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

The authors assume that a nonoscillatory solution to the half-linear equation $(r(t)\Phi(x'))+c(t)\Phi(x)=0,\;\Phi(x)=| x| ^{p-2}x,\;p>1,$ 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.

### MSC:

 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems 47N20 Applications of operator theory to differential and integral equations
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### References:

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