Georgian Mathematical Journal最新文献

{"title":"BV capacity and perimeter in abstract Wiener spaces and applications","authors":"Guiyang Liu, He Wang, Yu Liu","doi":"10.1515/gmj-2023-2081","DOIUrl":"https://doi.org/10.1515/gmj-2023-2081","url":null,"abstract":"This paper is devoted to introducing and investigating the bounded variation capacity and the perimeter in the abstract Wiener space <jats:italic>X</jats:italic>, thereby discovering some related inequalities. Functions of bounded variation in an abstract Wiener space <jats:italic>X</jats:italic> have been studied by many scholars. As the continuation of this research, we define the corresponding BV capacity <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>cap</m:mi> <m:mi>H</m:mi> </m:msub> <m:mo>⁡</m:mo> <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:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2081_eq_0438.png\" /> <jats:tex-math>{operatorname{cap}_{H}(,cdot,)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> (now called abstract Wiener BV capacity) and investigate its properties. We also investigate some properties of sets of finite γ-perimeter, with γ being a Gaussian measure. Subsequently, the isocapacitary inequality associated with <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msub> <m:mi>cap</m:mi> <m:mi>H</m:mi> </m:msub> <m:mo>⁡</m:mo> <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:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2081_eq_0438.png\" /> <jats:tex-math>{operatorname{cap}_{H}(,cdot,)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is presented and we are able to show that it is equivalent to the Gaussian isoperimetric inequality. Finally, we prove that every set of finite γ-perimeter in <jats:italic>X</jats:italic> has mean curvature in <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>L</m:mi> <m:mn>1</m:mn> </m:msup> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>γ</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2081_eq_0347.png\" /> <jats:tex-math>{L^{1}(X,gamma)}</jats:tex-math> </jats:alternatives> </jats:inline-formula>.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138517380","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}

{"title":"On 〈s〉-generalized topologies","authors":"Jacek Hejduk, Mehmet Kucukaslan, Anna Loranty","doi":"10.1515/gmj-2023-2096","DOIUrl":"https://doi.org/10.1515/gmj-2023-2096","url":null,"abstract":"In this paper, we focus our attention on an outer Lebesgue measure and density-type generalized topologies connected with this measure and with nondecreasing and unbounded sequences of positive reals. Some properties of such generalized topologies and continuous functions connected with this space are presented.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504240","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}

{"title":"The second nonlinear mixed Lie triple derivations on standard operator algebras","authors":"Nadeem ur Rehman, Junaid Nisar, Bilal Ahmad Wani","doi":"10.1515/gmj-2023-2086","DOIUrl":"https://doi.org/10.1515/gmj-2023-2086","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">𝒜</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2086_eq_0304.png\" /> <jats:tex-math>{mathcal{A}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a standard operator algebra containing the identity operator <jats:italic>I</jats:italic> on an infinite dimensional complex Hilbert space <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"script\">ℋ</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2086_eq_0308.png\" /> <jats:tex-math>{mathcal{H}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> which is closed under adjoint operation. Suppose that <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>ϕ</m:mi> <m:mo>:</m:mo> <m:mrow> <m:mi mathvariant=\"script\">𝒜</m:mi> <m:mo>→</m:mo> <m:mi mathvariant=\"script\">𝒜</m:mi> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2086_eq_0329.png\" /> <jats:tex-math>{phi:mathcal{A}tomathcal{A}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is the second nonlinear mixed Lie triple derivation. Then ϕ is an additive <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mo>∗</m:mo> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2086_eq_0290.png\" /> <jats:tex-math>{ast}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-derivation.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138517379","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}

{"title":"Existence and uniqueness of solution for the nonlinear Brusselator system with Robin boundary conditions","authors":"Ghassan A. Al-Juaifri, Akil J. Harfash","doi":"10.1515/gmj-2023-2091","DOIUrl":"https://doi.org/10.1515/gmj-2023-2091","url":null,"abstract":"The system of Brusselator-type reaction-diffusion equations (RDs) on open bounded convex domains <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi mathvariant=\"script\">𝒟</m:mi> <m:mo>⊂</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>d</m:mi> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2091_eq_0354.png\" /> <jats:tex-math>{mathcal{D}subsetmathbb{R}^{d}}</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 stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>d</m:mi> <m:mo>≤</m:mo> <m:mn>3</m:mn> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2091_eq_0269.png\" /> <jats:tex-math>{(dleq 3)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> with Robin boundary conditions (Rbcs) has been mathematically analyzed. The Faedo–Galerkin approach is used to demonstrate the global existence and uniqueness of a weak solution to the system. The weak solution’s higher regularity findings are constructed under more regular conditions on the initial data. In addition, continuous dependence on the initial conditions has been proved.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504234","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}

{"title":"Bilinear multipliers on weighted Orlicz spaces","authors":"Rüya Üster","doi":"10.1515/gmj-2023-2099","DOIUrl":"https://doi.org/10.1515/gmj-2023-2099","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi mathvariant=\"normal\">Φ</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2099_eq_0238.png\" /> <jats:tex-math>{Phi_{i}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be Young functions and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mi>ω</m:mi> <m:mi>i</m:mi> </m:msub> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2099_eq_0296.png\" /> <jats:tex-math>{omega_{i}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> be weights on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>d</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2099_eq_0267.png\" /> <jats:tex-math>{mathbb{R}^{d}}</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>i</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mn>3</m:mn> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2099_eq_0356.png\" /> <jats:tex-math>{i=1,2,3}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. A locally integrable function <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>m</m:mi> <m:mo>⁢</m:mo> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mi>ξ</m:mi> <m:mo>,</m:mo> <m:mi>η</m:mi> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2099_eq_0359.png\" /> <jats:tex-math>{m(xi,eta)}</jats:tex-math> </jats:alternatives> </jats:inline-formula> on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:msup> <m:mi>ℝ</m:mi> <m:mi>d</m:mi> </m:msup> <m:mo>×</m:mo> <m:msup> <m:mi>ℝ</m:mi> <m:mi>d</m:mi> </m:msup> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2099_eq_0266.png\" /> <jats:tex-math>{mathbb{R}^{d}timesmathbb{R}^{d}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is said to be a bilinear multiplier on <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi>ℝ</m:mi> <m:mi>d</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2099_eq_0267.png\" /> <jats:tex-math>{mathbb{R}^{d}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> of type <jats:inline-formul","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504237","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}

{"title":"Calculation of Reynolds equation for the generalized non-Newtonian fluids and its asymptotic behavior in a thin domain","authors":"Mohamed Dilmi, Aissa Benseghir, Mourad Dilmi, Hamid Benseridi","doi":"10.1515/gmj-2023-2090","DOIUrl":"https://doi.org/10.1515/gmj-2023-2090","url":null,"abstract":"Three-dimensional boundary-value problem describing a generalized non-Newtonian fluid with nonlinear Tresca friction type in a thin domain <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mi mathvariant=\"normal\">Ω</m:mi> <m:mi>ε</m:mi> </m:msup> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2090_eq_0267.png\" /> <jats:tex-math>{Omega^{varepsilon}}</jats:tex-math> </jats:alternatives> </jats:inline-formula> are considered. We study the asymptotic behavior when one dimension of the fluid domain tends to zero. We prove some weak convergence of the velocity and the pressure of the fluid. Then the limit problem in two-dimensional domain and the specific Reynolds equation are obtained.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504235","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}

{"title":"On the standing wave in coupled fractional Klein–Gordon equation","authors":"Zhenyu Guo, Xin Zhang","doi":"10.1515/gmj-2023-2089","DOIUrl":"https://doi.org/10.1515/gmj-2023-2089","url":null,"abstract":"Abstract The aim of this paper is to deal with the standing wave problems in coupled nonlinear fractional Klein–Gordon equations. First, we establish the constrained minimizations for a single nonlinear fractional Laplace equation. Then we prove the existence of a standing wave with a ground state using a variational argument. Next, applying the potential well argument and the concavity method, we obtain the sharp criterion for blowing up and global existence. Finally, we show the instability of the standing wave.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504236","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}

{"title":"New quantum integral inequalities for left and right log-ℏ-convex interval-valued functions","authors":"Haiyang Cheng, Dafang Zhao, Guohui Zhao, Delfim F. M. Torres","doi":"10.1515/gmj-2023-2088","DOIUrl":"https://doi.org/10.1515/gmj-2023-2088","url":null,"abstract":"We introduce the concept of quantum integration for interval-valued functions and establish new <jats:italic>q</jats:italic>-Hermite–Hadamard and <jats:italic>q</jats:italic>-Hermite–Hadamard–Fejér inequalities for left and right <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>log</m:mi> <m:mo>⁢</m:mo> <m:mtext>-</m:mtext> <m:mo>⁢</m:mo> <m:mi>h</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2088_eq_0218.png\" /> <jats:tex-math>{mathrm{log}text{-}h}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-convex interval-valued functions. Our results generalize the known ones in the literature and serve as a foundation for future studies in inequalities for interval-valued functions and interval differential equations. We illustrate our results with examples.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504239","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}

{"title":"On geometrical characteristics and inequalities of new bicomplex Lebesgue Spaces with hyperbolic-valued norm","authors":"Erdem Toksoy, Birsen Sağır","doi":"10.1515/gmj-2023-2093","DOIUrl":"https://doi.org/10.1515/gmj-2023-2093","url":null,"abstract":"In this work, it is assumed that the norm over bicomplex numbers is the hyperbolic (<jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>𝔻</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2093_eq_0265.png\" /> <jats:tex-math>{mathbb{D}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-valued) norm. In this paper, we provide an overview of bicomplex Lebesgue spaces and investigate some of their geometric properties, including <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>𝔹</m:mi> <m:mo>⁢</m:mo> <m:mi>ℂ</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2093_eq_0260.png\" /> <jats:tex-math>{mathbb{B}mathbb{C}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-convexity, <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>𝔹</m:mi> <m:mo>⁢</m:mo> <m:mi>ℂ</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2093_eq_0260.png\" /> <jats:tex-math>{mathbb{B}mathbb{C}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-strict convexity, and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mi>𝔹</m:mi> <m:mo>⁢</m:mo> <m:mi>ℂ</m:mi> </m:mrow> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2093_eq_0260.png\" /> <jats:tex-math>{mathbb{B}mathbb{C}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-uniform convexity. Moreover, the basic inequalities such as <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>𝔻</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2093_eq_0265.png\" /> <jats:tex-math>{mathbb{D}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-Hölder’s inequality and <jats:inline-formula> <jats:alternatives> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>𝔻</m:mi> </m:math> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2093_eq_0265.png\" /> <jats:tex-math>{mathbb{D}}</jats:tex-math> </jats:alternatives> </jats:inline-formula>-Minkowski inequality for bicomplex Lebesgue spaces are presented, used to show geometric properties.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504241","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}

{"title":"Fractal Mellin transform and non-local derivatives","authors":"Alireza Khalili Golmankhaneh, Kerri Welch, Cristina Serpa, Palle E. T. Jørgensen","doi":"10.1515/gmj-2023-2094","DOIUrl":"https://doi.org/10.1515/gmj-2023-2094","url":null,"abstract":"Abstract This paper provides a comparison between the fractal calculus of fractal sets and fractal curves. There are introduced the analogues of the Riemann–Liouville and Caputo integrals and derivatives for fractal curves, which are non-local derivatives. Moreover, the concepts analogous to the fractional Laplace operator to address fractal non-local differential equations on fractal curves are defined. Additionally, in the paper it is introduced the fractal local Mellin transform and fractal non-local transform as tools for solving fractal differential equations. The results are supported with tables and examples to demonstrate the findings.","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2023-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138517378","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}