In this paper, we introduce the notion of σ-symmetric amenability of Banach algebras and investigate some hereditary properties of them. We also apply our results to several abstract Segal algebras and group algebras.
{"title":"σ-symmetric amenability of Banach algebras","authors":"Lin Chen, Mohammad Javad Mehdipour, Jun Li","doi":"10.1515/gmj-2024-2011","DOIUrl":"https://doi.org/10.1515/gmj-2024-2011","url":null,"abstract":"In this paper, we introduce the notion of σ-symmetric amenability of Banach algebras and investigate some hereditary properties of them. We also apply our results to several abstract Segal algebras and group algebras.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140299183","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is concerned with constructions and characterizations of matrix equalities that involve mixed products of Moore–Penrose inverses and group inverses of two matrices. We first construct a mixed reverse-order law (AB)†=B∗(A∗ABB∗)#A∗{(AB)^{{dagger}}=B^{ast}(A^{ast}ABB^{ast})^{#}A^{ast}}, and show that this matrix equality always holds through the use of a special matrix rank equality and some matrix range operations, where A and B are two matrices of appropriate sizes, (⋅)∗{(,cdot,)^{ast}}, (⋅)†{(,cdot,)^{{dagger}}} and (⋅)#{(,cdot,)^{#}} mean the conjugate transpose, the Moor
本文关注涉及两个矩阵的摩尔-彭罗斯倒数和群倒数的混合乘积的矩阵等式的构造和特征。我们首先构建了一个混合逆序律 ( A B ) † = B ∗ ( A ∗ A B B ∗ ) # A ∗ {(AB)^{{dagger}}=B^{ast}(A^{/ast}ABB^{/ast})^{#}A^{/ast}} ，并通过证明这个矩阵相等总是成立的，来说明这个矩阵相等是正确的。 通过使用特殊的矩阵秩相等和一些矩阵范围运算，证明这个矩阵相等总是成立的，其中 A 和 B 是两个适当大小的矩阵，( ⋅ ) ∗ {(,cdot,)^{ast}} ，( ⋅ ) † {(,cdot,)^{ast}} 。 , ( ⋅ ) † {(cdot,)^{{dagger}} 和 ( ⋅ ) # {(cdot,)^{#}} 分别指矩阵的共轭转置、摩尔-彭罗斯逆和群逆。然后，我们给出了这一等式的各种变化形式，并推导出它们成立的必要条件和充分条件。特别是特别是，我们展示了一个有趣的事实，即两个反序规律( A B ) † = B † A † {(AB)^{{dagger}}=B^{{dagger}}A^{{dagger}}} 和 ( A ∗ A B B∗ ) # = ( B B ∗ ) # ( A A ) # {(A^{ast}ABB^{ast})^{#}=(BB^{ast})^{#}(A^{ast}A)^{#}} 是等价的。
{"title":"Constructions and characterizations of mixed reverse-order laws for the Moore–Penrose inverse and group inverse","authors":"Yongge Tian","doi":"10.1515/gmj-2024-2016","DOIUrl":"https://doi.org/10.1515/gmj-2024-2016","url":null,"abstract":"This paper is concerned with constructions and characterizations of matrix equalities that involve mixed products of Moore–Penrose inverses and group inverses of two matrices. We first construct a mixed reverse-order law <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>A</m:mi> <m:mo></m:mo> <m:mi>B</m:mi> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>†</m:mo> </m:msup> <m:mo>=</m:mo> <m:mrow> <m:msup> <m:mi>B</m:mi> <m:mo>∗</m:mo> </m:msup> <m:mo></m:mo> <m:msup> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:msup> <m:mi>A</m:mi> <m:mo>∗</m:mo> </m:msup> <m:mo></m:mo> <m:mi>A</m:mi> <m:mo></m:mo> <m:mi>B</m:mi> <m:mo></m:mo> <m:msup> <m:mi>B</m:mi> <m:mo>∗</m:mo> </m:msup> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mi mathvariant=\"normal\">#</m:mi> </m:msup> <m:mo></m:mo> <m:msup> <m:mi>A</m:mi> <m:mo>∗</m:mo> </m:msup> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2016_eq_0486.png\" /> <jats:tex-math>{(AB)^{{dagger}}=B^{ast}(A^{ast}ABB^{ast})^{#}A^{ast}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and show that this matrix equality always holds through the use of a special matrix rank equality and some matrix range operations, where <jats:italic>A</jats:italic> and <jats:italic>B</jats:italic> are two matrices of appropriate sizes, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mo rspace=\"4.2pt\" stretchy=\"false\">(</m:mo> <m:mo rspace=\"4.2pt\">⋅</m:mo> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>∗</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2016_eq_0509.png\" /> <jats:tex-math>{(,cdot,)^{ast}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mo rspace=\"4.2pt\" stretchy=\"false\">(</m:mo> <m:mo rspace=\"4.2pt\">⋅</m:mo> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mo>†</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2016_eq_0510.png\" /> <jats:tex-math>{(,cdot,)^{{dagger}}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mo rspace=\"4.2pt\" stretchy=\"false\">(</m:mo> <m:mo rspace=\"4.2pt\">⋅</m:mo> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> <m:mi mathvariant=\"normal\">#</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2016_eq_0508.png\" /> <jats:tex-math>{(,cdot,)^{#}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> mean the conjugate transpose, the Moor","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140299182","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a linear constant coefficient ODE depending on a parameter, when this parameter approaches zero, the solution set converges to the solution set of the limit differential equation if the leading coefficient does not vanish. The situation is very subtle in the singular case, i.e., in the case when this coefficient becomes zero. The solution set then may even collapse completely. In this note, a formalism is developed in which the solution set of a linear constant coefficient ODE always depends continuously on the equation coefficients.
{"title":"Degeneration phenomenon in linear ordinary differential equations","authors":"Vakhtang Lomadze","doi":"10.1515/gmj-2024-2007","DOIUrl":"https://doi.org/10.1515/gmj-2024-2007","url":null,"abstract":"Given a linear constant coefficient ODE depending on a parameter, when this parameter approaches zero, the solution set converges to the solution set of the limit differential equation if the leading coefficient does not vanish. The situation is very subtle in the singular case, i.e., in the case when this coefficient becomes zero. The solution set then may even collapse completely. In this note, a formalism is developed in which the solution set of a linear constant coefficient ODE always depends continuously on the equation coefficients.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139918169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the present paper, we examine the perturbation of continuous frames and Riesz-type frames in Hilbert C*{C^{*}}-modules. We extend the Casazza–Christensen general perturbation theorem for Hilbert space frames to continuous frames in Hilbert C*{C^{*}}-modules. We obtain a necessary condition under which the perturbation of a Riesz-type frame of Hilbert C*{C^{*}}-modules remains to be a Riesz-type frame. Also, we examine the effect of duality on the perturbation of continuous frames in Hilbert C*{C^{*}}-modules, and we prove that if the operator frame of a continuous frame F is near to the combination of the synthesis operator of a continuous Bessel mapping G and the analysis operator of F, then G is a continuous frame.
在本文中，我们研究了希尔伯特 C * {C^{*}} 模块中连续框架和里兹型框架的扰动。 -模块中的连续帧和里兹型帧的扰动。我们将希尔伯特空间帧的卡萨扎-克里斯滕森一般扰动定理推广到希尔伯特 C * {C^{*}} 模块中的连续帧。 -模块中的连续帧。我们得到了一个必要条件，在这个条件下，希尔伯特 C * {C^{*}} 模块的李斯型帧的扰动仍然是一个李斯型帧。 -模块的里兹型框架的扰动仍然是里兹型框架的必要条件。此外，我们还考察了对偶性对希尔伯特 C * {C^{*}} 模块中连续帧的扰动的影响。 -模块的扰动的影响，并证明如果连续帧 F 的算子帧接近于连续贝塞尔映射 G 的合成算子与 F 的分析算子的组合，那么 G 就是一个连续帧。
{"title":"On perturbation of continuous frames in Hilbert C *-modules","authors":"Hadi Ghasemi, Tayebe Lal Shateri","doi":"10.1515/gmj-2023-2111","DOIUrl":"https://doi.org/10.1515/gmj-2023-2111","url":null,"abstract":"In the present paper, we examine the perturbation of continuous frames and Riesz-type frames in Hilbert <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2111_eq_0167.png\" /> <jats:tex-math>{C^{*}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-modules. We extend the Casazza–Christensen general perturbation theorem for Hilbert space frames to continuous frames in Hilbert <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2111_eq_0167.png\" /> <jats:tex-math>{C^{*}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-modules. We obtain a necessary condition under which the perturbation of a Riesz-type frame of Hilbert <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2111_eq_0167.png\" /> <jats:tex-math>{C^{*}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-modules remains to be a Riesz-type frame. Also, we examine the effect of duality on the perturbation of continuous frames in Hilbert <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>C</m:mi> <m:mo>*</m:mo> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2111_eq_0167.png\" /> <jats:tex-math>{C^{*}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-modules, and we prove that if the operator frame of a continuous frame <jats:italic>F</jats:italic> is near to the combination of the synthesis operator of a continuous Bessel mapping <jats:italic>G</jats:italic> and the analysis operator of <jats:italic>F</jats:italic>, then <jats:italic>G</jats:italic> is a continuous frame.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139918315","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we extend Hardy’s inequality to infinite tensors. To do so, we introduce Cesàro tensors ℭ{mathfrak{C}}, and consider them as tensor maps from sequence spaces into tensor spaces. In fact, we prove inequalities of the form ∥ℭxk∥t,1≤U∥x∥lpk|mathfrak{C}x^{k}|_{t,1}leq U|x|_{l_{p}}^{k} (k=1,2k=1,2), where x is a sequence, ℭxk{mathfrak{C}x^{k}} is a tensor, and ∥⋅∥t,1{|cdot|_{t,1}}, ∥⋅∥lp
在本文中，我们将哈代不等式扩展到无限张量。为此，我们引入 Cesàro 张量 ℭ {mathfrak{C}} ，并将其视为从序列空间到张量空间的张量映射。 ，并将它们视为从序列空间到张量空间的张量映射。事实上，我们证明了形式为 ∥ ℭ x k ∥ t , 1 ≤ U ∥ x ∥ l p k|mathfrak{C}x^{k}|_{t,1}leq U|x|_{l_{p}}^{k} 的不等式。 ( k = 1 , 2 k=1,2 ), 其中 x 是一个序列，ℭ x k {mathfrak{C}x^{k}} 是一个张量，并且 ∥ ⋅ ∥ t , 1 {|cdot|_{t,1}} , ∥ ⋅ ∥ l p {|cdot|_{l_{p}}} 分别是张量规范和序列规范。常数 U 与 x 无关，我们寻求 U 的最小值。
{"title":"A generalization of Hardy’s inequality to infinite tensors","authors":"Morteza Saheli, Davoud Foroutannia, Sara Yusefian","doi":"10.1515/gmj-2024-2006","DOIUrl":"https://doi.org/10.1515/gmj-2024-2006","url":null,"abstract":"In this paper, we extend Hardy’s inequality to infinite tensors. To do so, we introduce Cesàro tensors <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ℭ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2006_eq_0150.png\" /> <jats:tex-math>{mathfrak{C}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, and consider them as tensor maps from sequence spaces into tensor spaces. In fact, we prove inequalities of the form <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mrow> <m:mo>∥</m:mo> <m:mrow> <m:mi>ℭ</m:mi> <m:mo></m:mo> <m:msup> <m:mi>x</m:mi> <m:mi>k</m:mi> </m:msup> </m:mrow> <m:mo>∥</m:mo> </m:mrow> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>≤</m:mo> <m:mrow> <m:mi>U</m:mi> <m:mo></m:mo> <m:msubsup> <m:mrow> <m:mo>∥</m:mo> <m:mi>x</m:mi> <m:mo>∥</m:mo> </m:mrow> <m:msub> <m:mi>l</m:mi> <m:mi>p</m:mi> </m:msub> <m:mi>k</m:mi> </m:msubsup> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2006_eq_0106.png\" /> <jats:tex-math>|mathfrak{C}x^{k}|_{t,1}leq U|x|_{l_{p}}^{k}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (<jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2006_eq_0107.png\" /> <jats:tex-math>k=1,2</jats:tex-math> </jats:alternatives> </jats:inline-formula>), where <jats:italic>x</jats:italic> is a sequence, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ℭ</m:mi> <m:mo></m:mo> <m:msup> <m:mi>x</m:mi> <m:mi>k</m:mi> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2006_eq_0149.png\" /> <jats:tex-math>{mathfrak{C}x^{k}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a tensor, and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>∥</m:mo> <m:mo>⋅</m:mo> <m:msub> <m:mo>∥</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2006_eq_0155.png\" /> <jats:tex-math>{|cdot|_{t,1}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>∥</m:mo> <m:mo>⋅</m:mo> <m:msub> <m:mo>∥</m:mo> <m:msub> <m:mi>l</m:mi> <m:mi>p</m:mi> </m:msub> </m:msub> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139918167","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Busemann–Petty problem of arbitrary measure for symmetric star bodies is proposed and studied by Zvavitch, which is a generalization of the classical Busemann–Petty problem. In this paper, we study the Busemann–Petty-type problem for homogeneous measure for general star bodies.
{"title":"Busemann--Petty-type problem for μ-intersection bodies","authors":"Chao Li, Gangyi Chen","doi":"10.1515/gmj-2024-2009","DOIUrl":"https://doi.org/10.1515/gmj-2024-2009","url":null,"abstract":"The Busemann–Petty problem of arbitrary measure for symmetric star bodies is proposed and studied by Zvavitch, which is a generalization of the classical Busemann–Petty problem. In this paper, we study the Busemann–Petty-type problem for homogeneous measure for general star bodies.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139918317","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
David Barilla, Martin Bohner, Giuseppe Caristi, Fariba Gharehgazlouei, Shapour Heidarkhani
In this paper, we consider a fractional p-Laplacian elliptic Dirichlet problem that possesses one control parameter and has a Lipschitz nonlinearity order of p-1{p-1}. The multiplicity of the weak solutions is proved by means of the variational method and critical point theory. We investigate the existence of at least three solutions to the problem.
{"title":"Fractional p-Laplacian elliptic Dirichlet problems","authors":"David Barilla, Martin Bohner, Giuseppe Caristi, Fariba Gharehgazlouei, Shapour Heidarkhani","doi":"10.1515/gmj-2024-2008","DOIUrl":"https://doi.org/10.1515/gmj-2024-2008","url":null,"abstract":"In this paper, we consider a fractional <jats:italic>p</jats:italic>-Laplacian elliptic Dirichlet problem that possesses one control parameter and has a Lipschitz nonlinearity order of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>p</m:mi> <m:mo>-</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2008_eq_0274.png\" /> <jats:tex-math>{p-1}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. The multiplicity of the weak solutions is proved by means of the variational method and critical point theory. We investigate the existence of at least three solutions to the problem.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-02-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139918316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper studies the relationship between the existence of periodic solutions of systems of dynamic equations on time scales and their corresponding systems of differential equations. We have established that, for a sufficiently small graininess function, if a dynamic equation on a time scale has an asymptotically stable periodic solution, then the corresponding differential equation will also have a periodic solution. A converse result has also been obtained, where the existence of a periodic solution of a differential equation implies the existence of a corresponding solution on time scales, provided that the graininess function is sufficiently small.
{"title":"On the correspondence between periodic solutions of differential and dynamic equations on periodic time scales","authors":"Viktoriia Tsan, Oleksandr Stanzhytskyi, Olha Martynyuk","doi":"10.1515/gmj-2024-2003","DOIUrl":"https://doi.org/10.1515/gmj-2024-2003","url":null,"abstract":"This paper studies the relationship between the existence of periodic solutions of systems of dynamic equations on time scales and their corresponding systems of differential equations. We have established that, for a sufficiently small graininess function, if a dynamic equation on a time scale has an asymptotically stable periodic solution, then the corresponding differential equation will also have a periodic solution. A converse result has also been obtained, where the existence of a periodic solution of a differential equation implies the existence of a corresponding solution on time scales, provided that the graininess function is sufficiently small.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-02-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139771538","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we deal with the notion of Va{V_{a}}-deformed free convolution, introduced in [A. D. Krystek and L. J. Wojakowski, Associative convolutions arising from conditionally free convolution, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 2005, 3, 515–545], from a point of view related to the theory of Cauchy–Stieltjes kernel (CSK) families and their corresponding variance functions. We determine the formula for variance function under a power of Va{V_{a}}-deformed free convolution. Then we provide an approximation of elements of the CSK family generated by Va{V_{a}}-deformed free Poisson distribution.
本文将讨论 V a {V_{a}} 变形自由卷积的概念。 -变形自由卷积的概念。D. Krystek 和 L. J. Wojakowski, Associative convolutions arising from conditionally free convolution, Infin.Dimens.Anal.Quantum Probab.Relat.Top.8 2005, 3, 515-545], 从与 Cauchy-Stieltjes 核（CSK）族及其相应方差函数理论相关的角度出发。我们确定了 V a {V_{a}} 的幂下的方差函数公式。 -变形自由卷积下的方差函数公式。然后，我们提供了由 V a {V_{a}} 变形自由泊松分布生成的 CSK 族元素的近似值。 -变形自由泊松分布产生的 CSK 族元素的近似值。
{"title":"V_a -deformed free convolution and variance function","authors":"Raouf Fakhfakh","doi":"10.1515/gmj-2024-2004","DOIUrl":"https://doi.org/10.1515/gmj-2024-2004","url":null,"abstract":"In this paper, we deal with the notion of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>V</m:mi> <m:mi>a</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2004_eq_0134.png\" /> <jats:tex-math>{V_{a}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-deformed free convolution, introduced in [A. D. Krystek and L. J. Wojakowski, Associative convolutions arising from conditionally free convolution, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 8 2005, 3, 515–545], from a point of view related to the theory of Cauchy–Stieltjes kernel (CSK) families and their corresponding variance functions. We determine the formula for variance function under a power of <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>V</m:mi> <m:mi>a</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2004_eq_0134.png\" /> <jats:tex-math>{V_{a}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-deformed free convolution. Then we provide an approximation of elements of the CSK family generated by <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>V</m:mi> <m:mi>a</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2024-2004_eq_0134.png\" /> <jats:tex-math>{V_{a}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-deformed free Poisson distribution.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139658777","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"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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