General variational inclusions involving difference of operators

Muhammad Aslam Noor,Khalida Inayat Noor,Rabia Kamal

doi:10.1186/1029-242x-2014-98

Muhammad Aslam Noor, Khalida Inayat Noor + Show 1 more

Open Access

https://doi.org/10.1186/1029-242x-2014-98

Copy DOI

Abstract

In this paper, we introduce and consider a new class of general variational inclusions involving the difference of operators in a Hilbert space. We establish the equivalence between the general variational inclusions and the fixed point problems as well as with a new class of resolvent equations using the resolvent operator technique. We use this alternative formulation to discuss the existence of a solution of the general variational inclusions. We again use this alternative equivalent formulation to suggest and analyze a number of iterative methods for finding a zero of the difference of operators. We also discuss the convergence of the iterative method under suitable conditions. Our methods of proofs are very simple as compared with other techniques. Several special cases of these problems are also considered. The results proved in this paper may be viewed as a refinement and an improvement of the known results in this area.

Highlights

Variational inclusions involving the difference of operators provide us with a unified, natural, novel, and simple framework to study a wide class of problems arising in DC programming, prox-regularity, multicommodity network, image restoring processing, tomograpy, molecular biology, optimization, pure and applied sciences, see [ – ] and the references therein
We have shown that the odd-order and nonsymmetric obstacle problems arising in various branches of pure and applied sciences are a special case of the general variational inclusions and can be treated in the unified framework of the general variational inclusions
Motivated and inspired by the research activities going on in this field, we introduce and consider a new class of variational inclusions involving the difference of operators, which is called the general variational inclusion

Summary

Introduction

Variational inclusions involving the difference of operators provide us with a unified, natural, novel, and simple framework to study a wide class of problems arising in DC programming, prox-regularity, multicommodity network, image restoring processing, tomograpy, molecular biology, optimization, pure and applied sciences, see [ – ] and the references therein. We use the resolvent operator technique to establish the equivalence between the general mixed variational inclusions and fixed point problem, which is Lemma . This equivalent formulation is used to suggest and analyze a new Mann-type iterative method for solving the general variational inclusions, see Algorithm . The resolvent equations approach is used to suggest and analyze a number of new iterative methods for solving the general variational inclusions and related optimization problems.

Results

Conclusion