Periodic solutions and the avoidance of pull–in instability in non–autonomous micro–electro–mechanical systems

The paper proves existence of a T-periodic solution for x¨ + c x˙ + x − α|x|x = V(t)^2/(1−x)^2 under the condition V_M^2 ≤ A_α by (i) deriving a general lower/upper-solution existence criterion for x¨ + Cx˙ + h(x) − V(t)^2/(1−x)^2 = 0 (Theorem 2.1) and (ii) specializing to h(x)=x−α|x|x using the maximizer of f(x)=(x−αx|x|)(1−x)^2 to show that h(x)=V_M^2/(1−x)^2 has a root in [0,1), leading to Theorem 3.1 with A_α as stated . The candidate solution replicates this structure: it computes the same maximizer and A_α, then constructs constant sub/supersolutions and invokes a standard periodic BVP theorem (Nagumo/Bernstein-type growth) to conclude existence between them. Minor issues in the candidate write-up include a swapped label of lower/upper solutions relative to the paper’s convention, and an over-strong claim that f_α(x)>0 on (0,1) (not true when α>1; positivity holds on [0,β] with β=min(1,1/α) as in the paper). These do not affect the core argument. Overall, both are correct and essentially the same proof, with the paper citing De Coster–Habets and the model citing a closely related existence framework.