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In this paper, we consider hybrid algorithms for finding common elements of the set of common fixed points of two families quasi-ϕ-non-expansive mappings and the set of solutions of an equilibrium problem. We establish strong convergence theorems of common elements in uniformly smooth and strictly convex Banach spaces with the property (K).

In this paper, some new iterative schemes for approximating the common element of the set of fixed points of strongly relatively nonexpansive mappings and the set of zero points of maximal monotone operators in a real uniformly smooth and uniformly convex Banach space are proposed. Some weak convergence theorems are obtained, which extend and complement some previous work.

In this paper, we continue to discuss the properties of iterates generated by a strict pseudocontraction or a finite family of strict pseudo-contractions in a real q-uniformly smooth Banach space. The results presented in this paper are interesting extensions and improvements upon those known ones of Marino and Xu [Marino, G., Xu, H. K.: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl., 324, 336–349 (2007)]. In order to get a strong convergence theorem, we modify the normal Mann’s iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in C. This result extends a recent result of Kim and Xu [Kim, T. H., Xu, H. K.: Strong convergence of modified Mann iterations. Nonl. Anal., 61, 51–60 (2005)] both from nonexpansive mappings to λ-strict pseudo-contractions and from Hilbert spaces to q-uniformly smooth Banach spaces.

In this paper, some iterative schemes for approximating the common element of the set of zero points of maximal monotone operators and the set of fixed points of relatively nonexpansive mappings in a real uniformly smooth and uniformly convex Banach space are proposed. Some strong convergence theorems are obtained, to extend the previous work.

In this paper, most of classical and modern convergence theorems of iterative schemes for nonexpansive mappings are presented and the main results in the paper generalize and improve the corresponding results given by many authors.

Let E be a real uniformly smooth Banach space and T : E → E a strong pseudocontraction with a bounded range. We prove that the Mann and Ishikawa iteration procedures are T -stable. Some related results deal with the stability of these procedures for the iteration approximation of solutions of nonlinear equations involving accretive operators. Our results improve and/or extend those corresponding results announced by Osilike.

In the present note, we give a short proof of a theorem due to Xu and Roach.

In this paper, another new iterative algorithm is proposed for approximating a common element of the common fixed point set for a family of quasi-φ-asymptotically nonexpansive mappings and the solution set of equilibrium problems in Banach spaces. Then a strong convergence theorem is proved by using the propsoed algorithm, which generalizes and improve the recent related results.