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Cubo (Temuco)

On-line version ISSN 0719-0646

Cubo vol.23 no.3 Temuco Dec. 2021

http://dx.doi.org/10.4067/S0719-06462021000300469 

Articles

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

Abdelhai Elazzouzi1 
http://orcid.org/0000-0002-6952-3112

Khalil Ezzinbi2 
http://orcid.org/0000-0001-5334-2264

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. abdelhai.elazzouzi@usmba.ac.ma

2Département de Mathématiques, Faculté des Sciences Semlalia, Université Cadi Ayyad, B.P. 2390, Marrakesh, Morocco. ezzinbi@uca.ac.ma .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. mohammed.kriche@usmba.ac.ma

ABSTRACT

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

RESUMEN

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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Acknowledgement

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.

References

[1] M. Adimy and K. Ezzinbi, “Local existence and linearized stability for partial functional differential equations”, Dyn. Syst. Appl., vol. 7, no. 3, pp. 389-404, 1998. [ Links ]

[2] W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector valued Laplace transforms and Cauchy problems, Monographs in Mathematics, vol. 96, Basel: Birkhauser Verlag, 2001. [ Links ]

[3] W. Arendt , A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, B. Neubrander and U. Schlotterbeck, One-parameter semigroup of positive operators, Lecture Notes in Mathematics, vol. 1184, Berlin: Springer-Verlag, 1984. [ Links ]

[4] T. Burton, Stability and periodic solutions of ordinary differential equation and functional differential equations, New York: Academic Press, 1985. [ Links ]

[5] S. N. Chow and J. K. Hale, “Strongly limit-compact maps”, Funkcial. Ekvac., vol. 7, pp. 31-38, 1974. [ Links ]

[6] G. Da Prato and E. Sinestrari, “Differential operators with nondense domains”, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), vol. 14, no. 2, pp. 285-344, 1987. [ Links ]

[7] N. Dunford and J. T. Schwartz, Linear operators, Part I, New York: John Wiley & Sons Inc, 1987. [ Links ]

[8] A. Elazzouzi andK. Ezzinbi , “Ultimate boundedness and periodicity for some partial functional differential equations with infinite delay”, J. Math. Anal. Appl., vol. 329, no. 1, pp. 498-514, 2007. [ Links ]

[9] K. Ezzinbi , “A survey on new methods for partial functional differential equations and applications”, Afr. Mat., vol. 31, no. 1, pp. 89-113, 2020. [ Links ]

[10] K. Ezzinbi and M. Taoudi, “Periodic solutions and attractiveness for some partial functional differential equations with lack of compactness”, Proc. Amer. Math. Soc., vol. 149, no. 3, pp. 1165-1174, 2021. [ Links ]

[11] K. J. Engel andR. Nagel , One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics, vol. 194, New York: Springer-Verlag, 2000. [ Links ]

[12] J. Hale and O. Lopes, “Fixed point theorems and dissipative processes”, J. Differential Equations, vol. 13, no. 2, pp. 391-402, 1973. [ Links ]

[13] H. R. Henriquez, “Periodic solutions of quasi-linear partial functional differential equations with unbounded delay”, Funkcial. Ekvac. , vol. 37, no. 2, pp. 329-343, 1994. [ Links ]

[14] Y. Hino, T. Naito, N. Van Minh and J. Son Shin, Almost periodic solutions of differential equations in Banach spaces, Stability and Control: Theory, Methods and Applications, vol. 15, London: Taylor & Francis, 2002. [ Links ]

[15] Y. Hino , S. Murakami and T. Yoshizawa, “Existence of almost periodic solutions of some functional differential equations in a Banach space”, Tohoku Math. J., vol. 49, no. 1, pp. 133-147, 1997. [ Links ]

[16] G. M. N’Guerekata, Hui-Sheng Ding and W. Long, “Existence of pseudo almost periodic solutions for a class of partial functional differential equations”, Electron J. Differential Equations , vol. 2013, no. 104, 14 pages, 2013. [ Links ]

[17] M. Kostić, Almost periodic and almost automorphic type solutions to integro-differential equations, Berlin: W. de Gruyter, 2019. [ Links ]

[18] B. M. Levitan, Almost periodic functions (in Russian), Moscow: Gosudarstv. Izdat. Tehn.-Teor. Lit., 1953. [ Links ]

[19] R. D. Nussbaum, “The radius of essential spectrum”, Duke Math. J., vol. 37, pp. 473-478, 1970. [ Links ]

[20] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, New York: Springer-Verlag , 1983. [ Links ]

[21] M. Schechter, Principles of functional analysis, New York: Academic Press , 1971. [ Links ]

[22] J. S. Shin andT. Naito , “Semi-Fredholm operators and periodic solution for linear functionaldifferential equations”, J. Differential Equations , vol. 153, no. 2, pp. 407-441, 1999. [ Links ]

[23] C. C. Travis and G. F. Webb, “Existence and stability for partial functional differential equations”, Trans. Amer. Math. Soc., vol. 200, pp. 395-418, 1974. [ Links ]

[24] J. Massera, “The existence of periodic solutions of differential equations”, Duke Math. J. , vol. 17, pp. 457-475, 1950. [ Links ]

[25] J. Wu, Theory and applications of partial functional differential equations, Applied Mathematical Sciences, vol. 119, New York: Springer-Verlag , 1996. [ Links ]

[26] T. Yoshizawa , Stability theory by Liapunov’s second method, Publications of the Mathematical Society of Japan, vol. 9, Tokyo: Math. Soc. Japan, 1966. [ Links ]

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

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