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An existence theorem for a fixed point of an α-nonexpansive mapping of a nonempty bounded, closed and convex subset of a uniformly convex Banach space has been recently established by Aoyama and Kohsaka with a non-constructive argument. In this paper, we show that appropriate Ishikawa iterate algorithms ensure weak and strong convergence to a fixed point of such a mapping. Our theorems are also extended to CAT ( 0 ) spaces. AMS Subject Classification:54E40, 54H25, 47H10, 37C25.

Let H be a real Hilbert space, and let C be a nonempty closed convex subset of H. Let α > 0 , and let A be an α-inverse strongly-monotone mapping of C into H. Let T be a generalized hybrid mapping of C into H. Let B and W be maximal monotone operators on H such that the domains of B and W are included in C. Let 0 < k < 1 , and let g be a k-contraction of H into itself. Let V be a γ ¯ -strongly monotone and L-Lipschitzian continuous operator with γ ¯ > 0 and L > 0 . Take μ , γ ∈ R as follows: 0 < μ < 2 γ ¯ L 2 , 0 < γ < γ ¯ − L 2 μ 2 k . Suppose that F ( T ) ∩ ( A + B ) − 1 0 ∩ W − 1 0 ≠ ∅ , where F ( T ) and ( A + B ) − 1 0 , W − 1 0 are the set of fixed points of T and the sets of zero points of A + B and W, respectively. In this paper, we prove a strong convergence theorem for finding a point z 0 of F ( T ) ∩ ( A + B ) − 1 0 ∩ W − 1 0 , where z 0 is a unique fixed point of P F ( T ) ∩ ( A + B ) − 1 0 ∩ W − 1 0 ( I − V + γ g ) . This point z 0 ∈ F ( T ) ∩ ( A + B ) − 1 0 ∩ W − 1 0 is also a unique solution of the variational inequality 〈 ( V − γ g ) z 0 , q − z 0 〉 ≥ 0 , ∀ q ∈ F ( T ) ∩ ( A + B ) − 1 0 ∩ W − 1 0 . Using this result, we obtain new and well-known strong convergence theorems in a Hilbert space. In particular, we solve a problem posed by Kurokawa and Takahashi (Nonlinear Anal. 73:1562-1568, 2010). MSC:47H05, 47H10, 58E35.

In this paper, let E be a reflexive and strictly convex Banach space which either is uniformly smooth or has a weakly continuous duality map. We consider the hybrid viscosity approximation method for finding a common fixed point of an infinite family of nonexpansive mappings in E. We prove the strong convergence of this method to a common fixed point of the infinite family of nonexpansive mappings, which solves a variational inequality on their common fixed point set. We also give a weak convergence theorem for the hybrid viscosity approximation method involving an infinite family of nonexpansive mappings in a Hilbert space. MSC:47H17, 47H09, 47H10, 47H05.

The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points of an asymptotically strict pseudocontractive mapping in the intermediate sense and the set of solutions of a variational inequality problem for a monotone and Lipschitz continuous mapping. We introduce an extragradient-like iterative algorithm that is based on the extragradient-like approximation method and the modified Mann iteration process. We establish a strong convergence theorem for two sequences generated by this extragradient-like iterative algorithm. Utilizing this theorem, we also design an iterative process for finding a common fixed point of two mappings, one of which is an asymptotically strict pseudocontractive mapping in the intermediate sense and the other taken from the more general class of Lipschitz pseudocontractive mappings. 1991 MSC: 47H09; 47J20.

The hybrid steepest-descent method introduced by Yamada (2001) is an algorithmic solution to the variational inequality problem over the fixed point set of nonlinear mapping and applicable to a broad range of convexly constrained nonlinear inverse problems in real Hilbert spaces. Lehdili and Moudafi (1996) introduced the new prox-Tikhonov regularization method for proximal point algorithm to generate a strongly convergent sequence and established a convergence property for it by using the technique of variational distance in Hilbert spaces. In this paper, motivated by Yamada's hybrid steepest-descent and Lehdili and Moudafi's algorithms, a generalized hybrid steepest-descent algorithm for computing the solutions of the variational inequality problem over the common fixed point set of sequence of nonexpansive-type mappings in the framework of Banach space is proposed. The strong convergence for the proposed algorithm to the solution is guaranteed under some assumptions. Our strong convergence theorems extend and improve certain corresponding results in the recent literature.

Recently, Kawasaki and Takahashi (J. Nonlinear Convex Anal. 14:71-87, 2013) defined a broad class of nonlinear mappings, called widely more generalized hybrid, in a Hilbert space which contains generalized hybrid mappings (Kocourek et al. in Taiwan. J. Math. 14:2497-2511, 2010) and strict pseudo-contractive mappings (Browder and Petryshyn in J. Math. Anal. Appl. 20:197-228, 1967). They proved fixed point theorems for such mappings. In this paper, we prove fixed point theorems for widely more generalized hybrid non-self mappings in a Hilbert space by using the idea of Hojo et al. (Fixed Point Theory 12:113-126, 2011) and Kawasaki and Takahashi fixed point theorems (J. Nonlinear Convex Anal. 14:71-87, 2013). Using these fixed point theorems for non-self mappings, we proved the Browder and Petryshyn fixed point theorem (J. Math. Anal. Appl. 20:197-228, 1967) for strict pseudo-contractive non-self mappings and the Kocourek et al. fixed point theorem (Taiwan. J. Math. 14:2497-2511, 2010) for super hybrid non-self mappings. In particular, we solve a fixed point problem. MSC:Primary 47H10; secondary 47H05.