Georgian Mathematical Journal最新文献

{"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":"231 1","pages":""},"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":"22 2","pages":""},"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 &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msub&gt; &lt;m:mi mathvariant=\"normal\"&gt;Φ&lt;/m:mi&gt; &lt;m:mi&gt;i&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2099_eq_0238.png\" /&gt; &lt;jats:tex-math&gt;{Phi_{i}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be Young functions and &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msub&gt; &lt;m:mi&gt;ω&lt;/m:mi&gt; &lt;m:mi&gt;i&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2099_eq_0296.png\" /&gt; &lt;jats:tex-math&gt;{omega_{i}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; be weights on &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msup&gt; &lt;m:mi&gt;ℝ&lt;/m:mi&gt; &lt;m:mi&gt;d&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2099_eq_0267.png\" /&gt; &lt;jats:tex-math&gt;{mathbb{R}^{d}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;i&lt;/m:mi&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mn&gt;3&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2099_eq_0356.png\" /&gt; &lt;jats:tex-math&gt;{i=1,2,3}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;. A locally integrable function &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;m&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;ξ&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;η&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2099_eq_0359.png\" /&gt; &lt;jats:tex-math&gt;{m(xi,eta)}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; on &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:msup&gt; &lt;m:mi&gt;ℝ&lt;/m:mi&gt; &lt;m:mi&gt;d&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;m:mo&gt;×&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mi&gt;ℝ&lt;/m:mi&gt; &lt;m:mi&gt;d&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2099_eq_0266.png\" /&gt; &lt;jats:tex-math&gt;{mathbb{R}^{d}timesmathbb{R}^{d}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; is said to be a bilinear multiplier on &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msup&gt; &lt;m:mi&gt;ℝ&lt;/m:mi&gt; &lt;m:mi&gt;d&lt;/m:mi&gt; &lt;/m:msup&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2099_eq_0267.png\" /&gt; &lt;jats:tex-math&gt;{mathbb{R}^{d}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; of type &lt;jats:inline-formul","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"21 5","pages":""},"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":"22 1","pages":""},"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":"21 6","pages":""},"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":"21 3","pages":""},"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":"21 1","pages":""},"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":"97 1","pages":""},"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}

{"title":"Reproducing kernel Banach space defined by the minimal norm property and applications to partial differential equation theory","authors":"Tomasz Łukasz Żynda","doi":"10.1515/gmj-2023-2095","DOIUrl":"https://doi.org/10.1515/gmj-2023-2095","url":null,"abstract":"It it well known that a Hilbert space &lt;jats:italic&gt;V&lt;/jats:italic&gt; of functions defined on &lt;jats:italic&gt;U&lt;/jats:italic&gt; is a reproducing kernel Hilbert space if and only if for any &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;z&lt;/m:mi&gt; &lt;m:mo&gt;∈&lt;/m:mo&gt; &lt;m:mi&gt;U&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2095_eq_0240.png\" /&gt; &lt;jats:tex-math&gt;{zin U}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, in the set &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:msub&gt; &lt;m:mi&gt;V&lt;/m:mi&gt; &lt;m:mi&gt;z&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;m:mo&gt;:=&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;{&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;f&lt;/m:mi&gt; &lt;m:mo&gt;∈&lt;/m:mo&gt; &lt;m:mi&gt;V&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;f&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;z&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;}&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2095_eq_0132.png\" /&gt; &lt;jats:tex-math&gt;{V_{z}:={fin Vmid f(z)=1}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, if non-empty, there is exactly one element with minimal norm and there is a direct connection between the reproducing kernel and such an element. In this paper, we define reproducing kernel Banach space as a space which satisfies this property and the reproducing kernel of it using this relation. We show that this reproducing kernel share a lot of basic properties with the classical one. The notable exception is that in Banach spaces the equality &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;K&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;z&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;w&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mover accent=\"true\"&gt; &lt;m:mrow&gt; &lt;m:mi&gt;K&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;w&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;z&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;¯&lt;/m:mo&gt; &lt;/m:mover&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_gmj-2023-2095_eq_0105.png\" /&gt; &lt;jats:tex-math&gt;{K(z,w)=overline{K(w,z)}}&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; does not have to be true without assumptions that &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;K&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;z&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;w&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;≠&lt;/m:mo&gt; &lt;m:mn&gt;0&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;K&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo ","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":"21 4","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138504238","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":"Localization operators and inversion formulas for the Dunkl–Weinstein–Stockwell transform","authors":"Fethi Soltani, Ibrahim Maktouf","doi":"10.1515/gmj-2023-2077","DOIUrl":"https://doi.org/10.1515/gmj-2023-2077","url":null,"abstract":"Abstract We define and study the Stockwell transform &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msub&gt; &lt;m:mi mathvariant=\"script\"&gt;S&lt;/m:mi&gt; &lt;m:mi&gt;g&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;/m:math&gt; mathscr{S}_{g} associated to the Dunkl–Weinstein operator &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msub&gt; &lt;m:mi mathvariant=\"normal\"&gt;Δ&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mi&gt;k&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;β&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msub&gt; &lt;/m:math&gt; Delta_{k,beta} and prove a Plancherel theorem and an inversion formula. Next, we define a reconstruction function &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msub&gt; &lt;m:mi&gt;f&lt;/m:mi&gt; &lt;m:mi mathvariant=\"normal\"&gt;Δ&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;/m:math&gt; f_{Delta} and prove Calderón’s reproducing inversion formula for the Dunkl–Weinstein–Stockwell transform &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msub&gt; &lt;m:mi mathvariant=\"script\"&gt;S&lt;/m:mi&gt; &lt;m:mi&gt;g&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;/m:math&gt; mathscr{S}_{g} . Moreover, we define the localization operators &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:msub&gt; &lt;m:mi mathvariant=\"script\"&gt;L&lt;/m:mi&gt; &lt;m:mi&gt;g&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;σ&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; mathcal{L}_{g}(sigma) associated to this transform. We study the boundedness and compactness of these operators and establish a trace formula. Finally, we introduce and study the extremal function &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:msubsup&gt; &lt;m:mi&gt;F&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mi&gt;η&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;k&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;∗&lt;/m:mo&gt; &lt;/m:msubsup&gt; &lt;m:mo lspace=\"0.278em\" rspace=\"0.278em\"&gt;:=&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;η&lt;/m:mi&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mi&gt;I&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:msubsup&gt; &lt;m:mi mathvariant=\"script\"&gt;S&lt;/m:mi&gt; &lt;m:mi&gt;g&lt;/m:mi&gt; &lt;m:mo&gt;∗&lt;/m:mo&gt; &lt;/m:msubsup&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:msub&gt; &lt;m:mi mathvariant=\"script\"&gt;S&lt;/m:mi&gt; &lt;m:mi&gt;g&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:mn&gt;1&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:msubsup&gt; &lt;m:mi mathvariant=\"script\"&gt;S&lt;/m:mi&gt; &lt;m:mi&gt;g&lt;/m:mi&gt; &lt;m:mo&gt;∗&lt;/m:mo&gt; &lt;/m:msubsup&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:mi&gt;k&lt;/m:mi&gt; &lt;m:mo stretchy=\"false\"&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; F^{ast}_{eta,smash{k}}:=(eta I+mathscr{S}^{ast}_{g}mathscr{S}_{g})^{-1}mathscr{S}^{ast}_{g}(k) , and we deduce best approximate inversion formulas for the Dunkl–Weinstein–Stockwell transform &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msub&gt; &lt;m:mi mathvariant=\"script\"&gt;S&lt;/m:mi&gt; &lt;m:mi&gt;g&lt;/m:mi&gt; &lt;/m:msub&gt; &lt;/m:math&gt; mathscr{S}_{g} on the Sobolev space &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mrow&gt; &lt;m:msubsup&gt; &lt;m:mi mathvariant=\"script\"&gt;H&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mi&gt;k&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;β&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mi&gt;s&lt;/m:mi&gt; &lt;/m:msubsup&gt; &lt;m:mo&gt;⁢&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo stretchy=\"false\"&gt;(&lt;/m:mo&gt; &lt;m:m","PeriodicalId":55101,"journal":{"name":"Georgian Mathematical Journal","volume":" 84","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135340610","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}