Perturbations of nonlinear differential equations

Fennell, R. E.; Proctor, T. G.

doi:10.1090/S0002-9947-1973-0361309-1

Perturbations of nonlinear differential equations
HTML articles powered by AMS MathViewer

by R. E. Fennell and T. G. Proctor PDF

Trans. Amer. Math. Soc. 185 (1973), 401-411 Request permission

Abstract:

Scalar and vector comparison techniques are used to study the comparative asymptotic behavior of the systems (1) $x’ = f(t,x)$ and (2) $y’ = f(t,y) + g(t,y)$. Conditions are given which allow bounds for the solutions of (2) to be obtained assuming a knowledge of the solutions of (1) and which guarantee the generalized asymptotic equivalence of (1) and (2). Examples are given.

References

Fred Brauer and James S. W. Wong, On asymptotic behavior of perturbed linear systems, J. Differential Equations 6 (1969), 142–153. MR 239213, DOI 10.1016/0022-0396(69)90122-3
Fred Brauer, Perturbations of nonlinear systems of differential equations. IV, J. Math. Anal. Appl. 37 (1972), 214–222. MR 287098, DOI 10.1016/0022-247X(72)90269-7
L. P. Burton and William M. Whyburn, Minimax solutions of ordinary differential systems, Proc. Amer. Math. Soc. 3 (1952), 794–803. MR 51385, DOI 10.1090/S0002-9939-1952-0051385-X
R. E. Fennell and T. G. Proctor, On asymptotic behavior of perturbed nonlinear systems, Proc. Amer. Math. Soc. 31 (1972), 499–504. MR 287099, DOI 10.1090/S0002-9939-1972-0287099-3
T. G. Hallam and J. W. Heidel, The asymptotic manifolds of a perturbed linear system of differential equations, Trans. Amer. Math. Soc. 149 (1970), 233–241. MR 257486, DOI 10.1090/S0002-9947-1970-0257486-0
T. G. Hallam, G. Ladas, and V. Lakshmikantham, On the asymptotic behavior of functional differential equations, SIAM J. Math. Anal. 3 (1972), 58–64. MR 315247, DOI 10.1137/0503006
Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
Athanassios G. Kartsatos, On the relationship between a nonlinear system and its nonlinear perturbation, J. Differential Equations 11 (1972), 582–591. MR 296443, DOI 10.1016/0022-0396(72)90068-X
G. Ladas, V. Lakshmikantham, and S. Leela, On the perturbability of the asymptotic manifold of a perturbed system of differential equations, Proc. Amer. Math. Soc. 27 (1971), 65–71. MR 274888, DOI 10.1090/S0002-9939-1971-0274888-3
Gangaram S. Ladde, Asymptotic equivalence and functional differential systems, An. Şti. Univ. “Al. I. Cuza" Iaşi Secţ. I a Mat. (N.S.) 18 (1972), 331–342 (English, with Romanian summary). MR 340760

Differential and integral inequalities

J. A. Marlin and R. A. Struble, Asymptotic equivalence of nonlinear systems, J. Differential Equations 6 (1969), 578–596. MR 252768, DOI 10.1016/0022-0396(69)90010-2