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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
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References:

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