Set-valued integral equations of fractional-orders

Abstract

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. The set-valued integral equations (integral inclusions) arises in the study of control system. 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.