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

Shamshad Husain,Vishnu Narayan Mishra,Sanjeev Gupta

doi:10.1155/2013/738491

Abstract

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.

Highlights

Variational inclusions, as the generalization of variational inequalities, have been widely studied in recent years
The resolvent operator technique for the study of variational inclusions has been widely used by many authors
Using new a resolvent operator technique, we prove the existence of solutions and suggest an iterative algorithm for the generalized set-valued variational-like inclusions

Summary

Introduction

Variational inclusions, as the generalization of variational inequalities, have been widely studied in recent years. ⟨P (x) − P (y) , η (x, y)⟩ ≥ (−γ1) 󵄩󵄩󵄩󵄩P (x) − P (y)󵄩󵄩󵄩󵄩 2, (5) ∀x, y ∈ X, (v) λP-Lipschitz continuous, if there exists a constant λP > 0 such that

Results

Conclusion