Nonexpansive Mappings In Banach Spaces Research Articles

Generalized Krasnoselskii–Mann-Type Iteration for Nonexpansive Mappings in Banach Spaces

The Krasnoselskii–Mann iteration plays an important role in the approximation of fixed points of nonexpansive mappings, and it is well known that the classic Krasnoselskii–Mann iteration is weakly convergent in Hilbert spaces. The weak convergence is also known even in Banach spaces. Recently, Kanzow and Shehu proposed a generalized Krasnoselskii–Mann-type iteration for nonexpansive mappings and established its convergence in Hilbert spaces. In this paper, we show that the generalized Krasnoselskii–Mann-type iteration proposed by Kanzow and Shehu also converges in Banach spaces. As applications, we proved the weak convergence of generalized proximal point algorithm in the uniformly convex Banach spaces.

Approximating Fixed Points of a General Class of Nonexpansive Mappings in Banach Spaces with Applications

In this paper, we present some fixed point results for a general class of nonexpansive mappings which are not necessarily continuous on their domains. We show that this class properly contains many other classes of nonexpansive type mappings. Some illustrative examples have been presented and various prominent iteration processes have been compared for different choices of parameters and initial guesses. We also present an application of our results to nonlinear integral equations.

A new hybrid projection algorithm for equilibrium problems and asymptotically quasi $$\phi $$ ϕ -nonexpansive mappings in Banach spaces

In this paper, we propose a new hybrid projection algorithm for finding a common element of the solution set of equilibrium problems and the fixed point set of asymptotically quasi $$\phi $$ -nonexpansive mappings in strictly convex and uniformly smooth Banach spaces. A numerical example is given to illustrate the convergence of the proposed hybrid projection algorithms.

Generalized equilibrium and fixed point problems for Bregman relatively nonexpansive mappings in Banach spaces

In this paper, we introduce and study a hybrid iterative method for finding a common solution of a generalized equilibrium problem and a fixed point problem for a Bregman relatively nonexpansive mapping in reflexive Banach spaces. We prove that the sequences generated by the hybrid iterative algorithm converge strongly to a common solution of these problems. Further, we give some consequences of the main result. Finally, we discuss a numerical example to demonstrate the applicability of the iterative algorithm.

A strong convergence theorem for solving the split feasibility and fixed point problems in Banach spaces

In this paper, we introduce a new parallel iterative method for finding a common solution of the multiple-set split feasibility and fixed point problems concerning left Bregman strongly nonexpansive mappings in Banach spaces.

Strong convergence theorems for equilibrium problems and fixed point problems in Banach spaces

The paper proposes two new iterative methods for solving pseudomonotone equilibrium problems involving fixed point problems for quasi- $$\phi $$ -nonexpansive mappings in Banach spaces. The proposed algorithms combine the extended extragradient method or the linesearch method with the Halpern iteration. The strong convergence theorems are established under standard assumptions imposed on equilibrium bifunctions and mappings. The results in this paper have generalized and enriched existing algorithms for equilibrium problems in Banach spaces.

New hybrid method for equilibrium problems and relatively nonexpansive mappings in Banach spaces

In this paper, applying hybrid projection method, a new modified Ishikawa iteration scheme is presented for finding a common element of the solution set of an equilibrium problem and the set of fixed points of relatively nonexpansive mappings in Banach spaces. A numerical example is given and the numerical behaviour of the sequences generated by this algorithm is compared with several existence results in literature to illustrate the usability of obtained results.

Approximating endpoints of multi-valued nonexpansive mappings in Banach spaces

The purpose of this paper is to introduce a modified Ishikawa iteration for multi-valued mappings and show that the proposed iteration pattern will converge to an endpoint of a certain multi-valued nonexpansive mapping.

Iteration process for solving a fixed point problem of nonexpansive mappings in Banach spaces

In this work, we introduce a new iteration process for solving the fixed point problem of a finite family of nonexpansive mappings. We then prove, in Banach spaces, the strong convergence theorem under some mild conditions. Finally, we give some numerical results to support our main result.

A new iterative method for multivalued nonexpansive mappings in Banach spaces with application

A new iterative method for multivalued nonexpansive mappings in Banach spaces with application

Strong convergence theorem of split equality fixed point for nonexpansive mappings in Banach spaces

Strong convergence theorem of split equality fixed point for nonexpansive mappings in Banach spaces

Some sufficient conditions for fixed points of multivalued nonexpansive mappings in Banach spaces

Some sufficient conditions for fixed points of multivalued nonexpansive mappings in Banach spaces

Modified general iterative algorithm for an infinite family of nonexpansive mappings in Banach spaces

In this paper we introduce an iterative method for finding a common fixed point of an infinite family of nonexpansive mappings in q-uniformly real smooth Banach space which is also uniformly convex. We proved strong convergence of the proposed iterative algorithms to the unique solution of a variational inequality problem.

Hybrid method for the equilibrium problem and a family of generalized nonexpansive mappings in Banach spaces

Hybrid method for the equilibrium problem and a family of generalized nonexpansive mappings in Banach spaces

Hybrid iterative algorithm for an infinite families of closed, uniformly asymptotic regular and uniformly Bregman totally quasi-D-asymptotically nonexpansive mappings in Banach spaces

Hybrid iterative algorithm for an infinite families of closed, uniformly asymptotic regular and uniformly Bregman totally quasi-D-asymptotically nonexpansive mappings in Banach spaces

Iterative algorithms for solving variational inequalities and fixed point problems for asymptotically nonexpansive mappings in Banach spaces

The purpose of this paper is to study some iterative algorithms for finding a common element of the set of solutions of systems of variational inequalities for inverse-strongly accretive mappings and the set of fixed points of an asymptotically nonexpansive mapping in uniformly convex and 2-uniformly smooth Banach space or uniformly convex and q-uniformly smooth Banach space. Strong convergence theorems are obtained under suitable conditions. We also give some numerical examples to support our main results. The results obtained in this paper improve and extend the recent ones announced by many others in the literature.

A strong convergence theorem for finding a common fixed point of a finite family of Bregman nonexpansive mappings in Banach spaces which solves a generalized mixed equilibrium problem

In this paper, we study an iterative method for finding a common fixed point of a finite family of Bregman strongly nonexpansive mappings in real reflexive Banach spaces. Moreover, we prove a strong convergence theorem for finding common fixed points which solve a generalized mixed equilibrium problem. As a special case, we solve mixed variational inequality problems.

On fixed points of fundamentally nonexpansive mappings in Banach spaces

We first obtain some properties of a fundamentally nonexpansive self-mapping on a nonempty subset of a Banach space and next show that if the Banach space is having the Opial condition, then the fixed points set of such a mapping with the convex range is nonempty. In particular, we establish that if the Banach space is uniformly convex, and the range of such a mapping is bounded, closed and convex, then its the fixed points set is nonempty, closed and convex.

PROXIMAL POINT ALGORITHM FOR A COMMON OF COUNTABLE FAMILIES OF INVERSE STRONGLY ACCRETIVE OPERATORS AND NONEXPANSIVE MAPPINGS WITH CONVERGENCE ANALYSIS

In this paper, we investigate and analyze a proximal point algorithm via viscosity approximation method with error. This algorithm is introduced for finding a common zero point for a countable family of inverse strongly accretive operators and a countable family of nonexpansive mappings in Banach spaces. Our result can be extended to some well known results from a Hilbert space to a uniformly convex and 2−uniformly smooth Banach space. Finally, we establish the strong convergence theorems for the proximal point algorithm. Also, some illustrative numerical examples are presented.

Properties of Fixed Point Sets of Monotone Nonexpansive Mappings in Banach Spaces

ABSTRACTIn this work, we discuss the properties of the common fixed points set of a commuting family of monotone nonexpansive mappings. In particular, we show that under suitable assumptions, this set is a monotone nonexpansive retract.