Existence for nonconvex integral inclusions via fixed points. (English) Zbl 1113.45014

Cernea, Aurelian

Arch. Math., Brno 39, No. 4, 293-298 (2003).

The author investigates solvability of the integral inclusion of the form
\[ x(t)=\lambda (t)+\int _0^t f(t,s,u(s))\,ds,\quad u(t)\in F(t,V(x)(t)), \tag{*} \]
where \(\lambda \: I=[0,T]\to \mathbb R^n\), \(F\: I\times X \to {\mathcal P}(X)\), \(f\: I\times I\times X\to X\), \(V\: C(I,X)\to C(I,X)\) are given mappings, \(X\) is a Banach space and \({\mathcal P}(X)\) is the family of nonempty subsets of \(X\). Additional conditions on the functions \(\lambda ,f,F\) are given (not including a convexity assumption on the values of \(F\)) which guarantee the existence of a solution \(x\) of (*) whose \(C(I,X)\) norm satisfies certain Filippov-type inequality.

Reviewer: Ondřej Došlý (Brno)

MSC:

45N05	Abstract integral equations, integral equations in abstract spaces
45G10	Other nonlinear integral equations

Keywords:

contractive set-valued maps; Banach space; Filippov-type inequality