Existence of Fixed Points in a Class of Convex Sets

Abstract

There are many well-known results about existence of fixed points for nonexpansive mappings defined on convex sets of Banach spaces which are $\tau$-compact, where $\tau$ is usually a topology coarser than the norm topology. In this paper, we prove that it is possible to remove the $\tau$-compactness assumption and still obtain some fixed point results in a large framework. As particular cases, we can consider certain families of non-$w^{\ast }$-compact subsets in ${\ell }_1$, the James space $J_p$, the trace space ${\mathcal{C}}_1({\ell }_2)$ and some other dual nonreflexive Banach spaces and we prove that „local" stability results for nonexpansive mappings can be achieved. In order to reach our purposes, we will make use of two new geometric coefficients $a_{\tau }(X)$ and $t_{\tau }(C)$. Since the problem of the existence of fixed points for a nonexpansive mapping can equivalently be redefined assuming that the mapping is asymptotically regular, we will first focus on asymptotically regular mappings which satisfy a rather weak Lipschitz condition according to the value of the fraction $\frac{a_{\tau }(X)}{t_{\tau }(C)}$. Furthermore, we will show some examples proving that our results are sharp and can be used to obtain „global" stability results for the fixed point property.