In this short paper, we prove fixed point theorems for nonexpansive mappings
whose domains are unbounded subsets of Banach spaces. These theorems are
generalizations of Penot's result in [Proc. Amer. Math. Soc., 131 (2003),
2371--2377].Comment: 7 page

Suzuki, Tomonari

English

arXiv

In this short paper, we prove fixed point theorems for nonexpansive mappings whose domains are unbounded subsets of Banach spaces. These theorems are generalizations of Penot's result in [4]

Le Matematiche (Dipartimento di Matematica e Informatica, Università degli Studi di Catania)

LE MATEMATICHEVol. LVIII (2003)  Fasc. I, pp. 149157FIXED POINT THEOREMS FOR ASYMPTOTICALLYCONTRACTIVE MAPPINGSTOMONARI SUZUKIIn this short paper, we prove �xed point theorems for nonexpansivemappings whose domains are unbounded subsets of Banach spaces. Thesetheorems are generalizations of Penots result in [4].1. Introduction.Let C be a closed convex subset of a Banach space E , and let T be anonexpansive mapping on C , i.e., �Tx − T y� ≤ �x − y� for all x , y ∈C . Weknow that T has a �xed point in the case that E is uniformly convex and Cis bounded; see Browder [1] and Go¨hde [2]. Kirk [3] extended these result tothe case that C is weakly compact and has normal structure. We note that suchdomain C of T is a bounded subset. Recently, Penot proved the following in[4]: T has a �xed point in the case that E is uniformly convex, C is unbounded,and T is asymptotically contractive, i.e.,lim supy∈C�y�→∞�T x0 − T y��x0 − y� < 1Entrato in redazione il 28 Maggio 2004.2000 Mathematics Subject Classi�cation: Primary 47H10 - 47H09.Key words and phrases: Nonexpansive mapping, Asymptotically contractive mapping,Fixed point.150 TOMONARI SUZUKIfor some x0 ∈C .In this paper, we prove �xed point theorems for nonexpansive mappingswhose domains are unbounded subsets of Banach spaces. These theorems aregeneralizations of Penots result in [4].2. Conditions for Mappings.In this section, let T be a nonexpansive mapping on a nonempty closedconvex subset C of a Banach space E . We discuss the following conditions forT :(C1) There exists r ∈ (0, 1) such that for every x1 ∈ C , there exists η > 0satisfying�T x1 − T y� ≤ r �x1 − y�for all y ∈C with �y� > η;(C2) there exist r ∈ (0, 1), x0 ∈C and η > 0 such that�T x0 − T y� ≤ r �x0 − y�for all y ∈C with �y� > η;(C3) for each λ > 0 and for each x1 ∈C , there exists η > 0 satisfying�T x1 − T y� ≤ �x1 − y� − λfor all y ∈C with �y� > η;(C4) there exists x0 ∈C for each λ > 0, there exists η > 0 satisfying�T x0 − T y� ≤ �x0 − y� − λfor all y ∈C with �y� > η;(C5) there exists λ > 0 such that for each x1 ∈C , there exists η > 0 satisfying�T x1 − T y� ≤ �x1 − y� − λfor all y ∈C with �y� > η;(C6) there exist x0 ∈C and η > 0 such that�Tx0 − T y� ≤ �x0 − y� − �T x0 − x0�for all y ∈C with �y� > η.We obtain the following.FIXED POINT THEOREMS FOR ASYMPTOTICALLY. . . 151Proposition 1. (C1) ⇔ (C2) ⇒ (C3) ⇔ (C4)⇒ (C5)⇒ (C6) holds.Proof. It is obvious that (C1) ⇒ (C2), (C3) ⇒ (C4), and (C3) ⇒ (C5). We�rst prove (C2) ⇒ (C1). We assume (C2), i.e., there exist r � ∈ (0, 1), x0 ∈ Cand η� > 0 such that �T x0 − T y� ≤ r � �x0 − y� for all y ∈C with �y� > η�.Put r = (1+ r �)/2. We let x1 ∈C be �xed and putη = max�η�, �x1� + �x1 − x0� + �T x1 − T x0�r − r ��.Then for y ∈C with �y� > η, we have�x1 − x0� + �T x1 − T x0� ≤ (r − r �) (η− �x1�)≤ (r − r �) (�y� − �x1�)≤ (r − r �) �x1 − y�and hence�T x1 − T y� ≤ �T x1 − T x0� + �T x0 − T y�≤ �T x1 − T x0� + r ��x0 − y�≤ �T x1 − T x0� + r ��x1 − x0� + r ��x1 − y�≤ �T x1 − T x0� + �x1 − x0� + r ��x1 − y�≤ r�x1 − y�.This implies (C1). We can similarly prove (C2) ⇒ (C4) and (C4)⇒ (C3). We�nally show (C5) ⇒ (C6). We assume (C5), i.e., there exists λ > 0 such thatfor each x1 ∈C , there exists η > 0 satisfying �T x1 − T y� ≤ �x1 − y� − λ forall y ∈C with �y� > η. We putd = infx∈C �T x − x�and assume d > 0. Then there exists x1 ∈C such that �T x1 − x1� < d + λ/2.For such x1, we choose η > 0 satisfying �T x1 − T y� ≤ �x1 − y� − λ for ally ∈C with �y� > η. For each t ∈ (0, 1), since a mapping x �→ (1− t)T x + t x1on C is contractive, there exists yt ∈C such thatyt = (1− t)T yt + t x1.Since d ≤ �T yt − yt� = t�T yt − x1�≤ t ��T yt − T x1� + �T x1 − x1��≤ t ��yt − x1� + �T x1 − x1��≤ t ��yt� + �x1� + �Tx1 − x1��,152 TOMONARI SUZUKIwe have �yt� > η for some small t > 0. So, we have�x1 − yt� + �yt − T yt� = �x1 − T yt�≤ �x1 − T x1� + �T x1 − T yt�≤ �x1 − T x1� + �x1 − yt� − λ≤ d + λ/2+ �x1 − yt� − λand hence�yt − T yt� ≤ d − λ/2.This contradicts to the de�nition of d . Therefore we obtain d = 0. We canchoose x0 ∈C with �T x0 − x0� < λ. Then there exists η > 0 such that�T x0 − T y� ≤ �x0 − y� − η< �x0 − y� − �T x0 − x0�for all y ∈C with �y� > η. This completes the proof. �We can easily prove the following.Proposition 2. Suppose that C is unbounded. Then the followingare equivalentto (C1) and (C2):(i) T is asymptotically contractive;(ii) for every x1 ∈C,lim supy∈C�y�→∞�T x1 − T y��x1 − y� < 1holds.And the following are equivalent to (C3) and (C4):(i) there exists x0 ∈C such thatlimy∈C�y�→∞��T x0 − T y�− �x0 − y��= −∞;(ii) for every x1 ∈C,limy∈C�y�→∞��T x1 − T y�− �x1 − y��= −∞holds.FIXED POINT THEOREMS FOR ASYMPTOTICALLY. . . 1533. Suf�cient and Necessary Condition.In this section, we discuss about the suf�cient and necessary condition fornonexpansive mappings having a �xed point.Lemma 1. Let C be a closed convex subset of a Banach space E and let T bea nonexpansive mapping on C. Suppose that (C6), i.e., there exist x0 ∈ C andη > 0 such that�T x0 − T y� ≤ �x0 − y� − �Tx0 − x0�for all y ∈ C with �y� > η. Then there exists ρ > 0 such that T (D) ⊂ D,where D = {y ∈C : �y − x0� ≤ ρ}.Proof. We putρ = η+ �x0� + �Tx0 − x0� > 0.Then in the case of y ∈ D and �y� ≤ η, we have�T y − x0� ≤ �T y − T x0� + �T x0 − x0�≤ �y − x0� + �T x0 − x0�≤ �y� + �x0� + �T x0 − x0�≤ η+ �x0� + �T x0 − x0�= ρ.In the case of y ∈ D and �y� > η, we have�T y− x0� ≤ �T y− T x0� + �T x0 − x0�≤ �y − x0�≤ ρ.Therefore we obtain the desired result. �A closed convex subset C of a Banach space E is said to have the�xed point property for nonexpansive mappings (FPP, for short) if for everybounded closed convex subset D of C , every nonexpansive mapping on D hasa �xed point. Similarly, C is said to have the weak �xed point property fornonexpansive mappings (WFPP, for short) if for every weakly compact convexsubset D of C , every nonexpansive mapping on D has a �xed point. LetE∗ be the dual of E . Then a closed convex subset C of E∗ is said to havethe weak∗ �xed point property (with respect to E ) for nonexpansive mappings(W∗FPP, for short) if for every weakly∗ compact convex subset D of C , every154 TOMONARI SUZUKInonexpansivemapping on D has a �xed point. So, by the results of Browder [1]and Go¨hde [2], every uniformly convex Banach space has FPP. Also, by Kirksresult [3], every Banach space with normal structure has WFPP. We recall thata closed convex subset C of a Banach space E is locally weakly compact if andonly if every bounded closed convex subset of C is weakly compact. So, everyclosed convex subset of a re�exive Banach space is locally weakly compact.Using Lemma 1, we obtain the following propositions.Proposition 3. Let C be a closed convex subset of a Banach space E . Assumethat C has FPP. Let T be a nonexpansive mapping on C. Then the followingare equivalent:(i) T has a �xed point in C;(ii) T satis�es (C6).Proof. We �rst show (ii) implies (i). We suppose that (ii), i.e., there exist x0 ∈Cand η > 0 such that�T x0 − T y� ≤ �x0 − y� − �Tx0 − x0�for all y ∈ C with �y� > η. By Lemma 1, there exists ρ > 0 such thatT (D) ⊂ D, whereD = {x ∈C : �x − x0� ≤ ρ}.So, by the assumption, there exists z0 ∈ D such that T z0 = z0 . Conversely,let us prove that (i) implies (ii). Let x0 be a �xed point of T . Since T isnonexpansive, we have�T x0 − T y� ≤ �x0 − y� = �x0 − y� − �Tx0 − x0�for all y ∈C . This implies (C6). This completes the proof. �Proposition 4. Let C be a closed convex subset of a Banach space E . Assumethat C is locally weakly compact and has WFPP. Let T be a nonexpansivemapping on C. Then T has a �xed point in C if and only if T satis�es (C6).Proposition 5. Let E be a Banach space and let E∗ be the dual of E . Let Cbe a weakly∗ closed convex subset of E∗ . Assume that C has W∗FPP. Let T bea nonexpansive mapping on C. Then T has a �xed point in C if and only if Tsatis�es (C6).As a direct consequence, we have the following.FIXED POINT THEOREMS FOR ASYMPTOTICALLY. . . 155Theorem 1. Let E be a Banach space and let E∗ be the dual of E . Assumethat either of the following:(i) C is a closed convex subset of E and has FPP.(ii) C is a closed convex subset of E , which is locally weakly compact andhas WFPP.(iii) C is A weakly∗ closed convex subset of E∗ and has W∗FPP.Let T be a nonexpansive mapping on C. Suppose that C is unbounded, and Tis asymptotically contractive. Then T has a �xed point.Remark. (ii) implies (i).Theorem 2. (Penot [4]) Let C be a unbounded closed convex subset of auniformly convex Banach space E . Let T be a nonexpansive mapping on C.Suppose that T is asymptotically contractive. Then T has a �xed point.4. Examples.In Proposition 1, we prove (C1)⇒ (C3) ⇒ (C5) ⇒ (C6). In this section,we give three examples which show that the inverse of the above implicationsdo not hold in general.Example 1. Put E = R and C = [1,∞). De�ne a nonexpansive mapping Ton C byT x = x − log(x )for all x ∈C . Then T satis�es (C3) and does not satisfy (C1).Proof. Sincelimy∈C�y�→∞�T1− T y��1− y� = limy→∞y − log(y)− 1y − 1 = 1,T does not satisfy (C1) by Proposition 2. Sincelimy∈C�y�→∞��T1− T y�− �1− y�� = limy→∞��y − log(y)− 1�− �y − 1��= limy→∞− log(y) = −∞,T satis�es (C3) by Proposition 2. �156 TOMONARI SUZUKIExample 2. Let E = c0 be the Banach space consisting of all real sequencesconverging to 0 with supremum norm. De�ne a closed convex subset C of Eby C = {x ∈ E : 0 ≤ x (n) ≤ n for all n ∈N}.De�ne a nonexpansive mapping T on C by(T x )(n) = max{0, x (n)− 2}for n ∈N. Then T satis�es (5) and does not satisfy (3).Proof. Put λ = 3 and x1 = 0 ∈ C . It is clear that T x1 = 0. Fix η > 0 andchoose n ∈N with η < n and 2 ≤ n. Put y ∈C byy(k) =� n, if k = n,0, if k �= n.Then �y� = n > η and(T y)(k) =� n − 2, if k = n,0, if k �= n.So, we have�T x1 − T y� = �T y� = n − 2 > n − λ = �y� − λ = �x1 − y� − λ.Therefore T does not satisfy (C3). We next put λ = 1 and �x x1 ∈ C . Thenthere exists n1 ∈N such that 0 ≤ x1(n) < 1 for all n ∈N with n ≥ n1. By thede�nition of T , (T x1)(n) = 0 for n ∈ N with n ≥ n1. Put η = n1 + 5, and�x y ∈C with �y� > η. We choose n2 ∈ N with y(n2) = �y�. Then from thede�nition of C , we haven1 < n1 + 5 = η < �y� = y(n2) ≤ n2.It is clear that �y� = y(n2) > 2. For n ∈N with n < n1, we have| (T x1)(n)− (T y)(n) | ≤ n < n1 < n1 + 3 = η − 2.On the other hand, for n ∈N with n ≥ n1, we have| (T x1)(n) − (T y)(n) | = | (T y)(n) | == max{y(n)− 2, 0} ≤ �y� − 2 = y(n2) − 2.FIXED POINT THEOREMS FOR ASYMPTOTICALLY. . . 157Since n1 < n2, η− 2 < �y� − 2 = y(n2)− 2, and| (T x1)(n2)− (T y)(n2) | = max{y(n2) − 2, 0} = y(n2)− 2,we have�T x1 − T y� = y(n2)− 2.So, we obtain�T x1 − T y� = y(n2)− 2≤ y(n2) − x1(n2)− λ≤ �x1 − y� − λ.This implies (C5). This completes the proof. �Exmple 3. Put E = R and C = [1,∞). De�ne a nonexpansive mapping T onC by T x = xfor all x ∈C . Then T satis�es (C6) and does not satisfy (C5).Proof. Since�T x − T y� = �x − y� = �x − y� − �x − T x�for all x , y ∈C , T satis�es (C6). And from the �rst equality, T does not satisfy(C5). �REFERENCES[1] F.E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Nat.Acad. Sci. USA, 54 (1965), pp. 10411044.[2] D. Go¨hde , Zum Prinzip def kontraktiven Abbildung, Math. Nachr., 30 (1965),pp. 251258.[3] W.A. Kirk, A �xed point theorem for mappings which do not increase dis-tances, Amer. Math. Monthly, 72 (1965), pp. 10041006.[4] J.P. Penot, A �xed-point theorem for asymptotically contractive mappings, Proc.Amer. Math. Soc., 131 (2003), pp. 23712377.Department of Mathematics,Kyushu Institute of Technology1-1, Sensuicho, TobatakuKitakyushu 804-8550, (JAPAN)e-mail: suzuki-t@mns.kyutech.ac.jp

Fixed point theorems for asymptotically contractive mappings

In this short paper, we prove fixed point theorems for nonexpansive mappings whose domains are unbounded subsets of Banach spaces. These theorems are generalizations of Penot's result in [4].<br /

Tomonari Suzuki

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Fixed point theorems for asymptotically contractive mappings

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