Throughout this paper, we denote by N, Z, Q and R the sets of all positive integers, all integers, all rational numbers and all real numbers, respectively. We put R+ = [0,∞)n and ej = (0, 0, · · · , 0, 0, (j) 1 , 0, 0, · · · , 0) ∈ R for j ∈ N with 1 ≤ j ≤ n. Let C be a subset of a Banach space E, and let T be a nonexpansive mapping on C, i.e., ‖Tx−Ty‖ ≤ ‖x− y‖ for all x, y ∈ C. We know that T ...

We present several new results on the asymptotic behavior of firmly nonexpansive mappings in Banach spaces and in the Hubert ball. Let D be a subset of a (real) Banach space X. Recall that a mapping T: D -» X is said to be firmly nonexpansive [2, 4] if for each x and y in D, the convex function /: [0,1] -> [0, oo) defined by f{s) = \(\-s)x + sTx-((l-s)y + sTy) \ is nonincreasing. Note that T is...

We construct random iterative processes for weakly contractive and asymptotically nonexpansive random operators and study necessary conditions for the convergence of these processes. It is shown that they converge to the random fixed points of these operators in the setting of Banach spaces. We also proved that an implicit random iterative process converges to the common random fixed point of a...

In 1977, Baillon and Haddad proved that if the gradient of a convex and continuously differentiable function is nonexpansive, then it is actually firmly nonexpansive. This result, which has become known as the Baillon-Haddad theorem, has found many applications in optimization and numerical functional analysis. In this note, we propose short alternative proofs of this result and strengthen its ...

We introduce an explicit iterative scheme for computing a common fixed point of a sequence of nearly nonexpansive mappings defined on a closed convex subset of a real Hilbert space which is also a solution of a variational inequality problem. We prove a strong convergence theorem for a sequence generated by the considered iterative scheme under suitable conditions. Our strong convergence theore...

We study convergences of Mann and Ishikawa iteration processes for mappings of asymptotically quasi-nonexpansive type in Banach spaces. 1. Introduction and preliminaries. Let D be a nonempty subset of a real Banach space X and T : D → D a nonlinear mapping. The mapping T is said to be asymptotically quasi-nonexpansive (see [5]) if F(T) = ∅ and there exists a sequence {k n } in [0, ∞) with lim n...

*Correspondence: [email protected] School of Science, Southwest University of Science and Technology, Mianyang, Sichuan 621010, China Abstract In this paper, we prove the demiclosed principle for total asymptotically nonexpansive nonself mappings in hyperbolic spaces. Then we obtain convergence theorems of the mixed Agarwal-O’Regan-Sahu type iteration for total asymptotically nonexpansive non...

We introduce two modifications of the Mann iteration, by using the hybrid methods, for equilibrium and fixed point problems for an infinite family of asymptotically nonexpansive mappings in a Hilbert space. Then, we prove that such two sequences converge strongly to a common element of the set of solutions of an equilibrium problem and the set of common fixed points of an infinite family of asy...

In this paper we obtain new effective results on the Halpern iterations of nonexpansive mappings using methods from mathematical logic or, more specifically, proof-theoretic techniques. We give effective rates of asymptotic regularity for the Halpern iterations of nonexpansive self-mappings of nonempty convex sets in normed spaces. The paper presents another case study in the project of proof m...

We prove Browder’s type strong convergence theorems for infinite families of nonexpansive mappings. One of our main results is the following: let C be a bounded closed convex subset of a uniformly smooth Banach space E. Let {Tn : n ∈ N} be an infinite family of commuting nonexpansive mappings on C. Let {αn} and {tn} be sequences in (0,1/2) satisfying limn tn = limn αn/t n = 0 for ∈ N. Fix u ∈ C...