Shamshad Husain

Generalized -Cocoercive Operators and Generalized Set-Valued Variational-Like Inclusions

We investigate a new class of cocoercive operators named generalized -cocoercive operators in Hilbert spaces. We prove that generalized -cocoercive operator is single-valued and Lipschitz continuous and extends the concept of resolvent operators associated with -cocoercive operators to the generalized -cocoercive operators. Some examples are given to justify the definition of generalized -cocoercive operators. Further, we consider a generalized set-valued variational-like inclusion problem involving generalized -cocoercive operator. In terms of the new resolvent operator technique, we give the approximate solution and suggest an iterative algorithm for the generalized set-valued variational-like inclusions. Furthermore, we discuss the convergence criteria of iterative algorithm under some suitable conditions. Our results can be viewed as a generalization of some known results in the literature.

Graph convergence for the H(⋅,⋅)-mixed mapping with an application for solving the system of generalized variational inclusions

In this paper, we investigate a class of accretive mappings called theH(⋅,⋅)-mixed mappingsin Banach spaces. We prove that the proximal-point mapping associated with theH(⋅,⋅)-mixed mapping issingle-valued and Lipschitz continuous. Some examples are given to justify thedefinition of H(⋅,⋅)-mixed mapping.Further, a concept of graph convergence concerned with theH(⋅,⋅)-mixed mapping isintroduced in Banach spaces and some equivalence theorems betweengraph-convergence and proximal-point mapping convergence for theH(⋅,⋅)-mixed mappingssequence are proved. As an application, we consider a system of generalizedvariational inclusions involving H(⋅,⋅)-mixed mappingsin real q-uniformly smooth Banach spaces. Using the proximal-pointmapping method, we prove the existence and uniqueness of solution and suggest aniterative algorithm for the system of generalized variational inclusions.Furthermore, we discuss the convergence criteria for the iterative algorithmunder some suitable conditions.MSC: 47J19, 49J40, 49J53.

Algorithm for Solving a New System of Generalized Variational Inclusions in Hilbert Spaces

We introduce and study a new system of generalized variational inclusions involving -cocoercive and relaxed -cocoercive operators, which contain the systems of variational inclusions and the systems of variational inequalities, variational inclusions, and variational inequalities as special cases. By using the resolvent technique for the -cocoercive operators, we prove the existence of solutions and the convergence of a new iterative algorithm for this system of variational inclusions in Hilbert spaces. An example is given to justify the main result. Our results can be viewed as a generalization of some known results in the literature.

Algorithm for Solving a New System of Generalized Nonlinear Quasi-Variational-Like Inclusions in Hilbert Spaces

We introduce and study a new system of generalized nonlinear quasi-variational-like inclusions with -cocoercive operator in Hilbert spaces. We suggest and analyze a class of iterative algorithms for solving the system of generalized nonlinear quasi-variational-like inclusions. An existence theorem of solutions for the system of generalized nonlinear quasi-variational-like inclusions is proved under suitable assumptions which show that the approximate solutions obtained by proposed algorithms converge to the exact solutions.

Gap functions and error bounds for generalized extended mixed vector equilibrium problem

Equilibrium problems are very important mathematical models and are closely related with fixed point problems, variational inequalities and nash equilibrium problems. Gap functions and error bounds which play a vital role in algorithms design, are two much-addressed topics of vector equilibrium problems. This paper deals with generalized extended mixed vector equilibrium problems. Without any scalarization approach, the gap functions and their regularized versions for generalized extended mixed vector equilibrium problems are first obtained. Then, in the absence of the projection operator method, some error bounds for generalized extended mixed vector equilibrium problems are established in terms of these regularized gap functions. The results obtained in this paper are in more simpler form from the computational point of view.

H(., ., .)-\(\eta \)-Proximal-Point Mapping with an Application

In this article, we introduce the new notion of accretive mapping known as H(., ., .)-\(\eta \)-mixed accretive mapping in q-uniformly smooth Banach spaces. It is the generalization of the H(., .)-accretive mapping, introduced and studied by Zou and Huang [25]. Then, we will introduce the proximal-point mapping related with H(., ., .)-\(\eta \)-mixed accretive mapping and discuss its Lipschitz continuity. We design an iterative algorithm for solving the system of variational inclusions by utilizing the proximal-point method and prove the convergence of iterative sequences generated by the algorithm. Few examples are considered to illustrate the introduced proximal-point mapping.

Mixed Cocoercive Operators with an Application for Solving Variational Inclusions in Hilbert Spaces

We investigate a class of new -mixed cocoercive operators in Hilbert spaces. We extend the concept of resolvent operators associated with -cocoercive operators to the -mixed cocoercive operators and prove that the resolvent operator of -mixed cocoercive operator is single valued and Lipschitz continuous. Some examples are given to justify the definition of -mixed cocoercive operators. Further, by using resolvent operator technique, we discuss the approximate solution and suggest an iterative algorithm for the generalized mixed variational inclusions involving -mixed cocoercive operators in Hilbert spaces. We also discuss the convergence criteria for the iterative algorithm under some suitable conditions. Our results can be viewed as a generalization of some known results in the literature.