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Periodic Perturbations of Linear Systems at Resonance

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Abstract

Second-order linear Hamiltonian systems at resonance with periodic nonlinearity is investigated. An existence result of solutions for such systems is obtained by means of variational methods, saddle point theorem, and an index theory for second-order linear Hamiltonian systems. Meanwhile, two examples and two extensions are presented.

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References

  1. D. Bonheure, C. Fabry, and D. Ruiz, Problem at resonance for equations with periodic nonlinearities, Nonlinear Anal., 55 (2003), 557–581.

    Article  MathSciNet  MATH  Google Scholar 

  2. Y. Bu, Multiplicity of solutions to asymptotically linear seond-order ordinary differential system, Ann. Differential Equations, 27(3) (2011), 283–292.

    MathSciNet  MATH  Google Scholar 

  3. A. Cañada and D. Ruiz, Periodic perturbations of a class of resonant problems, Calc. Var., 23 (2005), 281–300.

    Article  MathSciNet  MATH  Google Scholar 

  4. A. Cañada and F. Roca, Existence and multiplicity of solutions of some conservative pendulum-type equations with homogeneous Dirichlet conditions, Differential Integral Equations, 10(6) (1997), 1113–1122.

    MathSciNet  MATH  Google Scholar 

  5. A. Cañada and F. Roca, Qualitative properties of some noncoercive functionals arising from nonlinear boundary value problems, Nonlinear Anal., 37 (1999), 861–873.

    Article  MathSciNet  MATH  Google Scholar 

  6. A. Cañada, Nonlinear ordinary boundary value problems under a combined effect of periodic and attractive nonlinearities, J. Math. Anal. Appl., 243 (2000), 174–189.

    Article  MathSciNet  MATH  Google Scholar 

  7. A. Cañada and A. Ureña, Solvability of some pendulum-type equations with linear damping and homogeneous Dirichlet conditions, Nonlinear Anal., 49 (2002), 1149–1165.

    Article  MathSciNet  MATH  Google Scholar 

  8. A. Cañada and D. Ruiz, Resonant nonlinear boundary value problems with almost periodic nonlinearity, Bull. Belg. Math. Soc. Simon Stevin, 9 (2002), 193–204.

    MathSciNet  MATH  Google Scholar 

  9. A. Cañada and D. Ruiz, Asymptotic analysis of oscillating parametric integrals and ordinary boundary value problems at resonance, J. Math. Anal. Appl., 313 (2006), 218–233.

    Article  MathSciNet  MATH  Google Scholar 

  10. A. Cañada, Multiplicity results near the principal eigenvalue for boundary-value problems with periodic nonlinearity, Math. Nachr., 280 (2007), 235–241.

    Article  MathSciNet  MATH  Google Scholar 

  11. S. Chen and S. Li, Splitting lemma in the infinity and a strong resonant problem with periodic nonlinearlity, Nonlinear Anal., 65 (2006), 567–582.

    Article  MathSciNet  MATH  Google Scholar 

  12. S. Chen and S. Li, Splitting lemma at infinity and a strong resonant problem with periodic nonlinearlity, Calc. Var., 27(1) (2006), 105–123.

    Article  MathSciNet  Google Scholar 

  13. Y. Dong, Index theory, nontrivial solutions and asymptotically linear second-order Hamiltonian systems, J. Differential Equations, 214 (2005), 233–255.

    Article  MathSciNet  MATH  Google Scholar 

  14. Y. Dong, Index theory for linear selfadjoint operator equations and nontrivial solutions for asymptotically linear operator equations, Calc. Var., 38 (2010), 75–109.

    Article  MathSciNet  MATH  Google Scholar 

  15. J. Gámez and J. Ruiz-Hidalgo, Bifurcation from infinity and resonance results at high eigenvalues in dimension one, J. Funct. Spaces Appl., 2012, Article ID 284696, 11 pages.

    Google Scholar 

  16. N. Ghoussoub, Duality and perturbation methods in critical point theory, Cambridge Tracts in Math., 107, Cambridge University Press, Cambridge (1993).

    Book  MATH  Google Scholar 

  17. P. Girg, F. Roca, and S. Villegas, Semilinear Sturm-Liouville problem with periodic nonlinearity, Nonlinear Anal., 61 (2005), 1157–1178.

    Article  MathSciNet  MATH  Google Scholar 

  18. D. Lupo and S. Solimini, A note a resonance problem, Proc. Edinb. Math. Soc., 102A (1986), 1–7.

    MathSciNet  MATH  Google Scholar 

  19. I. Main, Vibrations and waves in physics, Cambridge University Press, Cambridge (1993).

    Book  MATH  Google Scholar 

  20. A. Nayfeh and D. Mook, Nonlinear oscillations, Wiley, NewYork (1979).

    MATH  Google Scholar 

  21. P. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional Conference Series in Mathematics, 65 (1986), AMS, Providence, RI.

    Google Scholar 

  22. R. Schaaf and K. Schimit, A class of nonlinear Sturm-Liouville problems with infinitely many solutions, Trans. Amer. Math. Soc., 306 (1988), 853–859.

    Article  MathSciNet  MATH  Google Scholar 

  23. R. Schaaf and K. Schimit, Perodic perturbations of linear problems at resonance on convex domains, Rocky Mountain J. Math., 20(4) (1990), 1119–1131.

    Article  MathSciNet  MATH  Google Scholar 

  24. R. Schaaf and K. Schimit, Asymptotic behavior of positive solution branches of elliptic problems with linear part at resonance, Z. Angew. Math. Phys., 43 (1992), 645–676.

    Article  MathSciNet  MATH  Google Scholar 

  25. S. Solimini, On the solvability of some elliptic partial differential equations with the linear part at resonance, J. Math. Annal. Appl., 117 (1986), 138–152.

    Article  MathSciNet  MATH  Google Scholar 

  26. A. Ureña, Nondegeneracy of elliptic problems with periodic nonlinearities in the plane, Z. Angew. Math. Phys., 58 (2007), 391–415.

    Article  MathSciNet  MATH  Google Scholar 

  27. J. Ward, A boundary value with a periodic nonlinearity, Nonliear Anal., 10 (1986), 207–213.

    Article  MathSciNet  MATH  Google Scholar 

  28. X. Wu and C. Tang, Some existence theorems for elliptic resonant problems, J. Math. Anal. Appl., 264 (2001), 133–146.

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. School of Mathematical Science, Nanjing Normal University, Nanjing, 210023, P. R. China

    Yucheng Bu

  2. Danyang Normal School, Zhenjiang College, Zhenjiang, 212300, P. R. China

    Yucheng Bu

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  1. Yucheng Bu
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Bu, Y. Periodic Perturbations of Linear Systems at Resonance. Indian J Pure Appl Math 50, 225–238 (2019). https://doi.org/10.1007/s13226-019-0320-0

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  • DOI: https://doi.org/10.1007/s13226-019-0320-0

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