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.


45J05 Integro-ordinary differential equations
45G10 Other nonlinear integral equations
26A33 Fractional derivatives and integrals
Full Text: DOI


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