Pub Date : 2023-01-01DOI: 10.14232/ejqtde.2023.1.23
P. Amster, C. Rogers
Here, a proto-type Ermakov–Painlevé I equation is introduced and a homogeneous Dirichlet-type boundary value problem analysed. In addition, a novel Ermakov–Painlevé I system is set down which is reducible by an involutory transformation to the autonomous Ermakov–Ray–Reid system augmented by a single component Ermakov–Painlevé I equation. Hamiltonian such systems are delimited
{"title":"On a Dirichlet boundary value problem for an Ermakov–Painlevé I equation. A Hamiltonian EPI system","authors":"P. Amster, C. Rogers","doi":"10.14232/ejqtde.2023.1.23","DOIUrl":"https://doi.org/10.14232/ejqtde.2023.1.23","url":null,"abstract":"Here, a proto-type Ermakov–Painlevé I equation is introduced and a homogeneous Dirichlet-type boundary value problem analysed. In addition, a novel Ermakov–Painlevé I system is set down which is reducible by an involutory transformation to the autonomous Ermakov–Ray–Reid system augmented by a single component Ermakov–Painlevé I equation. Hamiltonian such systems are delimited","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66585691","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 : 2023-01-01DOI: 10.14232/ejqtde.2023.1.33
M. Borsuk, Damian Wiśniewski
In this paper we consider the Dirichlet problem for quasi-linear second-order elliptic equation with the m ( x ) -Laplacian and the strong nonlinearity on the right side in an unbounded cone-like domain. We study the behavior of weak solutions to the problem at infinity and we find the sharp exponent of the solution decreasing rate. We show that the exponent is related to the least eigenvalue of the eigenvalue problem for the Laplace–Beltrami operator on the unit sphere.
{"title":"The Dirichlet problem in an unbounded cone-like domain for second order elliptic quasilinear equations with variable nonlinearity exponent","authors":"M. Borsuk, Damian Wiśniewski","doi":"10.14232/ejqtde.2023.1.33","DOIUrl":"https://doi.org/10.14232/ejqtde.2023.1.33","url":null,"abstract":"In this paper we consider the Dirichlet problem for quasi-linear second-order elliptic equation with the m ( x ) -Laplacian and the strong nonlinearity on the right side in an unbounded cone-like domain. We study the behavior of weak solutions to the problem at infinity and we find the sharp exponent of the solution decreasing rate. We show that the exponent is related to the least eigenvalue of the eigenvalue problem for the Laplace–Beltrami operator on the unit sphere.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66586382","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}
在这篇论文中，我们看《薛定谔的跟踪连接在系统解决方案来说 { − Δ u + ( V 1 + λ 1 ) u = | u | 2 ∗ − 2u + | u | p 1 − 2 u + β r 1 | u | r 1− 2 u | v | r 2 我 n R N ,− Δ v + ( V 2 + λ 2 ) v = | v | 2 ∗ − 2 v + |v | p 2 − 2 v + β r 2 | u | r 1 | v| r 2 − 2 v 我 n R N ,玩得prescribed团∫ R N u - 2 = a 1 > 0和∫ R N v - 2 = a 2 > 0,哪里λ1 , 美国第二λ∈R威尔rise Lagrange multipliers 3 N⩾,∗= 2 N / ( <
Pub Date : 2023-01-01DOI: 10.14232/ejqtde.2023.1.13
T. Cardinali, Giulia Duricchi
In this note we prove the existence of mild solutions for nonlocal problems governed by semilinear second order differential inclusions which involves a nonlinear term driven by an operator. A first result is obtained in suitable Banach spaces in the lack of compactness both on the fundamental operator, generated by the linear part, and on the nonlinear multivalued term. This purpose is achieved by combining a fixed point theorem, a selection theorem and a containment theorem. Further we provide another existence result in reflexive spaces by using the classical Hahn–Banach theorem and a new selection proposition, proved here, for a multimap guided by an operator. This setting allows us to remove some assumptions required in the previous existence theorem. As a consequence of this last result we obtain the controllability of a problem driven by a wave equation on which an appropriate perturbation acts.
{"title":"Further study on second order nonlocal problems monitored by an operator","authors":"T. Cardinali, Giulia Duricchi","doi":"10.14232/ejqtde.2023.1.13","DOIUrl":"https://doi.org/10.14232/ejqtde.2023.1.13","url":null,"abstract":"In this note we prove the existence of mild solutions for nonlocal problems governed by semilinear second order differential inclusions which involves a nonlinear term driven by an operator. A first result is obtained in suitable Banach spaces in the lack of compactness both on the fundamental operator, generated by the linear part, and on the nonlinear multivalued term. This purpose is achieved by combining a fixed point theorem, a selection theorem and a containment theorem. Further we provide another existence result in reflexive spaces by using the classical Hahn–Banach theorem and a new selection proposition, proved here, for a multimap guided by an operator. This setting allows us to remove some assumptions required in the previous existence theorem. As a consequence of this last result we obtain the controllability of a problem driven by a wave equation on which an appropriate perturbation acts.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66585115","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 : 2022-07-04DOI: 10.14232/ejqtde.2023.1.32
J. Nishiguchi
The method and the formula of variation of constants for ordinary differential equations (ODEs) is a fundamental tool to analyze the dynamics of an ODE near an equilibrium. It is natural to expect that such a formula works for delay differential equations (DDEs), however, it is well-known that there is a conceptual difficulty in the formula for DDEs. Here we discuss the variation of constants formula for DDEs by introducing the notion of a mild solution, which is a solution under an initial condition having a discontinuous history function. Then the principal fundamental matrix solution is defined as a matrix-valued mild solution, and we obtain the variation of constants formula with this function. This is also obtained in the framework of a Volterra convolution integral equation, but the treatment here gives an understanding in its own right. We also apply the formula to show the principle of linearized stability and the Poincaré–Lyapunov theorem for DDEs, where we do not need to assume the uniqueness of a solution.
{"title":"Mild solutions, variation of constants formula, and linearized stability for delay differential equations","authors":"J. Nishiguchi","doi":"10.14232/ejqtde.2023.1.32","DOIUrl":"https://doi.org/10.14232/ejqtde.2023.1.32","url":null,"abstract":"The method and the formula of variation of constants for ordinary differential equations (ODEs) is a fundamental tool to analyze the dynamics of an ODE near an equilibrium. It is natural to expect that such a formula works for delay differential equations (DDEs), however, it is well-known that there is a conceptual difficulty in the formula for DDEs. Here we discuss the variation of constants formula for DDEs by introducing the notion of a mild solution, which is a solution under an initial condition having a discontinuous history function. Then the principal fundamental matrix solution is defined as a matrix-valued mild solution, and we obtain the variation of constants formula with this function. This is also obtained in the framework of a Volterra convolution integral equation, but the treatment here gives an understanding in its own right. We also apply the formula to show the principle of linearized stability and the Poincaré–Lyapunov theorem for DDEs, where we do not need to assume the uniqueness of a solution.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":" ","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45437429","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 : 2022-05-02DOI: 10.14232/ejqtde.2022.1.31
H. Walther
For a differential equation with a state-dependent delay we show that the associated solution manifold X f of codimension 1 in the space C 1 ( [ − r , 0 ] , R ) is an almost graph over a hyperplane, which implies that X f is diffeomorphic to the hyperplane. For the case considered previous results only provide a covering by 2 almost graphs.
{"title":"On the solution manifold of a differential equation with a delay which has a zero","authors":"H. Walther","doi":"10.14232/ejqtde.2022.1.31","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.31","url":null,"abstract":"For a differential equation with a state-dependent delay we show that the associated solution manifold \u0000 \u0000 X\u0000 f\u0000 \u0000 of codimension 1 in the space \u0000 \u0000 C\u0000 1\u0000 \u0000 (\u0000 [\u0000 −\u0000 r\u0000 ,\u0000 0\u0000 ]\u0000 ,\u0000 \u0000 R\u0000 \u0000 )\u0000 is an almost graph over a hyperplane, which implies that \u0000 \u0000 X\u0000 f\u0000 \u0000 is diffeomorphic to the hyperplane. For the case considered previous results only provide a covering by 2 almost graphs.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42357355","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 : 2022-04-20DOI: 10.14232/ejqtde.2023.1.9
Equations Bruno S. V. Ara'ujo, R. Demarque, L. Viana
In this paper, we present a new Carleman estimate for the adjoint equations associated to a class of super strong degenerate parabolic linear problems. Our approach considers a standard geometric imposition on the control domain, which can not be removed in general. Additionally, we also apply the aforementioned main inequality in order to investigate the null controllability of two nonlinear parabolic systems. The first application is concerned a global null controllability result obtained for some semilinear equations, relying on a fixed point argument. In the second one, a local null controllability for some equations with nonlocal terms is also achieved, by using an inverse function theorem.
{"title":"Carleman inequality for a class of super strong degenerate parabolic\u0000 operators and applications","authors":"\t\tEquations\t\t\tBruno S. V. Ara'ujo, R. Demarque, L. Viana","doi":"10.14232/ejqtde.2023.1.9","DOIUrl":"https://doi.org/10.14232/ejqtde.2023.1.9","url":null,"abstract":"In this paper, we present a new Carleman estimate for the adjoint\u0000 equations associated to a class of super strong degenerate parabolic linear\u0000 problems. Our approach considers a standard geometric imposition on the\u0000 control domain, which can not be removed in general. Additionally, we also\u0000 apply the aforementioned main inequality in order to investigate the null\u0000 controllability of two nonlinear parabolic systems. The first application\u0000 is concerned a global null controllability result obtained for some\u0000 semilinear equations, relying on a fixed point argument. In the second one,\u0000 a local null controllability for some equations with nonlocal terms is also\u0000 achieved, by using an inverse function theorem.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"29 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74888891","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 : 2022-01-01DOI: 10.14232/ejqtde.2022.1.10
J. Džurina, B. Baculíková
This paper is devoted to the study of the oscillatory behavior of half-linear functional differential equations with deviating argument of the form (E)(r(t)(y′(t))α)′=p(t)yα(τ(t)).. We introduce new technique based on monotonic properties of nonoscillatory solutions to offer new oscillatory criteria for (E). We will show that presented results essentially improve existing ones even for linear differential equations.
本文研究了具有(E) (r (t) (y ' (t))形式的半线性泛函微分方程的振荡行为。α) ' = p (t) y α (τ (t))。我们引入了基于非振荡解单调性的新技术，给出了(E)的新的振荡判据。我们将证明，即使对于线性微分方程，所提出的结果本质上也改进了现有的结果。
{"title":"Oscillation of half-linear differential equations with mixed type of argument","authors":"J. Džurina, B. Baculíková","doi":"10.14232/ejqtde.2022.1.10","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.10","url":null,"abstract":"<jats:p>This paper is devoted to the study of the oscillatory behavior of half-linear functional differential equations with deviating argument of the form <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\" display=\"block\"> <mml:mtable displaystyle=\"true\"> <mml:mlabeledtr> <mml:mtd id=\"mjx-eqn-Eabs\"> <mml:mrow> <mml:mtext>(</mml:mtext> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>E</mml:mi> </mml:mrow> <mml:mtext>)</mml:mtext> </mml:mrow> </mml:mtd> <mml:mtd> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mi>r</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo stretchy=\"false\">(</mml:mo> <mml:msup> <mml:mi>y</mml:mi> <mml:mo>′</mml:mo> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:msup> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>α<!-- α --></mml:mi> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>′</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>p</mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:msup> <mml:mi>y</mml:mi> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>α<!-- α --></mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>τ<!-- τ --></mml:mi> <mml:mo stretchy=\"false\">(</mml:mo> <mml:mi>t</mml:mi> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo stretchy=\"false\">)</mml:mo> <mml:mo>.</mml:mo> </mml:mtd> </mml:mlabeledtr> </mml:mtable> </mml:math>. We introduce new technique based on monotonic properties of nonoscillatory solutions to offer new oscillatory criteria for <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\" xmlns=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow class=\"MathJax_ref\" href=\"#mjx-eqn-Eabs\"> <mml:mrow> <mml:mtext>(</mml:mtext> <mml:mrow class=\"MJX-TeXAtom-ORD\"> <mml:mi>E</mml:mi> </mml:mrow> <mml:mtext>)</mml:mtext> </mml:mrow> </mml:mrow> </mml:math>. We will show that presented results essentially improve existing ones even for linear differential equations.</jats:p>","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66583666","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 : 2022-01-01DOI: 10.14232/ejqtde.2022.1.12
D. Dragičević
The main purpose of this paper is to establish a global smooth linearization result for two classes of nonautonomous dynamics with discrete time. More precisely, we consider a nonlinear and nonautonomous dynamics given by a two-sided sequence of maps as well as variational systems whose linear part is contractive, and under suitable assumptions we construct C 1 conjugacies between the original dynamics and its linear part. We stress that our dynamics acts on a arbitrary Banach space. Our arguments rely on related results dealing with autonomous dynamics.
{"title":"Global smooth linearization of nonautonomous contractions on Banach spaces","authors":"D. Dragičević","doi":"10.14232/ejqtde.2022.1.12","DOIUrl":"https://doi.org/10.14232/ejqtde.2022.1.12","url":null,"abstract":"The main purpose of this paper is to establish a global smooth linearization result for two classes of nonautonomous dynamics with discrete time. More precisely, we consider a nonlinear and nonautonomous dynamics given by a two-sided sequence of maps as well as variational systems whose linear part is contractive, and under suitable assumptions we construct C 1 conjugacies between the original dynamics and its linear part. We stress that our dynamics acts on a arbitrary Banach space. Our arguments rely on related results dealing with autonomous dynamics.","PeriodicalId":50537,"journal":{"name":"Electronic Journal of Qualitative Theory of Differential Equations","volume":"1 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66583745","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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