## On a disconjugacy criterion for second-order dynamic equations on time scales.(English)Zbl 1014.34023

A time scale is any nonempty closed subset of $${\mathbb{R}}$$. A theory of dynamic equations on time scales $${\mathbb{T}}$$ was introduced in order to unify the differential ($${\mathbb{T}}={\mathbb{R}}$$) and difference ($${\mathbb{T}}={\mathbb{Z}}$$) equation theories, and to explain the discrepancies between them. This paper contains a sufficient condition for the disconjugacy of the second-order linear dynamic equation $[p(t)y^\Delta(t)]^\Delta+q(t)y(\sigma(t)) =0\tag{*}$ on the time scales interval $$[a,\sigma^2(b)]$$. The disconjugacy of (*) is defined as the nonexistence of a nontrivial solution $$y$$ to (*) with at least two generalized zeros in $$[a,\sigma^2(b)]$$, whereas a point $$t$$ is a generalized zero of $$y$$ if either $$y(t_0)=0$$ or $$y(\rho(t_0))y(t_0)<0$$ (with the exception at $$t=a$$, where the latter inequality is not needed). The following theorem is the main result of this paper:
Let $$q_+(t):=\max\{q(t),0\}$$. Suppose either $\left[ \int_a^{\sigma^2(b)} \frac{\Delta t}{p(t)} \right] \int_a^{\sigma(b)} q_+(t)\Delta t<0$ or the equality holds in the above condition, in which case we suppose that there is an interval $$[\alpha,\beta]\subset[a,b]$$ containing at least three points such that $$q_+(t)>0$$ for all $$t\in[\alpha,\beta]$$. Then equation (*) is disconjugate on $$[a,\sigma^2(b)]$$.
The proof is based on the mean value theorem on time scales (which is also proven in this paper) and on a relation between the monotonicity and the $$\Delta$$-derivative of a function on $${\mathbb{T}}$$.

### MSC:

 34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations 39A11 Stability of difference equations (MSC2000) 93A10 General systems
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