Quasi-nonexpansive Mappings In Banach Spaces Research Articles

Iterative algorithm for approximating fixed points of multivalued quasinonexpansive mappings in Banach spaces

Let E be a strictly convex real Banach space and let Dsubseteq E be a nonempty closed convex subset of E. Let T_{i}: {D}longrightarrow mathcal{P}({D}), i=1,2,3,ldots be a countable family of quasinonexpansive multivalued maps that are continuous with respect to the Hausdorff metric, mathcal{P}(D) is the family of proximinal and bounded subsets of D. Supposing that the family has at least one common fixed point, we show that a Krasnoselskii–Mann-type sequence converges strongly to a common fixed point. Our result generalizes and complements some important results for single-valued and multivalued quasinonexpansive maps.

A self-adaptive method for split common null point problems and fixed point problems for multivalued Bregman quasi-nonexpansive mappings in Banach spaces

In this paper, we propose a self-adaptive algorithm for solving the split common null point problem and the fixed point problem for multivalued Bregman quasi-nonexpansive mappings in Banach spaces. We prove that the sequence generated by our iterative scheme converges strongly to a common solution of the above-mentioned problems under some suitable conditions. We also apply our main result to split feasibility problems in Banach spaces. Finally, numerical examples are given to support our main theorem. The results presented in this paper improve and extend many recent results in the literature.

Viscosity approximation methods for quasi-nonexpansive mappings in Banach spaces

In this article, we present viscosity approximation methods for finding a common point of the set of solutions of a variational inequality problem and the set of fixed points of a multi-valued quasinonexpansive mapping in a Banach space. We also discuss some examples to illustrate facts and study the convergence behaviour of the iterative schemes presented herein, numerically.

Approximation of fixed points for a semigroup of Bregman quasi-nonexpansive mappings in Banach spaces

Approximation of fixed points for a semigroup of Bregman quasi-nonexpansive mappings in Banach spaces

Generalized split feasibility problem for multi-valued Bregman quasi-nonexpansive mappings in Banach spaces

In this paper, the notion of generalized split feasibility problem (GSFP) is studied in p-uniformly convex Banach spaces. Some special cases of the GSFP are highlighted. A self-adaptive step-size iterative algorithm which converges strongly to solution of the GSFP is proved. The implementation of the method is demonstrated with two numerical examples. Our method does not require prior information of operator norms. Our results extend, improve and enrich recently announced related results in the literature.

New iterative schemes for solving variational inequality and fixed points problems involving demicontractive and quasi-nonexpansive mappings in Banach spaces

Abstract In this paper, we suggest and analyze a new iterative method for finding a common element of the set of fixed points of a quasi-nonexpansive mapping and the set of fixed points of a demicontractive mapping which is the unique solution of some variational inequality problems involving accretive operators in a Banach space. We prove the strong convergence of the proposed iterative scheme without imposing any compactness condition on the mapping or the space. Finally, applications of our theorems to some constrained convex minimization problems are given.

A modified Mann iterative scheme based on the generalized explicit methods for quasi-nonexpansive mappings in Banach spaces

In this paper, we introduce and study a new iterative algorithm which is a combination of a modified Mann iterative scheme and a generalized explicit methods (GEM) for finding a common fixed points of an infinite family of quasi-nonexpansive mappings in Banach spaces. Under suitable conditions, some strong convergence theorems for finding a common fixed points of an infinite family of quasi-nonexpansive mappings are obtained without imposing any compactness assumption. Presented results improve and generalize many known results in the current literature.

A modified Krasnosellkii-Mann algorithms for computing fixed points of multivalued quasi-nonexpansive mappings in Banach spaces

It is well known that Krasnoselskii-Mann iteration of nonexpansive mappings find application in many areas of mathematics and know to be weakly convergent in the infinite dimensional setting. In this paper, we introduce and study an explicit iterative scheme by a modified Krasnoselskii-Mann algorithm for approximating fixed points of multivalued quasi-nonexpansive mappings in Banach spaces. Strong convergence of the sequence generated by this algorithm is established. There is no compactness assumption. The results obtained in this paper are significant improvement on important recent results.

The general split equality problem for Bregman quasi-nonexpansive mappings in Banach spaces

The purpose of this paper is to propose and study an algorithm for solving the general split equality problem governed by Bregman quasi-nonexpansive mappings in Banach spaces. Under some mild conditions, we established the norm convergence of the proposed algorithm. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.

Multidirectional hybrid algorithm for the split common fixed point problem and application to the split common null point problem.

In this article, a new multidirectional monotone hybrid iteration algorithm for finding a solution to the split common fixed point problem is presented for two countable families of quasi-nonexpansive mappings in Banach spaces. Strong convergence theorems are proved. The application of the result is to consider the split common null point problem of maximal monotone operators in Banach spaces. Strong convergence theorems for finding a solution of the split common null point problem are derived. This iteration algorithm can accelerate the convergence speed of iterative sequence. The results of this paper improve and extend the recent results of Takahashi and Yao (Fixed Point Theory Appl 2015:87, 2015) and many others .

A Modified Iterative Algorithm for Split Feasibility Problems of Right Bregman Strongly Quasi-Nonexpansive Mappings in Banach Spaces with Applications

In this paper, we present a new iterative scheme for finding a common element of the solution set F of the split feasibility problem and the fixed point set F ( T ) of a right Bregman strongly quasi-nonexpansive mapping T in p-uniformly convex Banach spaces which are also uniformly smooth. We prove strong convergence theorem of the sequences generated by our scheme under some appropriate conditions in real p-uniformly convex and uniformly smooth Banach spaces. Furthermore, we give some examples and applications to illustrate our main results in this paper. Our results extend and improve the recent ones of some others in the literature.

Iterative Scheme for System of Equilibrium Problems and Bregman Asymptotically Quasi-Nonexpansive Mappings in Banach Spaces

In this paper, we presented a new iterative scheme for finding common fixed points of two Bregman asymptotically quasi-nonexpansive mappings together with the solutions of equilibrium problems in Banach spaces. Furthermore, the convergence theorem and its deduced theorems are also proved and presented. Our results extend and improve the recent ones of some others in the literature.

Approximating a common fixed point of finite family of asymptotically quasi-nonexpansive mappings in Banach spaces

The purpose of this paper is to prove convergence of a one-step iterative algorithm to approximate a common fixed point of finite family of asymptotically quasi-nonexpansive mappings in a uniformly convex Banach space by assuming some control conditions on the parameters. Our results extend and improve the corresponding results in the literature.

Weak and Strong Convergence Theorems for Finite Families of Asymptotically Quasi-Nonexpansive Mappings in Banach Spaces

A finite-step iteration sequence for two finite families of asymptotically nonexpansive mappings is introduced and the weak and strong convergence theorems are proved in Banach space. The results presented in the paper generalize and unify some important known results of relevant scholars.

Some convergence theorems for total asymptotically nonexpansive mappings in Banach spaces

Abstract The purpose of this paper is to establish some new approximation theorems of common fixed points for a countable family of total asymptotically quasi-nonexpansive mappings in Banach spaces which generalize and improve the corresponding theorems of Chidume et al. [J. Math. Anal. Appl. 333 (2007), 128–141], [J. Math. Anal. Appl. 326 (2007), 960–973], [Internat. J. Math. Math. Sci. 2009, Article ID 615107, 17 pp.] and others.

Convergence theorems of implicit iterates with errors for generalized asymptotically quasi-nonexpansive mappings in Banach spaces

In this paper, we prove that an implicit iterative process with er-rors converges strongly to a common xed point for a nite family of generalizedasymptotically quasi-nonexpansive mappings on unbounded sets in a uniformlyconvex Banach space. Our results unify, improve and generalize the correspond-ing results of Ud-din and Khan [4], Sun [21], Wittman [23], Xu and Ori [26] andmany others.

Weak convergence theorems for general equilibrium problems and variational inequality problems and fixed point problems in banach spaces

In this paper, we introduce two new iterative algorithms for finding a common element of the set of solutions of a general equilibrium problem and the set of solutions of the variational inequality for an inverse-strongly monotone operator and the set of common fixed points of two infinite families of relatively nonexpansive mappings or the set of common fixed points of an infinite family of relatively quasi-nonexpansive mappings in Banach spaces. Then we study the weak convergence of the two iterative sequences. Our results improve and extend the results announced by many others.

A generalization and improvement of Chidume theorems for total asymptotically nonexpansive mappings in Banach spaces

Abstract The purpose of this article is to establish some new approximation theorems of common fixed points for a countable family of total asymptotically quasi-nonexpansive mappings in Banach spaces which generalize and improve the corresponding theorems of Chidume et al. and others. 2000 AMS Subject Classification: 47J05; 47H09; 49J25.

Approximation of Common Fixed Points for Families of Mappings in Banach Space

In this paper. We introduce a general iterative method for the family of mappings and prove the strong convergence of the new iterative scheme in Banach space. The new iterative method includes the iterative scheme of Khan and Domlo and Fukhar-ud-din [Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 341 (2008) 1–11]. The results generalize the corresponding results.

Strong convergence theorems for fixed point problems and generalized equilibrium problems of three relatively quasi-nonexpansive mappings in Banach spaces

The purpose of this paper is to introduce a new hybrid projection algorithm for finding a commom element of the set of common fixed points of three relatively quasi-nonexpansive mappings and the set of solutions of a generalized equilibrium problem in Banach space. Our results improve and extend the corresponding results announced by many others.