ISSN 0304 - 9892 ·.J

JfiBnBbha

·VOLUME 24

Special Volume to Honour Professor J.N. Kapur on J;iis 70th Birthday

1994

Published by :

The Vijnana Parishad of India . DAYANAND VEDIC POSTGRADUATE COLLEGE

Bundelkhand University ORAI, U.P., INDIA

JNANABHA

H.M. Srivastava University of Victort.·, Victoria, B.C., Canada

EDITORS

AND R.C. Singh Chandel

D. V. Postgraduate College Orai, U.P., India

Editorial Advisory Board Chief Advisor: J.N. Kapur (Delhi)

R.G. Buschman (Loramie, WY) L. Carlitz (Durham, NC) K.L. Chung (Stanford) L.S. Kothari (Delhi) I. Massabo (Rende, Italy) K.M. Saksena (Kanpur) R.C. Mehrotra (,Allahabad) S.P. Singh (St. John's) C. Prasad (Allahabad) L.J. Slater (Cambridge, U.K.) B.E. Rhoades (Bloomington, JN) K.N. Srivastava (Bhopal) D. Roux (Milano, Italy) A.B. Tayler* (Oxford, U.K) H.K. Srivastava (Delhi) V.K. Verma (Delhi)

The Vijnana Parishad of India (Society for Applications of Mathematics)

(Registered under the Societies Registratio:p..Act XXI of 1860) Office: D.V. Postgraduate College, Ora~OOO, U.P., India

President: Vice-Presidents :

Secretary-Treasurer: Foreign Secretary :

R. D. Agrawal (Vidisha) R. K. Saxena (Jodhpur) V. P. Saxena (Gwalior) G. C. Sharma (Agra)

COUNCIL J.N. Kapur (Delhi) B.S. Rajput (Srinagar) M.K. Singal (Meerut) G.S. Niranjan (Principal, D. V. Postgraduate College, Orai) R.C. Singh Chandel (Orai) H.M. Srivastava (Victoria)

Members M. P. Singh (Delhi) S. L. Singh (Hardwar) R. P. Singh (Bhopal) P.R. Subramanian (Madras)

*Born: September 5, 1931 ; Di'ed (of cancer) : January 28, 1995

This volume of JN AN AB HA

is being dedicated to honour Professor J.N. Kapur

on his 70th Birthday

PROFESSOR JAGAT NARAIN KAPUR Born : September 7, 1923

Felicitation on his 10th Birthday : September 7, 1993

Jnanabha, Vol 24, 1994 (Dedicated to Professor J.N. Kapur on his 70th Birthday)

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Jnanabha, Vol24, 1994 (Dedicated to Professor J.N. Kapur on his 70th Birthday)

PROFESSOR J.N. KAPUR : MAN AND MATHEMATICIAN

By R.C. Singh Chandel

Secretary, Vijnana Parishad of India, D.V. Postgraduate College, Orai - 285001 U.P., India

Professor J.N. Kapur (Jagat Narain Kapur), today needs no introduction to the academic community of India. However, to make formality we may say that simple, unassuming, energetic, dynamic, witty and over mailing, a varasious reader, an aloquent speaker, a prolific writer, an out standing researcher, an excellant teacher, an able administrator, an eminent educationist, encouraging his followers, a true and devoted salesman of Mathematics; these are only some of the qualities of Professor Kapur as a man.

Professor Kapur was born in Delhi on September 7, 1923. He w::iR ed11c::ifa~d ::it D.A.V. High School, ·Daryaganj and at Hindu College, University of Delhi. He had a first class first throughout. He beat the Delhi University records in B.A. (Hons) and M.A. examinations by wide margins securing 91% and 97% marks respectively. He also won three gold medals for his brilliant performance.

Soon after passing his M.A., Professor Kapur joined Hindu College Delhi as a l.ec.t.umr, where he served for fifteen years and miLaLliHlw<l himHelf as a teacher of repute. In 1957, he got Ph. D. from Delhi University. In 1959 he joined the University of Delhi as a Reader. In 1961 he left Delhi to join I.LT. , to organise Mathematics Department. His stay at Kanpur (1961- 1970) was a period of intense research activity. By his imagination, foresight, understanding and hard work, he established to Department at I.LT. as one of the strongest schools of Mathematics in the Country. On 31st December, 1970, Professor Kapur took up an assignment as ·Vice-Chancellor of Meerut University Meerut at so early an age of forty seven. He produced himself as good adminil:ltratur a8 a maUwrnaticio.n. During the period of his Vice-Chancellun;hip, he visited all the 56 Colleges affiliated to the Meerut Univen;iLy many times and talked with students, teachers, managements and public to solve their problems. He made the atmosphere of the University peaceful and brought the University to the top. It was due to his serious efforts that the right kind of academic atmosphere prevailed and the academic activities of the University were consequently at their peak during the period he was the Vice-Chancellor.

Dudng his Vicr~-Chirncnllnrship, hr~ continued his academic work also with same sprite and when he found that his administrative <lutieH were interfering with his academic work, he resigned during

4

the second term of Vice-Chancellorship of Meerut University and Joined back IIT, Kanpur in August, 1974.

In September, 1983 his 60th birthday was celebrated in different part of India. He was retired but was requested to continue for three years more at I.I.T., Kanpur. He was only the Professor of the presetigious Institutie to have been given this honour.

In October, 1986 he returned to Delhi and joined I.I.T., Delhi, Delhi University as an Honourary Visiting Professor for a period of two years.

He was also selected as an Indian National Science Academy (INSA) Senior Scientist for 1987, 1988 and 1989. In October 1988, he joined the School of Computer and System Sciences of Jawahar Lal Nehru University, New Delhi as Honourary Professor for life. Associated with Professor Kapur since 1970 through "JANABHA" a Journal of "VIJNANA PARISHAD OF INDIA" whose I am one of the Editors and Prof. Kapur is its Chief Advisor. He always en courages us. The "VIJNAN P ARISHAD OF INDIA" in its present form is due to inspiration of Professor Kapur. He may be treated as founder President of present "VIJNANA P ARISHAD OF INDIA" because before him Parishad was an unregistered body having no President.

There are millions of students who appear in university examination. Out of these, there are a few thousands who get first division in all positions in all their examinations. Out of these, three may be a few who beat earlier university records in both their first and second degree examinations. Out of these there is only one who beat these earlier records by wide margins of 4% and 15% marks. This man is Professor Kapur.

There are tens of thousand of persons who get Ph.D. degrees. A number of their theses are based on ten or more published papers. There was one thesis of more than 900 pages submitted for Ph.D. degree and this was based on 30 published papers. This was submitted by Prof. Kapur.

Thero are tens of thousand of mathP.maticians who publish research papers. Out of these, a few hundred publish more than 100 research papers. Prof. Kapur is one of them, having published more that 400 research papers.

There are research workers who work in one area all their life. Some work in two or three areas. Very few worked in ten or more different areas. Prof. Kapur is one of them.

There are many research guides who gude students for Ph.D. degree. There are some who p,11ide more than 30 studunts. Prof. Kapur is one of them.

There are many persons who write general scholarly, articles on educational matters. There are a fow who publish more tha11 500 such articles. Prof. Kapur is one of them.

There are amny authors who write dozens of non-ficion books. Prof. Kapur is one of them, having writt0n more than 60 books, nrnre than GO of which :ire innovative in nnturu.

5

There are a large number of authors whose books are published locally. There is a smaller number of authors whose books are published by international publishers. Prof. Kapur is one of them. Four of his books have already been published in USA and Canada.

There are persons who have written books for primary level or secondary level or senior secondary level or undergraduate level or post graduate level or reserach level or general books or books on education. Prof. Kapur has written books at all these levels.

There are tens of thousands of scientists in the country. Out of these about a thousand are Fellows of one or the other of the three National Science Academies. There are about 200 who are Fellows of all the three Academies. Proff. kapur is one of them and is quite senior among them.

There are many mathematicians who have been pres~ilent of Indian mathematical Society or Calcutta Mathematical Society or Bharat Ganita Parishad or Indian Society of Theoretical and Applied Mechanics or indian Society of Agricultural Statistics or Mathematical Section of Indian Science Congress or Physical Science Section of National Academy of Sciences or of Associations of mathematics Teachers of India or of Vijnana Parishad of India or of Mathematical Association of India or of Indian Society of Industrial and Applied Mathematics. Only Prof. Kapur has been the president of all of them.

He has also been Vice-President of Operation Research Society of India, Indian Society for History of Mathematics, and Society of Scientific Values. This also reflects his wide interest in Pure and Applied Mathematics, Statics Theoretical and Applied Mechanics, Operations Research, Mathematics Education, History of Mathematics and Scientific Values.

There arc many Indian mathematicians who have been visiting professors abroad and out of thorn, some have been visiting professors in developed countries like USA, Canada and Australia. Prof. Kapur has been a full visisitng professor in eight universities of developed countries. He has been a visiting professor not only in departments of Pure and Applied Mathematics but also in departments of Management Science, Industrial Engineering, School of Business and System Engineering. At present he is honorary professor in a school of Computer and System Sciences. This also shows the wide range of his interests.

There are many persons who worked as editors of journals or on editorial boards of journals. Prof. Kapur has worked as editor of four journals and has been on the editorial board of two dozen other journals.

Many persons have directed summer schools organised by the UGC or NCERT. Some have directed a number of them. Prof. Kapur has directed about 30 of them. What is the more important is that he is one of few persons who have organised the summer schools on their own initiative.

6

There have been thousands of Vice-Chancellors of Indian Universities, but only a few who got this position before they were even 4 7 years old and fewer still who were requested to accept this position. Infact his Cancellor got levave of absence for him from IIT/K without his even asking for it. He was a Vice Chancellor who stood firmly for 100% fairness and impartiality in all admissions, appointments and examinations. It was he who conducted the CPMT examination in UP for four years without a single complaint. There were plenty of complaints before and after his time, but none during his time. In fact in an extraodinary resolution, the UP government congratulated him on his outstanding work which looked like a miracle at that time.

Again many eminent persons like their birthdays to be celebrated. However, when his students collected some funds for celebrating his 50th birthday, he prohibited them from doing so and these funds became the nucleus for a mathematical Sciences Trust Society to work for the development of Mathematical Sciences in India.

When the UP government wanted to give honararium for the work he did for CPMT, he refused to accept it and at his instuuce the UP governement donated the amount to MSTS. He gave the royality of most of his books to the Trust. The Trust has published about 30 books so far and gives prizes worth about Rs. 20,000 per year to mathematical Olympia! winners and to all those who stand first in Mathematics in Board and University examinations all over the country.

Many persons can claim to have done something for mathematics educations research by using government money. He does not use government money. He believes in giving und not in taking. He believes that the country will develop when the people depend on their own resources rather than on government resources for serving the country.

Many persons in India have personal libraries. Some mathematicians and Scientists in India have their personal libriaries. But few have a library of the size of Prof. Kapur's Library. His library consists of more than 5000 books, 1000 journals and 200 Ph.D. theses.

Prof. Kapur has won many awards including GP Chatte:r:ji awnrd of ISCA, Education Minister's Gold medal of N.A.S.C .. Yop;iji Maharnj Ccnoterary Award and best paper awards in many internnti01rnl conferences. He has won honours inclufling Visiting Professorships, Fellowships of Academies, PreRi<lent.ships of Societies, Vice- Chaueellership etc. but he haR mwer asked for them. Ile hw; never hinted to anybody that he should be considered for any one of these.

He believes in the joy of work. He enjoys reading, writing, lecturing, oevering. He neither expects appereciation or reward for his work. If there is anything he expects from his students. Friends and admires, it is only that thRy may give their best Lo mathematics science country and man-kind.

7

Prof. Kapur has many firsts to his credit. He was the first to organise the summer schools in India, of course with the cooperation of others. He was the first head of the mathematics department of IIT, Kanpur and he built this department to international stadards in a short time. He was the first author to write books on Mathematical Statics, Mathematical Modelling, Mathematical Models in Biology and Medicine, Maximum Entropy Models and Entropy Optimisation Priciples, Biographies of Indian Mathematicians and Olympiad Problems. In many cases his books were the first of their kind in the world.

PROFESSOR J.N. KAPUR AT A GALANCE NEME DATE OF BIRTH AFFILIATIONS

ADDRESS

EDUCATION

POSITIONS HELD 1944-59

1959-1961

1961-86

1986-88

Jagat Narain Kapur 7-9-1923 Honorary Professor School of Computer and Systems Sciences, Jawahar Lal Nehru University, New Delhi - 110067 & Honorary Director MathP.matical Science Trust Society, New Delhi C-766, New Friends Colony, New Delhi, 110065 India (001) 6832290. B.A. (Horn;;.) Mathematics, 1942, Delhi University, India First Class First M.A. Mathematics, 1944, Delhi University, India First Class First Ph.D. Mathematics, 1957, Delhi University, India Certificate course in Statistics (ICAR) First Class First in first batch. Broke Previomi Delhi University Records in both B.A. (Hons.) and M.A. examinations by marginR of 4 percent and 15 percent marks hy securing 91 and 97 percent marks respectively. Won Three Gold Medals and Six Scholorships.

Senior Lecturer, Hindu College (Graduate School) Delhi University, India 1945-1959 Reader (Associate Professor) Instituti~ of Postgraduate Studies, Delhi University, India, 1959-1961 Professor, Indian Institute of Technology, Kanpur, India 1961- 1986. Visiting Professor , Mathematics Department, IIT, Delhi & Delhi University,

8

1987-89 Senior Scientist, Indian National Science Academy,

1988-90 Adjunct Professor, Waterloo University Canada

1988 upto date Honly. Professor, school of Computer and Systems Science, Jawahar Lal Nehru University, New Delhi.

VISI'f[NG PROFESSOR l'niversity /\rkan~.;as

Country USA

Carnegie.Mellon USA Siena Italy Manitoba

W aterloCJ \7\TntC>rloo

Manitoba

Manitoba New South Wales Flinders Carleton Waterloo Waterloo I.I.'T'. Delhi Waterloo Delhi W:i1.r'rloo Waterloo

Canada

Canada Canada Canada

Canada Australia

Australia Canada Canada Camirla India Canada lndiu Canada Canada

Year 1969 1969-70 1970 1980-81

1981-82 1982 1983

1984 1984

1984 1985 1985 1086 986-88 1987 1987-88 1988-89 1990, 91 & 1992

Department Mathematics Mathematics Mathematics Business and Acturial Mathematics Applied Mathematics System Engineering Actuarial and Management Sciences Industrial Engineering Applied Mathematics

Mathematical Sciences Busim~ss School SyRtems Design Enginnering Syo;tem Design Engineering Mathematics System Design Engineering MathematicR Systems Design Engineering Syo;tems Design Eng-ineering

FELLOWSHIPS TN PROFESSIONAL ORGANISATIONS

Indian Academy of Science nation;1! i\.cademy of Sei(-mces Institute of Mathematics and iLs Applications, UK

YEAR OF ELECTIONS F.A.Sc. F.N.A.Sc. f<'.I.M.A.

1fl65 1965 1966

lndi:rn National Science Ac;:irlemy F.N.A. 1969 PRESU)J!.:N'I'8H1P OF PROFESSIONAL ORGANISATIONS Indian Science Congress Associotion (Mathematics Section) HbaraL Gmiil.a Parishad

1968

1969

Indian Mathematical Society Calcutta Mathematical Society

1971,1972 1975

Association of Mathematics Teachers of India 1977-90 Indian Society of Theroretical and Applied Mechanics 1978 Mathematical Association of India 1979* Indian Society of Agricultural Statistics 1984 National Academy of Sciences (Physical Sciences Sect.) 1984 Mathematical Sciences Trust Society 1973* Vijnana Parishad of India 1985* Indian Society of Industrial and Applied Mathematics 1993* Indian National Commission on History of Sciences (Modern Period) CHIEF EDITOR The Mathematics Seminar Bulletin, Mathematical Association of India The Mthematics Student (IMS) The Mathematical Education (UGC) National Council of Educational Research and Training Upper Primary mathematics Text Books Indira Gandhi National Open University Text books on Mathematics MEMBERSHIP (PAST OR CURRENT)

1993

1963-68 1969* 1968-70 1984*

1987-89

1987-89

• of Editorial Boards of Twenty Indian and Foreign Journals

9

• (Chief Advisor : Jnanabha Published by Vijnana Parishad of India, since 1970)

• of 25 professional societies in the worded • International Commission of Mathematics Instruction (India's

Representative) • Indian National Commission on History of Sciences • Excutive Committee and Council, Indian Science Congress

Association. • Council, Sectional Committee and Basic Sciences Committee of

INSA

• Council, of National Academy of Sciences • Organizing Committee of Fourth International Conference on

MathenrnticH l Mocl1~ll inp,-.

• Organizing Committee of First and Second International Congress of 'reaching of Mathematical Modelling.

• International Conforence on Tra11::;vuruLiu11 (Chairmann, Mathematics Committee)

10

• Central Board of Secondary Education (Chairman, Mathematics Committee)

• national Council of Teacher Education • university Grants Commission Mathematics Panel • Innovations Committee of National Council of Educational

Research and Training, India • Review Committee of Regional Colleges of Education (Chairman) • Council of Indian Society of History of Mathematics (V ce­

President)

• Council of Operations Research Society of India ( VIce-President) • Science Education Forum of Indian Sceince Congress (Convenor) • Parmar Institute of Mathematial Sciences Shimla, India

(Honorary Dicector) • Indian Mathematical Society (Academic Secretary') • Council of Society of Scientific Values (Vice-Chairman) • National Commiitee on Mathematical Sciences (DST) (Chairman) • N atiorni l Committee on MathemaLical Education and Research

(DST)

• Governing bodies of nine U.P. Engineering Colleges. PUBLICATIONS : RESEARCH PAPERS (ABOUT 4000 PUBLISHED PAGES)

Internal Ballistics of Orthodox Guns (21), Internal Ballistics of Special Guns and Rockets (15), Form Function for Multitubular Charge (8), International Ballistics of Cpmposite and. Moderated Charges (14), General Fluid Flows (15), Compressible Fluid Flows (9), General Non-Newtonian Fluid Flows (27), Non-Newtoni:m Fluids in Inl0t Regionr. (H), Contlucting Non-Newtonian Fluid Flows (3), Megneto Hydrodynamics (20), Heat Transfer (2), Uencral Ppulation Dynamics (8), Difference Equation Population Models (6), Population Models with Time Delays (8), Prey Predator and Competition Models (7), Age-Structured Population Models (16), Mathematical Bioeconomics (14) Bio-mechanics (4), Compartment Analysis (3), Stochastic Processes (14), Measures of Information and Their Properties (50), Entropy Optimization Principles (16), Maximum Entropy Principle in Statistics Statistical Mechanics, and Operations Research (19). MEP Models in Marketing, Political Science, Economics Business, Search Theory, Population Dynamics, Pattern Recognition, Image processing, Fexible Manufacturing Systems, Coding Theory and Regional and Urban Planning (16), General Information Theory (9), Financial Mathematics Decision Theory and Social Sciences (7), Flexible Manufcaturing Systems (12), Fibonacci Numbers (12). Geometry (23), Innovation Deffusion Models (2) Survey Papers (9), Miscellaneous (10), Measures of Information nnd Their Applications ( 60), Statistics ( 4).

PUBLICATIONS : GENERAL ARTICLES (ABOUT 10000 PAGES)

11

Mathematical Education (178), Expository Mathematics (135), Applications of Mathematics (55), Mathematics Problem (12), Education (22) Higher Education (86), Vice-Chancellor Addresses (34), Convocation Address (6), Articles/Book Reviews (115), General Articles (18), Editorial (30), Scientific Values (6). PUBLICATIONS BOOKS: (ABOUT 16,000 PAGES)

Advanced Level Books (13), Expository Mathematics (23), Mathematics Olympiad Problem Solving (6), Text Books (12), Mathematics Education (17), Higher Edication (3), Bcok Edited (10).

ADMINISTRATIVE EXPERIENCE Vice-Chancellor, Meerut University, India (1971-74). Head of

postgraduate mathematics Deptt. (32 Years), Acting Director, Indian Institute of Technology, Kanpur, India on different occasions for approximately 250 days Director of Thirty Summer Schools of Mathematir.s, Organizer of a dozen national and international Conferences. AWARDS N ationul Academy Gold Medal for best research 1980 National Lecturer, University Gransts Commission 1982-83 G.P. Chatterjee Award, Indian Science Congress Association 1988 Distinguished Service Award, 1983 Mnthcmalic.:ul A1:1sociation of India Platinum Jubilee Lectureship, Indian Science 1988 Congress Association Best paper Award, Administrative Science 1986 Congress i\ssnr.i::ition Best paper Award, Administrative Science 1986 Association, Canada Felicitation by Yogiji Maharaj Trust, Gandhigram 1992 · UNIVERSITIES VISITED

USA (39), Canada (12), Italy (2), UK (12), West Germany (9), Australia (12), Netherlands <::n, Singapore (1), Iran (~), Thailand (1), Bangladesh (1). MATHEMATICS EDUCATION PROJECTS VISI'J'F.D

USA (14), UK (8); NethP.rl::inds (2), Italy (3), Australia (8). SEMINARS/LECTURES

Given in India (about (500), Given in other countries (about 200)

An Invitation to Contribute and Subscribe to

I JOURNAL OF MATHEMATICS

(An International Journal devoted to Research in Mathematics and Mathematical Sciences)

( ADVISORY BOARl2_j

R.P. Agarwal (Lucknow, INDIA) AP. Dwivedi (Kanpur, INDIA) K.M. Saksena (Kanpur, !NOIA) S.L. Shukla (Bakewar, !NOIA) Vikramaditya Singh (Kanpur, !NOIA) D.K. Sinha (Shanti Niketan, !NOIA)

( EDITORIAL BOARD .)

Mihir B. Banerjee

Department of Mathematics Himachal Pradesh University Summer Hill Shimla-171 005, !NOIA

Sanford S. Miller

Department of Mathematics State University of New York College at Brockport Brockport, New York 14420, U.S.A.

R.S. Pathak

Dapartment of Mathematics Banaras Hindu University Varanasi - 221 005, INDIA

G.C. Sharma

Department of Mathematics Institute of Basic Scienr.e.s Khandari Agra - 282 002, !NOIA

Por details please write to :

Satya Dea

Department of Mathematics and Computer Science Rani Ourgawati University Jabalpur - 482 001, !NOIA

Katsuyuki Nishimoto

College of Engineering Nihon University, Koriyama Fukushima-ken JAPAN 963

11.R. Roy Department of Mathematics Jadavpur University Calcutta - 700 032, !NOIA

H.M. Srivastava

Department of Mathematics & Stats. University ul Victoria, P.O. Bux 3045, Victoria, B.C. CANADA VBW 3P4

VINOD KUMAR Editor, Epsilon Journal of Mathematics

Department of Mathematics Cl1rist Church College

Kanpur - 208 001, INDIA

Jiianabha, Vol 24, 1994 Wcdiwted to Pro/('.ssor J.N. Kapur on his 70th Birthday)

AN OPERATIONAL CALCULUS FOR THE INDEX zF1 - TRANSFORM

By N. Hayek and B.J. Gonzalez

Departamento de Analisis Matematico, Universidad de La Laguna, 38271 La Laguna (Tenerife), Canary Islands, Spain

(Receiuecl: September 1,1993; Revised: Februw:y 28,1994)

ABSTRACT · In this paper an operational calculus for the index 2F 1 -

transform is developed. Furthermore, this calculus is applied to solve certain type of differential equations involving generalized functions.

I. Introduction The index ? 1-transform for real-valued functions is defined in

[3] (see also [4], by the formula:

where

F('C) = r F (µ, ex, 'C, x)f(x)dx, 0

... (1.1)

F(~t, ex 'C, x) = 2F 1(µ + ~ + i'C, µ + ~ - i'C; µ + 1; - x)xa, ... (J .2)

2F1 being the Gauss hypergeometric function. Here ex and µ are complex parameters and 'C is a positive real parameter.

This integral transform was extended to certain spaces of generalized functions (see [I)]). For it, a testing function space (namely, U ) wull introduced in the following manner: u umooth function a, ~l, a. <)>(x) on 0 < x < 00, belongs to U if and only if a,µ,a

Yk ~1 (<11) = sup I (2x + l)a xµ12 - a (x + 1)1112 Ak <jl(x) J < oo •.• (1.3) , a, , ex 0 < x <

00 x

with a E ro, }), µ, ex E c, k = 0, 1, 2, .. . and where Ax denotes the differential operntor :

A :=x<x-µ(x+l)-µD xµ+l(x+l)µ+lD x-a. . .. (1.4) x x x

The topology in U is induced by the countable family of • u, µ,a

seminorms {y, } . It turns out that the kernel function F(µ, a, t, x) 1l, a, ~l, ex

is in Ur, 11

rv· As it is usual, the dual space of U . is denoted by I I rt,,",('(

U' , and I denotes the real interval (0, 00). a,p,cx

14

The transform F(t) of the generalized function f E U' is then a,µ,a defined as follows :

2:F/f) = F('t) = < f(x), F(µ, a, t, x) > , t > 0 ... (1.5)

and for a E [O, %) and f E E'(I) the following inversion formula holds :

< f, <!>) = lim (JN S(µ, 't) G(µ, a, t, x) F(t)d't, <j>(x) ) ... (1.6) N---c; oo 0

with Re a> 0, Re µ > 0, ~ < Re (µ-'a) < ~ and Re C% -a) < - %· Here,

and

S(µ, 't) = 2

2 't sh n 't r(µ +' l + i't) r (µ + l - i't)

nl(µ + 1) 2 2

G( ) _ µ - a F (1 · i · . 1· ) µ, a, 't, X - x 2 I z + lt, z - l 't ,µ + , - x .

Further more, the following uniqueness theorem was proved : if f,gE E'(I) and 2 :F1[f] = 2 :F1fg], thenf=g.

In [5] was also established that for every f E U' there exists a,µ,a a non- negative integer 't such that ·

2:F1(f) = F('t) = 0 ('t2r-Re µ -i)' 't ~ oo ... (1.7) On the other hand, the following relation holds :

2:F1 (<Axff)= (- l)K [~ + i\2 + 't2 r 2:F1(/), KEN. ... (1.8)

A'x being the adjoint operator of Ax defined by (1.4).

In this paper we consider the operational equation : P(A' )u =g, x

where g E E' (I), P is any polynomial different from zero in (- oo, 0) and a, µ E R .

Our aim is to find a generalized function u. E £' (I) satisfying the above operational equation. Spaces 'IX.I) and E(l) and their duals D'(l) and c'(/)have its usual meaning [7].

2. The operational equation Let us consider the following equation :

P (A' )u =g, x ... (2.1)

where g E E'(I), P(z) denotes an arbitrary polynomial without zeros in - 00 < z :s; 0 and just as in (1.8) Ax' is the differential operator

Ax' =x-a Dxxµ+ \x + l)µ+ 1 Dxxa-µ (x + 1)- µ'

which is the adjoint of Ax.

15

. By applying the generalized index 2:r1-transform to both sides of (2.1) and using (1.8),we get

2

P (-(µ+ t) -12) U('c) = G(1),

where U(1) and G(1) represent the generalized index 2:F1-tranform of 2

u(x) and g(x), respectively. Now, since P (- ( µ + i) - 12 J t= O,by means of the inversion formula (1.6), and for any<)>~ v}J it is fJund that

< u, <)>) = lim ( t S(µ, 1) G(µ, a, 1, x) G(1)2 _d1, c)>(x))

N ~ = 0 p (- (µ + t) _ 12) ... (2.2)

Thus, we have obtained formally a solution of (2.1). Now we must prove that (2.2) is certainly a generalized function and that it satisfies the equation (2.1). For this, the following Lemma is required.

Lemma 2.1 For x E [a, b] and µ > 0, there exists a T1 > 0 such

that V 1 ~ T1

I zF 1 (t + i1, t - i1 ; µ + 1; - x )I ~ B 1

and there also exists a T 2 > 0, such that. V ·r. > T 2

1

I H' (1 . 1 . . 1· )I < B - -Z" 1 2 + l1, 2 - l1 ' µ + ' - x - 2 1 2

with B1 > 0 and B2 > 0.

Proof. Starting from the integral representation [6] (p. 248).

F ( r:i.. • _ ) _ r(y) 4 [ K ( 2s ) J (2 ) a. + R _yd 2 1a,..,,y, x -r(a)r(p)xa.+~/2 0

a.-~ --IX y-1 ss P s.

which is valid for Re a> 0, Rep> 0, and K, J being the well-known Bessel functions, we can write :

F (1 . 1 " 1 )- f(µ+ 1) 4 -112[K (2s)J 2) -11d ?. 1 2 + L't:, z - L't:, µ + , - X - l . l . X 2· . . I ( S S S

· f(li + 11)f(2 - vr) o "Cz 'IX µ

Hence, for a suitable T 1 > 0, if 1 ~ T 1 one has

Thus, for 1 ~ T1,

I K2ti (¥xJI ~Ko P~J · I H' (1 . 1 . . 1· I < B Z" 1 2 + 1.1, 2 - 1.1, µ + ' - x - 1

with B 1 > 0.

On the other hand, for 1 -) ()0,

16

IK (2s_.)! < ,,, -7TT _.!_ 2,i '-lX _ 1v1 1 e T 2

holds for certain M1

> 0 (see [2] 7.14.2 (69)).

Moreover, Jr

r(1+i'T(1-ic)=~hnc. Then for T 2 T 2 :

12F 1(±+i1,t-i1;µ+l;-x)I <;,M

21- 112e-wrchn1( s-µ IJ~/2s)!ds<;,B2t-t

B2 being a positive constant ' D

Now, if we again consider the equation (2.1) it results, after taking a polynomial Q(z) of degree r + µ + 1 without zeros on - co< z:::; 0, r being the integer given in (1.7), that the convergence of the right- hand side of (2.2) can be established as follows. First, we have

<fN, ~rj ) .S(µ;r) G(µ, a, i-, x) -·· - 2 2 di-, <j>(x) = 0 p (- (~l + ~) - "C )

< JN S(µ, i-)G(µ, a, i-,x)G(i-) di- <j>(x) ) = Q (A') I 2 . 2 ' x 0 p (- (µ + ~)2 - i-2) Q (- (µ I- 2) - i; )

< fN S(µ, i-) qjµ,~ x) G (i-~ 2 - di-, Q (Ax) <j>(x) ) . 0 p (- (µ + ~)2 - i-2) Q (- (µ + ~) - "C )

This follows by making use of :

A'x G (µ,a, i-, x) = -[(µ + ~)2 + T2] G (µ,a, i-, x).

Thus, if the support of <t> is eontainod in [a,, 6J, tlie expression (2.2) cau be written as

lim JN -----~~' T)G(l! -- -·--·di- t G(µ, a, i-, x) <j>(x) dx N--'Joo 0 P(-(µ+~)2-"C2)Q(-(µ+1)2-T2) a

... (2.3)

Then, by invoking Lemma 2.1 we can find suitable constants C,D,E,N1 and N 2 such that

t /-----~(µ1_T)GJ!l_1;-·---·-1 clT t. IG (µ,a, T,x) <)i(x) ldx <;, o P(-(µ+ 2)2-12)Q(-(~t+~)2 12) a

17

< c JN! I S( ) 1) I d - 0 p (- (µ + ~)2 - 12) Q (- (µ + ~)2 - 12)

1 +

D (2 I G(1)S(µ, 1) d + Ni p (- (µ + ~)2 _ 12) Q (- (µ + ~)2 _ 12)

1·

~ 1 2F1 (~+i1,~-i1;µ+l;-x)I dx+

E r 12' - ~ I S(µ, 1) I d N2 Ip (- (µ + ~)2 - 12) Q (- (µ + lf.2)2 - 12) I

1·

Now, it is not difficult to prove the boundedness of the first and the second integrals above. For the third one, taking into account that , for 1 -7 oo,

I S(µ., 1) I :::;; M12~1 + 1, M > 0,

this integral converges as N -7 ""·

Therefore,the integral (2.3) exists and thus, by the completeness of Tl (I), there exists f E Tl (1) such that

lim <JN G(1) )

S(µ, 1) G(µ, a, 1, x) 2 2 d1, <jl(x) = < f, <jl >. N -~ "' 0 P ( - (µ I ;) ·-· 't )

... (2.4)

The generalized function f determined in (2.4) is the restriction

of u E £'(/) to 'IJ' (1). In view of the continuity of the differentiation and the multiplication by x and by .! in 'If(l), one can show that, for any

<jl E ~I)

lim N->~

< P(A' ) ( S(µ, -r) G(µ, a, 1, x) G(-r) 2 2

d-r, <)>(x) ) = < P(A' ) f, $ > x 0 p (- (µ + 1) - 't ) x

and from this expression and the above inversion formula, it finally follows that

< g, <jl > = ( P (A'x) f; <jl)

1'his result proves th;:it the eenernlizd function f E 'IJ' (1) is the

restriction of u E £'(/) to 'iJ(l) and that it satisfies the equation (2.1)

Acknowledgements The authors thank Professor H.M. Srivastava for his kind

suggestions.

REFERENCES Ill A.Erdelyi, W. Magnus, F.Oberhettingcr and F.G.Tricomi, Higher Transcendental

Functions, Vol. I, McGraw-Hill, New York, 1953. [21 /\.. Erdclyi, W. Magmrn,F. Obnrhcttingcr, F.C. Tricomi,Tablca of' Integral

Transforms, Vols.I and II, McGraw-Hill,New York,I954.

18

[31 N.Hayek, B.J.Gonzalez and E.R. Negrin, Una clase de transformada indice relacionada con la de Olevskii, Actas XIV Jornadas Hispano-Lusas de Matematicas,Univ. La Laguna Vol. I, (1989), 401-405.

[41 N. Hayek, B.J. Gonzalez and E.R. Negrin, Abelian theorems for the index zF1-transform, Reu. Teen. Fae. lngr. Uniu. Zulia 15 (1992), 167-171.

[5] N. Hayek and B.J. Gonzalez, The index zF1-transform of generalized functions,Comment. Math. Univ.Carolinae, 34 (1993),657-671.

[6]

[7]

B. van der Pol and H. Bremmer. Press, New York, 1964.

AH. Zemanian, Generalized Pubishers,New York,1968.

Operational Calculus, Cambridge University

Integral Transformations, Interscience

0

Jnanabha, Vol24, 1994 (Dedicated to Professor J.N. Kapur on his 70th Birth.day)

MULTIDIMENSIONAL FRACTIONAL DERIVATIVES OF THE MULTIPLE HYPERGEOMETRIC FUNCTIONS OF SEVERAL

VARIABLES By

R.C. Singh Chandel and P.K. Vishwakarma

Department of Mathematics

D.V.Postgraduate College, Orai-285001, Uttar Pradesh, India (Received: October 21, 1990; Revised: February 26,1992)

ABSTRACT In the present paper, multidimensional fractional derivatives of

the multiple hypergeometric functions of several variables have been obtained.

1. INTRODUCTION Motivated by the earlier work of Srivastava and Goyal (11],

Srivastava, Chandel and Vishwarkarma (12] derived various fractional derivatives iuvolviug Lhe generalized mulliple liypergeumeLric fuudion of Srivastava and Daoust (10] and the multivariable H-function of Srivastava and Panda [13]. They also discussed the special cases of these results involving the multiple hypergeometric functions of several variables and their confluent forms defined by Lauricella [8], Exton [5],[6]), Chandel [1], Chandel and Gupta [2], and Karlsson [7].

In the present paper, we apply the same techniqueR in order to derive the multidimensional fractional derivatives involving the multiple hypergeometric functions P..J:.>, P..J;l, P../jl and P..j} of Lauricella

[8] and their confluent forms dn) <I>(n) 'l'(n) <I>(n) (k)E(n) (k)E(n) of ~1 ' 2 ' 2 ' 3 ' (1) D ' (2) D '

Exton ('(5] [6]) (k)E(n) of Chandcl [1] (k)p(n) (k)p(,n) (k)p(n) of Chllildcl ' (1) C ' AC' BD ' AD

d G t [21 d th · fl t fi (k)m(n) (k)n.(n) an up a , an eir con uen orms (l)'!'A<;' (2 )'-!'AC'

~~j<P~~' m<l>~_b and (k)p(c'i of KarlHson f7l.

(2.1)

2. FRACTIONAL DERIVATIVES Making an appeal Lu Lhe formula [9,p.67]

Dµ I 'Al = r(A. + l) 'A - µ R (~) - 1 x x l(A. _ µ + l) x , e /\, > ,

we derive the folloing fractional derivatives involving the above multiple hypergeometric functions :

l n

'A - µ 'A p 'A. - 1 n) . (2.2) D 1 i ... D " ,, IT x 1 p<A [a, µ1, ... , µ. , x x J n

[ II .i = 1

20

(2.3)

(2.4)

cl, ... , en; z1x1' .. ., znxn]}

n r(\J -I, J) Xµj-

1 ?An) [a, A

1, ... , A ; c

1, .. ., c ; z

1x

1, ... z x ],

(~!. J n n n n .I

=TI j=l

lz1x1 I+ ... + jz11x

11I<1, Re('.\)> 0, i = 1, ... , n.

A-fl le-µ Di i ... D" ,,

x1

x 1 n le. - 1 (n)

IT ~/ FA [a, b1, .. ., b11

; Al' ... A11

;

j=l

z1xl' ... ,znxn))

n r(A) µ -1 = ,I1 ~ x/ F;_l [a, b1, .. ., bn; µ1' .. ., µn; z 1xl' ... , zn:x),

;=l J

I z1x1 I 1 ... + I z x I < 1, Re(A.) > 0, . i = 1, ... , n. n n 1.

D~1 µ

1 ... D~" -µ,, l n.

1 n j = l

A. - l rln) b b . x.1 .l'B [µl, ... ,µn, ]'"'' n'

J

c; z1x1, ... , znx,.))

n f'(A.) µ -1

= rr f'(:.) x/ p.~i) [Al' ... , A.,n' bl, .. ., bn; c; z1x1, ... , znxn], J=l J

max j i21x1 j, .. ., lz11x11 I}< 1, Re(~)> 0, j = 1, ... , n.

l n

'A - fl le - µ le - 1 n) (2.5) D xl 1 ... Dx" " IT XjJ . F1 [ a1, .. ., a

11,

1 " .i=l

(2.6)

17

= IT }=1

µl'"'µn;c;z 1x1, .. .,z x j n. Tl

r(/...,.) fl - l "" ) J ~ xJ l'B~n [ a 1, ... a , !c

1, .. ., /...,

11; c; z1xl' .. ., z,,x

11 ,

r(µ) J 17

max {I zlx11112, .. ., I z,?n 11/2} < 1, Re (\) > 0, i = l, ... , n.

I. -fl le fl l 11 l D x1 i ... D/ ,, IJ x>- 1 Ff!:·) [a, b; A.,l' .. ., \i;zlxl' .. ., znx,) l " j= 1

21

n f(;\,.) µ - l rr f(µI.) x/ . F~) [a, b; µ1' ... , µn; ZlXl' ... ,Zn xn), j=l J

lz1x1 1112 + ... + lz x 1112 < 1, Re(;\,.)> 0, i = 1, ... , n. n n 1.

(2.7) ), - fl ).. -- µ

DI 1 ... D II II

x x I II

n ).. -1 ) IT x/ Ji'.{;) [a, µ1, ... , µn; c; z1x 1, ... , znx,,J

j = 1

n I(;\,.) µ - 1

= rr f(:) x/ PF» [a, Al' ... An; c; zlxl, ... , zllx,), J = 1 .I

max j lz1x1 I, ... , lznxn I}< 1, Re(\)> 0, i = 1, ... , n.

(2.8) f..- f..-1 11

A.-1 I µI n µII j (k) (n) . '· DX ... DX rr xj (l)ED ra, Al' ... , All, c, c, I n }- l

z1xl' ... , znxn])

n r·~ . (J\,) µ - 1 (k) ( ) I . , ]

= TI ~ x.1 . EDn [a, A1, ... A ; c, c; z1x1, ... , znxn , . f(µ.) .I (i) n

.I= 1 .I

Re('A.)>0 lz.x.I <r., i=l, ... ,n; r1 = ... =rk, l ' l l l

rk+l = ... =rn,rk+rn=l.

(2.9) I n f.. - µ A. - µ A.. - 1 (k) (n) '

D 1 1 ... D n II l TI x.J (,.,)ED [a, a, p.1' ... , µn x x .I ,, 1 n }= 1

c; zh, ... , znxn])

n f('A.) p -1 k

TI ~ J ( .)E(n) [ ' ~ ~ · c· z x z x ] = X · (2) D a a' J\,1' ··· J\,n' ' 1 1' ... , n n '

' - 1 f(µ,.) J )- . Re('A.)>0 lz.x.I <r., i=l, ... ,n; r1 = ... =rk,

l ' l l 1.

r k + 1 - ... rn, r k . r n - r k + r n.

(2.10) l n A. -1 A. µ A. - µ J (k) (n) , .

Dx1 l ... Dx11 II n x. (l)EC [a, a' "'1' ... , "'n' I 11 • } • •

J=l

; z1xl' ... , z,,x"I)

22

- rrn f'(~) µ) - 1 (k)E(n) [ I b . J - . f'(µ.) xj (1) C a a' µ1, .. ., µn' zlx1, .. ., znxn '

J = 1 J

Re (A)> 0, \z?i \ < ri, i == 1, ... , n; (~ + ... + ,r;:;;)2

+cw-~-+-;-+ ... + 1T,Y = i.

n

(2.11) A. -µ A. -µ

D 1 1 ••• D" II

x x TI A.. - 1 (k)r:!.n) b x/ rAC [a, b, bk+ 1, ... , n;

I I j= 1

Al' ... , An; zh, ... , znxnJl

n f'(~) µJ-1 (k)rin) . . x] == n -- x. rA'c[a,b,bk+l'""b ,µl, ... µn' 2 1xl'"" 2 n n' . f'(µ.) J n

J = 1 J

Re (Iv.)> 0, j = 1, .. ., n. J

(212) D A.k+l-µk+l DAI!-µ/! j nn A.-1 (k)rin) [abµ µ. ' x ... x :K_J"'.J rAC ' ' k+l'"" n'

k+I n }=k+l

==

(2.13)

c,, ... ,en; zl' ... , z,, zk + i"k + l' ... , znxnJl

n f'(lv.) µ-1 (k)rl) ~ . ·z n _..:_,[__ x .J Pfc [a , b, Ak + 1' .. ., f\,n 'cl, .. ., en, 1' .. ., f'(µ.) J j=k+l J

2 k' 2 k + 1 xn + l' .. ., 2 nxn],

Re (Iv) > 0, i == k + 1, .. ., n ;

(\21\112+ ... + lzk\112)2+ \zk+lxk+ll + ... + \znxn\ < 1.

A.-t A.- jn A.-1 I ~I n µn j (k) n) . DX ... DX nxj ~D[a,Al'"""'n'

1 n j= l

c; ck +l' .. ., en ;z,x,, ... ,znxnJ)

n f'(tv.) ~t - 1 k)rri ) ] = TI . __ :__,r_ xJ C Pfn [a, A.1' .. ., lvn; c; ck+

1, .. ., en; z1x1' .. ., z

11x

11 ,

. ' f'(~t.) .! 1 = J. .1

Re (Iv.)> 0, i = 1, .. ., n. /,

23

max{lz1x1 1, ... , lzkxkl)+ lzk+lxk+l+ ... + lznx) <l

(2.14)

l

n A. - A.--µ A.-1

D k+t µ11+1 D 11 11 IT xi (k)-nin) [ab b ·c· x ··· x j l'An ' 1' ... , n' ' k+J II =k+l

'-k+l' ... , A,,;z1, ... , zk'zk+i"k+l' ... ,z,,x,,l)

n r(A) µJ - 1 (k)-nin) b . . z ... z ' IT ~ x. l'AD [a' bl' ... , n 'c, µk + l' ... , µ11' 1' ' k r(µ.) .! .i=k+l .!

(2.15)

(2.16)

2 k + lxk + 1' ... , 2 nxn],

max{lz1I, ... , lzkll+ lzk+lxk+ll + ... + lznxnl <1,

Re (A.)> 0, i"" 1, ... , n. !

l

n A..-1 (k) n) . IT x/ p(BD [a, µk+ 1' ... , µn' bl,

.! = k + 1

A. -µ A. -µ Dk+l k+l ... D" II

xk + 1 xn

... , bn; c; z1, .. ., zk' zk + 1xk + 1' .. ., znxn])

n r(A-.) µ -1

IT ·-·-1 x J (k)ri.n) [a ~ ~ b b · c· z z re.) J l'BD '/\.,k+l' ... ,/\.,n' 1, ... , n' ., 1, ... , k' j=k+l µJ

2 k + lxk + 1' ... , 2 nxn],

max {iz1 I, ... , lzk I, lz11 + 1x1l+ 1 1,. . ., lznxn I}< 1,

Re(\)> 0, i = k + 1, .. ., n.

A-1 A.-µ112

A-1 I ~ 1 11 II j (h) 11) Dx ... P,, IT xj F'Rn [a, ak + 1' ... ,an,

I II j = l

- x 11 ~l1' .. ., µn; e; 2 1xl, ... , ""n n J

ll r(A,.) µ - 1 = IT _r __ (:...,L x J (11JF,B11L!' [a 'a, 1' .. ., a ' A1, ... , A ; c; z1x1, ... , z x, J,

. ~l -) .J _, 11 + . n . n n 1

.! = 1 .I

max { lz 1x1 1, .. ., lz11x,Jl<l, Re(A)>O, i=l,. . .,n.

A. p A. -p 1 n l l 11 11 A. - l (h) ;-i,_n) • (2.17) DX ... D,, n ~i' 1 ('![)fa, b, ~LI, .. ., ~lk, c,

l II .i= 1

24

A,+ 1' ... ,A,,; z,x,, ... , z,,x,,l)

n l(/c) µ - 1 ]

I1 J J (k)-nfn) [ b 'l 'l . -. µ µ · z x z x , = -- x. l'CD a, 'l\,l, ... ,l\,k'l' k+l''"' n' 1 l'"'' n n . f(µ.) J . J = 1 J

Re (/c.) > 0, i = 1, ... , n. l

(2.18) A-µ A -µ

D 1 1 ... D" II x x

I

n A -1 rr j ";;(ll) [ µ b b . xj -1 µ1, ... , n' 1, ... , n-l'

j=l

c;_ z ,x,, ... , z,,x,,l)

n f'(A.) µ - 1 ) J fl ___:___I__ J =(n ['l 'l b b · c· z x ... z x , = X- -1 1\,1'""1\,n' 1'"" n-1'' 11' 'nn . 1(µ.) J

J = 1 J

Re (A .. ) '> 0, i - 1, .. ., n. l

(2.19) f n-1

A - . 'A -µ A.-1 D 1 µ1 D ,, _ 1 II _ 1 . n x. , x ... x l J

1 11-l j=l

-(n) [a a µ µ l; =-1 1' ... , n' l' ... , n -

c; z1x 1, .. ., z 1x 1, z ]) n - n - n

n I l'(A.) µ - I - n _}_ j =(n) [a 'l 'l . c· - . f(µ.) xj -1 1' .. ., an, l\.l' .. ., l\.n-1' '

J = 1 J

z1x 1, .. .,z 1x 1,z], n - n -· n

Re O·) > 0, i = 1, .. ., n - l.

(2.20) !..-µ I._-µ

1J1 1 ... vn--1 11-1 :I\ X11 - I [

n-1 A -1 n x/

.i= 1

<P~n) [~ll' ... , µn - 1;

r. . -· ... ?' x .. l fl ) .<· r' 1' ... ~ n - 1 n l' ~ n

n - 1 r(A,.) µ _ 1 = _IT r(µ/.) x/ <l>~n) [A,l' ... , A,n -1; c; zlxl' .. ., zn- lxn. - l' zn],

.I= 1 J

Re (A..)> 0, i = 1, ... , n - l. /,

25

(2.21) '!'~') [a, A1, ... , An; zi"I, ... , z nxn]) 1

n /..- /..- f..-1

D 1 µ1 ... D II µII I1 x j x x J

1 II j = 1

n l(A.) µ -1 = IT __J_ xJ qi(

2n) [a, µ

1, ... , µ ; z

1x

1, ... , z x ].

. !(µ.) J n n n J = 1 J

Re (A.)> 0, i = 1, ... , n. L

(2.22) A ·- µ A - µ { n A. - 1 )

1 1 II II J (n) . . D ... D TI x. <P2

[µ1

, ... , µ , c, z1x

1, ... , z x ]

x x J n nn 1 II j = 1

n l(A) µ _ 1 _ IT _._! J r1-.(n) ['"' '"' · c· z x z x ] - X · '¥2 rel' · · ·' re ' ' 1 1' · · ·' ' . f(µ.) J n n n

J = 1 J

Re (A) > 0, i = 1, ... , n.

ln-1

1'-µ A-µ 1'-1 (2 20) D 1 1 1J n -1 n 1 IT x j qi(n) [a µ ... µ - .

. x . . . x .i D ' 1' ' n - 1' '

(2.24)

(2.25)

1 n-1 .i=l

c; zi"I, ... , zn _ 1xn _ 1, zn])

11 - l f(A) . 1

I1 _:_J:_ µ - (n) [ 1 '"' · c· z x x .1 <PD a, rel, .. ., ren - 1' - ' ' 1 1' . l(µ.) J J = 1 J

Re (Ai)> 0, i = 1, ... , n - 1.

... , z 1 x 1' z ], n- n- n

A-µ 1'-µl" A-1 ) 1 1 n 11 j (k) (n) • . Dx ... Dx IT x.i (l)<PAC [a, b, Al' .. ., A,,, z1x1, ... , z,,x,,]

l II j = 1

n l(A.) µ -1 - fl -·-1 x .i (k)r1-.(n) [a b· µ µ · z x z x ] - ( . (l)'YAC ' ' 1' ... , ' 1 1' ... , ' r 1[.) .I ll n n

j = 1 ~ )'

Re (\) > 0, i = 1, ... , n.

A- !n 1'-1 1 µ1 A - ~1 i (11)< (n) • Dx ... Dxn II IT xf (2) I>AC [a, bk+ 1' .. ., b,, 'Al'

1 II .I= 1

'"I • .,, x - x Jl n n n ... ' /\, ' ,., 1 1' .. ., ,, J

n f(A.)

TI __:__r_ µ - 1 (k) I>(n) [ b b · µ µ · z x z X ] = f( .) X'JJ (2)< AC a, k + 1' ... , n' 1' ... , n' 1 1' ... , n n' .i=l µ}

26

Re (A)> 0, i == 1, ... , n.

(2.26) fi. -µ

D\ +I - µk +I ... D II II x x

ft+ 1 II

" fi. - 1

TI j (h\p(n) [ I[ · xj (2) AC a, µh + 1' ... , 'n'

j = k + 1

cl, ... ,cn; 2 1' ... ,zk' 2 k+lxk+l' ... ,znxn]

n r(lc)

n _ _j__ µJ - 1 (k) (n) [ 1 1 . .

. r(µ.) xj (2)<l>AC a, rck + l' ... , rcn 'cl, ... , en' 2 1' ... zk, .1=k + 1 .I

2 k + 12 k + l' ... , 2 nxn],

Re (A.)> 0, i == k + 1, ... , n. l

(2.27) J..-µ A·-µ!n fi.-1 l D 1 1 D " n TI x J (k)<J/n) [a µ ... µ · c· z x ... z x J x ... x j (1) AD ' 1' ' n' ' 1 1' ' n n

1 n j= 1

nn f(~) µJ - 1 (k) (n) • . J == -- x. (l)<l>AD [a, !c1, ... ,A, , c, z1x1, ... , z

11xn,

. r(µ.) .I n .I= 1 .I

(2.28)

Re(/,.,.)> 0, i = 1, ... , n.

l fi. -µ ! n A -1 l A - µ 11 n J (k) (n) . . Dx1 1 ... Dx TI x1 (l)<l>BD [a, µ1, ... , µ", c, z 1x1, ... , znxn)

1 n j=I

nn r('J) µJ- 1 (k) (n) . .. ] == . r(µ.) xj (l)<l>BD [a, A1, ... , An, c, z1x1, ... , znxn ,

.I= 1 .I

Re (/,.,.) > 0, i == 1, .. ., n. l

(2.29) ! n

fi.J- 1 (k) (n) TI xj (?.J<l>BD [a, ~1k 1 1' ... , µ,,, J=k+l

fi. -µ fi. -µ Dk+l k+1 ... D" II

x n xk + 1

, b1, ... ,bn;c;z1, .. .,zk,zk+r'k+I' .. .,znx.J)

n r(A.) µ -1 __ .!_ X J (k)<l>(n) [ 1 1 b b . f(µ.) j (2) BD a, l'vk + l' .. ., l'vn' l' .. ., n'

.I

= TI j=kll

c; z1, .. ., zk, zk + 1xk + 1' ... , z 11x,),

Re (\) > 0, i == k + 1, ... , n.

(2.30) D\ -µ1 ... D\-1 -µ11-1 xl x

11 ! n

.B fi. - l ; (kJ,, . .<nl [ a

~,· (2) ''sn a, k , 1' ... , un,

n f(A.) --~' f(µi) rr

.i=l

27

µ 1, .. ., µ ; c; z 1x1, ... , z x ]l n n n

_µ - 1 (k),n(/1) [ ~ ~ . Xj.i (2)'!-'BD a, ak + 1' ... ,an, 1\,1' ... , l\,n'

c; z 1x 1, ... , z x ], n n

Re C\) > 0, i == 1, ... , n.

ACKNOWLEDGEMENTS The authors are thankful to the Council of Science and

Technology (Uttar Pradesh, India) for providing financial assistance. They are also thankful to Professor H.M. Srivastava ( University of Victoria, Victoria, B.C., Canada) for his valuable suggestions.

REFERENCES [1] R.C.S. Chandel,On some multiple hypergeometric functions related to Lauricella

functions, Jiianabha Sect. A 3 (1973), 119-136 ; Errata and Addenda, ibid. 5 (1975), 177-180.

[2] R.C.S. ChandPl :oind A.K.Gupta, Multiple hypergeometric functions related to Lauricella's functions, Jnanabha 16 (1986), 195-209.

[3] R.C.S.Chandel and P.K. ViRhwakarma, Karlsson's multiple hypergeometric function and its confluent forms, Jiianabha 19 (1989) 173-185.

[41 R.C.S. Chandel and P.K. Vishwakarma, Fractional integration and integral representations of Karlsson's multiple hypergeumelru: function and Its conilucnl formo, Jnanabha 20 (Hl90), 101-110.

[5] II. Exton, On two multiple hypergeometric funclions related to Lauricella's F!Jl Jnanabha Sect. A 2 (1992), 59-7::l),

[61 H. Exton, Multiple llypergeometric Functinn..q and Applica.tiuni;, .John Wiley ancl 8ons, New York, 197G.

[7] P.W. Karlsson, On intermediate Lauricella functions, Jiian.abha· 16 (1986), 211-222.

[SJ G. Lauricella, 8ullc funzioni ipetgeomciriche a piu va1iabili, Rend. Gire. M£1t. Palermo 7 (1893), 111-158.

[9] K .. B. Oldham and J. Spanier, The Fractional Calculu.s, Academic Press, New York and London, 1974.

[10] H.M. Srivastava and M.C.Daoust, Certain generalized Neumann expansions associated with the Kampe' de Feriet function. Nederl. Akad. Wetensch.Proc. Ser. A 72 = Indag. Math. 31 (1969), 449-4117.

1111 H.M. Srivastava and S.P. Goyal, Fractional derivatives of the H- function of several variables, J. Math, Anal. Appl. 112 (1985), 641-651.

[12] H.M. Srivastava, R.C.S. Chandel and R.K. Vishwakarama,Fractional derivatives of certain generalized hypergeometric functions of several variables, J. Math. Anal. Appl. 184 (1994),560-572.

1131 H.M. Srivastava and Il. Panda, Some bilateral generating fon"t.ionR for a class of generalized hypergeometric polynomials, J. Reine Angew. Math. 283/284 (1976), 265-274.

Jiianabha,Vol24, 1994 (Dedicated to Professor J.N. Kapur on his 70th Birthday)

CERTAIN GENERATING FUNCTIONS FOR THE KONHAUSER'S POLYNOMIALS za(x; k)

n

S.N. Singh and L.S. Singh Department of Mathematics and Statistics, Avadh University

Faizabad, 224001, U.P., India (Received: March 10, 1990; Revised: April 20, 1994; Final form: 3cptember15,1994.)

ABSTRACT In this paper, we derive some generating relations for one set

Z~(x; k) of the Konhauser biorthogonal polynomials. For these

polynomials we ·have established some generating relations involving Kampe de Feriet's double hypergeometric function and Srivastava's general class of triple hypergeometric functions. These generating relations are deduced from certain well known results involving generalized hypergeometric polynomials.

' ' 1. INTRODUCTION

Konhauser [SJ discussed two polynomial sets ~(x; k) and

Z~(x; k) which are biorthogonal with respect to the weight function

(x)'1e-x over the interval (0, oo), where a> - 1 and k is a positive integer ~(x; k) is a polynomial of degree n in xk.

The biorthoguuulity rclutiuu i::; given by [3,p.303]

[ _ -.:ra r(km +a+ 1) xae x .I ·ex· k) za (x·k) dx = 0 o ,:: ,. p ' m ' . m. ! nm'

... (1.1)

where bnm is the K;ronecker delta.

Fork = 1, these polynomials reduce to the Laguerre polynomials L~~\x) and for k = 2, these polynomials were studied by Spencer and

Fano /9 I in certain calculaions involving the penetration of garnrnu rays through matter and were subsequently discussed by Freiser [7}.

These polynomialwwere further investigated by Prabhakar (5,6], Srivastava [llJ to [15], Patil and Thakare [4], Agarwal and Manocha [1 l and, Srivastava and Singh [10].

One set z:~(x; k) of the Konhauser's biorthogonal polynomials is given explicitly by [3].

30

~ . (n) xkj .L.., (- 1)' j l(k-

1=0

za(X- k) = JJ_kn +a+ 1) 11 ' n

... (1.2)

a> - 1 and k is a positive integer. An immediate consequence of (1.2) is the formula

Z~(x; k) = n 1 n . 1Fk (xlkl (a+l)k [-n; l

· (tik;a+l); ... (1.3)

µµ+1 µ+A--1 where l'i(A, µ) represents the array of parametes i -A-, ... , A

The Kampe de Feriet's double hypergeometric function (in the contracted notation of Burchnall and Chaundy [2, p. 112] is defined by

P.,2) X = L m + n m n

[

(a): (b): (c): l = [(a)] [(b)] [(c)) xmyn

(e):(g):(h): ,y m,n=O [(e)]m+n[(g)]m[(h)]nm!n!

... (1.4)

and a general class of triple hypergeometric functions, due to H.M.Srivastav11, is defined by [11, p. 428)

[

(a):: (b); (b'); (b"): (c); (c'); (c"); l ...,,(3) J1' x,y,z

(e) :: (g) ; (g'); (g"): (h); (h'); (h");

= L [(a)]l+m+n [(b)]l+mf(~')Jm+n[(~?n+l. l,m,n=O [(e)Jl+m+n [(g)]l+m[(g )]m+n[(g ))]n+l

[(c)Jl [(c')J m l(c")] n xi ym Zn

[(h)]l[(h')]m[(h")]11

• l ! m ! n ! · ... (1.5)

In the definitions of (1.4) and (1.5),as wAll as in what follffW8, (a) and (b) abbreviate the sequence of ~1 parameter::; a

1, ... , aA and the

n ' product (a1)

11, .. ., (aA)n. respectively.

Here, we deduce some generating relationH for the polynomials Z'"(x; k) from some well known results for certain generalized n hypergeometric polynomials (if, e.e;.) Sriv~rnt:wa and Manocha (16)

involving the function P..2> (x, y) and P..3l (x,y,z). All the generating relations, would reduce, when k = 1, to known results for Laguerre polynomials.

2. GENERATING RELATIONS The generating relations that we deduce here from certain well

known re::mlts are :

31

2. n=O

[(a)] n

[(e)) (ex+ 1) z~ (x; k)yn n kn

= p2) [(a) : ; -; (e): (ex+ l)lk, ... , (ex+ k)lk; -;

y, - jcx)k lk. y1} (2.1)

ex+l 1-ex+k) = [(a)]n (1- -k )n ... ( k n zrx-kn (x; k)yn

L [(e)]n (ex+ 1 kn) kn n n=O

[

(a); - ; 1 - (ex+ l)lk, ... , 1 - (ex+ k)lk; =F2)

(e): -; ; y, - {(xi k/ . y}l

... (2.2)

~ [(a)) [(b")l {(a)+ n, (c') _ m n F - L ((e)) ((g")l (a+l) A+C' E+H h'

n = O n n kn (e) + n, ( ); y l Z:: (x; k)z'

r(a) :: -; (b"): ; (c); -; 1

- .F"-3l { k } y, z, - (x/k) . . Z

(e) :: - ; -; (g") : (ex+ l)lk, ... , (ex+ k)lk; (h'); - ; ... (2.3)

and

"' [(a)] [(b")] (1 - ex+ 1) ex+ k I, n n k n ... (1- ---k-)n

n = o [(e)n [(g")] (ex+ 1- kn) n kn

F y . za - kn (x; k )Zn

[

(a)+ n, (c'); l A+E' E+II _(e)+n, (h'); n

'{ [(a)::-;-; (b"): -; (c'); 1 - (ex+ l)lk, ... 1 - (ex+ k)lk; l F') x, z, - {(xlk)k . z))

(e) :: -·; - ; (g") -; (h'); ------------------; ... (2.4)

Dcrivutions of (2.1) and (2.2) : From Srivu::iLuvu urnl Munudw (16,p.194), and Rainville [18,p. 321, we have

32

L [(e)] n [(h)]n H+B+IFC+G

= [(a)] [(c)] Jl_ n, 1- (h) - n, (b):

n = O n n 1 - (c) - n, (g);

(a): (b); (c);

= p2J y, [(- l)]c-h-1. xy

(e): (g); (h);

l yn x -

n!

... (2.5)

Now the result (2.1) would follow at once if we interpret (2.5) in the light of (1.3).

In deducing (2.2), from the known result (2.5) we make use of

(a+l-kn) [-n; l a-kn kn k Zn (x; k) = n 1 1Fk (xlk) .

· ~(k; a+ 1 - kn);

Derivations of (2.3) and (2.4) :

In case uf the function J;fi'i) Cr, y, z), we recall from Srivastava and Manocha l16,p.157] that

[(a)] [(b")]. [(c. ")] [(a)+ n, (c'); ~ n n n F · .{....J [(e)] l(g")) [(h")J A+C F:+H' ,

n = O /1t n n (e) + n, (h );

[

- n, (c), 1 - (h") - n; F "

.C+H"+l H+C (h),1-(c")-n;

[

(a) :: -; -; (b") : (c); (c'); (c"); =F3)

(e) :: -; -; (g") : (h); (h'); (h"); y, z, xz l

y]

l·z17

x ;!

... (2.6)

The specialised forms of (2.6) lead us to the desired generating relations (2.3) and (2.4).

ACKNOWLEDCTEMENT The authors are thankful to Professor H.M. Srivastava for his

v;iluablo help and suggestions.

REFERENCES fl I A.K. Arrarwal and H.L. Ma11udia, A note of' Konhauser sets of biorthogonal

polynomials, Nederl. Akad. Wetensch. Proc. Ser. A. 83, lndag.Math. 42 (1980), 113-118.

33

[2] J.L.Burchnall and T.W. Chaundy, Expansions of Appell's double hypergeometric functions.II, Quart. J. Math. Oxford Ser. 12 (1941), 112-128.

[3] J.D.E. Konhauser, Biorthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math. 21 (1967), 303-314.

[41 K.R. Patil and N.K. Thakare, Multilinear generating function for the Konhauser-biorthogonal polynomial sets., SIAM J. Math. Anal. 9 (1978), 921-923.

[5] T.R. Prabhakar,On a set of polynomials suggested by Laguerre polynomials, Pacific J. Math. 35 (1970), 213-219.

[61 T.R. Prabhakar,On the other set of the biorthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math. 37 (1971), 801-804.

[7] S. Preiser, An investigation of biorthogonal polynomials derivable from ordinary differential equations of the third order, J. Math. Anal. 4 (1962), 38-64.

[8] E.D. Rainville, Special Functions, Macmillan, New York 1960. [9] L.Spencer and U.Fano, Penetration and diffusion of X-rays : Calculation of

special distributions by polynomials expansion, J. Res. Nat. Bur. Standard 46 (1951), 446-461.

(10] A.N. Srivastava and S.N. Singh, On the Konhauser polynomials yj~l (x; k), Indian J. Pure Appl. Math. 10, (1979), 1121-1126.

(11] H.M. Srivastava,Generalized Neumann expansions involving hypergeometric functions. Proc. Cambridge Philos. Soc. 63 (1967), 425-492.

(12] H.M. Srivastava,On the Kanhauser sets of biorthogonal polynomials suggested by the Laguerre polynomials. Pacific J, Math. 43 (1973), 489-492.

(13] H.M.Srivastava, A note on the Konhauser sets of biorthogonal polynomials suggested by the Laguerre polynomials, Pacific J. Math. 90 (1980), 197-200.

[14] H.M. Srivastava, Some biorthogonal polynomials suggested by the Laguerre polynomials, Pacific J, Math. 58 (1982) 235-250.

(15] H.M. Srivastava, A note on a multilinear generating function for the Konhauser biorthogonal polynomials, SIAM J. Math.Anal. 14 (1983), 369-371.

[16] H.M. Srivastava and H.L. Manocha, A Treatise on Generating Functions, Halsted Press,Wiley, New York, 1984.

OD

Jnanabha, Vol24, 1994 (Dedicated to Professor J.N. Kapur on his 70th Birthday)

INTEGRALS ASSOCIATED WITH GAUSS'S HYPERGEOMETRIC SERIES, MULTIV ARIABLE

H-FUNCTION AND A GENERAL CLASS OF POLYNOMIALS By

R.K.Saxena, Chena Ram and O.P. Dave Department of Mathematics and Statistics,

Jai Narain Vyas University, Jodhpur-342001, Rajasthan, India

(Received: July 15,1993)

ABSTRACT A new class of integrals associated with hypergeometric series,

the multivariable H-function and a general class of polynomials are evaluated. The results obtained are of general character and include the integrals given by Sharma and Rathie [9] and Arora and Rathie [l] etc.

1. INTRODUCTION The object of thi8 paper jg to ovnluntc r,omc integrals involving -

product of hypergeometric series,the multivariable H-function due to Srivastava and Panda [11] and a general class of polynomials due to Srivastava [10]. The integrals evaluated in this paper extend the results of Sharma and Rathie f9l and Arora and Rathie [l] etc.

The · multivariable H-function introduced and studied by Srivastava and Panda [11] will be defined and represented in the following contracted form [12, pp.251-252, Eqns.] (C.l) - (C.3)]:

•· 0 . . . lzl . A(ll A<rl . (1) c<i> . . <r> c<rJ) l , n. m1, n

1, ... , m, n - (a.. . , ... , _ ) . (c. , _ ) , ... , (c. , .

1 H r r • J J J 1,p J J 1,p J J ,p : - (1) (r). , . (1) · (1).

1 . Ir) Ir) r

_p,q:p1,q1; ... ;p.q z (b1:B

1, ... ,B

1J1

.td1

,D1

J1 ; ... ;\d:,D.)1 r r r _ _ , q , <1

1 .1 .1 , 11,.

. r . r

-1 J f . II II ~. - -- ... \jl <s·, ... , s) <1>· <s·) z., ds1 ... ds ... <i.D

(2mv)r L L i r . i i . i r 1 r i=l i=l

wh(~re 1v = -0, m.

/. n.

l

II r (d~i) - D~i) S·) II r (1 - c~i) + dil S·) J J l - J J l ._, ._,

J- J - __ ,, __________ ., ____ ,,_, __ _ <1>/si) = q p. l ' II r (1- d~i) + D~i) S·) II r (c~i) - dil I;.)

J J l J J l

.i=m;+l j=n;+l

\;/ (i =- 1, ... , r)

... (l.2)

36

n r

n rel-a.+" A(i)S·) J L-i J l

\jf csl' ·· ., sr> = ·= 1 i = 1

q r p r

n rc1-b.+" B(i)S·) n r (a.- " A(i)S·) J L-i;t .I L-i.1i

j=l i=l j=n+l i=l

... (1.3)

A detailed account of the multivariable H-function can be found in the monograh by Srivastava et al. [12).

Following Srivastava [10), the general class of polynomials is defined as

[j3/a] (- 13) S~[x] = L u !au Ap, u (x)u; 13 = 0, 1, 2, ... . .. (1.4)

u=O

where a is an arbitrary positive integer and the co-efficients Ap,u (13, u > 0) are arbitrary constants, real or complex. On suitably

specializing the coefficients AJ3,u' si[x] yeilds a number of known polynomials as its special cases [13,pp.158-161]. For convenience, the H-function of r complex variables defined by (1.1) will be denoted by the contracted notation H[z1, ~ .. , zr].

2.INTEGRALS The following results are to be established here:

First Integral :

fl p-11 P[l bl 1-2p-1F[ s:.1 i;:. 2· t(l+a) ] o t ( ~ t) . +at+ ( - t) 2 I v,u, 2<v + u + ), 1 +at+ b(l - t)

where

. S~ [xRk]H[z1R~1, .. ., zrR~r]dt

2v+o-2r-1rc~+§ + l) 2 2

- (v - 0)(1 + a)P (1 + b)P + l l'(v) r (0)

[j3/a] ( A) ~ I-' au A (x)u k, u ! 13, ll

zt=O

V 1 () HO,n+3;m1,n1; .•. ;m,,n, . r-+-r- . { · [zl~l (2 2) (2) p+3:q+3;p,.q, ; ... ;pcq, z; 12

V 0 1 0, n + 3; ml, nl; ... ; m r' n r . • -r - r -+- H .

Jzl Jlj

(2) (2 2) p+3,q+3;p,.q,; ... ;pcq z; J2

. .. (2.1)

37

4(t+at)(l+b)(l-t)·Vii::{l 2, ... ,r} R-R= 2' ' - i [l + at + b(l - t)]

... (2.2)

v 8 I 1 = (1 - p - ku; A1, ... , Ar) ; (1 - p - ku + 2 + 2 ; A1, .. ., Ar);

(~ _§_ p ku· ~ ~ ) · (a · A(l) A(r) l · 2 - 2 - - ' Jl.l' ... , Jl.r ' j' j ,. . ., j ji, p'

(c(l), cl1\ ; ... ; (c(r), c<r\ .... (2.3) j } ,pl } j ,pr

I = (b. · B(l)· ... · B(r)) · (d(lJ D<1l) · ... · (d(r) D(rl) · 2 J ' J ' ' J l, q' J ' J l, q 1, ' .I ' J l, q /

v 8 v (1 - p - ku + 2 - 2; A1, ... , \); <2- p -ku; A,1, ... , \);

1 () (2 + 2 - p - ku; Al' .. ., \). ... (2.4)

v () J 1 = (1- p - ku; Al' .. .,\); (1- p -ku + 2 + 2; Al' .. .,\);

(_§_ ~ p ku·A A,)·(a ·A(l) · A(r)) · 2 - 2 - - ' 1' .. ., ,. ' j ' j ' ... , j 1, p'

(c\ll, c\1\ . ; ... ; (c(r), c<r))l ... (2.5) j j ,pl j j ,pr

J - (b · B(ll ·B<rl) · (d(l) D<1l) · · (d(r) D(rl) · 2- ., . , .. ., . 1 ' . ' . 1 , .. ., . ' . 1 ' ./ ./ ./ ,q } ./ ,ql J j ,qr

(l - p - ku - ~ + % ; A.1' .. ., \); <%- p - ku; Al' .. ., \);

1 v (2 + 2- p - ku; Al' .. ., \). . .. (2.6)

The (sufficient) conditions of validity of (2.1) arc given below : (i) The constants a and b are such that nons of the expressions

1 +a, 1 + b, 1 +at+ (1 - t), where. 0 $ t $ 1, lR ZP.ro. . .. (2. 7) (ii) Re(r) > o, Re(2{p - v - o) > o, k ~ o. .. . (2,8)

,. (iii) R.e{p + ku + ). 'A ~) > 0, 'A ~ 0 ... (2.9)

- /,I. J i=l

and S· = min [Re(d\il /D\il)] where u = 0, 1, .. ., [~/a] t i 5') 5' m 1 J

t

'\/ i r (1, 2, .. ., r) j = 1, 2, ... , m,. .

. (iv) n > o, I arg z ·I -::: -21

n n. ; vii:: 11, 2, ... , rJ ... c2.10J /. l l

where

38

p n. /. pi q

Q- -I A(i) + I c<il - I c<il -I BCil . -l .! .! .! .!

j=n+l )=l )= n + 1 I

j=l

m. qi I

+ I, Dyl - I, nyl > O; vi c: {1, 2, .. ., r) j= 1 .i= mi+ 1

Second Integral : 1 f tP-\1 - tf- 2 [ 1 +at+ b(l - t)r 2P + 1

2F1 [v, o; ~(v + 8);

0

t(l +a) a k A. A.

1 +at+ b(l - t)l S~ [xR ]H[z1R 11, ... , zrRrr] dt

2v+o-2pr(~+~) . 2 2

== (l + a)P(l + b)P- 1 f'(v) f'(o)

[~/a] (-A) f.I au A (x)u I u 1 ~,u

. u=O 1"'1 El] · · ·m n ,, 0 n + 3 ; ml, Ill' ... , r' r • v 1 !3..H' : . j r(2 + 2J f<2l P +a."+ a;p,. "'' ... ; P,. • z, E2

l · Jz1 G1]} 0 n+3·m n ; ... ;mr,nr . b 1 ' . ' l' 1 .

+ f(~) 1(2 + 2) Hp+ :3, q+ 3;pl,ql; ... ;pr' q z; G2

where R, R1, .. ., Rr are defined by (2.2).

v () E 1 = (2 - p - ku ; A1, .. ., \); (2 - p - ku + 2 + 2; /...1, .. ., P.);

(l 1-~ - ~ - p -ku; Al' .. ;,\); (aJ ;A)1l, .. ,Ay\ ,p;

(c(l), C(l))l ; ... ; (c\r>, drJ)l .! .I ,pl .I .I ,pr

E "' (b · 13(l) B(r)) · (d(l)) · · .(d(r) D(r)) · 2 .. ' ' ' ' .. ., ' 1 ' ' ] ' ... , . ' . 1 ' j j .1 ,q .I ,qi . .I .I ,qr

(2 p - ku + *- ~; A.1 , .. ., "}..,.); (1 + -2v - p - ku; A.1, .. ., \); ,_, ""'

3 0 (2 + 2 - P - ku; A.l, .. ., /..)

v 0 G 1 = (2 - p - ku; A1, ... , A.,.); (2 - p - ku + 2 + 2; A.1, ... , A.r);

... (2.11)

... (2.12)

... (2.13)

39

( 1 + ~ - ~ - p - ku ; A.1, ... , \); (ai ; Aj1l, ... , AY\ , P ;

(c(1l, c<l\ ; ... ; (c(r), dr\ ... (2.14) J J ,pl J J ,pr

G2

= (b.; B(1l, ... , B(r\ ; (d(l), n<l\ ; ... ; (d(rl, n<r\ ; J J ) ,q ) ) ,ql ) ) ,qr

v 0 0 (2 - p - ku - 2 + 2; A.1, ... , \); (1+2 - p - ku; A.1, ... , \);

3 v (2 + 2- P - ku; A.1' ... , A.r) ... (2.15)

The (sufficient) conditions of validity of (2.11) are given below : (i) The conditions (2.7) and (2.10) hold.

(ii) Re(p) > 1, Re(2p -v - o) > 2; k ~ 0. r

(iii) Re(p + ku + :E /\.. (;.) > 1; A.. 2: 0 'V' j, i = 1 i i J.

... (2.lG)

whem

~· = min [Re(d~i) /D\il)] 'I/ i £ {1, 2, ... , r} i 1 ~j~ m. J J

'

... (2.17)

Third Integral :

f1t/2 o. ew<2P + l)G (sin 0)P (cos O)P -· l 2F 1 [v, o; ~(v + o + 2) ewe cos O]

. S~[xTt]H[z 1~1, ••• , zr~r]d0 v 0

e(w(p + ll/2l1t r c2

+ 2

+ 1)

= 22p-v-o+ 1 r(v) r(o) rev -8)

[~/ex] (- ~)

L . (u) 7u A~, u(x)u u=O

0 n + 3; m n · .. : m n 1 1 1z I l {r (v + 1)/2) r (- o/2) H ' 1• 1' ' r, r :

P + 3, q + 3 ; P1' q1, ... ; P , q z. I r r 2

V o 1 O,n.+3;m1,n.1; ... ;mr,nr1Z~ Jl]l - r(-) r(- + --)H :

2 2 2 p+3,q+3;p1,q1; ... ;pr,q Zr J2 J ... (2.18)

where Il' 12' J 1 and J 2 are defined by (2.3), (2.4), (2.5), and (2.6)

respectively and w = ...r-:T. Also

40

4e2w0 sin 8 cos e . vi £ {l, 2, ... , r} T = T. = · wn/2 ' l e

... (2.19)

The (sufficient) conditions for the validity of the integral (2.18) are given below :

(i) Re(p) > 0, Re(2p - v - 8) > 0, k 2. O; r

(ii) Re(p + ku + I: A.. S·) > O; A,. 2. 0 V j ; i = 1 l l l

where<;. is defined by (2.17). l

Fourth Integral :

J:12 ew<2P - l)O (sin 8)P - 2 (cos 8)P - 1

2F

1 [v, 8; ( v ; 8) ; ewe cos 8]

. Sp[xrk]H[z 1T~1, ... , zrT>]de

ewn(p - 1)/2 r (~ + ~) [~/a] (- ~) = 2 2 " au A (x)u

22p- v- 3 r(v) r(8) u~O (u) ! ~. u

v 1 (i 0, n + 3; m , n1; ... , m , n

{ [

z1E1] . f(2 + 2} r (2) H r r r :

. p+3,q+3;p1,q1; ... ;p,,q, z; E12z G ]}

v (i 1 0, n + 3; m1

, n ; ... ; m , n 1 1 + I'(2) r(2 + 2)H r r r : ••• (2.20)

P + 3, q + 3; P1' qi; ... ; P , q z. G r r 2

where T, T 1, ... , Trare defined by (2.19) and E 1, E 2, G1 and G2 are defined by (2.12), (2.13), (2.14) and (2.15) respectively.

The (sufficient) conditions for the validity of the integral (2.20) are given below :

(i) Re(p) > 1, Ue(2p - v - o) > 2, k2. 0. r

(ii) Re(p + ku - 2 + I: A,.<;.)> O; A.. 2. 0 \i j i = 1 l l l

where Si is defined by (2.17) ..

(iii) ThP. (2.16) alfio hold. Fifth Integral:

fn/2 . · ·

ew<2P + l)e (sin 8)P - 1 (C08 8)P F [v o· .!.(v + 0 + 2)· () ~·:.'.:2'.1 I '2 '

ew(O-n/2) sin 8) S(f lX7wl H [z 1T~1, ... , 2,.T>J de

.... (2.21)

ewitp/2 r (v /2 + 8/2 + 1)

22p - v - 8 + 1 (v _ 8) f(v) 1(8)

[p/a] (- ~) "\:"' au A (x)u L,, u ! p, u

u=O

l V 1 8 O,n+3;m ,n ; ... ;m.,n rzl 111 re-+-) r (-) H I I r r :

2 2 2 p+3,q+3;p,q; ... :p,q z. l I I r r r 2

y 8 1 O,n+3;m,n; ... ;m,nfzlJlj) - r (-) r (- + -) H I I r r :

2 2 2 p+3,q+3;p ,q; ... ;p ,q z. J I I r r 2

where T, T1, T2, .. ., T,. are defined by (2.19).

41

... (2.22)

and 11, 12, J 1 and J 2 are defined by (2.3), (2.4), (2.5) and (2.6)

respectively.

The (sufficient) conditions for the validity of the integral (2.22) are given below :

(i) Re(p) > O; Re(2p - v - 8) > O; k ~ 0.

(ii) The condition (2.16) holds.

Sixth Integral :

fit/2

ew(Zp - l)e (sin e)P - 1 (COR fl)P - 2 F [v o· .!(v + 8): 0 . 2 l ' '2 ..

ew<e -it!Z) sin 9] S~ [X'.Z*] H fz 1T~1, ... , zrT>]de

e1(/rrp/?. r c~ + ~) [p/cxJ <-Pl = 2 2 "\:"' au A (x)u

22p - v - 8 r(v) r(8) u~O u ! p, u

{ v 1 8 0, n + 3; m , n ; ... ; m, n Jzl E.ll . re-+-) r (-) H I 1 r r :

2 2 2 p + 3, q + 3, p 1' q 1; ... ; P,, q z. E · · r 2

V 8 1 0, n + 3; m. , n ; ... ; m 'nJ .zl Glll + (-) r (- + -) H 1 1 r , :

2 2 2 p+3,q+3,pi,q ; ... ;p,q z· G 1 r r 2

... (2.23)

where T, Ti are defined by (2.19) and E 1, E 2, G 1 and by (2.12),(2.13),(2.14) and (2.15) respectively.

G2 are defined

The (sufficient) conditions of validity of the integral (~.~8) are given below :

(i) Re(p) > 1, He(2p - v - 8) > 2; k 2 0.

(ii) The condition::; (2.16) and (2.21) also hold.

42

Seventh Integral : 1

fo tcr- l (1-t)cr-µ- l ?1 [v, -v; µ; t] s~ [xtk (1- t)k]

. H[z 1R 1; ... ; zrR~dt

·2- zv- 1 I'( ) l~/aJ (- ~) lrel!:2 - _2v + _21) = µ L au A (x)u

. r (µ - v) u 1 ~. u rcr: . _) u=O 2 +z+z

rz 1 L 1j rel!: - ~) 0, n + 3; m , n ; ... ; m, n 2 2

.H 1 1 r r : ----

p+S,q+Z;pI,ql; ... ;pr,qr z~ Lz re~+~)

0, n + 3; m , n ; ... ; m , n 1 1 .H I I r r : Iz L*])

p +a, q + 2;pI, qI; ... ;pr' IJ :z; L;

where Ri = [t(l - tJ"; ; V i c {1, 2, ... , r)

L 1 ..- (1 - cr - ku; A.1, ... , \) ; (1 - cr + µ - ku; A1, ... ,Ar);

(1- cr + ~ + i-ku; A.1, ... , \); (aj; Aj1)); ... , AY\.p . (c~l), dl\ ; ... ; (c(r), c<r))l

} } ,pl J } ,pr

L2 = (b .; B(1), ... , B~r\ ; (d~l), DP\ . ; ... ; (d~r>, D\r)\ ; J } J ,q J } ,qI } J ,qr

... (2.24)

... (2.25)

... (2.26)

(l ~ 2a + µ + v - 2ku; 2A1, ... , 2\); (1- O" + ~ - ~ ~ ku; A.1, .. ., Ar)

... (2.27)

L~ = (1 - cr ·. ku; Al' .. ., f..) ; (1- cr + µ - ku; Al' ... , \);

(1 O" ku + µ + ~ · A. A. )· (a ·A (1) A (r) ) · 2 - - 2 2 ' 1' ... , r ' j' j ' ... , .i l,p'

. (c~1 ), c<1))1 ; ... ; (c1rl, dr\ J .I ,pl J .I ,pr

L * - (b . B(l) B(r)) . (d(l) D(l)) . . (d(r),D(r)) . 2 - ., . ' ... , . 1 ' . ' . 1 ' ... , . . 1 '

.I .I J 'q .I .I 'q 1 } .I 'qr

... (2.28)

(1 - 2cr + µ + v - 2ku; 2A.1, .. ., 2\); (~ - a+~ -"i- ku; A1, .. ., I.)

... (2.29)

43

The (sufficient) conditions of validity of (2.24) are given below: (i) Re(CT) > 0, Re(µ)> 0, Re(CT - µ) > 0, k 2 0. . .. (2.30)

r r

(ii) Re(CT+ku+ I: A-S-)>0,Re(CT-µ+ku+ I:A.S-)>1; i=l l l i=l l /.

... (2.31)

\ 2 0 \:/ j; u = 0, 1, ... , (~/o:) and j = 1, 2, ... , mr.

and d(i)

S· = min [Re(__l__(·))], \:/ i E 1, 2, ... , r) ... (2.32) i 1 s,· s m D.i

. i J

(iii) The condition (2.10) also holds.

Eighth Integral :

I~ t<J- 1 c1-tf-µ- 1 zF1 [v, -1-v; µ;tis~ [xti c1- t)k1

. H[z1R 1; ... ; zrRr]dt

= 2- 2v - 2 r(µ) [I] (- ~)au A (x)u {2 r.(~ - ~ + ~) reµ - v) - u i ~, u rcr: .:.... =-)

U-O 2+2+2

[zl L3] µ n!:!: - ~) O,n+4;m

1,n

1; ... ;m,n 2 2 H r r • _

. p+4,q+3;pl,qI; .. ,;pr,qr z: L1 rc¥-+~+1) . f' I ~ 2

[z L *j l 0, n I 3; m , n

1; ... ; m , n 1 1

.H 1 I I :

P + 3, q + 2;p" q,; ,,;p,, •, z; L; J ... (2.33)

where R 1, .. ., Rrare defined by (2.25) and Lr and L; are defined by (2.28) and (2.29) respectively.

L 3 = (1- u -ku; A1, . ., Ar); (1- cr + ~L - Im; lvl' ... , l..r);

(~ - CT - ku; A1, . ., Ar); (1 - CT+~+~ - ku; Al' ... , \);

(. . A(l) A<r>) . ( (ll c<ll) . ( .. (rl c<rl) a., . ' .. ., . 1 ' c. ' . 1 ' .. ., c. ' . 1 ) ) ) ,p ) ) ,pl ) ) ,pr ... (2.34)

L = (b · B(lJ. · B(r)) · (d<1l n<1l) · · (d(r) n<r» · 4 }' j ' .. ., j l. q' j ' j l, qi.' .. ., j ' j l, q:

44

(1 - 2CT + µ + V - 2ku; 2Al' ... , 2Ar); (1- Ci+~ - ku; A1, ... ,Ar)

.(~ _ ~ - Ci - ku; A1, ... , Ar).

The (sufficient) conditions of validity of (2.33) are given below : (i) The conditions (2.10), (2.30) and (3.31) also hold.

Ninth Integral : 1 f O t<1

-1(1 - t)cr- µ - l

2F

1 (V, 1 -v; µ; t] S~ [x tk(l - t)k]

.H[z1R 1; ... ; zrR)dt

2- 2vr(µ) l~/o:J (- ~) = I: aµ A (x)u

r(µ - v) u = o u ! ~, u

{r cl:':.-~+.!.) · 1z1 L5] 2 2 2 0, n + 3; m , n ; ... ; m, n -----H 1 1 r r •

ql:':.+Y_l) p+3,q+2;p1,q1; ... ;p,,q z: L 2 2 2 r 6

nl:':. - Y) Jz1 L;j) 2 2 0,n+3;m 1,n1; ... ;m,n + H r r :

nl:':.+Y) p+3,q+2;p1,q1; ... ;p,q z. L* ·2 ~ r r 6

where R 1, R 2 , ... , R,. defined by (2.25).

L!l = (2 - a - ku; Al' ... , A.,.); (1 - cr + µ. - ku; A.1' .. ., Ar);

(1- CT+~+~ - ku; A1, .. ., A,J; (ai Aj1>· .. Ay»l,p;

(c\I>,cCl\ ; ... ; (c\r), cCr\ J J ,pl J. J .P,

L - ( b · B(l) s<r» · ( d(l) ,.D(l)) · · (d(r) v<r» · 6 - j' j ' .. ., j I, q' j j l, q/ ... _, j ' j l,q:

. .. (2.35)

... (2.36)

... (2.37)

l:':. v . (1 -- 2cr + µ + v - 2ku; !d'A.1' ... , 2\);.(l - er+

2 -2- ku; A.l' ... ,A.,.)

L;;;;, (2 - CT - ku; A.1, .. ., A.,.); (l - CT+µ - ku; A.l' ... , A.,.);

(' a+l:':.+~+~ ku·A. A.)·(a·A(l) A(r)) · 2 - 2 2 2 - ' 1' .. ., r' )' j ' .. ., j l,p'

(c(l) ,cCI» l ' ... , (c\r>,c~r\ .I .I ,pl .I J .P,

... (2.38)

... (2.39)

45

L * - (b · B(l) B(r)) · (d(ll n<1l) · · (d(r) D(rl) · 6 - ., . ' ... , . 1 ' . ' . 1 ' ... , . ' . 1 ' J J J ,q j J ,ql J J ,qr

(1 - 2cr + µ + v - 2ku; 2/..,1' ... , 2\); (~ - CT+~ - ~ - ku; t..,1, ... , \)

... (2.40)

The (sufficient) conditions of validity of (2.36) are given below: (i) The conditions (2.10), (2.30) and (2.31) hold . Proof of (2.1) : To evaluate the integral (2.1), we first express

the multivariable H-function in terms of multiple Mellin-Barnes type contour integral (1.1), then on using (1.4) and interchanging the order of ~.-integrals and t-integral which is permissible under the conditions

t

stated on account of the absolute (and uniform) convergence of the integrals and finally evaluating the t-integral with the help of known result [(9,pp.26,Eq. (2.1)], the result readily follows.

The remaining integrals (2.11), (2.18), (2.20), (2.22) and (2.23) can be proved in the same way by employing the integrals [9, pp.26~28, Eqs. (2.2) to (2.6)], respectively.The following integrals (2.24), (2.33) and (2.36), can be established by using the results [1,pp.84-86, Eqs. (5.1) to (5.3)) respectively.

3. SPECIAL CASES (1) If we have n = p = q = 0, the multivariable H-function in(2.1) breaks up into products of r, H-functions and following integral is obtained:

f 1 _ 2 _ 1 [ 1 . t( 1 + a) J tP-

1(1·-t)P[l+at+b(l-t)J P ,f1 v,o;3(v+o+ 2), l+at 1 b(l t)

()

r l (c<i), c<i» } m n , J J l,p. 0: k i' j RI\.· I t

.S13 [xR] TI H 2 i i' (d~i>,D(i)) i = 1 pi' qi J J 1, qi

2v+o-2p-1r(~+~+l) [j3/o:J (-~) "' 2 2 . L ·~A u (x)u

(V - 8)(1 + a)P (1 + b)P + r(v) r(8) u = 0 u . j3,

j rzl I' 1] V 1 8 0,3;ml,nl; ... ;mr,nr .

re;-+;-) r (-;--) H : , 2 2 2 3, 3;pl, qi; ... ;pr' qr Zr ] 2

r(v 8 1 0 3· m rz J' ] - 9) r(- + -) H ' ' i' n 1; ···; m ·' n 1 1 -'"" 2 2 I r .

3, 3·p q. . . , 1' 1' ... ,p q . r' r Z J' r 2

. .. (3.1)

46

where R = R 1 = R 2 = ... = Ri are defined by (2.2)

I' 1 = (1 - p - ku; Al' ... , Ar); (1 - p - ku + ~ + ~; A1, ... , \)

v o k · A A )· (lJ C(l) · · ( (rJ cCr)) (3 2) . (Z - Z - p - u, l' ... , r' (cj ' j )l,p1' ... , cj ' j l,pr ... .

I , - (d(l) D(l)) · · (d(r) D(r)) · (1 k ~ .§.. ~ ~ )· 2 - . ' . 1 ' ... ,. . ' . 1 ' - p - u + 2 - 2' "'1' ... , /\, ' J J 'q I J J 'qr r

v 1 () ' <-z- p - ku; A1, ... , \); (2 + 2 - p - ku; A1, ... , \) ... (3.3)

J' 1 = (1- p - ku; A1, ... , \); (1- p - ku + ~ + ~; A1, ... , \)

(-20 - _2v - p - ku; "-1, ... ,A); (c(l>, c(l\ ; ... ; (c(r>, dr\ ... (3.4) J J ,pl j J ,pr

J' = (dP) D~1)) · ... · (d~r) D~r)) · · (1 - p - ku - ~ + _§_ · 2 J ' J 1,q/ ' J ' J l,q/ 2 2'

where

0 . 1 v . A.1, ... , Ar); (2 - p - ku; A.1, ... , \); (2 + 2 - p - ku; A.1' ... , \) ..• (3.5)

The (sufficient) conditions of validity of (3.1) are given below : (i) The conditions (2.7),(2.8) and (2.9) hold.

(ii) H: > 0, I arg z. j <-217t n*; Vi f (1, 2, ... , r).

l l l .

m. l qi n.

I P;

n~ = ~ D~il - } D(i) + ~ di) - ~ d 1l > 0 ... (3.6) l ,L.., J ~ J ,L..,J £.., J

j=l j=mi+l i=l i=n;+l

Similarly from the remaining integrals, integrals involving product of r, H-functions can also be derived but for the sake of brevity, they are not presented here. A detailed account of the Jl- function is available from the monograph of Mathai and Saxena [7]. (2) lf WA take r-:- l; p = 0 in (3.1) it gives 1 f tP - l (1 0

fl

t)P ll + ut + b(l - t)] 2P - 1 2F

1 [v, 8; i (v + o + 2);

t~1 +a) . m , n [ I (c, C) ] . 1 +-;;f+b(.i - t)] HP

1

'. q1

1 2 1Ri (;, ~)t,p1 dt

.I J 1,ql

C)v+o-2r-1r.v R l) ..:., l-+·--+

= (V _:8)(1 ~~)P (l + b~P + 1 r(v) q8) jr(i +~)I(~)

[

I Ill +3 [ m,n+3 1 y 8 1 m,n

. H 1 1 zl I II - 1(2) rc2 + 2) H l l zl pI+3,qI+3 2 p1+3,q1+3

where

R. = 4(t + at)(l + b)(l - t) 1 [l +at+ b(l - t)] 2

/

Jtl J II

2

Ii"= (1 - p, Al); (1 - p + _2v + _28; Al); (~2 - _28 - p, Al)(c ., C .)1 J J , pl

It= (dj, D)1, qi; (1- p +~-%,Al); (~ - p, Al);(~+%- p, Al)

Jt = (1- p, Al), (1- p +-2v +-28; Al); (-2~>.- ~2 - p, Al)(c., C.)l J J ,pl

J211 =(di' D)1, q I; (1 - p - ~ + ~' Al); (~ - p, Al); (~ + ~ - p, Al)

The (sufficient) conditions of validity of (3.7) are given below:

47

... (3.7)

.... (3.8)

... (3.9)

.... (3.10)

... (3.11)

(i) The constants a and b are such that none of the expressions 1 +a, 1 + b, 1 +at+ b(l - t) where 0 ::; t::; 1 is zero.

(ii) Re(p) > 0, Re(2p - v - o) > 0. (iii) Re(p + A1 min [Re(d./D.]) > O; A1 ;:::: 0.

ls;js;m1

J J

(iv) .Q > 0, 1 I arg zl) < 2 7t.O.

nl P1

where .Q = I: C. - !: C. + }=l J j=n +1 1

1

m 1

I: )=l

ql

D.- I: D.>O. J }=m + 1 J

1

ACKNOWLEDGEMENTS The first author is grateful to the University Grants

Commission of India, for providing financial support for this work. The authros wish to thank Professor H.M. Srivastava of the University of Victoria, Victoria, Canada for giving useful suggestions in this paper.

REFERENCES Ill Arora,Kamal and Rathie, A.K. ; Some theorems contiguous to Whipple and

Dixon theorems for the series aF2 (l); J. Fractional Calculus, Vol 1 (1992), 79-86.

[2J Bailey, W.N. ; Generalized Hypergeometric Series, Cambridge University Press, Cambridge (1935).

(3] Lavoie, J.L.; Notes on a paper by J.B.Miller, J. Austral.Math.Soc., Ser. B. 29 (1987), 216-220.

[41 Lavoie, J.L.; Summation formulas for the series aF2 (1), Math. Computation, 49 (1987), No. 179,269-274.

[5] Lavoie, J.L. ; Gronedeen, F. and Rathie, A.K.; Generalization of Watson's theorem on the sum of a aF2. Indian. J. Math. 34 (1992), 23-32.

[6J MacRobert, T.M.; Beta function formulae and Integrals involving E-functions, Math. Annalaen, 142 (1961), 450-452.

48

[7] Mathai, AM.and Saxena,R.K; The H-function with Applications in Statistics and other Disciplines.John Wiley and Sons, New York (1978).

[81 Rainville, E.D.; Special fun.ctions,Macmillan, New York (1960). [9] Sharma, G. and Rathie, AK.; Integrals of a new series of hypergeometrical

series,Vijnan.a Parishad An.usan.dhan. Patrika, 34, No.1-2,(1991),24-30. [10] Srivastava, H.M.;A contour integral involving Fox's H- function,lndian. J. Math.,

14 (1972),1-6.

[11) Srivastava,H.M. and Panda,R.;Some bilateral generating functons for a class of generalized hypergeometric polynomials, J. Reine Angew.Math. 2831284 (1976), 2652-74.

(12] Srivastava,H.M., Gupta,K.C. and Goyal, S.P.; The H-functions of one and Two Variables with Applications, South Asian Publishers, New Delhi (1982).

(13] H.M. Srivastava and N.P. Singh; The integration of certain products of the multivariable H-function with a general class of polynomials, Rend.Circ. Mat. Palermo 32(2), (1983), 157-187.

Jiianabha, Vol24, 1994 (Dedicated to Professor J.N. Kapur on /us 70th Birthday)

COMMON FIXED POINTS OF WEAKLY COMMUTING MAPPINGS

By RP.Pant

Department of Mathematics, Kumaun University (D.S.B.Campus)

Nainital-263 002, U.P., India (Receiued: October 15,1993)

ABSTRACT A common fixed point theorem for two sequences of

:::;elf-mappiug:::;, re:::;pedively weakly comrnuLing wiLh Lwu given self-mapping of a complete metric space and satisfying a Meir and Kflfllfir typfl contr::ir.tiVfl conclition iP, ohtninncl.'l'hn AXiP,tflncfl of the fixed point has been established assuming the continuity of only one of the mappings. Our work generalizes several well known results on co11Ln-1divH 11rnppi11p,s.

1. INTRODUCTION The study of common fixed points of contractive type mappings

has emerged as an area of vigorous research activity and a number of interesting results have been reported.Majority of these results either deal with commuting mappings or with more generalized concept of weak commutativity of mappings introduced by Sessa [8]. Jungeck [2] introduced the notion of compatibility of mappings, also called asymptotic commutativity by Tivari and Singh[9] in an independent formulation. It was clainwd that weak commutativity implies compatibility [3], [9] but not conversely [7]. However, in a review of [3] (Mathematical Review 89 h : 54030) Singh has shown the existence uf a wnakly mmm11tine pair of mappings satisfyine a r.ontractive condition for which there exists no sequence of points satisfying the condition of compatibility.

In this paper we obtain a common fixed point theorem for two sequences of selfmappings respectively weakly commuting with two selfrnappings and satisfying a Meir and Keeler type contractive condition. The theorem assumes the continuitv of onlv one of the mappings. 'l'he mapping condition studied by us" is a genveralization of the mapping condition 22 of Rhoades [6]. Our work generalizes the re:::;ults due to Fisher Ill, Pant [11, Park and Rhoades [5] and a number of other results.

50

2ofiE§ULTS If (X, d) be a metric space, two selfmappings F' and G of X are

called weakly commuting provided d(FGx, GFx) S: d(Fx, Gx) for each x in X.

Theoiremo Let {P) and {Q), i,j == 1, 2, 3, ... , be sequences of J

selfmappings and let S and T be selfmappings of a complete metric space (X, d) satisfying the conditions ·

Given 2 > 0, there exists an h(2) > 0, h(2) being a nondecreasing function of 2, such that for all x, y in X

2 S: max {d(Sx, Ty), d(P1x, Sx), , T:y),

[d(P?, J:y) + cl(Q_p Sx)]/21 < 2 + h

:=:;, d(P1x, Q_fY) < 2,

Pix== Ql whenever Pix== Sx, QJY ==Ty.

... (1)

... (2)

Let the range of T contain the range of each Pi and the range of S contain the range of each QJ" If each Pi weakly commutes with S and each Qj weakly commutes with T and if one of the mappings

{P), {Q), S or T be continuous then the mappings {P), iQ), S and

T, i,j == 1, 2, 3, ... ,have a unique common fixed point which is also the unique common fixed point of Pi and S and of Q_i and T.

Proof. First, with the helµ of (1), we note that for all x, y in X such that Pix :F Sx, Ql :F Ty, i,j == 1, 2, 3, ... ,

d(Pix, Q_;Y) < max{d(Sx, Ty), d(Pix, Sx), d(QjY, Ty),

[d(Pix, T:Y) I· d(Qjy, Sx)l/2}. . .. (3)

Secondly, the nondecreasing character of h(t) implies that given t > 0, there exists to> 0, such that to< t <to+ h(t0) or equivalently

max (d(Sx, Ty), d(Pix, Sx), d(QjY, Ty),

[d(Pix, Ty)+ d(QjY, Sx)]/2) == t

--> d(Pix, Q_p) <- <:0 , £0 < c. . .. (·1)

Let us arbitrarily select a pair of integers i and j and let x0 be any

point in X. Choose a sequence of points {xn: n = 0, 1, ?., ... }in X defined

by P1x 211 =T,,:2n+ 1 and Q}-~'2n 1 1 =Sx2n+ 2 .This can be done since the

ranges of T and S respectively contain the ranges of Pi and Q_i" We

can assume that Pix'2n "/:. Qf 211 + 1 and Qf'l.n 1 1:F Pix2n + 2 for every valun uf' n, otherwise Llie existence of the fixed point is easy to establish. Then from (3) we obtain

d(Pix'ln' Ql?.n + 1) < d(Q/'2n 1), pi1~4n) ... (5)

Si

and d(Q x? 1, P.x2 ) < d(Px 2 2, Q x 2 1) . .f "....// - l ll I II - .J ll --

... (6)

Similarly, for every integer p > 0 d(Qx Px ) < d(Px Qx. )

./ 2n + l' 1 2(n +p)+ 2 · 1 2n' ./ 2(n +p)+ 1

+ d(P?2n' Q/2n + 1) ... (7)

and d(Px2 , Q x., 1) < d(Q x2 1, Px2( )) 1 n .I ,,(n+p)+ .I n- z n-t-p

+d(Qx2 1,Px

2). . .. (8)

.J n. - 1 n

In view of (5) and (6) we claim that lim d(P1x 211 , Q/211 + 1) = 0 = n ---t=

lim d(Q_r 211 + 1, Pix 211 + 2 ). For, if not, suppose for instance II. ·-7

lim d(Pix2n, Q_r2n + 1) = r, r > 0. Then given h > 0, there exists a n-'t= ..

positive integer N 1, sJch that for each integer m ::'.'. N 1 we have

r :::; d(Pix2m' l/.l2m + 1) < r + h .... (9)

or r:::; max {d(Sx2m + 2 , Tx2m + 1), d(Pix2m + 2, Sx2m + 2),

d(Qf2m + 1' Tx2m + l)'

[d(P?2m + 2' Tx2m + 1) + d(Qf2m + 1' Sx2m + 2)/2) < r + h .

... (10)

Selecting h in (10) in accordance with (1), for each integer m ::'.'. N 1, we

obtain d(Pix2m + 2, Qj"(.2m + 1) < r and so d(Pix2m + 2, Q_r2m + 3) < r,

which contradicts (9). Therefore

lim d(Pix2n' Qf2n + 1) = 0 = lim d(Q/x2n + 1' Pix2n + 2). n-'t= n-'7=.

... (11)

Also by virtue of inequalities (7), (8) and equation (11), it follows that

lim d(Pix2n' Qf2(n + p) + 1> = lim d(Qf2n + 1' Pix2(n + p) + 2>· n-;oo n-700

Now,if possible, suppose lim d(Pix2n, Qf2(n + p) + 1) = r, r > 0. n _,, oo

Then in view of this and equation (11), given h > 0 there exists a positive integer N

2 such that for each integer m ::'.'. N

2, we have

r:::; d(Pix2m' C/.l2cm + p) + 1) = d(Tx2m + 1' Sx2(m + p) + 2) < r +hi 4, ... (12)

r::::; d(Q_l2m + J' P?2(m + p) + 2) = d(Sx'2m + 2' Tx2(m + p) + 3) < r +hi 4, ... (13)

d(P.x., , Qx., 1) < h.14, /. an .J ,,111 + ·

52

and d(Ql2m _ 1, Pix2171 ) < h/4.

Therefore for each m ?.N2we have

r :S: max (d(Sx Tx ) d(P.x Sx ) 2m + 2' 2(m + p) + 3 ' 1 2m + 2' 2m + 2 '

d(Q x Tx c) [d(P.x Tx ) .J 2(m+p)+3' 2(m+p)+3' 1 2m+2' 2(m+p)+3

+ d(Q x2

( ) 3

, Sx2 2)] /2)) < r +hi 4 .... (14)

J m+p+ m+

Selecting h in (14) in accordance with (1), for each m?. N 2 , we have

d(Pix 2171 + 2, Q1x 2(m +pl+ 3) < r, which contradicts ( 12). Hence

{P1x0, Q/1, .. ., P1x 211 , Q/211 + 1, ... ) is a Cauchy sequence in the complete metric space X and so has a limit point z in X. Also the sequences (Pix2 ) = {Tx 211 + 1 ) and {Qj x 2n + 1) = (Sx2n + 2) converge to the point z.

Let us now suppose that the mapping S is continuous. Then since P. and S commute weakly, the sequences {P.Sx2 ) and {SSx2 }

1 • 1 n n converge to Sz. We claim that z = Sz. For if z '/:. Sz, then the inequality

d(P.Sx2 , Qx9 1 1) <max {d(SSx

2 , Tx2 1

), d(P.Sx2 , SSx2 ), 1 n J ~n · n n + 1 n n·

d(Qf?.n+l' Tx2n 11), fd(PiSx2n' 'l'.'r::'!,n+1)

+ d(Q/2n + 1' SSx2n)] /2}

on letting n ~ oo and in view of (4) leads to d(z, Sz) < d(z, Sz), contradiction. Therefore 2 ·--0 8.z:. Similarly the .iuequaliLy

d(Piz' Q/'2n + 1) <max {d(Sz, Tx2n -t 1), d(Piz, Sz),

d( Q.Jx2n + l' 7:~:2n + 1),

[d(Pz, Tx., 1) + d(Q~v:2 + 1, Sz)]/2} · l w11+ .J n

yields z = P;z. This means that thfff8 cxi8ts a puiuL 2·0 in X such that

Sz = z = Pi;c = Tz0 , as the range of Pi is contained in the range of T.

Morevoer, the inequality.

d(P/, Q_/0) max (d(Sz, Tz 0), d(Pt, Sz), d(Qf 0, Tz0),

[d(Pl" Tzo) I d(QfO' S.z)]/2}

yields z = Qf 0 =-- Tz0 . This equation, in view of weak commutativity of (l; and T, implies Q_f = Tz. Finally Qf = Tz together with (3) leads to

;:c = (}_/ :-::: Tz.

We have therefore proved that z is a common fixed point of P., CJ, Sand T for an arbitrarily chosen pair of integers i andj.

I ./

Now, if pm;sible suppose w is a second common fixed point of P ,_

and S. Then d(w, z) = d(P.w, C/.7) <'.max {d(Sw, T;:;), d(P.w, Sw), d(Qz, 1'z),

I J ' I .J

53

[d(P w, Tz) + d(Q z, Sw)]/2) l J

= d(w, z),

a contradiction. Hence z is the unique common fixed point of Pi and S. Similarly z is also the unique common fixed point of Qj and T.

Now if we keep i fixed and vary j, we find that z is the unique common fixed point of Q. and T for every value of j. On the other

J hand, since z = Q z = Tz, if we keep j fixed and vary i it can be easily

J shown that z is the unique common fixed point of P and S for every

/.

value of i. Using similar type of arguments we can prove that z is the

unique common fixed point of P ., Q ., S and T if. P. is assumed to be l J l

continuous, for some value of i, instead of S. Similarly z is the unique common fixed point of P., Q., S and T if

l J

either T or Q .is continuous instead of S or P .. respectively. J l •

'l'his completes the proof of the theorem. Remark 1. In the above theorem if we take P. = P = Q. for each

/. ./

i and each j and S = T, we obtain Theorem 4 of Park and Rhoades (5] as a particular case of our theorem. In that case we can drop the assumption on nondecreasing character of h(E)

Remark 2. In this theorem if we take P. = P and Q. = Q for l J

each i and each ; and if we take max{d(Sx, Ty, d(Px, Sx), d(Qy, Ty), [d(Px, Ty)+ d(Qy,'SxJ/2) =max {d(Sx, Ty), d(P'x, S'x), d(Qy, 1~y)), we

obtain the Theorem of Pant 141. Further by choosing h(E) = 2(1-c) c/c, 0 < c < 1, we shall obtain Theorem 2 of Fisher [1] as :=! RpA~i :i] ~:l R0.

We now give an example of mappings satisfying the conditions of the above theorem and having a unique common'fixed point.

Example. Let X = [U, 1] und let d be the usual metric on X. Define selfmappings P., Q., S and T onX,i,j = 1, 2, 3, .. ., as follows

l .!

Pix= ix/(6i + 1), x =/. 1, f'.1 = il(l~i + 1) /,

Qx=O J , x =/. 1, 9i1' =j/(6j + 1)

Sx =x/2, x =t l, Sl = 1/4 Tx =x for each x in X.

Thrm {Pi}, 19il, S and T satisfy all the conditions of the theorem have a unique common fixed point x = 0.

REFERENCES

~mcl

[lJ B. Fisher, Common fixed points of four rnappin~s. Bull. Inst. Math., Acad . .':iinica 11 (1983),103-Jrn.

34

Ui G. ,Jungck, Compatible mappings and common fixed points. lnternat. J. Math.i'vlath.Sci. 9 (1986). 771-779.

13) G.Jungck. Common fixed points for commuting and compatible maps on compacta. Proc.AmerJvfoth. Soc. 103 (1988J.977-98:3.

141 RP. Pant. Common fixed points of two pairs of commuting mappings, Indian J Pure Appl. Math. 17 (1986), 187-192.

151 S. Park aml B.E.Rhoades, Meir and Keeler type contractive condit.ions, Math. -lapon. 26-I <1981),13-20.

161 B.E. l'{hoades,A comparison of various definitions of contractive mappings, Tran,,. Amer. Math,. Soc. 226 (1977), 257-290.

171 B.E. Rhoades. S. Park and KB. Moon, On generalizations of the Meir-Keeler type contraction maps, J. Math. Anal. Appl. 146 ( 1990). 482-494.

181 S. Sessa. On a weak commutativity condition of rni1ppings in fixed point considerations, Pub/. Inst. Math. 32 <1982J. l49-J5:'l.

191 B.M.L. Tivari and S.L.Singh,A note on recent generalizations of .Jungck contraction principle. J. Uttar Pradesh Gout. Colleges Acod. Soc. 3119.'iGJ,1 :3-18.

00

,Jnanabha, Vol 24. 1:Ji1 (Di:!dicated tn Prof;~ssur J.l\l. .!(1 ::'x.1r on hi~ 70th F~irthday)

COIV!U\l[(JN JFT:;(JB:f» POii\'TS 01F' TWO PAJIRS OF ()i 1F1

fVIl~:JPP JII~J (;, &3

.J.M.C. Jos~r1i and: H.P. Pant

I)epa1trn.ent of ";:(uri,1<.~_un {Jn.iversity

(D1.S.B.(jrunpus)

Nainital - 263002, U.P., India (Rcceiucd ·October 21.1.993!

11JB;S1LftAcc1r A common fixed point theorem for two pairs of sequences of

pairwise weakly commuting selfmappings of a complete metric space satisfying Meir and Keeler type contractive condition is obtained. The existence of the fixed point has been established under the assumption of continuity of only one of the mappings. Our work generalizes several fixed point theorems concerning contractive mappings.

1. INTRODUCTION

The study of common fixed points of mappings satisfying some contractive type condition has been at the centre of vigorous research activity and a number of interesting results have been obtained by various authors. Most of these results either deal with commuting mappings or assume the notion of weak commutativity of mappings introduced by Sessa [8]. Jungck [~] introduced the notion of compatibility of mappings, also called asymptotic commutativity by Tivari and Singh [9] in an independent formulation. It was claimed that weak commutativity implies compatibility [3], [9] but not conversely [7]. However, in a review of [3] (Mathematical Review 89 h : 54030) Singh has produced an example to show the existence of a weakly commuting pairs of mappings satisfying a contractive condition for which there exists no sequence of points satisfying the condition of compatibility.

In this paper we obtain a common fixed point theorem for two pairs of sequences of pairwise weekly commuting mappings satisfying a Meir and Keeler type contractive condition. We have assumed the couLiuuiLy uf u11ly u1w uf Llw 111uµµi11gs. Thu muvµiug cu11JiLiun studied by us is a generalization of mapping condition 10 of Rhoades [6]. Our work generalizes the results duo to Fisher [1], Pant [4], Pa~:, and Bae [5] nncl eorno other rmmltt>.

2.RESULTS If (X, d) b0 a mflt:ric Hpacn, twil m~lfm:-1ppinGR F :mrl G of X are

callrnl weakly commuting provided d(FCiX, OFx) <; d(Fx, Ox) for uach x inX.

56

Theorem. Let {P.), {S.), {Q.} and (T), i,j == 1, 2, 3, ... , be l I J J

sequences of mapings of a complete metric space (X, d) into itself satisfying the conditions :

Given c > 0, there exists an h(c) > 0, h(c) being nondecreasing function of c, such that for all x, y in X

cs max {d(S.x, T.y), d(Px, S.x), d(Q .y, Ty)) < c + h l ./ I I .f .f

=::::> d(Pix, Qj>') < c ... (1)

P x ==Q'j>' whenever P x == S x, Q .y == T v. I l l J .f

... (2)

Let P and S commute weakly and Q and T commute weakly l l J J

for every value of i and j respectively. If the ranges of T and S. J l

respectively contain the ranges of P and Q. for every value if i andj l J

and if one of the mappings is continuous then the sequences {P.), {S.), {Q.} and {T.} have a unique common fixed point which is also

l l J J the unique common fixed point of (P.} and {S.} and of the sequences

l l

{Q) and IT/

Proof First, with the help of (1), we note that for all x, y in X such that Pix'# Six, Qj>' '# Tj>', i,j == 1, 2, 3, ... ,

d(P?, Qj>') <max {d(S?, Tj>'), d(Pix, Six), d(Qj>', T_f!)). . .. (3)

Secondly, the nondecreasing character of h(E) implies that given c > 0, thAre existi:; c0 > 0, such that c0 < E < c0 + h(c0) or equivalently.

max{d(S/t:, Tpr'), d(P;x, Six), d(Q_fY· T;v)l '~' F.

=::::> d(Pix, Qj>') < i::0, i::0o < c. . .. (4)

Let us arbitrarily select a pair of integers i and j and let x0 be

any point in X, Choose a sequence of points (x : n == 0, 1, 2, ... ) in X n

defined by P.x2

==Tx2 1 and Qx

2 1 ==Sx9 9 .This can be done

1 n J n + J n + 1 -n + ~ since the ranges of T. and S. respectively contain the ranges of

J l

P. and Q.. We can assume that P.x2

;tQ.x., Ll and l .J /. n J s-n·,

Q :~., + 1 'f Px2 2 for every value of n, otherwise the existence of tlrn J ,,n 1 n +

common fixetl point is em;y to cstabfo;h. Then from (8), we obtain d(P.x,, , Qx

2 + l) <, d(Qx 2 1, P.x

2 ), ... (5)

1 ,,,n .1 n . J -fl. 1 n

and d(C}_/2n - l' Pix2n) < d(Pix2n - 2' CJ./2n - 1). ··· (6)

Also for any integer p > 0

d(Qx2 + 1,Px2( 1 )+z)<max{d(P.x2 ,Q.x2( + )+l) .f on l II -- fl l //. J n fl ,

d(P.x,1 , Qxn + 1)} I -//. J 4/l '

... (7)

57

and d(P?2n' Q/2(n + p) + 1)

<max {d(Q x 2 1, P.x2 ), d(Q x 2 1, P.x2( ))}. ... (8) J n- l n .J n- l n+p

In view of (5) and (6) we claim that lim d(Pix2n, Q.f2n + 1) = 0 n ---7=

= lim d(Q.f2n + 1, Pix2n + 2). For, if not, suppose for instance n --7 oo

lim d(Pix2n, Qjx2n + 1) = r, r > 0. Then given h > 0 there exists a n ----7 =

positive integer N such that for each integer m ;::: N, we have

r::; d(Pi x 2m, Qj x 2111 + 1) < r + h ... (9)

or r::; max {d(S.x2 2, Tx2 1

), d(P.x2 2, S.x2 2),

1. m+ .J m+ l m+ l m+

d(Qx2 1,Tx2 1

))<r+h. . .. (10) .J m + .J m +

Selecting h in (10) in accordance with (1), for each m;::: N we obtain d(Pix2111 + 2, Q;X2m + 1) < r and so d(Pix2111 + 2, Q;x2111 + 3) < r, which contradicts (9). Therefore

lim d(P .. x9 ,Qx9 _1_1)=0= lim d(Q.x 9 __ 1,P.x9 + 9 )

i ~n J LJn J LJn I i LJn LJ n~~ n~oo

... (11)

Also, in view of (7) (8) and (11), it follows that

lim d(Pix2n' Q/2(n + p) + 1) '-' lim d(Q/2n + 1' I'/2(n + p) + 2l· n-'>= n-'>=

Following similar argument, as used to evaluate limit (11),it follows easily that

lim d(P.x,, ,Qx,,( ) 1)=0=lim d(Qx., 1,P.x,,( + )+ 9 ). r. LJn. J ,, n I p I J ,,n. I I. LJ n. p ~ n-)oo n )oo

Hence {Pix0, Q/1, ... , Pix2n, Q/2n + 1, ... } is a Cauchy sequence in

the complete metric space X and so has a limit z in X.

Also, thP. 8P.<J.11P.n~P. {Pp:2n = T_((.2n + 1l Fmn l~f2n + 1 = Six2n + zl converge to z.

Let us now suppose that the mapping S. is continuous. Then, I.

since P. and~- commute weakly, the sequences {P.S.x2

) and {SB.x2 l i i ii n ii n coverage to S .z. We claim that z = S .z. For if z -=F S .z then the

l l !

inequality

d(P/:;?2n' Q/2n + 1) < max {d(Si8 /2n' T/2n + 1),

d(Pi8 ?2n' 8 i8 ?2n), d(Qf2n + 1' Tf2n + l)l

on letting n -7 oo and in view of (4) leads to d<;:,, Siz) < d(z, Siz), a contradiction. Therefore z = Siz. Similarly the inequality

d(P.z, Qx2 1 1) < max{d(S.z, T Xn 1 1), d(P.z, S.z),

r J n - ·. 1 J .c.n -- i i

d(Q_/211 + 1' Tfzn + 1ll

58

yields z ==PF This means that there exists a point 2 0 in X such that

S/ == z = Piz = Tl0, as the !·ange of P1 is contained in the range of Tj. Moreover, the inequality

d(P{, Qf0 ) <max {d(Siz, T~z0), d(F/, 8t), d(Q/0, 1~z 0 ))

yields z == Qfo == Tfo· This equation, in vie'N of weak commutativity of Q and T, implies Q z = Tz. Finally, the relation Qr.;;= T z together

J ./ .I .I . .I with (3) leads to z == Q_f == T f.

We have therefore proved that z is a common fixed point of Pi,

Q , .S. and T for the arbitrarily chosen pair of integers i and j. J I .J

Now, if possible, suppose w is a second common fixed point of P and S .. Then

I I

d(w, z):::: d(P.w, Qz) l J

<max {d(Siw, Tf), d(Piw, Siw), d(Ql, Tl))

== d(w, z),

a contradiction. Hence z is the unique common fixed point of Pi and Si. Similarly z is the unique common fixed point of Q. and TJ ..

.J

Now, if we keep i fixed and vary j, we find that z is the unique common fixed point of Q. and T. for every valuo of j. On the other

.1 .J hand, since Q z :::: z = T z, if we keep j fixed and vary i and also drop

.J ./ . the assumption of continuity of S. we can easily show that z is the

/.

unique common fixed point of Pi and 8 1 for every value of i.

Using similar type of arguments we can prove that z is the unique common fixed poi11t of P., Q., S., :md T. if P is assumed to be

/. j' l .J l

contmuous instead of S .. l

Similarly z is the unique common fixed point of Pi' Q1, S1 and T1 if either Qj or ~·is continuous instead of Si or P1 respectively.

This completes the proof of the theorem. Remark l. In the above theorem if we take

P. = P, Q. = Q, S. =AS, and T.""' T for each i and j, we uuLain /. J l J

Lhe thr~orem of Pant [2].

Remark 2. In the above theorem if we take P. = P, Q. = Q, S. = S · l .J I

and T.i == T for each i and j ancl let h(e) == 2(1 - c) £/ c, 0 < c < 1, then we

obtain Theorem 2 of Fisher fl].

Remark 3. In the above theorem if WP tnlrn P.""Q.=a anil 8 ,....T.==/"for!Wt)ryvalueol"iand1"and!et

I ',/ h l .f '

59

max (d(S?, 1~y), d(P1x, S1x), d(Q}', T;Y)) = d(S1x, T?'),

then we can drop the assumption on nondecreasing character of h(E) and we obtain the Theorem 2.4 of Park and Bae [3].

We now give an example of sequences of mappings satisfying the conditions of the above theorem and having a unique common fixed point.

Example. Let X"' [O, 1] and let d be the usual metric on X Defin0 selfmappings P, Q, S and T on X, i,j = 1, 2, 3, ... , as follows

I J l J

P? = ix/(6i + 1), x i:- l,

Qx = 0, .!

Sx = ix/(2i + 1). I

x t:-1,

xi:- l,

T. x = x for each x in X . .l

p 1=il(l2i+1) I

Ql =j/(6j + 1) .!

Sl==il(4i+l) I

Then {P.}, (Q }, {S.) and {T.} satisfy all the conditions of the l .I l .I

theorem and have a unique common fixed point x = 0.

REFERENCES [lj B. Fisher, Common fixed points of four mappings, Bull. Inst. Math., Acad. Sinica

11 (1983), 103-113. 121 G. Jungck, Compatible mappings and common fixed points, Internat. J. Math.

Math.Soci. 9 (1986),771-779. 1:31 G. Jungck, Common fixed points for commuting and compatible maps on

compacts, Proc. Amer. Math. Soc. 103 (1988), 977-983. [4] R.P. Pant, Common fixed points of two pairs of commuting mappings, Indian J.

Pure Appl. Math. 17 (1986),187-192. [5] S. Park and J. S. Bae, Extension of a fixed point theorem of Meir and Keeler,

Ark. Math. 19 (1981), 223-228. 16] B.E. Rhoades, A comparison of various definitions of contractive mappings,

Trans. Amer. Math. Soc. 226 (1977),257-290. [7] B.E. Rhoades, S. Park and K. B. Moon, On generalizations of the Meir-Keeler

type contraction maps, .f. Math. Anal. Appl. 146 (1990), 482-494. [81 S. Sessa, On a weak commutativity condition of mappings in fixed point

considfmitions, Pub/. Inst. Math. 32 (1982),149-153. 191 B.M.L. Tivari and S. L. Singh, A note on recent generahzat10ns of .Jungck

contraction principle, J. Uttar Pradesh Govt. Colleges Acad. Soc. 3 (1986), 13-18.

CJO

Jiianabha, Vol24, 1994 (Dedicated tu Professor J.N. Kapur on his 70th Birthday)

INTEGRALS INVOLVING A GENERAL CLASS OF MULTIV ARIABLE POLYNOMIALS, JACOBI POLYNOMIALS,

AND FOX'S H-FUNCTION R.S. Pareek

Department of Mathematics, University of Rajasthan,

Jaipur-302004, Rajasthan, India (Received: November 23, 1992; Revised: March 2, 1993!

ABSTRACT Here we evaluate four integrals involving various products of

general class of multivariable polynomials introduced earlier by H.M. Srivastava and M. Garg, Jacobi ploynomials, and Fox's H- function. Our results are quiLe general in character and provide interesting unifications and generalizations of a large number of (known or new) intf~er::il formulae.

1. Introduction Recently Kalla, Conde and Luke [2], Kalla [3] and Kant and

Koul [4) have established certain integrals involving Jacobi polynomials, generalized .JacuLi fu11cLim1H, a geueral dass of polynomials and 1-"ox'o H function. In an attempt to unify and extend these results we have evaluated four finite integrals involving the product of Jacobi polynomials, a g-oneral class of multivariablo polynomials :rnc'I Fox's H-function. The technique followed i1:1 essentially that of Kalla et al.([21,rBD.

The general class of polynomials sm lxJ introduced by Srivastava n

[7],p.l, Eq. (1)) has further been generalized to a multivurinblc polynomial in the following m::inner (Srivastava and G·arg [8] :

mk 1 ... 1mksn 1 1 r r

sm ' ... , m (x x) = n 1 ' 1' ... , r 2: (- n)m k + ... + m k 1 1 r r k

1, ... ,kr"'O

k k x11 ... X,.r

A(n ; kl, ... , kr) kl ! ... k, ! ... (Ll)

where m 1, ... , m,. are arbitrary positive integers and the coefficients A(n ; k 1, ... ,hr) (n, ki ~ 0, i = 1, ... , r) are arbitrary constants real or complex.

To facilitate the derivation of our main integrals given in the next.. section we shall require an integral contained in the following

62

Len1ma if ~L, v, yi'' oi (i == 1, ... , r) are all non-negative real numbers

(not all zero simultaneously), Re (ex)> - 1, Re(~)> - 1,

Re(p)+~t min [Re(b/B)]>-1,Re(o)+v min [Re(b/B)l>-1, l <;,j :; M J J l <;,j :; M J .I

A> 0, 8 > 0, I arg(z) I < i An, then

lj (t -- x)P x 0 p(a., Bl (1 - z' x) 8 111 t'. 'm,(zl(t -· x) yl x'\ .. ., z .(t - x?ri\) • Q ll Tl I

[

(a,A)1

~ Hlvf· N z(t - x)µxv .· J .! ,p dx

I,Q (b,B)IQ J .I

p+cr+l =t

111 k + ... + m k $ n 1 1 r r II (- n)m k + + m k A(n ; kl, .. ., k )

1 1 ... r r r

kl! ... kr ! :t I:

w=O k1

, .. .,kr=0

r(cx + u + 1) (- u) (a.+~+ u + l) w w r(cx+w+l)u!w!

r

<-Vt>w IT i = 1

(z. tY + 8)k l l '· i

M,N+2 II

p + 2, Q + 1 ztµ + v

r r I (-p- L y.k.,µ),(-0-w- I, ok,v),(a.,A.)1 i=l I l i=I l l J J ,p

r

(b.,B.) 1 Q,(1-p-cr-w- L (y.+8.)k,µ+v) J J ' i-=1 l l l

.... (1.2)

=tp+cr+l m/1

1 + ... +m/1,.sn

r u M I: £.: r

w~O h=n R-0

(-n)m.k. + .. +m,.k,A(11·k , .. .,k,) J j ., 1

hl ... kl'! k1

, .. ., /11

= n

/' !'

n (z .. tY; '3;/i I~ (a+ u + 1) (- u) (a+~ 1- u -1 1) 1 (1 + p + :Z yk + µ C,

1,R)

I ' W W . l l l l t

i = 1 i= /'

r (<x + w + 1) u ! w ! R ! r (2 + p I (J I w + i:: 1 (Y; + 8)k; I (~l + v)C,h, R)

,. r(l + cr + w + . I: 8/?. i + vsh. R)

z-1 /'():. (I, w µ+v~ -·····----- ,. ·-·-·--.. --- "'h,R) z z , t) (z t )'-.,h,R

r(l + cr + w + . I: 8/li + vsh,R) I 1

m/J+ ... +m/1,.<;11

l: 11,, .. ., Ii,.= 0

II ,~ ,,..,

w =0

M I:

h=O I: g(w, h, R) ... (say)

R=O

... (1.3)

... (1.4)

bh +R where sh, R = --B --, (R = 0, 1, 2, ... )

h

M N

rI r(b - B 'f;h R) I1 f'(l - a -AJ 'f;h R) .J .J ' .J ,

)=1 )=1 " j;t h f( l; ) = ----'----------------,,,-·--·--·--------------h,R Q ,

11 r(l-b+B'f;hR) fi f'(a-A'f;hR)Bh .! .! ' .J .J ,

./=M+ 1 N P A1

A=0 LA- 5~ + L 1=l .i j=N+l j=l

Q p

and o = L B. - L. A . . j=l .J ;=l J

j=N + 1

Q

B - I B .! j=Mcl J

63

... (1.5)

.. (1.6)

... (1.7)

... (1.8)

The integral (1.2) is a special case of an integral recently established by the author ([6], p.10, Eq. (2.1). The value of the integral in the form given by (1.3) can be obtained easily by using the following series expansion for Fox's H-function due to Braaksma [1], see also Srivastava et al. ([9], p.12, Eq. (2 .. 2.4))

M,N M,N[ (a.,A.)1 pl M = E, H [x] =H x (b.J -i) ' = I: I: f('f;1 R)x il,R

P, Q P, Q j' j 1, Q h = 1 R = 0 1

'

... (1.9)

where S1i, R and f('f;h, R) are defined by (1.5) and (1.6) respectively. MN

H ' [x] stands for the well-known .Fox's H-fi.mction. For details of P,Q

this function one can refer to the book by Srivastava et al. [9].

2. Main Integrals Let µ, v, y., 8. are all non-negative real numbers (not all zero

l /.

simultaneously), 8 > O,A > 0, I arg (z) I <;}An, A and o being given by (1.7) and (1.8), respectively.

Re(p) + µ min {Re(b/Bi)} > - 1 l-;;,j-;;,M . .

Re(cr) + v min {Re(b/B)l > - 1, l <;; '5M .

and

F(x) = (t - x)P xa P~1a., f3J (1 -z' x) S~1 1' ... , m,. (z 1 (t - x)Y1 x 1\ .. .,

. MN[ (a.,A.)1pl z,.(t-x)Yr:x6,.)li' z(t--x)PxV .! .I ' _

P. (-i b.i, B) l, C(

... (2.1)

64

then

t mk+ +mk<n 1 1 ,. ,. - ll M

(iJ J F(x) log (t - x) dx = 0

I I k

1, ... ,h,.=0 w=O

I I g(w, h, R) h=l R=O

[log t + 1Jf ( 1 + p + i ~ 1

r

Y;k; +µsh.,.) \Jf(2 + p + 0 + w + I (yi + 8) i = 1

+ ki + (µ + vls1z, m] ... (2.2)

m1k

1+ ... +m,.k,.<;,n

11 M

(ii) r F(x) log x dx = 0

I I 2.: 2.: g(w, h,R). k

1, .. ., h,. = O w = O h = 1 R = 0

,. [log t + \jf (l + 0 + i ~

1 o.k.+v'f:,h R-\jf(2+p+2+w+ 2.: (y+o.)k.

I /. , i = l l l l

+ (µ + v )Sh, R)] ... (2.3)

r m. lk 1 + ... + m},.-:;, n u M .t 2.: (iii) J. F(x) log [(t - x)x] dx = .t .t

0 k , .. .,k =0 w=O h=l R=O 1 r

g(w, h,R)

[ logt2 +\j/(l+p+ ~ y.k.+~L(;1 R)+\Jl(l+cr+w+ ~ o.Ji.+v(,h.) i = l l l l, i = l I l , /

- 2\jf (2 + p + 0 + w + ~ (y. + o)k + (µ + v) (;1 R)] ... (2.4) i = l l l l I,

[ l m. k + ... + m k -:;, n. ,1. t -x 1 1 ,. ,. lt "'

(iv) f F(x) log -- dx = 2.: 2: L 2.: g(w, h, R) 0 x Ii . .,., h = 0 w · 0 h ~ 1 R - 0

l r

[ r r l \ll(l+p+ L y.k.qt(;1 R)-ljf(l f-CT+w+ L o.k.+v<;11 R)

i =]. I I I., i = l t t , ... (2.5)

whore \jl(z);;... ~(~; and ff(W, h, R) and 'Sh,R are given by (1.4) and

(1.5) respectively. Also the series occurring on the right-hand side of integrals (2.2) through (2.5) converge absolutely.

Proof : The results in (?..2) and (2.::1) can be obtaiue<l by taking the partial derivatives of both sides of (U:l) with respect to p and cr. respectively. The integral (2.4) is obtained by adding (2.2) and (2.3) and the integral (2.5) is obtained by subtracting (2.3) from (2.2).

3. Particular Cases At the outset, we should remark that our integral form11la0 me

quite general iu chHracter. Indeed, these ruHults c:an suitably be

65

specialized to a number of known and new integrals involving a large spectrum of various classes of polynomials and elementary special functions. However, we mention here only a few special cases of (2.2). (i) If we put

E U'

IT Ce)k e'+ .. +k,e;n TI (u ')1 ., ¢' ..

(r)

U (r)

TI (u ) (rl i = 1 1 .I . .I= 1

A(n; h1, ... ,hr) =·-Dn----J I]'/ ./ .I k qi.

j = 1 r , .. (3.1) (r)

IT v v (d.)k , k Ir) TI ( ') TI ( (r))

. .! 1 ti + ... + /j vi " s' . . . uj k s(r) j=l .i=l 1.1 )=1 r.J

in (2.1) we arrive easily at the following interesting integral formula

r P cr n1 RJ 1 + E: U', ... , [Ji'l [ [- n : m 1, ... , m ],

(t-x) x r,tx," (1-z'x) log (t-x) F r 0 11 D : V', .. ., 0rl

[(e) : 8', ... , 9(r)] : [(U') : <p'] ; ... ; [(U(rl) : cp(r)J (t )Y 8 [(d) : t', .. ., 't(r)] : [(V') : s'l : ... : [(y(r)) : s(r)] Z1 - x 1 x 1, .. .,

y 0 ] MN[ µ l(aj,A)l,p} zr(t - x) r x r H z(t - x) x (b. B ·) .x

PQ .J' J 1, Q

mlkl + ... + m,.k,. s; n u M °" [

k1, .. .,~r=O w:O h:l R:O n(w,h,R) logt+

,. 'l'(l+p+ ;~i Y;k;+µs1i,R)-'1'(2+p+cr+w+ ;~i (Y;+l\)k;+Cµ+v)sh,R)

I ... (3.2)

where T](w, h, .H) is obtained by putting the value of A(n; k 1, .. ., kr)

from (3.1) in g (w, h, R). (ii) Again if we take m. = 0, z. -j 0 (i = 2, .. ., r) (~.~) arnl replace

/. /.

A(n; k1, 0, ... , 0) by An, k therein, we easily arrive at the following integral

rt (t - x)P xcr pa,~) (1 - z'x) sm fz (t - x)Y x 0] log (t - x) Jo u n 1

(M, N) [ (a., A\ I'] H z(t - x)µ xv .I .I .. • dx

PQ (bj'l3_)1,Q

[nlml " M ~ (-n) A f'(o:+u+l)(-u) (o:+R+ +l) l: 1: 1: :E tp+cr+1 ___ _!7.1}1 n,k w I-' u . w

hO w Oh=OR=O k!r(o:+w+l)u!w!R!

I' (l + p + yk + µl;h R) [' (l + O" + w + '6k +VS ) [ -----(~ +-lJ + cr + w ~ (y + '6)/l + (~l + V) ~(h-,-id:JL log l + 111 (l + p + yk + µ~h, R)

66

' •(2 + p + a+ w + (y + O)k + (µ ")sh, R) }((h,R)(z,t' 0 hf (~z't!'' (zt'' ' '>Sh, R)

... (3.3)

where [nlmJ(-n) k

S m m A k (3 4) . (x) == I: --k 1- ,x ··· · n h = 0 . . n, ti

is a general class of polynomials introduced by Srivastava ([7], p.l, Eq. (1). This general class of polynomials include almost all the wellknown orthogonal polynomial and generalized hypergeometric polynomials available in the literature. For example

(iii) Ifwe putz == l,A ==_r(l ±_!_l±An)__ ... (3.5) 1 n, '1 n ! rc1 - fl + Ak)

in (3.3) , replace y and o by y/A, and o//..., respectively, we get the following interesting integral :

~ (t -x)P XO' Pua, Pl (1 - z'x) log (t - x) zr;: [(t - x)Y x 0 ; /...,]

H z(t - x)µ xv 1' 1 l, P dx == L :I: L L M~ l(a. A) l [nlniJ u M

P Q (bj, B)1, Q k = o w = O h = o R = o

1 (- n)mk r(l + TJ +A.ii) r (a+ u + 1) (- u)w (a+ p + u + l)w

~+cr+ . -k ! n ! r(l - r1 + Ak) r (a+ w + l) u ! w ! R !

rc1 + P + t k + µsh, R> rc1 + a+ w + f k + v sh, R)

0 log t +

rc2 + p +a+ w + ( Y ~ ) k + (~t + v) s1i, R)

IJI (1 + p + f:k + r1 sh,R)-'I' (2 + p + <H w + (1: "} + (µ +vJ Sh,nl]

firJ'.. ) t(Y + o)k ·I..,'t·)w (~tµ + V) S (3 6) '"'h,R .· (2~' "' "· 11 ... •

where Z';:(x; /...,) stands for the Kohnauser biort.hogonal polynomial (Konhauscr [5]).

(iv) If we take u == 0, t = 1 in (3.3) and replacing x by 1 ~ y

Lo

therein wt.: easily gHt I.he integral roc.cntly given by KanL and Kaul ([4], p. 6, Eq. (15). The other integrals (16),(17) and (18) due to them are also special cases of our main integrals.

Acknowledgements Tho author is thankful to Dr. S.P. Goyal (Associate Proti~sRnr,

DeparLment of Mathematics, University of Rajasthan, Jaipur) for his kind lrnlp anrl R"Jlidance rlurin~ thr>. preparation uf Lhill paper.He is also

67

thankful to Professor H.M. Srivastava (University of Victoria, Canada) for his valuable suggestions.

REFERENCES [11 B.L.J. Braasksma, Asymptotic expansions and analytic continuations for a class

of Barnes-integrals, Compositio Math. 15 (1963),339- 341.

[21 S.L. Kalla,S. Conde and Y.L. Luke, Integrals of Jacobi functions, Math. Comp. 38 (1982) 207-214.

[31 S.L. Kalla, Integrals of generalized Jacobi functions, Proc. Nat. Acad. Sci. India Sect .A 58 (1988), 123-128.

[41 Shashi Kant and C.L. Kou!, Integrals involving Fox's H-function .. Proc. Indian Acad. Sci. Math. Sci., 101 (1991), 37-41.

[51 J.D.E.Konhauser, Biorthogonal polynomials suggested by Laguerre polynomials, Pacific J. Math. 21 (1967), 303-314.

[61 R.S. Pareek, Certain multiple integrals involving a general class of multivariable polynomials and II-functions with applications.,Ganita Sandesh, 6 (1992) 9-15.

[7] H.M. Srivastava, A contour integral involving Fox's H-function. Indian J. Math. 14 (1972), 1-6.

[8] H.M. Srivastava and M. Garg,Some integrals involving a general class of polynomial, and the multivariable H-function. Rev. Roum(J,ine Phys. 32 (1987), 685-692.

[9] H.M. Srivastava, KC. Gupta and S.P. Goyal, The H-Functions of one and Two Variables with Applications. South Asian Publishers, New Delhi and Madras,1982.

[JO

Jiianabha, Vol24, 1994 (Dedicated to Professor J.N. Kapur on his 70th Birthday)

MAGNETOHYDRODYNAMIC TEMPERATURE DISTRIBUTION OF TWO IMMISCIBLE VISCOUS LIQUIDS BETWEEN TWO

PARALLEL PLATES By

Pushpendra Kumar, N.P.Singh* and Ajay Kumar Singh Department of Mathematics

C.L.Jain (PG) College, Firozabad-283203, India (Receiued: Nouember 28, 1993)

ABSTRACT This paper is concerned with unsteady flow of two conducting,

incompressible,viscous, immiscible liquids between two parallel plates in the presence of uniform magnetic field applied perpendicular to the Dow region. In Lhe aualy::;i::;, iL ha::; been a::;::;umed LhaL Lhe upper plate is moving with transient velocity while the lower plate is fixed. Velocity of the liquids and temperature at the plates have been obtained, following SoundalgekaT et al. (1990). The effect on magueLic field, Reynolds number on the velocities of the lower and upper liquids and the temperature distribution of the lower and upper liquids for different values of Eckert number have been studied with the help of tables.

1. Introduction Heat transfer in unsteady flows of immiscible viscous liquids in

porous channel assumes importance due to its important applications in ground water hydrology, aero-dynamics and petroleum industry. However, this aspect has received little attention in the literature. Hartmann (1937) has done the pioneer work in the field of magnetohydrodynamics.

Temperature distribution in poiseuille flow between two parallel flat plates has been studied by Dube (1970). Mathur et al. (1972), Sacheti and Bhatt (1973) have considered the unsteady flow between two plates when lower plate is oscillating and upper plate is moving uniformely in their own planes in the presence and absence of a magnetic field respectively. Mathur and Jain (1981) have studied the magnetohydrodynamic temperature distribution for an um;Leu<ly laminar flow between two parallel plates.Sacheti et al. (1989) have discussed heat transfer in steady flow of immiscible fluids in a channel bounded below by a naturally permeable wall. Bhargava and Sacheti (1989) have investigated heat transfer in generalised Couette flow of

H.N. 236, Durga Nagar, * Address for correspondence : Firozabad - 283203, (UP)

70

two immiscible Newtonian fluids through a porous channel. Pal and Pal (1989) have investigated the transient temperature distribution in skin and subcutanceous tissues with improved variations in bio-physical parameters under various environmental conditions. Recently, Soundalgekar et al. (1990) have considered heat transfer in MHD unsteady stagnation point flow with variable wall temperature.

In the present paper, unsteady flow of two conducting, incompressible, viscous liquids between two parallel plates in presence of uniform magnetic field has been studied. Velocity of the liquids and temperature at the plates have been obtained, following Soundalgekar et al. (1990). The effect on magnetic field, Reynolds number on the velocities of the lower and upper liquids and the temperature distribution of the lower and upper liquids for different values of Eckert number have been studied with the help of tables.

2. Formulation of the Problem Let us consider the laminar flow of two incompressible

immiscible viscous, electrically, conducting liquids between two parallel plates. A constant magnetic field B0 is applied perpendicular

to the flow region. The depth betwen the plates is 2d. The x-axis is assumed in the line flow along the central line which passes through the midway of the parallel plates and the y-axis is taken perpendicular to it. It is further assumed that the lower plate is fixed and the upper plate is moving with transient velocity U. The governing equations of motion and energy for the present configuration are :

du. a2u. (). _i == v. __ i - _!_ B2 u. dt t dJ'2 pi 0 l

. .. (2.1)

- 2T ( ]2

rlT; == J( ri .21

+ ~ ~ui dt p.c .1.., p.c rly

l p VJ l p (i c- 1, 2) ... (2.2)

where in these equations, vi is thn kinematic coefficient of vi:,;cosity, ui

is the velocity of the liquids, ai is the electrical conductivity of the

liquids, Pi is the density of the liquids, cp is the specific heat at constant pressure, K is the thermal conductivity, t is the time µi is the coefficient of visco~ity, B 0 is the magnitude of the magnetic field and

the temperture is denoted by Ti.

The boundary conditions for the lower liquid are

u 1. :U0, T~==Td at ~=OJ u 1 - 0, T1 - T0 at y - - d

... (2.3)

'l'he boundary conditions for the upper liquid are

u 2 = U 0' ll2= u u 0 '

T 2 =Td at y=O

T 2 =T0 at y=d

We introduce the following non-dimensional variables-u. t T - T 0 u;'' =yf, ./ =~, t'' = d' rt= T 1

-T .. 0 d 0

71

.. (2.4)

Using the above non-dimensional variables, the equations (2.1) and (2.2) after neglecting the asterisks over them are reduced to

du. 1 a2u.

-~ = -- __ i -M2u. (Jt Ri Cly2 1 1

. .. (2.5)

and 2 [ ·i2 ()T. l Cl T P Clu L l. r l -=----+- --

Clt R. E ()"2 R. Cly L C. J L

... (2.6)

I

where,

vi 1 K 1 d = R. (Reynolds number), -. c- = E (Eckert number).

L µL p C.

'

[~]1/2 U5 B 0 _!:_ =Mi (Hartmann number), (T _ T =Pr (Prandtl number)

Pi · d o)cP

The non-dimensional boundary conditions for lower liquid are

l ' 1 u = 1 T = 1 at y = 0 l u

1 =0, T

1 =0 at y=-1 ··· (2·7)

The non-dimensional boundary conditions for the upper liquid are

u2 = 1, T2 = 1 at y = O ) U T 0 t 1

... (2.8) u2 = ' 2 = a Y =

3. Solution of the Problem Following Soundalgekar et al. (1990), we assume

u = u(O)(y) + E e- nt u(l)(y) l L /. L

T. = T,O) (y) + E e- nt T,1) (y) l /. L

... (3.1)

Substituting the values of u. and T. in the equations (2.5) and L l

(2.6), we obtain. ()2u(O)

·~-- ·- M 2 R. u(O) (y) = 0 ... (3.2) fJy~ I /. I

72

a2u(l) I 2 (1) ------R(M- -n)u (YJ==O

oy2 I I I

... (3.3)

2 02r;oJ (au;oJ l --+PE --- ==0 cy2 r Ci dy

... (3.4)

()27-(lJ ou(O) dlpl --

1- + nR E T. 1>(y) == - 2P E __ z__ • -

1--ay2 I c

1 I r Ci dy ()y

... (3.5)

Taking i == 1 and i == 2, the solutions of equations (3.2) and (3.3) under the boundary conditions (2.7) and (2.8), are

sinh b1 (1 + y) u(O) == ------

1 sinh b1

0 sinh b2 (1-y) + U sinh biY

u< ) == -2 sinh b

2

u<1l = u<1l == 0 1 2

where b1 ==M1 ~' b2 =M2{R~.

... (3.6)

... (3.7)

... (3.8)

'I'aking i == 1 and i == 2, the solution of P.quation (3.1) and (3.5) with the help of equations (3.6) and (3. 7) under the boundary conditions (2.7) and (2.8), are

pr Ef' bi [ 2 cosh 2b (1 + y)l r pr EC bi 1"-P> = 1 - i L + - 1 - -~ 1 - _1 -1 '> 2 'I ~ 2 sinh~ h

1 4bi 2 Rinh b1

r 1 1 cosh 2b1]] pr Ec1 b~ cosh 2bl

12 + -2 - 2 y + 2 2 ... (3.9)

4b1 4b1 8b1 sinh b1

2 O) PrEc2 b2 [co8h 2h-}' {U2 ·} y2 r u2 1 }

r\~ =1- . 2

-2

-2

-Ucoshb2 +··-1-

2 +-

2--Ucoshb2_

' smh b2

4b2

2 L

+ cosh 2b2 (1 -y) _ U sinh 62 sinh 2b.y'_l

8·b2 41 2 2 -'2

b P E [ {( l I 2 r c cosh 2b 2 1 u2 1 l - y - 2

--- - - - + U cosh b y - __ :.J · sinh2 b 4b2 2 2 2 2

2 2

73

+ 4~~ {(~ - 26- cosh b2 )y -(~ - cosh b2 )}

_ z {!!2 l } U sinh 2b2 sinh b2] 2 2 + 2 - U cosh b2 + y 4b2

2

... (3.10)

and rOJ = T:lJ - 0 1 2 - ... (3.11)

On putting the values of u(O) u(OJ u(l) u(ll T:0J T:OJ T\1l and 1'2' 1'2' l' 2' 1

V22) in equation (3.1), we obtain

sinh b1

(1 + y) ll = --------

] sinh b1

sinh b2 (1 - y) + U sinh b';!)' lJ,2 =

sinh b2

_ P,J\ bi [y2 cosh 2b1 (1 + y)l T 1 - 1 - ') 0 + 9 +

2 sinh-b "" 4b~ 1 1

PrEc,bl Jl 1 cosh 26, . P,.Ec1bl cosh 261 l 2 } 2

2 sinh~b 1 I 4bi 4bJ: } '"'l """' "l l - •) 2 + -,, - ,, ' ·I·

and

P,.Ec2 b~ [cosh 26'2)' { u2 } T 2 = 1 - . 2 --2- 2 - u. cosh 62 2 smh 6

2 46

2

y2 J u2 1 } cm;h 2h2 (1 -y) + ~ r~ + ~ - TT r.osh h2 +-----Rh',!,-­

~

l t/·PE

_ Usinhb2 sinh2b2 y -y- 2 __ '~~lcosh2b2 4b2 sinh2 b 4b2

2 2 2

lf[l- U2

+ U cosh b _}- _!} + _g__ {(Q--1 - cosh b )y

2 2 2 2 462 2 2U 2 • 2

- (~ - cosh bl~ i [: + ~ U cosh b2)

... (3.12)

... (3.13)

... (3.14)

74

+ Y -~~ ~inh 2b2 sinh b2]

4b2 2

... (3.15)

Table-I Velocity of the lower liquid defined in equation (3.12) for different values of Hartmann number M 1.

··-

y M 1 =1.0

0 1.00000 - 0.1 0.81112 - 0.2 0.65499 - 0.3 0.52506 - 0.4 0.41619 - 0.5 0.32403 - 0.6 0.24487 - 0.7 0.17554 - 0.8 0.11325 - 0.9 0.05551 - 1.0 0.00000

-- --

Table-II

M1 =3.0 M 1 =5.0

---r-----···-······ 1.00000 0.54880 0.30118 0.16526 0.09065 0.04966 0.02709 0.01459 0.00748 0.00316 0.00000

1.00000 0.36788 0.13544 0.04979 0.01832 0.00674 0.00248 0.00091 0.00034 0.00011 0.00000

Velocity of the lower liquid defined in equation (3.12) for different values of Reynolds number R 1.

y R 1 =2.0 R 1 ==4.0 R 1 =6.0 -- ··--

0 1.00000 1.00000 l.OOUUU - 0.1 0.85030 0.81112 0.77903 - 0.2 0.71763 0.65499 0.60503 - 0.3 0.59934 0.52506 0.46752 - 0.4 0.49305 0.41619 0.35819

0.5 0.39664 0.32403 0.27048 - 0.6 0.30818 0.24487 0.19907 - 0.7 0.22589 0.17fifl4 0.1 :i966 - U.8 0.14812 0.11325 0.08868 - 0.9 0.07333 0.05551 0.04304 - l.O 0.00000 0.00000 0.00000

·~-------·- . ------·-··---·- -·- .1....------- ---.--·-- --~----

Table-III Velocity of the upper liquid defined in equation (3.13) for different values of Hartmann number M2 at U = 2.0

M =3r::) M ="r.> J 2 ·' 2 ,,., ···-~-~-·---~- ------ -

y

75 ------------- 1------- T 0 1.00000 1.00000 1.00000

0.1 0.78015 0.47786 0.38528 0.2 o.63997 o.23152 I 0.14914 o.3 I o.56513 1 0.11874 0.05955 0.4 I 0.54799 I 0.07449 I 0.02847 o.5 o.586~1 I 0.01322

1 0.02537

0.6 0.685c)5 I 0.11427 ' 0.04718 ,

0.8 1.11030 I 0.45555 0.29687 0.9 1.47960 0.95280 0.77009

0.7 0.85491 I 0.22108 0.11535 J' L-------~--- 2.00000 __ J 2.00000 2.00000

Table-IV

Velocity of the upper liquid defined in equation (3.13) for different values of Reynolds number R 2 at U = 2.0

r- y R2 = 2.5 r 2 = 4.5 R 2 = 6.5

0 1.00000 1.00000 1.00000 0.1 0.87478 0.78015 0.71608 0.2 0.79907 0.63997 0.53812 0.3 0.76844 0.56513 0.43988 0.4 0.78125 0.54799. 0.40674 0.5 0.83828 0.58681 0.44384 0.6 0.94256 0.68555 0.52512 0.7 1.10020 0.85491 0.69415 0.8 1.32000 1.11030 0.96593 0.9 l.61430 1.4 7960 1.38080 1.0 2.00000 2.00000 2.00000

Table-V Temperature distribution of the lower liquid defined in equation (3.14) for different values of Eckert number Ec at Pr= 0. 71

l

E =0.20 E = 0.30 cl cl

y EC = 0.10 1 ---

0.0 1.00000 1.00000 1.00000 - 0.1 0.90478 0.90956 0.91434 - 0.2 0.80744 0.81489 0.82234 - 0.3 0.70866 0.71733 0.72598 - 0.4 0.60886 0.61744 0.62659 - 0.5 0.50834 0.51673 0.52507 - 0.6 0.40730 0.41467 0.42197 - 0. 7 0.30643 0.31183 0.31771 - 0.8 0.2041fi 0.20838 0.21253 - 0.9 0.10217 0.10442 0.10658 - l.O 0.00004 0.00006 0.00008

-·--> ·--·"-·- -~ ..

76

Table-VI Temperature distribution of the upper liquid defined in equation (3.15) for different values of Eckert number E at P = 0. 71 and u = 2.0 c

2 r 1----;-----··,.--- - - ·-------·--· .

f- _ -~ ____ J FC,,o 015 -+ E'i~o~25 -~ O 0.99593 I 0.99320

0.1 0.67094 0.51770 0.2 0.34495 0.04126 0.3 0.02376 - 0.42745 0.4 - 0.28540 - 0.87623 0.5 - 0.56901 - 1.28600 0.6 - 0.81289 - 1.62230 0.7 - 0.97224 - 1.82110 0.8 -. 0.97184 - 1.76220 0.9 - 0.69042 - 1.21770 1.0 - 1.14060 - 1.90170

4. Discussion

Ec 1 ~ 0.35 ~~~1 0.99045 0.36484

- 0.26212 - 0.87821 - 1.46660 - 2.00020 - 2.43090 - 2.66920 - 2.54700 -1.74450 - 2.66200

The numerical values of the velocity of the lower liquid for diffemnt values of Hartmauu number M1 and Reynolds number R 1

with y have been listed in Table-I and Table II, respectively. A study of these two tables shows that the velocity of the lower liquid decreases as the intensity of magnetic field M

1 increases and similar

in the case when Reynolds number R1

increases. Besides, we observe

that on increasing y the velocity of the lower liquid decreases in hoth the tablet:J.

The numerical values of the velocity of the upper liquid for different values of Hartmann number M

2 and Reynolds number R 2

with y at U = 2.0 have been listed in Table-UT and 'fable IV, respectively. It is observed from these tables that the velocity of the upper liquid decreases as the intensity of the m::ignctic field M2 iucreases. Again when Reynolds number R2 increases the velocity also

decreases for given value of paramdn H is intercoting to note Llwt when y increases the velocity of the upper liquid decreases while it increases again as y increases from y = 0.5 toy = 1.0.

'fhe numerical values for the temperature of the lower liquid for difforcnt values uf Eckert number E at P = . 7l are shovm in Table-V. c ,.

I

We observe that the temperature of the lower liquid increases as the Eckert number Ee increases.Besides on increasing y temperature of

1

the lower liquid decreases. The numerical values for the temperature oft.hi:> upper liquid for

different values of the ~ckort number E at P = 0.71 and U = 2.0 arc (" I'

2

77

shown in Table-VI. Here we observe that the temperature of the upper liquid decreases as the Eckert number E increases.Besides on

.c2 increasing y temperature of the upper liquid decreases.

REFERENCES

Ill S.K. Bhargaya and N.C. Sacheti, Heat transfer in generalised Couette flow of two immiscible Newtonian fluid through a porous channel: Use of Brinkman model, Ind. Journal of"Technology, 27, (1989), 211.

12] S.N. Dube, Temperature distribution in poiseuille flow beh>'een two parallel flat plates, Communicated by M. Sengupta, F.N.I.l (2) (1970).

J:iJ J. Hartmann, Theory of laminar flow of an electrically conducting liquid in a homogeneous magnetic field, K. Dansk. Viden.Seds, Mat. Phys. Medd., 15 (6) (1937).

[4] A.K. Mathur, N.C. Sacheti and B.S. Bhatt, Unsteady hydromagnetic felow between parallel plates, Math. Student, 40 (1) (1972), 145

[5] A.K. Mathur and N.C. Jain, Magnetohydrodynamic temperature distribution for an unsteady laminar flow between two parallel plates, Proceedings of Dr. Ray Memorial Symposium on Fluid Dynamics and Allied Tnpir.s. (I HK l ), H l.

[6] D.S. Pal and S. Pal, Transient temperature distribution in skin and subcutanceous tissues with improved variations in bio- physical parameters imder various environmental conditions, Indian Journal of Tachnology, 27, (1080), 4:12.

[7] A.K.Sacheti and B.S. Bhatt,Unsteady flow of a viscous incompressible fluid between parallel plates, DefSci. Jour 23, (1973), 247.

ISJ N.C. Sacheti, S.K.Bhargava and B.S.Bhatt, First Carribbeen Conference on Fluid Dynamics, The University of West Indies, St. Augustine, Trinidad, West Indies, (1989), 8.

[9] V.M. Soundalgekar, T.V. Ramana Murty aud 1-l.H. 'l'11kh11r, 11"111. Trant41i~r in MHD unsteady stagnation point flow with variable wall temperature,lndian Jour. Pure Appl. Math. 21 (4), (1990) 381.

00

Jnanabha, Vol 24, 1994 (Dedicated to Professor J.N. Kapur on his 70th Birthday)

SOME WEAKER FORMS OF FUZZY CONTINUOUS MAPPINGS

Sunder Lal and Pushpendra Singh

Department of Mathematics, Institure of Basic Science Agra University, Khandari; Agra - 282002, U.P

(Received: January 30, 1994)

ABSTRACT Some weaker forms of fuzzy continuous mappings have been

introduced and studied. Results pertaining to some of their preservation properties are also discussed.

1. INTRODUCTION Using semi open sets in topological spaces, several weaker forms

of continuity pave been studied in the literature. For example, Bhamini [2] defined a mapping f: X---? Y to be almost semi continuous if the inverse image of regularly open sets in Y are semi open in X. She [2] calls f to be weakly semi continuous if for every open set U of Y, .{ \U) c S - int r 1 (<.:lU). Here S-int denotes the semi interior operator. Singal and Yadav [7] call f to be slightly semicontinuous if r 1(U) is semi open in X for every clopen subset U of Y. Here we study these mappings in the fuzzy setting. However, we use a more suggestive terminology for these mappings. We call a mapping f: X---? Y to be l(a., ~) if the inverse image of a sot of type ~ is a set of type a.. For example, a mapping f: X ~ Y is l(u, clo) if the inverse image of each clopen set in Y is open in X. Bhamini's almost semi continuous mapping in our terminology is I (so, ro) and slightly semi continuous mapping of Singal and Yadav is I (so, clo). Here we study some of these mapping in the fuzzy setting.

2. PRELIMINARIES Throughout this paper X and Y mean fuzzy topological spaces. I

denotes the closed unit interv::i I. The definitions of fuzzy sets, fuzzy topological spaces and other concepts about fuucLiuns can be found inl3,8,9J.

Ld U Le a fuzzy set of X. U is said to be fozzy semi open Get of X if there exists a fuzzy open set V such that Vs Us clV. ComplemeuL of a fuzzy semi open set is called fuzzy semi closed. S-int U (Semi interior of U) is defined as the suprimum of all fuzzy semi-open sets contained in U and S-clU (semi closure of U) as the infimum of all fuzzy semi closed sets containing U.

80

A fuzzy set U of a fuzzy srace X is called a fuzzy regular open set of X if int clU = U, and a fuzzy regular closed set of X if cl int U = u.

A fuzzy set U of Xis called fuzzy clopen (resp. fuzzy semi clopen) set of X if U is fuzzy open and fuzzy closed (resp. fuzzy semi open and

fuzzy semi closed). A fuzzy set U of Xis said to be fuzzy o''' - open if U can be written in the form U = V U., where U. are fuzzy clopen

I. I

sets.Complement of a fuzzy o* - openset is called fuzzy o'~ closed.

A mapping f: X ~ Y from a fuzzy space X to fuzzy space is called a, (i) fuzzy open mapping (1], if f(A) is fuzzy open set of Y, for each fuzzy open set A of X; (ii) fuzzy closed mapping [ 1), if f(A) is fuzzy closed set of Y, for each fuzzy closed set A of X; (iii) fuzzy presemi open mapping, if f(A) is fuzzy semiopen set of Y for each fuzzy semi open A of X.

A fuzzy space X is ::;aid to be (i) fuzzy semi compact (resp. fuzzy slightly compat) if every fuzzy semi open (resp. fuzzy clopen) cover of X has a finite subcover; (ii) fuzzy strongly S-closed if every fuzzy semi open c.over of X has a finite subfamily such that semi closures of whose members cover X.

A fuzzy space Xis said to he (i) fuzzy SP.mi T0

, if every fuzzy

set A of X can be written in the form A = v A U . . , where U .. lJ IJ

i c I j F '';

arc fuzzy semi open or fuzzy semi closed sets; (ii) fuzzy semi T1, if

every fuzzy set A of X can be written in the form A "" 'v' U., where U. i F I l l

are fuzzy 8emi closed RP.ts; (iii) fuzzy semi T2

, if every fuzzy set A of

X can be written in the form A = v A U .. = v A S - cIU. ., where U.. are fuzzy

i E I i E ,, !) i E I j E J. l/ IJ • I I

semi open sets, (iv) fuzzy S-regular, if every fuzzy open set U can bci written in the form U == V TT., where U. ore fuzzy semi open sets

i E I l l

with S - cl Ui-:::: U; (u) fuzzy S-normal, if for any fm:zy closes set Kand

fuzzy open ::;et U such that K-:::: U, there exists a fuzzy V such that K..;, S - int V-:::: S - clV $ U.

A fuzzy ::;paee (X, -r) is called a fuzzy semi regular ::;pace [1] if the collection of all fuzzy regular open sets of X forms a base for fuzzy topology.

A fuzzy space X is called product related [1) to a fuzzy space Y if for rmy fuzzy 8ot U of X and \l uf Y whenever A' < U and B' < V implies (A' x 1) v (l x B') 2 U x V, where A is fozzy open set of X and B is fuzzy open :,;et of Y, there exists a fuzzy open set A

1 of Kand a fuzzy

81

open set B 1 ofY such that A1

' 2:: U or B1

':::: V and (A1

' x l) v (1 x B 1')

=(A' x l) v (1 x B').

3. Fuzzy I(so, ro) MAPPINGS 3.1 Definition : A mapping f: X -~ Y from a fuzzy space X to a

fuzzy space Y is said to be fuzzy l(so, ro) if inverse image of every fuzzy regular open set of Y is fuzzy semi open set of X.

3.2 Remark : Fuzzy l(so, ro) mappings are a generalization of the concept of almost semi continuous mapping studied by Bhamini (2).

3.3 Theorem : For a mapping f: X ~ Y from a fuzzy space X to a fuzzy space Y, the following are equivalent : (a) f is fuzzy l(so, ro); (b) the inverse image of every fuzzy closed set of Y is fuzzy semi closed set of X; (c) [ 1(U)) S: S - int ([ 1 (int clU)) for every fuzzy open set U of Y; (d) S - cl ([ 1 (cl int V)) s: [ 1 (V) for every fuzzy closed set VofY.

Proof : (a)<=> (b), because for any ti.Izzy set U of Y, we have [

1(U') = [ 1(U))'. (a)=> (c). Let U be a fuzzy open set of Y. Then U <!: int clU and hence [ 1(U) ~ [ 1 (int clU). Since int clU is a fuzzy rP.e11 ll'lr open set of Y, [ 1( int c/[J) is a fuzzy semi open set of X. Thus [ 1(U) s [ 1 (int clU) = S ·-int [ 1 (int clU) . (c) <=> (d). Taking complement of both sides of (c) we get (d), and vice versa. (d) => (b). Let U be a fozzy regular closed set of Y. By (d) we have

s - cl <r 1 (cl int U)) = s -cl r 1(u) sr 1crl). Thus S cl [ 1(U) = [ \U), which is fuzzy semi closed. 3.4 theorem. If f: X ~ Y be a fuzzy !(so, ro) mapping, then for

every fuzzy open set U uf Y, S - cl r 1(U) :S [ \clU) Proof: Let f:X ~ Y be a fuzzy l(so, ro) mapping from a fuzzy

space X to a fuzzy space Y. Then [ 1(U) s: [ 1 (clU) for every fuzzy set U of Y. If U is fuzzy open, then clU is fuzzy regular closed. Hence r 1(clU) is fuzzy semi closed set of x. Thus we have s - r:t r 1 cu)~ r 1 cctu).

Clearly, every fuzzy 1(0, ro) mapping as well as every fuzzy l(so, o) mapping is fuzzy !(so, ro). But the converse need not be true as is shown by tho following examples :

3.5 Example : Let Ul' U2 awl U3

be fuzzy set::; uf I defined as follows:

for each x E J,

U1(x)=x, Osxsl;

U~(x)=l-x, OS:xS:l;

82

U3(x) =x, O:s;xl/2;

= 0, l/2 < x $ l. Consider fuzzy topologies 't

1 = {O, U1 /\ U2, l} and

't2 = {O, U1, U2

, U1

/\ U2

, U1 v U2

, 1} on I and the mapping

f: (J, 't1) ~ (1, 't2) defined by f(x) = x, for each x E J. It is clear that

U1, U2 , U1 v U2 and U1 /\ U2 being fuzzy open and fuzzy closed are

fuzzy regular open sets in (/, 't2). It is obvious that inverse image of

every fuzzy regular open set in (/, 't2) is fuzzy semi open in (J, 't1),

because in (/,'t1),cl(U1 AU2)=1 and U1 AU2 :s;{ 1 (U1)$l,

U1 /\ U2 $f 1(U2

) $ 1, U1

/\ U2$f 1 (U

1 /\ U

2) $ l. Hence the mapping

is fuzzy /(so, ro). The mapping is not fuzzy l(o, ro), because none of T 1(U1), T 1(U2) and T 1 (U1 v U2) is open in(/, 't1).

3.6 Example: We refer to fuzzy sets as defined in Example 3.5 Consider the fuzzy topologies 't1 = {O, U 1 A U2, l} and

't2 = {O; U1, U2 , U~; U1 v U2, U1 A U2 , I} on I and the mapping

f: (l, 't1) ~ (/, t 2) defined by f(x) =x for every x EI It is clear that the

mapping is fuzzy !(so, ro). Also, because 0 is the only fuzzy open set contained in T 1(U3), T 1(U3) = U3 is not fuzzy semi open set of (I, t 1).

Hence {is not a fuzzy !(so, o) mapping. 3.7 Remark : Fuzzy l(o, ro) and fuzzy !(so, o) mappings have

been studied under the names fuzzy almost continuous and fuzzy semi continuous [l].

3.8 Theorem : If f: X ~ Y is a fuzzy /(so, ro) Mapping from a fuzzy space X to fuzzy semi regular space Y. Then f is fuzzy ](so, o).

Proof: Let Ube a fuzzy open set of Y. Then U = v U., where i E I l

Ui are fuzzy regular open sets of Y; since Y is fuzzy semi regular. Now, using Theorem 3.3 (c), we get

r 1 <U) = r 1 < v u.) = v r 1 cu.) ielt iel t

$ y S-intclU.)= y S,-intr 1 (U.) iel t iel t

$8-int( y f 1 (U.)-=S- int{ 1(U),1 i e I t '

which shows that f' 1(U) is fuzzy semi open set of X. Hence f is fuzzy l(so, o).

I' I

83

3.9 Theorem : Let f: X -7 Y be a fuzzy l(so, so) and fuzzy pre semi open mapping and let g : Y -7 Z be any mapping. Then gof: X -7 Z is fuzzy l(so, ro) if g is fuzzy l(so, ro).

Proof : The if part is obvious. To prove the only if part, let gof: X -7 Z be a fuzzy I (so, ro) mapping. Let U be a fuzzy regular open set of z. then (gof)- I (U) = r 1 (g"- I(U)) is fuzzy semi open set of X and hence f(f I(g- \U)) = g- I(U) is fuzzy semi open set of Y. Therefore g is fuzzy l(so, ro).

3.10 Remark : Fuzzy l(so, so) mappings have been studied under the name fuzzy irresolute by Mukherjee and Sinha [6].

3.11 Theorem : Let (1

: X1 -7 Y1 and (2 : X2 -7 Y2 be fuzzy

l(so, ro) mappings. If Y1

is product related to Y2

, then the product

mapping f1 xf2 : XI xX2 -7 YI x Y2 is fussy l(so, ro).

Proof: Let Xl' X2, YI and Y2 be fuzzy spaces such that l'."I is

product related to Y2. Let U = v (U. x U.) be a fuzzy open set ·of i J

YI x Y2, where U{ and ~ are open sets of YI and Y2 respectively.

Since fI and f2 are fuzzy l(so, ro) mappings, we have

(fI x f2)- l (U) = v lfl I (Ui) x f;. I (~)}

~/y {S - int fi 1 (int cl Ui) x S - int t;. 1 (int cl ~;)}

~ v {S - int Cf! I (int cl Ui) x f;. I (int cl Uj>)}

~ S - int {v ((1

x {.2)- I (int cl U. x int cl U.)}

l .l

= S - int {v (1 x/2)- I (int cl (Ui x l:{,))} ~ S - int {if.I x f 0 )- I (int cl (v (U. x U.)))}

~ i J

= S int (fI x {2)- I (int clU).

Thus by theorem 3.3 (c), (1

x f?. is fuzzy !(so, ro).

3.12 Theorem : Let X ;:incl Y be fuzzy spaces such that X is product related to Y and let f: X -> Y be a mapping. If the graph mapping g : X -iX x Y of f is fuzzy l(so, ro ), then f is also fuzzy l(so, ro).

!Proof : Suppose that g : X.....:; Xx Y is fuzzy .!(so, ro) mapping and U.is a fuzzy open set of Y. We have

f} (U) = 1 Af l (U) =g- l (1 x U)

~ S - intg- 1 (int cl (1 x U))

= S - intg 1 {int (1 x clU))

= S - int g- 1 h x int cl U)

84

== s - int r 1 (int clU).

Hence by Theorem 3.3 (c), f is fuzzy I (so, ro).

4. FUZZY WEAKLY SEMI CONTINUOUS MAPPINGS 4.1 Definition : A mapping f: X---'> Y from a fuzzy space X to a

fuzzy space Y is said to be fuzzy weakly semi continuous if for every fuzzy open set U of Y,

r 1 (U) ::::; s - int r 1 (clU).

4.2 Remark : Fuzzy weakly semi continuous mappings are the genralization of weakly semi continuous mappings intorduced by Bhamini [2).

4.3 Definition [l] : A mapping f: X---'> Y from a fuzzy space X to a fuzzy space Y is called fuzzy weakly continuous mapping if for each fuzzy open set U of Y, r 1 (U)::::; int r 1 (clU).

Clearly, every fuzzy weakly continuous as well as every fuzzy l(so, ro) mapping is fuzzy weakly semi continuous. For the converse, we have

4.4 Example : LP.t X= {x,y,z) and U1

, U2 be fuzzy sets of X

defined as follows : TT1(x) = 0.2, U1(y) == 0.4, U1(;::) = 0.3;

Uix) = 0.4, Uiy) = O.fi, Tlz(Z) = 0.3;

Let T1 = (0, U2, 1) and T2 = {O, U1

, 11. Let f: (X, T1) -> (X, T2) be

the identity, mapping. By easy computations it is clear that in (X, T2), cl U1 = U/, int clTT1 == U1 and in (X, 't'1), S - i.ut r 1 (U1') = U2'

and S - int r 1 (U1) ""0. Hence r 1 ( U1) ::::; S - int r 1 (clU 1), which

shows that mapping is fuzzy weakly semi continuous. However, r 1 (ll1)::. int r 1 (int cZU1)showiug that the mapping is not fuzzy

l(so, ro).

4.5 Example : We take fuzzy sets U1

, U2 and fuzzy topologies

as defined in previous example, and f: (X, 't'2) ~ (X, 't'1) the identity

mapping. It is clear that in (X, 't'1), clU2 = U2', and in

(x, 't2), int r 1 (U2') = U1 and S - int r 1 (U2') = U

2'. Hence r 1(U2)

~ 8 - int f 1 (U,,') showing that the mapping is fuzzy weakly semi ~ .

continuous. However, r 1 (U2) >int r 1 ( U2'), hence the mapping is not fuzzy weakly continuous.

4.6 Theorem : Let f: (X, 't 1) ---'> (X, 't2) be a fuzzy weakly semi

continuous mapping from a fuzzy space X to a fuzzy regular space Y. Then /is fuzzy l(so, o).

85

Proof : Let U be a fuzzy open set of Y. Since Y is fuzzy regular space U = v U., where U. are fuzzy open sets of Y with cl U. :::; U.

i EI l l l

Now, since f is fuzzy weakly semi continuous, we have.

r 1 CU)= r 1 c v u.) = v r 1cu.) i EI l i EI l

:::; v S - int r 1(cl U.) i EI l

:::; s - int ( v r 1 (clU.)) = s - int r \U), i EI l

which shows that [ 1(U) is a fuzzy semi open set of X. Thus, f is fuzzy l(so, o).

4. 7 Theorem : Let X1, X2, Y 1 and Y 2 be fuzzy spaces such that

Y1 is product related to Y2. Then, the product

f1 x f2 : xl x x2 ~ y 1 x y2 of fuzzy weakly semi continuous mapping

{1 : X1 ~ Y1 and {2 : X2 ~ Y2 is fuzzy weakly semi continuous.

Proof : Let {1 : X1 ~ Y1 and {2 : X2 ~ Y2 be fuzzy weakly semi

continuous mappings such that fuzzy space Y1 is product related to Y2 .

Let U = v (U. x U.) be a fuzzy open set of Y1 x Y 2, where U. and U. l J l J

are fuzzy open sets of Y1 and Y2 respectively. We have, by fuzzy

weakly semi continuity of {1 and {2

({1 x f2r 1 (U) = v {(i 1 (Ui) x f2 1 (U)l

s v {S - int fl 1 (clUi) x S - int f; 1 (clU)}

s v {S - int ifi. 1 (cl Ui) x f2 1 (clU))l

s s - int {v f1 x f2)- 1 (cl ui x cl ~)}

= s - int {v (f1 x f,1T 1 (cl (Ui x U))}

= s - int ({1 x f2r 1 (clU).

'l'hus by the definition of fuzzy weakly semi continuity, f]. x f~ is

fuzzy weakly semi continuous. 4.8 Theorem : Let X and Y be product related fuzzy spaces and

f: X ~ Ybe a mapping. If the graph function I( : X ~ X x Y of f is fuzzy weakly semi

continuous, then f is also fuzzy weakly semi continuous. Proof : Let X and Y be fuzzy spaces such that X is product

related to Y and let f: X ~ Y he· a mapping. Suppm;e that the graph

86

function g : X ~Xx Y off is fuzzy weakly semi continuous. Then for any fuzzy open set U of Y, we have

r 1(U) = 1 /\ r \U) =g- 1 (1 x U) =:::; S - intg- 1 (cl (1 x U)

= s - int g- 1 (1 x clU) = s - int r 1 (clU).

Thus by the definition, f is fuzzy weakly semi continuous. 5. :Fuzzy J(so, clo) MAPPINGS

Slightly semi continuous mappings in general topology have been introduced by Singal and Yadav [7]. Here we generalize this notion to the fuzzy setting.

5.1 Definition : A mapping f: X ~ Y from a fuzzy topological space X to a fuzzy space Y is said to be fuzzy l(so, clo) if inverse image of every fuzzy clopen set of Y is fuzzy semi open set of X.

5.2 Theorem : For a mapping f: X ~ Y from a fuzzy space X to a fuzzy spce Y, the following are equivalent :

(a) f is fuzzy l(so, clo); (b) Inverse image of every fuzzy clopen set of Y is fuzzy semi

closed set of X. (c) Inverse image of every fuzzy clopen set of Y is fuzzy semi

clopen set of X.

(d) Inverse image of every fuzzy o* -open set of Y is fuzzy sellli · open set of X.

(e) Inverse image of every fuzzy s* -closed set of Y is fuzzy semi closed set of X.

Proof: It is obvious. The following implications hold.

Fuzzy 1(0, clo) =>Fuzzy l(so, clo) =>Fuzzy weakly semi continuity We now show that none of the above implications is reversible.' 5.2 Example : Let U 1 and U 2 be fuzzy sets of I defined as

·· follows : for each x E I, U1(x) = 2x,

=1, U2(x) =2x,

0 =:;.x=:;·:1;2:

112 $x <'1; o:.::;.x ::;;·;1/4

, = 1- 2x, l/4 ~ x1<:: Jh; =0, lh $x$ l.

Consider fuzzy topologies Tl= {O, u2'!.1} and '.T2 = {0, ul, U// ul v U1', ul /\·U1', 1} on I and the mapping r: (1, Tl)~ (I, T2), defined

by /(x) = x, for every x E /. It is clear · that Ul' U/, U1 v U1' and U1 /\ U/ are fuzzy clopon sets in (J, T2). Ii!t1,1(1/:r1)

87

we have clU2 =U'2,U2 5'rI(UI) 5'U2',U2 5'r\U1')5'

U2',rI(UivUI') =U

2' andrICU1 AUI')=U2. Thus the inverse

image of every fuzzy clopen set in (l, -r2) is fuzzy semi open in (l, 'tI).

Hence f is fuzzy !(so, clo). However, the inverse image of fuzzy clopen set UI in (/, -r

2) is not fuzzy open in (J, -rI). Hence the mapping is not

fuzzy l(o, clo).

5.3 Remark : l(o clo) maps in general topology have been studied under the name slightly continuous maps [5].

5.4 Example : Let X = {x, y, z), and Ul' U2, VI, V2 and V 3 be

fuzzy sets of X defined as follows : U 1(x) = 0.1, UI (y) = 0.4,

U2(x) = 0.3, U2 (y)= 0.3,

V'i(x) = 0.4;

V2(x) = 0.6,

V3(X) = 0.8,

V 1 (y)=0.5,

v2 CY>= o.5,

v3 CY>= o.6,

U1(z) = 0.2;

U2(z)= 0.3;

VI(z) =0.3;

V2(z)= 0;7;

V3(z) = 0.8;

Let -r1 = {O, U2, 1} and -r2 = {O, Ul' VI, V2, V3, l}.

If we define f (X, -r1) ~ (X, -r2) to be identity mapping, then it is

easy to ' see that the inverse image of fuzzy clopen sets VI and 'V2 in (l/'t2) is fuzzy semi open in(/, 'tI). Thus the mapping f is fuzzy !(so, clo). It is clear that in (/, -r2), cl U1 = V3'

and:in (1,-r1),S-intr 1 cV3')=0. Hence U1 is a fuzzy open set in

(1, -r2) with. f" I(U1) > S- int{ 1 (clU1). This shows that the mapping f is not fuzzy weakly semi continuous.

5.5 Theorem : If the graph function of a function is fuzzy l(so,lo), then the function itself is fuzzy ](so, clo).

Proof 1: Let f: X ~ Y be a mapping from a fuzzy space X to a fuzzy.· space Y such that the graph function g : X ~Xx Y is fuzzy l(so, cld~. Ll:lt Ube a fuzzy clopen set of Y. Then 1 x U is a fuzzy clopen set of JC<X. Y. For each x E Xwe have.

g"I I (1 x U) (x) = (1 x U) g (x) = (1 x U) (x, f(x))

=min (l(IX),' U(f(x))) = 1 /\ r 1 CU) (x) = r 1 (U) (x),

which is filzzy· semi open,. because g is fuzzy l(so, clo). Hence f is also fuzzy l(sa;>clo).

5~6!Remork : Since the intersection of two fuzzy· clopennsets is fuzzy clopon, the family of all fuzzy clopcn sets of a fuzzy topolOgical

*' space (X, 't)'.forms a base for a fuzzy topology 't · on X.

88

5.7 Definition : Fuzzy topology 1* generated by the set of all fuzzy clopen sets of r;x, 1) is called the fuzzy 0- dimensionalization of'!.

A fuzzy topological space (X, 1) is fuzzy 0-dimensional if 1* = 1.

5.8 Theorem: A mapping f: (X, 11)--'> (Y, 1

2) is fuzzy l(so, clo) if

f: (X, 11) ~ (Y, 1;) is fuzzy /(so, o).

The proof is obvious from Theorem 5.2 (e).

5.9 Corollary : If f: X-..:; Y is a fuzzy !(so, clo) mapping and Y is a fuzzy 0-dimensional space, then f is fuzzy !(so, o).

5.10 Theorem : If f: X ~ Y is a fuzzy !(so, so) fuzzy presemi open mapping from a fuzzy space X on to fuzzy space Y and g : Y--'> Z is any mapping, then gof is fuzzy l(so, clo) iff g is fuzzy !(so, clo).

Proof: Suppose thatgof:X ~z is fuzzy l(so,clo) mapping. Let Ube. a fuzzy clopen set of Z. Then (goff 1 (U)==r 1 (g- 1 (U)) is fuzzy semi open set of X. Since f is fuzzy pre semi open and surjective, hence f(f 1 (g- 1 (U))) = g- 1 (U)is fuzzy semi open set of Y. Thus g is fuzzy I(so, clo).

The converse is obvious.

5.11 Theorem : A fuzzy I(so, clu) imagH of a fozzy strongly S'-closed space is fuzzy slightly compact.

Proof: Let f: X ~ Y be a fuzzy l(so, clo) mapping from u fuzzy strongly S-clused space X on to a fuzzy space Y. Let {U)i e 1 be a fuzzy

clopen cover of Y. Since f is fuzzy l(so,clo) {r 1 (U)l is a fuzzy semi

clopen cover of X. Since the space iR fuz:.::y strongly S-close<l, there exists a finite subset K of I such that

v r 1 (U.) = lx iE K l

From the surjoctivity off, we have

re v r 1 <U·>> -= v r<r \U.)) = v u. = iy. ieK t ieK t teK t ,

Thus Y is fuzzy sliehtly compact. Since fuzzy SP.mi compact space is fuzzy strongly S-closed, the

following corollary is immediate : 5.12 Corollary : A fuzzy I(so, clo) image of a fuzzy semi

compact spacP. is fuzzy slightly compact.

. 5.13 Theorem : Let f: X ~ Y be fuzzy [(so, clo) injective mapping from a fuzzy space X to a fuzzy 0-dimensional space Y. ThP.n X is fuzzy semi P whenever Y i8 fuzzy P, where P E (T 0, T 1, T 2) [ 4].

Proof : Since Y is fuzzy 0-dimensional and l is fuzzy !(so, clo) then by Corollary 5.9, f is fuzzy I(so, u).

89

Suppose Y is fuzzy T0

. Let A be a fuzzy set of X. Then

f(A) == v I\ U .. , where U.. are fuzzy open or fuzzy closed sets of iE/ j=EJ lj lj

. I

Y Since f is fuzzy l(so, o) and injective, we have

A==r 1 crcA))==r 1 c v I\ u.)== v I\ r 1 cu .. ), i E I j E J lj i E I j E J tj

I I

where [ 1 (Ui) are fuzzy semi open or fuzzy semi closed sets of X. Thus X is fuzzy semi T 0 .

Next, let Y be fuzzy T1

. If A is a fuzzy set of X, then f(A) can be

written in the form f(A) == v U., where U. are fuzzy closed sets of Y. i EI t t

Since f is fuzzy l(so, o) injection. We have

A== r 1 crcA))) == r 1 c v u.) == v. r 1 cu.), iEI t. iEI 1

where [ 1 (Ui) are fuzzy semi closed sets of X. Thus X is fuzzy semi

T1.

Finally, suppose that Y is fuzzy T2

space. Let A. be a fuzzy set of

X. Then f(A) being fuzzy set of Y can be written as f(A) == v I\ U .. == v I\ cl U . ., where U .. are fuzzy semi

i e / J E ,r u i E / J E J. 11 u ' /.

open sets of Y. Since f is injective and l(so, 0), we have

A== r l(f(A)) == r 1 c v I\ u..) == r i l. v ' ' i E I J E "i lj i E I J E J.

l

I\ cl l) u .. J'·

= v I\ r 1 cc1 u .. ) c. v I\ s - cl r 1 cu..) IJ i E I J E J. l)

' i E Ij E Ji

where/- 1 (U .. ) . are fuzzy open seb; of X . lj' •

Hence A =A== v I\ r 1(U..) == v I\ S-cl r 1 (U .. ). . I' J l/ ' I . J IJ IE JE . !E JE i

'fhrn; X iH fuzzy KP.mi T0 •

"' 5.14 Theorem: Let f: X ~ Y be a fuzzy l(so, clo) injection from

a fuzzy space X to fuzzy 0-dimensional space Y. (a) If Y is fuz:t:.y n~gular [4] and/ is fuz:t:.y opP.n (or fuzzy closed),

then Xis fuzzy S-rcgular. (b) If Y is fuzzy normal [ 4] and f is fuzzy closed, then X is fuzzy

S-normal.

90

Proof: (a) Let A be a fuzzy open set of X. Since f is :fuzzy open mapping , f(A) is fuzzy open set of Y. Since Y is fuzzy:r:egul~r, we have f(A) = v U., where U. are fuzzy open sets of Y with cl U. :s;f(A). By

i E I l l l

Corollary 5.9, f is fuzzy l(so, o) and injective. We therefore have

A= r 1 (f(A)) = r 1 ( v U.) = v r 1 (U.),

iel 1 iel i

where [ 1 (Ui) are fuzzy serrii open sets of X with S =cl [ 1(Ui)

:s; r- 1 (cl Ui) s A. Thus Xis fuzzy S-regular.

. (b) Le~Kand''U be,' respectively fuzzy closed and fuzzy open: sets of X such that Ifs U. Thell f(K) is fuzzy closed and f(U')' is fuzzy\open set of Y such that f(K) sf(U')' . Since Y is fuzzy normal, there exists a fuzzy set V of Y: such that

ffK)~'s int. V·~ dVs:.f(U')'. BfiCorollarj7:5.9,fis fuzzy\I(so, o)and ~njective; we have

K~fi 1 lf(K))$r11(:hit V) i:;; s - irtt r 1(V) :s;S:'- cl r 1 (Vf

s~n 1 ( czvy ~ f! 1(f(rf')'). t: tJ. Thus Xis· fil.tey.'·s':norm.aL!.

REFERENCES . !if Ki.i{:.'Aia:d; 1 oi1 1 fu~ey semi ~ntinuity; rJziY\almosf'continuity and fuzzy weakiy

contihiiity; JJMdths. Anal Appl. 82 (1981), 14-32.::. (2)' M.P!Billunitii; 1Ph:D.Thesis, De1hi University, 1983. (3) C.L. Chiing; Fhzzytopological spa>ces, J. Maths. Anal Appl. 2"ii (1968),182-190. (4) B. Hutt'On:and•I.i&illy;Separation axioms in fuzzy topological spaces, Fuzzy.Sets'

and Systems 3:(1980), '93-104.· [5] R.C. Jain, Ph;D. Thesis,sMeerut Univetsity; 1981. 161 M:N. Mukherjee and s::P. Sinha/ frees'Olute and almost open functions between

fuzzy topological spaces, Ftizzy~ets and:Systems 29 (1989)i ·381-::1$8. [7] A.R Sirigal and,D.S. Ymlav, A generalization of semi contlritious: mappings, J,

Bihar Mdtlts.Sdc:< 11 (1987), 1 - 9; [8Ji Tuna HatiC Yal\iac, Fuzzy sets and functions ·on fuzzy space.s, J) Mtith. Anal.

Appl. 12G!(l987>,i409-423. 191 L.A. Zadeh; Fuzzy RP.t,<J,lriform.and Colttrol 8 (1965}, 338"353.

00

Jnanabha, Vol24, 1994 (Dedicated to Professor J.N. Kapur on his 70th Birthday)

cE:RtAIN :R:EsVLAs FOR A GENER.AL cuss oF: POL YNOM:tAiS, :KONHAUSER B''1oRTHOGONAL1.

POLYNOMIALS AND THE l\UJLTIVARIABLE H-FUNCTION S.P. Goyal

Department of Mathematics University ofRajasthan Jaipur - 302004, Inida

and Sunil Saxena, Department of mathematics Podd~r College, Nawalgarh - 333042, Inidia

(Received: March 12, 1993; Revised: November 3, 1994)

ABSTRACT In this paper first we evaluate· ari: it;ttegral irtvolvirtg ·th~· pfoduct t

of a ' general classes of polxnortF.a!s, Konh~user . bibrthogonal • polynomials and :the multivru·fable HJfuhi.itiOri. This integral !is then:· employed to

1 establish an expansion• fdrmufa • for:tP,e '-Product of general··

class of'polynomials and the(mtiltivariable'-H-ft.ilictidri i1Fla•"series of biOrthogonal ·polynomials. The ·results establi~hkd:l h~re 1 ' are •quite general in character· and a number •of (khown aiitl new results Whlch follow as special cases of our results are discussed• bii~fly ..

1: Intfoductfon~' K.6I1hkiuser' ([5], p. 303, 304) has considered 'followirig· proti of r

biorthogonal polynomials:

xki . . r (a+kh+ lj I.n (- li/c: j') t·(a+ kj + 1} z~ (x; k) = ni ! j = 0 ... (1.1)

and

1 n w w .. 'l • , -- ~ - . (W U+.J+1- ··, ~ (x, k) - 1 I. 1 I. ( 1)' ( -, )( k )n ... (1.2)1,

n.w=Ow·j~O 1

which[ were actu~Hy suggested by tllie Laguerre polyp.omiaJs,, k being a p:ositive integer:- .Indeeddfor k = 1, each ofi these pol)1l1omfals reduces to the Lagul:lrre polynomials L~~>(x ); and their. special' cases when k = 2: were'.encounteredr..earlier.1by Sp1mcer and Hmo 1 [9] hi certain analytical!] caleuMtionscinvoNing the penet:tation of 'Gamma ray~ through matter anddwere studied subsequently>by\Preiser!f6Ji,(See also Srivastava~{lO] for comments:c onr ther~ claimed i generaliliations of F the .Konhauser polynomials by Rl::Uzandd.. [7, p. uon..

Srivastava. 0111,, p.l, Eq. (1)) inttoduced ah gr=meral\i elm;:; uU ppl:Ynomials

92

[NIM] (-N)M. S%[x] = l: --~AN .xl, N= 0, 1, 2, ... ... (1.3)

j=O J. ,J

where Mis arbitrary positive integer and AN . (N,j 2:: 0) are arbitrary . ,j

constants, real or complex. By suitably specializing the coefficients AN ., the general class of

,j

polynomials can be reduced to classical orthogonal polynomials, Bessel polynomials and generalized hypergeometric polynomials (see, for example, Srivastava [ll] and Srivastava and Singh [12]).

The multivariable H-function occuring in this paper has been defined by Srivastava and Panda ((13], p. 271, Eq. (4.1)). We shall use the following contracted notation (Srivastava et al. (14], p. 251, Eq. (C.l))

0 N · m n 1; ... ; mu, nu · ' . 1' ' .

[

zl

H[zl, ... , zul =HP, Q' :pl' ql; ... ;pu, qu Z..u

( ., (u)) ( , ') ( (u) .Ju)) l a.;a .. , ... ,a. 1.: c.,y. 1 ; .... ; c. ,T .. 1 1 1 ,p 1 1 ,pl .I 1 l,pu

b 1:1 , 1~(u)) (d, 0 ') (d(ul o(u)) ( .; '' ... , p. 1 Q: . 'u. 1 ; ... ; .. ' u. 1

1 1 1 ' 1 1 ,qi 1 1 ,qu

... (1.4)

to denote the H-function of u complex variables z 1, ... , zu. Here all the G.reek letters are assumed to be positive real numbers for standardization purposes, the definition of the multivariable H­function will, however, be meaningful even if some of these quantities are zero. The details of these quantities are zero. The details of this function can be found in the papers and book referred to above.

Raizadu ((7), p.64, Eq. (2.1.2)) has introduced the generalised polynomial set defined by the following Rodirigues type formula

Su, 13, t lx· r s q A B k l] =(Ax+ B)- ex (1 - tr)- f3/t n ' ' ' ' ' ' ' TJ: 7 n [(Ax+ B)a + qn (I_ 'CXr)(- f3/t) + sn]

' where the differential operator Tk, z is defined as

T.k, l =xl (k +xD)

... (1.5)

'" (1.6)

The explicit form of this generalized polynomial set is ([7], p. 71, Eq. (2.3.4))

sex, f3, t [x· r s q A B m ll l] = Bqn x1<m + n) (1 - ur)sn zrn + n fl ' ' ' ' ' ' ' '

m+n. p m+n o :E l: l: I:

p=O t=Oo=Oi=O

(- 1)0 (- p)t (- 0\ (a)0 (a - qn)i

P ! o!iT t ! (i - a - S). l

93

(- ~ - sn) (i + k +rt) ( -=_'.t_{_f ~)15 ... (1.7) 1: p l m+n 1-'t:Xr B

Taking A= 1, B = 0 and 1: = 0 in (1.7), we arrive at the following polynomial set :

lim s~, p, 1: [x; r, sq, 1, 0 'm k, l] '-; 0

= s~, p, 0 [x; r, q, 1, 0, m, k, l]

=xqn+l(m+n)zm+nm~n f -~=p)t(a+qn+k+rtj (~xrf p=Ot=Op!t! l fn+n

... (1.8)

The above-mentioned polynomial set (1.8) is also general in nature and contains known polynomials due to Gould and Hopper [2), Singh and Srivastava [8), Chatterjee [1] and Krall and Frink [3] as its special cases.

· 2. Main Integral : In this section we evaluate the following general integral

~ (x - S)µ- l e-T]X ~ (x; k) s~, ~' 0 (y(x "'."" s)P; r, q, 1, 0, m, k, l)

s~ [z(x - S)cr] H[zl(x - sf1, ... , zh (x - S)crh] dx

m+A. p [NIM] n w ·;..;;e-llsem+A.yqA.+l(m+A.)+rp L L· L L L

p=O f=O 15=0 w=O i=O

-~f',---,--_M_15 ___ i A V + q11, + + (- I)n z() SP Sw (-p)i-N) (-w) ( ~ k rfl p ! f ! () ! i ! n ! w ! N, c5 l + A.

µ' H. • t [z1, ... , z1 l f.,u,p, t

where µ' = µ + q'A,p + pl(m +'A,)+ uo + prp and

m' 0 t [zl, ... , z,] t,o,p, t

r

' ~ 1 0, B + 2 : A

1, B ; ... ; A

1, B

1 =H i ' ' :

C + 2, D + 1 : C , D ; ... ; C1

, D1

• 1 1 I I 7.

'h

(1- µ'; cr, ... ,ah), cr crh

' 1 -) l+i+a-µ +n;-,;, ... , k' ( k

1 + i +CJ. - µ' • (jl (jh I (h) .

k ' k ' ... , k ), (a., a.' ... , a. >1. c ' J J .I

(b . (~ / l{(h )) • . , p·' ... ,I-'· 1 D .

.1 .1 J '

... (2.1)

... (2.2)

94

~~Yl ~01 0l l (d:, o' 1 1 !,cl , J ·j >1,C j'

J )1,Dl; ... ;(ih),S(h» h J J l,Dh

The (sufficient) conditions of validity of (2.1) are (i) k, s and r are positive integers. (ii) Re(ll) > 0, Re( a+ 1) > 0,

h

... (2.3)

Re(µ+ cro) + L cr. min Re[d~j) /oV)] > 0, (8 = 1, .. ., [N !Ml) j=l J lSiSA. 1 1

J

Ciii) n. > o, I arg z ·I < 112 n. n J I I

where C D A. D.

n. = - .!: a<!> - !: ~</> + f. oV) - f. tP> J i=B+l I i=l I i=l I i=A+I 1

j

B. C. J

+ !: Y./> -i = 1 I

J .fl) . -!: y; u - 1, 2, ... , h) i=B.+l I

.. (2.4)

J

Proof To evaluate the integral (2.1), we first express the multi­

variable H-function involved in the left-hand side in terms of multiple Mellin-Barnes tzye contour integral with tli,e help• of (1.4), general class of pqlynom+als and. geJ?.~ralised polynozpial s~t. i:q series given by (1.3) &n;d, q.s) r~~pectiv~ly and change t.l;le qrder opnt~grations, wich is ,per~i~~}.bl~ ~nder the conditions stated with; (2.3), we get the left-ban<! Rid~ ,of (2.1) as

1 I• \ .• ~·. · • ,

I .. : ,,.. ·.WifA. p .[NIMJ1'1(-p)f" '. (. ' A_,, : 't' 't'., ·'t' yp . '.; L.l'"t ,: fJ .... J ..... ----,--{, ~

(J=O.f=O a~o, P · ·

(-N)~g 'v.f-q/..+k'+rf)· •.· --AN,n ( l m+A.

zm + A .zO yqA. + l(m I A) + 1p

~ J ... J 0(tl) ... 0(th) \jf(tl' .. ., th) (2m) L 1 Lh

{

. h

( (x-s)µ+qA.p+lp(m+A.)+cro+prp+ j:1

cr/1 -1

e-rix ~ (x; k) dx} zi1 ... i~ dt1 ... dth ... (2.5)

Now for f:)valuation of the, inner x-integral, using the {[10], p. 43, Eq. (3.9)), changing the order of integration and summation involved therein and expressing the multiple contour integral as the multi-variable II-function, we' easily get the right-had side'of (2.1).

r.J (i. ·. r . t 2 1 • , ~

.F· 1< t~--

(J 'i!

'•

95

3. Special cases of (2.1)

. . - - - - . r(l + p + gN) (1) Tak.mg z -M -1, O"-g,AN, 15- N ! r (l + p +go)

and using the following relationship

s1 [ (x - sfl ~ zR, [x - s); g) in (2.1), we get the follwoing

interesting integral :

~ (x - S)µ - 1 e- T)X ~ (x; k) zi ((x - S) ; g)

s~, ~. 0 (y(x - S)P; r, q, 1, 0, m, k, l) H[zl (x - S)cr1, ... , zh(x -sfh] dx

m+A p N n w =e-ri~zm+"-yqA.+(m+n)+rp L L L L L

p=O f=O S=O w=O i=O

(-p)f(-N)s (-w)i f(l + ~ +gN) v +q 'A+k +rf p ! f ! o ! n ! w ! N ! f (1 +~+go)(---- l )m +"-

n p w rrtl" ) (- 1) ~ S n·. ~ t [z 1, ... , z1 J •.. (3.1 i,u,p, t

where µ" = µ + q'Al + pl(m 1 A) + gb + prp. . .. (3.2)

(ii) Taking M = 1, N = 0, AN 0 = 1 in (2.1), then Sij[z(x - S)CfJ reduce to '

one and the integral (2.1) takes the following form

( (x - S)µ - 1 e- rix ~ (x; k) sx· ~, 0 (y(x - l;)P ; r, q, 1, 0, m, k, l)

Hf~ 1 (x ,._ 1;)0 1, •.. , z,,(x - ~YTh.l dx ··.:• ''· . ". '· m+A p n w

= e-11~ zm +A. yq"- +, l(m + A.)+ rp :E , .', :E :E :E '· p=O f=O w=O i=O

(-p)r(-w)i v+q'A+k+rf'1

. ~,: ,'. ;·,,,

p ! f ! n ! i ! w ! ( l )~.+ r;l- 1r~~rt i;wm o,p, t [zl, ... , zh]

where µ"' = µ + q'A,p + pl(m + /..,) + prp.

4. Expansion theorem Suppose (i) k and r are positive intege1·s, (ii1 a1 > 0, a> 0, µ > O; Re(ot) > - 1,

. . h

Re(~ +!a~) +.L a1

min ~e(dy) /of))> - 1 i=l l<i:5A. . J

(o = 1, ... ,IN/Ml) (iii)Q >0 largz.I <l/20.n:(j=l, ... ,h)

J .I J

t•

... (3.3)

96

(.Q. is defined by (2..1)), J

then

xµ- 1 s~, ~, 0 (y xP; r, q, 1, 0, m, k, l) s% [z xcr] H[zlxP1, .. ., zh Xcrh]

oo [ m +A. p [JV/Ml n w 2: zm +A. yqA. + l(m + A.J + rp 2: 2: 2: I: . 2:

n=O p=O f=O o=O w=O i=O

(- P)/-N)Mo (-w)i v + qA, + k + rf p 0 (- l)n p!f!o!w!i! ( l )m+A.AN,os z r(l+a+kn)

IP1o t[z 1, .. .,zh] Z (x;k) ... (4.1) '+a. l a. l, u,p, n

Proof: Let

xµ- ls~,~, 0 (yxP; r, q, l, 0, m, k, l) s~ [zxcr] H[zlxcr1, ... , zh x<\]

= 2: k za. (x· k) n n '

n=O ... (4.2)

Multiplying both sides of (4.2) by e-x xa. ~ (x; k) and intergrating with respect to x from 0 to oo, we get

[e-xxµ+a.- 1 W(x·k)Sv,~,o(yxP·r q l 0 m kl) 0 k ' A. ' ' • • ' • •

s% [zx<YJ H[z1xa1, ... , zh x'\] dx

= ~ k [ e-x xa. za. (x; k) ya (x;k) dx ... (4.3) n = 0 n 0 .n v

using the integral (2.1) and the following orthogonal property ([5], p . . 303)

Joo xa. e-x ya (x· k) za. (x· k) dx = r (a+ kn+ l) 0 v ' n ' nl vn 0 '! • '

(where Re( a)> - 1, k, v, n are pm:rit.ivc integers. Also~\ n is the

weH-known Kronecker delta function) in (4.3), we find that

m+A. p K = zm +A. yqt.. + l(m +'A)+ rp 2: 2:

Tl p=O f=O

[NIM]

L 0=0

Tl w

L t w=O i=O

(- p){_ (- N)Mii (- w)i .v + qA, + k + r{) A i;;P ----·-·····-( l m+A. N,ii p!f!o!i!w! ,

(·-· l)n zo

r(l + a + kil) . , + (X . H [zl, ... , zh]

i, o,p, t

... (4.4)

... (4.5)

97

where H~' 8, ~-. 1 [z l, ... , z h] can be defined similarly to (2.3).

Substituting the value of K11

from (4.5) in (4.2), we arrive at the required expansion formula (4.1).

5. Special Cases of (4.1) : Using the substitutions in (4.ll as mentioned with the special cases of (2.1 ), we arrive easily at the following expansion formulas

xµ - l z~ (x; g) S~· 1;, 0 (y xP; r, q, 1, 0, m, k, l) H [zlx01, ... , zh x 0

1i]

• m+A. p N 11 10

= zm +A yqA. + l(m + /,) + rp L L L L L L n=O p=O /=0 o=O w=O i=O

(-p)l(-N)o(-w)i v+qA+k+rf r(l+~+gN) (-1)'1 p --·---.. ·--------· (----··----···---) -------·----- ·----------- ~

p If! O ! w Ii I l m +A. N ! r (1 +~+go) r (1 +a+ kn)

, +a [ ] Hµ zl' ... , Zn i, 8,p, t

... (5.1)

and

xµ- l S~· /;;, O (yxP; r, q, 1, 0, k, m, Z) H[z1x0 1, ... , zh x 0 1t] =

m+A. p n w = zm +A. y qA. + l(m +A.) 2: 2: L 2: . L

n=O p=O /=0 w=O t=O

(-p)l(-w)i (v+qA;k+rf)m+A.-r" (~-~)~ '---' [,,P

µ'+a H. 0 t [z 1, ... ,z1 ] i, ,p,. i

... (5.2)

The conditions of validity for (5.1) and (5.2) are easly obtainable from those mentioned with the main expansion theorem (4.1).

A number of other integrals and expansion formulas involving product of elementary special functions of one and more variables can be obtained from (2.1), (3.1), (3.2), (4.1), (5.1) and (5.2) as special cases. This can be done by specializing the parameters of th0 m11 lti-v::iriahle H-function in a suitable manner.

ACKNOWLEDGEMENTS The authors are thankful to Prof. H.M. Srivastava (University of

Victoria, Chanda) for making valuable suggestions for the improvement of the paper. The first author (S.P.G.) is thankfol to University Grants Commission, New Delhi for providing necessary financial assistance to carry out the work.

REFERENCES Ill S.K. Chattei:jea, Quelgues functions generatica de polynomes d' Hermite du

point de I' alg-cbrn de Lie, C.R. A.cad. 8ci. Paris 8cr. A 268 (1969), 600-602. 1?-1 H W no11lrl rmrl A.'1' Hoppnr Opnrntionnl formulas connected wit.h two

gcncrnlizntiono of Hermite polynorninlri, Duhe Moth. J 29 ( 1869), 55.

98

[3] H.L. Krall and 0. Frink, On a new class of polynomials: The Bessel polynomials, Trans. Amer. math. Soc. 65 (1949), 110-115.

[4] J.D.E. Konhauser, Some properties of biorthogonal polynomials, J. Math. Anal. Appl. 11 (1965), 242-260.

[5J J.D.E. Konhauser, Biorthogonal polynomials suggested by Laguerre polynomials, Pacific J. Math. 21 (1967), 303-304

[6] S. Preiser, An investigation of biorthogonal polynomials derivable from ordinary differential equations of the third order, J. Math. Anal. Appl, 4 (1962), 38-64

[7] S.K. Raizada, A Study of Unified Representations of Special Functions of Mathematical Physics and Their use in Statistical and Boundary Value Problems, Ph.D. thesis, Bundelkhand Univ., Jhansi, India, 1991.

[8] R.P. Singh and K.N. Srivastava, A note on generalization of Laguerre and Humbert polynomials, Ricerca (Napoli) (2) 14 (1963), Settembre-dicember, 11-21.

[9] L. Spencer, and U. Fano, Penetration and diffusion of X-Rays. Calculation of spatial distribution by polynomials expansion, J. Res. Nat. Bur. Standards 46 (1951), 446 461

[IOI H.M. Srivastava, Some families of dual and triple series equations involving the Konhauser birothogonal polynomials, Ganita 43, (1992), 75-84.

[11] H.M. Srivastava, A contour integral involving Fox's H-function, Indian J.Math. 14, (1972), 1-6.

[12] H.M. Srivastava and N.P. Singh, The integration of certain products of the multivariable H-function with a general class of polynomials, Rend. Circ. Mat. Palermo (2) 32, (1983) 157-187.

[13] H.M. Srivastava and R. Panda, Some bilateral generating functions for a generalizRd hypergcomctric polyuomiall:l, J. Reine Angew. Math. 283/284, (1976), 265-274.

[14] H.M. Srivastava, K.C. Gupta and S.P. Goyal : The H-Functions of One and Two Variables with Applications, South Assian Publishers, New Delhi, 1982.

Jnanabha, Vol24, 1994 (Dedicated to Professor J.N. Kapu.r on his 70th Birthday)

A GENERALIZED STUDY OF A VISCOUS INCOMPRESSIBLE FLUID FLOW THROUGH VARIOUS CROSS SECTIONS OF A TUBE

By

N. Yadav

Department of Mathematics and Computer Science

National University of Lesotho Lesotho (SOUTHERN AFRICA)

!Receiued: September 9, 1993; Reuised: December 29, 1994)

ABSTRACT An entirely new technique has been developed to study the

velocity diGrribution of viocouo fluidt: flowing through tubes whose cross sections are any rectilinear figure of n-sides.

1. INTRODUCTION The number of cases of viscous incompressible fluids flowing through channels of various cross section viz, rectangle, equilateral triabgle, right angle triangle, between parallel plates, elliptic section of tubes [1] have been discussed by various investigators. Besides these, the velocity distribution over Llw cross sediou of tube has also been determined for section bounded by (i) confocal ellipses (ii) a line and a parabola (iii) two confocal parabolas and part of their common axis (iv) two confocal parabolas etc. by different investigators [1]. In this paper an entirely new technique has been developed to study the velocity distribution when the croG~i section is any rcdilinon.r figure of n sides.

2. PRELIMINARIES In case of steady motion along a pipe of uniform cross section, if z-axis is taken parallel to the length of the pipe;

u = v = O; dW dW --""'o and -----0. az at So N avier Stokes equations reduce to n f1ingle equation

. ~ 2 -fl?: - - +fl v w ·- 0. dz

(2.1)

Assuming z to be zero and the pressure gradient Tz to be constant

(= h), the equation (2.1) reduces to,

2 h (2.2) V w = - - = -K (say). µ

100

Hence, finally we get,

'i72w V'2w (2.3) --+---=-K

ax2 C1y2

which has a particular solution

K Ax2 +By2

<2 A) w = -2-iC+B-

If <P(x, y) is also a solution of V'2cp(x, y) = 0. Then (2.3) has a general solution

(2.5) K Ax2 +By2

w=<p(x,y)-2Af.B

When the liquid touches the wall of the pipe, w = 0 and we have

. K A-x:2 + Ry2 (2.G) ()J(x, y) - -2 . _ = 0

which gives the boundry of the cross sedion of the pipe for flow through which (2.5) holds. <p(x, y) is circular harmonics which are known to be of one of the froms

of the harmonics rn cos (n8), ~ sin (n8) cos (nB), sin (nB) , log r and 8 ~ ,n

or their combinations. In cartesian coordinates the harmonics are given by xn _ nc

2 xn -2y2 + nc

4xn -4y4 ... and nc

1xn - ly _ nC;fXn -3y3 ... ,

which correspond to rncos(n8) and rnsin(n8). Dividing them by 2 2 . cos (n8) sin (n8)

(x + y r' we get the forms correspondmg to and . . rn rn

The harmonics of the type log r and e in polar coordinates are given

by log (x2 + y 2 ) and tan- 1 (,t) in cartesian coordinates. x

These harmonics result from the complex transformation,

w ""zn and w =log z,

where n is positive or negative integer. Unlimited number of harmonics ::ire derived from tho transformation w ::.:. f(z).

First case of interest occurs when <p(x, y) is at most of second degree in (x, y). In U1is case

(2.7) 2 2

_ . 2 2 K Ax +By w - a 2(x -y ) + 2b2xy +a 1x + b1y + a0 - 2 A+ B ·

Thn houmlary is given by

(2.8) [ KA lr.2 [ KB }2

a2 - 2(A + B) f + 2bzxy - a2 + ?.(A+ B) + alx + b,y +an= 0

101

which is a conic section. If equation (2.8) is written as,

2 2 (2.9) ax + by + 2hxy + 2gx + 2fy + c = 0, then

(2.10) w = ax2 + 2hxy + 2gx + 2fy + by2 + c

gives the velocity function admissible. The point of maximum velocity coincides with the centre because they are given by the same set of equations. If (x', y') is the centre, then the velocity at centre is,

(2.11) WC =gx' + fy' + C.

Constants a, b and Kare related by the equations;

2 K V' w = 2a + 2b = - K, a + b = - 2 . For special values of the constants the general form (2.10) assumes one of the following forms :

(2.12) w = - ~ (x2 + y2 - a 2)

when the cross section is the circle, x2 + y2 = a 2

K a2b2 [x2 Y2 l (2.13) w = - 2 2 2 2 + 2 - 1

a + b a b when the cross section is ellipse

x2 y2 2 +--z=l

a b

- K a2b2 [x2 y2 l (2.14) w - 2 2 2 2 - 2 - 1

b -a a b when the cross sectiion is hyperbola,

x2 y2 ----1 a2 b2 ...

This is not a closed curve therefore channels must be constructed instead of pipes. Flow in parabolic channels may be studied by the form

(2.15) w = - ~ (x2 - 4y)

while the flow between two parallel planes may be formulated in the forms

. K 2 2 (2.16) w = ~ (x - a )

102

(2.17) w == ~ (x2 - mxy) or more symmetrical form

(2.18) w == k (x2 - m2y2) 2(1-m2)

gives the flow in V-shaped channels of two intersecting planes. These special forms and many others have been studied in detail by many authors. Harmonics of the type cosh (nx) cos (ny) in series with suitable coefficients have been employed to form velocity fuction for flow when the cross section is rectangle x ==±a, y == ± b, and also when it is right angled triangle. Third degree harmonics have been employed for equilateral triangle. But the case of any triangle or any parallelogram or other rectilinear figure have not been solved as yet.

In what follows, we propose to clevelope an entirely new technique and find out the velocity distribution when the cross section is any rectilinear figure of n-sides. We shall limit our selves to quadratic form of the velocity function and shall not use higher degree harmonics nor harmonics like cosh (nx) cosh (ny).

3. BOUNDARY OF THE RECTILINEAR FIGURE OF n-SIDES The polar equation of straight line is (3.1) pcos(8·-a)==p. If there arc n lines, their equation may be written as, (3.2) r r.os (8 - a,.)== p, (r""' 1, 2, 3, ... , n)

If P,. is the angular coordinate of the point of intersection of rth and

(r + l)th lines then P,. satisfies,

(3.3) cos (j) - a. )p 1 = cos CP - a 1)p . r r r+ r r+ r

The ,-th line extends from e==r:i. toe==r:i..

Pr - 1 Pr

Since the figure is closed, P0 == Pn The figure under consideration may be said to have a boundary given by,

(3.4) r ~os (8 - a,.)= P,. ; P,. _ 1 < e < P,. . This boundary may be represented more elegantly by means of functions which we shall call <P (8, ~W); P < W c(l (8, PW), have the following definition :

<I>(8, Pl3'l == 1 for p < 8 < W <P(f·), f{fY) == 0 for W < 8 < 2rc + p. Hence the whole boundary is given by,

,,

r=n (3.5) 'L

r=l [p cos (8 - a) - pr] <I> (8, ~r- l~r) = 0.

The same boundary is also given by, r=n

(3.6) r: 1

A,. [p cos (8 - a,.) - p) <jJ (8, ~r _ 1 ~,.) = 0

where 'A,. are arbitrary constants.

103

In what follows a quadratic form is preferable for the boundary, so in place of (3.5) or (3.6) we shall denote the boundary by,

r=n (3.7) 'L [p;- p2 cos2 (8 - a) <I> (8, ~r _ 1 ~,.) = 0

r=l

r=n (3.8) 1: A, [p2 - p2 cos2 (8 - a )l <I> (8, ~ 1 ~ ) = 0 r r r r- r

r=l

for ~ 1 < 8 < ~ in each case. r- r

In fact p 2 = p2 cos2 (8 - ex), represents pair of lines bolh being parallel and on opposit sides of the origin at distance p. Therefore (::!.'/) or (3.8) represent two equal rectilinear figures each of which is obtainable by rotating the other round the origin through 180°. This fact will have to he taken into account in calculating total flow per unit time.

It should be noted that functions like <jl(8, PW) arc always possible to construct and these may be analytically expressed for 0 < 8 < 21t as,

s = 00

<l> (8, ~W) = 'L C8

cos (s8) s=O

where C8

are the Fourier coefficients calculated in the usual manner [2].

ThP. vnhm;:; of <I> (fl, ~W) relevant for this calculation are :

<!> = 0 for 0 < 8 < p <!> = 1 for 13 < l:l < W <!> = 0 for w < e < 2rt

Therefore

') J7t C = !:!.. <I> (0, PW> cos (s0) de s 7t 0

2 JP' = - COR (.<:fl) dfl 7t p

- ~ (sin sW - sin sp) and STI

104

15W w_r\ c = - de = L..=.£ . o n ~ n

Thus equation (3.8) written in full is,

(3.9) ':i;"A [p2 -p2 cos2 (8-a)J[~,.-~r-i+"i= r=l

1 1 1 TI s=l

2(sin s~,. - sins~,._ 1) I · STI

4. DETERMINATION OF THE VELOCITY DISTRIBUTION Here we shall be studying the velocity distribution given by,

r=nK (4.1) w= ,.: 12[p;-p

2 cos2 (8-ex,.)J¢(8,~,._ 1 ~,.)

For ~,. _ 1 < 8 < ~,.

w = ~ [p; -p2 cos2 (0 - a,.) J

w = K2

[p2 - x2 cos2a -y2 sin2 a - 2..<y cos ex sin ex J r r r r r

so that,

w = ~ [- 2 cos2 a,. - 2 sin

2 ex,. J = - K.

Also w = 0 gives the boundary considered in 3 i.e. r=n

I: [p; -p2 cos2 (8 - ex,.)] <I> (8, B,. _ 1 ~,.) == 0. ,. = l

Thus equations ( 4.1) give the form of velocity function for steady flow of an incompressible viscous fluid in pipe whose cross section is rectilinear fieiire of n sides. So far we have taken P,. to be quite arbitrary but continuity considerations impose limitations on the value of P,. which we now

proceed to di cuss. The velocity function should be continuous from both the Hides of the line fl = ~,., so,

(4.2) 2

p; -p2 cos2 <B,. - a,.) = 2

p; + 1 - p2 cos2 CB,. a,.+ 1) K[ , J K.[ J

for all valueR of p.

This is possible only if

P,.=Pr+l and B,.-a,.=a,.+ 1 -~,.· So we get,

(4.:l) p 1 =p2 =p:J = ... ""Pn and every

105

(4.3) P1 =p2 =p 3 = ... =p17

and every

(4.4) ex/' + ex/' _:'.:_}

~,.=-~2

In fact two conditions (4.3) and (4.4) are identical, when one is satisfied, the other will also be satisfied. Under these conditions w takes the form,

r = n K [ J [ ex,._ 1 + ex,. ex,. + a,.+ 1 J (4.5) W=,.:l Z p 2 -p2 cos2 (8--ex,.) 8,--

2-- 2

or more simply for

ex,._ 1 + ex,. ex,. + ex,.+ 1 ---~----- < 8 <

2 2

(4.6) w = ~ [p2 - p2 cos2 (8- a,.)J

The boundary of the cross secti0n of the pipe being formed of lines,

(4.7) p cos (8 - a,.)= ±p; r = 1, 2, 3, ... n.

The rectilinear figure given by (4.7) has the following properties :

(1) A circle can be inscribed in the figure touching all the sides.

(2) Origin or p = 0, is the centre of the circle. (3) p is the radius of the in-circle. So that all the boundary lines are equidistant from the origin and all such figures can be described round a circle.

If L is the perimeter of the rectilinear figure then the perimeter of a similar figure indside with sides at distance of S from the outP.r side is,

L(p-S,) p

So that the whole area of cross section is given by,

(4.8) JtJ J..,111-i-) L

A = V:' -"- dS = !:::E_ 0 p 2

and total flow per unit time,

Q = r E [P2 - CP - S)2J L(JJ - S) dS = !SbE~ . . o2 p 8

This must be Lhe flow iu µair represented by (3.7) or (3.8) therefore for a single pipe we get,

'l (4 9) Q = Klp'

• lfi

Mean velocity

106

(4.10) Q KLp 3 2 Kp2

U=-=---=-A 16 Lp 8 .

Maximum velocity

(4.11)

(4.12)

(4.13)

Kp2 c=--

2

U Kp 2 2 p k------­- KA - 8 KLp - 4L

k' = _c_ = 2Kp2 = p

KA 2KLp L.

5. SOME PARTICULAR CASES OF VISCOUS FLOW THROUGH TRIANGULAR AND RECTANGULAR CROSS

SECTIONS Triangles are figures with smailest numbers of sides that can be

described round a circle. The general formula for total flow per unit time for rectilinear

figures in section 4 has been obtained as,

KLp3 Q=-

16 If S is the area and s is the perimeter then

s d p =-an L =2s s

Therefore,

(5.l) Q = 2[(s s3 = J{S3 16 s 8s2

It will be interesting to compare this result (5.1) with two known solutions for triangles given by Boussinesq. Boussinesq formulae for equilateral triangle given by,

.Y = ± '13x, .Y = b, gives,

Kb4 (5.2) Q = 60 "3

(5.3)

Here, in this case, h2

s ... 1rr· s - ..JSu. Therefore from (5.1) we get

K b6 1 Kb4

Q = 8 3·{3 3b2 = 72 {:f .

Hence, (5.2) and (5.3) give values of Q in the ratio of 6:5.

107

Solution obtained by Proudman and others for rightangled triangle given by,

x =a, y =a, x + y = 0, gives

K [4 00 l (5.4) Q=4 3a5- 2: lf 5 coth(Na)

a n =0

where 2Na = (2n + l)n. In this case

a2 S = 2; s = (1 + l!'12)a.

Therefore, Q from (5.1) is given by,

(5.5) K 2a6 Ka4

Q= = 64(3 + 2'12)a2 186.496 ·

Thus formula (5.1) for Q in case of triangular cross section does not agree with the two known solutions. But when the number of sides is made indefinitely great and the rectilinear figure becomes a circle of radius a;

S = na2 and s = 1ra

then from (5.1) we get,

K 3 6 k 4 (o 6) Q = ~ = !S:_E_

. 8n2a2 8

which agrees with the known solution for pipe of circular cross section. This result (5.6) may be taken as a clue to the correctness of the solution obtained in previous section. However, the resu It can be experimentally checked for the correctness of the formulae of previous section.

In case of a square where sides are 2a ;

S=4a2,s=4a; Q is given by

64K a6 Ka4 Q= =-

128 a. 2 2

and k = 312 , k' = * .

In similar way total flow for other cuGcG of rectangle may be calculated. The result of this section can be easily extended to the cases when the cross section of the pipe is made up of arc of conic sections instead of straight lines.

REFERENCES Ill H. Bateman, H.L. Dryden and F.D. Murnnghan, Hydrodynamics, Dover

Publications (1956) [21 G.I'. 'l'ubluv, Fu111h•1 St'ric'.o;, T'rent.irn ITall, Inc (JG62)

Jnanabha, Vol 24, 1994 (Dedicated to Professor J.N. Kapur on his 70th Birthday)

TWO FIXED POINT THEOREMS FOR NONSELF MAPPINGS By

B.E. Rhoades Department of Mathematics, Indiana university

Bloomington, Indiana 47405, U.S.A. !Received: September 2, 1994; Revised: December 15, 1994)

ABSTRACT N adim Assad [l] established two fixed point theorems

for nonself maps. In this paper we generalize his contractive definitions and establish the corresponding fixed point theorems.

Nadim Assad [l] established a fixed point theorem for a nonself map satisfying the condition

(1) d(Tx, Ty)::; b(x, y) [d(x, Tx) + d(y, Ty)]+ c(x, y) min{d(x, Ty), d(y, Tx)),

where b and c are decreasing functions from R+ ~ [O, 1) such that 2b(t) + c(t) < 1 for all t > 0. He also established a fixed point theorem for K compact, T continuous, and strict inequality in (1) with 2b(t) + c(t) = 1.

The purpose of this paper is to generalize Assad's definition [l), and then prove the corresponding fixed point theorems.

LeL x0 e X,x 1' :=Tx0 If x1'e K, set x 1'=x1. OLherwise, dwut-:1e

x1 E ()K such that d(x0

, x1') = d(x

0, x

1) + d(x

1, x'1). Then x1 EK and Tx1

is defined. Suppose that, for n 2:: 1, {x0, x1, ... } and {x0', x1', ... } have

been chosen so that, for 1 s. i s. n, (i) x.'=Tx.

1,

l l -

(ii) x'. = x. if x.' t: K, or I I I

(iii) xi E ()Kand satisfies the relation

d(x 1,x') ... =d(x 1,x)+d(x,x').

l l / ... /. /. /.

Let x' 1 = Tx . If x' 1 E K, set x 1 = x' 1. Otherwise choose n+ n n+ n+ n+ xn+lEdK so that d(x

11,x'n-t·l)=d(x

11,x

11+l)+d(x

11+l'x'11 +l). The

scquecne (x } so constructed shall be called the general orbit of T at x0. n

Let P ::'.'."'. (x. E (x ) : x = x.') and Q := [x. E (x } : x. :;t x.'). Note that, L n 1. 1. i n i L

if x c Q, then x 1 c P. II . fl I

110

Theorem 1. Let X be a Banach s.vace, K a nomempty closed subset of X, T : K---+ X satisfying the condition that x E CJK implies that Tx E K and, for all x, y E X,

(2) d(Tx, Ty) s; b(x, y) max {d(x, Tx), d(y, Ty))+ c(x, y) min {d(x, Ty), d(y, Tx))

where b and c are decreasing functions from R+---+ [O, 1) such that b(t) + c(t) < 1 for all t > 0. Then T has a unique fixed point.

Proof. Let x 0 E X, {x,) a general orbit of x 0 , 111

:= d(x11

, x11

+ 1). Without

any loss of generality we may assume that T17

> 0 for each n. For, if

there exists a value of n such that T17

= 0, then xn = x11

+ 1. If x 11 E CJK, then x'

11 + 1 E K so that x'

11 + 1 = xn + 1. Thus

x11

= x 11 + 1 = Tx11

and x11

is a fixed point of T. If x11 ~ CJK, then x

11 EK

and either x' + 1 E K or x' 1 ~ K. If x' + 1 E K, then, as before, n 11+ 11 x

11=x

11+ 1 = Tx

17, and x

11 is a fixed point of T. If x'

11 + 1 ~ K, then there

exists an x11

+ 1 E (JK such that d(x11

, x11

+ 1) d(x11

+ 1, x'n + 1), and

xn = xn + 1 is impossible.

We shall first show that 't 1, <max (1 , 't 1}. n+ n n-

Case (a). Suppose that xn + 1, xn + 2 E P. Then, from (2),

'tn + 1 = d(x11

+ 1, x17

+ 2)::;; b(d(xn, x11

+ 1)) max {d(x11

, Tx), d(xn + 1, Txn + 1)}

+ c(d(x11

, x17

+ 1) min {d(xn, Txn + 1), d(x11

+ 1, Tx11

))

and 't 1 < 't . n+ n

Case (b). Suppose that x 1 E P, x 2 E Q. Then d(x + 1, x' + 2) = n+ n+ n n

d(xn 1 1, x11 +) + d(x

11 + 2, x'n + 2), which implies that

1 < d(x x' ) = d(x' x' ) == drTx Tx ·) n + 1 n + 1' n + 2 n + 1' · n + 2 ' n n +I

:S b(d(x , x 1

) max (d(x , Tx ), d(.-r;. 1, Tx: + 1)} n n+ · n n n+ n

+ c(d(x , x + 1)) min [d(x , Tx + 1), u(x + 1, Txn)) n n n n n .

= b(d(x , x + 1) max {1 , d(x + 1, x' + 2)}, n n n n n

which implies that 1 1

< 't . n + n

Case (c). Suppose that xn + 1 E Q, x11

+ 2 E P. Since no two consecutive

x'11s can lie in Q, xn c: P.

Proposition 1. [I} Let x, y E X, X a Banach space, andµ:= 'Ax+ (1 - A)y, where 0 ::;; A, ::::: 1. Then, for any w E X, llw - µII :::; max { llw -x II, llw -y Ill.

URing Proposition 1, t , 1 s; max {d(x' 1, x + 2, d(x , x11

+ ·»l· ll·r n+ n //. . ~

Suppose that the maximum iR d(x' . 1, x . 2). II+. II+

111

From ( 1) it follows that

Tn + 1::; d(xn + 1' x'n + 2) = d(x'n + 1' x'n + 2) = d(Txn, Txn + 1)

::; b(d(x , x 1

) max {d(x Tx ), d(x 1

Tx 1

)} n n+ n n n+ n+

+ c(d(x , x 1

)) min {d(x , Tx 1), d(x 1, Tx )) n n+ n n+ n+ n

= b(111

) max {d(xn, x'n + 1), 111

+ 1) + c(T,) min {d(x11

, xn + 2), d(xn + 1, x' 11

+ 1)}

::; b(T ) max {d(x x' 1), T 1) + c(T ) d(x , x 2) n n n+ n+ n n n+

::; b(T ) max {d(x , x' 1), T 1) + c(T ) d(x'n + 1, x 2). n n n+ n+ n x+

If max {d(xn, x' 11

+ 1), Tn + 1) = d(x11

, x' n + 1), then

d(x' 11

+ 1, x11

+ 1)::; b(T,) d(x11

, x'n + 1) + c(T11

) d(x' 11

+ 1, x11

+ 2),

or b(T )

't + 1 ::; 1 ~ , d(x , x' + 1) < 't 1, since x E P, x 1 E Q. n -c 't n n n- n n+ n

If max {d(xn, x' n + 1, •n + 1) = •n + 1, then

(3) d(x' n + 1' xn + z) ::::; b('tn)•n + 1 1 c('tn) d(x' n + 1' xn + 2)

or

b('t ) 't < n T n+1- 1-c('t) n+ l'

n

a contradiction

Suppose now that max {d(x' 1; x + 9 ), d(x , x + 9 )} is d(x , x + "). n+ n ,_, n n .l..l n n '-' Then

't ::; d(x x ) = d(Tx Tx ) n+l n' n.+2 n-·1' n+l

::; b(d(x 1, x 1) max {d(x 1, Tx 1), d(x + 1, Tx + 1)} n- n+ n- n- n n

c(d(xn _ 1, xn + 1)) min {d(xn _ 1, Txn + 1), d(xn + 1, Txn _ 1)}

(4) ::; b(d(xn -1' xn + 1)) max {'tn - 1' 'tn + 1) + c(d(xn -1' xn + l))•n·

. If max Crn _ 1, •n + 1) = •n _ 1, then we have

•n + 1::;; b(d(xn -1' xn + l)) •n - 1 + c(d(xn - 1' xx+ l))'tn

< [b(d(d(xn -1' xn + 1)) + c(d(xn - 1' xn + ,))]•n -1 < •11 -1'

since x 1, x E P. n- n

If max {'tn -. l' Tn + 1} = •n + 1, then JI

c(d(xn -- 1' xn + 1)) T < ·· · --- ---------·· T < T n + 1 - 1 - b(d(x , x )) n n - 1 ·

n--l n+l

112

Therefore, in all cases, T 1 <ma~~ {T , t

1).

n + 11 11-

The remaining cases, x 1

, x 2

E Q, cannot occur. II+ ll +'

Next we shall show that {T } converges to zero. There are two n

possibilities. Either {x } possesses a subsequence {x (' J) with the 11 11 a

property that x (k) 1, x (! l ') E P, or it doesn't. n + n :: +...,

Suppose such a sequence exists.

Fact l. T 1 <max {T , T 1

) implies that T 1 ::; max {L k' T k 1) n+ n n- n+ n-. n- -for each 0 ::; k ::; n.

Proof. The result is trivially true fork = 0. Assume the induction hypothesis. Since

'n - h ::; max {Ln - k -- l' 'n - k - )

'n + 1 <max {tn -k' tn - k - 1) <max {max { 'n - k -1' 'n -k _ ), 'n -k - 1)

=max {tn-k-1' 1n-k-2).

It then folluwa that

(Pi) 1 n(k) ~ max { tn(h - lJ 1 l' tn(h - 1) + 2).

Consider tn(h - l) + 1. Since xn(k - l) + 1, xn(k - l) + 2 E P, 'tn (k - l) + 1

<'tn(k-l)" For tn(k-l) 121 since xn(k-l)+ 2 f" P, if xn(k-l)+:~E P, then

'n(k l) < tn(h _ l) 1 1 by Case (a). lf xn(k _ 11 + 3 E:: Q, then +2 .

'n(h _ l) + 2 < 'n(k - l) + 1 by Case (b). Therefore tn(h) < 'n(k _ l), and {\} converges. Call the limit 't.

Suppose T > 0. From (5),

(6) 1 n(k + 1)::; 1n(h) + 1 = d(xn(h) + l' xn(k) + 2)::; 1n(k)"

Therefore limk d(x (I 1 1, x (' l 2) = 't. ' n I + //.ti +

From Case (a), d(x (I) 1 ,x 'J+''::;;b('t ('))'t (k)" Siuc:c t (')2t, · /1 1 ·I n.(11 "' n 11 n n 11.

and b is decreasing, 't !., 1 ~ b('t)'t (k)" Taking the limit as n ---;. 00

n(..)+ n yiPli!s t::; b(r)t, a contradiction. rl'lierefore 't = 0.

F'act 2. For each j su/Ticienlly large there exists k = kU) such that n(k) ~j:::;; n(k + 1). Using Fact l, 'i::;; 'tn(k)"

Proof. Dy inductlon. If .J ""n(k), then 'J = tn(k)" Tf j = n(k) + 1, then

~i < 'J- 1 = 'n(k) by Case (a). If j = n(k) + 2, then ~<max {'tn(k) + 1, 1 11(h)}

by what we have just provided.

Suppose that the result is true for j = n(k) + i. Then j + 1 ::; n(k + 1). If j + 1 = n(h + l), then we have equality. If j + 1 < n(k + 1), then

t. 1 1 <max (1 ·('l . T (' l .. 1} <max {1(k)'1 (k)) = 't (k)' .J -- . n 11' ·I ', 11 11~ + t ·- n . n n \,

by the induction hypothesis. Therefore lim T. = 0 .

.I

113

Now suppose that no such sequence exists. Then, for all n sufficiently large, one must have, for two consecutive values of n, x E P and x 1 E Q .. n n+

Let {x (.)} be the subsequence of {x } consisting of points in Q. n i n Since no two consecutive points can be in Q, xn(i) + 1 E P. Since no two

consecutive points are in P, xn(i) + 2 E Q; i.e., n(i) + 2 = n(i + 1). Hence ,

also, n(i - 1) = n(i) - 2. Thus the subsequence {xn(i)) is either

{x2n) or {x2n + 1).

Suppose {x2n} c Q. Then x2n + 1 E P and, by Case (c), t 2n < t 2n _ 2.

Therefore h2

n} is monotone decreasing and has a limit. Call it t.

For each n, by Proposition 1,

t2n = d(x2n' Xzn + 1) s; max {d(x2n - 1, x2n + 1), d(x' 2n' x2n + 1)} ·

Thus there must exist at least one infinite subsequence of {2n} for which either

(7) 't2n(h) s; d(x2n(h) - 1' x2n(h) + 1)

or

(8) 't2n(h) s; d(x' 2n(h)' x2n(h) + 1>· If (8) is true, then, by Case (c),

(9) 't2n(h) s; d(x' 2n(h) + 1) < 't2n(h) - 2·

If t > 0, from the first part of Case (c),

't2n(h) s; d(x' 2n(h) -1' x' 2n(h)) s; b(t2n(h) - 2) t2n(h)- 2.

from Case (b). For all h sufficiently large, t 2n(h) _ 1 > t/2. Thus, for

each such h,

t2n(h) ~ b(t/Z) ·r2n(h) - 1 ·

Taking the limit as h --) oo yields ts; b(t/2)t, a contradiction. Therefore t = o. By Fact 2, lim ·r.; = o.

If (7) is satisfied, the, by the second part of Case (c),

(lO) t2n(h) < [b(d(x2n(h) - 2' x2n(h)) + c(d(x2n(h) - 2' x2n(h))]T2n(h) - 2.

The sequence {d(x2n(h) -· 2, x 2n(h))} is bounded. Hence it has a convRrgent i:;uLt·mqucmce. Without loss of generality we may assume that limh (d(x 2n(h) _ 2, x 2n(hl) == p. Also, {t2n(h)} is monotone decreasing, hence convergent. Call the limit t.

114

If p > 0, choose h so large that (d(x2n(h) _ 2, x 2n(hi> > p/2. Then taking the limit of (10) as h ~ oo yields

'! :::; [b(p/2) + c(p/2)] '! < '!,

a contradiction. Therefore p = 0.

Finally, 12n(h) - 2 - d(x2n(h) - 2' x2n(h)) :::; d(x2n(h) - 1' x2n(h)) < d(x2n(h) - 1' x' 2n(h)) < 1 2n(h) - 2·

Hence limh d(x2n(h)- 1, x 2n(h)).

Suppose {x2 1) c Q. Then x

2 1 E P for all n sufficiently large. n+ n+

By Case (c), 12 1

< 12 1

and {12 1

) is monotone decreasing and n+ n- n+

has a limit '! 2 0.

For each n, by Proposition 1,

12n + 1=d(x2n+1' x2n + 2:::; max {d(x2n' x' 2n + 2), d(x'2n + l' x2n + 2)}

Then there exists at least one infinite subsequence of {2n + 1) for which either

(ll) 12n(h) + 1 :::; d(x2n(h)' x2n(h) + 2)

or

(l2) 12n(h) + 1:::; d(x' 2n(h) + l' x2n(h) + 2).

If (12) holds, then, by Case (c), 12n(h) + 1 < 12n(h) _ l'

and it follows that lim "n = 0.

If (11) holds then, as in case (5), lim '! = 0. n

We shall now show that {xn) is Cauchy.

Suppose it is not Cauchy. Then there exist an £ > 0 and two subsequences {p(n)} and {q(n)} such that, for all n, p(n) > q(n) 2 n, d(xp(n), xq(ni>;::: £and d(xp(n)- 1, xq(ni> < £.

Proof. The first inequality is true by the negation of the definition of a Cauchy sequence. The second inequality follows by the following argument. Since {'!n} is monotone decreasing with limit zero, there exists an N such that n > N implies that "n < c:. Let n be the smallest integer for which p(n) > q(n) 2N. Then p(n) > q(n) + 1, for p(n) = q(n) + 1 implies that d(xp(n)' xq(ni) < £, a contradiction. Since d(xp(n) + 1, xy(ni) < £, choose tho smallest integur r > p(n) + l such that d(x,., xq(ni);::: £. Then, with r := p(n), we have d(xp(n)' xq(ni)

2 £ and d(xp(n) _ 1, xq(ni) < £. Now em poly the same argument to p(n + 1), q(n + 1), etc.We then have, with Sn:= d(xp(n)' xq(ni)'

£:::; 8 n $ d(xp(n)' xq(n) -1) + d(xp(n)- 1' xq(n» < 1p(ni> + £ ·

115

Taking the limit as n ~ 00 , yields lim d(x ( ) 1, x r· .)) = c.. p n - q n

Using the triangular inequality, - 1 - 1 + x < d(x x . ) < 1 + 1 + s

p(n) q(n) n - p(n) + l' q(n) + 1 - p(n) q(n) n'

- 1 + s < d(x x ) < 1 + s p(n) n - p(n) + l' q(n) - p(n) n'

- 1 + s < d(x x ) < s + 1 q(n) - 1 n - p(n), q(n) - 1 - n q(n) - l'

- 1 - T + S ::::; d(x X ) ::::; S + T + T . p(n) - 1 q(n) - 1 n p(n) - l' q(n) - 1 n q(n) - 1 p(n) - 1

Taking the limit as n ~ = in each of the above inequalties yields

lim d(xp(n) + 1, xq(n) + 1) = lim d(xp(n) + 1, xq(n)) = lim d(xp(n)' xq(n)- 1)

= lim d (x ( ) 1, x ( ) 1) = c.. pn- qn-

Also,

Sn::::; d(xp(n)' xq(n) + 1) + "Cq(n)::::; Sn+ 21q(n)

and Sn::::; d(x(p(n)- l' Xq(n)) +- "Cp(n)-1)::::; Sn+ 2"Cp(n) -1

= lim d(xp(n)- l' xq(n) = £.

Case (A). Suppose that xp(n) + 1 , x q(n) + 1 E P. Then

d(xp(n) + 1, xq(n) + 1) d(Txp(n)' Txq(n)

::::; b(d)xp(n)' xq(n))) max {d(xp(n), Txp(n))' d(xq(n)' Txq(n) }

+ c(d(xp(n)' xq(n))) min [d(xp(n)' Txq(n)' d(xq(n)' Txp(n)l

::::; b(c.) max {tp(n)' 'l:q(n)} + c(E) min {d(xp(n)' xq(n)+ 1), d(xq(n), Xp(n) + 1)}.

Case (B). xp(n)+ 1 F P, r.q(n.)+ 1 F Q. 'Then :x:qfn) c- P, and

d(x x ) = d(Tx Tx ) p(n) + 1' q(n) p(n.)' q(n) - 1

< h(d(.xp(n.)' xq(n) _ 1)) max {d(xp(n)' Txp(n), dxp(n)- 1, 1'x11(n) _ 1))

+c(d(x ( )' x ( ) 1)) min {d(x ( )'Tr; ( . 1), d(x ( ) 1), d(.x ( ) 1, Tx ( ) 1)} · p n q n - p n q n) - q n - q n - p n -

Case (C). xp(n) + 1 E Q, xq(n) + 1 E P.

d(xp(n)' xq(n) 11) = d(xq(n) _ .), Txfl(n))

::::; b(d(xp(n) _ 1, xq(n)) max {d(xp(n)' xq(n), dxq(n)- l' Txq(n) _ 1)}

I c(d(xp(n}' xq(n) _ 1)) min {d(xp(n)' Txq(n) _ 1, Txp(n.) _ 1)}

Case (D). xp(n.) + 1, xq(n.) + 1 E Q.

d(x x ) = d(Tx Tx ) p(n)' q(n) p(n) - 1' q(n) - 1

::::; b(d(x x )) max {d(x XA d(x x )) p(n) - l' q(n) - 1 p(n) - 1,' p(n), q(n) - 1' q(n)

I c(d(xp(n} , xq(n) _ 1)) min {d(xp(n)- l xq(n)), dxq(n)- l' xp(n)l ·-1

116

Using the facts that

lim d(xp(n)' x q(n) _ 1) = lim d(xp(n) _ 1. x q(n)) == lim d(xp(n) _ 1, xq(n) _ 1) == e,

we have, for all n sufficiently large, upon adding the inequalities in Cases (a) - (d),

d(xp(n) + l' xq(n) + 1) + d(xp(n) + 1' xq(n)) + d(xp(n)' xq(n) + 1) + d(xp(n)' xq(n))

:s; b(e) max {tp(n)' tq(n)l + c(e) min {d(xp(n)' xq(n) + 1), d(xq(n)' xp(n) + 1)

+ b(e/2) max (tp(n)' tq(n)} + c(e/2) min {d(xp(n)' xq(ni>' d(xq(n)- l' xp(n))

+ b(£/2) max {1 ( ) 1, T ( )} + c(£/2) min {d(x ( ) 1, x ( J 1), d(x ( )' x < )) pn- qn pn- qn+ qn pn

+ b(£/2) max {Tp(n)-l' Tq(n)- l} +c(£/2) min ld(xp(n)- l'xq(n)), d(xq(nJ-l' xp(n)))

Taking the limit as n ~"°gives 4£:::; c(c)e + c(c/2) 3£ < 4e,

a contradiction. Therefore {x } is Cauchy, hence convergent to a point n z.

We shall now show that z is a fixed point of T. Since t,,, > 0 for

all n, there exists a subsequence {xh(n)} of {xn} such that xh(n)-:/:- z. In

turn, there must be a subsequence of {xh(n)} consisting only of points of

P or points of Q. Without lmss of g~~nerality we shall a81mme twu cases.

Case (E), {xh(n)} c Q, and Case (F), fxh(n} c P.

For Case (E),xh(n) E Q implies thatxh(n) + 1 E P. Define Pn := d(z, x).

d(xh(n) + 1, Tz) ~ b(ph(ni> max {Th(n)' d(z, Tz)} + c(ph(ni>min {d(xh(n)' Tz), Ph(n)+ 11.

Fix i:: > 0. ThP.n there exists an N ::mch that n > N implies that p h(n) < e .

Therefore, c(ph(ni> > c(e), or - c(ph(n)) < - c(e). Then

d(xh(n) + 1, Tz) ~ (1 - c(£)). max {'t'h(n)' d(z, Tz)I +min (d(xh(,;)' Tz), ph(n) + 1).

Taking the limit as n ~ "° yields d(z, Tz) s (1- c(e))d(z, Tz), which implies that z = Tz.

For Case (Ji'), we get

d('>;.' h(n) + 1, Tz) =- d(Txh(n)' Tz)

::; b(d(xh(n)' z)) max {d(xh(n)' x' h(n) + 1), d(z, Tz)}

+ c(p h(n)'z)) min {d(xh(n)' Tz), d(z, x' h(n) + 1)1.

Fix E > 0 and choose n large enough so that ph(n) < E. Then

(13) d(x' h(n) + 1, Tz) s; (1 - c(e)) max {d(xh(n), x' h(n) + 1), d(z, Tz)}

+min {d(xh(n)' Tz), d(z, x' h(n) + 1)}.

:,.·,

''··

'::'

117

From Proposition 1, d(z, x'h(n) + 1) $max {(d, xh(nl' d(z, xhui) + 1)). Taking the limit of (13) as n --7 oo yields d(z, Tz)::;; (1 - c(E)) d(z, Tz),

which implies that z = Tz. The uniqueness of the fixed point follows from the Lemma of (2],

since definition (2) is a special case of (24).

Theorem 2. Let K be a nomempty compact subset of a Banach space X, T : K --7 X, T continuous and such that x E oK implies Tx E K. Suppose that, for each x,y EK, x -:t=y,

d(Tx, Ty), b(x, y) max {d(x, Tx), d(y, Ty)}+ c(d(x, y) min {d(x, Ty), d(y, Tx)},

where b and c are nonnegative decreasing functions from W --7 [O, 1) such that b(t) + c(t) $ 1 for all t > O.Then T has a unique fixed point.

Proof. Let fxn) be a general orbit of x0. By following the first part of the proof of Theorem 1, there arc two changes. One is that $ is replaced every where by strict inequality. The other is that c('t,) =!- l. For, if c('tn) = 1, then, from inequality (3), b('tn) = 0, and one obtains the

contradiction d(x'n + 1, xn + 2) < d(x' n + 1, xn + 2). In inequality (4) one obtains the inequality 'tn + 1 <'tn_1·

Since X is compact Ix } has a convergent subsequence Ix ( .)} . Call n nt the limit z. Then lxn(i)} has one of the following three properties :

. · (P 1) lxn(i)} has a subsequence lxn(k)) such that xn(k) + 1 and xn(k) + 2 E P,

or no such subsequence exists. Therefore, for all k sufficiently large, one must have either

. (P2) for all n sufficently large, the subsequence lx2n} c Q, or

· (P 3) for all n sufficiently large, the subsequence {x2n} E P.

If condition (P 1) is satisfied, then one obtains inequality (6).

Taking the limit aB Tl > '"' yields, miiug Uw continuity of T,

(14) d(z, Tz) == d(Tz, r2z) = d(z, Tz).

If condition (P2) is satisfied, then ('t2n/ is monotonP. decreasing,

hence convergent. If (8) is satisfied, (14) follows from (9) by taking the limit as h --7 oo,

If (7) is satisfied, then, by the trianir,ular inequality, 't - d(x x ) < d(x x ) < d(x x' )

?.n(h) 2n(h) -- 1' 2n(h) + 1 - 2n(h) 1' 2n(h) - 2n(h) -1' 2n(h)·

= d(Tx2n(h) - 2' Tx2n(h) - 1) $ 't2n(h) - 2'

118

by Case (b). Taking the limit as h ~ 00 yields limh d(x2n(h) - l' x2n(h) + 1) = O and limh d(x2n(h) - l' x' 2n(hi) = limh T2n(h)'

which implies (14). The proof for (P 3) is similar.

Suppose that z ;t. Tz. Then, from (14),

d(z, Tz) = d(Tz, T 2z) < b(z, Tz) max {d(z, Tz) d(Tz, T 2z)} < d(z, Tz),

a contradiction. Therefore. z = Tz.

Uniqueness of the fixed point follows in Theorem l. Theorems 3.1 and 4.1 of [1] are special cases, respectively, of

Theorems 1 and 2 of this paper. REFERENCES

[lj Nadim A. Assad, On some nonself mappings in Banach spaces, Math. Japonica 29 (1988), 501-515.

[2] B.E. Rhoades, A comparison of various definitions of contractive mappings, Trans. Amer. Math Soc. 216 (1977), 257-290.

Jn8n8bhrr, Vol 24, 1994 ,IJ,·dil'(t/cd lu Pro/~s,,'<JI' J1V. 1''u;J11r ()/1 /1is 701/i !Jirtltduvl

COMMON FIXED POJNT THEOREMS FOR DEl\'~lFYJNG MAPPINGS

By

B.E. Rhoades Department of Mathematics, Indiana University

Bloomington, Indiana 47405, U.S.A 1Recei1•ed: l'-11!l't't11ii1 r :.!!. J.')9.J, Rl'l'ised. lJcccmhl'I' I. 199·/i

ABSTRACT

Several fixed point theorems for multivalued compatible maps are proved. As corollaries we obtain several known fixed puint theorems for densifying map:::;.

In a recent paper Diviccaro, Khan and Sessa /41 established the full ow i11g Lheurem.

Theorem J. Letl and g be two densifying and r:ommuting maps of' a bounded complete metric space (X, d) such that

(1) d(j'x, fy) <max {d(gx, gy), d(gx, fi), d(gy, fy), d(gx, /'.,v), d(f5.)', fx))

fiJ1· all x, y in X 1:Jw.:h that the right lwnd 1:Jide u/ (1) i1:J positive and fX r;;;f{X. Then f and g have u unique common fixed point.

The purpose of this note is to prove the aboe resl1lt by rcplncinp; thu hyputho::1is uf cornmuting Ly thu weaker £W8umplio11 of compatibility. This goal will be realized by first proving some fixed point theorems for multivalued maps, and then obtaining the desired result as a corollary.

We first state some familiar definitions. For a metric sp<1ctc (X, d), l a sclfrnap of X, A a bounded subset of X, u.(A) denotes the measure of monocompactness of A; i.e., the infimum of all f > 0 such LhaL A admits a finite covering consisting of subseLs wilh dia11wLer:; lc:1::i thnn c.

The following well-known properties of u hold : (il 0 s cz(A) s diam A,

(ii) u.(A) = 0 iff A is precompact,

(iii) (X(A U B) max max (u(A), u.(B)l for rrny bounded subsets A, B oLY,

(iul A <;;;; B implies that (t.(A.)::; u.(B).

/'is snicl to be clunsifying if f is continuous nncL for any boundl'cl n11r1 pn·c11rnpact suhsl:l /\ 11f'X, WL' l1;1H~ o.(ji/\) ,· cz(/\).

: 2c

Let B1X1 dcnr:k tlw sd uf bl)LJJ1dt:d sul>8ets of 8 compkte lllLtric sp~ice (X, cf) and clefinC' <1 function b: B(X) x 8(X)----:; 10, =I b.v ()1A.. BJ= sup (d(a, h): of' A, b E Bi. H is immecli8te that o satisf"i!:·s the triangulClr inequnlity ancl that C\(J\, B) = 0 iJT A= B ={a). Let. I be a sdfinap of X. F: X -7 B(X). F and I are s<iid to be o-compatiblc iff x E B(X) for x E X and C\(!Fx . Fix ) ----:; 0 wlwnever {.-c } is a sequence

II II II

such that Ix ----:; t and IF:r I----:; (t) for sonw I in X. (See, e.g .. ,Jungck n n · ·

and Rhoades [6]).

Theorem 2. Let F be a contin1w11s mapping of" a complete metric space (X, d) into B(X), I a co11timw11s selfinap of X Such that

(2) o(Fx, Fy):::; c max {d(lx, ly), 8(lx, FxJ, o(ly, Fy), o(lx, Fy), o(ly, Fx))

for all x, y in X, where 0:::; c:::; 1. If" FX s;;; IX and F and I ore o-compatible, then F and I have a l/,lll(jlle common fixed point z and further Fz == (z}.

Proof. The proof follows along the lines of the proof of theorem 1 of Fisher [5]. One defines a sequence {x) by x 0 EX, y 1 co ~F'x0 . Since

FX ~IX, choose x1 so that Ix1 = y 1. In general xn is chosen so that

lxn = )'11

, y11

E F:t11

_1. Then, as in Fisher, it follows that Ixn converges to a point. z, and the sets {Fx

1) converge to the set {z).

. I

From (2),

8(Flx , Fx ) :::; c max I 8<I2x , h ), 8(J2x , Flx ), o(lx , Fx. ), n n · 11 n n 11 . 11· 11

o(J2xn' Fx,), o(lx11

, Flx11

))

(3) ~ c max [8(1Fx 1

, Ix ). o(!Fx 1, Flx ) o(Ix , Fx ). fl 11·' ll - fl /I. II

b(IFxn _ 1, Fxn)' 8(lx11

, Flxn)).

From the definition of o-compatibility, o(IFx11

_ 1, Flx11

) -? 0. Since F and I are continuous, it then follows that lim o(IFx

11 _ 1, Ix,,) = lim 8(Flx

11, lx

11) = o(Fz, z) and lim 'D(IFx

11 _ 1, Fx

11) =

limo (Flxn, Fx11

) = 1'i(Fz, :?). Taking the limit. uf (3) as n -~ =_, we obtain

o(Fz, z):::; c max {8(F:::, .z), 8(Fz, Fz)l = o(F:<:, Fz). Again usmg (2),

o(F'lxn, Flx) ~ c max {d(1 2x11

, I 2x), '6(!2xll, Flx,)).

Taking the limit as n-? oo yield:; o(Fz, F'z) $co (Fz, Fz), from which it follows that Fz is the Hingleton set lzl.

~incc FX i;;;; IX, ihere exiAh: n point w such that lw == z. LT.sing (2),

BtFx11

, Fw) $ c max kl(lx,,, lw), o (lx11

, P.r,,J. 6(/w, Fwl, o(lx11

, Fw), '6(/w, Px11

1l.

Taking the limit 38 II ----:; 00 gives o(z, Fw) $ co(z, Fw), which impli(·:-; th;d Pz = izl == lw. From Uw o-rnmpatibility of I and Fit follows thut

12 .i

(z)=Fz=Fl1c=IF1c=:l:zJ, and·z is a fixed pnint uf I. (\>ndiL1n :21 forces the common fix(d pciint to be uniquv.

Theoren1 3. Let F be o continlli'llS 111uµping of n 1 <1m;:r1cl l/f('fric space (X, d) into BtX), Io conti11um1s sc/jiuup o{X such rho!

14,! fi(F.1, Fy) 01Fr. Fri< 1,1m; id1h. /y), ()\h. Fri. <i1lv, /·\ i. ()1jx, 1-\ J. 61fr. Fr;.

for oil x, y in X f(Jr which the right-!1wl-sidc of (4; is pc,::-:itice. If FX <;;;;IX and F ond I ore 8-compotiblc. then F one! I lwl'c o 11ni(//IC common fixed point 2 011d fii.rther J<z = (z).

Proof. Suppose there exists a 0:::; c < 1 such that F and I :-rn!idy inequality (2l for all r, yin X for which the right had side of inequ;tlity (2! is positive. If the right-hancl-sidl' of (2) is zero it follows tlwt \\"(' must have Fx = Fy =(fr)= Uy), whid1 forces the left had side uf (2J to be zero, so (2) is satisfied for all x, y in X and the result follrnvs from Theorem 2.

If no such number c exists, then there is an increasing sequence k ), wiLh limiL 1, and sequences {x }, lv ) such that

11 If ·11

. ~<fx11 , fe'.y11

);:: c,, max (d(lxl'ly11

), 8(Jx,,, Fx11

), 5(J.v11

, J;~>'J 8(1x,,, F_v11

), 8(ly11

• FY11

Jl.

Since X is compact, we may assume, without loss of generality, that the sequecnes {x

11) and LY) converge to points x and y respectively.

Letteniug n Lernl Lu infinity in the above rnequality leads to, since F and I are continuous,

o(Fx, Fy) .2 max (d(!x, Iy), o(Lt, Fx), o(ly, Ji:V1, 'O!_lx, Jt:v!, o(ly, Ft)):

Using the above inequality, along with (4) it follows that we must have

F:x = Fy ={Ix)= {ly). From the 6-compahb1Jity of 1 and F, we then have Flx = IFx. Thus F 2x =FIX= Fix = IFx = (l2x).

Suppose that Flx *Pc Then, from (4),

o(Flx, Fx) <max fd(I2x, Ix), o(I2x, Flx), 8(lx, Fx), 'ft(I2x, Fx), o(lx, Fb:))

= o(Fix, Fx),

a contradiction. Therefore Fix= Fx, and Ix is a fixed point of F. With Ix= z it follows that {Iz) = (!2x). =Fix= Ix= z, and z is also a fixed point of I. Definition (4) forces uniqueness of the common fixed point.

Corollary 1. Let f and g be continuous selfmaps of a bounded metric space (X, d) satis/vin!{ inequality ( Jj for all x_,y in X for which the right-hand-side of W is positive. If fX r;;;,gX and f and g are compatible, then f and g haue a unique common fixed point z.

Proof. Define A= O(x0), the orbit of x, for some x 0 in X. Since X is

bounded, so is A. Moreover, A b:0 l U {f(A)} U (g(A)}. Thus a(A) = max ((/(A), <X(~(A.)}. Sinct! f nnd g nrc clen:.1ifying nnd X is complete, it follm.vs that A is compact. Now apply Tlll'orern 3.

Cornll.::rry 2, Let (X, dl he o curnp!ete liletric ,"pace, fa dcn.<·1j011g cel/inap u/ X satisfri11µ

122

d1fr, /\·) < nrnx id(:r. Y). c/(.r. {cl, d(v, j'v), r/(1. /'rJ, d(y. /rl)

/(Jr coch x. y in X, :c T-y fl /i;r some x 0 in X, the sequence \.r) defined

hy l:r 0.x 1 =f(x0 ),x2 =f!x 1) ... )·is hounded, then/ has a 11111r111e ji.-ed

jJOl!l I.

Proof. \Vith A.= 0(:r), A is bounded, and the result folio\\:-; from Corollary 1 by setting g = f'.

Theorem 4. Let /; g be se!f!naps of' a complete metric space (X cl), l poc.sessing a unique fixed point 2 and g commuting u:ith l Then z is the unique comnwn fixed point off and g.

Proof. gz = g/z = fgz, and gz is also a fixed point of l Since the fixed point off is unique. gz = z .. Suppose that w is also a common fixed point of land g. TJwn w is a fixed point off. Since f has a unique fixed point, Ii'= z.

Remarks. 1. Theorem;; 1, 2, and Corollary 2 of Fisher [5] are special cases of Theorems 2,3, and Corollary 1, respectively, of this paper.

2. Theorem 4 of Diviccaro; Khan and Sessa [4] is a special case of Corollary 1.

3. Theorem 1 of Achari [l], Theorem 3 of Chatterjee 12], and Theorem 1 of Chattopadhyay [3] are special cases of Corollary 2.

4. Theorem 2 of Achari [l] is a special case of Theorem 4.

REFERENCES 111 J. Achari. Some results on clensifying mappings, Bull. 1vlath. de Ro111111111ie. 24

( 1982), 23-30.

121 B. Chatte1jee, Remark on some theorems ofiseki, 1ndian J. Pure Appl. Moth. 10 ( 1979). 158-160.

1:31 B. Chattopadhyay. A fixed point theorem of a densifying mapping on a bounded complete metric space, Indian ,!. Pure Appl. Math. 9 (1978), 320-323.

141 l'd.L. Diviccaro, M.S. Khan, and S. Sessa, Common fixed point tlwornms for Jpnsi(ying mappings, Raduui Mat. 6 (1990), 295-301.

/51 B. Fisher. Common fixed points of mappings and set-valued mapping". Rostur:/1 Mui.It. Ko/loq. 18(1981). 69-77.

IGI G. ,Jungck and B.E. rnwades. Some fixed point theorems for compatible m:tJrn. 1111. J. M.ath. & Alath . .'3ci. l6(J99:J). 417-428.

·i:-1

Jnanabha, Vol24, 1994 (Dedicated to Professor J.N. Kapur on /us 70th Birthday)

SOME REMARKS ON A FIXED POINT IN A-METRIC SPACE by

A.K. Agrawal and Abrar Ahmed,

Department of Mathematics,

Sahu Jain College, Najibabad-246 763, U.P. (Received: December 10, 1994)

ABSTRACT In this paper, we follow Rawat and Shahu, [3] and generalise

the result using Patil and Achari [2], and Dubey [1] for pair of mappings in 2-metric space.

Theorem 1: Let f, g be a pair of self maps of a complete metric space X. If

there exist positive real numbers 0 ::;; ~ y, 8 < 1 satisfying a + ~ < 1, ~ + 8 < 1, and if (1) Let

(1) (d(fu1

, gu2

))2 ::;; a d(u1

, fu3

) d(u2

, gu4

) + ~d(u 1 , u2

)2

+ yd(u2, gu4) d(u3, gu6) + Od(u4, fit 5 )2

for all u.i, i = 1, 2, ... , 6 F X then f; g have a common unique fixed point.

Proof : Let x, y E X and points

Then

if

If we put

and

U1=gfx, U2=fgy, U3=gy, ll4=fx, ll5=X, U.5=Y

(difgfx, gfgy))2 ::;; (a+ ~) (d(gfx, fgy)) 2

. 2 . a + ~ < 1 = k , then 0 < k < 1 d(j'gfx, gfgy) :; d(gfx, fgy

x = fx x = vx fur every n = 1 2 3 n n - l' n + 1 ° n · ' '

x = xn _ 3, y = x11

_ 2, then

d(x11 , xn + 1) = d(fgf xn - 3, gfg xn - 2)

s kd(Ifx11

_ 3, fgxn _ 2)

::;; kd(x11

_ 1, xn)

::;; k2d(xn - 2> xn - 1)

s K1d(xu, x 1)

124

where K < 1. Therefore K1' ~ 0 as n ~ oo.

So the sequence defined above is Cauchy sequence in a complete metric space X. There exists a point z in X such that x ~ z, n ~ 00

n

ofg.

Let u 1 =u3 =u5 =z, u 2 =u4 =u6 =xn

(d(jz, xn + 1))2

= (d(fz, gxn))2

::; cul(z, fz) d(xn, gxn) + ~d(z, xn)2

2 + yd(xn, gxn)d(z, gxn) + &l(xn, fz)

n ~ oo

2 A,.J 2 (d(jz, z)) ::; cul(z, fz)d (xn, xn + 1) + pu-(z, xn)

n ~ oo

2 + yd(xn, xn + 1)d(z,xn + 1) + 8d(xn, fz) ,

(d(fz, z))2::; o:.O + j3.0 + y.O + &l(z, tzl,

(d(fz, z)l::; o(d(z, fz)f A contradiction 0 < <> < l. Hence fr.= z i.e. z is u fixed point off. Similarly z is also a fixed

To claim the uniqueness we say that

(d(z, w))2 = (d(fz, gw))2

::; a.d(z, fz)d(w, gw) + ~(z, w)2 + yd(w, gw) d(z, gw) + 8d(w, fz) 2

d(z, w)2

:::; CP 1-()) d(z, w)2

p + 0 < 1, which is a contradiction. Hence z = w.

Theorem 2: If { = g, we get, 1~1hich we give without proof.

(d(fu 1 , fu 2))2 ~a d(u 1• {l.f.3 )d (u.2, fi14J + j3d(u1, ui''

· + yd(u2, fU1 )d(u3, fr.tG) + &.l(u4, fu.5) 2

0 < uI> u2, ... , u6 EX< 1 then f has a fixed point. We state and prove the following theorem which is direct application of the theorem 1 for a family of mappings.

Theorem 3: Let {fk} (k = 1, 2, 3, ... , n) be a family of mapping of a complete

metric space X into itself. If fk satisfies th.P. conditionr:

(i) ( 1 fr· fn. commute with every fk.

(ii) (d(fl f2 ... fnuI, fn - I ··· f~ f~u2))2

::; ad(ul' f1 f2 ··· fn u3)d (uz, fn fn -1 ··· f2 f1 u4) + ~d(ul, uz)2

+ yd(u2,fn fn - 1 · ·· f {1u4) d(u3,fn fn - 1 ··· f1 u6)

+ &l,(u4J1 f2 ··· fn U5) 2

125

u 1 ... u 6 in X and 0 ::; a, ~ ... < 1 then {fk} have a common fixed point.

Proof: From (ii) If we put ( 1 ... fn = f, fn .. .f1 = g

Then it takes the form (1) by theorem 1 f, g have a unique fized point of z. i.e. fz = z =gz for any fk,fk(z) = fkz by a view of (1)

f(k, k z) = fk z so fk(z) is fized point off and z is a fized point of z.

By putting u 1 = u 3 = u5

= fkz, u2 = u 4 = u 6 =z in (1)

d(jkz, z)2 = d(ffkz, gz)2

~ ad(jkz, ft/,z) d(z, gz) + !'Jd(fkz, z)2

I yd(z, gz) difk z, gz) + &i(z, ffk z)2

2 2 d(fkz, z) ::; (~ + o)d (z, fkz) ' (~ + 0 < 1).

difk2, z)2

::; (~ + o) difk2, zf A contradiction. Hence fkz = z i.e. z is a fixed point of a family of

mappings {fk).

REFERENCES [1] Praveen Dubey, A result on common fixed point and sequence of mappings,

Garula, 2 (1991), 134. [2] P.T. Patil and J. Achari. A note on common fized points in 2-metric spaces, The

mathematics Education, 22 (1988), 126, 129. [3] Pratima Rawat and P.L. Sahu Common fized point theorem for 3-mappings, Acta

Sciencia. lndica, XVM 4(1989, 351.

Jiianabha, Vol24, 1994 (Dedicated to Professor J.N. Kapur on his 70th Birthday)

RAYLEIGH-TAYLOR INSTABILITY OF ROTATING OLDROYDIAN VISCOELASTIC FLUIDS IN POROUS MEDIUM

IN PRESENCE OF AV ARIABLE MAGNETIC FIELD

P.KUMAR

Department of Mathematics, Himachal Pradesh UnivC)rsity, Summer Hill, Shimla-171 005, Inida

(Received: December 15, 1994)

ABSTRACT The Rayleigh-Taylor instability of Oldroydian viscoelastic fluid

in porous medium in the presence of uniform rotation and variable magnetic field is considered. The magnetic field, the viscosity and the density are assumed to be exponentially varying. For stable density stratification, the system is found to be stable for distrubances of all wave numbers. The magnetic field stablizes the potentially unstable stratification for small wave- length perturbations which are .otherwise unstable. The long wave length perturbations remain unstable and are not stablized by magnetic field. Rotation does not affect the stability or instability, as such, of a stratification.

1. Introduction A detailed account of the instability of the plane interface

between two Newtonian fluids, under varying assumptions of hydrodynamics and hydromagnetics, has been given by Chandrasekhar [1]. Bhatia [2] has considered the Rayleigh-Taylor instability of two viscous superposed conducting fluids in the presence of a uniform horizontal magnetic field. Sharma [3] has studierl the instability of the plane interface between two oldroydian viscoelastic superposed conducting fluids in the presece of a uniform . magnetic field. Bhatia and Steiner [4] have studied the problem of thermal instability of a Maxwellian viscoelastic fluid in the presence of rotation and have found that the rotation has a destabilizing effect in contrast to the stabilizing effect on Newtonian fluid. Bhatia arid Steiner [5] have also com:idorod the problem of thermal inotability of a Maxwellian fluid in hyuromugudic::.> auu have fouuu that the muguelic field has stabilizing effect on viscoelastic fluid just as in the case of Newtonian fluid. Eltayeb [6] has stuied the convective instability in a rapidly rotating oldroydian viscoelm1tic fluid. The medium has boen considered to be non-porous in all the above studies.

When the fluid slowly percolates through the pores of a macroscopically homogeneous and isotropic porous medium, the gross effect

128

is represented by Darcy's law according to which the usual viscous term in the equations of fluid motion is replaced by the resistance term - (µ/ k

1)u;7 where µ is the viscosity of the fluid, k

1 the permeability of the

medium and ~the filter velocity of the fluid. Lapwood [7] has studied the sability of convective flow in hydromagnetics in a porous medium using Rayleigh's procedure. The Rayleigh instability of a thermal boundary layer in flow through porous medium has been considered by Wooding [8]. Oldroyd [9] proposed a theoretical model for a class of viscoelastic fluids. An experimental demonstration by Toms and Strawbridge [10] reveals that a dilute solution of methyl methacrylate in n-butyl acetate agrees well with the theoretical model of Oldroyd fluid.

The present paper attempts to study the stability of the plane interface separating two incompressible superposed rotating Oldroydian viscoelastic fluids in porous medium in presence of a variable magnetic field. The instability of such viscoelastic fluids in porous medium may find applications in geophysics.

2. Perturbation Equations

Let T .. , z .. , e .. , µ,A, A0{< 'A.), p, o . ., v., x. and dd denote re:::1pectivcly

IJ IJ lJ l) /. l t the total stress te1wor, the shear stress tensor, the rate-of-strain tensor, the viscosity, the stress relaxation time, the strain retardation time, the isotropic pressure, the Kronecker delta, the velocity vector, the position vector and the mobile operator. Then the Oldroydian viscoelastic fluid is described by the comititutive relations

T iJ = - poi) + zii'

(i + A1t) ziJ == 2µ(1 + Ao:i)eU, ... (1)

_.!(avi ~1 e ij - 2 OX· + dX · .

J l

Relations of the type (1) were proposed and studied by Oldroyd [9]. Oldroyd [9] also showed that many rheological equations of state of general validity, reduce to (1) when linearized. Ao= 0 yields the fluid to

be Maxwellian whereas A= A0 = 0 gives the Newtonian viscmrn fl11i<l,

Consider a static state in which an incompressible Oldroydian viscoelastic fluid is arranged in horizontal strata in porous medium and the pressure p and the density p are funtions of the vertical coordinate z onlL The system is acted on by a vario.blc horizontal magnetic field H(H0(z), 0, 0), a uniform rotation IT(O, 0, .Q) and a

gravity force gfo, 0, - g). The character of the equilibrium of this initial static state is determined, as usual, by mipposing that the tiystem is slightly disturbed and then by following its further evolution.

Let Vfu, u, w), op, op and li!(h , h , h ) denote respectively the " y· z

perturbations in fluid velocity (0, 0, 0), fluid density p, fluid pressure p

129

and the magnetic field H(H0(z), 0, 0). Then the linearized

hydromagnetic perturbation equations of rotating Odroydian viscoelastic fluid in porous medium are

1; ( 1 + A;t J= ( 1 + A~t )[- v op+ g7 op+-;- ci?+ IT)

+ ~; j(Vx'f0xE+(VxE)xnj]-J:1(1+A0:t)u; ... (2)

v. i?= 0,

E an!= (H. V)i?- ci?: V)H, at v. nt= 0,

a E dt op= - w(Dp),

... (3)

... (4)

... (5)

... (6)

where E is the medium porosity, µe the magnetic permeability and D = d!dz. Equation (6) results from the fact that the density of every particle remains unchanged as we follow it with its motion.

Analyzing the disturbances into normal modes, we seek solutions whose dependence on x, y and tis given by

exp (ikx--r; + iky.Y + nt), ... (7)

where kx, ky are horizontal wave numbers, k'1. = k~ + k; and n is a complex constant.

For perturbations of the form (7), Eqs. (2)-(6) give

Q ~ E (1 + An)nu = - (1 + An)ikx bp + (1+An)41T~ h2 (DH)

2p.Q J!_ .. +(l+A.n) v-k (l+A,0n)u, (8)

E 1

;c1 + An)nv - - (1 + An)iky 8p + (1 + Antf:_ (ikxhy -ikyhx)

- (1 +An) 2P.Q u - l!_k (l + A0n)v, (9) E l

Q µ~ (1 + An)nw = - (1 +An) [Dop + g8p] + (1 +An) -

4 .

E n

[(ik~z -Dhx) - hx V:i -f(l + A.0n)w, ... (10) 1

ikxu + ikyv + Dw = 0, ... (11)

ikxhx + ikyhy + Dhz = 0, ... (12)

E nhx = ikJizt - w(Dll), ... ( 13)

130

Enhy == ik/Iv, ... (14)

E nhz == ik:flw, ... ( 15)

En bp = - w Dp. . .. (16)

Multiplying Eqs. (8) and (9) by by - ik , - ik respectively, x y

adding and using Eqs. (11), (13)-15, we obtain

2-n(l + A.n )Dw == - k2(1 + An )bp - _g_k (1 + A0n) Dw E 1

2 2 2pQ µ/f µ/fk

- (1 +An)/;+ (1 + A.n)k~ -4-i; + (1 +An)

4_-u w(DH), ... (17)

E I y nnE JldLE

where I; the z-component of vorticity, is given by

I' dV dU .k .k ':> dX - (bl == l xV - l y U.

Multipying Eqs. (8) and (9) by - ik and + ik , respectively, y x

adding and using Eqs. (11), (13)-(15), we obtain

/;== 2(l+A.n)QDw 2 2' ... (18) EV · k xVA

(1 + A.n)n + (1 + A0n)-k + (11 +An) --1 n

where V1 == µeH2/41tp is square of the Alfven velocity and v(==µ/p) stands for kinematic viscosity. Eliminating op between Eqs. (10) and (17), using (18) and the relation

i 2(1 + A.n )k Q n } ik2u = -(k Dw + k 1;) = k + + (1 + l.n)k2~A Dw,

x y x 2 EV x (1 + "11.)n + (1 + A.0n)n k ·

1

... (19)

we get after simplification

[(1 + A11.)n + (l +A n)-EVJ[D(pDW)-k2pw] + fgk2(-l_+_~}J (Dp)w o k. n

!

+ 4(1 + 'An)2 Q2n[-··· j~=-D.::.:w'---------i (1+ An)n2 + (1 + 'A0n)n ~~ + (1 + 'An)k!v!

2

+ (1 +'An) µ4 .. l!_kx [D(H2Dw) -k2H 2w] == 0. . .. (20)

. 1tn

3. The Case of Exponentially Varying Density, Viscosity and Magnetic Field

Assume the stratifications in density, viscosity and Assume the stratifications in density, viscosity and magnetic field of the form

i31

_ Pz , _ !)z H2 _ H2 _Hz f:lz (21) p - Poe ' µ -- µoe ' - - o e ' ... where p0, µ0, H 0 and ~ are constants. Equations (21) imply that the coefficient of kinematic viscosity v and the Alfven velocity are constant every where. Using the stratification of the form (21), Eq. (20) trnasforms to

[

EVo (l+'An) 2 2 (1 + 'An)n + (1 + 'A0n) hi+ n VAkx

4(1 + 'An)2Q

2n ]D2 + w

2 EVo ·· 2 2 (1 + 'An)n + (1 + 'A0n)n k + (1 + A.n)V_Akx

1

[

EVo (1 +'An) + (l+A-n)n+(l+'A0n)k;+ n V!k~ a

+ 4(l+A.n1 Q n ~Dw ... (22) 2 2 ]

(1 + f...n)ns + (1 + A.0n kvo + (1 + A.n)V!k! 1

[(1 ,,_) (l 'I )EVo (l+"An)V2k2 A.(1+/...n)}2 -0 - +Ni n + +/\,on k + A x - g.., w - ' 1 n n

µo T>'J. µ/f5 where v0 =- and V_A = -

4- are constants.

Po n:po

The general solution of Eq. (22) is

w =A1eq1z +A2

eq2z, ..• (23)

where A 1, A2 are two arbitrary constants and q 1, q?. are the roots of the equation

[(l + 'An)n + (1 +'Aon) :vo + (1 +n 'An) V!k!

1

4(1 + A.n)20.

2n ] 2 + q

2 EVo ~ ::! (1 + /...n)n + (1 + 'A0n k + (1 + A.n)VA.kx

1

l Ev (1 A ) + (1 + A.n)n + (1 + A.0n) -k

1° +

1n n V!k!

4(1 + A.n)2

Q2n ] A. + ..,q

EVo 2 2 (1 + 'Afi)ns + (1 + A-0n -k + (1 + A.n)VA_kx

"1

132

[(1 1 ) (l 1 ) EVo (1 +An) V2k2 r:i. (1 + An)]k2 _ 0 - + An n + + Aon -k + A x - g 1-1 - • 1 n n

... (24)

If the fluid is supposed to be confined between two rigid planes at z = 0 and z = d, then the vanishing of w at z = 0 is satisfied by the choice

w =A (eql z - eq2z),

while the vanishing of w at z = d requires exp (q 1 - q2)d = 1,

which imply that

(q1 - qz)d = 2imn,

where m is an integer. Eq. (24) gives

... (25)

... (26)

... (27)

q1, 2 = [- ~/2[(1 + l.n)n + (1 + A,,n) ::o + (l: l.n) V!k;

· 4(1 + Ni)'2.0.'2n } +~~~~~--'--~~'---~~~~~~

2 EVo 2 2 (1 + lvi)n + (l + A0n)n k + (1 + An)V.Akx

1

lJ f EVu (l+An) 2 2 ±2j_betasup2(l+/..n)n1-(l+A.n)~+ · n V.Akx

+-····· . 4(1 + An)2.Q

2n l2

2 EVo .-, 2 (1 + /..11.)n + (1 + A0n k I (1 + A.n)VA.kx

. 1

+ 4k2Ici + A.n)n +(~+Aon) EkVo + (l +An) v!k; 1 1 n

+ ... 4(1 + A.n)20

2n ]

2 EVo ·" <> (1 + /...n)n + (l + A.0n -k + (1 + An)V,4k;

'l

· l (l + l.n)n + (1 +Aon) ::o + (1 +n l.n) v;.k; -g~ (l : "") ir2 J

[

i= v0 (1 +An) · (1 + An)n + (1 + A.0n) -k + Vlk;

1 n

133

4(1 + An)2D.

2n ]-

1

+ -(l_+_'An-)n_2_+_( 1---"-+-A_o_n-'E-kv_o_+_(l_+_A_n_)V_A_2 k-; 1

... (28)

Inserting the values of q1

, q2

from (28) in Eq. (27) and

simplifying, we obtain 6 5 4 3 2 A 6n +A5n +A4n +A3n +A2n +A1n +A0 = 0, ... (29)

where A 6 =AA2, A 5 = 2AA[l + A0],

2 2 E Vo 2 2 2 2 2 2

A 4 = [A{l+A0 ~2-+2(A+A0)}+B{A Q }+A {2kxvAvAA-c)], k1

EV 0 EV 0

[

2 2 l A3 = 2A{l +Ao ki 0 1+2B{AQ~) + A.t(2 +Ao k1°)(2k;v!A- C) '

A,= HE :~i) + B{Q2

} + (2k~~A - CJ j 1 +(A+ A,) ::0

) + A2k~~(k:~A - G)l 2 2 2 2 FVQ 2 2

A 1 = 2'Akx VA (kx VAA - C) + k (2kxvAA - C), 1

2 2 2 2 Ao = kx VA (kx VA_ A - C),

where, we have put

A= p2d2 + 4m2,.2 + 4k2d2,

B = p2d2 + 4m2rt2,

c = 4k2d 2gp.

... (30)

Eq. (29) is the dispersion relation studying the effect of rotation and the variable (exponential) horizontal magnetic field on the stability of stratified (exponentially varying density, visr.mlity) oldroydian viscoelastic field in porous medium.

~ For stable stratification p < 0, Eq. (29) does not have any change of sign and so has no positve root of n and the system is always stable for disturbances of all wave numbers.

For unstable stratification and if

2 4gpk2 VA< --

2 , ... (31)

2 21t 2 2 ([3 + 4m 2 + 4k )kx d

134

the constant term in Eq. (29) is negative, therefore, it has at least one positive real root and hence, the system is unstable for all wave numbers satisfying the inequality

2 2 2 2 2 k

2 g13 sec 8 13 d + 4m n < 2 - 2 '

VA 4d ... (32)

where 8 is the angle between kx and k. (i.e. kx = k cos 8).

If 13 > 0 (unstable stratification) and also

2 4@k2

VA_> ( 2 2 } ' 132 + 4m n: + 4k2 2

d2 x

... (33)

then Eq. (29) has no positive root and so the system is stable.

Thus, for unstable density stratification and magnetic field such that

2

4j)k } ' v! > ( 2 4m2n:2 + 4k2 ; 13 + d2

... (34)

the system is unstable for all wave unmbers satisfying 2 2 2 2 2

k 2 A sec 0 13 d + 4m n <gp--2-- 2

VA. 4d

Also, it is clear from Eq. (29) that rotation does not affect the stability or instability , as such, of a stratification.

ACKNOWLEDGEMENT The author is bigly thankful to Prof. R.C. Sharma, F.N.A. Sc.,

Department of Mathematics, H.P. University, Sh.imla for his valuable assistance and suggestions in the preparation of the paper.

REFERENCES [1) S. Chadrasekhar, Hydrodynamics and Hydromagnetic Stability, Dover

publication, New York, 1981. (2) P.K. Bhatia, Nuovo Cimento, 19B, (1974), 161. [31 R.C. Sharma. J. Muth. Phys. 8r.f, 12, (19'/8), 603. 141 P.K. Bhatia and J.M. Steiner, Z. Angew Math. Mech., 52 (1972), 321. 151 P.K. Bhatia and J.M. Steiner. J. Math. Anal. Appl., 41 (1973), 271. 161 I.A. Eltayeb, z. Angew Math. Mech., 55 (1975), 599. 171 E.R. Lapwood, Proc. Camh. Phil. Soc., 44 (1918) 508. 181 R.A. Wooding, J. Fluid Mech., 9 (1960), 183. [91 J.G. Oldroyd, Proc. Roy. Soc. (London), A245 (1958), 278. 1101 B.A. Toms and D.J. Strabridge, Trans. Faraday Soc., 49 (1953), 1225.

Jnanabha,Vol24, 1994 (Dedicated to Professor J.N. Kapur on his 70th Birthday)

FREE CONVECTIVE MHD FLOW OF A DUSTY VISCOELASTIC LIQUID IN POROUS MEDIUM PAST AN OSCILLATING

INFI~"'I'I'E POUOUS PLATE By

!\LP. Singh" and Ajay Kumar Singh C.L. Jain Postgraduate College, Firozabad-283202, U.P., India

(Received : November 15, 1993)

ABSTRACT Free convective laminar flow of an electrically conducting

incompressible dusty viscoelastic liquid (Walters liquid model-B) embedded with non-conducting dust particles through a porous medium past an infinite porous plate subject to a slightly sinusoidal transverse velocity distribution has been analysed. The mathematical analysis is presented for the hydromagnetic boundary layer flow without taking into account the induced magnetic field. Expressions for velocity field, temperature distribution, skin- friction and heat transfer have been obtained and discussed with the help of graphs.

1. INTRODUCTION. The study of the flow through porous media is of principal interest due to its applications in geophysics in the assessment of geothermal resources, in design of under ground energy storage systems, oil recovery, soil sciences, astrophysics, nuclear power reactor engineering and so on. Lighthill [1] has studied laminar skin-friction and heat transfer to fluctuating in stream velocity. Stability laminar flow of a dusty gas has been studied by Saffman [5]. Mukherjee and Mukherjee [3] have investigated a problem on unsteady flow of dusty viscous fluid due to time dependent tangential stress applied at the surface. Free convection effect on the Stoke's problem for an infinite vertical plate in a dusty fluid has been discussed by Ramamurthy [4]. Mishra [2] has investigated a problem on MHD flow of an oscillating plate in absence of pressure gradienl. Sharma and Sharma [6] have studied free convection non-Newtonian flow past an infinite plate with suction and constant heat flux. Unsteady free convection MHD dusty flow between heated porous vertical plates has been studied by Subeydar [10]. Recently, Singh and Rana [7] have discussed three dimensional flow and heat transfer th1·011eh a porous medium. More, recently Singh et al [8) have investigated heat transfer in three dimensional MHD flow past a porous plate.

*Address for correspondence: 236 Durga Nagar, Firozabad-283202 U.P.

136

The object of the present paper is to analyse the effect of free convective laminar flow of an electrically conducting incompressible dusty viscoelastic liquid - (Walters liquid model-B) embedded with non-conducting dust particles through a porous medium past an infinite porous plate subject to a slightly sinusoidal transverse velocity distribution. Velocity field, temperature distribution, skin-friction and heat transfer have been obtained and discussed with the help of graphs.

2. FORMULATION OF THE PROBLEM. Let us consider the laminar flow of an electrically conducting, incompressible, dusty, viscoelastic (Walters liquid model-B) liquid in a porous medium past an infinite porous oscillating flat plate. The plate executes oscillations in its own plane with frequency n decreasing exponentially with time. Let x-axis is along the flow of liquid in the plane of the plate and y-axis perpendicular to it. The dust particles are assumed to be solid, spherical, non-conducting and symmetrically distributed in the flow region.,; A constant velocity v0 (v < 0 for suction and a> 0 for injection)

normal to the plate is applied. A uniform magnetic field B

0 is applied perpendicular to the flow

region. Since the liquid is viscoelastic, the induced magnetic field has been neglected in comparison with the applied magnetic field. It is assumed that the temperature of the plate remains steady at T w and

free strel:\m temperature at T. Then the governing boundary layer equations:' ~r a two-dimensional unsteady, laminar, incompressible <lusty visc~lastic liquid (Walters liquid model-B) can be expressed as

~l - 2 KN dU du. A (T T () u. (j B2 v 1 0 . --v --:-- =g >-' - )+v --- u--u+--(v-u) dt 0 dy x 00 ay2 p 0 k p

[ 03u iJ;:iu}

- k ay2at - aya

Ju mat =K1 (u -v) ... (2.2)

ar ar 'A a2r ., ~ - VO a= - --:----;;- ... (....,,3) - . !)' pep dy"'

The boundary conditionR are :

u=v=v0 (1+Ee-nt), T=Tw(l+Ee-nt) aty=O ... (2.4)

u == v => v 0, '1'------:) T 00

as y ---> oo

where u is the velocity of the liquid, v is the velocity of the dust particles, cflis the specific heat at const:mt pressure of the liquid, P is the coefficient of volume expansion, g is the acceleration dim to x gravity, v is the kinem::itic viscosity of the liquid, K is the elastic

137

constant, p is the density of the liquid, t is the time, A, is the thermal conductivity of the liquid, Tis the temperature in boundary layer, B 0 is the induction of magnetic field, k is the permeability of the medium, T

00 is the temperature of the liquid for away from the plate, 0 is the

electrical conductivity of the liquid, m is the mass of dust particules, N 0 is the number density of dust particles and .v0 is the suction velocity.

We introduce the following non-dimensional quantities -

yvo ' - u *- - u - ' y - V ' VO

. 2 u.,._ u f'- vo

vo' - 4v '

Kv 2 T-T c K' = _o T'' = --""-- J:l:E = p (Prandtl number)

2' T-T' A, ' v w 00

2 1 + --: (Permeability parameter), VO k k

crv B2 0

--2-· = M (Hartmann number),

puo

vg (T -T )r:i. x w ooP ::i · - G (Grashoff number).

UO Using these non-dimensional quantities, the equations (2.1),

(2.2) and (2.3) after ignoring the stars over them reduces to

·;Pu au i l i au i a3u a3u 2 +;i...-CM+k)u+ (u-u)--4 ~,t-K( 4 2 - 3 )=-GT ay v .)' w a ay at ay

and

au w-=u-v vt <PT + P aT _ P aT = 0 ay2 ay 4 at

.... (2.5)

... (2.6)

... (2.7)

where mN

0 l = -- (mass concentration of dust particles), and p

" w = mv~o/ 4vK1

.

The non-diemensional boundary conditions are

u = u = 1 + E e- nt, T = 1 + E e-nt at y = 0

u = u -> 1 , T -7 0 as y -> oo

3. SOLUTION OF THE PROBLEM. Following LightMll [1], we assume

- - '\

... (2.8)

138

T(y, t) = [l - f 1(y)] + E e- nt [1--- f~,,{J)]

u(y, t) = g(y) + E e- nt g 2(y)

u(y, t) =F1(y) + E e-nt F2(y)

... (3.1)

... (3.2)

... (3.3)

Substituting (3.1) - (3.3) in the equations (2.6) and (2.7)_. after comparing the harmonic terms we get

fi° + pf1' = 0 ... (3.4)

l +,, f.' nf. n p'2 + 2 +4 2 =4 gl =F1

... (3.5)

... (3.6)

g2 F 2 = l . . .. (3.7)

( -nw)

Using relations (3.6), (3.7) and (3.1) - (3.3) in the equation (2.5), we obtain

- 1 Kgt'+gt+g1'-(M+k)g1 =Cf1-l)G ... (3.8)

Kg ,,, (l Kn) ,, , M 1 n ln ). . (f. l)G 2 + + ·4 g 2 + g 2 - ( + k - 4 - 1 _ wn ~ 2 = 2 -

The transformed boundary conditions are :

f 1=f2 =0,g1=a2=l, F1=F2=l aty=O

f1 =f2 ~1, f=g1 ~1, g 2 =F2 => 0 as y ~ 00

... (3.9)

... (3.10)

The soluiton of equations (3.4) and (3.5) under the boundary condtions (3.10), we have

f1 =1- exp (-Py)

f 2 = 1 - exp (-PH?J)

where H 2 =~ [ 1 + (l -~)112] Following Soundalgckar [9], we assume

gl =go1 +Kgn + O(K2)

g2 =go2 + Kgl2 + O(K2)

FI= Fol+ KF11 + O(K2)

F2 =Fo2+KF12+ O(K2)

... (3.11)

... (3.12)

... (3.13)

... (3.14)

... (3.15)

... (3.16)

Substituting (3.13) - (3.16) and using (3.11) and (3.12) in the equations (3.6) - (3.9), we obtain

fol =gOl' f11 =gll

F = go2 02 (1- wn)

f -~!'B_ 12 - (l - wn)

" ' (M 1 gOl +gol - +k)go1=-Ge-PY

" ' (M 1) ,,, gll +qll - +k gll =-gOl

,, , M 1 n Zn ae-PH,; go2 + qo2 - ( + K -4 - (l _ wn) g02 = -

G ,, , M 1 n ln · n ,, · 12 +g12 -( +k4-(l-2n))g12=-4go2 -go2

The corresponding boundary conditions are

g01=l, g11=0, go2=l, gi2=0,

F 01 =1, F 11 =0, {02 =1, F 12 = 0, at y = O .

go1 ~ l, Fo1 ~gll =go2 =g12 ='Fo2 =Fu =F12 ~ 0

asy ~ oo.

139

... (3.17)

... (3.18)

... (3.19)

... (3.20)

... (3.21)

... (3.22)

... (3.23)

. .. (3.24)

On solving equations (3.17) to (3.23) under the boundary conditions (3.24), after substituting in the equations (3.13) - (3.16), we have

g1=1 + Gb1(e-H.¥-e-PY) + KGbf p3(e-H.t>' -e-PY) ... (3.25)

g 2 = (1 + Gb3 + KGb3b5

)e-HaY - b3G(l + Kb

5) e-PH.zy ... (3.26)

F1

=1 + Gb1(e-H.t>' -e-PY) +KGbip3(e-H.l-e-PY) ... (3.27)

F 2 = (1 + Gb7

+ KGb9b3) e- Hil - (Gb

7 + KGb?.b9)e-PHz.v ... (3.28)

Hence on putting the values of g 1,g2,F1,F2'f1 and {2 in the

equations (3.1), (3.2) and (3.3) we obtain the velocity of the liquid, particle and temperature of the liquid

u = 1 + Gb1(1 +Kb

1p 3) (e-H.? -e-PY) + E (1 + Gb3 +KGb3b5)

e- (HaY + nt) - E Gb3(1 + Kb

5) e- (PH,zl + nt) •.. (3.29)

v = 1 + Gb1 (1 + Kb1p 3)(e-lf.? - e- PY)+ E (1 + Gb7 + KGb3b9)

- e- <H1l + nt) - E G(b7

+ Kb3b

9) e- (PH.zy + nt) ..• (3.30)

T = P.- PY+ i=: P.- (Pl12y + 11t)

where, ... (3.31)

140

1/2

H4 =~[1+{1+4(M+ ~)} 1/2 l[ { ( l_ !!:.._ n Jil H6 = 2 1 + 1+4 M + K- 4 1-wn ~

l

b, =[P(P-1)-(M + ~ T' -1

b3 = [PH2(PH2- l)-~M +i-~ -1 ~nwn JJ 2 2 ( !!:.. b5 = b3P H 2 PH2 - 4

b7 = b3 1 (l - 2n)

bg = b7b5.

4. SKIN-FRICTION AND HEAT TRANSFER. Skin-friction 't1 for the liquid is given by

'ti=(~l=O

l

::: blG(l + H . .blp3) (P - H4) - E I (1 + Gb3 + Kb3b5G)H6

- (Gb3 +KGb3b5) PH2]e-nt ... (4.1)

Skin-friction 't2 of the dust particles is given by

'tz = (avl ()y = 0

't2 = Gb1 (1 + Kb 1p3) (P -H4) - E [ (1 + b7G + KGb3b9) H 6

- (b7

+ Kb3b

9) GPH

2J e-nt ... (4.2)

Heat transfer in tl'\rms of NuRRHlt number N is given by (1.

N _,_ -(()1'1 a dy =0

= P(l + E IJ2) e- nt ... (4.3)

5. DISCUSSION. Velocity profiles of the liquid and dust particles for varius values of magnetic field (M = 0.2, 1.2 and 2.2) at P"' .71, G - 5.0, l = 0.3, w = 0.2, K = 1.0, E = 0.4, t = 1.0, n = 0.1 and

·fh = 0.25 are shown in figure l. From the figure it is obvious that the " velocity of the liquid and particles decreases as the intensity of

magnetic field M increases. We also observe that the velocity of the

. · 1 (t_lli

141

' FIG.l VELDCfTY PROFlES OF THE LIQUID ANO DUST PARTlliS

>'

FOR VARIOUS VAWES OF MAGt4ETK:.: FIELD (HARTMANN

NUMBER M)

(P =0·71, G::; 5 0, 1=03/ w=O 2, K=10, E: =0 4, t=l·O, n= 0·1 and k =0 25) .

1 8.

~~-, M:02 -- - ~----, I __...- ~~ "---'...._ / "-12

& ,//_,... '',,...._ ""~~~- ''"- -22 / '...._ ' M_ / .....-::-- --~" ...._

~ /; ~~ ~'~;)/ ', ~ '-, ',,

~"-:::: 1s1,11

, · ., y1:;_._

"' u ~ , -~-,." ~""

1·6 •.

1 7

13.

LIQUID VELOCITY (u)

2 12 :> PAfHICLE VEl.OClfY ( v) .... Ii 6 _, w > 1·1

10·

0 02 01.

y ... ···-·-·····

06 08 10 1·2 1-4 16

···--·--...

142

FIG.2 VELOCITY PROFILES OF THE LIQUID AND DUST PARTICLE

FOR VARIOUS VALUES OF POROSITY PARAMETER k

(P=0·71, G:::S·O, l=03, w=02,K=l0,€:::04,t=10_,n::. 01

anq M::: 1 2)

1 s -

1 7 ·--- .. ·- k = 0 30

/ ::- - -- ----- ----------- - -... '·· {"" 7-/ -- -- ~ -"--._ ·-.. .... . k = 0 2 5 ~ ----- -- --- -,_ ·, '

, ,::---- ~""'' ~--::, 1 ,_,'° / / - - -- - - ' •<'-,

16 {; // '' ~-,..::.---. ' t ----~ ----- ' · ... :--... ~ / -·- ~---<:-' ., ..... --..... :___...._ . f / ---------::-~,, '-.... '-...::::, ' ' ',., ------: ',

................ ":::---- .. J ' JS!- "', '<-:_- __ -.......:::<'- 1 .... ___

~.................... I I -....., I ~~ II -----~-'-1 4

13

> ;J

i:: 1 2 ·-u 0 iii >

11

1-0

0 02

.. ,

LIQUID VELOCITY(u)

PARTICLE VELOCITY ( v)

l ___ _;

0·4 06 08 10 1 2 11. 1 6

y -- .. ·- ------·------------ ----·-------~

143

FIG.3 VARIATION OF SKIN-FRACTION rACTOR WITH TIME FOi"l

DIFFERENT VALUES OF GRASHOr F NUMBER G,

(P=071, l=OJ, W=0·2, K=1 0, E =0·4,n= 01, I<= 0·25 and M:12)

32

Z B

2£

1 0

-::_ I 6 .. ,, z !? n u a .._ ,i O·B•·

~

0£

0

SK,.J-rnf.CTION ( I1 }

s1:1N-FRACTION "{ r 2 I

---- -- - --- - --- -=-=-=--=--:::: Gd

- _:-_:::..:::~:_:::: ... ::::::~=-=--=- G~S

Gd

<------L.~--.L----'-----'

2 £ 6 8 10

t------

FIG.4 VARIATION OF HEAT TRANSFER WITH FREQUENCY

FOR DIFFERENT VAWES OF TIME I

( p :C>71, €: 0·4)

10·7Jr-----]:0·12 - ......__----===----~- "' ::1 ----- ---- Id ffi ------f:j

~ 071 a I: -t w T 070

10 ----'----'---~----'

02 O·J 0·£ 05 0·6 " _______ ___, ..

144

liquid and particles increases mi y increases from y = 0.0 to y = 0.4 after that the velocity of the liquid particles decreases from y = 0.4 to y = 1.6 for all given magnitudes of magnetic intensity.

Figure 2 shows the velocity profiles of the liquid and dust particles for various values of porosity parameter (k = .20, .25 and .30) at P = .71, G = 5.0, l = 0.3, w = 0.2, K = 1.0, E = 0.4, t = 1.0, n = 0.1 and M = 1.2. From the figure, we observe that the valocity of the liquid and dust particles increases as the porosity parameter k increases. Besides, we observe that the velocity of the liquid and particles increases as y increases from y = 0.0 to y = .4 after that it decreases from y = .4 to y = 1.6 for all the given values of the porosity parameter.

Variation of skin-friction factor with time for different values of Grashoff number (G = 3.0, 5.0 and 7.0) at P = .71, l = 0.3, w = 0.2, K = 1.0, E = 0.4, n = 0.1, k = 0.25 and M = 1.2 are shown in figure 3. From the figure, it is obvious that the skin-friction factor of the liquid and dust particles increases as the Grashoff number increases. Besides, we observe that on increasing y the variation of skin-friction factor increases.

Figure 4 indicates the variation of heat transfer with frequency for different values of time (t = 1.0, 2.0 and 3.0) at P = .71, E = 0.4. From the figure, we observe that the heat transfer decreases as time increases. It is also obvious that the heat transfer decreases as frequency n increases.

REFERENCES [1] M"'T. Lighthill, Prof!. Floy. Soc. 224A (19G4J, 1.

[2] B.B. Mishra, The Mathematics Education 25, (2) (1991), 73. [3] S. Mukherjee and S. Mukherjee, Acta Ciencia lnclicu 12m (4), (1986), 279. [4] V. Ramamurthy, J. Math Phys. Sci. 24 (5), (1950), 297.

l51 P.G. Saffinan, Jour. Fluid Mech. 13 (1962), 120. (6] P.R. Shurmu and M.K. Sharma, Actu Ciencia Indica 17 (m), (4), (HHHJ, 6~9 ('/] K.D. Singh and 8.K. Rana, Indian J.Pure Appl. Math. 23 (7), (1992), 905. [8] K.D. Singh, Khcm Chand and S.K. Rana, Indian J. Pure Appl. Math. 24 (1993),

327. [9] V.M. Soundalgekar, Chem. Engg. 8r.i. 26 (1971), 2043. [IO] S. Subeydar, Ph.D. Thesis, Agra Univim;ity, Agra (1992), 178.

.J.Danabha, Vol 24, 1994 !Dediwted to Profe.s.snr JN. Kapur on his 70th Birthday)

G/G/l/N QUEUEING MODEL WITH LCF'S-P/R SERVICE POLICY

By Madhu clain and R.P. Ghimire

Department of Mathematics, D.A.V. Postgraduate College Delwadun-248001, U.P., India.

!Rcceiued: April 10, 1994!

ABSTRACT This paper is devoted to the G/G/1/N queueing model in which

customers are served under last-come-first--servod (LCFS) queue discipline and arbitrary reoturting service policy (R). 'rl1P. irip1d. nf lhP is stationary with rate 'A. The service time of customers depends on the queue size (i.e. number of customen; present in the system). For steady-state, queue size distribution has been obtained explicitly.

I.INTRODUCTION. Que1rnine system with l::ist come··first served (LCFS) service disciµliue µlays vital role in practice however in a limited area, for example : stocks ;:ire oftfm refilled but alfw worlrnd off at the top. In LCFS P oervice discipline an arriving customer gets service immediately and a pre·empted customer restarts service when all cw;tomers arriving after him leave the system. Some authors have made efforts to tackle the queueing problems having LCFS service discipline. Kelly (5) introduced LCFS-P/R queue discipline for l\!I/G/1 system. Yamazaki (8) analysed the G/G/1 queue with LCFS service disciplme and later on, he [9) obtained invariance relations for GI/G/1 queue with LCFS and preemptive resume restarting service policy. Santhikumar and Sumita (7) discussed GI/Giil model 1 by imposing LCFS-preemptive restarting policy and proved that the stationary system queue length diRtrihnt.ion jRut R[t:er a departure irn;tant is geometric. Fakinos [~) obtained the expressions for GI/GI/I queue with LCFS- P/resume and service time distribution rlf~pending on the system queue length. Yamazaki (10) genrali2etl. iL fur GI/GI/1/K system with LCFS-P!H service discipline where H is a restarting plicy which may depend on the history of premption of the restarting customern. Miym:Rwa [6] extended the work of Ymnuzaki [10) by imposing simple rejection rule but without any restriction (i.e. arriving customers are accepted without depending on queue size). Fakios (3, 4) obtained some useful results for G/G/1 modP.l nndm· thP. last-come-first served service discipline when the customers where drawn from infinite population and interarrival times and service times under stationary Rtochastic processes. DUk [l) studied a finite

146

LCFS buffer queueing model with batch input and non-exponential service.

In may practical applications, the customers are often drawn from finite population for example; telecommunication, computer and manufacturing systems etc. The present day demands motivate us to investigate a G/G/l/N model under LCFS/P service discipline For solution purpose the system is considered at arrival epochs and in continuous time. The steady state queue size distribution has been obtained in section 2. Section 3 deals with special case. In the last section 4, the conclusion has been drawn.

2. THE MODEL To describe our model we use the following notations :

)=

b=

c=

cr = N=

QK=

Q(t) =

Q(n) =

rn,pn =

rn,Pn =

b* =

b* =

s = n

s = n

w = n

n (x):::: n

Numebr of custormers waiting behind each moving customer from position n - 1 to position n.

Mean value of random variable x , n

Mean value of the busy period. 1- b/c.

Total number of customers present in the system. Queue size immediately before Kth arrival epocks.

Queue size at time t. Number of customer in queue n. Limiting probabilities distributions of queue size.

Tails probabilities of r n and p n respectively.

Mean actual service time at conditional probability.

Mean service time at unconditional probability. Mean system time at conditional probability.

Mean system time unconditional probability.

Mean waiting time for type n customers.

Distribution function of service time.

The stochastic processes {Q(t) : t?: 0} and {QK: K = 1, 2, ... , Nl have limiting probability distributions :

p = lim p { Q (t) = n) n t --'t 00

and rn = lim p {Q(n) = n), (n = 0, 1, 2, ... , N}. K--'t oo

The corresponding tail probabilities are N

A p = 1: P· n . /.

t=n ... (1)

147

N r =Lr. n . i

... (2) i=n

Under LCFS/P discipline the relation between limiting probability distributions p and r are given by (Yamazaki [10])

n n

Pn = P,/n-1' n = 1, 2, .. .,N

where p = 'Ab is traffic intensity. n n

and n N K

r = n 0 I I: n 0. n j=l J K=Oj=l .!

... (3)

... (4)

We assume that in equilibrium the arrival processes for queue n is a stationary point process with rate A, = 'AP 1. The mean service n n-

time (b*) and mean system time (s ) are given respectively by n n

N-n b* = I: b* .

n . 0

n+.1 .J=

rn-l+j I\

rn -1

N-n r . - ~ n-l+J s = ,;.., s . n .

0 ·n +.1

.1= r n- l

In equilibrium for non-empty queue n

p(Q(n) > 0) =An b~

so that (5) implies N-n

p{Q(l)2:n)=AP i; b''' rn-l+j n-1 n+· "

j=O J rn-1

I\ N-n p = L * . r

n .i = 0 Pn + J n - 1 +.i' Also

We note that I\ /\

fln=pn-Pn+l

(n "" 1, 2, .. ., N)

=p* r 1 (n=l,2, .. .,N) n n

where p* ='Ab* is new traffic intensity. n n

Using Little's result to queue n, we get E [Q (n)] ='A s , n n

Also

. .. (5)

... (6)

... (7)

.. (8)

... (9)

... (10)

148

N-n

E[Q(n)] = 2: p {Q(n) ::?:K} k=l

N-n = :E p{Q(l) ;::: n + K - 1)

K= 1

N-n /\

= L Pn+K-1 K=l

From equation (6) and (ll), equation (10) yields

so that

and

N-n N-n /\

L p =A I: s . . 0 n + J . 0 n +J

.1= .! =

/\

Pn=snrn-l

r . . n - i +J

A _A =A(s r -s r) Pn Pn+l n n-1 n+l n

Now from eqmitiou::; (9) and (11), we get

(sn - b~) rn = rn 1

5n + 1

Denoting

... (11)

... (12)

... (13)

... (14)

... (15)

s -b" w .,..,. =··n n=_n_ (n= 1 2 N) (16) v ' ' ' .•• , .••

n 8n+l 8n+l

Equation (15) reduces to

r =er r 1 = [~] r 1, (n = 1, 2, ... , N) ... (17) n n n- s n-n+l

From normalizing condition N L rK= 1 au<l equation (17), we get

K=O

N K r 0 L 11 er.= 1 ... (18)

K=OJ=l .I

Thus limiting probability cfo.:;t.ribution r n is givon by

n N K r =CT cr./ L 11 er., (n=0,1,2, ... ,N) ... (19)

n J=l .! K=OJ=l .I

Also equation (9) becomes n-1

* P 11 =p·,, 11 .i l

II Jt cr./ L 11

.I k=OJ=I CT)' (n = 0, 1, ... , N) ... (20)

149

The limiting probability when there is no customer in the system is givE>n by

N

Po= 1- I: II= 1

3. SPECIAL CASE

n-l N K

p* n CT / I: n CT· 11

)=1 J K=O)=l J

.... (21)

In the special case when service requirement is independent of queue size, B (x) = B(x), b* = b'', s = s, CT =CT and p* = p*, n n n n n (n = 1, 2, ... , N), we have

1 - p* Po=-

1- *N+ 1 -p 1 - CT

r =-----0 1-~+l

where= 1- b/c

so that

P = 11 - b''!c"') P 1 n \ n -

and

= (l-h"'! *)llh*/ * rn r 0 . r . c

From equation (9), we get

P ( 1 - CT) * n - 1 ( 1 2 N) = _N 1 p CT n ,.... ' ' .. ., n 1-CJ.+

4. CONCLUSION

... (22)

... (23)

... (24)

... (25)

... (26)

The Rteady state queue size distribution ohtnim~cl explicitly for considered G/G/1/N model is of great utilizaton due to its applications in finite buffer situations in industrial problems. The last-come firnt-served queue discipline with vreemption and arbitrary restarting policy are common in inventory and production systems.

ACKNOWLEDGEMEN'l' This research is supported by DST Grant No. SR/OY/M00/91.

REFERENCES Ill Van N.M. Dijk, A LCF'S finite buffer model with batch input and

non-exponential services, Stochastic Processes and Their Applications 33 (1989) 12:3-129.

[21 D. Fakinos, The single server queue with service depending on queue size and with the preemptive resume last-come-first-served queue discipline, ,J. Appl. Prob. 24 (1987), 758-767.

[::JI D. [i'akinos, An applicaton of Little's result to the G/G/1 (LCF'S/P) queue, ,I. Appl. Res. Soc. 39, No. 2 (1988), 209-213.

[41 P. FakinoR., The G/C:/l ~)LCFS/l'J queue with service depending on queue size, Eurp. J. Operat. Res. 59 (1992), :103-307.

150

[5] F.P. Kelly, The departure process from a queueing system Math. Proc. Camb. Phil. Soc. 80 (1976), 283-285

[61 M. Miyazawa, On the system queue length distributions of LCFS-P queues with arbitrary acceptance and restarting policies, J. Appl. Prob. 29 (1992), 430-440.

[71 I.G. Santhikumar and U. Sumita on G/G/1 queues with LIFO-P service discipline, J. Operat. Res. Soc. Japan. 29 (1986) 220-230.

[8] G. Yamazaki, The G/G/1 queue with last-come-first-served. Ann. Inst. Statist. Math., 34 (1982) 599-644.

191 G. Yamazaki, Invariance relations of GVG!l queueing systems with preemptive-resume last-come-first-served queue discipline, J. Operat. Res. Soc. Japan, 27 (1984), 338-346

[101 G. Yamazaki, Invariance relatins in single server queues with LCFS service discipline, Ann. Inst. Statist. Math., 42 (1990), 475-488.

Jnanabha, Vol24, 1994 (Dedicated to Professor J.N. Kapur on his 70th Birthday)

MEROMORPHIC MULTIVALENT FUNCTIONS WITH FIXED COEFFICIENTS

S.R. Kulkarni Department of Mathematics Willingdon College, Sangli

416415, India.

and S.B. Joshi Department of Mathematics

W alchand College of Enginnering

Sangli 416415, India.

(Received: December 25, 1994)

ABSTRACT In the present paper we have obtained coefficient inequality for

the class QZ (p, A, B) which consists of functions of the form

1 (B -A)k 00

f(z) = - + zP + L a zP + n (a 2 0) zP (2 +A + B) n = 1 P + n p + n

which are regular and p-valent in the punctured disc

u* = {z : 0 < I z I < 1) . .Further we have shown that the class QZ (p' A, B) is closed under arithmetic mean and convex linear combinations. Lastly we have obtained the radius of convexity for the class

QZ(p, A, B). Various results obtained in the present paper are shown to be sharp.

1. Introduction Let Q(p) denotes the class of functions of the type

1 00

f(z) = + L a. zP+n zP n 0 p+n

.... ( 1.1)

which are regular and p-valent * ~ u· =: {z: 0 < lz I < 1). Further, let Q (p)

m the punctured disc

dcnotcG the GUbcluGG of Q(p) consisting of functions of the form

f(z) = 1 + ~ a .? + n and satisf)ring a 2 0.

zP n-0 pin p+n . .. (1.2)

In [2] Uralegaddi-Ganigi introduced a class Q*(p, A, B) consisting of functions of type ( 1.2) and satisfying the condition

I . zf (z)lf(z) + p I < 1 I z I < 1 ... (1.3) Bzf(z)lf(z) +Ap '

where

-l:S;A<B'.':'.l, A+B20.

152

Meromorphic univalent functions have been rather extensively studied by several authors like Uralegaddi [3] and Uralegaddi- Ganigi [2]. We begin by recalling the following lemma clue to U ralegacldi-Ganigi [2].

Lemma l Let the function f(z) be defined by (1.2l. Then the

function f(z) is in the class Q;;(p, A, B) if and only if

2: ((l+B)n+(2+A+B)p)a 5op(B-A). . .. (1.4) n=O p+n

In view of Lemma 1, we observe that f(z) given by (1.2) in the

class Q;1(p, A, B) satisfy

(B-A) a <------­

P - (2 +A+ B) .

Hence we may take

_ (B-A)k < < a P - (2 +A + B) ' 0 - k - 1.

... (1.5)

... (1.6)

Let Q;(p, A, B) denotes the subclass of Q;(p, A, B) consisting of

functions of the form

1 (B - A)k [I ~ 1J+ I! -+ · ·-·-z + ,;., a z-zP (2 +A + B) n = 1 P + n

((z) .... (1.7)

where a 2 0 and 0 s k ::; 1.

p+n

In the present paper we have obtairn~d coefficient inequality for

the class Q~ (p, A, B). Further we show th;t the class QZ (p, A, R) is

closed under arithmetic mean and convex linear combinations. Lastly

WP. hnvc obtained radiu:.; of convexity fur the class Q~(p, A, H). Various

results obtained in this paper are shown to be sharp. Techniques used ure similar to I.hose of Silverman and .Silvia [l].

Main Results !.>.. Coefficient Inequality

Theorem 1 Let the function f(z) be defined by (1.7). Then rn m

Lhe cla88 Q~'(p, A, lJ) if and only if

2: {(l+B)n+(2+A+B)p)a s;p(B-A)(l-k). n=I p+n

... (2.1)

1'hc result is sharp. Proof Putting

153

a =_@__~A) h JJ 2+A+B' O<k:S:l ... (2.2)

in (1.4) and simplifying we get result. The result is sharp for the function

1 (B~-AJ.~-/(z) = -·i~ + (2+ A+ B)

(B - ~~Lpi_~_=-!~L ____ 71' + 11 n ?'. J. . .. (2.3) + ((1--:+~B)n-~~ (2 +A+ B)pl -·~ ' z

Co:rroU.ary Let tho f..:n:tlon /(z) given by (1.7) be in the elf:·

Q;;(p, A, B). Then

/ (B --A) p(l -- h) _ a '- -·-·---·--------------- --------- 17 > I p + 11 - ((1 + B)n + (2 +A+ B)p)' · - ,_.

... (2.4)

3~ Closure r_r']bleorexn

In this section we shall show that the das~: Q~'(p, A, B) is closed

under arithmetic mean and convex linear combinations. Theorem 2 Let

f(~)-J + (B-A)k .P i ,_,p+n ._,ff .i,_, - ,,P (2+A+B)" + up+nJ \ap+n,j~)

w · n-1 ... (2.5)

be in the class Q%(p, A, B) for everyj = 1, 2, .,., m, then the fond.ion

1 rB-A)k TJ "" b 11+n · a(z)"' -- + ~ z + L z (b ?'. 0) 5 z1J (2 +A + B) n = 1 p + n JJ + n

is also a member ofQ~'(p,A,B), where

1 m b =--- L a ..

p+n Jn j=l p+n,J

... (2.6)

Proof Since f(z) E Qz(p, A, B) it follows from Theorem 1 that .I

L: {(l 1 R)n+(2+A+B)p}a .:£(b-A)p(l-k) n = 1 JJ + n,J ·

... (2.7)

for every j = 0, 1, 2, .... , m. Hence

2: { 1 + B)n + (2 +A + B)p} b n=l . p+n

[ m l ,, L ((l+B)n+(2+A+B)p} l/m 2.: a .

n"l j=O p+n,J

1 Ill

----I I ((l+R)n.+(2+A+B)p}a . m .i o n = 1 P + n,J

~ (B A)p (1 - h)

154

S:(B-A)p(l-h)

and the result follows. Theorem 3 Let

_ 1 (B -A)h P f(z) - zP + (2 +A + B) z

and

z =l_+ (B-A)h zP+ (B-A)p(l-h) zP+n n>l. fp+n() zP (2+A+B) {(2+A+B)p+(l+B)n} ' ( - )

Then f(z) E Q*(p, A, B) if and only if it can be expressed in the form

+-(z) = L 'A f (z). I' n=-lp+np+n

where

'A ;:::: 0 and L 'A = 1. p+n n=-lp+n

Proof Let

Since

f(z) = n =L-1 \ + n fp + n (z),

1 (B-A)h p i (B-A)p(l-k) zP+n f(z)= zP+(2+A+B)

2 + n=l{(i+B)n+(2+A+B)p} .

L n=l

(B-A)p(l-h) 'A +n {(l+B)n+(2+A+B)p}

{(1 + B)n + (2 +A +B)p) (B -A) p(l -k)

= L 'A =1-'A :S:l. n=l p+n p

Hence, by Theorem 1, f(z) E QZ (p, A, B).

Conversely, suppose that

1 (B -A)k 00

f(z) = -- + + L a zP + n (a ::?'. 0) zP (2 +A + B) n = 1 p + n p + n

is in the class QZ(p, A, B). Then by using (2.4) we get

< (B-A)p(l -k) > l ap+n -{(l+B)n+(2+A+B)p)' n_ ·

Setting .

'A = _{(1 + B)n + (2 _+A+ B)p) a (n;:::: l). 1i+11 (B-A)p(l -k) p+n

and

A. =1- L: A. p n=l p+n

we have

fi(z) = L: A f (z). p+n p+n n=-l 4. Radius of Convexity Theorem 4

155

Let f(z) defined by (1.7) be in class Q*k(p, A, B). Then f(z) is mermorphically p-valent convex in 0 < I z I < r = r(p, A, B, K), is the largest value for which

3p2(B-A)kr2P (p+n)(3p+n)(B-A)p(l-k) 2p+n< 2

(2+A+B) + {(l+B)n+(2+A+B)p) r -P'

(n = 1, 2, ... ).

The result is sharp for the function 1 (B-A)k p (B-A)p(l-k) p+n f (z)--+ z + z

p+n - 2P (2+A+B) {(l+B)n+(2+A+B)p) '

for some n.

Proof It is sufficient to show that

l{zf(z);c;/f(z) I Sp for 0 < lzl r(p,A,B,k).

Note that

for 0 <

I [zf (z)]' +pf (z) I

f(z)

2p2(B-A)kr2p+ ~ (p+n)(2p+n)a r2p+n (2 +A + B)

/1 = 1 P + n

< . Sp

- _ (B-A)kp r2p_ ~ (p+n)a r2p+n p (2 +A + B)

11 1 P + n

lz I Sr if

3 2 _JlL= ,4)k 2p p 2 +A + B) r + L: (p + n) (3p + n) a r2P + n < 2 n=l p+n -P ·

Since f(z) E Q*k(p, A, B), we may take

(R - A) p(l - k) a = · · ··-·-·-·-------·· . .E._c:.''c__ L: A. < 1

p+n {(l+B)n+(2+A+B)p} 'n=I p+n - ·

For each fixed r, choose an integer n = n(r) for which

156

(p + n) (3p + n) r 2P + 71 ·s m · 1 Th --- i axuna. en (2 +A+ B) '

2p +n I: (p+n)(3 +n)a r2p+n<{p+n)(3p+n)(B-A)(l-k)r

n=l p p+n - ((l+B)n+(2+A+B)p}

Now find the value r 0

= r0

(p, A, B, k) and corresponding n(r 0

) so that

'}

'.}_p~(B -A)k r2p + (p + n) (3p + n) (B -A) (p(l - k)_ r2p + n = 2

(2 +A +B) o {(1 +B)n + (2 +A +B)p} o p

This is the value of r0

for which f(z) is convex m

0< lzl <r0 =r0 (p,A,B,k).

REFERENCES Ill H. Silverman, and E.M. Silvia. Fixed coefficient for subclasses of starlike

funclions, Houston J. Math. 7 (1981), 129-136. (2] B.A. Uralegaddi, and M.D. Ganigi, Meromorphic multivalent functions with

positive coefficients, Nep. Math. Sci. Rep 11(2), (1986), 95-102. [3] B.A. Uralegaddi, Meromorphically starlike functions with positive and fixed

second coefficients, Kyungpook Math. J. 29(1), (1989), 64-68.

l

Jiianabha, Vol 24, 1994 (Dedicated to Professor J.N. Kapur on his 70th Birthday)

ELECTROGRA VITATIONAL INSTABILITY OF A FLUID CYLINDER UNDER MODULATING VARYING ELECTRIC

FIELD ON UTILIZING THE ENERGY PRINCIPLE Ahmed E. Radwan

Department of Mathematicfi, Faculty of Science, Ain-Shams University, Abbassia, Cairo, Eg<ypt

(Receiued: Nouember 20, 1994)

ABSTRACT The electrogravitational instability of a fluid circular jet

dispersed in a self-gravitating tenuous medium of negligible motion pervaded by modulating general varying electric field has been developed on utilizing the Lagranei:rn energy principle. The fundamental equations describing the problem are deriving and solved in the unperturbed and perturbed state:,,;, the total changes in the electric, kinetic and self-gravitating energies are computed. 'rlrn Lagrangian energy principle technique has been used and a result it is found that the system is governing by a second order integro-differential Mathieu euqation. Several categories have been :rn;:ilysed via this equation. The gravitational force is only destabilizing for small axisymmetric perturbation. 'l'he internal electric field penetrated the fluid cylnder has no direct influence on the stability of the fluid while the exterior longitudinal and transvernc P.lP.ctric fields are being stabilizing or destabilizing according to restrictions. Resonance domains are appeared due t.o the field pc:rioflicit.y and in some regions the stability conditions depend only on the field frequency. The electric field frequency is stabilizing in a small ::ixi··Rymmetric region o.nd deRt.ahilir;ing otherwise. The amplitude of the modulating electric field could be fully stabilizing, under certain restrictions, nnd suppressing tho deotubilizing character of thf~ ut.lwr physical parameters and hence stability ariRes. ..

Numeruous rP.ported works could be recovered with appropriate choices as limiting cases.

1. Introduction The in stability of cylindrical fluid column endowed with surfrico

tension or/and acted upon external forces such as electrodynamic or electromagnetic forces has been involved in several texts by a lot of researchers. Referring to these pioneering works see.

{ [2] ,[3] ,[4] ,[6] ,[9] ,[13], [14], [15] ,[16], [21] ,}

The response of the axisymmetric instability of a self­gravit.::iting cylinder ambient with self-gravitating vacuum wm; due to Chandrasekhar and Fermi [51J. It has a correlation with

158

understanding the dynamical behaviour of the spiral arms of galaxies and sun spots. Chandrashekhar l4J studied such a problem in details for different cases in several literature. He [ 4] summarised his results along with those of others for different problems of different configuration models. Radwan [17] has extended that work [5] by studying the stability of a self- gravitating fluid cylinder dispersed in a self -gravitating fluid of different density. In the works [5] and [17] it is used totally different techniques where in the latter we have used the principle of energy.

The influence of the electrodynamic force on the self-gravitating fluid cylinder has been examined for first time by Radwan [18] . In this recent work [18] both the self-gravitating fluid and the surrounding vacuum are assumed to be pervaded by uniform (constant) electric fields. The aim of the present work is to investigate the electro-dynamic stability of a self-gravitating fluid cylinder such that the electric fields not only varying but also modulating with the same periodicity. This will be done for all axisymmetric and non-axisymmetric modes of perturbation, on using the Lagrangian energy principle technique.

2. Formulation of the Problem Consider a dielectric self-gravitating fluid cylinder (of radius R0)

ambient with a tenuous dielectric medium of negligible motion. c1 is the dielectric constant of the fluid matter and idem Ee for the .surrouuding medium where from now on the superscripts i and e indicate interior and exterior the fluid jet. The fluid is assumed to be non-viscous, incompreRsible and of uniform mass density p. The fluid cylinder is being pervaded by the modulating electric field

E~ = (0, 0, l)R0

cos (wt) (1)

while the surrounding medium is assumed to be penetrated by the modulating general varying electric field.

~ = (0, ~R0 r -1, a)E0

cos(w t) ... (2)

where E0 and w is the intensity and frequency of the electric field and

u, ~ are parameters satisf}ring certain conditions. The components of Eh and R0 arc considered along the utilizing cyliudric:al polar coordinates (r, q>, z) with the z-axis coinciding with the axis of the fluid cylinder. The fluid is acting up on the intertia, electrodynamic, pretJtmre gradient awl self- gravitating forces while the surrounding medium of the fluid is acting up on the self-g-ravitating and electrodynamic forces. We assume,initially, that there are no surface chargm; at the boundary surface aud l.herefore the surface charge density will be zero during the perturbation. We also assume that the quasi- static approximation is valid for the problem under consideration.

159

The fundamental basic equation required for describing and analysing such kind of problems are coming out from the combination of the ordinary hydrodynamic equations together with those of Newtonian gravitational theory and with Maxwell's electrodynamics equations. For the problem under consideration, these basic equations may be formulated as follows.

In the fluid region : The gravitational electrofluid dynamic vector equation of motion

au i i i l p(--::;- + (u.V')u) = (c /2) V'(E . E) - V - pV'<P ... (3) at P ·

The equation of continuity expressing the conversation of mass v. u = 0. . .. (4)

The self-gravitating Poisson's equation 2 i V <P = 4nGp. . .. (5)

The Maxwell's electrodynamic equations

V . (Ei Ei) = 0, V x Ei = 0 ... (6), (7)

In the surrounding tenuous medium : The Laplace's equation for the gravitational potential

\/'2 <j>E = 0. . .. (8)

Maxwell's electrodynamic equations

V' . (Ee~) = 0, V' X Ff= 0. . .. (9), (10)

Here p and u are the fluid kinetic pressure and velocity vector, G is the gravitational constant, <Pi is the gravitational potential in the fluid region and idem <PE in the surrounding medium and E is the electric field intensity.

3. Linearization and Solutions In order to analyse such kind of study we assume that the fluid

boundary ::mrface is acting up on a sinusoidal deformation and consequently the location of the perturbed interface co11 lcl he given in the form.

r = R0

+ y(t) cos (kz) + mcj>) . .. (lla)

Here y(t) is, some function of time t, the surface diaplacement, k (any real number) is the longitudinal wavenumber and m (an integer) is the azimuthal wavenumber. 'T'he second term in the right side of (11) iB Lhe elevation of the surface wave measured from the unperturbed position. Based on the perturbation form (11 a), for small departure from the equilibrium state every physical variable quantity Q(r, cj>, z) could be expressed as

Q(r, c, z; t) = Q0(r) + y(t) Q

1(r, c, z) ... (llbl

i60

where Q 1 is the change in Q clue t,J a perturbation.

It is intended to investigate the stability of this problem using an analytical perturbation technique on the basis of the Lagrangian energy principle. It is noted that the Lagrangian function L is constructed as

L=Q -W -V 1 1 1

... (12)

and the Lagra' .31>:ond order differential equation is being

1i (~~ )-- ~~ = 0

where y is the Lag· u.n,c'.-ian variable for the present problem and where the clot over y means that y is being differentiated with respect to time. The physical quantities .01, W1 and V 1 are the changes in Q, W

and V due to the perturbation of the boundary surface (11 a) with Q is the total kinetic energy, W is the total electrical energy and V is the gravitational potential energy. One have to refer here that the quantities with subscript 1 mean quantities in the perturbation state while those with subscript o mean their value in equilibrium state.

Tn the perturbation state, the basic electrodynamic equations (6), (7), (9) and (10) degenerate to

V x E~" = 0, V . (£ E~e) = 0 ... (14a, b) (15a, b)

From the viewpoint of the vector analysis, equations (14,a,b) moan that E~ e could be derived from scalar electrical potentials.

Ei1·, e = - V \jfi, e ... (16a, b) l

Combining these vector equations together with equations (15 a, b), the electrical potentfrlls 'l'I e satisf)r 1h~ L::ipl<1ce's oquntion:.i.

v2 \jf~ e = 0 ... (17a, b)

In cylindrical IJOlar coordinate (r, cp, z) those equations may be rewritten as

[ d\lfi,el [ d\ i,el [ d\lfi,el ·- l a i - 1 a - , "1 - 1 a 1

r . dr ,. ---a;:- + r . a-;p ,. ~ + r dz ,. ~ = 0 ... (18a, b)

By revorting to the linear perturbation technique and based on the space time (r, <jl, z; t) -dependence, tho perturbed quantities \jf~, \j/~, \jf~ efi etc. can be expressed as cos (kz + mcp) time an amplitude

functon of r v~z.,

\If~ e (r, c, z; t) = y(t) \jl~ e (r) cos (kz + mcp) ... (19a, b)

Inserting (19 a, b) intu (18 ;1, h) we obtain

161

- 1 d \jf 1 . 2, 2 - 2 i e l d i. el r dr rd--;:- - (k + m r ) 'J!} = 0 ... (20a, b)

The solutions of he second order ordinary differential equations (20 a,b) are given in terms of Bessel's functions with imaginary argument. Under the present circumstances, the non-singular for \jf~

as r -7 0 interior the fluid cylinder and for efi as r -7 oo exterior it are

given by

\j/1

1· (r) = A 1 (t) I (kr)

m

\Ve1' (r) = Ae (t) K (kr).

m

... (21)

... (22)

where A 1 and Ae are time dependent functions of integrations to be determined; lm(kr) and Kn/kr) are the modified Bessel's functions of first and second kind of order m.

The basic gravitational potential equations (5) and (8) reduce to the following equations in the unperturbed state

2 i V <t>0

= 4n:Gp ... (23)

v2 <I>~= o. . .. (24)

These equations are solved on using the unperturbed simplifications of longitudinal and azimuthal symmetries CJ/CJz = 0 and o/CJc = 0. The ::;ulutiorn; are matched across the boundry surface at r = R 0 . Apart from the singular solutions the solution for <I>~

as r -7 0 inLerior Lhe fluid cylinder and that one for <I>~ exterior the

cylinder as r -7 oo are given by

<I>~= nGpr2 ... (25)

<I>~= n:p GR~+ 2nGp R~ log(r!R0

)2 + C

0 ... (26)

where C0 is an additive constant with which we need not be further concerned.In the perturbed state, the basic equations (5) and (8) degenerate to the Laplace's equations

v2 <I>~ c = 0. ... (27a, b)

These equations could be solved on using similar steps as those which have already been used for solving equations (17 a, b). The non-singular solutions for <I>~ and <I>~ are given by

ct/1• (r, <P, z; t) =Bi y(t) I (kr) cos (kz + m<(l)

m

<l)e1' (r, <p, z; t) =Be y(t) K (kr) coo (kz 1 m<jl)

111

... (28)

... (29)

,J.

., ,,,~

~ )·' ,.""·. l • ~I \11 ~Ji I

162

where Bi and Be are constants of integration to be identified. It is worthwhile to mention here that we have originally considered departures from an unperturbed right cylindrical shape of an incompressible fluids. For this reason the argument of the sinusoidal acting wave sin(kz + m<j> + nn/2) where n is an integer appeared as cos(kz + m<j>) in the solutions (21),(22),(28)and (29).

Now, since the Lagrangian coordinate y is a function of time t, each element of the fluid will execute a motion. Such a motion may be derived from the Lagrangian displacement

u == ai:,;at. . .. C30)

However, taking the divergence of the perturbed equation of motion which resulting equation (3), linearizing and using the incompressibility condition (V' . u 1 == 0), we obtain

2 i i/2 i V' SI== 0, sl == P/P +<I> -(f. ) (E.E)r ... (31, 32)

1'he non-singular solution of (31) is given by

S1(r, <j>, z; t) == d y(,t) ln/kr) coR (k.z + m<j>)

where ci is an unspecified constant of integr::ition. 4. Boundary Conditions

... (33)

The solutions (19 a,b), (21),(22),(25) to (29) and (33) of the fundamental equations (3) to (10) must satisfy certain boundary conditions across the perturbed interface (11 a) at r == R0. These

appropriate boundary conditions could be stated as follows . . \

4.1. The electrodynamic conditions ('1) The electric potential 'I' must be continuous across the pert.urhed h,iterface (11 a) at r = R0 . 'l'his condition, on using (21) and (22), yields

\ A e(t) = [I (x)IK (x)] Ai(t) ... (34) m m

· w~ere x(=kR0

) is the dimensionless longitudinal wavenumber.

fo) \The nomral component of the electric displacement must be also .• coniinuons across the perturbed lmundary surface (11 a) at r = R0 i.e. ' '

N. (ci Ei - ee~) = 0. . .. (;~5) Here kr is, the outward unit vector normal to the perturbed interface (11 a), given by

\ i <j> = V' F(r, <j>, z; t)I I V'F(r, c, z; t) I ... (36)

where F(r, c, z; t) == 0 is the equation of the boundary surface given by ' F(r, </J, z; t) = r - R

0 - y(,t) cos (kz + m<j>) ... (37)

Thui:;

N~N0 +N1

163

= (1, 0, 0) + (0, - mR0

, k) y(t) cos (hz + m<jl) ... (38)

By the use of (1),(2),(11 a), (16 a,b), (19 a,b), (21), (22) and (38) for the condition (35), we obtain

£e K '(x) A 3 - h '(x) Ai= (ik·x (£i - £e) - imk~lJE0 cos wt ... (39) m m

Solving (34) and (39) for N(t) and Ae(t) we get

[iX(£i = £e) - i~£e]K (x) A 1(t) = . m E cos(w t)

k[£el (x)K '(x) - £11 '(x)K (x)] O m m m m

... (40)

ix(£i - £e) - i~£e] I (x) Ae(t) = , . m E cos (wt).

k[£el (x)K (x) - £il '(x)K (x)] o m m m m

. .. (41)

4.2. Self-gravitating Conditons (i) The gravitational potential <I> must be continuous across the

perturbed fluid interface (11 a) at r = R 0. This condition gives

Bi I (x) =Be K (x). . .. (42) m m

(ii) The derivative. of the self-gravitating potential <I> must also be continuous across the surface (11 a) r = R0. This condition, on using

(11 a), (25),(26), (28) and (29) yields

Bil '(x) - Be K '(x) = 47tp Glk. m m

. .. (43)

Solving (42) and (43) for Bi and Bewe get

Bi= 41tpGR0Km(x), Be= 41tpGRa1m(x) ... (44)

where uRe have been made of the Wronskian

W (1 (x), K (x)) =I (x) K '(x) -I '(x) K (x) = - x - I ... (46) mm m m'm m''m·

in obtaining (44) and (45).

4.3. The Kinematic Condition The normal component of the velocity of the fluid particles mmit

be compatible with tho velocity of the perturbed surface (11 a) at r =R0 i.e.

d1' ulr = (Jt ·

... (47)

By the use of the linearized form of the equation of motion (3)

()uI p Tt=-VS1 '" (48)

and equations (30) Lo (33) we obtain

164

~I= - (d Ip) y(t) kl '(kr) cos(hz + m<jl). m

. .. (49)

By an appeal to (11 a) and (47) to (49), the coefficient C' is completely determined and the perturbed velocity vector of the fluid is being

UI = [Ro2 /(x I '(x))] ady(t) \7 [I (hr) cos (kz + m<jl)] ... (50) m t m

5o Computation of Kinetic and Poetntial Energies The change in the total kinetic energy .Q

1 (per unit length) of the

fluid jet associated with the motion specified by (30) is given by

1 JR o J2rr J2rr dkz D1

= :zP (u.u)1

r dr -2

de O kz= O Q n

... (51)

2

= (npR21(2k2)) [I '(x)r 2 (dy(t)J J (y) 0 · m dt m

where the integral Jm(y) is defined by

... (52)

J (y) = r [(I '(y))2 + (1+m2y- 2) (I (y))2

] y dy m o m m

... (53)

This integration hm; been carried out on using the identity (which follows from Bessel's equations)

dd (y f (y) f '(y)] = y[f '(y))2 + (1 + m'J y- 2) (f (y))2] ... (54) y m m · m m

where fm(y) stands for. both the functions Im(x) and Km(x), and

therefore

J (y) I R = x I (x) 1 '(x) m r= m m ... (55)

0

Consequently

.Ql-npRO 2xlm'(x) dt . . .. (fifi) _ 4 l Im (x) J (1:Jfil_J2

.

Now suppose that t e amplitude y of the surface deformation (11 a) is increased by the increment 8y, then due to this infinitisimal iucrement in the amplitude of deformation the change 8V in the gravitational potential energy can be determined by evaluating the work done during the displacement uf the m;1tter required to produce the change in y. Tu evaluate thi::; work it is necessary to specify quantitavely the redistribution which does take place. Arbitrary deformation of an incompressible fluid can be thought of as resulting from the Lagrangian displacement 1;

1 applied to each point of the fluid

medium. We assume that the perturbed motion is irrotational and this is in fact because we have considered in t.he unperturbed t:tate that

165

the fluids are incompressible and non-viscous. Therefore, the Lagrangian displacement of the fluid could be expressed as

S1 = \7 Gl. . .. (57)

Combining equation (57) together with the incompressible condition, we find that the displacement poetential G1 satisfies Laplace's

equation

\7 2G = 0. 1

. .. (58)

The solution of this equation on using steps similar to those used for solving equations (17 a,b) is given by

G1

=C1

y(t)Jm(kr).cos(kz+m¢). . .. (59)

The constant of integration C 1 could be determined by applying the

condition states that the radial component of s1must reduce to

y(t) cos (kz + m¢) at r = R0 . Therefore

R2 s

1 = I ? ·y(t) V [/ (kr) cor; (kz + me)j.

x (x) m m

... (60)

Hence the corresponding displacement os1 which must be

applied to each point of the fluid in order to increase the amplitude of tho deformation by oy is given by

R~ oy(t) . . . osl = xl '(x) \7 [Jm(kr) cos (kz + m¢)]. ... (61)

m

Now, due to that additional deformation 8y, the change in the total self-gravitating potential energy oV1 (per unit length) can be

identified by integrating the work done by the displacements 8s1 in

the gravitational potential <I>~. This means that

JR + -y(t) cos (kz +me) .

8V1 = 2n:p « 0 (os1

. \7 <I>~) r dr » ()

... (62)

where the angular brackets mean that the quantity enclosed should be avorageu over c anu z. Substituting from (28), (44) and (Gl) into (62) and carrying out the required integration we get

4 2 2pK (x) 8V = 2 y(oy) pGR n: [p - -----'~ -- J (y)] ... (63)

l 0 x I (x) m m

where Jm(Y) is given by (53) or rather by (55), Inserting (55) into (63)

and integrating· the resulting expression from zero to y we get

V1

= 2 n:2 a p2 R4 (T (x) K (x) - -~) y2 ... (64) 0 m m ~

166

which gives the required change in the total self-gravitating potential energy due to the deformation (11 a).

Now,our duty is to determine the change in the total electrical energy W 1 of the dielectric gravitational fluid cylinder dispersed in a

dielectric tenuous medium of negligible motion.

We consider an electrostatic field E~ which has been established

in the dielectic tenuous medium of dielectric constant Ee. We assume that a dielectric body (a fluid cylinder) of a dielectric constant £i has been submerged into the field while the sources of E~ are maintained

constant. Hence the electric energy of the dielectric body (fluid cylinder) in the electric field is given (by Stratton [20] as

W le ifr.£i 0

= 2 (£ - £) (n0

. e0

) r dr ... (65)

where E~ is the modified field diRtrihution when the dielectric fluid column has been submerged in the medium. If the fluid cylinder is deformed and the electric fielif distribution now becomes, .El, then the electric energy of the deformed fluid cylinder is given by

W- ~(ct! ih JfJ (~. E~) r dr d<j> dkz. . .. (66)

Therefore, the changc i 11 the electric energy duo to the performed deformation (11 a) i::; being

W1

= W - W0

. . .. (67)

Substituting from (1),(2),(16 a), (19 a), (21) anc1 (40) into (65) and (66) an<l c:Rrrying out tho requil'e<l integration::; we obtain

:l :l e i 2 W0

=(E0

a /2)(£ -E)R0

, at r=R0

and

( e i) J 2rr J2rr JR 3 • w = e ~ f. . 0 (E . Et) r dr dhz d<j>. ~ 0 kz=U U O l

(69)

Carrying out the integrationR in (69) and substituting from (68) and (69) into (67), the total change in the electric energy (per unit length) due to the deformation (11 a) of the fluid cylinder is given by

32 2 { i (! () } rtR0E

0 co::; wt [x e - e (x + m~w· I (x) K (x) 1:v m · m e 132 ( ) y =---------- - . . . +£ ... 70

1 2 x(Ei I '(x)K (x) - EeK '(x)I (x) m m m m

6. Characteristic Value Problem By the use of (56),(64) and (70) for (12), the Lagrangian function

is construce<l and moreover if we use the Lagrangi:m equation (13) we obtain

167

x l '(x)l - l:Cx) (lm(x) Km(x) - ~)y

2 d Y _ 4nGp dt2

2 m m m en. (7 ) E 0

2y z i (x) [ [x( <X£e - £i) + m~£e] 2

l K 2]

+ ~2 cos (wt) l ( ) . - e 1-1 = 0 ... 1 pR

0 m x [£il '(x)K (x) - £el K ']

m m m m

Equation (71) is an integro-differential equation governing the surface displacement y of the perturbation and may be rewritten in the following brief form

2 !!:._]_ 2 2 2 + (b - h cos ll)Y = 0

d11 ... (72)

where ll and b are defined by

4nGp x 1m'(x) [l ] 11=wt,b= 2 lm(x) 2-lm(x)Km(x) ... (73,74]

2 _ -E~ [xlm'(x)){(x£i-£e(<XX+m~))2 lm(x)Km(x) e 2} h - 2 2 l . - £ ~ ... (75)

pR0

w m (x) x[i::il (x)K (x) - Eel (x)K '(x)] m m m m

The second order ordinary differential equation (72) has the canonical form

where

2 !f:...:1 +(a - 2q cos 211) Y= 0 dll

2

a= b- !:..12 2 i '

7. Limiting Cases

q=::lh2 4 .

... (76)

... (77)

Under appropriate choices we can obtain lhe fulluwing limiting cases :

If we put ~ = 0 and G = 0 in equation (71) we recover the dispersion relation of Reynolds [19] (eqn (44)) if we neglect the surface leu::,;iuu arn.l charges there and assume that fluid cylinder is ambient with vacuum medium.

If we suppose that G = 0, ~ = 0 and w = 0, equation (71) degenerates to that of N { symbol f }ayyar and Murty [12] (eqn (23) ) if we neglect the capillary force there.

Inserting ~ = 0 and G = 0 in equation (71) we recover the characteristic equation of Abouelmagd and Nayyar [1] (eqn (2.7)) if we ueg1ect the surface tension influence there.

J:i'ollowing Cha;idrashekhar and Fermi [5] we,: postull'ltE:l that y- exp (O' t) where cr 1s the growthrate, and mo,reqyey )'lf,l assµ,m(l here.

,I(•· ,,,,. ""·'" ., ..... '.

168

that E 0 = 0, ~ = 0, ex= 0, w = 0 and simultaneously m = O; equation (72)

yields

2 x1

1(x)

CT == 4n:p G Io(x) ((lo(x) Ko) -i), ... (78)

where use has been made of the relation J 0'(x) =11(x). The dispersion relation (78) has been derived for first time by Chandrashekhar and Fermi [5] for the aim of investigating the self-gravitating fluid cylinder dispersed into a self-gravitaing vacuum. Indeed they [5] have utilized a totally different technique from that used here. Their perturbation technique [5] is mainly depends on representing the solenoidal vector fields in terms of poloidal and toroidal vector fields, and that analysis is only valid for axisymmetric perturbation mode m = 0 but not for those of non- axisymmetric m :;t: 0.

If we postulate y - exp (CT t) and we moreover assume that E0 == 0, m :;t: 0, ~ = 0, ex= 0 and w = 0, the characteristic equation (72)

yields

2 - xlm'(x) 1 CT - 4n:Gp

1 ( ). (1 (x) K (x) - -). x m m 2

m

... (79)

This dispen;ion relation is derived and discussed by Chandrasekhar (4] on utilizing the noraml mode analysis, and did mention the correlation of such study with understanding the dynamical behaviour of the spiral arms of galaxies.

If we postulate that y::. exp (cr t) and we put ~ = 0 and w = 0, we obtain from (72) an electrogravitational dispersion relation of a fluid cylinder pervaded by uniform electric field (0, 0, E

0) di1:>persed in

gravitating medium penetrated by the longitudinal uniform electric field (0, 0, exE

0), see Radwan (18].

If we assume that y - exp(cr t) and simultaneou1:>ly we suppose that w = 0, a - 0 and m ;;::-o, the charactcriAt.ic equation (72) degenerntes t.o a dispersion relation of a :,;elf-gravitating tluid cylinder pervaded by (0, 0, E

0) and surrounded by the varying electric field

(0, B R0

r- 1 R0

, 0).

Other stability criteria could be derived in the cases : (i) As G - 0, E

0 ¢ 0, B..;.. 0, u = 0, w - 0 and m = 0.

(ii) As G == 0, E 0 ,c 0, B == 0, ex :;t: 0, w = 0 and m =0.

(iii) The cases (i) and (ii) as m;:::: 0. 8. Stability Discussions The integro-differential c~quation (72) or rather its canonical

form (76) is the Mathieu second order differential equation and its

169

solution is given in terms of Mathieu functions. These transcendental functios (their behaviour, numerical data, .... etc.) are studied in several text books, see for example Morse and Feshback. [11] and Mclachlan [10]. We have to mention here that our model of the gas-core fluid cylinder is being stable if the solution of equation (72) is periodic and this could be occurred under certain restrictions. Tbese appropriate restrictions are cL·pending on the relationship which correlate the parameters q ~rnd \Y. In numerical studies presenting q on a horizontal axis and CY. as vertical axis, it is found that the (q, cx)-plane is classified into diffl'rent categories. These categories arc corresponding to stable and unstable regions bounded the characteristic curves of the Mathieu functions, see Ince [8] or/and Mclachlan [10). Therefore we predict that the model of the gas-core fluid jet is stable if the solution of the Mathieu differential equation (72) is obtained such that the point (q, ex) lies interior or/and on the boundaries of a stable domain in the stable regions and vice versa. Hence the condition for stability degenerates to the problem of the boundary regions of Mathieu function. Mclachlan /10/ gave the explicit condition

. 2 n: u

I 1

1/2

D(O) sm 2-- < 1 ... (79)

for stability where D(O) is the Hill's determinant. However the . analysis of this condition is useless and rather cumbersome because D(O) is infinite.

The numerical discussions and investigations concerning the stable and unstable regions of the characteristic curves of the Mathieu functions reveal, for very 8mall values of q, thl'lt thP. first imstablP. region is bounded by the curves

(j= 1 ±q ... (80)

while the bmmnnry r.urves of the other unstable regions which are higher than the first unstable regions are totally different from (80). However, in this re8pect, Morse ancl Feshhrir.h [·11] hnvr. frnmn ont nn approximate relation for identifying the (in-) stability states. That aproximate relation is, valid only for very small values of h2 being

(h4 - 16(1- b)h2 + 32b(l - b)) 2 0 ... (81)

If the restrictions (81) are satisfied the model must be stable and vice versa where the equality is corresponding to the marginal (neutral) stability. One have to refer here that I h 2 i is very ::;mall as the periodic field frequency w is very high. The inequality (81) is quadratic in h 2 and could be expressed as

2 2 (h - (X1)(h (X

2)?:: 0. . .. (82)

with

<x1

==8(1 h) /) nnd <X2

• 8(1-b)+D ... (83), (84)

170

are the two roots of the equality in the restrictions (81) where

D = (32 (1 - b)(2 - 3b)} 112 . . .. (85)

It is found more convenient to study and write down some properties of the modified Bessel functions before carrying out the instability and oscillation investigations of the present problem.

Consider the recurrence relation (cf. Gradshteyn and Ryzhik [7], of the modified Bessel functions

2 J '(x) =I 1(x) +I

1(x) ... (86)

m m+ m-

2 K '(x) = -K 1(x) -K

1(x).

m m+ m-... (87)

Also consider the facts, for every non-zero real value of x, that I (x) is positively definite and monotonic increasing while K (x) is m m monotonic decreasing but never negative i.e.

I (x) > 0, K (x) > 0 ... (88, 89) m m

By the use of the relations (86) and (87) and the inequalities (88) and (89) we can show, for every x -t 0, that

get

I ' (x) > 0, K '(x) < 0. m · m · · . .. (90, 91)

Consequently for any positive values of £i and £e and x =/.: 0, we

I (x) K (x) > 0 Ill /fl

x(l '(x)/1 (x)) > 0 m m

[ci I '(x) K (x) - f..e I (x) K '(x)] > 0. m m m m

... (92)

... (93)

. .. (94)

However, it is found numerically concerning the inequality (92), (see Chandrasekhar [4)), that

I (x) K (x) < 1. as m 2 1. m m 2 ... (95)

In the axisymmetric mode of peryurbation m = 0, the value of the compound functions I0(x) K0(x) may be larger or smaller than 1 and that depends on the x value. Moreover it is found that

10(x) Kf) s 1 as 1.0668 s x < = ... (96)

I0(x) K

0(x)'?.1 as 0 < x < 11.0668 ... (97)

In view of the foregoing relations and inequalities, using (75), we can show that

h2 < 0 provided that

{x[eil '(x)K (x) - eel (x)K '(x)] Ee p2J < (lxf.i - f.e(ax + m j))J 2 I (x)K (x)) mm m. m .. m·m

1,

171

Now, by an appeal to the inequalities (92), (93) and (95) to (98) for discussing the dispersion relation (79), we may identify the self-gravitating force influence. Following Chandrasekhar's analysis [4], it is found that the model is purely self-gravitating stable in the modes m 2 1 for all x ,c 0 and also in the mode m = 0 in the domains 1.0668::::: x < =. It is only self-gravitating unstable in the axisymmetric modem= 0 in the domain 0 < x < 1.0668.

In order to determine the influences of the different acting electrodynamic forces on the present model with neglecting the periodicity of the basic electric fields, we would have to use the stability criterion

pR20

02

x I '(x) [ [x(£i 1£e) - (ax+ m~)J 2 I (x)K (x) l

---=~11_1_ ~2_ . m m ... (99) E 2

0 £e I,)x) x[(£11£e) I '(x)K (x) -I (x)K '(x)]

m m m m

which is coming out from (71) by comiidering y ::_exp( CT t) and G = 0. The quantity R 0(p)112/(E0(er.) 112) has a unit of time therefore, the

relation (99) is a dimensionless equaiton. In equation (99) there is no any term free from the parameters a and ~- This means that the interior electric field E~ has no influence on the stabilty of the present

problem. ThP. inf11rnnr.P. of thP. :::ixi~l P.XtP.rior P.lP.ctric field (Eg)," is

represented by the term, including a in equation (99) as~= 0,

x 2 [(e1 /£e) - a] 2 I '(x)K (x) m m ... (100)

I (x)K '(x) - (fi /f!)I '(x)K (x)] m m m m

By the use of the inequalities (89), (90), and (94), we find that the axial exterior electric field is stablizing

ei > 0, £e > 0, a> 0. . .. (101)

This result is valied for all values of x ,c 0 in the axisymmetric mode m = 0 and also in those of non-axisymmetric m :/:. 0.

The effect of the azimuthal exterior electric field (~)<!> is

represented by the terms, including~ in equation (99) as u = 0,

x I '(x) "r m2

I (x) K (x) l m P"' 1- m m ... (102) lm(x) . x[(e1 lee) I '(x) K (x) -I (x)K '(x)] . .

m m m m

In the axisymmetric mode m = 0, it is purely destabilizing rltw to its contriLutiuu p2(x I

1(x)II

0(x)) where we have used the relation

10'(.x)- ll(x).

In the non-axisymmetric perturbation modes m 2 1, the exterior azimuthal electric field has a strong stabilizing influence if, for each non-zero real value of x, the restrictions

172

(m 2I (x)K (x)) :s;x[(£il£e)l '(x)K (x)-l (x)K '(x)] ni rn n1 rn 1n rn

... (103)

are satisfied and vice versa where use has been made of the inequalities (88) to (CJ4).

We may conclude here that the electrodynamic forces (with general variable electric field) acting on the present model are stabilizing if

x(£il '(x)K (x) - £el (x)K '(x))~2 :s; [x£i - £e(o:x + m~)] 2 I (x)K (x) m m m m m m

.... (104)

where the equalities are associated witht he neutral stabilities. Indeed these general restrictions are deduced without paid any atention for the magnitude of E 0. The effect of the latter will be determined later

through the investigation of the integro- differential equation (72) or rather its canonical form (76) and this will be our present scope.

Now, let us return to the general characteristic Mathieu differential equation (72). Following Morse and Feshbach [20] principles with taking into accounts the relations (83) to (85) for the inequalities (81) and (82), the (in-) stability investigations and discussions should be carried out in the following different cases.

The case 0 < b < (2/3) In this case we have b > 0 and simultaneously 3b < 2 we can

show that D is real. Hence we can prove that a 1 is positive, note that

if a1

is positive, then a 2 is also positive because a 2 > a 1 as follows. If

a 1 is negative, we have from (83) that (8(1 - b) - D) < 0 and on using

(85) we get b > l which is a contradiction to the present postulates that 0 < b < (2/3) of the our case. Therefore, a

1 must be positive and

a 2 is so. Now since a 1 and a 2 are real and positive we predict, on

using (98), that each of (h2 - a1

) and (h2 - a 2) is negative, Hence the

product (h2 - a 1)(h2 u2) is positive llefi.niLe and this shows that the

restrictions (81) are statisfied under the conditions 0 < b and 3b < 2. The case (2/3) < b < 1. This case is restricted by the two conditions (2/3) < b and b < 1

simultaneously. Using these together with (98) we can prove that the two roots a 1 and a 2 are complex. Consequently the stability

restrictions (81) arc indentically satisfied. Summarizing the results of the foregoing cases 0 < b < (2/3) and

(2/3) < b < 1, we see that the RfahiJity restrictions (81) arc 8Utisficd and hence the model at hand is stable if b is bounded by

0 < b < 1. ... (105)

173

Combining (74) and (105), we conclude that the modulating dielectric self-gravitating fluid cylinder is stable if the frequency of the periodic electric field is larger than a critical value we such that

[

xl '(x) l w 2 = 4rrGp /

1 (!.. - I (x) K (x)) > 0 ... (106a)

c (x) 2 m m m

This restriction is independent of the amplitude of the applied electric field. Therefore, upon choosing appropriate value of W the model could be in stabilizing state. Hence an investigation of the right side in the equality (106a) is leading to find out exac~ly where are the domains of stability and those of instability. This is our present scope.

form The equality in (106a) may be rewritten in the non-dimension

2 w 4rrGp

xl '(x) I I (x) K (x)). m (-- m ~ 2 m ... (106b)

m

By the use of the inequalities (93) and (95) for this relation, we conclude that the electirc field frequency is strongly destabilizing in all non-axit:;y mmetric perturbation modes m ;::: 1. Also in using the inequality (93) in thA :u:isymmetic morlA m = 0 of perturbation and taking into account the inequalities (96) and (97) for disscusing the relation (106 b) we deduce t.hr: following. The electric field frequency w has strung stabilizing influence in the doman 0 < x:::; 1.0668 1.0668 while it has a destabilizing effect in the doman 1.0668 < x < 00 upon axisymmetirc perturbation.

Therfore, we conclude that the frequency of the electric field is stabilizing for axisymmetric mode in the doman 0 < x ~ 1.0668 only and destablizing otherwise.

The case b > 1. In this case we have b > 1 so 3b > 2 and we can also show that

the determinant D is being real. Moreover we can show that a 1 is

negative here as follows. If a1

is positive, using (83), we get

8(1- b) ::o:D. ... (107)

From the view point of (85), the restrictions (107) may be rewritten as

( 64(1 - b)2);::: ( 32(1- b) (2 - 3b)) ... (108)

from which we obtain b < 1. The latter result is a contradiction to the supposition of the present case that b > 1, hence a 1 must be negative. In similar steps we can show that a 2 is positive, hence on using the inequality (98) we have

(h2 --aJ<o ... (109)

174

for all values of b (> 1). By the u.:;e or (109), the stability restricitions (82) would be satisfied only if

h 2 > (X - 1

or alternatively

with

2R2 2 p w E~::::: 2 ° [8(1-b)+DJ

F (x) m

xl '(x) 1 x£i - £e(a.x + m~)] 2 I (x)K (x) )

F2 (x) = __ n_i _ . m m _ fe~2 m Im(x) x[rrl '(x)K (x) - £el (x)K '(x)]

m m m m

where use has been made of (75) and (83).

The case b < 0

... (110)

... (111)

... (112)

This case in which b is assumed to be negative looks like the previous case in which b > 1 where we can show that

a1

= - ve and a2

- + ve ... (113) (114)

C:or1:,;1~q11t\nt.ly t.hl' stability restrictions (H~) Are sAtistiecl 1mcler the validity of the inequalities (109) and (110).

Therefore, we conclude that for the cases b < 0 and b > 1, the dielectric fluid cylinder will be stable if there exist a critical value Eg of the electirc field intensity E0 such that E

0::::: Eg whtm=i

wR p 0 1/2

Ee= F (x) (8(1 - b) + D) . . .. (115) m

It i::; wurLhwhile to mention here that the quantity (1/ E~/(pR~))- l/2 has a unit of "time" and therefore we may formulate

the characteristic relation (1 lfi) in ;:i dimensionless form. Thence by giving appropriate values for the occurred different parameters in regular steps of the wavenumber x, we could find out the critical value of E i.e. Ec

0 above which the instability character of the model is

p completely suppressed and stability then arises and sets in.

REFERENCES Ill A.M. Abeuelmagd and N.K. Nayyar, J. Phys. A, 3 (1970), 296. 121 L. Baker, Phys. Fluid Mech., 26 (1983), 391. [3) D. Callebaut and A. Radwan, Proc. Eurnp. Phys. Soc. (Germany) lOD (198H), 11. [41 S. Chandrasekhar, Hydredynamic and Hydromagnetic Stability, Dover, NY.

1981.

175

[5] S. Chandrasekhar and E. Fermi, Astrophys. J., 118 (1953), 116.

[6) P.G. Drazin and W.H. Reid, Hydrodynamic Stability, Cambridge Univ. Press London. 1980.

[7] I. Gradshteyn and I. Ryzhik, Tables of Integrals, Series and Products, Academic Press Inc., London, 1980.

[8) E.L. Ince, Characteristic Numbers of Mathieu Equation, P.R.S.E., 46 (1976], 20. [9] A. Larraza and S. Putterman, J. Fluid Mech., 148 (1984), 443. [10) N. Mclachlan, Theory and Aplications of Mathieu Functions, Dover, N.Y., 1964. [11] P. Morse and H. Feshbach, Methods of Theoretical Physics, Part 1, Mc Graw

Hill, N.Y., 1953. [12] N.K. Nayyar and G.S. Murty, Proc. Phys. Sec. London, 75 (1960), 369. [13] J.W. Rayleigh (3rd Lord), The Theory of Sound, Dover, N.Y., 1945. [14] A. Radwan and D. Callebaut, Int. Con{ on Magnetic Fluids, Tokyo and Sendai,

Japan 4 (1986), 19. [15] A. Radwan, Nueve Cimento, 90 (1987), 1233. See also A. Radwan and S. Elazab,

Simon Stevin, 61 (1987), 293. [16] A. Radwan, J. Magn. Magn. Matr., 72 (1988), 219;

Nuovo Cimento, 103B (1989), l;

Astrophys. Space Sci. 172 (1990), 305;

Plasma Phys. (UK.). 44 (1991), 455. [17] A. Radwan, Arch. Mech. (Poland) 43 (1990), 31. [18] A. Radwan, J. Fluid Dynamics (U.S.A.) (1992), (In Press) [19] J.M. Reynolds, Phys. Fluids, 8 (1963), 161. [20] J.A. Stratton, Electromagnetic Theory, Mc Graw-Hill, London, 1947; See also

N. Nayyar and G. Murty, Proc. Phys. Soc., London, 75 (1960), 369. [21] H. Yeh. J. Fluid Mech., 152 (1985), 479.

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CONTENTS +Rft:;it, ~ ~. ~ ~ s"fo ~ f.6T ~

- B. B. Lal

Professor J.N. Kapur: Man and Mathematician . - R.C. Singh Chandel

An Operational Calculus for the Index 2li'i-Transform - N.Hayek and B.J. Genzalez

Multidimensional Practional Derivatives of Multiple Hyper­geometric Functions of Several Variabes

R.C. Singh Chandel and P.K Vishwakarma

Certain Generating Func~ions for the ~onhauser's Polynomials - S. N. Singh and L.S. Singh

Integrals Associated with Gauss's Hypergeometric Series, Multi­variable H-Function and General Class of Polynomials

- R.K Saxena, Chenna Ram and O.P. Dave

Common Fixed Points of Weakly Commutating Mappings -R.P. Pant

Common Fixed points of two Pairs of Squences of Mappings - J.M. C. Joshi and R.P. Pant

Integrals Involving a General Class of Multivariable Polynomials, Jacobi Polynomials and Fox's H-Function

- R.S. Pareek

Magnetohydrodynamic Temperature Distribution of Two Immis­cible Viscous Liquids between Two Parallel Plates -

- Pushpendra Kumar, N.P. Singh and Ajay Kumar

Some Weaker Forms of Fuzzy Continuous Mappings - Sunder Lal and Pushpendra Singh

Certain Results for a General Class of Polynomials, Konhauser Biorthgonal Polynomials and the Multivariable H-Function

- S.P. Goyal and Sunil Saxena

A Generalized Study of a Viscous Incompressible Fluid Flow Through Various Cross Sections of a Tube -

-N. Yadaua

Two Fixed Point Theorems For Nonself Mappings - B.E. Rhoades

Common Fixed Point Theorems for Densifying Mappings - B.E. Rhoades

Some Remarks on a Fixed Point in Metric Space - A.K Agrawal and Abrar Ahmed

Reyleigh - Taylor Instability of Rotating oldroydian Viscoelastic. Fluids in Porous Mediu.m in Presence of A Variable Magnetic Field - . .

P. Kumar

Free Convective MHD Flow of a Dusty Visco Elastic Liquid in Porous Medium Past an Oscillating Infinite Porous Plate

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