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Periodic solutions to parameter-dependent equations with a \(\phi \)-Laplacian type operator

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Abstract

We study the periodic boundary value problem associated with the \(\phi \)-Laplacian equation of the form \((\phi (u'))'+f(u)u'+g(t,u)=s\), where s is a real parameter, f and g are continuous functions, and g is T-periodic in the variable t. The interest is in Ambrosetti–Prodi type alternatives which provide the existence of zero, one or two solutions depending on the choice of the parameter s. We investigate this problem for a broad family of nonlinearities, under non-uniform type conditions on g(t, u) as \(u\rightarrow \pm \infty \). We generalize, in a unified framework, various classical and recent results on parameter-dependent nonlinear equations.

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References

  1. Amann, H., Ambrosetti, A., Mancini, G.: Elliptic equations with noninvertible Fredholm linear part and bounded nonlinearities. Math. Z. 158, 179–194 (1978)

    Article  MathSciNet  Google Scholar 

  2. Ambrosetti, A., Prodi, G.: On the inversion of some differentiable mappings with singularities between Banach spaces. Ann. Mat. Pura Appl. 93, 231–246 (1972)

    Article  MathSciNet  Google Scholar 

  3. Bereanu, C.: Multiple periodic solutions of some Liénard equations with \(p\)-Laplacian. Bull. Belg. Math. Soc. Simon Stevin 15, 277–285 (2008)

    MathSciNet  MATH  Google Scholar 

  4. Bereanu, C., Mawhin, J.: Existence and multiplicity results for some nonlinear problems with singular \(\phi \)-Laplacian. J. Differ. Equ. 243, 536–557 (2007)

    Article  MathSciNet  Google Scholar 

  5. Bereanu, C., Mawhin, J.: Multiple periodic solutions of ordinary differential equations with bounded nonlinearities and \(\phi \)-Laplacian. NoDEA Nonlinear Differ. Equ. Appl. 15, 159–168 (2008)

    Article  MathSciNet  Google Scholar 

  6. Cid, J.A., Torres, P.J.: On the existence and stability of periodic solutions for pendulum-like equations with friction and \(\phi \)-Laplacian. Discrete Contin. Dyn. Syst. 33, 141–152 (2013)

    Article  MathSciNet  Google Scholar 

  7. De Coster, C., Habets, P.: Two-point boundary value problems: lower and upper solutions, vol. 205 of Mathematics in Science and Engineering, Elsevier B. V. Amsterdam (2006)

  8. Fabry, C., Mawhin, J., Nkashama, M.N.: A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations. Bull. Lond. Math. Soc. 18, 173–180 (1986)

    Article  MathSciNet  Google Scholar 

  9. Feltrin, G., Zanolin, F.: An application of coincidence degree theory to cyclic feedback type systems associated with nonlinear differential operators. Topol. Methods Nonlinear Anal. 50, 683–726 (2017)

    MathSciNet  MATH  Google Scholar 

  10. Folland, G.B.: Real Analysis. Modern Techniques and their Applications. Pure and Applied Mathematics (New York). Wiley, New York (1984)

    MATH  Google Scholar 

  11. Furi, M., Pera, M.P.: On unbounded branches of solutions for nonlinear operator equations in the nonbifurcating case. Boll. Un. Mat. Ital. B 1, 919–930 (1982)

    MathSciNet  MATH  Google Scholar 

  12. García-Huidobro, M., Manásevich, R., Zanolin, F.: Strongly nonlinear second-order ODEs with unilateral conditions. Differ. Integral Equ. 6, 1057–1078 (1993)

    MathSciNet  MATH  Google Scholar 

  13. Hale, J.K.: Ordinary Differential Equations, 2nd edn. Robert E. Krieger Publishing Co., Inc., Huntington, NY (1980)

    MATH  Google Scholar 

  14. Kristály, A., Rădulescu, V. D., Varga, C. G.: Variational principles in mathematical physics, geometry, and economics. Qualitative analysis of nonlinear equations and unilateral problems, vol. 136 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge (2010)

  15. Manásevich, R., Mawhin, J.: Periodic solutions for nonlinear systems with \(p\)-Laplacian-like operators. J. Differ. Equ. 145, 367–393 (1998)

    Article  MathSciNet  Google Scholar 

  16. Mawhin, J.: The periodic Ambrosetti-Prodi problem for nonlinear perturbations of the \(p\)-Laplacian. J. Eur. Math. Soc. 8, 375–388 (2006)

    Article  MathSciNet  Google Scholar 

  17. Mawhin, J.: Resonance problems for some non-autonomous ordinary differential equations, in: Stability and bifurcation theory for non-autonomous differential equations, vol. 2065 of Lecture Notes in Math., Springer, Heidelberg, pp. 103–184 (2013)

    Google Scholar 

  18. Mawhin, J., Rebelo, C., Zanolin, F.: Continuation theorems for Ambrosetti-Prodi type periodic problems. Commun. Contemp. Math. 2, 87–126 (2000)

    MathSciNet  MATH  Google Scholar 

  19. Mawhin, J., Szymańska-Dȩbowska, K.: Second-order ordinary differential systems with nonlocal Neumann conditions at resonance. Ann. Mat. Pura Appl. 195, 1605–1617 (2016)

    Article  MathSciNet  Google Scholar 

  20. Obersnel, F.: Classical and non-classical sign-changing solutions of a one-dimensional autonomous prescribed curvature equation. Adv. Nonlinear Stud. 7, 671–682 (2007)

    Article  MathSciNet  Google Scholar 

  21. Omari, P.: Nonordered lower and upper solutions and solvability of the periodic problem for the Liénard and the Rayleigh equations. Rend. Istit. Mat. Univ. Trieste 20(suppl.), 54–64 (1988)

    MathSciNet  Google Scholar 

  22. Ortega, R.: Stability of a periodic problem of Ambrosetti-Prodi type. Differ. Integral Equ. 3, 275–284 (1990)

    MathSciNet  MATH  Google Scholar 

  23. Pucci, P., Serrin, J.: The maximum principle. Progress in Nonlinear Differential Equations and their Applications, vol. 73. Birkhäuser Verlag, Basel (2007)

    Book  Google Scholar 

  24. Sovrano, E.: Ambrosetti-Prodi type result to a Neumann problem via a topological approach. Discrete Contin. Dyn. Syst. Ser. S 11, 345–355 (2018)

    MathSciNet  MATH  Google Scholar 

  25. Sovrano, E., Zanolin, F.: Ambrosetti-Prodi periodic problem under local coercivity conditions. Adv. Nonlinear Stud. 18, 169–182 (2018)

    Article  MathSciNet  Google Scholar 

  26. Villari, G.: Soluzioni periodiche di una classe di equazioni differenziali del terz’ordine quasi lineari. Ann. Mat. Pura Appl. 73, 103–110 (1966)

    Article  MathSciNet  Google Scholar 

  27. Ward, J.R.: Periodic solutions of ordinary differential equations with bounded nonlinearities. Topol. Methods Nonlinear Anal. 19, 275–282 (2002)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors thank the anonymous referee for the useful remarks that inspired the discussion in Sect. 5.

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Authors and Affiliations

  1. Department of Mathematics, Computer Science and Physics, University of Udine, Via delle Scienze 206, 33100, Udine, Italy

    Guglielmo Feltrin & Fabio Zanolin

  2. Istituto Nazionale di Alta Matematica “Francesco Severi” at Department of Mathematics and Geosciences, University of Trieste, Via A. Valerio 12/1, 34127, Trieste, Italy

    Elisa Sovrano

Authors
  1. Guglielmo Feltrin
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  2. Elisa Sovrano
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  3. Fabio Zanolin
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Correspondence to Fabio Zanolin.

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Work partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The first author is supported by the project ERC Advanced Grant 2013 n. 339958 “Complex Patterns for Strongly Interacting Dynamical Systems—COMPAT”

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Feltrin, G., Sovrano, E. & Zanolin, F. Periodic solutions to parameter-dependent equations with a \(\phi \)-Laplacian type operator. Nonlinear Differ. Equ. Appl. 26, 38 (2019). https://doi.org/10.1007/s00030-019-0585-3

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