Set-valued integral equations of fractional-orders

摘要

The topic of fractional calculus (derivative and integral of arbitrary orders) is enjoying growing interest not only among Mathematicians, but also among physicists and engineers (see 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 18, 19, 20, 23, 24, 25). The set-valued integral equations (integral inclusions) arises in the study of control system (see 21, 22, 26). In this paper we prove the existence of locally bounded variation solution of a Volterra type set-valued integral equation of arbitrary (not necessarily integer) order. The proof will be based on the measure of weak noncompactness and the existence of Caratheodory selectors. As a consequence we study the initial value problem for some set-valued differential and integro-differential equations. The corresponding single-valued problems will be firstly considered.