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Geometrical conditions in product spaces
Nonlinear Analysis 46 (2001) 1063 – 1071
www.elsevier.com/locate/na
Geometrical conditions in product spaces G. Marinoa; ∗ , P. Pietramalaa , Hong-Kun Xub; 1 a Dipartimento
di Matematica, Universita della Calabria, Arcavacata di Rende, 87036 Cosenza, Italy of Mathematics, University of Durban-Westville, Private Bag X54001, Durban 4000, South Africa
b Department
Received 21 June 1999; accepted 12 January 2000
Keywords: Weak normal structure; Nonexpansive mapping; Fixed point property; Product space; Property (k); Property (M )
1. Introduction Let K be a nonempty subset of a Banach space X . Recall that a mapping T : K → K is nonexpansive if Tx − Ty ≤ x − y for x; y ∈ K. A Banach space X is said to have the weak 2xed point property (WFPP) if, for every nonempty weakly compact convex subset K of X , each nonexpansive mapping T : K → K has a 7xed point. In 1965 Kirk [12] proved that a weak normal structure implies WFPP. (Recall that a Banach space X has a weak normal structure (WNS) if every weakly compact convex subset K of X consisting of more than one point has a nondiametral point; i.e., a point x ∈ K such that sup{x − y: y ∈ K} ¡ diam(K).) It is, however, not easy to detect if a given weakly compact convex subset of a Banach space has a weak normal structure. So over the last 30 years, considerable research has been done towards 7nding geometrical conditions on the underlying Banach space which are su=cient to imply WNS; for example, Opial’s property [28], uniform convexity in every direction (cf. [30]), k-uniform convexity [32], and nearly uniform convexity [8]. More recently, some more geometrical conditions have been discovered [3,4,6,11,29–31,34] which imply either WNS or WFPP. The problem of whether the WFPP (resp. WNS) is preserved under direct sum of two (or more) Banach spaces, each of which has WFPP (resp. WNS), is an old one. ∗
Corresponding author. E-mail addresses: [email protected] (G. Marino), [email protected] (P. Pietramala), hkxu@pixie. udw.ac.za (H.-K. Xu). 1
Supported in part by NRF (South Africa).
0362-546X/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 0 0 ) 0 0 1 5 3 - X
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G. Marino et al. / Nonlinear Analysis 46 (2001) 1063 – 1071
The answer to this problem depends clearly upon the norm of the product space. We shall consider here only the p-norm for p ∈ [1; ∞] though many of the results in this paper remain valid in a slightly more general framework. Let X and Y be two Banach spaces and p ∈ [1; ∞]. We use X ⊕p Y to denote the product space X ⊕ Y equipped with the norm (x; y) := (xp + yp )1=p
if p ¡ ∞;
(x; y) := max{x; y} if p = ∞: Suppose that X and Y both have a property (). According to Kirk [16], there are basically two ways to formulate permanence of () in the product space X ⊕p Y , either 1. when does X ⊕p Y have property ()? or 2. when does a rectangle K1 ⊕p K2 , with K1 ⊂ X and K2 ⊂ Y , have property ()? If is WNS, the 7rst positive remarkable answer to Question 1 above is due to Belluce et al. [1]. In 1984, Landes [23,24] showed that WNS is preserved in X ⊕p Y for 1 ¡ p ≤∞, but not preserved in X ⊕1 Y . This motivates the investigation of this question: What additional conditions on the spaces X and Y are su=cient for the product space X ⊕1 Y to have WNS? Some partial answers to the question have been obtained in [5]. On the other hand, if is the WFPP, the situation is much more delicate and many questions remain open because Questions 1 and 2 are fundamentally diLerent and much more is known about the second than the 7rst. In fact, there exist very few and very partial results about Question 1 [5,15,20,21,33] while Question 2 for 1 ≤ p ¡ ∞ was solved positively by Kirk and Martinez-Yanez [17] (see [11] for a slightly more general result). If X and Y have WFPP and K1 ⊂ X and K2 ⊂ Y are weakly compact convex subsets of X and Y , respectively, then each nonexpansive mapping T : K1 ⊕p K2 → K1 ⊕p K2 has a 7xed point. Question 2, in general, still remains unsolved in K1 ⊕∞ K2 , but there are several and very interesting results in this direction (cf. [11,13,14,17 –19,22,33]). Finally, we want to mention a result obtained by Lin [26] when property () is the Browder–GNodhe (B–G) property: If X and Y have the B–G property and moreover X is uniformly convex and Y has the Schur property, then X ⊕p Y has the B–G property for 1 ≤ p ¡ ∞. Summarizing, we are interested in answering these two questions: • When does X ⊕1 Y have WNS or a stronger property? • When does X ⊕p Y , 1 ≤ p ≤ ∞, have WFPP or a stronger property? It is the purpose of this article to give partial answers to these questions. In particular, we shall show that if X and Y both have property (k), then X ⊕1 Y has WNS. The same is true if X has GGLD and Y has property (k). We also show that if X and Y both have property (M ), then X ⊕∞ Y has the WFPP. Throughout the paper we shall use the notations: * for weak convergence and → for strong convergence.
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2. Results Recall that a bounded sequence (x n ) in a Banach space X is called diametral if diam(x n ) ¿ 0 and lim diam(x n+1 ; co(x1 ; : : : ; x n )) = diam(x n ). The basic characterization of normal structure, due to Brodskii and Milman [2], is as follows: A bounded convex subset K of a Banach space X , has normal structure if and only if it does not contain any diametral sequence. So, if X fails to have weak normal structure, then, there exists a sequence (x n ) such that x n * 0 and limx n − x = diam(x n ) = 1 for all x ∈ co(x n ). In particular, since 0 ∈ co(x n ), we have limx n = 1. For more details, see [7]. A stronger condition than WNS was introduced by Sims [30]. Denition 1. A Banach space X has property (k) if there exists k ∈ (0; 1) such that whenever x n * 0, x n → 1 and lim inf x n − x ≤ 1, we have x ≤ k. Proposition 1. Let X and Y be Banach spaces with property (k). Then X ⊕1 Y; has the weak normal structure. Proof. Suppose that X ⊕1 Y fails to have the WNS. Then there exists a sequence (x n ; yn ) in X ⊕1 Y such that • (x n ; yn ) * 0. • (x n ; yn ) → 1. • lim dist((x n+1 ; yn+1 ); co((x1 ; y1 ); : : : ; (x n ; yn ))) = 1 = diam((x n ; yn )). Without loss of generality, we may assume that the following limits all exist: limx n =: A;
limyn =: B;
lim x n − xj =: Aj ; n
lim yn − yj =: Bj : n
(1)
It then follows that 1 = A + B = Aj + Bj ;
∀j ≥ 1:
(2)
Hence, we see that if Bj ¿ B; Aj ¡ A and if Bj ≤ B; Aj ≥ A. So we have a subsequence (nj ) of positive integers for which either Anj ≤ A for all j or Bnj ≤ B for all j. Since both X and Y have property (k), we obtain that either x nj ≤ kA for all j or ynj ≤ kB for all j. Taking the limit as j → ∞ yields either A ≤ kA or B ≤ kB. This is a contradiction. We can strengthen the conclusion of Proposition 1 by assuming the nonstrict Opial property on X and Y . This means that for any sequence (x n ), we have x n * 0 ⇒ lim sup x n ≤ lim sup x n − x
∀x ∈ X:
(3)
Proposition 2. Let X and Y be Banach spaces both with property (k) and the nonstrict Opial property. Then; for each p ∈ [1; ∞); X ⊕p Y has both property (k) and the nonstrict Opial property.
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Proof. For simplicity, we prove the case of p=1. Let (x n ; yn ) * 0; (x n ; yn )=x n + yn → 1, and lim sup(x n ; yn ) − (x; y) = lim sup(x n − x + yn − y) ≤ 1. We have to show that x + y ≤ k. By passing to suitable subsequences if necessary, we may assume that the following limits all exist: limx n =: A;
limyn =: B;
lim x n − x =: C; n
lim yn − y =: D: n
(4)
Then we have A+B=1
and
C + D ≤ 1:
By the nonstrict Opial property of X and Y , we see that A ≤ C and B ≤ D and hence we must have A = C and B = D. Now in view of property (k) of X and Y , we get x ≤ kA and y ≤ kB. Therefore, (x; y) = x + y ≤ kA + kB = k and X ⊕1 Y has property (k). We next show that X ⊕1 Y has the nonstrict Opial property. For this purpose we assume that (x n ; yn ) is a weakly null sequence in X ⊕1 Y . For any (x; y) ∈ X ⊕1 Y we have to show that, lim sup (x n ; yn ) ≤ lim sup (x n − x; yn − y). But through a suitable subsequence (n ) we have lim sup(x n ; yn ) = lim x n + lim yn ≤ lim x n − x + lim yn − y ≤ lim sup(x n − x + yn − y) = lim sup (x n − x; yn − y): Remark. Proposition 2 is not true in general for the case p = ∞. Indeed, if we take the product space ‘2 ⊕∞ R; (x n ; yn ) := (en ; 0), and (x; y) := (0; 1), then we have lim sup (x n ; yn ) − (x; y) = 1 = lim (x n ; yn ). However, (x; y) = 1 and thus property (k) does not hold for any k ∈ [0; 1). In spite of this, one can easily see that the nonstrict Opial property is preserved in the product space X ⊕p Y for 1 ≤ p ≤ ∞. More recently, in order to improve a result of Lin [27], Xu and Marino [34] introduced a uniform version of property (k). Denition 2. A Banach space X is said to have the uniform property (k) if there exists a constant k ∈ [0; 1) such that whenever (x n ); (ym ) ⊂ X satisfy the conditions: x n * 0; x n → 1 and lim supm lim supn x n − ym ≤ 1, we have lim supm ym ≤ k. By the same argument as used in the proof of Proposition 2 we have Proposition 3. Let X and Y be Banach spaces both having the uniform property (k) and the nonstrict Opial property. Then; for each p ∈ [1; ∞); X ⊕p Y has the uniform property (k) and the nonstrict Opial property. Another su=cient condition for weak normal structure was introduced by JimPenezMelado [9] in 1992.
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Denition 3. A Banach space X is said to have the generalized Gossez–Lami Dozo property (GGLD) if, for all weakly null sequences (x n ) in X such that lim x n = 1, we have lim supm lim supn x n − xm =: D(x n ) ¿ 1. It is shown in [34] that the uniform property (k) implies GGLD but property (k) does not imply GGLD. We have the following result. Proposition 4. Let X and Y be Banach spaces. Assume that X has GGLD and Y has property (k). Then X ⊕1 Y has WNS. Proof. Suppose that X ⊕1 Y fails to have WNS. Then there exists a sequence (x n ; yn ) ∈ X ⊕1 Y satisfying • (x n ; yn ) * 0. • x n + yn ≤ 1; lim(x n + yn ) = 1. • limn (x n − xj + yn − yj ) = 1 = diam((x n ; yn )) for j ≥ 1. Without loss of generality, we may assume that there exist the limits: limx n =: a; limyn =: b; limn x n −xm =: am , and limn yn −ym =: bm . If (x n ) does not converge to 0 in norm, then GGLD of X implies that a = lim x n ¡ lim sup lim sup x n − xm : n
m
n
Let m0 be an integer large enough so that a ¡ lim x n − xm n
∀m ≥ m0 :
Since, for m ≥ m0 , lim yn − ym = 1 − lim x n − xm ¡ 1 − a = b = lim yn ; n
n
n
property (k) of Y then yields ym ≤ k · lim yn n
∀m ≥ m0 :
Hence by taking limit as m → ∞, we get b ≤ kb which implies that b=0. We therefore have lim x n = 1 and lim supm lim supn x n − xm = 1. But this contradicts the GGLD of X . Next, if x n → 0, then we see that lim yn − ym = 1 − xm ≤ 1 = lim yn : n
n
It follows from property (k) of Y that ym ≤ k
∀m:
This however contradicts the fact that limn yn = 1. In 1993 Kalton [10] introduced the notion of property (M ). Denition 4. A Banach space X is said to have property (M ) provided weakly null types are constant on spheres about the origin. That is, if x n * 0 and u; v ∈ X such that x = y, then lim sup x n + u = lim sup x n + v.
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Garcia-Falset and Sims [6] proved that property (M ) implies WFPP. One may then ask this question: Is property (M ) preserved in product spaces? The answer is negative, as shown in the examples below. Example 1. Let p ∈ [1; ∞). Consider l2 ⊕p l2 . Put (u; v) := lim sup (en ; en ) − (u; v) n
= lim sup(en − up2 + en − vp2 )1=p ; n
u; v ∈ l2 ;
where (en ) is the standard unit basis of l2 . Then, (0; e1 ) = (2−1=p e1 ; 2−1=p e1 ) = 1: However, easy calculations show that (0; e1 ) = (1 + 2p=2 )1=p and (2−1=p e1 ; 2−1=p e1 ) = 21=p (1 + 2−2=p )1=2 : Hence, (0; e1 ) = (2−1=p e1 ; 2−1=p e1 ) for p = 2 and l2 ⊕p l2 does not have property (M ) if p = 2. Example 2. Let p = ∞ and consider l2 ⊕∞ l3 . Set (u; v) := lim sup (en ; en ) − (u; v) n
= lim sup max{en − u2 ; en − v3 }; n
u ∈ l2 ; v ∈ l3 ;
√ where (en ) √is the standard unit basis of l2 and l3 . Then we have (e1 ; 0) = 2 while (0; e1 ) = 3 2; nevertheless, (e1 ; 0) = (0; e1 ) = 1. Hence l2 ⊕∞ l3 fails to have property (M ). In spite of the last example, we have the following result. Proposition 5. Let X and Y be Banach spaces both having property (M ). Then X ⊕∞ Y has the weak 2xed point property for nonexpansive mappings. Proof. Suppose that X ⊕∞ Y fails to have the WFPP. Then we have a weakly compact convex subset K of X ⊕∞ Y with diam(K) = 1 and a 7xed point free nonexpansive mapping T : K → K with respect to which K is a minimal nonempty weakly compact convex invariant subset which contains an approximate 7xed point sequence (an ) with an * 0 and, by the Goebel–Karlovitz lemma, lim an − z = diam(K) = 1 for all z ∈ K. Let [X ⊕∞ Y ] := l∞ (X ⊕∞ Y )=c0 (X ⊕∞ Y ) be equipped with the quotient norm given canonically by [zn ] = lim sup zn : Let [K] := {[zn ]: zn ∈ K; ∀n}. Then [T ][x n ] := [Tx n ] is a well-de7ned nonexpansive mapping on [K]. Let W := {[wn ] ∈ [K]: [wn ] − [an ] ≤
1 3
and D[wn ] ≤ 23 };
G. Marino et al. / Nonlinear Analysis 46 (2001) 1063 – 1071
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where D[wn ] := lim supm lim supn wn − wm is well de7ned since for (wn − yn ) ∈ c0 (X ⊕∞ Y ) we have D(wn ) = D(yn ). Then W is [T ]−invariant, closed, convex and nonempty as [ 23 an ] ∈ W . For [wn ] ∈ W , we may extract a subsequence (wnk ) such that wnk * w0 ∈ K and lim wnk = [wn ]. Now, wnk = (x nk ; ynk ); x nk ∈ X; ynk ∈ Y , and without loss of generality, we may assume that there exist the limits: lim x nk =: A and lim ynk =: B so that [wn ] = limk wnk = max{A; B} = A. (Here we assume that A ≥ B since the proof given below works for the case B ≥ A as well.) Write w0 = (x0 ; y0 ). Then x nk * x0 and ynk * y0 . Moreover, wnk − ank * w0 , so x0 ≤ w0 ≤ lim inf wnk − ank ≤ [wnk ] − [ank ] ≤ 13 : Let d := limk x nk − x0 . We distinguish two cases. Case 1. d ¡ 35 . Then [wn ] = lim x nk = lim (x nk − x0 ) + x0 ≤ d + x0 ¡ 13=15: k
Case 2. d ≥ In this case,
k
3 5
¿ 13 ≥ x0 . let zk := (x0 − x nk )=d.
Then zn → 1. We also have
[wn ] = lim x nk = d lim (x0 =d) − zk = d (x0 =d) ≤ d sup k
k
v≤1
(v);
where (v) := lim supk zk − v for v ∈ X . Now by Garcia-Falset and Sims [6, Lemma 3:1] we get sup
v≤1
(v) = D(zk ):
It then follows that [wn ] ≤ d sup
v≤1
(v) = d D(zk )
= D(x nk ) ≤ max{D(x nk ); D(ynk )} = D(wnk ) ≤ D(wn ) = D[wn ] ≤ 23 : In summary, we get that any [wn ] ∈ W is away from one. More precisely, we have [wn ] ≤ 13=15. This however contradicts Lin’s extension [25] of the Goebel–Karlovitz lemma which assures that W contains elements of norm arbitrarily close to one. The above provides a partial answer to a question of Khamsi [11] for the case p=∞. Finally, we examine another property stronger than the weak normal structure, property (P) introduced by Tan and Xu [33]. Denition 5. A Banach space X is said to have Property (P) if, for any nonconstant sequence (x n ) in X , we have x n * 0 ⇒ lim inf x n ¡ diam(x n ): (Clearly, in this de7nition, ‘lim inf ’ can be replaced with ‘lim sup’.) Our 7nal result improves Corollary 2:1 and Theorem 2:3 of [33] in the case p = ∞. Proposition 6. Let X and Y be Banach spaces both with property (P). Then X ⊕∞ Y also has property (P).
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Proof. Let (x n ; yn ) be a weakly null sequence in X ⊕∞ Y . Then we have lim sup (x n ; yn ) = lim sup max{x n ; yn } = max{lim sup x n ; lim sup yn } ¡ max{diam(x n ); diam(yn )} = diam(x n ; yn ): Acknowledgements The authors thank the referee for his=her suggestions for the manuscript. References [1] P. Belluce, W.A. Kirk, E.F. Steiner, Normal structure in Banach spaces, Paci7c J. Math. 26 (1968) 433–440. [2] M.S. Brodskii, D.P. Milman, On the center of a convex set, Dolk. Akad. Nauk SSSR 59 (1948) 837–840. [3] D. van Dulst, B. Sims, Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK), Lecture Notes in Math. 991 (1983) 35–43. [4] J. Garcia-Falset, A. Jimenez-Melado, E. Llorens-Foster, Measures of noncompactness and normal structure in Banach spaces, Studia Math. 110 (1994) 1–8. [5] J. Garcia-Falset, E. Llorens-Foster, Normal structure and 7xed point property, Glasgow Math. J. 38 (1996) 29–37. [6] J. Garcia-Falset, B. Sims, Property (M ) and the weak 7xed point property, Proc. Amer. Math. Soc. 125 (1997) 2891–2896. [7] K. Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, New York, 1990. [8] R. HuL, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10 (1980) 743–749. [9] A. Jimenez-Melado, Stability of weak normal structure in James quasi-reSexive space, Bull. Austral. Math. Soc. 46 (1992) 367–372. [10] N.J. Kalton, M -ideals of compact operators, Illinoins J. Math. 37 (1987) 181–191. [11] M.A. Khamsi, On normal structure, 7xed point property and contractions of type ('), Proc. Amer. Math. Soc. 106 (1989) 995–1001. [12] W.A. Kirk, A 7xed point theorem for mappings which do not increase distances, Amer. Math. Soc. Monthly 72 (1965) 1004–1006. [13] W.A. Kirk, Nonexpansive mappings in product spaces, set-valued mappings and k-uniform rotundity, in: F.E. Browder (Ed.), Nonlinear Functional Analysis and its Applications Amer. Math. Soc. Symp. Pure Math. vol. 45 (2), 1986, pp. 51– 64. [14] W.A. Kirk, An iteration process for nonexpansive mappings with applications to 7xed point theory in product spaces, Proc. Amer. Math. Soc. 107 (1989) 411–415. [15] W.A. Kirk, Nonexpansive mappings in separable product spaces, Boll. U.M.I. 9-A (1995) 239–244. [16] W.A. Kirk, Some questions in metric 7xed point theory, in: T. Dominguez-Benavides (Ed.), Recent Advances on Metric Fixed Point Theory Proc. of the International Workshop on Metric Fixed Point Theory, Seville, Spain, September 1995; Universidad de Seville Press, Seville, 1996, pp. 73–97. [17] W.A. Kirk, C. Martinez-Yanez, Nonexpansive and locally nonexpansive mappings in product spaces, Nonlinear Anal. TMA 12 (1988) 719–725. [18] W.A. Kirk, Y. Stenfeld, The 7xed point property for nonexpansive mappings in certain product spaces, Houston J. Math. 10 (1984) 207–214.
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[19] T. Kuczumov, Fixed point theorems in product spaces, Proc. Amer. Math. Soc. 108 (1990) 727–729. [20] T. Kuczumov, S. Reich, M. Schmidt, A 7xed point property of l1 -product spaces, Proc. Amer. Math. Soc. 119 (1993) 457–463. [21] T. Kuczumov, S. Reich, M. Schmidt, A. Stachura, Strong asymptotic mormal structure and 7xed points in product spaces, Nonlinear Anal. TMA 21 (1993) 501–515. [22] T. Kuczumov, A. Stachura, Bruck’s retraction method, Fixed Point Theory and Applications, Proceedings of the International Conference on Marseille-Luminy, France, 1989, Pitman Research Notes in Mathematics Series, vol. 252, 1991, Pitman, London, pp. 285 –292. [23] T. Landes, Permanence properties of normal structure, Paci7c J. Math. 110 (1984) 125–143. [24] T. Landes, Normal structure and the sum-property, Paci7c J. Math. 123 (1986) 127–147. [25] P.L. Lin, Unconditional bases and 7xed points of nonexpansive mappings, Paci7c J. Math. 116 (1985) 69–76. [26] P.K. Lin, The Browder-GNohde property in product spaces, Houston J. Math. 13 (1987) 235–240. [27] P.K. Lin, Asymptotic behavior for asymptotically nonexpansive mappings, Nonlinear Anal. TMA 26 (1996) 1137–1141. [28] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967) 591–597. [29] S. Rolewicz, On U-uniform convexity and drop property, Studia Math. 87 (1987) 181–191. [30] B. Sims, A class of spaces with weak normal structure, Bull. Austral. Math. Soc. 50 (1994) 523–528. [31] B. Sims, Banach space geometry and the 7xed point property, in: T. Dominguez-Benavides (Ed.), Recent Advances on Metric Fixed Point Theory, Proceedings of the International Workshop on Metric Fixed Point Theory, Seville, Spain, September 1995; Universidad de Seville Press, Seville, 1996, pp. 137–160. [32] F. Sullivan, A generalization of uniformly rotund Banach spaces, Can. J. Math. 31 (1979) 628–636. [33] K.K. Tan, H.K. Xu, On 7xed points of nonexpansive mappings in product spaces, Proc. Amer. Math. Soc. 113 (1991) 983–989. [34] H.K. Xu, G. Marino, Uniform property (k) and its related properties, Bull. Austral. Math. Soc. 57 (1998) 93–107.
www.elsevier.com/locate/na
Geometrical conditions in product spaces G. Marinoa; ∗ , P. Pietramalaa , Hong-Kun Xub; 1 a Dipartimento
di Matematica, Universita della Calabria, Arcavacata di Rende, 87036 Cosenza, Italy of Mathematics, University of Durban-Westville, Private Bag X54001, Durban 4000, South Africa
b Department
Received 21 June 1999; accepted 12 January 2000
Keywords: Weak normal structure; Nonexpansive mapping; Fixed point property; Product space; Property (k); Property (M )
1. Introduction Let K be a nonempty subset of a Banach space X . Recall that a mapping T : K → K is nonexpansive if Tx − Ty ≤ x − y for x; y ∈ K. A Banach space X is said to have the weak 2xed point property (WFPP) if, for every nonempty weakly compact convex subset K of X , each nonexpansive mapping T : K → K has a 7xed point. In 1965 Kirk [12] proved that a weak normal structure implies WFPP. (Recall that a Banach space X has a weak normal structure (WNS) if every weakly compact convex subset K of X consisting of more than one point has a nondiametral point; i.e., a point x ∈ K such that sup{x − y: y ∈ K} ¡ diam(K).) It is, however, not easy to detect if a given weakly compact convex subset of a Banach space has a weak normal structure. So over the last 30 years, considerable research has been done towards 7nding geometrical conditions on the underlying Banach space which are su=cient to imply WNS; for example, Opial’s property [28], uniform convexity in every direction (cf. [30]), k-uniform convexity [32], and nearly uniform convexity [8]. More recently, some more geometrical conditions have been discovered [3,4,6,11,29–31,34] which imply either WNS or WFPP. The problem of whether the WFPP (resp. WNS) is preserved under direct sum of two (or more) Banach spaces, each of which has WFPP (resp. WNS), is an old one. ∗
Corresponding author. E-mail addresses: [email protected] (G. Marino), [email protected] (P. Pietramala), hkxu@pixie. udw.ac.za (H.-K. Xu). 1
Supported in part by NRF (South Africa).
0362-546X/01/$ - see front matter ? 2001 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 0 0 ) 0 0 1 5 3 - X
1064
G. Marino et al. / Nonlinear Analysis 46 (2001) 1063 – 1071
The answer to this problem depends clearly upon the norm of the product space. We shall consider here only the p-norm for p ∈ [1; ∞] though many of the results in this paper remain valid in a slightly more general framework. Let X and Y be two Banach spaces and p ∈ [1; ∞]. We use X ⊕p Y to denote the product space X ⊕ Y equipped with the norm (x; y) := (xp + yp )1=p
if p ¡ ∞;
(x; y) := max{x; y} if p = ∞: Suppose that X and Y both have a property (). According to Kirk [16], there are basically two ways to formulate permanence of () in the product space X ⊕p Y , either 1. when does X ⊕p Y have property ()? or 2. when does a rectangle K1 ⊕p K2 , with K1 ⊂ X and K2 ⊂ Y , have property ()? If is WNS, the 7rst positive remarkable answer to Question 1 above is due to Belluce et al. [1]. In 1984, Landes [23,24] showed that WNS is preserved in X ⊕p Y for 1 ¡ p ≤∞, but not preserved in X ⊕1 Y . This motivates the investigation of this question: What additional conditions on the spaces X and Y are su=cient for the product space X ⊕1 Y to have WNS? Some partial answers to the question have been obtained in [5]. On the other hand, if is the WFPP, the situation is much more delicate and many questions remain open because Questions 1 and 2 are fundamentally diLerent and much more is known about the second than the 7rst. In fact, there exist very few and very partial results about Question 1 [5,15,20,21,33] while Question 2 for 1 ≤ p ¡ ∞ was solved positively by Kirk and Martinez-Yanez [17] (see [11] for a slightly more general result). If X and Y have WFPP and K1 ⊂ X and K2 ⊂ Y are weakly compact convex subsets of X and Y , respectively, then each nonexpansive mapping T : K1 ⊕p K2 → K1 ⊕p K2 has a 7xed point. Question 2, in general, still remains unsolved in K1 ⊕∞ K2 , but there are several and very interesting results in this direction (cf. [11,13,14,17 –19,22,33]). Finally, we want to mention a result obtained by Lin [26] when property () is the Browder–GNodhe (B–G) property: If X and Y have the B–G property and moreover X is uniformly convex and Y has the Schur property, then X ⊕p Y has the B–G property for 1 ≤ p ¡ ∞. Summarizing, we are interested in answering these two questions: • When does X ⊕1 Y have WNS or a stronger property? • When does X ⊕p Y , 1 ≤ p ≤ ∞, have WFPP or a stronger property? It is the purpose of this article to give partial answers to these questions. In particular, we shall show that if X and Y both have property (k), then X ⊕1 Y has WNS. The same is true if X has GGLD and Y has property (k). We also show that if X and Y both have property (M ), then X ⊕∞ Y has the WFPP. Throughout the paper we shall use the notations: * for weak convergence and → for strong convergence.
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2. Results Recall that a bounded sequence (x n ) in a Banach space X is called diametral if diam(x n ) ¿ 0 and lim diam(x n+1 ; co(x1 ; : : : ; x n )) = diam(x n ). The basic characterization of normal structure, due to Brodskii and Milman [2], is as follows: A bounded convex subset K of a Banach space X , has normal structure if and only if it does not contain any diametral sequence. So, if X fails to have weak normal structure, then, there exists a sequence (x n ) such that x n * 0 and limx n − x = diam(x n ) = 1 for all x ∈ co(x n ). In particular, since 0 ∈ co(x n ), we have limx n = 1. For more details, see [7]. A stronger condition than WNS was introduced by Sims [30]. Denition 1. A Banach space X has property (k) if there exists k ∈ (0; 1) such that whenever x n * 0, x n → 1 and lim inf x n − x ≤ 1, we have x ≤ k. Proposition 1. Let X and Y be Banach spaces with property (k). Then X ⊕1 Y; has the weak normal structure. Proof. Suppose that X ⊕1 Y fails to have the WNS. Then there exists a sequence (x n ; yn ) in X ⊕1 Y such that • (x n ; yn ) * 0. • (x n ; yn ) → 1. • lim dist((x n+1 ; yn+1 ); co((x1 ; y1 ); : : : ; (x n ; yn ))) = 1 = diam((x n ; yn )). Without loss of generality, we may assume that the following limits all exist: limx n =: A;
limyn =: B;
lim x n − xj =: Aj ; n
lim yn − yj =: Bj : n
(1)
It then follows that 1 = A + B = Aj + Bj ;
∀j ≥ 1:
(2)
Hence, we see that if Bj ¿ B; Aj ¡ A and if Bj ≤ B; Aj ≥ A. So we have a subsequence (nj ) of positive integers for which either Anj ≤ A for all j or Bnj ≤ B for all j. Since both X and Y have property (k), we obtain that either x nj ≤ kA for all j or ynj ≤ kB for all j. Taking the limit as j → ∞ yields either A ≤ kA or B ≤ kB. This is a contradiction. We can strengthen the conclusion of Proposition 1 by assuming the nonstrict Opial property on X and Y . This means that for any sequence (x n ), we have x n * 0 ⇒ lim sup x n ≤ lim sup x n − x
∀x ∈ X:
(3)
Proposition 2. Let X and Y be Banach spaces both with property (k) and the nonstrict Opial property. Then; for each p ∈ [1; ∞); X ⊕p Y has both property (k) and the nonstrict Opial property.
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Proof. For simplicity, we prove the case of p=1. Let (x n ; yn ) * 0; (x n ; yn )=x n + yn → 1, and lim sup(x n ; yn ) − (x; y) = lim sup(x n − x + yn − y) ≤ 1. We have to show that x + y ≤ k. By passing to suitable subsequences if necessary, we may assume that the following limits all exist: limx n =: A;
limyn =: B;
lim x n − x =: C; n
lim yn − y =: D: n
(4)
Then we have A+B=1
and
C + D ≤ 1:
By the nonstrict Opial property of X and Y , we see that A ≤ C and B ≤ D and hence we must have A = C and B = D. Now in view of property (k) of X and Y , we get x ≤ kA and y ≤ kB. Therefore, (x; y) = x + y ≤ kA + kB = k and X ⊕1 Y has property (k). We next show that X ⊕1 Y has the nonstrict Opial property. For this purpose we assume that (x n ; yn ) is a weakly null sequence in X ⊕1 Y . For any (x; y) ∈ X ⊕1 Y we have to show that, lim sup (x n ; yn ) ≤ lim sup (x n − x; yn − y). But through a suitable subsequence (n ) we have lim sup(x n ; yn ) = lim x n + lim yn ≤ lim x n − x + lim yn − y ≤ lim sup(x n − x + yn − y) = lim sup (x n − x; yn − y): Remark. Proposition 2 is not true in general for the case p = ∞. Indeed, if we take the product space ‘2 ⊕∞ R; (x n ; yn ) := (en ; 0), and (x; y) := (0; 1), then we have lim sup (x n ; yn ) − (x; y) = 1 = lim (x n ; yn ). However, (x; y) = 1 and thus property (k) does not hold for any k ∈ [0; 1). In spite of this, one can easily see that the nonstrict Opial property is preserved in the product space X ⊕p Y for 1 ≤ p ≤ ∞. More recently, in order to improve a result of Lin [27], Xu and Marino [34] introduced a uniform version of property (k). Denition 2. A Banach space X is said to have the uniform property (k) if there exists a constant k ∈ [0; 1) such that whenever (x n ); (ym ) ⊂ X satisfy the conditions: x n * 0; x n → 1 and lim supm lim supn x n − ym ≤ 1, we have lim supm ym ≤ k. By the same argument as used in the proof of Proposition 2 we have Proposition 3. Let X and Y be Banach spaces both having the uniform property (k) and the nonstrict Opial property. Then; for each p ∈ [1; ∞); X ⊕p Y has the uniform property (k) and the nonstrict Opial property. Another su=cient condition for weak normal structure was introduced by JimPenezMelado [9] in 1992.
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Denition 3. A Banach space X is said to have the generalized Gossez–Lami Dozo property (GGLD) if, for all weakly null sequences (x n ) in X such that lim x n = 1, we have lim supm lim supn x n − xm =: D(x n ) ¿ 1. It is shown in [34] that the uniform property (k) implies GGLD but property (k) does not imply GGLD. We have the following result. Proposition 4. Let X and Y be Banach spaces. Assume that X has GGLD and Y has property (k). Then X ⊕1 Y has WNS. Proof. Suppose that X ⊕1 Y fails to have WNS. Then there exists a sequence (x n ; yn ) ∈ X ⊕1 Y satisfying • (x n ; yn ) * 0. • x n + yn ≤ 1; lim(x n + yn ) = 1. • limn (x n − xj + yn − yj ) = 1 = diam((x n ; yn )) for j ≥ 1. Without loss of generality, we may assume that there exist the limits: limx n =: a; limyn =: b; limn x n −xm =: am , and limn yn −ym =: bm . If (x n ) does not converge to 0 in norm, then GGLD of X implies that a = lim x n ¡ lim sup lim sup x n − xm : n
m
n
Let m0 be an integer large enough so that a ¡ lim x n − xm n
∀m ≥ m0 :
Since, for m ≥ m0 , lim yn − ym = 1 − lim x n − xm ¡ 1 − a = b = lim yn ; n
n
n
property (k) of Y then yields ym ≤ k · lim yn n
∀m ≥ m0 :
Hence by taking limit as m → ∞, we get b ≤ kb which implies that b=0. We therefore have lim x n = 1 and lim supm lim supn x n − xm = 1. But this contradicts the GGLD of X . Next, if x n → 0, then we see that lim yn − ym = 1 − xm ≤ 1 = lim yn : n
n
It follows from property (k) of Y that ym ≤ k
∀m:
This however contradicts the fact that limn yn = 1. In 1993 Kalton [10] introduced the notion of property (M ). Denition 4. A Banach space X is said to have property (M ) provided weakly null types are constant on spheres about the origin. That is, if x n * 0 and u; v ∈ X such that x = y, then lim sup x n + u = lim sup x n + v.
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Garcia-Falset and Sims [6] proved that property (M ) implies WFPP. One may then ask this question: Is property (M ) preserved in product spaces? The answer is negative, as shown in the examples below. Example 1. Let p ∈ [1; ∞). Consider l2 ⊕p l2 . Put (u; v) := lim sup (en ; en ) − (u; v) n
= lim sup(en − up2 + en − vp2 )1=p ; n
u; v ∈ l2 ;
where (en ) is the standard unit basis of l2 . Then, (0; e1 ) = (2−1=p e1 ; 2−1=p e1 ) = 1: However, easy calculations show that (0; e1 ) = (1 + 2p=2 )1=p and (2−1=p e1 ; 2−1=p e1 ) = 21=p (1 + 2−2=p )1=2 : Hence, (0; e1 ) = (2−1=p e1 ; 2−1=p e1 ) for p = 2 and l2 ⊕p l2 does not have property (M ) if p = 2. Example 2. Let p = ∞ and consider l2 ⊕∞ l3 . Set (u; v) := lim sup (en ; en ) − (u; v) n
= lim sup max{en − u2 ; en − v3 }; n
u ∈ l2 ; v ∈ l3 ;
√ where (en ) √is the standard unit basis of l2 and l3 . Then we have (e1 ; 0) = 2 while (0; e1 ) = 3 2; nevertheless, (e1 ; 0) = (0; e1 ) = 1. Hence l2 ⊕∞ l3 fails to have property (M ). In spite of the last example, we have the following result. Proposition 5. Let X and Y be Banach spaces both having property (M ). Then X ⊕∞ Y has the weak 2xed point property for nonexpansive mappings. Proof. Suppose that X ⊕∞ Y fails to have the WFPP. Then we have a weakly compact convex subset K of X ⊕∞ Y with diam(K) = 1 and a 7xed point free nonexpansive mapping T : K → K with respect to which K is a minimal nonempty weakly compact convex invariant subset which contains an approximate 7xed point sequence (an ) with an * 0 and, by the Goebel–Karlovitz lemma, lim an − z = diam(K) = 1 for all z ∈ K. Let [X ⊕∞ Y ] := l∞ (X ⊕∞ Y )=c0 (X ⊕∞ Y ) be equipped with the quotient norm given canonically by [zn ] = lim sup zn : Let [K] := {[zn ]: zn ∈ K; ∀n}. Then [T ][x n ] := [Tx n ] is a well-de7ned nonexpansive mapping on [K]. Let W := {[wn ] ∈ [K]: [wn ] − [an ] ≤
1 3
and D[wn ] ≤ 23 };
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where D[wn ] := lim supm lim supn wn − wm is well de7ned since for (wn − yn ) ∈ c0 (X ⊕∞ Y ) we have D(wn ) = D(yn ). Then W is [T ]−invariant, closed, convex and nonempty as [ 23 an ] ∈ W . For [wn ] ∈ W , we may extract a subsequence (wnk ) such that wnk * w0 ∈ K and lim wnk = [wn ]. Now, wnk = (x nk ; ynk ); x nk ∈ X; ynk ∈ Y , and without loss of generality, we may assume that there exist the limits: lim x nk =: A and lim ynk =: B so that [wn ] = limk wnk = max{A; B} = A. (Here we assume that A ≥ B since the proof given below works for the case B ≥ A as well.) Write w0 = (x0 ; y0 ). Then x nk * x0 and ynk * y0 . Moreover, wnk − ank * w0 , so x0 ≤ w0 ≤ lim inf wnk − ank ≤ [wnk ] − [ank ] ≤ 13 : Let d := limk x nk − x0 . We distinguish two cases. Case 1. d ¡ 35 . Then [wn ] = lim x nk = lim (x nk − x0 ) + x0 ≤ d + x0 ¡ 13=15: k
Case 2. d ≥ In this case,
k
3 5
¿ 13 ≥ x0 . let zk := (x0 − x nk )=d.
Then zn → 1. We also have
[wn ] = lim x nk = d lim (x0 =d) − zk = d (x0 =d) ≤ d sup k
k
v≤1
(v);
where (v) := lim supk zk − v for v ∈ X . Now by Garcia-Falset and Sims [6, Lemma 3:1] we get sup
v≤1
(v) = D(zk ):
It then follows that [wn ] ≤ d sup
v≤1
(v) = d D(zk )
= D(x nk ) ≤ max{D(x nk ); D(ynk )} = D(wnk ) ≤ D(wn ) = D[wn ] ≤ 23 : In summary, we get that any [wn ] ∈ W is away from one. More precisely, we have [wn ] ≤ 13=15. This however contradicts Lin’s extension [25] of the Goebel–Karlovitz lemma which assures that W contains elements of norm arbitrarily close to one. The above provides a partial answer to a question of Khamsi [11] for the case p=∞. Finally, we examine another property stronger than the weak normal structure, property (P) introduced by Tan and Xu [33]. Denition 5. A Banach space X is said to have Property (P) if, for any nonconstant sequence (x n ) in X , we have x n * 0 ⇒ lim inf x n ¡ diam(x n ): (Clearly, in this de7nition, ‘lim inf ’ can be replaced with ‘lim sup’.) Our 7nal result improves Corollary 2:1 and Theorem 2:3 of [33] in the case p = ∞. Proposition 6. Let X and Y be Banach spaces both with property (P). Then X ⊕∞ Y also has property (P).
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Proof. Let (x n ; yn ) be a weakly null sequence in X ⊕∞ Y . Then we have lim sup (x n ; yn ) = lim sup max{x n ; yn } = max{lim sup x n ; lim sup yn } ¡ max{diam(x n ); diam(yn )} = diam(x n ; yn ): Acknowledgements The authors thank the referee for his=her suggestions for the manuscript. References [1] P. Belluce, W.A. Kirk, E.F. Steiner, Normal structure in Banach spaces, Paci7c J. Math. 26 (1968) 433–440. [2] M.S. Brodskii, D.P. Milman, On the center of a convex set, Dolk. Akad. Nauk SSSR 59 (1948) 837–840. [3] D. van Dulst, B. Sims, Fixed points of nonexpansive mappings and Chebyshev centers in Banach spaces with norms of type (KK), Lecture Notes in Math. 991 (1983) 35–43. [4] J. Garcia-Falset, A. Jimenez-Melado, E. Llorens-Foster, Measures of noncompactness and normal structure in Banach spaces, Studia Math. 110 (1994) 1–8. [5] J. Garcia-Falset, E. Llorens-Foster, Normal structure and 7xed point property, Glasgow Math. J. 38 (1996) 29–37. [6] J. Garcia-Falset, B. Sims, Property (M ) and the weak 7xed point property, Proc. Amer. Math. Soc. 125 (1997) 2891–2896. [7] K. Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, New York, 1990. [8] R. HuL, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10 (1980) 743–749. [9] A. Jimenez-Melado, Stability of weak normal structure in James quasi-reSexive space, Bull. Austral. Math. Soc. 46 (1992) 367–372. [10] N.J. Kalton, M -ideals of compact operators, Illinoins J. Math. 37 (1987) 181–191. [11] M.A. Khamsi, On normal structure, 7xed point property and contractions of type ('), Proc. Amer. Math. Soc. 106 (1989) 995–1001. [12] W.A. Kirk, A 7xed point theorem for mappings which do not increase distances, Amer. Math. Soc. Monthly 72 (1965) 1004–1006. [13] W.A. Kirk, Nonexpansive mappings in product spaces, set-valued mappings and k-uniform rotundity, in: F.E. Browder (Ed.), Nonlinear Functional Analysis and its Applications Amer. Math. Soc. Symp. Pure Math. vol. 45 (2), 1986, pp. 51– 64. [14] W.A. Kirk, An iteration process for nonexpansive mappings with applications to 7xed point theory in product spaces, Proc. Amer. Math. Soc. 107 (1989) 411–415. [15] W.A. Kirk, Nonexpansive mappings in separable product spaces, Boll. U.M.I. 9-A (1995) 239–244. [16] W.A. Kirk, Some questions in metric 7xed point theory, in: T. Dominguez-Benavides (Ed.), Recent Advances on Metric Fixed Point Theory Proc. of the International Workshop on Metric Fixed Point Theory, Seville, Spain, September 1995; Universidad de Seville Press, Seville, 1996, pp. 73–97. [17] W.A. Kirk, C. Martinez-Yanez, Nonexpansive and locally nonexpansive mappings in product spaces, Nonlinear Anal. TMA 12 (1988) 719–725. [18] W.A. Kirk, Y. Stenfeld, The 7xed point property for nonexpansive mappings in certain product spaces, Houston J. Math. 10 (1984) 207–214.
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