Monotone iterative method for abstract impulsive integro-differential equations with nonlocal conditions in Banach spaces

Abstract

In this paper we use a monotone iterative technique in the presence of the lower and upper solutions to discuss the existence of mild solutions for a class of semilinear impulsive integro-differential evolution equations of Volterra type with nonlocal conditions in a Banach space E $$\left\{ \begin{gathered} u'(t) + Au(t) = f(t,u(t),Gu(t)) t \in J,t \ne t_k , \hfill \\ \Delta _{\left. u \right|_{t = t_k } } = u\left( {t_k^ + } \right) - u\left( {t_k^ - } \right) = I_k \left( {u\left( {t_k } \right)} \right), k = 1,2, \ldots ,m, \hfill \\ u(0) = g(u) + x_0 , \hfill \end{gathered} \right.$$ where A: D(A) ⊂ E → E is a closed linear operator and −A generates a strongly continuous semigroup T(t) (t ⩾ 0) on E, f ∈ C(J × E × E, E), J = [0, a], 0 < t 1 < t 2 < ... < t m < a, I k ∈ C(E, E), k = 1, 2, ..., m, and g constitutes a nonlocal condition. Under suitable monotonicity conditions and noncompactness measure conditions, we obtain the existence of the extremal mild solutions between the lower and upper solutions assuming that −A generates a compact semigroup, a strongly continuous semigroup or an equicontinuous semigroup. The results improve and extend some relevant results in ordinary differential equations and partial differential equations. Some concrete applications to partial differential equations are considered.