Fixed points, selections and common fixed points for nonexpansive-type mappings

Fixed points, selections and common fixed points for nonexpansive-type mappings Rafa Esp´ınola1, Pepa Lorenzo1, Adriana Nicolae2 1Departamento de An´alisis Matem´atico, Universidad de Sevilla, P.O. Box 1160, 41080 Sevilla, Spain 2Department of Mathematics, Babe¸s-Bolyai University, Kog˘alniceanu 1, 400084, Cluj-Napoca, Romania Manuscript correspondence: Departamento de An´alisis Matem´atico, Universidad de Sevilla, P.O.Box 1160, 41080 Sevilla, Spain email: [email protected] Phone: 0034 954 559 560 Fax: 0034 954 557 972 Email addresses: [email protected] (R. Esp´ınola), [email protected] (P. Lorenzo), [email protected]cluj.ro (A. Nicolae) 1 Abstract We study the existence of fixed points in the context of uniformly convex geodesic metric spaces, hyperconvex spaces and Banach spaces for single and multivalued mappings satisfying conditions that generalize the concept of nonexpansivity. Besides, we use the fixed point theorems proved here to give common fixed point results for commuting mappings. Key-words: fixed point, selection of multifunctions, generalized nonexpansive mappings, commuting mappings, metric space, Banach space. 1 Introduction In [26], T. Suzuki extends the concept of singlevalued nonexpansive mapping in the following way: a mapping fdefined on a subset Kof a Banach space is said to satisfy condition (C) if for x, y ∈Kwith (1/2)kx−f(x)k≤kx−yk, then kf(x)−f(y)k≤kx−yk. T. Suzuki [26] proves some basic properties and gives fixed point theorems and convergence results for mappings satisfying condition (C). Following [26], A. Razani and H. Salahifard [23] state part of T. Suzuki’s [26] results in the context of a complete CAT(0) space and generalize condition (C) to the multivalued case: a multivalued mapping Tdefined on subset of a CAT(0) space is said to satisfy condition (C) if for each x, y ∈Kand ux∈T(x) with (1/2)d(x, ux)≤d(x, y) there exists uy∈T(y) such that d(ux, uy)≤d(x, y). This condition is used in [23] to prove a fixed point theorem for multivalued mappings and some common fixed point results. Motivated by the results in [26], J. Garc´ıa-Falset, E. Llorens-Fuster and T. Suzuki consider in [7] two generalizations in the singlevalued case of condition (C) giving examples and establishing fixed point results. The purpose of this paper is to study condition (C) for multivalued mappings in the context of geodesic metric spaces (with special attention to the case of R-trees) and Banach spaces, and condition (C) for singlevalued mappings in the context of hyperconvex spaces. After some preliminary contents in Section 2, we begin Section 3 by studying the multivalued case in geodesic spaces. We assume condition (C) for multivalued mappings as in [23] where different results in this direction were obtained for CAT(0) spaces. In our work, we derive a technical lemma (Lemma 3.2) which is a multivalued version of the key fact which is behind the main results in [7, 26]. Our results are first obtained for as general as complete uniformly convex geodesic spaces and then particularized for more precise geometries. Since CAT(0) spaces are a particular class of uniformly convex geodesic spaces, we obtain more general results than those from [23]. Moreover, thanks mainly to Lemma 3.2, we fill in a gap in the proof of the main multivalued result in [23]. We continue Section 3 by introducing a new condition for multivalued mappings in the spirit of (C). We give examples showing that this condition is actually weaker than condition (C) and prove a selection theorem in R-trees for mappings satisfying this newly introduced condition from where a stronger fixed point result for multivalued mappings follows. This selection result resembles a very important one, see for instance [12, 25], for hyperconvex spaces (notice, see [14], that complete R-trees are hyperconvex) although the approach here is completely different as the proof relies on very particular properties of R-trees rather than on hyperconvexity. It is worthwhile to point out that R-trees find 2 a lot of applications in different areas as, for instance, the indexing of information or phylogenetics. We close Section 3 with an appendix where we study the existence of fixed points for singlevalued mappings with property (C) in hyperconvex metric spaces. It is very well-known (see [17, Chapter 13]) that nonexpansive self-mappings defined on nonempty bounded and closed hyperconvex spaces have fixed points. Therefore it is natural to wonder about this problem for mappings with condition (C). We first study the compact case providing a positive answer. For the more general case we need to introduce a new condition on the mapping under consideration. In particular it is shown that a 2-lipschitzian self-mapping with condition (C) defined on a nonempty closed and bounded hyperconvex space has a fixed point. This result is significant among the class of known results for mappings with condition (C) since it is the first one without compactness conditions for which neither the uniqueness of asymptotic centers nor anything similar to the Opial property is required (see Sections 2 and 4 for definitions). Therefore, this result follows through a completely new approach compared to those in [7, 23, 26] and implies new results even, for instance, in injective Banach spaces. In Section 4 we revisit the classical theory of nonexpansive multivalued mappings on Banach spaces to study it under condition (C). We show the existence of fixed points for such a mapping in a Banach space with the Opial property. The method of asymptotic centers allows us to establish the same result in a uniformly convex in every direction (UCED) Banach space. Moreover, if we also assume the continuity of the mapping we can prove the existence of fixed points in a Banach space for which the asymptotic center of a bounded sequence with respect to a bounded closed convex subset is nonempty and compact, that is, a counterpart of the Kirk-Massa theorem. Finally, in Section 5, we appeal to the fixed point theorems proved in this paper in order to give some common fixed point results for commuting mappings. 2 Preliminaries Let (X, d) be a metric space. A geodesic path from xto yis a mapping c: [0, l]⊆R→X with c(0) = x, c(l) = yand d(c(t), c(t0)) = |t−t0|for every t, t0∈[0, l]. The image c([0, l]) of cforms a geodesic segment which joins xand yand is not necessarily unique. If no confusion arises, we will use [x, y] to denote a geodesic segment joining xand y. (X, d) is a (uniquely) geodesic space if every two points x, y ∈Xcan be joined by a (unique) geodesic path. A point z∈Xbelongs to the geodesic segment [x, y] if and only if there exists t∈[0,1] such that d(z, x) = td(x, y) and d(z, y) = (1 −t)d(x, y), and we will write z= (1 −t)x+ty for simplicity. A subset Kof Xis convex if it contains any geodesic segment that joins every two points of it. In a geodesic space (X, d), the metric d:X×X→Ris convex if for any x, y, z ∈X one has d(x, (1 −t)y+tz)≤(1 −t)d(x, y) + td(x, z) for all t∈[0,1]. A geodesic space which metric is convex will be referred as a space with convex metric. A trivial example of a uniquely geodesic space with convex metric is a strictly convex Banach space. For more details about geodesic metric spaces one may check [2]. A geodesic space (X, d) is uniformly convex if for any r > 0 and ∈(0,2] there exists 3 δ∈(0,1] such that if a, x, y ∈Xwith d(x, a)≤r,d(y, a)≤rand d(x, y)≥r then d(1 2x+1 2y, a)≤(1 −δ)r. From the definition, it is easy to see that uniformly convex metric spaces are uniquely geodesic. A mapping δ: (0,∞)×(0,2] →(0,1] providing such a δ=δ(r, ) for a given r > 0 and ∈(0,2] is called a modulus of uniform convexity. The mapping δis monotone (resp. lower semi-continuous from the right) if for every fixed it decreases (resp. is lower semi-continuous from the right) with respect to r(see also [5], [18]). CAT(0) spaces in the sense of Gromov (see [2]) are uniformly convex metric spaces with convex metric. Let (X, d) be a metric space and let (xn)n∈Nbe a bounded sequence in X. For x∈X, define r(x, (xn)) = lim supn→∞ d(x, xn). The asymptotic radius of (xn)n∈Nis given by r((xn)) = inf {r(x, (xn)) : x∈X}, and the asymptotic center of (xn)n∈Nis the set A((xn)) = {x∈X:r(x, (xn)) = r((xn))}. Throughout this paper we will denote a uniformly convex metric space with monotone (or lower semi-continuous from the right) modulus of uniform convexity as a UC space. In [5], the authors prove that every bounded sequence in a UC space has a unique asymptotic center. A bounded sequence (xn)n∈Nin a complete UC space is regular if r((xn)) = r((xnk)) for every subsequence (xnk)k∈Nof (xn)n∈N. It is known that in a Banach space every bounded sequence contains a regular subsequence (see, for instance, [17], Chapter 2, Lemma 5.2). Since the proof has a metric nature we can conclude that every bounded sequence (xn)n∈Nin a complete UC space has a regular subsequence (xnk)k∈Nand thus every subsequence of (xnk)k∈Nhas the same asymptotic center as (xnk)k∈N. Let (X, d) be a metric space. Taking z∈Xand r > 0 we denote the closed ball centered at zwith radius rby e B(z, r). Given Ya nonempty subset of X, we define the distance of a point z∈Xto Yby dist(z, Y ) = infy∈Yd(z, y).The metric projection (or nearest point mapping)PYonto Yis the mapping PY(z) = {y∈Y:d(z, y) = dist(z, Y )},for every z∈X. If Yis additionally bounded, the diameter of Yis given by diamY= supx,y∈Yd(x, y). In this paper we also consider the following families of sets: P(X) = {Y⊆X:Yis nonempty}, Pb(X) = {Y⊆X:Yis nonempty and bounded}, Pb,cv(X) = {Y⊆X:Yis nonempty, bounded and convex}, Pcl,cv(X) = {Y⊆X:Yis nonempty, closed and convex}, Pb,cl,cv(X) = {Y⊆X:Yis nonempty, bounded, closed and convex}, 4 Pcp(X) = {Y⊆X:Yis nonempty and compact}, Pcp,cv(X) = {Y⊆X:Yis nonempty, compact and convex}. A metric space (X, d) is metrically convex if for any two distinct points x, y ∈X and any α, β > 0 such that d(x, y) = α+βthere exists z∈Xwith d(x, z) = αand d(y, z) = β.Xhas the binary intersection property if Ti∈Ie Bi6=∅for every collection of balls ( e Bi)i∈Isuch that any two of these balls intersect. A metric space (X, d) is hyperconvex if Ti∈Ie B(xi, ri)6=∅for every collection of points (xi)i∈Iin Xand positive numbers (ri)i∈Isuch that d(xi, xj)≤ri+rjfor any i, j ∈ I. Hyperconvexity is equivalent to the binary intersection property and the metric convexity. More about hyperconvex spaces can be found in [1, 12, 25] or in Chapter 13 of [17]. Given (X, d) a metric space and A⊆X, the number rx(A) = supy∈Ad(x, y) is called the radius of Arelative to x∈X. The radius of Ais r(A) = infx∈Xrx(A), the center of Ais the set C(A) = {x∈X:rx(A) = r(A)}and the admissible cover of Ais defined by cov(A) = T{e B:e Bis a closed ball and A⊆e B}. The set Ais said to be admissible if A= cov(A). For Xa hyperconvex space and A⊆X, cov(A) = Tx∈Xe B(x, rx(A)) and diam(A)=2r(A) (for details see Chapter 13 of [17]). An R-tree is a uniquely geodesic metric space Xsuch that if [y, x]∩[x, z] = {x}then [y, x]∪[x, z]=[y, z] for each x, y, z ∈X. From the definition it immediately follows that if x, y, z ∈X, then [x, y]∩[x, z]=[x, w] for some w∈X. Likewise, if Kis a closed and convex subset of an R-tree X, then for every x∈X,PK(x) is a singleton and for any y∈K,d(x, y) = d(x, PK(x)) + d(PK(x), y). A standard example of an R-tree is R2 endowed with the so-called river metric. For x= (x1, x2), y = (y1, y2)∈R2, the river metric (denoted by ρ) is defined by ρ(x, y) = |x2−y2|if x1=y1, |x2|+|y2|+|x1−y1|otherwise. It is known that R-trees are CAT(0) spaces and that a metric space is a complete R-tree if and only if it is hyperconvex and has unique geodesic segments (see [14]). More about the fixed point theory in R-trees can be found in [4, 15, 21, 22]. In [26], T. Suzuki considered the following generalized family of nonexpansive mappings in the setting of a Banach space. We will use in the sequel the norm notation, but the same definitions also hold when working in the metric setting (naturally, the norm will be replaced by the distance). Definition 2.1. Let Xbe a Banach space, K∈P(X)and f:K→X. Then fsatisfies condition (C)if 1 2kx−f(x)k ≤ kx−yk=⇒ kf(x)−f(y)k ≤ kx−yk, for all x, y ∈K. Obviously, every nonexpansive mapping meets condition (C). We next summarize some of the basic properties proved in [26] in relation to these mappings. The proofs of these results are metric in nature so the properties also apply in the metric case. Throughout this paper we denote the set of fixed points of a mapping fby Fix(f). 5 Lemma 2.2. Let Xbe a Banach space and K∈P(X). Assume that the mapping f:K→Xsatisfies condition (C). Then for each x, y ∈K, (i) if z∈Fix(f), then kz−f(x)k≤kz−xk, that is, fis quasinonexpansive; (ii) kf(x)−f(y)k ≤ kx−ykor kf2(x)−f(y)k≤kf(x)−yk; (iii) kx−f(y)k ≤ 3kf(x)−xk+kx−yk. Using these properties, T. Suzuki [26] proves fixed point theorems for mappings satisfying condition (C). In [7], the authors study two generalizations of condition (C) giving examples and establishing fixed point results. One of these conditions is the following. Definition 2.3. Let Xbe a Banach space, K∈P(X),f:K→Xand µ≥1. The mapping fsatisfies condition (Eµ)if for all x, y ∈K, kx−f(y)k ≤ µkf(x)−xk+kx−yk. Lemma 2.2, (iii) yields that condition (C) implies (E3), but Example 3 of [7] shows that (E3) does not imply (C). Other examples for different values of µare studied in [7]. In the next sections we will make use of the lemma below which is a special case of Proposition 2 in [9]. Lemma 2.4. Let Xbe a geodesic metric space with convex metric, α∈(0,1) and (xn)n∈Nand (yn)n∈Nbounded sequences in Xsuch that xn+1 = (1 −α)xn+αynand d(yn+1, yn)≤d(xn+1, xn)for every n∈N. Then limn→∞ d(xn, yn)=0. The following two theorems were proved in [23], but in the setting of a complete CAT(0) space. It is easy to see that these results hold in more general contexts. We will formulate the first result in the framework of a uniquely geodesic metric space. Theorem 2.5. Let Xbe a uniquely geodesic metric space and K∈Pcl,cv(X). Suppose f:K→Ksatisfies condition (C)and Fix(f)6=∅. Then Fix(f)is closed and convex. The proof of the second theorem only requires the uniqueness of the asymptotic center and the convexity of the metric. This is why we state this result under the hypothesis of a complete UC space with convex metric. Theorem 2.6. Let Xbe a complete UC space with convex metric and suppose K∈ Pb,cl,cv(X). If f:K→Ksatisfies condition (C)then Fix(f)is nonempty, closed and convex. In [23], the authors also extend Suzuki’s [26] condition (C) to the multivalued case in the following way. Definition 2.7. Let Xbe a metric space and K∈P(X). A mapping T:K→P(X)is said to satisfy condition (C)if for each x, y ∈Kand ux∈T(x)such that 1 2d(x, ux)≤d(x, y), there exists uy∈T(y)such that d(ux, uy)≤d(x, y). 6 The above condition is used in [23] to give a fixed point theorem for multivalued mappings and some common fixed point results. In the rest of this paper we use condition (C) for both single and multivalued mappings with the context distinguishing between the two cases. The same also holds for other conditions we make use of. 3 Fixed points and selections in geodesic spaces In this section we study the multivalued version of mappings with condition (C) in geodesic metric spaces. Following the singlevalued case, we introduce the next condition and prove that for µ= 3 it is a generalization of condition (C). Definition 3.1. Let Xbe a metric space, K∈P(X),T:K→P(X)and µ≥1. The mapping Tsatisfies condition (Eµ)if for each x, y ∈Kand ux∈T(x)there exists uy∈T(y)such that d(x, uy)≤µd(x, ux) + d(x, y). We prove next that a multivalued mapping which satisfies condition (C) also satisfies (E3). This property will constitute a key tool in proving our results. Lemma 3.2. Let Xbe a metric space, K∈P(X)and let T:K→P(K)satisfy condition (C). Then Tsatisfies condition (E3). Proof. Let x, y ∈Kand ux∈T(x). Because (1/2)d(x, ux)≤d(x, ux) there exists vx∈T(ux) such that d(ux, vx)≤d(x, ux).(1) We prove that either 1 2d(x, ux)≤d(x, y) (2) or 1 2d(ux, vx)≤d(ux, y) (3) holds. Suppose (1/2)d(x, ux)> d(x, y) and (1/2)d(ux, vx)> d(ux, y). Then, using (1) we obtain the following contradiction d(x, ux)≤d(x, y) + d(y, ux)<1 2d(x, ux) + 1 2d(ux, vx)≤d(x, ux). Hence, if (2) holds, then there exists uy∈T(y) such that d(ux, uy)≤d(x, y), so d(x, uy)≤d(x, ux) + d(ux, uy)≤d(x, ux) + d(x, y). If (3) holds, then there exists uy∈T(y) such that d(vx, uy)≤d(ux, y). Using again (1) we have that d(x, uy)≤d(x, ux) + d(ux, vx) + d(vx, uy)≤2d(x, ux) + d(ux, y)≤3d(x, ux) + d(x, y). Thus, the inequality holds in each of the two cases and we are done. 7 Definition 3.3. Let Xbe a metric space, K∈P(X)and T:K→P(X). We say that (xn)n∈N⊆Kis an approximate fixed point sequence for the mapping Tif for each n∈N there exists yn∈T(xn)such that limn→∞ d(xn, yn)=0. The next result provides an approximate fixed point sequence for a multivalued mapping satisfying condition (C). We use this result in the rest of the paper because many of our proofs rely on it. Proposition 3.4. Let Xbe geodesic metric space with convex metric, K∈Pb,cv(X) and T:K→P(K). If Tsatisfies condition (C), then Thas an approximate fixed point sequence. Proof. Let x1∈K,y1∈T(x1) and take x2= (1/2)x1+ (1/2)y1. Then (1/2)d(x1, y1) = d(x1, x2) so, by condition (C), there exists y2∈T(x2) such that d(y1, y2)≤d(x1, x2). Continuing in this vein, we can build the sequences (xn)n∈Nand (yn)n∈Nsuch that yn∈T(xn), xn+1 = (1/2)xn+ (1/2)ynand d(yn+1, yn)≤d(xn+1, xn) for every n∈N. Using Lemma 2.4 we obtain that limn→∞ d(xn, yn) = 0. Our first fixed point result for multivalued mappings is given for self-mappings on a compact set. Theorem 3.5. Let Xbe a geodesic space with convex metric and K∈Pcp,cv(X). Suppose T:K→Pcl(K)satisfies condition (C). Then Fix(T)6=∅. Proof. By Proposition 3.4, there exist two sequences (xn)n∈Nand (yn)n∈Nin Ksuch that yn∈T(xn) and limn→∞ d(xn, yn) = 0. Since Kis compact, we can find a subsequence (xnk)k∈Nof (xn)n∈Nsuch that (xnk)k∈Nconverges to some x∈K. Using Lemma 3.2, we have that for all k∈N dist(xnk, T(x)) ≤3d(xnk, ynk) + d(xnk, x). Taking the limit as k→ ∞ we obtain that dist(x, T (x)) = 0. Since T(x) is closed it follows that x∈T(x). In the following theorem we move the compactness condition from the domain to the images of the mapping. This theorem is actually an extension of Theorem 3.2 of [23] in the context of a complete UC space with convex metric. We also remove the convexity condition on the image sets of the mapping. Moreover, we obtain our results in a simple way as a consequence of Lemma 3.2 which avoids to go through a delicate point in the proof of Theorem 3.2 of [23]. Theorem 3.6. Let Xbe a complete UC space with convex metric and K∈Pb,cl,cv(X). Suppose T:K→Pcp(K)satisfies condition (C). Then Fix(T)6=∅. Proof. By Proposition 3.4, we can find the sequences (xn)n∈Nand (yn)n∈Nin Ksuch that yn∈T(xn) and limn→∞ d(xn, yn) = 0. As explained in Section 2, we may suppose that (xn)n∈Nis regular (otherwise choose a regular subsequence of it). Denote the unique asymptotic center of (xn)n∈Nby x. Let n∈N. Applying Lemma 3.2 for xn, x and yn respectively it follows that there exists zn∈T(x) such that d(xn, zn)≤3d(xn, yn) + d(xn, x). 8 Let (znk)k∈Nbe a subsequence of (zn)n∈Nthat converges to some z∈T(x). Then, for each k∈N, d(xnk, z)≤d(xnk, znk) + d(znk, z)≤3d(xnk, ynk) + d(xnk, x) + d(znk, z). Taking the superior limit as k→ ∞ and knowing that the asymptotic center of (xnk)k∈N is precisely xwe obtain that x=z∈T(x). Hence, the proof is complete. Remark 3.7. From the above proof it is immediate that in Theorem 3.6 we can drop the convexity of the metric and assume instead that the mapping admits an approximate fixed point sequence. In the next result we will consider the following new condition for multivalued mappings which will be shown to be weaker than condition (C). Definition 3.8. Let Xbe a metric space, K∈P(X)and T:K→P(X). The mapping Tsatisfies condition (C0)if for each x, y ∈Kand ux∈T(x)with d(x, ux) = dist(x, T(x)) and 1 2d(x, ux)≤d(x, y), there exists uy∈T(y)such that d(ux, uy)≤d(x, y). We prove next a selection theorem in R-trees for multivalued mappings satisfying condition (C0) and analyze afterwards the relation of (C0) to (C) and (E3) respectively. Theorem 3.9. Let Xbe an R-tree, K∈P(X)and T:K→Pcl,cv(X)a mapping which satisfies (C0). Then the mapping f:K→Xdefined by f(x) = PT(x)(x)for each x∈K is a selection of Tthat satisfies condition (C). Proof. Notice that the properties of R-trees (see Section 2) guarantee that fis welldefined. Let x, y ∈Ksuch that f(x)6=f(y) and (1/2)d(x, f(x)) ≤d(x, y). Consider p(x) = PT(y)(f(x)) and p(y) = PT(x)(f(y)). First, suppose p(x)6=f(y) and p(y)6=f(x). Since p(x) is the projection of f(x) onto T(y) it follows that d(f(x), f(y)) = d(f(x), p(x)) + d(p(x), f(y)), i.e., p(x)∈[f(x), f(y)]. Since T(y) is convex, [p(x), f(y)] ⊆T(y). This implies [f(x), f(y)] ∩[f(y), y] = {f(y)}because otherwise the minimality of f(y) would be contradicted. Thus, f(y)∈[f(x), y]. Similarly, f(x)∈[f(y), x]. Then f(x), f(y)∈[x, y] (otherwise supposing for example that z∈[x, f(y)]∩[f(y), y] with z6=f(y) we have that f(x)∈[z, f(y)] and f(y)∈[z, f(x)] which is false). Therefore, d(f(x), f(y)) ≤d(x, y). In fact, d(f(x), f(y)) = d(x, y)−dist(x, T(x)) −dist(y, T(y)). Now assume p(x) = f(y). Then d(f(x), f(y)) = dist(f(x), T(y)) and so, by condition (C0), d(f(x), f(y)) = dist(f(x), T(y)) ≤d(x, y). 9 Recall that Xis said to be uniformly convex in every direction (UCED, in short) if δz()>0 for all  > 0 and z∈Xwith kzk= 1, where δz() is the modulus of convexity of Xin the direction zdefined by δz() = inf 1−1 2kx+yk:kxk ≤ 1,kyk ≤ 1, x −y=z. Obviously, uniformly convex Banach spaces are UCED. It is known that in a UCED Banach space, the asymptotic center of a sequence with respect to a weakly compact convex set is a singleton. Hence, every regular sequence with respect to such a set is asymptotically uniform. Theorem 4.6. Let Kbe a weakly compact and convex subset of a UCED Banach space X. Suppose T:K→Pcp(K)is a mapping satisfying condition (C). Then Fix(T)6=∅. Proof. Let (xn)n∈Nand (yn)n∈Nbe two sequences in Ksuch that yn∈T(xn) and limn→∞ kxn−ynk= 0. Without loss of generality, me may assume that (xn)n∈Nis regular with respect to K. Let zbe the unique point in the asymptotic center of (xn)n∈N in K. By Lemma 3.2, for each n∈Nthere exists vn∈T(z) such that kxn−vnk ≤ 3kxn−ynk+kxn−zk. From the compactness of T(z) we can assume that (vn)n∈Nconverges to a point v∈T(z). It follows that lim sup n→∞ kxn−vk ≤ lim sup n→∞ kxn−zk. Since (xn)n∈Nis regular we conclude that v=z∈T(z). Dhompongsa et al. [3] have recently proved the Tinvariance of the asymptotic center in Kof an approximate fixed point sequence for T, when Tis a singlevalued mapping satisfying condition (C). We now state a result which can be seen as an adaptation of this fact to the multivalued case. Proposition 4.7. Let Kbe a weakly compact subset of a Banach space X. Suppose T:K→Pcp(K)satisfies condition (C)and (xn)n∈Nis an approximate fixed point sequence for T. Then, there exists a subsequence (zn)n∈Nof (xn)n∈Nsuch that T(x)∩A6=∅,for all x∈A:= A(K, (zn)). Proof. Since Tis a self-mapping we can build a subsequence (zn)n∈Nof (xn)n∈Nwhich is regular and asymptotically uniform with respect to K. Denote r(K, (zn)) by r. Taking any x∈Aand following the same argument as in the proof of the above theorem we obtain a sequence (vn)n∈N⊆T(x) norm convergent to a point v∈T(x) such that lim sup n→∞ kxn−vk ≤ lim sup n→∞ kxn−xk=r. This shows that v∈A, and so T(x)∩A6=∅. Now we are ready to prove an analogous result to the Kirk-Massa theorem [16] for mappings satisfying condition (C). 16 Theorem 4.8. Let Kbe a bounded, closed and convex subset of a Banach space X and T:K→Pcp,cv(K)be a continuous mapping with respect to the Pompeiu-Hausdorff distance satisfying condition (C). Suppose that each sequence in Khas a nonempty and compact asymptotic center relative to K. Then Fix(T)6=∅. Proof. According to the previous proposition we can take a sequence (xn)n∈Nin Ksuch that T(x)∩A6=∅,for all x∈A:= A(K, (xn)). Now we define the mapping ˜ T:A→Pcp,cv(A) by ˜ T(x) = T(x)∩A. Since Tis continuous, from Proposition 2.45 in [11] we know that the mapping ˜ Tis upper semi-continuous. Since T(x)∩Ais a compact convex set we can apply the Kakutani-Bohnenblust-Karlin theorem (see [10]) to obtain a fixed point for ˜ Tand hence for T. Remark 4.9. Recall that a multivalued mapping T:K→Pb(X) is said to be nonexpansive if H(T(x), T(y)) ≤ kx−ykfor all x, y ∈K, where Hdenotes the Pompeiu-Hausdorff distance. It is worth pointing out that another natural extension of the Suzuki’s condition (C) for a multivalued mapping T:K→P(X) is the following: for all x, y ∈K 1 2dist(x, T(x)) ≤ kx−yk=⇒H(T(x), T(y)) ≤ kx−yk. Obviously, a nonexpansive mapping meets the above condition. However, it is not clear if a mapping satisfying the above condition also satisfies (C). Still, if Ttakes compact values is easy to see that this new condition implies condition (C). Since in our theorems Tis assumed to be compact valued, such results generalize classical fixed point theorems for multivalued mappings (see [16],[19],[20]). 5 Common fixed points In our last section we will apply some of the fixed point theorems stated in previous sections to obtain results on the existence of common fixed point. Definition 5.1. Let Xbe a metric space and K∈P(X). Suppose f:K→Kand T:K→P(K). Then fand Tare commuting mappings if f(y)∈T(f(x)) for all x∈K and y∈T(x). We start by giving a lemma that will constitute a main tool in proving our results. Lemma 5.2. Let Xbe a metric space, K∈P(X),f:K→Ksatisfying condition (C) and with Fix(f)6=∅. Supppose T:K→P(K)is such that for every x, y ∈Fix(f), the set PT(y)(x)is a singleton. If fand Tcommute, then PT(y)(x)∈Fix(f)for all x, y ∈Fix(f). Proof. Let x, y ∈Fix(f) and denote PT(y)(x) by u. Because fmeets condition (C) and 0 = (1/2)d(x, f(x)) ≤d(x, u) we obtain that d(x, f(u)) = d(f(x), f(u)) ≤d(x, u) = dist(x, T(y)). But f(u)∈T(y) because fand Tcommute, y∈Fix(f) and u∈T(y). Hence, f(u) = uand the conclusion follows. 17 The following theorem is an extension of Theorem 4.2 of [23] in the setting of a UC space with convex metric. Notice that our approach is different in the second half of the proof from that of [23]. In particular, ours fills a gap in the proof of [23]. Notice also that this theorem extends some other results in the theory, see, for instance, [6, 24]. Theorem 5.3. Let Xbe a complete UC space with convex metric and K∈Pb,cl,cv(X). Suppose f:K→Kand T:K→Pcp,cv(K)satisfy condition (C). If fand Tcommute, then there exists z∈Ksuch that z=f(z)∈T(z). Proof. Using Theorem 2.6, it follows that Fix(f) is nonempty, closed and convex. Since the setting we work in is a UC space, the projection onto each compact and convex set is a singleton. By Lemma 5.2, PT(x)(x)∈T(x)∩Fix(f) for each x∈Fix(f) and so we can consider the mapping T(·)∩Fix(f) : Fix(f)→Pcp(Fix(f)). We show that this mapping satisfies condition (C). Let x, y ∈Fix(f), ux∈T(x)∩Fix(f) such that (1/2)d(x, ux)≤ d(x, y). Since Tfulfills (C), there exists vy∈T(y) such that d(ux, vy)≤d(x, y). Let uy stand for PT(y)(ux). According to Lemma 5.2, uy∈T(y)∩Fix(f). It is also clear that d(ux, uy)≤d(ux, vy)≤d(x, y). Thus, the mapping T(·)∩Fix(f) : Fix(f)→Pcp(Fix(f)) satisfies (C) which means, using Theorem 3.6, that there exists z∈Ksuch that z= f(z)∈T(z). Likewise, one can prove the following result in the framework of R-trees. Theorem 5.4. Let Xbe a bounded complete R-tree. Suppose f:X→Xand T:X→ Pcl,cv(X)satisfy conditions (C)and (C0)respectively. If fand Tcommute, then there exists z∈Ksuch that z=f(z)∈T(z). Proof. According to Theorem 2.6, it follows that Fix(f) is nonempty, closed and convex (and so also hyperconvex). This means that Fix(f) is in its own turn a complete R-tree. Since in an R-tree the projection onto each closed and convex set is a singleton we can apply Lemma 5.2 and so T(x)∩Fix(f)6=∅for each x∈Fix(f). Now consider the mapping T(·)∩Fix(f) : Fix(f)→Pcl,cv(Fix(f)). We show that this mapping satisfies condition (C0). Let x, y ∈Fix(f), ux∈T(x)∩Fix(f) such that d(x, ux) = dist(x, T(x)∩ Fix(f)) and (1/2)d(x, ux)≤d(x, y). Applying Lemma 5.2, PT(x)(x)∈T(x)∩Fix(f) which implies that dist(x, T (x)) = dist(x, T(x)∩Fix(f)), so d(x, ux) = dist(x, T(x)). Because Tsatisfies (C0), there exists vy∈T(y) such that d(ux, vy)≤d(x, y). Let uy stand for PT(y)(ux). According to Lemma 5.2, uy∈T(y)∩Fix(f). It is also clear that d(ux, uy)≤d(ux, vy)≤d(x, y). Thus, the mapping T(·)∩Fix(f) : Fix(f)→ Pcl,cv(Fix(f)) satisfies (C0) which means, using Corollary 3.11, that there exists z∈K such that z=f(z)∈T(z). 6 Acknowledgments We would like to thank Attapol Kaewkhao for noticing and letting us know about a mistake in a previous version of this paper. The research of the first two authors was partially supported by DGES, Grant MTM2009-10696-C02-01 and Junta de Andaluc´ıa, Grant FQM-127. The third author was supported by programs co-financed by the Sectoral Operational Programme Human Resources Development, Contract POS DRU 6/1.5/S/3 - “Doctoral studies: through 18 science towards society”. She would also like to express her appreciation to the Department of Mathematical Analysis and to the Institute of Mathematics of the University of Seville (IMUS) for their support. References [1] J.B. Baillon, Nonexpansive mapping and hyperconvex spaces, Contemp. Math. 72 (1988) 11-19. [2] M.R. Bridson, A. Haefliger, Metric spaces of non-positive curvature, Springer- Verlag, Berlin, 1999. [3] S. Dhompongsa, W. Inthakon, A. Kaewkhao, Edelstein’s method and fixed point theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 350 (2009) 12-17. [4] R. Esp´ınola, W.A. Kirk, Fixed point theorems in R-trees with applications to graph theory, Topology Appl. 153 (2006) 1046-1055. [5] R. Esp´ınola, A. Fern´andez-Le´on, B. Pi¸atek, Fixed points of single- and set-valued mappings in uniformly convex metric spaces with no metric convexity, Fixed Point Theory and Appl. 2010 (2010) Article ID 169837 16 pages. [6] R. Esp´ınola, N. Hussain, Common fixed points for multimaps in metric spaces, Fixed Point Theory and Appl., Volume 2010 (2010), Article ID 204981, 14 pages [7] J. Garc´ıa-Falset, E. Llorens-Fuster, T. Suzuki, Fixed point theory for a class of generalized nonexpansive mappings, doi:10.1016/j.jmaa.2010.08.069. [8] K. Goebel, On a fixed point theorem for multivalued nonexpansive mappings, Ann. Univ. Marie Curie-Sklodowska 29 (1975) 70-72. [9] K. Goebel, W.A. Kirk, Iteration processes for nonexpansive mappings, Contemp. Math. 21 (1983) 115-123. [10] K. Goebel, W.A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Univ. Press, 1990. [11] S. Hu, N. Papageorgiou, Handbook of Multivalued Analysis, vol. 1, Kluwer Academic Publishers, Dordrecht, 1997. [12] M.A. Khamsi, W.A. Kirk, C. Mart´ınez Y´a˜nez, Fixed point and selection theorems in hyperconvex spaces, Proc. Amer. Math. Soc. 128 (2000) 3275-3283. [13] W.A. Kirk, Nonexpansive mappings in product spaces, set-valued mappings and k-uniformly rotundity, (F. E. Browder, ed.) Amer. Math. Soc. Symp. Pure Math. 45 (1986) 51-64. [14] W.A. Kirk, Hyperconvexity of R-trees, Fund. Math. 156 (1998) 67-72. 19 [15] W.A. Kirk, Some recent results in metric fixed point theory, J. Fixed Point Theory Appl. 2 (2007), 195-207. [16] W.A. Kirk, S. Massa, Remarks on asymptotic and Chebyshev centers, Houston J. Math. 16 (1990) 357-364. [17] W.A. Kirk, B. Sims (eds.), Handbook of Metric Fixed Point Theory, Kluwer Academic Publishers, Dordrecht, 2001. [18] U. Kohlenbach, L. Leu¸stean, Asymptotically nonexpansive mappings in uniformly convex hyperbolic spaces, J. Eur. Math. Soc. 12 (2010) 71-92. [19] E. Lami Dozo, Multivalued nonexpansive mappings and Opial’s condition, Proc. Amer. Math. Soc. 38 (1973) 286-292. [20] T.C. Lim, A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space, Bull. Amer. Math. Soc. 80 (1974) 1123-1126. [21] J.T. Markin, Fixed points, selections and best approximation for multivalued mappings in R-trees, Nonlinear Anal. 67 (2007) 2712–2716. [22] B. Pi¸atek, Best approximation of coincidence points in metric trees, Ann. Univ. Mariae Curie-Sk lodowska Sect. A 62 (2008) 113–121. [23] A. Razani, H. Salahifard, Invariant approximation for CAT(0) spaces, Nonlinear Anal. 72 (2010) 2421-2425. [24] N. Shahzad, Fixed point results for multimaps in CAT(0) spaces, Topology Appl. 156 (2009), 997-1001. [25] R. Sine, Hyperconvexity and nonexpansive multifunctions, Trans. Amer. Math. Soc. 315 (1989) 755–767. [26] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008) 1088-1095. 20