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On nonconvex valued Volterra integral inclusions in Banach spaces. (English) Zbl 0822.45008
The author considers the Volterra integral inclusion $x(t)\in p(t)+ \int^ t_ 0 U(t, s)\text{ ext } F(s, x(s)) ds,\quad t\in [0, b],\tag{*}$ where $$\text{ext } F(s, x(s))$$ denotes the extremal points in a separable Banach space $$X$$. The existence of a solution $$x(\cdot)\in C([0, b], X)$$ to (*) is proven if several hypotheses hold. The result generalizes a previous existence theorem of the author in the same journal. The theorem is used to obtain a bang-bang principle for controlled Volterra integral equations.

##### MSC:
 45N05 Abstract integral equations, integral equations in abstract spaces 47H05 Monotone operators and generalizations 45G10 Other nonlinear integral equations
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