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Zbl 1199.34169
Došlý, O.; Ünal, M.
Conditionally oscillatory half-linear differential equations. (English)
[J] Acta Math. Hung. 120, No. 1-2, 147-163 (2008). ISSN 0236-5294; ISSN 1588-2632/e

The authors assume that a nonoscillatory solution to the half-linear equation $$(r(t)\Phi(x'))+c(t)\Phi(x)=0,\ \Phi(x)=\vert x\vert ^{p-2}x,\ p>1,$$ is known. Then they are able to construct a function $d$ such that the (perturbed) equation $$(r(t)\Phi(x'))+(c(t)+\lambda d(t))\Phi(x)=0$$ is conditionally oscillatory. They also establish an asymptotic formula for a solution of the perturbed equation in the critical case, i.e., when $\lambda$ equals the oscillation constant. These results are then used to obtain new (non)oscillation criteria, which extend previous results for perturbed half-linear Euler type and Euler-Weber type equations. The concepts of generalized Riccati equation and of principal solution, and the Schauder-Tychonoff fixed point theorem play an important role in the proofs.
[Pavel Rehák (Brno)]

MSC 2000:: *34C10 Qualitative theory of oscillations of ODE: Zeros, etc.
34A34 Nonlinear ODE and systems, general
47N20 Appl. of operator theory to differential and integral equations

Keywords: half-linear oscillation theory; conditionally oscillatory equation; oscillation and nonoscillation criteria; Riccati type equation

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