The method of upper and lower solutions for an integral inclusion of Volterra type (Q2782946)

This note is concerned with the existence of solutions for integral inclusions of Volterra type: NEWLINE\[NEWLINE y(t) \in g(t) + \int_0^t K(t,s)F(s,y(s)) ds\;\text{ for almost all }t \in [0,T]. \tag \(*\) NEWLINE\]NEWLINE Here \(g\) and \(K\) are single-valued, but \(F\) is multi-valued, and a solution is required to satisfy NEWLINE\[NEWLINE y(t) = g(t) + \int_0^t K(t,s)v(s) ds\;\text{ for almost all }t \in [0,T], \tag \(**\) NEWLINE\]NEWLINE with \(v(s) \subset F(s,y(s))\) for almost all \(s\). The authors get an upper or lower solution of (\(*\)) if they replace the equality sign in (\(**\)) by \(\geq\) or \(\leq\). An existence result is given where the right-hand side of (\(*\)) is interpreted as a condensing multi-valued map on the set of continuous functions on \([0,T]\) which lie between an upper and a lower solution of (\(*\)).