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Generalized hybrid mappings on CAT ( κ ) spaces

Journal of Inequalities and Applications20142014:403

https://doi.org/10.1186/1029-242X-2014-403

  • Received: 19 June 2014
  • Accepted: 18 September 2014
  • Published:

Abstract

In this paper, we obtain the demiclosed principle, fixed point theorems, and Δ-convergence theorems for the class of generalized hybrid mappings on CAT ( κ ) spaces with κ > 0 . Our results extend the results of Lin et al. (Fixed Point Theory Appl. 2011:49, 2011) and many others.

Keywords

  • fixed point
  • generalized hybrid mapping
  • Δ-convergence
  • CAT ( κ ) space

1 Introduction

For a real number κ, a CAT ( κ ) space is a geodesic metric space whose geodesic triangle is thinner than the corresponding comparison triangle in a model space with curvature κ. The precise definition is given below. The letters C, A, and T stand for Cartan, Alexandrov, and Toponogov, who have made important contributions to the understanding of curvature via inequalities for the distance function.

Fixed point theory in CAT ( κ ) spaces was first studied by Kirk [1, 2]. His works were followed by a series of new works by many authors, mainly focusing on CAT ( 0 ) spaces (see e.g., [318]). Since any CAT ( κ ) space is a CAT ( κ ) space for κ κ , all results for CAT ( 0 ) spaces immediately apply to any CAT ( κ ) space with κ 0 . However, there are only a few articles that contain fixed point results in the setting of CAT ( κ ) spaces with κ > 0 .

The concept of generalized hybrid mappings was introduced in Hilbert spaces by Kocourek et al. [19]. Later on, Lin et al. [10] defined a generalized hybrid mapping, which is more general than that of Kocourek et al. [19], in a CAT ( 0 ) space setting. This class of mappings properly contains the class of nonspreading mappings and the class of hybrid mappings; see [10] for more details. In [10], the authors also obtained the demiclosed principle, fixed point theorems as well as Δ-convergence theorems for generalized hybrid mappings in CAT ( 0 ) spaces. In this paper, we extend the results of Lin et al. [10] to the general setting of CAT ( κ ) spaces with κ > 0 .

2 Preliminaries

Let ( X , ρ ) be a metric space. A geodesic path joining x X to y X (or, more briefly, a geodesic from x to y) is a map c from a closed interval [ 0 , l ] R to X such that c ( 0 ) = x , c ( l ) = y , and ρ ( c ( t ) , c ( t ) ) = | t t | for all t , t [ 0 , l ] . In particular, c is an isometry and ρ ( x , y ) = l . The image c ( [ 0 , l ] ) of c is called a geodesic segment joining x and y. When it is unique this geodesic segment is denoted by [ x , y ] . This means that z [ x , y ] if and only if there exists α [ 0 , 1 ] such that
ρ ( x , z ) = ( 1 α ) ρ ( x , y ) and ρ ( y , z ) = α ρ ( x , y ) .
In this case, we write z = α x ( 1 α ) y . For D ( 0 , + ] , the space X is called a D-geodesic space if every two points of X with their distance smaller than D are joined by a geodesic segment. An ∞-geodesic space is simply called a geodesic space. The space X is said to be uniquely geodesic (D-uniquely geodesic) if there is exactly one geodesic segment joining x and y for each x , y X (for x , y X with ρ ( x , y ) < D ). A subset C of X is said to be convex if C includes every geodesic segment joining any two of its points. The set C is said to be bounded if
diam ( C ) : = sup { ρ ( x , y ) : x , y C } < .
Now we present the model spaces M κ n , for more details on these spaces the reader is referred to [20]. Let n N . We denote by E n the metric space R n endowed with the usual Euclidean distance. We denote by ( | ) the Euclidean scalar product in R n , that is,
( x | y ) = x 1 y 1 + + x n y n , where  x = ( x 1 , , x n ) , y = ( y 1 , , y n ) .
Let S n denote the n-dimensional sphere defined by
S n = { x = ( x 1 , , x n + 1 ) R n + 1 : ( x | x ) = 1 } ,

with metric d S n ( x , y ) = arccos ( x | y ) , x , y S n .

Let E n , 1 denote the vector space R n + 1 endowed with the symmetric bilinear form which associates to vectors u = ( u 1 , , u n + 1 ) and v = ( v 1 , , v n + 1 ) the real number u | v is defined by
u | v = u n + 1 v n + 1 + i = 1 n u i v i .
Let H n denote the hyperbolic n-space defined by
H n = { u = ( u 1 , , u n + 1 ) E n , 1 : u | u = 1 , u n + 1 > 0 } ,
with metric d H n such that
cosh d H n ( x , y ) = x | y , x , y H n .
Definition 2.1 Given κ R , we denote by M κ n the following metric spaces:
  1. (i)

    if κ = 0 then M 0 n is the Euclidean space E n ;

     
  2. (ii)

    if κ > 0 then M κ n is obtained from the spherical space S n by multiplying the distance function by the constant 1 / κ ;

     
  3. (iii)

    if κ < 0 then M κ n is obtained from the hyperbolic space H n by multiplying the distance function by the constant 1 / κ .

     
A geodesic triangle ( x , y , z ) in a geodesic space ( X , ρ ) consists of three points x, y, z in X (the vertices of ) and three geodesic segments between each pair of vertices (the edges of ). A comparison triangle for a geodesic triangle ( x , y , z ) in ( X , ρ ) is a triangle ¯ ( x ¯ , y ¯ , z ¯ ) in M κ 2 such that
ρ ( x , y ) = d M κ 2 ( x ¯ , y ¯ ) , ρ ( y , z ) = d M κ 2 ( y ¯ , z ¯ ) and ρ ( z , x ) = d M κ 2 ( z ¯ , x ¯ ) .

If κ 0 then such a comparison triangle always exists in M κ 2 . If κ > 0 then such a triangle exists whenever ρ ( x , y ) + ρ ( y , z ) + ρ ( z , x ) < 2 D κ , where D κ = π / κ . A point p ¯ [ x ¯ , y ¯ ] is called a comparison point for p [ x , y ] if ρ ( x , p ) = d M κ 2 ( x ¯ , p ¯ ) .

A geodesic triangle ( x , y , z ) in X is said to satisfy the CAT ( κ ) inequality if for any p , q ( x , y , z ) and for their comparison points p ¯ , q ¯ ¯ ( x ¯ , y ¯ , z ¯ ) , one has
ρ ( p , q ) d M κ 2 ( p ¯ , q ¯ ) .

Definition 2.2 If κ 0 , then X is called a CAT ( κ ) space if X is a geodesic space such that all of its geodesic triangles satisfy the CAT ( κ ) inequality.

If κ > 0 , then X is called a CAT ( κ ) space if X is D κ -geodesic and any geodesic triangle ( x , y , z ) in X with ρ ( x , y ) + ρ ( y , z ) + ρ ( z , x ) < 2 D κ satisfies the CAT ( κ ) inequality.

Now, we recall the concepts of comparison angle and upper (Alexandrov) angle (cf. [8]).

Definition 2.3 Let p, q, and r be three points in a geodesic space. The interior angle of ¯ ( p ¯ , q ¯ , r ¯ ) E 2 at p ¯ is called the comparison angle between q and r at p and will be denoted by ¯ p ( q , r ) .

Definition 2.4 Let X be a geodesic space and let c : [ 0 , a ] X and c : [ 0 , a ] X be two geodesic paths with c ( 0 ) = c ( 0 ) . Given t ( 0 , a ] and t ( 0 , a ] , we consider the comparison triangle ¯ ( c ( 0 ) ¯ , c ( t ) ¯ , c ( t ) ¯ ) and the comparison angle ¯ c ( 0 ) ( c ( t ) , c ( t ) ) in E 2 . The (Alexandrov) angle or the upper angle between the geodesic paths c and c is the number ( c , c ) defined by
( c , c ) : = lim sup t , t 0 + ¯ c ( 0 ) ( c ( t ) , c ( t ) ) .

The angle between the geodesic segments [ p , x ] and [ p , y ] will be denoted by p ( x , y ) . Notice that the Alexandrov angle coincides with the spherical angle on S n and the hyperbolic angle on H n .

In a CAT ( 0 ) space ( X , ρ ) , if x , y , z X then the CAT ( 0 ) inequality implies
( CN ) ρ 2 ( x , 1 2 y 1 2 z ) 1 2 ρ 2 ( x , y ) + 1 2 ρ 2 ( x , z ) 1 4 ρ 2 ( y , z ) .
This is the (CN) inequality of Bruhat and Tits [21]. This inequality is extended by Dhompongsa and Panyanak [22] to
( CN ) ρ 2 ( x , ( 1 α ) y α z ) ( 1 α ) ρ 2 ( x , y ) + α ρ 2 ( x , z ) α ( 1 α ) ρ 2 ( y , z )
for all α [ 0 , 1 ] and x , y , z X . In fact, if X is a geodesic space then the following statements are equivalent:
  1. (i)

    X is a CAT ( 0 ) space;

     
  2. (ii)

    X satisfies (CN);

     
  3. (iii)

    X satisfies (CN).

     
Let R ( 0 , 2 ] . Recall that a geodesic space ( X , ρ ) is said to be R-convex for R (see [23]) if for any three points x , y , z X , we have
ρ 2 ( x , ( 1 α ) y α z ) ( 1 α ) ρ 2 ( x , y ) + α ρ 2 ( x , z ) R 2 α ( 1 α ) ρ 2 ( y , z ) .
(1)

It follows from (CN) that a geodesic space ( X , ρ ) is a CAT ( 0 ) space if and only if ( X , ρ ) is R-convex for R = 2 . The following lemma is a consequence of Proposition 3.1 in [23].

Lemma 2.5 Let κ > 0 and ( X , ρ ) be a CAT ( κ ) space with diam ( X ) π / 2 ε κ for some ε ( 0 , π / 2 ) . Then ( X , ρ ) is R-convex for R = ( π 2 ε ) tan ( ε ) .

We now collect some elementary facts about CAT ( κ ) spaces. Most of them are proved in the setting of CAT ( 1 ) spaces. For completeness, we state the results in CAT ( κ ) with κ > 0 .

Lemma 2.6 ([[8], Proposition 3.5])

Let κ > 0 and ( X , ρ ) be a complete CAT ( κ ) space with diam ( X ) π / 2 ε κ for some ε ( 0 , π / 2 ) . Let x X and C be a nonempty closed convex subset of X. Then
  1. (i)

    the metric projection P C ( x ) of x onto C is a singleton;

     
  2. (ii)

    if x C and y C with y P C ( x ) , then P C ( x ) ( x , y ) π / 2 ;

     
  3. (iii)

    for each y C , ρ ( P C ( x ) , P C ( y ) ) ρ ( x , y ) .

     
Let { x n } be a bounded sequence in a CAT ( κ ) space ( X , ρ ) . For x X , we set
r ( x , { x n } ) = lim sup n ρ ( x , x n ) .
The asymptotic radius r ( { x n } ) of { x n } is given by
r ( { x n } ) = inf { r ( x , { x n } ) : x X } ,
and the asymptotic center A ( { x n } ) of { x n } is the set
A ( { x n } ) = { x X : r ( x , { x n } ) = r ( { x n } ) } .

It is well known from Proposition 4.1 of [8] that in a CAT ( κ ) space with diameter smaller than π 2 κ , A ( { x n } ) consists of exactly one point. We now give the concept of Δ-convergence and collect some of its basic properties.

Definition 2.7 ([6, 24])

A sequence { x n } in X is said to Δ-converge to x X if x is the unique asymptotic center of { u n } for every subsequence { u n } of { x n } . In this case we write Δ - lim n x n = x and call x the Δ-limit of { x n } .

Lemma 2.8 Let κ > 0 and ( X , ρ ) be a complete CAT ( κ ) space with diam ( X ) π / 2 ε κ for some ε ( 0 , π / 2 ) . Then the following statements hold:
  1. (i)

    [[8], Corollary  4.4] every sequence in X has a Δ-convergence subsequence;

     
  2. (ii)

    [[8], Proposition  4.5] if { x n } X and Δ - lim n x n = x , then x k = 1 conv ¯ { x k , x k + 1 , } , where conv ¯ ( A ) = { B : B A  and  B  is closed and convex } .

     

By the uniqueness of asymptotic centers, we can obtain the following lemma (cf. [[22], Lemma 2.8]).

Lemma 2.9 Let κ > 0 and ( X , ρ ) be a complete CAT ( κ ) space with diam ( X ) π / 2 ε κ for some ε ( 0 , π / 2 ) . If { x n } is a sequence in X with A ( { x n } ) = { x } and { u n } is a subsequence of { x n } with A ( { u n } ) = { u } and the sequence { ρ ( x n , u ) } converges, then x = u .

Definition 2.10 Let C be a nonempty subset of a CAT ( κ ) space ( X , ρ ) . A mapping T : C X is called a generalized hybrid mapping [10] if there exist functions a 1 , a 2 , a 3 , k 1 , k 2 : C [ 0 , 1 ) such that
  • (P1) ρ 2 ( T ( x ) , T ( y ) ) a 1 ( x ) ρ 2 ( x , y ) + a 2 ( x ) ρ 2 ( T ( x ) , y ) + a 3 ( x ) ρ 2 ( T ( y ) , x ) + k 1 ( x ) ρ 2 ( T ( x ) , x ) + k 2 ( x ) ρ 2 ( T ( y ) , y ) for all x , y C ;

  • (P2) a 1 ( x ) + a 2 ( x ) + a 3 ( x ) 1 for all x , y C ;

  • (P3) 2 k 1 ( x ) < 1 a 2 ( x ) and k 2 ( x ) < 1 a 3 ( x ) for all x C .

A point x C is called a fixed point of T if x = T ( x ) . We denote the set of all fixed points of T with F ( T ) .

3 Main results

3.1 Demiclosed principle

Theorem 3.1 Let κ > 0 and ( X , ρ ) be a complete CAT ( κ ) space with diam ( X ) π / 2 ε κ for some ε ( 0 , π / 2 ) . Let C be a nonempty closed convex subset of X, and T : C X be a generalized hybrid mapping with 2 k 1 ( x ) 1 a 2 ( x ) < R 2 for all x C where R = ( π 2 ε ) tan ( ε ) . Let { x n } be a sequence in C with Δ - lim n x n = z and lim n ρ ( x n , T ( x n ) ) = 0 . Then z C and z = T ( z ) .

Proof Since Δ - lim n x n = z , by Lemma 2.8, z C . Since T is a generalized hybrid mapping,
ρ 2 ( T ( x n ) , T ( z ) ) a 1 ( z ) ρ 2 ( z , x n ) + a 2 ( z ) ρ 2 ( T ( z ) , x n ) + a 3 ( z ) ρ 2 ( T ( x n ) , z ) + k 1 ( z ) ρ 2 ( T ( z ) , z ) + k 2 ( z ) ρ 2 ( T ( x n ) , x n ) a 1 ( z ) ρ 2 ( z , x n ) + a 2 ( z ) [ ρ ( T ( z ) , T ( x n ) ) + ρ ( T ( x n ) , x n ) ] 2 + a 3 ( z ) [ ρ ( T ( x n ) , x n ) + ρ ( x n , z ) ] 2 + k 1 ( z ) ρ 2 ( T ( z ) , z ) + k 2 ( z ) ρ 2 ( T ( x n ) , x n ) ,
yielding
lim sup n ρ 2 ( T ( x n ) , T ( z ) ) lim sup n ρ 2 ( z , x n ) + k 1 ( z ) 1 a 2 ( z ) ρ 2 ( z , T ( z ) ) .
This implies that
lim sup n ρ 2 ( x n , T ( z ) ) lim sup n [ ρ ( x n , T ( x n ) ) + ρ ( T ( x n ) , T ( z ) ) ] 2 lim sup n ρ 2 ( T ( x n ) , T ( z ) ) lim sup n ρ 2 ( z , x n ) + k 1 ( z ) 1 a 2 ( z ) ρ 2 ( z , T ( z ) ) .
(2)
On the other hand, by Lemma 2.5 we have
ρ 2 ( x n , 1 2 z 1 2 T ( z ) ) 1 2 ρ 2 ( x n , z ) + 1 2 ρ 2 ( x n , T ( z ) ) R 8 ρ 2 ( z , T ( z ) ) .
(3)
By (2) and (3), we get
lim sup n ρ 2 ( x n , 1 2 z 1 2 T ( z ) ) 1 2 lim sup n ρ 2 ( x n , z ) + 1 2 lim sup n ρ 2 ( x n , T ( z ) ) R 8 ρ 2 ( z , T ( z ) ) lim sup n ρ 2 ( x n , z ) + k 1 ( z ) 2 ( 1 a 2 ( z ) ) ρ 2 ( z , T ( z ) ) R 8 ρ 2 ( z , T ( z ) ) .
Thus
( R 8 k 1 ( z ) 2 ( 1 a 2 ( z ) ) ) ρ 2 ( z , T ( z ) ) lim sup n ρ 2 ( x n , z ) lim sup n ρ 2 ( x n , 1 2 z 1 2 T ( z ) ) 0 .

Since 2 k 1 ( z ) 1 a 2 ( z ) < R 2 , we get k 1 ( z ) 2 ( 1 a 2 ( z ) ) < R 8 and so ρ 2 ( z , T ( z ) ) = 0 . Hence z = T ( z ) . □

The following corollary shows that how we derive a result for CAT ( 0 ) spaces from Theorem 3.1.

Corollary 3.2 Let ( X , ρ ) be a complete CAT ( 0 ) space, C be a nonempty bounded closed convex subset of X, and T : C C be a generalized hybrid mapping. Let { x n } be a sequence in C with Δ - lim n x n = z and lim n ρ ( x n , T ( x n ) ) = 0 . Then z C and z = T ( z ) .

Proof It is well known that every convex subset of a CAT ( 0 ) space, equipped with the induced metric, is a CAT ( 0 ) space (cf. [20]). Then ( C , ρ ) is a CAT ( 0 ) space and hence it is a CAT ( κ ) space for all κ > 0 . Notice also that C is R-convex for R = 2 . Since C is bounded, we can choose ε ( 0 , π / 2 ) and κ > 0 so that diam ( C ) π / 2 ε κ . The conclusion follows from Theorem 3.1. □

3.2 Fixed point theorems

Theorem 3.3 Let κ > 0 and ( X , ρ ) be a complete CAT ( κ ) space with diam ( X ) π / 2 ε κ for some ε ( 0 , π / 2 ) . Let C be a nonempty closed convex subset of X, and T : C C be a generalized hybrid mapping with k 1 ( x ) = k 2 ( x ) = 0 for all x C . Then T has a fixed point.

Proof Fix x C and define x n : = T n ( x ) for n N . Suppose that A ( { x n } ) = { z } . Then by Lemma 2.8, z C . Since T is generalized hybrid and k 1 ( z ) = k 2 ( z ) = 0 ,
ρ 2 ( x n , T ( z ) ) a 1 ( z ) ρ 2 ( z , x n 1 ) + a 2 ( z ) ρ 2 ( T ( z ) , x n 1 ) + a 3 ( z ) ρ 2 ( x n , z ) .
Taking the limit superior on both sides, we get
lim sup n ρ 2 ( x n , T ( z ) ) a 1 ( z ) lim sup n ρ 2 ( z , x n 1 ) + a 2 ( z ) lim sup n ρ 2 ( T ( z ) , x n 1 ) + a 3 ( z ) lim sup n ρ 2 ( x n , z ) ( a 1 ( z ) + a 3 ( z ) ) lim sup n ρ 2 ( x n , z ) + a 2 ( z ) lim sup n ρ 2 ( x n , T ( z ) ) .

This implies by (P2) that lim sup n ρ 2 ( x n , T ( z ) ) lim sup n ρ 2 ( x n , z ) . But, since A ( { x n } ) = { z } , it must be the case that z = T ( z ) and the proof is complete. □

As a consequence of Theorem 3.3, we obtain:

Corollary 3.4 Let ( X , ρ ) be a complete CAT ( 0 ) space, C be a nonempty bounded closed convex subset of X, and T : C C be a generalized hybrid mapping with k 1 ( x ) = k 2 ( x ) = 0 for all x C . Then T has a fixed point.

3.3 Δ-Convergence theorems

We begin this section by proving a crucial lemma.

Lemma 3.5 Let κ > 0 and ( X , ρ ) be a complete CAT ( κ ) space with diam ( X ) π / 2 ε κ for some ε ( 0 , π / 2 ) . Let C be a nonempty closed convex subset of X, and T : C X be a generalized hybrid mapping with 2 k 1 ( x ) 1 a 2 ( x ) < R 2 for all x C where R = ( π 2 ε ) tan ( ε ) . Suppose { x n } is a sequence in C such that lim n ρ ( x n , T x n ) = 0 and { ρ ( x n , v ) } converges for all v F ( T ) , then ω w ( x n ) F ( T ) . Here ω w ( x n ) : = A ( { u n } ) where the union is taken over all subsequences { u n } of { x n } . Moreover, ω w ( x n ) consists of exactly one point.

Proof Let u ω w ( x n ) , then there exists a subsequence { u n } of { x n } such that A ( { u n } ) = { u } . By Lemma 2.8, there exists a subsequence { v n } of { u n } such that Δ - lim n v n = v C . By Theorem 3.1, v F ( T ) . By Lemma 2.9, u = v . This shows that ω w ( x n ) F ( T ) . Next, we show that ω w ( x n ) consists of exactly one point. Let { u n } be a subsequence of { x n } with A ( { u n } ) = { u } and let A ( { x n } ) = { x } . Since u ω w ( x n ) F ( T ) , { ρ ( x n , u ) } converges. Again, by Lemma 2.9, x = u . This completes the proof. □

Theorem 3.6 Let κ > 0 and ( X , ρ ) be a complete CAT ( κ ) space with diam ( X ) π / 2 ε κ for some ε ( 0 , π / 2 ) . Let C be a nonempty closed convex subset of X, and T : C X be a generalized hybrid mapping with F ( T ) . Let { α n } be a sequence in [ 0 , 1 ] and define a sequence { x n } in C by
{ x 1 C chosen arbitrary , x n + 1 : = P C ( ( 1 α n ) x n α n T ( x n ) ) , n N .
Let R = ( π 2 ε ) tan ( ε ) and suppose that
  1. (i)

    2 k 1 ( x ) 1 a 2 ( x ) < R 2 for all x C ,

     
  2. (ii)

    lim inf n α n [ ( 1 α n ) R 2 k 2 ( z ) 1 a 3 ( z ) ] > 0 for all z F ( T ) .

     

Then { x n } Δ-converges to an element of F ( T ) .

Proof Let z F ( T ) . Since T is generalized hybrid,
ρ 2 ( T ( x ) , z ) ρ 2 ( z , x ) + k 2 ( z ) 1 a 3 ( z ) ρ 2 ( T ( x ) , x ) for all  x C .
By Lemmas 2.5 and 2.6, we have
ρ 2 ( x n + 1 , z ) = ρ 2 ( P C ( ( 1 α n ) x n α n T ( x n ) ) , z ) ρ 2 ( ( 1 α n ) x n α n T ( x n ) , z ) ( 1 α n ) ρ 2 ( x n , z ) + α n ρ 2 ( T ( x n ) , z ) R 2 α n ( 1 α n ) ρ 2 ( x n , T ( x n ) ) ρ 2 ( x n , z ) + α n [ k 2 ( z ) 1 a 3 ( z ) R ( 1 α n ) 2 ] ρ 2 ( x n , T ( x n ) ) .
(4)
By (ii), there exist δ > 0 and N N such that
α n [ ( 1 α n ) R 2 k 2 ( z ) 1 a 3 ( z ) ] δ > 0 for all  n N .
Without loss of generality, we may assume that
α n [ ( 1 α n ) R 2 k 2 ( z ) 1 a 3 ( z ) ] > 0 for all  n N .
(5)
It follows from (4) and (5) that { ρ ( x n , z ) } is a nonincreasing sequence and hence lim n ρ ( x n , z ) exists. Again, by (4), we have
lim n α n [ ( 1 α n ) R 2 k 2 ( z ) 1 a 3 ( z ) ] ρ 2 ( x n , T ( x n ) ) = 0 .

This implies by (ii) that lim n ρ ( x n , T ( x n ) ) = 0 . By Lemma 3.5, ω w ( x n ) consists of exactly one point and is contained in F ( T ) . This shows that { x n } Δ-converges to an element of F ( T ) . □

Theorem 3.7 Let κ > 0 and ( X , ρ ) be a complete CAT ( κ ) space with diam ( X ) π / 2 ε κ for some ε ( 0 , π / 2 ) . Let C be a nonempty closed convex subset of X, and T : C X be a generalized hybrid mapping with F ( T ) . Let { α n } and { β n } be sequences in [ 0 , 1 ] and define a sequence { x n } in C by
{ x 1 C chosen arbitrary , x n + 1 : = P C ( ( 1 α n ) T ( x n ) α n T ( y n ) ) , y n : = P C ( ( 1 β n ) x n β n T ( x n ) ) .
Assume that
  1. (i)

    k 2 ( z ) = 0 for all z F ( T ) ,

     
  2. (ii)

    lim inf n α n > 0 and lim inf n β n ( 1 β n ) > 0 .

     

Then { x n } Δ-converges to an element of F ( T ) .

Proof Fix z F ( T ) . By (i), we have ρ ( T ( x ) , z ) ρ ( x , z ) for all x C . Let R = ( π 2 ε ) tan ( ε ) . By Lemmas 2.5 and 2.6, we have
ρ 2 ( y n , z ) = ρ 2 ( P C ( ( 1 β n ) x n β n T ( x n ) ) , z ) ρ 2 ( ( 1 β n ) x n β n T ( x n ) , z ) ( 1 β n ) ρ 2 ( x n , z ) + β n ρ 2 ( T ( x n ) , z ) R 2 β n ( 1 β n ) ρ 2 ( x n , T ( x n ) ) ρ 2 ( x n , z ) R 2 β n ( 1 β n ) ρ 2 ( x n , T ( x n ) ) ρ 2 ( x n , z ) .
(6)
This implies that
ρ 2 ( x n + 1 , z ) = ρ 2 ( P C ( ( 1 α n ) T ( x n ) α n T ( y n ) ) , z ) ρ 2 ( ( 1 α n ) T ( x n ) α n T ( y n ) , z ) ( 1 α n ) ρ 2 ( T ( x n ) , z ) + α n ρ 2 ( T ( y n ) , z ) R 2 α n ( 1 α n ) ρ 2 ( T ( x n ) , T ( y n ) ) ( 1 α n ) ρ 2 ( x n , z ) + α n ρ 2 ( y n , z ) R 2 α n ( 1 α n ) ρ 2 ( T ( x n ) , T ( y n ) ) ρ 2 ( x n , z ) R 2 α n ( 1 α n ) ρ 2 ( T ( x n ) , T ( y n ) ) ρ 2 ( x n , z ) .
Hence lim n ρ ( x n , z ) exists and
0 R 2 α n ( 1 α n ) ρ 2 ( T ( x n ) , T ( y n ) ) ρ 2 ( x n , z ) ρ 2 ( x n + 1 , z ) + α n [ ρ 2 ( y n , z ) ρ 2 ( x n , z ) ] .
So,
α n [ ρ 2 ( x n , z ) ρ 2 ( y n , z ) ] ρ 2 ( x n , z ) ρ 2 ( x n + 1 , z ) .
Since lim inf n α n > 0 , lim sup n [ ρ 2 ( x n , z ) ρ 2 ( y n , z ) ] = 0 . By (6), we have
R 2 β n ( 1 β n ) ρ 2 ( x n , T ( x n ) ) ρ 2 ( x n , z ) ρ 2 ( y n , z ) .

This implies by (ii) that lim n ρ ( x n , T ( x n ) ) = 0 . By Lemma 3.5, ω w ( x n ) consists of exactly one point and is contained in F ( T ) . This shows that { x n } Δ-converges to an element of F ( T ) . □

The following lemma is also needed (cf. [[10], Lemma 4.2]).

Lemma 3.8 Let κ > 0 and ( X , ρ ) be a complete CAT ( κ ) space with diam ( X ) π / 2 ε κ for some ε ( 0 , π / 2 ) . Let { x n } and { y n } be sequences in X with lim n ρ ( x n , y n ) = 0 . If Δ - lim n x n = x and Δ - lim n y n = y , then x = y .

Theorem 3.9 Let κ > 0 and ( X , ρ ) be a complete CAT ( κ ) space with diam ( X ) π / 2 ε κ for some ε ( 0 , π / 2 ) . Let C be a nonempty closed convex subset of X, and T , S : C X be a two generalized hybrid mappings with F ( T ) F ( S ) . Let { α n } and { β n } be a sequence in [ 0 , 1 ] and define a sequence { x n } in C by
{ x 1 C chosen arbitrary , x n + 1 : = P C ( ( 1 α n ) x n α n T ( y n ) ) , y n : = P C ( ( 1 β n ) x n β n S ( x n ) ) .
Let R = ( π 2 ε ) tan ( ε ) and suppose that
  1. (i)

    lim inf n α n ( 1 α n ) > 0 ,

     
  2. (ii)

    k 2 T ( z ) = 0 and lim inf n β n [ ( 1 β n ) R 2 k 2 S ( z ) 1 a 3 S ( z ) ] > 0 for all z F ( T ) F ( S ) .

     

Then { x n } Δ-converges to a common fixed point of S and T.

Proof Let z F ( T ) F ( S ) . Since k 2 T ( z ) = 0 , ρ ( T ( x ) , z ) ρ ( x , z ) for all x C . By Lemmas 2.5 and 2.6, we have
ρ 2 ( y n , z ) = ρ 2 ( P C ( ( 1 β n ) x n β n S ( x n ) ) , z ) ρ 2 ( ( 1 β n ) x n β n S ( x n ) , z ) ( 1 β n ) ρ 2 ( x n , z ) + β n ρ 2 ( S ( x n ) , z ) R 2 β n ( 1 β n ) ρ 2 ( x n , S ( x n ) ) ( 1 β n ) ρ 2 ( x n , z ) + β n [ ρ 2 ( x n , z ) + k 2 S ( z ) 1 a 3 S ( z ) ρ 2 ( S ( x n ) , x n ) ] R 2 β n ( 1 β n ) ρ 2 ( x n , S ( x n ) ) ρ 2 ( x n , z ) β n [ ( 1 β n ) R 2 k 2 S ( z ) 1 a 3 S ( z ) ] ρ 2 ( S ( x n ) , x n ) .
(7)
By (ii), there exist δ > 0 and N N such that
β n [ ( 1 β n ) R 2 k 2 S ( z ) 1 a 3 S ( z ) ] δ > 0 for all  n N .
Without loss of generality, we may assume that
β n [ ( 1 β n ) R 2 k 2 S ( z ) 1 a 3 S ( z ) ] > 0 for all  n N .
By (7), ρ ( y n , z ) ρ ( x n , z ) . Thus
ρ 2 ( x n + 1 , z ) = ρ 2 ( P C ( ( 1 α n ) x n α n T ( y n ) ) , z ) ρ 2 ( ( 1 α n ) x n α n T ( y n ) , z ) ( 1 α n ) ρ 2 ( x n , z ) + α n ρ 2 ( T ( y n ) , z ) R 2 α n ( 1 α n ) ρ 2 ( x n , T ( y n ) ) ( 1 α n ) ρ 2 ( x n , z ) + α n ρ 2 ( y n , z ) R 2 α n ( 1 α n ) ρ 2 ( x n , T ( y n ) ) ρ 2 ( x n , z ) R 2 α n ( 1 α n ) ρ 2 ( x n , T ( y n ) ) ρ 2 ( x n , z ) .
(8)
Hence lim n ρ ( x n , z ) exists and
lim n α n ( 1 α n ) ρ 2 ( x n , T ( y n ) ) = 0 .
By (i), lim n ρ 2 ( x n , T ( y n ) ) = 0 . It follows from (8) that
0 R 2 α n ( 1 α n ) ρ 2 ( x n , T ( y n ) ) ρ 2 ( x n , z ) ρ 2 ( x n + 1 , z ) + α n [ ρ 2 ( y n , z ) ρ 2 ( x n , z ) ] .
Thus
α n ( 1 α n ) [ ρ 2 ( x n , z ) ρ 2 ( y n , z ) ] ρ 2 ( x n , z ) ρ 2 ( x n + 1 , z ) .
Again, by (i), lim sup n [ ρ 2 ( x n , z ) ρ 2 ( y n , z ) ] = 0 . By (7), we have
β n [ ( 1 β n ) R 2 k 2 S ( z ) 1 a 3 S ( z ) ] ρ 2 ( x n , S ( x n ) ) ρ 2 ( x n , z ) ρ 2 ( y n , z ) .
This implies by (ii) that lim n ρ ( x n , S ( x n ) ) = 0 . Hence,
lim sup n ρ ( y n , x n ) = lim sup n ρ ( P C ( ( 1 β n ) x n β n S ( x n ) ) , P C ( x n ) ) lim sup n ρ ( ( 1 β n ) x n β n S ( x n ) , x n ) = lim sup n β n ρ ( S ( x n ) , x n ) = 0 .

So, lim n ρ ( y n , T ( y n ) ) = 0 . By Lemma 3.5, there exist u , v C such that ω w ( x n ) = { u } F ( S ) and ω w ( y n ) = { v } F ( T ) . This means that Δ - lim n x n = u and Δ - lim n y n = v . Hence, by Lemma 3.8, u = v and the proof is complete. □

Declarations

Acknowledgements

The authors thank Chiang Mai University for financial support.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai, 50200, Thailand

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© Nanjaras and Panyanak; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

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