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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 thebifunction in the equilibrium problem is the sum of two ones. We propose asplitting algorithm combining the gradient method for equilibrium problem andthe Mann iteration scheme for fixed points of nonexpansive mappings. At eachiteration of the algorithm, two strongly convex subprograms are required tosolve separately, one for each of the component bifunctions. Our main resultstates that, under paramonotonicity property of the given bifunction, thealgorithm converges to a solution without any Lipschitz type condition as wellas H\"older continuity of the bifunctions involved.

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