A general iterative method for addressing mixed equilibrium problems and optimization problems

Strong convergence theorems for equilibrium problems and fixed point problems: A new iterative method, some comments and applications

“…So, when we study the solution of MEP, we only need to study the solution of the equilibrium (1.1). This also shows that some "so-called" mixed equilibrium problem studied in [15][16][17] is still the equilibrium problem (1.1).…”

“…The problem MEP studied in [15][16][17] and the problem GEP studied in [18][19][20] are still the problem (1.1) studied in the literature [5][6][7][8][9][10][11][21][22][23][24] and others.…”

“…Remark 4.1 Recently, some authors introduced the following mixed equilibrium problem (MEP, for short) (see [15][16][17] and references therein) and generalized equilibrium problem (GEP, for short) (see [18][19][20] and references therein):…”

“…In [15][16][17], the authors gave some iterative methods for finding the solution of MEP when the bi-function f(x, y) admits the conditions (A1)-(A4) and the real-valued function satisfies the following condition:…”

Modified hybrid iterative methods for generalized mixed equilibrium, variational inequality and fixed point problems

“…In 2008, Peng and Yao [22] studied an iterative method for the GMEP (1.1) related to an α-inverse-strongly monotone mapping B, the VIP (1.5) for a monotone and Lipschitz continuous mapping F and a nonexpansive mapping S, and proved strong convergence to a point z ∈ GMEP(Θ, ϕ, B) ∩ VI(C, F) ∩ Fix(S). In 2010, by using the method of Yao et al [39], Jaiboon and Kumam [12] also introduced an iterative method related to optimization problem for the MEP (1.3), the VIP (1.5) for an α-inverse-strongly monotone mapping F and a sequence {S n } of nonexpansive mappings, and showed strong convergence to a point z ∈ ∩ ∞ n=1 Fix(S n ) ∩ MEP(Θ, ϕ) ∩ VI(C, F).…”