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Second-Order Differential Inclusions

Various aspects of the theory of second-order differential inclusions attract the attention of many researchers (see., e.g., [1, 2, 6, 12, 18, 42, 46, 47, 68, 70, 97]). In this chapter we consider the boundary value problem of form u00 2 Q.u/; u.0/ D u.1/ D 0;

(4.1)

for second-order differential inclusions which arises naturally from some physical and control problems. Using the method of guiding functions we study the existence of solutions of problem (4.1) in an one-dimensional and in Hilbert spaces.

4.1 Existence Theorem in an One-Dimensional Space By W02;2 Œ0; 1 we denote the subset of W 2;2 Œ0; 1 consisting of all functions vanishing at the end-points of Œ0; 1, i.e., W02;2 Œ0; 1 D fu 2 W 2;2 Œ0; 1W u.0/ D u.1/ D 0g: Define the continuous integral operator j W L2 Œ0; 1 ! C Œ0; 1 by Z .jf /.t/ D

1

G.t; s/f .s/ds; 0

where

 G.t; s/ D

t.s  1/ if 0  t  s; s.t  1/ if s  t  1:

Notice that the operator j in fact acts into W02;2 Œ0; 1 and, for any f 2 L2 Œ0; 1; the boundary value problem

V. Obukhovskii et al., Method of Guiding Functions in Problems of Nonlinear Analysis, Lecture Notes in Mathematics 2076, DOI 10.1007/978-3-642-37070-0 4, © Springer-Verlag Berlin Heidelberg 2013

105

106

4 Second-Order Differential Inclusions



u00 .t/ D f .t/ for a:e: t 2 Œ0; 1; u.0/ D u.1/ D 0

can be written in the form: u D jf (see, e.g., [72]). By applying the Arzela–Ascoli theorem, it is easy to see also that the operator j transforms bounded sets into a relatively compact ones. In this section we consider the existence of solutions to the following boundary value problem for the operator-differential inclusion 

u00 2 Q.u/; u.0/ D u.1/ D 0;

(4.2)

where QW C Œ0; 1 ! C.L2 Œ0; 1/ is a multimap satisfying the following conditions:   .Q1/ The composition j ı Q belongs to the class CJ C Œ0; 1I C Œ0; 1 . .Q2/ There are constants p; q > 0 such that p

kQ.u/k2  q.1 C kuk2 / for all u 2 C Œ0; 1, where kQ.u/k2 D supfkf k2 W f 2 Q.u/g: By a solution to problem (4.2) we mean a function u 2 W02;2 Œ0; 1 such that there is a function f 2 Q.u/ satisfying u00 .t/ D f .t/ for a:a: t 2 Œ0; 1: Remark 4.1. Let us mention that the class of multimaps Q satisfying condition .Q1/ is large enough. For example, for every CJ-multimap Q the multimap j ı Q is a CJ-multimap. Moreover, there are multimaps Q which are not CJ-multimaps while j ı Q are CJ-multimaps. For example, let F W Œ0; 1  R ! Kv.R/ be a L2 upper Carath´eodory multimap. It is well known that the superposition multioperator PF is well-defined, it is closed and has convex closed values. Set QW C Œ0; 1 ! Cv.L2 Œ0; 1/, Q.x/ D PF .x/. From Proposition 1.17 it follows that the multimap j ı Q is closed. It is clear that for every bounded subset U  C Œ0; 1 the set Q.U / is bounded in L2 Œ0; 1, therefore the set j.Q.U // is a relatively compact set in C Œ0; 1. Hence, j ı Q is an u.s.c. multimap with  compact convex values and so, it belongs to the class J C Œ0; 1I C Œ0; 1    CJ C Œ0; 1I C Œ0; 1 : The main