Asymptotic formulae for solutions of half-linear differential equations

journal article in Web of Science

en

We establish asymptotic formulae for regularly varying solutions of the half-linear differential equation $$(r(t)|y'|^{\alpha-1}\sgn y')'=p(t)|y|^{\alpha-1}\sgn y,$$ where $r,p$ are positive continuous functions on $[a,\infty)$ and $\alpha\in(1,\infty)$. The results can be understood in several ways: Some open problems posed in the literature are solved. Results for linear differential equations are generalized; some of the observations are new even in the linear case. A refinement on information about behavior of solutions in standard asymptotic classes is provided. A precise description of regularly varying solutions which are known to exist is given. Regular variation of all positive solutions is proved.

We establish asymptotic formulae for regularly varying solutions of the half-linear differential equation $$(r(t)|y'|^{\alpha-1}\sgn y')'=p(t)|y|^{\alpha-1}\sgn y,$$ where $r,p$ are positive continuous functions on $[a,\infty)$ and $\alpha\in(1,\infty)$. The results can be understood in several ways: Some open problems posed in the literature are solved. Results for linear differential equations are generalized; some of the observations are new even in the linear case. A refinement on information about behavior of solutions in standard asymptotic classes is provided. A precise description of regularly varying solutions which are known to exist is given. Regular variation of all positive solutions is proved.

half-linear differential equation; nonoscillatory solution; regular variation; asymptotic formula

31.01.2017

0096-3003

292

165–177

13