Abstract
In this work, we modify the inertial hybrid algorithm with Armijo line search using a parallel method to approximate a common solution of nonmonotone equilibrium problems in Hilbert spaces. A weak convergence theorem is proved under some continuity and convexity assumptions on the bifunction and the nonemptiness of the common solution set of Minty equilibrium problems. Furthermore, we demonstrate the quality of our inertial parallel hybrid algorithm by using image restoration, as well as its superior efficiency when compared with previously considered parallel algorithms.
Highlights
Let H be a real Hilbert space with the inner product ·, · and induced norm · and let F be an open convex subset of H
We focus on the common equilibrium problem (CEP), which is to find s ∈ C such that ψi(s, v) ≤ 0 for all v ∈ C, (1.6)
A weak convergence theorem is established under some suitable conditions imposed on the bifunction ψi
Summary
Let H be a real Hilbert space with the inner product ·, · and induced norm · and let F be an open convex subset of H. Where C is a nonempty closed and convex subset of F and ψ : F × F → R is a bifunction with ψ(s, s) = 0 for all s ∈ C. For the Minty equilibrium problem (MEP), it was introduced by Castellani and Giuli [9] in 2013. This problem is associated with the equilibrium problem (1.1), which is to find s ∈ C such that ψ(t, s) ≤ 0 for all t ∈ C. The solution set of the Minty equilibrium problem is represented as SM. This problem is associated with the equilibrium problem (1.1), which is to find s ∈ C such that ψ(t, s) ≤ 0 for all t ∈ C. (1.2)
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