Conditionally oscillatory half-linear differential equations
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.
scientific article; zbMATH DE number 5355153