First-order impulsive functional differential equations with periodic boundary value conditions
This paper is concerned with the existence of maximal and minimal solutions to the following class of first-order impulsive functional-differential equations with periodic boundary conditions \[ u'(t)=f(t,u(t),u(\theta(t))), \quad t\not=t_{k}, \quad t\in [0,T], \] \[ \Delta u(t_{k})=I_{k}(u(t_{k})), \quad k=1,\dots,p, \] \[ u(0)=u(T), \] with \(f\in C([0,T]\times \mathbb{R}^{2}, \mathbb{R})\), \(I_{k}\in C(\mathbb{R}, \mathbb{R})\), \(0<t_1<t_{2}<\dots<t_{p}<T\), \(\Delta u(t_{k})=u(t_{k}^{+})-u(t_{k})\), \(k=1,2,\dots, p\), and \(\theta\in C([0,T],[0,T])\) with \(0\leq\theta(t)\leq t\), \(\forall t\in [0,T]\). The results are obtained via a new concept of upper and lower solutions coupled with the monotone iterative technique. These results improve and complement those in \textit{Z. He} and \textit{W. Ge} [Appl. Math. Comput. 104, 51-63 (1999; Zbl 0930.34060)] and \textit{J. J. Nieto} and \textit{R. Rodrígue-López} [Comput. Math. Appl. 40, 433-442 (2000; Zbl 0958.34055)].
impulsive functional-differential equation
scientific article; zbMATH DE number 1843951