Continuous selections of set of mild solutions of evolution inclusions

• Main Authors: Annamalai Anguraj, Chinnagounder Murugesan
• Format: Article
• Language: English
• Published: Texas State University 2005-02-01
• Series: Electronic Journal of Differential Equations
• Subjects: Mild solutions; differential inclusions; integrodifferential inclusions.
• Online Access: http://ejde.math.txstate.edu/Volumes/2005/21/abstr.html

We prove the existence of continuous selections of the set valued map $xio mathcal{S}(xi)$ where $mathcal{S}(xi)$ is the set of all mild solutions of the evolution inclusions of the form $$displaylines{ dot{x}(t) in A(t)x(t)+int_0^tK(t,s)F(s,x(s))ds cr x(0)=xi ,quad tin I=[0,T], }$$ 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 $Iimes I$ with $tgeq s$ for all $t,sin I$ and $xi in X$.