Existence results for impulsive semilinear differential inclusions with nonlinear boundary conditions

Abstract

In this paper, we discuss the nonlinear boundary problem for first-order impulsive semilinear differential inclusions. We establish existence results by using Martelli's fixed point theorem with upper and lower solutions method. We find that by giving different definitions of lower and upper solutions we can get all existence results. We also present an example.

Mild Solutions of Second-Order Semilinear Impulsive Differential Inclusions in Banach Spaces

“…Notice that the explicit formula for the mild solution of Equation ( 15) can be obtained also as restriction to [a, b] of a mild solution defined in [0, T]. In fact, assume that x : [0, T] → E is a mild solution of Equation (12). Then, according to the definition, there exists f ∈ S 1 F, x such that Equation (13) holds for every t ∈ [0, T].…”

“…The investigation of impulsive differential equations and inclusions in infinite-dimensional Banach spaces has been undertaken by a lot of authors starting from the end of the last century-see, e.g., [9][10][11][12] and the references therein for the problems governed by first-order impulsive equations.…”