This problem is connected with the fixed point problem for nonlinear mappings, the problem of finding a zero point of an accretive operator and so on.
When N : L ( E, Y ) × L ( E, Y ) × L ( E, Y ) → L ( E, Y ) and H K × K K → Y are two single-valued mappings, the problem (1.4) includes some generalized variational inequality problems investigated in [8, 11, 16 19] as special cases.
The problem of finding a common element of Fix ( T ) ∩ Γ for asymptotically quasi-nonexpansive mappings is more general than that for nonexpansive mappings and the problem of finding a solution of the (SFP) in [[8], Theorem 5.7].
In the present paper, we study the equilibrium problems (1.1) and (1.2), the fixed point problem for a family of infinitely nonexpansive mappings, and the problem of zero points of maximal monotone mapping (1.3).
If T 1, T 2 : H → H are univariate mappings then the problem (SGMVI) reduced to the following.
Especially, the well-known convex feasibility problem reduces to finding a point in the intersection of the fixed point sets of a family of nonexpansive mappings and the problem of finding an optimal point that minimizes a given cost function over the set of common fixed points of a family of nonexpansive mappings.
(III) If A, B : H → H are single-valued mappings, then, from the problem (1.1), we have the following system of general nonlinear mixed variational inequalities problems: .
In brief, for suitable choices of the mappings presented in the problem (3.1) and the constants ρ, ν, one can obtain various new and previously known systems of variational inequalities and variational inclusions as special cases of the problem (3.1).
If A, B : H → H are single-valued mappings, then, from the problem (1.1), we have the following system of general nonlinear mixed variational inequalities problems: Find x*, y* ∈ H, such that (1.4).
See, for example, [1 22] and the following examples: Example 1.1 If F : B 1 → B 1 and G : B 2 → B 2 are two single-valued mappings, then, from the problem (1.1), we have the following problem: Find ( x, y ) ∈ B 1 × B 2 such that { 0 ∈ N 1 ( x, G ( y ) ) + M 1 ( x, x ), 0 ∈ N 2 ( F ( x ), y ) + M 2 ( y, y ).
Chang et al. [16] introduced some iterative processes based on the extragradient method for finding the common element of the set of fixed points of a family of infinitely nonexpansive mappings, the set of problem (1.2), and the set of solutions of a variational inequality problem for an -inverse strongly monotone mapping.
