
Existence and multiplicity of nontrivial solutions of Dirichlet problems for secondorder impulsive differential equation
 LI Xiaoyan, XU Man

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2016, 51(12):
2935.
doi:10.6040/j.issn.16719352.0.2016.078

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In this paper, we study the existence and multiplicity of nontrivial solutions of Dirichlet problems for secondorder impulsive differential equation{u″(t)+f(t,u(t))=0, t∈(0,1), t≠t_{i},Δu_{t=ti}=α_{i}u(t_{i}), i=1, 2,…,k,u(0)=u(1)=0,where α_{i}>1, i=1, 2,…,k are given constants, 0=t_{}0<t_{}1<t_{}2<…<t_{k}<t_{k+}1=1 are given impulsive points. Δu_{t=ti}=u(t^{+}_{i})u(t^{}_{i}), u(t^{+}_{i}), u(t^{}_{i}) denote the right and left limit of u at t=t_{i}, respectively. f∈C(［0,1］×R, R). The main results extend and improve some results on existence and multiplicity of nontrivial solutions of Dirichlet problems for secondorder impulsive differential equation. The proof of the main results are based on the LópezGómezs bifurcation theory established in 2001.