Four recent papers 1–4 present simple and elegant proofs of fixed point results for pointwise contractions, asymptotic pointwise contractions, and asymptotic nonexpansive mappings. Kirk and Xu 1 study these mappings in the context of weakly compact convex subsets of Banach spaces, respectively, in uniformly convex Banach spaces. Hussain and Khamsi 2 consider these problems in the framework of m...

This paper is concerned with an ergodic theorem for asymptotically nonexpansive mappings in the intermediate sense in Banach spaces.

In this paper, we introduce a class of total quasi-φ-asymptotically nonexpansive nonself mappings which contain several kinds of mappings as its special cases, and obtain some strong convergence theorems for this type of mappings in Banach spaces under some mild control conditions. The results presented in this paper improve and extend some recent corresponding results.

In this paper, we consider an iteration process for approximating common fixed points of two asymptotically quasinonexpansive mappings and we prove some strong and weak convergence theorems for such mappings in uniformly convex Banach spaces. Keywords—Asypmtotically quasi-nonexpansive mappings, Common fixed point, Strong and weak convergence, Iteration process.

Minimal invariant sets for nonexpansive mappings share some singular geometrical properties. Here we present some seemingly unknown ones.

Recently, Reich and Zaslavski have studied a new inexact iterative scheme for fixed points of contractive and nonexpansive multifunctions. In 2011, Aleomraninejad, et. al. generalized some of their results to Suzuki-type multifunctions.  The study of iterative schemes for various classes of contractive and nonexpansive mappings is a central topic in fixed point theory. The importance of Banach ...

and Applied Analysis 3 Lemma 2.2 cf., 4 . Let D be a nonempty subset of a reflexive, strictly convex, and smooth Banach space E. Let R be a retraction from E onto D. Then R is sunny and generalized nonexpansive if and only if 〈 x − Rx, JRx − Jy ≥ 0, 2.2 for all x ∈ E and y ∈ D. A generalized resolvent Jr of a maximal monotone operator B ⊂ E∗ × E is defined by Jr I rBJ −1 for any real number r >...

Let K be a convex subset of a uniformly convex Banach space. It is proved that if K is compact, then the fixed points of a continuous generalized nonexpansive self-mapping T on K can be approximated by the iterates of T, with t B (0,1), T,(x) = (1 t)x + tT(x), x e K; T, is asymptotically regular if T has a fixed point. Let (X, d) be a (nonempty) metric space. A function a of X X X into [0, oo) ...

In 2006, Espinola and Kirk made a useful contribution on combining fixed point theoryand graph theory. Recently, Reich and Zaslavski studied a new inexact iterative scheme for fixed points of contractive and nonexpansive multifunctions. In this paper, by using  the main idea of their work and the idea of combining fixed point theory on intuitionistic fuzzy metric spaces and graph theory, ...

In this paper, we study the strong convergence of the Halpern type algorithms for a strongly quasi-nonexpansive sequence of operators. These results extend the results of Saejung [11]. Some applications in infinite family of firmly quasi-nonexpansive mappings, multiparameter proximal point algorithm, constraint minimization and subgradient projection are presented.