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Zbl 1179.45010
Chang, Yong-Kui; Kavitha, V.; Arjunan, M.Mallika
Existence and uniqueness of mild solutions to a semilinear integrodifferential equation of fractional order. (English)
[J] Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 71, No. 11, A, 5551-5559 (2009). ISSN 0362-546X

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.
[Iulian Stoleriu (Iaşi)]

MSC 2000:: *45J05 Integro-ordinary differential equations
45G10 Nonsingular nonlinear integral equations
26A33 Fractional derivatives and integrals (real functions)
34A08

Keywords: integro-differential equations; fractional differential equations; existence; uniqueness; nonlocal initial condition; mild solutions; Banach contraction principle; fixed point theorem

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