Abstract
We introduce and study a new class of auxiliary problems for solving the equilibrium problem in Banach spaces. Not only the existence of approximate solutions of the equilibrium problem is proven, but also the strong convergence of approximate solutions to an exact solution of the equilibrium problem is shown. Furthermore, we give some iterative schemes for solving some generalized mixed variational-like inequalities to illuminate our results.
Highlights
IntroductionThere are several papers available in the literature which are devoted to the development of iterative procedures for solving some of these equilibrium problems in finite as well as infinite-dimensional spaces
Let X be a real Banach space with dual X∗, let K ⊂ X be a nonempty subset, and let f : K × K → R = (−∞, +∞) be a given bifunction
By the equilibrium problem introduced by Blum and Oettli in [1], we can formulate the following equilibrium problem of finding an x ∈ K such that f (x, y) ≥ 0, ∀y ∈ K, (1.1)
Summary
There are several papers available in the literature which are devoted to the development of iterative procedures for solving some of these equilibrium problems in finite as well as infinite-dimensional spaces. In [8], Iusem and Sosa presented some iterative algorithms for solving equilibrium problems in finite-dimensional spaces. They have established the convergence of the algorithms In [19], Chen and Wu introduced an auxiliary problem for the equilibrium problem (1.1). They showed that the approximate solutions generated by the auxiliary problem converge to the exact solution of the equilibrium problem (1.1) in Hilbert space. Our results extend and improve the corresponding results reported in [3, 4, 19]
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