Oscillation criteria for first-order dynamic equations with nonmonotone delays

Year 2021, , 318 - 325, 11.04.2021

Nurten Kılıç , Özkan Öcalan

https://doi.org/10.15672/hujms.674428

Cited By: 2

Abstract

 In this paper, we consider the first-order dynamic equation as the following:
$$x^{\Delta}(t)+\sum\limits_{i=1}^m p_i(t)x(\tau_i(t))=0,\,\,t\in[t_0,\infty)_{\mathbb{T}}$$

where $p_{i}\in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}},\mathbb{R}^{+}\right) ,$ $\tau _{i}\in C_{rd}\left( [t_{0},\infty )_{\mathbb{T}},\mathbb{T}\right) $ $(i=1,2,\ldots ,m)$ and $\tau_i(t)\leq t,\,\, \lim_{t\to\infty}\tau_i(t)=\infty$. When the delay terms $\tau_{i}(t)$ $(i=1,2,\ldots ,m)$ are not necessarily monotone, we present new sufficient conditions for the oscillation of first-order delay dynamic equations on time scales.

Keywords

dynamic equations, nonmonotone delay, oscillation, time scales

References

• [1] R.P. Agarwal and M. Bohner, An oscillation criterion for first order delay dynamic equations, Funct. Differ. Equ. 16 (1), 11-17, 2009.
• [2] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, 2001.
• [3] M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003.
• [4] M. Bohner, Some oscillation criteria for first order delay dynamic equations, Far East J. Appl. Math. 18 (3), 289-304, 2005.
• [5] G.E. Chatzarakis, R. Koplatadze and I.P. Stavroulakis, Oscillation criteria of first order linear difference equations with delay argument, Nonlinear Anal. 68, 994-1005, 2008.
• [6] G.E. Chatzarakis, R. Koplatadze and I.P. Stavroulakis, Optimal oscillation criteria for first order difference equations with delay argument, Pacific J. Math. 235, 15-33, 2008.
• [7] G.E. Chatzarakis, Ch.G. Philos and I.P. Stavroulakis, On the oscillation of the solutions to linear difference equations with variable delay, Electron. J. Differ. Equ. 2008 (50), 1-15, 2008.
• [8] G.E. Chatzarakis, Ch.G. Philos and I. P. Stavroulakis, An oscillation criterion for linear difference equations with general delay argument, Port. Math. 66 (4), 513-533, 2009.
• [9] 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.
• [10] L.H. Erbe and B.G. Zhang, Oscillation of first order linear differential equations with deviating arguments, Differential Integral Equations 1, 305-314, 1988.
• [11] L.H. Erbe and B.G. Zhang, Oscillation of discrete analogues of delay equations, Differential Integral Equations 2, 300-309, 1989.
• [12] L.H. Erbe, Qingkai Kong and B.G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1995.
• [13] N. Fukagai and T. Kusano, Oscillation theory of first order functional differential equations with deviating arguments, Ann. Mat. Pura Appl. 136, 95-117, 1984.
• [14] I. Győri and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, 1991.
• [15] S. Hilger, Ein MaXkettenkalkWul mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. thesis, Universtat Wurzburg, 1988.
• [16] S. Hilger, Analysis on measure chains -- a unified approach to continuous and discrete calculus, Results in Mathematics 18, 1856, 1990.
• [17] J. Jaroš and I.P. Stavroulakis, Oscillation tests for delay equations, Rocky Mountain J. Math. 29, 139-145, 1999.
• [18] C. Jian, Oscillation of linear differential equations with deviating argument, Math. Pract. Theor. 1, 32-41, 1991 (in Chinese).
• [19] B. Karpuz and Ö. Öcalan, New oscillation tests and some refinements for first-order delay dynamic equations, Turkish J. Math. 40 (4), 850-863, 2016.
• [20] B. Karpuz, Sharp oscillation and nonoscillation tests for linear difference equations, J. Difference Equ Appl. 23 (12), 1929-1942, 2017.
• [21] R.G. Koplatadze and T.A. Chanturija, Oscillating and monotone solutions of firstorder differential equations with deviating arguments, (Russian), Differentsial’nye Uravneniya 8, 1463-1465, 1982.
• [22] M.K. Kwong, Oscillation of first-order delay equations, J. Math. Anal. Appl. 156, 274-286, 1991.
• [23] G.S. Ladde, V. Lakshmikantham and B.G. Zhang, Oscillation theory of differential equations with deviating arguments, Monographs and Textbooks in Pure and Applied Mathematics, vol. 110, Marcel Dekker, Inc., New York, 1987.
• [24] Ö. Öcalan, U.M. Özkan and M.K. Yıldız, Oscillatory solutions for dynamic equations with non-monotone arguments, J. Math. Comput. Sci. 7 (4), 725-738, 2017.
• [25] Ö. Öcalan, Oscillation of first-order dynamic equations with nonmonotone delay, Math. Methods Appl. Sci. 43 (7), 3954-3964, 2020.
• [26] Ch.G. Philos and Y.G. Sficas, An oscillation criterion for first-order linear delay differential equations, Canad. Math. Bull. 41, 207-213, 1998.
• [27] Y. Şahiner and I.P. Stavroulakis, Oscillations of first order delay dynamic equations, Dynam. Systems Appl. 15 (3-4), 645-655, 2006.
• [28] J.S. Yu and Z.C. Wang, Some further results on oscillation of neutral differential equations, Bull. Aust. Math. Soc. 46, 149-157, 1992.
• [29] J.S. Yu, Z.C. Wang, B.G. Zhang and X.Z. Qian, Oscillations of differential equations with deviating arguments, PanAmerican Math. J. 2, 59-78, 1992.
• [30] B.G. Zhang and C.J. Tian, Oscillation criteria for difference equations with unbounded delay, Comput. Math. Appl. 35 (4), 19-26, 1998.
• [31] B.G. Zhang and C.J. Tian, Nonexistence and existence of positive solutions for difference equations with unbounded delay, Comput. Math. Appl. 36, 1-8, 1998.
• [32] B.G. Zhang and X. Deng, Oscillation of delay differential equations on time scales, Math. Comput. Modelling 36 (11-13), 1307-1318, 2002.
• [33] B.G. Zhang, X. Yan and X. Liu, Oscillation criteria of certain delay dynamic equations on time scales, J. Difference Equ. Appl. 11 (10), 933-946, 2005.
• [34] Y. Zhou and Y.H. Yu, On the oscillation of solutions of first order differential equations with deviating arguments, Acta Math. Appl. Sinica 15 (3), 288-302, 1999.