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.
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Objective & Hypothesis: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 $\tau_i(t)\leq t,\,\, \lim_{t\to\infty}\tau_i(t)=\infty$.
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Key Findings & Data: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 $\tau_i(t)\leq t,\,\, \lim_{t\to\infty}\tau_i(t)=\infty$.
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Conclusion & Impact: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.
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.
Kılıç, N., Öcalan, Ö. (2021). Oscillation criteria for first-order dynamic equations with nonmonotone delays. Hacettepe Journal of Mathematics and Statistics, Volume: 50 Issue: 2. https://doi.org/10.15672/hujms.674428
R.P. Agarwal and M. Bohner, An oscillation criterion for first order delay dynamic equations, Funct. Differ. Equ. 16 (1), 11-17, 2009.
M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhauser, Boston, 2001.
M. Bohner and A. Peterson, Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, 2003.
M. Bohner, Some oscillation criteria for first order delay dynamic equations, Far East J. Appl. Math. 18 (3), 289-304, 2005.
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.
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.
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.
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.
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.
L.H. Erbe and B.G. Zhang, Oscillation of first order linear differential equations with deviating arguments, Differential Integral Equations 1, 305-314, 1988.
L.H. Erbe and B.G. Zhang, Oscillation of discrete analogues of delay equations, Differential Integral Equations 2, 300-309, 1989.
L.H. Erbe, Qingkai Kong and B.G. Zhang, Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1995.
N. Fukagai and T. Kusano, Oscillation theory of first order functional differential equations with deviating arguments, Ann. Mat. Pura Appl. 136, 95-117, 1984.
I. Győri and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Clarendon Press, Oxford, 1991.
S. Hilger, Ein MaXkettenkalkWul mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. thesis, Universtat Wurzburg, 1988.
S. Hilger, Analysis on measure chains -- a unified approach to continuous and discrete calculus, Results in Mathematics 18, 1856, 1990.
J. Jaroš and I.P. Stavroulakis, Oscillation tests for delay equations, Rocky Mountain J. Math. 29, 139-145, 1999.
C. Jian, Oscillation of linear differential equations with deviating argument, Math. Pract. Theor. 1, 32-41, 1991 (in Chinese).
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.
B. Karpuz, Sharp oscillation and nonoscillation tests for linear difference equations, J. Difference Equ Appl. 23 (12), 1929-1942, 2017.
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.
M.K. Kwong, Oscillation of first-order delay equations, J. Math. Anal. Appl. 156, 274-286, 1991.
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.
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.
Ö. Öcalan, Oscillation of first-order dynamic equations with nonmonotone delay, Math. Methods Appl. Sci. 43 (7), 3954-3964, 2020.
Ch.G. Philos and Y.G. Sficas, An oscillation criterion for first-order linear delay differential equations, Canad. Math. Bull. 41, 207-213, 1998.
Y. Şahiner and I.P. Stavroulakis, Oscillations of first order delay dynamic equations, Dynam. Systems Appl. 15 (3-4), 645-655, 2006.
J.S. Yu and Z.C. Wang, Some further results on oscillation of neutral differential equations, Bull. Aust. Math. Soc. 46, 149-157, 1992.
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.
B.G. Zhang and C.J. Tian, Oscillation criteria for difference equations with unbounded delay, Comput. Math. Appl. 35 (4), 19-26, 1998.
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.
B.G. Zhang and X. Deng, Oscillation of delay differential equations on time scales, Math. Comput. Modelling 36 (11-13), 1307-1318, 2002.
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.
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.