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Zbl 1067.34077
Burton, T.A.
Fixed points, stability, and exact linearization. (English)
[J] Nonlinear Anal., Theory Methods Appl. 61, No. 5, A, 857-870 (2005). ISSN 0362-546X

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 {\it R. A. Smith} [Q. J. Math., Oxf. II. Ser. 12, 123--125 (1961; Zbl 0103.05604)] for the linear equation without a delay.

MSC 2000:: *34K20 Stability theory of functional-differential equations
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: asymptotic stability; contraction mappings; second-order delay equations

Citations: Zbl 0103.05604

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