Hybrid method for equilibrium problems and variational inclusions

Abstract

By providing a new iterative method our aim is finding a common element of the set of fixed points of two nonexpansive mappings, the set of solutions to a variational inclusion and the set of solutions of a generalized equilibrium problem in a real Hilbert space. We review the strong convergence of the new iterative method in the framework of Hilbert spaces. Finally, we show that our main result is a generalization for some known theorems in this field.

Iterative Methods of Weak and Strong Convergence Theorems for the Split Common Solution of the Feasibility Problems, Generalized Equilibrium Problems, and Fixed Point Problems

“…Let H be a real Hilbert space with the inner product 〈•, •〉 and norm ‖ • ‖ and C be a nonempty, closed, and convex subset of H. Recall that a mapping A: C ⟶ H is said to be monotone if 〈Au − Av, u − v〉 ≥ 0 for all u, v ∈ C [18,19]. A mapping A is said to be α-strongly monotone whenever there exists a positive real number α such that 〈Au − Av, u − v〉 ≥ α‖u − v‖ 2 for all u, v ∈ C. A mapping A is said to be α-inverse strongly monotone if there exists a positive real number α such that 〈Au − Av, u − v〉 ≥ α‖Au − Av‖ 2 for all u, v ∈ C. Recall that the classical variational inequality problem, which we denote by VI(C, A), is to find x ∈ C such that 〈Ax, y − x〉 ≥ 0, for all y ∈ C [16,17].…”

“…If F(x, y) � 0 for all x, y ∈ C, then GEP(F, A) is denoted by VI(C, A) � x * ∈ C: 〈Ax * , y − x * 〉 ≥ 0, ∀y ∈ C . is is the set of solutions of the variational inequality for A (see, for example, [15][16][17][18][19][20][21]). If C � H, then VI(H, A) � A − 1 (0) where A − 1 (0) � x ∈ H: Ax � 0 { }.…”

Strong Convergence Analysis of Iterative Algorithms for Solving Variational Inclusions and Fixed-Point Problems of Pseudocontractive Operators

“…where I − λf is a forward step and (I + λg) − 1 is a backward step with λ > 0. is algorithm is a splitting algorithm which solves the difficulty of calculating of the resolvent of f + g. Recently, there has been increasing interest for studying common solution problems relevant to (1) (see for example, [17][18][19][20][21][22][23][24][25][26][27]). Especially, Zhao, Sahu, and Wen [28] presented an iterative algorithm for solving a system of variational inclusions involving accretive operators.…”

On Ulam–Hyers–Rassias stability of a generalized Caputo type multi-order boundary value problem with four-point mixed integro-derivative conditions

“…(for example, refer to [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15]). Many researchers play an important role in different desirable developments on the existence criteria, and some results about the uniqueness for numerous fractional differential equations have been obtained (see for instance [7,[16][17][18][19][20][21][22][23][24]). On the other hand, the subject of stability is a very important notion in physics since most phenomena in the real world include this concept.…”