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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 {\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:
34K20Stability theory of functional-differential equations
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
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References:
[1] Ballieu, R. J.; Peiffer, K.: Attractivity of the origin for the equation x″+f(t,x,$x^{\prime})\vert x^{\prime}\vert \alpha x^{\prime}+g(x)=0$. J. math. Anal. appl. 65, 321-332 (1978) · Zbl 0387.34038
[2] Burton, T. A.: Stability and periodic solutions of ordinary and functional differential equations. (1985) · Zbl 0635.34001
[3] Burton, T. A.: Stability by fixed point theory or Liapunov theorya comparison. Fixed point theory 4, 15-32 (2003) · Zbl 1061.47065
[4] Burton, T. A.: Fixed points and stability of a nonconvolution equation. Proc. am. Math. soc. 132, 3679-3687 (2004) · Zbl 1050.34110
[5] Burton, T. A.; Hooker, J. W.: On solutions of differential equations tending to zero. J. reine angew. Math. 267, 151-165 (1974) · Zbl 0298.34042
[6] Hatvani, L.: Nonlinear oscillation with large damping. Dyn. syst. Appl. 1, 257-270 (1992) · Zbl 0763.34042
[7] Hatvani, L.: Integral conditions on the asymptotic stability for the damped linear oscillator with small damping. Proc. am. Math. soc. 124, 415-422 (1996) · Zbl 0844.34051
[8] Hatvani, L.; Krisztin, T.: Necessary and sufficient conditions for intermittent stabilization of linear oscillators by large damping. Differential integral equations 10, 265-272 (1997) · Zbl 0893.34045
[9] Hatvani, L.; Krisztin, T.; Totik, V.: A necessary and sufficient condition for the asymptotic stability of the damped oscillator. J. differential equations 119, 209-223 (1995) · Zbl 0831.34052
[10] Hatvani, L.; Totik, V.: Asymptotic stability of the equilibrium of the damped oscillator. Differential integral equations 6, 835-848 (1993) · Zbl 0777.34036
[11] Levin, J. J.; Nohel, J. A.: Global asymptotic stability for nonlinear systems of differential equations. Arch. rational mech. Anal. 5, 194-211 (1960) · Zbl 0094.06402
[12] Pucci, P.; Serrin, J.: Asymptotic stability for intermittently controlled nonlinear oscillators. SIAM J. Math. anal. 25, 815-835 (1994) · Zbl 0809.34067
[13] Smith, R. A.: Asymptotic stability of x″+$a(t)x^{\prime}+x=0$. Quart. J. Math. Oxford ser. 12, No. 2, 123-126 (1961) · Zbl 0103.05604
[14] Thurston, L. H.; Wong, J. S. W.: On global asymptotic stability of certain second order differential equations with integrable forcing terms. SIAM J. Appl. math. 24, 50-61 (1973) · Zbl 0279.34041
[15] Zhang, B.: On the retarded Li√©nard equation. Proc. am. Math. soc. 115, 779-785 (1992) · Zbl 0756.34075