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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.

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