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The Mann algorithm in a complete geodesic space with curvature bounded above

Abstract

The purpose of this paper is to prove two Δ-convergence theorems of the Mann algorithm to a common fixed point for a countable family of mappings in the case of a complete geodesic space with curvature bounded above by a positive number. The first one for nonexpansive mappings improves the recent result of He et al. (Nonlinear Anal. 75:445-452, 2012). The last one is proved for quasi-nonexpansive mappings and applied to the problem of finding a common fixed point of a countable family of quasi-nonexpansive mappings.

MSC:49J53.

1 Introduction

For a real number κ, a CAT(κ) space is defined by a geodesic metric space whose geodesic triangle is sufficiently thinner than the corresponding comparison triangle in a model space with curvature κ. The concept of these spaces has been studied by a large number of researchers. We know that any CAT(κ) space is a CAT( κ ) space for κ >κ (see [1]), thus all results for CAT(0) spaces can immediately be applied to any CAT(κ) with κ0. Moreover, CAT(κ) spaces with positive κ can be treated as CAT(1) spaces by changing the scale of the space. So we are interested in CAT(1) spaces.

One of the most important analytical problems is the existence of fixed points for nonlinear mappings. In the case for nonexpansive mappings in a CAT(κ) space was proved by Kirk [2, 3] for κ0, and by Espánola and Fernández-León [4] for κ>0. In the cases when at least one fixed point exists, it is natural to wonder whether such a fixed point can be approximated by iterations. There are many methods for approximating fixed points of a nonexpansive mapping T. One of the most successful methods is the Mann algorithm [5] which is defined in a geodesic space X by x 1 X and

x n + 1 = t n x n (1 t n )T x n for all nN,
(1.1)

where { t n } is a sequence in [0,1]. By using this algorithm, He et al. [6] proved the following result.

Theorem 1.1 Let X be a complete CAT(1) space and T:XX be a mapping. Let { x n } be the Mann algorithm (1.1) in X. Suppose that

  • (C0′) T is nonexpansive with F(T);

  • (C1′) d( x 1 ,F(T))<π/4;

  • (C2) n = 1 t n (1 t n )=.

Then the sequence { x n } Δ-converges to a fixed point of T.

Motivated by these results, we prove two Δ-convergence theorems of the Mann algorithm to a common fixed point for a countable family of mappings in complete CAT(1) spaces. The first one for nonexpansive mappings improves Theorem 1.1. The last one is proved for quasi-nonexpansive mappings and applied to the problem of finding a common fixed point of a countable family of quasi-nonexpansive mappings.

2 Preliminaries

Let X be a metric space with a metric d and let x,yX with d(x,y)=l. A geodesic path from x to y is an isometry c:[0,l]X such that c(0)=x and c(l)=y. The image of a geodesic path from x to y is called a geodesic segment joining x and y. Let r(0,]. If for every x,yX with d(x,y)<r, a geodesic from x to y exists, then we say that X is r-geodesic. Moreover, if such a geodesic is unique for each pair of points, then X is said to be r-uniquely geodesic.

A geodesic segment joining x and y is not necessarily unique in general. When it is unique, this geodesic segment is denoted by [x,y]. We write z[x,y] if and only if there exists t[0,1] such that d(z,x)=(1t)d(x,y) and d(z,y)=td(x,y). In this case, we will write z=tx(1t)y for simplicity. A geodesic triangle (x,y,z) consists of three points x,y,zX and geodesic segments [y,z], [z,x] and [x,z] joining two of them. We write w(x,y,z) if w[y,z][z,x][x,y].

To define a CAT(κ) space, we use the following notation called model space. For κ=0, the two-dimensional model space M κ 2 = M 0 2 is the Euclidean space R 2 with the metric induced from the Euclidean norm. For κ>0, M κ 2 is the two-dimensional sphere (1/ κ ) S 2 whose metric is a length of a minimal great arc joining each two points. For κ<0, M κ 2 is the two-dimensional hyperbolic space (1/ κ ) H 2 with the metric defined by a usual hyperbolic distance.

The diameter of M κ 2 is denoted by D κ , that is, D κ =π/ κ if κ>0 and D κ = if κ0. We know that M κ 2 is a D κ -uniquely geodesic space for each κR.

Let κR. For (x,y,z) in a geodesic space X satisfying that d(x,y)+d(y,z)+d(x,z)<2 D κ , there exist points x ¯ , y ¯ , z ¯ M κ 2 such that d(x,y)= d M κ 2 ( x ¯ , y ¯ ), d(y,z)= d M κ 2 ( y ¯ , z ¯ ) and d(x,z)= d M κ 2 ( x ¯ , z ¯ ). We call the triangle having vertices x ¯ , y ¯ and z ¯ in M κ 2 a comparison triangle of (x,y,z). Notice that it is unique up to an isometry of M κ 2 . For a specific choice of comparison triangles, we denote it by ( x ¯ , y ¯ , z ¯ ). A point p ¯ [ x ¯ , y ¯ ] is called a comparison point for p[x,y] if d(x,p)= d M κ 2 ( x ¯ , p ¯ ).

Let κR and X be a D κ -geodesic space. If for any x,y,zX with d(x,y)+d(y,z)+d(x,z)<2 D κ , for any p,q(x,y,z), and for their comparison points p ¯ , q ¯ ( x ¯ , y ¯ , z ¯ ), the inequality

d(p,q) d M κ 2 ( p ¯ , q ¯ )

holds, then we call X a CAT(κ) space. It is easy to see that all CAT(κ) spaces are D κ -uniquely geodesic; consider the triangle such that two of its vertices are identical.

Let { x n } be a bounded sequence in a metric space X. For xX, let r(x,{ x n }):= lim sup n d(x, x n ) and define the asymptotic radius r({ x n }) of { x n } by

r ( { x n } ) := inf x X r ( x , { x n } ) .

An element z of X is said to be an asymptotic center of { x n } if r(z,{ x n })=r({ x n }). We say that { x n } is Δ-convergent to xX if x is the unique asymptotic center of any subsequence of { x n }. The concept of Δ-convergence introduced by Lim in 1976 was shown by Kirk and Panyanak [7] in CAT(0) spaces to be very similar to the weak convergence in Banach space setting.

Remark 2.1

  1. (1)

    If { x n } is a sequence in a complete CAT(κ) space such that r({ x n })< D κ /2, then its asymptotic center consists of exactly one point.

  2. (2)

    Every sequence { x n } whose asymptotic radius is less than D κ /2 has a Δ-convergent subsequence (see [4, 7, 8]), that is, ω Δ ({ x n }):={xX:there exists { x n k }{ x n }such that { x n k }Δ-converges to x}.

We note that Remark 2.1(1) was proved by Dhompongsa et al. [9] for the case κ0, and by Espánola and Fernández-León [4] for the case κ>0.

Let X be a metric space with a metric d. A mapping T:XX is called nonexpansive if

d(Tx,Ty)d(x,y)for all x,yX.

A point xX is called a fixed point of T if x=Tx. We denote by F(T) the set of fixed points of T. The mapping T is called quasi-nonexpansive if F(T) and

d(Tx,p)d(x,p)for all xX and pF(T).

The following lemmas are essentially needed for our main results.

Lemma 2.2 ([[10], Lemma 1])

Let { a n } and { b n } be two sequences of nonnegative real numbers such that

a n + 1 a n + b n for all nN.

If n = 1 b n <, then lim n a n exists.

Lemma 2.3 ([[11], Lemma 5.4])

Let (x,y,z) be a geodesic triangle in a CAT(1) space such that d(x,y)+d(x,z)+d(y,z)<2π. Let u=tz(1t)x and v=tz(1t)y for some t[0,1]. If d(x,z)M, d(y,z)M, and sin((1t)M)sinM for some M(0,π), then

d(u,v) sin ( 1 t ) M sin M d(x,y).

Lemma 2.4 ([[12], Corollary 2.2])

Let (x,y,z) be a geodesic triangle in a CAT(1) space such that d(x,y)+d(x,z)+d(y,z)<2π. Let u=tx(1t)y for some t[0,1]. Then

cosd(u,z)sind(x,y)cosd(x,z)sin ( t d ( x , y ) ) +cosd(y,z)sin ( ( 1 t ) d ( x , y ) ) .

Lemma 2.5 ([[11], Lemma 3.1])

Let (x,y,z) be a geodesic triangle in a CAT(1) space such that d(x,y)+d(x,z)+d(y,z)<2π, and let t[0,1]. Then

cosd ( t x ( 1 t ) y , z ) tcosd(x,z)+(1t)cosd(y,z).

3 Main results

We start with some propositions which are common tools for proving the main results in the next two subsections.

Proposition 3.1 Let { x n } be a sequence of a complete CAT(1) space X such that r({ x n })<π/2. Suppose that lim n d( x n ,z) exists for all z ω Δ ({ x n }). Then { x n } Δ-converges to an element of ω Δ ({ x n }). Moreover, ω Δ ({ x n }) consists of exactly one point.

Proof Let x be the asymptotic center of { x n } and let { x n k } be any subsequence of { x n } with the asymptotic center y. We show that y=x and hence { x n } Δ-converges to x as desired. Since r({ x n k })r({ x n })<π/2, there exists a subsequence { x n k l } of { x n k } such that { x n k l } Δ-converges to z for some zX. Clearly, z ω Δ ({ x n }) and it follows from the assumption that lim n d( x n ,z) exists. Let uX. Then

lim n d ( x n , z ) = lim k d ( x n k , z ) = lim l d ( x n k l , z ) lim sup l d ( x n k l , u ) lim sup k d ( x n k , u ) lim sup n d ( x n , u ) .

Since

lim n d( x n ,z) lim sup n d( x n ,u)for all uX,

we have z=x. Since

lim k d( x n k ,z) lim sup k d( x n k ,u)for all uX,

we have z=y. This implies that y=x. □

Proposition 3.2 Let X be a complete CAT(1) space and { T n }:XX be a countable family of quasi-nonexpansive mappings with F:= n = 1 F( T n ). Let { x n } be a sequence in X such that d( x 1 ,F)<π/2 and

x n + 1 = t n x n (1 t n ) T n x n for all nN,

where { t n } is a sequence in [0,1]. Then

  1. (i)

    { x n } is well defined and r({ x n })<π/2. In particular, ω Δ ({ x n }) and d( x n + 1 ,p)d( x n ,p) for all nN whenever d( x 1 ,p)<π/2 and pF.

  2. (ii)

    If ω Δ ({ x n })F, then { x n } Δ-converges to an element of F.

Proof (i) Let pF be such that d( x 1 ,p)<π/2. Since d( T 1 x 1 ,p)d( x 1 ,p)<π/2, we have that d( x 1 , T 1 x 1 )<π. This implies that x 2 is well defined. It follows from Lemma 2.5 that

cos d ( x 2 , p ) = cos d ( t 1 x 1 ( 1 t 1 ) T 1 x 1 , p ) t 1 cos d ( x 1 , p ) + ( 1 t 1 ) cos d ( T 1 x 1 , p ) t 1 cos d ( x 1 , p ) + ( 1 t 1 ) cos d ( x 1 , p ) = cos d ( x 1 , p ) ,

and we have

d ( x 2 , p ) min { d ( x 1 , x 2 ) + d ( x 1 , p ) , d ( T x 1 , x 2 ) + d ( T x 1 , p ) } = min { ( 1 t 1 ) d ( x 1 , T x 1 ) + d ( x 1 , p ) , t 1 d ( x 1 , T x 1 ) + d ( T x 1 , p ) } min { 1 t 1 , t 1 } d ( x 1 , T x 1 ) + d ( x 1 , p ) 1 2 d ( x 1 , T x 1 ) + d ( x 1 , p ) < π 2 + π 2 = π .

Therefore, d( x 2 ,p)d( x 1 ,p)<π/2. Using mathematical induction, we can conclude that the sequence { x n } is well defined and

d( x n + 1 ,p)d( x n ,p)d( x 1 ,p)<π/2for all nN.

Then lim n d( x n ,p) exists which is less than π/2, and so r({ x n })<π/2. This implies that ω Δ ({ x n }).

(ii) Let u ω Δ ({ x n })F. Then there exists a subsequence { x n k } of { x n } such that { x n k } Δ-converges to u. Notice that r(u,{ x n k })=r({ x n k })r({ x n })<π/2. Thus there is NN such that d( x N ,u)<π/2. Similar to the first step, we have that d( x n + 1 ,u)d( x n ,u) for all nN. This implies that lim n d( x n ,u) exists. It follows immediately from Proposition 3.1 that { x n } Δ-converges to an element of F and the proof is finished. □

3.1 Countable nonexpansive mappings

The following concept is introduced by Aoyama et al. [13]. Let X be a complete metric space and { T n } be a countable family of mappings from X into itself with F:= n = 1 F( T n ). We say that ({ T n },T) satisfies AKTT-condition if

  • n = 1 sup{d( T n + 1 y, T n y):yY}< for each bounded subset Y of X;

  • Tx:= lim n T n x for all xX and F(T)=F.

Remark 3.3 Assume that ({ T n },T) satisfies AKTT-condition.

  1. (1)

    For each xX, we have { T n x} is a Cauchy sequence and hence the mapping T above is well defined.

  2. (2)

    If { x n } is bounded, then n = 1 d( T n + 1 x n , T n x n )<.

Theorem 3.4 Let X be a complete CAT(1) space and { T n } be a countable family of mappings from X into itself. Let { x n } be a sequence in X defined by x 1 X and

x n + 1 = t n x n (1 t n ) T n x n for all nN,

where { t n } is a sequence in [0,1]. Suppose that

  • (C0 n ) T n is nonexpansive for all nN and F:= n = 1 F( T n );

  • (C1) d( x 1 ,F)<π/2;

  • (C2) n = 1 t n (1 t n )=;

  • (C3′) ({ T n },T) satisfies AKTT-condition.

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

Proof We first show that lim n d( x n , T n x n ) exists. Using the nonexpansiveness of T n and the definition of { x n }, we obtain that

d ( x n + 1 , T n + 1 x n + 1 ) d ( x n + 1 , T n x n ) + d ( T n x n , T n x n + 1 ) + d ( T n x n + 1 , T n + 1 x n + 1 ) d ( x n + 1 , T n x n ) + d ( x n , x n + 1 ) + d ( T n x n + 1 , T n + 1 x n + 1 ) = d ( x n , T n x n ) + d ( T n x n + 1 , T n + 1 x n + 1 )

for all nN. It follows from Lemma 2.2 and n = 1 d( T n x n + 1 , T n + 1 x n + 1 )< that

lim n d( x n , T n x n )

exists.

Next, we show that lim n d( x n , T n x n )=0. Assume that lim n d( x n , T n x n )>0. Thus, without loss of generality, there is a positive real number A such that

Ad( x n , T n x n )<πfor all nN.

To get the right inequality of the preceding expression, let pF be such that d( x 1 ,p)<π/2. By Proposition 3.2(i), we have

d( x n , T n x n )d( x n ,p)+d( T n x n ,p)2d( x n ,p)<2d( x 1 ,p)<π.

Put A n :=d( x n , T n x n ) for all nN. By elementary trigonometry and Lemma 2.4, we get that

cos d ( x n + 1 , p ) sin A n = cos d ( t n x n ( 1 t n ) T n x n , p ) sin A n cos d ( x n , p ) sin ( t n A n ) + cos d ( T n x n , p ) sin ( ( 1 t n ) A n ) cos d ( x n , p ) ( sin ( t n A n ) + sin ( ( 1 t n ) A n ) ) ,

and it follows that

cos d ( x n + 1 , p ) cos d ( x n , p ) cos d ( x n , p ) ( sin ( t n A n ) + sin ( ( 1 t n ) A n ) sin A n 1 ) = 2 cos d ( x n , p ) sin ( t n A n / 2 ) sin ( ( 1 t n ) A n / 2 ) cos ( A n / 2 ) 2 cos d ( x 1 , p ) sin ( t n A / 2 ) sin ( ( 1 t n ) A / 2 ) cos ( A / 2 ) 2 t n ( 1 t n ) cos d ( x 1 , p ) sin 2 ( A / 2 ) cos ( A / 2 )

for all nN. Notice that cosd( x 1 ,p), cos(A/2), and sin(A/2) are positive. Consequently,

n = 1 t n (1 t n ) cos ( A / 2 ) 2 cos d ( x 1 , p ) sin 2 ( A / 2 ) n = 1 ( cos d ( x n + 1 , p ) cos d ( x n , p ) ) <,

which is a contradiction. Then we get that lim n d( x n , T n x n )=0 and hence

d ( x n , T x n ) d ( x n , T n x n ) + d ( T n x n , T x n ) d ( x n , T n x n ) + sup { d ( T n y , T y ) : y Y } 0 ,

where Y:={ x n }.

Finally, we show that { x n } Δ-converges to an element of F(T). To apply Proposition 3.2(ii), we show that ω Δ ({ x n })F(T). Let u ω Δ ({ x n }). Then there exists a subsequence { x n k } of { x n } such that { x n k } Δ-converges to u. Clearly, u is the unique asymptotic center of { x n k }. Using the nonexpansiveness of T and d( x n ,T x n )0, we get that

lim sup k d ( x n k , T u ) lim sup k d ( x n k , T x n k ) + lim sup k d ( T x n k , T u ) lim sup k d ( x n k , u ) .

This implies that Tu=u, that is, ω Δ ({ x n })F(T). This completes the proof. □

As an immediate consequence of Theorem 3.4, we obtain the following result.

Corollary 3.5 Let X be a complete CAT(1) space and T:XX be a mapping. Let { x n } be the Mann algorithm (1.1) in X. Suppose that

  • (C0′) T is nonexpansive with F(T);

  • (C1) d( x 1 ,F(T))<π/2;

  • (C2) n = 1 t n (1 t n )=.

Then the sequence { x n } Δ-converges to a fixed point of T.

Remark 3.6 Our Corollary 3.5 improves Theorem 3.1 of He et al. [6] (see Theorem 1.1) because (C1′) of Theorem 1.1 implies (C1) of Corollary 3.5. Moreover, (C1) is sharp in the sense that if d( x 1 ,F(T))=π/2, then we may construct the Mann algorithm for a nonexpansive mapping which is not Δ-convergent.

Example 3.7 Let S 2 be the unit sphere of the Euclidean space R 3 with the geodesic metric. Let T: S 2 S 2 be defined by

T(x,y,z)=(y,x,z)for all (x,y,z) S 2 .

Then T is nonexpansive and F(T)={(0,0,1),(0,0,1)}. Let { x n } be a sequence in S 2 defined by x 1 =(1,0,0) and

x n + 1 = 1 2 x n 1 2 T x n for all nN.

Then d( x 1 ,F(T))=π/2 and

x n + 1 = ( cos n π 4 , sin n π 4 , 0 ) for all nN.

It is easy to see that { x 8 n + 1 } has the unique asymptotic center which is {(1,0,0)} and { x 8 n + 3 } has the unique asymptotic center which is {(0,1,0)}. Hence, { x n } is not Δ-convergent.

3.2 Countable quasi-nonexpansive mappings

In this subsection, we give a supplement result to Theorem 3.4. Obviously, every nonexpansive mapping with a fixed point is quasi-nonexpansive. Moreover, if T is nonexpansive, then T is Δ-demiclosed [11], that is, if for any Δ-convergent sequence { x n } in X, its Δ-limit belongs to F(T) whenever lim n d( x n ,T x n )=0.

In the following theorem, we deal with quasi-nonexpansive mappings satisfying Δ-demiclosedness. This interesting class of mappings includes the metric projections [11]. However, there are many metric projections such that they are not nonexpansive.

Theorem 3.8 Let X be a complete CAT(1) space and { T n } be a countable family of mappings from X into itself. Let { x n } be a sequence in X defined by x 1 X and

x n + 1 = t n x n (1 t n ) T n x n for all nN,

where { t n } is a sequence in (0,1). Suppose that

  • (C0n) T n is quasi-nonexpansive for all nN and F= n = 1 F( T n );

  • (C1) d( x 1 ,F)<π/2;

  • (C2′) lim inf n t n (1 t n )>0;

  • (C3) there exists a mapping T:XX such that

    • { T n }converges uniformly to T on each bounded subset of X;

    • F(T)=F;

  • (C4) T is Δ-demiclosed.

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

Remark 3.9 Let us compare Theorems 3.4 and 3.8:

  1. (1)

    (C0 n ) (C0n);

  2. (2)

    (C3′) (C3);

  3. (3)

    (C0′) and (C3′) (C4);

  4. (4)

    (C2′) (C2).

Proof of Theorem 3.8 We first show that lim n d( x n , T n x n )=0. Let pF(T) be such that d( x 1 ,p)<π/2. We have that A:= lim n d( x n ,p) exists which is less than π/2 and r({ x n })<π/2 by Proposition 3.2(i). Put B:= lim sup n d( x n , T n x n ). Notice that B<π. Since lim inf n t n (1 t n )>0, we may assume that there exists a subsequence { n k } of such that lim k d( x n k , T n k x n k )=B and t n k t(0,1). Using the quasi-nonexpansiveness of T n and Lemma 2.4, we get that

cos d ( x n k + 1 , p ) sin d ( x n k , T n k x n k ) = cos d ( t n k x n k ( 1 t n k ) T n k x n k , p ) sin d ( x n k , T n k x n k ) cos d ( x n k , p ) sin ( t n k d ( x n k , T n k x n k ) ) + cos d ( T n k x n k , p ) sin ( ( 1 t n k ) d ( x n k , T n k x n k ) ) cos d ( x n k , p ) ( sin ( t n k d ( x n k , T n k x n k ) ) + sin ( ( 1 t n k ) d ( x n k , T n k x n k ) ) ) .

Letting k yields

cosAsinBcosA ( sin t B + sin ( 1 t ) B ) .

Using elementary trigonometry, we get that B=0. Hence it follows that

lim n d( x n , T n x n )=0.

By condition (C3), we get that

d ( x n , T x n ) d ( x n , T n x n ) + d ( T n x n , T x n ) d ( x n , T n x n ) + sup { d ( T n y , T y ) : y Y } 0 ,

where Y:={ x n }.

Finally, we show that ω Δ ({ x n })F(T). Let u ω Δ ({ x n }). Then there exists a subsequence { x n k } of { x n } such that { x n k } Δ-converges to u. It follows from the Δ-demiclosedness of T and d( x n ,T x n )0 that Tu=u, that is, ω Δ ({ x n })F(T). Hence the result follows from Proposition 3.2(ii). The proof is now finished. □

As an immediate consequence of Theorem 3.8, we obtain the following result.

Corollary 3.10 Let X be a complete CAT(1) space and T:XX be a mapping with F(T). Let { x n } be the Mann algorithm (1.1) in X. Suppose that

  • (C0) T is quasi-nonexpansive and Δ-demiclosed;

  • (C1) d( x 1 ,F(T))<π/2;

  • (C2′) lim inf n t n (1 t n )>0.

Then the sequence { x n } Δ-converges to a fixed point of T.

Question 3.11 We do not know whether the conclusion of Corollary 3.10 holds if (C2′) is replaced by the more general condition (C2).

Let { T n }:XX be a countable family of quasi-nonexpansive mappings with F:= n = 1 F( T n ). We next show how to generate a family { W n } and a mapping W satisfying (C3′) and (C4), and hence Theorem 3.8 is applicable.

Theorem 3.12 Let X be a complete CAT(1) space such that d(u,v)<π/2 for all u,vX, and let { T n }:XX be a countable family of quasi-nonexpansive mappings with F:= n = 1 F( T n ). Then there exist a family of quasi-nonexpansive mappings { W n }:XX and a quasi-nonexpansive mapping W:XX such that

  1. (i)

    ({ W n },W) satisfies AKTT-condition and F(W)= n = 1 F( W n )=F;

  2. (ii)

    W is Δ-demiclosed whenever T n is Δ-demiclosed for all nN.

To prove Theorem 3.12, we need the following lemmas.

Lemma 3.13 ([14])

Let X be a complete CAT(1) space such that d(u,v)<π/2 for all u,vX, and let S,T:XX be quasi-nonexpansive mappings with F(S)F(T). Then, for each 0<t<1, F(S)F(T)=F(tS(1t)T) and the mapping tS(1t)T is quasi-nonexpansive.

The following lemma is essentially proved in [11]. For the sake of completeness, we show the proof.

Lemma 3.14 Let X be a complete CAT(1) space such that d(u,v)<π/2 for all u,vX, and let S,T:XX be quasi-nonexpansive mappings with F(S)F(T). Let { x n } be a sequence of X. If d( x n , 1 2 S x n 1 2 T x n )0, then d( x n ,S x n )0 and d( x n ,T x n )0.

Proof Put W:= 1 2 S 1 2 T. By Lemma 3.13, we have that W is quasi-nonexpansive and F(W)=F(S)F(T). Let pF(W). By Lemma 2.4 and the quasi-nonexpansiveness of S and T, we get that

2 cos d ( W x n , p ) sin d ( S x n , T x n ) 2 cos d ( S x n , T x n ) 2 = cos d ( 1 2 S x n 1 2 T x n , p ) sin d ( S x n , T x n ) cos d ( S x n , p ) sin d ( S x n , T x n ) 2 + cos d ( T x n , p ) sin d ( S x n , T x n ) 2 2 cos d ( x n , p ) sin d ( S x n , T x n ) 2 .

This implies that

d(S x n ,T x n )=0orcos d ( S x n , T x n ) 2 cos d ( x n , p ) cos d ( W x n , p ) .

It follows from d( x n ,W x n )0 that cos d ( x n , p ) cos d ( W x n , p ) 1, that is, d(S x n ,T x n )0. Thus

d ( x n , S x n ) d ( x n , 1 2 S x n 1 2 T x n ) + d ( 1 2 S x n 1 2 T x n , S x n ) = d ( x n , 1 2 S x n 1 2 T x n ) + d ( S x n , T x n ) 2 0 .

Hence d( x n ,S x n )0. Similarly, d( x n ,T x n )0 and the proof is finished. □

We are now ready to prove Theorem 3.12.

Proof of Theorem 3.12 Put S n := 1 2 I 1 2 T n for all nN. We define a family of mappings { W n }:XX by

W 1 x = S 1 x ; W 2 x = 1 2 S 1 x 1 2 S 2 x ; W 3 x = 1 2 S 1 x 1 2 ( 1 2 S 2 x 1 2 S 3 x ) ; W n x = 1 2 S 1 x 1 2 ( 1 2 S 2 x 1 2 ( 1 2 ( 1 2 S n 1 x 1 2 S n x ) ) ) ;

It follows from Lemma 3.13 that W n is quasi-nonexpansive for all nN and n = 1 F( W n )= n = 1 F( T n ).

We first show that n = 1 sup{d( W n + 1 x, W n x):xX}<. For kN, put V k ( k ) := S k and

V n ( k ) := 1 2 S k 1 2 ( 1 2 S k + 1 1 2 ( 1 2 ( 1 2 S n 1 1 2 S n ) ) )

for all n>k. By Lemma 2.3, we have that

d ( W n + 1 x , W n x ) = d ( 1 2 S 1 x 1 2 V n + 1 ( 2 ) x , 1 2 S 1 x 1 2 V n ( 2 ) x ) ( 2 2 ) d ( 1 2 S 2 x 1 2 V n + 1 ( 3 ) x , 1 2 S 2 x 1 2 V n ( 3 ) x ) ( 2 2 ) 2 d ( 1 2 S 3 x 1 2 V n + 1 ( 4 ) x , 1 2 S 3 x 1 2 V n ( 4 ) x ) ( 2 2 ) n 1 d ( 1 2 S n x 1 2 S n + 1 x , S n x ) ( 2 2 ) n 1 π 2

for all xX and nN. Then

sup { d ( W n + 1 x , W n x ) : x X } ( 2 2 ) n 1 π 2

and the result follows. In particular, { W n x} is a Cauchy sequence for each xX. We now define the mapping W:XX by

Wx:= lim n W n xfor all xX.

Next, we show that F(W)= n = 1 F( T n ). It is easy to see that n = 1 F( T n )F(W). On the other hand, let q n = 1 F( T n )= n = 1 F( S n ) and pF(W). We prove that pF( T k ) for all kN. Let kN be given. For any n>k, it follows from Lemma 2.5 that

cos d ( q , W n p ) = cos d ( q , 1 2 S 1 p 1 2 ( 1 2 S 2 p 1 2 ( 1 2 ( 1 2 S n 1 p 1 2 S n p ) ) ) ) 1 2 cos d ( q , S 1 p ) + 1 2 cos d ( q , 1 2 S 2 p 1 2 ( 1 2 ( 1 2 S n 1 p 1 2 S n p ) ) ) 1 2 cos d ( q , S 1 p ) + + 1 2 k cos d ( q , S k p ) + + 1 2 n 1 cos d ( q , S n 1 p ) + 1 2 n 1 cos d ( q , S n p ) ( 1 2 + + 1 2 k 1 + 1 2 k + 1 + + 1 2 n 1 + 1 2 n 1 ) cos d ( q , p ) + 1 2 k cos d ( q , S k p ) = ( 1 1 2 k ) cos d ( q , p ) + 1 2 k cos d ( q , S k p ) .

Letting n yields W n pWp=p and

cosd(q,p)=cosd(q,Wp) ( 1 1 2 k ) cosd(q,p)+ 1 2 k cosd(q, S k p).

This implies that cosd(q,p)=cosd(q, S k p). It follows from Lemma 2.4 that

cos d ( q , p ) sin d ( p , T k p ) = cos d ( q , S k p ) sin d ( p , T k p ) = cos d ( q , 1 2 p 1 2 T k p ) sin d ( p , T k p ) cos d ( q , p ) sin d ( p , T k p ) 2 + cos d ( q , T k p ) sin d ( p , T k p ) 2 2 cos d ( q , p ) sin d ( p , T k p ) 2 .

Using elementary trigonometry, we get that d(p, T k p)=0. Since k is arbitrary, we have p n = 1 F( T n ). Hence (i) is proved.

Finally, we prove (ii). We assume that T n is Δ-demiclosed for all nN. We show that W is Δ-demiclosed. Let { x n }X be such that lim n d( x n ,W x n )=0 and { x n } Δ-converges to xX. It follows from the definitions of { W n } and { V n ( 2 ) } that

W n = 1 2 S 1 1 2 V n ( 2 ) for all n2.

Similar to the proof of the first and the second steps, we can define the quasi-nonexpansive mapping V:XX by

Vx:= lim n V n ( 2 ) xfor all xX

and F(V)= n = 2 F( T n ). This implies that

W= 1 2 S 1 1 2 VandF(W)=F( S 1 )F(V).

Then, by Lemma 3.14, we obtain that d( x n , S 1 x n )0 and d( x n ,V x n )0. Thus

d( x n , T 1 x n )=2d ( x n , 1 2 x n 1 2 T 1 x n ) =2d( x n , S 1 x n )0.

Since T 1 is Δ-demiclosed, we have xF( T 1 ). Continuing this procedure gives x n = 1 F( T n )=F(W). This completes the proof. □

References

  1. Bridson MR, Haefliger A Grundlehren der Mathematischen Wissenschaften 319. In Metric Spaces of Non-Positive Curvature. Springer, Berlin; 1999. (Fundamental Principles of Mathematical Sciences)

    Chapter  Google Scholar 

  2. Kirk WA: Geodesic geometry and fixed point theory. Colecc. Abierta 64. In Seminar of Mathematical Analysis. Univ. Sevilla Secr. Publ., Seville; 2003:195–225.

    Google Scholar 

  3. Kirk WA: Geodesic geometry and fixed point theory II. In International Conference on Fixed Point Theory and Applications. Yokohama Publ., Yokohama; 2004:113–142.

    Google Scholar 

  4. Espínola R, Fernández-León A:CAT(k)-Spaces, weak convergence and fixed points. J. Math. Anal. Appl. 2009, 353: 410–427. 10.1016/j.jmaa.2008.12.015

    MathSciNet  Article  Google Scholar 

  5. Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3

    Article  Google Scholar 

  6. He JS, Fang DH, López G, Li C:Mann’s algorithm for nonexpansive mappings in CAT(κ) spaces. Nonlinear Anal. 2012, 75: 445–452. 10.1016/j.na.2011.07.070

    MathSciNet  Article  Google Scholar 

  7. Kirk WA, Panyanak B: A concept of convergence in geodesic spaces. Nonlinear Anal. 2008, 68: 3689–3696. 10.1016/j.na.2007.04.011

    MathSciNet  Article  Google Scholar 

  8. Goebel K, Kirk WA Cambridge Studies in Advanced Mathematics 28. In Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.

    Chapter  Google Scholar 

  9. Dhompongsa S, Kirk WA, Sims B: Fixed points of uniformly Lipschitzian mappings. Nonlinear Anal. 2006, 65: 762–772. 10.1016/j.na.2005.09.044

    MathSciNet  Article  Google Scholar 

  10. Tan K-K, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. J. Math. Anal. Appl. 1993, 178: 301–308. 10.1006/jmaa.1993.1309

    MathSciNet  Article  Google Scholar 

  11. Kimura Y, Satô K: Halpern iteration for strongly quasinonexpansive mappings on a geodesic space with curvature bounded above by one. Fixed Point Theory Appl. 2013., 2013: Article ID 14

    Google Scholar 

  12. Kimura Y, Satô K: Convergence of subsets of a complete geodesic space with curvature bounded above. Nonlinear Anal. 2012, 75: 5079–5085. 10.1016/j.na.2012.04.024

    MathSciNet  Article  Google Scholar 

  13. Aoyama K, Kimura Y, Takahashi W, Toyoda M: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Anal. 2007, 67: 2350–2360. 10.1016/j.na.2006.08.032

    MathSciNet  Article  Google Scholar 

  14. Kimura, Y, Satô, K: Image recovery problem on a geodesic space with curvature bounded above by one (submitted)

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Acknowledgements

The authors thank the referee for their suggestions which enhanced the presentation of the paper. The first author is supported by Grant-in-Aid for Scientific Research No. 22540175 from Japan Society for the Promotion of Science. The second author is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education of Thailand. The last author is supported by the Thailand Research Fund through the Royal Golden Jubilee Ph.D. Program (Grant No. PHD/0188/2552) and Khon Kaen University under the RGJ-Ph.D. scholarship.

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Kimura, Y., Saejung, S. & Yotkaew, P. The Mann algorithm in a complete geodesic space with curvature bounded above. Fixed Point Theory Appl 2013, 336 (2013). https://doi.org/10.1186/1687-1812-2013-336

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Keywords

  • CAT(κ) space
  • fixed point
  • Mann algorithm
  • nonexpansive mapping
  • quasi-nonexpansive mapping
  • Δ-convergence