on theauxiliary operator to obtain an extension and a generalizationofTheorem 2 which is a fundamental extension of importantclassical and related results

2 Preliminaries

Let 119870 be a nonempty subset of a Banach space 119864 and 119879

119870 rarr 119870 a nonlinear mapping The mapping 119879 is said to beLipschitzian if for each 119899 isin N there exists a constant 119871

119899gt

0 such that 119879119899119909 minus 119879119899

119910 le 119871119899119909 minus 119910 for all 119909 119910 isin 119870

A Lipschitzian mapping is called uniformly 119871-Lipschitzian if119871119899

= 119871 for all 119899 isin N and asymptotically nonexpansive iflim119899rarrinfin

119871119899= 1

Next let 119870 be a nonempty subset of a Banach space 119864

and 119886119899 a fixed sequence in [0infin) with 119886

119899rarr 0 as 119899 rarr

infin A mapping 119879 119870 rarr 119870 is called nearly Lipschitzianmapping with respect to 119886

119899 if for each 119899 isin N there exists

a constant 119871119899ge 0 such that1003817100381710038171003817119879119899

119909 minus 119879119899

1199101003817100381710038171003817 le 119871119899(1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 + 119886119899) (1)

The infimum 120578(119879119899

) = sup119879119899119909minus119879119899

119910(119909minus119910+ 119886119899) 119909 119910 isin

119870 119909 = 119910 of constants 119871119899for which (1) holds is called nearly

Lipschitzian constant Nearly Lipschitzian operators withsequences (119886

119899 120578(119879119899

)) are classified in [1 2] as shown below

(a) nearly contraction if 120578(119879119899) lt 1 for all 119899 isin N

(b) nearly nonexpansive if 120578(119879119899) le 1 for all 119899 isin N

(c) nearly asymptotically nonexpansive if 120578(119879119899) ge 1 forall 119899 isin N and lim

119899rarrinfin120578(119879119899

) le 1

(d) nearly uniformly 119871-Lipschitzian if 120578(119879119899) le 119871 forall 119899 isin N

(e) nearly uniformly 119871-contraction if 120578(119879119899) le 119871 lt 1 forall 119899 isin N

Examples and a short survey of these classes of nearlyLipschitzian operators are listed above and related operatorsare illustrated in [1] (pp 655ndash656) where it is remarkedthat if 119870 is bounded then the asymptotically nonexpansivemapping 119879 is a nearly nonexpansive mapping Also it isobserved therein that a nearly asymptotically nonexpansivemapping reduces to asymptotically nonexpansive type if 119870 isbounded For details authors are referred to Agarwal et al [2]pp 259ndash263 especially the bibliographic notes and remarksthere in

3 Main Results

Our main results depend on Lemma 1 and the following newimportant inequality needed in the sequel which we shallprove using archimedean propertyWe are still sharpening anestimate for the parameter 120591 in Lemma 3 below

Lemma 3 Let 119881 be a normed linear space over a scaler fieldF (F is real or complex) Then for all distinct points 119909 119910 isin

119881 there exists 120591 isin R such that1003817100381710038171003817120572119909 minus 120573119910

1003817100381710038171003817 le [2 |120572| + 12059110038161003816100381610038161205731003816100381610038161003816]1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 (2)

for all 120572 120573 isin F

Proof As mentioned above the proof is a consequence ofArchimedean property of real numbers that if 119886 and 119887 arepositive real numbers then 119886 lt 119899119887 for some 119899 isin N Since119909 = 119910 we have

(120572 + 120573) (119909 minus 119910) = 120572119909 minus 120573119910 + 120573119909 minus 120572119909

= 120572119909 minus 120573119910 minus (120572119910 minus 120573119909)

997904rArr1003816100381610038161003816120572 + 120573

1003816100381610038161003816

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 =

1003817100381710038171003817120572119909 minus 120573119910 minus (120572119910 minus 120573119909)1003817100381710038171003817

ge1003817100381710038171003817120572119909 minus 120573119910

1003817100381710038171003817 minus1003817100381710038171003817120572119910 minus 120573119909

1003817100381710038171003817

997904rArr1003817100381710038171003817120572119909 minus 120573119910

1003817100381710038171003817 le1003816100381610038161003816120572 + 120573

1003816100381610038161003816

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 +

1003817100381710038171003817120573119909 minus 1205721199101003817100381710038171003817

=1003816100381610038161003816120572 + 120573

1003816100381610038161003816

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 +

1003817100381710038171003817(120572 + 1205731) 119909 minus 120572119910

1003817100381710038171003817

(for some 1205731isin F)

le1003816100381610038161003816120572 + 120573

1003816100381610038161003816

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 + |120572|

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 +

100381610038161003816100381612057311003816100381610038161003816 119909

le (2 |120572| + 12059110038161003816100381610038161205731003816100381610038161003816)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

(3)

Equation (3) follows from Archimedean property whileboundedness is inferred from the fact that 120573

1= 120573 minus 120572 for

arbitrary 120572 120573 isin F

Remark 4 It is important to make the following observa-tions

(1) If 120572 120573 isin R then (3) reduces to1003817100381710038171003817120572119909 minus 120573119910

1003817100381710038171003817 le 2 (|120572| +10038161003816100381610038161205731003816100381610038161003816)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 (4)

as verified below since 120572 120573 isin R if on the contrary(4) is not satisfied then from (3) we have |120573

1|119909 gt

|120573|119909minus119910which end up with a contradiction demon-strated belowSuppose |120573

1|119909 gt |120573|119909 minus 119910 Setting 120572 120573 gt 0 such

that 120573 minus 120572 gt 0 yields 120572 gt 1 whenever 120572 120573 gt 0 suchthat 120573 minus 120572 gt 0 which is a contradiction

(2) It is important to observe that if 119909 and 119910 were notdistinct in Lemma 3 then 120572 = 120573 would be a valid andnatural constraint However for 119909 = 119910 the problem istrivial

Lemma 5 Let119870 be a nonempty subset of a Banach space andlet 119879 119870 rarr 119870 be a nearly Lipschitzian map with sequence(119886119899 120578(119879119899

)) such that 120578infin(119879) = lim

119899rarrinfin[120578(119879119899

)]1119899

lt 1Thenthe auxiliary operator 119878 N times 119870 rarr 119870 defined by 119878(119899 119909) =

119879119899minus1

119909 + 119879119899

119909 minus 119879119899minus1

119909119909 has a fixed point in 119864

ISRN Applied Mathematics 3

Proof Given that 119878(119899 119909) = 119879119899minus1

119909 + 119879119899

119909 minus119879119899minus1

119909119909 where 119879is a nearly Lipschitzian map with sequence (119886

119899 120578(119879119899

)) wehave

119878 (119899 119909) minus 119878 (119899 + 1 119909)

le10038171003817100381710038171003817119879119899

119909 minus 119879119899minus1

11990910038171003817100381710038171003817

+10038171003817100381710038171003817

10038171003817100381710038171003817119879119899

119909 minus 119879119899minus1

11990910038171003817100381710038171003817119909 minus

10038171003817100381710038171003817119879119899+1

119909 minus 119879119899

1199091003817100381710038171003817100381711990910038171003817100381710038171003817

le 120578 (119879119899minus1

) (119879119909 minus 119909 + 119886119899minus1

)

+10038171003817100381710038171003817

10038171003817100381710038171003817119879 (119879119899

119909) minus 119879 (119879119899minus1

119909)10038171003817100381710038171003817119909 minus

10038171003817100381710038171003817119879119899

119909 minus 119879119899minus1

1199091003817100381710038171003817100381711990910038171003817100381710038171003817

le 120578 (119879119899minus1

) (119879119909 minus 119909 + 119886119899minus1

)

+10038171003817100381710038171003817120578 (119879) (

10038171003817100381710038171003817119879119899

119909 minus 119879119899minus1

11990910038171003817100381710038171003817+ 1198861) 119909 minus

10038171003817100381710038171003817119879119899

119909 minus 119879119899minus1

1199091003817100381710038171003817100381711990910038171003817100381710038171003817

le 120578 (119879119899minus1

) (119879119909 minus 119909 + 119886119899minus1

)

+10038171003817100381710038171003817[120578 (119879) 119886

1+ (120578 (119879) minus 1) 119879

119899

119909 minus 119879119899minus1

119909 ] 11990910038171003817100381710038171003817

le 120578 (119879119899minus1

) (119879119909 minus 119909 + 119886119899minus1

)

+ [120578 (119879) 1198861+ 120578 (119879

119899minus1

) (120578 (119879) minus 1) (119879119909 minus 119909 + 119886119899minus1

)]

times 119909

(5)

This gives 119878119899

119909 minus 119878119899+1

119909 le 120578(119879)1198861119909 + [1 + (120578(119879) minus

1)119909]120578(119879119899minus1

)(119879119909 minus 119909 + 119886119899minus1

) which yields 119878119899119909 minus 119878119899+1

119909 le

120578(119879)1198861119909 + [1 + (120578(119879) minus 1)119909](119889

0119909

+119872)120578(119879119899minus1

) where

119889119899119909

=10038171003817100381710038171003817119879119899+1

119909 minus 119879119899

11990910038171003817100381710038171003817 (6)

Using the hypothesis

120578infin(119879) = lim

119899rarrinfin

[120578 (119879119899

)]1119899

lt 1 (7)

together with the Root Test for convergence of series of realnumbers we obtain sum

infin

119899=1119878119899

119909 minus 119878119899+1

119909 lt infin which meansthe sequence 119878(119899 119909) is a Cauchy sequence and so has a limitpoint 119909lowast in 119864

We are left to show that the limit 119909lowast of 119909119899 = 119878(119899 119909) is

a fixed point of 119878(119899 sdot) for all 119899 isin N To achieve this itsuffices to prove that 119878(119899 sdot) is continuous which follows anapplication of Lemma 3 namely Let 119909 119910 isin 119870 then

1003817100381710038171003817119878 (sdot 119909) minus 119878 (sdot 119910)1003817100381710038171003817

=1003817100381710038171003817119909 + 119879119909 minus 119909 119909 minus 119910 minus

1003817100381710038171003817119879119910 minus 1199101003817100381710038171003817 119910

1003817100381710038171003817

le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 +1003817100381710038171003817119879119909 minus 119909 119909 minus

1003817100381710038171003817119879119910 minus 1199101003817100381710038171003817 119910

1003817100381710038171003817

=1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 +1003817100381710038171003817100381710038171198891119909

119909 minus 1198891119910

119910100381710038171003817100381710038171003817

le [1198891119909

+ 1205911198891119910

+ 1]1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

(by Lemma 3 and (8))

(8)

for some positive real number 120591 So given any 120598 we have 120575 =

120598(1198891119909

+ 1205911198891119910

+ 1) such that 119878119909 minus 119878119910 lt 120598 whenever 119909 minus

119910 lt 120575 for some 120591 gt 0 Therefore 119878 is continuous in 119909 and solim119899rarrinfin

119909119899= lim119899rarrinfin

119878(119899 119909) = 119878(119899 lim119899rarrinfin

119909119899) = 119909lowast

To apply Lemma 1 we need its extension for hemicontin-uous mappings given in the following form

Lemma 6 Let 119870 be a nonempty subset of a Banach spaceand let 119879 119870 rarr 119870 be hemicontinuous nearly Lipschitzianmapping Suppose that119879119899119906 = 119901 as 119899 rarr infin for some 119906 119901 isin 119870Then 119901 is an element of Fix(119879)

Proof Consider the following operator S N times 119864 rarr 119864

defined by

S (119899 119906) = 119906 +10038171003817100381710038171003817119879119899

119906 minus 119879119899minus1

11990610038171003817100381710038171003817119879119899minus1

119906 (9)

Clearly S restricted to 119870 reduces to the auxiliary operator 119878above at the fixed point of S We will show that given that 119879is hemicontinuous then S is a selfmap of 119870 for all 119899 that isS(sdot 119909) 119870 rarr 119870 since119870 is closed

ClearlyS restricted to119870 and119879 have common fixed pointset that is Fix(119879) = Fix(119878) (provided 119879 has a fixed point)and 119878(119899 + 1 119906) = 119906 + 119879

119899+1

119906 minus 119879119899

119906119879119899

119906 = 119906 + 119889119899119879119899

119906But from the last proof we verified that 119878 is a continuousmapping on 119870 and has asymptotic fixed point 119909lowast isin 119864 Alsoby hemicontinuity of 119879 and continuity of S the sequence119891119909119899119892 = 119891S(119899 119909

0119892) converges strongly to 119909 which means

that 119891S(119899 119909119899)119892 converges weakly to S(119899 119901) which means S

is demicontinuous on 119870By Lemma 1 we have that 119901 isin Fix(S) = Fix(119879)

Theorem 7 Let 119870 be a nonempty closed subset of a Banachspace 119864 and 119879 119870 rarr 119870 a hemicontinuous nearly Lipschit-zian mapping with sequence (119886

119899 120578(119879119899

)) Suppose 120578infin(119879) =

lim119899rarrinfin

[120578(119879119899

)]1119899


(a) 119879 has unique fixed point 119901 isin 119870



to 119901

(c) 119878(119899 1199090)minus119901 le [2119872+(120578(119879)+1)119909

0minus1198791199090]suminfin

119894=119899120578(119879119894

)

for all 119899 isin N where 119872 = sup119899isinN119886119899 and 119878(119899 119909) =

119879119899minus1

119909 + 119879119899

119909 minus 119879119899minus1

119909119909

Proof By Lemma 6 the auxiliary operator given by 119878119906 =

119906+119879119906minus119906119906 is a selfmap of119870 and together with Lemma 5weconclude that 119878 has a fixed point in 119870 which is also a fixedpoint of 119879 To prove (a) we are left to show that the fixedpoint is unique The proof of uniqueness and for (b) and (c)follow from the fact thatS is demicontinuous contraction sothat Theorem 2 applies

References

[1] D R Sahu ldquoFixed points of demicontinuous nearly Lipschitzianmappings in Banach spacesrdquo Commentationes MathematicaeUniversitatis Carolinae vol 46 no 4 pp 653ndash666 2005


[2] R P Agarwal D OrsquoRegan and D R Sahu Fixed Point Theoryfor Lipschitzian-Type Mappings with Applications Springer Sci-ence+Business New York NY USA 2009

[3] R E Bruck ldquoOn the convex approximation property andthe asymptotic behavior of nonlinear contractions in Banachspacesrdquo Israel Journal of Mathematics vol 38 no 4 pp 304ndash314 1981

[4] K Goebel and W A Kirk ldquoA fixed point theorem for transfor-mations whose iterates have uniformLipschitz constantrdquo StudiaMathematica vol 47 pp 135ndash140 1973

[5] K Goebel and W A Kirk Topics in Metric Fixed Point TheoryCambridge University Press Cambidge UK 1990

[6] D R Sahu and J S Jung ldquoFixed-point iteration pro-cesses for non-lipschitzian mappings of asymptotically quasi-nonexpansive typerdquo International Journal of Mathematics andMathematical Sciences vol 2003 no 33 pp 2075ndash2081 2003

[7] T D Benavides G L Acedo and H K Xu ldquoWeak uniformnormal structure and iterative fixed points of nonexpansivemappingsrdquo ColloquiumMathematicum vol 68 no 1 pp 17ndash231995

[8] F E Browder ldquoFixed point theorems for noncompactmappingsin Hilbert spacesrdquo Proceedings of the National Academy ofSciences of the United States of America vol 53 no 6 pp 1272ndash1276 1965

[9] K Goebel andW A Kirk ldquoA fixed point theorem for asymptot-ically nonexpansive mappingsrdquo Proceedings of the AmericanMathematical Society vol 35 no 1 pp 171ndash174 1972

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to 119909lowast

(c) 1198791198991199090minus119909lowast

le (1199090minus1198791199090119872)sum

infin

119894=119899120578(119879119894

) for all 119899 isin N

The aim of this work is appling Lemma 1 to obtain condi-tions for existence and uniqueness of asymptotic fixed pointof a new auxiliary operator and appling Theorem 2 on theauxiliary operator to obtain an extension and a generalizationofTheorem 2 which is a fundamental extension of importantclassical and related results

2 Preliminaries

Let 119870 be a nonempty subset of a Banach space 119864 and 119879

119870 rarr 119870 a nonlinear mapping The mapping 119879 is said to beLipschitzian if for each 119899 isin N there exists a constant 119871

119899gt

0 such that 119879119899119909 minus 119879119899

119910 le 119871119899119909 minus 119910 for all 119909 119910 isin 119870

A Lipschitzian mapping is called uniformly 119871-Lipschitzian if119871119899

= 119871 for all 119899 isin N and asymptotically nonexpansive iflim119899rarrinfin

119871119899= 1

Next let 119870 be a nonempty subset of a Banach space 119864

and 119886119899 a fixed sequence in [0infin) with 119886

119899rarr 0 as 119899 rarr

infin A mapping 119879 119870 rarr 119870 is called nearly Lipschitzianmapping with respect to 119886

119899 if for each 119899 isin N there exists

a constant 119871119899ge 0 such that1003817100381710038171003817119879119899

119909 minus 119879119899

1199101003817100381710038171003817 le 119871119899(1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 + 119886119899) (1)

The infimum 120578(119879119899

) = sup119879119899119909minus119879119899

119910(119909minus119910+ 119886119899) 119909 119910 isin

119870 119909 = 119910 of constants 119871119899for which (1) holds is called nearly

Lipschitzian constant Nearly Lipschitzian operators withsequences (119886

119899 120578(119879119899

)) are classified in [1 2] as shown below

(a) nearly contraction if 120578(119879119899) lt 1 for all 119899 isin N

(b) nearly nonexpansive if 120578(119879119899) le 1 for all 119899 isin N

(c) nearly asymptotically nonexpansive if 120578(119879119899) ge 1 forall 119899 isin N and lim

119899rarrinfin120578(119879119899

) le 1

(d) nearly uniformly 119871-Lipschitzian if 120578(119879119899) le 119871 forall 119899 isin N

(e) nearly uniformly 119871-contraction if 120578(119879119899) le 119871 lt 1 forall 119899 isin N

Examples and a short survey of these classes of nearlyLipschitzian operators are listed above and related operatorsare illustrated in [1] (pp 655ndash656) where it is remarkedthat if 119870 is bounded then the asymptotically nonexpansivemapping 119879 is a nearly nonexpansive mapping Also it isobserved therein that a nearly asymptotically nonexpansivemapping reduces to asymptotically nonexpansive type if 119870 isbounded For details authors are referred to Agarwal et al [2]pp 259ndash263 especially the bibliographic notes and remarksthere in

3 Main Results

Our main results depend on Lemma 1 and the following newimportant inequality needed in the sequel which we shallprove using archimedean propertyWe are still sharpening anestimate for the parameter 120591 in Lemma 3 below

Lemma 3 Let 119881 be a normed linear space over a scaler fieldF (F is real or complex) Then for all distinct points 119909 119910 isin

119881 there exists 120591 isin R such that1003817100381710038171003817120572119909 minus 120573119910

1003817100381710038171003817 le [2 |120572| + 12059110038161003816100381610038161205731003816100381610038161003816]1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 (2)

for all 120572 120573 isin F

Proof As mentioned above the proof is a consequence ofArchimedean property of real numbers that if 119886 and 119887 arepositive real numbers then 119886 lt 119899119887 for some 119899 isin N Since119909 = 119910 we have

(120572 + 120573) (119909 minus 119910) = 120572119909 minus 120573119910 + 120573119909 minus 120572119909

= 120572119909 minus 120573119910 minus (120572119910 minus 120573119909)

997904rArr1003816100381610038161003816120572 + 120573

1003816100381610038161003816

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 =

1003817100381710038171003817120572119909 minus 120573119910 minus (120572119910 minus 120573119909)1003817100381710038171003817

ge1003817100381710038171003817120572119909 minus 120573119910

1003817100381710038171003817 minus1003817100381710038171003817120572119910 minus 120573119909

1003817100381710038171003817

997904rArr1003817100381710038171003817120572119909 minus 120573119910

1003817100381710038171003817 le1003816100381610038161003816120572 + 120573

1003816100381610038161003816

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 +

1003817100381710038171003817120573119909 minus 1205721199101003817100381710038171003817

=1003816100381610038161003816120572 + 120573

1003816100381610038161003816

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 +

1003817100381710038171003817(120572 + 1205731) 119909 minus 120572119910

1003817100381710038171003817

(for some 1205731isin F)

le1003816100381610038161003816120572 + 120573

1003816100381610038161003816

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 + |120572|

1003817100381710038171003817119909 minus 1199101003817100381710038171003817 +

100381610038161003816100381612057311003816100381610038161003816 119909

le (2 |120572| + 12059110038161003816100381610038161205731003816100381610038161003816)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817

(3)

Equation (3) follows from Archimedean property whileboundedness is inferred from the fact that 120573

1= 120573 minus 120572 for

arbitrary 120572 120573 isin F

Remark 4 It is important to make the following observa-tions

(1) If 120572 120573 isin R then (3) reduces to1003817100381710038171003817120572119909 minus 120573119910

1003817100381710038171003817 le 2 (|120572| +10038161003816100381610038161205731003816100381610038161003816)1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 (4)

as verified below since 120572 120573 isin R if on the contrary(4) is not satisfied then from (3) we have |120573

1|119909 gt

|120573|119909minus119910which end up with a contradiction demon-strated belowSuppose |120573

1|119909 gt |120573|119909 minus 119910 Setting 120572 120573 gt 0 such

that 120573 minus 120572 gt 0 yields 120572 gt 1 whenever 120572 120573 gt 0 suchthat 120573 minus 120572 gt 0 which is a contradiction

(2) It is important to observe that if 119909 and 119910 were notdistinct in Lemma 3 then 120572 = 120573 would be a valid andnatural constraint However for 119909 = 119910 the problem istrivial

Lemma 5 Let119870 be a nonempty subset of a Banach space andlet 119879 119870 rarr 119870 be a nearly Lipschitzian map with sequence(119886119899 120578(119879119899

)) such that 120578infin(119879) = lim

119899rarrinfin[120578(119879119899

)]1119899

lt 1Thenthe auxiliary operator 119878 N times 119870 rarr 119870 defined by 119878(119899 119909) =

119879119899minus1

119909 + 119879119899

119909 minus 119879119899minus1

119909119909 has a fixed point in 119864



119909 + 119879119899

119909 minus119879119899minus1


119899 120578(119879119899

)) wehave

119878 (119899 119909) minus 119878 (119899 + 1 119909)

le10038171003817100381710038171003817119879119899

119909 minus 119879119899minus1

11990910038171003817100381710038171003817

+10038171003817100381710038171003817

10038171003817100381710038171003817119879119899

119909 minus 119879119899minus1

11990910038171003817100381710038171003817119909 minus

10038171003817100381710038171003817119879119899+1

119909 minus 119879119899

1199091003817100381710038171003817100381711990910038171003817100381710038171003817

le 120578 (119879119899minus1

) (119879119909 minus 119909 + 119886119899minus1

)

+10038171003817100381710038171003817

10038171003817100381710038171003817119879 (119879119899

119909) minus 119879 (119879119899minus1

119909)10038171003817100381710038171003817119909 minus

10038171003817100381710038171003817119879119899

119909 minus 119879119899minus1

1199091003817100381710038171003817100381711990910038171003817100381710038171003817

le 120578 (119879119899minus1

) (119879119909 minus 119909 + 119886119899minus1

)

+10038171003817100381710038171003817120578 (119879) (

10038171003817100381710038171003817119879119899

119909 minus 119879119899minus1

11990910038171003817100381710038171003817+ 1198861) 119909 minus

10038171003817100381710038171003817119879119899

119909 minus 119879119899minus1

1199091003817100381710038171003817100381711990910038171003817100381710038171003817

le 120578 (119879119899minus1

) (119879119909 minus 119909 + 119886119899minus1

)

+10038171003817100381710038171003817[120578 (119879) 119886

1+ (120578 (119879) minus 1) 119879

119899

119909 minus 119879119899minus1

119909 ] 11990910038171003817100381710038171003817

le 120578 (119879119899minus1

) (119879119909 minus 119909 + 119886119899minus1

)

+ [120578 (119879) 1198861+ 120578 (119879

119899minus1


)]

times 119909

(5)

This gives 119878119899

119909 minus 119878119899+1

119909 le 120578(119879)1198861119909 + [1 + (120578(119879) minus

1)119909]120578(119879119899minus1

)(119879119909 minus 119909 + 119886119899minus1


119909 le

120578(119879)1198861119909 + [1 + (120578(119879) minus 1)119909](119889

0119909

+119872)120578(119879119899minus1

) where

119889119899119909

=10038171003817100381710038171003817119879119899+1

119909 minus 119879119899

11990910038171003817100381710038171003817 (6)


120578infin(119879) = lim

119899rarrinfin

[120578 (119879119899

)]1119899

lt 1 (7)


infin

119899=1119878119899

119909 minus 119878119899+1




1003817100381710038171003817119878 (sdot 119909) minus 119878 (sdot 119910)1003817100381710038171003817


1003817100381710038171003817119879119910 minus 1199101003817100381710038171003817 119910

1003817100381710038171003817

le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 +1003817100381710038171003817119879119909 minus 119909 119909 minus

1003817100381710038171003817119879119910 minus 1199101003817100381710038171003817 119910

1003817100381710038171003817

=1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 +1003817100381710038171003817100381710038171198891119909

119909 minus 1198891119910

119910100381710038171003817100381710038171003817

le [1198891119909

+ 1205911198891119910

+ 1]1003817100381710038171003817119909 minus 119910

1003817100381710038171003817


(8)


120598(1198891119909

+ 1205911198891119910



119909119899= lim119899rarrinfin

119878(119899 119909) = 119878(119899 lim119899rarrinfin

119909119899) = 119909lowast




defined by

S (119899 119906) = 119906 +10038171003817100381710038171003817119879119899

119906 minus 119879119899minus1

11990610038171003817100381710038171003817119879119899minus1

119906 (9)



119899+1

119906 minus 119879119899

119906119879119899

119906 = 119906 + 119889119899119879119899






119899 120578(119879119899

)) Suppose 120578infin(119879) =

lim119899rarrinfin

[120578(119879119899

)]1119899





to 119901

(c) 119878(119899 1199090)minus119901 le [2119872+(120578(119879)+1)119909


119894=119899120578(119879119894

)


119879119899minus1

119909 + 119879119899

119909 minus 119879119899minus1

119909119909



References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909 + 119879119899

119909 minus119879119899minus1


119899 120578(119879119899

)) wehave

119878 (119899 119909) minus 119878 (119899 + 1 119909)

le10038171003817100381710038171003817119879119899

119909 minus 119879119899minus1

11990910038171003817100381710038171003817

+10038171003817100381710038171003817

10038171003817100381710038171003817119879119899

119909 minus 119879119899minus1

11990910038171003817100381710038171003817119909 minus

10038171003817100381710038171003817119879119899+1

119909 minus 119879119899

1199091003817100381710038171003817100381711990910038171003817100381710038171003817

le 120578 (119879119899minus1

) (119879119909 minus 119909 + 119886119899minus1

)

+10038171003817100381710038171003817

10038171003817100381710038171003817119879 (119879119899

119909) minus 119879 (119879119899minus1

119909)10038171003817100381710038171003817119909 minus

10038171003817100381710038171003817119879119899

119909 minus 119879119899minus1

1199091003817100381710038171003817100381711990910038171003817100381710038171003817

le 120578 (119879119899minus1

) (119879119909 minus 119909 + 119886119899minus1

)

+10038171003817100381710038171003817120578 (119879) (

10038171003817100381710038171003817119879119899

119909 minus 119879119899minus1

11990910038171003817100381710038171003817+ 1198861) 119909 minus

10038171003817100381710038171003817119879119899

119909 minus 119879119899minus1

1199091003817100381710038171003817100381711990910038171003817100381710038171003817

le 120578 (119879119899minus1

) (119879119909 minus 119909 + 119886119899minus1

)

+10038171003817100381710038171003817[120578 (119879) 119886

1+ (120578 (119879) minus 1) 119879

119899

119909 minus 119879119899minus1

119909 ] 11990910038171003817100381710038171003817

le 120578 (119879119899minus1

) (119879119909 minus 119909 + 119886119899minus1

)

+ [120578 (119879) 1198861+ 120578 (119879

119899minus1


)]

times 119909

(5)

This gives 119878119899

119909 minus 119878119899+1

119909 le 120578(119879)1198861119909 + [1 + (120578(119879) minus

1)119909]120578(119879119899minus1

)(119879119909 minus 119909 + 119886119899minus1


119909 le

120578(119879)1198861119909 + [1 + (120578(119879) minus 1)119909](119889

0119909

+119872)120578(119879119899minus1

) where

119889119899119909

=10038171003817100381710038171003817119879119899+1

119909 minus 119879119899

11990910038171003817100381710038171003817 (6)


120578infin(119879) = lim

119899rarrinfin

[120578 (119879119899

)]1119899

lt 1 (7)


infin

119899=1119878119899

119909 minus 119878119899+1




1003817100381710038171003817119878 (sdot 119909) minus 119878 (sdot 119910)1003817100381710038171003817


1003817100381710038171003817119879119910 minus 1199101003817100381710038171003817 119910

1003817100381710038171003817

le1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 +1003817100381710038171003817119879119909 minus 119909 119909 minus

1003817100381710038171003817119879119910 minus 1199101003817100381710038171003817 119910

1003817100381710038171003817

=1003817100381710038171003817119909 minus 119910

1003817100381710038171003817 +1003817100381710038171003817100381710038171198891119909

119909 minus 1198891119910

119910100381710038171003817100381710038171003817

le [1198891119909

+ 1205911198891119910

+ 1]1003817100381710038171003817119909 minus 119910

1003817100381710038171003817


(8)


120598(1198891119909

+ 1205911198891119910



119909119899= lim119899rarrinfin

119878(119899 119909) = 119878(119899 lim119899rarrinfin

119909119899) = 119909lowast




defined by

S (119899 119906) = 119906 +10038171003817100381710038171003817119879119899

119906 minus 119879119899minus1

11990610038171003817100381710038171003817119879119899minus1

119906 (9)



119899+1

119906 minus 119879119899

119906119879119899

119906 = 119906 + 119889119899119879119899






119899 120578(119879119899

)) Suppose 120578infin(119879) =

lim119899rarrinfin

[120578(119879119899

)]1119899





to 119901

(c) 119878(119899 1199090)minus119901 le [2119872+(120578(119879)+1)119909


119894=119899120578(119879119894

)


119879119899minus1

119909 + 119879119899

119909 minus 119879119899minus1

119909119909



References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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