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Fixed points, stability, and exact linearization. (English) Zbl 1067.34077

Summary: We study the scalar equation \(x''+ f(t, x, x')x'+ b(t)g(x(t- L))= 0\) by means of contraction mappings. Conditions are obtained to ensure that each solution \((x(t),x'(t))\to (0, 0)\) as \(t\to\infty\). The conditions allow \(f\) to grow as large as \(t\), but not as large as \(t^2\). This is parallel to the classical result of R. A. Smith [Q. J. Math., Oxf. II. Ser. 12, 123–125 (1961; Zbl 0103.05604)] for the linear equation without a delay.

MSC:

34K20 Stability theory of functional-differential equations
47H10 Fixed-point theorems

Citations:

Zbl 0103.05604
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References:

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