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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 $$(r(t)\Phi(x'))+c(t)\Phi(x)=0,\ \Phi(x)=\vert x\vert ^{p-2}x,\ p>1,$$ 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.

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
Full Text: DOI
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