Abstract In this article, we deal with the following p p -fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity: M ( [ u ] s , A p ) ( − Δ ) p , A s u + V ( x ) ∣ u ∣ p − 2 u = λ ∫ R N ∣ u ∣ p μ , s * ∣ x − y ∣ μ d y ∣ u ∣ p μ , s * − 2 u + k ∣ u ∣ q − 2 u , x ∈ R N , M({left[u]}_{s,A}^{p}){left(-Delta )}_{p,A}^{s}u+Vleft(x){| u| }^{p-2}u=lambda left(mathop{int }limits_{{{mathbb{R}}}^{N}}frac{{| u| }^{{p}_{mu ,s}^{* }}}{{| x-y| }^{mu }}{rm{d}}yright){| u| }^{{p}_{mu ,s}^{* }-2}u+k{| u| }^{q-2}u,hspace{1em}xin {{mathbb{R}}}^{N}, where 0 < s < 1 < p 0lt slt 1lt p , p s < N pslt N , p < q < 2 p s , μ * plt qlt 2{p}_{s,mu }^{* } , 0 < μ < N 0lt mu lt N , λ lambda , and k k are some positive parameters, p s , μ * = p N − p μ 2 N − p s {p}_{s,mu }^{* }=frac{pN-pfrac{mu }{2}}{N-ps} is the critical exponent with respect to the Hardy-Littlewood-Sobolev inequality, and functions V V and M M satisfy the suitable conditions. By proving the compactness results using the fractional version of concentration compactness principle, we establish the existence of nontrivial solutions to this problem.
摘要 本文处理了以下带有电磁场和 Hardy-Littlewood-Sobolev 非线性的 p p 分薛定谔-基尔霍夫方程：M ( [ u ] s , A p ) ( - Δ ) p , A s u + V ( x ) ∣ u ∣ p - 2 u = λ ∫ R N ∣ u ∣ p μ 、s * ∣ x - y ∣ μ d y ∣ u ∣ p μ , s * - 2 u + k ∣ u ∣ q - 2 u , x∈ R N , M({left[u]}_{s,A}^{p}){left(-Delta )}_{p、A}^{s}u+Vleft(x){| u| }^{p-2}u=lambda left(mathop{int }limits_{{mathbb{R}}}^{N}}frac{{| u| }^{p}_{mu ,s}^{* }}{{x-y| }^{mu }}{rm{d}}yright){| u| }^{p}_{mu 、s}^{* }-2}u+k{| u| }^{q-2}u,hspace{1em}xin {{mathbb{R}}}^{N}, where 0 < s < 1 < p 0lt slt 1lt p , p s < N pslt N , p < q < 2 p s , μ * plt qlt 2{p}_{s,mu }^{* }.0 < μ < N 0lt mu lt N , λ lambda , 和 k k 是一些正参数，p s , μ * = p N - p μ 2 N - p s {p}_{s,mu }^{* }=frac{pN-pfrac{mu }{2}}{N-ps} 是关于哈代-利特尔伍德-索博列夫不等式的临界指数，函数 V V 和 M M 满足合适的条件。通过利用分数版的集中紧凑性原理证明紧凑性结果，我们确定了此问题的非小解的存在性。
{"title":"On the p-fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity","authors":"Min Zhao, Yueqiang Song, D. D. Repovš","doi":"10.1515/dema-2023-0124","DOIUrl":"https://doi.org/10.1515/dema-2023-0124","url":null,"abstract":"Abstract In this article, we deal with the following p p -fractional Schrödinger-Kirchhoff equations with electromagnetic fields and the Hardy-Littlewood-Sobolev nonlinearity: M ( [ u ] s , A p ) ( − Δ ) p , A s u + V ( x ) ∣ u ∣ p − 2 u = λ ∫ R N ∣ u ∣ p μ , s * ∣ x − y ∣ μ d y ∣ u ∣ p μ , s * − 2 u + k ∣ u ∣ q − 2 u , x ∈ R N , M({left[u]}_{s,A}^{p}){left(-Delta )}_{p,A}^{s}u+Vleft(x){| u| }^{p-2}u=lambda left(mathop{int }limits_{{{mathbb{R}}}^{N}}frac{{| u| }^{{p}_{mu ,s}^{* }}}{{| x-y| }^{mu }}{rm{d}}yright){| u| }^{{p}_{mu ,s}^{* }-2}u+k{| u| }^{q-2}u,hspace{1em}xin {{mathbb{R}}}^{N}, where 0 < s < 1 < p 0lt slt 1lt p , p s < N pslt N , p < q < 2 p s , μ * plt qlt 2{p}_{s,mu }^{* } , 0 < μ < N 0lt mu lt N , λ lambda , and k k are some positive parameters, p s , μ * = p N − p μ 2 N − p s {p}_{s,mu }^{* }=frac{pN-pfrac{mu }{2}}{N-ps} is the critical exponent with respect to the Hardy-Littlewood-Sobolev inequality, and functions V V and M M satisfy the suitable conditions. By proving the compactness results using the fractional version of concentration compactness principle, we establish the existence of nontrivial solutions to this problem.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":"77 16","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139458306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yuankui Ma, Lingling Luo, Taekyun Kim, Hongze Li, Wenpeng Zhang
Abstract Dedekind sums and their generalizations are defined in terms of Bernoulli functions and their generalizations. As a new generalization of the Dedekind sums, the degenerate poly-Dedekind sums, which are obtained from the Dedekind sums by replacing Bernoulli functions by degenerate poly-Bernoulli functions of arbitrary indices are introduced in this article and are shown to satisfy a reciprocity relation.
{"title":"A study on a type of degenerate poly-Dedekind sums","authors":"Yuankui Ma, Lingling Luo, Taekyun Kim, Hongze Li, Wenpeng Zhang","doi":"10.1515/dema-2023-0121","DOIUrl":"https://doi.org/10.1515/dema-2023-0121","url":null,"abstract":"Abstract Dedekind sums and their generalizations are defined in terms of Bernoulli functions and their generalizations. As a new generalization of the Dedekind sums, the degenerate poly-Dedekind sums, which are obtained from the Dedekind sums by replacing Bernoulli functions by degenerate poly-Bernoulli functions of arbitrary indices are introduced in this article and are shown to satisfy a reciprocity relation.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":"18 12","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139456872","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Shahzaib Ashraf, Muhammad Sohail, Muhammad Shakir Chohan, Siriluk Paokanta, Choonkil Park
Abstract This article presents a higher-order circular intuitionistic fuzzy time series forecasting method for predicting the stock change index, which is shown to be an improvement over traditional time series forecasting methods. The method is based on the principles of circular intuitionistic fuzzy set theory. It uses both positive and negative membership values and a circular radius to handle uncertainty and imprecision in the data. The circularity of the time series is also taken into consideration, leading to more accurate and robust forecasts. The higher-order forecasting capability of this method provides more comprehensive predictions compared to previous methods. One of the key challenges we face when using the amount featured as a case study in our article to project the future value of ratings is the influence of the stock market index. Through rigorous experiments and comparison with traditional time series forecasting methods, the results of the study demonstrate that the proposed higher-order circular intuitionistic fuzzy time series forecasting method is a superior approach for predicting the stock change index.
{"title":"Higher-order circular intuitionistic fuzzy time series forecasting methodology: Application of stock change index","authors":"Shahzaib Ashraf, Muhammad Sohail, Muhammad Shakir Chohan, Siriluk Paokanta, Choonkil Park","doi":"10.1515/dema-2023-0115","DOIUrl":"https://doi.org/10.1515/dema-2023-0115","url":null,"abstract":"Abstract This article presents a higher-order circular intuitionistic fuzzy time series forecasting method for predicting the stock change index, which is shown to be an improvement over traditional time series forecasting methods. The method is based on the principles of circular intuitionistic fuzzy set theory. It uses both positive and negative membership values and a circular radius to handle uncertainty and imprecision in the data. The circularity of the time series is also taken into consideration, leading to more accurate and robust forecasts. The higher-order forecasting capability of this method provides more comprehensive predictions compared to previous methods. One of the key challenges we face when using the amount featured as a case study in our article to project the future value of ratings is the influence of the stock market index. Through rigorous experiments and comparison with traditional time series forecasting methods, the results of the study demonstrate that the proposed higher-order circular intuitionistic fuzzy time series forecasting method is a superior approach for predicting the stock change index.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":"103 22","pages":""},"PeriodicalIF":2.0,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139454073","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The aim of this article is to introduce α alpha - ψ psi -proximal contraction in the setting of ℱ-metric space and prove the existence of best proximity points for these contractions. As applications of our main results, we obtain coupled best proximity points on ℱ-metric space equipped with an arbitrary binary relation.
{"title":"Best proximity points in ℱ-metric spaces with applications","authors":"Durdana Lateef","doi":"10.1515/dema-2022-0191","DOIUrl":"https://doi.org/10.1515/dema-2022-0191","url":null,"abstract":"Abstract The aim of this article is to introduce α alpha - ψ psi -proximal contraction in the setting of ℱ-metric space and prove the existence of best proximity points for these contractions. As applications of our main results, we obtain coupled best proximity points on ℱ-metric space equipped with an arbitrary binary relation.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":" ","pages":""},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48167068","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
T. Saeed, M. Khan, Shah Faisal, H. Alsulami, M. Alhodaly
Abstract The Hermite-Hadamard inequality is regarded as one of the most favorable inequalities from the research point of view. Currently, mathematicians are working on extending, improving, and generalizing this inequality. This article presents conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in weighted and unweighted forms by using the idea of majorization and convexity together with generalized conformable fractional integral operators. They not only represent continuous and discrete inequalities in compact form but also produce generalized inequalities connecting various fractional operators such as Hadamard, Katugampola, Riemann-Liouville, conformable, and Rieman integrals into one single form. Also, two new integral identities have been investigated pertaining a differentiable function and three tuples. By using these identities and assuming ∣ f ′ ∣ | f^{prime} | and ∣ f ′ ∣ q ( q > 1 ) | f^{prime} {| }^{q}hspace{0.33em}left(qgt 1) as convex, we deduce bounds concerning the discrepancy of the terms of the main inequalities.
{"title":"New conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in terms of generalized conformable fractional operators via majorization","authors":"T. Saeed, M. Khan, Shah Faisal, H. Alsulami, M. Alhodaly","doi":"10.1515/dema-2022-0225","DOIUrl":"https://doi.org/10.1515/dema-2022-0225","url":null,"abstract":"Abstract The Hermite-Hadamard inequality is regarded as one of the most favorable inequalities from the research point of view. Currently, mathematicians are working on extending, improving, and generalizing this inequality. This article presents conticrete inequalities of the Hermite-Hadamard-Jensen-Mercer type in weighted and unweighted forms by using the idea of majorization and convexity together with generalized conformable fractional integral operators. They not only represent continuous and discrete inequalities in compact form but also produce generalized inequalities connecting various fractional operators such as Hadamard, Katugampola, Riemann-Liouville, conformable, and Rieman integrals into one single form. Also, two new integral identities have been investigated pertaining a differentiable function and three tuples. By using these identities and assuming ∣ f ′ ∣ | f^{prime} | and ∣ f ′ ∣ q ( q > 1 ) | f^{prime} {| }^{q}hspace{0.33em}left(qgt 1) as convex, we deduce bounds concerning the discrepancy of the terms of the main inequalities.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":" ","pages":""},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46742265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Cauchy problems with fractal-fractional differential operators with a power law, exponential decay, and the generalized Mittag-Leffler kernels are considered in this work. We start with deriving some important inequalities, and then by using the linear growth and Lipchitz conditions, we derive the conditions under which these equations admit unique solutions. A numerical scheme was suggested for each case to derive a numerical solution to the equation. Some examples of fractal-fractional differential equations were presented, and their exact solutions were obtained and compared with the used numerical scheme. A nonlinear case was considered and solved, and numerical solutions were presented graphically for different values of fractional orders and fractal dimensions.
{"title":"Analysis of Cauchy problem with fractal-fractional differential operators","authors":"N. Alharthi, A. Atangana, B. Alkahtani","doi":"10.1515/dema-2022-0181","DOIUrl":"https://doi.org/10.1515/dema-2022-0181","url":null,"abstract":"Abstract Cauchy problems with fractal-fractional differential operators with a power law, exponential decay, and the generalized Mittag-Leffler kernels are considered in this work. We start with deriving some important inequalities, and then by using the linear growth and Lipchitz conditions, we derive the conditions under which these equations admit unique solutions. A numerical scheme was suggested for each case to derive a numerical solution to the equation. Some examples of fractal-fractional differential equations were presented, and their exact solutions were obtained and compared with the used numerical scheme. A nonlinear case was considered and solved, and numerical solutions were presented graphically for different values of fractional orders and fractal dimensions.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":" ","pages":""},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49616299","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract Fractional Hahn boundary value problems are significant tools to describe mathematical and physical phenomena depending on non-differentiable functions. In this work, we develop certain aspects of the theory of fractional Hahn boundary value problems involving fractional Hahn derivatives of the Caputo type. First, we construct the Green function for an α th alpha {rm{th}} -order fractional boundary value problem, with 1 < α < 2 1lt alpha lt 2 , and discuss some important properties of the Green function. The solutions to the proposed problems are obtained in terms of the Green function. The uniqueness of the solutions is proved by various fixed point theorems. The Banach’s contraction mapping theorem, the Schauder’s theorem, and the Browder’s theorem are used.
{"title":"Some results on fractional Hahn difference boundary value problems","authors":"Elsaddam A. Baheeg, K. Oraby, M. Akel","doi":"10.1515/dema-2022-0247","DOIUrl":"https://doi.org/10.1515/dema-2022-0247","url":null,"abstract":"Abstract Fractional Hahn boundary value problems are significant tools to describe mathematical and physical phenomena depending on non-differentiable functions. In this work, we develop certain aspects of the theory of fractional Hahn boundary value problems involving fractional Hahn derivatives of the Caputo type. First, we construct the Green function for an α th alpha {rm{th}} -order fractional boundary value problem, with 1 < α < 2 1lt alpha lt 2 , and discuss some important properties of the Green function. The solutions to the proposed problems are obtained in terms of the Green function. The uniqueness of the solutions is proved by various fixed point theorems. The Banach’s contraction mapping theorem, the Schauder’s theorem, and the Browder’s theorem are used.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":" ","pages":""},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45551179","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Chonjaroen Chairatsiripong, D. Yambangwai, T. Thianwan
Abstract In this article, weak and strong convergence theorems of the M-iteration method for 𝒢-nonexpansive mapping in a uniformly convex Banach space with a directed graph were established. Moreover, weak convergence theorem without making use of Opial’s condition is proved. The rate of convergence between the M-iteration and some other iteration processes in the literature was also compared. Specifically, our main result shows that the M-iteration converges faster than the Noor and SP iterations. Finally, the numerical examples to compare convergence behavior of the M-iteration with the three-step Noor iteration and the SP-iteration were given. As application, some numerical experiments in real-world problems were provided, focused on image deblurring and signal recovering problems.
{"title":"Convergence analysis of M-iteration for 𝒢-nonexpansive mappings with directed graphs applicable in image deblurring and signal recovering problems","authors":"Chonjaroen Chairatsiripong, D. Yambangwai, T. Thianwan","doi":"10.1515/dema-2022-0234","DOIUrl":"https://doi.org/10.1515/dema-2022-0234","url":null,"abstract":"Abstract In this article, weak and strong convergence theorems of the M-iteration method for 𝒢-nonexpansive mapping in a uniformly convex Banach space with a directed graph were established. Moreover, weak convergence theorem without making use of Opial’s condition is proved. The rate of convergence between the M-iteration and some other iteration processes in the literature was also compared. Specifically, our main result shows that the M-iteration converges faster than the Noor and SP iterations. Finally, the numerical examples to compare convergence behavior of the M-iteration with the three-step Noor iteration and the SP-iteration were given. As application, some numerical experiments in real-world problems were provided, focused on image deblurring and signal recovering problems.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":" ","pages":""},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44610628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The study presented in this article involves q-calculus connected to fractional calculus applied in the univalent functions theory. Riemann-Liouville fractional integral of q-hypergeometric function is defined here, and investigations are conducted using the theories of differential subordination and superordination. Theorems and corollaries containing new subordination and superordination results are proved for which best dominants and best subordinants are given, respectively. As an application of the results obtained by the means of the two theories, the statement of a sandwich-type theorem concludes the study.
{"title":"Sandwich-type results regarding Riemann-Liouville fractional integral of q-hypergeometric function","authors":"A. Alb Lupaș, G. Oros","doi":"10.1515/dema-2022-0186","DOIUrl":"https://doi.org/10.1515/dema-2022-0186","url":null,"abstract":"Abstract The study presented in this article involves q-calculus connected to fractional calculus applied in the univalent functions theory. Riemann-Liouville fractional integral of q-hypergeometric function is defined here, and investigations are conducted using the theories of differential subordination and superordination. Theorems and corollaries containing new subordination and superordination results are proved for which best dominants and best subordinants are given, respectively. As an application of the results obtained by the means of the two theories, the statement of a sandwich-type theorem concludes the study.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":" ","pages":""},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44974129","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract The aim of this article is to construct univariate Bernstein-type operators ( ℬ m x G ) ( x , z ) left({{mathcal{ {mathcal B} }}}_{m}^{x}G)left(x,z) and ( ℬ n z G ) ( x , z ) , left({{mathcal{ {mathcal B} }}}_{n}^{z}G)left(x,z), their products ( P m n G ) ( x , z ) left({{mathcal{P}}}_{mn}G)left(x,z) , ( Q n m G ) ( x , z ) left({{mathcal{Q}}}_{nm}G)left(x,z) , and their Boolean sums ( S m n G ) ( x , z ) left({{mathcal{S}}}_{mn}G)left(x,z) , ( T n m G ) ( x , z ) left({{mathcal{T}}}_{nm}G)left(x,z) on elliptic region, which interpolate the given real valued function G G defined on elliptic region on its boundary. The bound of the remainders of each approximation formula of corresponding operators are computed with the help of Peano’s theorem and modulus of continuity, and the rate of convergence for functions of Lipschitz class is computed.
{"title":"Bernstein-type operators on elliptic domain and their interpolation properties","authors":"M. Iliyas, Asif Khan, M. Mursaleen","doi":"10.1515/dema-2022-0199","DOIUrl":"https://doi.org/10.1515/dema-2022-0199","url":null,"abstract":"Abstract The aim of this article is to construct univariate Bernstein-type operators ( ℬ m x G ) ( x , z ) left({{mathcal{ {mathcal B} }}}_{m}^{x}G)left(x,z) and ( ℬ n z G ) ( x , z ) , left({{mathcal{ {mathcal B} }}}_{n}^{z}G)left(x,z), their products ( P m n G ) ( x , z ) left({{mathcal{P}}}_{mn}G)left(x,z) , ( Q n m G ) ( x , z ) left({{mathcal{Q}}}_{nm}G)left(x,z) , and their Boolean sums ( S m n G ) ( x , z ) left({{mathcal{S}}}_{mn}G)left(x,z) , ( T n m G ) ( x , z ) left({{mathcal{T}}}_{nm}G)left(x,z) on elliptic region, which interpolate the given real valued function G G defined on elliptic region on its boundary. The bound of the remainders of each approximation formula of corresponding operators are computed with the help of Peano’s theorem and modulus of continuity, and the rate of convergence for functions of Lipschitz class is computed.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":" ","pages":""},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47069950","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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