Let C be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space E with dual space E * . A novel hybrid method for finding a solution of an equilibrium problem and a common element of fixed points for a family of a general class of nonlinear nonexpansive maps is constructed. The sequence of the method is proved to converge strongly to a common element of the family and a solution of the equilibrium problem. Finally, an application of our theorem complements, generalizes and extends some recent important results (Takahashi et al., Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces,

Let C be a nonempty closed and convex subset of a uniformly smooth and uniformly convex real Banach space with dual space E *. In this paper, a new iterative algorithm of Krasnoselskiitype is constructed and used to approximate a common element of a generalized mixed equilibrium problem and a common fixed point of a countable family of generalized-J-nonexpansive maps. Applications of our theorem, in the case of real Hilbert spaces, complement and extend the results of Peng and Yao, (

LetEbe a uniformly convex and uniformly smooth real Banach space, and letE* be its dual. LetA : E→ 2E*be a bounded maximal monotone map. Assume thatA−1(0) ≠ Ø. A new iterative sequence is constructed which convergesstronglyto an element ofA−1(0). The theorem proved complements results obtained on strong convergence ofthe proximal point algorithmfor approximating an element ofA−1(0) (assuming existence) and also resolves an important open question. Furthermore, this result is applied to convex optimization problems and to variational inequality problems. These results are achieved by combining a theorem of Reich on the strong convergence of the resolvent of maximal monotone mappings in a uniformly smooth real Banach space and new geometric properties of uniformly convex and uniformly smooth real Banach spaces introduced by Alber, with a technique of proof which is also of independent interest.

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