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 ⟨p,q⟩S≔∫0∞p*(x)WLA(x)q(x)dx+M∫0∞(p′(x))*W(x)q′(x)dx,{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, where WLA(x)= <
在本手稿中，我们研究了矩阵正交多项式的一些代数和微分性质，这些矩阵正交多项式与由 ⟨ p , q ⟩ S ≔ ∫ 0 ∞ p * ( x ) W L A ( x ) q ( x ) d x + M ∫ 0 ∞ ( p ′ ( x ) ) 定义的 Laguerre-Sobolev 右倍线性形式有关。 * W ( x ) q ′ ( x ) d x , {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、 其中 W L A ( x ) = e - λ x x A {{bf{W}}}_{{bf{L}}}^{bf{A}}}left(x)={e}^{-lambda x}{x}^{{bf{A}}} 是拉盖尔矩阵权重、 W {bf{W}} 是某个矩阵权重，p p 和 q q 是矩阵多项式，M {bf{M}} 和 A {bf{A}} 是矩阵，使得 M {bf{M}} 是非奇异矩阵，A {bf{A}} 满足谱条件，λ lambda 是实部为正的复数。
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.
{"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":null,"pages":null},"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}
Silvestru Sever Dragomir, Mohamed Jleli, Bessem Samet
In this article, we establish Hermite-Hadamard-type inequalities for the two classes of functions X±λ(Ω)={f∈C2(Ω):Δf±λf≥0}{X}_{pm lambda }left(Omega )={fin {C}^{2}left(Omega ):Delta fpm lambda fge 0}, where λ>0lambda gt 0 and ΩOmega is an open subset of R2{{mathbb{R}}}^{2}. We also obtain a characterization of the set X−λ(Ω){X}_{-lambda }left(Omega ). Notice that in the one-dimensional case, if
在本文中，我们为两类函数 X ± λ ( Ω ) = { f∈ C 2 ( Ω ) : Δ f ± λ f ≥ 0 } 建立了赫米特-哈达玛式不等式。 {X}_{pm lambda }left(Omega )={fin {C}^{2}left(Omega ):Delta fpm lambda fge 0} 其中 λ > 0 （lambda gt 0）和 Ω Omega 是 R 2 {{mathbb{R}}^{2} 的一个开放子集。我们还可以得到集合 X -λ ( Ω ) {X}_{-lambda }left(Omega ) 的特征。请注意，在一维情况下，如果 Ω = I Omega =I（R {mathbb{R}} 的一个开放区间）且 λ = ρ 2 lambda ={rho }^{2} ，ρ > 0 λ = ρ 2 lambda ={rho }^{2}, ρ > 0 λ = ρ 2 lambda ={rho }^{2}. ρ > 0 rho gt 0 ，那么 X λ ( Ω ) {X}_{lambda }left(Omega ) （resp. X - λ ( Ω ) {X}_{-lambda }left(Omega ) 类函数 f∈ C 2 ( I ) fin {C}^{2}left(I)，使得 f f 在 I I 上是三角函数 ρ rho -凸（或者说双曲函数 ρ rho -凸）。
{"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":null,"pages":null},"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}
In this note, we prove that the toroidal pseudo-differential operator is bounded from L∞(Tn){L}^{infty }left({{mathbb{T}}}^{n}) to BMO(Tn){rm{BMO}}left({{mathbb{T}}}^{n}) if the symbol belongs to the toroidal Hörmander class Sρ,δn(ρ−1)∕2(Tn×Zn){S}_{rho ,delta }^{nleft(rho -1)/2}left({{mathbb{T}}}^{n}times {{mathbb{Z}}}^{n}) with 0<ρ≤10lt rho le 1 and 0≤δ<10le
在本说明中，我们证明，如果符号属于环形霍曼德类 S ρ，则环形伪微分算子从 L ∞ ( T n ) {L}^{infty }left({{mathbb{T}}}^{n}) 到 BMO ( T n ) {rm{BMO}}left({{mathbb{T}}}^{n}) 是有界的、δ n ( ρ - 1 ) ∕ 2 ( T n × Z n ) {S}_{rho ,delta }^{nleft(rho -1)/2}left({{mathbb{T}}}^{n}times {{mathbb{Z}}}^{n}) with 0 <；ρ ≤ 1 0lt rho le 1 和 0 ≤ δ < 1 0le delta lt 1 。作为推论，我们得到了当 2 < p < 时 L p {L}^{p} 上环形伪微分算子的结果；∞ 2lt plt infty for symbols in the class S ρ , δ m ( T n × Z n ) {S}_{rho 、m≤ n ( ρ - 1 ) 1 2 - 1 p + n p min { 0 , ρ - δ } mle nleft(rho -1)left(phantom{rule[-0.75em]{}{0ex}},（frac{1}{2}-frac{1}{p}right）+frac{n}{p}min （left{0,rho -delta right}） .
Somayeh Mirzadeh, Hasan Barsam, Loredana Ciurdariu
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 XX (where XX 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.
在本研究中，我们讨论了如何最小化定义在 X X（其中 X X 是实局部凸拓扑向量空间）上的两个非正值递增、共垂和准凹（ICRQC）函数之差的问题。为此，我们首先给出了非正共射函数上支持集最小元素的不同特征。然后，我们提出了两个非正 ICRQC 函数之差的全局最小值的充分和必要条件。
{"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":null,"pages":null},"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}
We investigate third-order strongly nonlinear differential equations of the type (Φ(k(t)u″(t)))′=f(t,u(t),u′(t),u″(t)),a.e. on[0,T],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], where ΦPhi is a strictly increasing homeomorphism, and the non-negative function kk 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
我们研究的三阶强非线性微分方程类型为 ( Φ ( k ( t ) u ″ ( t ) ) ′ = f ( t , u ( t ) , u ′ ( t ) , u ″ ( t ) ) , a.e. on [ 0 , T ] , 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.关于}hspace{0.1em}hspace{0.33em}left[0,T]，其中Φ Phi是严格递增的同构，非负函数k k可能在度量为零的集合上消失。利用上下解法，我们证明了与上述方程相关的一些边界值问题的存在性结果。此外，我们还考虑了二阶整微分方程，如 ( Φ ( k ( t ) v ′ ( t ) ) ′ = f t , ∫ 0 t v ( s ) d s , v ( t ) , v ′ ( t ) , a.e.on [ 0 , T ] , left(Phi left(kleft(t)v^{prime} left(t)))^{prime} =fleft(t,underset{0}{overset{t}{int }}vleft(s){rm{d}}s,vleft(t),v^{prime} left(t)right),hspace{1em}hspace{0.1em}text{a.e.on}/hspace{0.1em}/hspace{0.33em}/left[0,T]，为此我们提供了各种边界条件的存在性结果，包括周期性条件、Sturm-Liouville 条件和 Neumann-type 条件。
{"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 <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0008_eq_001.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant=\"normal\">Φ</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>k</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo accent=\"true\">″</m:mo> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo accent=\"false\">′</m:mo> <m:mo>=</m:mo> <m:mi>f</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mi>u</m:mi> <m:mo accent=\"false\">′</m:mo> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:msup> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mo accent=\"true\">″</m:mo> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:mspace width=\"1em\"/> <m:mspace width=\"0.1em\"/> <m:mtext>a.e. on</m:mtext> <m:mspace width=\"0.1em\"/> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mi>T</m:mi> </m:mrow> <m:mo>]</m:mo> </m:mrow> <m:mo>,</m:mo> </m:mrow> </m:math> <jats:tex-math>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],</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-2024-0008_eq_002.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi mathvariant=\"normal\">Φ</m:mi> </m:math> <jats:tex-math>Phi </jats:tex-math> </jats:alternatives> </jats:inline-formula> is a strictly increasing homeomorphism, and the non-negative function <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2024-0008_eq_003.png\"/> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>k</m:mi> </m:math> <jats:tex-math>k</jats:tex-math> </jats:alternatives> </jats:inline-formula> 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 <jats:disp-formula> <jats:alternatives> <jats:graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graph","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":null,"pages":null},"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}
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.
我们为全局循环指数（基尔霍夫指数的一种变体）找到了新的上限，并讨论了达到这些上限的广泛图系。
{"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":null,"pages":null},"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}
For a given sequence a=(a1,…,an)∈Rna=left({a}_{1},ldots ,{a}_{n})in {{mathbb{R}}}^{n}, our aim is to obtain an estimate of En≔a1+an2−1n∑i=1nai{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|. Several classes of sequences are studied. For each class, an estimate of En{E}_{n} 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.
对于给定序列 a = ( a 1 , ... , a n ) ∈ R n a=left({a}_{1},ldots ,{a}_{n})in {{{mathbb{R}}}^{n}, 我们的目的是获得 E n ≔ a 1 + a n 2 - 1 n ∑ i = 1 n a i {E}_{n}:=left| {E}_{n}:=( a 1 , ... , a n ) 我们的目的是获得 E n ≔ a 1 + a n 2 - 1 n ∑ i = 1 n a i {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| .本文研究了几类序列。对于每一类序列，我们都得到了 E n {E}_{n} 的估计值。我们还引入了凸矩阵类，它是坐标上凸函数类的离散版本。对于这组矩阵，我们证明了新的离散赫米特-哈达玛不等式。我们获得的结果是连续情况下已知结果在离散情况下的扩展。
{"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":null,"pages":null},"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}
In this article, we study a p-Laplacian Schrödinger-Poisson system involving a parameter q≠0qne 0 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 q*>0{q}^{* }gt 0 such that only the trivial solution is admitted for q∈(q*,+∞).qin left({q}^{* },+infty ). On the other hand, there are two positive solutions existing for q∈(0,q0*+ε)qin left(0,{q}_{0}^{* }+varepsilon ), where ε>0varepsilon gt 0 and q
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