Abstract
This paper is devoted to the analysis of the oscillatory behavior of Euler type linear and half-linear differential equations. We focus on the so-called conditional oscillation, where there exists a borderline between oscillatory and non-oscillatory equations. The most complicated problem involved in the theory of conditionally oscillatory equations is to decide whether the equations from the given class are oscillatory or non-oscillatory in the threshold case. In this paper, we answer this question via a combination of the Riccati and Prüfer technique. Note that the obtained non-oscillation of the studied equations is important in solving boundary value problems on non-compact intervals and that the obtained results are new even in the linear case.
Highlights
1 Introduction In this paper, we study oscillatory properties of the half-linear differential equation r(t)Φ x + s(t)Φ(x) = 0, Φ(x) = |x|p–1 sgn x, p > 1, (1.1)
Euler type differential equations play an important role in solving non-linear BVP associated to equations with p-Laplacian
Our research is motivated by paper [3], in which an eigenvalue problem associated to the half-linear equation r(t)Φ x + s1(t) + λs2(t) Φ(x) = 0 is studied, and by [24, 48], in which the oscillation of a neutral half-linear differential equation is examined
Summary
Once we have a general conditionally oscillatory equation, we are able to obtain many oscillation or nonoscillation results by applying comparison theorems. Since the equation can be explicitly solved and its general solution is c1 x + c2 x log x, c1, c2 ∈ R, in the critical case, we can directly see that Eq (2.2) is non-oscillatory for γ = 1/4. The aim of this paper is to prove Theorem 4.3, we obtain a more general result, which implies new criteria for perturbed equations.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
