Electronic Journal of Differential Equations,
Vol. 2005(2005), No. 21, pp. 1-7.

Title: Continuous selections of set of mild solutions of evolution inclusions

Authors: Annamalai Anguraj (P.S.G. College of Arts & Science, Tamilnadu, India)
         Chinnagounder Murugesan (Gobi Arts & Science College, Tamilnadu, India)
 
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.