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Convergence and stability of generalized φ-weak contraction mapping in CAT(0) spaces

  • Kyung Soo Kim EMAIL logo
Published/Copyright: September 6, 2017
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Open Mathematics
From the journal Open Mathematics Volume 15 Issue 1

Abstract

The aim of this paper is to prove some fixed point results for generalized φ-weak contraction mapping and study a new concept of stability which is called comparably almost T-stable by using iterative schemes in CAT(0) spaces.

Keywords: CAT(0) space; Generalized φ-weak contraction mapping; Fixed point; Ishikawa iterative scheme; Comparably almost T-stable
MSC 2010: 54H25; 34D30; 54E35; 47H09; 47H10

1 Introduction and preliminaries

Let (X, d) be a metric space. A mapping T : X → X is a contraction if there exists a constant α ∈ (0, 1) such that

d(Tx,Ty)≤α⋅d(x,y),∀x,y∈X.

A mapping T : X → X is a φ-weak contraction if there exists a continuous and nondecreasing function φ : [0, ∞) → [0, ∞) with φ−1(0) = {0} and limt→∞φ(t) = ∞ such that

d(Tx,Ty)≤d(x,y)−φ(d(x,y)),∀x,y∈X. (1)

If X is bounded, then the infinity condition can be omitted.

The concept of the φ-weak contraction was introduced by Alber and Guerre-Delabriere [1] in 1997, who proved the existence of fixed points in Hilbert spaces. Later Rhoades [2] in 2001 extended the results of [1] to metric spaces.

Theorem l.1

([2]). Let (X, d) be a complete metric space, T : X → X be a φ-weak contractive self-map on X. T has a unique fixed point p in X.

Remark 1.2

Theorem 1.1 is one of generalizations of the Banach contraction principle because it takes φ(t) = (1 − α)t for α ∈ (0, 1), then φ-weak contraction contains contraction as special cases.

In 2016, Xue [3] introduced a new contraction type mapping as follows.

Definition 1.3

([3]). A mapping T : X → X is a generalized φ-weak contraction if there exists a continuous and nondecreasing function φ : [0, ∞) → [0, ∞) with φ(0) = 0 such that

d(Tx,Ty)≤d(x,y)−φ(d(Tx,Ty)),∀x,y∈X (2)

holds.

We notice immediately that if T : X → X is φ-weak contraction, then T satisfies the following inequality

d(Tx,Ty)≤d(x,y)−φ(d(Tx,Ty)),∀x,y∈X.

However, the converse is not true in general.

Example 1.4

([4]). Let X = (−∞, + ∞) be endowed with the Euclidean metric d(x, y) = |x−y| and let Tx=25x for each x ∈ X. Define φ(t) : [0, +∞) → [0, +∞) by φ(t)=43t. Then T satisfies (2), but T does not satisfy inequality (1). Indeed,

d(Tx,Ty)=|25x−25y|≤|x−y|−43⋅25|x−y|=d(x,y)−φ(d(Tx,Ty))

and

d(Tx,Ty)=|25x−25y|≥|x−y|−43|x−y|=d(x,y)−φ(d(x,y))

for all x, y ∈ X.

Example 1.5

([3]). Let X = [0, +∞) be endowed by d(x, y) = |x− y| and let Tx=x1+x for each x ∈ X. Define φ : [0, +∞) → [0, +∞) by φ(t)=t21+t. Then

d(Tx,Ty)=|x1+x−y1+y|=|x−y|(1+x)(1+y)≤|x−y|1+|x−y|=|x−y|−|x−y|21+|x−y|=d(x,y)−φ(d(x,y))

holds for all x, y ∈ X. So T is a φ-weak contraction. However, T is not a contraction.

Remark 1.6

The above examples show that the class of generalized φ-weak contractions properly includes the class of φ-weak contractions and the class of φ-weak contractions properly includes the class of contractions. In fact, let T : X → X be a contraction, there exists α ∈ (0, 1) such that

d(Tx,Ty)≤α⋅d(x,y),∀x,y∈X.

Then

d(Tx,Ty)≤α⋅d(x,y)=d(x,y)−(1−α)d(x,y)=d(x,y)−φ(d(x,y)),

where, φ(d(x, y)) = (1 −, α)d(x, y) . So, T is a φ-weak contraction. Moreover, let T be a φ-weak contraction, from property of φ, we have d(Tx, Ty) ≤ d(x, y) and

φ(d(Tx,Ty))≤φ(d(x,y)),

From (1),

d(Tx,Ty)≤d(x,y)−φ(d(x,y))≤d(x,y)−φ(d(Tx,Ty)),∀x,y∈X.

Therefore, T is a generalized φ-weak contraction.

In the meantime, if T is a φ-weak contractive self mapping for one mapping φ so we do not expect that the φ-weak contractivity should be satisfied with the same function φ. Let us suppose that T is a φ-weak contractive self mapping and consider

φ~(x)=min{φ(x/2);x/2}.

Then, if d(Tx,Ty)>12d(x,y), we have

d(Tx,Ty)≤d(x,y)−φ(d(Tx,Ty))≤d(x,y)−φ(12d(x,y))

on account of monotonocity of φ and finally

d(Tx,Ty)≤d(x,y)−φ~(d(x,y)).

On the other hand, if d(Tx,Ty)<12d(x,y), we get

d(Tx,Ty)<d(x,y)−12d(x,y)≤d(x,y)−φ~(d(x,y)).

So T is just the φ~ -weak contractive mapping. The continuity and monotonocity of φ~ follows directly from properties of min function, φ and the metric.

One of the most interesting aspects of metric fixed point theory is to extend a linear version of known result to the nonlinear case in metric spaces. To achieve this, Takahashi [5] introduced a convex structure in a metric space (X, d). A mapping W : X × X × [0, 1] → X is a convex structure in X if

d(u,W(x,y,λ))≤λd(u,x)+(1−λ)d(u,y)

for all x, y ∈ X and λ ∈ [0, 1]. A metric space with a convex structure W is known as a convex metric space which is denoted by (X, d, W). A nonempty subset K of a convex metric space is said to be convex if

W(x,y,λ)∈K

for all x, y ∈ K and λ ∈ [0, 1]. In fact, every normed linear space and its convex subsets are convex metric spaces but the converse is not true, in general (see, [5]).

Example 1.7

([6]). Let X = {(x1, x2) ∈ ℝ2 : x1 > 0, x2 > 0. For all x = (x1, x2), y = (y1, y2) ∈ X and λ ∈ [0, 1]. We define a mapping W : X × X × [0, 1] → X by

W(x,y,λ)=(λx1+(1−λ)y1,λx1x2+(1−λ)y1y2λx1+(1−λ)y1)

and define a metric d : X × X → [0, ∞) by

d(x,y)=|x1−y1|+|x1x2−y1y2|.

Then we can show that (X, d, W) is a convex metric space but not a normed linear space.

A metric space X is a CAT(0) space. This term is due to M. Gromov [7] and it is an acronym for E. Cartan, A.D. Aleksandrov and V.A. Toponogov. If X is geodesically connected, then every geodesic triangle in X is at least as ‘thin’ as its comparison triangle in the Euclidean plane (see, e.g., [8, p.159]). It is well known that any complete, simply connected Riemannian manifold nonpositive sectional curvature is a CAT(0) space. The precise definition is given below. For a thorough discussion of these spaces and of the fundamental role they play in various branches of mathematics, see Bridson and Haefliger [8] or Burago et al. [9].

Let (X, d) be a metric space. A geodesic path joining x ∈ X to y ∈ X (or, more briefly, a geodesic from x to y) is a mapping c from a closed interval [0, l] ⊂ ℝ to X such that c(0) = x, c(l) = y, and d(c(t), c(t′)) = |t − t′| for all t, t′ ∈ [0, l]. In particular, c is an isometry and d(x, y) = l. The image α 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 geodesic, and X is said to be uniquely geodesic if there is exactly one geodesic joining x and y for each x, y ∈ X. A subset Y ⊆ X is said to be convex if Y includes every geodesic segment joining any two of its points.

A geodesic triangle Δ (x1, x2, x3) is a geodesic metric space (X, d) consisting of three points x1, x2, x3 ∈ X (the vertices of Δ) and a geodesic segment between each pair of vertices (the edges of Δ). A comparison triangle for the geodesic triangle Δ (x1, x2, x3) in (X, d) is a triangle Δ(x1, x2, x3) = Δ (x1, x2, x3) in ℝ2 such that dℝ2(xi, xj) = d(xi, xj) for i, j ∈ {1, 2, 3}. Such a triangle always exists (see, [8]).

A geodesic metric space is said to be a CAT(0) space if all geodesic triangles of appropriate size satisfy the following CAT(0) comparison axiom.

Let Δ be a geodesic triangle in X and let Δ ⊂ ℝ2 be a comparison triangle for Δ. Then Δ is said to satisfy the CAT(0) inequality if for all x, y ∈ Δ and all comparison points x, y ∈ Δ,

d(x,y)≤d(x¯,y¯).

Complete CAT(0) spaces are often called Hadamard spaces (see, [10]). If x, y1, y2 are points of a CAT(0) space and if y0 is the midpoint of the segment [y1, y2], which we will denote by y1⊕y22, then the CAT(0) inequality implies

d2(x,y1⊕y22)≤12d2(x,y1)+12d2(x,y2)−14d2(y1,y2).

This inequality is the (CN) inequality of Bruhat and Tits [11]. In fact, a geodesic space is a CAT(0) space if and only if it satisfies the (CN) inequality (cf. [8, p.163]). The above inequality has been extended by [12] as

d2(z,αx⊕(1−α)y)≤αd2(z,x)+(1−α)d2(z,y)−α(1−α)d2(x,y), (CN*)

for any α ∈ [0, 1] and x, y, z ∈ X.

Let us recall that a geodesic metric space is a CAT(0) space if and only if it satisfies the (CN) inequality (see, [8, p.163]). Moreover, if X is a CAT(0) metric space and x, y ∈ X, then for any α ∈ [0, 1], there exists a unique point α x ⊕ (1 − α)y ∈ [x, y] such that

d(z,αx⊕(1−α)y)≤αd(z,x)+(1−α)d(z,y) (3)

for any z ∈ X and [x, y] = {αx ⊕ (1 − α)y : α ∈ [0, 1]}. In view of the above inequality, CAT(0) space has Takahashi’s convex structure

W(x,y,α)=αx⊕(1−α)y.

It is easy to see that for any x, y ∈ X and λ ∈ [0, 1],

d(x,(1−λ)x⊕λy)=λd(x,y),d(y,(1−λ)x⊕λy)=(1−λ)d(x,y).

As a consequence,

1⋅x⊕0⋅y=x,(1−λ)x⊕λx=λx⊕(1−λ)x=x.

Moreover, a subset K of CAT(0) space X is convex if for any x, y ∈ K, we have [x, y] ⊂ K (see, [1, 13, 14]).

From the results of Harder [15], Harder and Hicks [16, 17], where a concept of stable fixed point iterative scheme was introduced and studied, many other stability results for several fixed point iterative schemes and for various classes of nonlinear mappings were obtained.

Definition 1.8

([17]). Let (X, d) be a metric space, T : X → X be a self-mapping and x0 ∈ X. Assume that the iterative scheme

xn+1=f(T,xn),n≥0 (4)

converges to a fixed point p of T. Let {zn} be an arbitrary sequence in X and define

εn=d(zn+1,f(T,zn)),n≥0. (5)

The iterative scheme defined by (4) is said to be T-stable or stable with respect to T if and only if

limn→∞εn=0⇒limn→∞zn=p. (6)

Osilike [18] introduced a weaker concept of stability.

Definition l.9

([18]). Let (X, d) be a metric space, T : X → X be a self-mapping and x0 ∈ X. Assume that the iterative scheme (4) converges to a fixed point p of T. Let {zn} be an arbitrary sequence in X and defined by (5). The iterative scheme defined by (4) is said to be almost T-stable or almost stable with respect to T if and only if

∑n=0∞εn<∞⇒limn→∞zn=p. (7)

Remark 1.10

It is ovbious that any stable iterative scheme is also almost stable but the reverse is not true in generally, as shown in [19].

Now, we consider a new concept of stability.

Definition 1.11

Let (X, d) be a metric space, T : X → X be a self-mapping and x0 ∈ X. Assume that the iterative scheme (4) converges to a fixedpoint p of T. Let {zn} be an arbitrary sequence in X and defined by (5). The iterative scheme defined by (4) is said to be comparably almost T-stable or comparably almost stable with respect to T if and only if

∑n=0∞(θn+εn)<∞,θn≥0⇒limn→∞zn=p,limn→∞θn=0. (8)

Remark 1.12

It is ovbious that any almost stable iterative scheme is also comparably almost stable. In fact, let {xn} be an almost stable and ∑n=0∞(θn+εn)<∞,θn≥0. By the comparison test, we have ∑n=0∞εn<∞. From {xn} being an almost stable, we obtain limn→∞zn = p. Also, since ∑n=0∞(θn+εn)<∞, we have limn→∞(θn + εn) = 0. From ∑n=0∞εn<∞, we get limn→∞ εn = 0. Then limn→∞ θn = 0.

Moreover, if θn = 0 in (8), then (8) reduces to (7). So an almost stable iterative scheme is a special case of comparably almost stable iterative scheme.

The aim of this paper is to prove some fixed point results for generalized φ-weak contraction mapping and study a new concept of stability which is called comparably almost T-stable by using iterative schemes in CAT(0) spaces.

2 Convergence theorems of iterative schemes

Xue [3] proved the following very intersting fixed point theorem in complete metric space.

Theorem 2.1

([3]). Let (X, d) be a complete metric space and let T : X → X be a generalized φ-weak contraction. Then the Picard iterative scheme ([20])

xn+1=Txn

converges to the unique fixed point.

Theorem 2.2

Let T be a generalized φ-weak contractive self mapping of a closed convex subset K of a Banach space X. Then the Picard iterative scheme

xn+1=Txn

converges strongly to the fixed point p with the following error estimate:

∥xn+1−p∥≤Φ−1(Φ(∥x1−p∥−n)),

where Φ is defined by the antiderivative

Φ(t)=∫1φ(t)dt,Φ(0)=0

and Φ−1 is the inverse of Φ.

Proof

The proof is similar as [2, Theorem 2]. However, for completeness, we give a sketch of the proof. We can obtain convergence following from Theorem 2.1. To establish the error estimete, from (2) with λn = ∥xn − p∥,

λn+1=∥xn+1−p∥=∥Txn−p∥≤∥xn−p∥−φ(∥xn+1−p∥)=λn−φ(λn+1),

so, we have

φ(λn+1)≤λn−λn+1. (9)

Thus

Φ(λn)−Φ(λn+1)=∫λn+1λn1φ(t)dt=λn−λn+1φ(μn),

for some λn+1 < μn < λn. Since φ is nondecreasing, from (9),

Φ(λn)−Φ(λn+1)=λn−λn+1φ(μn)≥λn−λn+1φ(λn)≥1.

Thus

Φ(λn+1)≤Φ(λn)−1≤⋯≤Φ(λ1)−n.

This completes the proof of Theorem 2.2. □

We shall now investigate the convergence of other iterative scheme applied to T.

Theorem 2.3

Let (X, d) be a CAT(0) space, K be a closed convex subset of X, T be a generalized φ-weak contractive self mapping of K. Then the Ishikawa iterative scheme ([21])

xn+1=(1−αn)xn⊕αnTYn,yn=(1−βn)xn⊕βnTxn,n≥1 (10)

satisfies

  1. 0 ≤ αn, βn ≤ 1,

  2. ∑n=1∞αnβn=∞,

converges to the unique fixed point p of T.

Proof

From Theorem 2.1 and Theorem 2.2, T has a unique fixed point. Take it p. Using (10) and (3),

d(yn,p)=d((1−βn)xn⊕βnTxn,p)≤(1−βn)d(xn,p)+βnd(Txn,p)≤(1−βn)d(xn,p)+βn[d(xn,p)−φ(d(Txn,p))]=d(xn,p)−βnφ(d(Txn,p)). (11)

From (11), we have

d(xn+1,p)=d((1−αn)xn⊕αnTyn,p)≤(1−αn)d(xn,p)+αnd(Tyn,p)≤(1−αn)d(xn,p)+αn[d(yn,p)−φ(d(Tyn,p))]≤(1−αn)d(xn,p)+αn[d(xn,p)−βnφ(d(Txn,p))]−αnφ(d(Tyn,p))≤d(xn,p)−αnβnφ(d(Txn,p))≤d(xn,p). (12)

Therefore {d(xn, p)} is a nonnegative nonincreasing sequence, which converges to a limit L ≥ 0. Suppose that L > 0. For notational convenience, let λn = d(xn, p). Since {d(xn, p)} is a nonincreasing sequence, we have λn ≥ L, i.e.,

d(xn,p)≥d(xn+1,p)≥⋯≥L,∀n∈N. (13)

Most of all, we want to show that

d(Txn,p)≥L,∀n∈N.

It is sufficient to show that there exists n1 ∈ ℕ such that

d(xn1,p)≤d(Txn,p),n≥1.

Suppose that d(Txn, p) < L. Then

d(xn1,p)>d(Txn,p),∀n1∈N. (14)

Since limn→∞ d(xn, p) = L and (14), for ε2 = L − d(T xn, p) > 0, there exists N ∈ ℕ with d(xN, p) < d(Txn, p) + ε4 such that

|d(xn,p)−L|≤|L−d(Txn,p)|+|d(Txn,p)−d(xn,p)|=L−d(Txn,p)+d(xn,p)−d(Txn,p)≤ε2+d(xN,p)−d(Txn,p)<ε2+ε4<ε

for n ≥ N. On the other hand, from (14), we obtain

d(xN,p)<d(Txn,p)+ε4=d(Txn,p)+12(L−d(Txn,p))=12(L+d(Txn,p))<12(L+d(xN,p)),

i.e.,

d(xN,p)<L.

This is a contradiction to (13). Therefore

d(Txn,p)≥L. (15)

From (12), (13) and (15), it follows that, for any fixed integer N ∈ ℕ,

∑n=N∞αnβnφ(L)≤∑n=N∞αnβnφ(d(Txn,p))≤∑n=N∞(d(xn,p)−d(xn+1,p))≤d(xN,p).

This is a contradiction to the condition (ii). Therefore

limn→∞d(xn,p)=L=0.

This completes the proof of Theorem 2.3. □

Corollary 2.4

Let (X, d) be a CAT(0) space, K be a closed convex subset of X, T be a generalized φ-weak contractive self mapping of K. Then the Mann iterative scheme ([22])

xn+1=(1−αn)xn⊕αnTxn,n≥1 (16)

satisfies

  1. 0 ≤ αn ≤ 1,

  2. ∑n=1∞αn=∞,

converges to the unique fixed point p of T.

Proof

If βn = 0 in Ishikawa iterative scheme (10), then it reduces to the Mann iterative scheme (16). So the proof is similar to that of Theorem 2.3, and will be omitted. □

3 Stability of fixed paint iterative schemes

Lemma 3.1

([1]). Let {λn} and {γn} be two sequences of nonnegative real numbers and {σn} be a sequence of positive numbers satisfying the recursive inequality

λn+1≤λn−σnφ(λn)+γn,n≥1,

where φ : [0, ∞) →[0, ∞) is a continuous and nondecreasing function with φ(0) = 0. If

∑n=1∞σn=∞andlimn→∞γnσn=0,

then {λn} converges to 0 as n → ∞.

Theorem 3.2

Let (X, d) be a CAT (0) space and T : X → X be a generalized φ-weak contractive self mapping satisfying (2). Let x1 be an arbitrary in X and define

xn+1=(1−αn)xn⊕αnTyn,yn=(1−βn)xn⊕βnTxn,n≥1,

where {αn} and {βn} are sequences in [0, 1] satisfying

  1. ∑n=1∞ αnβn = ∞,

  2. αn(1+βn) ≤ 1.

Let {zn} be any given scheme in X and define a scheme {εn} by

sn=(1−βn)zn⊕βnTzn,εn=d(zn+1,(1−αn)zn⊕αnTsn),n≥1.

Then

  1. if ∑n=1∞ (θn+εn) < ∞, then the Ishikawa iterative scheme {xn} is comparably almost stable with respect to T, where

    θn=φ(d(zn,p))−αnβnφ(d(Tzn,p))−αnφ(d(Tsn,p)),

  2. if the scheme {zn} converges to fixed point of T, then limn → ∞εn = 0.

Proof

Let p denote the unique fixed point of T. We have the following estimate

d(zn+1,p)≤d(zn+1,(1−αn)zn⊕αnTsn)+d((1−αn)zn⊕αnTsn,p)≤εn+(1−αn)d(zn,p)+αnd(Tsn,p)≤(1−αn)d(zn,p)+αn[d(sn,p)−φ(d(Tsn,p+εn≤(1−αn)d(zn,p)+αn[(1−βn)d(zn,p)+βnd(Tzn,p)]−αnφ(d(Tsn,p))+εn≤(1−αnβn)d(zn,p)+αnβn[d(zn,p)−φ(d(Tzn,p−αnφ(d(Tsn,p))+εn=d(zn,p)−αnβnφ(d(Tzn,p))−αnφ(d(Tsn,p))+εn=d(zn,p)−φ(d(zn,p))+θn+εn, (17)

where θn = φ(d(zn, p)) − αnβnφ(d(Tzn, p)) − αnφ(d(Tsn, p)). Since

0≤d(Tsn,p)≤d(sn,p)−φ(d(Tsn,p))≤d(sn,p)=d((1−βn)zn⊕βnTzn,p)≤(1−βn)d(zn,p)+βnd(Tzn,p)≤d(zn,p), (18)

from the condition (ii), we obtain

θn=φ(d(zn,p))−αnβnφ(d(Tzn,p))−αnφ(d(Tsn,p))≥φ(d(zn,p))−αnβnφ(d(zn,p))−αnφ(d(zn,p))≥(1−αnβn−αn)φ(d(zn,p))≥0.

Since ∑n=1∞ (θn + εn) < ∞, by (17) and Lemma 3.1, we get

limn→∞d(zn,p)=0.

It follows that

limn→∞zn=p. (19)

Since ∑n=1∞ (θn + εn) < ∞, we have limn → ∞(θn+εn) = 0. From (19),

0≤d(Tzn,p)≤d(zn,p)−φ(d(Tzn,p))≤d(zn,p)→0 as n→∞,

therefore

limn→∞d(Tzn,p)=0. (20)

Similarly, from (18) and (19), we have

limn→∞d(Tsn,p)=0. (21)

Since φ is continuous, from (19)-(21), we obtain

limn→∞θn=limn→∞[φ(d(zn,p))−αnβnφ(d(Tzn,p))−αnφ(d(Tsn,p))]=0.

Hence {xn} is a comparably almost T-stable on X.

Next, suppose that limn → ∞zn = p. From (18), we obtain

εn=d(zn+1,(1−αn)zn⊕αnTsn)≤d(zn+1,p)+d((1−αn)zn⊕αnTsn,p)≤d(zn+1,p)+(1−αn)d(zn,p)+αnd(Tsn,p)≤d(zn+1,p)+d(zn,p)→0as n→∞.

This completes the proof. □

Corollary 3.3

Let (X, d) be a CAT (0) space and T : X → X be a generalized φ-weak contractive self mapping satisfying (2). Let x1 be an arbitrary in X and define

xn+1=(1−αn)xn⊕αnTxn,n≥1,

where {αn} is sequence in [0, 1] satisfying ∑n=1∞ αn = ∞. Let {zn} be any given scheme in X and define a scheme {εn} by

εn=d(zn+1,(1−αn)zn⊕αnTzn),n≥1.

Then

  1. if ∑n=1∞ (θn + εn) < ∞, then the Mann iterative scheme {xn} is comparably almost stable with respect to T, where

    θn=φ(d(zn,p))−αnφ(d(Tzn,p)),

  2. if the scheme {zn} converges to fixed point of T, then limn → ∞εn = 0.

Proof

If βn = 0 in Ishikawa iterative scheme in Theorem 3.2, then it reduces to the Mann iterative scheme in Corollary 3.3. So the proof is similar to that of Theorem 3.2, and will be omitted. □

4 Some open problems

Let S be a semigroup. We denote by B(S) be the space of all bounded real-valued functions defined on S with supremum norm. For each s ∈ S, we define the left and right translation operators ls and rs on B(S) by

(lsf)(t)=f(st) and (rsf)(t)=f(ts)

for each t ∈ s and f ∈ B(S), respectively. Let X be a subspace of B(S) containing 1. An element μ in the dual space X* of X is said to be a mean on X if ∥μ∥ = μ(1) = 1. For s ∈ S, we can define a point evaluation δs by δs(f) = f(s) for each f ∈ X. It is well known that μ is mean on X if and only if

infs∈Sf(s)≤μ(f)≤sups∈Sf(s)

for each f ∈ X. Each mean on X is the weak*-limit of convex combination of point evaluations.

Let X be a translation invariant subspace of B(S) (i.e., lsX ⊂ X and rsX ⊂ X for each s ∈ S) containing 1. Then a mean μ on X is said to be left invariant (resp. right invariant) if

μ(lsf)=μ(f)(resp.μ(rsf)=μ(f))

for each s ∈ s and f ∈ X. A mean μ on X is said to be invariant if μ is both left and right invariant (see, [23-26]). X is said to be left (resp. right) amenable if X has a left (resp. right) invariant mean. X is amenable if X is left and right amenable. The semigroup S is called amenable if B(S) has an invariant mean (see, [27-29]). Moreover, S is amenable when S is a commutative semigroup or a solvable group. However, the free group or semigroup of two generators is not left or right amenable.

Let S be a semigroup and F(T) denote the fixed point set of T. Then 𝔍 = {Ts : s ∈ S} is called a representation of S if Te = I and Tst = TsTt for each s, t ∈ S. We denote by F(𝔍) the set of common fixed points of {Ts : s ∈ S}, i. e.,

F(J)=⋂s∈SF(Ts)=⋂s∈S{x∈C:Tsx=x}.

Open Problem 1

It will be interesting to obtain a generalization of both Theorem 2.3 and Theorem 3.2 to commutative, amenable semigroups as in the case of general metric spaces.

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 triangle in a model space with curvature κ.

For κ = 0, the 2-dimensional model space Mκ2=M02 is the Euclidean space ℝ2 with the metric induced from the Euclidean norm. For κ > 0, Mκ2 is the 2-dimensional sphere 1κS2 whose metric is length of a minimal great arc joining each two points. For κ < 0, Mκ2 is the 2-dimensional hyperbolic space 1−κH2 with the metric defined by a usual hyperbolic distance. For more details about the properties of CAT (κ) spaces (see, [8, 30, 31]).

Open Problem 2

It will be interesting to obtain a generalization of both Theorem 2.3 and Theorem 3.2 to CAT (κ) space.

  1. Competing interests

    The author declares to have no competing interests.

Acknowledgement

The author would like to thank the referee for his/her valuable comments and suggestions which improved the presentation of this paper. This work was supported by Kyungnam University Research Fund, 2017.

References

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Received: 2016-9-13
Accepted: 2017-7-6
Published Online: 2017-9-6

© 2017 Kim

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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Keywords for this article
CAT(0) space; Generalized φ-weak contraction mapping; Fixed point; Ishikawa iterative scheme; Comparably almost T-stable
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