Integral Inclusions
Chapter
Abstract
This chapter studies integral inclusions in Banach spaces. In particular we discuss the Volterra integral inclusion
and the Hammerstein integral inclusion Here F : [0, T] × E → E is a multivalued map with nonempty compact values; E is a real Banach space. In section 8.2 we present some existence results for (8.1.1) and (8.1.2) when F is a Carathéodory multifunction of u.s.c. or l.s.c. type satisfying some measure of noncompactness assumption. The theory of differential inclusions, usually when dim E < ∞, has received a lot of attention over the last twenty years or so. In this chapter a mixture of old and new ideas are presented so that a general existence theory for multivalued equations can be obtained. The ideas in this chapter were adapted from Deimling [4], Frigon [6] and O’Regan [13]. In particular the technique used in this chapter relies on Ky Fan’s or Schauder’s Fixed Point Theorem [16] together with a trick introduced in [9] and a result of Fitzpatrick and Petryshyn [5].
$$ y(t){\rm{ }} \in {\rm{ g(t) + }}\smallint _0^t k(t,s){\rm{ F(s,y(s))ds for t }} \in {\rm{ [0,T]}} $$
(8.1.1)
$$ y(t){\rm{ }} \in {\rm{ g(t) + }}\smallint _0^T k(t,s){\rm{ F(s,y(s))ds for t }} \in {\rm{ [0,T]}} $$
(8.1.2)
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