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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))(0,0) as t. 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.
34K20Stability theory of functional-differential equations
47H10Fixed point theorems for nonlinear operators on topological linear spaces