Abstract
Two results about the multiplicity of nontrivial periodic bouncing solutions for sublinear damped vibration systems −ẍ = g(t)ẋ + f(t, x) are obtained via the Generalized Nonsmooth Saddle Point Theorem and a technique established by Wu Xian and Wang Shaomin. Both of them imply the condition “f ≥ 0” required in some previous papers can be weakened, furthermore, one of them also implies the condition about \({\partial F(t,x)\over{\partial t}}\) required in some previous papers, such as “\(\vert{\partial F(t,x)\over{\partial t}}\vert\leq\sigma_{0}F(t,x)\)” and “\(\vert{\partial F(t,x)\over{\partial t}}\vert\leq C(1+F(t,x))\)”, is unnecessary, where F(t, x) ≔ ∫ x0 f(t, s) ds, and σ0, C are positive constants.
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Supported by the National Natural Science Foundation of China (Grant No. 12171355) and Elite Scholar Program in Tianjin University, P. R. China
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Wang, S.Q., Guo, F. Multiplicity of Periodic Bouncing Solutions for Sublinear Damped Variation Systems via Nonsmooth Variational Methods. Acta. Math. Sin.-English Ser. 39, 1332–1350 (2023). https://doi.org/10.1007/s10114-023-1166-2
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DOI: https://doi.org/10.1007/s10114-023-1166-2
Keywords
- Damped vibration systems
- generalized Nonsmooth Saddle Point Theorem
- sublinear conditions
- periodic bouncing solutions
- multiplicity