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Zbl 1220.45009
Yan, Zuomao
Existence of solutions for nonlocal impulsive partial functional integrodifferential equations via fractional operators.
(English)
[J] J. Comput. Appl. Math. 235, No. 8, 2252-2262 (2011). ISSN 0377-0427

The paper deals with the following partial functional integro-differential equation: $$\multline x'(t) = Ax(t) + F\left(t,\,x(\sigma_1(t)),\,\dots,\,x(\sigma_n(t)),\int_0^t h(t,\,s,\,x(\sigma_{n+1}(s)))ds\right),\\ t\in [0,\,b],\,t\ne t_k,\; k=\overline{1,\,m}, \endmultline$$ 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.
[Iulian Stoleriu (Iaşi)]
MSC 2000:
*45J05 Integro-ordinary differential equations
45G10 Nonsingular nonlinear integral equations
26A33 Fractional derivatives and integrals (real functions)

Keywords: impulsive partial functional integro-differential equations; fixed point; analytic semigroup; nonlocal conditions; Cauchy problem; mild solution; Leray-Schauder alternative

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