January  2018, 38(1): 293-309. doi: 10.3934/dcds.2018014

Limit periodic upper and lower solutions in a generic sense

1. 

Universitá di Milano, via Saldini 50, 20133 Milano, Italy

2. 

NCMIS, RCSDS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

* Corresponding author: Zhe Zhou

Received  February 2017 Revised  July 2017 Published  September 2017

Fund Project: The second author is partially supported by the Key Lab of Random Complex Structures and Data Science, Chinese Academy of Sciences (Grant No. 2008DP173182) and the National Natural Science Foundation of China (Grant No. 11301512 and No. 11671382).

The method of upper and lower solutions is a main tool to prove the existence of periodic solutions to periodic differential equations. It is known that, in general, the method does not extend to the almost periodic case. The aim of the present paper is to show that, however, something interesting survives: if the classical assumptions of the method are satisfied, then the expected existence result holds generically in the limit periodic framework.

Citation: Massimo Tarallo, Zhe Zhou. Limit periodic upper and lower solutions in a generic sense. Discrete & Continuous Dynamical Systems, 2018, 38 (1) : 293-309. doi: 10.3934/dcds.2018014
References:
[1]

A. I. AlonsoR. Obaya and R. Ortega, Differential equations with limit-periodic forcings, Proc. Amer. Math. Soc., 131 (2003), 851-857.  doi: 10.1090/S0002-9939-02-06692-3.  Google Scholar

[2]

H. Brezis, Analyse Fonctionnelle -Théorie et Applications Masson, Paris, 1983.  Google Scholar

[3]

W. A. Coppel, Dichotomies in Stability Theory Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, 1978.  Google Scholar

[4]

C. Corduneanu, Almost Periodic Functions 2nd English ed. , AMS Chelsea Publishing, New York, 1989. Google Scholar

[5]

C. De Coster and P. Habets, Two-point Boundary Value Problems: Lower and Upper Solutions Mathematics in Science and Engineering 205, Elsevier, Amsterdam, 2006.  Google Scholar

[6]

A. Fink, Almost Periodic Differential Equations Springer, New York/Berlin, 1974.  Google Scholar

[7]

B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations Cambridge Univ. Press, Cambridge, 1982.  Google Scholar

[8]

W. Magnus and S. Winkler, Hill's Equation Dover, New York, 1979.  Google Scholar

[9]

P. Martinez–AmoresJ. MawhinR. Ortega and M. Willem, Generic results for the existence of nondegenerate periodic solutions of some differential systems with periodic nonlinearities, J. Differential Equation, 91 (1991), 138-148.  doi: 10.1016/0022-0396(91)90135-V.  Google Scholar

[10]

R. Ortega, The pendulum equation: from periodic to almost periodic forcings, Differential Integral Equations, 22 (2009), 801-814.   Google Scholar

[11]

R. Ortega and M. Tarallo, Almost periodic upper and lower solutions, J. Differential Equation, 193 (2003), 343-358.  doi: 10.1016/S0022-0396(03)00130-X.  Google Scholar

[12]

K. Scmitt and J. R. Ward, Almost periodic solutions of nonlinear second order differential equations, Res. Math., 21 (1992), 190-199.  doi: 10.1007/BF03323078.  Google Scholar

[13]

S. Smale, An infinite dimensional version of Sard's theorem, Amer. J. Math., 87 (1965), 861-866.  doi: 10.1142/9789812792822_0005.  Google Scholar

show all references

References:
[1]

A. I. AlonsoR. Obaya and R. Ortega, Differential equations with limit-periodic forcings, Proc. Amer. Math. Soc., 131 (2003), 851-857.  doi: 10.1090/S0002-9939-02-06692-3.  Google Scholar

[2]

H. Brezis, Analyse Fonctionnelle -Théorie et Applications Masson, Paris, 1983.  Google Scholar

[3]

W. A. Coppel, Dichotomies in Stability Theory Lecture Notes in Mathematics, Vol. 629, Springer-Verlag, 1978.  Google Scholar

[4]

C. Corduneanu, Almost Periodic Functions 2nd English ed. , AMS Chelsea Publishing, New York, 1989. Google Scholar

[5]

C. De Coster and P. Habets, Two-point Boundary Value Problems: Lower and Upper Solutions Mathematics in Science and Engineering 205, Elsevier, Amsterdam, 2006.  Google Scholar

[6]

A. Fink, Almost Periodic Differential Equations Springer, New York/Berlin, 1974.  Google Scholar

[7]

B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations Cambridge Univ. Press, Cambridge, 1982.  Google Scholar

[8]

W. Magnus and S. Winkler, Hill's Equation Dover, New York, 1979.  Google Scholar

[9]

P. Martinez–AmoresJ. MawhinR. Ortega and M. Willem, Generic results for the existence of nondegenerate periodic solutions of some differential systems with periodic nonlinearities, J. Differential Equation, 91 (1991), 138-148.  doi: 10.1016/0022-0396(91)90135-V.  Google Scholar

[10]

R. Ortega, The pendulum equation: from periodic to almost periodic forcings, Differential Integral Equations, 22 (2009), 801-814.   Google Scholar

[11]

R. Ortega and M. Tarallo, Almost periodic upper and lower solutions, J. Differential Equation, 193 (2003), 343-358.  doi: 10.1016/S0022-0396(03)00130-X.  Google Scholar

[12]

K. Scmitt and J. R. Ward, Almost periodic solutions of nonlinear second order differential equations, Res. Math., 21 (1992), 190-199.  doi: 10.1007/BF03323078.  Google Scholar

[13]

S. Smale, An infinite dimensional version of Sard's theorem, Amer. J. Math., 87 (1965), 861-866.  doi: 10.1142/9789812792822_0005.  Google Scholar

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