where problem (3.1) is called fuzzy general nonlinear set-valued mixed quasi-variational inclusions with ( A, η ) -accretive mappings.
In this paper, the existing theorems and methods for finding solutions of systems of general nonlinear set-valued mixed variational inequalities problems in Hilbert spaces are studied.
The property of the resolvent operator J φ presented in Lemma 1.6 plays an important role in developing the numerical methods for solving the system of general nonlinear set-valued mixed variational inequalities problems.
For each fixed positive real numbers, ρ and η, we consider the following so-called system of general nonlinear set-valued mixed variational inequalities problems: Find x*, y*∈ H, u* ∈ Ay*, v* ∈ Bx*, such that (1.1).
A new class of fuzzy general nonlinear set-valued mixed quasi-variational inclusions frameworks for a perturbed Ishikawa-hybrid quasi-proximal point algorithm using the notion of ( A, η ) -accretive is developed.
The equivalent formulation (1.5) enables us to suggest an explicit iterative method for solving the system of general nonlinear set-valued mixed variational inequalities problem (1.1), as we show in the next section.
We consider the fuzzy general nonlinear set-valued mixed quasi-variational inclusions with ( A, η ) -accretive mappings (FGNSVMQVI): Find x ∈ X and b k ∈ X ( k ∈ C ) such that S ˆ k x ( b k ) ≥ a k ( x ) and 0 ∈ F ( b 1, f ( b 2 ) ) + G ( b 3, b 4 ) + g ( x ), (3.1).
Inspired and motivated by recent research work in this field, in this paper, a fuzzy general nonlinear set-valued mixed quasi-variational inclusions framework for a perturbed Ishikawa-hybrid quasi-proximal point algorithm using the notion of ( A, η ) -accretive is developed.
Taking a k ( x ) = 1 ( k ∈ C ) for all x ∈ X, problem (3.1) is equivalent to the following problem: Find x ∈ X and b k ∈ S k ( x ) ( k ∈ C ) such that 0 ∈ F ( b 1, f ( b 2 ) ) + G ( b 3, b 4 ) + g ( x ), (3.2). which is called a new class of general nonlinear set-valued mixed quasi-variational inclusions with ( A, η ) -accretive mappings (GNSVMQVI).
Let (K Hto2^{H}) be a set-valued mapping such that (K x)) is closed convex set in H for any element x of H. Now, we consider the set-valued mixed quasi-variational inequality problem, denoted by SVMQVIP, which consists in finding (xin H) such that (uin T x)), (xin K x)) and bigllangle A u,u), y-x bigrrangle +f x,y -f x,y -f x,x geq0orall yin K(x).
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