Abstract
In this paper, we introduce a new iterative algorithm for solving the split equality generalized mixed equilibrium problems. The weak and strong convergence theorems are proved for demi-contractive mappings in real Hilbert spaces. Several special cases are also discussed. As applications, we employ our results to get the convergence results for the split equality convex differentiable optimization problem, the split equality convex minimization problem, and the split equality mixed equilibrium problem. The results in this paper generalize, extend, and unify some recent results in the literature.
Highlights
The equilibrium problem has been extensively studied, beginning with Blum and Oettli [ ] where they proposed it as a generalization of optimization and variational inequality problem
F λx + ( – λ)z, y –C \ { }, ∀y, z ∈ C, λ ∈ (, ], where F : C × C −→ H is the set-valued mapping with the condition F(λx + ( – λ)z, x) ⊇ { }, and [·, z) denotes the line-segment excluding the point z
Motivated by the recent above work of Moudafi et al [ ], Ma et al [ ], Ma et al [ ] and Chidume et al [ ], in this paper, we introduce a new iterative algorithm for solving the split equality generalized mixed equilibrium problem ( . ) for demi-contractive mappings
Summary
The equilibrium problem has been extensively studied, beginning with Blum and Oettli [ ] where they proposed it as a generalization of optimization and variational inequality problem. (Opial’s lemma [ ]) Let H be a real Hilbert space and {μn} be a sequence in H such that there exists a nonempty set W ⊂ H satisfying the following conditions: (i) for every μ ∈ W , limn→∞ μn – μ exists; (ii) any weak cluster point of the sequence {μn} belongs to W .
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