Impulsive problems for fractional evolution equations and optimal controls in infinite dimensional spaces - Wang, JinRong; Zhou, Yong; Wei, Wei (Topological Methods in Nonlinear Analysis, 2011)
In this paper, a class of impulsive fractional evolution equationsand optimal controls in infinite dimensional spaces is considered. A suitable concept of a $PC$-mild solution is introduced and a suitable operator mapping is also constructed. By using a $PC$-typeAscoli-Arzela theorem, the compactness of the operator mapping isproven. Applying a generalized Gronwall inequality andLeray-Schauder fixed point theorem, the existence and uniqueness ofthe $PC$-mild solutions is obtained. Existence of optimal pairs forsystem governed by impulsive fractional evolution equations is alsopresented. Finally, an example illustrates the applicability of ourresults.