Existence Results for Some Initial and Boundary Value Problems Without Growth Restriction Author(s): Marlène Frigon and Donal O'Regan Source: Proceedings of the American Mathematical Society, Vol. 123, No. 1 (Jan., 1995), pp. 207-216 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/2160628 Accessed: 21/07/2010 11:43 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=ams. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact
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PROCEEDINGSOF THE AMERICANMATHEMATICAL SOCIETY Volume 123, Number 1, January1995
EXISTENCE RESULTS FOR SOME INITIAL AND BOUNDARY VALUE PROBLEMS WITHOUT GROWTH RESTRICTION MARLENEFRIGON AND DONAL O'REGAN (Communicatedby Hal L. Smith) In this paper,using the SchauderFixed Point Theorem,we establish some existence results or initial and boundaryvalue problems for differential equations withouth growth restrictionon the right member. ABSTRACT.
1. INTRODUCTION
In this paper, we give existence results for initial and boundary value problems without growth restrictionon the right member. The paper is divided into two sections. In ?2 we examine second-orderproblems of the form: (py')'(t) = f(t, y(t), p(t)y'(t))
(B)
a.e. t E [0, 1],
aOy(0) - bolim p(t)y'(t) = ro, aly(1) + b, limp(t)y'(t) = rl, (-+1
where p e C[0, 1] n C1(0, 1) with p(t) > 0 on t e (0, 1), and f: [0, 1] x -2 IR is a Caratheodoryfunction. By a solution to (B), we mean a function y E C[O, 1] n CI(0, 1) such that py' is absolutely continuous on [0, 1], and which satisfies the differential equation and boundary conditions. Boundary value problemsof the form (B) have been extensively treated in the literature. In most of these papers, f satisfiesa growth condition in y' . The resultspresented in this section are obtained without any growth assumptionon f, and generalize some results deduced from the location of the zeros of the nonlinearity;see, for example, [3, 5, 6, 9]. They rely on the notion of upper and lower surfaces for (B), that is, surfaces having a particularform and on which the function f has a given sign. This notion was introduced in [3] in the particular case where p =1 .
In ?3 we examine the existence of solutions for a first-orderinitial value Received by the editors April 22, 1993. 1991 MathematicsSubjectClassification.Primary34B15. Reaserchsupportedin part by NSREC-Canada. ?
207
1994 AmericanMathematicalSociety 0002-9939/94 $1.00 + $.25 per page
MARLENEFRIGON AND DONAL O'REGAN
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problem of the form:
y'(t) = f(t, y(t)) a.e. t E [0, T], y(O) = r where f: [0, T] x IR-+ IR is a Caratheodory function. Here, by a solution to (I), we mean an absolutely continuous function on [0, T] which satisfies the differential equation and the initial condition. The existence theorems presented here generalize some known results based on the notion of upper and lower solutions for the initial value problem; see [1, 6, 8]. In this paper, we prove the existence of a solution for the problem (I) between two functions s1 and so which are not necessarily absolutely continuous or continuous, without using the derivative or the Dini derivates of those functions. All our results are obtained via the Schauder Fixed Point Theorem. First of all, let us establish some notation. Let p E C[O, 1]nfC1(O, 1) be such that p(t) > 0 on t E (0, 1). We define the Banach space K1[0, 1] by {y E C[O, 1] n C1 (0, 1) JPY'E C[O, 1] } with the norm Ily 1vI = max{ Ily11o, 1py'Hllo}
where jjyjjo=max{jy(t)jIt E [0, 1]}, and we define K2[0, 1] by {y eK1[0, 1] I
py' E W 01[0, 1] }I. We denote Kb[0, 1] = {y E KI[0, l] Iy satisfies boundary conditions aoy(O) - bolimt op(t)y'(t) = ro, a1y(l) + b1 limt ip(t)y'(t) = r1 }, = r} E C[0,l]jIY(O) Kb2[0,1] = K2[0,~ 1] n Kb[?, 1]; and Cr[0,l1]={y For the sake of completeness, we give the following two results which will be used later. The second one is a Maximum Principle.
Lemma 1.1. Let x: [0, 1] -+ R be an absolutelycontinuousfunction, and U c [0, 1] be a measurableset such that x(U) is negligeable. Then x'(t) = 0 a.e. te U. Lemma 1.2. Let p E C[c, d] n C1(c, d) be such that p(t) > 0 for all t E (c, d); and let y E K2[c, d] be such that (py')'(t) > 0 a.e. t E (c, d), aoy(c) - bolimt cp(t)y'(t) < 0, and aly(d) + b1limt dp(t)y'(t) < 0, where max{ao, al} > 0 aj, bi > 0, max{ai, bi} > 0, i = 0,1 . Then y(t) < 0 for all t E [c, d].
Proof. Define
n
y(t)2 Gk) j0
if y(t) > 0, otherwise.
Then (py')'(t) G(t) > 0 a.e. t E (c, d). Integratingfrom c to d and using the integration by parts formula gives: d
p (t)y'(t) G'(t) d t > 0.
G(d) limp(t)y'(t) - G(c) limp(t )y'(t) t--+d
t-~C
We deduce that y(t) < 0 for all t E [c, d]. 2.
BOUNDARY
O
VALUE PROBLEMS
In this section, we consider the boundary value problem (B) where aj, bi > 0, max{aj, bi} > 0, i = 0, 1, max{ao, al} > 0, and p and f satisfy the following assumption:
(H) peC[O, 1]nC1(0, 1), p(t)>0 forall te(0, 1), f :[0, 1]xIR2-+IR is a Caratheodory function, that is: (i) t | - f(t, y, q) is measurable for all
INITIALAND BOUNDARYVALUEPROBLEMS
209
(y, q) e R2; (ii) (y, q) 1- f(t, y, q) is continuous for a.e. t e [0, 1]; (iii) for any r > 0, there exists hr E L1[O, 1] such that If(t, y, q)l < hr(t) a.e. t E [0, 1], and for all /y/< r, IqI< r. Moreover, f1 p-1(s) Jf hr(x) dxds
