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Existence of solutions for integrodifferential inclusions in Banach spaces. (English) Zbl 0824.45017

Two existence theorems are derived for the integrodifferential inclusion \(\dot x\in F(t, x(t), V(x(t)))\), \(x(0)= x_ 0\), in a separable Banach space \(X\). It is assumed that \(F\) is a multifunction such that \((t, x, y)\to F(t, x, y)\) is weakly measurable (or graph measurable) and \((x, y)\to F(t, x, y)\) is upper (or lower) semicontinuous from \(X\times X\) into \(X_ w\) (\(X_ w\) denotes the space \(X\) with the weak topology). Moreover, \(V\) tands for the Volterra integral operator and the map \(F\) satisfies some other assumptions involving the measure of noncompactness.

MSC:

45N05 Abstract integral equations, integral equations in abstract spaces
45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations
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