Pub Date : 2024-03-13DOI: 10.1186/s13661-024-01844-4
Zhenfeng Zhang, Tianqing An, Weichun Bu, Shuai Li
In this paper, we study fractional $p_{1}(x,cdot )& p_{2}(x,cdot )$ -Laplacian Schrödinger-type equations for Robin boundary conditions. Under some suitable assumptions, we show that two solutions exist using the mountain pass lemma and Ekeland’s variational principle. Then, the existence of infinitely many solutions is derived by applying the fountain theorem and the Krasnoselskii genus theory, respectively. Different from previous results, the topic of this paper is the Robin boundary conditions in $mathbb{R}^{N}setminus overline{Omega}$ for fractional order $p_{1}(x,cdot )& p_{2}(x,cdot )$ -Laplacian Schrödinger-type equations, including concave-convex nonlinearities, which has not been studied before. In addition, two examples are given to illustrate our results.
{"title":"Existence and multiplicity of solutions for fractional (p_{1}(x,cdot )& p_{2}(x,cdot ))-Laplacian Schrödinger-type equations with Robin boundary conditions","authors":"Zhenfeng Zhang, Tianqing An, Weichun Bu, Shuai Li","doi":"10.1186/s13661-024-01844-4","DOIUrl":"https://doi.org/10.1186/s13661-024-01844-4","url":null,"abstract":"In this paper, we study fractional $p_{1}(x,cdot )& p_{2}(x,cdot )$ -Laplacian Schrödinger-type equations for Robin boundary conditions. Under some suitable assumptions, we show that two solutions exist using the mountain pass lemma and Ekeland’s variational principle. Then, the existence of infinitely many solutions is derived by applying the fountain theorem and the Krasnoselskii genus theory, respectively. Different from previous results, the topic of this paper is the Robin boundary conditions in $mathbb{R}^{N}setminus overline{Omega}$ for fractional order $p_{1}(x,cdot )& p_{2}(x,cdot )$ -Laplacian Schrödinger-type equations, including concave-convex nonlinearities, which has not been studied before. In addition, two examples are given to illustrate our results.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140126823","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}
Pub Date : 2024-03-08DOI: 10.1186/s13661-024-01843-5
Salah Boulaaras, Rashid Jan, Abdelbaki Choucha, Aderrahmane Zaraï, Mourad Benzahi
We examine a Kirchhoff-type equation with nonlinear viscoelastic properties, characterized by distributed delay, logarithmic nonlinearity, and Balakrishnan–Taylor damping terms (elastic membrane equation). Under appropriate hypotheses, we establish the occurrence of solution blow-up.
{"title":"Blow-up and lifespan of solutions for elastic membrane equation with distributed delay and logarithmic nonlinearity","authors":"Salah Boulaaras, Rashid Jan, Abdelbaki Choucha, Aderrahmane Zaraï, Mourad Benzahi","doi":"10.1186/s13661-024-01843-5","DOIUrl":"https://doi.org/10.1186/s13661-024-01843-5","url":null,"abstract":"We examine a Kirchhoff-type equation with nonlinear viscoelastic properties, characterized by distributed delay, logarithmic nonlinearity, and Balakrishnan–Taylor damping terms (elastic membrane equation). Under appropriate hypotheses, we establish the occurrence of solution blow-up.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140076497","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}
Pub Date : 2024-03-07DOI: 10.1186/s13661-024-01842-6
Yunze Shao, Junjie Du, Xiaofei Li, Yuru Tan, Jia Song
Over the years, the research of backward stochastic differential equations (BSDEs) has come a long way. As a extension of the BSDEs, the BSDEs with time delay have played a major role in the stochastic optimal control, financial risk, insurance management, pricing, and hedging. In this paper, we study a class of BSDEs with time-delay generators driven by Caputo fractional derivatives. In contrast to conventional BSDEs, in this class of equations, the generator is also affected by the past values of solutions. Under the Lipschitz condition and some new assumptions, we present a theorem on the existence and uniqueness of solutions.
{"title":"Caputo fractional backward stochastic differential equations driven by fractional Brownian motion with delayed generator","authors":"Yunze Shao, Junjie Du, Xiaofei Li, Yuru Tan, Jia Song","doi":"10.1186/s13661-024-01842-6","DOIUrl":"https://doi.org/10.1186/s13661-024-01842-6","url":null,"abstract":"Over the years, the research of backward stochastic differential equations (BSDEs) has come a long way. As a extension of the BSDEs, the BSDEs with time delay have played a major role in the stochastic optimal control, financial risk, insurance management, pricing, and hedging. In this paper, we study a class of BSDEs with time-delay generators driven by Caputo fractional derivatives. In contrast to conventional BSDEs, in this class of equations, the generator is also affected by the past values of solutions. Under the Lipschitz condition and some new assumptions, we present a theorem on the existence and uniqueness of solutions.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140054525","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}
Pub Date : 2024-03-01DOI: 10.1186/s13661-024-01841-7
A. Razani, Giovany M. Figueiredo
In this paper, a semipositone anisotropic p-Laplacian problem $$ -Delta _{overrightarrow{p}}u=lambda f(u), $$ on a bounded domain with the Dirchlet boundary condition is considered, where $A(u^{q}-1)leq f(u)leq B(u^{q}-1)$ for $u>0$ , $f(0)<0$ and $f(u)=0$ for $uleq -1$ . It is proved that there exists $lambda ^{*}>0$ such that if $lambda in (0,lambda ^{*})$ , then the problem has a positive weak solution $u_{lambda}in L^{infty}(overline{Omega})$ via combining Mountain-Pass arguments, comparison principles, and regularity principles.
{"title":"Positive solutions for a semipositone anisotropic p-Laplacian problem","authors":"A. Razani, Giovany M. Figueiredo","doi":"10.1186/s13661-024-01841-7","DOIUrl":"https://doi.org/10.1186/s13661-024-01841-7","url":null,"abstract":"In this paper, a semipositone anisotropic p-Laplacian problem $$ -Delta _{overrightarrow{p}}u=lambda f(u), $$ on a bounded domain with the Dirchlet boundary condition is considered, where $A(u^{q}-1)leq f(u)leq B(u^{q}-1)$ for $u>0$ , $f(0)<0$ and $f(u)=0$ for $uleq -1$ . It is proved that there exists $lambda ^{*}>0$ such that if $lambda in (0,lambda ^{*})$ , then the problem has a positive weak solution $u_{lambda}in L^{infty}(overline{Omega})$ via combining Mountain-Pass arguments, comparison principles, and regularity principles.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140007631","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}
Pub Date : 2024-02-29DOI: 10.1186/s13661-024-01839-1
Xinyu Cui, Shengbin Fu, Rui Sun, Fangfang Tian
This paper focuses on the long time behavior of the solutions to the Cauchy problem of the three-dimensional compressible magneto-micropolar fluids. More precisely, we aim to establish the optimal rates of temporal decay for the highest-order spatial derivatives of the global strong solutions by the method of decomposing frequency. Our result can be regarded as the further investigation of the one in (Wei, Guo and Li in J. Differ. Equ. 263:2457–2480, 2017), in which the authors only provided the optimal rates of temporal decay for the lower-order spatial derivatives of the perturbations of both the velocity and the micro-rotational velocity.
本文主要研究三维可压缩磁介质流体的考奇问题解的长时间行为。更确切地说，我们旨在通过分解频率的方法建立全局强解的最高阶空间导数的最佳时间衰减率。我们的结果可以看作是对《差分方程》（Wei, Guo and Li in J. Differ. Equ. 263:2457-2480, 2017）中结果的进一步研究，在该文中，作者只提供了速度和微旋转速度扰动的低阶空间导数的最优时间衰减率。
{"title":"Optimal decay-in-time rates of solutions to the Cauchy problem of 3D compressible magneto-micropolar fluids","authors":"Xinyu Cui, Shengbin Fu, Rui Sun, Fangfang Tian","doi":"10.1186/s13661-024-01839-1","DOIUrl":"https://doi.org/10.1186/s13661-024-01839-1","url":null,"abstract":"This paper focuses on the long time behavior of the solutions to the Cauchy problem of the three-dimensional compressible magneto-micropolar fluids. More precisely, we aim to establish the optimal rates of temporal decay for the highest-order spatial derivatives of the global strong solutions by the method of decomposing frequency. Our result can be regarded as the further investigation of the one in (Wei, Guo and Li in J. Differ. Equ. 263:2457–2480, 2017), in which the authors only provided the optimal rates of temporal decay for the lower-order spatial derivatives of the perturbations of both the velocity and the micro-rotational velocity.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140007656","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}
This paper focuses on introducing and examining the class of generalized strongly n-polynomial convex functions. Relationships between these functions and other types of convex functions are explored. The Hermite–Hadamard inequality is established for generalized strongly n-polynomial convex functions. Additionally, new integral inequalities of Hermite–Hadamard type are derived for this class of functions using the Hölder–İşcan integral inequality. The results obtained in this paper are compared with those known in the literature, demonstrating the superiority of the new results. Finally, some applications for special means are provided.
本文主要介绍和研究广义强 n 多项式凸函数。本文探讨了这些函数与其他类型凸函数之间的关系。建立了广义强 n 多项式凸函数的 Hermite-Hadamard 不等式。此外，还利用荷尔德-İşcan 积分不等式为这一类函数导出了新的 Hermite-Hadamard 型积分不等式。本文获得的结果与文献中已知的结果进行了比较，证明了新结果的优越性。最后，还提供了一些特殊手段的应用。
{"title":"Generalized strongly n-polynomial convex functions and related inequalities","authors":"Serap Özcan, Mahir Kadakal, İmdat İşcan, Huriye Kadakal","doi":"10.1186/s13661-024-01838-2","DOIUrl":"https://doi.org/10.1186/s13661-024-01838-2","url":null,"abstract":"This paper focuses on introducing and examining the class of generalized strongly n-polynomial convex functions. Relationships between these functions and other types of convex functions are explored. The Hermite–Hadamard inequality is established for generalized strongly n-polynomial convex functions. Additionally, new integral inequalities of Hermite–Hadamard type are derived for this class of functions using the Hölder–İşcan integral inequality. The results obtained in this paper are compared with those known in the literature, demonstrating the superiority of the new results. Finally, some applications for special means are provided.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139969721","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}
Pub Date : 2024-02-23DOI: 10.1186/s13661-024-01837-3
Zelong Yu, Zhanbing Bai, Suiming Shang
This paper provides a class of upper and lower solution definitions for second-order coupled systems by transforming the fourth-order differential equation into a second-order differential system. Then, by constructing a homotopy parameter and utilizing the maximum principle, we propose an upper and lower solutions method for studying a class of second-order coupled systems with Dirichlet boundary conditions and obtain an existence result.
{"title":"Upper and lower solutions method for a class of second-order coupled systems","authors":"Zelong Yu, Zhanbing Bai, Suiming Shang","doi":"10.1186/s13661-024-01837-3","DOIUrl":"https://doi.org/10.1186/s13661-024-01837-3","url":null,"abstract":"This paper provides a class of upper and lower solution definitions for second-order coupled systems by transforming the fourth-order differential equation into a second-order differential system. Then, by constructing a homotopy parameter and utilizing the maximum principle, we propose an upper and lower solutions method for studying a class of second-order coupled systems with Dirichlet boundary conditions and obtain an existence result.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139950813","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}
Pub Date : 2024-02-23DOI: 10.1186/s13661-024-01836-4
Yongkuan Cheng, Yaotian Shen
In this paper, we consider a model problem arising from a classical planar Heisenberg ferromagnetic spin chain: $$ -Delta u+V(x)u-frac{u}{sqrt{1-u^{2}}}Delta sqrt{1-u^{2}}=c vert u vert ^{p-2}u,quad xin mathbb{R}^{N}, $$ where $2< p<2^{*}$ , $c>0$ and $Ngeq 3$ . By the cutoff technique, the change of variables and the $L^{infty}$ estimate, we prove that there exists $c_{0}>0$ , such that for any $c>c_{0}$ this problem admits a positive solution. Here, in contrast to the Morse iteration method, we construct the $L^{infty}$ estimate of the solution. In particular, we give the specific expression of $c_{0}$ .
在本文中，我们考虑了一个由经典平面海森堡铁磁自旋链产生的模型问题：$$ -Delta u+V(x)u-frac{u}{sqrt{1-u^{2}}}Delta sqrt{1-u^{2}}=c vert u vert ^{p-2}u,quad xin mathbb{R}^{N}, $$其中$2< p0$，$Ngeq 3$。通过截断技术、变量变化和 $L^{infty}$ 估计，我们证明存在 $c_{0}>0$ ，这样对于任意 $c>c_{0}$ 问题都有一个正解。在这里，与莫尔斯迭代法不同，我们构建了解的 $L^{infty}$ 估计值。我们特别给出了 $c_{0}$ 的具体表达式。
{"title":"Quasilinear Schrödinger equations with superlinear terms describing the Heisenberg ferromagnetic spin chain","authors":"Yongkuan Cheng, Yaotian Shen","doi":"10.1186/s13661-024-01836-4","DOIUrl":"https://doi.org/10.1186/s13661-024-01836-4","url":null,"abstract":"In this paper, we consider a model problem arising from a classical planar Heisenberg ferromagnetic spin chain: $$ -Delta u+V(x)u-frac{u}{sqrt{1-u^{2}}}Delta sqrt{1-u^{2}}=c vert u vert ^{p-2}u,quad xin mathbb{R}^{N}, $$ where $2< p<2^{*}$ , $c>0$ and $Ngeq 3$ . By the cutoff technique, the change of variables and the $L^{infty}$ estimate, we prove that there exists $c_{0}>0$ , such that for any $c>c_{0}$ this problem admits a positive solution. Here, in contrast to the Morse iteration method, we construct the $L^{infty}$ estimate of the solution. In particular, we give the specific expression of $c_{0}$ .","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139950811","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}
Pub Date : 2024-02-23DOI: 10.1186/s13661-024-01835-5
Po-Chun Huang, Bo-Yu Pan
This article investigates the local well-posedness of Turing-type reaction–diffusion equations with Robin boundary conditions in the Sobolev space. Utilizing the Hadamard norm, we derive estimates for Fokas unified transform solutions for linear initial-boundary value problems subject to external forces. Subsequently, we demonstrate that the iteration map, defined by the unified transform solutions and incorporating nonlinearity instead of external forces, acts as a contraction map within an appropriate solution space. Our conclusive result is established through the application of the contraction mapping theorem.
{"title":"The Robin problems for the coupled system of reaction–diffusion equations","authors":"Po-Chun Huang, Bo-Yu Pan","doi":"10.1186/s13661-024-01835-5","DOIUrl":"https://doi.org/10.1186/s13661-024-01835-5","url":null,"abstract":"This article investigates the local well-posedness of Turing-type reaction–diffusion equations with Robin boundary conditions in the Sobolev space. Utilizing the Hadamard norm, we derive estimates for Fokas unified transform solutions for linear initial-boundary value problems subject to external forces. Subsequently, we demonstrate that the iteration map, defined by the unified transform solutions and incorporating nonlinearity instead of external forces, acts as a contraction map within an appropriate solution space. Our conclusive result is established through the application of the contraction mapping theorem.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139950804","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}
Pub Date : 2024-02-22DOI: 10.1186/s13661-024-01834-6
Khansa Hina Khalid, Akbar Zada, Ioan-Lucian Popa, Mohammad Esmael Samei
In this work, we analyze a q-fractional jerk problem having anti-periodic boundary conditions. The focus is on investigating whether a unique solution exists and remains stable under specific conditions. To prove the uniqueness of the solution, we employ a Banach fixed point theorem and a mathematical tool for establishing the presence of distinct fixed points. To demonstrate the availability of a solution, we utilize Leray–Schauder’s alternative, a method commonly employed in mathematical analysis. Furthermore, we examine and introduce different kinds of stability concepts for the given problem. In conclusion, we present several examples to illustrate and validate the outcomes of our study.
{"title":"Existence and stability of a q-Caputo fractional jerk differential equation having anti-periodic boundary conditions","authors":"Khansa Hina Khalid, Akbar Zada, Ioan-Lucian Popa, Mohammad Esmael Samei","doi":"10.1186/s13661-024-01834-6","DOIUrl":"https://doi.org/10.1186/s13661-024-01834-6","url":null,"abstract":"In this work, we analyze a q-fractional jerk problem having anti-periodic boundary conditions. The focus is on investigating whether a unique solution exists and remains stable under specific conditions. To prove the uniqueness of the solution, we employ a Banach fixed point theorem and a mathematical tool for establishing the presence of distinct fixed points. To demonstrate the availability of a solution, we utilize Leray–Schauder’s alternative, a method commonly employed in mathematical analysis. Furthermore, we examine and introduce different kinds of stability concepts for the given problem. In conclusion, we present several examples to illustrate and validate the outcomes of our study.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2024-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139927645","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}
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