Oscillation criteria for delay equations

Authors:
M. Kon, Y. G. Sficas and I. P. Stavroulakis

Journal:
Proc. Amer. Math. Soc. **128** (2000), 2989-2997

MSC (1991):
Primary 34K15; Secondary 34C10

Published electronically:
April 28, 2000

MathSciNet review:
1694869

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

This paper is concerned with the oscillatory behavior of first-order delay differential equations of the form

(1) |

where is non-decreasing, for and . Let the numbers and be defined by

It is proved here that when and all solutions of Eq. (1) oscillate in several cases in which the condition

holds, where is the smaller root of the equation .

**1.**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**, 10.1137/0518005**2.**J. Chao, On the oscillation of linear differential equations with deviating arguments,*Math. in Practice and Theory***1**(1991), 32-40.**3.**Q. Chuanxi and G. Ladas,*Oscillations of neutral differential equations with variable coefficients*, Appl. Anal.**32**(1989), no. 3-4, 215–228. MR**1030096**, 10.1080/00036818908839850**4.**Y. Domshlak, Sturmian Comparison Method in investigation of the behavior of solutions for Differential-Operator Equations, ``Elm", Baku, USSR, 1986 (Russian).**5.**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**, 10.1080/00036819608840464**6.**Jozef Džurina,*Oscillation of second-order differential equations with mixed argument*, J. Math. Anal. Appl.**190**(1995), no. 3, 821–828. MR**1318602**, 10.1006/jmaa.1995.1114**7.**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**, 10.1090/conm/131.2/1175863**8.**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**, 10.1090/S0002-9939-1995-1242082-1**9.**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****10.**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**, 10.1007/BF01773379**11.**J. Jaros and I. P. Stavroulakis, Oscillation tests for delay equations,*Rocky Mountain J. Math.***29**(1999), 197-207. CMP**99:12****12.**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****13.**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****14.**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**, 10.1007/BF02254685**15.**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****16.**Man Kam Kwong,*Oscillation of first-order delay equations*, J. Math. Anal. Appl.**156**(1991), no. 1, 274–286. MR**1102611**, 10.1016/0022-247X(91)90396-H**17.**Gerasimos Ladas,*Sharp conditions for oscillations caused by delays*, Applicable Anal.**9**(1979), no. 2, 93–98. MR**539534**, 10.1080/00036817908839256**18.**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****19.**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****20.**G. Ladas and I. P. Stavroulakis,*On delay differential inequalities of first order*, Funkcial. Ekvac.**25**(1982), no. 1, 105–113. MR**673706****21.**Bing Tuan Li,*Oscillations of delay differential equations with variable coefficients*, J. Math. Anal. Appl.**192**(1995), no. 1, 312–321. MR**1329426**, 10.1006/jmaa.1995.1173**22.**Bingtuan Li,*Oscillation of first order delay differential equations*, Proc. Amer. Math. Soc.**124**(1996), no. 12, 3729–3737. MR**1363175**, 10.1090/S0002-9939-96-03674-X**23.**A. D. Myshkis, Linear homogeneous differential equations of first order with deviating arguments,*Uspehi Mat. Nauk***5**2 (36) (1950), 160-162 (Russian).**24.**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**, 10.4153/CMB-1998-030-3**25.**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**, 10.1017/S0004972700011758**26.**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**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
34K15,
34C10

Retrieve articles in all journals with MSC (1991): 34K15, 34C10

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

DOI:
https://doi.org/10.1090/S0002-9939-00-05530-1

Keywords:
Oscillation,
delay differential equations

Received by editor(s):
December 4, 1998

Published electronically:
April 28, 2000

Dedicated:
Dedicated to Professor V. A. Staikos on the occasion of his 60th birthday

Communicated by:
Hal L. Smith

Article copyright:
© Copyright 2000
American Mathematical Society