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Article The Inversion Results for the Limit q-bernstein Operator(Springer Basel Ag, 2018) Ostrovska, SofiyaThe limit q-Bernstein operator B-q appears as a limit for a sequence of the q-Bernstein or for a sequence of the q-Meyer-Konig and Zeller operators in the case 0 < q < 1. Lately, various features of this operator have been investigated from several angles. It has been proved that the smoothness of f is an element of C[0, 1] affects the possibility for an analytic continuation of its image B-q f. This work aims to investigate the reciprocal: to what extent the smoothness of f can be retrieved from the analytical properties of B-q f.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: 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: 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: 17Citation - Scopus: 16The Approximation by q-bernstein Polynomials in the Case q ↓ 1(Springer Basel Ag, 2006) Ostrovska, SLet B-n (f, q; x), n = 1, 2, ... , 0 < q < infinity, be the q-Bernstein polynomials of a function f, B-n (f, 1; x) being the classical Bernstein polynomials. It is proved that, in general, {B-n (f, q(n); x)} with q(n) down arrow 1 is not an approximating sequence for f is an element of C[0, 1], in contrast to the standard case q(n) up arrow 1. At the same time, there exists a sequence 0 < delta(n) down arrow 0 such that the condition 1 <= q(n) <= delta(n) implies the approximation of f by {B-n(f, qn; x)} for all f is an element of C[0, 1].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 Citation - WoS: 3Citation - Scopus: 2The Approximation of All Continuous Functions on [0,1] by q-bernstein Polynomials in the Case q → 1+(Springer Basel Ag, 2008) Ostrovska, SofiyaSince for q > 1, the q-Bernstein polynomials B-n,B-q(f;.) are not positive linear operators on C[0, 1], their convergence properties are not similar to those in the case 0 < q = 1. It has been known that, in general, B-n,B-qn(f;.) does not approximate f is an element of C[0, 1] if q(n) -> 1(+), n ->infinity, unlike in the case q(n) -> 1(-). In this paper, it is shown that if 0 <= q(n) - 1 = o(n(-1)3(-n)), n -> infinity, then for any f is an element of C[0, 1], we have: B-n,B-qn(f; x) -> f(x) as n -> infinity, uniformly on [ 0,1].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 Citation - WoS: 1Automatic Boundedness of Adjointable Operators on Barreled Vh-Spaces(Springer Basel Ag, 2022) Ay, SerdarWe consider the space of adjointable operators on barreled VH (Vector Hilbert) spaces and show that such operators are automatically bounded. This generalizes the well known corresponding result for locally Hilbert C*-modules. We pick a consequence of this result in the dilation theory of VH-spaces and show that, when barreled VH-spaces are considered, a certain boundedness condition for the existence of VH-space linearisations, equivalently, of reproducing kernel VH-spaces, is automatically satisfied.Article Moment Determinacy Versus q-moment Determinacy of Probability Distributions(Springer Basel Ag, 2021) Ostrovska, Sofiya; Turan, MehmetSince the classical moment problem is an important issue deeply connected to various mathematical disciplines, its q-analogue based on the notion of q-moments has emerged in the study of q-distributions. For a wide class of probability distributions, both of these problems can be considered. The aim of this work is to establish a connection between the two moment problems. In this paper, the class A of probability distributions possessing finite moments of all orders and support on (0, infinity) is examined. For each q is an element of(0,1), a distribution P is an element of A can be characterized with respect to moment-determinacy as well as q-moment determinacy. It is proved that the properties of P regarding these characterizations may differ, and that the q-moment determinacy of P may depend on the value of q.
