Abstract In this article, we define the following: Let N 0 {{mathbb{N}}}_{0} be the set of all nonnegative integers and D = ( d i ) i ∈ N 0 D={left({d}_{i})}_{iin {{mathbb{N}}}_{0}} a family of additive mappings of a ∗ ast -ring R R such that d 0 = i d R {d}_{0}=i{d}_{R} . D D is called a Jordan ( α , β ) left(alpha ,beta ) -higher ∗ ast -derivation (resp. a Jordan triple ( α , β ) left(alpha ,beta ) -higher ∗ ast -derivation) of R R if d n ( a 2 ) = ∑ i + j = n d i ( β j ( a ) ) d j ( α i ( a ∗ i ) ) {d}_{n}left({a}^{2})={sum }_{i+j=n}{d}_{i}left({beta }^{j}left(a)){d}_{j}left({alpha }^{i}left({a}^{{ast }^{i}})) (resp. d n ( a b a ) = ∑ i + j + k = n d i ( β j + k ( a ) ) d j ( β k ( α i ( b ∗ i ) ) ) d k ( α i + j ( a ∗ i + j ) ) {d}_{n}left(aba)={sum }_{i+j+k=n}{d}_{i}left({beta }^{j+k}left(a)){d}_{j}left({beta }^{k}left({alpha }^{i}left({b}^{{ast }^{i}}))){d}_{k}left({alpha }^{i+j}left({a}^{{ast }^{i+j}})) ) for all a , b ∈ R a,bin R and each n ∈ N 0 nin {{mathbb{N}}}_{0} . We show that the two notions of Jordan ( α , β ) left(alpha ,beta ) -higher ∗ ast -derivation and Jordan triple ( α , β ) left(alpha ,beta ) -higher ∗ ast -derivation on a 6-torsion free semiprime ∗ ast -ring are equivalent.
在本文中，我们定义如下:设N为0 {{mathbb{N}}}＿{0} 为所有非负整数的集合，且D= (d1) i∈n0 D={left（{d}＿{I}）}＿{Iin {{mathbb{N}}}＿{0}} A *的一组可加映射 ast -环R R使得d0 = i d R {d}＿{0}= 1{d}＿{r} 。D D被称为约当(α， β) left（alpha ，beta ) -较高* ast - derivative(衍生)Jordan三重(α， β) left（alpha ，beta ) -较高* ast 如果d n (a 2) =∑i + j = n d i (β j (a)) d j (α i (a * i)) {d}＿{n}left（{a}＾{2}）={sum }＿{i+j=n}{d}＿{I}left（{beta }＾{j}left(a)){d}＿{j}left（{alpha }＾{I}left（{a}＾{{ast }＾{I}})(回答;回答D n (a b a) =∑I + j + k = n D I (β j + k (a)) D j (β k (α I (b∗I))) D k (α I + j (a∗I + j))) {d}＿{n}left(aba)={sum }＿{i+j+k=n}{d}＿{I}left（{beta }＾{j+k}left(a)){d}＿{j}left（{beta }＾{k}left（{alpha }＾{I}left（{b}＾{{ast }＾{I}}））){d}＿{k}left（{alpha }＾{i+j}left（{a}＾{{ast }＾{i+j}})))对于所有a,b∈R a,bin R和每个n∈n0nin {{mathbb{N}}}＿{0} 。我们证明了Jordan (α， β)的两个概念 left（alpha ，beta ) -较高* ast -衍生和Jordan三重(α， β) left（alpha ，beta ) -较高* ast 6-无扭转半素数*上的导数 ast -环是等价的。
{"title":"Jordan triple (α,β)-higher ∗-derivations on semiprime rings","authors":"O. H. Ezzat","doi":"10.1515/dema-2022-0213","DOIUrl":"https://doi.org/10.1515/dema-2022-0213","url":null,"abstract":"Abstract In this article, we define the following: Let N 0 {{mathbb{N}}}_{0} be the set of all nonnegative integers and D = ( d i ) i ∈ N 0 D={left({d}_{i})}_{iin {{mathbb{N}}}_{0}} a family of additive mappings of a ∗ ast -ring R R such that d 0 = i d R {d}_{0}=i{d}_{R} . D D is called a Jordan ( α , β ) left(alpha ,beta ) -higher ∗ ast -derivation (resp. a Jordan triple ( α , β ) left(alpha ,beta ) -higher ∗ ast -derivation) of R R if d n ( a 2 ) = ∑ i + j = n d i ( β j ( a ) ) d j ( α i ( a ∗ i ) ) {d}_{n}left({a}^{2})={sum }_{i+j=n}{d}_{i}left({beta }^{j}left(a)){d}_{j}left({alpha }^{i}left({a}^{{ast }^{i}})) (resp. d n ( a b a ) = ∑ i + j + k = n d i ( β j + k ( a ) ) d j ( β k ( α i ( b ∗ i ) ) ) d k ( α i + j ( a ∗ i + j ) ) {d}_{n}left(aba)={sum }_{i+j+k=n}{d}_{i}left({beta }^{j+k}left(a)){d}_{j}left({beta }^{k}left({alpha }^{i}left({b}^{{ast }^{i}}))){d}_{k}left({alpha }^{i+j}left({a}^{{ast }^{i+j}})) ) for all a , b ∈ R a,bin R and each n ∈ N 0 nin {{mathbb{N}}}_{0} . We show that the two notions of Jordan ( α , β ) left(alpha ,beta ) -higher ∗ ast -derivation and Jordan triple ( α , β ) left(alpha ,beta ) -higher ∗ ast -derivation on a 6-torsion free semiprime ∗ ast -ring are equivalent.","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":"41839843","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 prove that if a self-affine arc γ ∈ R 2 gamma in {{mathbb{R}}}^{2} does not satisfy weak separation condition, then it is a segment of a parabola or a straight line. If a self-affine arc γ gamma is not a segment of a parabola or a line, then it is a component of the attractor of a Jordan multizipper with the same set of generators.
摘要证明了自仿射弧γ∈r2 gamma in {{mathbb{R}}}^{2}不满足弱分离条件，则它是抛物线或直线的一段。如果自仿射弧γ γ不是抛物线或直线的一段，则它是具有同一组生成器的Jordan多拉链的吸引子的一个分量。
{"title":"On the structure of self-affine Jordan arcs in ℝ2","authors":"A. Tetenov, Allanazar Kutlimuratov","doi":"10.1515/dema-2022-0228","DOIUrl":"https://doi.org/10.1515/dema-2022-0228","url":null,"abstract":"Abstract We prove that if a self-affine arc γ ∈ R 2 gamma in {{mathbb{R}}}^{2} does not satisfy weak separation condition, then it is a segment of a parabola or a straight line. If a self-affine arc γ gamma is not a segment of a parabola or a line, then it is a component of the attractor of a Jordan multizipper with the same set of generators.","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":"48719458","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 Kirchhoff model is derived from the vibration problem of stretchable strings. This article focuses on the long-time dynamics of a class of higher-order coupled Kirchhoff systems with nonlinear strong damping. The existence and uniqueness of the solutions of these equations in different spaces are proved by prior estimation and the Faedo-Galerkin method. Subsequently, the family of global attractors of these problems is proved using the compactness theorem. In this article, we systematically propose the definition and proof process of the family of global attractors and enrich the related conclusions of higher-order coupled Kirchhoff models. The conclusions lay a theoretical foundation for future practical applications.
{"title":"The asymptotic behaviors of solutions for higher-order (m1, m2)-coupled Kirchhoff models with nonlinear strong damping","authors":"Penghui Lv, Guoguang Lin, Xiaojun Lv","doi":"10.1515/dema-2022-0197","DOIUrl":"https://doi.org/10.1515/dema-2022-0197","url":null,"abstract":"Abstract The Kirchhoff model is derived from the vibration problem of stretchable strings. This article focuses on the long-time dynamics of a class of higher-order coupled Kirchhoff systems with nonlinear strong damping. The existence and uniqueness of the solutions of these equations in different spaces are proved by prior estimation and the Faedo-Galerkin method. Subsequently, the family of global attractors of these problems is proved using the compactness theorem. In this article, we systematically propose the definition and proof process of the family of global attractors and enrich the related conclusions of higher-order coupled Kirchhoff models. The conclusions lay a theoretical foundation for future practical applications.","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":"46954030","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}
M. Vivas-Cortez, Martin Patricio Árciga, Juan Carlos Najera, J. E. Hernández
Abstract The fundamental objective of this article is to investigate about the boundary value problem with the uses of a generalized conformable fractional derivative introduced by Zarikaya et al. (On generalized the conformable calculus, TWMS J. App. Eng. Math. 9 (2019), no. 4, 792–799, http://jaem.isikun.edu.tr/web/images/articles/vol.9.no.4/11.pdf). In the development of the this article, by using classical methods of fractional calculus, we find a definition of the generalized fractional Wronskian according to the fractional differential operator defined by Zarikaya, a fractional version of the Sturm-Picone theorem, and in addition, the stability criterion given by the Hyers-Ulam theorem is studied with the use of the aforementioned fractional derivatives.
{"title":"On some conformable boundary value problems in the setting of a new generalized conformable fractional derivative","authors":"M. Vivas-Cortez, Martin Patricio Árciga, Juan Carlos Najera, J. E. Hernández","doi":"10.1515/dema-2022-0212","DOIUrl":"https://doi.org/10.1515/dema-2022-0212","url":null,"abstract":"Abstract The fundamental objective of this article is to investigate about the boundary value problem with the uses of a generalized conformable fractional derivative introduced by Zarikaya et al. (On generalized the conformable calculus, TWMS J. App. Eng. Math. 9 (2019), no. 4, 792–799, http://jaem.isikun.edu.tr/web/images/articles/vol.9.no.4/11.pdf). In the development of the this article, by using classical methods of fractional calculus, we find a definition of the generalized fractional Wronskian according to the fractional differential operator defined by Zarikaya, a fractional version of the Sturm-Picone theorem, and in addition, the stability criterion given by the Hyers-Ulam theorem is studied with the use of the aforementioned fractional derivatives.","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":"46178643","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 In this article, a numerical solution for the Rosenau-Kawahara equation is considered. A new conservative numerical approximation scheme is presented to solve the initial boundary value problem of the Rosenau-Kawahara equation, which preserves the original conservative properties. The proposed scheme is based on the finite difference method. The existence of the numerical solutions for the scheme has been shown by Browder fixed point theorem. The priori bound and error estimates, as well as the conservation of discrete mass and discrete energy for the finite difference solutions, are discussed. The discrepancies of discrete mass and energy are computed and shown by the curves of these quantities over time. Unconditional stability, second-order convergence, and uniqueness of the scheme are proved based on the discrete energy method. Numerical examples are given to show the effectiveness of the proposed scheme and confirm the theoretical analysis.
{"title":"A novel conservative numerical approximation scheme for the Rosenau-Kawahara equation","authors":"Xin-tian Pan, Lu-ming Zhang","doi":"10.1515/dema-2022-0204","DOIUrl":"https://doi.org/10.1515/dema-2022-0204","url":null,"abstract":"Abstract In this article, a numerical solution for the Rosenau-Kawahara equation is considered. A new conservative numerical approximation scheme is presented to solve the initial boundary value problem of the Rosenau-Kawahara equation, which preserves the original conservative properties. The proposed scheme is based on the finite difference method. The existence of the numerical solutions for the scheme has been shown by Browder fixed point theorem. The priori bound and error estimates, as well as the conservation of discrete mass and discrete energy for the finite difference solutions, are discussed. The discrepancies of discrete mass and energy are computed and shown by the curves of these quantities over time. Unconditional stability, second-order convergence, and uniqueness of the scheme are proved based on the discrete energy method. Numerical examples are given to show the effectiveness of the proposed scheme and confirm the theoretical analysis.","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":"43806192","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 In this article, a class of cyclic (noncyclic) operators are defined on Banach spaces via concept of measure of noncompactness using some abstract functions. The best proximity point (pair) results are manifested for the said operators. The obtained main results are applied to demonstrate the existence of optimum solutions of a system of fractional differential equations involving ( k , ψ ) left(k,psi ) -Hilfer fractional derivatives.
{"title":"Global optimum solutions for a system of (k, ψ)-Hilfer fractional differential equations: Best proximity point approach","authors":"P. Patle, M. Gabeleh, M. de La Sen","doi":"10.1515/dema-2022-0253","DOIUrl":"https://doi.org/10.1515/dema-2022-0253","url":null,"abstract":"Abstract In this article, a class of cyclic (noncyclic) operators are defined on Banach spaces via concept of measure of noncompactness using some abstract functions. The best proximity point (pair) results are manifested for the said operators. The obtained main results are applied to demonstrate the existence of optimum solutions of a system of fractional differential equations involving ( k , ψ ) left(k,psi ) -Hilfer fractional derivatives.","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":"42883235","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 In this article, we have established some new bounds of Fejér-type Hermite-Hadamard inequality for k k -fractional integrals involving r r -times differentiable preinvex functions. It is noteworthy that in the past, there was no weighted version of the left and right sides of the Hermite-Hadamard inequality for k k -fractional integrals for generalized convex functions available in the literature.
{"title":"Weighted Hermite-Hadamard inequalities for r-times differentiable preinvex functions for k-fractional integrals","authors":"F. Zafar, S. Mehmood, A. Asiri","doi":"10.1515/dema-2022-0254","DOIUrl":"https://doi.org/10.1515/dema-2022-0254","url":null,"abstract":"Abstract In this article, we have established some new bounds of Fejér-type Hermite-Hadamard inequality for k k -fractional integrals involving r r -times differentiable preinvex functions. It is noteworthy that in the past, there was no weighted version of the left and right sides of the Hermite-Hadamard inequality for k k -fractional integrals for generalized convex functions available in the literature.","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":"48308078","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 classification of the phase portraits is one of the classical and difficult problems in the qualitative theory of polynomial differential systems in R 2 {{mathbb{R}}}^{2} , particularly for quadratic systems. Even with the hundreds of studies on the topology of real planar quadratic vector fields, fully characterizing their phase portraits is still a difficult problem. This paper is devoted to classifying the phase portraits of two polynomial vector fields with two usual invariant algebraic curves, by investigating the geometric solutions within the Poincaré disc. One can notice that these systems yield 26 topologically different phase portraits.
{"title":"Phase portraits of two classes of quadratic differential systems exhibiting as solutions two cubic algebraic curves","authors":"R. Benterki, Ahlam Belfar","doi":"10.1515/dema-2022-0218","DOIUrl":"https://doi.org/10.1515/dema-2022-0218","url":null,"abstract":"Abstract The classification of the phase portraits is one of the classical and difficult problems in the qualitative theory of polynomial differential systems in R 2 {{mathbb{R}}}^{2} , particularly for quadratic systems. Even with the hundreds of studies on the topology of real planar quadratic vector fields, fully characterizing their phase portraits is still a difficult problem. This paper is devoted to classifying the phase portraits of two polynomial vector fields with two usual invariant algebraic curves, by investigating the geometric solutions within the Poincaré disc. One can notice that these systems yield 26 topologically different phase portraits.","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":"48541728","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 Let G G be a group and R R be a G G -graded commutative ring with nonzero unity 1. In this article, we introduce the concept of graded weakly 1-absorbing primary ideals which is a generalization of graded 1-absorbing primary ideal. A proper graded ideal P P of R R is said to be a graded weakly 1-absorbing primary ideal of R R if whenever nonunit elements x , y , z ∈ h ( R ) x,y,zin hleft(R) such that 0 ≠ x y z ∈ P 0ne xyzin P , then x y ∈ P xyin P or z n ∈ P {z}^{n}in P , for some n ∈ N nin {mathbb{N}} . Several properties of graded weakly 1-absorbing primary ideals are investigated.
{"title":"Graded weakly 1-absorbing primary ideals","authors":"M. Bataineh, R. Abu-Dawwas","doi":"10.1515/dema-2022-0214","DOIUrl":"https://doi.org/10.1515/dema-2022-0214","url":null,"abstract":"Abstract Let G G be a group and R R be a G G -graded commutative ring with nonzero unity 1. In this article, we introduce the concept of graded weakly 1-absorbing primary ideals which is a generalization of graded 1-absorbing primary ideal. A proper graded ideal P P of R R is said to be a graded weakly 1-absorbing primary ideal of R R if whenever nonunit elements x , y , z ∈ h ( R ) x,y,zin hleft(R) such that 0 ≠ x y z ∈ P 0ne xyzin P , then x y ∈ P xyin P or z n ∈ P {z}^{n}in P , for some n ∈ N nin {mathbb{N}} . Several properties of graded weakly 1-absorbing primary ideals are investigated.","PeriodicalId":10995,"journal":{"name":"Demonstratio Mathematica","volume":"56 1","pages":""},"PeriodicalIF":2.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67144060","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 In this article, we study the regularity criteria of the weak solutions to the Boussinesq equations involving the horizontal component of velocity or the horizontal derivatives of the two components of velocity in anisotropic Lorentz spaces. This result reveals that the velocity field plays a dominant role in regularity theory of the Boussinesq equations.
{"title":"Regularity criteria via horizontal component of velocity for the Boussinesq equations in anisotropic Lorentz spaces","authors":"R. Agarwal, Ahmad M. Alghamdi, S. Gala, M. Ragusa","doi":"10.1515/dema-2022-0221","DOIUrl":"https://doi.org/10.1515/dema-2022-0221","url":null,"abstract":"Abstract In this article, we study the regularity criteria of the weak solutions to the Boussinesq equations involving the horizontal component of velocity or the horizontal derivatives of the two components of velocity in anisotropic Lorentz spaces. This result reveals that the velocity field plays a dominant role in regularity theory of the Boussinesq equations.","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":"46151945","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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