Kiguradze-type Oscillation Theorems for Second Order Superlinear Dynamic Equations on Time Scales

Baoguo, Jia ; Erbe, Lynn ; Peterson, Allan

Canadian mathematical bulletin, Tome 54 (2011) no. 4, pp. 580-592

Résumé

Consider the second order superlinear dynamic equation $$(*)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{x}^{\Delta \Delta }}(t)+p(t)f(x(\sigma (t)))=0$$ where $p\,\in \,C(\mathbb{T},\,\mathbb{R})$ , $\mathbb{T}$ is a time scale, $f\,:\,\mathbb{R}\,\to \,\mathbb{R}$ is continuously differentiable and satisfies ${{f}^{'}}(x)>0$ , and $x\,f\,(x)\,>\,0$ for $x\,\ne \,0$ . Furthermore, $f(x)$ also satisfies a superlinear condition, which includes the nonlinear function $f(x)\,=\,{{x}^{\alpha }}$ with $\alpha \,>\,1$ , commonly known as the Emden–Fowler case. Here the coefficient function $p(t)$ is allowed to be negative for arbitrarily large values of $t$ . In addition to extending the result of Kiguradze for $\left( * \right)$ in the real case $\mathbb{T}\,=\,\mathbb{R}$ , we obtain analogues in the difference equation and $q$ -difference equation cases.

Détail

DOI : 10.4153/CMB-2011-034-4

Mots-clés : 34K11, 39A10, 39A99, Oscillation, Emden–Fowler equation, superlinear