The aim of this short survey is to collect and combine basic notions and results in the fixed point theory in the context of $b$-metric spaces. It is also aimed to show that there are still enough rooms for several researchers in this interesting direction and a huge application potential.
{"title":"A Short Survey on the Recent Fixed Point Results on $b$-Metric Spaces","authors":"E. Karapınar","doi":"10.33205/CMA.453034","DOIUrl":"https://doi.org/10.33205/CMA.453034","url":null,"abstract":"The aim of this short survey is to collect and combine basic notions and results in the fixed point theory in the context of $b$-metric spaces. It is also aimed to show that there are still enough rooms for several researchers in this interesting direction and a huge application potential.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45112850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present work, our aim of this study is generalization and extension of the theory of interpolation of two dimensional functions to functionals or operators by means of Urysohn type nonlinear operators. In accordance with this purpose, we introduce and study a new type of Urysohn type nonlinear operators. In particular, we investigate the convergence problem for nonlinear operators that approximate the Urysohn type operator in two dimensional case. The starting point of this study is motivated by the important applications that approximation properties of certain families of nonlinear operators have in signal-image reconstruction and in other related fields. We construct our nonlinear operators by using a nonlinear form of the kernels together with the Urysohn type operator values instead of the sampling values of the function.
{"title":"Approximation Results for Urysohn Type Two Dimensional Nonlinear Bernstein Operators","authors":"H. Karsli","doi":"10.33205/CMA.453027","DOIUrl":"https://doi.org/10.33205/CMA.453027","url":null,"abstract":"In the present work, our aim of this study is generalization and extension of the theory of interpolation of two dimensional functions to functionals or operators by means of Urysohn type nonlinear operators. In accordance with this purpose, we introduce and study a new type of Urysohn type nonlinear operators. In particular, we investigate the convergence problem for nonlinear operators that approximate the Urysohn type operator in two dimensional case. The starting point of this study is motivated by the important applications that approximation properties of certain families of nonlinear operators have in signal-image reconstruction and in other related fields. We construct our nonlinear operators by using a nonlinear form of the kernels together with the Urysohn type operator values instead of the sampling values of the function.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44473461","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present article, we study the approximation of difference of operators and find the quantitative estimates for the difference of Lupas operators with Lupas-Sz asz operators and Lupas-Kantorovich operators in terms of modulus of continuity. Also, we find the quantitative estimate for the difference of Lupas-Kantorovich operators and Lupas-Szasz operators.
{"title":"Differences of Operators of Lupaş Type","authors":"Vijay Gupta","doi":"10.33205/CMA.452962","DOIUrl":"https://doi.org/10.33205/CMA.452962","url":null,"abstract":"In the present article, we study the approximation of difference of operators and find the quantitative estimates for the difference of Lupas operators with Lupas-Sz asz operators and Lupas-Kantorovich operators in terms of modulus of continuity. Also, we find the quantitative estimate for the difference of Lupas-Kantorovich operators and Lupas-Szasz operators.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47457503","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we introduce Kantorovich type modification of $(p, q)$-Meyer-K o nig-Zeller operators. We estimate rate of convergence of proposed operators using modulus of continuity and Lipschitz class functions. Further, we obtain the statistical convergence and local approximation results for these operators. In the last section, we estimate the rate of convergence of $(p, q)$-Meyer-K o nig-Zeller Kantorovich operators by means of Matlab programming.
本文介绍了$（p，q）$-Meyer-K o nig-Zeller算子的Kantorovich型修改。我们使用连续模和Lipschitz类函数来估计所提出的算子的收敛速度。此外，我们还得到了这些算子的统计收敛性和局部逼近结果。在最后一节中，我们通过Matlab编程估计了$（p，q）$-Meyer-K o nig-Zeller-Kantorovich算子的收敛速度。
{"title":"Approximation Properties of Kantorovich Type Modifications of $(p, q)-$Meyer-König-Zeller Operators","authors":"Ramapati Maurya, Honey Sharma, Cheeena Gupta","doi":"10.33205/CMA.436071","DOIUrl":"https://doi.org/10.33205/CMA.436071","url":null,"abstract":"In this paper, we introduce Kantorovich type modification of $(p, q)$-Meyer-K o nig-Zeller operators. We estimate rate of convergence of proposed operators using modulus of continuity and Lipschitz class functions. Further, we obtain the statistical convergence and local approximation results for these operators. In the last section, we estimate the rate of convergence of $(p, q)$-Meyer-K o nig-Zeller Kantorovich operators by means of Matlab programming.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48521536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The purpose of this paper is to construct a general class of operators which has known Baskakov-Sza sz-Stancu that preserving constant and $e^{2ax}, a>0$ functions. We scrutinize a uniform convergence result and analyze the asymptotic behavior of our operators, as well. Finally, we discuss the convergence of corresponding sequences in exponential weighted spaces and make a comparison about which one approximates better between classical Baskakov-Sza sz-Stancu operators and the recent operators.
{"title":"Approximation by Baskakov-Szász-Stancu Operators Preserving Exponential Functions","authors":"Murat Bodur, Övgü Gürel Yılmaz, A. Aral","doi":"10.33205/CMA.450708","DOIUrl":"https://doi.org/10.33205/CMA.450708","url":null,"abstract":"The purpose of this paper is to construct a general class of operators which has known Baskakov-Sza sz-Stancu that preserving constant and $e^{2ax}, a>0$ functions. We scrutinize a uniform convergence result and analyze the asymptotic behavior of our operators, as well. Finally, we discuss the convergence of corresponding sequences in exponential weighted spaces and make a comparison about which one approximates better between classical Baskakov-Sza sz-Stancu operators and the recent operators.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2018-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45165976","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Euler--Maclaurin (EM) summation formula is used in many theoretical studies and numerical calculations. It approximates the sum $sum_{k=0}^{n-1} f(k)$ of values of a function $f$ by a linear combination of a corresponding integral of $f$ and values of its higher-order derivatives $f^{(j)}$. An alternative (Alt) summation formula was recently presented by the author, which approximates the sum by a linear combination of integrals only, without using high-order derivatives of $f$. It was shown that the Alt formula will in most cases outperform, or greatly outperform, the EM formula in terms of the execution time and memory use. In the present paper, a multiple-sum/multi-index-sum extension of the Alt formula is given, with applications to summing possibly divergent multi-index series and to sums over the integral points of integral lattice polytopes.
{"title":"Approximating sums by integrals only: multiple sums and sums over lattice polytopes","authors":"I. Pinelis","doi":"10.33205/cma.1102689","DOIUrl":"https://doi.org/10.33205/cma.1102689","url":null,"abstract":"The Euler--Maclaurin (EM) summation formula is used in many theoretical studies and numerical calculations. It approximates the sum $sum_{k=0}^{n-1} f(k)$ of values of a function $f$ by a linear combination of a corresponding integral of $f$ and values of its higher-order derivatives $f^{(j)}$. An alternative (Alt) summation formula was recently presented by the author, which approximates the sum by a linear combination of integrals only, without using high-order derivatives of $f$. It was shown that the Alt formula will in most cases outperform, or greatly outperform, the EM formula in terms of the execution time and memory use. In the present paper, a multiple-sum/multi-index-sum extension of the Alt formula is given, with applications to summing possibly divergent multi-index series and to sums over the integral points of integral lattice polytopes.","PeriodicalId":36038,"journal":{"name":"Constructive Mathematical Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2017-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43063326","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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