Recent trends in differential equations

581 Pages·1992·16.87 MB·English

by  Agarwal R.P. (ed.)

RECENT
TRENDS IN
DIFFERENTIAL
EQUATIONS
WORLD SCIENTIFIC SERIES IN APPUCABLE ANALYSIS
Editor: R. P. Agarwal
Vol. 1: Recent Trends in Differential Equations
erf. R. P. Agarwal
Forthcoming
Vol. 2: Contributions in Numerical Mathematics
erf. R. P. Agarwal
World Scientific Series in Applicable Analysis
Volume 1
Editor
R. P. Agarwal
Dept of Mathematics
National University of Singapore
RECENT
TRENDS IN
DIFFERENTIAL
EQUATIONS
World Scientific
Singapore • New Jersey • London • Hong Kong
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RECENT TRENDS IN DIFFERENTIAL EQUATIONS
Copyright © 1992 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form
or by any means,electronic or mechanical, including photocopying, recording or any
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written permission from the Publisher.
ISBN 981-02-0963-0
Printed in Singapore by Utopia Press.
V
A TRIBUTE TO PROFESSOR V. LAKSHMIKANTHAM ON
HIS 65 - th BIRTHDAY
Mathematical achievements of Lak till the year 1985 have been categorized
by S.Leela, Yu.A.Mitropol'skii and A.A.Martynyuk in "A survey of collected
works of V.Lakshmikantham, Differential'nye Uravneniya, 22(1986),555-572."
In the last six years his scientific activities have grown exponentially. Infact, to
date he has 250 research publications and 15 monographs to his credit, and su­
pervised 36 Ph.D. students. He has lectured at more than 150 research centers
and International Conferences. He has also edited 8 conference proceedings
and currently he is either the Editor or an Associate editor of 12 Scientific
Journals.
At a very young age when his peer group was receiving formal education
in schools, he only had a informal schooling at home, where he learnt about
Poetry and other classics in Telugu literature. And although at a later stage
his talent found expression in Mathematics, his creative ability in poetry hasn't
been completely overshadowed. This is evident even now when Lak has a few
spare moments.
Because of the family responsibilities and other obligations, he started
his working life as a Bank clerk. He left his job after six years to continue
further studies and joined the Osmania University as a Laboratory assistant
in the Chemistry department. During this period as a private student he
received his undergraduate and post-graduate degrees. It is after his Masters
degree in 1955, his research proclivities came to the fore and within a short
period of 2 years, his research career bloomed inspite of limited nourishing
environment and facilities at the Osmania University. During 1955-60, he
served as a lecturer in the department of Mathematics of Osmania University,
and published 12 research papers in several diverse fields.
In 1960, he left Osmania for University of California at Los Angeles, and
got the attention of Professors Bellman and Coddington. After one year at
UCLA, he went to the University of Wiscosin Mathematics Research Center,
Madison, where along with other researchers like Brauer, Driver and Nohel in
vi
Stability theory, he produced some significant results in Lyapunov methodol­
ogy. Activities in the Lyapunov functions and their applications to Ordinary as
well as Partial differential equations continued for several years. In fact, dur­
ing this period a close friend asked him "Lak, Why Lyapunov functions again
and again?" He promptly replied "You people still don't see the importance
of Lyapunov functions." 1962 - 63 was his yet another prolific period with the
Research Institute for Advanced Studies, Baltimore, Maryland, group.
In the year 1963 he joined University of Alberta as an Associate Profes­
sor. But, within a short period of one year he returned to India and joined
Marathwada University as a Professor and Chairman of the Mathematics
department. During his short stay there, he supervised six Ph.D. students
S.G.Deo, A.A.Kayande, G.S.Ladde, S.Leela, N.Muley and G.Shendge in spite
of the strict academic environment (learning was possible there as hundred
years back in India, particularly as in Gurukulas). During this period he often
used to ask students "Do you know any Mathematics?" Once he did get the
reply "Yes, I know er.". He thought for a while and replied to him "You know
33% of the known functions, because all the known functions are permutations
and combinations of ex,xn, and sin a:".
His next move was to the University of Rhode Island where he was the
Chairman and Professor of Mathematics until 1973. During this period Aca­
demic Press not only published his famous two volume monograph " Differ­
ential and Integral Inequalities : Theory and Applications", but also a mono­
graph on Differential Equations in Abstract Spaces.
Professor Lak spent following 15 years as the Chairman of the Department
of Mathematics at the University of Texas at Arlington. He attracted a large
number of well - known Mathematicians and transformed Arlington as a hub
of tremendous amount of Mathematical activities. He arranged 8 International
Conferences on Nonlinear Analysis and, in addition, wrote approximately one
monograph a year!
In the year 1989 Professor Lak has joined the Florida Institute of Technol­
ogy (F.I.T.), Melbourne, Florida as the chairman of the Department of Applied
Mathematics. Since he joined F.I.T. a large number of Russian Mathemati­
cians have visited his department resulting in significant amount of Mathe­
matical activity. More so, he is organizing the World Congress of Nonlinear
Analysts, at Tampa, Florida in August, 1992. This is likely to be attended by
more than 1000 nonlinear scientists. All this has given prominence to F.I.T.
as a well known place for research in Mathematics.
A tribute to a teacher could not be complete without expressing the kind of
influence he exerted on his students. And in this direction Lak has had nothing
short of the Midas touch. His encouraging advice to students induces them to
bring out the very best and infuses them with varied creative aspirations. And
there can be no let up in their efforts. For, in him, they have a role model who
continues to inspire them with his boundless creative energy and enthusiasm,
vii
the stage of his career and life notwithstanding.
Watching him speedily complete a task on hand, only to take on yet an­
other, one begins to wonder about the spring of motivation he draws from.
Could it be his subconscious that eggs him on to make up for the lost time in
those early years of his life? One muses.
If ever a list were to be compiled of people guided by the dictum "Simple
Living, High Thinking", Lak would easily qualify as one of the prime candi­
dates. Totally oblivious to his appearance, he, perhaps, feels secure in con­
viction that one's real appeal lies in accumulated knowledge and other inner
strengths.
In view of all above we have all the reasons to celebrate and wish him many
more prolific and enjoyable years to come.
R.P.Agarwal
S.Leela
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ix
CONTRIBUTORS
N.U.Ahmed (Canada) Man Kam Kwong (U.S.A.)
D.D.Ang (Vietnam) G.Ladas (U.S.A.)
O.Arino (France) K.Jaya Lakshmi (India)
D.Bainov (Bulgaria) B.S.Lalli (Canada)
A.Chaljub - Simon (France) B.Lani - Wayda (Germany)
K.W.Chang (Canada) R.R.Lassiter (U.S.A.)
Shui - Nee Chow (U.S.A.) L.L.Littlejohn (U.S.A.)
C.Corduneanu (U.S.A.) G.X.Liu (Canada)
A.DeBortoli (U.S.A.) J.Mawhin (Belgium)
K.Deimling (Germany) A.Ben M'Barek (Morocco)
V.Dragan (Romania) W.Mckinney (U.S.A.)
M.S.P.Eastham (Bahrain) A.B.Mingarelli (Canada)
A.Elbert (Hungary) M.Z.Nashed (U.S.A.)
F.W.Eloe (U.S.A.) F.Neuman (Czechoslovakia)
L.II.Erbe (Canada) J.J.Nieto (Spain)
W.D.Evans (U.K.) E.S.Noussair (Australia)
W.N.Everitt (U.K.) A.Peterson (U.S.A.)
H.I. Freed man (Canada) V.Petrov (Bulgaria)
K.Gopalsamy (Australia) V.Sree Hari Rao (India)
S.R.Grace (Egypt) Donal O'Regan (Ireland)
C.P.Gupta (U.S.A.) J.Ridenhour (U.S.A.)
I.Gyori (Hungary) K.Schmitt (U.S.A.)
A.Halanay (Romania) S.W.Schultz (U.S.A.)
T.G.Hallam (U.S.A.) V.Seda (Czechoslovakia)
Xue - Zhong He (China) I.P.Stavroulakis (Greece)
J.Henderson (U.S.A.) Da - Qing Sun (China)
W.Huang (U.S.A.) C.A.Swanson (Canada)
A.Ivanov (Kiev) Y.Takeuchi (Japan)
J.Jaros (Czechoslovakia) W.Trench (U.S.A.)
J.Jaworska (U.S.A.) G.Vidossich (Italy)
S.K.Kaul (Canada) P.Volkmann (Germany)
J.Kato (Japan) L.K.Vy (Vietnam)
W.Kelley (U.S.A.) Hans - Otto Walther (German
Y.Kitamura (Japan) S.Wang (Canada)
H.W.Knobloch (Germany) Z.Wang (China)
A.M.Krall (U.S.A.) G.F.Webb (U.S.A.)
T.Kusano (Japan) Xu Yuantong (China)
K.H.Kwon (Korea) Yagu Zhao (U.S.A.)
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