A splitting algorithm for finding fixed points of nonexpansive mappings and solving equilibrium problems

Abstract

We consider the problem of finding a fixed point of a nonexpansive mapping, which is also a solution of a pseudo-monotone equilibrium problem, where the bifunction in the equilibrium problem is the sum of two ones. We propose a splitting algorithm combining the gradient method for equilibrium problem and the Mann iteration scheme for fixed points of nonexpansive mappings. At each iteration of the algorithm, two strongly convex subprograms are required to solve separately, one for each of the component bifunctions. Our main result states that, under paramonotonicity property of the given bifunction, the algorithm converges to a solution without any Lipschitz type condition as well as Hölder continuity of the bifunctions involved.Mathematics Subject Classification (2010). 47H05, 47H10, 90C33.

Viscosity methods for nonexpansive and monotone mappings in Hilbert spaces

“…This theory has witnessed an explosive growth in theoretical advances and applications. Recently, numerical methods have been introduced and studied for solutions of equilibrium problem (1.2); see [3,6,9,10,17,18] and the references therein.…”

On fixed point approach to equilibrium problem

“…After the appearance of the paper by Blum and Oettli [11], the problem (EP) has attracted much attention of many authors and a lot of algorithms have been developed for solving the problem where the bifunction f have monotonic properties. These algorithms are based upon different methods such as penalty and gap functions [7,8,9,23,24,25,29,33], regularization [3,20,28,34,35], extragradient methods [10,19,26,37,39,40,41,44,46,47,48], splitting technique [2,15,32]. A comprehensive referencelist on algorithms for the equilibrium problem can be found in the interesting monograph [6].…”

“…(MEP) By considering this mixed form one can employ special structures of each φ and ϕ in subgradient splitting algorithms, where the bifunction f (x, y) can be expressed by the sum of two bifunctions f 1 (x, y)+ f 2 (x, y) and the iterates are defined by taking the proximal mappings of each f 1 and f 2 separately, see [2,3,15,35,32].…”