In this paper, we study the qualitative behavior of the rational recursive equation begin{equation*} x_{n+1}=frac{x_{n-4}}{pm1pm x_{n}x_{n-1}x_{n-2}x_{n-3}x_{n-4}}, quad n in mathbb{N}_{0}:={0}cupmathbb N, end{equation*} where the initial conditions are arbitrary nonzero real numbers. The main goal of this paper, is to obtain the forms of the solutions of the nonlinear fifth-order difference equations, where the initial conditions are arbitrary positive real numbers. Moreover, we investigate stability, boundedness, oscillation and the periodic character of these solutions. The results presented in this paper improve and extend some corresponding results in the literature.
本文研究了初始条件为任意非零实数的有理递推方程begin{equation*} x_{n+1}=frac{x_{n-4}}{pm1pm x_{n}x_{n-1}x_{n-2}x_{n-3}x_{n-4}}, quad n in mathbb{N}_{0}:={0}cupmathbb N, end{equation*}的定性性质。本文的主要目的是得到初始条件为任意正实数的非线性五阶差分方程的解的形式。此外，我们还研究了这些解的稳定性、有界性、振荡性和周期性。本文的结果改进和推广了文献中一些相应的结果。
{"title":"Dynamical behavior of one rational fifth-order difference equation","authors":"B. Oğul, D. Şi̇mşek","doi":"10.15330/cmp.15.1.43-51","DOIUrl":"https://doi.org/10.15330/cmp.15.1.43-51","url":null,"abstract":"In this paper, we study the qualitative behavior of the rational recursive equation begin{equation*} x_{n+1}=frac{x_{n-4}}{pm1pm x_{n}x_{n-1}x_{n-2}x_{n-3}x_{n-4}}, quad n in mathbb{N}_{0}:={0}cupmathbb N, end{equation*} where the initial conditions are arbitrary nonzero real numbers. The main goal of this paper, is to obtain the forms of the solutions of the nonlinear fifth-order difference equations, where the initial conditions are arbitrary positive real numbers. Moreover, we investigate stability, boundedness, oscillation and the periodic character of these solutions. The results presented in this paper improve and extend some corresponding results in the literature.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"40 11","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-04-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72631811","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 goal of this paper is to find some important Einstein manifolds using conformal Ricci solitons and conformal Ricci almost solitons. We prove that a Kenmotsu metric as a conformal Ricci soliton is Einstein if it is an $eta$-Einstein or the potential vector field $V$ is infinitesimal contact transformation or collinear with the Reeb vector field $xi$. Next, we prove that a Kenmotsu metric as gradient conformal Ricci almost soliton is Einstein if the Reeb vector field leaves the scalar curvature invariant. Finally, we have embellished an example to illustrate the existence of conformal Ricci soliton and gradient almost conformal Ricci soliton on Kenmotsu manifold.
{"title":"A study on conformal Ricci solitons and conformal Ricci almost solitons within the framework of almost contact geometry","authors":"S. Dey","doi":"10.15330/cmp.15.1.31-42","DOIUrl":"https://doi.org/10.15330/cmp.15.1.31-42","url":null,"abstract":"The goal of this paper is to find some important Einstein manifolds using conformal Ricci solitons and conformal Ricci almost solitons. We prove that a Kenmotsu metric as a conformal Ricci soliton is Einstein if it is an $eta$-Einstein or the potential vector field $V$ is infinitesimal contact transformation or collinear with the Reeb vector field $xi$. Next, we prove that a Kenmotsu metric as gradient conformal Ricci almost soliton is Einstein if the Reeb vector field leaves the scalar curvature invariant. Finally, we have embellished an example to illustrate the existence of conformal Ricci soliton and gradient almost conformal Ricci soliton on Kenmotsu manifold.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"8 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89674471","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}
Pub Date : 2023-04-10DOI: 10.15330/cmp.15.1.246-259
A. Serdyuk, I. V. Sokolenko
We find two-sided estimates for Kolmogorov, Bernstein, linear and projection widths of the classes of convolutions of $2pi$-periodic functions $varphi$, such that $|varphi|_2le1$, with fixed generated kernels $Psi_{bar{beta}}$, which have Fourier series of the form $$sumlimits_{k=1}^infty psi(k)cos(kt-beta_kpi/2),$$ where $psi(k)ge0,$ $sumpsi^2(k)
{"title":"Asymptotic estimates for the widths of classes of functions of high smothness","authors":"A. Serdyuk, I. V. Sokolenko","doi":"10.15330/cmp.15.1.246-259","DOIUrl":"https://doi.org/10.15330/cmp.15.1.246-259","url":null,"abstract":"We find two-sided estimates for Kolmogorov, Bernstein, linear and projection widths of the classes of convolutions of $2pi$-periodic functions $varphi$, such that $|varphi|_2le1$, with fixed generated kernels $Psi_{bar{beta}}$, which have Fourier series of the form $$sumlimits_{k=1}^infty psi(k)cos(kt-beta_kpi/2),$$ where $psi(k)ge0,$ $sumpsi^2(k)<infty, beta_kinmathbb{R}$. It is shown that for rapidly decreasing sequences $psi(k)$ (in particular, if $limlimits_{krightarrowinfty}{psi(k+1)}/{psi(k)}=0$) the obtained estimates are asymptotic equalities. We establish that asymptotic equalities for widths of this classes are realized by trigonometric Fourier sums.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"107 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-04-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74703455","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 solution of the investigated problem with an unknown coefficient in the equation was constructed by using the method of separation of variables. The properties of the induced spectral problem for the second-order differential equation with involution are studied. The dependence on the equation involutive part of the spectrum and its multiplicity as well as the structure of the system of root functions and partial solutions of the problem were investigated. The conditions for the existence and uniqueness of the solution of the inverse problem have been established. To determine the required coefficient, Volterra's integral equation of the second kind was found and solved.
{"title":"Inverse problems of determining an unknown depending on time coefficient for a parabolic equation with involution and periodicity conditions","authors":"Y. Baranetskij, I. Demkiv, A. Solomko","doi":"10.15330/cmp.15.1.5-19","DOIUrl":"https://doi.org/10.15330/cmp.15.1.5-19","url":null,"abstract":"The solution of the investigated problem with an unknown coefficient in the equation was constructed by using the method of separation of variables. The properties of the induced spectral problem for the second-order differential equation with involution are studied. The dependence on the equation involutive part of the spectrum and its multiplicity as well as the structure of the system of root functions and partial solutions of the problem were investigated. The conditions for the existence and uniqueness of the solution of the inverse problem have been established. To determine the required coefficient, Volterra's integral equation of the second kind was found and solved.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"102 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2023-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75811136","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}
Pub Date : 2022-12-30DOI: 10.15330/cmp.14.2.464-474
F. Çitak
In this paper, basic concepts of soft set theory was mentioned. Then, bipolar soft Lie algebra and bipolar soft Lie ideal were defined with the help of soft sets. Some algebraic properties of the new concepts were investigated. The relationship between the two structures were analyzed. Also, it was proved that the level cuts of a bipolar soft Lie algebra were Lie subalgebras of a Lie algebra by the new definitions. After then, soft image and soft preimage of a bipolar soft Lie algebra/ideal were proved to be a bipolar soft Lie algebra/ideal.
{"title":"A new kind of soft algebraic structures: bipolar soft Lie algebras","authors":"F. Çitak","doi":"10.15330/cmp.14.2.464-474","DOIUrl":"https://doi.org/10.15330/cmp.14.2.464-474","url":null,"abstract":"In this paper, basic concepts of soft set theory was mentioned. Then, bipolar soft Lie algebra and bipolar soft Lie ideal were defined with the help of soft sets. Some algebraic properties of the new concepts were investigated. The relationship between the two structures were analyzed. Also, it was proved that the level cuts of a bipolar soft Lie algebra were Lie subalgebras of a Lie algebra by the new definitions. After then, soft image and soft preimage of a bipolar soft Lie algebra/ideal were proved to be a bipolar soft Lie algebra/ideal.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"67 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73814575","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}
Pub Date : 2022-12-30DOI: 10.15330/cmp.14.2.485-492
Ö. Acar, E. Erdoğan, A. S. Özkapu
This study is devoted to investigate the problem whether the existence and uniqueness of integral type contraction mappings on orthogonal metric spaces. At the end, we give an example to illustrative our main result.
研究正交度量空间上的积分型收缩映射是否存在唯一性问题。最后，给出了一个算例来说明我们的主要结果。
{"title":"Generalized integral type mappings on orthogonal metric spaces","authors":"Ö. Acar, E. Erdoğan, A. S. Özkapu","doi":"10.15330/cmp.14.2.485-492","DOIUrl":"https://doi.org/10.15330/cmp.14.2.485-492","url":null,"abstract":"This study is devoted to investigate the problem whether the existence and uniqueness of integral type contraction mappings on orthogonal metric spaces. At the end, we give an example to illustrative our main result.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"28 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81795648","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}
Pub Date : 2022-12-30DOI: 10.15330/cmp.14.2.429-436
Kh.O. Sukhorukova, M. Zarichnyǐ
The functor of $ast$-measures of compact support on the category of ultrametric spaces and non-expanding maps is introduced in the previous publication of the authors. In the present note, we prove that this functor determines a monad on this category. The monad structure allows us to define the tensor product of $ast$-measures. We consider some applications of this notion to equilibrium theory.
{"title":"On $ast$-measure monads on the category of ultrametric spaces","authors":"Kh.O. Sukhorukova, M. Zarichnyǐ","doi":"10.15330/cmp.14.2.429-436","DOIUrl":"https://doi.org/10.15330/cmp.14.2.429-436","url":null,"abstract":"The functor of $ast$-measures of compact support on the category of ultrametric spaces and non-expanding maps is introduced in the previous publication of the authors. In the present note, we prove that this functor determines a monad on this category. The monad structure allows us to define the tensor product of $ast$-measures. We consider some applications of this notion to equilibrium theory.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"3 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79492525","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}
Pub Date : 2022-12-30DOI: 10.15330/cmp.14.2.475-484
M. Omaba
A new generalization of the Katugampola generalized fractional integrals in terms of the Mittag-Leffler functions is proposed. Consequently, new generalizations of the Hermite-Hadamard inequalities by this newly proposed fractional integral operator, for a positive convex stochastic process, are established. Other known results are easily deduced as particular cases of these inequalities. The obtained results also hold for any convex function.
{"title":"Generalized fractional inequalities of the Hermite-Hadamard type via new Katugampola generalized fractional integrals","authors":"M. Omaba","doi":"10.15330/cmp.14.2.475-484","DOIUrl":"https://doi.org/10.15330/cmp.14.2.475-484","url":null,"abstract":"A new generalization of the Katugampola generalized fractional integrals in terms of the Mittag-Leffler functions is proposed. Consequently, new generalizations of the Hermite-Hadamard inequalities by this newly proposed fractional integral operator, for a positive convex stochastic process, are established. Other known results are easily deduced as particular cases of these inequalities. The obtained results also hold for any convex function.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80013707","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}
Pub Date : 2022-12-30DOI: 10.15330/cmp.14.2.493-503
I. Klevchuk
The aim of the present paper is to investigate of some properties of periodic solutions of a nonlinear autonomous parabolic systems with a periodic condition. We investigate parabolic systems of differential equations using an integral manifolds method of the theory of nonlinear oscillations. We prove the existence of periodic solutions in an autonomous parabolic system of differential equations with weak diffusion on the circle. We study the existence and stability of an arbitrarily large finite number of cycles for a parabolic system with weak diffusion. The periodic solution of parabolic equation is sought in the form of traveling wave. A representation of the integral manifold is obtained. We seek a solution of parabolic system with the periodic condition in the form of a Fourier series in the complex form and introduce a norm in the space of the coefficients in the Fourier expansion. We use the normal forms method in the general parabolic system of differential equations with retarded argument and weak diffusion. We use bifurcation theory for delay differential equations and quasilinear parabolic equations. The existence of periodic solutions in an autonomous parabolic system of differential equations on the circle with retarded argument and small diffusion is proved. The problems of existence and stability of traveling waves in the parabolic system with retarded argument and weak diffusion are investigated.
{"title":"Existence and stability of traveling waves in parabolic systems of differential equations with weak diffusion","authors":"I. Klevchuk","doi":"10.15330/cmp.14.2.493-503","DOIUrl":"https://doi.org/10.15330/cmp.14.2.493-503","url":null,"abstract":"The aim of the present paper is to investigate of some properties of periodic solutions of a nonlinear autonomous parabolic systems with a periodic condition. We investigate parabolic systems of differential equations using an integral manifolds method of the theory of nonlinear oscillations. We prove the existence of periodic solutions in an autonomous parabolic system of differential equations with weak diffusion on the circle. We study the existence and stability of an arbitrarily large finite number of cycles for a parabolic system with weak diffusion. The periodic solution of parabolic equation is sought in the form of traveling wave. A representation of the integral manifold is obtained. We seek a solution of parabolic system with the periodic condition in the form of a Fourier series in the complex form and introduce a norm in the space of the coefficients in the Fourier expansion. We use the normal forms method in the general parabolic system of differential equations with retarded argument and weak diffusion. We use bifurcation theory for delay differential equations and quasilinear parabolic equations. The existence of periodic solutions in an autonomous parabolic system of differential equations on the circle with retarded argument and small diffusion is proved. The problems of existence and stability of traveling waves in the parabolic system with retarded argument and weak diffusion are investigated.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"61 4 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90122018","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}
Pub Date : 2022-12-30DOI: 10.15330/cmp.14.2.453-463
V. Babenko, N. Parfinovych, D. Skorokhodov
We solve the problem of the best approximation of closed operators by linear bounded operators in Hilbert spaces under assumption that the operator transforms orthogonal basis in Hilbert space into an orthogonal system. As a consequence, sharp additive Hardy-Littlewood-Pólya type inequality for multiple closed operators is established. We also demonstrate application of these results in concrete situations: for the best approximation of powers of the Laplace-Beltrami operator on classes of functions defined on closed Riemannian manifolds, for the best approximation of differentiation operators on classes of functions defined on the period and on the real line with the weight $e^{-x^2}$, and for the best approximation of functions of self-adjoint operators in Hilbert spaces.
{"title":"The best approximation of closed operators by bounded operators in Hilbert spaces","authors":"V. Babenko, N. Parfinovych, D. Skorokhodov","doi":"10.15330/cmp.14.2.453-463","DOIUrl":"https://doi.org/10.15330/cmp.14.2.453-463","url":null,"abstract":"We solve the problem of the best approximation of closed operators by linear bounded operators in Hilbert spaces under assumption that the operator transforms orthogonal basis in Hilbert space into an orthogonal system. As a consequence, sharp additive Hardy-Littlewood-Pólya type inequality for multiple closed operators is established. We also demonstrate application of these results in concrete situations: for the best approximation of powers of the Laplace-Beltrami operator on classes of functions defined on closed Riemannian manifolds, for the best approximation of differentiation operators on classes of functions defined on the period and on the real line with the weight $e^{-x^2}$, and for the best approximation of functions of self-adjoint operators in Hilbert spaces.","PeriodicalId":42912,"journal":{"name":"Carpathian Mathematical Publications","volume":"53 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86725372","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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