A Survey on the Oscillation of Difference Equations with Constant Delays
В этом обзоре представлены необходимые и достаточные условия для колебания всех решений дифференциальных уравнений с запаздыванием с одной или несколькими постоянными аргументами, в терминах характеристического уравнения. Явные необходимые и достаточные условия (в терминах постоянного коэффициента и одного постоянного аргумент) также представлены в случае одного постоянного аргумент. В случае нескольких аргументов даются явные, но только достаточные условия. В этом случае результаты также распространяются на уравнения с переменными коэффициентами. Ключевые слова: Колебание, запаздывание, разностные уравнения
DOI:
https://doi.org/10.26577/JMMCS-2019-2-21Keywords:
Oscillation, Delay, Difference EquationsAbstract
In this survey, necessary and sufficient conditions for the oscillation of all solutions of delay
difference equations with one or several constant arguments, in terms of the characteristic equation,
are presented. Explicit necessary and sufficient conditions (in terms of the constant coefficient and
constant argument only) are also presented in the case of one constant argument. In the case of
several arguments explicit but sufficient conditions only are given. In this case the results are also
extended to equations with variable coefficients.
References
1. P. Agarwal, Difference Equations and Inequalities, Marcel Dekker, 1992.
2. R. P. Agarwal, M. Bohner, S. R. Grace and D. O’ Regan, Discrete Oscillation Theory, Hindawi Publishing Corporation, New York, 2005.
3. O. Arino, I. Gyori and A. Jawhari, Oscillation Criteria in Delay Equations, J. Differential Equations 53 (1984), 115-123.
4. E. Braverman and B. Karpuz, On oscillation of differential and difference equations with non-monotone delays, Appl. Math. Comput., 218 (2011), 3880−3887.
5. G. E. Chatzarakis and I. P. Stavroulakis, Oscillations of first order linear delay difference equations, Austral. J.
Math.Anal. Appl. 3 (2006), no.1, Art.14, pp. 1-11.
6. M.P. Chen and Y.S. Yu, Oscillations of delay difference equations with variable coefficients, Proc. First Intl. Conference on Difference Equations, (Edited by S.N. Elaydi et al), Gordon and Breach 1995, pp. 105-114.
7. Y. Domshlak, Discrete version of Sturmian Comparison Theorem for non-symmetric equations, Dokl. Azerb. Acad. Sci. 37 (1981), 12-15 (in Russian).
8. L. H. Erbe and B. G. Zhang, Oscillation of discrete analogues of delay equations, Differential Integral Equations 2 (1989), pp 300−309.
9. I. Gyori and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, 1991.
10. B.R. Hunt and J.A. Yorke, When all solutions of x′(t) = −Σ
qi(t)x(t − Ti(t)) = 0 Oscillate, J. Differential Equations 53 (1984), 139-145.
11. B. Karpuz, Sharp oscillation and nonoscillation tests for linear difference equations, J. Difference Equ. Appl., 23 (2017). no.12, 1229-1242.
12. G. Ladas, Recent developments in the oscillation of delay difference equations, In International Conference on Differential Equations, Stability and Control, Dekker, New York, 1990.
13. G. Ladas, Ch.G. Philos and Y.G. Sficas, Sharp conditions for the oscillation of delay difference equations, J. Appl. Math. Simulation, 2 (1989), 101-112.
14. G. Ladas and I.P. Stavroulakis, On Delay Differential Inequalities of First order, Funkcial. Ekvac.25 (1982), no.9,
637-640.
15. G. Ladas and I.P. Stavroulakis, Oscillations caused by several retarded and advanced arguments, J. Differential Equations 44 (1982), no.1 134-152.
16. V. Lakshmikantham and D. Trigiante, Theory of difference equations: Numerical methods and applications, Mathematics in Science and Engineering, 181, Academic Press, Boston, MA, 1998.
17. G.M.Moremedi and I.P. Stavroulakis, Oscillation conditions for difference equations with a monotone or non-monotone argument, Discr. Dyn. Nat. Soc., Vol. 2018, Article ID 9416319, 13 pages.
18. G.M. Moremedi, I.P. Stavroulakis abd Zh.Kh. Zhunussova, Necessary and Sufficient Conditions for Oscillations of Functional Differential Equations, J. Math. Mech. Comput. Sci., 99 (2018), No 3, 12-23.
19. J.H. Shen and I.P. Stavroulakis, Oscillation criteria for delay difference equations, Electron. J. Diff. Eqns. Vol. 2001 (2001), no.10, pp. 1-15.
20. I.P. Stavroulakis, Oscillations of delay difference equations, Comput. Math. Applic., 29 (1995), 83-88..
21. I.P. Stavroulakis, Oscillation criteria for first order delay difference equations, Mediterr. J. Math. 1 (2004), 231-240;
22. X.H. Tang and Y.B. Deng, Oscillation of difference equations with several delays, Hunan Daxue Xuebao 25 (1998),no.2, 1-3.
23. X.H. Tang and J.S. Yu, A further result on the oscillation of delay difference equations, Comput. Math. Applic., 38 (1999), 229-237.
24. X.H. Tang and R.Y. Zhang, New oscillation criteria for delay difference equations, Comput. Math. Applic.,42 (2001), no.10-11. 1319-1330.
25. J.S. Yu, B.G. Zhang and Z.C. Wang, Oscillation of delay difference equations, Appl. Anal., 53 (1994), 117-124.
2. R. P. Agarwal, M. Bohner, S. R. Grace and D. O’ Regan, Discrete Oscillation Theory, Hindawi Publishing Corporation, New York, 2005.
3. O. Arino, I. Gyori and A. Jawhari, Oscillation Criteria in Delay Equations, J. Differential Equations 53 (1984), 115-123.
4. E. Braverman and B. Karpuz, On oscillation of differential and difference equations with non-monotone delays, Appl. Math. Comput., 218 (2011), 3880−3887.
5. G. E. Chatzarakis and I. P. Stavroulakis, Oscillations of first order linear delay difference equations, Austral. J.
Math.Anal. Appl. 3 (2006), no.1, Art.14, pp. 1-11.
6. M.P. Chen and Y.S. Yu, Oscillations of delay difference equations with variable coefficients, Proc. First Intl. Conference on Difference Equations, (Edited by S.N. Elaydi et al), Gordon and Breach 1995, pp. 105-114.
7. Y. Domshlak, Discrete version of Sturmian Comparison Theorem for non-symmetric equations, Dokl. Azerb. Acad. Sci. 37 (1981), 12-15 (in Russian).
8. L. H. Erbe and B. G. Zhang, Oscillation of discrete analogues of delay equations, Differential Integral Equations 2 (1989), pp 300−309.
9. I. Gyori and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, 1991.
10. B.R. Hunt and J.A. Yorke, When all solutions of x′(t) = −Σ
qi(t)x(t − Ti(t)) = 0 Oscillate, J. Differential Equations 53 (1984), 139-145.
11. B. Karpuz, Sharp oscillation and nonoscillation tests for linear difference equations, J. Difference Equ. Appl., 23 (2017). no.12, 1229-1242.
12. G. Ladas, Recent developments in the oscillation of delay difference equations, In International Conference on Differential Equations, Stability and Control, Dekker, New York, 1990.
13. G. Ladas, Ch.G. Philos and Y.G. Sficas, Sharp conditions for the oscillation of delay difference equations, J. Appl. Math. Simulation, 2 (1989), 101-112.
14. G. Ladas and I.P. Stavroulakis, On Delay Differential Inequalities of First order, Funkcial. Ekvac.25 (1982), no.9,
637-640.
15. G. Ladas and I.P. Stavroulakis, Oscillations caused by several retarded and advanced arguments, J. Differential Equations 44 (1982), no.1 134-152.
16. V. Lakshmikantham and D. Trigiante, Theory of difference equations: Numerical methods and applications, Mathematics in Science and Engineering, 181, Academic Press, Boston, MA, 1998.
17. G.M.Moremedi and I.P. Stavroulakis, Oscillation conditions for difference equations with a monotone or non-monotone argument, Discr. Dyn. Nat. Soc., Vol. 2018, Article ID 9416319, 13 pages.
18. G.M. Moremedi, I.P. Stavroulakis abd Zh.Kh. Zhunussova, Necessary and Sufficient Conditions for Oscillations of Functional Differential Equations, J. Math. Mech. Comput. Sci., 99 (2018), No 3, 12-23.
19. J.H. Shen and I.P. Stavroulakis, Oscillation criteria for delay difference equations, Electron. J. Diff. Eqns. Vol. 2001 (2001), no.10, pp. 1-15.
20. I.P. Stavroulakis, Oscillations of delay difference equations, Comput. Math. Applic., 29 (1995), 83-88..
21. I.P. Stavroulakis, Oscillation criteria for first order delay difference equations, Mediterr. J. Math. 1 (2004), 231-240;
22. X.H. Tang and Y.B. Deng, Oscillation of difference equations with several delays, Hunan Daxue Xuebao 25 (1998),no.2, 1-3.
23. X.H. Tang and J.S. Yu, A further result on the oscillation of delay difference equations, Comput. Math. Applic., 38 (1999), 229-237.
24. X.H. Tang and R.Y. Zhang, New oscillation criteria for delay difference equations, Comput. Math. Applic.,42 (2001), no.10-11. 1319-1330.
25. J.S. Yu, B.G. Zhang and Z.C. Wang, Oscillation of delay difference equations, Appl. Anal., 53 (1994), 117-124.
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Published
2019-07-02
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Mathematics
