二阶脉冲微分方程三点边值问题

Three-point Boundary Value Problems for Second-order Impulsive Ordinary Differential Equations

研究了一类具有边值条件u(0)=0,u(1)-αu(η)=b形如u″+a(t)f(u)=0,-Δu′(tk)=Ik(u(tk))(k=1,2,…,m)的二阶脉冲微分方程三点边值问题解的存在性。在合适的假设条件下,利用Schauder不动点定理讨论了该脉冲微分方程解的存在性,并在此基础上通过相关引理给出了方程至少存在一个正解和无解的充分条件,即存在ε*>0,使得当0<b<ε*时,所考虑的脉冲微分方程边值问题至少存在一个正解;另外,当b>ε*时,边值问题无解。

This paper investigates boundary value problems for a class of second-order three-point impulsive ordinary differential e- quations u＂＋a（t） f（u） = O, --△ u＇ （tk） = Ik （u（tk） ） （k = 1,2, ..., m） with boundary value conditions u（O） = 0, u（ 1 ） -- au（η ） = b. The existence of solutions is discussed by the Sehauder＇s fixed point theorem. A sufficient condition on the existence of positive solutions is obtained through relevant lemmas under suitable assumption conditions, that is, there exists a scalar ε＊〉Osuch that the consid- ered impulsive ordinary differential equations boundary problem has at least one solution for 0〈b〈ε＊ and no solution for b〉ε＊ , re- spectively. Some existing corresponding results are improved and extended.