*(English)*Zbl 0812.47058

This article deals with a mapping $T$ of a nonempty set $C$ of a Banach space $\mathbb{X}$ into itself for which the inequalities

hold. The following four results are presented:

(1) if $\mathbb{X}$ has uniformly normal structure, $C$ is a bounded set and $sup{k}_{n}<N{\left(\mathbb{X}\right)}^{1/2}$ ($N\left(\mathbb{X}\right)=inf\{\text{diam}\phantom{\rule{4.pt}{0ex}}E/\text{rad}\phantom{\rule{4.pt}{0ex}}E$: $E$ is bounded closed convex set of $\mathbb{X}\}$) and there exists a nonempty bounded closed convex set $E$ containing weak $\omega $-limit set of $T$ at $E$ then $T$ has a fixed point in $E$;

(2) if $\mathbb{X}$ is uniformly smooth, ${k}_{n}\to 1$, ${x}_{n}$ $(n=1,2,\cdots )$ is a fixed point of

${x}_{n}-T{x}_{n}\to 0$ then ${x}_{n}$ converges strongly to a fixed point of $T$;

(3) if $\mathbb{X}$ is a Banach space with a weakly continuous duality map, $C$ is a weakly compact convex subset, ${k}_{n}\to 1$ then $T$ has a fixed point and moreover if $T$ is weakly asymptotically regular at some $x\in C$ then ${T}^{n}x$ converges weakly to a fixed point of $T$;

(4) if the Maluta constant $D\left(\mathbb{X}\right)<1$, $C$ is a closed bounded convex set, ${k}_{n}\to 1$, $T$ is weakly asymptotically regular on $C$ then $T$ has a fixed point.

##### MSC:

47H10 | Fixed point theorems for nonlinear operators on topological linear spaces |

47H09 | Mappings defined by “shrinking” properties |