Oscillation criteria for delay equations
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- by M. Kon, Y. G. Sficas and I. P. Stavroulakis PDF
- Proc. Amer. Math. Soc. 128 (2000), 2989-2997 Request permission
Abstract:
This paper is concerned with the oscillatory behavior of first-order delay differential equations of the form \begin{eqnarray} x’ (t)+p(t)x({\tau }(t))=0, \quad t\geq t_{0}, \end{eqnarray} where $p, {\tau } \in C([t_{0}, \infty ), \mathbb {R}^+), \mathbb {R}^+=[0, \infty ), \tau (t)$ is non-decreasing, $\tau (t) <t$ for $t \geq t_{0}$ and $\lim _{t{\rightarrow }{\infty }} \tau (t) = \infty$. Let the numbers $k$ and $L$ be defined by \[ k=\liminf _{t{\rightarrow }{\infty }} \int _{\tau (t)}^{t}p(s)ds \quad \mbox {and} \quad L=\limsup _{t{\rightarrow }{\infty }} \int _{\tau (t)}^{t}p(s)ds. \] It is proved here that when $L<1$ and $0<k \leq \frac {1}{e}$ all solutions of Eq. (1) oscillate in several cases in which the condition \[ L>2k+\frac {2}{{\lambda }_{1}}-1 \] holds, where ${\lambda _1}$ is the smaller root of the equation $\lambda =e^{k \lambda }$.References
- O. Arino, G. Ladas, and Y. G. Sficas, On oscillations of some retarded differential equations, SIAM J. Math. Anal. 18 (1987), no. 1, 64–73. MR 871821, DOI 10.1137/0518005
- J. Chao, On the oscillation of linear differential equations with deviating arguments, Math. in Practice and Theory 1 (1991), 32-40.
- Q. Chuanxi and G. Ladas, Oscillations of neutral differential equations with variable coefficients, Appl. Anal. 32 (1989), no. 3-4, 215–228. MR 1030096, DOI 10.1080/00036818908839850
- Y. Domshlak, Sturmian Comparison Method in investigation of the behavior of solutions for Differential-Operator Equations, “Elm", Baku, USSR, 1986 (Russian).
- Y. Domshlak and I. P. Stavroulakis, Oscillations of first-order delay differential equations in a critical state, Appl. Anal. 61 (1996), no. 3-4, 359–371. MR 1618248, DOI 10.1080/00036819608840464
- Jozef Džurina, Oscillation of second-order differential equations with mixed argument, J. Math. Anal. Appl. 190 (1995), no. 3, 821–828. MR 1318602, DOI 10.1006/jmaa.1995.1114
- Hyo Chul Myung and Arthur A. Sagle, Quadratic differential equations and algebras, Proceedings of the International Conference on Algebra, Part 2 (Novosibirsk, 1989) Contemp. Math., vol. 131, Amer. Math. Soc., Providence, RI, 1992, pp. 659–672. MR 1175863, DOI 10.1090/conm/131.2/1175863
- Á. Elbert and I. P. Stavroulakis, Oscillation and nonoscillation criteria for delay differential equations, Proc. Amer. Math. Soc. 123 (1995), no. 5, 1503–1510. MR 1242082, DOI 10.1090/S0002-9939-1995-1242082-1
- L. H. Erbe and B. G. Zhang, Oscillation for first order linear differential equations with deviating arguments, Differential Integral Equations 1 (1988), no. 3, 305–314. MR 929918
- Nobuyoshi Fukagai and Takaŝi Kusano, Oscillation theory of first order functional-differential equations with deviating arguments, Ann. Mat. Pura Appl. (4) 136 (1984), 95–117. MR 765918, DOI 10.1007/BF01773379
- J. Jaroš and I. P. Stavroulakis, Oscillation tests for delay equations, Rocky Mountain J. Math. 29 (1999), 197-207.
- R. G. Koplatadze, Zeros of solutions of first-order differential equations with retarded argument, Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 14 (1983), 128–135 (Russian, with English and Georgian summaries). MR 741460
- R. G. Koplatadze and T. A. Chanturiya, Oscillating and monotone solutions of first-order differential equations with deviating argument, Differentsial′nye Uravneniya 18 (1982), no. 8, 1463–1465, 1472 (Russian). MR 671174
- R. Koplatadze and G. Kvinikadze, On the oscillation of solutions of first-order delay differential inequalities and equations, Georgian Math. J. 1 (1994), no. 6, 675–685. MR 1296574, DOI 10.1007/BF02254685
- Erwin Kozakiewicz, Conditions for the absence of positive solutions of a first order differential inequality with a single delay, Arch. Math. (Brno) 31 (1995), no. 4, 291–297. MR 1390588
- Man Kam Kwong, Oscillation of first-order delay equations, J. Math. Anal. Appl. 156 (1991), no. 1, 274–286. MR 1102611, DOI 10.1016/0022-247X(91)90396-H
- Gerasimos Ladas, Sharp conditions for oscillations caused by delays, Applicable Anal. 9 (1979), no. 2, 93–98. MR 539534, DOI 10.1080/00036817908839256
- G. Ladas, V. Lakshmikantham, and J. S. Papadakis, Oscillations of higher-order retarded differential equations generated by the retarded argument, Delay and functional differential equations and their applications (Proc. Conf., Park City, Utah, 1972) Academic Press, New York, 1972, pp. 219–231. MR 0387776
- G. Ladas, Y. G. Sficas, and I. P. Stavroulakis, Functional-differential inequalities and equations with oscillating coefficients, Trends in theory and practice of nonlinear differential equations (Arlington, Tex., 1982) Lecture Notes in Pure and Appl. Math., vol. 90, Dekker, New York, 1984, pp. 277–284. MR 741515
- G. Ladas and I. P. Stavroulakis, On delay differential inequalities of first order, Funkcial. Ekvac. 25 (1982), no. 1, 105–113. MR 673706
- Bing Tuan Li, Oscillations of delay differential equations with variable coefficients, J. Math. Anal. Appl. 192 (1995), no. 1, 312–321. MR 1329426, DOI 10.1006/jmaa.1995.1173
- Bingtuan Li, Oscillation of first order delay differential equations, Proc. Amer. Math. Soc. 124 (1996), no. 12, 3729–3737. MR 1363175, DOI 10.1090/S0002-9939-96-03674-X
- A. D. Myshkis, Linear homogeneous differential equations of first order with deviating arguments, Uspehi Mat. Nauk 5 $N^{0}$ 2 (36) (1950), 160-162 (Russian).
- Ch. G. Philos and Y. G. Sficas, An oscillation criterion for first order linear delay differential equations, Canad. Math. Bull. 41 (1998), no. 2, 207–213. MR 1624266, DOI 10.4153/CMB-1998-030-3
- Jian She Yu and Zhicheng Wang, Some further results on oscillation of neutral differential equations, Bull. Austral. Math. Soc. 46 (1992), no. 1, 149–157. MR 1170449, DOI 10.1017/S0004972700011758
- J. S. Yu, Z. C. Wang, B. G. Zhang, and X. Z. Qian, Oscillations of differential equations with deviating arguments, Panamer. Math. J. 2 (1992), no. 2, 59–78. MR 1160129
Additional Information
- M. Kon
- Affiliation: Department of Mathematics, Boston University, Boston, Massachusetts 02215
- Email: mkon@math.bu.edu
- Y. G. Sficas
- Affiliation: Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
- I. P. Stavroulakis
- Affiliation: Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
- Email: ipstav@cc.uoi.gr
- Received by editor(s): December 4, 1998
- Published electronically: April 28, 2000
- Communicated by: Hal L. Smith
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 128 (2000), 2989-2997
- MSC (1991): Primary 34K15; Secondary 34C10
- DOI: https://doi.org/10.1090/S0002-9939-00-05530-1
- MathSciNet review: 1694869
Dedicated: Dedicated to Professor V. A. Staikos on the occasion of his 60th birthday