## 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

Zbl 0103.05604
Full Text:

### References:

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