Research Article
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Year 2023, Volume: 6 Issue: 2, 161 - 182, 31.07.2023
https://doi.org/10.33773/jum.1233884

Abstract

References

  • F. S. Acharya, Controllability of Second Order Semilinear Impulsive Partial Neutral Functional Differential Equations with Infinite Delay, International Journal of Mathematical Sciences and Applications, pp.207-218 (2013).
  • R. Benkhalti, K. Ezzinbi, Existence and stability in the α-norm for some partial functional differential equations with infinite delay, Differential and Integral Equations, Vol.19, No.5, pp.545-572 (2006).
  • J. Bochenek, An abstract nonlinear second order differential equation, Annales Polonici Mathematici, pp.155-166 (1991).
  • G. Da Prato, E. Giusi, Una caratterizzazione dei generatori di funzioni coseno astratte, Bollettino dell’Unione Matematica Italiana, Vol.22, pp.357-362 (1967).
  • H.O Fattorini, Ordinary differential equations in linear topological spaces, II, Journal Differential Equations, Vol.6, pp.50-70 (1969).
  • H.O Fattorini, Ordinary Differential Equations in Linear Topological Spaces, Journal of Differential Equations, Vol.5, pp.72-105 (1968).
  • J.K Hale, J. Kato, Phase space for retarded equations with infnite delay, Funkcial. Ekvack, Vol.21, pp.11-41 (1978).
  • R. Ameziane Hassani, A. Blali, A. EL Amrani, K. Moussaouja, Cosine families of operators in a class of Fr´echet spaces, Proyecciones Journal of Mathematics, pp.103-118 (2018).
  • J.M. Jeong, E.Y. Ju, S.H. Cho, Control problems for semilinear second order equations with cosine families, Advances in Difference Equations, pp.1-13 (2016).
  • D.N. Pandey, S. Das, N. Sukavanam, Existence of Solution for a Second-Order Neutral Differential Equation with State Dependent Delay and Non-instantaneous Impulses, International Journal of Nonlinear Science, pp.145-155 (2014).
  • M. Sova, Cosine operator functions, Rozprawy Matematiyczne, Vol.49, pp.1-47 (1966).
  • M. Sova, Semigroups and cosine functions of normal operators in Hilbert spaces, Casopis Pro Pestov´an´ı Matematiky, Vol.93, pp.437-458 (1968).
  • C.C. Travis, G.F. Webb, An Abstract Second Order Semilinear Second order Volterra Integrodifferential Equation, SIAM Journal of Mathematical Analysis, Vol.10, pp.412-424 (1979).
  • C.C. Travis, G.F. Webb, Cosine families and abstract nonlinear second order differential equations, Transaction of Acta Mathematica Academiae Seientiarum Hungaricae Tomus, Vol.32, No.(3-4), pp.75-96 (1978).
  • I. Zabsonr´e, M. Napo, Existence and Regularity of Solutions for Some Nonlinear Second Order Differential Equation in Banach Spaces, Recent Studies in Differential Equations, pp.29-48 (2020).

LOCAL EXISTENCE AND REGULARITY OF SOLUTIONS FOR SOME SECOND ORDER DIFFERENTIAL EQUATION WITH INFINITE DELAY

Year 2023, Volume: 6 Issue: 2, 161 - 182, 31.07.2023
https://doi.org/10.33773/jum.1233884

Abstract

In this work, we study the existence and regularity of solutions for some second order differential equations with infinite delay in Banach spaces. We suppose that the undelayed part admits a cosine operator in the sense given by Da Prato and Giusi, [ G. Da Prato and E. Giusi, \emph{Una caratterizzazione dei generatori di funzioni coseno astratte}, Bollettino dell'Unione Matematica Italiana, 22, 357-362, (1967)]. The delayed part is assumed to be locally Lipschitz. Firstly, we show the existence of the mild solutions. Secondly, we give sufficient conditions ensuring the existence of strict solutions.

References

  • F. S. Acharya, Controllability of Second Order Semilinear Impulsive Partial Neutral Functional Differential Equations with Infinite Delay, International Journal of Mathematical Sciences and Applications, pp.207-218 (2013).
  • R. Benkhalti, K. Ezzinbi, Existence and stability in the α-norm for some partial functional differential equations with infinite delay, Differential and Integral Equations, Vol.19, No.5, pp.545-572 (2006).
  • J. Bochenek, An abstract nonlinear second order differential equation, Annales Polonici Mathematici, pp.155-166 (1991).
  • G. Da Prato, E. Giusi, Una caratterizzazione dei generatori di funzioni coseno astratte, Bollettino dell’Unione Matematica Italiana, Vol.22, pp.357-362 (1967).
  • H.O Fattorini, Ordinary differential equations in linear topological spaces, II, Journal Differential Equations, Vol.6, pp.50-70 (1969).
  • H.O Fattorini, Ordinary Differential Equations in Linear Topological Spaces, Journal of Differential Equations, Vol.5, pp.72-105 (1968).
  • J.K Hale, J. Kato, Phase space for retarded equations with infnite delay, Funkcial. Ekvack, Vol.21, pp.11-41 (1978).
  • R. Ameziane Hassani, A. Blali, A. EL Amrani, K. Moussaouja, Cosine families of operators in a class of Fr´echet spaces, Proyecciones Journal of Mathematics, pp.103-118 (2018).
  • J.M. Jeong, E.Y. Ju, S.H. Cho, Control problems for semilinear second order equations with cosine families, Advances in Difference Equations, pp.1-13 (2016).
  • D.N. Pandey, S. Das, N. Sukavanam, Existence of Solution for a Second-Order Neutral Differential Equation with State Dependent Delay and Non-instantaneous Impulses, International Journal of Nonlinear Science, pp.145-155 (2014).
  • M. Sova, Cosine operator functions, Rozprawy Matematiyczne, Vol.49, pp.1-47 (1966).
  • M. Sova, Semigroups and cosine functions of normal operators in Hilbert spaces, Casopis Pro Pestov´an´ı Matematiky, Vol.93, pp.437-458 (1968).
  • C.C. Travis, G.F. Webb, An Abstract Second Order Semilinear Second order Volterra Integrodifferential Equation, SIAM Journal of Mathematical Analysis, Vol.10, pp.412-424 (1979).
  • C.C. Travis, G.F. Webb, Cosine families and abstract nonlinear second order differential equations, Transaction of Acta Mathematica Academiae Seientiarum Hungaricae Tomus, Vol.32, No.(3-4), pp.75-96 (1978).
  • I. Zabsonr´e, M. Napo, Existence and Regularity of Solutions for Some Nonlinear Second Order Differential Equation in Banach Spaces, Recent Studies in Differential Equations, pp.29-48 (2020).
There are 15 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Research Article
Authors

Miailou Napo This is me

Mohamado Kıema This is me

Issa Zabsonre 0000-0002-2968-8481

Publication Date July 31, 2023
Submission Date January 13, 2023
Acceptance Date July 29, 2023
Published in Issue Year 2023 Volume: 6 Issue: 2

Cite

APA Napo, M., Kıema, M., & Zabsonre, I. (2023). LOCAL EXISTENCE AND REGULARITY OF SOLUTIONS FOR SOME SECOND ORDER DIFFERENTIAL EQUATION WITH INFINITE DELAY. Journal of Universal Mathematics, 6(2), 161-182. https://doi.org/10.33773/jum.1233884
AMA Napo M, Kıema M, Zabsonre I. LOCAL EXISTENCE AND REGULARITY OF SOLUTIONS FOR SOME SECOND ORDER DIFFERENTIAL EQUATION WITH INFINITE DELAY. JUM. July 2023;6(2):161-182. doi:10.33773/jum.1233884
Chicago Napo, Miailou, Mohamado Kıema, and Issa Zabsonre. “LOCAL EXISTENCE AND REGULARITY OF SOLUTIONS FOR SOME SECOND ORDER DIFFERENTIAL EQUATION WITH INFINITE DELAY”. Journal of Universal Mathematics 6, no. 2 (July 2023): 161-82. https://doi.org/10.33773/jum.1233884.
EndNote Napo M, Kıema M, Zabsonre I (July 1, 2023) LOCAL EXISTENCE AND REGULARITY OF SOLUTIONS FOR SOME SECOND ORDER DIFFERENTIAL EQUATION WITH INFINITE DELAY. Journal of Universal Mathematics 6 2 161–182.
IEEE M. Napo, M. Kıema, and I. Zabsonre, “LOCAL EXISTENCE AND REGULARITY OF SOLUTIONS FOR SOME SECOND ORDER DIFFERENTIAL EQUATION WITH INFINITE DELAY”, JUM, vol. 6, no. 2, pp. 161–182, 2023, doi: 10.33773/jum.1233884.
ISNAD Napo, Miailou et al. “LOCAL EXISTENCE AND REGULARITY OF SOLUTIONS FOR SOME SECOND ORDER DIFFERENTIAL EQUATION WITH INFINITE DELAY”. Journal of Universal Mathematics 6/2 (July 2023), 161-182. https://doi.org/10.33773/jum.1233884.
JAMA Napo M, Kıema M, Zabsonre I. LOCAL EXISTENCE AND REGULARITY OF SOLUTIONS FOR SOME SECOND ORDER DIFFERENTIAL EQUATION WITH INFINITE DELAY. JUM. 2023;6:161–182.
MLA Napo, Miailou et al. “LOCAL EXISTENCE AND REGULARITY OF SOLUTIONS FOR SOME SECOND ORDER DIFFERENTIAL EQUATION WITH INFINITE DELAY”. Journal of Universal Mathematics, vol. 6, no. 2, 2023, pp. 161-82, doi:10.33773/jum.1233884.
Vancouver Napo M, Kıema M, Zabsonre I. LOCAL EXISTENCE AND REGULARITY OF SOLUTIONS FOR SOME SECOND ORDER DIFFERENTIAL EQUATION WITH INFINITE DELAY. JUM. 2023;6(2):161-82.