Electronic Journal of Differential Equations,
Vol. 2010(2010), No. 166, pp. 1-10.

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

Authors: Xiaojing Wang (Huaiyin Normal Univ., Jiangsi, China)
         Chuanzhi Bai  (Huaiyin Normal Univ., Jiangsi, China)

Abstract:
 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.

Submitted November 1, 2010. Published November 17, 2010.
Math Subject Classifications: 34A37, 34B15.
Key Words: Impulsive integro-differential equation; 
           coupled lower-upper quasi-solutions; 
	   monotone iterative technique.