Nearly Contraction Mapping Principle for Fixed Points of Hemicontinuous Mappings

We extend the application of nearly contraction mapping principle introduced by Sahu (2005) for existence of fixed points of demicontinuous mappings to certain hemicontinuous nearly Lipschitzian nonlinear mappings in Banach spaces. We have applied certain results due to Sahu (2005) to obtain conditions for existence—and to introduce an asymptotic iterative process for construction—of fixed points of these hemicontractions with respect to a new auxiliary operator. 1. Introduction In this paper, we have applied certain results due to Sahu [1] on nearly contraction mapping principle to obtain conditions for existence of fixed points of certain hemicontinuous mappings and introduced an asymptotic iterative process for construction of fixed points of these hemicontinuous mappings with respect to a new auxiliary operator. Our results are important generalizations and an extension of important and fundamental aspect of a branch of asymptotic theory of fixed points of non-Lipschitzian nonlinear mappings in real Banach spaces. Let and be real Banach spaces and a nonempty subset of . A mapping is said to be (see, e.g., [2]) (i)demicontinuous if whenever a sequence converges strongly to it implies that the sequence converges weakly to ;(ii)hemicontinuous if whenever a sequence converges stronly on a line to it implies that the sequence converges weakly to , that is, as Asymptotic fixed point theory which has been studied by so many authors [1, 3–6] has a fundamental role in nonlinear functional analysis concerning existence and construction of fixed points of Lipschitzian mappings, -uniformly Lipschitzian mappings, and non-Lipschitzian mappings among other classes of operators (see, e.g., [5, 7–9]). A very important branch of the theory of asymptotic fixed point relates to the important class of asymptotically nonexpansive mappings which have been studied by various authors in specific types of Banach spaces. Motivated by the need to relax continuity condition inherent in asymptotic nonexpansiveness of asymptotically nonexpansive mappings in certain applications, Sahu [1] considered and introduced the nearly contraction mapping principle into the study of asymptotic fixed point theory concerning nearly Lipschitzian mappings and obtained the following results among others. Lemma 1. Let be a nonempty subset of a Banach space, and let be hemicontinuous. Suppose that as for some Then, is an element of , the set of fixed point of . Theorem 2. Let be a nonempty closed subset of a Banach space and a demicontinuous nearly Lipschitzian mapping with sequence . Suppose .