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Perturbations of the half-linear Euler — Weber type differential equation. (English) Zbl 1107.34030

Summary: We investigate oscillatory properties of the half-linear second-order differential equation

r (t) Φ (x ' ) ' +c(t)Φ(x)=0,Φ(x)=|x| p-2 x,p>1,

viewed as a perturbation of another half-linear differential equation of the same form

r (t) Φ (x ' )]+c ˜(t)Φ(x)=0·(*)

The obtained oscillation and nonoscillation criteria are formulated in terms of the integral [c(t)-c ˜(t)]×h p (t)dt, where h is a function which is close to the principal solution of (*), in a certain sense. A typical model of (*) in applications is the half-linear Euler-Weber differential equation with the critical coefficients

Φ ( x ' ) ' +γp t p +μ p t p log 2 tφ(x)=0,γ p :=p-1 p p ,μ p :=1 2p-1 p p-1 ,

and we establish oscillation and nonoscillation criteria for perturbations of this equation. Some open problems and perspectives of the further research along this line are formulated, too.


MSC:
34C11Qualitative theory of solutions of ODE: growth, boundedness