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Periodic solutions of second order equations with asymptotical non-resonance

  • * Corresponding author: Dingbian Qian

    * Corresponding author: Dingbian Qian 
This work was supported by National Natural Science Foundation of China (No.11671287, No.61573228)
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  • 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.

    Mathematics Subject Classification: Primary: 34C25, 34B15; Secondary: 54D25.

    Citation:

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  •   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.
      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.
      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.
      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.
      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.
      T. Ding  and  W. Ding , Resonance problem for a class of Duffing's equations, Chin. Ann. of Math. -B, 6 (1985) , 427-432. 
      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.
      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.
      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.
      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.
      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.
      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. 
      D. Qian , Time-maps and Duffing equations across resonance, Science in China A, 23 (1993) , 471-479. 
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