# American Institute of Mathematical Sciences

September  2018, 38(9): 4715-4726. doi: 10.3934/dcds.2018207

## Periodic solutions of second order equations with asymptotical non-resonance

 1 College of Information Science and Engineering, Shandong Agricultural University, Taian 271018, China 2 School of Mathematical Sciences, Soochow University, Suzhou 215006, China

* Corresponding author: Dingbian Qian

Received  January 2018 Revised  April 2018 Published  June 2018

Fund Project: This work was supported by National Natural Science Foundation of China (No.11671287, No.61573228).

This paper deals with the periodic solutions of second order equations with asymptotical non-resonance. Using the point of view that the force is a perturbation, we can think that, asymptotically, the solutions of forced non-autonomous equation behave as those of the autonomous equation. Then, under a sharp integral condition, we prove that the periodic solution of non-autonomous equation can be estimated by using time map of autonomous equation. The existence of periodic solutions is thus proved via qualitative analysis and topological degree theory. The main result in this paper generalize a existence result obtained by Capietto, Mawhin and Zanolin.

Citation: Xuelei Wang, Dingbian Qian, Xiying Sun. Periodic solutions of second order equations with asymptotical non-resonance. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4715-4726. doi: 10.3934/dcds.2018207
##### References:
 [1] A. Capietto, J. Mawhin and F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc., 329 (1992), 41-72.  doi: 10.1090/S0002-9947-1992-1042285-7.  Google Scholar [2] 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.  doi: 10.1007/BF01295308.  Google Scholar [3] T. Ding, An infinite class of periodic solutions of periodically perturbed Duffing equations at resonance, Proc. Amer. Math. Soc., 86 (1982), 47-54.  doi: 10.1090/S0002-9939-1982-0663864-1.  Google Scholar [4] T. Ding, R. Iannacci and F. Zanolin, Existence and multiplicity results for periodic solutions of semilinear Duffing equations, J. Differential Equations, 105 (1993), 364-409.  doi: 10.1006/jdeq.1993.1093.  Google Scholar [5] T. Ding and F. Zanolin, Periodic solutions of Duffing's equations with superquadratic potential, J. Differential Equations, 97 (1992), 328-378.  doi: 10.1016/0022-0396(92)90076-Y.  Google Scholar [6] T. Ding and W. Ding, Resonance problem for a class of Duffing's equations, Chin. Ann. of Math. -B, 6 (1985), 427-432.   Google Scholar [7] L. Fernandes and F. Zanolin, Periodic solutions of a second order differential equation with one-sided growth restrictions on the restoring term, Arch. Math., 51 (1988), 151-163.  doi: 10.1007/BF01206473.  Google Scholar [8] A. Fonda and F. Zanolin, On the use of time-maps for the solvability of nonlinear boundary value problems, Arch. Math., 59 (1992), 245-259.  doi: 10.1007/BF01197322.  Google Scholar [9] D. Leach, On Poincaré's perturbation theorem and a theorem of W.S. Loud, J. Differential Equations, 7 (1970), 34-53.  doi: 10.1016/0022-0396(70)90122-1.  Google Scholar [10] W. Loud, Periodic solutions of nonlinear differential equations of Duffing type, In: Proc. United States - Japan Seminar on Differential and Functional Equations, Benjamin, New York, 1967,199–224.  Google Scholar [11] J. Mawhin, Recent trends in nonlinear boundary value problems, In: VII Intern. Konferenz über Nichtlineare Schwingungen (Berlin 1975), Band I. 2, Abhandlungen der AdW, Akademie Verlag, Berlin, 1977, 51–70.  Google Scholar [12] Z. Opial, Sur les solutions périodiques de l'équation différentielle $x''+g(x) = p(t)$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys., 8 (1960), 151-156.   Google Scholar [13] D. Qian, Time-maps and Duffing equations across resonance, Science in China A, 23 (1993), 471-479.   Google Scholar

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##### References:
 [1] A. Capietto, J. Mawhin and F. Zanolin, Continuation theorems for periodic perturbations of autonomous systems, Trans. Amer. Math. Soc., 329 (1992), 41-72.  doi: 10.1090/S0002-9947-1992-1042285-7.  Google Scholar [2] 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.  doi: 10.1007/BF01295308.  Google Scholar [3] T. Ding, An infinite class of periodic solutions of periodically perturbed Duffing equations at resonance, Proc. Amer. Math. Soc., 86 (1982), 47-54.  doi: 10.1090/S0002-9939-1982-0663864-1.  Google Scholar [4] T. Ding, R. Iannacci and F. Zanolin, Existence and multiplicity results for periodic solutions of semilinear Duffing equations, J. Differential Equations, 105 (1993), 364-409.  doi: 10.1006/jdeq.1993.1093.  Google Scholar [5] T. Ding and F. Zanolin, Periodic solutions of Duffing's equations with superquadratic potential, J. Differential Equations, 97 (1992), 328-378.  doi: 10.1016/0022-0396(92)90076-Y.  Google Scholar [6] T. Ding and W. Ding, Resonance problem for a class of Duffing's equations, Chin. Ann. of Math. -B, 6 (1985), 427-432.   Google Scholar [7] L. Fernandes and F. Zanolin, Periodic solutions of a second order differential equation with one-sided growth restrictions on the restoring term, Arch. Math., 51 (1988), 151-163.  doi: 10.1007/BF01206473.  Google Scholar [8] A. Fonda and F. Zanolin, On the use of time-maps for the solvability of nonlinear boundary value problems, Arch. Math., 59 (1992), 245-259.  doi: 10.1007/BF01197322.  Google Scholar [9] D. Leach, On Poincaré's perturbation theorem and a theorem of W.S. Loud, J. Differential Equations, 7 (1970), 34-53.  doi: 10.1016/0022-0396(70)90122-1.  Google Scholar [10] W. Loud, Periodic solutions of nonlinear differential equations of Duffing type, In: Proc. United States - Japan Seminar on Differential and Functional Equations, Benjamin, New York, 1967,199–224.  Google Scholar [11] J. Mawhin, Recent trends in nonlinear boundary value problems, In: VII Intern. Konferenz über Nichtlineare Schwingungen (Berlin 1975), Band I. 2, Abhandlungen der AdW, Akademie Verlag, Berlin, 1977, 51–70.  Google Scholar [12] Z. Opial, Sur les solutions périodiques de l'équation différentielle $x''+g(x) = p(t)$, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys., 8 (1960), 151-156.   Google Scholar [13] D. Qian, Time-maps and Duffing equations across resonance, Science in China A, 23 (1993), 471-479.   Google Scholar
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