Let X and Y be Banach spaces, and P: X -> Y a Gateaux differentiate operator having closed graph. Suppose that there is a continuous function c: [0, oo) -> (0, oo) satisfying dP x (B(0;ϊ))2B(0;c(\\χ\\)). Then it is shown that for any K > 0 (possibly K = oo), P(B(Q; K)) contains B(P(0); f*c(s) ds). Similar results are obtained for local expansions and locally strongly φ-accretive operators. These results extend a number of known theorems by giving the precise geometric estimations for normal

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... ability of Px = y. 289 290 JONG SOOK BAE AND SANGSUK YIE 2. A fixed point theorem. In this section we give a fixed point theorem which is a basic tool in proving theorems in the next section. Actually our theorem is based on the following well-known Caristi-Kirk-Browder fixed point theorem [5] , which is an equivalent formulation of Ekeland's minimization theorem [8, 9] . THEOREM 2.1. Let (M, d) be a complete metric space and φ be a lower semicontinuous (l.s.c.) function from M to R U {oo}, ^ oo, bounded from below. Let g be a self map ofM satisfying, Proof of Theorem 2.2. Now we construct a new function φ: M -> [0, oo], which is Φ oo, l.s.c. and satisfies (2.1), so that by applying Theorem 2.1, g has a fixed point in M. If K = oo and / 0°° c(s) ds = oo, then Park and Bae [11] showed that the equality rd(x o / J d(x 0 / J d(x 0 ,x) gives φ which is a desired one. Therefore we may assume that K < oo (if K = oo and j™c(s)ds < oo, then the similar method well do). Now J d(x 0 ,x) J d(x 0 ,x) fd(x o ,y) + φ(y) -I c(s) ds. J d(χ 0 ,y)