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The existence of periodic solutions for nonlinear systems of first-order differential equations at resonance

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Abstract

The nonlinear system of first-order differential equations with a deviating argument

$$\dot x(t) = Bx(t) + F(x(t - \tau )) + p(t)$$

is considered, where x(t)εR 2, τεR, BεR 2×2, F is bounded and p(t) is continuous and 2π-periodic. Some sufficient conditions for the existence of 2π-periodic solutions of the above equation, in a resonance case, by using the Brouwer degree theory and a continuation theorem based on Mawhin's coincidence degree are obtained. Some applications of the main results to Duffing's equations are also given.

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References

  1. Hale J K:Ordinary Differential Equations[M]. New York: Wiley Interscience, 1969.

    Google Scholar 

  2. Nagle R K. Nonlinear boundary value problems for ordinary differential equations with a small parameter[J].SIAM J Math Analysis, 1978,9(3):719–729.

    Article  MathSciNet  Google Scholar 

  3. Mawhin J. Landesman-Lazter's type problems for nonlinear equations[A]. In:Conferenze Seminario Matematica[C]. Di Bari: Dell Universita, 1977,147.

    Google Scholar 

  4. Fucik S.Solvability of Nonlinear Equations and Boundary Value Problems[M]. Dordrecht, Holland: D. Reidel Publishing, 1980.

    Google Scholar 

  5. Nagle R K, Sinkala Z. Existence of 2π-periodic solutions for nonlinear systems of first-order ordinary differential equations at resonance[J].Nonlinear Analysis (TMA), 1995,25(1):1–16.

    Article  MATH  MathSciNet  Google Scholar 

  6. MA Shi-wang, WANG Zhi-cheng, YU Jian-she. Coincidence degree and periodic solutions of Duffing equations[J].Nonlinear Analysis (TMA), 1998,34(2):443–460.

    Article  Google Scholar 

  7. Lazer A C, Leach D E. Bounded perturbations of forced harmonic oscillations at resonance[J].Ann Mat Pura Appl, 1969,82(1):49–68.

    Article  MATH  MathSciNet  Google Scholar 

  8. Schuur J D. Perturbation at resonance for a fourth order ordinary differential equation[J].J Math Anal Appl, 1978,65(1):20–25.

    Article  MATH  MathSciNet  Google Scholar 

  9. DING Tong-ren. Nonlinear oscillations at a point of resonance [J].Sci China Series A, 1982,(1): 1–13. (in Chinese)

    Google Scholar 

  10. HAO Dun-yuan, MA Shi-wang. Semilinear Duffing equations crossing resonance points[J].J Differential Equations, 1997,133(1):98–116.

    Article  MathSciNet  Google Scholar 

  11. Deimling K.Nonlinear Functional Analysis[M]. New York: Springer-Verlag, 1985.

    Google Scholar 

  12. Mawhin J.Topological degree methods in nonlinear boundary value problems CBMS[J]. Providence RI: Amer Math Soc, 1979,40.

    Google Scholar 

  13. Mawhin J. Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mapping in locally convex topological vector spaces[J].J Differential Equations, 1972,12(2): 610–636.

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Applied Mathematics, Shanghai Jiaotong University, 200030, Shanghai, P R China

    MA Shi-wang Associate Professor, Doctor

  2. Department of Applied Mathematics, Human University, 410082, Changsha, P R China

    MA Shi-wang Associate Professor, Doctor, Wang Zhi-cheng & Yu Jian-she

Authors
  1. MA Shi-wang Associate Professor, Doctor
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  2. Wang Zhi-cheng
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  3. Yu Jian-she
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Additional information

Communicated by LI Ji-bin

Foundation item: the National Natural Science Foundation of China (19801014, 19971026, 19831030)

Biography: MA Shi-wang (1965-)

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Shi-wang, M., Zhi-cheng, W. & Jian-she, Y. The existence of periodic solutions for nonlinear systems of first-order differential equations at resonance. Appl Math Mech 21, 1282–1291 (2000). https://doi.org/10.1007/BF02459250

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  • DOI: https://doi.org/10.1007/BF02459250

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