and Applied Analysis 3 Let X be a smooth Banach space. We always use φ : X × X → R to denote the Lyapunov functional defined by φ ( x, y ) ‖x‖ − 2〈x, Jy〉 ∥∥y∥∥2, ∀x, y ∈ X. 1.7 It is obvious from the definition of the function φ that (‖x‖ − ∥∥y∥∥)2 ≤ φ(x, y) ≤ (‖x‖ ∥∥y∥∥)2, ∀x, y ∈ X. 1.8 Following Alber 4 , the generalized projection ΠC : X → C is defined by ΠC x arg inf y∈C φ ( y, x ) , ∀x ∈ ...

The equilibrium problem and its generalizations had a great influence in the development of some branches pure applied sciences. problems theory provides natural novel approach for arising nonlinear analysis, physics engineering, image reconstruction, economics, finance, game optimization. In recent times, there were many methods order to solve generalizations. Some authors proposed iterative s...

In this paper, strong convergence theorems of Ishikawa type implicit iteration process with errors for a finite family of asymptotically nonexpansive in the intermediate sense and asymptotically quasi-pseudocontractive type mappings in normed linear spaces are established by using a new analytical method, which essentially improve and extend some recent results obtained by Yang [Convergence the...

The purpose of this paper is to introduce a new hybrid projection method based on modified Mann iterative scheme by the generalized f-projection operator for a countable family of relatively quasi-nonexpansive mappings and the solutions of the system of generalized mixed equilibrium problems. Furthermore, we prove the strong convergence theorem for a countable family of relatively quasi-nonexpa...

In this paper‎, ‎strong convergence theorems of Ishikawa type implicit iteration‎ ‎process with errors for a finite family of asymptotically‎ ‎nonexpansive in the intermediate sense and asymptotically‎ ‎quasi-pseudocontractive type mappings in normed linear spaces are‎ ‎established by using a new analytical method‎, ‎which essentially‎ ‎improve and extend some recent results obtained by Yang‎ ‎...

and Applied Analysis 3 where {an}, {bn}, {αn} are appropriate sequences in 0, 1 . The theory of three-step iterative scheme is very rich, and this scheme, in the context of one or more mappings, has been extensively studied e.g., see Khan et al. 6 , Plubtieng and Wangkeeree 7 , Fukhar-ud-din and Khan 5 , Petrot 13 , and Suantai 14 . It has been shown in 15 that three-step method performs better...

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) ...