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Generalized hybrid mappings on CAT(κ) spaces

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.

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 κ0. 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 xX to yX (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)=(1α)ρ(x,y)andρ(y,z)=αρ(x,y).

In this case, we write z=αx(1α)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,yX (for x,yX 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 nN. 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 κR, 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 ¯ )andρ(z,x)= d M κ 2 ( z ¯ , x ¯ ).

If κ0 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 κ0, 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,zX 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 ) (1α) ρ 2 (x,y)+α ρ 2 (x,z)α(1α) ρ 2 (y,z)

for all α[0,1] and x,y,zX. 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,zX, we have

ρ 2 ( x , ( 1 α ) y α z ) (1α) ρ 2 (x,y)+α ρ 2 (x,z) R 2 α(1α) 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=(π2ε)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 xX 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 xC and yC with y P C (x), then P C ( x ) (x,y)π/2;

  3. (iii)

    for each yC, 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 xX, 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 xX 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:BA 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:CX 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,yC;

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

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

A point xC 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:CX be a generalized hybrid mapping with 2 k 1 ( x ) 1 a 2 ( x ) < R 2 for all xC where R=(π2ε)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 zC and z=T(z).

Proof Since Δ- lim n x n =z, by Lemma 2.8, zC. 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:CC 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 zC 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:CC be a generalized hybrid mapping with k 1 (x)= k 2 (x)=0 for all xC. Then T has a fixed point.

Proof Fix xC and define x n := T n (x) for nN. Suppose that A({ x n })={z}. Then by Lemma 2.8, zC. 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:CC be a generalized hybrid mapping with k 1 (x)= k 2 (x)=0 for all xC. 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:CX be a generalized hybrid mapping with 2 k 1 ( x ) 1 a 2 ( x ) < R 2 for all xC where R=(π2ε)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 vF(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 =vC. By Theorem 3.1, vF(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:CX 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=(π2ε)tan(ε) and suppose that

  1. (i)

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

  2. (ii)

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

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

Proof Let zF(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 xC.

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 NN such that

α n [ ( 1 α n ) R 2 k 2 ( z ) 1 a 3 ( z ) ] δ>0for all nN.

Without loss of generality, we may assume that

α n [ ( 1 α n ) R 2 k 2 ( z ) 1 a 3 ( z ) ] >0for all nN.
(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:CX 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 zF(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 zF(T). By (i), we have ρ(T(x),z)ρ(x,z) for all xC. Let R=(π2ε)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:CX 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=(π2ε)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 zF(T)F(S).

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

Proof Let zF(T)F(S). Since k 2 T (z)=0, ρ(T(x),z)ρ(x,z) for all xC. 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 NN such that

β n [ ( 1 β n ) R 2 k 2 S ( z ) 1 a 3 S ( z ) ] δ>0for all nN.

Without loss of generality, we may assume that

β n [ ( 1 β n ) R 2 k 2 S ( z ) 1 a 3 S ( z ) ] >0for all nN.

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,vC 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. □

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The authors thank Chiang Mai University for financial support.

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Nanjaras, B., Panyanak, B. Generalized hybrid mappings on CAT(κ) spaces. J Inequal Appl 2014, 403 (2014). https://doi.org/10.1186/1029-242X-2014-403

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