Opial modulus and stability of the fixed point property

Nonlinear Analysis 39 (2000) 341 – 349

www.elsevier.nl/locate/na

Opial modulus and stability of the ÿxed point property A. JimÃenez-Meladoa; ∗; 1 , E. Llorens-Fuster b; 2 a

Departamento de AnÃalisis MatemÃatico, Facultad de Ciencias, Universidad de MÃalaga, Campus de Teatinos, 29071 MÃalaga, Spain b Departamento de Anà alisis MatemÃatico, Facultad de MatemÃaticas, 46100 Burjassot, Valencia, Spain Received 30 January 1997; accepted 10 September 1997

Keywords: Fixed point property; Nonexpansive mappings; Opial modulus

1. Introduction and preliminary results Let C be a nonempty subset of a Banach space X , with norm k · k. A mapping T : C → X is called nonexpansive if kT (x) − T (y)k ≤ kx − yk for all x; y ∈ C. We say that X has the weak ÿxed point property (WFPP) if every nonexpansive mapping T : C → C deÿned on a nonempty convex and weakly compact subset K of X has a ÿxed point. Perhaps, the most classical result in this ÿeld is that of Kirk [11], which asserts that every Banach space with normal structure has the WFPP. Since then, much eorts has been directed to ÿnding geometrical conditions on a Banach space X which guarantee the WFPP for X . Although Alspach [2] showed that L1 [0; 1] lacks the WFPP, an earlier result of Day et al. [5] states that there exists a renorming of L1 [0; 1] with the WFPP. Hence, the WFPP is not invariant under topological isomorphisms. Nevertheless, many stability results have been established for the WFPP in terms of the Banach–Mazur distance and other coecients of Banach spaces. Recently, GarcÃa–Falset [7] showed that a Banach space Y has the WFPP if there exists a Banach space X such that d(X; Y ) R(X )¡2, where d(X; Y ) is the Banach– ∗ 1 2

Corresponding author. Partially supported by D.G.I.C.Y.T. PB94-1496 and by a grant from La Junta de AndalucÃa. Partially supported by D.G.I.C.Y.T., PB93-117-CO2-02.

0362-546X/99/$ - see front matter ? 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 3 6 2 - 5 4 6 X ( 9 8 ) 0 0 1 7 3 - 4

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Mazur distance from X to Y and R(X ) is the supremum of the numbers lim inf kx+xn k, the supremum being taken over all weakly null sequences (xn ) in the unit ball of X and all points x in the unit ball of X . In the proof of this result, GarcÃa–Falset uses a nice trick combined with “nonstandard” techniques, a method introduced in Fixed Point Theory by Maurey [13] and afterwards developed by Khamsi, Lin and Borwein and Sims, among others (see [1]). Inspired by the proof of [7], DomÃnguez–Benavides [6] has improved the stability constant 2R(X )−1 . In order to do this, he deÿned for a Banach space X and a nonnegative real number a, the coecient R(a; X ) = sup{lim inf kx + xn k}; where the supremum is taken over all x ∈ X with kxk ≤ a and all weakly null sequences (xn ) in the unit ball of X such that   D(xn ) = lim sup lim sup kxn − xm k ≤ 1: n

m

It is shown in [6] that a Banach space Y has the WFPP if there exists a Banach space X such that   1+a : a≥0 : d(X; Y )¡M (X ) = sup R(a; X ) It turns out that√for 1¡p¡∞, M (lp ) = (1 + 21=(p−1) )(p−1)=p and that for a Hilbert space X , M (X ) = 3. Hence, the result of [6] improves all previously known stability results in Hilbert spaces and lp , 1¡p¡∞ (see [3, 4, 9, 10, 14]). In this paper we improve the stability constant M (X ) for a certain class of Banach spaces. To do this, we use the ideas arising from [6], but we give an elementary proof. 2. The results To state our main result we need some deÿnitions. Recall that a Banach space X has the Uniform Opial Property [15] if for every c¿0 there exists r = r(c)¿0 such that 1 + r ≤ lim inf kx + xn k for every x ∈ X with kxk ≥ c and every weakly null sequence (xn ) in X with lim inf kxn k ≥ 1. If X is a Banach space, we consider its modulus of Opial, rX (c), c ≥ 0, as in [12] by setting rX (c) = inf {lim inf kx + xn k − 1}: The inÿmum is taken over all x ∈ X with kxk ≥ c and all weakly null sequences (xn ) in X with lim inf kxn k ≥ 1. In order to simplify the statement of the following theorem, we deÿne, for B ≥ 1, the number CX (B) := sup{c ≥ 0: rX (c) ≤ B − 1}:

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Since the following well-known inequality holds for all c ≥ 0 (see [16]), c − 1 ≤ rX (c) ≤ c; then CX (B) ≥ B − 1. Theorem 1. Let X be a Banach space with norm k · k and denote by rX its modulus of Opial. Let | · | be a norm on X equivalent to k · k and let B be a positive real number such that for all x ∈ X; kxk ≤ |x| ≤ Bkxk: Let Y be the Banach space X with the new norm | · | and let C(B) = CX (B). If B satisÿes the following inequality:   1+a : a≥0 ; B¡ sup R( Ba C(B); X ) then Y has the ÿxed point property. Proof. First, let us observe that we can suppose that there exists a¿0 such that 1+a ¿B: R((a=B)C(B); X ) Suppose that Y lacks the WFPP. Then there exists a nonempty, convex and weakly compact set C ⊂ X and a ÿxed point free | · |-nonexpansive mapping T : C → C. By Zorn’s lemma, C contains a subset K which is minimal for the properties of being nonempty, convex, weakly compact and T -invariant. Since T has no ÿxed point in K, d = diam(K)¿0 and we may assume that d = 1. Standard arguments show that K contains an approximate ÿxed point sequence (afps) for T , i.e., a sequence (xn ) in K such that lim |xn − T (xn )| = 0. Since every subsequence of (xn ) is again an afps for T , we may assume that (xn ) is itself a weakly convergent afps. By translating K if necessary, we may also assume that 0 ∈ K, K is minimal for T and (xn ) is weakly null. Since K is minimal for T , the well-known Goebel–Karlovitz lemma (see [8]) implies that if (wn ) is an afps for T in K, then (wn ) is diametral, i.e., lim |wn − x| = 1 for every x ∈ K. In particular, since 0 ∈ K, lim |wn | = 1. This implies that for each ”¿0 there exists ¿0 such that, if x ∈ K and |x − T (x)|¡ then |x|¿1 − ”. Since   C(B) B R a ; X ¡1 1+a B we can choose ”¿0 such that   B C(B) ; X ¡1 − ”: R a 1+a B Let ¿0 be a real number such that |x|¿1 − ” for all x ∈ K with |x − T (x)|¡, and let ¿0 be a real number such that ¡ min{1; }.

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For each positive integer n, the Banach contraction principle ensures the existence of zn ∈ K such that zn = (1 − )T (zn ) + txn ; where t = 1=(1 + a). Since for every positive integer n we have |zn −T (zn )| ≤ ¡, we see that |zn |¿1−” for every positive integer n. On the other hand, from the triangle inequality and the | · |-nonexpansiveness of T we obtain that |zn − xn | ≤ (1 − )|zn − xn | + (1 − )|T (xn ) − xn | + (1 − t)|xn | and so |zn − xn | ≤

1− |T (xn ) − xn | + (1 − t)|xn |:

This last inequality implies that lim sup |zn − xn | ≤ 1 − t. In a similar way, we obtain that |zn − zm | ≤ t for all positive integers n, m. Then we have D(zn ) ≤ t. Since K is weakly compact, there exists a subsequence (znk ) of (zn ) such that lim sup |zn | = lim |znk | and such that (znk ) is weakly convergent, say to y ∈ K. By passing to subsequences, if necessary we may assume that the following limits exist: lim kznk − yk;

lim kznk − y − xnk k;

lim kxnk − znk k;

lim kznk k:

Fix k ∈ N . Since the sequence (znk − znr ) converges weakly to znk − y, as r → ∞, then |znk − y| ≤ lim inf |znk − znr |; r

so that lim sup |znk − y| ≤ D(znk ) ≤ D(zn ) ≤ t: Hence, we have that lim kznk − yk ≤ lim sup |znk − y| ≤ t: Since (xnk ) is an afps for T , we have that lim |xnk | = 1, by the Goebel–Karlovitz lemma. Thus, 1 = lim |xnk | ≤ lim sup |xnk − znk + y| + lim sup |znk − y| ≤ lim sup |xnk − znk + y| + t: From this we get that lim kxnk − znk + yk ≥

1 1−t lim sup |xnk − znk + y| ≥ : B B

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Since t¡1, we can consider the sequence (uk ) deÿned by uk =

B (−xnk + znk − y): 1−t

Then (uk ) is weakly convergent to 0 and lim kuk k ≥ 1. From the deÿnition of rX we also have that

 

B B

y ≥ 1 + rX kyk : lim inf uk + 1−t 1−t Using that uk + [B=(1 − t)]y = (znk − xnk )B=(1 − t), we obtain

B B B

y = lim kznk − xnk k ≤ lim sup |znk − xnk | ≤ B: lim inf uk +

1−t 1−t 1−t B kyk) and, consequently, kyk ≤ [(1 − t)=B]C(B). Hence, we have that B − 1 ≥ rX ( 1−t Now, ÿx any ¿0. Since lim kznk − yk ≤ t, there exists a subsequence (yk ) of (znk ) such that kyk − yk¡t +  for every positive integer k. On the other hand, we have       yk zn t yk − y =D ≤D ≤ ¡1 D t+ t+ t+ t+

and also that

y 1 − t C(B) 1 − t C(B) C(B)

=a :

t +  ≤ t +  B ≤ t B B From the above facts we conclude that



 

yk

= lim yk − y + y ≤ R a C(B) ; X : lim

t+

t +  t +  B and, since ¿0 is arbitrary,   C(B) 1 R a ;X : lim kyk k ≤ 1+a B Finally, we get that

  B C(B) R a ;X ; 1 − ” ≤ lim |znk | = lim |yk | ≤ 1+a B

which is a contradiction. Corollary 1. Let X be a Hilbert space with norm k · k. Suppose that | · | is a norm on X such that kxk ≤ |x| ≤ Bkxk for all x ∈ X p √ If B¡ 2 + 2; then (X; | · |) has the WFPP.

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√ Proof. Since k · k is an√euclidean norm; we have that rX (c) = 1 + c2 − 1 for all c ≥ 0 and, therefore, C(B) = B2 − 1. On the other hand, r 1 + a2 ; R(a; X ) = 2 for all a ≥ 0. So  r  1 B2 − 1 C(B) ;X = : + a2 R a B 2 B2 By the previous theorem, the result will follow if we have       1+a 1+a : a ≥ 0 = sup q : a≥0 ; B¡ sup a 2  1  R( B C(B); X ) 2 B −1 2 + a B2 i.e., if there exists a ≥ 0 such that q

1+a 1 2

+ a2 (B2 − 1)=B2

¿B

or, equivalently, if there exists a ≥ 0 such that F(a) := 2

(1 + a)2 + a2 ¿B2 : 1 + 2a2

√ Observe that F attains its maximum value at a0 = 2=2. So the corollary follows if B satisÿes √ ! 2 ¿B2 ; F 2 p √ and this happens if B¡ 2 + 2. In order to obtain a similar result for ‘p when 1¡p¡∞ and p 6= 2 we will need to ÿnd the maximum of the positive real function Fp (a) := 2

(1 + a)p + ap 1 + 2ap

deÿned for a ≥ 0. It is easy to see that 2 = Fp (0) = lim Fp (a): a→∞

On the other hand, for nonnegative a Fp0 (a) =

2p (ap−1 + (a + 1)p−1 − 2(a(a + 1))p−1 ): (2ap + 1)2

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Since the term 2p=(2ap + 1)2 is positive, the sign of the derivative Fp0 (a) is the same as the sign of the continuous function Gp (a) := ap−1 + (a + 1)p−1 − 2(a(a + 1))p−1 : We have Gp

 1=(p−1) ! 1 1 = ¿0: 2 2

Moreover, for p¿2  r ! p−1  r !  r 1=(p−1) 1=(p−1) √ 1 1  = 1 + (1 − 2)  + 1 Gp  2 2 2 r ¡

√ 1 + (1 − 2) 2

r

1 +1 2

!p−1

! 1 + 1 = 0: ¡ 2 q Therefore, there exists ap in the open interval (( 12 )1=(p−1) ; ( 12 )1=(p−1) ), for which Fp0 (ap ) = 0. Using arguments from elementary Calculus, it is straightforward to see that ap is the unique maximum of Fp . When 1¡p¡2, one can also ÿnd a similar interval. We omit its argument for the sake of brevity. r

√ 1 + (1 − 2) 2

r

Corollary 2. Let p¿1; and let k · k denote the standard norm on ‘p . Assume that | · | is a norm on ‘p such that; for all x ∈ ‘p ; kxk ≤ |x| ≤ Bkxk: If B¡cp := (Fp (ap ))1=p ; then (‘p ; | · |) has the WFPP. Proof. Since (see [16]) r‘p (c) = (1+cp )1=p −1 for all c ≥ 0, then C‘p (B) = C(B) = (Bp − 1)1=p . On the other hand, (see [6]), for all a ≥ 0   1=p  1 ap (Bp − 1) C(B) ; ‘p = + : R a B 2 Bp By the previous Theorem, the result will follow if there exists a ≥ 0 such that 1+a R(a C(B) B ; ‘p )

=

i.e., (1 + a)p ¿Bp

1+a

( 12



+

ap (Bp −1) 1=p ) Bp

1 Bp − 1 + ap 2 Bp

¿B;

 ;

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which is equivalent to 2

(1 + a)p + ap ¿Bp : 1 + 2ap

The left-hand side is Fp (a) and attains its maximum value at some ap ¿( 12 )1=(p−1) . Thus, the corollary follows if B satisÿes Fp (ap )¿Bp ; which is our hypothesis. Remark 1. To compute the constant cp in the above corollary seems to be hard; especially if p is not a natural number. Nevertheless; it is clear that    r ! 1=p  1=(p−1)    1=(p−1) !!1=p   1 1  Fp ; Fp  : cp ≥ max   2 2   The right-hand side in this inequality yields bounds of stability in ‘p which are greater than M (‘p ); at least for p¡6. Remark 2. Let X be a Banach space with Opial modulus rX . In [12] the authors proved that rX is continuous on [0; ∞) and that; for 0¡c1 ≤ c2 ; rX (c2 ) − rX (c1 ) ≤ (c2 − c1 )

rX (c1 ) + 1 : c1

Now; if rX satisÿes the condition rX (1) = 0; we then have for c ≥ 1 c − 1 ≤ rX (c) ≤ (c − 1)(rX (1) + 1) = c − 1: Therefore; for those Banach spaces X with rX (1) = 0 we have CX (B) = B (B ≥ 1). It is obvious that; for such spaces; the stability bound given by our theorem is M (X ). Spaces verifying rX (1) = 0 are; for instance; X = ‘2 ⊕ R (see [12, Remark 3.1]), and the Bynum spaces ‘p; ∞ (1¡p¡∞). References [1] A.G. Aksoy, M.A. Khamsi, Nonstandard Methods in Fixed Point Theory, Universitext, Springer, Berlin, 1990. [2] D.E. Alspach, A ÿxed point free nonexpansive map, Proc. Amer. Math. Soc. 82 (1981) 423– 424. [3] J. Bernal, F. Sullivan, Banach spaces that have normal structure and are isomorphic to a Hilbert space, Proc. Amer. Math. Soc. 90 (1984) 550 –554. [4] W.L. Bynum, Normal structure coecients for Banach spaces, Paciÿc J. Math. 86 (1980) 427– 436. [5] M.M. Day, R.C. James, S. Swaminathan, Normed linear spaces which are uniformly convex in every direction, Canad. J. Math. 23 (1971) 1051–1059. [6] T. DomÃnguez Benavides, Stability of the ÿxed point property for nonexpansive mappings, to appear.

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[7] J. GarcÃa Falset, The ÿxed point property in spaces with NUS property, preprint. [8] K. Goebel, W.A. Kirk, Topics in metric ÿxed point theory, Cambridge Univ. Press, Cambridge, 1990. [9] A. JimÃenez-Melado, E. Llorens-Fuster, Stability of the ÿxed point property for nonexpansive mappings, Houston J. Math. 18 (2) (1992) 251–257. [10] M.A. Khamsi, On the stability of the ÿxed point property in ‘p , Revista Colombiana de Mat. 28 (1994) 1– 6. [11] W.A. Kirk, A ÿxed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (1965) 1004 –1006. [12] P.K. Lin, K.K. Tan, H.K. Xu, Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings, Nonlinear Anal. TMA 24 (1995) 929 –946. [13] B. Maurey, Points ÿxes des contractions sur un convexe fermÃe de L1 , Semin Anal. Fonct. ExposÃe VIII (1980) 18. [14] S. Prus, On Bynum’s ÿxed point theorem, Atti Sem. Mat. Fis. Univ. di Modena 38 (1990) 535 –545. [15] S. Prus, Banach spaces with the uniform Opial property, Nonlinear Analysis TMA 18 (1992) 697–704. [16] H.K. Xu, Geometrical coecients of Banach spaces and nonlinear mappings, preprint.