Boundary Value Problems最新文献

{"title":"A new weighted fractional operator with respect to another function via a new modified generalized Mittag–Leffler law","authors":"Sabri T. M. Thabet, Thabet Abdeljawad, Imed Kedim, M. Iadh Ayari","doi":"10.1186/s13661-023-01790-7","DOIUrl":"https://doi.org/10.1186/s13661-023-01790-7","url":null,"abstract":"Abstract In this paper, new generalized weighted fractional derivatives with respect to another function are derived in the sense of Caputo and Riemann–Liouville involving a new modified version of a generalized Mittag–Leffler function with three parameters, as well as their corresponding fractional integrals. In addition, several new and existing operators of nonsingular kernels are obtained as special cases of our operator. Many important properties related to our new operator are introduced, such as a series version involving Riemann–Liouville fractional integrals, weighted Laplace transforms with respect to another function, etc. Finally, an example is given to illustrate the effectiveness of the new results.","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":"60 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135350479","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 nonlinear fractional Choquard equation with indefinite potential and general nonlinearity","authors":"Fangfang Liao, Fulai Chen, Shifeng Geng, Dong Liu","doi":"10.1186/s13661-023-01786-3","DOIUrl":"https://doi.org/10.1186/s13661-023-01786-3","url":null,"abstract":"Abstract In this paper, we consider a class of fractional Choquard equations with indefinite potential $$ (-Delta )^{alpha}u+V(x)u= biggl[ int _{{mathbb{R}}^{N}} frac{M(epsilon y)G(u)}{ vert x-y vert ^{mu}},mathrm{d}y biggr]M( epsilon x)g(u), quad xin {mathbb{R}}^{N}, $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>−</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>α</mml:mi> </mml:msup> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>V</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:mrow> <mml:mo>[</mml:mo> <mml:msub> <mml:mo>∫</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>N</mml:mi> </mml:msup> </mml:msub> <mml:mfrac> <mml:mrow> <mml:mi>M</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ϵ</mml:mi> <mml:mi>y</mml:mi> <mml:mo>)</mml:mo> <mml:mi>G</mml:mi> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>x</mml:mi> <mml:mo>−</mml:mo> <mml:mi>y</mml:mi> <mml:msup> <mml:mo>|</mml:mo> <mml:mi>μ</mml:mi> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mspace /> <mml:mi>d</mml:mi> <mml:mi>y</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> <mml:mi>M</mml:mi> <mml:mo>(</mml:mo> <mml:mi>ϵ</mml:mi> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> <mml:mi>g</mml:mi> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> <mml:mo>,</mml:mo> <mml:mspace /> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mi>R</mml:mi> <mml:mi>N</mml:mi> </mml:msup> <mml:mo>,</mml:mo> </mml:math> where $alpha in (0,1)$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:math> , $N> 2alpha $ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>N</mml:mi> <mml:mo>></mml:mo> <mml:mn>2</mml:mn> <mml:mi>α</mml:mi> </mml:math> , $0<mu <2alpha $ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mn>0</mml:mn> <mml:mo><</mml:mo> <mml:mi>μ</mml:mi> <mml:mo><</mml:mo> <mml:mn>2</mml:mn> <mml:mi>α</mml:mi> </mml:math> , ϵ is a positive parameter. Here $(-Delta )^{alpha}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mo>−</mml:mo> <mml:mi>Δ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mi>α</mml:mi> </mml:msup> </mml:math> stands for the fractional Laplacian, V is a linear potential with periodicity condition, and M is a nonlinear reaction potential with a global condition. We establish the existence and concentration of ground state solutions under general nonlinearity by using variational methods.","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135435849","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":"Computing Dirichlet eigenvalues of the Schrödinger operator with a PT-symmetric optical potential","authors":"Cemile Nur","doi":"10.1186/s13661-023-01787-2","DOIUrl":"https://doi.org/10.1186/s13661-023-01787-2","url":null,"abstract":"Abstract We provide estimates for the eigenvalues of non-self-adjoint Sturm–Liouville operators with Dirichlet boundary conditions for a shift of the special potential $4cos ^{2}x+4iVsin 2x$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mn>4</mml:mn> <mml:msup> <mml:mo>cos</mml:mo> <mml:mn>2</mml:mn> </mml:msup> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:mn>4</mml:mn> <mml:mi>i</mml:mi> <mml:mi>V</mml:mi> <mml:mo>sin</mml:mo> <mml:mn>2</mml:mn> <mml:mi>x</mml:mi> </mml:math> that is a PT-symmetric optical potential, especially when $|c|=|sqrt{1-4V^{2}}|<2$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo>|</mml:mo> <mml:mi>c</mml:mi> <mml:mo>|</mml:mo> <mml:mo>=</mml:mo> <mml:mo>|</mml:mo> <mml:msqrt> <mml:mrow> <mml:mn>1</mml:mn> <mml:mo>−</mml:mo> <mml:mn>4</mml:mn> <mml:msup> <mml:mi>V</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:msqrt> <mml:mo>|</mml:mo> <mml:mo><</mml:mo> <mml:mn>2</mml:mn> </mml:math> or correspondingly $0leq V<sqrt {5}/2$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mn>0</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>V</mml:mi> <mml:mo><</mml:mo> <mml:msqrt> <mml:mn>5</mml:mn> </mml:msqrt> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:math> . We obtain some useful equations for calculating Dirichlet eigenvalues also for $|c|geq 2$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo>|</mml:mo> <mml:mi>c</mml:mi> <mml:mo>|</mml:mo> <mml:mo>≥</mml:mo> <mml:mn>2</mml:mn> </mml:math> or equally $Vgeq sqrt{5}/2$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>V</mml:mi> <mml:mo>≥</mml:mo> <mml:msqrt> <mml:mn>5</mml:mn> </mml:msqrt> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> </mml:math> . We discuss our results by comparing them with the periodic and antiperiodic eigenvalues of the Schrödinger operator. We even approximate complex eigenvalues by the roots of some polynomials derived from some iteration formulas. Moreover, we give a numerical example with error analysis.","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":"128 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135591274","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":"Analytical mechanics methods in finite element analysis of multibody elastic system","authors":"Maria Luminita Scutaru, Sorin Vlase, Marin Marin","doi":"10.1186/s13661-023-01784-5","DOIUrl":"https://doi.org/10.1186/s13661-023-01784-5","url":null,"abstract":"Abstract The study of multibody systems with elastic elements involves at the moment the reevaluation of the classical methods of analysis offered by analytical mechanics. Modeling this system with the finite element method requires obtaining the motion equation for an element in the circumstances imposed by a multibody system. The paper aims to present the main analysis methods used by researchers, to make a comparative analysis, and to show the advantages or disadvantages offered by different methods. For the presentation of the main methods (namely Lagrange’s equations, Gibbs–Appell’s equations, Maggi’s formalism, Kane’s equations, and Hamilton’s equations) a unified notation is used. The paper provides a critical evaluation of the studied applications that involved some of these methods, highlighting the reason why it was decided to use them. Also, the paper identifies potential research areas to explore.","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":"249 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135596572","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":"Remarks on a fractional nonlinear partial integro-differential equation via the new generalized multivariate Mittag-Leffler function","authors":"Chenkuan Li, Reza Saadati, Joshua Beaudin, Andrii Hrytsenko","doi":"10.1186/s13661-023-01783-6","DOIUrl":"https://doi.org/10.1186/s13661-023-01783-6","url":null,"abstract":"Abstract Introducing a new generalized multivariate Mittag-Leffler function which is a generalization of the multivariate Mittag-Leffler function, we derive a sufficient condition for the uniqueness of solutions to a brand new boundary value problem of the fractional nonlinear partial integro-differential equation using Banach’s fixed point theorem and Babenko’s technique. This has many potential applications since uniqueness is an important topic in many scientific areas, and the method used clearly opens directions for studying other types of equations and corresponding initial or boundary value problems. In addition, we use Python which is a high-level programming language efficiently dealing with the summation of multi-indices to compute approximate values of the generalized Mittag-Leffler function (it seems impossible to do so by any existing integral representation of the Mittag-Leffler function), and provide an example showing applications of key results derived.","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":"124 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135697329","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 generalized Halanay inequalities and relative applications to neural networks with variable delays","authors":"Chunsheng Wang, Han Chen, Runpeng Lin, Ying Sheng, Feng Jiao","doi":"10.1186/s13661-023-01773-8","DOIUrl":"https://doi.org/10.1186/s13661-023-01773-8","url":null,"abstract":"Abstract The asymptotic behavior of solutions for a new class of generalized Halanay inequalities is studied via the fixed point method. This research provides a new approach to the study of the stability of Halanay inequality. To make the application of fixed point method in stability research more flexible and feasible, we introduce corresponding functions to construct an operator according to different characteristics of coefficients. The results obtained in this paper are applied to the stability study of a neural network system, which has high value in application. Moreover, three examples and simulations are given to illustrate the results. The conclusions in this paper greatly improve and generalize the relative results in the current literature.","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135386379","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 well posedness of solutions for the 2D magnetomicropolar boundary layer equations in an analytic framework","authors":"Xiaolei Dong","doi":"10.1186/s13661-023-01782-7","DOIUrl":"https://doi.org/10.1186/s13661-023-01782-7","url":null,"abstract":"Abstract In this paper, we prove the existence and uniqueness of solutions to the 2D magnetomicropolar boundary layer equations on the half-plane by using the classical bootstrap argument in an analytic framework.","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":"81 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135817745","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":"Initial boundary value problem for a viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term: decay estimates and blow-up result","authors":"Billel Gheraibia, Nouri Boumaza","doi":"10.1186/s13661-023-01781-8","DOIUrl":"https://doi.org/10.1186/s13661-023-01781-8","url":null,"abstract":"Abstract In this paper, we study the initial boundary value problem for the following viscoelastic wave equation with Balakrishnan–Taylor damping and a delay term where the relaxation function satisfies $g'(t)leq -xi (t)g^{r}(t)$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msup> <mml:mi>g</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:mo>≤</mml:mo> <mml:mo>−</mml:mo> <mml:mi>ξ</mml:mi> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> <mml:msup> <mml:mi>g</mml:mi> <mml:mi>r</mml:mi> </mml:msup> <mml:mo>(</mml:mo> <mml:mi>t</mml:mi> <mml:mo>)</mml:mo> </mml:math> , $tgeq 0$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>t</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>0</mml:mn> </mml:math> , $1leq r< frac{3}{2}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mn>1</mml:mn> <mml:mo>≤</mml:mo> <mml:mi>r</mml:mi> <mml:mo><</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>3</mml:mn> </mml:mrow> <mml:mn>2</mml:mn> </mml:mfrac> </mml:math> . The main goal of this work is to study the global existence, general decay, and blow-up result. The global existence has been obtained by potential-well theory, the decay of solutions of energy has been established by introducing suitable energy and Lyapunov functionals, and a blow-up result has been obtained with negative initial energy.","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":"39 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135015932","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 fundamental solution and blow-up problem of an anisotropic parabolic equation","authors":"Huashui Zhan","doi":"10.1186/s13661-023-01780-9","DOIUrl":"https://doi.org/10.1186/s13661-023-01780-9","url":null,"abstract":"Abstract This paper is devoted to the study of anisotropic parabolic equation related to the $p_{i}$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:msub> <mml:mi>p</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math> -Laplacian with a source term $f(u)$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:math> . If $f(u)=0$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:math> , then the fundamental solution of the equation is constructed. If there are some restrictions on the growth order of u in the source term, the initial energy $E(0)$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>E</mml:mi> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>)</mml:mo> </mml:math> is positive and has a super boundedness, which depends on the Sobolev imbedding index, then the local solution may blow up in finite time.","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135395699","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":"Applying periodic and anti-periodic boundary conditions in existence results of fractional differential equations via nonlinear contractive mappings","authors":"Sumati Kumari Panda, Velusamy Vijayakumar, Kottakkaran Sooppy Nisar","doi":"10.1186/s13661-023-01778-3","DOIUrl":"https://doi.org/10.1186/s13661-023-01778-3","url":null,"abstract":"Abstract We introduce a notion of nonlinear cyclic orbital $(xi -mathscr{F})$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;ξ&lt;/mml:mi&gt; &lt;mml:mo&gt;−&lt;/mml:mo&gt; &lt;mml:mi&gt;F&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:math&gt; -contraction and prove related results. With these results, we address the existence and uniqueness results with periodic/anti-periodic boundary conditions for: 1. The nonlinear multi-order fractional differential equation $$ mathcal{L}(mathcal{D})theta (varsigma )=sigma bigl(varsigma , theta ( varsigma ) bigr), quad varsigma in mathscr{J}=[0,mathscr{A}], mathscr{A}&gt;0, $$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mi&gt;L&lt;/mml:mi&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;D&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;mml:mi&gt;θ&lt;/mml:mi&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;ς&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;mml:mo&gt;=&lt;/mml:mo&gt; &lt;mml:mi&gt;σ&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;ς&lt;/mml:mi&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mi&gt;θ&lt;/mml:mi&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;ς&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;/mml:mrow&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mspace /&gt; &lt;mml:mi&gt;ς&lt;/mml:mi&gt; &lt;mml:mo&gt;∈&lt;/mml:mo&gt; &lt;mml:mi&gt;J&lt;/mml:mi&gt; &lt;mml:mo&gt;=&lt;/mml:mo&gt; &lt;mml:mo&gt;[&lt;/mml:mo&gt; &lt;mml:mn&gt;0&lt;/mml:mn&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mi&gt;A&lt;/mml:mi&gt; &lt;mml:mo&gt;]&lt;/mml:mo&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;mml:mi&gt;A&lt;/mml:mi&gt; &lt;mml:mo&gt;&gt;&lt;/mml:mo&gt; &lt;mml:mn&gt;0&lt;/mml:mn&gt; &lt;mml:mo&gt;,&lt;/mml:mo&gt; &lt;/mml:math&gt; where $$begin{aligned} &amp;mathcal{L}(mathcal{D})=gamma _{w} ,{}^{c} mathcal{D}^{delta _{w}}+ gamma _{w-1} ,{}^{c} mathcal{D}^{delta _{w-1}}+cdots+gamma _{1} ,{}^{c} mathcal{D}^{delta _{1}}+gamma _{0} ,{}^{c} mathcal{D}^{delta _{0}}, &amp;gamma _{flat}in mathbb{R}quad (flat =0,1,2,3,ldots,w), qquad gamma _{w} neq 0, &amp;0leq delta _{0}&lt; delta _{1}&lt; delta _{2}&lt; cdots&lt; delta _{w-1}&lt; delta _{w}&lt; 1; end{aligned}$$ &lt;mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"&gt; &lt;mml:mtable&gt; &lt;mml:mtr&gt; &lt;mml:mtd /&gt; &lt;mml:mtd&gt; &lt;mml:mi&gt;L&lt;/mml:mi&gt; &lt;mml:mo&gt;(&lt;/mml:mo&gt; &lt;mml:mi&gt;D&lt;/mml:mi&gt; &lt;mml:mo&gt;)&lt;/mml:mo&gt; &lt;mml:mo&gt;=&lt;/mml:mo&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;γ&lt;/mml:mi&gt; &lt;mml:mi&gt;w&lt;/mml:mi&gt; &lt;/mml:msub&gt; &lt;mml:mspace /&gt; &lt;mml:msup&gt; &lt;mml:mrow /&gt; &lt;mml:mi&gt;c&lt;/mml:mi&gt; &lt;/mml:msup&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;D&lt;/mml:mi&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;δ&lt;/mml:mi&gt; &lt;mml:mi&gt;w&lt;/mml:mi&gt; &lt;/mml:msub&gt; &lt;/mml:msup&gt; &lt;mml:mo&gt;+&lt;/mml:mo&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;γ&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;w&lt;/mml:mi&gt; &lt;mml:mo&gt;−&lt;/mml:mo&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:msub&gt; &lt;mml:mspace /&gt; &lt;mml:msup&gt; &lt;mml:mrow /&gt; &lt;mml:mi&gt;c&lt;/mml:mi&gt; &lt;/mml:msup&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;D&lt;/mml:mi&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;δ&lt;/mml:mi&gt; &lt;mml:mrow&gt; &lt;mml:mi&gt;w&lt;/mml:mi&gt; &lt;mml:mo&gt;−&lt;/mml:mo&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:mrow&gt; &lt;/mml:msub&gt; &lt;/mml:msup&gt; &lt;mml:mo&gt;+&lt;/mml:mo&gt; &lt;mml:mo&gt;⋯&lt;/mml:mo&gt; &lt;mml:mo&gt;+&lt;/mml:mo&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;γ&lt;/mml:mi&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:msub&gt; &lt;mml:mspace /&gt; &lt;mml:msup&gt; &lt;mml:mrow /&gt; &lt;mml:mi&gt;c&lt;/mml:mi&gt; &lt;/mml:msup&gt; &lt;mml:msup&gt; &lt;mml:mi&gt;D&lt;/mml:mi&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;δ&lt;/mml:mi&gt; &lt;mml:mn&gt;1&lt;/mml:mn&gt; &lt;/mml:msub&gt; &lt;/mml:msup&gt; &lt;mml:mo&gt;+&lt;/mml:mo&gt; &lt;mml:msub&gt; &lt;mml:mi&gt;γ&lt;/mml:mi&gt; &lt;mml:mn&gt;0&lt;/mml:mn&gt; &lt;/mml:msub&gt; &lt;mml:mspace /&gt; &lt;mml:msup&gt; &lt;mml:mrow /&gt; &lt;mml:mi&gt;c&lt;/mm","PeriodicalId":55333,"journal":{"name":"Boundary Value Problems","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135741336","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}