Abstract The main goal of this paper is to classify the global phase portraits of seven quadratic polynomial differential systems, exhibiting as invariant algebraic curves seven well-known algebraic curves of degree six. We prove that these systems have five topologically different phase portraits in the Poincarédisc.
{"title":"Global Phase Portraits of Quadratic Polynomial Differential Systems Having as Solution Some Classical Planar Algebraic Curves of Degree 6","authors":"R. Benterki, Ahlam Belfar","doi":"10.2478/tmmp-2022-0010","DOIUrl":"https://doi.org/10.2478/tmmp-2022-0010","url":null,"abstract":"Abstract The main goal of this paper is to classify the global phase portraits of seven quadratic polynomial differential systems, exhibiting as invariant algebraic curves seven well-known algebraic curves of degree six. We prove that these systems have five topologically different phase portraits in the Poincarédisc.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"81 1","pages":"129 - 144"},"PeriodicalIF":0.0,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45081778","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}
Abstract In this paper, we deal with the properties (a)R-star-semi-Lindelöf and (a)M-star-semi-Lindelöf in (a)topological spaces. These properties are interesting as every (a)Rs-separable space is (a)R-star-semi-Lindelöf and every (a)s-semi-Lindelöf space is (a)R-star-semi-Lindelöf but not every (a)R-star-semi-Lindelöf space is (a)Rs-separable or (a)s-semi-Lindelöf. It is shown that if an (a)topological space X is the union of countably many (a)-open and (a)Rstar-semi-Lindelöf subspaces, then X is (a)R-star-semi-Lindelöf. Similar results are obtained in the context of (a)M-star-semi-Lindelöf spaces. Further, suitable and required counterexamples are given.
{"title":"Selective Version of Star-Semi-Lindelöfness in (a) Topological Spaces","authors":"Sheetal Luthra, Harsh V. S. Chauhan, B. Tyagi","doi":"10.2478/tmmp-2022-0002","DOIUrl":"https://doi.org/10.2478/tmmp-2022-0002","url":null,"abstract":"Abstract In this paper, we deal with the properties (a)R-star-semi-Lindelöf and (a)M-star-semi-Lindelöf in (a)topological spaces. These properties are interesting as every (a)Rs-separable space is (a)R-star-semi-Lindelöf and every (a)s-semi-Lindelöf space is (a)R-star-semi-Lindelöf but not every (a)R-star-semi-Lindelöf space is (a)Rs-separable or (a)s-semi-Lindelöf. It is shown that if an (a)topological space X is the union of countably many (a)-open and (a)Rstar-semi-Lindelöf subspaces, then X is (a)R-star-semi-Lindelöf. Similar results are obtained in the context of (a)M-star-semi-Lindelöf spaces. Further, suitable and required counterexamples are given.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"81 1","pages":"39 - 56"},"PeriodicalIF":0.0,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45130763","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}
Abstract It is well known that if the surface f : [−1, 1] × [−1, 1] → ℝ has a finite area, then the total variations of both sections fx(y)= f (x, y)and f y(x) = f (x, y)of f are finite almost everywhere in [−1, 1]. In the note it is proved that these variations can be infinite on residual subsets of [−1, 1].
摘要众所周知，如果曲面f:[−1,1]x[−1,1]→∈有一个有限的面积，那么f的两个截面fx(y)= f (x, y)和f (x) = f (x, y)在[−1,1]内几乎处处都是有限的。在注释中证明了这些变化在[−1,1]的残差子集上是无限的。
{"title":"Remark on a Theorem of Tonelli","authors":"W. Wilczyński","doi":"10.2478/tmmp-2022-0006","DOIUrl":"https://doi.org/10.2478/tmmp-2022-0006","url":null,"abstract":"Abstract It is well known that if the surface f : [−1, 1] × [−1, 1] → ℝ has a finite area, then the total variations of both sections fx(y)= f (x, y)and f y(x) = f (x, y)of f are finite almost everywhere in [−1, 1]. In the note it is proved that these variations can be infinite on residual subsets of [−1, 1].","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"81 1","pages":"89 - 92"},"PeriodicalIF":0.0,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43771223","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}
Abstract In this paper, we prove the equivalence between a K-functional and a modulus of smoothness generated by Laguerre-Bessel operator on 𝕂=[0,+∞[×[0,+∞[. mathbb{K} = [0, + infty [ times [0, + infty [.
{"title":"Modulus of Smoothness and K-Functionals Constructed by Generalized Laguerre-Bessel Operator","authors":"L. Rakhimi, R. Daher","doi":"10.2478/tmmp-2022-0008","DOIUrl":"https://doi.org/10.2478/tmmp-2022-0008","url":null,"abstract":"Abstract In this paper, we prove the equivalence between a K-functional and a modulus of smoothness generated by Laguerre-Bessel operator on 𝕂=[0,+∞[×[0,+∞[. mathbb{K} = [0, + infty [ times [0, + infty [.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"81 1","pages":"107 - 116"},"PeriodicalIF":0.0,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42721143","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}
Abstract We study the numerical radius of bounded operators on direct sum of a family of Hilbert spaces with respect to the ℓp-norm, where 1 ≤ p ≤∞. We propose a new method which enables us to prove validity of many inequalities on numerical radius of bounded operators on Hilbert spaces when the underling space is a direct sum of Hilbert spaces with ℓp-norm, where 1 ≤ p ≤ 2. We also provide an example to show that some known results on numerical radius are not true for a space that is the set of bounded operators on ℓp-sum of Hilbert spaces where 2
{"title":"Numerical Radius of Bounded Operators with ℓp-Norm","authors":"Sadaf Fakri Moghaddam, A. Mirmostafaee","doi":"10.2478/tmmp-2022-0012","DOIUrl":"https://doi.org/10.2478/tmmp-2022-0012","url":null,"abstract":"Abstract We study the numerical radius of bounded operators on direct sum of a family of Hilbert spaces with respect to the ℓp-norm, where 1 ≤ p ≤∞. We propose a new method which enables us to prove validity of many inequalities on numerical radius of bounded operators on Hilbert spaces when the underling space is a direct sum of Hilbert spaces with ℓp-norm, where 1 ≤ p ≤ 2. We also provide an example to show that some known results on numerical radius are not true for a space that is the set of bounded operators on ℓp-sum of Hilbert spaces where 2","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"81 1","pages":"155 - 164"},"PeriodicalIF":0.0,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48710774","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}
Abstract In this article, we mainly study the interpolations between arithmetic mean and geometric mean—power mean, Heron mean and Heinz mean. First, we obtain the improvement and reverse improvement of arithmetic-power mean inequalities by the convexity of the function. We show that the proof of Heron mean inequality due to Yang and Ren: [Some results of Heron mean and Young’s inequalities, J. Inequal. Appl. 2018 (2018), paper no, 172], is not substantial. In addition, we also obtain Heron-Heinz mean inequalities for t ∈ ℝ. Further corresponding operator versions and generalizations are also established.
{"title":"Some Inequalities Involving Interpolations Between Arithmetic and Geometric Mean","authors":"Hongliang Zuo, Yuwei Li","doi":"10.2478/tmmp-2022-0007","DOIUrl":"https://doi.org/10.2478/tmmp-2022-0007","url":null,"abstract":"Abstract In this article, we mainly study the interpolations between arithmetic mean and geometric mean—power mean, Heron mean and Heinz mean. First, we obtain the improvement and reverse improvement of arithmetic-power mean inequalities by the convexity of the function. We show that the proof of Heron mean inequality due to Yang and Ren: [Some results of Heron mean and Young’s inequalities, J. Inequal. Appl. 2018 (2018), paper no, 172], is not substantial. In addition, we also obtain Heron-Heinz mean inequalities for t ∈ ℝ. Further corresponding operator versions and generalizations are also established.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"81 1","pages":"93 - 106"},"PeriodicalIF":0.0,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48899808","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}
Fatima Si Bachir, S. Abbas, Maamar Benbachir, M. Benchohra
Abstract In this work we deal with a uniqueness result of solutions for a class of fractional differential equations involving the Caputo-Fabrizio derivative. We provide a result on the global convergence of successive approximations.
{"title":"Successive Approximations for Caputo-Fabrizio Fractional Differential Equations","authors":"Fatima Si Bachir, S. Abbas, Maamar Benbachir, M. Benchohra","doi":"10.2478/tmmp-2022-0009","DOIUrl":"https://doi.org/10.2478/tmmp-2022-0009","url":null,"abstract":"Abstract In this work we deal with a uniqueness result of solutions for a class of fractional differential equations involving the Caputo-Fabrizio derivative. We provide a result on the global convergence of successive approximations.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"81 1","pages":"117 - 128"},"PeriodicalIF":0.0,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42529852","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}
Abstract This article is devoted to sets having the Moran structure. The main attention is given to topological, metric, and fractal properties of certain sets whose elements have restrictions on using digits or combinations of digits in own representations.
{"title":"Some Fractal Properties of Sets Having the Moran Structure","authors":"S. Serbenyuk","doi":"10.2478/tmmp-2022-0001","DOIUrl":"https://doi.org/10.2478/tmmp-2022-0001","url":null,"abstract":"Abstract This article is devoted to sets having the Moran structure. The main attention is given to topological, metric, and fractal properties of certain sets whose elements have restrictions on using digits or combinations of digits in own representations.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"81 1","pages":"1 - 38"},"PeriodicalIF":0.0,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42740617","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}
Abstract The aim of this paper is to establish some generalized integral inequalities for convex functions of 2-variables on the co-ordinat. Then, we will give a generalized identity and with the help of this integral identity, we will investigate some integral inequalities connected with the right hand side of the Hermite-Hadamard type inequalities involving Riemann integrals and Riemann-Liouville fractional integrals.
{"title":"On the Generalized Inequalities for Co-Ordinated Convex Functions","authors":"M. Sarıkaya","doi":"10.2478/tmmp-2022-0004","DOIUrl":"https://doi.org/10.2478/tmmp-2022-0004","url":null,"abstract":"Abstract The aim of this paper is to establish some generalized integral inequalities for convex functions of 2-variables on the co-ordinat. Then, we will give a generalized identity and with the help of this integral identity, we will investigate some integral inequalities connected with the right hand side of the Hermite-Hadamard type inequalities involving Riemann integrals and Riemann-Liouville fractional integrals.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"81 1","pages":"69 - 80"},"PeriodicalIF":0.0,"publicationDate":"2022-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47084044","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}
Abstract The system of nonlinear neutral difference equations with delays in the form { Δ(yi(n)+pi(n)yi(n−τi))=ai(n)fi(yi+1(n))+gi(n),Δ(ym(n)+pm(n)ym(n−τm))=am(n)fm(y1(n))+gm(n),[left{ begin{array}{l} Delta ({y_i}(n) + {p_i}(n){y_i}(n - {tau _i})) = {a_i}(n){f_i}({y_{i + 1}}(n)) + {g_i}(n), Delta ({y_m}(n) + {p_m}(n){y_m}(n - {tau _m})) = {a_m}(n){f_m}({y_1}(n)) + {g_m}(n), end{array} right.] for i = 1, . . . , m − 1, m ≥ 2, is studied. The sufficient conditions for the existence of an asymptotically periodic solution of the above system, are established. Here sequences (pi(n)), i = 1,..., m, are bounded away from -1. The presented results are illustrated by theoretical and numerical examples.
{"title":"Existence of The Asymptotically Periodic Solution to the System of Nonlinear Neutral Difference Equations","authors":"E. Schmeidel, M. Zdanowicz","doi":"10.2478/tmmp-2021-0025","DOIUrl":"https://doi.org/10.2478/tmmp-2021-0025","url":null,"abstract":"Abstract The system of nonlinear neutral difference equations with delays in the form { Δ(yi(n)+pi(n)yi(n−τi))=ai(n)fi(yi+1(n))+gi(n),Δ(ym(n)+pm(n)ym(n−τm))=am(n)fm(y1(n))+gm(n),[left{ begin{array}{l} Delta ({y_i}(n) + {p_i}(n){y_i}(n - {tau _i})) = {a_i}(n){f_i}({y_{i + 1}}(n)) + {g_i}(n), Delta ({y_m}(n) + {p_m}(n){y_m}(n - {tau _m})) = {a_m}(n){f_m}({y_1}(n)) + {g_m}(n), end{array} right.] for i = 1, . . . , m − 1, m ≥ 2, is studied. The sufficient conditions for the existence of an asymptotically periodic solution of the above system, are established. Here sequences (pi(n)), i = 1,..., m, are bounded away from -1. The presented results are illustrated by theoretical and numerical examples.","PeriodicalId":38690,"journal":{"name":"Tatra Mountains Mathematical Publications","volume":"79 1","pages":"149 - 162"},"PeriodicalIF":0.0,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45506094","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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