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Boundedness and stability for the damped and forced single well Duffing equation

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  • By using differential inequalities we improve some estimates of W.S. LOUD for the ultimate bound and asymptotic stability of the solutions to the Duffing equation $ u''+ c{u'} + g(u)= f(t)$ where $c>0$, $f $ is measurable and essentially bounded, and $g$ is continuously differentiable with $g'\ge b>0$.
    Mathematics Subject Classification: Primary: 34C11, 34C15,34C25, 34C27, 34D05; Secondary: 34D23, 34D30.


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  • [1]

    L. Amerio, Soluzioni quasi periodiche, o limitate, di sistemi differenziali non lineari quasi periodici, o limitati, Ann. Mat. Pura. Appl., 39 (1955), 97-119.


    M. Biroli, Sur les solutions bornés et presque péiodiques des éuations et inéuations d'éolution, Ann. Mat. Pura Appl., 93 (1972), 1-79.


    M. L. Cartwright and J. E. Littlewood, On non-linear differential equations of the second order, Ann. Math., 48 (1947), 472-494.


    C. M. Dafermos, Almost periodic processes and almost periodic solutions of evolution equations, in "Proccedings of a University of Florida International Symposium," Academic Press, 1977, 43-57.


    C. Fitouri, "Thesis Dissertation," ch.2, University of Zürich, 2008.


    C. Fitouri and A. Haraux, Sharp estimates of bounded solutions to some semilinear second order dissipative equations, J. Math. Pures Appl., 92 (2009), 313-321.


    A. Haraux, "Nonlinear Evolution Equations: Global Behavior of Solutions," Springer-Verlag, New York, 1981.


    A. Haraux, "Systèmes Dynamiques Dissipatifs et Applications," Masson,Paris, 1991.


    A. Haraux, On the double well Duffing equation with a small bounded forcing term, Rend. Accad. Naz. Sci. XL Mem. Mat. Appl.,29 (2005), 207-230.


    A. Haraux, Sharp estimates of bounded solutions to some second-order forced dissipative equations, J. Dynam. Differential Equations, 19 (2007), 915-933.


    W. S. Loud, On periodic solutions of Duffing's equation with damping, Journal of Mathematics and Physics,34 (1955), 173-178.


    W. S. Loud, Boundedness and convergence of solutions of x''+cx' +g(x) = e(t), Duke Math. J., 24 (1957), 63-72.


    W. S. Loud, Periodic solutions of x''+cx' +g(x) = f(t), Mem. Amer. Math. Soc., 31 (1959), 1-57.


    Ph. Souplet, Uniqueness and nonuniqueness results for the antiperiodic solutions of some second-order nonlinear evolution equations, Nonlinear Analysis T.M.A., 26 (1996), 1511-1525.doi: 10.1016/0362-546X(95)00012-K.

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