Random fixed points of increasing compact random maps

Ismat Beg

Address. Department of Mathematics, Lahore University of Management Sciences, 54792 Lahore, PAKISTAN

E-mail: beg@lums.edu.pk

Abstract. Let $(\Omega ,\Sigma )$ be a measurable space, $% (E,P)$ be an ordered separable Banach space and let $[a,b]$ be a nonempty order interval in $E$. It is shown that if $f:\Omega \times \lbrack a,b]\rightarrow E$ is an increasing compact random map such that $a\leq f(\omega ,a)$ and $f(\omega ,b)\leq b$ for each $\omega \in \Omega $ then $f$ possesses a minimal random fixed point $\alpha $ and a maximal random fixed point $\beta $.

AMSclassification. 47H07, 47H40, 47H10, 60H25

Keywords. Random fixed point, random map, measurable space, ordered Banach space