Electron. J. Diff. Equ., Vol. 2010(2010), No. 166, pp. 1-10.

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

Xiaojing Wang, Chuanzhi Bai

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.

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Xiaojing Wang
Department of Mathematics, Huaiyin Normal University
Huaian, Jiangsi 223300, China
email: wangxj2010106@sohu.com
Chuanzhi Bai
Department of Mathematics, Huaiyin Normal University
Huaian, Jiangsi 223300, China
email: czbai8@sohu.com

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