## On some existence results of mild solutions for nonlocal integrodifferential Cauchy problems in Banach spaces.(English)Zbl 1242.45010

Various existence results of a mild solution of the integro-differential equation $u'(t)=Au(t)+f\Bigl(t,u(t),\int_0^tk(t,s,u(s))ds\Bigr)\quad(0<t<b)$ with a nonlocal initial value condition $u(0)=g(u)+u_0$ are obtained. Here, $$A$$ is the generator of a $$C_0$$-semigroup. Recall that a mild solution is a function $$u$$ which formally satisfies the corresponding variation-of-constants formula. The main hypotheses are some growth estimates for $$k$$, $$f$$, and $$g$$ (leading to a-priori bounds), compactness of either $$f$$ or of the semigroup, and either compactness of $$g$$ or that $$g$$ is Lipschitz with a sufficiently small constant. The proofs use Schauder’s or Schaefer’s fixed point theorem.

### MSC:

 45J05 Integro-ordinary differential equations 34G20 Nonlinear differential equations in abstract spaces 45N05 Abstract integral equations, integral equations in abstract spaces 45G10 Other nonlinear integral equations
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