Cocoercive Mappings Research Articles

A simple proof for Imnang’s algorithms

In this paper, a simple proof of the convergence of the recent iterative algorithm by relaxed (u, v)-cocoercive mappings due to Imnang (J. Inequal. Appl. 2013:249, 2013) is presented.

Algorithmic and analytical approach of solutions of a system of generalized multi-valued nonlinear variational inclusions

The main contributions of this paper is twofold. First, our primary concern is to suggest a new iterative algorithm using the P-?-proximal-point mapping technique and Nadler?s technique for finding the approximate solutions of a system of generalized multi-valued nonlinear variational-like inclusions. Under some appropriate conditions imposed on the parameters and mappings involved in the system of generalized multi-valued nonlinear variational-like inclusions, the strong convergence of the sequences generated by our proposed iterative algorithm to a solution of the aforesaid system is proved. Second, the H(.,.)-?-cocoercive mapping considered in [R. Ahmad, M. Dilshad, M. Akram, Resolvent operator technique for solving a system of generalized variational-like inclusions in Banach sapces, Filomat 26(5)(2012) 897- 908] is investigated and analyzed, and the fact that under the assumptions imposed on H(., .)-?-cocoercive mapping, every H(.,.)-?-cocoercive mapping is P-?-accretive and is not a new one is pointed out. At the same time, some important comments on H(.,.)-?-cocoercive mapping and the results given in the above-mentioned paper are stated.

A Simple proof for the algorithms of relaxed $(u, v)$-cocoercive mappings and $alpha$-inverse strongly monotone mappings

In this paper, a simple proof is presented for the Convergence of the algorithms for the class of relaxed $(u, v)$-cocoercive mappings and $alpha$-inverse strongly monotone mappings. Based on $alpha$-expansive maps, for example, a simple proof of the convergence of the recent iterative algorithms by relaxed $(u, v)$-cocoercive mappings due to Kumam-Jaiboon is provided. Also a simple proof for the convergence of the iterative algorithms by inverse-strongly monotone mappings due to Iiduka-Takahashi in a special case is provided. These results are an improvement as well as a refinement of previously known results.

Generalized monotone mapping and resolvent equation technique with an application

The objective of this paper is to study generalized monotone mapping, which is the addition of cocoercive mapping and monotone mapping. First resolvent operator is obtained and discussion of its few properties. Then we give the resolvent equation associated with the resolvent operator and find a solution to a variational-like inclusion problem.

Convergence Analysis of a Three-Step Iterative Algorithm for Generalized Set-Valued Mixed-Ordered Variational Inclusion Problem

This manuscript aims to study a generalized, set-valued, mixed-ordered, variational inclusion problem involving H(·,·)-compression XOR-αM-non-ordinary difference mapping and relaxed cocoercive mapping in real-ordered Hilbert spaces. The resolvent operator associated with H(·,·)-compression XOR-αM-non-ordinary difference mapping is defined, and some of its characteristics are discussed. We prove existence and uniqueness results for the considered generalized, set-valued, mixed-ordered, variational inclusion problem. Further, we put forward a three-step iterative algorithm using a ⊕ operator, and analyze the convergence of the suggested iterative algorithm under some mild assumptions. Finally, we reconfirm the existence and convergence results by an illustrative numerical example.

Convergence of Some Iterative Algorithms for System of Generalized Set-Valued Variational Inequalities

In this article, we consider and study a system of generalized set-valued variational inequalities involving relaxed cocoercive mappings in Hilbert spaces. Using the projection method and Banach contraction principle, we prove the existence of a solution for the considered problem. Further, we propose an iterative algorithm and discuss its convergence. Moreover, we establish equivalence between the system of variational inequalities and altering points problem. Some parallel iterative algorithms are proposed, and the strong convergence of the sequences generated by these iterative algorithms is discussed. Finally, a numerical example is constructed to illustrate the convergence analysis of the proposed parallel iterative algorithms.

Strong and weak convergence theorems for general mixed equilibrium problems and general variational inequality problems and fixed point problems for two nonexpansive semigroups in Hilbert spaces

In this paper, we introduce some iterative algorithms for finding a common element of the set of solutions of the general mixed equilibrium problem and the set of solutions of a general variational inequality for two cocoercive mappings and the set of common fixed points of two nonexpansive semigroups in Hilbert space. We obtain both strong and weak convergence theorems for the sequences generated by these iterative processes in Hilbert spaces. Our results improve and extend the results announced by many others.

Generalized monotone mapping and resolvent equation technique with an application

The objective of this paper is to study generalized monotone mapping, which is the addition of cocoercive mapping and monotone mapping. First resolvent operator is obtained and discussion of its few properties. Then we give the resolvent equation associated with the resolvent operator and find a solution to a variational-like inclusion problem.

General Iterative Scheme for Split Mixed Equilibrium Problems, Variational Inequality Problems and Fixed Point Problems in Hilbert Spaces

The purpose in this paper is to study the strong convergence of general iterative scheme to find a common element of the set of a finite family of nonexpansive mappings, the set of solutions of variational inequalities for a relaxed cocoercive mapping and the set of solutions of split mixed equilibrium problem. Our results extend recent results announced by many others.

Extensions of Saeidi's Propositions for Finding a Unique Solution of a Variational Inequality for $(u,v)$-cocoercive Mappings in Banach Spaces

Let $C$ be a nonempty closed convex subset of a real Banach space $E$, let $B: C rightarrow E $ be a nonlinear map, and let $u, v$ be positive numbers. In this paper, we show that the generalized variational inequality $V I (C, B)$ is singleton for $(u, v)$-cocoercive mappings under appropriate assumptions on Banach spaces. The main results are extensions of the Saeidi's Propositions for finding a unique solution of the variational inequality for $(u, v)$-cocoercive mappings in Banach spaces.

System of Extended General Variational Inequalities for Relaxed Cocoercive Mappings in Hilbert Space

In this manuscript, we study a system of extended general variational inequalities (SEGVI) with several nonlinear operators, more precisely, six relaxed ( α , r ) -cocoercive mappings. Using the projection method, we show that a system of extended general variational inequalities is equivalent to the nonlinear projection equations. This alternative equivalent problem is used to consider the existence and convergence (or approximate solvability) of a solution of a system of extended general variational inequalities under suitable conditions.

A Neurodynamic Model to Solve Nonlinear Pseudo-Monotone Projection Equation and Its Applications.

In this paper, a neurodynamic model is given to solve nonlinear pseudo-monotone projection equation. Under pseudo-monotonicity condition and Lipschitz continuous condition, the projection neurodynamic model is proved to be stable in the sense of Lyapunov, globally convergent, globally asymptotically stable, and globally exponentially stable. Also, we show that, our new neurodynamic model is effective to solve the nonconvex optimization problems. Moreover, since monotonicity is a special case of pseudo-monotonicity and also since a co-coercive mapping is Lipschitz continuous and monotone, and a strongly pseudo-monotone mapping is pseudo-monotone, the neurodynamic model can be applied to solve a broader classes of constrained optimization problems related to variational inequalities, pseudo-convex optimization problem, linear and nonlinear complementarity problems, and linear and convex quadratic programming problems. Finally, several illustrative examples are stated to demonstrate the effectiveness and efficiency of our new neurodynamic model.

Fixed point problems and a system of generalized nonlinear mixed variational inequalities

In this paper, we introduce and consider a new system of generalized nonlinear mixed variational inequalities involving six different nonlinear operators and discuss the existence and uniqueness of solution of the aforesaid system. We use three nearly uniformly Lipschitzian mappings ( ) to suggest and analyze some new three-step resolvent iterative algorithms with mixed errors for finding an element of the set of fixed points of the nearly uniformly Lipschitzian mapping , which is the unique solution of the system of generalized nonlinear mixed variational inequalities. The convergence analysis of the suggested iterative algorithms under suitable conditions is studied. In the final section, an important remark on a class of some relaxed cocoercive mappings is discussed. MSC:47H05, 47J20, 49J40, 90C33.

Viscosity iterative method for a new general system of variational inequalities in Banach spaces

In this paper, we study a new iterative method for finding a common element of the set of solutions of a new general system of variational inequalities for two different relaxed cocoercive mappings and the set of fixed points of a nonexpansive mapping in real 2-uniformly smooth and uniformly convex Banach spaces. We prove the strong convergence of the proposed iterative method without the condition of weakly sequentially continuous duality mapping. Our result improves and extends the corresponding results announced by many others. MSC: 46B10; 46B20; 47H10; 49J40

GENERAL NONLINEAR RANDOM SET-VALUED VARIATIONAL INCLUSION PROBLEMS WITH RANDOM FUZZY MAPPINGS IN BANACH SPACES

This paper is dedicated to study a new class of general nonlinear random A-maximal <TEX>$m$</TEX>-relaxed <TEX>${\eta}$</TEX>-accretive (so called (A, <TEX>${\eta}$</TEX>)-accretive [49]) equations with random relaxed cocoercive mappings and random fuzzy mappings in <TEX>$q$</TEX>-uniformly smooth Banach spaces. By utilizing the resolvent operator technique for A-maximal <TEX>$m$</TEX>-relaxed <TEX>${\eta}$</TEX>-accretive mappings due to Lan et al. and Chang's lemma [13], some new iterative algorithms with mixed errors for finding the approximate solutions of the aforesaid class of nonlinear random equations are constructed. The convergence analysis of the proposed iterative algorithms under some suitable conditions are also studied.

Viscosity approximation methods based on generalized contraction mappings for a countable family of strict pseudo-contractions, a general system of variational inequalities and a generalized mixed equilibrium problem in Banach spaces

In this paper, we introduce viscosity approximation methods based on generalized contraction mappings for finding a set of common fixed points of a countable family of strict pseudo-contraction mappings, the common element of the set solutions of a general system of variational inequalities with Lipschitzian and relaxed cocoercive mappings and the set of solutions of a generalized mixed equilibrium problem. Furthermore, strong convergence theorems of the purposed iterative process are established in the framework of Banach spaces. The results presented in this paper improve and extend many recent important results.

Strong Convergence of a General Iterative Algorithm for Mixed Equilibrium, Variational Inequality and Common Fixed Points Problems

The aim of this paper, is to introduce and study a general iterative algorithm concerning the new mappings which the sequences generated by our proposed scheme converge strongly to a common element of the set of solutions of a mixed equilibrium problem, the set of common fixed points of a finite family of nonexpansive mappings and the set of solutions of the variational inequality for a relaxed cocoercive mapping in a real Hilbert space. In addition, we obtain some applications by using this result. The results obtained in this paper generalize and refine some known results in the current literature.

A Viscosity Hybrid Steepest Descent Method for Equilibrium Problems, Variational Inequality Problems, and Fixed Point Problems of Infinite Family of Strictly Pseudocontractive Mappings and Nonexpansive Semigroup

In this paper, modifying the set of variational inequality and extending the nonexpansive mapping of hybrid steepest descent method to nonexpansive semigroups, we introduce a new iterative scheme by using the viscosity hybrid steepest descent method for finding a common element of the set of solutions of a system of equilibrium problems, the set of fixed points of an infinite family of strictly pseudocontractive mappings, the set of solutions of fixed points for nonexpansive semigroups, and the sets of solutions of variational inequality problems with relaxed cocoercive mapping in a real Hilbert space. We prove that the sequence converges strongly to a common element of the above sets under some mild conditions. The results shown in this paper improve and extend the recent ones announced by many others.

A viscosity scheme for mixed equilibrium problems, variational inequality problems and fixed point problems

In this paper, we study a new general iterative scheme for finding a common element of the set of solutions of finite general mixed equilibrium problems, the set of solutions of finite variational inequalities for cocoercive mappings, the set of solutions of common fixed points of an infinite family of nonexpansive mappings and the set of solutions of fixed points of a nonexpansive semigroup in Hilbert space. Strong convergence theorems are obtained under appropriate conditions.

Generalized Monotone Mapping with an Application for Solving a Variational Inclusion Problem

The aim of this paper is to define a generalized monotone mapping, which is the sum of symmetric cocoercive mapping and symmetric monotone mapping. The resolvent operator associated with generalized monotone mapping is defined and some of its properties are shown. We solve a variational inclusion problem using these new concepts. For illustration, some examples are given.