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Conditionally oscillatory half-linear differential equations. (English) Zbl 1199.34169

The authors assume that a nonoscillatory solution to the half-linear equation

(r(t)Φ(x ' ))+c(t)Φ(x)=0,Φ(x)=|x| p-2 x,p>1,

is known. Then they are able to construct a function d such that the (perturbed) equation

(r(t)Φ(x ' ))+(c(t)+λd(t))Φ(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 λ 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.

MSC:
34C10Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory
34A34Nonlinear ODE and systems, general
47N20Applications of operator theory to differential and integral equations
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