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.