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Asymptotic Behavior for a Class of Nonlinear Impulsive Neutral Delay Differential Equations

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Abstract:

This paper is concerned with the
nonlinear impulsive neutral delay differential equation with
positive and negative coefficients,

{\begin{cases} [x (t) - c (t) x (t - τ)]^{'} + p (t) f (x (t - δ)) - q (t) f (x (t - σ)) = 0, t \geq t_{0}, t \neq t_{k}, \\ x (t_{k}) = b_{k} x (t_{k}^{-}) \\ \disp + (1 - b_{k}) (\int_{t_{k} - δ}^{t_{k}} p (s + δ) f (x (s)) d s - \int_{t_{k} - σ}^{t_{k}} q (s + σ) f (x (s)) d s), \\ k = 1, 2, 3, \dots . \end{cases}

$\left\{ \begin{array}{ll} [x(t)-c(t)x(t-\tau)]'+p(t)f(x(t-\delta))-q(t)f(x(t-\sigma))=0,\quad t\geq t_0,~ t\not= t_k, \\ x(t_k)= b_kx(t_k^-)\\ \disp\qquad \quad\ \ +(1-b_k)\left(\int^{t_k}_{t_k-\delta}p(s+\delta)f(x(s)){\rm d}s- \int^{t_k}_{t_k-\sigma}q(s+\sigma)f(x(s)){\rm d}s\right),\\ \hskip 9.7cm k=1,2,3,\cdots. \end{array} \right.$
Sufficient conditions are obtained for every solution of
the above equation tending to a constant as

t \to \infty

$t\to \infty$ .

Key words: Asymptotic behavior, Liapunov functional, Neutral delay differential equation, Impulse

CLC Number:

34K20

WEI Geng-Ping, SHEN Jian-Hua. Asymptotic Behavior for a Class of Nonlinear Impulsive Neutral Delay Differential Equations[J].Acta mathematica scientia,Series A, 2010, 30(3): 753-763.

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