On a Multivalued Iterative Equation of Order n

Because of no Lipschitz condition for upper semi-continuous (usc for short) multifunctions and some other technical difficulties, only the second order polynomial-like iterative equation with multifunctions was discussed but the general case of order n remains open. In this paper we consider the general case for a special class of multifunctions, called unblended multifunctions. We investigate the set of all jumps for iterates of those multifunctions and consider the piecewise Lipschitz condition. Then we prove the existence of usc multi-valued solutions for a modified form of this equation, which gives the existence of usc multi-valued solutions for this equation of general order n in the inclusion sense.