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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)+ 0 t B(t-s)u(s)ds+f(t,u(t)),0tT,(1)
u(0)=u 0 +g(u),(2)

in a Banach space X, where A:D(A)XX is a densely defined, closed linear operator that generates a C 0 -semigroup {T(t),t[0,T]}, {B(t),t[0,T]} is a family of continuous linear operators from D(A) into X, the function f:[0,T]×XX is continuous and the operator g:C([0,T]×X)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.

45N05Abstract integral equations, integral equations in abstract spaces
45J05Integro-ordinary differential equations
45G10Nonsingular nonlinear integral equations