Point Problems In Banach Spaces Research Articles

Strong Convergence of an Iterative Method for Solving Generalized Mixed Equilibrium Problems and Split Feasibility Problems

In this paper, we investigate iterative methods for solving generalized mixed equilibrium problems, split feasibility problems, and fixed point problems in Banach spaces. We introduce a new extragradient algorithm using the generalized metric projection and prove a strong convergence theorem for finding a common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions to the split feasibility problem, the set of fixed points of a resolvent operator, and the set of solutions of the generalized mixed equilibrium problem. The algorithm is analyzed in a real 2-uniformly convex and uniformly smooth Banach space, taking into account computational errors. A numerical example is provided to illustrate the applicability and performance of the proposed method.

Solving split equality equilibrium and fixed point problems in Banach spaces

In this paper, we introduce a new algorithm for approximating a common solution of Split Equality Generalised Mixed Equilibrium Problem (SEGMEP) and Split Equality Fixed Point Problem (SEFPP) for two infinite families of closed uniformly Li-Lipschitz continuous and Ki-Lipschitz continuous and uniformly quasi-ϕ-asymptotically nonexpansive mappings (i ∈ N and ϕ is the Lyapunov functional) in Banach spaces. Under standard and mild assumption of monotonicity and lower semicontinuity of the SEGMEP associated mappings, we establish the strong convergence of the scheme without imposing any compactness type conditions on either the operators or the spaces considered. We apply our result to approximate the solution of Split Equality Convex Minimization Problem (SECMP) and Split Equality Variational Inclusion Problem (SEVIP). A numerical example is presented to illustrate the performance and implementability of our method. Our results extend, generalize and complement several related works in the literature.

An inertial shrinking projection self-adaptive algorithm for solving split variational inclusion problems and fixed point problems in Banach spaces

Abstract In this article, we study the split variational inclusion and fixed point problems using Bregman weak relatively nonexpansive mappings in the p p -uniformly convex smooth Banach spaces. We introduce an inertial shrinking projection self-adaptive iterative scheme for the problem and prove a strong convergence theorem for the sequences generated by our iterative scheme under some mild conditions in real p p -uniformly convex smooth Banach spaces. The algorithm is designed to select its step size self-adaptively and does not require the prior estimate of the norm of the bounded linear operator. Finally, we provide some numerical examples to illustrate the performance of our proposed scheme and compare it with other methods in the literature.

Cyclic projection methods for solving the split common zero point problem in Banach spaces

Cyclic projection methods for solving the split common zero point problem in Banach spaces

Inertial Method for Solving Pseudomonotone Variational Inequality and Fixed Point Problems in Banach Spaces

In this paper, we introduce a new iterative method that combines the inertial subgradient extragradient method and the modified Mann method for solving the pseudomonotone variational inequality problem and the fixed point of quasi-Bregman nonexpansive mapping in p-uniformly convex and uniformly smooth real Banach spaces. Under some standard assumptions imposed on cost operators, we prove a strong convergence theorem for our proposed method. Finally, we perform numerical experiments to validate the efficiency of our proposed method.

Inertial hybrid gradient method with adaptive step size for variational inequality and fixed point problems of multivalued mappings in Banach spaces

We propose in this article a new inertial hybrid gradient method with self-adaptive step size for approximating a common solution of variational inequality and fixed point problems for an infinite family of relatively nonexpansive multivalued mappings in Banach spaces. Unlike in many existing hybrid gradient methods, here the projection onto the closed convex set is replaced with projection onto some half-space which can easily be implemented. We incorporate into the proposed algorithm inertial term and self-adaptive step size which help to accelerate rate of convergence of iterative schemes. Moreover, we prove a strong convergence theorem without the knowledge of the Lipschitz constant of the monotone operator and we apply our result to find a common solution of constrained convex minimization and fixed point problems in Banach spaces. Finally, we present a numerical example to demonstrate the efficiency of our algorithm in comparison with some recent iterative methods in the literature.

Two new algorithms to solve the system of generalized mixed equilibrium and fixed point problems in Banach spaces

Two new algorithms to solve the system of generalized mixed equilibrium and fixed point problems in Banach spaces

A strongly convergent algorithm for solving multiple set split equality equilibrium and fixed point problems in Banach spaces

AbstractIn this article, using an Halpern extragradient method, we study a new iterative scheme for finding a common element of the set of solutions of multiple set split equality equilibrium problems consisting of pseudomonotone bifunctions and the set of fixed points for two finite families of Bregman quasi-nonexpansive mappings in the framework of p-uniformly convex Banach spaces, which are also uniformly smooth. For this purpose, we design an algorithm so that it does not depend on prior estimates of the Lipschitz-type constants for the pseudomonotone bifunctions. Furthermore, we present an application of our study for finding a common element of the set of solutions of multiple set split equality variational inequality problems and fixed point sets for two finite families of Bregman quasi-nonexpansive mappings. Finally, we conclude with two numerical experiments to support our proposed algorithm.

Mixed Variational-like Inclusion Involving Yosida Approximation Operator in Banach Spaces

This paper deals with the study of mixed variational-like inclusions involving Yosida approximation operator (MVLIYAO) and η-proximal mapping in Banach spaces. It is investigated that (MVLIYAO) is equivalent to fixed point problems in Banach spaces. Using this equivalence, a new iterative algorithm is proposed to find the solution of (MVLIYAO). A numerical example is provided to support our main result by using the MATLAB program.

On split feasibility problem for finite families of equilibrium and fixed point problems in Banach spaces

Abstract In this article, motivated by the works of Ali Akbar and Elahe Shahrosvand [Split equality common null point problem for Bregman quasi-nonexpansive mappings, Filomat 32 (2018), no. 11, 3917–3932], Eskandani et al. [A hybrid extragradient method for solving pseudomonotone equilibrium problem using Bregman distance, J. Fixed Point Theory Appl. 20 (2018), 132], B. Ali and M. H. Harbau [Convergence theorems for Bregman K-mappings and mixed equilibrium problems in reflexive Banach spaces, J. Funct. Spaces (2016) Article ID 5161682, 18 pages], and some other related results in the literature, we introduce a hybrid extragradient iterative algorithm that employs a Bregman distance approach for approximating a split feasibility problem for a finite family of equilibrium problems involving pseudomonotone bifunctions and fixed point problems for a finite family of Bregman quasi-asymptotically nonexpansive mappings using the concept of Bregman K-mapping in reflexive Banach spaces. Using our iterative algorithm, we state and prove a strong convergence result for approximating a common solution to the aforementioned problems. Furthermore, we give an application of our main result to variational inequalities and also report a numerical example to illustrate the convergence of our method. The result presented in this article extends and complements many related results in the literature.

An inertial method for a solution of split equality of monotone inclusion and the f$$ f $$‐fixed point problems in Banach spaces

In this paper, we propose an inertial algorithm for solving split equality of monotone inclusion and ‐fixed point of Bregman relatively ‐nonexpansive mapping problems in reflexive real Banach spaces. Using the Bregman distance function, we prove a strong convergence theorem for the algorithm produced by the method in real reflexive Banach spaces. In addition, we provide some applications of our method and give numerical results to demonstrate the applicability and efficiency of the proposed method.

Inertial iterative method with self-adaptive step size for finite family of split monotone variational inclusion and fixed point problems in Banach spaces

Abstract In this paper, we propose and study a new inertial iterative algorithm with self-adaptive step size for approximating a common solution of finite family of split monotone variational inclusion problems and fixed point problem of a nonexpansive mapping between a Banach space and a Hilbert space. This method combines the inertial technique with viscosity method and self-adaptive step size for solving the common solution problem. We prove a strong convergence result for the proposed method under some mild conditions. Moreover, we apply our result to study the split feasibility problem and split minimization problem. Finally, we provide some numerical experiments to demonstrate the efficiency of our method in comparison with some well-known methods in the literature. Our method does not require prior knowledge or estimate of the operator norm, which makes it easily implementable unlike so many other methods in the literature, which require prior knowledge of the operator norm for their implementation.

Dynamical technique for split common fixed point problem in Banach spaces

Dynamical technique for split common fixed point problem in Banach spaces

A New Self-Adaptive Method for the Multiple-Sets Split Common Null Point Problem in Banach Spaces

A New Self-Adaptive Method for the Multiple-Sets Split Common Null Point Problem in Banach Spaces

A totally relaxed self-adaptive algorithm for solving a variational inequality and fixed point problems in Banach spaces

A totally relaxed self-adaptive algorithm for solving a variational inequality and fixed point problems in Banach spaces

A self-adaptive Tseng extragradient method for solving monotone variational inequality and fixed point problems in Banach spaces

Abstract In this paper, we introduce a self-adaptive projection method for finding a common element in the solution set of variational inequalities (VIs) and fixed point set for relatively nonexpansive mappings in 2-uniformly convex and uniformly smooth real Banach spaces. We prove a strong convergence result for the sequence generated by our algorithm without imposing a Lipschitz condition on the cost operator of the VIs. We also provide some numerical examples to illustrate the performance of the proposed algorithm by comparing with related methods in the literature. This result extends and improves some recent results in the literature in this direction.

On split equality equilibrium, monotone variational inclusion and fixed point problems in Banach spaces

In this paper, we propose and study a new algorithm for finding a common element of the sets of solutions of split equality pseudomonotone equilibrium, split equality monotone variational inclusion and fixed point problems for Bregman relatively nonexpansive mappings in [Formula: see text]-uniformly convex and uniformly smooth Banach spaces. Our iterative method uses stepsizes which do not require prior knowledge of the operator norm. We apply our result to solve split equality variational inequality and split equality convex minimization problems. The result present in this paper unifies and extends several existing results in the literature.

An iterative algorithm with inertial technique for solving the split common null point problem in Banach spaces

In this work, we introduce a modified inertial algorithm for solving the split common null point problem without the prior knowledge of the operator norms in Banach spaces. The strong convergence theorem of our method is proved under suitable assumptions. We apply our result to the split feasibility problem, split equilibrium problem and split minimization problem. Finally, we provide some numerical experiments including compressed sensing to illustrate the performances of the proposed method. The result presented in this paper improves and generalizes many recent important results in the literature.

On Weak Well-posedness of the Nearest Point and Mutually Nearest Point Problems in Banach Spaces

On Weak Well-posedness of the Nearest Point and Mutually Nearest Point Problems in Banach Spaces

Strong convergence theorems for variational inequalities and fixed point problems in Banach spaces

In this paper, using a sunny generalized non-expansive retraction which is different from the metric projection and generalized metric projection in Banach spaces, we present a retractive iterative algorithm of Krasnosel’skii-type, whose sequence approximates a common solution of a mono-variational inequality of a finite family of η-strongly-pseudo-monotone-type maps and fixed points of a countable family of generalized non-expansive-type maps. Furthermore, some new results relevant to the study are also presented. Finally, the theorem proved complements, improves and extends some important related recent results in the literature.