Multiplicity of Periodic Bouncing Solutions for Sublinear Damped Variation Systems via Nonsmooth Variational Methods

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 ∂F(t,x)∂t required in some previous papers, such as “∣∂F(t,x)∂t∣≤σ0F(t,x)” and “∣∂F(t,x)∂t∣≤C(1+F(t,x))”, is unnecessary, where F(t, x) ≔ ∫f(t, s) ds, and σ, C are positive constants.