Pub Date : 2024-01-19DOI: 10.1186/s13661-024-01819-5
Lixin Sheng, Weimin Hu, You-Hui Su
In this paper, we investigate the existence of mild solutions as well as optimal controls for non-autonomous impulsive evolution equations with nonlocal conditions. Using the Schauder’s fixed-point theorem as well as the theory of evolution family, we prove the existence of mild solutions for the concerned problem. Furthermore, without the Lipschitz continuity of the nonlinear term, the optimal control result is derived by setting up minimizing sequences twice. An example is given of the application of the results.
{"title":"Existence and optimal controls of non-autonomous for impulsive evolution equation without Lipschitz assumption","authors":"Lixin Sheng, Weimin Hu, You-Hui Su","doi":"10.1186/s13661-024-01819-5","DOIUrl":"https://doi.org/10.1186/s13661-024-01819-5","url":null,"abstract":"In this paper, we investigate the existence of mild solutions as well as optimal controls for non-autonomous impulsive evolution equations with nonlocal conditions. Using the Schauder’s fixed-point theorem as well as the theory of evolution family, we prove the existence of mild solutions for the concerned problem. Furthermore, without the Lipschitz continuity of the nonlinear term, the optimal control result is derived by setting up minimizing sequences twice. An example is given of the application of the results.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"13 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139510218","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 : 2024-01-19DOI: 10.1186/s13661-024-01825-7
Mati ur Rahman, Mei Sun, Salah Boulaaras, Dumitru Baleanu
In this manuscript, our primary objective is to delve into the intricacies of an extended nonlinear Schrödinger equation. To achieve this, we commence by deriving a dynamical system tightly linked to the equation through the Galilean transformation. We then employ principles from planar dynamical systems theory to explore the bifurcation phenomena exhibited within this derived system. To investigate the potential presence of chaotic behaviors, we introduce a perturbed term into the dynamical system and systematically analyze the extended nonlinear Schrödinger equation. This investigation is further enriched by the presentation of comprehensive two- and 3D phase portraits. Moreover, we conduct a meticulous sensitivity analysis of the dynamical system using the Runge–Kutta method. Through this analytical process, we confirm that minor fluctuations in initial conditions have only minimal effects on solution stability. Additionally, we utilize the complete discrimination system of the polynomial method to systematically construct single traveling wave solutions for the governing model.
{"title":"Bifurcations, chaotic behavior, sensitivity analysis, and various soliton solutions for the extended nonlinear Schrödinger equation","authors":"Mati ur Rahman, Mei Sun, Salah Boulaaras, Dumitru Baleanu","doi":"10.1186/s13661-024-01825-7","DOIUrl":"https://doi.org/10.1186/s13661-024-01825-7","url":null,"abstract":"In this manuscript, our primary objective is to delve into the intricacies of an extended nonlinear Schrödinger equation. To achieve this, we commence by deriving a dynamical system tightly linked to the equation through the Galilean transformation. We then employ principles from planar dynamical systems theory to explore the bifurcation phenomena exhibited within this derived system. To investigate the potential presence of chaotic behaviors, we introduce a perturbed term into the dynamical system and systematically analyze the extended nonlinear Schrödinger equation. This investigation is further enriched by the presentation of comprehensive two- and 3D phase portraits. Moreover, we conduct a meticulous sensitivity analysis of the dynamical system using the Runge–Kutta method. Through this analytical process, we confirm that minor fluctuations in initial conditions have only minimal effects on solution stability. Additionally, we utilize the complete discrimination system of the polynomial method to systematically construct single traveling wave solutions for the governing model.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"141 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139495476","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 : 2024-01-19DOI: 10.1186/s13661-024-01823-9
Marin Marin, Sorin Vlase, Denisa Neagu
Our study is dedicated to a mixture composed of a dipolar elastic medium and a viscous Moore–Gibson–Thompson (MGT) material. The mixed problem with initial and boundary data, considered in this context, is approached from the perspective of the existence of a solution to this problem as well as the uniqueness of the solution. Considering that the mixed problem is very complex, both from the point of view of the basic equations and that of the initial conditions and the boundary data, the classical methods become difficult. That is why we preferred to transform it into a problem of Cauchy type on a conveniently constructed Hilbert space. In this way, we immediately proved both the existence and uniqueness of the solution, with techniques from the theory of semigroups of linear operators.
{"title":"On a composite obtained by a mixture of a dipolar solid with a Moore–Gibson–Thompson media","authors":"Marin Marin, Sorin Vlase, Denisa Neagu","doi":"10.1186/s13661-024-01823-9","DOIUrl":"https://doi.org/10.1186/s13661-024-01823-9","url":null,"abstract":"Our study is dedicated to a mixture composed of a dipolar elastic medium and a viscous Moore–Gibson–Thompson (MGT) material. The mixed problem with initial and boundary data, considered in this context, is approached from the perspective of the existence of a solution to this problem as well as the uniqueness of the solution. Considering that the mixed problem is very complex, both from the point of view of the basic equations and that of the initial conditions and the boundary data, the classical methods become difficult. That is why we preferred to transform it into a problem of Cauchy type on a conveniently constructed Hilbert space. In this way, we immediately proved both the existence and uniqueness of the solution, with techniques from the theory of semigroups of linear operators.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"89 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139510420","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 : 2024-01-18DOI: 10.1186/s13661-024-01821-x
Rafik Guefaifia, Tahar Bouali, Salah Boulaaras
In this paper, using variational methods introduced in the previous study on fractional elliptic systems, we prove the existence of at least three weak solutions for an elliptic nonlinear system with a p-Laplacian ψ-Hilfer operator.
{"title":"Three solutions for fractional elliptic systems involving ψ-Hilfer operator","authors":"Rafik Guefaifia, Tahar Bouali, Salah Boulaaras","doi":"10.1186/s13661-024-01821-x","DOIUrl":"https://doi.org/10.1186/s13661-024-01821-x","url":null,"abstract":"In this paper, using variational methods introduced in the previous study on fractional elliptic systems, we prove the existence of at least three weak solutions for an elliptic nonlinear system with a p-Laplacian ψ-Hilfer operator.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"3 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139495408","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 : 2024-01-18DOI: 10.1186/s13661-024-01820-y
Gunaseelan Mani, Maria A. R. M. Antony, Zoran D. Mitrović, Ahmad Aloqaily, Nabil Mlaiki
In this paper, we propose the notion of extended neutrosophic rectangular metric space and prove some fixed point results under contraction mapping. Finally, as an application of the obtained results, we prove the existence and uniqueness of the Caputo fractional differential equation.
{"title":"A fixed point result on an extended neutrosophic rectangular metric space with application","authors":"Gunaseelan Mani, Maria A. R. M. Antony, Zoran D. Mitrović, Ahmad Aloqaily, Nabil Mlaiki","doi":"10.1186/s13661-024-01820-y","DOIUrl":"https://doi.org/10.1186/s13661-024-01820-y","url":null,"abstract":"In this paper, we propose the notion of extended neutrosophic rectangular metric space and prove some fixed point results under contraction mapping. Finally, as an application of the obtained results, we prove the existence and uniqueness of the Caputo fractional differential equation.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"25 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139495435","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 : 2024-01-18DOI: 10.1186/s13661-023-01818-y
Fatih Hezenci, Hüseyin Budak, Hasan Kara, Umut Baş
In this current research, we focus on the domain of tempered fractional integrals, establishing a novel identity that serves as the cornerstone of our study. This identity paves the way for the Milne-type inequalities, which are explored through the framework of differentiable convex mappings inclusive of tempered fractional integrals. The significance of these mappings in the realm of fractional calculus is underscored by their ability to extend classical concepts into more complex, fractional dimensions. In addition, by using the Hölder inequality and power-mean inequality, we acquire some new Milne-type inequalities. Moreover, the practicality and theoretical relevance of our findings are further demonstrated through the application of specific cases derived from the theorems.
{"title":"Novel results of Milne-type inequalities involving tempered fractional integrals","authors":"Fatih Hezenci, Hüseyin Budak, Hasan Kara, Umut Baş","doi":"10.1186/s13661-023-01818-y","DOIUrl":"https://doi.org/10.1186/s13661-023-01818-y","url":null,"abstract":"In this current research, we focus on the domain of tempered fractional integrals, establishing a novel identity that serves as the cornerstone of our study. This identity paves the way for the Milne-type inequalities, which are explored through the framework of differentiable convex mappings inclusive of tempered fractional integrals. The significance of these mappings in the realm of fractional calculus is underscored by their ability to extend classical concepts into more complex, fractional dimensions. In addition, by using the Hölder inequality and power-mean inequality, we acquire some new Milne-type inequalities. Moreover, the practicality and theoretical relevance of our findings are further demonstrated through the application of specific cases derived from the theorems.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"36 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139495782","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 : 2024-01-17DOI: 10.1186/s13661-024-01824-8
Rong Guo, Xuan Leng
This paper is concerned with the existence and uniqueness of global attractors for a class of degenerate parabolic equations with memory on $mathbb{R}^{n}$ . Since the corresponding equation includes the degenerate term $operatorname{div}{a(x)nabla u}$ , it requires us to give appropriate assumptions about the weight function $a(x)$ for studying our problem. Based on this, we first obtain the existence of a bounded absorbing set, then verify the asymptotic compactness of a solution semigroup via the asymptotic contractive semigroup method. Finally, the existence and uniqueness of global attractors are proved. In particular, the nonlinearity f satisfies the polynomial growth of arbitrary order $p-1$ ( $pgeq 2$ ) and the idea of uniform tail-estimates of solutions is employed to show the strong convergence of solutions.
{"title":"Dynamical behavior of a degenerate parabolic equation with memory on the whole space","authors":"Rong Guo, Xuan Leng","doi":"10.1186/s13661-024-01824-8","DOIUrl":"https://doi.org/10.1186/s13661-024-01824-8","url":null,"abstract":"This paper is concerned with the existence and uniqueness of global attractors for a class of degenerate parabolic equations with memory on $mathbb{R}^{n}$ . Since the corresponding equation includes the degenerate term $operatorname{div}{a(x)nabla u}$ , it requires us to give appropriate assumptions about the weight function $a(x)$ for studying our problem. Based on this, we first obtain the existence of a bounded absorbing set, then verify the asymptotic compactness of a solution semigroup via the asymptotic contractive semigroup method. Finally, the existence and uniqueness of global attractors are proved. In particular, the nonlinearity f satisfies the polynomial growth of arbitrary order $p-1$ ( $pgeq 2$ ) and the idea of uniform tail-estimates of solutions is employed to show the strong convergence of solutions.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"15 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139482273","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 : 2024-01-10DOI: 10.1186/s13661-023-01815-1
Mohamed Abdelwahed, Nejmeddine Chorfi
This work deals with the spectral element discretization of the time-dependent Stokes problem in two- and three-dimensional domains. The boundary condition is defined on the normal component of the velocity and the tangential components of the vorticity. The discretization related to the time variable is processed by a Backward Euler method. We prove through a detailed numerical analysis the well-posedness of the full discrete problem.
{"title":"Spectral element discretization of the time-dependent Stokes problem with nonstandard boundary conditions","authors":"Mohamed Abdelwahed, Nejmeddine Chorfi","doi":"10.1186/s13661-023-01815-1","DOIUrl":"https://doi.org/10.1186/s13661-023-01815-1","url":null,"abstract":"This work deals with the spectral element discretization of the time-dependent Stokes problem in two- and three-dimensional domains. The boundary condition is defined on the normal component of the velocity and the tangential components of the vorticity. The discretization related to the time variable is processed by a Backward Euler method. We prove through a detailed numerical analysis the well-posedness of the full discrete problem.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"4 12 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139409898","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 : 2024-01-08DOI: 10.1186/s13661-023-01813-3
Mingli Hong, Feng Zhou, Chunyou Sun
We consider dynamics of a semilinear heat equation on time-varying domains with lower regular forcing term. Instead of requiring the forcing term $f(cdot )$ to satisfy $int _{-infty}^{t}e^{lambda s}|f(s)|^{2}_{L^{2}},ds
{"title":"Continuity and pullback attractors for a semilinear heat equation on time-varying domains","authors":"Mingli Hong, Feng Zhou, Chunyou Sun","doi":"10.1186/s13661-023-01813-3","DOIUrl":"https://doi.org/10.1186/s13661-023-01813-3","url":null,"abstract":"We consider dynamics of a semilinear heat equation on time-varying domains with lower regular forcing term. Instead of requiring the forcing term $f(cdot )$ to satisfy $int _{-infty}^{t}e^{lambda s}|f(s)|^{2}_{L^{2}},ds<infty $ for all $tin mathbb{R}$ , we show that the solutions of a semilinear heat equation on time-varying domains are continuous with respect to initial data in $H^{1}$ topology and the usual $(L^{2},L^{2})$ pullback $mathscr{D}_{lambda}$ -attractor indeed can attract in the $H^{1}$ -norm, provided that $int _{-infty}^{t}e^{lambda s}|f(s)|^{2}_{H^{-1}(mathcal{O}_{s})},ds< infty $ and $fin L^{2}_{mathrm{loc}}(mathbb{R},L^{2}(mathcal{O}_{s}))$ .","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"108 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139398129","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 : 2024-01-05DOI: 10.1186/s13661-023-01816-0
Jianwen Zhou, Yueting Yang, Wenbo Wang
In this paper, we are concerned with the Kirchhoff-type variable-order fractional Laplacian problems involving critical exponents and logarithmic nonlinearity. By using the constraint variational method, we show the existence of one least energy sign-changing solution. Moreover, we show that this energy is strictly larger than twice the ground energy.
{"title":"Sign-changing solutions for Kirchhoff-type variable-order fractional Laplacian problems","authors":"Jianwen Zhou, Yueting Yang, Wenbo Wang","doi":"10.1186/s13661-023-01816-0","DOIUrl":"https://doi.org/10.1186/s13661-023-01816-0","url":null,"abstract":"In this paper, we are concerned with the Kirchhoff-type variable-order fractional Laplacian problems involving critical exponents and logarithmic nonlinearity. By using the constraint variational method, we show the existence of one least energy sign-changing solution. Moreover, we show that this energy is strictly larger than twice the ground energy.","PeriodicalId":49228,"journal":{"name":"Boundary Value Problems","volume":"39 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139105139","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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