LOCAL EXPANSIONS AND ACCRETIVE MAPPINGS

Let X and Y be complete metric spaces with Y metrically convex, let D c X be open, fi u 0 X, and let d(u) d(Uo,U) for all u E D.


INTRODUCTION
This paper may be viewed as a sequel to that of Kirk and SchDneberg [i].We first prove a general theorem for "local expansions" and we then apply this result in special settings to the study of the existence of zeros of the locally strongly accre- tive and -accretive mappings.In the interest of attaining the generality readily offered by our techniques, we formulate our results for set-valued mappings even though some of our assumptions (e.g., continuity, as opposed to semicontinuity) might seem stringent for such mappings.The results themselves, however, represent extensions of those of [i] even in the point-valued case.
Results similar to those obtained here may be found in Ray and Walker [2] and in Torrej6n [3] however, the methods employed are different.Torrej6n relies on differ- ential inequalities, while Ray and Walker use the Brezis-Browder order principle to prove a refined version of the Caristi-Ekeland minimization principle, and this in turn is used to obtain, among other things, a Banach space version of the surjectivlty part of our Theorem 2.1.On the other hand, Torrej6n obtains our Theorem 2.1 under the assumptions that X is a Banach space and Y is a complete and metrically con- vex metric space.While it is likely that the methods of Ray-Walker and of Torrej6n could be modified to attain the generality we obtain, our approach, which is a refine- ment of the argument of Kirk-SchSneberg [1], seems more direct and more in the spirit of the original work of Browder [h, h].In particular, Browder uses an argument (cf. [4, Theorem 4.9]) roughly like the one we use below to show that a local expansion from a complete metric space X to a metric space Y is, under suitable connected- ness hypotheses, actually a covering map of X onto Y.
For the most part, we use standard notation.B(x;r) denotes the closed ball centered at a point x of a metric space with radius r > 0. We shall use S(Y)   and CY) to denote, respectively, the family of nonempty bounded closed subsets and the family of nonempty compact subsets of a metric space Y, and we assign to these families the usual Hausdorff metric (denoted by H).For a Banach space X, the map- ping J X 2 X* denotes the usual normalized duality mapping" (xl [j x*.ll ll:llxll, <x, >:IIxll Also, for a subset A of X, we use AI to denote inf[llxll: x A].
Finally, if X and Y are metric spaces, then a set-valued mapping f" X 2 x is said to be closed if for Ix ] in X, the conditions x x, Yn T(x ), and n n n yn y imply y T(x).
2. A THEOREM ON LOCAL EXPANSIONS.
for all u,v N.Then, given y Y, the following are equivalent.
(b) There exists x 0 D such that for each x XD, In particular, if D X, then f is surjective.
PROOF.Since (a) --> (b) is trivial, we suppose (b) holds and show that the assumption y f(D) leads to a contradiction.For each x D, let r(x) sup[r (0,1): B(x;r) h and dist(f(u By assumption, r(x) > 0 for each x D, and moreover if We define a sequence [Un]c D as follows.Let u I x 0, t I 0, and select w I f(u I) and a path " [0,i] Y joining w I and y (with (0) w I) such that the length, %(F), of F satisfies %(F) m inf[%(w,y) :w f(x0)]+.

APPLICATIONS TO ACCRETIVE MAPPINGS
Let X be a real Banach space and D c X. We recall that a mapping A: X-2 X is said to be accretive if for each x,y D, u A(x), v A(y): (u-v,x-y>+ m sup[(u-v,j> j J(x,y)} 0.
Therefore A:D 2 X is accretive if for each i > 0, J m (I+A) -1 is a non- expansive mapping of (I+kA)(D) onto D. If (I+A)(D) X for some (hence all) % > 0, then A is said to be m-accretive.
Finally, A: D 2 X is said to be strongly accretive if A-cl is accretive for some c > 0.
For our first application we require the following version of Deimling's domain invariance theorem of [6].SchSneberg's modification (see [7]) of the Crandall-Pazy proof ([8]) of this result carries over from point-valued mappings to set-valued map- pings without essential change.
THEOREM 3.1 (of.[7]).Let X be a Banach space, (X) the nonempty bounded closed subsets of X, and H the Hausdorff metric on (X).Suppose U c X is open, and let T: U-(X) be continuous (relative to H) and satisfy for some c>O, This theorem can be proved as follows.Let x I U and Yl T(Xl)" Choose r > 0 and p > 0 so that B(Xl;r+p) c U. Fix y B(Yl;Cr) and define R U (X) as in (ii).
It must be shown that there exists x U such that 0 x+R(x); thus, 0 T(x)-y and y E T(x), from which B(Yl;Cr c T(U).
We now prove the analog of Theorem 3 of [i].
THEOREM 3.2.Let X be a Banach space with D an open subset of X, let c" [0, ) [0, ) be a continuous nonincreasing function for which c(s)ds +, and suppose T:D-S(X) is continuous on D and locally strongly accretive on D in the following sense" Each point z 6 D has a neighborhood N such that for each x,y N, if u T(x) and v T(y), then for some j J(x,y), <u-v,j) > c(max[ll xlI,IIYlI])II x-Yll 2. (*) Then the following are equivalent: (a') 0 T(D).
(b') There exists x 0 D such that IT(x0) g IT(x)l for each x 8D.
Also the local -accretive assumption on T of Theorem 3.3 implies that T is io- cally strongly -accretive in the sense of Definition 2.1 of Downing and Ray.Thus by Theorem 2.1 of [9], T maps open subsets of D onto open sets in Y.The result now follows from Theorem 2.1 as in the proof of Theorem 3.2.
Our final application of the above development is a global result patterned after the approach of [10].
THEOREM 3.4.Let X be a Banach space with D X bounded and open, let A" D---/(X) be continuous and accretive, and suppose there exists z D such that IA(E)I < inf[IA(x)l "x Then there exists a (single-valued) nonexpansive mapping f:D--D whose fixed points are zeros of A.