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A Stability theorem for the nonlinear differential equation x″+p(t)g(x)h(x′) = 0

Published online by Cambridge University Press:  09 April 2009

A. G. Kartsatos
A. G. Kartsatos
Affiliation:
Department of Mathematics Wayne State University Detroit, Michigan U.S.A.
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K. W. Chang generalizing a result of Lazer [3], proved in [4] the following THEOREM 1. Suppose that f:I → R+ = (0, + ∞), I = ([t0, + ∞), t0 0, is a non-decreasing function whose derivatives of orders ≧ 3 exist are continuous on ([t0, + ∞). Moreover, limt→+∞f(t) = +∞ and for some α, 1≧ α ≧ 2, and F =f-1/α then every solution x(t) of the equation tends to zero as t - + ∞.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 10 , Issue 1-2 , August 1969 , pp. 169 - 172
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Bihari, L., Oscillation and monotonity theorems concerning non-linear differential equations of the second order, Acta Math. Acad. Sci. Hung., 9 (1958), 83–104.CrossRefGoogle Scholar
[2]Cesari, L., Asymptotic behaviour and stability problems in ordinary differential equations (Second edition, Springer-Verlag, Berlin, 1963) sect. 5.5.CrossRefGoogle Scholar
[3]Lazer, A. C., A stability result for the differential equation y″+p(t)y=0, Michigan Math. J., 12 (1965), 193–196.CrossRefGoogle Scholar
[4]Chang, K. W., A stability result for the linear differential equation x″+f(t)x=0, J. Austral. Math. Soc., 7 (1967), 7–8.CrossRefGoogle Scholar