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Article Citation - WoS: 15Citation - Scopus: 14Generalized Transportation Cost Spaces(Springer Basel Ag, 2019) Ostrovska, Sofiya; Ostrovskii, Mikhail I.The paper is devoted to the geometry of transportation cost spaces and their generalizations introduced by Melleray et al. (Fundam Math 199(2):177-194, 2008). Transportation cost spaces are also known as Arens-Eells, Lipschitz-free, or Wasserstein 1 spaces. In this work, the existence of metric spaces with the following properties is proved: (1) uniformly discrete metric spaces such that transportation cost spaces on them do not contain isometric copies of l(1), this result answers a question raised by Cuth and Johanis (Proc Am Math Soc 145(8):3409-3421, 2017); (2) locally finite metric spaces which admit isometric embeddings only into Banach spaces containing isometric copies of l(1); (3) metric spaces for which the double-point norm is not a norm. In addition, it is proved that the double-point norm spaces corresponding to trees are close to l(infinity)(d) of the corresponding dimension, and that for all finite metric spaces M, except a very special class, the infimum of all seminorms for which the embedding of M into the corresponding seminormed space is isometric, is not a seminorm.Article Citation - WoS: 17Citation - Scopus: 27Fixed Point Theorems in New Generalized Metric Spaces(Springer Basel Ag, 2016) Karapinar, Erdal; Karapınar, Erdal; O'Regan, Donal; Roldan Lopez de Hierro, Antonio Francisco; Shahzad, Naseer; Karapınar, Erdal; Mathematics; MathematicsThe aim of our paper is to present new fixed point theorems under very general contractive conditions in generalized metric spaces which were recently introduced by Jleli and Samet in [Fixed Point Theory Appl. 2015 (2015), doi:10.1186/s13663-015-0312-7]. Although these spaces are not endowed with a triangle inequality, these spaces extend some well known abstract metric spaces (for example, b-metric spaces, Hitzler-Seda metric spaces, modular spaces with the Fatou property, etc.). We handle several types of contractive conditions. The main theorems we present involve a reflexive and transitive binary relation that is not necessarily a partial order. We give a counterexample to a recent fixed point result of Jleli and Samet. Our results extend and unify recent results in the context of partially ordered abstract metric spaces.Article Citation - WoS: 48Citation - Scopus: 46On Common Fixed Points in the Context of Brianciari Metric Spaces(Springer Basel Ag, 2017) Aydi, Hassen; Karapinar, Erdal; Zhang, DongIn this paper, we introduce the concept of generalized ()-contractions and generalized ()-Meir-Keeler-contractions in the setting of Brianciari metric spaces. We prove some common fixed point results for such contractions. An example is presented making effective the new concepts and results.Article On the Convergence of the q-bernstein Polynomials for Power Functions(Springer Basel Ag, 2021) Ostrovska, Sofiya; Ozban, Ahmet YasarThe aim of this paper is to present new results related to the convergence of the sequence of the complex q-Bernstein polynomials {B-n,B-q(f(alpha); z)}, where 0 < q not equal 1 and f(alpha) = x(alpha), alpha >= 0, is a power function on [0, 1]. This study makes it possible to describe all feasible sets of convergence K for such polynomials. Specifically, if either 0 < q < 1 or alpha is an element of N-0, then K = C, otherwise K = {0} boolean OR {q(-j)}(j=0)(infinity). In the latter case, this identifies the sequence K = {0} boolean OR {q(-j)}(j=0)(infinity) as the 'minimal' set of convergence for polynomials B-n,B-q(f; z), f is an element of C[0, 1] in the case q > 1. In addition, the asymptotic behavior of the polynomials {B-n,B-q(f(alpha); z)}, with q > 1 has been investigated and the obtained results are illustrated by numerical examples.Article Citation - WoS: 13Citation - Scopus: 13Coincidence Point Theorems on Quasi-Metric Spaces Via Simulation Functions and Applications To g-metric Spaces(Springer Basel Ag, 2018) Lopez de Hierro, A. F. Roldan; Karapinar, E.; O'Regan, D.In this paper, we present some coincidence point results in the framework of quasi-metric spaces using contractive conditions involving simulation functions. As consequences, we are able to particularize these results to a variety of situations including G-metric spaces. The results presented in this paper generalize and extend several comparable results in the existing literature. In addition, some examples are given.Article Citation - WoS: 4Citation - Scopus: 3On Orlicz-Power Series Spaces(Springer Basel Ag, 2010) Karapinar, Erdal; Zakharyuta, VyacheslavIn this manuscript, we investigate the isomorphisms of Orlicz-Kothe sequence spaces and quasidiagonal isomorphisms of Cartesian products of Orlicz-power series spaces.Article Citation - WoS: 69Citation - Scopus: 70An Approach To Best Proximity Points Results Via Simulation Functions(Springer Basel Ag, 2017) Karapinar, Erdal; Khojasteh, FarshidIn this paper, we investigate of the existence of the best proximity points of certain mapping defined via simulation functions in the frame of complete metric spaces. We consider the uniqueness criteria for such mappings. The obtained results unify a number of the existing results on the topic in the literature.Article Citation - WoS: 11Citation - Scopus: 15The q-versions of the Bernstein Operator: From Mere Analogies To Further Developments(Springer Basel Ag, 2016) Ostrovska, SofiyaThe article exhibits a review of results on two popular q-versions of the Bernstein polynomials, namely, the LupaAY q-analogue and the q-Bernstein polynomials. Their similarities and distinctions are discussed.Article The Saturation of Convergence for the Complex q-durrmeyer Polynomials(Springer Basel Ag, 2025) Gurel, Ovgu; Ostrovska, Sofiya; Turan, MehmetThe aim of this paper is to establish a saturation result for the complex q-Durrmeyer polynomials (Dn,qf)(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(D_{n,q}f)(z)$$\end{document}, where q is an element of(0,1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \in (0,1)$$\end{document}, f is an element of C[0,1].\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f \in C[0,1].$$\end{document} It is known that the sequence {(Dn,qf)(z)}n is an element of N\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{(D_{n,q}f)(z)\}_{n \in {\mathbb {N}}}$$\end{document} converges uniformly on any compact set in C\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}$$\end{document} to the limit function (D infinity,qf)(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(D_{\infty ,q}f)(z)$$\end{document}, which, therefore, is entire. Previously, the rate of this convergence has been estimated as O(qn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(q<^>n)$$\end{document}, n ->infinity.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \rightarrow \infty . $$\end{document} In the present article, this result is refined to derive Voronovskaya-type formula and to demonstrate that this rate is o(qn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$o(q<^>n)$$\end{document}, n ->infinity\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \rightarrow \infty $$\end{document} on a set possessing an accumulation point if and only if f takes on the same value at all qj\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q<^>j$$\end{document}, j is an element of N0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j \in {\mathbb {N}}_{0}$$\end{document}.Article Citation - WoS: 1Citation - Scopus: 1On a Quadratic Eigenvalue Problem and Its Applications(Springer Basel Ag, 2013) Atalan, Ferihe; Guseinov, Gusein ShWe investigate the eigenvalues and eigenvectors of a special quadratic matrix polynomial and use the results obtained to solve the initial value problem for the corresponding linear system of differential equations.
