The aim of this paper is to investigate hybrid algorithm for a common zero point of the sum of two monotone operators which is also a fixed point of a family of countable quasi-nonexpansive mappings. We point out two incorrect proof in paper (Hecai in Fixed Point Theory Appl 2013:11, 2013). Further, we modify and generalize the results of Hecai's paper, in which only a quasi-nonexpansive mappin...

A generalization of Halpern’s iteration is investigated on a compact convex subset of a smooth Banach space. The modified iteration process consists of a combination of a viscosity term, an external sequence, and a continuous nondecreasing function of a distance of points of an external sequence, which is not necessarily related to the solution of Halpern’s iteration, a contractive mapping, and...

The purpose of this paper is to provide the new demicloseness principle– τ (weakly) demicloseness principle. We prove that if X is a Banach space with locally uniformly τ -Opial condition, where τ is a Hausdorff topology on X, C is a nonempty τ compact subset of X, and T : C → C is a asymptotically nonexpansive type mapping. If {xα} is a net in C which converges to x in the sense of τ topology ...

In this paper, we introduce a class of totally quasi-φ-asymptotically nonexpansive nonself multi-valued mapping to modify the Halpern-Mann-type iteration algorithm for a totally quasi-φ-asymptotically nonexpansive nonself multi-valued mapping, which has the strong convergence under a limit condition only in the framework of Banach spaces. Our results are applied to study the approximation probl...

Let E be a real reflexive Banach space which has uniformly Gâteaux differentiable norm. Let K be aclosed convex subset of E which is also a sunny nonexpansive retract of E, and T : K → E be nonexpansive mapping satisfying the weakly inward condition and F (T ) = {x ∈ K, Tx = x} 6= ∅, and f : K → K be a contractive mapping. Suppose that x0 ∈ K, {xn} is defined by { xn+1 = αnf(xn) + (1− αn)((1− δ...

Recently, S. Reich and S. Simons provided a novel proof of the Kirszbraun-Valentine extension theorem using Fenchel duality and Fitzpatrick functions. In the same spirit, we provide a new proof of an extension result for firmly nonexpansive mappings with an optimally localized range. Throughout this paper, we assume that X is a real Hilbert space, with inner product p = 〈· | ·〉 and induced norm...

The purpose of this paper is to introduce two implicit iteration schemes for approximating fixed points of nonexpansive mapping T and a finite family of nonexpansive mappings {Ti}i 1, respectively, in Banach spaces and to prove weak and strong convergence theorems. The results presented in this paper improve and extend the corresponding ones of H.-K. Xu and R. Ori, 2001, Z. Opial, 1967, and oth...

We propose a strongly convergent algorithm for finding a common point in the solution set of a class of pseudomonotone equilibrium problems and the set of fixed points of nonexpansive mappings in a real Hilbert space. The proposed algorithm uses only one projection and does not require any Lipschitz condition for the bifunctions. AMS 2010 Mathematics subject classification: 65 K10, 65 K15, 90 C...