New extragradient methods with non-convex combination for pseudomonotone equilibrium problems with applications in Hilbert spaces

Abstract

In the literature, the most authors modify the viscosity methods or hybrid projection methods to construct the strong convergence algorithms for solving the pseudomonotone equilibrium problems. In this paper, we introduce some new extragradient methods with non-convex combination to solve the pseudomonotone equilibrium problems in Hilbert space and prove the strong convergence for the constructed algorithms. Our algorithms are very different with the existing ones in the literatures. As the application, the fixed point theorems for strict pseudo-contraction are considered. Finally, some numerical examples are given to show the effectiveness of the algorithms.

An Explicit Strong Convergence Iterative Scheme for Solving Equilibrium Problems

“…If we take S is a κ-strict pseudo-contraction and weakly continuous then f (x, y) = x − Sx, y − x satisfies the conditions (h1)-(h4) (see [29]) and 2c 1 = 2c 2 = 3−2κ 1−κ . As a consequence of results in Section 3 we have the following fixed point theorems.…”

Approximations of an Equilibrium Problem without Prior Knowledge of Lipschitz Constants in Hilbert Spaces with Applications

“…If we consider that mapping S is weakly continuous and a κ-strict pseudocontraction, then f (u, y) = u − Su, y − u satisfies the conditions (c1)-(c4) (see [43]) and 2c 1 = 2c 2 = 3−2κ 1−κ . The values of y n and z n in Algorithm 1 can be written as follows:…”

“…Condition (L4) could be exempted when L is monotone. Indeed, this condition, which is a particular case of Condition (c3), is only used to prove (43). Without Condition (L4), inequality (42) can be obtained by imposing monotonocity on L. In that case,…”

An Accelerated Extragradient Method for Solving Pseudomonotone Equilibrium Problems with Applications

“…The bifunction f is Lipschitz-type continuous with constants c 1 = c 2 = 2, satisfying the conditions (Ψ1)-(Ψ4). The solution set is EP( f , K) = {(u 1 , 1, 1, 1, 1) : u 1 > 1} (see [48] for more details). In addition, to estimate the optimal values of the control parameters, two experiments are performed by assuming the variation of the control parameters λ, λ 0 and inertial factor θ.…”