Abstract
In this article, we propose a new modified extragradient-like method to solve pseudomonotone equilibrium problems in real Hilbert space with a Lipschitz-type condition on a bifunction. This method uses a variable stepsize formula that is updated at each iteration based on the previous iterations. The advantage of the method is that it operates without prior knowledge of Lipschitz-type constants and any line search method. The weak convergence of the method is established by taking mild conditions on a bifunction. In the context of an application, fixed-point theorems involving strict pseudo-contraction and results for pseudomonotone variational inequalities are considered. Many numerical results have been reported to explain the numerical behavior of the proposed method.
Highlights
Let C be a nonempty, closed and convex subset of a real Hilbert space H and R, N be the sets of real numbers and natural numbers, respectively
A bifunction f : H × H → R is said to be Lipschitz-type continuous on C if there exist two positive constants c1, c2 such that f ( z1, z3 ) ≤ f ( z1, z2 ) + f ( z2, z3 ) + c1 k z1 − z2 k2 + c2 k z2 − z3 k2, ∀ z1, z2, z3 ∈ C
Let C be a nonempty, closed and convex subset of a real Hilbert space H and a normal cone of C at z1 ∈ C is defined by: NC (z1 ) = {z3 ∈ H : hz3, z2 − z1 i ≤ 0, ∀z2 ∈ C}
Summary
Let C be a nonempty, closed and convex subset of a real Hilbert space H and R, N be the sets of real numbers and natural numbers, respectively. The iterative sequence generated from the above-mentioned method provides a weak convergent iterative sequence and in order to operate it, prior information regarding the Lipschitz-type constants is required. These Lipschitz-type constants are mostly unknown or hard to compute. Vinh and Muu proposed an inertial iterative algorithm in [39] to solve a pseudomonotone equilibrium problem. Their main contribution is the availability of an inertial effect in the algorithm that is used to improve the convergence rate of the iterative sequence.
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