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A class of nonlocal impulsive problems for integrodifferential equations in Banach spaces. (English) Zbl 1209.34095

The paper deals with the nonlinear integrodifferential impulsive equation with nonlocal conditions

x ' (t)=Ax(t)+ft,x(t), 0 t k(t,s,x(s))ds,tJ=[0,b],tt i ,x(0)=g(x)+x 0 ,Δx(t i )=I i (x(t i )),i=1,2,,p,0=t 0 <t 1 <<t p <t p+1 =b,(P)

where A:D(A)XX is the generator of a strongly continuous semigroup {T(t),t0} on a Banach space X, f:J×X×XX, k:J×J×XX, g:PC(J,X)X and I i :XX, i=1,2,,p are given functions which satisfy some suitable assumptions, Δx(t i )=x(t i + )-x(t i - ), x(t i + )=lim h0 + x(t i +h) and x(t i - )=lim h0 - x(t i +h) are respectively the right and left limits of x(t) at t=t i . By using a generalization of the Ascoli-Arzela theorem and some fixed point theorems such as Schaefer’s fixed point theorem and Krasnosel’skii’s fixed point theorem, the authors study the existence and uniqueness of PC-mild solutions for problem (P).

MSC:
34K30Functional-differential equations in abstract spaces
45J05Integro-ordinary differential equations
34K45Functional-differential equations with impulses
47N20Applications of operator theory to differential and integral equations
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