Oscillation criteria for delay equations
Authors:
M. Kon, Y. G. Sficas and I. P. Stavroulakis
Journal:
Proc. Amer. Math. Soc. 128 (2000), 29892997
MSC (1991):
Primary 34K15; Secondary 34C10
Published electronically:
April 28, 2000
MathSciNet review:
1694869
Fulltext PDF Free Access
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Abstract: This paper is concerned with the oscillatory behavior of firstorder delay differential equations of the form    (1)  where is nondecreasing, 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 .
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 O. Arino, G. Ladas and Y. G. Sficas, On oscillations of some retarded differential equations, SIAM J. Math. Anal. 18 (1987), 6473. MR 88c:34088
 2.
 J. Chao, On the oscillation of linear differential equations with deviating arguments, Math. in Practice and Theory 1 (1991), 3240.
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 Q. Chuanxi and G. Ladas, Oscillations of Neutral Differential Equations with Variable Coefficients, Applicable Anal. 32 (1989), 215228. MR 90m:34142
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 Y. Domshlak, Sturmian Comparison Method in investigation of the behavior of solutions for DifferentialOperator Equations, ``Elm", Baku, USSR, 1986 (Russian).
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 Y. Domshlak and I. P. Stavroulakis, Oscillations of firstorder delay differential equations in a critical state, University of Ioannina, T. R. 257 November 1995, Applicable Anal. 61 (1996), 359371. MR 99a:34184
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 J. Dzurina, Oscillation of secondorder differential equations with mixed argument, J. Math. Anal. Appl. 190 (1995), 821828. MR 95k:34100
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 Á. Elbert and I. P. Stavroulakis, Oscillations of first order differential equations with deviating arguments, University of Ioannina, T. R. 172 1990, Recent trends in differential equations 163178, World Sci. Ser. Appl. Anal., 1, World Sci. Publishing Co. (1992). MR 93g:34019
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 9.
 L. H. Erbe and B. G. Zhang, Oscillation for first order linear differential equations with deviating arguments, Differential Integral Equations 1 (1988), 305314. MR 89e:34116
 10.
 N. Fukagai and T. Kusano, Oscillation theory of first order functional differential equations with deviating arguments, Ann. Mat. Pura Appl. 136 (1984), 95117. MR 86b:34135
 11.
 J. Jaros and I. P. Stavroulakis, Oscillation tests for delay equations, Rocky Mountain J. Math. 29 (1999), 197207. CMP 99:12
 12.
 R. G. Koplatadze, On zeros of solutions of first order delay differential equations, Proceedings of I. N. Vekua Institute of Applied Mathematics 14 (1983), 128135 (Russian). MR 85g:34062
 13.
 R. G. Koplatadze and T. A. Chanturija, On oscillatory and monotonic solutions of first order differential equations with deviating arguments, Differential'nye Uravnenija 18 (1982), 14631465 (Russian). MR 83k:34069
 14.
 R. G. Koplatadze and G. Kvinikadze, On the oscillation of solutions of first order delay differential inequalities and equations, Georgian Math. J. 1 (1994), 675685. MR 95j:34103
 15.
 E. Kozakiewicz, Conditions for the absence of positive solutions of a first order differential inequality with a single delay, Archivum Mathematicum, 31 (1995), 291297. MR 97c:34133
 16.
 M. K. Kwong, Oscillation of first order delay equations, J. Math. Anal. Appl. 156 (1991), 274286. MR 92b:34082
 17.
 G. Ladas, Sharp conditions for oscillations caused by delays, Applicable Anal. 9 (1979), 9398. MR 80h:34094
 18.
 G. Ladas, V. Lakshmikantham and L. S. Papadakis, Oscillations of higherorder retarded differential equations generated by the retarded arguments, Delay and Functional Differential Equations and their Applications, Academic Press, New York, 1972 219231. MR 52:8615
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 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, Tx. 1982), 277284, Lecture Notes in Pure and Appl. Math., 90 Marcel Dekker, New York, 1984. MR 85f:34128
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 G. Ladas and I. P. Stavroulakis, On delay differential inequalities of first order, Funkcial. Ekvac. 25 (1982), 105113. MR 83k:34070
 21.
 B. Li, Oscillations of delay differential equations with variable coefficients, J. Math. Anal. Appl. 192 (1995), 312321. MR 96c:34152
 22.
 B. Li, Oscillation of first order delay differential equations, Proc. Amer. Math. Soc. 124 (1996), 37293737. MR 97b:34078
 23.
 A. D. Myshkis, Linear homogeneous differential equations of first order with deviating arguments, Uspehi Mat. Nauk 5 2 (36) (1950), 160162 (Russian).
 24.
 Ch. G. Philos and Y. G. Sficas, An oscillation criterion for first order linear delay differential equations, Canad. Math. Bull. 41 (1998), 207213. MR 99b:34119
 25.
 J. S. Yu and Z. C. Wang, Some further results on oscillation of neutral differential equations, Bull. Austral. Math. Soc. 46 (1992), 149157. MR 93e:34099
 26.
 J. S. Yu, Z. C. Wang, B. G. Zhang and X. Z. Qian, Oscillations of differential equations with deviating arguments, PanAmerican Math. J. 2 (1992), 5978. MR 93e:34100
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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:
http://dx.doi.org/10.1090/S0002993900055301
PII:
S 00029939(00)055301
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
