Abstract
In this paper, we investigate periodic solutions of a beam equation with jumping nonlinearity. Such a nonlinearity describes the restoring force acting on the bridge due to the possible slackening of the stays of the suspension bridge. By analyzing the beam operator, we find that there is a strictly positive eigenfunction. Thus, for the external force which is a linear combination of the positive eigenfunction and the eigenfunction corresponding to the first negative eigenvalue, using the critical point theory and Brouwer degree theory, we obtain some results on the multiplicity of periodic solutions when the nonlinearity crosses the first negative eigenvalue.
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References
Amann H (1979) Saddle points and multiple solutions of differential equations. Math Z 169:127–166
Ambrosetti A, Rabinowitz PH (1973) Dual variational methods in critical point theory and applications. J Funct Anal 14:349–381
An Y (2002) Periodic solutions of a piecewise linear beam equation with damping and nonconstant load. Electron J Differ Equ 81:1–12
An Y, Zhong C (2003) Periodic solutions of a nonlinear suspension bridge equation with damping and nonconstant load. J Math Anal Appl 279:569–579
Berkovits J, Drábek P, Leinfelder H, Mustonen V, Tajčová G (2000) Time-periodic oscillations in suspension bridges: existence of unique solutions. Nonlinear Anal Real World Appl 1:345–362
Bochicchio I, Giorgi C, Vuk E (2012) Long-term dynamics of the coupled suspension bridge system. Math Models Methods Appl Sci 22:1–22
Choi QH, Jung T (1991) On periodic solutions of the nonlinear suspension bridge equation. Differ Integral Equ 4:383–396
Choi QH, Jung T (1997) A semilinear beam equation with nonconstant loads. Nonlinear Anal 30:5515–5525
Choi QH, Jung T (1999) A nonlinear suspension bridge equation with nonconstant load. Nonlinear Anal 35:649–668
Choi YS, Jen KC, McKenna PJ (1991) The structure of the solution set for periodic oscillations in a suspension bridge model. IMA J Appl Math 47:283–306
Choi QH, Jung T, McKenna PJ (1993) The study of a nonlinear suspension bridge equation by a variational reduction method. Appl Anal 50:73–92
Drábek P, Holubová G (1999) Bifurcation of periodic solutions in symmetric models of suspension bridges. Topol Methods Nonlinear Anal 14:39–58
Humphreys LD (1997) Numerical mountain pass solutions of a suspension bridge equation. Nonlinear Anal 28:1811–1826
Humphreys LD, McKenna PJ (1999) Multiple periodic solutions for a nonlinear suspension bridge equation. IMA J Appl Math 63:37–49
Lazer AC, McKenna PJ (1986) Critical point theory and boundary value problems with nonlinearities crossing multiple eigenvalues. Commun Partial Differ Equ 11:1653–1676
Lazer AC, McKenna PJ (1988) A symmetry theorem and applications to nonlinear partial differential equations. J Differ Equ 72:95–106
Lazer AC, McKenna PJ (1989a) Existence, uniqueness, and stability of oscillations in differential equations with asymmetric nonlinearities. Trans Am Math Soc 315:721–739
Lazer AC, McKenna PJ (1989b) A semi-Fredholm principle for periodically forced systems with homogeneous nonlinearities. Proc Am Math Soc 106:119–125
Lazer AC, McKenna PJ (1990) Large-amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis. SIAM Rev 32:537–578
McKenna PJ, Walter W (1987) Nonlinear oscillations in a suspension bridge. Arch Ration Mech Anal 98:167–177
Rabinowitz PH (1986) Minimax methods in critical point theory with applications to differential equations. In: CBMS regional conference series in mathematics, vol 65. American Mathematical Society, Providence
Acknowledgements
This work is partially supported by NSFC Grants (12225103, 12071065 and 11871140) and the National Key Research and Development Program of China (2020YFA0713602 and 2020YFC1808301).
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Sun, X., Ji, S. Periodic solutions of a beam equation with jumping nonlinearity. Comp. Appl. Math. 43, 143 (2024). https://doi.org/10.1007/s40314-024-02657-y
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DOI: https://doi.org/10.1007/s40314-024-02657-y