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Existence for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators. (English) Zbl 1178.34071

Summary: We study the existence of mild solutions for the following system in a general Banach space X (with norm ·):

d dt[x(t)-F(t,x(h 1 (t)))]Ax(t)+ 0 t K(t,s)G(s,x(h 2 (s)))ds,tJ-{t 1 ,,t m },whereJ=[0,a],Δx| t k =I k (x(t k - )),k=1,,m,x(0)=g(x)X·(1·1)

Here A is the infinitesimal generator of a compact analytic semigroup T(t(, t>0, G is a multi-valued map and Δx| t=t k =x(t k + )-x(t k - ), where x(t k - ) and x(t k + ) represent the left and right limits of x(t) and t=t k , respectively. Let K:DR, D={(t,s)J×J:ts} and F,G,g,I k (k=1,,m) and h 1 ,h 2 are given functions.

By using a fixed point theorem for multi-valued maps due to Dhage, a main existence theorem is established. Finally, we present an example to illustrate this main theorem.

34G25Evolution inclusions
47N20Applications of operator theory to differential and integral equations