The Asymptotic Behavior of Firmly Nonexpansive Mappings

We present several new results on the asymptotic behavior of firmly nonexpansive mappings in Banach spaces and in the Hubert ball. Let D be a subset of a (real) Banach space X. Recall that a mapping T: D -» X is said to be firmly nonexpansive [2, 4] if for each x and y in D, the convex function /: [0,1] -> [0, oo) defined by f{s) = \(\-s)x + sTx-((l-s)y + sTy) \ is nonincreasing. Note that T is firmly nonexpansive if and only if it is the resolvent (/ + AY ' for some accretive operator A c X X X, and that any linear projection of norm 1 is firmly nonexpansive. It is known [5] that firmly nonexpansive mappings have several remarkable properties which are not shared by all nonexpansive mappings. Our purpose here is to present several new results on the asymptotic behavior of these mappings. We solve, in particular, a problem which was left open in [5] (Corollary 1), use a new geometric property of Banach spaces (Theorem 2), and present Hubert ball analogues of some of our results (Theorem 3 and its corollaries). Rather unexpectedly, no range condition is assumed in Theorem 2. All of our results have their roots in the following triple equality. Theorem 1. Let. D be a subset of a Banach space X and T: D —> X a firmly nonexpansive mapping. If T can he iterated at x e D, then for all k > 1, lim \T', + XxT"x\= lim \Tn + kx T"x\/k = lim \T"x/n\. n —* oc n —* oc n —» oo Proof. Since T is nonexpansive, it is clear that the limits F = lim \T" + Xx T"x\ and R = lim \Tn + kx T"x\ n —» oc n —* oc exist, and that R < kL. Also, the second equality follows from the first just as in [1, Theorem 2.1]. Therefore all we have to show is that R > kL. To this end we use induction on k. Since the case Ac = 1 is clear, we assume that R = JL for all Received by the editors July 1, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 47H09. The first author was partially supported by the Fund for the Promotion of Research at the Technion. ©1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page 246 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use ASYMPTOTIC BEHAVIOR 247 1 N(e). Since T is firmly nonexpansive, we also have \Xn+l ~ Xn + k + l I ̂ I*« ~*~ Xn+l ~ \Xn + k ~*~ Xn + k+l) \/¿ < \xn — xn + k + l\/2 + \x„+i — xn + k\/2. Hence \Xn ~ xn + k + l I ̂ ¿\Xn + l ~ xn + k + l\~\xn+l ~ xn + k I > 2/fc(L e) -(A: 1)(L + e) = (k + 1)1 -(3Ä: l)e, and the result follows. Recall that a mapping T: D -» X is said to be asymptotically regular [3] aliefl if Fcan be iterated at x and lim„_>00(7''';c T" + lx) = 0. Our first corollary solves a problem which was left open in [5, p. 465]. Corollary 1. A firmly nonexpansive mapping which has a fixed point is asymptotically regular at each point where it can be iterated. The next result identifies the common limit of Theorem 1. Corollary 2. Let T: D —► D he firmly nonexpansive, and set d = inf{|j> — Ty\: y G D). Then for each x in D,lim„^x\T" + lx T"x\ = d. Recall that a closed convex subset C of a Banach space has the fixed point property for nonexpansive mappings (FPP for short) if every nonexpansive T: C -> C has a fixed point. Corollary 3. Let C be a closed convex subset of a Banach space X and T: C -> C a firmly nonexpansive mapping. If each bounded closed convex subset of X has the FPP, then T is fixed point free if and only if lim n _ ,J7'r''x\ = oo for all x in C. This result improves upon [5, Theorem 2.4(c)] because each bounded closed convex subset of a uniformly convex Banach space does indeed have the FPP. It cannot, however, be obtained by the approach of [5] because in general not every firmly nonexpansive mapping is strongly nonexpansive. The next result is essentially known (cf. [7, p. 53]). It can now be viewed as a consequence of Corollary 1 and the nonlinear mean ergodic theorem. Corollary 4. Let C be a closed convex subset of a uniformly convex Banach space with a Fréchet differentiable norm. If a firmly nonexpansive T: C —» C has a fixed point, then for each x in C the sequence of iterates { T"x} converges weakly to a fixed point of T. In order to continue we need a new geometric property of (infinite-dimensional) Banach spaces. First recall that the duality map J from X into the family of nonempty closed convex subsets of its dual A'* is defined by /(■*){x* g X*: (x,x*) =\xf =|**|2}. License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 248 SIMEON REICH AND ITAI SHAFRIR To each functional x* in the unit sphere oí X* there corresponds a face F of the unit sphere U of X, namely J\x*). We shall say that the norm of X is locally uniformly Fréchet differentiable (LUF) if for each face F of U the limit lim (Ijc + ty\ -\x\)/t f-0 is attained uniformly for all y in U and x in F. It is clear that X is LUF whenever A* is uniformly convex or X has a Fréchet-differentiable norm and compact faces. It can also be shown that if X is reflexive and locally uniformly convex, then X* is LUF. We shall use the following fact. Lemma 1. Let {xn} be a sequence in Xfor which there exist x G X with jjc| = 1 and a sequence x* c J(x) such that limB_00|x„| = lim„^00(A;„, x*). If X* is LUF, then {x,,} converges. For reflexive X, the converse of Lemma 1 is also true. Now let D be a subset of a Banach space X and T: D -* X a nonexpansive mapping. If F can be iterated at x G D, then L = limn_x\T"x\/n exists. We shall also need the following lemma from [9]. Lemma 2. There exists a functional z(x) g X* such that \z(x)\ = L and ((*Tmx)/m,z{x)) > L2