Stability for Nonautonomous Linear Differential Systems with Infinite Delay
Tóm tắt
We study the stability of general n-dimensional nonautonomous linear differential equations with infinite delays. Delay independent criteria, as well as criteria depending on the size of some finite delays are established. In the first situation, the effect of the delays is dominated by non-delayed diagonal negative feedback terms, and sufficient conditions for both the asymptotic and the exponential asymptotic stability of the system are given. In the second case, the stability depends on the size of some bounded diagonal delays and coefficients, although terms with unbounded delay may co-exist. Our results encompass DDEs with discrete and distributed delays, and enhance some recent achievements in the literature.
Tài liệu tham khảo
Arino, O., Gőri, I., Pituk, M.: Asymptotically diagonal delay differential systems. J. Math. Anal. Appl. 204, 701–728 (1996)
Berezansky, L., Braverman, E.: New stability conditions for linear differential equations with several delays. Abstract Appl. Anal. ID 1785668, 1–19 (2011)
Berezansky, L., Braverman, E.: Solution estimates for linear differential equations with delay. Appl. Math. Comput. 372, 124962 (2020)
Berezansky, L., Diblík, J., Svoboda, Z., Smarda, Z.: Exponential stability of linear delayed differential systems. Appl. Math. Comput. 320, 474–484 (2018)
Berezansky, L., Diblík, J., Svoboda, Z., Smarda, Z.: Exponential stability tests for linear delayed differential systems depending on all delays. J. Dyn. Differ. Equ. 31, 2095–2108 (2019)
Berman, A., Plemmons, R.: Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York (1979)
Coppel, W.A.: Dichotomies in Stability Theory. Lecture Notes in Mathematics, vol. 629. Springer, Berlin (1978)
Diekmann, O., Gyllemberg, M.: Equations with infinite delay: blending the abstract and the concrete. J. Differ. Equ. 252, 819–851 (2012)
Driver, R.D.: Linear differential systems with small delays. J. Differ. Equ. 21, 148–166 (1976)
Faria, T.: Stability and extinction for Lotka-Volterra systems with infinite delay. J. Dyn. Differ. Equ. 22, 299–324 (2010)
Faria, T., Huang, W.: Special solutions for linear functional differential equations and asymptotic behaviour. Differ. Integral Equ. 18, 337–360 (2005)
Faria, T., Obaya, R., Sanz, A.M.: Asymptotic behaviour for a class of non-monotone delay differential systems with applications. J. Dyn. Differ. Equ. 30, 911–935 (2018)
Faria, T., Oliveira, J.J.: Local and global stability for Lotka-Volterra systems with distributed delays and instantaneous feedbacks. J. Differ. Equ. 244, 1049–1079 (2008)
Fehér, A., Márton, L., Pituk, M.: Approximation of a linear autonomous differential equation with small delay. Simmetry 11, 1299 (2019)
Garab, A., Pituk, M., Stavroulakis, I.P.: A sharp oscillation criterion for a linear delay differential equation. Appl. Math. Lett. 93(2019), 58–65 (2019)
Gopalsamy, K.: Stability and Oscillation in Delay Differential Equations of Population Dynamics. Kluwer Academic, Dordrecht (1992)
Györi, I., Horváth, L.: Sharp estimation for the solutions of delay differential and Halanay type inequalities. Discrete Contin. Dyn. Syst. Ser. A 37, 3211–3242 (2017)
Györi, I., Horváth, L.: Sharp estimation for the solutions of inhomogeneous delay differential and Halanay type inequalities. Electron. J. Qual. Theory Differ. Equ. 2018. Paper No. 54, 1–18 (2018)
Haddock, J.R., Kuang, Y.: Asymptotic theory for a class of nonautonomous delay differential equations. J. Math. Anal. Appl. 168, 147–162 (1992)
Hale, J.K., Verduyn Lunel, S.M.: Introduction to Functional Differential Equations. Springer, New-York (1993)
Hatvani, L.: Asymptotic stability of nonautonomous functional differential equations with distributed delays, 2016. Electron. J. Differ. Equ. 302, 1–16 (2016)
Hino, Y., Murakami, S., Naito, T.: Functional Differential Equations with Infinite Delay. Springer, New-York (1991)
Hofbauer, J., So, J.W.-H.: Diagonal dominance and harmless off-diagonal delays. Proc. Am. Math. Soc. 128, 2675–2682 (2000)
Johnson, R.A., Sell, G.R.: Smoothness of spectral subbundles and reducibility of quasiperiodic linear differential systems. J. Differ. Equ. 41, 262–288 (1981)
Krisztin, T.: On stability properties for one-dimensional functional-differential equations. Funkcial. Ekvac. 34, 241–256 (1991)
Kuang, Y.: Delay Differential Equations with Applications in Population Dynamics. Academic Press, New York (1993)
Murakami, S., Naito, T.: Fading memory spaces and stability properties for functional differential equations with infinite delay. Funkcial. Ekvac. 32, 91–105 (1989)
Ngoc, P.H.A., Tinh, C.T.: Explicit criteria for exponential stability of time-varying systems with infinite delay. Math. Control Signals Syst. 28(1), 30 (2016)
Ngoc, P.H.A., Tran, T.B., Tinh, C.T., Huy, N.D.: Novel criteria for exponential stability of linear nonautonomous functional differential equations. J. Syst. Sci. Complex. 32(2), 479–495 (2019)
Rudin, W.: Real and Complex Analysis, 3rd edn. McGraw-Hill Book Co., New York (1987)
Sacker, R.J., Sell, G.R.: A spectral theory for linear differential systems. J. Differ. Equ. 27, 320–358 (1978)
So, J.W.-H., Yu, J.S., Chen, M.-P.: Asymptotic stability for scalar delay differential equations. Funkcial. Ekvac. 39(1), 1–17 (1996)
So, J.W.-H., Tang, X.H., Zou, X.: Global attractivity for nonautonomous linear delay systems. Funkcial. Ekvac. 4, 25–40 (2004)
Yoneyama, T.: The 3/2 stability theorem for one-dimensional delay-differential equation with unbounded delay. J. Math. Anal. Appl. 165, 133–143 (1992)