Pub Date : 2026-01-12DOI: 10.1016/j.aml.2026.109874
Chenyu Zhang , Junying Cao , Shuying Zhai
We present a fast and effective method for image inpainting based on a modified viscous Cahn–Hilliard equation. By employing the second-order operator time-splitting method, the original problem is discretized into two subproblems based on the distinct properties of each part of the model, where the linear subproblem is handled by a Crank–Nicolson (CN) finite difference scheme, and the nonlinear subproblem is treated explicitly with the second-order strong stability preserving Runge–Kutta (SSP-RK) method. Theoretical analysis indicates that the stability of the algorithm depends on the viscosity coefficient , and it is progressively strengthened as increases. Numerical experiments are presented to verify the robustness and accuracy of the proposed method.
{"title":"A fast explicit hybrid numerical method for image inpainting using the viscous Cahn–Hilliard model","authors":"Chenyu Zhang , Junying Cao , Shuying Zhai","doi":"10.1016/j.aml.2026.109874","DOIUrl":"10.1016/j.aml.2026.109874","url":null,"abstract":"<div><div>We present a fast and effective method for image inpainting based on a modified viscous Cahn–Hilliard equation. By employing the second-order operator time-splitting method, the original problem is discretized into two subproblems based on the distinct properties of each part of the model, where the linear subproblem is handled by a Crank–Nicolson (CN) finite difference scheme, and the nonlinear subproblem is treated explicitly with the second-order strong stability preserving Runge–Kutta (SSP-RK) method. Theoretical analysis indicates that the stability of the algorithm depends on the viscosity coefficient <span><math><mi>α</mi></math></span>, and it is progressively strengthened as <span><math><mi>α</mi></math></span> increases. Numerical experiments are presented to verify the robustness and accuracy of the proposed method.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"176 ","pages":"Article 109874"},"PeriodicalIF":2.8,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145956602","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-12DOI: 10.1016/j.aml.2026.109873
Xiaoping Zhai
The construction of global solutions for the compressible magnetohydrodynamic equations without magnetic diffusion in remains a challenging open problem. In an effort to address this issue, we study the Cauchy problem for the compressible magnetohydrodynamic equations with fractional dissipation on the magnetic field. We show that for any , the system admits a unique global solution in Besov spaces under the assumption of small initial data. Moreover, we develop a Lyapunov-type energy argument that yields time-decay estimates of the solutions without imposing additional smallness assumptions on the low-frequency part of the initial data.
{"title":"Global solutions to 3D MHD equations with fractional dissipation","authors":"Xiaoping Zhai","doi":"10.1016/j.aml.2026.109873","DOIUrl":"10.1016/j.aml.2026.109873","url":null,"abstract":"<div><div>The construction of global solutions for the compressible magnetohydrodynamic equations without magnetic diffusion in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mn>3</mn></mrow></msup></math></span> remains a challenging open problem. In an effort to address this issue, we study the Cauchy problem for the compressible magnetohydrodynamic equations with fractional dissipation <span><math><msup><mrow><mrow><mo>(</mo><mo>−</mo><mi>Δ</mi><mo>)</mo></mrow></mrow><mrow><mi>α</mi></mrow></msup></math></span> on the magnetic field. We show that for any <span><math><mrow><mn>0</mn><mo>≤</mo><mi>α</mi><mo><</mo><mn>1</mn></mrow></math></span>, the system admits a unique global solution in Besov spaces under the assumption of small initial data. Moreover, we develop a Lyapunov-type energy argument that yields time-decay estimates of the solutions without imposing additional smallness assumptions on the low-frequency part of the initial data.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"176 ","pages":"Article 109873"},"PeriodicalIF":2.8,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145956603","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-12DOI: 10.1016/j.aml.2026.109872
Cong Lin, Xiaoling Zou, Jingliang Lv
With the continuous development of the theory of stochastic biological population models, the study of the persistence of predator–prey models has become a very meaningful topic. This paper proves the sufficient conditions for the existence of a unique ergodic invariant measure for a three-dimensional stochastic predator–prey model with Markov switching, and subsequently proposes an approximate numerical method to verify the conditions of the two-dimensional boundary measure integration, thereby providing strong numerical support for the survival analysis of the model.
{"title":"Invariant measure and numerical simulations for a stochastic predator–prey model — An example of verifying two-dimensional boundary measure integration","authors":"Cong Lin, Xiaoling Zou, Jingliang Lv","doi":"10.1016/j.aml.2026.109872","DOIUrl":"10.1016/j.aml.2026.109872","url":null,"abstract":"<div><div>With the continuous development of the theory of stochastic biological population models, the study of the persistence of predator–prey models has become a very meaningful topic. This paper proves the sufficient conditions for the existence of a unique ergodic invariant measure for a three-dimensional stochastic predator–prey model with Markov switching, and subsequently proposes an approximate numerical method to verify the conditions of the two-dimensional boundary measure integration, thereby providing strong numerical support for the survival analysis of the model.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"176 ","pages":"Article 109872"},"PeriodicalIF":2.8,"publicationDate":"2026-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145956604","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-09DOI: 10.1016/j.aml.2026.109871
Ziqing Yuan
Nonlinear boundary value problems involving the fractional Laplacian operator are studied to establish precise conditions for the existence of solutions. By combining the method of sub- and super-solutions with fractional comparison principles, we prove the existence of nonnegative solutions and derive sharp estimates for the critical parameters. These results provide a theoretical foundation for analyzing threshold phenomena in applications such as disease transmission dynamics, where nonlocal interactions play a crucial role.
{"title":"Existence and parameter estimation for nonlinear boundary value problems involving fractional Laplacian","authors":"Ziqing Yuan","doi":"10.1016/j.aml.2026.109871","DOIUrl":"10.1016/j.aml.2026.109871","url":null,"abstract":"<div><div>Nonlinear boundary value problems involving the fractional Laplacian operator are studied to establish precise conditions for the existence of solutions. By combining the method of sub- and super-solutions with fractional comparison principles, we prove the existence of nonnegative solutions and derive sharp estimates for the critical parameters. These results provide a theoretical foundation for analyzing threshold phenomena in applications such as disease transmission dynamics, where nonlocal interactions play a crucial role.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"176 ","pages":"Article 109871"},"PeriodicalIF":2.8,"publicationDate":"2026-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145956605","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-07DOI: 10.1016/j.aml.2026.109870
Yiqiu Du, Yu Su
The Schrödinger–Bopp–Podolsky system arises in the second order gauge theory for the electromagnetic theory. When the exponent of power nonlinearity involved in the system is (this is a special mass subcritical case), it presents a new phenomenon due to the “conflict” between the Laplacian and Bopp–Podolsky term. In this case, Ramos–Siciliano [Zeitschrift für angewandte Mathematik und Physik, 74 (2023)] proved that there exists a constant such that for any , the system has a ground state solution. However, they just showed the existence of the constant . It is important for us to understand the constant . Hence, we estimate the constant by the best constant of the Gagliardo–Nirenberg–Coulomb inequality. Moreover, we show the existence of ground state solution.
{"title":"Ground state solution of Schrödinger–Bopp–Podolsky system in the mass subcritical case","authors":"Yiqiu Du, Yu Su","doi":"10.1016/j.aml.2026.109870","DOIUrl":"10.1016/j.aml.2026.109870","url":null,"abstract":"<div><div>The Schrödinger–Bopp–Podolsky system arises in the second order gauge theory for the electromagnetic theory. When the exponent of power nonlinearity involved in the system is <span><math><mrow><mi>p</mi><mo>∈</mo><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mfrac><mrow><mn>10</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>)</mo></mrow></mrow></math></span> (this is a special mass subcritical case), it presents a new phenomenon due to the “conflict” between the Laplacian and Bopp–Podolsky term. In this case, Ramos–Siciliano [Zeitschrift für angewandte Mathematik und Physik, 74 (2023)] proved that there exists a constant <span><math><mrow><msup><mrow><mi>c</mi></mrow><mrow><mo>∗</mo></mrow></msup><mo>></mo><mn>0</mn></mrow></math></span> such that for any <span><math><mrow><mi>c</mi><mo>></mo><msup><mrow><mi>c</mi></mrow><mrow><mo>∗</mo></mrow></msup></mrow></math></span>, the system has a ground state solution. However, they just showed the existence of the constant <span><math><msup><mrow><mi>c</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span>. It is important for us to understand the constant <span><math><msup><mrow><mi>c</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span>. Hence, we estimate the constant <span><math><msup><mrow><mi>c</mi></mrow><mrow><mo>∗</mo></mrow></msup></math></span> by the best constant of the Gagliardo–Nirenberg–Coulomb inequality. Moreover, we show the existence of ground state solution.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"176 ","pages":"Article 109870"},"PeriodicalIF":2.8,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145956607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2026-01-02DOI: 10.1016/j.aml.2025.109868
Ruisheng Qi
In this paper, we consider strong convergence of a novel fully discrete finite element approximation of stochastic PDEs with non-globally Lipschitz coefficients and multiplicative noise in space dimension . The discretization in space is the standard finite element method and the discretization in time is a tamed drift semi-implicit scheme. This scheme makes the nonlinearity be solved explicitly while being unconditionally stable. Under regularity assumptions, we establish the optimal strong convergence rates in both space and time for the considered scheme.
{"title":"Strong convergence of a fully discrete finite element approximation of non-Lipschitz SPDEs with multiplicative noise","authors":"Ruisheng Qi","doi":"10.1016/j.aml.2025.109868","DOIUrl":"10.1016/j.aml.2025.109868","url":null,"abstract":"<div><div>In this paper, we consider strong convergence of a novel fully discrete finite element approximation of stochastic PDEs with non-globally Lipschitz coefficients and multiplicative noise in space dimension <span><math><mrow><mi>d</mi><mo>≤</mo><mn>3</mn></mrow></math></span>. The discretization in space is the standard finite element method and the discretization in time is a tamed drift semi-implicit scheme. This scheme makes the nonlinearity be solved explicitly while being unconditionally stable. Under regularity assumptions, we establish the optimal strong convergence rates in both space and time for the considered scheme.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"176 ","pages":"Article 109868"},"PeriodicalIF":2.8,"publicationDate":"2026-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145894456","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the long-term multivalued random dynamical behavior for a non-autonomous Benjamin–Bona–Mahony equation on a 3D unbounded channel with a non-Lipschitz diffusion coefficient and a cubic polynomial growth vector field. Our main results are the existence of multivalued non-autonomous random dynamical systems and strongly compact random attractors in . Such results for quadratic polynomial vector fields have been proved by Chen, Wang, Wang and Zhang (Math. Ann, 386 (2023) 343–373) and Chen, Wang and Zhang (SIAM J. Math. Anal. 56 (2024) 254–274) by the spectral decomposition method. In this paper, we prove that such results are also valid for a weak integrability condition on the time-dependent external forcing and the polynomial vector fields of cubic growth which involving a critical Sobolev embedding. The famous energy balance equation method developed by Ball (Discrete Contin. Dyn. Syst., 10 (2004) 31–52) is used to deal with the non-compact embedding problems and the non-applicability of the spectral decomposition method for cubic polynomial vector fields.
{"title":"Multivalued random dynamics of colored noise driven BBM equations on 3D unbounded channels with cubic polynomial vector fields","authors":"Ruiyi Xu, Linsong Chen, Liguang Zhou, Xuping Zhang","doi":"10.1016/j.aml.2025.109869","DOIUrl":"10.1016/j.aml.2025.109869","url":null,"abstract":"<div><div>We study the long-term multivalued random dynamical behavior for a non-autonomous Benjamin–Bona–Mahony equation on a 3D unbounded channel with a <em>non-Lipschitz diffusion coefficient</em> and a <em>cubic polynomial growth vector field</em>. Our main results are the existence of multivalued non-autonomous random dynamical systems and strongly compact random attractors in <span><math><msubsup><mrow><mi>H</mi></mrow><mrow><mn>0</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>. Such results for <em>quadratic polynomial vector fields</em> have been proved by Chen, Wang, Wang and Zhang (Math. Ann, 386 (2023) 343–373) and Chen, Wang and Zhang (SIAM J. Math. Anal. 56 (2024) 254–274) by the spectral decomposition method. In this paper, we prove that such results are also valid for a weak integrability condition on the time-dependent external forcing and the polynomial vector fields of <em>cubic</em> growth which involving a <em>critical</em> Sobolev embedding. The famous energy balance equation method developed by Ball (Discrete Contin. Dyn. Syst., 10 (2004) 31–52) is used to deal with the non-compact embedding problems and the non-applicability of the spectral decomposition method for cubic polynomial vector fields.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"176 ","pages":"Article 109869"},"PeriodicalIF":2.8,"publicationDate":"2026-01-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145894455","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-30DOI: 10.1016/j.aml.2025.109861
Seung-Yeal Ha , Jaemoon Lee , Qinghua Xiao , Fanqin Zeng
We study the weak flocking of the Cucker–Smale–Fokker–Planck (in short, CS–FP) equation with a degenerate diffusion coefficient. When the communication weight function has a positive lower bound, weak flocking occurs asymptotically. In contrast, when the communication weight function tend to zero, it is not known whether weak flocking occurs or not. In this paper, we revisit this delicate situation in which the communication weight function tends to zero at infinity and the noise amplitude also decays to zero sufficiently fast. To bypass the difficulty, we use the method of effective domain by identifying a time-varying region in which the total mass outside of it decays to zero sufficiently fast. Moreover, we show that if the communication weight function and the noise amplitude both have a suitable polynomial decay, then weak flocking occurs.
我们研究了具有简并扩散系数的cucker - small - fokker - planck（简称CS-FP）方程的弱群。当通信权函数的下界为正时，渐近出现弱簇。相反，当通信权函数趋于零时，不知道是否发生弱群集。在本文中，我们重新审视了这种微妙的情况，即通信权函数在无穷远处趋于零，噪声幅度也足够快地衰减到零。为了克服这个困难，我们使用有效域的方法，通过确定一个时变区域，在该区域外的总质量衰减到零的速度足够快。此外，我们还证明了如果通信权函数和噪声幅值都有合适的多项式衰减，则会发生弱群集。
{"title":"A priori estimates on the weak flocking of the Cucker–Smale–Fokker–Planck equation","authors":"Seung-Yeal Ha , Jaemoon Lee , Qinghua Xiao , Fanqin Zeng","doi":"10.1016/j.aml.2025.109861","DOIUrl":"10.1016/j.aml.2025.109861","url":null,"abstract":"<div><div>We study the weak flocking of the Cucker–Smale–Fokker–Planck (in short, CS–FP) equation with a degenerate diffusion coefficient. When the communication weight function has a positive lower bound, weak flocking occurs asymptotically. In contrast, when the communication weight function tend to zero, it is not known whether weak flocking occurs or not. In this paper, we revisit this delicate situation in which the communication weight function tends to zero at infinity and the noise amplitude also decays to zero sufficiently fast. To bypass the difficulty, we use the method of effective domain by identifying a time-varying region in which the total mass outside of it decays to zero sufficiently fast. Moreover, we show that if the communication weight function and the noise amplitude both have a suitable polynomial decay, then weak flocking occurs.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"176 ","pages":"Article 109861"},"PeriodicalIF":2.8,"publicationDate":"2025-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145894474","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-30DOI: 10.1016/j.aml.2025.109860
Ahmed Alsaedi , Nikolaos S. Papageorgiou , Vicenţiu D. Rădulescu
We consider a nonlinear Dirichlet problem with gradient dependence. The features of this paper are twofold: (i) the problem is driven by a general nonlinear nonhomogeneous differential operator with Uhlenbeck–Lieberman structure; (ii) the reaction blows-up at the origin and it is gradient dependent. Using a topological approach based on fixed point theory, we show that for all small values of there are “eigenvalues” of the problem with smooth corresponding eigenfunctions.
{"title":"Small perturbations of convective singular eigenvalue problems","authors":"Ahmed Alsaedi , Nikolaos S. Papageorgiou , Vicenţiu D. Rădulescu","doi":"10.1016/j.aml.2025.109860","DOIUrl":"10.1016/j.aml.2025.109860","url":null,"abstract":"<div><div>We consider a nonlinear Dirichlet problem with gradient dependence. The features of this paper are twofold: (i) the problem is driven by a general nonlinear nonhomogeneous differential operator with Uhlenbeck–Lieberman structure; (ii) the reaction blows-up at the origin and it is gradient dependent. Using a topological approach based on fixed point theory, we show that for all small values of <span><math><mrow><mi>λ</mi><mo>></mo><mn>0</mn></mrow></math></span> there are “eigenvalues” of the problem with smooth corresponding eigenfunctions.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"176 ","pages":"Article 109860"},"PeriodicalIF":2.8,"publicationDate":"2025-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145894475","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-12-27DOI: 10.1016/j.aml.2025.109859
Andreea Gruie, Călin Şerban
We prove that, for any continuous , the non-homogeneous discrete Dirichlet problem where is a potential homeomorphism, is solvable iff . Our approach relies on the Legendre–Fenchel transform and Brouwer’s fixed point theorem.
{"title":"Non-homogeneous discrete Dirichlet problem with singular ϕ-Laplacian","authors":"Andreea Gruie, Călin Şerban","doi":"10.1016/j.aml.2025.109859","DOIUrl":"10.1016/j.aml.2025.109859","url":null,"abstract":"<div><div>We prove that, for any continuous <span><math><mrow><mi>f</mi><mo>:</mo><mi>Z</mi><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>]</mo></mrow><mo>×</mo><msup><mrow><mrow><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup><mo>)</mo></mrow></mrow><mrow><mi>T</mi></mrow></msup><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></mrow></math></span>, the non-homogeneous discrete Dirichlet problem <span><math><mrow><mi>Δ</mi><mrow><mo>[</mo><mi>ϕ</mi><mrow><mo>(</mo><mi>Δ</mi><mi>u</mi><mrow><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><mo>]</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>u</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mo>…</mo><mo>,</mo><mi>u</mi><mrow><mo>(</mo><mi>T</mi><mo>)</mo></mrow><mo>)</mo></mrow><mspace></mspace><mrow><mo>(</mo><mi>n</mi><mo>∈</mo><mi>Z</mi><mrow><mo>[</mo><mn>1</mn><mo>,</mo><mi>T</mi><mo>]</mo></mrow><mo>)</mo></mrow><mo>;</mo><mspace></mspace><mi>u</mi><mrow><mo>(</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mi>A</mi><mo>,</mo><mi>u</mi><mrow><mo>(</mo><mi>T</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>=</mo><mi>B</mi><mo>,</mo></mrow></math></span> where <span><math><mrow><mi>ϕ</mi><mo>:</mo><msub><mrow><mi>B</mi></mrow><mrow><mi>a</mi></mrow></msub><mo>→</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>N</mi></mrow></msup></mrow></math></span> is a potential homeomorphism, is solvable iff <span><math><mrow><mrow><mo>|</mo><mi>A</mi><mo>−</mo><mi>B</mi><mo>|</mo></mrow><mo><</mo><mrow><mo>(</mo><mi>T</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mi>a</mi></mrow></math></span>. Our approach relies on the Legendre–Fenchel transform and Brouwer’s fixed point theorem.</div></div>","PeriodicalId":55497,"journal":{"name":"Applied Mathematics Letters","volume":"176 ","pages":"Article 109859"},"PeriodicalIF":2.8,"publicationDate":"2025-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145845079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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