Abstract
We propose an implicit iterative scheme and an explicit iterative scheme for finding a common element of the set of solutions of system of equilibrium problems and a constrained convex minimization problem by the general iterative methods. In the setting of real Hilbert spaces, strong convergence theorems are proved. Our results improve and extend the corresponding results reported by Tian and Liu (2012) and many others. Furthermore, we give numerical example to demonstrate the effectiveness of our iterative scheme.
Highlights
Let H be a real Hilbert space with inner product ⟨, ⟩ and induced norm ‖ ⋅ ‖
Let {Fk} be a countable family of bifunctions from C × C to R, where R is the set of real numbers
Optimization, and economics reduce to finding a solution of the equilibrium problem
Summary
Let H be a real Hilbert space with inner product ⟨, ⟩ and induced norm ‖ ⋅ ‖. Many methods have been proposed to solve the equilibrium problem (2); see [2,3,4] and the references therein. Ceng et al [8] proposed implicit and explicit iterative schemes for finding the approximate minimizer of a constrained convex minimization problem and proved that the sequences generated by their schemes converges strongly to a solution of the constrained convex minimization problem. Tian and Liu [9] proposed implicit and explicit composite iterative algorithms for finding a common solution of an equilibrium problem and a constrained convex minimization problem; strong convergence theorems are obtained in [9]. Further we obtain strong convergence theorems for finding a common element of the set of solutions of a constrained convex minimization problems and the set of solutions of the equilibrium problem.
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