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Cubo (Temuco)

On-line version ISSN 0719-0646

Cubo vol.23 no.3 Temuco Dec. 2021 


On the periodic solutions for some retarded partial differential equations by the use of semi-Fredholm operators

Abdelhai Elazzouzi1

Khalil Ezzinbi2

Mohammed Kriche3 

1Département de Mathématiques, Laboratoire des Sciences de l’Ingénieur (LSI), Faculté Polydisciplinaire de Taza, Université Sidi Mohamed Ben Abdellah (USMBA) - Fes, BP. 1223, Taza, Morocco.

2Département de Mathématiques, Faculté des Sciences Semlalia, Université Cadi Ayyad, B.P. 2390, Marrakesh, Morocco. .Département de Mathématiques, Laboratoire des Sciences de l’Ingénieur (LSI), Faculté Polydisciplinaire de Taza, Université Sidi Mohamed Ben Abdellah (USMBA) - Fes, BP. 1223, Taza, Morocco.

3 Département de Mathématiques, Laboratoire des Sciences de l’Ingénieur (LSI), Faculté Polydisciplinaire de Taza, Université Sidi Mohamed Ben Abdellah (USMBA) - Fes, BP. 1223, Taza, Morocco.


The main goal of this work is to examine the periodic dynamic behavior of some retarded periodic partial differential equations (PDE). Taking into consideration that the linear part realizes the Hille-Yosida condition, we discuss the Massera’s problem to this class of equations. Especially, we use the perturbation theory of semi-Fredholm operators and the Chow and Hale’s fixed point theorem to study the relation between the boundedness and the periodicity of solutions for some inhomogeneous linear retarded PDE. An example is also given at the end of this work to show the applicability of our theoretical results.

Keywords and Phrases: Hille-Yosida condition; Integral solutions; Semigroup; Semi-Fredholm operators; Periodic solution, Poincaré map


El principal objetivo de este trabajo es examinar el comportamiento dinámico periódico de algunas ecuaciones diferenciales parciales (EDP) periódicas con retardo. Tomando en consideración que la parte lineal cumple la condición de Hille-Yosida, discutimos el problema de Massera para esta clase de ecuaciones. Especialmente usamos la teoría de perturbaciones de operadores semi-Fredholm y el teorema de punto fijo de Chow y Hale para estudiar la relación entre el acotamiento y la periodicidad de soluciones para algunas EDP no homogéneas lineales con retardo. Se entrega un ejemplo al final de este trabajo para mostrar la aplicabilidad de los resultados teóricos.

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The authors would like to thank the anonymous referees for their constructive comments and valuable suggestions, which are helpful to improve the quality of this paper.


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Accepted: October 07, 2021; Received: March 25, 2021

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