## The existence and multiplicity of solutions for an impulsive differential equation with two parameters via a variational method.(English)Zbl 1198.34037

The authors consider the impulsive boundary value problem for an ordinary differential equation of second order
\begin{aligned} &-u''(t) +cu(t) = \lambda g(t,u(t)) \quad \text{a.e.}\;t \in[0,T], t\neq t_k,\\ &\triangle u'(t_k) = I_k(u(t_k)), \quad k = 1,\ldots, p-1,\\ &u(0) = u(T), \quad u'(0) = u'(T),\end{aligned}
where $$0 < t_1 < \ldots < t_{p-1} < T$$; $$c, \lambda \in {\mathbb R}$$, $$\lambda \neq 0$$; $$g : [0,T] \times {\mathbb R} \to {\mathbb R}$$ is a continuous function; $$I_k : {\mathbb R} \to {\mathbb R}$$ are continuous. Some new criteria to guarantee the existence of at least one solution, two solutions and infinitely many solutions according to the values of the pair $$(c,\lambda)$$ are given. The results are obtained by using variational methods and critical point theory.

### MSC:

 34B37 Boundary value problems with impulses for ordinary differential equations 34B08 Parameter dependent boundary value problems for ordinary differential equations 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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