A method of L-quasi-upper and lower solutions for boundary value problems of impulsive differential equation in Banach spaces (Q1800666)

This paper is devoted to the study of the existence of solutions of the following periodic impulsive problem in Banach spaces: \[ \begin{aligned} &u'(t)=f(t,u(t),u(t)), \, t \in [0,1], \; t \neq t_k; \\ &u(t_k^+)-u(t_k^-)=I_k(u(t_k)), \; k=1, \dots,m; \, u(0)=u(1),\end{aligned} \] where \(f\in C([0,1] \times E \times E,E)\), \(0 < t_0 < \cdots < t_m <1\) and \(I_k \in C(E,E)\), with \(E\) a Banach space. Under suitable monotonicity assumptions on \(f\) and \(I_k\), the authors deduce the existence of coupled maximal and minimal L-quasi-solutions, which can be obtained by a monotone iterative procedure starting from the coupled L-quasi-upper and lower solutions. The result holds from the expression, via Green's function, of the solution of the related linear problems and suitable inequalities. Under stronger assumptions, it is proved that the considered problem has at least one solution lying between the maximal and minimal L-quasi-solutions.