Nonline~ , Ana,ysis. Theor y , Mer hods & Apphcat r ons, Vol 5, No. 5, pp. 509-516. 0362.546X,81r 050509-08 SO2.00~ 0 Printed m Great Britain 0 1981 Pergamon Press Ltd zyxwvutsr HIGHER ORDER DIFFERENTIABILITY OF MULTIFUNCTIONS J. SCHINAS and M. BOUDOURIDES Department of‘ Mathematxs, Democritus University of Thrace, Xanthl, Greece (Receiwd 2 1 February 1980; revised 12 August 1980) Key words: Multifunctions, upper semi-continuity, homogeneity, Hausdorff metric, differentiation of multifunctions, higher order differentials of multifunctions, differentiable selections of multifunctions. 1. INTRODUCTION THE PROBLEM of differentiability of multifunctions has been investigated by a number of authors, notably Banks and Jacobs [ 11,Lasota and Strauss [6] and De Blasi [4]. The purpose of the present paper is to employ De Blasi’s technique of differentiation in order to obtain higher differentials of multifunctions. Our definition of higher order differentiability is based upon the introduction of a sequence of spaces of multihomogeneous upper semicontinuous multifunctions, each of them endowed with a suitable metric. The latter generalizes the well known Hausdorff metric. In Section 2 we introduce the notation and the definitions to be used in the sequel. Section 3 contains the main results concerning properties of higher differentials. 2. NOTATION AND PRELIMINARIES Let Y be a Banach space with norm (I. I( and 9(Y) its power set. By 9#(Y) (resp. U( Y), U,(Y), .X(Y), X,(Y)) we denote the family of all nonempty bounded (resp. bounded closed, bounded closed convex, compact, compact convex) subsets of Y Also by S(a; r) (,?(a; I)) we denote the open (closed) ball centred at a E Ywith radius I > 0. We write S (S, in place of S(0; l), (S(O; 1)). For any nonempty A, B E 9(Y) and s E R we define A + B = {u+ ~:uEA,~EB}, sA = {sa:a~A}. Given any A, B E W(Y) we define d,*(A, B) = inf {t > 0: A c B + t,S> &(A, B) = max {dg(A, B), dg(B, A)} = inf (t > 0: A c B + tS, B c A + tS}. 509