On Second-order Multivalued Impulsive Functional Differential Inclusions in Banach Spaces

Differential equations arise in many real world problems such as physics, population dynamics, ecology, biological systems, biotechnology, industrial robotics, pharmacokinetics, optimal control, and so forth. Much has been done under the assumption that the state variables and system parameters change continuously. However, one may easily visualize situations in nature where abrupt changes such as shock, harvesting, and disasters may occur. These phenomena are shortterm perturbations whose duration is negligible in comparison with the duration of the whole evolution process. Consequently, it is natural to assume, in modelling these problems, that these perturbations act instantaneously, that is, in the form of impulses. For more details on this theory and on its applications we refer to the monographs of Baı̆nov and Simeonov [2], Lakshmikantham, Baı̆nov, and Simeonov [19], and Samoilenko and Perestyuk [24]. However, very few results are available for impulsive differential inclusions; see for instance, the papers of Benchohra and Boucherif [4, 5], Erbe and Krawcewicz [12], and Frigon and O’Regan [14]. Very recently an extension to functional differential equations of first order with impulsive effects has been done by Yujun [10] by using the coincidence degree theory, and by Benchohra and Ntouyas [7] with the aid of Schaefer’s theorem. These results have been also generalized to the multivalued case by the authors in [6] by combining the a priori bounds and the Leray-Schauder