  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 H\"older continuity of the bifunctions involved.
