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.


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
Full Text: DOI


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