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The fixed point property for mappings admitting a center
Nonlinear Analysis 66 (2007) 1257–1274 www.elsevier.com/locate/na
The fixed point property for mappings admitting a center J. Garc´ıa-Falset a,∗ , E. Llorens-Fuster a , S. Prus b a Departament d’An`alisi Matem`atica, Universitat de Val`encia, Dr. Moliner 50, 46100, Burjassot, Val`encia, Spain b Department of Mathematics, M. Curie-Sklodowska University, 20-031 Lublin, Poland
Received 2 December 2005; accepted 12 January 2006
Abstract We introduce a class of nonlinear continuous mappings in Banach spaces which allow us to characterize the Banach spaces without noncompact flat parts in their spheres as those that have the fixed point property for this type of mapping. Later on, we give an application to the existence of zeroes for certain kinds of accretive operators. c 2006 Elsevier Ltd. All rights reserved. Keywords: Fixed points; Nonlinear operators; Nearly strictly convex Banach spaces
1. Introduction In this paper we introduce a large class of continuous mappings (which we call J-type mappings) that, in some sense, include the (quasi) nonexpansive mappings. (See definitions below.) If C is a closed bounded and convex subset of a normed space (X, .) and T : C → X is a nonexpansive mapping with a fixed point y0 ∈ C, then it is obvious that, for every x ∈ C, T (x) − y0 ≤ x − y0 .
(1)
But the above inequality may be satisfied even for a nonexpansive fixed point free mapping, of course for some y0 ∈ C. For example, let us consider the affine Beals’ mapping defined on the ∗ Corresponding author.
E-mail addresses: [email protected] (J. Garc´ıa-Falset), [email protected] (E. Llorens-Fuster), [email protected] (S. Prus). c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.01.016
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unit ball B of the classical sequence space c0 by T (x 1 , x 2 , . . .) = (1, x 1 , x 2 , . . .). If we take y0 = 2e1 = (2, 0, . . .), we can easily see that T satisfies inequality (1) in B. Indeed, for every x = (x 1 , x 2 , . . .) ∈ B, T (x) − y0 = (−1, x 1 , x 2 , . . .) = 1 ≤ 2 − x 1 = (x 1 − 2, x 2 , x 3 , . . .) = x − y0 . We will show that it is still possible to get some information about T from this fact, which leads us to distinguish any point y0 ∈ X for which inequality (1) holds. We shall refer to such a point as a center for T . Our aim here is to study the class of all mappings admitting a center. It turns out that this class contains all contractions defined in closed sets of Banach spaces and even all the so called quasi-nonexpansive mappings (i.e. those for which every fixed point is a center) introduced by Tricomi for real functions and further studied by Diaz and Metcalf [9] and Dotson [10] for mappings in Banach spaces. It is not hard to see that the class of quasinonexpansive mappings properly contains the class of nonexpansive maps having fixed points, although there are continuous not quasi-nonexpansive mappings that admit a center. In Section 2 we state some notation. In Section 3 we define and give several examples of J-type mappings. Section 4 is devoted to some fixed point theorems for J-type mappings and we also give a characterization, via a fixed point theorem for J-type mappings, of a geometrical property of the Banach spaces, which was introduced in 1973 by Bruck [5] and which we call property (C). In Section 5 we discuss the scope of property (C) and list several classes of Banach spaces enjoying this property. The aim of Section 6 is to give some insight into the relationship between nonexpansive and J-type mappings, and finally in Section 7 we make use of the above results to prove an existence theorem for certain integral equations and establish the existence of zeroes for certain kinds of accretive operators. 2. Preliminaries Throughout this paper we assume that (X, .) is a real Banach space. As usual, we will denote by B[x, r ] and S[x, r ] the closed ball and the sphere of the Banach space (X, .) with radius r and center x ∈ X, respectively. Given a sequence (x n ) in X, we will write x n x whenever (x n ) is weakly convergent to x ∈ X. If C is a nonempty subset of X and y ∈ X, dist(y, C) := inf{y − x : x ∈ C}. If the infimum is attained for some x 0 in C, then we call this point a nearest point to y0 in C. The set of all nearest points in C to y0 will be denoted by PC (y0 ). It is well known that if C is a weakly compact set, then PC (y) = ∅ for each y ∈ X. The (set-valued) mapping PC is often called the nearest point mapping. We say that C ⊂ X is a Chebyshev set whenever PC (y) is (nonempty and) a singleton for every y ∈ X. As usual, given two Banach spaces (X, . X ) and (Y, .Y ), we will denote by X ⊕1 Y the product space X × Y endowed with the norm (x, y)1 := x X + yY . In the same way, X ⊕∞ Y means the product space X × Y endowed with the norm (x, y)∞ := max{x X , yY }. Recall that a mapping T : C → X is said to be nonexpansive if, for each x, y ∈ C, T (x) − T (y) ≤ x − y. If, in addition, there exists k ∈ [0, 1) such that T (x) − T (y) ≤ kx −y, then T is said to be a (strict) contraction. The space (X, .) has the fixed point property (FPP) if every nonexpansive self-mapping of each nonempty bounded closed convex subset C of X has a fixed point. If the same property holds for every weakly compact convex subset of X, we
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say that (X, .) has the weak fixed point property (WFPP). It has been known from the outset of the study of this property that it depends strongly on “nice” geometrical properties of the space. Recall also that a Banach space (X, .) is strictly convex whenever x = y = 1 and x = y imply 12 (x + y) < 1. In other words, (X, .) is strictly convex if nonempty closed convex subsets of its spheres are singletons (and therefore compact sets). In [3], a natural generalization of strict convexity was studied in the following terms. Definition 1. A Banach space (X, .) is said to be nearly strictly convex (NSC) if its unit sphere S does not contain noncompact closed and convex sets. Obviously, every strictly convex space is NSC, but the converse is not true. 3. A broad class of mappings In order to proceed, we give the following definition. Definition 2. Let C be a bounded closed convex subset of a Banach space X. We say that y0 ∈ X is a center for the mapping T : C → X if, for each x ∈ C, y0 − T (x) ≤ y0 − x . We say that T : C → X is a J-type mapping whenever it is continuous and it has some center y0 ∈ X. In this case, by Z (T ) we denote the set of all centers of the mapping T . Of course, if a mapping T : C → X has a center y0 ∈ C, then trivially T (y0 ) = y0 . Thus, fixed point results for J-type mappings are only nontrivial provided they have a center y0 ∈ C. A similar condition as the one that defines J-type mappings has been considered in [11]. Let us point out that T : C → C is quasi-nonexpansive provided that T has at least one fixed point in C and every fixed point is a center for T . Nevertheless, the concept of J-type mapping only needs one center which, in fact, does not need to be a fixed point. Several immediate consequences of the definition of a center are the following. (1) If a mapping T : C → X has a center y0 ∈ X and r ∈ (0, 1), then the mapping Tr := r I d + (1 − r )T has the same center y0 . (2) If a mapping T : C → C has a center y0 ∈ X, then all its iterates T n : C → C have the same center y0 . (3) If a mapping T : C → X has a center y0 ∈ X, then so does the restriction of T to every subset of C. (4) If a mapping T : C → X has a center y0 ∈ X, then the mapping T˜ : C − {y0 } → X given by T˜ (x − y0 ) = T (x) − y0 admits the center 0 X . (5) Although each fixed point of a nonexpansive mapping is a center for this mapping, for general lipschitzian mappings a fixed point does not need to be a center. Example 3. Let us consider the mapping T : 12 , 2 → 12 , 2 given by T (x) = 1x . Since the derivative T (x) = −1 is bounded on C := 12 , 2 , the mapping T is lipschitzian on C. It is x2 obvious that x 0 := 1 is the unique fixed point of T , but x 0 is not a center for T because T 1 − 1 = 1 > 1 − 1 = 1 . 2 2 2
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Furthermore, if y0 > 2, then |T (2) − y0 | = |1/2 − y0 | > |2 − y0 |, and thus y0 is not a center for T . Finally, if y0 < 12 , then |T (1/2) − y0 | = |2 − y0 | > |1/2 − y0 |. Therefore, T does not have centers, in spite of the fact that it has a fixed point in C. (6) There are expansive even non-lipschitzian J-type mappings: The mappings Tn : [0, 1] → [0, 1] given by Tn (x) = x n admit the point y0 = −a ≤ 0 as a center. Of course, for n ≥ 2, Tn is not nonexpansive on [0, 1]. Note that every Tn has two fixed points in [0, 1], namely y0 = 0 and y1 = 1. While y0 is a center for Tn , y1 is not a center for Tn and hence such mappings cannot be quasi-nonexpansive. √ On the other hand, the non-lipschitzian mapping T : [0, 1] → [0, 1] given by T (x) = x admits the fixed point y1 = 1 as a center. (7) The above examples also show that a J-type mapping may admit more than one center. 3.1. Instances of J-type mappings 3.1.1. Nonexpansive mappings with fixed points As we pointed out in the introduction, every nonexpansive mapping T : C → X of a bounded closed convex subset C of a Banach space X having a fixed point is a J-type mapping. (In particular, every contractive self-mapping of C is nonexpansive with a fixed point, as well as many Kannan-type mappings etc; see for instance [14].) Moreover, every fixed point y0 of such T is a center for T . There are well known nonexpansive mappings with the set of their fixed points not being a singleton. This is another instance of a J-type mapping that admits more than one center. It is well known that the fixed point set of a nonexpansive mapping does not need to be convex but is always closed. Now let us show that, for a J-type mapping T : C → X, the same fact holds for the set Z (T ) of its centers. Indeed, if z ∈ Z (T ), then we can find a sequence (z n ) in Z (T ) such that z n → z. Therefore, for every x ∈ C, we have T (x) − z = lim T (x) − z n ≤ lim x − z n = x − z, n→∞
n→∞
and consequently z ∈ Z (T ). In fixed point theory, a long list of geometric conditions on the norm of (X, .) has been studied which guarantee that a nonexpansive self-mapping of a set C under the above conditions has a fixed point (see [13,15]). 3.1.2. Alternate convexically nonexpansive mappings are J-type mappings In [18], the following definition was given. Definition 4. Let C be a bounded closed convex subset of a Banach space X. A mapping T : C → C is called alternate convexically nonexpansive (ACN) if, for each n ∈ N and x 1 , . . . , x n , y ∈ C, n n (−1)i+1 (−1)i+1 T (x i ) − T (y) ≤ xi − y . i=1 n n i=1 It is obvious that ACN mappings are nonexpansive. Moreover, if T : C → C is (the restriction of) a linear nonexpansive mapping, then it is ACN. Nevertheless, with the following easy example, we will show that there are affine contractive mappings without this property.
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Example 5. Let T : [0, 1] → [0, 1] be the mapping given by T (x) = 12 (x + 1). It is obvious that T is an affine and contractive mapping, but T is not ACN. (Take, for instance x 1 = 0, x 2 = 1/2, x 3 = 1 and y = 0.) The main result in [18] is: Theorem 6. Let C be a weakly compact convex subset of a strictly convex Banach space (X, .). Then every ACN mapping T : C → C has a fixed point. The non-ACN mapping given in Example 5 is a J-type mapping, because T is 1/2 contractive on [0, 1]. Next, we will see that the class of ACN mappings is strictly contained in the class of J-type mappings. Moreover, this fact does not depend on the existence of fixed points for such mappings. Theorem 7. Let (X, .) be a Banach space and let C be a nonempty convex closed bounded subset of X. If T : C → C is an ACN mapping, then T admits the center 0 X ∈ X. Proof. As T is an ACN mapping, it is nonexpansive. Then, by using standard methods (see [2]), we can find a sequence (x n ) in C such that (a) x n − T (x n ) → 0 (b) x n+1 − x n → 0. For this sequence, Condition (a) ensures that n n (−1)i+1 (−1)i+1 T (x i ) − x i = 0. lim n→∞ n n i=1 i=1 Moreover, Condition (b) implies that n (−1)i+1 x i = 0. lim n→∞ n i=1
Since T is ANC, for y ∈ C, we have n n (−1)i+1 (−1)i+1 T (y) = lim T (x i ) − T (y) ≤ lim x i − y = y. n→∞ n→∞ n n i=1 i=1 Thus, 0 X is a center for T .
Since we have seen above that there are expansive J-type mappings, the class of ANC mappings (which is contained in the class of nonexpansive mappings) is strictly smaller than the class of J-type mappings. 3.1.3. Mappings lying in spheres If (X, .) is a Banach space such that there exist nontrivial closed convex sets C contained in some sphere S[y0 , r ] of this space, then every continuous mapping T : C → C admits as a center the same center y0 of the sphere because, for every x ∈ C, y0 − T (x) = r = y0 − x. As a particular case, we may consider the following two examples.
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Example 8. A famous counterexample in metric fixed point theory for nonexpansive mappings is the Alspach mapping (see [1]) defined in a subset of the unit sphere of the classical space (L 1 [0, 1], .1 ), i.e. T : K → K given by
⎧ 1 ⎪ ⎪ t ∈ 0, ⎨min{2 f (2t), 2}, 2
T f (t) := ⎪ 1 ⎪ ⎩max{2 f (2t − 1) − 2, 0}, t ∈ ,1 2 where K := { f ∈ L 1 [0, 1] : 0 ≤ f ≤ 2, f 1 = 1}. It is well known that K is a weakly compact convex subset of the unit sphere of (L 1 [0, 1], .1 ) and T is a fixed point free isometry of K . Since K is contained in the unit sphere, T admits 0 L 1 [0,1] as a center. This example shows that the existence of a center for a mapping is not a sufficient condition to ensure that this mapping has a fixed point, even when the domain of the mapping is a weakly compact convex set. The following example is defined in the standard Banach space (1 , .1 ), where x1 = ∞ n=1 |x n |. Example 9. Consider the set S1+ := x = (x n ) ∈ 1 :
∞
x n = 1, x n ≥ 0 (n = 1, 2, . . .) .
n=1
It is obvious that S1+ is a closed convex subset of the unit sphere of 1 . Moreover, the linear right shift S : S1+ → S1+ given by S(x 1 , x 2 . . .) = (0, x 1 , x 2 , . . .) is .1 nonexpansive and has no fixed points in S1+ , in spite of the fact that it admits the center y0 = 01 . Furthermore, if y = (yn ) ∈ 1 is any vector with nonnegative coordinates, then −y is a center for S because, for every x ∈ S1+ , S(x) + y1 = y1 + (x 1 + y2 ) + (x 2 + y3 ) + · · · = (y1 + x 1 ) + (y2 + x 2 ) + · · · = x + y1 . 4. Fixed point results The first result of this section is closely inspired by Theorem 1 of [23]. Proposition 10. Let (X, .) be a real Banach space, C be a closed subset of X, and T : C → X be a J-type mapping. Suppose there exists x 0 ∈ C such that, for every n ≥ 0, T n (x 0 ) ∈ C. If lim dist(T n (x 0 ), Z (T )) = 0,
n→∞
then T has a fixed point x ∗ and the sequence of Picard iterates (T n (x 0 )) converges to x ∗ . Proof. First we will show that (T n (x 0 )) is a Cauchy sequence. Given > 0, there exists n 0 such that, for n ≥ n 0 dist(T n (x 0 ), Z (T )) < . 2 This means that we can find z 0 ∈ Z (T ) such that T n0 (x 0 ) − z 0 < . 2
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Since z 0 is a center for T , for m, n ≥ n 0 we have T n (x 0 ) − T m (x 0 ) ≤ T n (x 0 ) − z 0 + T m (x 0 ) − z 0 ≤ T n0 (x 0 ) − z 0 + T n0 (x 0 ) − z 0 < . Because C is a closed set in a Banach space, there exists x ∗ ∈ C such that T n (x 0 ) → x ∗ . Finally, let us see that x ∗ is a fixed point for T . Indeed, since lim dist(T n (x 0 ), Z (T )) = 0,
n→∞
it is not difficult to see that dist(x ∗ , Z (T )) = 0 and hence x ∗ ∈ Z (T ). As we have seen above, Z (T ) is a closed set, and then x ∗ is a center for T that belongs to the domain C of T . Therefore, x ∗ is a fixed point of T . Remark 11. In Section 3 we have seen that, if T : C → X is a J-type mapping with a center y ∈ X and C is a convex set, then the mappings Tr : C → X given by Tr := r I d + (1 − r )T admit y as a center. Therefore, Proposition 10 holds if we replace T by Tr and we assume the convexity for the set C. On the other hand, it is well known (see [20]) that, if T : C → C is nonexpansive, then these mappings Tr are always asymptotically regular, that is, for every x ∈ C lim Trn+1 (x) − Trn (x) = 0.
n→∞
For J-type mappings we can obtain a similar property in the framework of uniformly convex Banach spaces. Recall that a Banach space (X, .) is said to be uniformly convex provided that, if (x n ), (yn ) are sequences in the unit ball B X with limn 21 x n + yn = 1, then limn x n − yn = 0. Proposition 12. Let (X, .) be a uniformly convex Banach space, and let C be a closed convex subset of X. If T : C → C is a J-type mapping, then every mapping Tr : C → C is asymptotically regular. Proof. Let z ∈ X be a center for T . Since z is also a center for Tr , it is clear that, for each x ∈ C, the sequence (Trn (x) − z) is decreasing and therefore convergent to a real number, say a. We may assume that a > 0 (otherwise the conclusion is obvious). Since z ∈ Z (T ) ∩ Z (Tr ), we have T (Trn (x)) − z ≤ Trn (x) − z. Hence 1 [T (T n (x)) − z] ≤ 1 [T n (x) − z] → 1. r a a r On the other hand Trn+1 (x) − z = r Trn (x) + (1 − r )T (Trn (x)) − r z − (1 − r )z = r (Trn (x) − z) + (1 − r )(T (Trn (x)) − z).
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Consequently, 1 n+1 1 n n [T a r (x) − z] = a [r (Tr (x) − z) + (1 − r )(T (Tr (x)) − z)] → 1. Since (X, .) is uniformly convex, we derive that (Trn (x) − z) − (T (Trn (x)) − z) → 0. Then Trn+1 (x) − Trn (x) = r Trn (x) + (1 − r )T (Trn (x)) − Trn (x) = (1 − r )(Trn (x) − z) − (T (Trn (x)) − z) → 0
which completes the proof.
Proposition 10 works in every Banach space, but under a strong condition on the mapping T . Next, we will give a similar result in the framework of Banach spaces satisfying Opial’s condition. Recall that a Banach space (X, .) has the Opial condition if, for every weakly null sequence (x n ), the inequality lim inf x n < lim inf x n − x n→∞
n→∞
holds for all nonzero x ∈ X. Proposition 13. Let (X, .) be a Banach space with the Opial condition, and let C be a weakly compact subset of X. Let T : C → X be a J-type mapping. Suppose there exists x 0 ∈ C such that, for every n ≥ 0, T n (x 0 ) ∈ C and ww (x 0 ) ⊂ Z (T ). Then T has a fixed point x ∗ and the Picard iterates sequence (T n (x 0 )) is weakly convergent to x ∗ . Here ww (x 0 ) := {y ∈ X : T n j (x 0 ) y, (n j ) strictly increasing sequence in N}. Proof. Since C is a weakly compact set, ww (x 0 ) is nonempty, and moreover it is contained in C ∩ Z (T ). Therefore, each point in ww (x 0 ) is a fixed point of T . The proof will be complete if we show that ww (x 0 ) is a singleton. We assume, for a contradiction, that there exist two different points, y1 = w − lim j T n j (x 0 ), y2 = w − lim j T m j (x 0 ), in ww (x 0 ). Since y1 , y2 ∈ Z (T ), we know that the sequences (T n (x 0 ) − y1 ) and (T n (x 0 ) − y2 ) are decreasing, and hence they are convergent. From the Opial condition, we derive that lim T n (x 0 ) − y1 = lim T n j (x 0 ) − y1
n→∞
j →∞
< lim T n j (x 0 ) − y2 j →∞
= lim T n (x 0 ) − y2 . n→∞
Interchanging the roles of y1 and y2 , we have the desired contradiction.
This proposition recaptures the well known result due to Opial [22] concerning the weak convergence of the Picard iterates of a nonexpansive mapping, as well as the generalization to quasi-nonexpansive mappings given in [26].
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A generalization of the nearly strictly convexity could be the following. Definition 14. A Banach space (X, .) has property (C) (or has compact faces; see [5])) whenever the weakly compact convex subsets of its unit sphere are compact sets. It is quite easy to see that the classical space (c0 , .∞ ) fails to have this property. Moreover, in Section 5 we will give several examples of non-NSC Banach spaces enjoying property (C). Theorem 16 below will provide a nonlinear characterization of the Banach spaces with this property. Next, we give some fixed point results for (all) J-type mappings, but in the framework of the Banach spaces with property (C). Definition 15. We say that a Banach space (X, .) has the J-weak fixed point property (JWFPP) if every J-type self-mapping of every weakly compact convex subset C of X has a fixed point. The following theorem gives us a nonlinear characterization of the Banach spaces with property (C). Theorem 16. Let (X, .) be a Banach space. The following conditions are equivalent: (i) (X, .) has property (C). (ii) (X, .) has the J-WFPP. (iii) For every weakly compact convex subset K of (X, .) every continuous self-mapping T with a center at 0 X has a fixed point in K . Proof. (i) ⇒ (ii) Let C be a weakly compact convex subset of X and let T : C → C be a mapping with a center y0 ∈ X. If y0 ∈ C, then it is a fixed point of T and nothing needs to be proved. If y0 ∈ C, since C is weakly compact convex, the set K = PC (y0 ) := {x ∈ C : x − y0 = dist(y0 , C) =: d} is nonempty. Moreover, K = C ∩ B[y0, d] = C ∩ S[y0 , d] is closed and convex. For every x ∈ K , T (x) ∈ C and then d ≤ T (x) − y0 ≤ x − y0 = d, because y0 is a center of T . Hence T (x) ∈ K , which implies that K is T -invariant. Since (X, .) has property (C) and K lies in a sphere, K is a compact convex set. The conclusion now follows from Schauder’s theorem. (ii) ⇒ (iii) (Obvious). (iii) ⇒ (i). Suppose, by contradiction, that (X, .) does not have property (C). Then we may find a weakly compact convex subset K of the unit sphere of (X, .) that is not compact. By Klee’s theorem (see [21] or Theorem 19.3 in [13] for a stronger version), there exists a continuous, fixed point free (in fact lipschitzian) mapping T : K → K . Moreover, by paragraph 3 of Section 3.1.3, 0 X is a center of such a mapping and thus T should have a fixed point which is a contradiction. More can still be said about reflexive Banach spaces. Corollary 17. Let (X, .) be a reflexive Banach space with property (C). Let K be a convex closed subset of X. If a continuous mapping T : K → K admits a center y0 ∈ X, then T has a fixed point in K .
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Proof. The set C := K ∩ B[y0, dist(y0 , K )] is nonempty, weakly compact, convex and T -invariant. Corollary 18. Let C be a Chebyshev set in a Banach space (X, .). If a continuous mapping T : C → C admits a center y0 ∈ X, then PC (y0 ) is a fixed point of T . In [13], p. 116, the following property is defined. Definition 19. A Banach space (X, .) has the fixed point property for spheres if each nonempty, closed, and convex subset of the unit sphere has the fixed point property for nonexpansive mappings. Of course, reflexive Banach spaces with property (C) have the fixed point property for spheres. By the same arguments that we used in the proof of Theorem 16, it is easy to obtain the following result. Corollary 20. Let (X, .) be a Banach space with the fixed point property for spheres. Let C be a weakly compact convex subset of X. Then every J-type nonexpansive self-mapping T of C has a fixed point. Let us point out that, in this case, we do not need to use Schauder’s theorem. 4.1. Set-valued J-type mappings The concept of a center may easily be extended to mappings taking weakly compact convex values. Definition 21. Let C be a bounded closed convex subset of a Banach space X. We say that y0 ∈ X is a center for a mapping T : C → BC(X) if, for each x ∈ C, H (T (x), {y0}) ≤ y0 − x , where H stands for the well known Hausdorff metric in BC(X), the set of all nonempty bounded and closed subsets of X. We will say that T : C → BC(X) is a J-type mapping whenever it is upper semicontinuous and has some center y0 ∈ X. Of course, if a mapping T : C → BC(X) has a center y0 ∈ C, then trivially H (T (y0 ), {y0 }) = 0, that is, T (y0 ) = {y0 }, which means that y0 is a stationary point for T . By using a similar technique as in the proof of Theorem 16 along with the Kakutani–Bohnenblust–Karlin Theorem, we now obtain the following result. Theorem 22. Let C be a weakly compact convex subset of a Banach space with property (C). Then, every J-type set-valued mapping T : C → BC(C), taking convex values has a fixed point. 5. On property (C) in Banach spaces 1. As we mentioned above nearly strictly convex Banach spaces (which, in turn, include strictly convex Banach spaces and many others) have property (C). 2. Recall that a Banach space (X, .) has the Kadec property if its weak and norm topologies are the same on the unit sphere of the space. (In particular, every finite dimensional Banach space has this property. However, it is well known that there exist Banach spaces of this class that are
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not strictly convex). Clearly, the spaces with the Kadec property are also instances of spaces with property (C). 3. As usual, we say that a Banach space (X, .) has the Kadec–Klee property (also called the Radon–Riesz property) if, for every sequence (x n ) in X, the following implication holds: xn x ⇒ x n → x. x n → x It is also obvious that the Kadec–Klee property is a particular case of property (C). Moreover, it is easy to see that the Kadec property implies the Kadec–Klee property. Nevertheless, in [27] Troyanski gave an example of a Banach space that has the Kadec–Klee property but fails to have the Kadec property. 4. Recall that a Banach space is said to be nearly uniformly convex (NUC) if, for any > 0, there exists δ > 0 such that, for any sequence (x n ) in B X with sep((x n )) > , one has that dist(0, co ({x n })) < 1 − δ. Here, sep((x n )) := inf{x n − x m : m = n}. For more instances of NUC spaces, one can see [19] or [16]. It was seen in [3] that if (X, .) is NUC, then it is NSC and the converse, in general, is false. There, it was also shown that reflexive Kadec–Klee Banach spaces, that is, spaces enjoying the so called drop property (see [24]) are NSC. In this sense, it is interesting to notice that if (X, .) is a reflexive Banach space, then (X, .) is NSC if and only if (X, .) has property (C). Nevertheless, if (X, .) is not reflexive, the above equivalence fails. To see this, it is enough to consider the classical Banach space (1 , .1 ), since this space enjoys the Kadec property (hence it has property (C)) and, however, it is not NSC, because S1+ ⊂ S[01 , 1] is a noncompact closed convex set. Next, we give an example of a Banach space with property (C) and without any other of the above conditions. Example 23. In c0 we give the norm . D defined for x = (x n ) ∈ c0 by ∞ x n 2 . x D := x∞ + n n=1 The space (c0 , . D ) is strictly convex (see [13], p. 52), but it does not have the Kadec–Klee property. To see this, it is enough to consider the sequence (en + e1 ), where (en ) is the standard Schauder basis in c0 . Obviously, (X, .) := (1 , .1 ) ⊕1 (c0 , . D ) does not have the Kadec–Klee property and is not NSC, since it contains subspaces isometric to (1 , .1 ) and (c0 , . D ). In spite of this, (X, .) enjoys property (C). Indeed, let A be a weakly compact convex subset of the unit sphere of (X, .). We claim that if (x 1 , y1 ), (x 2 , y2 ) ∈ A, then there exists λ ∈ R such that y2 = λy1 . Otherwise, for every λ ∈ R, y2 = λy1 . Since (c0 , . D ) is strictly convex, we have that y1 + y2 y1 D + y2 D 2 < 2 D
and, since A is a convex set, x 1 + x 2 y1 + y2 x 1 + x 2 y1 + y2 , 1= = 2 + 2 . 2 2 1
D
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Consequently, x 1 + x 2 y1 + y2 x 1 1 + x 2 1 y1 D + y2 D < , + =1 1= 2 2 2 2 which is a contradiction, and thus there exists λ ∈ R such that y2 = λy1 , as claimed. Now, it is not difficult to see that A is compact. Indeed, consider a sequence {(x n , yn )} in A. Then there exists a sequence of real numbers {λn } and an element y0 ∈ (c0 , . D ) such that yn = λn y0 . On the other hand, since A is weakly compact, we may assume that {(x n , yn )} is weakly convergent to (x 0 , λ0 y0 ) ∈ A. This means that x n x 0 in 1 and λn y0 → λ0 y0 in c0 . Since 1 is a Schur space, we have that x n → x 0 , which implies that {(x n , yn )} is norm convergent to (x 0 , λ0 y0 ) ∈ A. Remark 24. The argument used in Example 23 shows that, if (X, .) is a nonreflexive, nonNCS, Schur space and (Y, .) is a strictly convex Banach space without the Kadec–Klee property, then (X, .) ⊕1 (Y, .) has property (C) but it is not NCS and does not have the Kadec–Klee property. On the other hand, if X, Y are two Banach spaces with property (C) and X is not a Schur space, then X ⊕∞ Y does not have property (C). To this end, we consider a weakly convergent sequence (x n ) in the unit ball of X which is not norm-convergent and y0 ∈ Y with y0 = 1. Then it is clear that K := {(x, y0 ) : x ∈ co{x n }} is a weakly compact convex subset of the unit sphere of X ⊕∞ Y which cannot be norm compact. Finally, it is clear that property (C) is not a topological property. Indeed, (c0 , .∞ ) does not have property (C), while the space (c0 , . D ) enjoys it. In this context, we give the following result, closely inspired by Corollary 2 in [24]. Proposition 25. Let (X, .) be a reflexive infinite dimensional Banach space. Then, given > 0, there exists an equivalent norm . on X such that (X, . ) lacks property (C) and such that, for every x ∈ X, x ≤ x ≤ (1 + )x . Proof. Let f ∈ X ∗ with f = 1, and let x 0 ∈ X with x 0 = 1 be such that f (x 0 ) = 1. Consider the norm on X given by x := max{x, (1 + )| f (x)|}. Let (x n ) be a sequence in V := ker( f ) ∩ B[0 X , 1]. Since (X, .) is reflexive, we may suppose that x n y0 ∈ V . Moreover, we may also suppose that x n − y0 ≥ η > 0. Let yn := 2(1+) (x n − y0 ) and let vn := yn +
1 1+ x 0 .
We have that
vn ≤ yn +
1 1 ≤ (x n + y0 ) + ≤ 1. 1+ 2(1 + ) 1+
On the other hand, (1 + )| f (vn )| = (1 + ) f
1 x 0 = 1. 1+
1 1 x 0 and 1+ x 0 = 1. Then it is easy to see that Hence vn = 1. Moreover, vn 1+ K := co{vn : n = 1, . . .} is a weakly compact convex subset of the unit sphere of (X, . ) which is not compact. Consequently, (X, . ) fails to have property (C).
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Remark 26. One may compare this result with the well known Zizler theorem about renormings: every separable Banach space admits an equivalent norm that is uniformly convex in every direction (UCED) and hence has property (C). Moreover, this equivalent norm can be chosen to be as close as desired to the original norm in the Banach–Mazur sense. These results show that property (C) has a strong geometrical (isometric but nonisomorphic) nature. 6. Nonexpansive J-type mappings The relationship between nonexpansive and J-type mappings is not quite clear. The aim of this section is to summarize some facts about this topic. If K is closed, convex and unbounded, then there exist easy examples of nonexpansive, even contractive mappings without centers. Example 27. Let f : [1, +∞) → [1, +∞) be given by x2 + 1 . x It easy to check that | f (x) − f (y)| < |x − y| for all x, y ∈ [1, ∞) and that f is fixed point free. By Corollary 17, it is clear that f does not admit any center in R. f (x) =
However, if we consider nonexpansive self-mappings of weakly compact convex subsets of Banach spaces, then it is not so easy to find nonexpansive mappings without centers. We know a long list of additional requirements on the geometry of the space under which every such mapping has a fixed point, and hence is a J-type mapping. Moreover, the most important example of nonexpansive fixed point free mapping in this context, namely the Alspach mapping (see Example 8), admits zero as a center and hence it is also a J-type mapping. Nevertheless, a modification of this example gives us a nonexpansive self-mapping of a weakly compact convex set that does not have a center. Example 28. Let T be the Alspach mapping considered on the whole interval C = f ∈ L 1 [0, 1] : 0 ≤ f ≤ 2 . Given f ∈ C, we put S( f ) = 2χ[0,1] − T f , where χ[0,1] is the characteristic function of the interval [0, 1]. Then S : C → C is an isometry and Sine [25] proved that S does not have a fixed point. We also put R( f ) = min{2, max{0, f }} for f ∈ L 1 [0, 1]. Then R( f ) is the unique point nearest to f in C. Assuming that y0 ∈ L 1 [0, 1] is a center for S, we see that S(R(y0 )) = R(y0 ) (compare with the proof of Theorem 16), which contradicts Sine’s result. The general theory of nonexpansive mappings in Banach spaces has a long list of famous open problems. Among many others, one can point out the question of whether either reflexive strictly convex or reflexive Kadec–Klee Banach spaces have the FPP. We may add one more general problem to this list: Question 1. Does property (C) imply the FPP for reflexive Banach spaces? Bearing in mind Theorem 16, one way to solve this problem could be to find an affirmative answer for the following question, which is a particular case of the first one.
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Question 2. In weakly compact convex subsets of either strictly convex or Kadec–Klee Banach spaces, is every nonexpansive mapping a J-type mapping? If (X, .) is a Banach space without the WFPP, then there exists a weakly compact convex subset C of X and a fixed point free nonexpansive mapping T : C → C. If T admits a center y0 ∈ X, then y0 is a center for the restriction of T to every subset of C. In particular, it is well known that there exists a weakly compact convex T -invariant subset M of C which is minimal for T and which has positive diameter, say d. We claim that such a minimal set must be contained in a sphere centered at y0 . Otherwise, let ρ := dist(y0 , M) > 0. If there exists some x ∈ M for which x − y0 > ρ, then the set
1 B y0 , (x − y0 + ρ) ∩ M 2 is nonempty, T -invariant, weakly compact, convex, and strictly smaller than M, which is a contradiction with the fact that M is minimal. Summarizing, one has. Proposition 29. If T is a nonexpansive fixed point free mapping leaving invariant a weakly compact convex set and with a center y0 ∈ X, then all its minimal subsets lie in a sphere centered in y0 . Properties of minimal invariant sets for nonexpansive mappings were also studied in [17]. Let a Banach space (X, .) have property (C) and T : C → C be a nonexpansive mapping, where C ⊂ X is convex and weakly compact. An argument from [17] shows that if M1 , M2 are minimal convex weakly compact T -invariant subsets of C, then diam M1 = diam M2 . Here, we need not assume that T has a center. 7. Applications 7.1. Application to an integral equation In this section we consider the functional-integral equation: y(t) = φ t, K (t, w)g(w, y(w)) dμ(w) , t ∈ Ω , Ω
(2)
where (X, .) is a Banach space and (Ω , Σ , μ) is a probability measure space, and we look for the existence of a solution y ∈ L p (μ, X) of (2). In [7], the author studied the existence of solutions of a particular case of this kind of equations. Here, we will use our Theorem 16 in order to obtain a similar result. Definition 30. Let (Ω , Σ , μ) be a finite measure space and let (X, .) be a Banach space. A continuous mapping f : Ω × X → X is said to be a k-0 X -J-type mapping with respect to the second variable if f (r, x) ≤ kx for each (r, x) ∈ Ω × X. Throughout the following we assume the next three hypotheses:
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(a) φ : Ω ×X → X (resp. g : Ω ×X → X) is a k-0 X -J-type mapping (resp. l-0 X -J-type mapping) with respect to the second variable and the map w → φ(w, y(w)) (resp. w → g(w, y(w))) belongs to L p (μ, X) for each y ∈ L p (μ, X). (b) |K (t, s)| ≤ M for each (t, s) ∈ Ω × Ω and K (t, .) : Ω → R is measurable and kl M ≤ 1. (c) There exist x 0 ∈ L p (μ, X) and r > 0 such that φ ., K (., w)g(w, y(w)) dμ(w) − x 0 ≤ r Ω
p
whenever y − x 0 p ≤ r . Theorem 31. Let (X, .) be a Banach space and let (Ω , Σ , μ) be a probability measure space and consider the Bochner space L p (μ, X) with 1 < p < +∞. Then, (1) If the conditions (a) and (b) are satisfied, then 0 L p (μ,X ) is a solution of equation (2). (2) Assume that the hypotheses (a),(b),(c) are satisfied. Suppose that (X, .) is reflexive and that L p (μ, X) has property (C). Then Eq. (2) has at least one solution y ∈ B[x 0, r ] ⊂ L p (μ, X). Proof. Let us introduce the operator T : L p (μ, X) → L p (μ, X) defined by K (t, w)g(w, y(w)) dμ t ∈ Ω . (T y)(t) := φ t, Ω
It is clear that if T has a fixed point y, then y is a solution of the our integral equation. (1) From Conditions (a) and (b) it follows that T (0 L p (μ,X ) ) = 0 L p (μ,X ) . (2) Since L p (μ, X) is reflexive, the closed ball K := B[x 0, r ] of L p (μ, X) is a weakly compact convex subset of this space. Thus, by Hypothesis (c), we know that K is a weakly compact convex T -invariant subset. By Hypothesis (a) we have that φ is a k-0 X -J-type mapping with respect to the second variable and g is a l-0 X -J-type mapping with respect to the second variable, which allows us to affirm that T is a J-type mapping. Indeed, since φ is a k-0 X -J-type mapping with respect to the second variable, we have p T y p = (T y)(t) p dμ(t) Ω p φ t, dμ(t) K (t, w)g(w, y(w)) dμ(w) = Ω Ω p K (t, w)g(w, y(w)) dμ(w) dμ(t) ≤ kp Ω
Ω
by Jensen’s inequality and Hypothesis (b) g(w, y(w)) p dμ(w) dμ(t). ≤ (k M) p Ω
Ω
Since g is a l-0 X -J-type mapping with respect to the second variable, we obtain y(w) p dμ(w) dμ(t) ≤ y p . T (y) p ≤ (k Ml) p Ω
Ω
The last inequality is a consequence of the assumption that (Ω , Σ , μ) is a probability measure space and Hypothesis (b).
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Finally, since T is a J-type mapping, and moreover we have assumed that L p (μ, X) enjoys property (C), by Theorem 16 we conclude that T has a fixed point in K . Remark 32. It is well known that, for 1 < p < +∞, the Bochner space L p (μ, X) is reflexive if and only if (X, .) is reflexive. Moreover, Day showed in [8] that, for 1 < p < +∞, the Bochner space L p (μ, X) is strictly convex if and only if (X, .) is strictly convex. In [18], Harandi showed that, under some conditions labelled (1), (2) and (3) in his article, the mapping T introduced in the proof of Theorem 31 is AC N. Then, by Theorem 7, we can conclude that 0 L p (μ,X ) is a center of this mapping T . Since T is well defined on L p (μ, X), we know that 0 L p (μ,X ) is a solution of Eq. (2). Therefore, in order to obtain a solution of Eq. (2), we do not need any additional condition on X. 7.2. Application to the accretivity Let (X, .) be a Banach space. A mapping A : X → 2 X will be called an operator on X. The effective domain of A is denoted by D(A) and its range by R(A). An operator A on X is said to be accretive if the inequality x − y ≤ x − y + λ(z − w) holds for all λ ≥ 0, z ∈ Ax, and w ∈ Ay. If, in addition, R(I + λA) is for one λ, hence for all λ, precisely X, then A is called m-accretive (we refer the reader to [4,6] for background material on accretivity). Among the problems treated in the theory of accretive operators, one of the most studied is that of determining when A has a zero, i.e. 0 ∈ R(A) (see, for instance, [12] to find properties which imply the existence of zeroes for accretive operators). Suppose that A is an accretive operator on X and 0 ∈ R(A). This means that there exists x 0 ∈ D(A) such that 0 ∈ Ax 0 . Therefore, by the definition of such operators, we obtain: x − x 0 ≤ x − x 0 + λu
for all (x, u) ∈ G(A).
(3)
Therefore, Inequality (3) is a necessary condition in order to obtain the existence of zeroes of accretive operators. In this section we will study the problem when Condition (3) is, in fact, a sufficient condition. Theorem 33. Let (X, .) be a Banach space and let A : D(A) → 2 X be an accretive operator on X satisfying Condition (3) for some x 0 ∈ X. Then (a) If x 0 ∈ R(I + A), then 0 ∈ R(A). (b) If (X, .) is a reflexive space with the fixed point property for spheres and co(D(A)) ⊆ R(I + A), then 0 ∈ R(A). Proof. Since the operator A is accretive, it is well known that the resolvent: g := (I + A)−1 : R(I + A) → D(A) is a single-valued nonexpansive mapping. We claim that g is a J-type mapping. For an arbitrary y ∈ R(I + A), there exists x ∈ D(A) such that x = g(y). Hence, y ∈ x + Ax, which means that there exists u ∈ Ax such that y = x + u. Therefore, using Condition (3) for λ = 1, we obtain g(y) − x 0 = x − x 0 ≤ x − x 0 + u = y − x 0 . Consequently, g is a J-type mapping, as claimed.
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(a) If x 0 ∈ R(I + A), then g(x 0 ) = x 0 and hence x 0 ∈ x 0 + Ax 0 , which implies that 0 ∈ Ax 0 . (b) We may consider the set C := co(D(A)) ∩ B[x 0, dist(x 0 , co(D(A)))]. Clearly, this set is convex. It is also nonempty and weakly compact, because the space (X, .) is reflexive. The point x 0 is a center for g, so C is also g-invariant. Thus, since g : C → C is a J-type nonexpansive mapping and (X, .) has the fixed point property for spheres, we can apply Corollary 20 to obtain the conclusion. Remark 34. Notice that every m-accretive operator satisfying Condition (3) fulfills Condition (a) of the above theorem, and hence it has a zero. On the other hand, it is still unknown whether a reflexive Banach space with the fixed point for spheres enjoys the FPP for nonexpansive mappings. However, by Corollary 20, this holds for J-type nonexpansive mappings. Therefore in part (b) of the above result, the fact that g is a J-type mapping is essential in order to obtain the conclusion. It is also interesting to note that, in part (b) of the above theorem, the hypothesis that (X, .) is reflexive and has the fixed point property for spheres cannot be omitted. Indeed, it suffices to consider the operator A : B[0c0 , 1] → c0 defined by A(x 1 , x 2 , . . .) = (x 1 − 1, x 2 − x 1 , x 3 − x 2 , . . .). It is not difficult to see that A is an accretive operator, 0 ∈ R(A), and A satisfies Condition (3) for x 0 = (2, 0, 0, . . .) and B[0c0 , 1] ⊂ R(I + A). Acknowledgement The first and second authors were partially supported by DGES, grant BFM 2003-03893-C0202. References [1] D.E. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82 (1981) 423–424. [2] J.B. Baillon, R. Sch¨oneberg, Asymptotic normal structure and fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 81 (1981) 257–264. [3] J. Banas, On drop property and nearly uniformly smooth Banach spaces, Nonlinear Anal. 14 (11) (1990) 927–933. [4] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, 1976. [5] R.E. Bruck, Properties of fixed-point sets of nonexpansive mappings in Banach spaces, Trans. Amer. Math. Soc. 179 (1973) 251–262. [6] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, 1990. [7] M.A. Daswish, Weakly singular functional-integral equation in infinite dimensional Banach spaces, Appl. Math. Comput. 136 (2003) 123–129. [8] M.M. Day, Strict convexity and smoothness of normed spaces, Trans. Amer. Math. Soc. 78 (1955) 516–528. [9] J.B. D´ıaz, F.T. Metcalf, On the set of subsequential limit points of successive approximations, Trans. Amer. Math. Soc. 135 (1969) 459–485. [10] W.G. Dotson, On the Mann iterative process, Trans. Amer. Math. Soc. 149 (1970) 65–73. [11] J. Garcia-Falset, Fixed points for mappings with the range type condition, Houston J. Math. 28 (2002) 143–158. [12] J. Garcia-Falset, S. Reich, Zeroes of accretive operators and the asymptotic behavior of nonlinear semigroups, Houston J. Math. (in press). [13] K. Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Univ. Press, 1990.
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[14] K. Goebel, W.A. Kirk, T. Shimi, A fixed point theorem in uniformly convex spaces, Boll. Unione Mat. Ital. 7 (1973) 67–75. [15] K. Goebel, W.A. Kirk, Classical theory of nonexpansive mappings, in: W.A. Kirk, B. Sims (Eds.), Handbook of Metric Fixed Point Theory, Kluwer A. P., 2001. [16] K. Goebel, T. Se¸kowski, The modulus of noncompact convexity, Ann. Univ. Mariae Curie-Sklodowska 29 (38) (1984) 41–48. [17] K. Goebel, B. Sims, More on minimal invariant sets for nonexpansive mappings, Nonlinear Anal. 47 (2001) 2667–2681. [18] A.A. Harandi, A fixed point result in strictly convex Banach spaces, Acta Math. Hungar. 105 (1–2) (2004) 139–143. [19] R.E. Huff, Banach spaces which are nearly uniformly convex, Rocky Mountain J. Math. 10 (1980) 743–749. [20] S. Ishikawa, Fixed points and iterations of nonexpansive mappings in Banach spaces, Proc. Amer. Math. Soc. 5 (1976) 65–71. [21] V. Klee, Some topological properties of convex sets, Trans. Amer. Math. Soc. 78 (1955) 30–45. [22] Z. Opial, Weak convergence of the sequence of succesive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967) 595–597. [23] W.V. Petryshyn, T.E. Williamson Jr., A necessary and sufficient condition for the convergence of a sequence of iterates for quasi nonexpansive mappings, Bull. Amer. Math. Soc. 78 (6) (1972) 1027–1031. [24] S. Rolewicz, On drop property, Studia Math. 85 (1987) 27–35. [25] R. Sine, Remarks on the example of Alspach, in: Nonlinear Analysis and Applications, in: Lecture Notes in Pure and Appl. Math., vol. 80, Dekker, New York, 1982, pp. 237–241. [26] M.A. Smyth, Aspects of the fixed point theory for some metrically defined maps, Ph.D. Dissertation, University of Newcastle, Australia, 1994. [27] S. Troyanski, On a property of the norm which is close to local uniform rotundity, Math. Ann. 271 (1985) 305–313.
The fixed point property for mappings admitting a center J. Garc´ıa-Falset a,∗ , E. Llorens-Fuster a , S. Prus b a Departament d’An`alisi Matem`atica, Universitat de Val`encia, Dr. Moliner 50, 46100, Burjassot, Val`encia, Spain b Department of Mathematics, M. Curie-Sklodowska University, 20-031 Lublin, Poland
Received 2 December 2005; accepted 12 January 2006
Abstract We introduce a class of nonlinear continuous mappings in Banach spaces which allow us to characterize the Banach spaces without noncompact flat parts in their spheres as those that have the fixed point property for this type of mapping. Later on, we give an application to the existence of zeroes for certain kinds of accretive operators. c 2006 Elsevier Ltd. All rights reserved. Keywords: Fixed points; Nonlinear operators; Nearly strictly convex Banach spaces
1. Introduction In this paper we introduce a large class of continuous mappings (which we call J-type mappings) that, in some sense, include the (quasi) nonexpansive mappings. (See definitions below.) If C is a closed bounded and convex subset of a normed space (X, .) and T : C → X is a nonexpansive mapping with a fixed point y0 ∈ C, then it is obvious that, for every x ∈ C, T (x) − y0 ≤ x − y0 .
(1)
But the above inequality may be satisfied even for a nonexpansive fixed point free mapping, of course for some y0 ∈ C. For example, let us consider the affine Beals’ mapping defined on the ∗ Corresponding author.
E-mail addresses: [email protected] (J. Garc´ıa-Falset), [email protected] (E. Llorens-Fuster), [email protected] (S. Prus). c 2006 Elsevier Ltd. All rights reserved. 0362-546X/$ - see front matter doi:10.1016/j.na.2006.01.016
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unit ball B of the classical sequence space c0 by T (x 1 , x 2 , . . .) = (1, x 1 , x 2 , . . .). If we take y0 = 2e1 = (2, 0, . . .), we can easily see that T satisfies inequality (1) in B. Indeed, for every x = (x 1 , x 2 , . . .) ∈ B, T (x) − y0 = (−1, x 1 , x 2 , . . .) = 1 ≤ 2 − x 1 = (x 1 − 2, x 2 , x 3 , . . .) = x − y0 . We will show that it is still possible to get some information about T from this fact, which leads us to distinguish any point y0 ∈ X for which inequality (1) holds. We shall refer to such a point as a center for T . Our aim here is to study the class of all mappings admitting a center. It turns out that this class contains all contractions defined in closed sets of Banach spaces and even all the so called quasi-nonexpansive mappings (i.e. those for which every fixed point is a center) introduced by Tricomi for real functions and further studied by Diaz and Metcalf [9] and Dotson [10] for mappings in Banach spaces. It is not hard to see that the class of quasinonexpansive mappings properly contains the class of nonexpansive maps having fixed points, although there are continuous not quasi-nonexpansive mappings that admit a center. In Section 2 we state some notation. In Section 3 we define and give several examples of J-type mappings. Section 4 is devoted to some fixed point theorems for J-type mappings and we also give a characterization, via a fixed point theorem for J-type mappings, of a geometrical property of the Banach spaces, which was introduced in 1973 by Bruck [5] and which we call property (C). In Section 5 we discuss the scope of property (C) and list several classes of Banach spaces enjoying this property. The aim of Section 6 is to give some insight into the relationship between nonexpansive and J-type mappings, and finally in Section 7 we make use of the above results to prove an existence theorem for certain integral equations and establish the existence of zeroes for certain kinds of accretive operators. 2. Preliminaries Throughout this paper we assume that (X, .) is a real Banach space. As usual, we will denote by B[x, r ] and S[x, r ] the closed ball and the sphere of the Banach space (X, .) with radius r and center x ∈ X, respectively. Given a sequence (x n ) in X, we will write x n x whenever (x n ) is weakly convergent to x ∈ X. If C is a nonempty subset of X and y ∈ X, dist(y, C) := inf{y − x : x ∈ C}. If the infimum is attained for some x 0 in C, then we call this point a nearest point to y0 in C. The set of all nearest points in C to y0 will be denoted by PC (y0 ). It is well known that if C is a weakly compact set, then PC (y) = ∅ for each y ∈ X. The (set-valued) mapping PC is often called the nearest point mapping. We say that C ⊂ X is a Chebyshev set whenever PC (y) is (nonempty and) a singleton for every y ∈ X. As usual, given two Banach spaces (X, . X ) and (Y, .Y ), we will denote by X ⊕1 Y the product space X × Y endowed with the norm (x, y)1 := x X + yY . In the same way, X ⊕∞ Y means the product space X × Y endowed with the norm (x, y)∞ := max{x X , yY }. Recall that a mapping T : C → X is said to be nonexpansive if, for each x, y ∈ C, T (x) − T (y) ≤ x − y. If, in addition, there exists k ∈ [0, 1) such that T (x) − T (y) ≤ kx −y, then T is said to be a (strict) contraction. The space (X, .) has the fixed point property (FPP) if every nonexpansive self-mapping of each nonempty bounded closed convex subset C of X has a fixed point. If the same property holds for every weakly compact convex subset of X, we
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say that (X, .) has the weak fixed point property (WFPP). It has been known from the outset of the study of this property that it depends strongly on “nice” geometrical properties of the space. Recall also that a Banach space (X, .) is strictly convex whenever x = y = 1 and x = y imply 12 (x + y) < 1. In other words, (X, .) is strictly convex if nonempty closed convex subsets of its spheres are singletons (and therefore compact sets). In [3], a natural generalization of strict convexity was studied in the following terms. Definition 1. A Banach space (X, .) is said to be nearly strictly convex (NSC) if its unit sphere S does not contain noncompact closed and convex sets. Obviously, every strictly convex space is NSC, but the converse is not true. 3. A broad class of mappings In order to proceed, we give the following definition. Definition 2. Let C be a bounded closed convex subset of a Banach space X. We say that y0 ∈ X is a center for the mapping T : C → X if, for each x ∈ C, y0 − T (x) ≤ y0 − x . We say that T : C → X is a J-type mapping whenever it is continuous and it has some center y0 ∈ X. In this case, by Z (T ) we denote the set of all centers of the mapping T . Of course, if a mapping T : C → X has a center y0 ∈ C, then trivially T (y0 ) = y0 . Thus, fixed point results for J-type mappings are only nontrivial provided they have a center y0 ∈ C. A similar condition as the one that defines J-type mappings has been considered in [11]. Let us point out that T : C → C is quasi-nonexpansive provided that T has at least one fixed point in C and every fixed point is a center for T . Nevertheless, the concept of J-type mapping only needs one center which, in fact, does not need to be a fixed point. Several immediate consequences of the definition of a center are the following. (1) If a mapping T : C → X has a center y0 ∈ X and r ∈ (0, 1), then the mapping Tr := r I d + (1 − r )T has the same center y0 . (2) If a mapping T : C → C has a center y0 ∈ X, then all its iterates T n : C → C have the same center y0 . (3) If a mapping T : C → X has a center y0 ∈ X, then so does the restriction of T to every subset of C. (4) If a mapping T : C → X has a center y0 ∈ X, then the mapping T˜ : C − {y0 } → X given by T˜ (x − y0 ) = T (x) − y0 admits the center 0 X . (5) Although each fixed point of a nonexpansive mapping is a center for this mapping, for general lipschitzian mappings a fixed point does not need to be a center. Example 3. Let us consider the mapping T : 12 , 2 → 12 , 2 given by T (x) = 1x . Since the derivative T (x) = −1 is bounded on C := 12 , 2 , the mapping T is lipschitzian on C. It is x2 obvious that x 0 := 1 is the unique fixed point of T , but x 0 is not a center for T because T 1 − 1 = 1 > 1 − 1 = 1 . 2 2 2
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Furthermore, if y0 > 2, then |T (2) − y0 | = |1/2 − y0 | > |2 − y0 |, and thus y0 is not a center for T . Finally, if y0 < 12 , then |T (1/2) − y0 | = |2 − y0 | > |1/2 − y0 |. Therefore, T does not have centers, in spite of the fact that it has a fixed point in C. (6) There are expansive even non-lipschitzian J-type mappings: The mappings Tn : [0, 1] → [0, 1] given by Tn (x) = x n admit the point y0 = −a ≤ 0 as a center. Of course, for n ≥ 2, Tn is not nonexpansive on [0, 1]. Note that every Tn has two fixed points in [0, 1], namely y0 = 0 and y1 = 1. While y0 is a center for Tn , y1 is not a center for Tn and hence such mappings cannot be quasi-nonexpansive. √ On the other hand, the non-lipschitzian mapping T : [0, 1] → [0, 1] given by T (x) = x admits the fixed point y1 = 1 as a center. (7) The above examples also show that a J-type mapping may admit more than one center. 3.1. Instances of J-type mappings 3.1.1. Nonexpansive mappings with fixed points As we pointed out in the introduction, every nonexpansive mapping T : C → X of a bounded closed convex subset C of a Banach space X having a fixed point is a J-type mapping. (In particular, every contractive self-mapping of C is nonexpansive with a fixed point, as well as many Kannan-type mappings etc; see for instance [14].) Moreover, every fixed point y0 of such T is a center for T . There are well known nonexpansive mappings with the set of their fixed points not being a singleton. This is another instance of a J-type mapping that admits more than one center. It is well known that the fixed point set of a nonexpansive mapping does not need to be convex but is always closed. Now let us show that, for a J-type mapping T : C → X, the same fact holds for the set Z (T ) of its centers. Indeed, if z ∈ Z (T ), then we can find a sequence (z n ) in Z (T ) such that z n → z. Therefore, for every x ∈ C, we have T (x) − z = lim T (x) − z n ≤ lim x − z n = x − z, n→∞
n→∞
and consequently z ∈ Z (T ). In fixed point theory, a long list of geometric conditions on the norm of (X, .) has been studied which guarantee that a nonexpansive self-mapping of a set C under the above conditions has a fixed point (see [13,15]). 3.1.2. Alternate convexically nonexpansive mappings are J-type mappings In [18], the following definition was given. Definition 4. Let C be a bounded closed convex subset of a Banach space X. A mapping T : C → C is called alternate convexically nonexpansive (ACN) if, for each n ∈ N and x 1 , . . . , x n , y ∈ C, n n (−1)i+1 (−1)i+1 T (x i ) − T (y) ≤ xi − y . i=1 n n i=1 It is obvious that ACN mappings are nonexpansive. Moreover, if T : C → C is (the restriction of) a linear nonexpansive mapping, then it is ACN. Nevertheless, with the following easy example, we will show that there are affine contractive mappings without this property.
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Example 5. Let T : [0, 1] → [0, 1] be the mapping given by T (x) = 12 (x + 1). It is obvious that T is an affine and contractive mapping, but T is not ACN. (Take, for instance x 1 = 0, x 2 = 1/2, x 3 = 1 and y = 0.) The main result in [18] is: Theorem 6. Let C be a weakly compact convex subset of a strictly convex Banach space (X, .). Then every ACN mapping T : C → C has a fixed point. The non-ACN mapping given in Example 5 is a J-type mapping, because T is 1/2 contractive on [0, 1]. Next, we will see that the class of ACN mappings is strictly contained in the class of J-type mappings. Moreover, this fact does not depend on the existence of fixed points for such mappings. Theorem 7. Let (X, .) be a Banach space and let C be a nonempty convex closed bounded subset of X. If T : C → C is an ACN mapping, then T admits the center 0 X ∈ X. Proof. As T is an ACN mapping, it is nonexpansive. Then, by using standard methods (see [2]), we can find a sequence (x n ) in C such that (a) x n − T (x n ) → 0 (b) x n+1 − x n → 0. For this sequence, Condition (a) ensures that n n (−1)i+1 (−1)i+1 T (x i ) − x i = 0. lim n→∞ n n i=1 i=1 Moreover, Condition (b) implies that n (−1)i+1 x i = 0. lim n→∞ n i=1
Since T is ANC, for y ∈ C, we have n n (−1)i+1 (−1)i+1 T (y) = lim T (x i ) − T (y) ≤ lim x i − y = y. n→∞ n→∞ n n i=1 i=1 Thus, 0 X is a center for T .
Since we have seen above that there are expansive J-type mappings, the class of ANC mappings (which is contained in the class of nonexpansive mappings) is strictly smaller than the class of J-type mappings. 3.1.3. Mappings lying in spheres If (X, .) is a Banach space such that there exist nontrivial closed convex sets C contained in some sphere S[y0 , r ] of this space, then every continuous mapping T : C → C admits as a center the same center y0 of the sphere because, for every x ∈ C, y0 − T (x) = r = y0 − x. As a particular case, we may consider the following two examples.
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Example 8. A famous counterexample in metric fixed point theory for nonexpansive mappings is the Alspach mapping (see [1]) defined in a subset of the unit sphere of the classical space (L 1 [0, 1], .1 ), i.e. T : K → K given by
⎧ 1 ⎪ ⎪ t ∈ 0, ⎨min{2 f (2t), 2}, 2
T f (t) := ⎪ 1 ⎪ ⎩max{2 f (2t − 1) − 2, 0}, t ∈ ,1 2 where K := { f ∈ L 1 [0, 1] : 0 ≤ f ≤ 2, f 1 = 1}. It is well known that K is a weakly compact convex subset of the unit sphere of (L 1 [0, 1], .1 ) and T is a fixed point free isometry of K . Since K is contained in the unit sphere, T admits 0 L 1 [0,1] as a center. This example shows that the existence of a center for a mapping is not a sufficient condition to ensure that this mapping has a fixed point, even when the domain of the mapping is a weakly compact convex set. The following example is defined in the standard Banach space (1 , .1 ), where x1 = ∞ n=1 |x n |. Example 9. Consider the set S1+ := x = (x n ) ∈ 1 :
∞
x n = 1, x n ≥ 0 (n = 1, 2, . . .) .
n=1
It is obvious that S1+ is a closed convex subset of the unit sphere of 1 . Moreover, the linear right shift S : S1+ → S1+ given by S(x 1 , x 2 . . .) = (0, x 1 , x 2 , . . .) is .1 nonexpansive and has no fixed points in S1+ , in spite of the fact that it admits the center y0 = 01 . Furthermore, if y = (yn ) ∈ 1 is any vector with nonnegative coordinates, then −y is a center for S because, for every x ∈ S1+ , S(x) + y1 = y1 + (x 1 + y2 ) + (x 2 + y3 ) + · · · = (y1 + x 1 ) + (y2 + x 2 ) + · · · = x + y1 . 4. Fixed point results The first result of this section is closely inspired by Theorem 1 of [23]. Proposition 10. Let (X, .) be a real Banach space, C be a closed subset of X, and T : C → X be a J-type mapping. Suppose there exists x 0 ∈ C such that, for every n ≥ 0, T n (x 0 ) ∈ C. If lim dist(T n (x 0 ), Z (T )) = 0,
n→∞
then T has a fixed point x ∗ and the sequence of Picard iterates (T n (x 0 )) converges to x ∗ . Proof. First we will show that (T n (x 0 )) is a Cauchy sequence. Given > 0, there exists n 0 such that, for n ≥ n 0 dist(T n (x 0 ), Z (T )) < . 2 This means that we can find z 0 ∈ Z (T ) such that T n0 (x 0 ) − z 0 < . 2
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Since z 0 is a center for T , for m, n ≥ n 0 we have T n (x 0 ) − T m (x 0 ) ≤ T n (x 0 ) − z 0 + T m (x 0 ) − z 0 ≤ T n0 (x 0 ) − z 0 + T n0 (x 0 ) − z 0 < . Because C is a closed set in a Banach space, there exists x ∗ ∈ C such that T n (x 0 ) → x ∗ . Finally, let us see that x ∗ is a fixed point for T . Indeed, since lim dist(T n (x 0 ), Z (T )) = 0,
n→∞
it is not difficult to see that dist(x ∗ , Z (T )) = 0 and hence x ∗ ∈ Z (T ). As we have seen above, Z (T ) is a closed set, and then x ∗ is a center for T that belongs to the domain C of T . Therefore, x ∗ is a fixed point of T . Remark 11. In Section 3 we have seen that, if T : C → X is a J-type mapping with a center y ∈ X and C is a convex set, then the mappings Tr : C → X given by Tr := r I d + (1 − r )T admit y as a center. Therefore, Proposition 10 holds if we replace T by Tr and we assume the convexity for the set C. On the other hand, it is well known (see [20]) that, if T : C → C is nonexpansive, then these mappings Tr are always asymptotically regular, that is, for every x ∈ C lim Trn+1 (x) − Trn (x) = 0.
n→∞
For J-type mappings we can obtain a similar property in the framework of uniformly convex Banach spaces. Recall that a Banach space (X, .) is said to be uniformly convex provided that, if (x n ), (yn ) are sequences in the unit ball B X with limn 21 x n + yn = 1, then limn x n − yn = 0. Proposition 12. Let (X, .) be a uniformly convex Banach space, and let C be a closed convex subset of X. If T : C → C is a J-type mapping, then every mapping Tr : C → C is asymptotically regular. Proof. Let z ∈ X be a center for T . Since z is also a center for Tr , it is clear that, for each x ∈ C, the sequence (Trn (x) − z) is decreasing and therefore convergent to a real number, say a. We may assume that a > 0 (otherwise the conclusion is obvious). Since z ∈ Z (T ) ∩ Z (Tr ), we have T (Trn (x)) − z ≤ Trn (x) − z. Hence 1 [T (T n (x)) − z] ≤ 1 [T n (x) − z] → 1. r a a r On the other hand Trn+1 (x) − z = r Trn (x) + (1 − r )T (Trn (x)) − r z − (1 − r )z = r (Trn (x) − z) + (1 − r )(T (Trn (x)) − z).
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Consequently, 1 n+1 1 n n [T a r (x) − z] = a [r (Tr (x) − z) + (1 − r )(T (Tr (x)) − z)] → 1. Since (X, .) is uniformly convex, we derive that (Trn (x) − z) − (T (Trn (x)) − z) → 0. Then Trn+1 (x) − Trn (x) = r Trn (x) + (1 − r )T (Trn (x)) − Trn (x) = (1 − r )(Trn (x) − z) − (T (Trn (x)) − z) → 0
which completes the proof.
Proposition 10 works in every Banach space, but under a strong condition on the mapping T . Next, we will give a similar result in the framework of Banach spaces satisfying Opial’s condition. Recall that a Banach space (X, .) has the Opial condition if, for every weakly null sequence (x n ), the inequality lim inf x n < lim inf x n − x n→∞
n→∞
holds for all nonzero x ∈ X. Proposition 13. Let (X, .) be a Banach space with the Opial condition, and let C be a weakly compact subset of X. Let T : C → X be a J-type mapping. Suppose there exists x 0 ∈ C such that, for every n ≥ 0, T n (x 0 ) ∈ C and ww (x 0 ) ⊂ Z (T ). Then T has a fixed point x ∗ and the Picard iterates sequence (T n (x 0 )) is weakly convergent to x ∗ . Here ww (x 0 ) := {y ∈ X : T n j (x 0 ) y, (n j ) strictly increasing sequence in N}. Proof. Since C is a weakly compact set, ww (x 0 ) is nonempty, and moreover it is contained in C ∩ Z (T ). Therefore, each point in ww (x 0 ) is a fixed point of T . The proof will be complete if we show that ww (x 0 ) is a singleton. We assume, for a contradiction, that there exist two different points, y1 = w − lim j T n j (x 0 ), y2 = w − lim j T m j (x 0 ), in ww (x 0 ). Since y1 , y2 ∈ Z (T ), we know that the sequences (T n (x 0 ) − y1 ) and (T n (x 0 ) − y2 ) are decreasing, and hence they are convergent. From the Opial condition, we derive that lim T n (x 0 ) − y1 = lim T n j (x 0 ) − y1
n→∞
j →∞
< lim T n j (x 0 ) − y2 j →∞
= lim T n (x 0 ) − y2 . n→∞
Interchanging the roles of y1 and y2 , we have the desired contradiction.
This proposition recaptures the well known result due to Opial [22] concerning the weak convergence of the Picard iterates of a nonexpansive mapping, as well as the generalization to quasi-nonexpansive mappings given in [26].
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A generalization of the nearly strictly convexity could be the following. Definition 14. A Banach space (X, .) has property (C) (or has compact faces; see [5])) whenever the weakly compact convex subsets of its unit sphere are compact sets. It is quite easy to see that the classical space (c0 , .∞ ) fails to have this property. Moreover, in Section 5 we will give several examples of non-NSC Banach spaces enjoying property (C). Theorem 16 below will provide a nonlinear characterization of the Banach spaces with this property. Next, we give some fixed point results for (all) J-type mappings, but in the framework of the Banach spaces with property (C). Definition 15. We say that a Banach space (X, .) has the J-weak fixed point property (JWFPP) if every J-type self-mapping of every weakly compact convex subset C of X has a fixed point. The following theorem gives us a nonlinear characterization of the Banach spaces with property (C). Theorem 16. Let (X, .) be a Banach space. The following conditions are equivalent: (i) (X, .) has property (C). (ii) (X, .) has the J-WFPP. (iii) For every weakly compact convex subset K of (X, .) every continuous self-mapping T with a center at 0 X has a fixed point in K . Proof. (i) ⇒ (ii) Let C be a weakly compact convex subset of X and let T : C → C be a mapping with a center y0 ∈ X. If y0 ∈ C, then it is a fixed point of T and nothing needs to be proved. If y0 ∈ C, since C is weakly compact convex, the set K = PC (y0 ) := {x ∈ C : x − y0 = dist(y0 , C) =: d} is nonempty. Moreover, K = C ∩ B[y0, d] = C ∩ S[y0 , d] is closed and convex. For every x ∈ K , T (x) ∈ C and then d ≤ T (x) − y0 ≤ x − y0 = d, because y0 is a center of T . Hence T (x) ∈ K , which implies that K is T -invariant. Since (X, .) has property (C) and K lies in a sphere, K is a compact convex set. The conclusion now follows from Schauder’s theorem. (ii) ⇒ (iii) (Obvious). (iii) ⇒ (i). Suppose, by contradiction, that (X, .) does not have property (C). Then we may find a weakly compact convex subset K of the unit sphere of (X, .) that is not compact. By Klee’s theorem (see [21] or Theorem 19.3 in [13] for a stronger version), there exists a continuous, fixed point free (in fact lipschitzian) mapping T : K → K . Moreover, by paragraph 3 of Section 3.1.3, 0 X is a center of such a mapping and thus T should have a fixed point which is a contradiction. More can still be said about reflexive Banach spaces. Corollary 17. Let (X, .) be a reflexive Banach space with property (C). Let K be a convex closed subset of X. If a continuous mapping T : K → K admits a center y0 ∈ X, then T has a fixed point in K .
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Proof. The set C := K ∩ B[y0, dist(y0 , K )] is nonempty, weakly compact, convex and T -invariant. Corollary 18. Let C be a Chebyshev set in a Banach space (X, .). If a continuous mapping T : C → C admits a center y0 ∈ X, then PC (y0 ) is a fixed point of T . In [13], p. 116, the following property is defined. Definition 19. A Banach space (X, .) has the fixed point property for spheres if each nonempty, closed, and convex subset of the unit sphere has the fixed point property for nonexpansive mappings. Of course, reflexive Banach spaces with property (C) have the fixed point property for spheres. By the same arguments that we used in the proof of Theorem 16, it is easy to obtain the following result. Corollary 20. Let (X, .) be a Banach space with the fixed point property for spheres. Let C be a weakly compact convex subset of X. Then every J-type nonexpansive self-mapping T of C has a fixed point. Let us point out that, in this case, we do not need to use Schauder’s theorem. 4.1. Set-valued J-type mappings The concept of a center may easily be extended to mappings taking weakly compact convex values. Definition 21. Let C be a bounded closed convex subset of a Banach space X. We say that y0 ∈ X is a center for a mapping T : C → BC(X) if, for each x ∈ C, H (T (x), {y0}) ≤ y0 − x , where H stands for the well known Hausdorff metric in BC(X), the set of all nonempty bounded and closed subsets of X. We will say that T : C → BC(X) is a J-type mapping whenever it is upper semicontinuous and has some center y0 ∈ X. Of course, if a mapping T : C → BC(X) has a center y0 ∈ C, then trivially H (T (y0 ), {y0 }) = 0, that is, T (y0 ) = {y0 }, which means that y0 is a stationary point for T . By using a similar technique as in the proof of Theorem 16 along with the Kakutani–Bohnenblust–Karlin Theorem, we now obtain the following result. Theorem 22. Let C be a weakly compact convex subset of a Banach space with property (C). Then, every J-type set-valued mapping T : C → BC(C), taking convex values has a fixed point. 5. On property (C) in Banach spaces 1. As we mentioned above nearly strictly convex Banach spaces (which, in turn, include strictly convex Banach spaces and many others) have property (C). 2. Recall that a Banach space (X, .) has the Kadec property if its weak and norm topologies are the same on the unit sphere of the space. (In particular, every finite dimensional Banach space has this property. However, it is well known that there exist Banach spaces of this class that are
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not strictly convex). Clearly, the spaces with the Kadec property are also instances of spaces with property (C). 3. As usual, we say that a Banach space (X, .) has the Kadec–Klee property (also called the Radon–Riesz property) if, for every sequence (x n ) in X, the following implication holds: xn x ⇒ x n → x. x n → x It is also obvious that the Kadec–Klee property is a particular case of property (C). Moreover, it is easy to see that the Kadec property implies the Kadec–Klee property. Nevertheless, in [27] Troyanski gave an example of a Banach space that has the Kadec–Klee property but fails to have the Kadec property. 4. Recall that a Banach space is said to be nearly uniformly convex (NUC) if, for any > 0, there exists δ > 0 such that, for any sequence (x n ) in B X with sep((x n )) > , one has that dist(0, co ({x n })) < 1 − δ. Here, sep((x n )) := inf{x n − x m : m = n}. For more instances of NUC spaces, one can see [19] or [16]. It was seen in [3] that if (X, .) is NUC, then it is NSC and the converse, in general, is false. There, it was also shown that reflexive Kadec–Klee Banach spaces, that is, spaces enjoying the so called drop property (see [24]) are NSC. In this sense, it is interesting to notice that if (X, .) is a reflexive Banach space, then (X, .) is NSC if and only if (X, .) has property (C). Nevertheless, if (X, .) is not reflexive, the above equivalence fails. To see this, it is enough to consider the classical Banach space (1 , .1 ), since this space enjoys the Kadec property (hence it has property (C)) and, however, it is not NSC, because S1+ ⊂ S[01 , 1] is a noncompact closed convex set. Next, we give an example of a Banach space with property (C) and without any other of the above conditions. Example 23. In c0 we give the norm . D defined for x = (x n ) ∈ c0 by ∞ x n 2 . x D := x∞ + n n=1 The space (c0 , . D ) is strictly convex (see [13], p. 52), but it does not have the Kadec–Klee property. To see this, it is enough to consider the sequence (en + e1 ), where (en ) is the standard Schauder basis in c0 . Obviously, (X, .) := (1 , .1 ) ⊕1 (c0 , . D ) does not have the Kadec–Klee property and is not NSC, since it contains subspaces isometric to (1 , .1 ) and (c0 , . D ). In spite of this, (X, .) enjoys property (C). Indeed, let A be a weakly compact convex subset of the unit sphere of (X, .). We claim that if (x 1 , y1 ), (x 2 , y2 ) ∈ A, then there exists λ ∈ R such that y2 = λy1 . Otherwise, for every λ ∈ R, y2 = λy1 . Since (c0 , . D ) is strictly convex, we have that y1 + y2 y1 D + y2 D 2 < 2 D
and, since A is a convex set, x 1 + x 2 y1 + y2 x 1 + x 2 y1 + y2 , 1= = 2 + 2 . 2 2 1
D
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Consequently, x 1 + x 2 y1 + y2 x 1 1 + x 2 1 y1 D + y2 D < , + =1 1= 2 2 2 2 which is a contradiction, and thus there exists λ ∈ R such that y2 = λy1 , as claimed. Now, it is not difficult to see that A is compact. Indeed, consider a sequence {(x n , yn )} in A. Then there exists a sequence of real numbers {λn } and an element y0 ∈ (c0 , . D ) such that yn = λn y0 . On the other hand, since A is weakly compact, we may assume that {(x n , yn )} is weakly convergent to (x 0 , λ0 y0 ) ∈ A. This means that x n x 0 in 1 and λn y0 → λ0 y0 in c0 . Since 1 is a Schur space, we have that x n → x 0 , which implies that {(x n , yn )} is norm convergent to (x 0 , λ0 y0 ) ∈ A. Remark 24. The argument used in Example 23 shows that, if (X, .) is a nonreflexive, nonNCS, Schur space and (Y, .) is a strictly convex Banach space without the Kadec–Klee property, then (X, .) ⊕1 (Y, .) has property (C) but it is not NCS and does not have the Kadec–Klee property. On the other hand, if X, Y are two Banach spaces with property (C) and X is not a Schur space, then X ⊕∞ Y does not have property (C). To this end, we consider a weakly convergent sequence (x n ) in the unit ball of X which is not norm-convergent and y0 ∈ Y with y0 = 1. Then it is clear that K := {(x, y0 ) : x ∈ co{x n }} is a weakly compact convex subset of the unit sphere of X ⊕∞ Y which cannot be norm compact. Finally, it is clear that property (C) is not a topological property. Indeed, (c0 , .∞ ) does not have property (C), while the space (c0 , . D ) enjoys it. In this context, we give the following result, closely inspired by Corollary 2 in [24]. Proposition 25. Let (X, .) be a reflexive infinite dimensional Banach space. Then, given > 0, there exists an equivalent norm . on X such that (X, . ) lacks property (C) and such that, for every x ∈ X, x ≤ x ≤ (1 + )x . Proof. Let f ∈ X ∗ with f = 1, and let x 0 ∈ X with x 0 = 1 be such that f (x 0 ) = 1. Consider the norm on X given by x := max{x, (1 + )| f (x)|}. Let (x n ) be a sequence in V := ker( f ) ∩ B[0 X , 1]. Since (X, .) is reflexive, we may suppose that x n y0 ∈ V . Moreover, we may also suppose that x n − y0 ≥ η > 0. Let yn := 2(1+) (x n − y0 ) and let vn := yn +
1 1+ x 0 .
We have that
vn ≤ yn +
1 1 ≤ (x n + y0 ) + ≤ 1. 1+ 2(1 + ) 1+
On the other hand, (1 + )| f (vn )| = (1 + ) f
1 x 0 = 1. 1+
1 1 x 0 and 1+ x 0 = 1. Then it is easy to see that Hence vn = 1. Moreover, vn 1+ K := co{vn : n = 1, . . .} is a weakly compact convex subset of the unit sphere of (X, . ) which is not compact. Consequently, (X, . ) fails to have property (C).
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Remark 26. One may compare this result with the well known Zizler theorem about renormings: every separable Banach space admits an equivalent norm that is uniformly convex in every direction (UCED) and hence has property (C). Moreover, this equivalent norm can be chosen to be as close as desired to the original norm in the Banach–Mazur sense. These results show that property (C) has a strong geometrical (isometric but nonisomorphic) nature. 6. Nonexpansive J-type mappings The relationship between nonexpansive and J-type mappings is not quite clear. The aim of this section is to summarize some facts about this topic. If K is closed, convex and unbounded, then there exist easy examples of nonexpansive, even contractive mappings without centers. Example 27. Let f : [1, +∞) → [1, +∞) be given by x2 + 1 . x It easy to check that | f (x) − f (y)| < |x − y| for all x, y ∈ [1, ∞) and that f is fixed point free. By Corollary 17, it is clear that f does not admit any center in R. f (x) =
However, if we consider nonexpansive self-mappings of weakly compact convex subsets of Banach spaces, then it is not so easy to find nonexpansive mappings without centers. We know a long list of additional requirements on the geometry of the space under which every such mapping has a fixed point, and hence is a J-type mapping. Moreover, the most important example of nonexpansive fixed point free mapping in this context, namely the Alspach mapping (see Example 8), admits zero as a center and hence it is also a J-type mapping. Nevertheless, a modification of this example gives us a nonexpansive self-mapping of a weakly compact convex set that does not have a center. Example 28. Let T be the Alspach mapping considered on the whole interval C = f ∈ L 1 [0, 1] : 0 ≤ f ≤ 2 . Given f ∈ C, we put S( f ) = 2χ[0,1] − T f , where χ[0,1] is the characteristic function of the interval [0, 1]. Then S : C → C is an isometry and Sine [25] proved that S does not have a fixed point. We also put R( f ) = min{2, max{0, f }} for f ∈ L 1 [0, 1]. Then R( f ) is the unique point nearest to f in C. Assuming that y0 ∈ L 1 [0, 1] is a center for S, we see that S(R(y0 )) = R(y0 ) (compare with the proof of Theorem 16), which contradicts Sine’s result. The general theory of nonexpansive mappings in Banach spaces has a long list of famous open problems. Among many others, one can point out the question of whether either reflexive strictly convex or reflexive Kadec–Klee Banach spaces have the FPP. We may add one more general problem to this list: Question 1. Does property (C) imply the FPP for reflexive Banach spaces? Bearing in mind Theorem 16, one way to solve this problem could be to find an affirmative answer for the following question, which is a particular case of the first one.
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Question 2. In weakly compact convex subsets of either strictly convex or Kadec–Klee Banach spaces, is every nonexpansive mapping a J-type mapping? If (X, .) is a Banach space without the WFPP, then there exists a weakly compact convex subset C of X and a fixed point free nonexpansive mapping T : C → C. If T admits a center y0 ∈ X, then y0 is a center for the restriction of T to every subset of C. In particular, it is well known that there exists a weakly compact convex T -invariant subset M of C which is minimal for T and which has positive diameter, say d. We claim that such a minimal set must be contained in a sphere centered at y0 . Otherwise, let ρ := dist(y0 , M) > 0. If there exists some x ∈ M for which x − y0 > ρ, then the set
1 B y0 , (x − y0 + ρ) ∩ M 2 is nonempty, T -invariant, weakly compact, convex, and strictly smaller than M, which is a contradiction with the fact that M is minimal. Summarizing, one has. Proposition 29. If T is a nonexpansive fixed point free mapping leaving invariant a weakly compact convex set and with a center y0 ∈ X, then all its minimal subsets lie in a sphere centered in y0 . Properties of minimal invariant sets for nonexpansive mappings were also studied in [17]. Let a Banach space (X, .) have property (C) and T : C → C be a nonexpansive mapping, where C ⊂ X is convex and weakly compact. An argument from [17] shows that if M1 , M2 are minimal convex weakly compact T -invariant subsets of C, then diam M1 = diam M2 . Here, we need not assume that T has a center. 7. Applications 7.1. Application to an integral equation In this section we consider the functional-integral equation: y(t) = φ t, K (t, w)g(w, y(w)) dμ(w) , t ∈ Ω , Ω
(2)
where (X, .) is a Banach space and (Ω , Σ , μ) is a probability measure space, and we look for the existence of a solution y ∈ L p (μ, X) of (2). In [7], the author studied the existence of solutions of a particular case of this kind of equations. Here, we will use our Theorem 16 in order to obtain a similar result. Definition 30. Let (Ω , Σ , μ) be a finite measure space and let (X, .) be a Banach space. A continuous mapping f : Ω × X → X is said to be a k-0 X -J-type mapping with respect to the second variable if f (r, x) ≤ kx for each (r, x) ∈ Ω × X. Throughout the following we assume the next three hypotheses:
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(a) φ : Ω ×X → X (resp. g : Ω ×X → X) is a k-0 X -J-type mapping (resp. l-0 X -J-type mapping) with respect to the second variable and the map w → φ(w, y(w)) (resp. w → g(w, y(w))) belongs to L p (μ, X) for each y ∈ L p (μ, X). (b) |K (t, s)| ≤ M for each (t, s) ∈ Ω × Ω and K (t, .) : Ω → R is measurable and kl M ≤ 1. (c) There exist x 0 ∈ L p (μ, X) and r > 0 such that φ ., K (., w)g(w, y(w)) dμ(w) − x 0 ≤ r Ω
p
whenever y − x 0 p ≤ r . Theorem 31. Let (X, .) be a Banach space and let (Ω , Σ , μ) be a probability measure space and consider the Bochner space L p (μ, X) with 1 < p < +∞. Then, (1) If the conditions (a) and (b) are satisfied, then 0 L p (μ,X ) is a solution of equation (2). (2) Assume that the hypotheses (a),(b),(c) are satisfied. Suppose that (X, .) is reflexive and that L p (μ, X) has property (C). Then Eq. (2) has at least one solution y ∈ B[x 0, r ] ⊂ L p (μ, X). Proof. Let us introduce the operator T : L p (μ, X) → L p (μ, X) defined by K (t, w)g(w, y(w)) dμ t ∈ Ω . (T y)(t) := φ t, Ω
It is clear that if T has a fixed point y, then y is a solution of the our integral equation. (1) From Conditions (a) and (b) it follows that T (0 L p (μ,X ) ) = 0 L p (μ,X ) . (2) Since L p (μ, X) is reflexive, the closed ball K := B[x 0, r ] of L p (μ, X) is a weakly compact convex subset of this space. Thus, by Hypothesis (c), we know that K is a weakly compact convex T -invariant subset. By Hypothesis (a) we have that φ is a k-0 X -J-type mapping with respect to the second variable and g is a l-0 X -J-type mapping with respect to the second variable, which allows us to affirm that T is a J-type mapping. Indeed, since φ is a k-0 X -J-type mapping with respect to the second variable, we have p T y p = (T y)(t) p dμ(t) Ω p φ t, dμ(t) K (t, w)g(w, y(w)) dμ(w) = Ω Ω p K (t, w)g(w, y(w)) dμ(w) dμ(t) ≤ kp Ω
Ω
by Jensen’s inequality and Hypothesis (b) g(w, y(w)) p dμ(w) dμ(t). ≤ (k M) p Ω
Ω
Since g is a l-0 X -J-type mapping with respect to the second variable, we obtain y(w) p dμ(w) dμ(t) ≤ y p . T (y) p ≤ (k Ml) p Ω
Ω
The last inequality is a consequence of the assumption that (Ω , Σ , μ) is a probability measure space and Hypothesis (b).
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Finally, since T is a J-type mapping, and moreover we have assumed that L p (μ, X) enjoys property (C), by Theorem 16 we conclude that T has a fixed point in K . Remark 32. It is well known that, for 1 < p < +∞, the Bochner space L p (μ, X) is reflexive if and only if (X, .) is reflexive. Moreover, Day showed in [8] that, for 1 < p < +∞, the Bochner space L p (μ, X) is strictly convex if and only if (X, .) is strictly convex. In [18], Harandi showed that, under some conditions labelled (1), (2) and (3) in his article, the mapping T introduced in the proof of Theorem 31 is AC N. Then, by Theorem 7, we can conclude that 0 L p (μ,X ) is a center of this mapping T . Since T is well defined on L p (μ, X), we know that 0 L p (μ,X ) is a solution of Eq. (2). Therefore, in order to obtain a solution of Eq. (2), we do not need any additional condition on X. 7.2. Application to the accretivity Let (X, .) be a Banach space. A mapping A : X → 2 X will be called an operator on X. The effective domain of A is denoted by D(A) and its range by R(A). An operator A on X is said to be accretive if the inequality x − y ≤ x − y + λ(z − w) holds for all λ ≥ 0, z ∈ Ax, and w ∈ Ay. If, in addition, R(I + λA) is for one λ, hence for all λ, precisely X, then A is called m-accretive (we refer the reader to [4,6] for background material on accretivity). Among the problems treated in the theory of accretive operators, one of the most studied is that of determining when A has a zero, i.e. 0 ∈ R(A) (see, for instance, [12] to find properties which imply the existence of zeroes for accretive operators). Suppose that A is an accretive operator on X and 0 ∈ R(A). This means that there exists x 0 ∈ D(A) such that 0 ∈ Ax 0 . Therefore, by the definition of such operators, we obtain: x − x 0 ≤ x − x 0 + λu
for all (x, u) ∈ G(A).
(3)
Therefore, Inequality (3) is a necessary condition in order to obtain the existence of zeroes of accretive operators. In this section we will study the problem when Condition (3) is, in fact, a sufficient condition. Theorem 33. Let (X, .) be a Banach space and let A : D(A) → 2 X be an accretive operator on X satisfying Condition (3) for some x 0 ∈ X. Then (a) If x 0 ∈ R(I + A), then 0 ∈ R(A). (b) If (X, .) is a reflexive space with the fixed point property for spheres and co(D(A)) ⊆ R(I + A), then 0 ∈ R(A). Proof. Since the operator A is accretive, it is well known that the resolvent: g := (I + A)−1 : R(I + A) → D(A) is a single-valued nonexpansive mapping. We claim that g is a J-type mapping. For an arbitrary y ∈ R(I + A), there exists x ∈ D(A) such that x = g(y). Hence, y ∈ x + Ax, which means that there exists u ∈ Ax such that y = x + u. Therefore, using Condition (3) for λ = 1, we obtain g(y) − x 0 = x − x 0 ≤ x − x 0 + u = y − x 0 . Consequently, g is a J-type mapping, as claimed.
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(a) If x 0 ∈ R(I + A), then g(x 0 ) = x 0 and hence x 0 ∈ x 0 + Ax 0 , which implies that 0 ∈ Ax 0 . (b) We may consider the set C := co(D(A)) ∩ B[x 0, dist(x 0 , co(D(A)))]. Clearly, this set is convex. It is also nonempty and weakly compact, because the space (X, .) is reflexive. The point x 0 is a center for g, so C is also g-invariant. Thus, since g : C → C is a J-type nonexpansive mapping and (X, .) has the fixed point property for spheres, we can apply Corollary 20 to obtain the conclusion. Remark 34. Notice that every m-accretive operator satisfying Condition (3) fulfills Condition (a) of the above theorem, and hence it has a zero. On the other hand, it is still unknown whether a reflexive Banach space with the fixed point for spheres enjoys the FPP for nonexpansive mappings. However, by Corollary 20, this holds for J-type nonexpansive mappings. Therefore in part (b) of the above result, the fact that g is a J-type mapping is essential in order to obtain the conclusion. It is also interesting to note that, in part (b) of the above theorem, the hypothesis that (X, .) is reflexive and has the fixed point property for spheres cannot be omitted. Indeed, it suffices to consider the operator A : B[0c0 , 1] → c0 defined by A(x 1 , x 2 , . . .) = (x 1 − 1, x 2 − x 1 , x 3 − x 2 , . . .). It is not difficult to see that A is an accretive operator, 0 ∈ R(A), and A satisfies Condition (3) for x 0 = (2, 0, 0, . . .) and B[0c0 , 1] ⊂ R(I + A). Acknowledgement The first and second authors were partially supported by DGES, grant BFM 2003-03893-C0202. References [1] D.E. Alspach, A fixed point free nonexpansive map, Proc. Amer. Math. Soc. 82 (1981) 423–424. [2] J.B. Baillon, R. Sch¨oneberg, Asymptotic normal structure and fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 81 (1981) 257–264. [3] J. Banas, On drop property and nearly uniformly smooth Banach spaces, Nonlinear Anal. 14 (11) (1990) 927–933. [4] V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, 1976. [5] R.E. Bruck, Properties of fixed-point sets of nonexpansive mappings in Banach spaces, Trans. Amer. Math. Soc. 179 (1973) 251–262. [6] I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, 1990. [7] M.A. Daswish, Weakly singular functional-integral equation in infinite dimensional Banach spaces, Appl. Math. Comput. 136 (2003) 123–129. [8] M.M. Day, Strict convexity and smoothness of normed spaces, Trans. Amer. Math. Soc. 78 (1955) 516–528. [9] J.B. D´ıaz, F.T. Metcalf, On the set of subsequential limit points of successive approximations, Trans. Amer. Math. Soc. 135 (1969) 459–485. [10] W.G. Dotson, On the Mann iterative process, Trans. Amer. Math. Soc. 149 (1970) 65–73. [11] J. Garcia-Falset, Fixed points for mappings with the range type condition, Houston J. Math. 28 (2002) 143–158. [12] J. Garcia-Falset, S. Reich, Zeroes of accretive operators and the asymptotic behavior of nonlinear semigroups, Houston J. Math. (in press). [13] K. Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Univ. Press, 1990.
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