
Journal of Convex Analysis 18 (2011), No. 3, 673686 Copyright Heldermann Verlag 2011 On a Multivalued Iterative Equation of Order n Bing Xu Yangtze Center of Mathematics and Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P. R. China Kazimierz Nikodem Dept. of Mathematics, University of BielskoBiala, 43309 BielskoBiala, Poland Weinian Zhang Yangtze Center of Mathematics and Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P. R. China matzwn@126.com Because of no Lipschitz condition for upper semicontinuous (usc for short) multifunctions and some other technical difficulties, only the second order polynomiallike 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 multivalued solutions for a modified form of this equation, which gives the existence of usc multivalued solutions for this equation of general order n in the inclusion sense. Keywords: Iteration, functional equation, multifunction, upper semicontinuity, unblended. MSC: 39B12, 37E05, 54C60 [ Fulltextpdf (156 KB)] for subscribers only. 