Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-20T12:23:39.942Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Stability and Boundedness of Differential Systems†

Published online by Cambridge University Press:  24 October 2008

V. Lakshmikantham
V. Lakshmikantham
Affiliation:
Osmania University, Hyderabad and University of California, Los Angeles
Get access Rights & Permissions [Opens in a new window]

Extract

Consider the differential systems

where A(t), g(t, y) and g(t, y) are operators acting in the real Banach space E, A(t) is an unbounded, closed, linear operator for each t in 0 ≤ t < ∞ and x0, y0 belong to the domain of definition of the operator A (t0). Let ‖x‖ denote the norm of an element x ε: E and R(λ, t) the resolvent of A(t). Here and in the following the prime denotes the right-hand derivative.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 58 , Issue 3 , July 1962 , pp. 492 - 496
Copyright
Copyright © Cambridge Philosophical Society 1962

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

(1)Cesari, L., Asymptotic behaviour and stability problems in ordinary differential equations. (Springer; Berlin-Göttingen-Heidelberg, 1959.)Google Scholar
(2)Kamke, E., Differentialgleichungen reeller Funktionen. (Akademische Verlagsgesellschaft; Leipzig, 1930.)Google Scholar
(3)Kato, T., Integration of the equation of evolution in a Banach space. J. Math. Soc. Japan, 5 (1953), 208–304.Google Scholar
(4)Kato, T., On linear differential equations in Banach spaces. Comm. Pure Appl. Math. 9 (1956), 479–486.CrossRefGoogle Scholar
(5)Krasnoselskii, M. A., Krein, S. G. and Soboleveskii, P. E., On differential equations with unbounded operators in Banach spaces. Dokl. Akad. Nauk SSSR (N.S.), 111 (1956), 19–22 (Russian).Google Scholar
(6)Lakshmikanth, V. On the boundedness of solutions of non-linear differential equations. Proc. American Math. Soc. 8 (1957), 1044–1048.Google Scholar
(7)Yoshizawa, T., Liapunov's function and boundedness of solutions. Funkcial. Ekvac. 2 (1959), 95–142.Google Scholar
(8)Yoshizawa, T., Stability and boundedness of systems. Arch. Rational Mech. Anal. 6 (1960), 409–421.CrossRefGoogle Scholar