Equivalence of Variational Inequalities with Wiener-Hopf Equations

Abstract

Abstract.We show that a variational inequality is equivalent to a generalized Wiener-Hopf equation in the sense that, if one of them has a solution so does the other one. Moreover, their solutions can be transformed to each other by a simple formula. Applications are considered.

Solvability of generalized nonlinear symmetric variational inequalities

“…where P K is the metric projection onto K. This is the original form of the Wiener-Hopf condition, as introduced by [16] in connection with the variational inequality (1.6). See also [1,11,13].…”

Wiener-Hopf Equations Technique for General Variational Inequalities Involving Relaxed Monotone Mappings and Nonexpansive Mappings

“…In this paper, using essentially the projection technique, we show that the general variational inequalities are equivalent to the general Wiener-Hopf equations, whose origin can be traced back to Shi [8]. It has been shown [4,[8][9][10] that the Wiener-Hopf equations are more flexible and general than the projection methods.…”

“…In this paper, using essentially the projection technique, we show that the general variational inequalities are equivalent to the general Wiener-Hopf equations, whose origin can be traced back to Shi [8]. It has been shown [4,[8][9][10] that the Wiener-Hopf equations are more flexible and general than the projection methods. Noor [4,9] has used the Wiener-Hopf equations technique to study the sensitivity analysis and the dynamical systems as well as to suggest and analyze several iterative methods for solving variational inequalities.…”

Moudafi’s viscosity approximation method for monotone operators and a bifunction

“…This theory has witnessed an explosive growth in theoretical advances and applications across all disciplines of pure and applied sciences. There are a substantial number of numerical methods including projection method and its variant forms, Wiener-Hopf equations, auxiliary principle and descent for solving various classes of variational inequalities and complementarity problems; see [1,2,3,4,5,6] and the references therein. It is well known that the projection methods, Wiener-Hopf equations techniques and auxiliary principle E-mail address: zhangml@henu.edu.cn.…”