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Nonoscillation criteria for second-order nonlinear differential equations with decaying coefficients

✍ Scribed by Jitsuro Sugie

Publisher: John Wiley and Sons
Year: 2008
Tongue: English
Weight: 158 KB
Volume: 281
Category: Article
ISSN: 0025-584X
DOI: 10.1002/mana.200510702

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✦ Synopsis

Abstract

This paper is concerned with the problem of deciding conditions on the coefficient q (t) and the nonlinear term g (x) which ensure that all nontrivial solutions of the equation (|x ′|^α–1^x ′)′ + q (t)g (x) = 0, α > 0, are nonoscillatory. The nonlinear term g (x) is not imposed no assumption except for the continuity and the sign condition xg (x) > 0 if x ≠ 0. In our problem, it is important to examine the relation between the decay of q (t) and the growth of g (x). Our main result extends some nonoscillation theorem for the generalised Emden–Fowler equation. Proof is given by means of some Liapunov functions and phase‐plane analysis. A simple example is includes to show that the monotonicity of g (x) is not essential in our problem. Finally, elliptic equations with the m ‐Laplacian operator are discussed as an application to our results. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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