In this study, we give a modification of Mellin convolution-type operators. In this way, we obtain the rate of convergence with the modulus of the continuity of the m m th-order Mellin derivative of function f f , but without the derivative of the operator. Then, we express the Taylor formula including Mellin derivatives with integral remainder. Later, a Voronovskaya-type theorem is proved. In the last part, we state order of approximation of the modified operators, and the obtained results are restated for the Mellin-Gauss-Weierstrass operator.
在本研究中，我们给出了梅林卷积型算子的一种修正方法。通过这种方法，我们得到了函数 f f 的 m m th 阶梅林导数连续性模数的收敛率，但没有算子的导数。然后，我们用带积分余数的梅林导数来表示泰勒公式。随后，我们证明了沃罗诺夫斯卡娅式定理。在最后一部分，我们说明了修正算子的近似阶数，并对梅林-高斯-韦尔斯特拉斯算子重述了所得结果。
{"title":"On the generalized Mellin integral operators","authors":"Cem Topuz, Firat Ozsarac, Ali Aral","doi":"10.1515/dema-2023-0133","DOIUrl":"https://doi.org/10.1515/dema-2023-0133","url":null,"abstract":"\u0000 In this study, we give a modification of Mellin convolution-type operators. In this way, we obtain the rate of convergence with the modulus of the continuity of the \u0000 \u0000 \u0000 \u0000 m\u0000 \u0000 m\u0000 \u0000 th-order Mellin derivative of function \u0000 \u0000 \u0000 \u0000 f\u0000 \u0000 f\u0000 \u0000 , but without the derivative of the operator. Then, we express the Taylor formula including Mellin derivatives with integral remainder. Later, a Voronovskaya-type theorem is proved. In the last part, we state order of approximation of the modified operators, and the obtained results are restated for the Mellin-Gauss-Weierstrass operator.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140517691","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":null,"pages":null},"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 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":null,"pages":null},"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}
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":null,"pages":null},"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 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":null,"pages":null},"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}
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":null,"pages":null},"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}
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":null,"pages":null},"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}
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":null,"pages":null},"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 We introduced a new general class of Prešić-type operators, by enriching the known class of Prešić contractions. We established conditions under which enriched Prešić operators possess a unique fixed point, proving the convergence of two different iterative methods to the fixed point. We also gave a data dependence result that was finally applied in proving the global asymptotic stability of the equilibrium of a certain k-th order difference equation.
{"title":"Asymptotic stability of equilibria for difference equations via fixed points of enriched Prešić operators","authors":"M. Pacurar","doi":"10.1515/dema-2022-0185","DOIUrl":"https://doi.org/10.1515/dema-2022-0185","url":null,"abstract":"Abstract We introduced a new general class of Prešić-type operators, by enriching the known class of Prešić contractions. We established conditions under which enriched Prešić operators possess a unique fixed point, proving the convergence of two different iterative methods to the fixed point. We also gave a data dependence result that was finally applied in proving the global asymptotic stability of the equilibrium of a certain k-th order difference equation.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42992990","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 purpose of this article is to describe the properties of the pair of solutions of several systems of Fermat-type partial differential difference equations. Our theorems exhibit the forms of finite order transcendental entire solutions for these systems, which are some extensions and improvement of the previous theorems given by Xu, Cao, Liu, etc. Furthermore, we give a series of examples to show that the existence conditions and the forms of transcendental entire solutions with finite order of such systems are precise.
{"title":"The study of solutions for several systems of PDDEs with two complex variables","authors":"Yi-Hui Xu, Xiao Lan Liu, H. Xu","doi":"10.1515/dema-2022-0241","DOIUrl":"https://doi.org/10.1515/dema-2022-0241","url":null,"abstract":"Abstract The purpose of this article is to describe the properties of the pair of solutions of several systems of Fermat-type partial differential difference equations. Our theorems exhibit the forms of finite order transcendental entire solutions for these systems, which are some extensions and improvement of the previous theorems given by Xu, Cao, Liu, etc. Furthermore, we give a series of examples to show that the existence conditions and the forms of transcendental entire solutions with finite order of such systems are precise.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":null,"pages":null},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43594261","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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