Nonlinear boundary value problems for first order differential equation with impulses (Q1032028)

The authors consider the boundary value problem \[ \begin{aligned} &u'(t) = f(t,u(t)),\quad 0 < t < T,\;t \neq t_k,\\ &\triangle u(t_k) = I_k(u(t_k)),\quad k = 1,2,\dots,m,\\ &g(u(0),u(T)) = 0, \end{aligned} \] where \(0 < t_1 < \dots < t_m < T\), \(f\) is continuous on \(([0,T]\setminus\{t_1,\dots,t_m\})\times{\mathbb R}\), \(I_k\) and \(g\) are continuous functions. New sufficient conditions ensuring the existence of a solution are presented. The results are obtained by using lower and upper solutions method (the existence of a pair of well-ordered lower and upper solutions is assumed). Note that a similar problem has been investigated in the paper \textit{I. Rachůnková} and \textit{J. Tomeček} [Math. Notes, Miskolc 3, No.~1, 59--69 (2002; Zbl 1017.34007)].