492 Pages·1978·5 MB·English

ACADEMIC PRESS RAPID MANUSCRIPT REPRODUCTION Proceedings of an International Symposium on Nonlinear Equations in Abstract Spaces, held at The University of Texas at Arlington, Arlington, Texas, June 8-10, 1977 NONLINEAR EQUATIONS IN ABSTRACT SPACES Edited by V. Lakshmikantham Department of Mathematics University of Texas at Arlington Arlington, Texas ACADEMIC PRESS New York San Francisco London 1978 A Subsidiary of Harcourt Brace Jovanovich, Publishers COPYRIGHT © 1978, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. NO PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER. ACADEMIC PRESS, INC. Ill Fifth Avenue, New York, New York 10003 United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London NWl 7DX Library of Congress Cataloging in Publication Data International Symposium on Nonlinear Equations in Abstract Spaces, University of Texas at Arlington, 1977. Nonlinear equations in abstract spaces. "Proceedings of an International Symposium on Nonlinear Equations in Abstract Spaces, held at the University of Texas at Arlington, Arlington, Texas, June 8-10, 1977.'* 1. Differential equations, Nonlinear-Numerical solutions—Congresses. 2. Volterra equations—Numerical solutions—Congresses. 3. Banach spaces-Congresses. I. Lakshmikantham, V. Π. Title. QA371.I553 1977 515'.35 78-8412 ISBN 0-12-434160-8 PRINTED IN THE UNITED STATES OF AMERICA LIST OF CONTRIBUTORS Numbers in parentheses indicate the page on which authors* contributions begin. •Indicates the author who presented the paper at the conference, ADOMIAN, G. (3), Center for Applied Mathematics, The University of Georgia, Athens, Georgia 30602 AHMAD, SHAIR (25), Department of Mathematics, Oklahoma State Uni versity, Stillwater, Oklahoma 74074 AMES, W. F. (43), Center for Applied Mathematics, The University of Georgia, Athens, Georgia 30602 ENGL, HEINZ W. (67), Institut für Mathematik, Kepler-Universität, Linz, Austria FITZGIBBON, W. E. (81), Department of Mathematics, University of Houston, Houston, Texas 77004 GOLDSTEIN, JEROME A. (95), Department of Mathematics, Tulane Uni versity, New Orleans, Louisiana 70118 KARTSATOS, ATHANASSIOS G. (105), Department of Mathematics, University of South Florida, Tampa, Florida 33620 KIFFE, THOMAS (365), Department of Mathematics, Texas A&M Univer sity, College Station, Texas 77843 KOBAYASHI, YOSHIKAZU (113), Faculty of Engineering, Niigata Univer sity, Nagaoka, Japan LAKSHMIKANTHAM, V. (117, 125), Department of Mathematics, The University of Texas at Arlington, Arhngton, Texas 76019 LAZER, ALAN C* (25), Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221 LEELA, S.* (125), Department of Mathematics, SUNY College at Genesco, Genesco, New York 14454 McKENNA, P. J. (375), Department of Mathematics, The University of Wyoming, Laramie, Wyoming 82071 MARTIN, ROBERT H., JR. (135), Department of Mathematics, North Carolina State University, Raleigh, North Carolina 27607 MITCHELL A. R. (387), Department of Mathematics, The University of Texas at Arlington, Arlington, Texas 76019 vii viii LIST OF CONTRIBUTORS MOAURO, v.* (125, 149), Istituto Matemático "R. Caccioppoli" deirUni- versitá di Napoli, Via Mezzocannone, Naples, Italy, and The University of Texas at Arlington, Arlington, Texas 76019 MONTROLL, ELLIOTT W. (161), Physics Department, University of Rochester, Rochester, New York 1(X)21 NAGLE, KENT (405), Department of Mathematics, University of South Florida, Tampa, Florida 33620 NASHED, M. Z. (217), Department of Mathematics, University of Delaware, Newark, Delaware 19711 NEUBERGER, J. W. (253), Department of Mathematics, North Texas State University, Denton, Texas 76203 ODEN, J. T. (265), Texas Institute for Computational Mechanics, The Uni versity of Texas at Austin, Austin, Texas 78712 PAYNE, FRED R. (417), Aerospace Engineering, The University of Texas at Arlington, Arhngton, Texas 76019 PETRYSHYN, W. V. (275), Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903 REICH, SIMEON (317), Department of Mathematics, University of Southern California, Los Angeles, California 90007 SALVADORI, L. (149), Istituto Matemático **G. Castelnuovo" delPUni- versitá di Roma, Rome, Italy, and The University of Texas at Arlington, Arhngton, Texas 76019 SCALIA, M. (149), Istituto Matemático *Ό. Castelnuovo" deirUniversitá di Roma, Rome, Italy SHOWALTER, R. Ε. (327), Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712 SINGH, K. L. (439), Department of Mathematics, Texas A&M University, College Station, Texas 77843 SMITH, CHRIS* (387), Department of Mathematics, The University of Texas at Arhngton, Arlington, Texas 76019 STECHER, MICHAEL* (365), Department of Mathematics, Texas A&M University, College Station, Texas 77843 TRAVIS, C. C. (331), Health and Safety Research Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37830 VAUGHN, R. L. (463), Department of Mathematics, The University of Texas at Arhngton, Arhngton, Texas 76019 WARD, JAMES R. (469), Department of Mathematics, Pan American Uni versity, Edinburg, Texas 78539 WEBB, G. F.* (331), Department of Mathematics, Vanderbilt University, Box 1085, Nashville, Tennessee 37235 WEISSLER, FRED B. (479), Department of Mathematics, The University of Texas at Austin, Austin, Texas 78712 Preface An International Symposium on Nonlinear Equations in Abstract Spaces was held at The University of Texas at Arhngton, June 8-10, 1977. The purposes of the symposium were to highlight some of the recent advances in abstract nonlinear equations, to stimulate discussions, and to share ideas for future research in this area. The present volume consists of the proceedings of the symposium. It in cludes papers that were delivered as invited talks and research reports as well as contributed papers. The theme of the symposium was the solvability of nonlinear equations, such as Volterra integral equations, ordinary differential equations, and differential equations with retarded argument, in Banach spaces. There is a group of papers dealing with boundary value problems using such techniques as nonlinear superposition, alternative methods, and fixed points of monotone mappings in ordered Banach spaces. Another group of papers is concerned with the application of nonlinear semigroup theory in solving Volterra integral equations and semilinear partial differential equa tions. A third group of papers deals with existence theorems for nonlinear evolution equations in Banach spaces. There are also some applications of the previous results to problems arising in nonlinear elasticity, turbulence in Newtonian fluids, classical mechanics, and mathematical models in social phenomena. I wish to express my special thanks to my colleagues and friends Bill Beeman, Steve Bernfeld, Dorothy Chestnut, Jerome Eisenfeld, Mike Lord, A. R. Mitchell, R. W. Mitchell, and Bennie Williams for assisting me in planning and organizing the symposium; to my secretaries Mrs. Gloria Brown, Debbie Green, and Mrs. Kandy Dyer for their assistance during the confer ence; and to Mrs. Mary Ann Grain and Mrs. Sandra Weber for excellent typing of the proceedings. IX NONLINEAR EQUATIONS IN ABSTRACT SPACES NEW RESULTS IN STOCHASTIC EQUATIONS THE NONLINEAR CASE G. Adomian Ce.nteJl 6ott Applie.d Ma;the.ma:ti..v.. UniVeJr.J.lUY 06 GeoJtg-<a Although basic laws generally lead to nonlinear differential and integral equations in many areas, linear approximations are usually employed for mathematical tractability and the use of superposition. Where nonlinear systems must be solved, perturba tion theory is classically used which can be a severe restrictio~ A further substantial difficulty is that real systems generally involve uncertainties and fluctuations and are consequently to be 1-6 considered as stochastic systems. The objective then must be to consider nonlinear stochastic systems to allow more realistic modelling. 7-18 Previous papers have studied objectives, limitations, and necessary restrictive assumptions for the various methods for solution of linear stochastic differential equations arising in physical problems. An iterative procedure was developed using 4 the concept of stochastic Green's functions and allowing treat ment of a wide class of problems without many of the limitations inherent in other methods or restrictive assumptions involving 9 highly special processes. It has been shown that the method is applicable to partial differential equations and wave propagation 6 in randomly varying media ,9 without an assumption of monochro maticity, which ignores the spectral spreading, and without the 10 closure approximations necessary in the hierarchy method. The Copyright ©1978byAcademicPress,Inc. Allrightsofreproductioninanyformreserved. 3 ISBN0-12-434160-8 4 G, Adomian latter approximations clearly ignore the random fluctuations in certain operators depending on the level of the hierarchy in which it is made. What is involved is that an ensemble average of the form <l?L- ·1P...HI,-1Py>, (where L is the deterministic part of the stochastic operator and R is the random part) is separated -I -I into <HL R...RL H><y>. The procedure is justified if H is sufficiently small. Essentially it is equivalent then to dropping terms after some power of E depending again on the level. Thus 1 writing <HL- Py> = <HL-1R><y> is equivalent to dropping terms of (/ in the so-called method of "smoothing perturbation".19 A variety of other approximate techniques for stochastic wave propa gation are similar to the smoothing perturbation method. Thus , 20 14 15 21 diagram methods, , , , parabolic equation approximation, etc., all involve the assumption of a near negligible randomness. The iterative procedure on the other hand is independent of the 8 presence of a small parameter, and it has been shown that the re- sults of perturbation theory are obtained where perturbation theory is applicable, i.e., where the random fluctuations are actual1y sma11. As Lerche and Parker19, and ear1ier, Krai'Chnan22, l5 23 Frisch , Herring , and others, have pointed out, the first order smoothing gives incorrect results in general. As in the hierarchy method, it is simply incorrect to approximate the equations to be solved. Extensive computer results in preparation for publica- ti,on24 W'iII show compar'isons and the superi'or vaI'1d'ity 0f the iterative procedure. As Lerche and Parker stated, the only way to be sure of the result is to solve the exact equations with a systematic mathematical approximation scheme. This has been our approach and it checks in the deterministic limit; it checks in the perturbation case and computer results show it corresponds extremely closely (as closely as desired, depending of course on computing sufficient terms) with the correct analytical solution (where that solution has been possible to obtain) while the other methods show larger and larger deviation as time progresses. A general expression was obtained for the stochastic Green's Stochastic Equations 5 functions which yield the mean solution or the two point correla tion of the solution. Relationships have been found between the various stochastic Green's functions for different statistical measures and also between stochastic Green's functions, random 8 Green's functions, and ordinary Green's functions . The stochas tic Green's function is determined as a kernel of an integral ex pression yielding directly the desired statistical measure of the solution process - in our case, either the expectation of the solution or the correlation or covariance of the solution. The long-term objective is to develop methods of calculating the sto chastic Green's function for the desired statistical measure (e.g., correlation or covariance) directly from knowledge of the given statistics (statistical measures) of the coefficients with 7 out the intermediate step of solving the equation. ,8 The Nontine~ Caoe. Recently, this work was significantly . 25-27 extended to nonlinear stochastic differential equat~ons mak- ing the results substantially more valuable in real physical prob lems. This paper reviews the status of newer work and a symmet rized form for the terms of the iteration which appears particu larly useful. The equation considered is .f.y + IJ(y,y,... ) = x(t) where x(t) is a stochastic process on a suitable index space T and probability space m,F,)1), .f. is a linear stochastic dif- n v v L ferentia1 operator of nth order given by .f. Qv(t,w)d /dt , v=O the v = O,l,.•. ,n - 1 are possibly stochastic processes on statistically independent of x(t), and N a nonli mm l. near stochastic term of the form N b~(t,w)(y()1))m)1 where )1=0 is the )1th derivative and b are stochastic processes j.; for ).l = O,l,...,m on T x ~. The coefficients Q and b)1 are possibly stochastic pro v cesses defined on the same probability space as x(t) purely for 6 G. Adomian convenience in notation. As pointed out in an earlier paper, there is no difficulty in allowing a to be defined on ~a or v even ~ x(t) on ~x' etc. Hence no restriction is implied a ' v here. As assumed in the earlier papers, an > 0 (we can let it conveniently be 1), and randomness may occur in all other terms aO""'~_l' although it would normally be only one or two which would be random. The coefficient processes av,b~ are assumed statistically independent from the input process x(t) for all allowed ~,v. Further, the processes av'e are of class on T for wE(~,F,~) almost surely for allowed ~,v We assume n £ = L + R where L = <£>, i.e., L = L <a (t,w»dv/dtV and n-1 v=o V R = L u (t,w)dV/dtV where * <a (t,w» exists and is contin V=O V V uous on T. We have Ly + Ry + N(y,y, ...) = x(t) Ly = x - Ry - N(y,y, ... ) Since L is assumed invertible Y = L-1x - L-1Ry - L-1N(y,y., ...) or in terms of the Green's function t(t,T) for L, to It Y = I ~(t,T)X(T)dT - 0 ~(t,T)R[Y(T)JdT -I: Mt,T)N(y(T),Y(T), ...)dT 00 i Let y L (-1) Y.' We get I:i=o -z, I: ~(t,T)R[YO(T)-Y1(T)+Y2(T) y = t(t,T)X(T)dT - ... JdT -I: t(t,T)N(YO(T) - Y1(T) + Y2(T).•.)dT In order to proceed with the iteration we must assume the form of * av <av> + uv(t,w) <Uv> - 0

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