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

Guseinov, G. Sh.; Kaymakçalan, B.

J. Comput. Appl. Math. 141, No. 1-2, 187-196 (2002).

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

Reviewer: Roman Hilscher (East Lansing)

MSC:

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

Keywords:

time scale measure chain; disconjugacy; generalized zero; mean value theorem

References:

[1]	Agarwal, R.P., Difference equations and inequalities, (1992), Marcel Dekker New York · Zbl 0784.33008
[2]	Agarwal, R.P.; Bohner, M., Basic calculus on time scales and some of its applications, Results math., 35, 3-22, (1999) · Zbl 0927.39003
[3]	Ahlbrandt, C.D.; Peterson, A.C., Discrete Hamiltonian systems: difference equation, continued fractions, and Riccati equations, (1996), Kluwer Academic Publishers Boston · Zbl 0860.39001
[4]	Atıcı, F.M.; Guseinov, G.Sh., Criteria for the stability of second order difference equations with periodic coefficients, Commun. appl. anal., 3, 503-515, (1999) · Zbl 0933.39013
[5]	Atıcı, F.M.; Guseinov, G.Sh.; Kaymakçalan, B., On Lyapunov inequality in stability theory for Hill’s equation on time scales, J. inequal. appl., 5, 603-620, (2000) · Zbl 0971.39005
[6]	B. Aulbach, S. Hilger, Linear dynamic processes with inhomogeneous time scale; in: Nonlinear Dynamics and Quantum Dynamical Systems, Gaussig, 1990, Mathematical Research, Vol. 59, Akademie Verlag, Berlin, 1990, pp. 9-20. · Zbl 0719.34088
[7]	Bohner, M., On disconjugacy for sturm – liouville difference equations, J. difference equations appl., 2, 227-237, (1996) · Zbl 0856.39019
[8]	M. Bohner, S. Clark, J. Ridenhour, Lyapunov inequalities for time scales, J. Inequal. Appl., to appear. · Zbl 1088.34503
[9]	Bohner, M.; Eloe, P.W., Higher order dynamic equations on measure chains, J. math. anal. appl., 246, 639-656, (2000) · Zbl 0957.34033
[10]	Borg, G., On a liapounoff criterion of stability, Amer. J. math., 71, 67-70, (1949) · Zbl 0031.30604
[11]	Brown, R.C.; Hinton, D.B., Opial’s inequality and oscillation of 2nd order equations, Proc. amer. math. soc., 125, 1123-1129, (1997) · Zbl 0866.34026
[12]	Cheng, S.S., Lyapunov inequalities for differential and difference equations, Fasc. math., 23, 25-41, (1991) · Zbl 0753.34017
[13]	Clark, S.; Hinton, D.B., Discrete Lyapunov inequalities, Dynam. systems appl., 8, 369-380, (1999) · Zbl 0940.39013
[14]	Clark, S.; Hinton, D.B., A Liapunov inequality for linear Hamiltonian systems, Math. inequal. appl., 1, 201-209, (1998) · Zbl 0909.34033
[15]	Coppel, W., Disconjugacy, Lecture notes in mathematics, Vol. 220, (1971), Springer Berlin
[16]	Hartman, P., Difference equations: disconjugacy, principal solutions, Green’s functions, complete monotonicity, Trans. amer. math. soc., 246, 1-30, (1978) · Zbl 0409.39001
[17]	Hilger, S., Analysis on measure chains; a unified approach to continuous and discrete calculus, Results math., 18, 18-56, (1990) · Zbl 0722.39001
[18]	M.G. Krein, On certain problems on the maximum and minimum of characteristic values and the Lyapunov zones of stability, Prikl. Mat. Meh. 15 (1951) 323-348, Amer. Math. Soc. Translations, Ser. 2, Vol. 1, Providence, RI, 1955, pp. 163-187.
[19]	Lakshmikantham, V.; Sivasundaram, S.; Kaymakçalan, B., Dynamic systems on measure chains, (1996), Kluwer Academic Publishers Dordrecht · Zbl 0869.34039
[20]	M.A. Liapounoff, Problème général de la stabilité du mouvement, Ann. Fac. Sci. Univ. Toulouse, Ser. 2, Vol. 9, 1907, 203-475, reprinted in the Ann. Math. Stud. No. 17, Princeton, 1949.
[21]	Peterson, A.; Ridenhour, J., A disconjugacy criterion of W.T. reid for difference equations, Proc. amer. math. soc., 114, 459-468, (1992) · Zbl 0744.34037
[22]	Reid, W.T., Oscillation criteria for linear differential systems with complex coefficients, Pacific J. math., 6, 733-751, (1956) · Zbl 0071.30203
[23]	Reid, W.T., A matrix Liapunov inequality, J. math. anal. appl., 32, 424-434, (1970) · Zbl 0208.11303
[24]	S.G. Topal, Rolle’s and generalized mean value theorems on time scales, submitted for publication.
[25]	Wintner, A., On the nonexistence of conjugate points, Amer. J. math., 71, 368-380, (1951) · Zbl 0043.08703