Impulsive boundary-value problems for first-order integro-differential equations

Wang, Xiaojing ; Bai, Chuanzhi

Electronic journal of differential equations, Tome 2010 (2010)

Résumé

This article concerns boundary-value problems of first-order nonlinear impulsive integro-differential equations:

$\displaylines{ y'(t) + a(t)y(t) = f(t, y(t), (Ty)(t), (Sy)(t)), \quad t \in J_0, \cr \Delta y(t_k) = I_k(y(t_k)), \quad k = 1, 2, \dots , p, \cr y(0) + \lambda \int_0^c y(s) ds = - y(c), \quad \lambda \le 0, }$

where $J_0 = [0, c] \setminus \{t_1, t_2, \dots , t_p\}, f \in C(J \times \mathbb{R} \times \mathbb{R} \times \mathbb{R}, \mathbb{R}), I_k \in C(\mathbb{R}, \mathbb{R}), a \in C(\mathbb{R}, \mathbb{R})$ and $a(t) \le 0$ for $t \in [0, c]$. Sufficient conditions for the existence of coupled extreme quasi-solutions are established by using the method of lower and upper solutions and monotone iterative technique. Wang and Zhang [18] studied the existence of extremal solutions for a particular case of this problem, but their solution is incorrect.

Détail

Classification : 34A37, 34B15
Keywords: impulsive integro-differential equation, coupled lower-upper quasi-solutions, monotone iterative technique