Electron. J. Diff. Equ., Vol. 2009(2009), No. 155, pp. 1-7.

Existence and uniqueness of solutions to fractional semilinear mixed Volterra-Fredholm integrodifferential equations with nonlocal conditions

Mohammed M. Matar

Abstract:
In this article we study the fractional semilinear mixed Volterra-Fredholm integrodifferential equation
$$ \frac{d^{\alpha }x(t)}{dt^{\alpha }} =Ax(t)+f\Big(t,x(t), \int_{t_0}^tk(t,s,x(s))ds,\int_{t_0}^{T}h(t,s,x(s))ds\Big) , $$
where $t\in [t_0,T]$, $t_0\geq 0$, $0<\alpha <1$, and $f$ is a given function. We prove the existence and uniqueness of solutions to this equation, with a nonlocal condition.

Submitted September 12, 2009. Published December 1, 2009.
Math Subject Classifications: 45J05, 26A33, 34A12.
Key Words: Fractional integrodifferential equations; mild solution; nonlocal condition; Banach fixed point.

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Mohammed M. Matar
Department of Mathematics, Al-Azhar University of Gaza
P. O. Box 1277, Gaza, Palestine
email: mohammed_mattar@hotmail.com

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