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

Periodic solutions of first order systems

J. R. Ward 1,

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.
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 & 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.  Google Scholar

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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.  Google Scholar

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A. C. Lazer and D. E. Leach, Bounded perturbations of forcedharmonic oscillators at resonance, Ann. Mat. Pura Appl. (4), 82 (1969), 49-68.  Google Scholar

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J. Mawhin, "Topological Degree Methods in Nonlinearboundary Value Problems," CBMS Regional Conference Series in Mathematics, 40. American Mathematical Society, Providence, R.I., 1979.  Google Scholar

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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.  Google Scholar

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P. Omari and F. Zanolin, Remarks on periodic solutions for firstorder nonlinear differential systems, Boll. Un. Mat. Ital. B (6), 2 (1983), 207-218.  Google Scholar

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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.  Google Scholar

show all references

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.  Google Scholar

[2]

M. Farkas, "Periodic Motions," Applied MathematicalSciences, Springer-Verlag, New York, 104, 1994.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

E. M. Landesman and A. C. Lazer, Nonlinear perturbations oflinear elliptic boundary value problems at resonance,, J. Math. Mech., 19 (): 609.   Google Scholar

[6]

A. C. Lazer and D. E. Leach, Bounded perturbations of forcedharmonic oscillators at resonance, Ann. Mat. Pura Appl. (4), 82 (1969), 49-68.  Google Scholar

[7]

J. Mawhin, "Topological Degree Methods in Nonlinearboundary Value Problems," CBMS Regional Conference Series in Mathematics, 40. American Mathematical Society, Providence, R.I., 1979.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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