Positive Solutions of Differential, Difference and Integral Equations Positive Solutions of Differential, Difference and Integral Equations by Ravi P. Agarwal Department ofM athematics. National University of Singapore. Singapore Donal O'Regan Department ofM athematics. National University ofI reland. Galway. Ireland and Patricia J. Y. Wong Division ofM athematics. Nanyang Technological University. Singapore SPRINGER-SCIENCE+BUSINESS MEDIA, B.V. A C.I.P. Catalogue record for this book is available from the Library of Congress. ISBN 978-90-481-5153-0 ISBN 978-94-015-9171-3 (eBook) DOI 10.1007/978-94-015-9171-3 Printed on acid-free paper AU Rights Reserved © 1999 Springer Science+-Business Media Dordrecht Originally published by Kluwer Academic Publishers in 1999 No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner Dedicated to our mothers Godavari Agarwal Eileen O'Regan Phuah Liew Hong Contents Preface ix Ordinary Differential Equations 1. First Order Initial Value Problems 1 2. Second Order Initial Value Problems 11 3. Positone Boundary Value Problems 19 4. Semi-positone Boundary Value Problems 29 5. Semi-infinite Interval Problems 40 6. Mixed Boundary Value Problems 47 7. Singular Boundary Value Problems 63 8. General Singular and Nonsingular Boundary Value Problems 86 9. Quasilinear Boundary Value Problems 106 10. Delay Boundary Value Problems 110 11. Coupled System of Boundary Value Problems 119 12. Higher Order Sturm-Liouville Boundary Value Problems 131 13. (n,p) Boundary Value Problems 190 14. Focal Boundary Value Problems 210 15. General Focal Boundary Value Problems 222 16. Conjugate Boundary Value Problems 236 Difference Equations J 7. Discrete Second Order Boundary Value Problems 261 18. Discrete Higher Order Sturm-Liouville Boundary Value Problems 279 19. Discrete (n,p) Boundary Value Problems 315 20. Discrete Focal Boundary Value Problems 325 21. Discrete Conjugate Boundary Value Problems 353 Integral and Integrodifferential Equations 22. Volterra Integral Equations 370 23. Hammerstein Integral Equations 381 24. First Order Integrodifferential Equations 386 References 395 Authors Index 412 Subject Index 417 Preface In analyzing nonlinear phenomena many mathematical models give rise to problems for which only nonnegative solutions make sense. Therefore, since the publication of the monograph Positive Solutions of Operator Equa tions in the year 1964 by the academician M.A. Krasnosel'skiit, hundreds of research articles on the theory of positive solutions of nonlinear problems have appeared. In this vast field of research, particular attention has been focused on the Sturm-Liouville boundary value problem y" + >..f(x, y, yl) = 0, a < x < b, >.. > 0 { oy(a) + (3yl(a) = Cl = '"Yy(b) - tSy'(b) C2. For example the special case y" + >..q(x)g(y) = 0, 0 < x < 1 { y(O) = 0 = y(l) and its n-dimensional analog du + >..q(x)g(u) { ulan = 0 whose positive solutions have been studied and constructed arise in a vari ety of interesting applications such as gas diffusion through porous media, thermal self-ignition of a chemically active mixture of gases in a vessel. With q(x) = 1 the cases g(y) = 0(1 + (3 - y)e-'Y/Y b large), g(y) = e-1/lyl = and g(y) (3(r - y )e-k/(1+y), respectively, arise in models in catalytic theory, chemically reacting systems and adiabatic tubular reactors. The special case q(x) = q(constant» 0, g(y) = eUY occurs in applications in volving the diffusidn of heat generated by positive temperature-dependent = sources. For instance, when 1 it arises in the analysis of Joule losses (J' in electrically conducting solids as well as in frictional heating. This model was proposed by Gelfand in the year 1959. The last twenty years have witnessed hundreds of research articles guar anteeing the existence of the positive solutions to = y"+>..q(X)g(x,y) 0, O 0, x E (0,1) rely upon the form and/or properties ofthe function g(x, y). The cases when x 9 is superlinear, sublinear, 9 2': O(Positone), 9 :::; O(Semi-Positone), and 9 is singular at y = 0 have been discussed extensively. In this classification two significant examples are: the Emden-Fowler problem y" + )..q(x)y-1' = 0, 0 < x < 1, 'Y > 0 { y(O) = 0 = y(l) which appears in various branches of fluid dynamics; and the Thomas Fermi problem { y" - x-1/2y3/2 = 0, 0 < x < 1 y(O) = 1, y(l) = 0 which was derived in the year 1927 for determining the electrical potential in an atom. The boundary value problem { y " + -N--1Y, +)..g ( x, y ) = 0, 0 < x < 1, N > 2 y'(O) =x 0, 'Yy(l) - 8y'(1) = C2 finds applications in combustion theory, and models various physiological processes. For example, when N = 3, 'Y > 0, 8 = -1, g(x,y) = -ay/(y + k), a > 0, k > 0 it arises in the study of steady-state oxygen diffusion in a cell with Michaelis-Menten kinetics. The solution y is a measure of the oxygen tension and x = 1 corresponds to the boundary of the cell membrane. If g(x, y) = ke-{3y, k > 0, (3 > 0 heat conduction in the human brain is modeled. Here 9 is the heat production rate per unit volume, y is the absolute temperature and x is the radial distance from the center. Evidently in these models only positive solutions are significant. The positive solutions of boundary value problems on semi-infinite in tervals such as y"+)..q(x)g(x,y) = 0, O 0 and Q g: [0, T] x R - R continuous. Their technique involves transforming the = = problem with singularities at y 0 and t 0 to a problem which is only o. singular at t = Here, we shall follow Agarwal and O'Regan [30] to examine the general singular problem (1.1), i.e. when f: [O,T] x (0,00) - R continuous, and the nonlinearity f is allowed to change sign. Our theory as a special case includes the results proved in [70]. n By a solution to (1.1) we mean a function y E C[O, T] C1 (0, T] which satisfies the differential equation on (0, T) and the stated initial condition. 1.2. Preliminary Results. To prove our main results we shall need the following: Definition 1.1. Let -00 < a < b < 00. A collection of real valued functions A = {Ii: [a, b] - R} is said to be R. P. Agarwal et al., Positive Solutions of Differential, Difference and Integral Equations © Springer Science+Business Media Dordrecht 1999 2 Chapter 1 1. uniformly bounded (bounded) if there exists a constant M > 0 with Ifi(t)1 ~ M for all t E [a, b] and all fi E A, and 2. equicontinuous if for every 10 > 0 there exists 8 = 8(10) > 0 such that ifto,tlE[a,b] then Ito-tIl <8 implies Ifi(tO)-li(tdl<€ for all fi E A. Theorem 1.1 (Arzela-Ascoli Theorem) [91,237]. A subset A of C([a, b], JR) is relatively compact if and only if it is bounded and equicon tinuous. Theorem 1.2 (Existence Principle) [247]. With respect to the initial value problem yl = q(t)g(t,y), 0 < t < T (1.2) = y(O) 0 suppose that g: [0, T] x JR JR is continuous, and --t (i) q E C(O, T] with q> 0 on (0, T] and JoT q(s)ds < 00. (a) Assume there is a constant M, independent of >., with Ilyll sup ly(t)1 =I M tE[O,T] for any solution y to yl = >.q(t)g(t,y), 0< t < T (1.3}.>. = y(o) 0 for each >. E (0,1). Then (1.2) has a solution. (b) Assume there exists a constant M such that Ig( t, y) I < M for t E [0, T] and y E JR. Then (1.2) has a solution. 1.3. Existence Results. Througout this chapter we will assume f(t,y) = f(y) for t E [0, fl. Our first main result is embodied in the following: Theorem 1.3. Suppose in addition to condition (i) the following hold: (ii) f: [O,T] x (0,00) --t JR is continuous, ° (iii) If(t,y)1 ~ g(y) + h(y) on [O,T] x (0°,0 0) with 9 > continuous and nonincreasing on (0,00), hi9 ? non decreasing on (0,00), (iv) let n E {2, 3, ... } and associated with each n, we have a constan°t = Pn such that {Pn} is a nonincreasing sequence with limn Pn -+DO and such that for Tin ~ t ~ T we have f(t,Pn)? 0,

Positive Solutions of Differential, Difference and Integral Equations

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