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 \({\mathbb{T}}\).


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


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