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Common fixed point results in CAT(0) spaces
Nonlinear Analysis 74 (2011) 1835–1840
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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Common fixed point results in CAT(0) spaces Ali Abkar ∗ , Mohammad Eslamian Department of Mathematics, Imam Khomeini International University, Qazvin 34149, Iran
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Article history: Received 28 August 2010 Accepted 28 October 2010 MSC: 47H10 47H09
abstract Let X be a complete CAT(0) space, T be a generalized multivalued nonexpansive mapping, and t be a single valued quasi-nonexpansive mapping. Under the assumption that T and t commute weakly, we shall prove the existence of a common fixed point for them. In this way, we extend and improve a number of recent results obtained by Shahzad (2009) [7,12], Shahzad and Markin (2008) [6], and Dhompongsa et al. (2005) [5]. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Common fixed point Generalized nonexpansive mapping Multivalued mapping CAT(0) space Quasi-nonexpansive mapping
1. Introduction A metric space X is a CAT(0) space if it is geodesically connected, and if every geodesic triangle in X is at least as thin as its comparison triangle in the Euclidean plane. It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space. Other examples are pre-Hilbert spaces, R-trees (see [1]), the complex Hilbert ball with a hyperbolic metric (see [2]), and many others. For more information on these spaces and on the fundamental role they play in geometry we refer the reader to [1]. Fixed point theory in CAT(0) spaces was first studied by Kirk (see [3,4]). He proved that every nonexpansive (single valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then the fixed point theory for single valued and multivalued mappings in CAT(0) spaces has been rapidly developed. In 2005, Dhompongsa et al. [5] obtained a common fixed point result for commuting mappings in CAT(0) spaces. Shahzad and Markin [6] studied an invariant approximation problem and provided sufficient conditions for the existence of z ∈ K ⊂ X such that d(z , y) = dist(y, K ) and z = t (z ) ∈ T (z ) where y ∈ X , T and t are commuting nonexpansive mappings. Shahzad [7] also proved a common fixed point and invariant approximation result in a CAT(0) space in which t and T are not necessarily commuting. In 2008, Suzuki [8] introduced a condition which is weaker than nonexpansiveness and stronger than quasinonexpansiveness. Suzuki’s condition which was named by him the condition (C ) reads as follows: A mapping T on a subset K of a Banach space X is said to satisfy the condition (C ) if 1
‖x − Tx‖ ≤ ‖x − y‖ ⇒ ‖Tx − Ty‖ ≤ ‖x − y‖, x, y ∈ K . 2 He then proved some fixed point and convergence theorems for such mappings. Motivated by this result, Garcia-Falset et al. in [9] introduced two kinds of generalization for the condition (C ) and studied both the existence of fixed points and their ∗
Corresponding author. Tel.: +98 9123301709; fax: +98 281 3780040. E-mail addresses: [email protected], [email protected] (A. Abkar), [email protected] (M. Eslamian).
0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.10.056
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A. Abkar, M. Eslamian / Nonlinear Analysis 74 (2011) 1835–1840
asymptotic behavior. Very recently, the current authors used a modified Suzuki condition for multivalued mappings, and proved some fixed point theorems for multivalued mappings satisfying this condition in Banach spaces [10,11]. In this paper we consider a CAT(0) space X , together with two mappings t and T , where T belongs to the new class of generalized nonexpansive multivalued mappings (in the sense of Suzuki) and t is a single valued quasi-nonexpansive mapping. We shall establish a common fixed point for these two mappings. Our result improves a number of very recent results of Shahzad [7,12], as well as those of Shahzad and Markin [6], and of Dhompongsa et al. [5]. 2. Preliminaries Let (X , d) be a metric space. A geodesic path joining x ∈ X and y ∈ X is a map c from a closed interval [0, r ] ⊂ R to X such that c (0) = x, c (r ) = y and d(c (t ), c (s)) = |t − s| for all s, t ∈ [0, r ]. In particular, the mapping c is an isometry and d(x, y) = r. The image of c is called a geodesic segment joining x and y which when unique is denoted by [x, y]. For any x, y ∈ X , we denote the point z ∈ [x, y] such that d(x, z ) = α d(x, y) by z = (1 − α)x ⊕ α y, where 0 ≤ α ≤ 1. The space (X , d) is called a geodesic space if any two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X . A subset K of X is called convex if K includes every geodesic segment joining any two points of itself. A geodesic triangle △(x1 , x2 , x3 ) in a geodesic metric space (X , d) consists of three points in X (the vertices of △) and a geodesic segment between each pair of points (the edges of △). A comparison triangle for △(x1 , x2 , x3 ) in (X , d) is a triangle △(x1 , x2 , x3 ) := △(x1 , x2 , x3 ) in the Euclidean plane R2 such that dR2 (xi , xj ) = d(xi , xj ) for i, j ∈ {1, 2, 3}. A geodesic metric space X is called a CAT(0) space if all geodesic triangles of appropriate size satisfy the following comparison axiom: Let △ be a geodesic triangle in X and let △ be its comparison triangle in R2 . Then △ is said to satisfy the CAT(0) inequality if for all x, y ∈ △ and all comparison points x, y ∈ △, d(x, y) ≤ dR2 (x, y). The following properties of a CAT(0) space are useful (see [1]): (i) A CAT(0) space X is uniquely geodesic. (ii) For any x ∈ X and any closed convex subset K ⊂ X there is a unique closest point to x ∈ K . A notion of △-convergence in CAT(0) spaces based on the fact that in Hilbert spaces a bounded sequence is weakly convergent to its unique asymptotic center has been studied in [13]. Let (xn ) be a bounded sequence in X and K be a nonempty bounded subset of X . We associate this sequence with the number r = r (K , {xn }) = inf{r (x, {xn }) : x ∈ K }, where r (x, {xn }) = lim sup d(xn , x), n→∞
and the set A = A(K , {xn }) = {x ∈ K : r (x, {xn }) = r }. The number r is known as the asymptotic radius of {xn } relative to K . Similarly, the set A is called the asymptotic center of {xn } relative to K . In a CAT(0) space, the asymptotic center A = A(K , {xn }) of (xn ) consists of exactly one point whenever K is closed and convex. A sequence (xn ) in a CAT(0) space X is said to be △-convergent to x ∈ X if x is the unique asymptotic center of every subsequence of (xn ). Notice that given (xn ) ⊂ X such that (xn ) is △-convergent to x and given y ∈ X with x ̸= y, lim sup d(x, xn ) < lim sup d(y, xn ). n→∞
n→∞
Thus every CAT(0) space X satisfies the Opial property. Lemma 2.1 ([13]). Every bounded sequence in a complete CAT (0) space has a △-convergent subsequence. Lemma 2.2 ([14]). If K is a closed convex subset of a complete CAT (0) space and if (xn ) is a bounded sequence in K , then the asymptotic center of (xn ) is in K . Lemma 2.3 ([15]). Let (X , d) be a CAT (0) space. For x, y ∈ X and t ∈ [0, 1], there exists a unique point z ∈ [x, y] such that d(x, z ) = td(x, y) and
d(y, z ) = (1 − t )d(x, y).
We use the notation (1 − t )x ⊕ ty for the unique point z of the above lemma. Definition 2.4. A point x ∈ K is called a fixed point of T if Tx = x, we shall denote by F (T ) the set of all fixed points of T . Definition 2.5 ([8]). Let T be a mapping on a subset K of a CAT(0) space X . T is said to satisfy condition (C ) if 1 2
d(x, Tx) ≤ d(x, y) ⇒ d(Tx, Ty) ≤ d(x, y),
x, y ∈ K .
A. Abkar, M. Eslamian / Nonlinear Analysis 74 (2011) 1835–1840
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In [9], Garcia-Falset et al. introduced two generalizations of the condition (C ) in a Banach space. In the following, we state their conditions in the framework of a CAT(0) space. Definition 2.6. Let T be a mapping on a subset K of a CAT(0) space X and µ ≥ 1. T is said to satisfy condition (Eµ ) if d(x, Ty) ≤ µd(x, Tx) + d(x, y),
x, y ∈ K .
We say that T satisfies condition (E ) whenever T satisfies the condition (Eµ ) for some µ ≥ 1. Definition 2.7. Let T be a mapping on a subset K of a CAT(0) space X and λ ∈ (0, 1). T is said to satisfy condition (Cλ ) if
λd(x, Tx) ≤ d(x, y) ⇒ d(Tx, Ty) ≤ d(x, y),
x, y ∈ K .
Notice that if 0 < λ1 < λ2 < 1 then the condition (Cλ1 ) implies the condition (Cλ2 ). The following example shows that the class of mappings satisfying the conditions (E ) and (Cλ ) for some λ ∈ (0, 1) is broader than the class of mappings satisfying the condition (C ). Example ([9]). For a given λ ∈ (0, 1) define a mapping T on [0, 1] by T (x) =
x ,
2 1 + λ
2+λ
x ̸= 1
,
x = 1.
Then the mapping T satisfies condition (Cλ ) but it fails condition (Cλ′ ) whenever 0 < λ < λ′ . Moreover T satisfies condition (Eµ ) for µ = 2+λ . 2 Definition 2.8. A mapping T on a subset K of a CAT(0) space X is called quasi-nonexpansive if F (T ) ̸= ∅ and d(T (x), z ) ≤ d(x, z ) for all x ∈ K and z ∈ Fix(T ). Lemma 2.9 ([9]). Assume that a mapping T satisfies the condition (E ) and has a fixed point. Then T is a quasi-nonexpansive mapping. But the converse is not true. Theorem 2.10 ([16]). Let K be a nonempty bounded closed convex subset of a complete CAT (0) space X . Suppose T : K → K is a quasi-nonexpansive mapping. Then F (T ) is closed and convex. Let (X , d) be a geodesic metric space. We denote by CB(X ) the collection of all nonempty closed bounded subsets of X , we also write KC (X ) to denote the collection of all nonempty compact convex subset of X . Let H be the Hausdorff metric with respect to d, that is,
H (A, B) := max sup dist(x, B), sup dist(y, A) , x∈A
y∈B
for all A, B ∈ CB(X ) where dist(x, B) = infy∈B d(x, y). Let T : X → 2X be a multivalued mapping. An element x ∈ X is said to be a fixed point of T , if x ∈ Tx. Definition 2.11. A multivalued mapping T : X → CB(X ) is said to be nonexpansive provided that H (Tx, Ty) ≤ d(x, y),
x, y ∈ X .
Suzuki’s condition can be modified to incorporate multivalued mappings. This was done by the current authors in [10]. Definition 2.12. A multivalued mapping T : X → CB(X ) is said to satisfy condition (C ) provided that 1 2
dist(x, Tx) ≤ d(x, y) ⇒ H (Tx, Ty) ≤ d(x, y),
x, y ∈ X .
We now state the multivalued analogs of the conditions (E ) and (Cλ ) in the following manner: Definition 2.13. A multivalued mapping T : X → CB(X ) is said to satisfy condition (Eµ ) provided that dist(x, Ty) ≤ µdist(x, Tx) + d(x, y),
x, y ∈ X .
We say that T satisfies condition (E ) whenever T satisfies (Eµ ) for some µ ≥ 1.
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Definition 2.14. A multivalued mapping T : X → CB(X ) is said to satisfy condition (Cλ ) for some λ ∈ (0, 1) provided that
λ dist(x, Tx) ≤ d(x, y) ⇒ H (Tx, Ty) ≤ d(x, y),
x, y ∈ X .
Lemma 2.15. Let T : X → CB(X ) be a multivalued nonexpansive mapping, then T satisfies the condition (E1 ). We now provide an example of a generalized nonexpansive multivalued mapping satisfying the conditions (Cλ ) and (E ) which is not a nonexpansive multivalued mapping. Example. Define a mapping T on [0, 5] by
T (x) =
0,
{1}
x 5
,
x ̸= 5 x = 5.
Let x, y ∈ [0, 5). It is easy to see that
x − y ≤ ‖x − y‖. 5
H (Tx, Ty) =
If x ∈ [0, 4] and y = 5, then H (Tx, Ty) = 1 ≤ 5 − x = ‖x − y‖. In case that x ∈ (4, 5) and y = 5, we have dist(x, Tx) = Therefore 1 2
dist(x, Tx) =
4x 10
>
16 10
4x . 5
> 1 > ‖x − y‖.
Moreover 1 2
dist(y, Ty) = 2 > 1 > ‖x − y‖.
These inequalities show that the mapping T satisfies the condition (Cλ ) for λ = 12 . It is not difficult to see that T satisfies the condition (E ). Now we shall see that the mapping T is not nonexpansive. To see this we take x = 29 and y = 5. Then we have H (Tx, Ty) = 1 >
1 2
= ‖x − y‖.
The following lemma is a consequence of Proposition 2 proved by Goebel and Kirk [17]. Lemma 2.16. Let {zn } and {wn } be two bounded sequences in a CAT (0) space X , and let 0 < λ < 1. If for every natural number n we have zn+1 = λwn ⊕ (1 − λ)zn and d(wn+1 , wn ) ≤ d(zn+1 , zn ), then limn→∞ d(wn , zn ) = 0. 3. A common fixed point Definition 3.1. Let K be a nonempty subset of a CAT(0) space X , t : K → K , and T : K → 2K are said to commute weakly if t (∂K T (x)) ⊂ T (t (x)) for all x ∈ K , where ∂X Y denotes the relative boundary of Y ⊂ X . Theorem 3.2. Let K be a nonempty closed convex bounded subset of a complete CAT (0) space X . Suppose T : K → K satisfies the conditions (E ) and (Cλ ) for some λ ∈ (0, 1). Then T has a fixed point in K . Proof. Define a sequence {xn } ∈ K by x1 ∈ K and xn+1 = λT (xn ) ⊕ (1 − λ)xn for all n ∈ N. Then we have
λd(xn , T (xn )) = d(xn , xn+1 ) for all n ∈ N. By the condition (Cλ ), we have d(T (xn ), T (xn+1 )) ≤ d(xn , xn+1 ). Now we can apply Lemma 2.16 to conclude that limn→∞ d(xn , T (xn )) = 0. Let A({xn }) = {w}, by Lemma 2.2 we have w ∈ K . Since T satisfies the condition (E ) we have d(xn , T (w)) ≤ µd(xn , T (xn )) + d(xn , w) for some µ ≥ 1. Taking limit superior on both sides in the above inequality, we obtain lim sup d(xn , T (w)) ≤ lim sup d(xn , w). n→∞
n→∞
By the uniqueness of the asymptotic center, we obtain T (w) = w .
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By using Theorem 3.2 along with Lemma 2.9 and Theorem 2.10 we obtain the following corollary. Corollary 3.3. Let K be a nonempty closed convex bounded subset of a complete CAT (0) space X . Suppose T : K → K satisfies the conditions (E ) and (Cλ ) for some λ ∈ (0, 1). Then F (T ) is a nonempty closed and convex set. Now the time is ripe to state the main result of this section. Theorem 3.4. Let K be a nonempty closed convex bounded subset of a complete CAT (0) space X . Let t : K → K be a quasinonexpansive single valued mapping, and let T : K → KC (K ) be a multivalued mapping satisfying the conditions (E ) and (Cλ ) for some λ ∈ (0, 1). If t and T commute weakly, then they have a common fixed point, i.e. there exists a point z ∈ K such that z = t (z ) ∈ T (z ). Proof. According to Theorem 2.10, it follows that Fix(t ) is a nonempty closed convex subset of X . We show that for x ∈ Fix(t ), T (x)∩ Fix(t ) ̸= ∅. To see this, let x ∈ Fix(t ), since t and T commute weakly, we have t (∂K T (x)) ⊂ T (t (x)) = T (x). Let y ∈ ∂K T (x) be the unique closest point to x. Since t is quasi-nonexpansive, we have d(t (y), x) ≤ d(y, x). Now by the uniqueness of y as the closest point to x, we get t (y) = y. Therefore T (x) ∩ Fix(t ) ̸= ∅ for x ∈ Fix(t ). Now we find an approximate fixed point sequence in Fix(t ) for T . Take x0 ∈ Fix(t ), since T (x0 ) ∩ Fix(t ) ̸= ∅, we can choose y0 ∈ T (x0 ) ∩ Fix(t ). Define x1 = (1 − λ)x0 ⊕ λy0 . Since Fix(t ) is convex, we have x1 ∈ Fix(t ). Let y1 ∈ ∂K T (x1 ) be chosen in such a way that d(y0 , y1 ) = dist(y0 , T (x1 )). We see that y1 ∈ Fix(t ). Since t is quasi-nonexpansive, we have d(ty1 , y0 ) ≤ d(y1 , y0 ) and hence by the uniqueness of y1 as the unique closest point to y0 we get t (y1 ) = y1 (note that ty1 ∈ t (∂K T (x1 )) ⊂ T (t (x1 )) = T (x1 )). Similarly, put x2 = (1 − λ)x1 ⊕ λy1 , again we choose y2 ∈ ∂K T (x2 ) in such a way that d(y1 , y2 ) = dist(y1 , T (x2 )). By the same argument, we get y2 ∈ Fix(t ). In this way we will find a sequence {xn } in Fix(t ) such that xn+1 = (1 − λ)xn ⊕ λyn where yn ∈ T (xn ) ∩ Fix(t ) and d(yn−1 , yn ) = dist(yn−1 , T (xn )). Therefore for every natural number n ≥ 1 we have
λ d(xn , yn ) = d(xn , xn+1 ) from which it follows that
λ dist(xn , T (xn )) ≤ λd(xn , yn ) = d(xn , xn+1 ),
n ≥ 1.
Since T satisfies the condition (Cλ ) we have H (T (xn ), T (xn+1 )) ≤ d(xn , xn+1 ),
n ≥ 1,
hence for each n ≥ 1 we have d(yn , yn+1 ) = dist(yn , T (xn+1 )) ≤ H (T (xn ), T (xn+1 )) ≤ d(xn , xn+1 ). We now apply Lemma 2.16 to conclude that limn→∞ d(xn , yn ) = 0 where yn ∈ T (xn ). By Lemma 2.1, the bounded sequence (xn ) in Fix(t ) has a △-convergent subsequence, hence by passing to a subsequence we can assume that △ − limn xn = v . We note that by Lemma 2.2, v ∈ Fix(t ). For each n ≥ 1, we choose zn ∈ ∂K T (v) such that d(xn , zn ) = dist(xn , T (v)). Moreover zn ∈ Fix(t ) for all natural numbers n ≥ 1. Indeed, since t and T commute weakly and zn ∈ ∂K T (v), we have t (∂K T (v)) ⊂ T (t (v)) = T (v). Hence t (zn ) ∈ T (v). Since t is quasi-nonexpansive, we have d(t (zn ), xn ) ≤ d(xn , zn ) and hence by the uniqueness of the closest point xn , we get t (zn ) = zn ∈ Fix(t ). Since T (v) is compact, the sequence {zn } has a convergent subsequence {znk } with limk→∞ znk = w ∈ T (v). Because znk ∈ Fix(t ) for all nk , and Fix(t ) is closed, we obtain that w ∈ Fix(t ). By the condition (E ), we have for some µ ≥ 1, dist(xnk , T (v)) ≤ µ dist(xnk , T (xnk )) + d(xnk , v). Note that d(xnk , w) ≤ d(xnk , znk ) + d(znk , w) ≤ µ dist(xnk , T (xnk )) + d(xnk , v) + d(znk , w).
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These entail lim sup d(xnk , w) ≤ lim sup d(xnk , v). k→∞
k→∞
From the Opial property of CAT(0) space X , we have that v = w ∈ T (v). Consequently v = t v ∈ T (v).
As a corollary we obtain the following result of Shahzad [7]. Corollary 3.5. Let K be a nonempty closed convex bounded subset of a complete CAT (0) space X . Let t : K → K be a quasinonexpansive single valued mapping, and let T : K → KC (K ) be a nonexpansive multivalued mapping. If t and T commute weakly, then they have a common fixed point, i.e. there exists a point z ∈ K such that z = t (z ) ∈ T (z ). Proof. By Lemma 2.16 the mapping T satisfies the condition E1 . We also note that T satisfies the condition (Cλ ) for all λ ∈ (0, 1). So the result follows from Theorem 3.4. Corollary 3.6. Let K be a nonempty closed convex bounded subset of a complete CAT (0) space X , t : K → K , and T : K → KC (K ) a single valued and a multivalued mapping, both satisfying the conditions (E ) and (Cλ ) for some λ ∈ (0, 1). Assume that t , T commute weakly, then there exists a point z ∈ K such that z = t (z ) ∈ T (z ). Proof. By Corollary 3.3, Fix(t ) is nonempty, closed and convex. So the result follows from Theorem 3.4.
Corollary 3.7. Let K be a nonempty bounded closed convex subset of a complete CAT (0) space X , and let T : K → KC (K ) be a multivalued mapping satisfying the conditions (E ) and (Cλ ) for some λ ∈ (0, 1). Then T has a fixed point. Let K be a nonempty subset of a CAT(0) space X . A point z ∈ X is said to be a center for the mapping t : K → X if for each x ∈ K , d(z , t (x)) ≤ d(z , x). The set of all centers of the mapping t is denoted by Z (t ). Theorem 3.8. Let X be a complete bounded CAT (0) space, and assume that t : X → X is a mapping, and T : X → KC (X ) is a multivalued mapping satisfying the conditions (E ) and (Cλ ) for some λ ∈ (0, 1). Assume that t , T commute weakly, and ∅ ̸= T (x) ∩ Fix(t ) ⊂ Z (t ) and that Fix(t ) is closed and convex, then there exists a point z ∈ X such that z = t (z ) ∈ T (z ). Proof. The proof essentially goes along the same lines as in the proof of Theorem 3.4.
References [1] M. Bridson, A. Haefliger, Metric Spaces of Nonpositive Curvature, Springer-Verlag, Berlin, 1999. [2] K. Goebel, S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Marcel Dekker, Inc., New York, 1984. [3] W.A. Kirk, Geodesic geometry and fixed point theory. II, in: International Conference on Fixed Point Theory and Applications, Yokohama Publ., Yokohama, 2004, pp. 113–142. [4] W.A. Kirk, Geodesic geometry and fixed point theory, in: Seminar of Mathematical Analysis, Malaga, Seville, 2002–2003, in: Colec. Abierta, vol. 64, Univ. Sevilla Secr. Publ., Seville, 2003, pp. 195–225. [5] S. Dhompongsa, A. Kaewkhao, B. Panyanak, Lim’s theorem for multivalued mappings in CAT(0) spaces, J. Math. Anal. Appl. 312 (2005) 478–487. [6] N. Shahzad, J. Markin, Invariant approximation for commuting mappings in hyperconvex and CAT(0) spaces, J. Math. Anal. Appl. 337 (2008) 1457–1464. [7] N. Shahzad, Fixed point results for multimaps in CAT(0) spaces, Topology Appl. 156 (2009) 997–1001. [8] T. Suzuki, Fixed point theorems and convergence theorems for some generalized nonexpansive mappings, J. Math. Anal. Appl. 340 (2008) 1088–1095. [9] J. Garcia-Falset, E. Llorens-Fuster, T. Suzuki, Fixed point theory for a class of generalized nonexpansive mappings, J. Math. Anal. Appl. 375 (2011) 185–195. [10] A. Abkar, M. Eslamian, A fixed point theorem for generalized nonexpansive multivalued mappings, Fixed Point Theory (in press). [11] A. Abkar, M. Eslamian, Fixed point theorems for Suzuki generalized nonexpansive multivalued mappings in Banach spaces, Fixed Point Theory Appl. 2010 (2010) Article ID 457935, 10 pp. [12] N. Shahzad, Invariant approximations in CAT(0) spaces, Nonlinear Anal. 70 (2009) 4338–4340. [13] W.A. Kirk, B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (2008) 3689–3696. [14] S. Dhompongsa, W.A. Kirk, B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear Convex Anal. 8 (2007) 35–45. [15] S. Dhompongsa, B. Panyanak, On △-convergence theorems in CAT(0) space, Comput. Math. Appl. 56 (2008) 2572–2579. [16] P. Chaoha, A. Phon-on, A note on fixed point sets in CAT(0) spaces, J. Math. Anal. Appl. 320 (2006) 983–987. [17] K. Goebel, W.A. Kirk, Iteration processes for nonexpansive mapping, Contemp. Math. 21 (1983) 115–123.
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Nonlinear Analysis journal homepage: www.elsevier.com/locate/na
Common fixed point results in CAT(0) spaces Ali Abkar ∗ , Mohammad Eslamian Department of Mathematics, Imam Khomeini International University, Qazvin 34149, Iran
article
info
Article history: Received 28 August 2010 Accepted 28 October 2010 MSC: 47H10 47H09
abstract Let X be a complete CAT(0) space, T be a generalized multivalued nonexpansive mapping, and t be a single valued quasi-nonexpansive mapping. Under the assumption that T and t commute weakly, we shall prove the existence of a common fixed point for them. In this way, we extend and improve a number of recent results obtained by Shahzad (2009) [7,12], Shahzad and Markin (2008) [6], and Dhompongsa et al. (2005) [5]. © 2010 Elsevier Ltd. All rights reserved.
Keywords: Common fixed point Generalized nonexpansive mapping Multivalued mapping CAT(0) space Quasi-nonexpansive mapping
1. Introduction A metric space X is a CAT(0) space if it is geodesically connected, and if every geodesic triangle in X is at least as thin as its comparison triangle in the Euclidean plane. It is well known that any complete, simply connected Riemannian manifold having nonpositive sectional curvature is a CAT(0) space. Other examples are pre-Hilbert spaces, R-trees (see [1]), the complex Hilbert ball with a hyperbolic metric (see [2]), and many others. For more information on these spaces and on the fundamental role they play in geometry we refer the reader to [1]. Fixed point theory in CAT(0) spaces was first studied by Kirk (see [3,4]). He proved that every nonexpansive (single valued) mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then the fixed point theory for single valued and multivalued mappings in CAT(0) spaces has been rapidly developed. In 2005, Dhompongsa et al. [5] obtained a common fixed point result for commuting mappings in CAT(0) spaces. Shahzad and Markin [6] studied an invariant approximation problem and provided sufficient conditions for the existence of z ∈ K ⊂ X such that d(z , y) = dist(y, K ) and z = t (z ) ∈ T (z ) where y ∈ X , T and t are commuting nonexpansive mappings. Shahzad [7] also proved a common fixed point and invariant approximation result in a CAT(0) space in which t and T are not necessarily commuting. In 2008, Suzuki [8] introduced a condition which is weaker than nonexpansiveness and stronger than quasinonexpansiveness. Suzuki’s condition which was named by him the condition (C ) reads as follows: A mapping T on a subset K of a Banach space X is said to satisfy the condition (C ) if 1
‖x − Tx‖ ≤ ‖x − y‖ ⇒ ‖Tx − Ty‖ ≤ ‖x − y‖, x, y ∈ K . 2 He then proved some fixed point and convergence theorems for such mappings. Motivated by this result, Garcia-Falset et al. in [9] introduced two kinds of generalization for the condition (C ) and studied both the existence of fixed points and their ∗
Corresponding author. Tel.: +98 9123301709; fax: +98 281 3780040. E-mail addresses: [email protected], [email protected] (A. Abkar), [email protected] (M. Eslamian).
0362-546X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.na.2010.10.056
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asymptotic behavior. Very recently, the current authors used a modified Suzuki condition for multivalued mappings, and proved some fixed point theorems for multivalued mappings satisfying this condition in Banach spaces [10,11]. In this paper we consider a CAT(0) space X , together with two mappings t and T , where T belongs to the new class of generalized nonexpansive multivalued mappings (in the sense of Suzuki) and t is a single valued quasi-nonexpansive mapping. We shall establish a common fixed point for these two mappings. Our result improves a number of very recent results of Shahzad [7,12], as well as those of Shahzad and Markin [6], and of Dhompongsa et al. [5]. 2. Preliminaries Let (X , d) be a metric space. A geodesic path joining x ∈ X and y ∈ X is a map c from a closed interval [0, r ] ⊂ R to X such that c (0) = x, c (r ) = y and d(c (t ), c (s)) = |t − s| for all s, t ∈ [0, r ]. In particular, the mapping c is an isometry and d(x, y) = r. The image of c is called a geodesic segment joining x and y which when unique is denoted by [x, y]. For any x, y ∈ X , we denote the point z ∈ [x, y] such that d(x, z ) = α d(x, y) by z = (1 − α)x ⊕ α y, where 0 ≤ α ≤ 1. The space (X , d) is called a geodesic space if any two points of X are joined by a geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X . A subset K of X is called convex if K includes every geodesic segment joining any two points of itself. A geodesic triangle △(x1 , x2 , x3 ) in a geodesic metric space (X , d) consists of three points in X (the vertices of △) and a geodesic segment between each pair of points (the edges of △). A comparison triangle for △(x1 , x2 , x3 ) in (X , d) is a triangle △(x1 , x2 , x3 ) := △(x1 , x2 , x3 ) in the Euclidean plane R2 such that dR2 (xi , xj ) = d(xi , xj ) for i, j ∈ {1, 2, 3}. A geodesic metric space X is called a CAT(0) space if all geodesic triangles of appropriate size satisfy the following comparison axiom: Let △ be a geodesic triangle in X and let △ be its comparison triangle in R2 . Then △ is said to satisfy the CAT(0) inequality if for all x, y ∈ △ and all comparison points x, y ∈ △, d(x, y) ≤ dR2 (x, y). The following properties of a CAT(0) space are useful (see [1]): (i) A CAT(0) space X is uniquely geodesic. (ii) For any x ∈ X and any closed convex subset K ⊂ X there is a unique closest point to x ∈ K . A notion of △-convergence in CAT(0) spaces based on the fact that in Hilbert spaces a bounded sequence is weakly convergent to its unique asymptotic center has been studied in [13]. Let (xn ) be a bounded sequence in X and K be a nonempty bounded subset of X . We associate this sequence with the number r = r (K , {xn }) = inf{r (x, {xn }) : x ∈ K }, where r (x, {xn }) = lim sup d(xn , x), n→∞
and the set A = A(K , {xn }) = {x ∈ K : r (x, {xn }) = r }. The number r is known as the asymptotic radius of {xn } relative to K . Similarly, the set A is called the asymptotic center of {xn } relative to K . In a CAT(0) space, the asymptotic center A = A(K , {xn }) of (xn ) consists of exactly one point whenever K is closed and convex. A sequence (xn ) in a CAT(0) space X is said to be △-convergent to x ∈ X if x is the unique asymptotic center of every subsequence of (xn ). Notice that given (xn ) ⊂ X such that (xn ) is △-convergent to x and given y ∈ X with x ̸= y, lim sup d(x, xn ) < lim sup d(y, xn ). n→∞
n→∞
Thus every CAT(0) space X satisfies the Opial property. Lemma 2.1 ([13]). Every bounded sequence in a complete CAT (0) space has a △-convergent subsequence. Lemma 2.2 ([14]). If K is a closed convex subset of a complete CAT (0) space and if (xn ) is a bounded sequence in K , then the asymptotic center of (xn ) is in K . Lemma 2.3 ([15]). Let (X , d) be a CAT (0) space. For x, y ∈ X and t ∈ [0, 1], there exists a unique point z ∈ [x, y] such that d(x, z ) = td(x, y) and
d(y, z ) = (1 − t )d(x, y).
We use the notation (1 − t )x ⊕ ty for the unique point z of the above lemma. Definition 2.4. A point x ∈ K is called a fixed point of T if Tx = x, we shall denote by F (T ) the set of all fixed points of T . Definition 2.5 ([8]). Let T be a mapping on a subset K of a CAT(0) space X . T is said to satisfy condition (C ) if 1 2
d(x, Tx) ≤ d(x, y) ⇒ d(Tx, Ty) ≤ d(x, y),
x, y ∈ K .
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In [9], Garcia-Falset et al. introduced two generalizations of the condition (C ) in a Banach space. In the following, we state their conditions in the framework of a CAT(0) space. Definition 2.6. Let T be a mapping on a subset K of a CAT(0) space X and µ ≥ 1. T is said to satisfy condition (Eµ ) if d(x, Ty) ≤ µd(x, Tx) + d(x, y),
x, y ∈ K .
We say that T satisfies condition (E ) whenever T satisfies the condition (Eµ ) for some µ ≥ 1. Definition 2.7. Let T be a mapping on a subset K of a CAT(0) space X and λ ∈ (0, 1). T is said to satisfy condition (Cλ ) if
λd(x, Tx) ≤ d(x, y) ⇒ d(Tx, Ty) ≤ d(x, y),
x, y ∈ K .
Notice that if 0 < λ1 < λ2 < 1 then the condition (Cλ1 ) implies the condition (Cλ2 ). The following example shows that the class of mappings satisfying the conditions (E ) and (Cλ ) for some λ ∈ (0, 1) is broader than the class of mappings satisfying the condition (C ). Example ([9]). For a given λ ∈ (0, 1) define a mapping T on [0, 1] by T (x) =
x ,
2 1 + λ
2+λ
x ̸= 1
,
x = 1.
Then the mapping T satisfies condition (Cλ ) but it fails condition (Cλ′ ) whenever 0 < λ < λ′ . Moreover T satisfies condition (Eµ ) for µ = 2+λ . 2 Definition 2.8. A mapping T on a subset K of a CAT(0) space X is called quasi-nonexpansive if F (T ) ̸= ∅ and d(T (x), z ) ≤ d(x, z ) for all x ∈ K and z ∈ Fix(T ). Lemma 2.9 ([9]). Assume that a mapping T satisfies the condition (E ) and has a fixed point. Then T is a quasi-nonexpansive mapping. But the converse is not true. Theorem 2.10 ([16]). Let K be a nonempty bounded closed convex subset of a complete CAT (0) space X . Suppose T : K → K is a quasi-nonexpansive mapping. Then F (T ) is closed and convex. Let (X , d) be a geodesic metric space. We denote by CB(X ) the collection of all nonempty closed bounded subsets of X , we also write KC (X ) to denote the collection of all nonempty compact convex subset of X . Let H be the Hausdorff metric with respect to d, that is,
H (A, B) := max sup dist(x, B), sup dist(y, A) , x∈A
y∈B
for all A, B ∈ CB(X ) where dist(x, B) = infy∈B d(x, y). Let T : X → 2X be a multivalued mapping. An element x ∈ X is said to be a fixed point of T , if x ∈ Tx. Definition 2.11. A multivalued mapping T : X → CB(X ) is said to be nonexpansive provided that H (Tx, Ty) ≤ d(x, y),
x, y ∈ X .
Suzuki’s condition can be modified to incorporate multivalued mappings. This was done by the current authors in [10]. Definition 2.12. A multivalued mapping T : X → CB(X ) is said to satisfy condition (C ) provided that 1 2
dist(x, Tx) ≤ d(x, y) ⇒ H (Tx, Ty) ≤ d(x, y),
x, y ∈ X .
We now state the multivalued analogs of the conditions (E ) and (Cλ ) in the following manner: Definition 2.13. A multivalued mapping T : X → CB(X ) is said to satisfy condition (Eµ ) provided that dist(x, Ty) ≤ µdist(x, Tx) + d(x, y),
x, y ∈ X .
We say that T satisfies condition (E ) whenever T satisfies (Eµ ) for some µ ≥ 1.
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Definition 2.14. A multivalued mapping T : X → CB(X ) is said to satisfy condition (Cλ ) for some λ ∈ (0, 1) provided that
λ dist(x, Tx) ≤ d(x, y) ⇒ H (Tx, Ty) ≤ d(x, y),
x, y ∈ X .
Lemma 2.15. Let T : X → CB(X ) be a multivalued nonexpansive mapping, then T satisfies the condition (E1 ). We now provide an example of a generalized nonexpansive multivalued mapping satisfying the conditions (Cλ ) and (E ) which is not a nonexpansive multivalued mapping. Example. Define a mapping T on [0, 5] by
T (x) =
0,
{1}
x 5
,
x ̸= 5 x = 5.
Let x, y ∈ [0, 5). It is easy to see that
x − y ≤ ‖x − y‖. 5
H (Tx, Ty) =
If x ∈ [0, 4] and y = 5, then H (Tx, Ty) = 1 ≤ 5 − x = ‖x − y‖. In case that x ∈ (4, 5) and y = 5, we have dist(x, Tx) = Therefore 1 2
dist(x, Tx) =
4x 10
>
16 10
4x . 5
> 1 > ‖x − y‖.
Moreover 1 2
dist(y, Ty) = 2 > 1 > ‖x − y‖.
These inequalities show that the mapping T satisfies the condition (Cλ ) for λ = 12 . It is not difficult to see that T satisfies the condition (E ). Now we shall see that the mapping T is not nonexpansive. To see this we take x = 29 and y = 5. Then we have H (Tx, Ty) = 1 >
1 2
= ‖x − y‖.
The following lemma is a consequence of Proposition 2 proved by Goebel and Kirk [17]. Lemma 2.16. Let {zn } and {wn } be two bounded sequences in a CAT (0) space X , and let 0 < λ < 1. If for every natural number n we have zn+1 = λwn ⊕ (1 − λ)zn and d(wn+1 , wn ) ≤ d(zn+1 , zn ), then limn→∞ d(wn , zn ) = 0. 3. A common fixed point Definition 3.1. Let K be a nonempty subset of a CAT(0) space X , t : K → K , and T : K → 2K are said to commute weakly if t (∂K T (x)) ⊂ T (t (x)) for all x ∈ K , where ∂X Y denotes the relative boundary of Y ⊂ X . Theorem 3.2. Let K be a nonempty closed convex bounded subset of a complete CAT (0) space X . Suppose T : K → K satisfies the conditions (E ) and (Cλ ) for some λ ∈ (0, 1). Then T has a fixed point in K . Proof. Define a sequence {xn } ∈ K by x1 ∈ K and xn+1 = λT (xn ) ⊕ (1 − λ)xn for all n ∈ N. Then we have
λd(xn , T (xn )) = d(xn , xn+1 ) for all n ∈ N. By the condition (Cλ ), we have d(T (xn ), T (xn+1 )) ≤ d(xn , xn+1 ). Now we can apply Lemma 2.16 to conclude that limn→∞ d(xn , T (xn )) = 0. Let A({xn }) = {w}, by Lemma 2.2 we have w ∈ K . Since T satisfies the condition (E ) we have d(xn , T (w)) ≤ µd(xn , T (xn )) + d(xn , w) for some µ ≥ 1. Taking limit superior on both sides in the above inequality, we obtain lim sup d(xn , T (w)) ≤ lim sup d(xn , w). n→∞
n→∞
By the uniqueness of the asymptotic center, we obtain T (w) = w .
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By using Theorem 3.2 along with Lemma 2.9 and Theorem 2.10 we obtain the following corollary. Corollary 3.3. Let K be a nonempty closed convex bounded subset of a complete CAT (0) space X . Suppose T : K → K satisfies the conditions (E ) and (Cλ ) for some λ ∈ (0, 1). Then F (T ) is a nonempty closed and convex set. Now the time is ripe to state the main result of this section. Theorem 3.4. Let K be a nonempty closed convex bounded subset of a complete CAT (0) space X . Let t : K → K be a quasinonexpansive single valued mapping, and let T : K → KC (K ) be a multivalued mapping satisfying the conditions (E ) and (Cλ ) for some λ ∈ (0, 1). If t and T commute weakly, then they have a common fixed point, i.e. there exists a point z ∈ K such that z = t (z ) ∈ T (z ). Proof. According to Theorem 2.10, it follows that Fix(t ) is a nonempty closed convex subset of X . We show that for x ∈ Fix(t ), T (x)∩ Fix(t ) ̸= ∅. To see this, let x ∈ Fix(t ), since t and T commute weakly, we have t (∂K T (x)) ⊂ T (t (x)) = T (x). Let y ∈ ∂K T (x) be the unique closest point to x. Since t is quasi-nonexpansive, we have d(t (y), x) ≤ d(y, x). Now by the uniqueness of y as the closest point to x, we get t (y) = y. Therefore T (x) ∩ Fix(t ) ̸= ∅ for x ∈ Fix(t ). Now we find an approximate fixed point sequence in Fix(t ) for T . Take x0 ∈ Fix(t ), since T (x0 ) ∩ Fix(t ) ̸= ∅, we can choose y0 ∈ T (x0 ) ∩ Fix(t ). Define x1 = (1 − λ)x0 ⊕ λy0 . Since Fix(t ) is convex, we have x1 ∈ Fix(t ). Let y1 ∈ ∂K T (x1 ) be chosen in such a way that d(y0 , y1 ) = dist(y0 , T (x1 )). We see that y1 ∈ Fix(t ). Since t is quasi-nonexpansive, we have d(ty1 , y0 ) ≤ d(y1 , y0 ) and hence by the uniqueness of y1 as the unique closest point to y0 we get t (y1 ) = y1 (note that ty1 ∈ t (∂K T (x1 )) ⊂ T (t (x1 )) = T (x1 )). Similarly, put x2 = (1 − λ)x1 ⊕ λy1 , again we choose y2 ∈ ∂K T (x2 ) in such a way that d(y1 , y2 ) = dist(y1 , T (x2 )). By the same argument, we get y2 ∈ Fix(t ). In this way we will find a sequence {xn } in Fix(t ) such that xn+1 = (1 − λ)xn ⊕ λyn where yn ∈ T (xn ) ∩ Fix(t ) and d(yn−1 , yn ) = dist(yn−1 , T (xn )). Therefore for every natural number n ≥ 1 we have
λ d(xn , yn ) = d(xn , xn+1 ) from which it follows that
λ dist(xn , T (xn )) ≤ λd(xn , yn ) = d(xn , xn+1 ),
n ≥ 1.
Since T satisfies the condition (Cλ ) we have H (T (xn ), T (xn+1 )) ≤ d(xn , xn+1 ),
n ≥ 1,
hence for each n ≥ 1 we have d(yn , yn+1 ) = dist(yn , T (xn+1 )) ≤ H (T (xn ), T (xn+1 )) ≤ d(xn , xn+1 ). We now apply Lemma 2.16 to conclude that limn→∞ d(xn , yn ) = 0 where yn ∈ T (xn ). By Lemma 2.1, the bounded sequence (xn ) in Fix(t ) has a △-convergent subsequence, hence by passing to a subsequence we can assume that △ − limn xn = v . We note that by Lemma 2.2, v ∈ Fix(t ). For each n ≥ 1, we choose zn ∈ ∂K T (v) such that d(xn , zn ) = dist(xn , T (v)). Moreover zn ∈ Fix(t ) for all natural numbers n ≥ 1. Indeed, since t and T commute weakly and zn ∈ ∂K T (v), we have t (∂K T (v)) ⊂ T (t (v)) = T (v). Hence t (zn ) ∈ T (v). Since t is quasi-nonexpansive, we have d(t (zn ), xn ) ≤ d(xn , zn ) and hence by the uniqueness of the closest point xn , we get t (zn ) = zn ∈ Fix(t ). Since T (v) is compact, the sequence {zn } has a convergent subsequence {znk } with limk→∞ znk = w ∈ T (v). Because znk ∈ Fix(t ) for all nk , and Fix(t ) is closed, we obtain that w ∈ Fix(t ). By the condition (E ), we have for some µ ≥ 1, dist(xnk , T (v)) ≤ µ dist(xnk , T (xnk )) + d(xnk , v). Note that d(xnk , w) ≤ d(xnk , znk ) + d(znk , w) ≤ µ dist(xnk , T (xnk )) + d(xnk , v) + d(znk , w).
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These entail lim sup d(xnk , w) ≤ lim sup d(xnk , v). k→∞
k→∞
From the Opial property of CAT(0) space X , we have that v = w ∈ T (v). Consequently v = t v ∈ T (v).
As a corollary we obtain the following result of Shahzad [7]. Corollary 3.5. Let K be a nonempty closed convex bounded subset of a complete CAT (0) space X . Let t : K → K be a quasinonexpansive single valued mapping, and let T : K → KC (K ) be a nonexpansive multivalued mapping. If t and T commute weakly, then they have a common fixed point, i.e. there exists a point z ∈ K such that z = t (z ) ∈ T (z ). Proof. By Lemma 2.16 the mapping T satisfies the condition E1 . We also note that T satisfies the condition (Cλ ) for all λ ∈ (0, 1). So the result follows from Theorem 3.4. Corollary 3.6. Let K be a nonempty closed convex bounded subset of a complete CAT (0) space X , t : K → K , and T : K → KC (K ) a single valued and a multivalued mapping, both satisfying the conditions (E ) and (Cλ ) for some λ ∈ (0, 1). Assume that t , T commute weakly, then there exists a point z ∈ K such that z = t (z ) ∈ T (z ). Proof. By Corollary 3.3, Fix(t ) is nonempty, closed and convex. So the result follows from Theorem 3.4.
Corollary 3.7. Let K be a nonempty bounded closed convex subset of a complete CAT (0) space X , and let T : K → KC (K ) be a multivalued mapping satisfying the conditions (E ) and (Cλ ) for some λ ∈ (0, 1). Then T has a fixed point. Let K be a nonempty subset of a CAT(0) space X . A point z ∈ X is said to be a center for the mapping t : K → X if for each x ∈ K , d(z , t (x)) ≤ d(z , x). The set of all centers of the mapping t is denoted by Z (t ). Theorem 3.8. Let X be a complete bounded CAT (0) space, and assume that t : X → X is a mapping, and T : X → KC (X ) is a multivalued mapping satisfying the conditions (E ) and (Cλ ) for some λ ∈ (0, 1). Assume that t , T commute weakly, and ∅ ̸= T (x) ∩ Fix(t ) ⊂ Z (t ) and that Fix(t ) is closed and convex, then there exists a point z ∈ X such that z = t (z ) ∈ T (z ). Proof. The proof essentially goes along the same lines as in the proof of Theorem 3.4.
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