Infinitely many periodic solutions of Duffing equations under integral condition

Nannan Zheng, Zaihong Wang



In this paper, we study the multiplicity of periodic solutions of a Duffing equation $$ x''+g(x)=p(t). $$% By using the generalized Poincaré-Birkhoff fixed point theroem, we prove that this equation has infinitely many periodic solutions provided $g$ satisfies a kind of integral condition and the related time map satisfies oscillating condition.


Duffing equation; periodic solution; Poincaré-Birkhoff fixed point theorem

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A. Capietto, J. Mawhin and F. Zanolin, A continuation theorem for periodic boundary value problems with oscillatory nonlinearities, Nonlinear Differential Equations Appl. 2 (1995), 133–163.

T. Ding, An infinite class of periodic solutions of periodically perturbed Duffing equation at resonance, Proc. Amer. Math. Soc. 86 (1982), 47–54.

T. Ding, R. Iannacci and F. Zanolin, Existence and multiplicity results for periodic solutions of semilinear Duffing equation, J. Differential Equations 105 (1993), 364–409.

M.A. Krasnosel’skiı̆, The Operator of Translation along the Trajectories of Differential Equations, Amer. Math. Soc., Providence, R. I., 1968.

D.E. Leach, On Poincaré’s perturbation theorem and a theorem of W.S. Loud, J. Differential Equations 7 (1970), 34–53.

J. Mawhin, Recent trends in nonlinear boundary value problems, Proc. 7th Int. Conf. Nonlinear Oscillations (Berlin) (G. Schmidt, ed.), Akademie–Verlag, Berlin (1977), band 1, 51–70.

Z. Opial, Sur les solutions périodiques l’équation différentielle x00 + g(x) = p(t), Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys. 8 (1960), 151–156.

D. Qian, Time-maps and Duffing equations across resonance, Scientia Sinica Ser. A 23 (1993), 471–479. (in Chinese)

C. Rebelo, A note on the Poincaré–Birkhoff fixed point theorem and periodic solutions of planar systems, Nonlinear Anal. 29 (1997), 291–311.

R. Reissig, Contraction mappings and periodically perturbed nonconservative systems, Atti Accad. Naz. Lincei Rend. Cl. Sci. 58 (1975), 696–702.

X. Wang, D. Qian and Y. Sun, Periodic solutions of second order equations with asymptotical non-resonance, Discrete Contin. Dyn. Syst. 38 (2018), 4715–4726.

Z. Wang, Multiplicity of Periodic Solutions of Semilinear Duffing’s Equation at Resonance, J. Math. Anal. Appl. 237 (1999), 166–187.


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