Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-05-16T14:43:07.128Z Has data issue: false hasContentIssue false
  • English
  • Français

II.—The Stability of Solutions of Non-linear Difference-differential Equations*

Published online by Cambridge University Press:  14 February 2012

E. M. Wright
E. M. Wright
Affiliation:
University of Aberdeen
Get access Rights & Permissions [Opens in a new window]

Synopsis

Poincaré, Liapounoff, Perron and others have proved theorems about the order of smallness, as the independent variable tends to + ∞, of solutions of differential equations with non-linear perturbation terms. A similar theory exists for difference equations. By a simple use of transforms, we here extend the theorems, with suitable modifications, to difference-differential equations. The results are an essential step in the development of a general theory of non-linear equations of this type.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 63 , Issue 1 , 1950 , pp. 18 - 26
Copyright
Copyright © Royal Society of Edinburgh 1950

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

References to Literature

Bellman, R., 1947. “On the boundedness of solutions of non-linear differential and difference equations”, Trans. Amer. Math. Soc., LXII, 357–386.CrossRefGoogle Scholar
Bellman, R., 1949. “On the existence and boundedness of solutions of non-linear differential-difference equations”, Annals of Math., L, 347–355.CrossRefGoogle Scholar
Cesari, L., 1939. “Sulla stabalità delle soluzioni delle equazioni differenziali lineari”, Ann. Scuola norm, super. Pisa., VIII, 131–148.Google Scholar
Levinson, N., 1946. “The asymptotic behaviour of a system of linear differential equations”, Amer. Journ. Math., LXVIII, 1–6.Google Scholar
Liapounoff, A., 1907. “Problème générale de la stabilité du mouvement”, Ann. Fac. Sci. Univ. Toulouse (2), IX, 203–475.CrossRefGoogle Scholar
Perron, O., 1918. “Ein neuer Existenzbeweis fuer die Integrale eines Systems gewoehnlicher Differentialgleichungen”, Math. Ann., LXXVIII, 378–384.Google Scholar
Perron, O., 1928. “Ueber Stabilitaet und asymptotisches Verhalten der Loesungen eines Systems endlicher Differenzengleichungen”, Journ. fuer Math., CLXI, 41–64.Google Scholar
Perron, O., 1929. “Ueber Stabilitaet und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen”, Math. Zeits., XXIX, 129–160.CrossRefGoogle Scholar
Pitt, H. R., 1944. “On a class of integro-differential equations”, Proc. Camb. Phil. Soc., XL, 199–211.CrossRefGoogle Scholar
Pitt, H. R., 1947. “On a class of integro-differential equations. II”, Proc. Camb. Phil. Soc., XLIII, 153–163.CrossRefGoogle Scholar
Poincaré, H., 1892. Les méthcdes nouvelles de la mécanique céleste, Paris.Google Scholar
Weyl, H., 1946. “Comment on the preceding paper”, Amer. Journ. Math., LXVIII, 7–12.Google Scholar
Widder, D. V., 1941. The Laplace transform, Princeton, Chapter II.Google Scholar
E. M., Wright, 1946. “The non-linear difference-differential equation”, Quart. Journ. Math., XVII, 245–252.Google Scholar
E. M., Wright, 1948. “The linear difference-differential equation with asymptotically constant coefficients”, Amer. Journ. Math., LXX, 221–238.Google Scholar
E. M., Wright, 1949 a. “The linear difference-differential equation with constant coefficients”, Proc Roy. Soc. Edin., A, LXII, 387–393.Google Scholar
E. M., Wright, 1949 b. “Perturbed functional equations”, Quart. Journ. Math., xx, 155–165.Google Scholar