# American Institute of Mathematical Sciences

January  2013, 33(1): 381-389. doi: 10.3934/dcds.2013.33.381

## Periodic solutions of first order systems

 1 Department of Mathematics, University of Alabama at Birmingham, Birmingham, AL 35294, United States

Received  August 2011 Revised  January 2012 Published  September 2012

Let $f\in C(% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{m},% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{m})$ and $p\in C([0,T],% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{m})$ be continuous functions. We consider the $T$ periodic boundary value problem (*) $u^{\prime}(t)=f(u(t))+p(t),$ $u(0)=u(T).$ It is shown that when $f$ is a coercive gradient function, or the bounded perturbation of a coercive gradient function, and the Brouwer degree $d_{B}(f,B(0,r),0)\neq0$ for large $r$, there is a solution for all $p.$ A result for bounded $f$ is also obtained.
Citation: J. R. Ward. Periodic solutions of first order systems. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 381-389. doi: 10.3934/dcds.2013.33.381
##### References:
 [1] A. Capietto, J. Mawhin and F. Zanolin, Continuation theoremsfor periodic perturbations of autonomous systems, Trans. Amer. Math. Soc., 329 (1992), 41-72. doi: 10.1090/S0002-9947-1992-1042285-7. [2] M. Farkas, "Periodic Motions," Applied MathematicalSciences, Springer-Verlag, New York, 104, 1994. [3] M. A. Krasnosel'skiĭ, "The Operator Oftranslation Along the Trajectories of Differential Equations," Translations of Mathematical Monographs, Translated from the Russian by ScriptaTechnica. American Mathematical Society, Providence, R. I., 19, 1968. [4] M. A. Krasnosel'skiĭ and P. P. Zabreĭko, "Geometricalmethods of Nonlinear Analysis," Translated from the Russian by Christian C. Fenske. Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 263, 1984. [5] E. M. Landesman and A. C. Lazer, Nonlinear perturbations oflinear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1969/1970), 609-623. [6] A. C. Lazer and D. E. Leach, Bounded perturbations of forcedharmonic oscillators at resonance, Ann. Mat. Pura Appl. (4), 82 (1969), 49-68. [7] J. Mawhin, "Topological Degree Methods in Nonlinearboundary Value Problems," CBMS Regional Conference Series in Mathematics, 40. American Mathematical Society, Providence, R.I., 1979. [8] J. Mawhin and J. R. Jr. Ward, Guiding-like functions forperiodic or bounded solutions of ordinary differential equations, Discrete Contin. Dyn. Syst., 8 (2002), 39-54. [9] P. Omari and F. Zanolin, Remarks on periodic solutions for firstorder nonlinear differential systems, Boll. Un. Mat. Ital. B (6), 2 (1983), 207-218. [10] N. Rouche and J. Mawhin, "Ordinary Differential Equations. Stability and Periodic Solutions," Translated from the French andwith a preface by R. E. Gaines. Surveys and Reference Works in Mathematics, 5. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980.

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##### References:
 [1] A. Capietto, J. Mawhin and F. Zanolin, Continuation theoremsfor periodic perturbations of autonomous systems, Trans. Amer. Math. Soc., 329 (1992), 41-72. doi: 10.1090/S0002-9947-1992-1042285-7. [2] M. Farkas, "Periodic Motions," Applied MathematicalSciences, Springer-Verlag, New York, 104, 1994. [3] M. A. Krasnosel'skiĭ, "The Operator Oftranslation Along the Trajectories of Differential Equations," Translations of Mathematical Monographs, Translated from the Russian by ScriptaTechnica. American Mathematical Society, Providence, R. I., 19, 1968. [4] M. A. Krasnosel'skiĭ and P. P. Zabreĭko, "Geometricalmethods of Nonlinear Analysis," Translated from the Russian by Christian C. Fenske. Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, Berlin, 263, 1984. [5] E. M. Landesman and A. C. Lazer, Nonlinear perturbations oflinear elliptic boundary value problems at resonance, J. Math. Mech., 19 (1969/1970), 609-623. [6] A. C. Lazer and D. E. Leach, Bounded perturbations of forcedharmonic oscillators at resonance, Ann. Mat. Pura Appl. (4), 82 (1969), 49-68. [7] J. Mawhin, "Topological Degree Methods in Nonlinearboundary Value Problems," CBMS Regional Conference Series in Mathematics, 40. American Mathematical Society, Providence, R.I., 1979. [8] J. Mawhin and J. R. Jr. Ward, Guiding-like functions forperiodic or bounded solutions of ordinary differential equations, Discrete Contin. Dyn. Syst., 8 (2002), 39-54. [9] P. Omari and F. Zanolin, Remarks on periodic solutions for firstorder nonlinear differential systems, Boll. Un. Mat. Ital. B (6), 2 (1983), 207-218. [10] N. Rouche and J. Mawhin, "Ordinary Differential Equations. Stability and Periodic Solutions," Translated from the French andwith a preface by R. E. Gaines. Surveys and Reference Works in Mathematics, 5. Pitman (Advanced Publishing Program), Boston, Mass.-London, 1980.
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