Monotone iterative technique for addressing impulsive integro-differential equations in Banach spaces

In this paper, we use a monotone iterative technique in the presence of lower and upper solutions to discuss the existence of solutions for the initial value problem of the impulsive integro-differential equation of Volterra type in a Banach space E {u'(t) = f(t, u (t), Tu(t)), t is an element of J, t not equal t(k), {Delta u vertical bar(t=tk) = I-k(u(t(k))), k = 1, 2, ...., m, {u(0) = x(0), where f is an element of C(J x E x E, E), J = [0, a], 0 < t(1) < t(2) < ... < t(m) < a, and I-k is an element of C(E, E), k = 1, 2,..., m. Under wide monotone conditions and the noncompactness measure condition of nonlinearity f, we obtain the existence of extremal solutions and a unique solution between lower and upper solutions. Our result improves and extends some relevant results in abstract differential equations. (c) 2005 Elsevier Ltd. All rights reserved.