Abstract
AbstractIn this paper, we first propose a weak convergence algorithm, called the linesearch algorithm, for solving a split equilibrium problem and nonexpansive mapping (SEPNM) in real Hilbert spaces, in which the first bifunction is pseudomonotone with respect to its solution set, the second bifunction is monotone, and fixed point mappings are nonexpansive. In this algorithm, we combine the extragradient method incorporated with the Armijo linesearch rule for solving equilibrium problems and the Mann method for finding a fixed point of an nonexpansive mapping. We then combine the proposed algorithm with hybrid cutting technique to get a strong convergence algorithm for SEPNM. Special cases of these algorithms are also given.
Highlights
Throughout the paper, unless otherwise is stated, we assume that H and H are real Hilbert spaces endowed with inner products and induced norms denoted by ·, · and·, respectively, whereas H refers to any of these spaces
To find a solution of split feasible problem (SFP) in finite-dimensional Hilbert spaces, a basic scheme proposed by Byrne [ ], called the CQ-algorithm, is defined as follows: xk+ = PC xk + γ AT (PQ – I)Axk, where I is the identity mapping, and PC is projection mapping onto C
Xu [ ] investigated the SFP setting in infinite-dimensional Hilbert spaces
Summary
Throughout the paper, unless otherwise is stated, we assume that H and H are real Hilbert spaces endowed with inner products and induced norms denoted by ·, · and. Many researchers have been proposed numerical algorithms for finding a common element of the set of solutions of monotone equilibrium problems and the set of fixed points of nonexpansive mappings; see, for example, [ – ] and the references therein. This paper focuses mainly on a split equilibrium problem and nonexpansive mapping involving pseudomonotone and monotone equilibrium bifunctions in real Hilbert spaces. It should be noticed that, under the monotonicity assumption on f and g, the solution sets Sol(C, f ) and Sol(C, g) of the equilibrium problems EP(C, f ) and EP(Q, g) are closed convex sets whenever f and g are lower semicontinuous and convex with respect to the second variables. The following lemmas are well known in the theory of monotone equilibrium problems.
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