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.