Nonlinear zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Andysir. Theory, Mcrhods & Applicmiom, Vol. 12, No. 12. pp. 141%1428, 1988. 0362-546x/88 53.00 + .w Printed in Great Eintain. Q 1988 Pergam0n Rcss plc zyxwvutsrq APPROXIMATE MEAN VALUE THEOREM FOR UPPER SUBDERIVATIVES DARIUSZ ZAGRODNY Institute of Mathematics, Technical University of L&i& Mdt, AI Politechniki 11, Poland (Received 5 April 1987; received for publication 1 March 1988) Key words and phrases: Rockafellar’s upper subderivatives, mean value theorem, sufficient conditions for minimum, lower semicontinuous functions. 1. INTRODUCTION THE PURPOSE of this paper is to give a mean value theorem for a lower semicontinuous function f on a Banach space, using the upper subderivatives defined by Rockafellar [l]. As will be shown by example (Section 4), it may happen that the subdifferential zyxwvutsrqponmlkjihgfedcbaZYXWV af(x) is empty for all x in the closed line segment [a, 61, where a and b belong to X. Therefore we cannot formulate the mean value theorem in such a form as that obtained in [2] and [3] for other generalized derivatives. We prove among other factors that there exists a sequence (xk) converging to some point x E [a, 61, x # b, such that the sets df(xJ are nonempty and limisup zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGF f t (xk; b - a) 3 f(b) -f(a). For a locally Lipschitzian function, this theorem is a generalization of Lebourg’s theorem (see [4, theorem 2.3.71). Finally, as an application of these results, we shall give a theorem on sufficient conditions for a minimum, which extends the theorem 3.1 presented by Chaney [5]. Throughout the paper, X denotes a real Banach space with dual space X*. For any a, b in X we denote by [a, b] the set {fa + (1 - f)blO s t c l}, and by B(x, r) the closed ball with the centre at x and radius r. The distance function dc of a nonempty closed subset C of X is defined by 4(-d = inf{lGr - YIIIY E Cl. 2. GENERALIZED DIRECTIONAL DERIVATIVES In this section we shall summarize those basic facts about generalized directional derivatives that will be used in the sequel. Let f be an extended-real-valued function on X, Following Rockafellar, the expression (y, cu) J rx will mean that y--x, ~-+f(X), cuZf(Y), 1413