# Qualitative Behavior of the Solutions to Delay and Difference Euations

### Abstract

It is noteworthy to observe that a first-order linear ordinary differential equation without delaydoes not possess oscillatory solutions. Therefore the investigation of oscillatory solutions is ofinterest for equations with delays or for the discrete analogue difference equations. Furthermore, themathematical modelling of several real-world problems leads to differential equations that dependon the past history rather than only the current state. In this article conditions are presentedsuch that all solutions of delay and difference equations are oscillatory while all solutions of thecorresponding ordinary differential equations without delay are, for example, decreasing and tendto zero. Equations with constant and variable arguments are investigated. Several examples ofdelay and difference equations with applications to many sectors of life are presented.### References

[1] G. E. Chatzarakis, R. Koplatadze and I. P. Stavroulakis, Oscillation criteria of first order linear difference equations with

delay argument, Nonlinear Anal., 68 (2008), 994–1005.

[2] G. E. Chatzarakis, R. Koplatadze, and I. P. Stavroulakis, Optimal oscillation criteria for first order difference equations

with delay argument, Pacific J. Math., 235 (2008), 15–33.

[3] Y. Domshlak, Discrete version of Sturmian Comparison Theorem for non-symmetric equations, Doklady Azerb. Acad. Sci.

37 (1981), 12–15.

[4] A. Elbert and I.P. Stavroulakis, Oscillations of first order differential equations with deviating arguments, Univ.of ioannina

TR No.172, 1990. Recent trends in differential equations 163–178, World Sci. Ser. Appl. Anal.,1, World Sci. Publishing

Co.(1992).

[5] L.H. Erbe, Qingkai Kong and B.G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New

York, 1995.

[6] L.H. Erbe and B.G. Zhang, Oscillation of first order linear differential equations with deviating arguments, Differential

Integral Equations 1 (1988), 305-314.

[7] L.H. Erbe and B.G. Zhang, Oscillation of discrete analogues of delay equations, Differential Integral Equations 2 (1989),

300-309.

[8] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic

Publishers, 1992.

[9] I. Gyori and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford,

1991.

[10] J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1997.

[11] J. Jaroˇs and I.P. Stavroulakis, Oscillation tests for delay equations, Rocky Mountain J. Math., 29 (1999), 139-145.

[12] C. Jian, Oscillation of linear differential equations with deviating argument, Math. in Practice and Theory, 1 (1991),

32-41 (in Chinese).

[13] B. Karpuz, Sharp oscillation and nonoscillation tests for linear difference equations, J. Difference Equ. Appl., 23 (2017).

no.12, 1229-1242.

[14] M. Kon, Y.G. Sficas and I.P. Stavroulakis, Oscillation criteria for delay equations, Proc. Amer. Math. Soc. 128 (2000),

2989-2997.

[15] R.G. Koplatadze and T.A. Chanturija, On the oscillatory and monotonic solutions of first order differential equations

with deviating arguments, Differentsial’nye Uravneniya, 18 (1982), 1463-1465.

[16] R.G. Koplatadze and G. Kvinikadze, On the oscillation of solutions of first order delay differential inequalities and

equations, Georgian Math. J. 1 (1994), 675-685.

[17] M.K. Kwong, Oscillation of first order delay equations, J. Math. Anal. Appl., 156 (1991), 374-286.

[18] G. Ladas, Recent developments in the oscillation of delay difference equations, International Conference on Differential

Equations, Stability and Control, Marcel Dekker, New York, 1990.

[19] G. Ladas, V. Laskhmikantham and J.S. Papadakis, Oscillations of higher-order retarded differential equations generated

by retarded arguments, Delay and Functional Differential Equations and Their Applications, Academic Press, New York,

1972, 219-231.

[20] G. Ladas, Ch.G. Philos and Y.G. Sficas, Sharp conditions for the oscillation of delay difference equations, J.Appl.Math.

Simul.,2(1989), 101-112.

[21] G.S. Ladde, V. Lakshmikantham and B.G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments,

Marcel Dekker, New York, 1987.

[22] A.D. Myshkis, Linear homogeneous differential equations of first order with deviating arguments, Uspekhi Mat. Nauk, 5

(1950), 160-162.

[23] Ch.G. Philos and Y.G. Sficas, An oscillation criterion for first-order linear delay differential equations, Canad. Math. Bull.

41 (1998), 207-213.

[24] Y.G. Sficas and I.P. Stavroulakis, Oscillation criteria for first-order delay equations, Bull. London Math. Soc., 35 (2003),

239-246.

[25] I.P. Stavroulakis, Oscillation Criteria for First Order Delay Difference Equations, Mediterr. J. Math. 1 (2004), 231-240.

[26] C. Sturm, Sur les equations differentielles lineaires du second ordre, J. Math. Pures Appl. 1 (1836), 106-186

[27] X.H. Tang and J.S. Yu, A further result on the oscillation of delay difference equations, Comput. Math. Applic., 38 (1999),

229-237.

[28] J.S. Yu and Z.C. Wang, Some further results on oscillation of neutral differential equations, Bull. Austral. Math. Soc., 46

(1992), 149-157.

[29] J.S. Yu, Z.C. Wang, B.G. Zhang and X.Z. Qian, Oscillations of differential equations with deviting arguments,

Panam.Math.J., 2 (1992), 59-78.

[30] J.S. Yu, B.G. Zhang and Z.C. Wang, Oscillation of delay difference equations, Applicable Anal., 53 (1994), 117-124.

[31] Y. Zhou and Y.H.Yu, On the oscillation of solutions of first order differential equations with deviating arguments, Acta

Math. Appl. Sinica 15, no.3, (1999), 288-302.

delay argument, Nonlinear Anal., 68 (2008), 994–1005.

[2] G. E. Chatzarakis, R. Koplatadze, and I. P. Stavroulakis, Optimal oscillation criteria for first order difference equations

with delay argument, Pacific J. Math., 235 (2008), 15–33.

[3] Y. Domshlak, Discrete version of Sturmian Comparison Theorem for non-symmetric equations, Doklady Azerb. Acad. Sci.

37 (1981), 12–15.

[4] A. Elbert and I.P. Stavroulakis, Oscillations of first order differential equations with deviating arguments, Univ.of ioannina

TR No.172, 1990. Recent trends in differential equations 163–178, World Sci. Ser. Appl. Anal.,1, World Sci. Publishing

Co.(1992).

[5] L.H. Erbe, Qingkai Kong and B.G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New

York, 1995.

[6] L.H. Erbe and B.G. Zhang, Oscillation of first order linear differential equations with deviating arguments, Differential

Integral Equations 1 (1988), 305-314.

[7] L.H. Erbe and B.G. Zhang, Oscillation of discrete analogues of delay equations, Differential Integral Equations 2 (1989),

300-309.

[8] K. Gopalsamy, Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic

Publishers, 1992.

[9] I. Gyori and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford,

1991.

[10] J.K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York, 1997.

[11] J. Jaroˇs and I.P. Stavroulakis, Oscillation tests for delay equations, Rocky Mountain J. Math., 29 (1999), 139-145.

[12] C. Jian, Oscillation of linear differential equations with deviating argument, Math. in Practice and Theory, 1 (1991),

32-41 (in Chinese).

[13] B. Karpuz, Sharp oscillation and nonoscillation tests for linear difference equations, J. Difference Equ. Appl., 23 (2017).

no.12, 1229-1242.

[14] M. Kon, Y.G. Sficas and I.P. Stavroulakis, Oscillation criteria for delay equations, Proc. Amer. Math. Soc. 128 (2000),

2989-2997.

[15] R.G. Koplatadze and T.A. Chanturija, On the oscillatory and monotonic solutions of first order differential equations

with deviating arguments, Differentsial’nye Uravneniya, 18 (1982), 1463-1465.

[16] R.G. Koplatadze and G. Kvinikadze, On the oscillation of solutions of first order delay differential inequalities and

equations, Georgian Math. J. 1 (1994), 675-685.

[17] M.K. Kwong, Oscillation of first order delay equations, J. Math. Anal. Appl., 156 (1991), 374-286.

[18] G. Ladas, Recent developments in the oscillation of delay difference equations, International Conference on Differential

Equations, Stability and Control, Marcel Dekker, New York, 1990.

[19] G. Ladas, V. Laskhmikantham and J.S. Papadakis, Oscillations of higher-order retarded differential equations generated

by retarded arguments, Delay and Functional Differential Equations and Their Applications, Academic Press, New York,

1972, 219-231.

[20] G. Ladas, Ch.G. Philos and Y.G. Sficas, Sharp conditions for the oscillation of delay difference equations, J.Appl.Math.

Simul.,2(1989), 101-112.

[21] G.S. Ladde, V. Lakshmikantham and B.G. Zhang, Oscillation Theory of Differential Equations with Deviating Arguments,

Marcel Dekker, New York, 1987.

[22] A.D. Myshkis, Linear homogeneous differential equations of first order with deviating arguments, Uspekhi Mat. Nauk, 5

(1950), 160-162.

[23] Ch.G. Philos and Y.G. Sficas, An oscillation criterion for first-order linear delay differential equations, Canad. Math. Bull.

41 (1998), 207-213.

[24] Y.G. Sficas and I.P. Stavroulakis, Oscillation criteria for first-order delay equations, Bull. London Math. Soc., 35 (2003),

239-246.

[25] I.P. Stavroulakis, Oscillation Criteria for First Order Delay Difference Equations, Mediterr. J. Math. 1 (2004), 231-240.

[26] C. Sturm, Sur les equations differentielles lineaires du second ordre, J. Math. Pures Appl. 1 (1836), 106-186

[27] X.H. Tang and J.S. Yu, A further result on the oscillation of delay difference equations, Comput. Math. Applic., 38 (1999),

229-237.

[28] J.S. Yu and Z.C. Wang, Some further results on oscillation of neutral differential equations, Bull. Austral. Math. Soc., 46

(1992), 149-157.

[29] J.S. Yu, Z.C. Wang, B.G. Zhang and X.Z. Qian, Oscillations of differential equations with deviting arguments,

Panam.Math.J., 2 (1992), 59-78.

[30] J.S. Yu, B.G. Zhang and Z.C. Wang, Oscillation of delay difference equations, Applicable Anal., 53 (1994), 117-124.

[31] Y. Zhou and Y.H.Yu, On the oscillation of solutions of first order differential equations with deviating arguments, Acta

Math. Appl. Sinica 15, no.3, (1999), 288-302.

Published

2018-08-29

How to Cite

IOANNIS, P..
Qualitative Behavior of the Solutions to Delay and Difference Euations.

**Journal of Mathematics, Mechanics and Computer Science**, [S.l.], v. 98, n. 2, p. 3-11, aug. 2018. ISSN 1563-0277. Available at: <http://bm.kaznu.kz/index.php/kaznu/article/view/494>. Date accessed: 20 feb. 2019. doi: https://doi.org/10.26577/jmmcs-2018-2-494.
Section

Mathematics

Keywords
oscillation, delay differential equations, difference equations