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.


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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