×

On the oscillation properties of first-order impulsive differential equations with a deviating argument. (English) Zbl 0880.34070

For first-order linear impulsive differential equations with a deviating argument oscillation Sturm-like theorems are proved. Various criteria for oscillation or nonoscillation of the solutions of these equations are found. The oscillatory properties of some concrete equations of the considered type are investigated. The following Sturm-like oscillation theorem from the paper gives a better idea of the considered equations and the obtained results. A brief explanation follows the theorem.
Theorem. Let the intervals \(J_n=(\xi_n,\eta_n)\) with \(\lim_{n\to\infty}\xi_n=+\infty\) be regular positive hemicycles of equation \((2)\) and on each of them let conditions \((a)\) and \((b)\) hold. Then all solutions of equation \((1)\) are oscillating.
The considered impulsive equation with deviating argument is of the type \[ L[x]:=x'(t)+\sum_{i=1}^m a_i(t)x[r_i(t)]=0\quad(t\neq t_j),\quad x(t_j^+)=\alpha_j(t_j^-).\tag{1} \] The solutions of (1) are compared with the solutions of \[ \tilde L[y]:=-y'(t)+\sum_{i=1}^m q_i'(t)b_i[q_i(t)]y[q_i(t)]=0\quad(t\neq t_j),\quad y(t_j^+)=\beta_j(t_j^-).\tag{2} \] The finite interval \(J=(\xi,\eta)\) is said to be a regular positive hemicycle of equation (2) if \(q_i(\eta)>\xi,\) \(q_i(\xi)<\eta\) \((i=1,2,\dots,m)\) and there exists a solution \(y(t)\) in the interval \(J\) such that \(y(\xi^+)=y(\eta^-)=0\), \(y(t)>0\) \((t\in J)\) and \(y(t)\leq 0\) \((t\in E\text{ ext}(J):=\left(\cup_{j=1}^m q_i(J)\right)\setminus J ).\) Conditions \((a)\) and \((b)\) are the following.
\((a)\) \(b_i(t)\geq0\) \((t\in q_i(J)\setminus J\), \(i=1,2,\dots m)\), \(a_i(t)\geq0\) \((t\in J\setminus q_i(J)\), \(i=1,2,\dots m)\), \(a_i(t)\geq b_i(t)\) \((t\in J\cap q_i(J)\), \(i=1,2,\dots m)\), \(1-\alpha_j\beta_j\geq0\) \((t_j\in J)\).
\((b)\) At least one of the inequalities \(1-\alpha_j\beta_j\geq0\) is strict or at least one of the remaining inequalities in condition \((a)\) is strict in some subinterval of the respective sets where they are supposed to be valid.
Reviewer: I.Ginchev (Varna)

MSC:

34K11 Oscillation theory of functional-differential equations
34C10 Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34A37 Ordinary differential equations with impulses
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] D.D.. Bainov, Yu. I. Domshlak and P.SS. Simeonov,Sturmian comparison theory for impulsive differential inequalities and equations, to appear. · Zbl 0856.34033
[2] D. D. Bainov, V. Lakshmikantham and P. S. Simeonov,Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. · Zbl 0719.34002
[3] D. D. Bainov and D. P. Mishev,Oscillation Theory for Neutral Differential Equations with Delay, Adam Hilger, Bristol, 1991. · Zbl 0747.34037
[4] D. D. Bainov and P. S. Simeonov,Impulsive Differential Equations: Periodic Solutions and Applications, Longman Scientific & Technical, Harlow, 1993. · Zbl 0815.34001
[5] D. D. Bainov and P. S. Simeonov,Systems with Impulse Effect: Stability, Theory and Applications, Ellis Horwood Series in Mathematics and its Applications, Ellis Horwood, Chichester, 1989. · Zbl 0676.34035
[6] A. I. Domoshnitsky,Extension of Sturm’s theorem to equations with a retarded argument, Differential Equations19 (1983), 1475–1482 (in Russian).
[7] Yu. I. Domshlak,Sturm-like Comparison Method in the Investigation of Solutions’ Behaviour for Differential-Operator Equations, Elm, Baku, 1986 (in Russian).
[8] Yu. I. Domshlak,Sturm-like comparison theorems for first and second order differential equations with deviations of the argument with variable sign, Ukrainian Mathematical Journal34 (1982), 158–163 (in Russian). · Zbl 0518.34065
[9] K. Gopalsamy and B. G. Zhang,On delay differential equations with impulses, Journal of Mathematical Analysis and Applications139 (1989), 110–122. · Zbl 0687.34065
[10] I. Györi and G. Ladas,Oscillation Theory of Delay Differential Equations, Clarendon Press, Oxford, 1991. · Zbl 0780.34048
[11] T. Kato and J. McLeod,The functional-differential equation x’ (t)+px(t)+qx({\(\lambda\)}t)=0, Bulletin of the American Mathematical Society77 (1971), 891–937. · Zbl 0236.34064
[12] G. S. Ladde, V. Lakshmikantham and B. G. Zhang,Oscillation Theory of Differential Equations with Deviating Argument, Marcel Dekker, New York, 1987. · Zbl 0832.34071
[13] A. D. Myshkis,Linear Differential Equations with Retarded Argument, Nauka, Moscow, 1972 (in Russian).
[14] A. Tomaras,Oscillatory behaviour of an equation arising from an industrial problem, Bulletin of the Australian Mathematical Society13 (1975), 255–260. · Zbl 0315.34091
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.