RESEARC H Open Access Fixed point theorems and Δ-convergence theorems for generalized hybrid mappings on CAT(0) spaces Lai-Jiu Lin 1* , Chih-Sheng Chuang 1 and Zenn-Tsun Yu 2 * Correspondence: maljlin@cc.ncue. edu.tw 1 Department of Mathematics, National Changhua University of Education, Changhua, 50058, Taiwan Full list of author information is available at the end of the article Abstract In this paper, we introduce generalized hybrid mapping on CAT(0) spaces. The class of generalized hybrid mappings contains the class of nonexpansive mappings, nonspreading mappings, and hybrid mappings. We study the fixed point theorems of generalized hybrid mappings on CAT(0) spaces. We also consider some iteration processes for generalized hybrid mappings on CAT(0) spaces, and our results generalize some results of fixed point theorems on CAT(0) spaces and Hilbert spaces. Keywords: nonexpansive mapping, fixed point, generalized hybrid mapping, CAT(0) spaces 1 Introduction Fixed point theory in CAT(0) spaces was firs t studied by Kirk [1,2]. He showed t hat every nonexpansive (single-valued) mapping defined on a bounded closed convex sub- set of a complete CAT(0) space always has a fixed point. Since then, the fixed point the ory for single-valued and multivalued mappings in CAT(0) spaces has been rapidly developed, and many papers have appeared (e.g., see [3-6] and related references.) Let (X, d) be a metric space. A geodesic path joining x Î X to y Î X (or, more briefly, a geodesic from x to y)isamapc from a closed interval [0, ℓ] ⊆ R to X such that c(0) = x, c(ℓ)=y,andd(c(t), c(t′)) = |t - t′|forallt, t′ Î [0, ℓ]. In particular, c is an isometry and d(x, y)=ℓ. The image a of c is called a geodesic (or metric) segment joining x and y. When it is unique, this geodesic is denoted by [x, y]. The space (X, d) is said to be a geodesic space if every two points of X are joined by a geodesi c, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y Î X.AsubsetY ⊆ X is said to be convex if Y includes every g eodesic segment joining any two of its points. A geodesic triangle Δ(x 1 , x 2 , x 3 ) in a geodesic space ( X, d) consists of three points x 1 , x 2 , and x 3 in X (the vertices of Δ and a geodesic segment between each pair of vertices (the edge of Δ). A comparison triangle for geode sic triangle Δ (x 1 , x 2 , x 3 )in(X, d)isa triangle ( x 1 , x 2 , x 3 ) := ( ¯ x 1 , ¯ x 2 , ¯ x 3 ) in the Euclidean plane E 2 such that d E 2 ( ¯ x i , ¯ x j )=d(x i , x j ) for i, j Î {1, 2, 3}. A geodesic space is said to be a CAT(0) space if all geodesic triangles of appropriate size satisfy the following comparison axiom. Lin et al. Fixed Point Theory and Applications 2011, 2011:49 http://www.fixedpointtheoryandapplications.com/content/2011/1/49 © 2011 Lin et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestrict ed use, distribution, and reprodu ction in any medium, provided the original work is properly cited. CAT(0): Let Δ be a geodesic triangle in X,andlet be a comparison triangle for Δ. Then, Δ is said to satisfy the CAT(0) inequality if for a ll x, y Î Δ and all comparison points ¯ x , ¯ y ∈ , d(x, y) ≤ d E 2 ( ¯ x, ¯ y ) . It is well known that any complete, simply con- nected Riemannian manifold hav ing nonpositive sectional curvature is a CAT(0) space. Other examples include Pre-Hilbert spaces [7], R-trees [8], the complex Hilbert ball with a hyperbolic metric [9], and many others. If x, y 1 , y 2 are points in a CAT(0) space, and if y 0 is the midpoint of the segment [y 1 , y 2 ], then the CAT(0) inequality implies d 2 (x, y 0 ) ≤ 1 2 d 2 (x, y 1 )+ 1 2 d 2 (x, y 2 ) − 1 4 d 2 (y 1 , y 2 ) . This is the (CN) inequality of Bruhat and Tits [10]. In fact, a geodesic space is a CAT(0) space if and only if it satisfies the (CN) inequality ([[7], p. 163]). In 2008, Dhompongsa and Panyanak [11] gave the follo wing result, and the proof is similar to the proof of remark in [[12], p. 374]. Lemma 1.1. [11] Let X be a CAT(0) space. Then, d (( 1 − t ) x ⊕ ty, z ) ≤ ( 1 − t ) d ( x, z ) + td ( y, z ) for all x, y, z Î X and t Î [0, 1]. By the above lemma, we know that CAT(0) space is a convex metric space. Indeed, a metric space X with a convex structure if there exists a mapping W : X × X × [0, 1] ® X such that d ( W ( x, y, t ) , z ) ≤ td ( x, z ) + ( 1 − t ) d ( y, z ) for all x, y, z Î X and t Î [0, 1] and call this space X a convex metric space [13]. Furthermore, Takahashi [13] has proved that d ( x, y ) = td ( x, W ( x, y, t )) + ( 1 − t ) d ( y, W ( x, y, t )) for all x, y, z Î X and t Î [0, 1] when X isaconvexmetricspacewithaconvex structure. So, we also get the following result, and it is proved in [11]. Lemma 1.2. [11] Let X beaCAT(0)space,andx, y Î X. For each t Î [0, 1], there exists a unique point z Î [x, y] such that d(x, z)=td(x, y) and d(y, z) = (1 - t)d(x, y). For convenience, from now on we will use the notation z =(1-t)x ⊕ ty. Therefore, we have: z = ( 1 − t ) x ⊕ ty ⇔ z ∈ [x, y], d ( x, z ) = td ( x, y ) ,andd ( y, z ) = ( 1 − t ) d ( x, y ). Let C be a nonempty closed convex subset of a CAT(0) space (X, d). A mapping T : C ® C is called a nonexpansive mapping if d(Tx, Ty) ≤ d(x, y)forallx, y Î C.A point x Î C iscalledafixedpointofT if Tx = x.LetF(T)denotethesetoffixed points of T. Now, we introduce the following nonlinear mappings on CAT(0) spaces. Definition 1.1. Let C be a nonempty closed convex subset of a CAT(0) space X.We say T : C ® X is a g eneralized hybrid mapping if there ar e functions a 1 , a 2 , a 3 , k 1 , k 2 : C ® [0, 1) such that Lin et al. Fixed Point Theory and Applications 2011, 2011:49 http://www.fixedpointtheoryandapplications.com/content/2011/1/49 Page 2 of 15 (P1) d 2 (Tx, Ty) ≤ a 1 ( x ) d 2 ( x, y ) + a 2 ( x ) d 2 ( Tx, y ) + a 3 ( x ) d 2 ( Ty,x ) + k 1 ( x ) d 2 ( Tx, x ) + k 2 ( x ) d 2 ( Ty,y ) for all x, y Î C; (P2) a 1 (x)+a 2 (x)+a 3 (x) ≤ 1 for all x, y Î C; (P3) 2k 1 (x) <1-a 2 (x) and k 2 (x) <1-a 3 (x) for all x Î C . Remark 1.1. In Definition 1.1, if a 1 (x)=1anda 2 (x)=a 3 (x)=k 1 (x)=k 2 (x)=0for all x Î C, then T is a nonexpansive mapping. In 20 08, Kohsaka and Takahashi [14] introduced nonspreading mappings on Banach spaces. Let C be a nonempty closed convex subset of a real Hilbert space H. A map- ping T : C ® C is said to be a nonspreading mapping if 2||Tx - Ty|| 2 ≤ ||Tx - y|| 2 +|| Ty - x|| 2 for all x, y Î C (for detail, refer to [15]). In 2010, Takahashi [16] introduced hybrid mapping on Hilbert spaces. Let C be a nonempty closed convex subset of a real Hilbert space H.AmappingT : C ® C is said to be hybrid if 3||Tx - Ty|| 2 ≤ ||x - y|| 2 +||Tx - y|| 2 +||x - Ty|| 2 for all x, y Î C. In 2011, Takahashi and Yao [17] also introduced two nonlinear mappings in Hilbert spaces. Let C be a nonempty closed convex subset of a real Hilbert space H. A map- ping T : C ® C is said to be TJ-1 if 2||Tx - Ty|| 2 ≤ ||x - y|| 2 +||Tx - y|| 2 for all x, y Î C. A mapping T : C ® C is said to be TJ-2 if 3||Tx - Ty|| 2 ≤ 2||Tx - y|| 2 +||Ty - x|| 2 for all x, y Î C. Now, we give the definitions of nonspreading mapping, TJ-1, TJ-2, hybrid mappings on CA T(0) spaces. In fact, these are special cases of generalized hybrid mappi ng on CAT(0) spaces. Definition 1.2.LetC be a nonempty closed convex subset of a complete CAT(0) space X.Then,T : C ® C is said to be a nonspreading mapping if 2d 2 (Tx, Ty) ≤ d 2 (Tx, y)+d 2 (Ty, x) for all x, y Î C. Definition 1.3.LetC be a nonempty closed convex subset of a complete CAT(0) space X. Then, T : C ® C is said to be hybrid i f 3d 2 (Tx, Ty) ≤ d 2 (x, y)+d 2 (Tx, y)+d 2 (x, Ty) for all x, y Î C. Definition 1.4.LetC be a nonempty closed convex subset of a complete CAT(0) space X. Then, T : C ® C is said to be TJ-1 if 2d 2 (Tx, Ty) ≤ d 2 (x, y)+d 2 (Tx, y) for all x, y Î C. Definition 1.5.LetC be a nonempty closed convex subset of a complete CAT(0) space X. Then, T : C ® C i s said to be TJ-2 if 3d 2 (Tx, Ty) ≤ 2d 2 (Tx, y)+d 2 (Ty, x)for all x, y Î C. On the other hand, we observe that construction of approximating fixed points of nonlinear mappings is an important subject in the theory of nonlinear mappings and its applicatio ns in a nu mber of applied areas. Let C be a nonempty closed convex sub- set of a real Hilbert space H, and let T, S : C ® C be two mappings. In 1953, Mann [18] gave an iteration process: x n+1 = α n x n + ( 1 − α n ) Tx n , n ≥ 0 , where the initial guess x 0 is taken in C arbitrarily, and {a n } is a sequence in the inter- val [0, 1]. Lin et al. Fixed Point Theory and Applications 2011, 2011:49 http://www.fixedpointtheoryandapplications.com/content/2011/1/49 Page 3 of 15 In 1974, Ishikawa [19] gave an iteration process which is defined recursively by ⎧ ⎪ ⎨ ⎪ ⎩ x 1 ∈ C chosen arbitrary, x n+1 := (1 − α n )x n + α n Ty n y n := (1 − β n )x n + β n Tx n where {a n } and {b n } are sequences in the interval [0, 1]. In 1986, Das and Debata [20] studied a two mappings’s iteration on the pattern of the Ishikawa iteration: ⎧ ⎪ ⎨ ⎪ ⎩ x 1 ∈ C chosen arbitrary, x n+1 := (1 − α n )x n + α n Ty n y n := (1 − β n )x n + β n Sx n (1:1) where {a n } and {b n } are sequences in the interval [0, 1]. In 2007, Agarwal et al. [21] introduced the following iterative process: ⎧ ⎪ ⎨ ⎪ ⎩ x 1 ∈ C chosen arbitrary, x n+1 := (1 − α n )Tx n + α n Ty n , y n := (1 − β n )x n + β n Tx n , (1:2) where the initial guess x 0 is taken in C arbitrarily, and {a n } and {b n } are sequences in the interval [0, 1]. In 2011, Khan and Abbas [22] modified (1.1) and (1.2) for two nonexpansive map- pings S and T in CAT(0) spaces as follows. ⎧ ⎪ ⎨ ⎪ ⎩ x 1 ∈ C chosen arbitrary, x n+1 := (1 − α n )x n ⊕ α n Ty n , y n := (1 − β n )x n ⊕ β n Sx n , (1:3) and ⎧ ⎪ ⎨ ⎪ ⎩ x 1 ∈ C chosen arbitrary, x n+1 := (1 − α n )Tx n ⊕ α n Ty n , y n := (1 − β n )x n ⊕ β n Tx n , (1:4) where the initial guess x 0 is taken in C arbitrarily, and {a n } and {b n } are sequences in the interval [0, 1]. Let D be a nonempty closed convex subset of a complete CAT(0) space (X, d). For each x Î X, there exists a unique element y Î D such that d(x, y) = min z ∈ D d(x, z ) [7]. In the sequel, let P D : X ® D be defined by P D (x)=y ⇔ d(x, y) = min z ∈ D d(x, z) . And we call P D the metric projection from the complete CAT(0) space X onto a nonempty closed convex subset D of X. Note that P D is a nonexpansive mapping [7]. Now, let C be a nonempty closed convex subset of a complete CAT(0) space X,let T, S : C ® X be two nonexpansive mappings, and we modified (1.3) and (1.4) as fol- lows: Lin et al. Fixed Point Theory and Applications 2011, 2011:49 http://www.fixedpointtheoryandapplications.com/content/2011/1/49 Page 4 of 15 ⎧ ⎪ ⎨ ⎪ ⎩ x 1 ∈ C chosen arbitrary, x n+1 := P C ((1 − α n )x n ⊕ α n Ty n ) , y n := P C ((1 − β n )x n ⊕ β n Sx n ), (1:5) and ⎧ ⎪ ⎨ ⎪ ⎩ x 1 ∈ C chosen arbitrary, x n+1 := P C ((1 − α n )Tx n ⊕ α n Ty n ) , y n := P C ((1 − β n )x n ⊕ β n Tx n ), (1:6) where the initial guess x 0 is taken in C arbitrarily, and {a n } and {b n } are sequences in the interval [0, 1]. In this paper, we study the fixed point theorems of gener aliz ed hybrid mappings on CAT(0) spaces. Next, we also consider iteration process (1.5), (1.6), or Mann’s type for generalized hybrid mappings on CAT(0) spaces, and our results improve or generalize recent results on fixed point theorems on CAT(0) spaces or Hilbert spaces. 2 Preliminaries In this paper, we need the following definitions, notations, lemmas, and related results. Lemma 2.1. [11] Let X be a CAT(0) space. Then, d 2 (( 1 − t ) x ⊕ ty, z ) ≤ ( 1 − t ) d 2 ( x, z ) + td 2 ( y, z ) − t ( 1 − t ) d 2 ( x, y ) for all t Î [0, 1] and x, y, z Î X. Definition 2.1 . Let {x n } be a bounded sequence in a CAT(0) space X, and let C be a subset of X. Now, we use the following notations: (i) r(x, {x n }) := lim sup n →∞ d(x, x n ) . (ii) r({x n }):=inf x ∈ X r(x, {x n } ) . (iii) r C ({x n }):=inf x ∈ C r(x, {x n } ) . (iv) A({x n }) := {x Î X : r(x,{x n }) = r({x n })}. (iv) A C ({x n }) := {x Î C : r (x,{x n }) = r C ({x n })}. Note that x Î X is called an asymptotic center of {x n }ifx Î A({x n }). It is known that in a CAT(0) space, A({x n }) consists of exactly one point [23]. Definition 2.2.[6]Let(X, d) be a CAT(0) space. A sequence {x n }inX is said to be Δ-convergent to x Î X if x is the unique asymptotic center of {u n } for every subse- quence {u n }of{x n }. That is, A({u n }) = {x} for e very subsequence {u n }of{x n }. In this case, we write - lim n x n = x and call x the Δ-limit of {x n }. In 200 8, Kirk and Panyanak [6] gave the following result for nonexpansive mappings on CAT(0) spaces. Theorem 2.1.[6]LetC beanonemptyclosedconvexsubsetofacompleteCAT(0) space X,andletT : C ® C beanonexpansivemapping.Let{x n }beabounded sequence in C with - lim n x n = x and lim n →∞ d(x n , Tx n )=0 . Then, x Î C and Tx = x. Lemma 2.2.[6]Let(X, d) be a CAT(0) space. Then, every bounded sequence in X has a Δ-convergent subsequence. Lin et al. Fixed Point Theory and Applications 2011, 2011:49 http://www.fixedpointtheoryandapplications.com/content/2011/1/49 Page 5 of 15 Lemma 2.3. [24] Let C be a nonempty closed convex su bset of a CAT(0) sp ace X.If {x n } is a bounded sequence in C, then the asymptotic center A({x n }) of {x n }isinC. Lemma 2.4. [11] Let C be a nonempty closed convex subset of a CAT(0) space (X, d). Let {x n }beaboundedsequenceinX with A({x n }) = {x}, and let {u n }beasubse- quence of {x n } with A({u n }) = {u}. Suppose that lim n →∞ d(x n , u ) exists. Then, x = u. Let {x n } be a bounded sequence in a CAT(0) space ( X, d), and let C beanonempty closed convex subset of X which contains {x n }. We denote the notation x n w iff (w)=inf x ∈ C (x) , where (x) := lim sup n →∞ d(x n , x ) . Then, we observe that A({x n })={x ∈ X : (x)=inf u∈X (u)},an d A C ({x n })={x ∈ C : (x)=inf u ∈ C (u)}. Remark 2.1. Let {x n } be a bounded sequence in a CAT(0) space (X, d), and let C be a nonempty closed convex subset of X which contains {x n }. If x n ⇀ w, then w Î C. Proof.Thereexist ¯ x ∈ X and ¯ y ∈ C such that A ( {x n } ) = { ¯ x } and A C ( {x n } ) = { ¯ y } .By Lemma 2.3, ¯ x = ¯ y . Hence, ( ¯ y)=( ¯ x) ≤ (w)=inf x ∈ C (x)=( ¯ y) . Hence, w Î A({x n }) and w = ¯ x ∈ C . □ Lemma 2.5. [25] Let C be a nonempty closed convex subset of a CAT(0) space (X, d), and let {x n } be a bounded sequence in C.If - lim n x n = x , then x n ⇀ x. Proposition 2.1 .LetC be a nonempty closed convex subset of a complete CAT(0) space (X, d), and let T : C ® X be a generalized hybrid mapping with F(T ) ≠ ∅. Then, F(T) is a closed convex subset of C. Proof.If{x n } is a sequence in F (T ) and lim n →∞ x n = x . Then, we have: d 2 (Tx, x n ) ≤ d 2 (x, x n )+ k 1 (x) 1 − a 2 ( x ) d 2 (Tx, x) . This implies that (1 − k 1 (x) 1 − a 2 ( x ) )d 2 (Tx, x) ≤ 0. Then, Tx = x and F(T) is a closed set. Next, we want to show that F(T)isaconvexset.Ifx, y Î F(T) ⊆ C and z Î [x, y], then there exists t Î [0, 1] such that z = tx ⊕ (1 - t)y. Since C is convex, z Î C. Lin et al. Fixed Point Theory and Applications 2011, 2011:49 http://www.fixedpointtheoryandapplications.com/content/2011/1/49 Page 6 of 15 Furthermore, d 2 (Tz, z) ≤ td 2 (Tz, x)+(1− t)d 2 (Tz, y) − t(1 − t)d 2 (x, y) ≤ td 2 (z, x)+ tk 1 (z) 1−a 2 (z) d 2 (Tz, z)+(1− t)d 2 (z, y)+ (1−t)k 1 (z) 1−a 2 (z) d 2 (Tz, z) − t(1 − t)d 2 (x, y ) ≤ t(1 − t) 2 d 2 (x, y)+ k 1 (z) 1−a 2 (z) d 2 (Tz, z)+t 2 (1 − t)d 2 (x, y) − t(1 − t)d 2 (x, y) ≤ k 1 (z) 1−a 2 ( z ) d 2 (Tz, z). Hence, Tz = z and F(T) is a convex set. □ Remark 2.2.LetC be a nonempty closed convex subset of a complete CAT(0) space (X, d), and let T : C ® X be any one of nonspreading mapping, TJ-1 mapping, TJ-2 mapping, and hybrid mapping. If F(T) ≠ ∅, then F(T) is a closed convex subset of C. 3 Fixed point theorems on complete CAT(0) space s The following theorem establishes a demiclosed principle for a generalized hybrid mapping on CAT(0) spaces. Theorem 3.1.LetC be a nonempty closed convex subset of a complete CAT(0) space X,andletT : C ® X be a generalized hybrid mapping. Let {x n }beabounded sequence in C with x n ⇀ x and lim n →∞ d(x n , Tx n )=0 . Then, x Î C and Tx = x. Proof.Sincex n ⇀ x,weknowthatx Î C and (x)=inf u ∈ C (u ) ,where (u) := lim sup n →∞ d(x n , u ) . Furthermore, we know that F(x) = inf{F (u):u Î X}. Since T is a generalized hybrid, d 2 (Tx n , Tx) ≤ a 1 (x)d 2 (x, x n )+a 2 (x)d 2 (Tx, x n )+a 3 (x)d 2 (Tx n , x)+k 1 (x)d 2 (Tx, x)+ k 2 (x)d 2 (Tx n , x n ) ≤ a 1 (x)d 2 (x, x n )+a 2 (x)(d(Tx, Tx n )+d(Tx n , x n )) 2 + a 3 (x)(d(Tx n , x n )+d(x n , x)) 2 +k 1 ( x ) d 2 ( Tx, x ) + k 2 ( x ) d 2 ( Tx n , x n ) . Then, we have: lim sup n→∞ d 2 (Tx n , Tx) ≤ lim sup n→∞ d 2 (x, x n )+ k 1 (x) ( 1 − a 2 ( x )) d 2 (x, Tx) . This implies that lim sup n→∞ d 2 (x n , Tx) ≤ lim sup n→∞ (d(x n , Tx n )+d(Tx n , Tx)) 2 ≤ lim sup n→∞ d 2 (Tx n , Tx) ≤ lim sup n→∞ d 2 (x, x n )+ k 1 (x) 1 − a 2 ( x ) d 2 (x, Tx) . Besides, by (CN) inequality, we have: d 2 (x n , 1 2 x ⊕ 1 2 Tx) ≤ 1 2 d 2 (x n , x)+ 1 2 d 2 (x n , Tx) − 1 4 d 2 (x, Tx) . Lin et al. Fixed Point Theory and Applications 2011, 2011:49 http://www.fixedpointtheoryandapplications.com/content/2011/1/49 Page 7 of 15 So, lim sup n→∞ d 2 (x n , 1 2 x ⊕ 1 2 Tx) ≤ 1 2 lim sup n→∞ d 2 (x n , x)+ 1 2 lim sup n→∞ d 2 (x n , Tx) − 1 4 d 2 (x, Tx) ≤ lim sup n→∞ d 2 (x n , x)+ k 1 (x) 2 ( 1 − a 2 ( x )) d 2 (x, Tx) − 1 4 d 2 (x, Tx) . So, ( 1 4 − k 1 (x) 2 ( 1 − a 2 ( x )) )d 2 (x, Tx) ≤ lim sup n→∞ d 2 (x n , x) − lim sup n→∞ d 2 (x n , 1 2 x ⊕ 1 2 Tx) . That is, ( 1 4 − k 1 (x) 2 ( 1 − a 2 ( x )) )d 2 (x, Tx) ≤ ((x)) 2 − (( 1 2 x ⊕ 1 2 Tx)) 2 ≤ 0 . Therefore, Tx = x. □ By Theorem 3.1 and Lemma 2.5, it is easy to get the conclusion. Corollary 3.1.LetC be a n onempty closed convex subset of a complete CAT(0) space X,andletT : C ® X be a generalized hybrid mapping. Let {x n }beabounded sequence in C with Δ-lim n x n = x and lim n →∞ d(x n , Tx n )=0 . Then, Tx = x. Theorem 3.1 generalizes Theorem 2.1 since the class of generalized hybrid mappings contains the class of nonexpansive mappings on CAT(0) spaces. Furthe rmore, we also get the following result. Corollary 3.2.LetC be a n onempty closed convex subset of a complete CAT(0) space X, and l et T : C ® X be any one of nonspreading mapping, TJ-1 mapping, TJ-2 mapping, and hybrid napping. Let {x n } be a bounded sequence in C with x n ⇀ x and lim n →∞ d(x n , Tx n )=0 . Then, Tx = x. Corollary 3.3. [14-17] Let C be a nonempty closed convex subset of a real H ilbert space H,andletT : C ® H be a a ny one of nonspreading mapping, hybrid mapping, TJ-1 mapping, and TJ-2 mapping. Let {x n }beasequenceinC with {x n }converges weakly to x Î C and lim n →∞ d(x n , Tx n )= 0 . Then, x Î C and Tx = x. Proof. For eac h x, y Î H,letd(x, y):=||x - y||. Clearly, a real Hilbert space H is a CAT(0) space, and C i s a nonempty closed convex subset of a CAT(0) space H, and T is generalized hybrid. Since {x n } converges weakly to x,{x n } is a bounded sequence. Since H is a real Hilbert space, lim sup n →∞ ||x n − x|| ≤ lim sup n →∞ ||x n − y||,foreachy ∈ C . This implies that x n ⇀ x. By Theorem 3.1, Tx = x and the proof is completed. □ Theorem 3.2.LetC be a nonempty closed convex subset of a complete CAT(0) space X, and let T : C ® C be a generalized hybrid mapping with k 1 (x)=k 2 (x)=0for all x Î C. Then, the following conditions are equivalent: (i) {T n x} is bounded for some x Î C; (ii) F(T) ≠ ∅. Lin et al. Fixed Point Theory and Applications 2011, 2011:49 http://www.fixedpointtheoryandapplications.com/content/2011/1/49 Page 8 of 15 Proof. Suppose that { T n x} is bounded for some x Î C. For each n Î N, let x n := T n x. Since {x n } is bounded, there exists ¯ x ∈ X such that A ( {x n } ) = { ¯ x } . By Lemma 2.3, ¯ x ∈ C . Furthermore, we have: d 2 ( x n , T ¯ x ) ≤ a 1 ( ¯ x ) d 2 ( ¯ x, x n−1 ) + a 2 ( ¯ x ) d 2 ( T ¯ x, x n−1 ) + a 3 ( ¯ x ) d 2 ( x n , ¯ x ). This implies that lim sup n→∞ d 2 (x n , T ¯ x) ≤ a 1 ( ¯ x) lim sup n→∞ d 2 ( ¯ x, x n−1 )+a 2 ( ¯ x) lim sup n→∞ d 2 (T ¯ x, x n−1 )+a 3 ( ¯ x) lim sup n→∞ d 2 (x n , ¯ x ) ≤ (a 1 ( ¯ x)+a 3 ( ¯ x)) lim sup n →∞ d 2 (x n , ¯ x)+a 2 ( ¯ x) lim sup n →∞ d 2 (x n , T ¯ x). Then ((T ¯ x)) 2 = lim sup n → ∞ d 2 (x n , T ¯ x) ≤ lim sup n → ∞ d 2 (x n , ¯ x)=(( ¯ x)) 2 . Since A ( {x n } ) = { ¯ x } , T ¯ x = ¯ x . This shows that F(T) ≠ ∅. It is easy to see that (ii) implies (i). □ By Theorem 3.2, it is easy to get the following results. Corollary 3.4.LetC be a n onempty closed convex subset of a complete CAT(0) space X,andletT : C ® C be any one of nonspreading mapping, TJ-1 mapping, TJ-2 mapping, hybrid mapping, and nonexpansive mapping. Then, {T n x } is bounded for some x Î C if and only if F(T) ≠ ∅. Corollary 3.5. [1,2] Let C be a nonempty bounded cl osed convex subset of a c om- plete CAT(0) space X, and let T : C ® C be a nonexpansive mapping. Then, F(T) ≠ ∅. Corollary 3.6. [14-17,26] Let C be a nonempty closed convex subset of a real Hilbert space H, and let T : C ® C be any one of nonspreading mapping, TJ-1 mapping, TJ- 2 mapping, hybrid mapping, and nonexpansive mapping. Then, {T n x } is bounded for some x Î C if and only if F(T) ≠ ∅. 4 Δ-convergent theorems In the sequel, we need the following lemmas. By Lemmas 2.2-2.4 and Theorem 3.1, and following the similar argument as in the proof of Lemma 2.10 in [11], we have the following result. Lemma 4.1. Let C be a nonempty closed convex subset of a complete CAT(0) space X,andletT : C ® X be a generalized hybrid mapping. If {x n }isaboundedsequence in C such that lim n →∞ d(x n , Tx n )=0 and {d(x n , v)} con verges for all v Î F (T ), then ω w (x n ) ⊆ F (T ), where ω w (x n ):=∪A({u n }) and {u n }isanysubsequenceof{x n }. Further- more, ω w (x n ) consists of exactly one point. Remark 4.1. The conclusion of Lemma 4.1 is still true if T : C ® X is any one of nonexpansive mapping, nonspreading mapping, TJ-1 mapping, TJ-2 mapping, and hybrid mapping. Theorem 4.1.LetC be a nonempty closed convex subset of a complete CAT(0) space X.LetT : C ® X be a generali zed hybrid mapping with F(T) ≠ ∅. Let {a n }bea sequence in [0, 1]. Let {x n } be defined by Lin et al. Fixed Point Theory and Applications 2011, 2011:49 http://www.fixedpointtheoryandapplications.com/content/2011/1/49 Page 9 of 15 x 1 ∈ C chosen arbitrary, x n+1 := P C ((1 − α n )x n ⊕ α n Tx n ) . Assume lim inf n→∞ α n [(1 − α n ) − k 2 (w) 1 − a 3 ( w ) ] > 0 for all w Î F(T). Then, {x n } Δ-con- verges to a point of F(T). Proof. Clearly, {x n } ⊆ C. Take any w Î F (T ) and let w be fixed. Then, d 2 (Tx, w) ≤ d 2 (w, x)+ k 2 (w) 1 − a 3 ( w ) d 2 (Tx, x ) for all x Î C. Hence, by Lemma 2.1, d 2 (x n+1 , w) = d 2 (P C ((1 − α n )x n ⊕ α n Tx n ), w) ≤ d 2 ((1 − α n )x n ⊕ α n Tx n , w) ≤ (1 − α n )d 2 (x n , w)+α n d 2 (Tx n , w) − α n (1 − α n )d 2 (x n , Tx n ) ≤ d 2 (x n , w)+α n [ k 2 (w) 1 − a 3 ( w ) − (1 − α n )]d 2 (Tx n , x n ). By assumption, there exists δ >0 and M Î N such that α n [(1 − α n ) − k 2 (w) 1 − a 3 ( w ) ] ≥ δ> 0 for all n ≥ M. Without loss of generality, we may assume that α n [(1 − α n ) − k 2 (w) 1 − a 3 ( w ) ] > 0 for all n Î N. Hence, {d(x n , w)} is decreasing, lim n →∞ d(x n , w ) exists, and {x n }is bounded. Then lim n→∞ α n [(1 − α n ) − k 2 (w) 1 − a 3 ( w ) ]d 2 (x n , Tx n )=0. This implies that lim n →∞ d(x n , Tx n )=0 . By Lemma 4.1, there exists ¯ x ∈ C such that ω w ( {x n } ) = { ¯ x}⊆F ( T ) . So, − lim n x n = ¯ x and the proof is completed. □ Theorem 4.2.LetC be a nonempty closed convex subset of a complete CAT(0) space X.LetT : C ® X be a generalized hybrid mapping w ith F(T) ≠ ∅.Let{a n }and {b n } be two sequences in [0, 1]. Let {x n } be defined as ⎧ ⎪ ⎨ ⎪ ⎩ x 1 ∈ C chosen arbitrary, x n+1 := P C ((1 − α n )Tx n ⊕ α n Ty n ) , y n := P C ((1 − β n )x n ⊕ β n Tx n ). Assume that: (i) k 2 (w) = 0 for all w Î F (T ); (ii) lim inf n → ∞ α n (1 − α n ) > 0 and lim inf n → ∞ β n (1 − β n ) > 0 . Lin et al. Fixed Point Theory and Applications 2011, 2011:49 http://www.fixedpointtheoryandapplications.com/content/2011/1/49 Page 10 of 15 [...]... behavior of contractions in Hilbert space Israel J Math 9, 235–240 (1971) doi:10.1007/BF02771588 doi:10.1186/1687-1812-2011-49 Cite this article as: Lin et al.: Fixed point theorems and Δ-convergence theorems for generalized hybrid mappings on CAT(0) spaces Fixed Point Theory and Applications 2011 2011:49 Submit your manuscript to a journal and benefit from: 7 Convenient online submission 7 Rigorous... Fixed point theorems for a class of nonlinear mappings related to maximal monotone operators in Banach spaces Arch Math 91, 166–177 (2008) doi:10.1007/s00013-008-2545-8 Iemoto, S, Takahashi, W: Approximating common fixed points of nonexpansive mappings and nonspreading mappings in a Hilbert space Nonlinear Anal 71, e2082–e2089 (2009) doi:10.1016/j.na.2009.03.064 Takahashi, W: Fixed point theorems for. .. 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Applications 2011, 2011:49 http://www.fixedpointtheoryandapplications.com/content/2011/1/49 Page 12 of 15 Corollary 4.1 Let C be a nonempty closed convex subset of a real Hilbert space H Let T : C ® X be any one of nonspreading mapping, nonexpansive mapping, hybrid mapping, TJ-1 mapping, and TJ-2 mapping Let PC be the metric projection from H onto C Suppose that F(T) ≠ ∅ Let {an} and {bn} be two sequences in... αn ) > 0 and lim inf βn (1 − βn ) > 0 Then, {xn} converges n→∞ n→∞ weakly to a point x of F(T) Proof For each x, y Î H, let d(x, y) := ||x - y|| Clearly, H is a CAT(0) space, and C is a nonempty closed convex subset of H Furthermore, tx ⊕ (1 - t)y = tx + (1 - t)y for all x, y Î C and t Î [0, 1] Since T is any one of nonspreading mapping, nonexpansive mapping, hybrid mapping, TJ-1 mapping, and TJ-2... Geodesic geometry and fixed point theory In Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), of Colección Abierta, vol 64, pp 195–225.University of Seville, Secretary of Publications, Seville, Spain (2003) 2 Kirk, WA: Geodesic Geometry and Fixed Point Theory II International Conference on Fixed Point Theory and Applications pp 113–142.Yokohama Publishers, Yokohama (2004) 3 Dhompongsa, S, Kaewkhao,... xn = x and the proof is completed □ x ¯ Remark 4.2 If 0 < a < b 0 Furthermore, the class of generalized n→∞ hybrid mappings contains the class of nonexpansive mappings in CAT(0) spaces Hence, Theorem 4.2 generalizes Theorem 1 in [22] Page 11 of 15 Lin et al Fixed Point Theory and Applications 2011, . al.: Fixed point theorems and Δ-convergence theorems for generalized hybrid mappings on CAT(0) spaces. Fixed Point Theory and Applications 2011 2011:49. Submit your manuscript to a journal and. hybrid mappings on CAT(0) spaces. We also consider some iteration processes for generalized hybrid mappings on CAT(0) spaces, and our results generalize some results of fixed point theorems on. mapping on CAT(0) spaces. The class of generalized hybrid mappings contains the class of nonexpansive mappings, nonspreading mappings, and hybrid mappings. We study the fixed point theorems of generalized