Fixed point theorems for nonexpansive mappings satisfying certain boundary conditions
W. A. Kirk · Proceedings of the American Mathematical Society · 1975
Let $K$ be a bounded closed convex subset of a Banach space $X$ with $\operatorname {int} K e \emptyset$, and suppose $K$ has the fixed point property with respect to nonexpansive self-mappings (i.e., mappings $U:K \to K$ such that $||U(x) - U(y)|| \leq ||x - y||,x,y \in K)$. Let $T:K \to X$ be nonexpansive and satisfy \[ \inf \{ ||x - T(x)||:x \in {\text { boundary }}K,T(x) otin K\} > 0.\] It is shown that if in addition, either (i) $T$ satisfies the Leray-Schauder boundary condition: there exists $z \in \operatorname {int} K$ such that $T(x) - z e \lambda (x - z)$ for all $x \in {\text { boundary }}K,\lambda < 1$, or (ii) $\inf \{ ||x - T(x)||:x \in K\} = 0$, is satisfied, then $T$ has a fixed point in $K$.