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In this paper we investigate the degree of approximation of a function belonging to the generalized Zygmund class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Z_{p}^{(\omega)}$\end{document}Zp(ω) (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p \ge1$\end{document}p≥1) by Hausdorff means of its Fourier series. We also deduce a corollary and mention a few applications of our main results.

Recently, Filipovic et al. [M. Filipovic, L. Paunovic, S. Radenovic, M. Rajovic, Remarks on 'Cone metric spaces and fixed point theorems of T-Kannan and T-Chatterjea contractive mappings', Math. Comput. Modelling. 54 (2011) 1467-1472] proved several fixed and periodic point theorems for solid cones on cone metric spaces. In this paper several fixed and periodic point theorems for T-contraction of two maps on cone metric spaces with solid cone are proved. The results of this paper extend and generalize well-known comparable results in the literature.

In this paper we show that by renorming an ordered Banach space, every cone $P$ can be converted to a normal cone with constant $K=1$ and consequently due to this approach every cone metric space is really a metric one and every theorem in metric space is valid for cone metric space automatically.

We obtain necessary conditions for a doubly triangular matrix A to have the property that a double series ΣΣ λmnbmn is summable |A|k whenever the series ΣΣbmn is bounded |A|k.

We prove the existence of coincidence points and common fixed points for multivalued f-weak contraction mappings which assume closed values only. As an application, related common fixed point, invariant approximation, random coincidence point and random invariant approximation results are also obtained. Our results provide extensions as well as substantial improvements of several well known results in the existing literature.

In 1930 Knopp presented the following matrix characterization for the core of ordinary sequences. If A is a nonnegative regular matrix then the core of [Ax] is contained in the core of [x], provided that [Ax] exists. Patterson in 1999 extended Knopp’s results to double sequences via four dimensional matrices. In a manner similar to the Knopp’s and Patterson’s results we present multidimensional extensions of Bustoz’s singular dimensional Gibbs phenomenon results. These results include a notion of what it means for a four dimensional matrix transformation to induce the double Gibbs phenomenon in s. In addition, necessary and sufficient conditions for a four dimensional matrix to induce the double Gibbs phenomenon is also presented.

In this paper, we introduce a new one-step iterative process to approximate the common fixed points of two multivalued nonexpansive mappings. We will also prove a strong convergence theorem in a uniformly convex Banach space under the multivalued version of so-called Condition (A′).

The existence of coincidence points and common fixed points for four mappings satisfying generalized contractive conditions without exploiting the notion of continuity of any map involved therein, in a cone metric space is proved. These results extend, unify and generalize several well known comparable results in the existing literature.

Using the setting of a generalized metric space, a fixed point theorem is proved for one map, and several fixed point theorems are proved for two maps. These results generalize several well known comparable results in the literature.

In this paper we extend a fixed point theorem of Imdad and Kumar, for a pair of non-self maps, to non-normal cone spaces.

We obtain sufficient conditions for the series ∑ a n λ n to be absolutely summable of order k by a triangular matrix.

We obtain sufficient conditions for the series ]]> ]]> \sum a_{n}\lambda_{n}$ to be absolutely summable of order $k$ by a triangular matrix.

The power series summability method is one of the most important and general methods in Summability Theory. In this work we will extend the power series method by considering the Burmann series method (i.e. f s ( z n ) = 1 f ( z n ) ∑ k = 0 ∞ s k b k [ h ( z n ) ] k , where f ( z n ) = ∑ k = 0 ∞ b k [ h ( z n ) ] k and | h ( z n ) | 1 ). Using the generalization we obtain a Tauberian theorem which contains, as special cases, some standard theorems for Abel, Borel, and Sonnenschein methods. We use the Abel and Borel matrix methods to illustrate this theorem.

We obtain a theorem on a local property of \N, p(n)\(k), summability of factored Fourier series and a summability factor theorem for Fourier series. (C) 2003 Elsevier Inc. All rights reserved.

We study the rate of approximation to functions inLpand, in particular, inLip(α,p)by weighted means of their Walsh-Fourier series, whereα>0and1≤p≤∞. For the casep=∞,Lpis interpreted to beCW, the collection of uniformlyWcontinuous functions over the unit interval[0,1). We also note that the weighted mean kernel is quasi-positive, under fairly general conditions.

In this article, we study coupled coincidence and coupled common fixed point theorems in ordered generalized metric spaces for nonlinear contraction condition related to a pair of altering distance functions. Our results generalize and modify several comparable results in the literature. 2000 MSC: 54H25; 47H10; 54E50.

In an earlier paper the first author [Ekrem Savas, Factors for | A |k summability of infinite series, Comput. Math. Appl. 53 (7) (2007) 1045-1049. [3]] obtained a summability factor theorem for absolute summability of the order k ? 1. In this paper we extend that result to doubly infinite matrices. © 2007 Elsevier Ltd. All rights reserved.

We obtain sufficient conditions for the series Sigma a(n), which is absolutely summable of order k by a weighted mean method, to be such that Sigma epsilon(n)a(n) is absolutely summable of order k by a triangular matrix. As corollaries we obtain a number of inclusion theorems. (C) 2003 Elsevier Inc. All rights reserved.

The convergence of Mann iteration is equivalent to the convergence of Ishikawa iterations, when T is an asymptotically nonexpansive and asymptotically pseudocontractive map.

Huang and Zhang [L.-G. Haung, X. Zhang, Cone metric spaces and fixed point theorems of contractive mappings, J. Math. Anal. Appl. 332 (2007) 1468–1476] proved some fixed point theorems in cone metric spaces. In this work we prove some fixed point theorems in cone metric spaces, including results which generalize those from Haung and Zhang’s work. Given the fact that, in a cone, one has only a partial ordering, it is doubtful that their Theorem 2.1 can be further generalized. We also show that these maps have no nontrivial periodic points.

WOS: 000267964200029 In this work it is shown that if C(alpha, beta) and C(gamma, delta) are double Cesaro, matrices with alpha >= gamma > -1, beta >= delta > -1, then any double series that is vertical bar C(gamma, delta) vertical bar(k)-summable is also vertical bar C(alpha, beta) vertical bar(k)-summable, k >= 1. (c) 2009 Published by Elsevier Ltd

This article has been retracted: please see Elsevier Policy on Article Withdrawal (http://www.elsevier.com/locate/withdrawalpolicy).This article has been retracted at the request of the Editor-in-Chief.This article has been retracted at the request of the editor as it duplicates significant parts of a paper that had already appeared in Appl. Math. Lett., 24 (2011) 97–102, doi:10.1016/j.aml.2010.08.025. One of the conditions of submission of a paper for publication is that authors declare explicitly that the paper is not under consideration for publication elsewhere. Re-use of any data should be appropriately cited. As such this article represents a severe abuse of the scientific publishing system. The scientific community takes a very strong view on this matter and apologies are offered to readers of the journal that this was not detected during the submission process.

We provide sufficient conditions for Picard iteration to converge faster than Krasnoselskij, Mann, Ishikawa, or Noor iteration for quasicontractive operators. We also compare the rates of convergence between Krasnoselskij and Mann iterations for Zamfirescu operators.

We use a Mann-type iteration scheme and the metric projection operator (the nearest-point projection operator) to approximate the solutions of variational inequalities in uniformly convex and uniformly smooth Banach spaces.

This article is concerned with coupled coincidence points and common fixed points for two mappings in metric spaces and cone metric spaces. We first establish a coupled coincidence point theorem for two mappings and a common fixed point theorem for two w-compatible mappings in metric spaces. Then, by using a scalarization method, we extend our main theorems to cone metric spaces. Our results generalize and complement several earlier results in the literature. Especially, our main results complement a very recent result due to Abbas et al.