Vatan Karakaya

Weighted statistical convergence and its application to Korovkin type approximation theorem

Abstract The concept of weighted statistical convergence was introduced and studied by Karakaya and Chishti (2009) [7] . In this paper, we modify the definition of weighted statistical convergence and find its relationship with the concept of statistical summability ( N ¯ , p n ) due to Moricz and Orhan (2004) [10] . We apply this new summability method to prove a Korovkin type approximation theorem by using the test functions 1 , e - x , e - 2 x . We apply the classical Baskakov operator to construct an example in support of our result.

Data dependence results of new multi-step and S-iterative schemes for contractive-like operators

In this paper, we prove that the convergence of a new iteration and S-iteration can be used to approximate the fixed points of contractive-like operators. We also prove some data dependence results for these new iteration and S-iteration schemes for contractive-like operators. Our results extend and improve some known results in the literature.MSC:47H10.

Difference sequence spaces derived by using a generalized weighted mean

Abstract In this work, we define new sequence spaces by combining a generalized weighted mean and a difference operator. Afterward, we investigate topological structures, which have completeness, A K -property, and A D -property. Also, we compute the α -, β - and γ -duals, and obtain bases for these sequence spaces. Finally, the necessary and sufficient conditions on an infinite matrix belonging to the classes ( c ( u , v , Δ ) : l ∞ ) and ( c ( u , v , Δ ) : c ) are obtained.

Some p -Type New Sequence Spaces and Their Geometric Properties

We introduce an -type new sequence space and investigate its some topological properties including and properties. Besides, we examine some geometric properties of this space concerning Banach-Saks type and Gurariis modulus of convexity.

Fixed Point of a New Three-Step Iteration Algorithm under Contractive-Like Operators over Normed Spaces

We introduce a new three-step iteration scheme and prove that this new iteration scheme is convergent to fixed points of contractive-like operators. Also, by providing an example, we show that our new iteration method is faster than another iteration method due to Suantai (2005). Furthermore, it is shown that this new iteration method is equivalent to some other iteration methods in the sense of convergence. Finally, it is proved that this new iteration method is -stable.

On Some Geometrical Properties of Generalized Modular Spaces of Cesáro Type Defined by Weighted Means

The main purpose of this paper is to introduce modular structure of the sequence space defined by Altay and Başar (2007), and to study Kadec-Klee () and uniform Opial properties of this sequence space on Köthe sequence spaces.

n-tuplet fixed point theorems for contractive type mappings in partially ordered metric spaces

In this paper, we study existence and uniquennes of fixed points of operator F:Xn→X where n is an arbitrary positive integer and X is partially ordered complete metric space.MSC: 47H10, 54H25, 54E50.

SOME GEOMETRIC PROPERTIES OF A NEW DIFFERENCE SEQUENCE SPACE INVOLVING LACUNARY SEQUENCES

Abstract In this paper, we define a new generalized difference sequence space involving lacunary sequence. Then, we examine k-NUC property and property (β) for this space and also show that it is not rotund where p = (pr) is a bounded sequence of positive real numbers with pr ≥ 1 for all r ∈ ℕ.

Jungck-Khan iterative scheme and higher convergence rate

In this paper, we continue the theme of analytical and numerical treatment of Jungck-type iterative schemes. In particular, we focus on a special case of Jungck-Khan iterative scheme introduced by Khan et al. [Analytical and numerical treatment of Jungck-type iterative schemes, Appl. Math. Comput. 231 (2014) 521–535] to get an insight in the strong convergence and data dependence results obtained therein. Our investigations show that this special case under different control conditions on parametric sequences provides higher convergence rate and better data dependence estimates as compared to the Jungck-Khan iterative scheme itself.

Erratum to: ‘n-tuplet fixed point theorems for contractive type mappings in partially ordered metric spaces’

*Correspondence: merturk3263@gmail.com 1Department of Mathematics, Yildiz Technical University, Davutpasa Campus, Esenler, Istanbul, Turkey Full list of author information is available at the end of the article Correction () Page , line : The statement ‘(ii) limr→t+ φ(r) ’ should be corrected as ‘(ii) limr→t+ φ(r) ’. () Page , line : The statement ‘. . . Condition  is satisfied.’ should be rewritten ‘. . . Condition  is satisfied and g is continuous.’ () Page , line : The statement ‘. . . and using (.)’ should be corrected as ‘. . . and using (.)’. () Page , line : The statement ‘≤ δj(k)+ + δl(k)+ + tk + n · φ( tk n )’ should be corrected as ‘≤ δj(k)+ + δl(k)+ + n · φ( tk n )’. () Page , line : The statement ‘From (.) and by . . .’ should be corrected as ‘From (.) and by . . . ’. () Page , line : ‘. . . now the assumption (b) holds.’ should be corrected as ‘. . . now the assumption (ii) holds.’ () Page , line  (line  in Corollary ) and Page , line  (line  in Corollary ): The statement ‘and there exist φ ∈ such that F ’ should be deleted. () Page , line  (line  in Corollary ) and Page , line  (line  in Corollary ): The statement ‘φ(F(x,x, . . . ,xn),F(y, y, . . . , yn))’ should be corrected as ‘d(F(x,x, . . . ,xn),F(y, y, . . . , yn))’. () Page , line  (line  in Corollary ) and Page , line  (line  in Corollary ): The statement ‘. . .F has the mixed g-monotone’ should be corrected as ‘. . .F has the mixed monotone’. That is, ‘g-’ should be deleted. () Page , line : ‘≤ φ( d(g(x),g(y))+d(g(x),g(y))+···+d(g(xn),g(yn)) n )’ should be corrected as ‘≤ φ( d(x,y)+d(x,y)+···+d(xn ,yn) n )’. () Page , line : ‘≤ n [d(g(x), g(y)) + d(g(x), g(y)) + · · · + d(g(xn), g(yn))]’ should be corrected as ‘≤ n [d(x, y) + d(x, y) + · · · + d(xn, yn)]’.