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

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.

### MSC:

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