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

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)