Existence and algorithm of solutions for general set-valued Noor variational inequalities with relaxed (μ,ν)-cocoercive operators in Hilbert spaces

Abstract

The purpose of this paper is to suggest and analyze a number of iterative algorithms for solving the generalized set-valued variational inequalities in the sense of Noor in Hilbert spaces. Moreover, we show some relationships between the generalized set-valued variational inequality problem in the sense of Noor and the generalized set-valued Wiener-Hopf equations involving continuous operator. Consequently, by using the equivalence, we also establish some methods for finding the solutions of generalized set-valued Wiener-Hopf equations involving continuous operator. Our results can be viewed as a refinement and improvement of the previously known results for variational inequality theory.Keywords General set-valued Noor variational inequality problem · The generalized set-valued Wiener-Hopf equations involving continuous operator · Relaxed (μ, ν)-cocoercive set-valued operator Mathematics Subject Classification (2000) 47J20 · 49J53

Hybrid iterative scheme by a relaxed extragradient method for solutions of equilibrium problems and a general system of variational inequalities with application to optimization

“…In particular, if A = B, then problem (1.11) reduces to finding (x * , y * ) ∈ C × C such that (1.12) which is defined by Verma [16] and Verma [17], and is called the new system of variational inequalities. Further, if x * = y * , then problem (1.12) reduces to the classical variational inequality VI(A, C ) which was originally introduced and studied by Stampacchia [28] in 1964 (see [2,4,5,7,26,27,29] for examples).…”

Systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces

“…This means, generally speaking, the class of system general nonlinear set-valued variational inequalities problems is more general and has had a great impact and influence in the development of several branches of pure, applied, and engineering sciences. For more information and results on the general variational inequalities problems, one may consult [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18].…”

Regularization and iterative method for general variational inequality problem in hilbert spaces

“…It is known that a class of nonsymmetric and odd-order obstacle, unilateral, and moving boundary value problems arising in pure and applied can be studied in the unified framework of general variational inequalities (e.g., [2] and the references therein). Observe that to guarantee the existence and uniqueness of a solution of the problem (1.1), one has to impose conditions on the operator A and g, see [3] for example in a more general case. By the way, it is worth noting that, if A fails to be Lipschitz continuous or strongly monotone, then the solution set of the problem (1.1) may be an empty one.…”

“…It is natural to consider a unified approach to these two different problems (e. g., [3][4][5][6][7][8]). Motivated and inspired by the research going in this direction, in this article, we present a method for finding a solution of the problem (1.1), which is related to the solution set of an inverse strongly monotone mapping and is as follows: Find u* H, g(u*) S(T) such that…”