Existence and approximation of solutions for generalized extended nonlinear variational inequalities

Abstract

In this paper, we consider a new class of generalized extended nonlinear quasi-variational inequality problems involving set-valued relaxed monotone operators and establish its equivalence with the fixed point problem. We study criteria for existence of their solutions. Iterative methods for finding approximate solutions are also proposed and analyzed. MSC: 47J20; 65K10; 65K15; 90C33

Variational Inequalities Approaches to Minimization Problems with Constraints of Generalized Mixed Equilibria and Variational Inclusions

“…Therefore, x * ∈ VI(C, ∇ f ) = Ξ. Furthermore, from (25), (26) and (28), we have that ∆ j t n x t n → x * , Λ m t n u t n → x * , u t n → x * and v t n → x * . First, we prove that…”

“…(iii) Letting ϕ = 0 and A = 0, GMEP (7) reduces to find x † ∈ C verifying (EP) : Θ(x † , y) ≥ 0, ∀y ∈ C, which was discussed in [18,19]. GMEP (7) has been discussed extensively in the literature; see e.g., [6,7,[20][21][22][23][24][25][26][27][28][29][30][31][32]. Now, we consider the minimization problem (CMP):…”

Projected Subgradient Algorithms for Pseudomonotone Equilibrium Problems and Fixed Points of Pseudocontractive Operators

“…Equilibrium problems have been studied extensively in the literature (see, e.g., [2][3][4][5]). Many problems, such as variational inequalities [6][7][8][9][10][11][12][13][14][15], fixed point problems [16][17][18][19][20][21], and Nash equilibrium in noncooperative games theory [1,22], can be formulated in the form of Equation (3).…”

Weighted Method for Uncertain Nonlinear Variational Inequality Problems

“…The above shows that the VIP can be transformed into a semi-infinite programming problem by a proper gap function. Postolache et al [4][5][6][7] made a contribution to the solution and application of variational inequalities.…”

“…In [6], the writers prove the existence and approximation of solutions for generalized extended nonlinear variational inequalities. In [7], the authors propose the variant extragradient-type method for monotone variational inequalities.…”