*(English)*Zbl 1049.47511

From the text: Theorem 1. Let $E$ be a uniformly convex Banach space with a uniformly Gateaux differentiable norm, $C$ a nonempty closed convex subset of $E$, and $T:C\to E$ a nonexpansive nonself-mapping. Suppose that $C$ is a nonexpansive retract of $E$, and that for each $u\in C$ and $t\in (0,1)$, the contraction ${G}_{t}$ defined by (1) has a (unique) fixed point ${x}_{t}\in C$. Then $T$ has a fixed point if and only if $\left\{{x}_{t}\right\}$ remains bounded as $t\to 1$ and in this case, $\left\{{x}_{t}\right\}$ converges strongly as $t\to 1$ to a fixed point of $T$.

Corollary 1 [H. K. Xu and X. M. Yin, Nonlinear Anal., Theory Methods Appl. 24, No.2, 223–228 (1995; Zbl 0826.47038)]. Let $H$ be a real Hilbert space, $C$ a nonempty closed convex subset of $H$, and $T:C\to H$ a nonexpansive nonself-mapping. Suppose that for some $u\in C$ and each $t\in (0,1)$, the contraction ${G}_{t}$ defined by (1) has a (unique) fixed point ${x}_{t}\in C$. Then $T$ has a fixed point if and only if $\left\{{x}_{t}\right\}$ converges strongly as $t\to 1$ to a fixed point of $T$.