共振条件下的二阶多点边值问题解的存在性和多解性

Existence and multiplicity results for a second-order multi-point boundary value problem at resonance

在共振条件m∑k=1a_k=1下,运用紧向量场方程的解集连通理论对二阶多点边值问题u″(t)=f(t,u(t))+e(t),t∈[0,1],u'(0)=0,u(1)=m∑k=1a_ku(η_k)建立了解的存在性和多解性结果。其中,f:[0,1]×R→R连续,e∈C([0,1],R),0<η_1<η_2<…<η_m<1,a_k>0(k=1,2,…,m)。

It is investigated that the existence and multiplicity results for a second-order multi-point boundary value problem at resonanceu″( t) = f( t,u( t)) + e( t),t ∈ [0,1],u'( 0) = 0,u( 1) =m∑k = 1a_ku( ηk)by the connectivity properties of solution set of param eterized fam ilies of com pact vector fields. where f: [0,1] ×R →R is continuous,e ∈ C( [0,1],R),0 < η_1< η_2< … < η_m< 1,a_k> 0( k = 1,2,…,m).