To overcome this difficulty, the author introduced a

nonexpansive mapping T, the fixed point set of which is coincident with the constraint set.

[1] Let C be a nonempty and weakly compact subset of a Hilbert space H and S: C [right arrow] K(H) a

nonexpansive mapping. Then I - S is demiclosed.

Many researchers studied the strong convergency theorems for solving the CCMP (1) using the sequence {[x.sub.n]} which is generated by algorithm (12) for their proposal on the gradient [nabla]g which is the class of

nonexpansive mapping and the class of L-Lipschitzian mapping (see [19-25]) and in case the gradient [nabla]g is the class of 1/L-ism mapping such that L > 0, Xu (2010) introduced the sequence {[x.sub.n]} which is generated by algorithm (12), and he proved that this sequence {[x.sub.n]} converges weakly to the minimizer of the CCMP (1) in the setting of infinite-dimensional real Hilbert space (see [15]) under some appropriate condition.

Recently, Shi and Chen [5] first considered the following Moudafi's viscosity iteration for a

nonexpansive mapping g : E [right arrow] E with 0 [not equal to] Fix(g) = {x | x = g(x)} and a contraction mapping f: E [right arrow] E in CAT(0) space X:

Let E be a reflexive, strictly convex and smooth Banach space and let T be a generalized

nonexpansive mapping from E into itself.

Theorem 2 can be applied to approximating the fixed point of a

nonexpansive mapping where the contraction constant [mu] = 1.

In 2010, Tian [10] introduced the following general iterative scheme for finding an element of set of solutions to the fixed point of

nonexpansive mapping in a Hilbert space.

In 2007, Takahashi and Takahashi [12] considered the equilibrium problem (1.2) and a

nonexpansive mapping by an iterative method.

In this section, we present and analyze the strong convergence of a new algorithm for finding a common solution of an equilibrium problem for a bifunction f : H x H [right arrow] R and a fixed point problem for a quasi -

nonexpansive mapping U : H [right arrow] H .In order to get the convergence of the algorithm, we consider the following assumptions:

Takahashi, "Strong convergence theorem for an equilibrium problem and a

nonexpansive mapping," in Nonlinear Analysis and Convex Analysis, W.

Kozlowski, "Fixed point iteration processes for asymptotic pointwise

nonexpansive mapping in modular function spaces," Fixed Point Theory and Applications, vol.

Let T : K [right arrow] CB(E) be a multi-valued

nonexpansive mapping. Then T satisfies condition ([E.sub.1]).