Abstract
Differential equations attract considerable attention in many applications. In particular, it was found out that half-linear differential equations behave in many aspects very similar to that in linear case. The aim of this contribution is to investigate oscillatory properties of the second-order half-linear differential equation and to give oscillation and nonoscillation criteria for this type of equation. It is also considered the linear Sturm- Liouville equation which is the special case of the half-linear equation. Main ideas used in the proof of these criteria are given and Hille-Nehari type oscillation and nonoscillation criteria for the Sturm-Liouville equation are formulated. In the next part, Hille-Nehari type criteria for the half-linear differential equation are presented. Methods used in this investigation are based on the Riccati technique and the quadratic functional, that are very useful instruments in proving oscillation/nonoscillation both for linear and half-linear equation. Conclude that there are given further criteria which guarantee either oscillation or nonoscillation of linear and half-linear equation, respectively. These criteria can be used in the next research in improving some conditions given in theorems of this paper.
Highlights
In this paper we investigate oscillatory properties of the half-linear second-order differential equation of the form r(t)Φ(x ) + c(t)Φ(x) = 0, (1)
Oscillation theory of (1) attracted considerable attention in the past years and it was shown that solutions of (1) behave in many aspects like those of the linear SturmLiouville differential equation (r(t)x ) + c(t)x = 0, (2)
Note that the term half-linear equations is motivated by the fact that the solution space of (1) has just one half of the properties which characterize linearity, namely homogeneity, but not additivity
Summary
In this paper we investigate oscillatory properties of the half-linear second-order differential equation of the form r(t)Φ(x ) + c(t)Φ(x) = 0,. Oscillation theory of (1) attracted considerable attention in the past years and it was shown that solutions of (1) behave in many aspects like those of the linear SturmLiouville differential equation (r(t)x ) + c(t)x = 0,. The aim of this paper is to present some results of the investigation oscillatory properties of equation (1) in comparison with that one of (2).
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