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Fixed points, stability, and exact linearization. (English) Zbl 1067.34077
Summary: We study the scalar equation ${x}^{\text{'}\text{'}}+f\left(t,x,{x}^{\text{'}}\right){x}^{\text{'}}+b\left(t\right)g\left(x\left(t-L\right)\right)=0$ by means of contraction mappings. Conditions are obtained to ensure that each solution $\left(x\left(t\right),{x}^{\text{'}}\left(t\right)\right)\to \left(0,0\right)$ 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 for nonlinear operators on topological linear spaces