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Variational methods to mixed boundary value problem for impulsive differential equations with a parameter. (English) Zbl 1189.34060

The authors consider impulsive boundary value problem for a differential equation of the second order of the form

-u '' (t)=λu(t)+f(t,u(t)),tt i ,t[0,T],-u ' (t i )=I i (u(t i )),i=1,2,,l,u ' (0)=0,u(T)=0,

where λ; 0<t 1 <<t l <T; I i for i=1,...,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.

34B37Boundary value problems for ODE with impulses
34B15Nonlinear boundary value problems for ODE
58E30Variational principles on infinite-dimensional spaces
58E05Abstract critical point theory