    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 & Continuous Dynamical Systems, 2013, 33 (1) : 381-389. doi: 10.3934/dcds.2013.33.381
##### References:
  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  M. Farkas, "Periodic Motions," Applied MathematicalSciences, Springer-Verlag, New York, 104, 1994. Google Scholar  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  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  E. M. Landesman and A. C. Lazer, Nonlinear perturbations oflinear elliptic boundary value problems at resonance,, J. Math. Mech., 19 (): 609. Google Scholar  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  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  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  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  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:
  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  M. Farkas, "Periodic Motions," Applied MathematicalSciences, Springer-Verlag, New York, 104, 1994. Google Scholar  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  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  E. M. Landesman and A. C. Lazer, Nonlinear perturbations oflinear elliptic boundary value problems at resonance,, J. Math. Mech., 19 (): 609. Google Scholar  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  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  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  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  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
  Marc Henrard. Homoclinic and multibump solutions for perturbed second order systems using topological degree. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 765-782. doi: 10.3934/dcds.1999.5.765  Anna Capietto, Walter Dambrosio. A topological degree approach to sublinear systems of second order differential equations. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 861-874. doi: 10.3934/dcds.2000.6.861  Jean Mawhin, James R. Ward Jr. Guiding-like functions for periodic or bounded solutions of ordinary differential equations. Discrete & Continuous Dynamical Systems, 2002, 8 (1) : 39-54. doi: 10.3934/dcds.2002.8.39  Huong Le Thi, Stéphane Junca, Mathias Legrand. First Return Time to the contact hyperplane for $N$-degree-of-freedom vibro-impact systems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021031  Kazuyuki Yagasaki. Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 387-402. doi: 10.3934/dcds.2011.29.387  Chungen Liu, Xiaofei Zhang. Subharmonic solutions and minimal periodic solutions of first-order Hamiltonian systems with anisotropic growth. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1559-1574. doi: 10.3934/dcds.2017064  Ahmed Y. Abdallah. Attractors for first order lattice systems with almost periodic nonlinear part. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1241-1255. doi: 10.3934/dcdsb.2019218  Juntao Sun, Jifeng Chu, Zhaosheng Feng. Homoclinic orbits for first order periodic Hamiltonian systems with spectrum point zero. Discrete & Continuous Dynamical Systems, 2013, 33 (8) : 3807-3824. doi: 10.3934/dcds.2013.33.3807  Yanqin Xiong, Maoan Han. Planar quasi-homogeneous polynomial systems with a given weight degree. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 4015-4025. doi: 10.3934/dcds.2016.36.4015  Montserrat Corbera, Claudia Valls. Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3209-3233. doi: 10.3934/dcdsb.2020225  Jaume Llibre, Y. Paulina Martínez, Claudio Vidal. Linear type centers of polynomial Hamiltonian systems with nonlinearities of degree 4 symmetric with respect to the y-axis. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 887-912. doi: 10.3934/dcdsb.2018047  Anna Go??biewska, S?awomir Rybicki. Equivariant Conley index versus degree for equivariant gradient maps. Discrete & Continuous Dynamical Systems - S, 2013, 6 (4) : 985-997. doi: 10.3934/dcdss.2013.6.985  Yong Liu, Jing Tian, Xuelin Yong. On the even solutions of the Toda system: a degree argument approach. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021075  Sihong Su. A new construction of rotation symmetric bent functions with maximal algebraic degree. Advances in Mathematics of Communications, 2019, 13 (2) : 253-265. doi: 10.3934/amc.2019017  Wenying Zhang, Zhaohui Xing, Keqin Feng. A construction of bent functions with optimal algebraic degree and large symmetric group. Advances in Mathematics of Communications, 2020, 14 (1) : 23-33. doi: 10.3934/amc.2020003  Xiaojun Chang, Yong Li. Rotating periodic solutions of second order dissipative dynamical systems. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 643-652. doi: 10.3934/dcds.2016.36.643  Maurizio Grasselli, Morgan Pierre. Convergence to equilibrium of solutions of the backward Euler scheme for asymptotically autonomous second-order gradient-like systems. Communications on Pure & Applied Analysis, 2012, 11 (6) : 2393-2416. doi: 10.3934/cpaa.2012.11.2393  Tatsien Li (Daqian Li). Global exact boundary controllability for first order quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1419-1432. doi: 10.3934/dcdsb.2010.14.1419  Elena-Alexandra Melnig. Internal feedback stabilization for parabolic systems coupled in zero or first order terms. Evolution Equations & Control Theory, 2021, 10 (2) : 333-351. doi: 10.3934/eect.2020069  Xiangjin Xu. Sub-harmonics of first order Hamiltonian systems and their asymptotic behaviors. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 643-654. doi: 10.3934/dcdsb.2003.3.643

2019 Impact Factor: 1.338