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.
📌 3 Maddede Bu Makale
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Amaç & Hipotez: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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Temel Bulgular & Veri: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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Sonuç & Katkı: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.
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