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

Yıl 2023, Cilt: 6 Sayı: 2, 161 - 182, 31.07.2023

Miailou Napo Mohamado Kıema Issa Zabsonre

https://doi.org/10.33773/jum.1233884

Öz

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.

Anahtar Kelimeler

Cosine family, second order differential equations, $\alpha$-norm, delay differential equation.

Kaynakça

• 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).