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 >...

and Applied Analysis 3 Moreover, PC is characterized by the following properties: 〈 x − PCx, y − PCx 〉 ≤ 0, ∥ ∥x − y∥∥2 ≥ ‖x − PCx‖ ∥ ∥y − PCx ∥ ∥ 2 , 2.4 for all x ∈ H and y ∈ C. We also need other sorts of nonlinear operators which are introduced below. Let T : H → H be a nonlinear operator. a T is nonexpansive if ‖Tx − Ty‖ ≤ ‖x − y‖ for all x, y ∈ H. b T is firmly nonexpansive if 2T − I is n...

The property of nonexpansivity (1-Lipschitz) is very important in the analysis of many optimization problems. In this paper we study a more general notion of nonexpansivity – Bregman nonexpansivity. We present a characterization of Bregman firmly nonexpansive operators in general reflexive Banach spaces. This characterization allows us to construct Bregman firmly nonexpansive operators explicit...

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.

In this paper, we prove some strong convergence approximation theorems of strongly relatively nonexpansive semi-group in the framework of Banach spaces. Using the concept of duality theorems, we obtain analogue results for strongly generalized nonexpansive semi-group and for semi-group of firmly generalized nonexpansive type. The results presented in this paper improve and extend some correspon...

In this paper, we prove the existence of a random fixed point of by using pointwise asymptotically nonexpansive random operator and the stability resultsof two iterative schemes for random operator.

Manuscript received January 01, 2014; revised March 28, 2014. This work was supported in part by King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. Cyril Dennis Enyi is with King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. (corresponding author: +966550782390; e-mail: cenyi@ kfupm.edu.sa). Mukiawa Edwin Soh is with King Fahd University of Petroleum and ...

Recently, S. Reich and S. Simons provided a novel proof of the Kirszbraun-Valentine extension theorem using Fenchel duality and Fitzpatrick functions. In the same spirit, we provide a new proof of an extension result for firmly nonexpansive mappings with an optimally localized range. Throughout this paper, we assume that X is a real Hilbert space, with inner product p = 〈· | ·〉 and induced norm...