## Existence of solutions for nonlocal impulsive partial functional integrodifferential equations via fractional operators.(English)Zbl 1220.45009

The paper deals with the following partial functional integro-differential equation:
$\begin{split} x'(t) = Ax(t) + F\left(t,\,x(\sigma_1(t)),\,\dots,\,x(\sigma_n(t)),\int_0^t h(t,\,s,\,x(\sigma_{n+1}(s)))ds\right),\\ t\in [0,\,b],\,t\neq t_k,\; k=\overline{1,\,m}, \end{split}$
where $$A$$ is the infinitesimal generator of a compact, analytic semigroup, $$t_k\in [0,\,b]$$, and $$F,\,h,\,\sigma_k$$ are some given functions. The equation is considered here together with the conditions:
$x(0) + g(x) = x_0\;\;\text{ and }\;\;\; x(t_k^+) - x(t_k^-) = I_k(x(t_k)),\;\; k=\overline{1,\,m}.$
It is shown that, under suitable conditions on the functions $$F,\,h,\,g,\,\sigma_k$$, and for any $$x_0\in X^{\alpha}$$, the above Cauchy problem has at least one mild solution on $$[0,\,b]$$. The proof of this result employs the Leray-Schauder alternative. The author also presents an illustrative example at the end of the paper.

### MSC:

 45J05 Integro-ordinary differential equations 45G10 Other nonlinear integral equations 26A33 Fractional derivatives and integrals
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### References:

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