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Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order. (English) Zbl 1179.45010
The paper deals with the existence and uniqueness of mild solutions to a semilinear integro-differential equation of fractional order. The problem studied here is: $$D^q x(t) + Ax(t) = f\left(t,\, x(t),\, \int_0^t e(t,\,s,\,x(s))\,ds\right),\quad t \in [0,\, a],\qquad x(0) + g(x) = x_0,$$ where $0<q<1$, $-A$ is the infinitesimal generator of a noncompact and analytic semigroup on a Banach space and $e,\, f$ and $g$ some functions. The initial data is taken from the Banach space $D(A^{\alpha})$, with $0< \alpha \leq 1$ with the norm $|x|_{\alpha} = |A^{\alpha} x|$. Under various conditions on the functions $e,\; f$ and $g$, the above problem admits a unique mild solution. The main techniques employed here are the Banach contraction principle and a fixed point theorem.

##### MSC:
 45J05 Integro-ordinary differential equations 45G10 Nonsingular nonlinear integral equations 26A33 Fractional derivatives and integrals (real functions) 34A08 Fractional differential equations
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##### References:
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