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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 Other nonlinear integral equations
26A33 Fractional derivatives and integrals
34A08 Fractional ordinary differential equations
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