Parallel schemes for solving a system of extended general quasi variational inequalities
Introduction
Variational inequality theory, which was introduced in sixties, has emerged as an interesting and fascinating branch of applicable mathematics with a wide range of applications in industry, finance, economics, optimization, social, regional, pure and applied sciences. This field is dynamic and is experiencing an explosive growth in both theory and applications; as a consequence, research techniques and problems are drawn from various fields. The ideas and techniques of variational inequalities are being applied in a variety of diverse areas of sciences and prove to be productive and innovative. It has been shown that this theory provides the most natural, direct, simple, unified and efficient framework for a general treatment of a wide class of unrelated linear and nonlinear problems, see, for example, [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31] and the references therein.
Among several iterative algorithms, the projection algorithm for finding numerical solutions of variational inequalities is a very useful and quite effective method. Projection method and its variants forms including the Wiener–Hopf equations represent important tools for finding the approximate solution of variational inequalities, the origin of which can be traced back to Lions and Stampacchia [14]. The main idea in this technique is to establish the equivalence between the variational inequalities and the fixed-point problem by using the concept of projection. This alternative formulation has played a significant part in developing various projection-type methods for solving variational inequalities. It is well known that the convergence of the projection methods requires that the operator must be strongly monotone and Lipschitz continuous. Using the projection operator technique, Noor et al. [26], [27] proposed some iterative methods for solving a system of extended general variational inequalities and their variant forms.
In this paper, inspired and motivated by the results of Noor et al. [26], [27], we consider a system of extended general quasi variational inequalities involving six nonlinear operators. Using projection technique, we prove that system of extended general quasi variational inequalities is equivalent to a system of nonlinear implicit projection equations. This alternative equivalent formulation help us to propose some parallel algorithms for solving a system of extended general quasi variational inequalities and its variant forms. We discuss the convergence of these methods under some mild conditions. Several special cases are also discussed. The obtained results unify a number of iterative methods in this direction. The comparison of these methods with other methods is a subject of future research.
Section snippets
Preliminaries and basic results
Let H be a real Hilbert space, whose norm and inner product are denoted by and , respectively. Let be two closed and convex sets in H.
For given nonlinear operators and point-to-set mappings and , which associate closed and convex valued sets and with elements , consider a problem of finding such thatwhere and are
Main results
In this section, we show that the system of extended general quasi variational inequalities (2.1) is equivalent to a system of fixed point problems. This alternative equivalent formulation is used to suggest parallel algorithms for solving (2.1) using the technique of Noor and Noor [24]. Lemma 3.1 The system of extended general quasi variational inequalities (2.1) has a solution, , if and only if satisfies the relationswhere
Conclusion
In this paper, we have introduced and considered a new system of extended general quasi variational inequalities. It has been shown that the new system of extended general quasi variational inequalities is equivalent to a system of fixed point problems. These alternative equivalent formulations have been used to propose and analyze several parallel schemes for solving a system of extended general quasi variational inequalities and their variant forms. Several special cases are also discussed.
Acknowledgement
Authors would like to thank Dr. S. M. Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Islamabad, Pakistan, for providing excellent research facilities. The authors are grateful to the referee and the Associate Editor for their valuable comments and suggestions. This research is supported by HEC Project NRPU No. 20-1966/R&D/11-2553.
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Awais Gul Khan is on study leave from the Department of Mathematics, GC University, Faisalabad, Pakistan.