The solutions for second order impulsive differential equations with dependence on the derivative terms in Banach spaces. (Chinese. English summary) Zbl 1389.34186
Summary: In this paper, we consider the existence and uniqueness of solutions for second-order impulsive differential equations with dependence on the first-order derivative
\[
\begin{aligned} - u''(t) &=f(t,u(t),\, u'(t)),\, t \neq t_k,\, t \in J=[0,1], \\ -\Delta u'|_{t=tk} &= I_k(u(t_k),\, u'(t_k)),\, k = 1, 2, \cdots, m,\\ u(0) &=\theta,\, u(1)=\theta \end{aligned}
\]
in Banach spaces, where, \(f \in C(J\times E \times E, E)\), \(I_k\in C(E \times E, E)\), \(k = 1, 2, \cdots, m\). We obtain the existence results of solutions and positive solutions by using the non-compactness measure and the Sadovskii fixed-point theorem. Moreover, we discuss the uniqueness of the solutions.
MSC:
34G20 | Nonlinear differential equations in abstract spaces |
34B37 | Boundary value problems with impulses for ordinary differential equations |
47N20 | Applications of operator theory to differential and integral equations |