Variational methods to mixed boundary value problem for impulsive differential equations with a parameter (Q730671)

The authors consider impulsive boundary value problem for a differential equation of the second order of the form \[ \begin{aligned} & -u''(t) = \lambda u(t) + f(t,u(t)),\;t \neq t_i, t \in [0,T], \\ & -\triangle u'(t_i) = I_i(u(t_i)), \;i = 1,2,\dots,l, \\ & u'(0) = 0, \;u(T) = 0, \end{aligned} \] where \(\lambda \in {\mathbb R}\); \(0 < t_1 < \dots < t_l < T\); \(I_i\) for \(i = 1,\ldots,l\) and \(f\) are continuous functions. Sufficient conditions for the existence of at least one solution, multiple and infinitely many solutions are obtained. The proofs are based on the critical point theory.