Some sufficient conditions on the existence and multiplicity of solutions for the damped vibration problems with impulsive effects ⎧ u”(t) + g(t)u’(t) + f(t,u(t)) = 0, a.e. t ∈ [0,T ⎨ u(0) = u(T) = 0 ⎩ $\Delta u^{\text{'}}\left(t_j\right)=u^{\text{'}}(t{⁺}_j-u^{\text{'}}\left(t{¯}_j\right)=I_j\left(u\left(t_j\right)\right)$, j = 1,...,p, are established, where $t₀=0<t₁<\cdots <t_p<t_{p+1}=T$, g ∈ L¹(0,T;ℝ), f: [0,T] × ℝ → ℝ is continuous, and $I_j:ℝ\to ℝ$, j = 1,...,p, are continuous. The solutions are sought by means of the Lax-Milgram theorem and some critical point theorems. Finally, two examples are presented to illustrate the effectiveness of our results....

In this paper, which is an extension of [4], we first show the existence of solutions to a class of Min Sup problems with linked constraints, which satisfy a certain property. Then, we apply our result to a class of weak nonlinear bilevel problems. Furthermore, for such a class of bilevel problems, we give a relationship with appropriate d.c. problems concerning the existence of solutions.