On the existence of minimal and maximal solutions of discontinuous functional Sturm-Liouville boundary value problems
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We apply a fixed point result for multifunctions to derive existence results for boundary value problems of Sturm-Liouville differential equations with nonlinearities that may involve discontinuous and functional dependencies. 1. Introduction The main goal of this paper is to study the solvability of the following Sturm-Liouville boundary value problem (BVP) −
d µ(t)u (t) = λg t,u,u(t),u (t) a.e. in J = t0 ,t1 , dt a0 u t0 − b0 u t0 = c0 , a1 u t1 + b1 u t1 = c1 ,
(1.1)
where g : J × C(J) × R × R → R. We are looking for solutions of (1.1) out of
Y = u ∈ C 1 (J) | µu ∈ AC(J) .
(1.2)
In Section 2, we give first an existence result for problems where the second, the functional argument u of g, is replaced in (1.1) by fixed functions v ∈ C(J), and study the dependence of solution sets of these problems on v. The so obtained results and a fixed point result for multifunctions proved recently in [7] are then used in Section 3 to derive existence results for minimal and maximal solutions of (1.1). Also in nonfunctional case we get new existence results. Because of weaker hypotheses than those assumed, for example, in [1, 3, 4, 5, 8, 9, 10], the fixed point results for single-valued operators do not apply. 2. Hypotheses and preliminaries 2.1. Hypotheses. Throughout this paper we assume that λ,a j ,b j ∈ R+ ,
a0 a1 + a0 b1 + a1 b0 > 0,
c j ∈ R, j = 0,1,
and that C(J) is ordered pointwise. Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:4 (2005) 403–412 DOI: 10.1155/JIA.2005.403
µ ∈ C J,(0, ∞) ,
(2.1)
404
Discontinuous functional Sturm-Liouville BVP’s
The function g : J × C(J) × R × R → R is assumed to satisfy the following hypotheses. (g0) (t,x, y) → g(t,v,x, y) is a Carath´eodory function, that is measurable in t and jointly continuous in (x, y), for each v ∈ C(J). (g1) |g(t,v,x, y)| ≤ p(t)max{|x|, | y |} + m(t) for all x, y ∈ R, for all v ∈ C(J), and for a.e. t ∈ J, where p,m ∈ L1+ (J). (g2) g(t, ·,x, y) is increasing for a.e. t ∈ J and for all x, y ∈ R. (g3) For each fixed v ∈ C(J), |g(t,v,x, y) − g(t,v,x,z)| ≤ pv (t)φv (| y − z|) for a.e. t ∈ J and for all x, y,z ∈ R, where pv ∈ L1+ (J), φv : R+ → R+ is increasing and 1 0+ (dx/φv (x)) = ∞. Notice that g can be discontinuous in its first and second arguments, and is monotone only with respect to its second, functional argument. It is also worth to notice that no lower or upper solutions are assumed to exist, and no Nagumo-type hypotheses are imposed on g. We are going to show that if λ is small enough, then the BVP (1.1) has under the above hypotheses a minimal solution u− and a maximal solution u+ in the sense that if u is any solution of (1.1), then u ≤ u− implies u = u− and u+ ≤ u implies u = u+ . 2.2. Auxiliary results. For the sake of completeness we recall in this section several auxiliary results whose proofs can be found, for example, in [1, 5]. Lemma 2.1. If q ∈ L1 (J), then the BVP d µ(t)u (t) = q(t) a.e. in J, dt a0 u t0 − b0 u t0 = c0 ,
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