A linearization method in oscillation theory of half-linear second-order differential equations

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ng inequalities for a certain function appearing in the half-linear version of Picone’s identity, we show that oscillatory properties of the half-linear second-order differential equation (r(t)Φ(x )) + c(t)Φ(x) = 0, Φ(x) = |x| p−2 x, p > 1, can be investigated via oscillatory properties of a certain associated second-order linear differential equation. This linear equation plays the role of a Sturmian majorant, in a certain sense, if p ≥ 2, and the role of a minorant if p ∈ (1,2]. 1. Introduction In this paper, we deal with oscillatory properties of the half-linear second-order differential equation 



r(t)Φ(x ) + c(t)Φ(x) = 0,

Φ(x) = |x| p−2 x, p > 1.

(1.1)

In the recent years, considerable similarity between oscillatory properties of (1.1) and its special case, the linear Sturm-Liouville equation 



r(t)x + c(t)x = 0,

(1.2)

has been found, see, for example, [1] and the references given therein. On the other hand, some natural differences were pointed out, mostly caused by the fact that the solution space of (1.1) has only one half of the properties which characterize linearity, namely homogeneity, but generally not additivity. This fact is also a motivation for the terminology half-linear equation. One of the important differences between (1.1) and (1.2) is missing transformation theory for half-linear equations. More precisely, the transformation x = h(t)y, where h is a differentiable function such that rh is also differentiable, applied to (1.2), gives the following (linear) identity: 









h(t) r(t)x + c(t)x = R(t)y  + C(t)y, Copyright © 2005 Hindawi Publishing Corporation Journal of Inequalities and Applications 2005:5 (2005) 535–545 DOI: 10.1155/JIA.2005.535

(1.3)

536

Linearization in half-linear oscillation theory

where 





C(t) = h(t) r(t)h (t) + c(t)h(t) .

R(t) = h2 (t)r(t),

(1.4)

In particular, if x is a solution of (1.2), y is a solution of the equation 



R(t)y  + C(t)y = 0.

(1.5)

Identity (1.3) can be verified by a direct differentiation and one can easily find that the important role is played by the linearity of the differential operator (r(t)x ) . Since the operator (r(t)Φ(x )) is no longer linear, the transformation (1.3) has no immediate halflinear extension. The transformation formula (1.3) for linear equation (1.2) is used in many oscillation criteria for this equation, with the following idea. Equation (1.2) is transformed into an “easier” equation (1.5), oscillatory properties of the easier equation (1.5) are studied and then the obtained results are “translated” back to original equation (1.2). The missing half-linear extension of (1.3) excludes the possibility to study (1.1) using this method. In this paper we try, in a certain sense, to eliminate this disadvantage of the qualitative theory of (1.1). We elaborate a method which enables to compare oscillatory properties of (1.1) with oscillatory properties of a certain associated linear equation of the form (1.2) and then to use the results of the deeply developed linear oscillation theory in t

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