Electron. J. Diff. Eqns., Vol. 2005(2005), No. 21, pp. 1-7.

Continuous selections of set of mild solutions of evolution inclusions

Annamalai Anguraj, Chinnagounder Murugesan

Abstract:
We prove the existence of continuous selections of the set valued map $\xi\to \mathcal{S}(\xi)$ where $\mathcal{S}(\xi)$ is the set of all mild solutions of the evolution inclusions of the form
$$\displaylines{ \dot{x}(t) \in A(t)x(t)+\int_0^tK(t,s)F(s,x(s))ds \cr x(0)=\xi ,\quad t\in I=[0,T], }$$
where $F$ is a lower semi continuous set valued map Lipchitzean with respect to $x$ in a separable Banach space $X$, $A$ is the infinitesimal generator of a $C_0$-semi group of bounded linear operators from $X$ to $X$, and $K(t,s)$ is a continuous real valued function defined on $I\times I$ with $t\geq s$ for all $t,s\in I$ and $\xi \in X$.

Submitted November 3, 2004. Published February 11, 2005.
Math Subject Classifications: 34A60, 34G20.
Key Words: Mild solutions; differential inclusions; integrodifferential inclusions.

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Annamalai Anguraj
Department of Mathematics
P.S.G. College of Arts & Science
Coimbatore - 641 014, Tamilnadu, India
email: angurajpsg@yahoo.com
Chinnagounder Murugesan
Department of Mathematics
Gobi Arts & Science College
Gobichettipalayam - 638 453, Tamilnadu, India

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