Eskandar Naraghirad

Existence of common fixed points using Bregman nonexpansive retracts and Bregman functions in Banach spaces

In this paper, we first introduce the concepts of Bregman nonexpansive retract and Bregman one-local retract and then use these concepts to establish the existence of common fixed points for Banach operator pairs in the framework of reflexive Banach spaces. No compactness assumption is imposed either on C or on T, where C is a closed and convex subset of a reflexive Banach space E and T:C→C is a Bregman nonexpansive mapping. We also establish the well-known De Marr theorem for a Banach operator family of Bregman nonexpansive mappings.MSC: Primary 06F30; 46B20, 47E10.

Approximating fixed points of α-nonexpansive mappings in uniformly convex Banach spaces and CAT(0) spaces

An existence theorem for a fixed point of an α-nonexpansive mapping of a nonempty bounded, closed and convex subset of a uniformly convex Banach space has been recently established by Aoyama and Kohsaka with a non-constructive argument. In this paper, we show that appropriate Ishikawa iterate algorithms ensure weak and strong convergence to a fixed point of such a mapping. Our theorems are also extended to CAT(0) spaces.AMS Subject Classification:54E40, 54H25, 47H10, 37C25.

Iterative Approaches to Find Zeros of Maximal Monotone Operators by Hybrid Approximate Proximal Point Methods

The purpose of this paper is to introduce and investigate two kinds of iterative algorithms for the problem of finding zeros of maximal monotone operators. Weak and strong convergence theorems are established in a real Hilbert space. As applications, we consider a problem of finding a minimizer of a convex function.

Strong convergence theorems for generalized nonexpansive mappings on star-shaped set with applications

AbstractIn this paper, we study attractive points for a class of generalized nonexpansive mappings on star-shaped sets and establish strong convergence theorems of the Halpern iterative sequences generated by these mappings in a real Hilbert space. We modify Halpern’s iterations for finding an attractive point of a mapping T satisfying condition (E) on a star-shaped set C in a real Hilbert space H and provide an affirmative answer to an open problem posed by Akashi and Takahashi in a recent work of (Appl. Math. Comput. 219(4):2035-2040, 2012) for nonexpansive and nonspreading mappings. Furthermore, we study the approximation of common attractive points of generalized nonexpansive mappings and derive a strong convergence theorem by a new iteration scheme for these mappings. As applications of our results, we study multiple sets split monotone inclusion problems for inverse strongly monotone mappings, multiple sets split optimization problems, and multiple sets split feasibility problems. Our results contain many original results on multiple sets split feasibility problem in the literature. Our results also improve and generalize many well-known results in the current literature. MSC:47H10, 37C25.

Bregman

Using Bregman functions, we introduce the new concept of Bregman generalized -projection operator , where is a reflexive Banach space with dual space is a proper, convex, lower semicontinuous and bounded from below function; is a strictly convex and Gâteaux differentiable function; and is a nonempty, closed, and convex subset of . The existence of a solution for a class of variational inequalities in Banach spaces is presented.

f

The Opial property of Hilbert spaces and some other special Banach spaces is a powerful tool in establishing fixed point theorems for nonexpansive and, more generally, nonspreading mappings. Unfortunately, not every Banach space shares the Opial property. However, every Banach space has a similar Bregman-Opial property for Bregman distances. In this paper, using Bregman distances, we introduce the classes of Bregman nonspreading mappings and investigate the Mann and Ishikawa iterations for these mappings. We establish weak and strong convergence theorems for Bregman nonspreading mappings.

-Projection Operator with Applications to Variational Inequalities in Banach Spaces

In this paper, we first introduce a new Halpern-type iterative scheme to approximate common fixed points of an infinite family of quasi-nonexpansive mappings and obtain a strongly convergent iterative sequence to the common fixed points of these mappings in a uniformly convex Banach space. We then apply our method to approximate zeros of an infinite family of accretive operators and derive a strong convergence theorem for these operators. It is important to state clearly that the contribution of this paper in relation with the previous works (see, for example, Yao et al. (Nonlinear Anal. 70:2332-2336, 2009)) is a technical method to establish a strong convergence theorem of Halpern type for a wide class of quasi-nonexpansive mappings. The method provides a positive answer to an old problem in fixed point theory and applications. Our results improve and generalize many known results in the current literature.MSC:47H10, 37C25.

Applications of Bregman-Opial Property to Bregman Nonspreading Mappings in Banach Spaces

We introduce and study a new system of generalized cocoercive operator inclusions in Banach spaces. Using the resolvent operator technique associated with cocoercive operators, we suggest and analyze a new generalized algorithm of nonlinear set-valued variational inclusions and establish strong convergence of iterative sequences produced by the method. We highlight the applicability of our results by examples in function spaces.