Existence principles for inclusions of Hammerstein type involving noncompact acyclic multivalued maps (Q2762924)

Drawing upon earlier work of the first author with \textit{M. Kamenski} [Nonlinear Anal. Theory, Methods, Appl. 42, 1101-1129 (2000; Zbl 0972.34049)] the authors study abstract Hammerstein integral inclusions \(u\in SG(u)\) where \(G\) is a multivalued Nemytskij operator and \(S\) is a nonlinear operator. In this situation it does not make sense to assume that \(N:=SG\) takes convex values, rather one has to admit acyclic values and to apply the Eilenberg-Montgomery fixed point theorem or one of its descendants. NEWLINENEWLINENEWLINEThe authors first prove an abstract fixed point theorem and then turn to Hammerstein inclusions: Let \(0<T<\infty\), \(E\) a real Banach space and \(1\leq p\leq\infty\), \(1<q<\infty\). Let \(I=[0,T]\) and let \(g\) be a mapping from \(I\times E\) to the subsets of \(E\) and consider the associated Nemyckij operator \(G\) mapping on \(L^{p}(I;E)\) defined by \(G(u)=\{w\in L^{q}(I;E)\mid w(s)\in g(s,u(s))\) almost everywhere on \(t\in I\}\) and a single-valued operator \(S:L^{q}(I;E)\to L^{p}(I;E)\). Let \(K\) be a closed convex subset of \(L^{p}(I;E)\) and \(U\) a relatively open subset of \(K\). The authors formulate a series of conditions guaranteeing that there is a \(u\in\overline{U}\) such that \(u\in SG(u)\). The crucial assumptions require that \(S\) be comparable to a Volterra operator in the following sense: There is a nonnegative kernel \(k\) in \(I\times I\) such that \(k(t,\cdot)\in L^{r}(I)\) (where \(r\) is conjugate to \(q\)) and \(t\mapsto\|k(t,\cdot)\|_r\) belongs to \(L^{p}(I)\) and \(|S(w_1)(t)-S(w_2)(t)\mid \leq\int_0^Tk(t,s)|w_1(s)-w_2(s)|ds\) almost everywhere on \(I\) for \(w_1,w_2\in L^{q}(I;E)\). NEWLINENEWLINENEWLINESecond, \(g\) is supposed to satisfy a certain growth condition and to assume nonempty compact convex values, \(g(t,\cdot)\) should be upper semicontinuous for almost all \(t\in I\) and each \(g(\cdot,x)\) should admit a strongly measurable selection. As to \(S\), it is supposed that \(S\) maps \(L^{q}(I;E)\) to \(K\) and that \(S\) is sequentially continuous from \(L^{q}(I;C)\) to \(L^{p}(I;E)\) for each compact convex subset \(C\) of \(E\) and that \(SG(u)\) is acyclic for each \(u\in K\). Moreover, there is a technical condition implying that \(SG\) is condensing (in fact, a slightly weaker form is used). This result is then applied to boundary value problems of the form \(u''(t)\in Au(t)+g(t,u(t))\) almost everywhere on \([0,T]\), \(u(0)=u(T)=0\).