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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 ' (t)=Au(t)+ft , u (t) , 0 t k (t,s,u(s)) d s(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:
45J05Integro-ordinary differential equations
34G20Nonlinear ODE in abstract spaces
45N05Abstract integral equations, integral equations in abstract spaces
45G10Nonsingular nonlinear integral equations