Boundedness and stability in nonlinear systems of differential equations using a modified variation of parameters formula

Youssef N. Raffoul

doi:10.56754/0719-0646.2501.037

DOI:

https://doi.org/10.56754/0719-0646.2501.037

Abstract

In this research we introduce a new variation of parameters for systems of linear and nonlinear ordinary differential equations. We use known mathematical methods and techniques including Gronwall’s inequality and fixed point theory to obtain boundedness on all solutions and stability results on the zero solution.

Keywords

System , Ordinary differential equations , Linear , Nonlinear , Fundamental matrix , Boundedness , Stability , New variation of parameters

Mathematics Subject Classification:

39A10 , 34A97

Youssef N. Raffoul Department of Mathematics, University of Dayton, Dayton, OH 45469-2316, USA. https://orcid.org/0000-0002-2154-9049

Pages: 37–55
Date Published: 2023-04-24
Vol. 25 No. 1 (2023)

M. Adivar and Y. N. Raffoul, “Existence of resolvent for Volterra integral equations on time scales”, Bull. Aust. Math. Soc., vol. 82, no. 1, pp. 139–155, 2010.

R. Bellman, Stability theory of differential equations, New York-Toronto-London: McGraw- Hill Book Company, Inc., 1953.

L. Berezansky and E. Braverman, “Exponential stability of difference equations with several delays: recursive approach”, Adv. Difference Equ., vol. 2009, Article ID 104310, 13 pages, 2009.

F. H. Brownell and W. K. Ergen, “A theorem on rearrangements and its application to certain delay differential equations”, J. Rational Mech. Anal., vol. 3, no. 5, pp. 565–579, 1954.

T. A. Burton, “Fixed points and stability of a nonconvolution equation”, Proc. Amer. Math. Soc., vol. 132, no. 12, pp. 3679–3687, 2004.

T. A. Burton, “Stability by fixed point theory or Liapunov theory: a comparison”, Fixed Point Theory, vol. 4, no. 1, pp. 15–32, 2003.

T. A. Burton, Stability by fixed point theory for functional differential equations, New York: Dover Publications, Inc., 2006.

T. A. Burton and T. Furumochi, “Fixed points in stability theory for ordinary and functional differential equations”, Dynam. Systems Appl., vol. 10, no. 1, pp. 89–116, 2001.

J. L. Goldberg, and A. J. Schwartz, Systems of ordinary differential equations: an introduction, Harper’s Series in Modern Mathematics, New York: Harper & Row, Inc., 1972.

P. Hartman, Ordinary differential equations, New York-London-Sydney: John Wiley & Sons, Inc., 1964.

P. Holmes, “A nonlinear oscillator with a strange attractor”, Phil. Trans. Roy. Soc. London Ser. A, vol. 292, no. 1394, pp. 419–448, 1979.

W. Kelley and A. Peterson, Difference equations an introduction with applications, San Diego: Academic Press, 2000.

A. Larraín-Hubach and Y. N. Raffoul, “Boundedness periodicity and stability in nonlinear delay differential equations”, Adv. Dyn. Syst. Appl., vol. 15, no. 1, pp. 29–37, 2020.

J. J. Levin, “The asymptotic behavior of the solution of a Volterra equation”, Proc. Amer. Math. Soc., vol. 14, no. 4, pp. 534–541, 1963.

J. J. Levin and J. A. Nohel, “On a nonlinear delay equation”, J. Math. Anal. Appl., vol. 8, no. 8, pp. 31–44, 1964.

R. K. Miller, Nonlinear Volterra integral equations, Mathematics Lecture Note Series, Menlo Park: W. A. Benjamin, Inc., 1971.

K. N. Murty and M. D. Shaw, “Stability analysis of nonlinear Lyapunov systems associated with an nth order system of matrix differential equations”, J. Appl. Math. Stochastic Anal., vol. 15, no. 2, pp. 141–150, 2002.

Y. Raffoul, “Exponential stability and instability in finite delay nonlinear Volterra integro-differential equations”, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., vol. 20, no. 1, pp. 95–106, 2013.

Y. N. Raffoul, Advanced differential equations, London: Elsevier/Academic Press, 2023.