An asymptotic theorem for a class of nonlinear neutral differential equations |
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| Authors: | Manabu Naito |
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| Institution: | (1) Department of Mathematics, Faculty of Science, Ehime University, Matsuyama, 790, Japan |
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| Abstract: | The neutral differential equation
![$$\frac{{d^n }}{{dt^n }}x(t) + x(t - \tau )] + \sigma F(t,x(g(t))) = 0,$$](/content/w0315nm15112gl12/10587_2004_Article_417640_TeX2GIFIE1.gif)
is considered under the following conditions: n  2,  > 0,  = ±1, F(t, u) is nonnegative on t
0,  ) × (0, ) and is nondecreasing in u  (0, infin;), and lim g(t) = as t  . It is shown that equation (1.1) has a solution x(t) such that
![$$\begin{gathered} \mathop {\lim }\limits_{t \to \infty } \frac{{x(t)}}{{t^k }}{\text{ exists and is a positive finite value if and only if}} \hfill \\ {\text{ }}\int_{{\text{t}}_{\text{0}} }^\infty {{\text{t}}^{{\text{n - k - 1}}} F(t,cg(t)]^k )} dt < \infty {\text{ for some c > 0}}{\text{.}} \hfill \\ \end{gathered} $$](/content/w0315nm15112gl12/10587_2004_Article_417640_TeX2GIFIE2.gif)
Here, k is an integer with 0  k n–1. To prove the existence of a solution x(t) satisfying (1.2), the Schauder-Tychonoff fixed point theorem is used. |
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