This article deals with a mapping of a nonempty set of a Banach space into itself for which the inequalities
hold. The following four results are presented:
(1) if has uniformly normal structure, is a bounded set and (: is bounded closed convex set of ) and there exists a nonempty bounded closed convex set containing weak -limit set of at then has a fixed point in ;
(2) if is uniformly smooth, , is a fixed point of
then converges strongly to a fixed point of ;
(3) if is a Banach space with a weakly continuous duality map, is a weakly compact convex subset, then has a fixed point and moreover if is weakly asymptotically regular at some then converges weakly to a fixed point of ;
(4) if the Maluta constant , is a closed bounded convex set, , is weakly asymptotically regular on then has a fixed point.