Banach空间含导数项的二阶脉冲微分方程的解

The Solutions for Second Order Impulsive Differential Equations with Dependence on the Derivative Terms in Banach Spaces

讨论了抽象空间中非线性项含一阶导数的二阶脉冲微分方程边值问题{-u″(t)=f(t,u(t),u'(t)),t≠tk,t∈J=[0,1],-Δu'|_(t=t_k)=I_k(u(t_k),u'(t_k)),k=1,2,…,m,u(0)=θ,u(1)=θ解的存在性与唯一性,其中f∈C(J×E×E,E),I_k∈C(E×E,E),k=1,2,…,m.通过选取恰当的工作空间及等价范数,在非线性项f(t,x,y)及脉冲函数Ik满足较一般的非紧性测度条件下,结合新的非紧性测度估计技巧与凝聚映射的Sadovskii不动点定理,得到解及正解的存在性结果.此外,进一步讨论该问题唯一解的存在性.

In this paper,we consider the existence and uniqueness solutions for second order impulsive differential equations with dependence on the first order derivative {-u″（t）=f（t,u（t）,u＇（t））,t≠tk,t∈J=[0,1], -Δu＇｜（t=tk）=Ik（u（tk）,u＇（tk））,k=1,2,…,m, u（0）=θ,u（1）=θ in Banach spaces,where,f∈ C（ J × E × E,E）,Ik∈C（ E × E,E）,k = 1,2,…,m. By choosing proper working space and equivalent norm,while the nonlinear term f（ t,x,y） and Ik（ x,y） satisfy more general non-compactness measure conditions,we obtain the existence results of solutions and positive solutions combining with the estimation skills of the non-compactness measure and the Sadovskii fixed-point theorem. Besides,we discuss the uniqueness of the solutions of this boundary value problem.