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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

$\left(r\left(t\right){\Phi }\left({x}^{\text{'}}\right)\right)+c\left(t\right){\Phi }\left(x\right)=0,\phantom{\rule{4pt}{0ex}}{\Phi }\left(x\right)={|x|}^{p-2}x,\phantom{\rule{4pt}{0ex}}p>1,$

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

$\left(r\left(t\right){\Phi }\left({x}^{\text{'}}\right)\right)+\left(c\left(t\right)+\lambda d\left(t\right)\right){\Phi }\left(x\right)=0$

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.

##### MSC:
 34C10 Qualitative theory of oscillations of ODE: zeros, disconjugacy and comparison theory 34A34 Nonlinear ODE and systems, general 47N20 Applications of operator theory to differential and integral equations
##### References:
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