## Nonoscillation for second order sublinear dynamic equations on time scales.(English)Zbl 1187.34128

Authors’ abstract: Consider the Emden-Fowler sublinear dynamic equation $x^{\Delta \Delta}(t)+p(t) f(x(\sigma(t)))=0, \tag{1}$
where $$p\in C(\mathbb T, \mathbb R)$$, $$\mathbb T$$ is a time scale, $$f(x)=\sum_{i=1}^m a_ix^{\beta_i}$$, where $$a_i>0$$, $$0<\beta_i<1$$, with $$\beta_i$$ the quotient of odd positive integers, $$1\leq i\leq m$$. When $$m=1$$, and $$\mathbb T=[a,\infty)\subset \mathbb R$$, $$(0.1)$$ is the usual sublinear Emden-Fowler equation which has attracted the attention of many researchers. In this paper, we allow the coefficient function $$p(t)$$ to be negative for arbitrarily large values of $$t$$. We extend a nonoscillation result of Wong for the second order sublinear Emden-Fowler equation in the continuous case to the dynamic equation (1). As applications, we show that the sublinear difference equation
$\Delta^2 x(n)+b(-1)^nn^{-c}x^{\alpha}(n+1)=0, \quad 0<\alpha<1,$
has a nonoscillatory solution, for $$b>0$$, $$c>\alpha$$, and the sublinear $$q$$-difference equation
$x^{\Delta \Delta}(t)+b(-1)^n t^{-c}x^{\alpha}(qt)=0, \quad 0<\alpha<1,$
has a nonoscillatory solution, for $$t=q^n \in \mathbb T=q_0^{\mathbb N}$$, $$q>1$$, $$b>0$$, $$c>1+\alpha$$.

### MSC:

 34N05 Dynamic equations on time scales or measure chains 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations

### Keywords:

Emden-Flower equation; sublinear; nonoscillation
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### References:

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