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Reflexivity is equivalent to the perturbed fixed point property for cascading nonexpansive maps in Banach lattices
Nonlinear Analysis 95 (2014) 414–420
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Reflexivity is equivalent to the perturbed fixed point property for cascading nonexpansive maps in Banach lattices Chris Lennard ∗ , Veysel Nezir Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
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Article history: Received 17 March 2013 Accepted 11 September 2013 Communicated by S. Carl MSC: primary 46B10 47H09 46B42 46B45 Keywords: Nonexpansive mapping Cascading nonexpansive mapping Reflexive Banach space Banach lattice Unconditional basis Symmetrically normed ideal of operators on an infinite-dimensional Hilbert space Affine mapping Fixed point property Closed, bounded, convex set Closed, convex hull
abstract Using a theorem of Domínguez Benavides and the Strong James’ Distortion Theorems, we prove that if a Banach space is a Banach lattice, or has an unconditional basis, or is a symmetrically normed ideal of operators on an infinite-dimensional Hilbert space, then it is reflexive if and only if it has an equivalent norm that has the fixed point property for cascading nonexpansive mappings. This new class of mappings strictly includes nonexpansive mappings. Also, using a theorem of Mil’man and Mil’man, we show that for the above classes of Banach spaces, reflexivity is equivalent to the fixed point property for affine cascading nonexpansive mappings. Further, we show that a cascading nonexpansive mapping on a closed, bounded, convex set in a Banach space always has an approximate fixed point sequence. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction In 1965, Browder [1] proved: [♠] (For every closed, bounded, convex (non-empty) subset C of a Hilbert space (X , ∥ · ∥), for all nonexpansive mappings T : C → C (i.e., ∥Tx − Ty∥ ≤ ∥x − y∥, for all x, y ∈ C ), T has a fixed point in C .) Soon after, also in 1965, Browder [2] and Göhde [3] (independently) generalised the result [♠] to uniformly convex Banach spaces (X , ∥ · ∥), e.g., X = Lp , 1 < p < ∞, with its usual norm ∥ · ∥p . Later in 1965, Kirk [4] further generalised [♠] to all reflexive Banach spaces X with normal structure: those spaces such that all non-trivial closed, bounded, convex sets C have a smaller radius than diameter. Spaces (X , ∥ · ∥) with the property of Browder [♠] became known as spaces with the ‘‘fixed point property for nonexpansive mappings’’ (FPP (n.e.)). After 48 years, it remains an open question as to whether or not every reflexive Banach space (X , ∥ · ∥) has the fixed point property for nonexpansive mappings. For more historical perspective on this open question and metric fixed point theory in general see, for example, the following sources: Goebel and Kirk [5], Goebel and Reich [6], Reich [7,8], Sims [9], Kirk and Sims [10] (and, in particular,
∗
Corresponding author. E-mail addresses: [email protected] (C. Lennard), [email protected] (V. Nezir).
0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.09.018
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Chapter 3, ‘‘Classical theory of nonexpansive mappings’’, by Goebel and Kirk), Goebel [11], Maurey [12], and Aksoy and Khamsi [13]. Reflexivity and its characterisation are also strongly related to the approximate fixed point property for nonexpansive mappings. We say that a convex subset C of a Banach space (X , ∥ · ∥) has the approximate fixed point property if every nonexpansive mapping T : K −→ K is such that inf{∥x − Tx∥ : x ∈ K } = 0 (i.e., T has an approximate fixed point sequence). Reich [14] proved that for all Banach spaces (X , ∥ · ∥), if X is reflexive then for all closed, convex subsets C of X , C has the approximate fixed point property if and only if C is linearly bounded. Shafrir [15] proved that the converse is also true; i.e., for a Banach space (X , ∥ · ∥), the following are equivalent: (1) X is reflexive; (2) for all closed convex subsets C of X , C has the approximate fixed point property if and only if C is linearly bounded. Recent related papers include Matoušková and Reich [16], and Domínguez Benavides [17]. Recently, Domínguez Benavides [18] proved the following intriguing result: (given a reflexive Banach space (X , ∥ · ∥), there exists an equivalent norm ∥ · ∥∼ on X such that (X , ∥ · ∥∼ ) has the fixed point property for nonexpansive mappings). This improves a theorem of van Dulst [19] for separable reflexive Banach spaces. In contrast to this result, the non-reflexive Banach space (ℓ1 , ∥ · ∥1 ), the space of all absolutely summable sequences with the absolute sum norm ∥ · ∥1 , fails ∞the fixed point property for nonexpansive mappings. For 1example, let C := {sequences (tn )n∈N : each tn ≥ 0 and n=1 tn = 1}. This is a closed, bounded, convex subset of ℓ . Let T : C → C be the right shift mapping on C ; i.e., T (t1 , t2 , t3 , . . .) := (0, t1 , t2 , t3 , . . .). T is clearly ∥ · ∥1 -nonexpansive (being an isometry) and fixed point free on C . Recently, in another significant development, Lin [20] provided the first example of a non-reflexive Banach space (X , ∥·∥) with the fixed point property for nonexpansive mappings. Lin verified this fact for (ℓ1 , ∥ · ∥1 ) with the equivalent norm ||| · ||| given by
|||x||| = sup k∈N
∞
8k 1 + 8k
|xn |,
for all x = (xn )n∈N ∈ ℓ1 .
n=k
What about (c0 , ∥ · ∥∞ ), the Banach space of real-valued sequences that converge to zero, with the absolute supremum norm ∥ · ∥∞ ? This is another non-reflexive Banach space. It also fails the fixed point property for nonexpansive mappings. E.g., let C := {sequences (tn )n∈N each tn ≥ 0, 1 = t1 ≥ t2 ≥ · · · ≥ tn ≥ tn+1 → 0, as n → ∞}. Let U: C → C be the natural right shift map; i.e., U (t1 , t2 , t3 , . . .) := (1, t1 , t2 , t3 , . . .). Then U is a ∥ · ∥∞ -nonexpansive (isometric, actually) mapping with no fixed points in the closed bounded convex set C . It is natural to ask whether there is a c0 -analogue of Lin’s theorem about ℓ1 . It remains an open question as to whether or not there exists an equivalent norm ∥ · ∥∼ on (c0 , ∥ · ∥∞ ) such that (c0 , ∥ · ∥∼ ) has the fixed point property for nonexpansive mappings. However, if we weaken the nonexpansive condition to ‘‘asymptotically nonexpansive’’, then the answer is ‘‘no’’. In 2000, Dowling, Lennard and Turett [21] showed that for every equivalent renorming ||| · ||| of (c0 , ∥ · ∥∞ ), there exists a closed, bounded, convex set C and an asymptotically nonexpansive mapping T : C → C (i.e., there exists a sequence (kn )n∈N in [1, ∞) such that kn −→ 1, and for all n ∈ N, for all x, y ∈ C , |||T n x − T n y||| ≤ kn |||x − y|||) such that T has no fixed point. n
In contrast to this, note that in 1972, Goebel and Kirk [22] proved that for all uniformly convex spaces (X , ∥ · ∥) (e.g., a Hilbert space), for every closed, bounded, convex set C ⊆ X , for all eventually asymptotically nonexpansive mappings T : C → C , T has a fixed point in C . We call a mapping T : C → C eventually asymptotically nonexpansive if there exists ν ∈ N and a sequence (kn )n≥ν in [1, ∞) such that kn −→ 1, and for all n ≥ ν , for all x, y ∈ C , ∥T n x − T n y∥ ≤ kn ∥x − y∥. n
In this paper, using the above-described theorem of Domínguez Benavides and the Strong James’ Distortion Theorems, we prove that if a Banach space is a Banach lattice, or has an unconditional basis, or is a symmetrically normed ideal of operators on an infinite-dimensional Hilbert space, then it is reflexive if and only if it has an equivalent norm that has the fixed point property for cascading nonexpansive mappings (see Definition 2.2). This new class of mappings strictly includes nonexpansive mappings. Cascading nonexpansive mappings are analogous to eventually asymptotically nonexpansive mappings, but examples show (see below) that neither of these two classes of mappings contain the other. Note that via Lin’s example (described above) of an equivalent renorming ||| · ||| of (ℓ1 , ∥ · ∥1 ) such that (ℓ1 , ||| · |||) has the fixed point property for nonexpansive mappings, in our theorem one cannot replace ‘‘cascading nonexpansive mappings’’ by ‘‘nonexpansive mappings’’. Also, using a theorem of Mil’man and Mil’man, we show that for Banach lattices, or Banach spaces with an unconditional basis, or symmetrically normed ideal of operators on an infinite-dimensional Hilbert space, reflexivity is equivalent to the fixed point property for affine cascading nonexpansive mappings. The theorem of Mil’man and Mil’man [23], mentioned above, states that a Banach space (X , ∥ · ∥) is reflexive if and only for every closed bounded convex subset C of X , and for all norm continuous affine mappings T : C −→ C , T has a fixed point in C . Another theorem analogous to this, and to the results in this paper, is due to Maurey [12] and Dowling and Lennard [24], who prove that a subspace Y of the Lebesgue function space (L1 [0, 1], ∥ · ∥1 ) is reflexive if and only if Y has the fixed point property for nonexpansive mappings. There are similar papers that characterise weakly compact convex sets among the closed bounded convex (or closed convex) sets in certain Banach spaces via a fixed point property. These include [25,26,17]. Finally, we prove that in any Banach space, for every closed bounded convex subset C and every cascading nonexpansive mapping T : C −→ C , T has an approximate fixed point sequence, i.e., infx∈C ∥Tx − x∥ = 0.
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On the other hand, it remains an open question as to whether or not every asymptotically nonexpansive mapping on a closed bounded convex set in a Banach space has an approximate fixed point sequence. We remark that in 1998 Kirk, Martinez Yañez and Shin [27], Theorem 4.1, discovered an interesting sufficient condition for a Banach space (X , ∥ · ∥) to be such that every asymptotically nonexpansive mapping on a closed bounded convex set in X has an approximate fixed point sequence. (This sufficient condition involves the ultrapowers of X .) We further remark that a significant part of this introduction comes from the Ph.D. thesis of the second author [28], written under the supervision of the first author. Also, the proofs of Theorem 3.1 and Corollary 3.5 ((1) H⇒ (3)) are similar to the proofs of Theorems 5.0.6, 5.0.7 and 5.2.2 ((1) H⇒ (2)) of [28]. 2. Preliminaries The symbols N and R denote the set of positive integers and the set of real numbers, respectively. Throughout this paper our scalar field is R. Also, N0 := {0} ∪ N is the set of all non-negative integers. Let (X , ∥ · ∥) be a Banach space and E ⊆ X . We will denote the closed, convex hull of E by co(E ). We will denote by c00 the vector space of all scalar sequences that have only finitely many non-zero terms. Also, for all n −1
n ∈ N, we define en ∈ c00 by en := (0, . . . , 0, 1, 0, . . . , 0, . . .). We will often write ‘‘closed bounded convex set’’ instead of ‘‘closed, bounded, convex set’’. Also, all closed bounded convex sets in this paper are assumed to be non-empty. Definition 2.1. Let C be a closed bounded convex subset of a Banach space (X , ∥ · ∥). A mapping U : C −→ C is said to be affine if for all λ ∈ [0, 1], for all x, y ∈ C , U (1 − λ) x + λ y = (1 − λ) U (x) + λ U (y).
Let C be a closed bounded convex subset of a Banach space (X , ∥ · ∥). Let T : C −→ C be a mapping. Let C0 := C and C1 := co T (C ) ⊆ C .
Clearly C1 is a closed bounded convex set in C . Let x ∈ C1 . Then Tx ∈ T (C1 ) ⊆ T (C ) ⊆ co T (C ) = C1 .
So, T maps C1 into C1 . Next, for all n ∈ N, we define Cn := co T (Cn−1 ) .
Clearly, for all n ∈ N, Cn is a closed bounded convex subset of C , T maps Cn into Cn and Cn ⊆ Cn−1 . Definition 2.2. Let (X , ∥ · ∥) be a Banach space and C be a closed bounded convex subset of X . Let T : C −→ C be a mapping and (Cn )n∈N0 be defined as above. We say that T is cascading nonexpansive if there exists a sequence (λn )n∈N0 in [1, ∞) such that λn −→ 1, and for all n ∈ N0 , for all x, y ∈ Cn , ∥Tx − Ty∥ ≤ λn ∥x − y∥. n
Note that every cascading nonexpansive mapping is norm-to-norm continuous, and every nonexpansive mapping is cascading nonexpansive. Cascading nonexpansive mappings arise naturally in Banach spaces (X , ∥ · ∥) that contain an isomorphic copy of ℓ1 or c0 (see Theorem 3.1). Examples of such spaces are Banach spaces isomorphic to a non-reflexive Banach lattice and nonreflexive Banach spaces with an unconditional basis. (See, for example, Lindenstrauss and Tzafriri [29] 1.c.5 and [30] 1.c.12.) Indeed, not containing an isomorphic copy of ℓ1 or c0 characterises reflexivity for both these classes of Banach spaces. Another class of Banach spaces of this type is the class of all symmetrically normed ideals of operators on an infinitedimensional Hilbert space. (See Dodds and Lennard [31], Proposition 4.5.) This is a very different class than the first two, since all symmetrically normed ideals, except the Hilbert–Schmidt operators, fail to have local unconditional structure. (See Lewis [32].) Definition 2.3. We say that a Banach space (X , ∥ · ∥) has the fixed point property for cascading nonexpansive mappings if for all closed bounded convex subsets C of X , for all cascading nonexpansive mappings T : C −→ C , T has a fixed point in C . Also, (X , ∥ · ∥) has the perturbed fixed point property for cascading nonexpansive mappings if there exists an equivalent norm ∥ · ∥∼ on X such that (X , ∥ · ∥∼ ) has the fixed point property for cascading nonexpansive mappings. Example 2.4 (Not all Cascading Nonexpansive Mappings are Eventually Asymptotically Nonexpansive). Let (γn )n∈N be a strictly increasing sequence in (0, 1) that converges to 1. Also assume that (γn+1 /γn )n∈N is a decreasing sequence in (1, 2]. An
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example of such a sequence is (γn := 8n /(1 + 8n ), for all n ∈ N), mentioned in the introduction above. Consider the Banach space ℓ1 , endowed with the equivalent norm
∥x∥∼ := sup γν ν∈N
∞
|xk |,
for all x ∈ ℓ1 .
k=ν
Let K := {x ∈ ℓ : each xj ≥ 0 and 1
∞
j =1
xj = 1}. Also, define the mapping R : K −→ K by Rx := (0, x1 , x2 , x3 , . . .), for all
x ∈ K . The set K0 := K is closed, bounded and convex in ℓ1 . Let Kn := co R(Kn−1 ) , for all n ∈ N. It is easy to check that each
n ∞ Kn = (0, . . . , 0, xn+1 , xn+2 , . . .) ∈ ℓ1 : each xj ≥ 0 and xj = 1 . j=n+1 n
n
Fix n ∈ N0 . Fix u, v ∈ Kn . Then u = (0, . . . , 0, xn+1 , xn+2 , . . .) and v = (0, . . . , 0, yn+1 , yn+2 , . . .), for some x, y as described n +1
n +1
above. Thus, Ru = (0, . . . , 0, xn+1 , xn+2 , . . .) and Rv = (0, . . . , 0, yn+1 , yn+2 , . . .). Then, ∥u − v∥∼ = sup γν ν≥n+1
∞
|xk − yk | and ∥Ru − Rv∥∼ = sup γν ν≥n+2
k=ν
∞
|xk − yk |.
k=ν−1
Therefore,
∥Ru − Rv∥∼ ≤ ∥u − v∥∼ sup
ν≥n+2
γν γn+2 = ∥u − v∥∼ . γν−1 γn+1
Also, en+1 , en+2 ∈ Kn and
∥Ren+1 − Ren+2 ∥∼ = ∥en+2 − en+3 ∥∼ = 2 γn+2 =
γn+2 ∥en+1 − en+2 ∥∼ . γn+1 γ
So, R is a cascading nonexpansive mapping on K , with best constants λn = γn+2 , for all n ∈ N0 . n+1 On the other hand, ∥e1 − e2 ∥∼ = 2 γ1 , and for all n ∈ N,
∥Rn e1 − Rn e2 ∥∼ = ∥en+1 − en+2 ∥∼ = 2 γn+1 =
γn+1 ∥e1 − e2 ∥∼ . γ1
γ
But nγ+1 −→ 1/γ1 > 1. Therefore, R is not eventually asymptotically nonexpansive on K . 1 n Example 2.5 (Not all Eventually Asymptotically Nonexpansive Mappings √ are Cascading Nonexpansive). Consider the Banach space (R, | · |) and the closed, bounded, convex set K0 := K := [0, 1/ 2]. Let a ∧ b := min{a, b}, for all a, b ∈ R. Also, let Q denote the set of all rational √ numbers √ and I := R \ Q be the set of all irrational numbers. We define the mapping U : K −→ K by setting Ux := 2 x ∧ (1/ 2 ), for all x ∈ Q ∩ K and Ux := 0, for all x ∈ I ∩ K . It is easy to check that Kn := co U (Kn−1 ) = K , for all n ∈ N. Further, 0, 1/2 ∈ K and
√ √ √ √ |U (0) − U (1/2)| = |0 − 1/ 2| = 1/ 2 = 2 (1/2) = 2 |0 − 1/2|. Thus, U fails to be a cascading nonexpansive mapping on K . On the other hand, for all n ≥ 2, for all x ∈ K , U n x = 0. Therefore, U is eventually asymptotically nonexpansive on K . Remark 2.6. Note that in the previous example, U is not a continuous mapping on K , and so U is not asymptotically nonexpansive. Recently, Łukasz Piasecki [33] has invented an asymptotically nonexpansive mapping T on a closed, bounded, convex set K in a Banach space (X , ∥ · ∥) such that T is not cascading nonexpansive. 3. Reflexivity iff the perturbed FPP for cascading nonexpansive mappings in Banach lattices Theorem 3.1. Let (X , ∥ · ∥) be a Banach space that contains an isomorphic copy of ℓ1 or c0 . Then there exists a closed bounded convex set C ⊆ X and an affine cascading nonexpansive mapping T : C −→ C such that T is fixed point free.
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Proof. Case 1. (X , ∥ · ∥) contains an isomorphic copy of ℓ1 . By the Strong James’ Distortion Theorem for ℓ1 [34,21], there exists a normalised sequence (xj )j∈N in X and a null sequence (εn )n∈N in (0, 1) such that for all n ∈ N, for all t = (tj )j∈N ∈ c00 ,
(1 − εn )
∞ j =n
∞ ∞ |tj | ≤ tj xj ≤ |t |. j =n j=n j
Define the closed bounded convex subset C of X by
∞
C := co{xj : j ∈ N} =
∞
tj xj : each tj ≥ 0 and
j =1
tj = 1 .
j =1
Further define T : C −→ C by
T
∞
:=
tj xj
j =1
∞
tj xj+1 .
j =1
The mapping T is affine and fixed point free. Inductively, we see that for all n ∈ N, Cn := co T (Cn−1 ) = T (Cn−1 ) is given by
Cn =
∞
tj xj : each tj ≥ 0 and
j=n+1
∞
tj = 1 .
j =n +1
Fix n ∈ N0 and fix x, y ∈ Cn . So, x = j=n+1 tj xj and y = Let αj := tj − sj , for all j ≥ n + 1. Then
∞
∞
j =n +1
sj xj , where each tj , sj ≥ 0 and
∞
j=n+1
tj =
∞
j=n+1
sj = 1.
∞ ∞ |αj |, αj xj ≥ (1 − εn+1 ) ∥x − y∥ = j=n+1 j=n+1 and since each ∥xk ∥ = 1,
∞ ∞ 1 |αj | ≤ αj xj+1 ≤ ∥Tx − Ty∥ = ∥x − y∥. j=n+1 j=n+1 (1 − εn+1 ) Consequently, T is cascading nonexpansive on C . Case 2. (X , ∥ · ∥) contains an isomorphic copy of c0 . By a strengthening of the Strong James’ Distortion Theorem for c0 ([21], Theorem 8), there exists a normalised sequence (xj )j∈N in X and a null sequence (εn )n∈N in (0, 1) such that for all n ∈ N, for all t = (tj )j∈N ∈ c00 ,
∞ tj xj ≤ (1 + εn ) max |tj |. max |tj | ≤ j =n j ≥n j≥n Define fk := x1 + · · · + xk , for all k ∈ N. Next we define the closed bounded convex subset C of X by
C := co{fk : k ∈ N} =
∞
tj xj : 1 = t1 ≥ t2 ≥ · · · ≥ tk ↓k 0 .
j =1
Also, we define the function T : C −→ C by
T
∞
tj xj
j =1
:= x1 +
∞
tj xj+1 .
j =1
The mapping T is affine and fixed point free. Inductively, it follows that for all n ∈ N, Cn := co T (Cn−1 ) = T (Cn−1 ) is given by
Cn =
∞
tj xj : 1 = t1 = t2 = · · · = tn+1 ≥ tn+2 ≥ · · · ≥ tk ↓k 0 .
j =1
Fix n ∈ N0 and fix x, y ∈ Cn . Hence, x = j=1 tj xj and y = j=1 sj xj , where 1 = t1 = · · · = tn+1 ≥ tn+2 ≥ · · · ≥ tk ↓k 0 and 1 = s1 = · · · = sn+1 ≥ sn+2 ≥ · · · ≥ sk ↓k 0. Let αj := tj − sj , for all j ∈ N. We have that
∞
∞ ∥x − y ∥ = αj xj ≥ max |αj |, j=n+2 j ≥ n +2
∞
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and
∞ ∞ ∞ ∥Tx − Ty∥ = αj xj+1 = αj xj+1 = αk−1 xk j =1 j=n+2 k=n+3 ≤ (1 + εn+3 ) max |αk−1 | = (1 + εn+3 ) max |αj | k≥n+3
j ≥ n +2
≤ (1 + εn+3 ) ∥x − y∥. Consequently, T is cascading nonexpansive on C .
Theorem 3.2. Let (X , ∥ · ∥) be a reflexive Banach space. Then there exists an equivalent norm ∥ · ∥∼ on X such that for every closed bounded convex subset C of X , for all ∥ · ∥∼ -cascading nonexpansive mappings T : C −→ C , T has a fixed point in C . Proof. By a theorem of Domínguez Benavides [18], there exists an equivalent norm ∥ · ∥∼ on X such that for every closed bounded convex subset E of X , for all ∥ · ∥∼ -nonexpansive mappings U : E −→ E, U has a fixed point in E. Fix an arbitrary closed bounded convex subset C of X . Let T : C −→ C be a ∥ · ∥∼ -cascading nonexpansive mapping. As above, let C0 := C and Cn := co T (Cn−1 ) , for all n ∈ N. By hypothesis there exists a sequence (λn )n∈N0 in [1, ∞) such that λn −→ 1, and for all n ∈ N0 , for all x, y ∈ Cn , ∥Tx − Ty∥∼ ≤ λn ∥x − y∥∼ . n
Now X is reflexive, and so C is weakly compact. By Zorn’s Lemma there exists a non-empty closed bounded convex set D ⊆ C such that D is a minimal invariant set for T .That is, T (D) ⊆ D, and if F is a non-empty closed bounded convex subset of D with T (F ) ⊆ F , then F = D. It follows that co T (D) = D. Let D0 := D and Dn := co T (Dn−1 ) , for all n ∈ N. Inductively, we see that for all n ∈ N, D = Dn ⊆ Cn . Therefore, by our hypotheses on T , for all x, y ∈ D, for all n ∈ N,
∥Tx − Ty∥∼ ≤ λn ∥x − y∥∼ . But λn −→ 1. Consequently, for all x, y ∈ D, n
∥Tx − Ty∥∼ ≤ ∥x − y∥∼ ; i.e., T is ∥ · ∥∼ -nonexpansive on D. By Domínguez Benavides [18], T has a fixed point in D ⊆ C .
From the above proof we also have the following result. Corollary 3.3. Let (X , ∥ · ∥) be a Banach space. Then the following are equivalent. (1) (X , ∥ · ∥) has the weak fixed point property for nonexpansive mappings; i.e., for every weakly compact convex subset C of X , for all nonexpansive mappings T : C −→ C , T has a fixed point. (2) (X , ∥ · ∥) has the weak fixed point property for cascading nonexpansive mappings, i.e., for every weakly compact convex subset C of X , for all cascading nonexpansive mappings T : C −→ C , T has a fixed point. Thus, all the results in the literature concerning spaces that have the weak fixed point property for nonexpansive mappings (see, for example, the texts [6,5,13,10,11]) are also true for cascading nonexpansive mappings. Combining Theorem 3.1, some remarks in the Preliminaries, and Theorem 3.2, we get the following fixed point property characterisation of reflexivity in Banach lattices, Banach spaces with an unconditional basis, and symmetrically normed ideals. Theorem 3.4. Let (X , ∥ · ∥) be a Banach lattice, or a Banach space with an unconditional basis, or a symmetrically normed ideal. Then the following are equivalent. (1) X is reflexive. (2) There exists an equivalent norm ∥ · ∥∼ on X such that for all closed bounded convex sets C ⊆ X and for all ∥ · ∥∼ -cascading nonexpansive mappings T : C −→ C , T has a fixed point in C . Note that via Lin’s example [20] of an equivalent renorming ||| · ||| of (ℓ1 , ∥ · ∥1 ) such that (ℓ1 , ||| · |||) has the fixed point property for nonexpansive mappings, we cannot replace ‘‘cascading nonexpansive mappings’’ by ‘‘nonexpansive mappings’’ in part (2) of Theorem 3.4. Corollary 3.5. Let (X , ∥ · ∥) be a Banach lattice, or a Banach space with an unconditional basis, or a symmetrically normed ideal. Then the following are equivalent. (1) X is reflexive. (2) There exists an equivalent norm ∥ · ∥∼ on X such that for all closed bounded convex sets C ⊆ X and for all ∥ · ∥∼ -cascading nonexpansive mappings T : C −→ C , T has a fixed point in C . (3) There exists an equivalent norm ∥·∥∼ on X such that for all closed bounded convex sets C ⊆ X and for all affine ∥·∥∼ -cascading nonexpansive mappings T : C −→ C , T has a fixed point in C . (4) For all equivalent norms ||| · ||| on X , for all closed bounded convex sets C ⊆ X and for all affine ||| · |||-cascading nonexpansive mappings T : C −→ C , T has a fixed point in C . (5) For all closed bounded convex sets C ⊆ X and for all affine norm-to-norm continuous mappings T : C −→ C , T has a fixed point in C .
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Proof. (1) ⇐⇒ (2). This is Theorem 3.4. (3) H⇒ (1). The statement (not(1)) implies (not(3)) by Theorem 3.1. (1) H⇒ (5). This is a theorem of Mil’man and Mil’man [23]. (5) H⇒ (4) H⇒ (3). This is clear. We remark that an analogous result to the equivalence of (1) and (4) in Corollary 3.5, involving a different kind of asymptotically nonexpansive map (semi-strongly asymptotically nonexpansive mappings), is proven in the Ph.D. thesis of the second author [28] (Theorem 5.2.2). Remark 3.6. In certain Banach spaces there exist cascading nonexpansive mappings on closed bounded convex sets that are not nonexpansive and fail to have a fixed point. For example, consider Lin’s [20] equivalent renorming ||| · ||| of (ℓ1 , ∥ · ∥1 ), and apply Theorem 3.1 to the space (ℓ1 , ||| · |||). Or, see Example 2.4. Nevertheless, in all Banach spaces, cascading nonexpansive mappings on closed bounded convex sets always have approximate fixed point sequences. This follows directly from [5] Lemma 20.1, which gives an upper estimate for the minimal displacement of Lipschitz mappings. Acknowledgements We thank Barry Turett for helpful discussions. The second author thanks the University of Pittsburgh, the Community College of Allegheny County and Point Park University for their financial support. We also thank the referees for their helpful suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]
F.E. Browder, Fixed-point theorems for noncompact mappings in Hilbert space, Proc. Natl. Acad. Sci. USA 53 (6) (1965) 1272–1276. F.E. Browder, Nonexpansive nonlinear operators in a Banach space, Proc. Natl. Acad. Sci. USA 54 (4) (1965) 1041–1044. D. Göhde, Zum Prinzip der kontraktiven Abbildung, Math. Nachr. 30 (1965) 251–258. W.A. Kirk, A fixed point theorem for mappings which do not increase distances, Amer. Math. Monthly 72 (9) (1965) 1004–1006. K. Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory, in: Cambridge Studies in Advanced Mathematics, vol. 28, Cambridge University Press, Cambridge, 1990. K. Goebel, S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, New York, 1984. S. Reich, The fixed point property for nonexpansive mappings I, Amer. Math. Monthly 83 (1976) 266–268. S. Reich, The fixed point property for nonexpansive mappings II, Amer. Math. Monthly 87 (1980) 292–294. B. Sims, The existence question for fixed points of nonexpansive maps, in: Lecture Notes, Kent State Univ., 1986. W.A. Kirk, B. Sims (Eds.), Handbook of Metric Fixed Point Theory, Kluwer Academic Publishers, Dordrecht, 2001. K. Goebel, Concise Course on Fixed Point Theorems, Yokohama Publishers, 2002. B. Maurey, Points fixes contractions de certains faiblement compacts de L1 , Seminaire d’Analyse Fonctionelle, 1980–1981, Centre de Mathématiques, École Polytech., Palaiseau, Exp. No. VIII, 1981. A.G. Aksoy, M.A. Khamsi, Nonstandard Methods in Fixed Point Theory, in: Universitet, Springer-Verlag, 1990. S. Reich, The almost fixed point property for nonexpansive mappings, Proc. Amer. Math. Soc. 88 (1983) 44–46. I. Shafrir, The approximate fixed point property in Banach and hyperbolic spaces, Israel J. Math. 71 (1990) 211–223. E. Matoušková, S. Reich, Reflexivity and approximate fixed points, Studia Math. 159 (2003) 403–415. T. Domínguez Benavides, The failure of the fixed point property for unbounded sets in c0 , Proc. Amer. Math. Soc. 140 (2) (2012) 645–650. T. Domínguez Benavides, A renorming of some nonseparable Banach spaces with the fixed point property, J. Math. Anal. Appl. 350 (2) (2009) 525–530. D. van Dulst, Equivalent norms and the fixed point property for nonexpansive mappings, J. Lond. Math. Soc. 25 (2) (1982) 139–144. P.K. Lin, There is an equivalent norm on ℓ1 that has the fixed point property, Nonlinear Anal. 68 (2008) 2303–2308. P.N. Dowling, C.J. Lennard, B. Turett, Some fixed point results in l1 and c0 , Nonlinear Anal. 39 (2000) 929–936. K. Goebel, W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972) 171–174. D.P. Mil’man, V.D. Mil’man, Some properties of non-reflexive Banach spaces, Mat. Sb. (N.S.) 65 (1964) 486–497 (in Russian). P.N. Dowling, C.J. Lennard, Every nonreflexive subspace of L1 [0, 1] fails the fixed point property, Proc. Amer. Math. Soc. 125 (2) (1997) 443–446. T. Domínguez Benavides, M.A. Japón, S. Prus, Weak compactness and fixed point property for affine mappings, J. Funct. Anal. 209 (1) (2004) 1–15. P.N. Dowling, C.J. Lennard, B. Turett, Weak compactness is equivalent to the fixed point property in c0 , Proc. Amer. Math. Soc. 132 (6) (2004) 1659–1666. W.A. Kirk, C. Martinez Yañez, S.S. Shin, Asymptotically nonexpansive mappings, Nonlinear Anal. TMA 33 (1) (1998) 1–12. V. Nezir, Fixed Point Properties for c0 -like Spaces, Ph.D. Thesis, University of Pittsburgh, 2012. J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces II: Function Spaces, in: Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 97, Springer-Verlag, 1979. J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces I: Sequence Spaces, in: Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 92, Springer-Verlag, 1977. P.G. Dodds, C.J. Lennard, Normality in trace ideals, J. Oper. Theory 16 (1986) 127–145. D.R. Lewis, An isomorphic characterization of the Schmidt class, Compos. Math. 30 (1975) 293–297. Ł. Piasecki, Personal communication, 2013. P.N. Dowling, W.B. Johnson, C.J. Lennard, B. Turett, The optimality of James’s distortion theorems, Proc. Amer. Math. Soc. 125 (1) (1997) 167–174.
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Reflexivity is equivalent to the perturbed fixed point property for cascading nonexpansive maps in Banach lattices Chris Lennard ∗ , Veysel Nezir Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
article
info
Article history: Received 17 March 2013 Accepted 11 September 2013 Communicated by S. Carl MSC: primary 46B10 47H09 46B42 46B45 Keywords: Nonexpansive mapping Cascading nonexpansive mapping Reflexive Banach space Banach lattice Unconditional basis Symmetrically normed ideal of operators on an infinite-dimensional Hilbert space Affine mapping Fixed point property Closed, bounded, convex set Closed, convex hull
abstract Using a theorem of Domínguez Benavides and the Strong James’ Distortion Theorems, we prove that if a Banach space is a Banach lattice, or has an unconditional basis, or is a symmetrically normed ideal of operators on an infinite-dimensional Hilbert space, then it is reflexive if and only if it has an equivalent norm that has the fixed point property for cascading nonexpansive mappings. This new class of mappings strictly includes nonexpansive mappings. Also, using a theorem of Mil’man and Mil’man, we show that for the above classes of Banach spaces, reflexivity is equivalent to the fixed point property for affine cascading nonexpansive mappings. Further, we show that a cascading nonexpansive mapping on a closed, bounded, convex set in a Banach space always has an approximate fixed point sequence. © 2013 Elsevier Ltd. All rights reserved.
1. Introduction In 1965, Browder [1] proved: [♠] (For every closed, bounded, convex (non-empty) subset C of a Hilbert space (X , ∥ · ∥), for all nonexpansive mappings T : C → C (i.e., ∥Tx − Ty∥ ≤ ∥x − y∥, for all x, y ∈ C ), T has a fixed point in C .) Soon after, also in 1965, Browder [2] and Göhde [3] (independently) generalised the result [♠] to uniformly convex Banach spaces (X , ∥ · ∥), e.g., X = Lp , 1 < p < ∞, with its usual norm ∥ · ∥p . Later in 1965, Kirk [4] further generalised [♠] to all reflexive Banach spaces X with normal structure: those spaces such that all non-trivial closed, bounded, convex sets C have a smaller radius than diameter. Spaces (X , ∥ · ∥) with the property of Browder [♠] became known as spaces with the ‘‘fixed point property for nonexpansive mappings’’ (FPP (n.e.)). After 48 years, it remains an open question as to whether or not every reflexive Banach space (X , ∥ · ∥) has the fixed point property for nonexpansive mappings. For more historical perspective on this open question and metric fixed point theory in general see, for example, the following sources: Goebel and Kirk [5], Goebel and Reich [6], Reich [7,8], Sims [9], Kirk and Sims [10] (and, in particular,
∗
Corresponding author. E-mail addresses: [email protected] (C. Lennard), [email protected] (V. Nezir).
0362-546X/$ – see front matter © 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.na.2013.09.018
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Chapter 3, ‘‘Classical theory of nonexpansive mappings’’, by Goebel and Kirk), Goebel [11], Maurey [12], and Aksoy and Khamsi [13]. Reflexivity and its characterisation are also strongly related to the approximate fixed point property for nonexpansive mappings. We say that a convex subset C of a Banach space (X , ∥ · ∥) has the approximate fixed point property if every nonexpansive mapping T : K −→ K is such that inf{∥x − Tx∥ : x ∈ K } = 0 (i.e., T has an approximate fixed point sequence). Reich [14] proved that for all Banach spaces (X , ∥ · ∥), if X is reflexive then for all closed, convex subsets C of X , C has the approximate fixed point property if and only if C is linearly bounded. Shafrir [15] proved that the converse is also true; i.e., for a Banach space (X , ∥ · ∥), the following are equivalent: (1) X is reflexive; (2) for all closed convex subsets C of X , C has the approximate fixed point property if and only if C is linearly bounded. Recent related papers include Matoušková and Reich [16], and Domínguez Benavides [17]. Recently, Domínguez Benavides [18] proved the following intriguing result: (given a reflexive Banach space (X , ∥ · ∥), there exists an equivalent norm ∥ · ∥∼ on X such that (X , ∥ · ∥∼ ) has the fixed point property for nonexpansive mappings). This improves a theorem of van Dulst [19] for separable reflexive Banach spaces. In contrast to this result, the non-reflexive Banach space (ℓ1 , ∥ · ∥1 ), the space of all absolutely summable sequences with the absolute sum norm ∥ · ∥1 , fails ∞the fixed point property for nonexpansive mappings. For 1example, let C := {sequences (tn )n∈N : each tn ≥ 0 and n=1 tn = 1}. This is a closed, bounded, convex subset of ℓ . Let T : C → C be the right shift mapping on C ; i.e., T (t1 , t2 , t3 , . . .) := (0, t1 , t2 , t3 , . . .). T is clearly ∥ · ∥1 -nonexpansive (being an isometry) and fixed point free on C . Recently, in another significant development, Lin [20] provided the first example of a non-reflexive Banach space (X , ∥·∥) with the fixed point property for nonexpansive mappings. Lin verified this fact for (ℓ1 , ∥ · ∥1 ) with the equivalent norm ||| · ||| given by
|||x||| = sup k∈N
∞
8k 1 + 8k
|xn |,
for all x = (xn )n∈N ∈ ℓ1 .
n=k
What about (c0 , ∥ · ∥∞ ), the Banach space of real-valued sequences that converge to zero, with the absolute supremum norm ∥ · ∥∞ ? This is another non-reflexive Banach space. It also fails the fixed point property for nonexpansive mappings. E.g., let C := {sequences (tn )n∈N each tn ≥ 0, 1 = t1 ≥ t2 ≥ · · · ≥ tn ≥ tn+1 → 0, as n → ∞}. Let U: C → C be the natural right shift map; i.e., U (t1 , t2 , t3 , . . .) := (1, t1 , t2 , t3 , . . .). Then U is a ∥ · ∥∞ -nonexpansive (isometric, actually) mapping with no fixed points in the closed bounded convex set C . It is natural to ask whether there is a c0 -analogue of Lin’s theorem about ℓ1 . It remains an open question as to whether or not there exists an equivalent norm ∥ · ∥∼ on (c0 , ∥ · ∥∞ ) such that (c0 , ∥ · ∥∼ ) has the fixed point property for nonexpansive mappings. However, if we weaken the nonexpansive condition to ‘‘asymptotically nonexpansive’’, then the answer is ‘‘no’’. In 2000, Dowling, Lennard and Turett [21] showed that for every equivalent renorming ||| · ||| of (c0 , ∥ · ∥∞ ), there exists a closed, bounded, convex set C and an asymptotically nonexpansive mapping T : C → C (i.e., there exists a sequence (kn )n∈N in [1, ∞) such that kn −→ 1, and for all n ∈ N, for all x, y ∈ C , |||T n x − T n y||| ≤ kn |||x − y|||) such that T has no fixed point. n
In contrast to this, note that in 1972, Goebel and Kirk [22] proved that for all uniformly convex spaces (X , ∥ · ∥) (e.g., a Hilbert space), for every closed, bounded, convex set C ⊆ X , for all eventually asymptotically nonexpansive mappings T : C → C , T has a fixed point in C . We call a mapping T : C → C eventually asymptotically nonexpansive if there exists ν ∈ N and a sequence (kn )n≥ν in [1, ∞) such that kn −→ 1, and for all n ≥ ν , for all x, y ∈ C , ∥T n x − T n y∥ ≤ kn ∥x − y∥. n
In this paper, using the above-described theorem of Domínguez Benavides and the Strong James’ Distortion Theorems, we prove that if a Banach space is a Banach lattice, or has an unconditional basis, or is a symmetrically normed ideal of operators on an infinite-dimensional Hilbert space, then it is reflexive if and only if it has an equivalent norm that has the fixed point property for cascading nonexpansive mappings (see Definition 2.2). This new class of mappings strictly includes nonexpansive mappings. Cascading nonexpansive mappings are analogous to eventually asymptotically nonexpansive mappings, but examples show (see below) that neither of these two classes of mappings contain the other. Note that via Lin’s example (described above) of an equivalent renorming ||| · ||| of (ℓ1 , ∥ · ∥1 ) such that (ℓ1 , ||| · |||) has the fixed point property for nonexpansive mappings, in our theorem one cannot replace ‘‘cascading nonexpansive mappings’’ by ‘‘nonexpansive mappings’’. Also, using a theorem of Mil’man and Mil’man, we show that for Banach lattices, or Banach spaces with an unconditional basis, or symmetrically normed ideal of operators on an infinite-dimensional Hilbert space, reflexivity is equivalent to the fixed point property for affine cascading nonexpansive mappings. The theorem of Mil’man and Mil’man [23], mentioned above, states that a Banach space (X , ∥ · ∥) is reflexive if and only for every closed bounded convex subset C of X , and for all norm continuous affine mappings T : C −→ C , T has a fixed point in C . Another theorem analogous to this, and to the results in this paper, is due to Maurey [12] and Dowling and Lennard [24], who prove that a subspace Y of the Lebesgue function space (L1 [0, 1], ∥ · ∥1 ) is reflexive if and only if Y has the fixed point property for nonexpansive mappings. There are similar papers that characterise weakly compact convex sets among the closed bounded convex (or closed convex) sets in certain Banach spaces via a fixed point property. These include [25,26,17]. Finally, we prove that in any Banach space, for every closed bounded convex subset C and every cascading nonexpansive mapping T : C −→ C , T has an approximate fixed point sequence, i.e., infx∈C ∥Tx − x∥ = 0.
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On the other hand, it remains an open question as to whether or not every asymptotically nonexpansive mapping on a closed bounded convex set in a Banach space has an approximate fixed point sequence. We remark that in 1998 Kirk, Martinez Yañez and Shin [27], Theorem 4.1, discovered an interesting sufficient condition for a Banach space (X , ∥ · ∥) to be such that every asymptotically nonexpansive mapping on a closed bounded convex set in X has an approximate fixed point sequence. (This sufficient condition involves the ultrapowers of X .) We further remark that a significant part of this introduction comes from the Ph.D. thesis of the second author [28], written under the supervision of the first author. Also, the proofs of Theorem 3.1 and Corollary 3.5 ((1) H⇒ (3)) are similar to the proofs of Theorems 5.0.6, 5.0.7 and 5.2.2 ((1) H⇒ (2)) of [28]. 2. Preliminaries The symbols N and R denote the set of positive integers and the set of real numbers, respectively. Throughout this paper our scalar field is R. Also, N0 := {0} ∪ N is the set of all non-negative integers. Let (X , ∥ · ∥) be a Banach space and E ⊆ X . We will denote the closed, convex hull of E by co(E ). We will denote by c00 the vector space of all scalar sequences that have only finitely many non-zero terms. Also, for all n −1
n ∈ N, we define en ∈ c00 by en := (0, . . . , 0, 1, 0, . . . , 0, . . .). We will often write ‘‘closed bounded convex set’’ instead of ‘‘closed, bounded, convex set’’. Also, all closed bounded convex sets in this paper are assumed to be non-empty. Definition 2.1. Let C be a closed bounded convex subset of a Banach space (X , ∥ · ∥). A mapping U : C −→ C is said to be affine if for all λ ∈ [0, 1], for all x, y ∈ C , U (1 − λ) x + λ y = (1 − λ) U (x) + λ U (y).
Let C be a closed bounded convex subset of a Banach space (X , ∥ · ∥). Let T : C −→ C be a mapping. Let C0 := C and C1 := co T (C ) ⊆ C .
Clearly C1 is a closed bounded convex set in C . Let x ∈ C1 . Then Tx ∈ T (C1 ) ⊆ T (C ) ⊆ co T (C ) = C1 .
So, T maps C1 into C1 . Next, for all n ∈ N, we define Cn := co T (Cn−1 ) .
Clearly, for all n ∈ N, Cn is a closed bounded convex subset of C , T maps Cn into Cn and Cn ⊆ Cn−1 . Definition 2.2. Let (X , ∥ · ∥) be a Banach space and C be a closed bounded convex subset of X . Let T : C −→ C be a mapping and (Cn )n∈N0 be defined as above. We say that T is cascading nonexpansive if there exists a sequence (λn )n∈N0 in [1, ∞) such that λn −→ 1, and for all n ∈ N0 , for all x, y ∈ Cn , ∥Tx − Ty∥ ≤ λn ∥x − y∥. n
Note that every cascading nonexpansive mapping is norm-to-norm continuous, and every nonexpansive mapping is cascading nonexpansive. Cascading nonexpansive mappings arise naturally in Banach spaces (X , ∥ · ∥) that contain an isomorphic copy of ℓ1 or c0 (see Theorem 3.1). Examples of such spaces are Banach spaces isomorphic to a non-reflexive Banach lattice and nonreflexive Banach spaces with an unconditional basis. (See, for example, Lindenstrauss and Tzafriri [29] 1.c.5 and [30] 1.c.12.) Indeed, not containing an isomorphic copy of ℓ1 or c0 characterises reflexivity for both these classes of Banach spaces. Another class of Banach spaces of this type is the class of all symmetrically normed ideals of operators on an infinitedimensional Hilbert space. (See Dodds and Lennard [31], Proposition 4.5.) This is a very different class than the first two, since all symmetrically normed ideals, except the Hilbert–Schmidt operators, fail to have local unconditional structure. (See Lewis [32].) Definition 2.3. We say that a Banach space (X , ∥ · ∥) has the fixed point property for cascading nonexpansive mappings if for all closed bounded convex subsets C of X , for all cascading nonexpansive mappings T : C −→ C , T has a fixed point in C . Also, (X , ∥ · ∥) has the perturbed fixed point property for cascading nonexpansive mappings if there exists an equivalent norm ∥ · ∥∼ on X such that (X , ∥ · ∥∼ ) has the fixed point property for cascading nonexpansive mappings. Example 2.4 (Not all Cascading Nonexpansive Mappings are Eventually Asymptotically Nonexpansive). Let (γn )n∈N be a strictly increasing sequence in (0, 1) that converges to 1. Also assume that (γn+1 /γn )n∈N is a decreasing sequence in (1, 2]. An
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417
example of such a sequence is (γn := 8n /(1 + 8n ), for all n ∈ N), mentioned in the introduction above. Consider the Banach space ℓ1 , endowed with the equivalent norm
∥x∥∼ := sup γν ν∈N
∞
|xk |,
for all x ∈ ℓ1 .
k=ν
Let K := {x ∈ ℓ : each xj ≥ 0 and 1
∞
j =1
xj = 1}. Also, define the mapping R : K −→ K by Rx := (0, x1 , x2 , x3 , . . .), for all
x ∈ K . The set K0 := K is closed, bounded and convex in ℓ1 . Let Kn := co R(Kn−1 ) , for all n ∈ N. It is easy to check that each
n ∞ Kn = (0, . . . , 0, xn+1 , xn+2 , . . .) ∈ ℓ1 : each xj ≥ 0 and xj = 1 . j=n+1 n
n
Fix n ∈ N0 . Fix u, v ∈ Kn . Then u = (0, . . . , 0, xn+1 , xn+2 , . . .) and v = (0, . . . , 0, yn+1 , yn+2 , . . .), for some x, y as described n +1
n +1
above. Thus, Ru = (0, . . . , 0, xn+1 , xn+2 , . . .) and Rv = (0, . . . , 0, yn+1 , yn+2 , . . .). Then, ∥u − v∥∼ = sup γν ν≥n+1
∞
|xk − yk | and ∥Ru − Rv∥∼ = sup γν ν≥n+2
k=ν
∞
|xk − yk |.
k=ν−1
Therefore,
∥Ru − Rv∥∼ ≤ ∥u − v∥∼ sup
ν≥n+2
γν γn+2 = ∥u − v∥∼ . γν−1 γn+1
Also, en+1 , en+2 ∈ Kn and
∥Ren+1 − Ren+2 ∥∼ = ∥en+2 − en+3 ∥∼ = 2 γn+2 =
γn+2 ∥en+1 − en+2 ∥∼ . γn+1 γ
So, R is a cascading nonexpansive mapping on K , with best constants λn = γn+2 , for all n ∈ N0 . n+1 On the other hand, ∥e1 − e2 ∥∼ = 2 γ1 , and for all n ∈ N,
∥Rn e1 − Rn e2 ∥∼ = ∥en+1 − en+2 ∥∼ = 2 γn+1 =
γn+1 ∥e1 − e2 ∥∼ . γ1
γ
But nγ+1 −→ 1/γ1 > 1. Therefore, R is not eventually asymptotically nonexpansive on K . 1 n Example 2.5 (Not all Eventually Asymptotically Nonexpansive Mappings √ are Cascading Nonexpansive). Consider the Banach space (R, | · |) and the closed, bounded, convex set K0 := K := [0, 1/ 2]. Let a ∧ b := min{a, b}, for all a, b ∈ R. Also, let Q denote the set of all rational √ numbers √ and I := R \ Q be the set of all irrational numbers. We define the mapping U : K −→ K by setting Ux := 2 x ∧ (1/ 2 ), for all x ∈ Q ∩ K and Ux := 0, for all x ∈ I ∩ K . It is easy to check that Kn := co U (Kn−1 ) = K , for all n ∈ N. Further, 0, 1/2 ∈ K and
√ √ √ √ |U (0) − U (1/2)| = |0 − 1/ 2| = 1/ 2 = 2 (1/2) = 2 |0 − 1/2|. Thus, U fails to be a cascading nonexpansive mapping on K . On the other hand, for all n ≥ 2, for all x ∈ K , U n x = 0. Therefore, U is eventually asymptotically nonexpansive on K . Remark 2.6. Note that in the previous example, U is not a continuous mapping on K , and so U is not asymptotically nonexpansive. Recently, Łukasz Piasecki [33] has invented an asymptotically nonexpansive mapping T on a closed, bounded, convex set K in a Banach space (X , ∥ · ∥) such that T is not cascading nonexpansive. 3. Reflexivity iff the perturbed FPP for cascading nonexpansive mappings in Banach lattices Theorem 3.1. Let (X , ∥ · ∥) be a Banach space that contains an isomorphic copy of ℓ1 or c0 . Then there exists a closed bounded convex set C ⊆ X and an affine cascading nonexpansive mapping T : C −→ C such that T is fixed point free.
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Proof. Case 1. (X , ∥ · ∥) contains an isomorphic copy of ℓ1 . By the Strong James’ Distortion Theorem for ℓ1 [34,21], there exists a normalised sequence (xj )j∈N in X and a null sequence (εn )n∈N in (0, 1) such that for all n ∈ N, for all t = (tj )j∈N ∈ c00 ,
(1 − εn )
∞ j =n
∞ ∞ |tj | ≤ tj xj ≤ |t |. j =n j=n j
Define the closed bounded convex subset C of X by
∞
C := co{xj : j ∈ N} =
∞
tj xj : each tj ≥ 0 and
j =1
tj = 1 .
j =1
Further define T : C −→ C by
T
∞
:=
tj xj
j =1
∞
tj xj+1 .
j =1
The mapping T is affine and fixed point free. Inductively, we see that for all n ∈ N, Cn := co T (Cn−1 ) = T (Cn−1 ) is given by
Cn =
∞
tj xj : each tj ≥ 0 and
j=n+1
∞
tj = 1 .
j =n +1
Fix n ∈ N0 and fix x, y ∈ Cn . So, x = j=n+1 tj xj and y = Let αj := tj − sj , for all j ≥ n + 1. Then
∞
∞
j =n +1
sj xj , where each tj , sj ≥ 0 and
∞
j=n+1
tj =
∞
j=n+1
sj = 1.
∞ ∞ |αj |, αj xj ≥ (1 − εn+1 ) ∥x − y∥ = j=n+1 j=n+1 and since each ∥xk ∥ = 1,
∞ ∞ 1 |αj | ≤ αj xj+1 ≤ ∥Tx − Ty∥ = ∥x − y∥. j=n+1 j=n+1 (1 − εn+1 ) Consequently, T is cascading nonexpansive on C . Case 2. (X , ∥ · ∥) contains an isomorphic copy of c0 . By a strengthening of the Strong James’ Distortion Theorem for c0 ([21], Theorem 8), there exists a normalised sequence (xj )j∈N in X and a null sequence (εn )n∈N in (0, 1) such that for all n ∈ N, for all t = (tj )j∈N ∈ c00 ,
∞ tj xj ≤ (1 + εn ) max |tj |. max |tj | ≤ j =n j ≥n j≥n Define fk := x1 + · · · + xk , for all k ∈ N. Next we define the closed bounded convex subset C of X by
C := co{fk : k ∈ N} =
∞
tj xj : 1 = t1 ≥ t2 ≥ · · · ≥ tk ↓k 0 .
j =1
Also, we define the function T : C −→ C by
T
∞
tj xj
j =1
:= x1 +
∞
tj xj+1 .
j =1
The mapping T is affine and fixed point free. Inductively, it follows that for all n ∈ N, Cn := co T (Cn−1 ) = T (Cn−1 ) is given by
Cn =
∞
tj xj : 1 = t1 = t2 = · · · = tn+1 ≥ tn+2 ≥ · · · ≥ tk ↓k 0 .
j =1
Fix n ∈ N0 and fix x, y ∈ Cn . Hence, x = j=1 tj xj and y = j=1 sj xj , where 1 = t1 = · · · = tn+1 ≥ tn+2 ≥ · · · ≥ tk ↓k 0 and 1 = s1 = · · · = sn+1 ≥ sn+2 ≥ · · · ≥ sk ↓k 0. Let αj := tj − sj , for all j ∈ N. We have that
∞
∞ ∥x − y ∥ = αj xj ≥ max |αj |, j=n+2 j ≥ n +2
∞
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419
and
∞ ∞ ∞ ∥Tx − Ty∥ = αj xj+1 = αj xj+1 = αk−1 xk j =1 j=n+2 k=n+3 ≤ (1 + εn+3 ) max |αk−1 | = (1 + εn+3 ) max |αj | k≥n+3
j ≥ n +2
≤ (1 + εn+3 ) ∥x − y∥. Consequently, T is cascading nonexpansive on C .
Theorem 3.2. Let (X , ∥ · ∥) be a reflexive Banach space. Then there exists an equivalent norm ∥ · ∥∼ on X such that for every closed bounded convex subset C of X , for all ∥ · ∥∼ -cascading nonexpansive mappings T : C −→ C , T has a fixed point in C . Proof. By a theorem of Domínguez Benavides [18], there exists an equivalent norm ∥ · ∥∼ on X such that for every closed bounded convex subset E of X , for all ∥ · ∥∼ -nonexpansive mappings U : E −→ E, U has a fixed point in E. Fix an arbitrary closed bounded convex subset C of X . Let T : C −→ C be a ∥ · ∥∼ -cascading nonexpansive mapping. As above, let C0 := C and Cn := co T (Cn−1 ) , for all n ∈ N. By hypothesis there exists a sequence (λn )n∈N0 in [1, ∞) such that λn −→ 1, and for all n ∈ N0 , for all x, y ∈ Cn , ∥Tx − Ty∥∼ ≤ λn ∥x − y∥∼ . n
Now X is reflexive, and so C is weakly compact. By Zorn’s Lemma there exists a non-empty closed bounded convex set D ⊆ C such that D is a minimal invariant set for T .That is, T (D) ⊆ D, and if F is a non-empty closed bounded convex subset of D with T (F ) ⊆ F , then F = D. It follows that co T (D) = D. Let D0 := D and Dn := co T (Dn−1 ) , for all n ∈ N. Inductively, we see that for all n ∈ N, D = Dn ⊆ Cn . Therefore, by our hypotheses on T , for all x, y ∈ D, for all n ∈ N,
∥Tx − Ty∥∼ ≤ λn ∥x − y∥∼ . But λn −→ 1. Consequently, for all x, y ∈ D, n
∥Tx − Ty∥∼ ≤ ∥x − y∥∼ ; i.e., T is ∥ · ∥∼ -nonexpansive on D. By Domínguez Benavides [18], T has a fixed point in D ⊆ C .
From the above proof we also have the following result. Corollary 3.3. Let (X , ∥ · ∥) be a Banach space. Then the following are equivalent. (1) (X , ∥ · ∥) has the weak fixed point property for nonexpansive mappings; i.e., for every weakly compact convex subset C of X , for all nonexpansive mappings T : C −→ C , T has a fixed point. (2) (X , ∥ · ∥) has the weak fixed point property for cascading nonexpansive mappings, i.e., for every weakly compact convex subset C of X , for all cascading nonexpansive mappings T : C −→ C , T has a fixed point. Thus, all the results in the literature concerning spaces that have the weak fixed point property for nonexpansive mappings (see, for example, the texts [6,5,13,10,11]) are also true for cascading nonexpansive mappings. Combining Theorem 3.1, some remarks in the Preliminaries, and Theorem 3.2, we get the following fixed point property characterisation of reflexivity in Banach lattices, Banach spaces with an unconditional basis, and symmetrically normed ideals. Theorem 3.4. Let (X , ∥ · ∥) be a Banach lattice, or a Banach space with an unconditional basis, or a symmetrically normed ideal. Then the following are equivalent. (1) X is reflexive. (2) There exists an equivalent norm ∥ · ∥∼ on X such that for all closed bounded convex sets C ⊆ X and for all ∥ · ∥∼ -cascading nonexpansive mappings T : C −→ C , T has a fixed point in C . Note that via Lin’s example [20] of an equivalent renorming ||| · ||| of (ℓ1 , ∥ · ∥1 ) such that (ℓ1 , ||| · |||) has the fixed point property for nonexpansive mappings, we cannot replace ‘‘cascading nonexpansive mappings’’ by ‘‘nonexpansive mappings’’ in part (2) of Theorem 3.4. Corollary 3.5. Let (X , ∥ · ∥) be a Banach lattice, or a Banach space with an unconditional basis, or a symmetrically normed ideal. Then the following are equivalent. (1) X is reflexive. (2) There exists an equivalent norm ∥ · ∥∼ on X such that for all closed bounded convex sets C ⊆ X and for all ∥ · ∥∼ -cascading nonexpansive mappings T : C −→ C , T has a fixed point in C . (3) There exists an equivalent norm ∥·∥∼ on X such that for all closed bounded convex sets C ⊆ X and for all affine ∥·∥∼ -cascading nonexpansive mappings T : C −→ C , T has a fixed point in C . (4) For all equivalent norms ||| · ||| on X , for all closed bounded convex sets C ⊆ X and for all affine ||| · |||-cascading nonexpansive mappings T : C −→ C , T has a fixed point in C . (5) For all closed bounded convex sets C ⊆ X and for all affine norm-to-norm continuous mappings T : C −→ C , T has a fixed point in C .
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Proof. (1) ⇐⇒ (2). This is Theorem 3.4. (3) H⇒ (1). The statement (not(1)) implies (not(3)) by Theorem 3.1. (1) H⇒ (5). This is a theorem of Mil’man and Mil’man [23]. (5) H⇒ (4) H⇒ (3). This is clear. We remark that an analogous result to the equivalence of (1) and (4) in Corollary 3.5, involving a different kind of asymptotically nonexpansive map (semi-strongly asymptotically nonexpansive mappings), is proven in the Ph.D. thesis of the second author [28] (Theorem 5.2.2). Remark 3.6. In certain Banach spaces there exist cascading nonexpansive mappings on closed bounded convex sets that are not nonexpansive and fail to have a fixed point. For example, consider Lin’s [20] equivalent renorming ||| · ||| of (ℓ1 , ∥ · ∥1 ), and apply Theorem 3.1 to the space (ℓ1 , ||| · |||). Or, see Example 2.4. Nevertheless, in all Banach spaces, cascading nonexpansive mappings on closed bounded convex sets always have approximate fixed point sequences. This follows directly from [5] Lemma 20.1, which gives an upper estimate for the minimal displacement of Lipschitz mappings. Acknowledgements We thank Barry Turett for helpful discussions. The second author thanks the University of Pittsburgh, the Community College of Allegheny County and Point Park University for their financial support. We also thank the referees for their helpful suggestions. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34]
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