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Existence of nonoscillatory solutions to neutral dynamic equations on time scales. (English) Zbl 1128.34043

The authors study the existence of a positive solution for the neutral functional dynamic equation on time scale
\[ [x(t)+p(t)x(g(t))]^{\Delta}+f(t,x(h(t))=0.\tag{1} \]
Here is one of the results of the paper.
Theorem. Equation (1) has an eventually positive solution \(x(t)\) with \(\lim_{t\rightarrow\infty}x(t)=a>0 \) if and only if there exists a constant \(K>0\) such that \[ \int_{t_0}^{\infty}\Delta s <\infty. \]

MSC:

34K11 Oscillation theory of functional-differential equations
34K40 Neutral functional-differential equations
39A10 Additive difference equations
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References:

[1] Agarwal, R.; Bohner, M.; O’Regan, D.; Peterson, A., Dynamic equations on time scales: A survey, J. comput. appl. math., 141, 1-26, (2002) · Zbl 1020.39008
[2] Bohner, M.; Peterson, A., Dynamic equations on time scales: an introduction with applications, (2001), Birkhäuser Boston · Zbl 0978.39001
[3] Bohner, M.; Peterson, A., Advances in dynamic equations on time scales, (2003), Birkhäuser Boston · Zbl 1025.34001
[4] Chen, Y.S., Existence of nonoscillatory solutions of nth order neutral delay differential equations, Funcial. ekvac., 35, 557-570, (1992) · Zbl 0787.34056
[5] Del Medico, A.; Kong, Q., Kamenev-type and interval oscillation criteria for second-order linear differential equations on a measure chain, J. math. anal. appl., 294, 621-643, (2004) · Zbl 1056.34050
[6] Erbe, L.; Peterson, A., Oscillation criteria for second-order matrix dynamic equations on a time scale, J. comput. appl. math., 141, 169-185, (2002) · Zbl 1017.34030
[7] Erbe, L.; Peterson, A.; Saker, S.H., Oscillation criteria for second-order nonlinear dynamic equations on time scales, J. London math. soc., 67, 701-714, (2003) · Zbl 1050.34042
[8] Mathsen, R.M.; Wang, Q.R.; Wu, H.W., Oscillation for neutral dynamic functional equations on time scales, J. difference equ. appl., 10, 651-659, (2004) · Zbl 1060.34038
[9] Saker, S.H., Oscillation of nonlinear dynamic equations, Appl. math. comput., 148, 81-91, (2004) · Zbl 1045.39012
[10] Zhu, Z.Q.; Wang, Q.R., Frequency measures on time scales with applications, J. math. anal. appl., 319, 398-409, (2006) · Zbl 1111.34029
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