Periodic solutions of first order systems

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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

Abstract
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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.

Keywords: coincidence degree, guiding functions, gradient systems, Periodic solutions, first order systems, homotopy., Brouwer degree.

Mathematics Subject Classification: 34C15, 34B2.

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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