Open Mathematics最新文献

{"title":"On Laguerre-Sobolev matrix orthogonal polynomials","authors":"Edinson Fuentes, Luis E. Garza, Martha L. Saiz","doi":"10.1515/math-2024-0029","DOIUrl":"https://doi.org/10.1515/math-2024-0029","url":null,"abstract":"In this manuscript, we study some algebraic and differential properties of matrix orthogonal polynomials with respect to the Laguerre-Sobolev right sesquilinear form defined by <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0029_eq_001.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <m:msub> <m:mrow> <m:mrow> <m:mo>⟨</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo>,</m:mo> <m:mi>q</m:mi> </m:mrow> <m:mo>⟩</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi mathvariant=\"bold\">S</m:mi> </m:mrow> </m:msub> <m:mo>≔</m:mo> <m:munderover> <m:mrow> <m:mrow> <m:mo>∫</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:munderover> <m:msup> <m:mrow> <m:mi>p</m:mi> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msubsup> <m:mrow> <m:mi mathvariant=\"bold\">W</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant=\"bold\">L</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant=\"bold\">A</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>q</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi mathvariant=\"normal\">d</m:mi> <m:mi>x</m:mi> <m:mo>+</m:mo> <m:mi mathvariant=\"bold\">M</m:mi> <m:munderover> <m:mrow> <m:mrow> <m:mo>∫</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:munderover> <m:msup> <m:mrow> <m:mrow> <m:mo stretchy=\"false\">(</m:mo> <m:mrow> <m:mi>p</m:mi> <m:mo accent=\"false\">′</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy=\"false\">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mo>*</m:mo> </m:mrow> </m:msup> <m:mi mathvariant=\"bold\">W</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi>q</m:mi> <m:mo accent=\"false\">′</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mi mathvariant=\"normal\">d</m:mi> <m:mi>x</m:mi> <m:mo>,</m:mo> </m:math> <jats:tex-math>{langle p,qrangle }_{{bf{S}}}:= underset{0}{overset{infty }{int }}{p}^{* }left(x){{bf{W}}}_{{bf{L}}}^{{bf{A}}}left(x)qleft(x){rm{d}}x+{bf{M}}underset{0}{overset{infty }{int }}{(p^{prime} left(x))}^{* }{bf{W}}left(x)q^{prime} left(x){rm{d}}x,</jats:tex-math> </jats:alternatives> </jats:disp-formula> where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0029_eq_002.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mrow> <m:mi mathvariant=\"bold\">W</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant=\"bold\">L</m:mi> </m:mrow> <m:mrow> <m:mi mathvariant=\"bold\">A</m:mi> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:msup> <","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"6 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141944107","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":"Matrix stretching","authors":"Vyacheslav Futorny, Mikhail Neklyudov, Kaiming Zhao","doi":"10.1515/math-2024-0031","DOIUrl":"https://doi.org/10.1515/math-2024-0031","url":null,"abstract":"We consider the tensor products of square matrices of different sizes and introduce the stretching maps, which can be viewed as a generalized matricization. Stretching maps conserve algebraic properties of the tensor product, but are not necessarily injective. Dropping the injectivity condition allows us to construct examples of stretching maps with additional symmetry properties. Furthermore, this leads to the averaging of the tensor product and possibly could be used to compress the data.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"198 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141944106","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":"Hermite-Hadamard-type inequalities for generalized trigonometrically and hyperbolic ρ-convex functions in two dimension","authors":"Silvestru Sever Dragomir, Mohamed Jleli, Bessem Samet","doi":"10.1515/math-2024-0028","DOIUrl":"https://doi.org/10.1515/math-2024-0028","url":null,"abstract":"In this article, we establish Hermite-Hadamard-type inequalities for the two classes of functions <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0028_eq_002.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mrow> <m:mo>±</m:mo> <m:mi>λ</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant=\"normal\">Ω</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>=</m:mo> <m:mrow> <m:mo stretchy=\"false\">{</m:mo> <m:mrow> <m:mi>f</m:mi> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi>C</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant=\"normal\">Ω</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>:</m:mo> <m:mi mathvariant=\"normal\">Δ</m:mi> <m:mi>f</m:mi> <m:mo>±</m:mo> <m:mi>λ</m:mi> <m:mi>f</m:mi> <m:mo>≥</m:mo> <m:mn>0</m:mn> </m:mrow> <m:mo stretchy=\"false\">}</m:mo> </m:mrow> </m:math> <jats:tex-math>{X}_{pm lambda }left(Omega )={fin {C}^{2}left(Omega ):Delta fpm lambda fge 0}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0028_eq_003.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>λ</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:math> <jats:tex-math>lambda gt 0</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0028_eq_004.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Ω</m:mi> </m:math> <jats:tex-math>Omega </jats:tex-math> </jats:alternatives> </jats:inline-formula> is an open subset of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0028_eq_005.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:msup> </m:math> <jats:tex-math>{{mathbb{R}}}^{2}</jats:tex-math> </jats:alternatives> </jats:inline-formula>. We also obtain a characterization of the set <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0028_eq_006.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>X</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mi>λ</m:mi> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant=\"normal\">Ω</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{X}_{-lambda }left(Omega )</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Notice that in the one-dimensional case, if <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmln","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"53 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886133","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":"Endpoint boundedness of toroidal pseudo-differential operators","authors":"Benhamoud Ramla","doi":"10.1515/math-2024-0023","DOIUrl":"https://doi.org/10.1515/math-2024-0023","url":null,"abstract":"In this note, we prove that the toroidal pseudo-differential operator is bounded from <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0023_eq_001.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi>L</m:mi> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">T</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{L}^{infty }left({{mathbb{T}}}^{n})</jats:tex-math> </jats:alternatives> </jats:inline-formula> to <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0023_eq_002.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">BMO</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">T</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{rm{BMO}}left({{mathbb{T}}}^{n})</jats:tex-math> </jats:alternatives> </jats:inline-formula> if the symbol belongs to the toroidal Hörmander class <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0023_eq_003.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msubsup> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mi>ρ</m:mi> <m:mo>,</m:mo> <m:mi>δ</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>ρ</m:mi> <m:mo>−</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∕</m:mo> <m:mn>2</m:mn> </m:mrow> </m:msubsup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">T</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> <m:mo>×</m:mo> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">Z</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>{S}_{rho ,delta }^{nleft(rho -1)/2}left({{mathbb{T}}}^{n}times {{mathbb{Z}}}^{n})</jats:tex-math> </jats:alternatives> </jats:inline-formula> with <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0023_eq_004.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>ρ</m:mi> <m:mo>≤</m:mo> <m:mn>1</m:mn> </m:math> <jats:tex-math>0lt rho le 1</jats:tex-math> </jats:alternatives> </jats:inline-formula> and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0023_eq_005.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:mi>δ</m:mi> <m:mo><</m:mo> <m:mn>1</m:mn> </m:math> <jats:tex-math>0le ","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"159 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-08-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141886132","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":"Characterizations of minimal elements of upper support with applications in minimizing DC functions","authors":"Somayeh Mirzadeh, Hasan Barsam, Loredana Ciurdariu","doi":"10.1515/math-2024-0030","DOIUrl":"https://doi.org/10.1515/math-2024-0030","url":null,"abstract":"In this study, we discuss on the problem of minimizing the differences of two non-positive valued increasing, co-radiant and quasi-concave (ICRQC) functions defined on <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0030_eq_001.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>X</m:mi> </m:math> <jats:tex-math>X</jats:tex-math> </jats:alternatives> </jats:inline-formula> (where <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0030_eq_002.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>X</m:mi> </m:math> <jats:tex-math>X</jats:tex-math> </jats:alternatives> </jats:inline-formula> is a real locally convex topological vector space). For this purpose, we first gave different characterizations of the upper support set’s minimal elements of non-positive co-radiant functions. Then, we presented sufficient and necessary conditions for the global minimizers of the differences of two non-positive ICRQC functions.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"73 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141785566","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":"Boundary value problems for integro-differential and singular higher-order differential equations","authors":"Francesca Anceschi, Alessandro Calamai, Cristina Marcelli, Francesca Papalini","doi":"10.1515/math-2024-0008","DOIUrl":"https://doi.org/10.1515/math-2024-0008","url":null,"abstract":"We investigate third-order strongly nonlinear differential equations of the type &lt;jats:disp-formula&gt; &lt;jats:alternatives&gt; &lt;jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0008_eq_001.png\"/&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"&gt; &lt;m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"normal\"&gt;Φ&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;k&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;t&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo accent=\"true\"&gt;″&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;t&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:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mo accent=\"false\"&gt;′&lt;/m:mo&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mi&gt;f&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;t&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:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;t&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo accent=\"false\"&gt;′&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;t&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo accent=\"true\"&gt;″&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;t&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:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mspace width=\"1em\"/&gt; &lt;m:mspace width=\"0.1em\"/&gt; &lt;m:mtext&gt;a.e. on&lt;/m:mtext&gt; &lt;m:mspace width=\"0.1em\"/&gt; &lt;m:mrow&gt; &lt;m:mo&gt;[&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mn&gt;0&lt;/m:mn&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mi&gt;T&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;]&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;left(Phi left(kleft(t){u}^{^{primeprime} }left(t)))^{prime} =fleft(t,uleft(t),u^{prime} left(t),{u}^{^{primeprime} }left(t)),hspace{1em}hspace{0.1em}text{a.e. on}hspace{0.1em}hspace{0.33em}left[0,T],&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:disp-formula&gt; where &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0008_eq_002.png\"/&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi mathvariant=\"normal\"&gt;Φ&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;Phi &lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; is a strictly increasing homeomorphism, and the non-negative function &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0008_eq_003.png\"/&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;k&lt;/m:mi&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;k&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; may vanish on a set of measure zero. Using the upper and lower solution method, we prove existence results for some boundary value problems associated with the aforementioned equation. Moreover, we also consider second-order integro-differential equations like &lt;jats:disp-formula&gt; &lt;jats:alternatives&gt; &lt;jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graph","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"5 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141778883","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 properties of soliton solution for the quasilinear Schrödinger system","authors":"Xue Zhang, Jing Zhang","doi":"10.1515/math-2024-0022","DOIUrl":"https://doi.org/10.1515/math-2024-0022","url":null,"abstract":"In this article, we consider the following quasilinear Schrödinger system: &lt;jats:disp-formula&gt; &lt;jats:alternatives&gt; &lt;jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0022_eq_001.png\"/&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"&gt; &lt;m:mfenced open=\"{\" close=\"\"&gt; &lt;m:mrow&gt; &lt;m:mtable displaystyle=\"true\"&gt; &lt;m:mtr&gt; &lt;m:mtd columnalign=\"left\"&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:mi&gt;ε&lt;/m:mi&gt; &lt;m:mi mathvariant=\"normal\"&gt;Δ&lt;/m:mi&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:mfrac&gt; &lt;m:mrow&gt; &lt;m:mi&gt;k&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:mfrac&gt; &lt;m:mi&gt;ε&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mo&gt;[&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;Δ&lt;/m:mi&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;]&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mfrac&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;m:mi&gt;α&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;α&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:mfrac&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;α&lt;/m:mi&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;β&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;/m:mtd&gt; &lt;m:mtd columnalign=\"left\"&gt; &lt;m:mi&gt;x&lt;/m:mi&gt; &lt;m:mo&gt;∈&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"double-struck\"&gt;R&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;N&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;/m:mtd&gt; &lt;/m:mtr&gt; &lt;m:mtr&gt; &lt;m:mtd columnalign=\"left\"&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:mi&gt;ε&lt;/m:mi&gt; &lt;m:mi mathvariant=\"normal\"&gt;Δ&lt;/m:mi&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:mfrac&gt; &lt;m:mrow&gt; &lt;m:mi&gt;k&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:mfrac&gt; &lt;m:mi&gt;ε&lt;/m:mi&gt; &lt;m:mrow&gt; &lt;m:mo&gt;[&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mi&gt;Δ&lt;/m:mi&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;]&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;m:mo&gt;=&lt;/m:mo&gt; &lt;m:mfrac&gt; &lt;m:mrow&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;m:mi&gt;β&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;α&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:mfrac&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;m:mi&gt;u&lt;/m:mi&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;α&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;m:mo&gt;∣&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;β&lt;/m:mi&gt; &lt;m:mo&gt;−&lt;/m:mo&gt; &lt;m:mn&gt;2&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mi&gt;v&lt;/m:mi&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;/m:mtd&gt; &lt;m:mtd columnalign=\"left\"&gt; &lt;m:mi&gt;x&lt;/m:mi&gt; &lt;m:mo&gt;∈&lt;/m:mo&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi mathvariant=\"double-struck\"&gt;R&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mi&gt;N&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;/m:mtd&gt; &lt;/m:mtr&gt; &lt;/m:mtable&gt; &lt;/m:mrow&gt; &lt;/m:mfenced&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;left{begin{array}{ll}-varepsilon Delta u+u+frac{k}{2}varepsilon left[Delta hspace{-0.25em}{| u| }^{2}]u=frac{2alpha }{alpha +beta }{| u| }^{alpha -2}u{| v| }^{beta },&amp; xin {{mathbb{R}}}^{N}, -varepsilon Delta v+v+frac{k}{2}varepsilon left[Delta hspace{-0.25em}{| v| }^{2}]v=frac{2beta }{alpha +beta ","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141785567","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":"Upper bounds for the global cyclicity index","authors":"José Luis Palacios","doi":"10.1515/math-2024-0016","DOIUrl":"https://doi.org/10.1515/math-2024-0016","url":null,"abstract":"We find new upper bounds for the global cyclicity index, a variant of the Kirchhoff index, and discuss the wide family of graphs for which the bounds are attained.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"34 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141718461","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 discrete inequalities for some classes of sequences","authors":"Mohamed Jleli, Bessem Samet","doi":"10.1515/math-2024-0021","DOIUrl":"https://doi.org/10.1515/math-2024-0021","url":null,"abstract":"For a given sequence <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0021_eq_001.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>a</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mrow> <m:mo>…</m:mo> </m:mrow> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>∈</m:mo> <m:msup> <m:mrow> <m:mi mathvariant=\"double-struck\">R</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> <jats:tex-math>a=left({a}_{1},ldots ,{a}_{n})in {{mathbb{R}}}^{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula>, our aim is to obtain an estimate of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0021_eq_002.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>E</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> <m:mo>≔</m:mo> <m:mfenced open=\"∣\" close=\"∣\"> <m:mrow> <m:mfrac> <m:mrow> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:mfrac> <m:mo>−</m:mo> <m:mfrac> <m:mrow> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:mfrac> <m:msubsup> <m:mrow> <m:mrow> <m:mo>∑</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msubsup> <m:msub> <m:mrow> <m:mi>a</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> </m:mrow> </m:mfenced> </m:math> <jats:tex-math>{E}_{n}:= left|hspace{-0.33em},frac{{a}_{1}+{a}_{n}}{2}-frac{1}{n}{sum }_{i=1}^{n}{a}_{i},hspace{-0.33em}right|</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Several classes of sequences are studied. For each class, an estimate of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0021_eq_003.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msub> <m:mrow> <m:mi>E</m:mi> </m:mrow> <m:mrow> <m:mi>n</m:mi> </m:mrow> </m:msub> </m:math> <jats:tex-math>{E}_{n}</jats:tex-math> </jats:alternatives> </jats:inline-formula> is obtained. We also introduce the class of convex matrices, which is a discrete version of the class of convex functions on the coordinates. For this set of matrices, new discrete Hermite-Hadamard-type inequalities are proved. Our obtained results are extensions of known results from the continuous case to the discrete case.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"35 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529761","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 fibering method approach for a Schrödinger-Poisson system with p-Laplacian in bounded domains","authors":"Jinfeng Xue, Libo Wang","doi":"10.1515/math-2024-0015","DOIUrl":"https://doi.org/10.1515/math-2024-0015","url":null,"abstract":"In this article, we study a &lt;jats:italic&gt;p&lt;/jats:italic&gt;-Laplacian Schrödinger-Poisson system involving a parameter &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0015_eq_001.png\"/&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;q&lt;/m:mi&gt; &lt;m:mo&gt;≠&lt;/m:mo&gt; &lt;m:mn&gt;0&lt;/m:mn&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;qne 0&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; in bounded domains. By using the Nehari manifold and the fibering method, we obtain the non-existence and multiplicity of nontrivial solutions. On one hand, there exists &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0015_eq_002.png\"/&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;q&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo&gt;*&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mo&gt;&gt;&lt;/m:mo&gt; &lt;m:mn&gt;0&lt;/m:mn&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;{q}^{* }gt 0&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; such that only the trivial solution is admitted for &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0015_eq_003.png\"/&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;q&lt;/m:mi&gt; &lt;m:mo&gt;∈&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:msup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;q&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo&gt;*&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:msup&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:mi&gt;∞&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;.&lt;/m:mo&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;qin left({q}^{* },+infty ).&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; On the other hand, there are two positive solutions existing for &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0015_eq_004.png\"/&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;q&lt;/m:mi&gt; &lt;m:mo&gt;∈&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mo&gt;(&lt;/m:mo&gt; &lt;m:mrow&gt; &lt;m:mn&gt;0&lt;/m:mn&gt; &lt;m:mo&gt;,&lt;/m:mo&gt; &lt;m:msubsup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;q&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mn&gt;0&lt;/m:mn&gt; &lt;/m:mrow&gt; &lt;m:mrow&gt; &lt;m:mo&gt;*&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:msubsup&gt; &lt;m:mo&gt;+&lt;/m:mo&gt; &lt;m:mi&gt;ε&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mo&gt;)&lt;/m:mo&gt; &lt;/m:mrow&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;qin left(0,{q}_{0}^{* }+varepsilon )&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt;, where &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0015_eq_005.png\"/&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:mi&gt;ε&lt;/m:mi&gt; &lt;m:mo&gt;&gt;&lt;/m:mo&gt; &lt;m:mn&gt;0&lt;/m:mn&gt; &lt;/m:math&gt; &lt;jats:tex-math&gt;varepsilon gt 0&lt;/jats:tex-math&gt; &lt;/jats:alternatives&gt; &lt;/jats:inline-formula&gt; and &lt;jats:inline-formula&gt; &lt;jats:alternatives&gt; &lt;jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0015_eq_006.png\"/&gt; &lt;m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;m:msubsup&gt; &lt;m:mrow&gt; &lt;m:mi&gt;q&lt;/m:mi&gt; &lt;/m:mrow&gt; &lt;m:mro","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"42 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141529641","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}