An itertive algorithm with error terms for solving a system of implicit n-variational inclusions

Zubair Khan,Syed Shakaib Irfan,M Firdosh Khan,P Shukla

doi:10.19139/soic-2310-5070-705

Abstract

A new system of implicit n-variational inclusions is considered. We propose a new algorithm with error terms for computing the approximate solutions of our system. The convergence of the iterative sequences generated by the iterative algorithm is also discussed. Some special cases are also discussed.

Highlights

Variational inclusions plays an important role in the generalization of classical variational inequalities
In 2007, Xia and Huang [29] studied variational inclusions with a general H-monotone operator in Banach spaces, Ahmad et al [3, 5, 7] considered resolvent operator technique to explain a system of generalized variational-like inclusions in Banach spaces, Verma [27] established and considered some new systems of variational inequalities in Hilbert spaces and generate some iterative algorithms for approximating the solutions of this system
By using resolvent operator technique, we propose a n-iterative algorithm with error terms for computing the approximate solutions of a new system of implicit nvariational inclusions

Summary

Historical Perspective and Prelude

Variational inclusions plays an important role in the generalization of classical variational inequalities. A new problem of much more interest which is called as system of variational inequalities (inclusions) were introduced and studied in the literature. As a generalization of some variational inequalities, Huang [16, 17] introduced Mann and Ishikawa type perturbed iterative algorithms for generalized non-linear implicit quasi-variational inclusions. In this paper we study and established a system of n-variational inclusions in real Hilbert spaces. Let H ∶ X → X be a relaxed Lipschitz continuous mapping, I ∶ X → X be an identity mapping and M ∶ X → 2X be a mutivalued, (I − H)- monotone mapping. Let I ∶ X → X be an identity mapping, H ∶ X → X be a r-relaxed Lipschitz continuous mapping and M ∶ X → 2X be a multivalued, (I − H)- monotone mapping.

Formulation of the Problem

Existence and Convergence Analysis

Conclusion