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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}^{\text{'}}\left(t\right)=Au\left(t\right)+f\left(t,u\left(t\right),{\int }_{0}^{t}k\left(t,s,u\left(s\right)\right)ds\right)\phantom{\rule{1.em}{0ex}}\left(0

with a nonlocal initial value condition

$u\left(0\right)=g\left(u\right)+{u}_{0}$

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 ODE in abstract spaces 45N05 Abstract integral equations, integral equations in abstract spaces 45G10 Nonsingular nonlinear integral equations