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Nonlocal problems for integrodifferential equations. (English) Zbl 1163.45010
The paper deals with the nonlocal Cauchy problem for the nonlinear integrodifferential equation
$u'(t)=Au(t)+\int_0^tB(t-s)u(s)\,ds+f(t,u(t)),\quad 0\leq t\leq T,\tag{1}$
$u(0)=u_0+g(u),\tag{2}$
in a Banach space $$X$$, where $$A:D(A)\subset X\to X$$ is a densely defined, closed linear operator that generates a $$C_0$$-semigroup $$\{T(t),\;t\in [0,T]\}$$, $$\{B(t),\;t\in [0,T]\}$$ is a family of continuous linear operators from $$D(A)$$ into $$X$$, the function $$f:[0,T]\times X\to X$$ is continuous and the operator $$g:C([0,T]\times X)\to X$$ is compact, which satisfy some additional assumptions. The authors prove that the resolvent operator $$R(t)$$ of equation $$(1)$$ is continuous in the uniform operator topology, for $$t>0$$, and then they establish the existence of mild solutions of the problem (1)–(2), by using Schaefer’s fixed point theorem.

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