New Trends in General Variational Inequalities
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New Trends in General Variational Inequalities Muhammad Aslam Noor1 · Khalida Inayat Noor1 · Michael Th. Rassias2,3,4
Received: 13 June 2020 / Accepted: 24 September 2020 / Published online: 6 October 2020 © The Author(s) 2020
Abstract It is well known that general variational inequalities provide us with a unified, natural, novel and simple framework to study a wide class of unrelated problems, which arise in pure and applied sciences. In this paper, we present a number of new and known numerical techniques for solving general variational inequalities and equilibrium problems using various techniques including projection, Wiener-Hopf equations, dynamical systems, the auxiliary principle and the penalty function. General variational-like inequalities are introduced and investigated. Properties of higher order strongly general convex functions have been discussed. The auxiliary principle technique is used to suggest and analyze some iterative methods for solving higher order general variational inequalities. Some new classes of strongly exponentially general convex functions are introduced and discussed. Our proofs of convergence are very simple as compared with other methods. Our results present a significant improvement of previously known methods for solving variational inequalities and related optimization problems. Since the general variational inequalities include (quasi) variational inequalities and (quasi) implicit complementarity problems as special cases, these results continue to hold for these problems. Some numerical results are included to illustrate the efficiency of the proposed methods. Several open problems have been suggested for further research in these areas.
B M.Th. Rassias
[email protected] M.A. Noor [email protected] K.I. Noor [email protected]
1
COMSATS University Islamabad, Park Road, Islamabad, Pakistan
2
Institute of Mathematics, University of Zurich, 8057, Zurich, Switzerland
3
Moscow Institute of Physics and Technology, Institutskiy per, d. 9, 141700 Dolgoprudny, Russia
4
Program in Interdisciplinary Studies, Institute for Advanced Study, 1 Einstein Dr, Princeton, NJ 08540, USA
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M.A. Noor et al.
Keywords Variational inequalities · Wiener-Hopf equations · Dynamical systems · Equilibrium problems
1 Introduction Variational inequalities theory, which was introduced by Stampacchia [173] and Ficchera [38] independently, has emerged as an interesting and fascinating branch of applied mathematics with a wide range of applications in industry, finance, economics, social, pure and applied sciences. Variational inequalities may be viewed as novel generalization of the variational principles, the origin of which can be traced back to Euler, Lagrange and the Bernoulli brothers. Variational principles have played a crucial and important role in the development of various fields of sciences and have appeared as a unifying force. The ideas and techniques of variational inequalities are being applied in a variety of diverse areas of sciences and prove to be productive a
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