Unstable manifolds of a class of delayed partial differential equations with nondense domain

Jendoubi, Chiraz

doi:10.4064/ap3913-11-2016

Cited by 9 publications

(8 citation statements)

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“…Many works have elaborated this field, see previous studies. 10,13,14,22,28 So, by a second initiative, we propose to study the conditional stability of periodic solution for (1) in the dichotomic case. We further prove that there exists a local stable manifold near the periodic solution.…”

Section: Definitionmentioning

confidence: 99%

“…In this section, we consider equations (1) and (13) in the dichotomic case. Let us first give the definition of such concept.…”

Section: Existence Uniqueness and Conditional Stability Of Periodicmentioning

confidence: 99%

“…Now, we prove the following lemma that gives respectively the form of bounded solutions for (1) and (13).…”

Section: U(t S)p(s)mentioning

confidence: 99%

“…1. Let w ∈ BC(R + , X 0 ) be the solution of equation (13). Then, w has the following representation:…”

Section: Lemma 2 Assume That (H 1 ) To (H 7 )mentioning

confidence: 99%

“…As w is a solution of Equation (13), it follows that w(t) − z(t) = U B (t, 0)(w(0) − z(0)) for all t ≥ 0. Setting = w(0) − z(0), it yields from the boundedness of z and w on [0, +∞) that ∈ P B (0)X 0 .…”

Section: Lemma 2 Assume That (H 1 ) To (H 7 )mentioning

confidence: 99%

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On the theory of periodic solution for some nondensely nonautonomous delayed partial differential equations

Jendoubi

2019

Math Methods in App Sciences

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We first study the Massera problem for the existence of a −periodic solution for some nondensely defined partial differential equation, where the autonomous linear part satisfies the Hille-Yosida condition and the delayed nonlinear part satisfies a locally Lipschitz condition. Second, inspired by an existing study, we prove in the dichotomic case, for = 1, the existence-uniqueness and conditional stability of the periodic solution. Moreover, we show the existence of a local stable manifold around such solution. Our theoretical results are finally illustrated by an application.

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Section: Definitionmentioning

confidence: 99%

“…In this section, we consider equations (1) and (13) in the dichotomic case. Let us first give the definition of such concept.…”

Section: Existence Uniqueness and Conditional Stability Of Periodicmentioning

confidence: 99%

“…Now, we prove the following lemma that gives respectively the form of bounded solutions for (1) and (13).…”

Section: U(t S)p(s)mentioning

confidence: 99%

“…1. Let w ∈ BC(R + , X 0 ) be the solution of equation (13). Then, w has the following representation:…”

Section: Lemma 2 Assume That (H 1 ) To (H 7 )mentioning

confidence: 99%

Section: Lemma 2 Assume That (H 1 ) To (H 7 )mentioning

confidence: 99%

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On the theory of periodic solution for some nondensely nonautonomous delayed partial differential equations

Jendoubi

2019

Math Methods in App Sciences

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Dichotomy and μ‐pseudo almost automorphic solutions for delayed partial functional differential equations in admissible spaces

Jendoubi

2020

Mathematische Nachrichten

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We prove the existence and uniqueness of μ‐pseudo almost automorphic solution for a delayed non‐autonomous partial functional differential equation in the exponential dichotomic case, where the nonlinear operator F satisfies the φ‐Lipschitz condition and φ belongs to some admissible spaces. We further prove the existence of an invariant stable manifold around the μ‐pseudo almost automorphic solution in that case. An application is given to illustrate our theory.

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On the Theory of Integral Manifolds for Some Delayed Partial Differential Equations with Nondense Domain

Jendoubi

2020

Ukr Math J

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Unstable manifolds of a class of delayed partial differential equations with nondense domain

Cited by 9 publications

References 0 publications

On the theory of periodic solution for some nondensely nonautonomous delayed partial differential equations

On the theory of periodic solution for some nondensely nonautonomous delayed partial differential equations

Dichotomy and μ‐pseudo almost automorphic solutions for delayed partial functional differential equations in admissible spaces

On the Theory of Integral Manifolds for Some Delayed Partial Differential Equations with Nondense Domain

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