KYUNGPOOK Math. J. 2019; 59(4): 665-678  
An Ishikawa Iteration Scheme for two Nonlinear Mappings in CAT(0) Spaces
Kritsana Sokhuma
Department of Mathematics, Faculty of Science and Technology, Phranakhon Rajabhat University, Bangkok 10220, Thailand
e-mail : k_sokhuma@yahoo.co.th
Received: December 9, 2017; Revised: December 10, 2018; Accepted: December 11, 2018; Published online: December 23, 2019.
© Kyungpook Mathematical Journal. All rights reserved.

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Abstract

We construct an iteration scheme involving a hybrid pair of mappings, one a single-valued asymptotically nonexpansive mapping t and the other a multivalued nonexpansive mapping T, in a complete CAT(0) space. In the process, we remove a restricted condition (called the end-point condition) from results of Akkasriworn and Sokhuma [1] and and use this to prove some convergence theorems. The results concur with analogues for Banach spaces from Uddin et al. [16].

Keywords: Ishikawa iteration, CAT(0) spaces, multivalued mapping, asymptotically nonexpansive mapping.
1. Introduction

Fixed point theory in CAT(0) spaces was first studied by Kirk [6, 8] who showed that every nonexpansive mapping defined on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. Since then, the existence problem of fixed point and the Δ–convergence problem of iterative sequences to a fixed point for nonexpansive mappings and asymptotically nonexpansive mappings in a CAT(0) space have been extensively developed with many papers published.

Let (X, d) be a geodesic metric space. 2K is denoted as the family of nonempty subsets of K, FB(K) is the collection of all nonempty closed bounded subsets of K and KC(K) is the collection of all nonempty compact convex subsets of K. A subset K of X is called proximinal if for each xX there exists an element kK such that

d(x,k)=dist(x,K)=inf{d(x,y):yK}.

The notation PB(K) is the collection of all nonempty bounded proximinal subsets of K.

Let H be the Hausdorff metric with respect to d such that

H(A,B)=max{supxAdist(x,B),supyBdist(y,A)},A,BFB(X),

where dist(x,B) = inf{d(x, y) : yB} is the distance from the point x to the subset B.

A mapping t : KK is said to be nonexpansive if

d(tx,ty)d(x,y) for all x,yK.

A point x is called a fixed point of t if tx = x.

A mapping t : KK is called asymptotically nonexpansive if there is a sequence {kn} of positive numbers with the property limnkn=1 such that

d(tnx,tny)knd(x,y) for all n1,x,yK.

A multivalued mapping T : KFB(K) is said to be nonexpansive if

H(Tx,Ty)d(x,y) for all x,yK.

A multivalued mapping T : KFB(K) is said to satisfy condition(E) if there exists μ ≥ 1 such that for each x, yK

dist(x,Ty)μ dist(x,Tx)+d(x,y).

Let T : KPB(K) be a multivalued mapping and define the mapping PT for each x by

PT(x):={yTx:d(x,y)=dist(x,Tx)}.

A point x is called a fixed point for a multivalued mapping T if xTx.

Then, IT is strongly demiclosed if for every sequence {xn} in K which converges to xK and such that limn→∞d(xn, Txn) = 0, then xT(x).

For every continuous mapping T : K → 2K, IT is strongly demiclosed but the converse is not true. Notice also that if T satisfies condition (E), then IT is strongly demiclosed.

The notation Fix(T) stands for the set of fixed points of a mapping T and Fix(t) ∩ Fix(T) stands for the set of common fixed points of t and T. A precise point x is called a common fixed point of t and T if x = txTx.

In 2009, Laokul and Panyanak [9] defined the iterative and proved the Δ–convergence for nonexpansive mapping in CAT(0) spaces as follows:

Let C be a nonempty closed convex subset of a complete CAT(0) space and t : CC be a nonexpansive mapping with Fix(t) := {xC : tx = x} ≠ ∅︀. Suppose {xn} is generated iteratively by x1C,

yn=βnxn(1βn)xn,xn+1=αntyn(1αn)xn.

for all n ∈ ℕ, where {αn} and {βn} are real sequences in [0, 1] such that one of the following two conditions is satisfied:

(i) αn ∈ [a, b] and βn ∈ [0, b] for some a, b with 0 < ab < 1,

(ii) αn ∈ [a, 1] and βn ∈ [a, b] for some a, b with 0 < ab < 1,

Then the sequence {xn} is Δ–convergent to a fixed point of t.

In 2010, Sokhuma and Kaewkhao [15] proved the convergence theorem for a common fixed point in Banach spaces as follows.

Let E be a nonempty compact convex subset of a uniformly convex Banach space X, and t : EE and T : EKC(E) be a single-valued nonexpansive mapping and a multivalued nonexpansive mapping, respectively. Assume in addition that Fix(t) ∩ Fix(T) ≠ ∅︀ and Tw = {w} for all w ∈ Fix(t) ∩ Fix(T). Suppose {xn} is generated iterative by x1E,

yn=(1βn)xn+βnzn,xn+1=(1αn)xn+αntyn,

for all n ∈ ℕ where znTxn and {αn}, {βn} are sequences of positive numbers satisfying 0 < aαn, βnb < 1. Then the sequence {xn} converges strongly to a common fixed point of t and T.

In 2013, Sokhuma [14] proved the convergence theorem for a common fixed point in CAT(0) spaces as follows.

Let K be a nonempty compact convex subset of a complete CAT(0) space X and t : KK and T : KFC(K) a single-valued nonexpansive mapping and a multivalued nonexpansive mapping respectively and Fix(t) ∩ Fix(T) ≠ ∅︀ satisfying Tw = {w} for all w ∈ Fix(t) ∩ Fix(T). Let {xn} is generated iterative by x1K,

yn=(1βn)xnβnzn,xn+1=(1αn)xnαntyn,

for all n ∈ ℕ where znTxn and {αn}, {βn} are sequences of positive numbers satisfying 0 < aαn, βnb < 1. Then the sequence {xn} converges strongly to a common fixed point of t and T.

In 2013, Laowang and Panyanak proved the convergence theorem for a common fixed point in CAT(0) spaces as follows.

Theorem 1.1.([10])

Let C be a nonempty bounded closed convex subset of a complete CAT(0) space X. Let f : CC be a pointwise asymtotically nonexpansive mapping, and g : CC a quasi-nonexpansive mapping, and let T : CKC(C) be a multivalued mapping satisfying conditions (E) and Cλ for some λ ∈ (0, 1). If f, g and T are pairwise commuting, then there exists a point zC such that z = f(z) = g(z) ∈ T(z).

In 2015, Akkasriworn and Sokhuma [1] proved the convergence theorem for a common fixed point in a complete CAT(0) space as follows.

Theorem 1.2

Let E be a nonempty bounded closed convex subset of a complete CAT(0) space X, t : EE and T : EFB(E) an asymptotically nonexpansive mapping and a multivalued nonexpansive mapping, respectively. Assume that t and T are commuting and Fix(t) ∩ Fix(T) ≠ ∅︀ satisfying Tw = {w} for all wFix(t) ∩ Fix(T) and n=1(kn1)<. Let {xn} be the sequence of the modified Ishikawa iterates defined by

yn=(1βn)xnβnzn,xn+1=(1αn)xnαntnyn,

for all n ∈ ℕ where znTtnxn and {αn}, {βn} ∈ [0, 1]. Then {xn} is Δ–convergent to a common fixed point of t and T.

In 2016, Uddin and Imdad [17] introduced the following iteration scheme:

Let K be a nonempty closed, bounded and convex subset of Banach space X, let f : KK be a single-valued nonexpansive mapping and let T : KFB(K) be a multivalued nonexpansive mapping with Fix(f) ∩ Fix(T) ≠ ∅︀ such that PT is a nonexpansive mapping. The sequence {xn} of the modified Ishikawa iteration is defined by

yn=αnzn+(1αn)xn,xn+1=βnfyn+(1βn)xn,

where x0K, znPT (xn) and 0 < aαn, βnb < 1. Then {xn} converges strongly to a common fixed point of f and T.

The Ishikawa iteration method was studied with respect to a pair of single-valued asymptotically nonexpansive mapping and a multivalued nonexpansive mapping. It also established the convergence theorem of a sequence from such process in a nonempty bounded closed convex subset of a complete CAT(0) space. A restricted condition (called end-point condition) in Akkasriworn and Sokhuma’s results was removed [1].

Here, an iteration method modifying the above was introduced and called the Ishikawa iteration method

Definition 1.3

Let K be a nonempty bounded closed convex subset of a complete CAT(0) space X, t : KK be a single-valued asymptotically nonexpansive mapping and T : KPB(K) be a multivalued nonexpansive mapping where PT (x) = {yTx : d(x, y) = dist(x, Tx)}. For fixed x1K the sequence {xn} of the Ishikawa iteration is defined by

yn=(1βn)xnβnzn,xn+1=(1αn)xnαntnyn,

for all n ∈ ℕ where znPT (tnxn) and {αn}, {βn} ∈ (0, 1).

2. Preliminaries

Relevant basic definitions followed previous research results and iterative methods were used frequently.

Let (X, d) be a metric space. A geodesic path joining xX to yX is a map c from a closed interval [0, s] ⊂ ℝ to X such that c(0) = x, c(s) = y, and d(c(t), c(u)) = |tu| for all t, u ∈ [0, s]. In particular, c is an isometry and d(x, y) = s. The image α of c is called a geodesic (or metric) segment joining x and y. When it is unique this geodesic segment 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, yX. A subset YX is said to be convex if Y includes every geodesic segment joining any two of its points.

A geodesic triangle Δ(x1, x2, x3) in a geodesic metric space (X, d) consists of three points x1, x2, x3 in 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 Δ̄(x1, x2, x3) := Δ(χ̄1, χ̄2, χ̄3) in the Euclidean plane such that for i, j ∈ {1, 2, 3}.

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

CAT(0): Let Δ be a geodesic triangle in X and let Δ̄ 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 χ̄, ȳ ∈ Δ̄, .

If x, y1, y2 are points in a CAT(0) space and

y0=12y112y2,

then the CAT(0) inequality implies that

d(x,y0)212d(x,y1)2+12d(x,y2)214d(y1,y2)2.

This is the (CN) inequality of Bruhat and Tits [3]. A geodesic space is a CAT(0) space if and only if it satisfies the (CN) inequality [2].

The following results and methods deal with the concept of asymptotic centres. Let K be a nonempty closed convex subset of a CAT(0) space X and {xn} be a bounded sequence in X. For xX, define the asymptotic radius of {xn} at x as the number

r(x,{xn})=limsupnd(xn,x).

Let

rr(K,{xn}):=inf{r(x,{xn}):xK}

and

AA(K,{xn}):={xK:r(x,{xn})=r}.

The number r and the set A are called the asymptotic radius and asymptotic centre of {xn} relative to K respectively.

If X is a complete CAT(0) space and K is a closed convex subset of X, then A(K, {xn}) consists of exactly one point. A sequence {xn} in CAT(0) space X is said to be Δ–convergent to xX if x is the unique asymptotic centre of every subsequence of {xn}. A bounded sequence {xn} is said to be regular with respect to K if for every subsequence {xn}, we get

r(K,{xn})=r(K,{xn}).

The definition of Δ–convergence is presented below.

Definition 2.1.([12, 8])

A sequence {xn} in a CAT(0) space X is said to be Δ–convergent to xX if x is the unique asymptotic centre of {un} for every subsequence {un} of {xn}. In this case,Δlimnxn=xand x is the Δ – limit of {xn}.

Some elementary facts about CAT(0) spaces which will be used in the proofs of the main results are stated. The following lemma can be found in [4, 5, 8].

Lemma 2.2.([8])

Every bounded sequence in a complete CAT(0) space has a Δ–convergent subsequence.

Lemma 2.3.([4])

If K is a closed convex subset of a complete CAT(0) space and {xn} is a bounded sequence in K, then the asymptotic centre of {xn} is in K.

Lemma 2.4.([5])

Let (X, d) be a CAT(0) space.

(i) For x, yX and u ∈ [0, 1], there exists a unique point z ∈ [x, y] such thatd(x,z)=ud(x,y)   and   d(y,z)=(1u)d(x,y).

he notation (1 − u)xty is used for the unique point z satisfying (2.2).

(ii) For x, y, zX and u ∈ [0, 1], d((1u)xuy,z)(1u)d(x,z)+ud(y,z).

The existence of fixed points for asymptotically nonexpansive mappings in CAT(0) spaces was proved by Kirk [7] as the following result.

Theorem 2.5

Let K be a nonempty bounded closed and convex subset of a complete CAT(0) space X and let t : KK be asymptotically nonexpansive. Then t has a fixed point.

Theorem 2.6.([13])

Let X be a complete CAT(0) space and K be a nonempty bounded closed and convex subset of X and t : KK be an asymptotically nonexpansive mapping. Then It is demiclosed at 0.

Corollary 2.7.([5])

Let K be a closed and convex subset of a complete CAT(0) space X and let t : KX be an asymptotically nonexpansive mapping. Let {xn} be a bounded sequence in K such that limnd(txn,xn)=0and Δlimnxn=w. Then tw = w.

Lemma 2.8.([11])

Let X be a complete CAT(0) space and let xX. Suppose {αn} is a sequence in [a, b] for some a, b ∈ (0, 1) and {xn}, {yn} are sequences in X such that limsupnd(xn,x)r,limsupnd(yn,x)r, and limnd((1αn)xnαnyn,x)=r for some r ≥ 0. Then limnd(xn,yn)=0.

The following fact is well-known.

Lemma 2.9

Let X be a CAT(0) space, K be a nonempty compact convex subset of X and {xn} be a sequence in K. Then,

dist(y,Ty)d(y,xn)+dist(xn,Txn)+H(Txn,Ty)

where yK and T be a multivalued mapping from K in to FB(K).

The important property can be found in [18].

Lemma 2.10

Let {an} and {bn} be two sequences of nonnegative numbers such that

an+1(1+bn)an,

for all n ≥ 1. If n=1bn converges, then limnan exists. In particular, if there is a subsequence of {an} which converges to 0 then limnan=0.

3. Main Results

The following lemmas play very important roles in this section.

Lemma 3.1

Let K be a nonempty bounded closed convex subset of a complete CAT(0) space X, T : KPB(K) be a multivalued mapping, and PT (x) = {yTx : d(x, y) = dist(x, Tx)}. Then the following are equivalent

(1) xFix(T), that is xTx;

(2) PT (x) = {x}, that is x = y for each yPT (x);

(3) xFix(PT), that is xPT (x).

Further, Fix(T) = Fix(PT).

Proof

(1) implies (2). Since xTx, then d(x, Tx) = 0. Therefore, for any yPT (x), d(x, y) = dist(x, Tx) = 0 and so x = y. That is, PT (x) = {x}.

(2) implies (3). Since PT (x) = {x}, then x ∈ Fix(PT) and hence xPT (x).

(3) implies (1). Since x ∈ Fix(PT), then xPT (x). Therefore, d(x, x) = dist(x, Tx) = 0 and so xTx by the closedness of Tx.

This implies that Fix(T) = Fix(PT).

Lemma 3.2

Let K be a nonempty bounded closed convex subset of a complete CAT(0) space X, t : KK and T : KPB(K) a single-valued asymptotically nonexpansive mapping and a multivalued nonexpansive mapping respectively with Fix(t) ∩ Fix(T) ≠ ∅︀ such that PT is nonexpansive and n=1(kn1)<. Let {xn} be the sequence of Ishikawa iterates defined by (1.2). Then limnd(xn,w) exists for all wFix(t) ∩ Fix(T).

Proof

Let x1K and w ∈ Fix(t)∩Fix(T), in the view of Lemma 3.1, wPT(w) = {w}. Now consider,

d(xn+1,w)=d((1αn)xnαntnyn,w)(1αn)d(xn,w)+αnd(tnyn,tnw)(1αn)d(xn,w)+αnknd(yn,w)=(1αn)d(xn,w)+αnknd((1βn)xnβnzn,w)(1αn)d(xn,w)+αnkn(1βn)d(xn,w)+αnknβnd(zn,w)(1αn)d(xn,w)+αnkn(1βn)d(xn,w)+αnknβndist(zn,PT(w))(1αn)d(xn,w)+αnkn(1βn)d(xn,w)+αnknβnH(PT(tnxn),PT(w))(1αn)d(xn,w)+αnkn(1βn)d(xn,w)+αnknβnd(tnxn,w)(1αn)d(xn,w)+αnkn(1βn)d(xn,w)+αnβnkn2d(xn,w)=[1+αn(1+βnkn)(kn1)]d(xn,w).

By the convergence of kn and αn, βn ∈ (0, 1), there exists some M >0 such that

d(xn+1,w)[1+M(kn1)]d(xn,w).

By condition n=1(kn1)< and Lemma 2.10, which implies that limnd(xn,w) exists.

Lemma 3.3

Let K be a nonempty bounded closed convex subset of a complete CAT(0) space X, t : KK and T : KPB(K) a single-valued asymptotically nonexpansive mapping and a multivalued nonexpansive mapping respectively with Fix(t) ∩ Fix(T) ≠ ∅︀ such that PT is nonexpansive and n=1(kn1)<. Let {xn} be the sequence of Ishikawa iterates defined by (1.2). Then limnd(tnyn,xn)=0.

Proof

Let x1K and w ∈ Fix(t) ∩ Fix(T), in view of Lemma 3.1, wPT (w) = {w}. From Lemma 3.2, limnd(xn,w)=c is set. Now consider,

d(yn,w)=d((1βn)xnβnzn,w)(1βn)d(xn,w)+βnd(zn,w)=(1βn)d(xn,w)+βndist(zn,PT(w))(1βn)d(xn,w)+βnH(PT(tnxn),PT(w))(1βn)d(xn,w)+βnd(tnxn,w)(1βn)d(xn,w)+βnknd(xn,w).

Notice that

d(tnyn,w)knd(yn,w)kn[(1βn)d(xn,w)+βnknd(xn,w)]=kn(1βn)d(xn,w)+βnkn2d(xn,w)=(knknβn+βnkn2)d(xn,w)=[kn+βnkn(kn1)]d(xn,w)[1+βnkn(kn1)]d(xn,w).

Then,

limsupnd(tnyn,w)limsupnknd(yn,w)limsupn[1+βnkn(kn1)]d(xn,w).

By kn → 1 as n→∞ and αn, βn ∈ (0, 1), which implies that

limsupnd(tnyn,w)limsupnd(yn,w)limsupnd(xn,w)=c.

Since c=limnd(xn+1,w)=limnd((1αn)xnαntnyn,w), it implies by Lemma 2.8 that limnd(tnyn,xn)=0.

Lemma 3.4

Let K be a nonempty bounded closed convex subset of a complete CAT(0) space X, t : KK and T : KPB(K) a single-valued asymptotically nonexpansive mapping and a multivalued nonexpansive mapping respectively with Fix(t) ∩ Fix(T) ≠ ∅︀ such that PT is nonexpansive and n=1(kn1)<. Let {xn} be the sequence of Ishikawa iterates defined by (1.2). Then limnd(xn,zn)=0.

Proof

Let x1K and w ∈ Fix(t) ∩ Fix(T), in view of Lemma 3.1, wPT (w) = {w}. Consider,

d(xn+1,w)=d((1αn)xnαntnyn,w)(1αn)d(xn,w)+αnd(tnyn,w)(1αn)d(xn,w)+αnknd(yn,w)

and hence

d(xn+1,w)d(xn,w)αnknd(yn,w)d(xn,w).

Therefore, by 0 < aαnb < 1, it follows that

(d(xn+1,w)d(xn,w)αn)+d(xn,w)knd(yn,w).

Thus,

liminfn{(d(xn+1,w)d(xn,w)αn)+d(xn,w)}liminfnknd(yn,w).

It follows that

climinfnd(yn,w).

Since limsupnd(yn,w)c, it follows that

c=limnd(yn,w)=limnd((1βn)xnβnzn,w).

Recall that

d(zn,w)=dist(zn,PT(w))H(PT(tnxn),PT(w))d(tnxn,w)knd(xn,w).

Hence,

limsupnd(zn,w)limsupnknd(xn,w)limsupnd(xn,w)=c.

Thus,

limnd(xn,zn)=0.

Lemma 3.5

Let K be a nonempty bounded closed convex subset of a complete CAT(0) space X, t : KK and T : KPB(K) a single-valued asymptotically nonexpansive mapping and a multivalued nonexpansive mapping respectively with Fix(t) ∩ Fix(T) ≠ ∅︀ such that PT is nonexpansive and n=1(kn1)<. Let {xn} be the sequence of Ishikawa iterates defined by (1.2). Then limnd(tnxn,xn)=0.

Proof

Consider,

d(tnxn,xn)d(tnxn,tnyn)+d(tnyn,xn)knd(xn,yn)+d(tnyn,xn)=knd(xn,(1βn)xnβnzn)+d(tnyn,xn)kn[(1βn)d(xn,xn)+βnd(xn,zn)]+d(tnyn,xn)=knβnd(xn,zn)+d(tnyn,xn).

Then,

limnd(tnxn,xn)limnknβnd(zn,xn)+limnd(tnyn,xn).

Hence, by Lemmas 3.3 and 3.4,limnd(tnxn,xn)=0.

Lemma 3.6

Let K be a nonempty bounded closed convex subset of a complete CAT(0) space X, t : KK and T : KPB(K) a single-valued asymptotically nonexpansive mapping and a multivalued nonexpansive mapping respectively with Fix(t) ∩ Fix(T) ≠ ∅︀ such that PT is nonexpansive and n=1(kn1)<. Let {xn} be the sequence of Ishikawa iterates defined by (1.2). Then limnd(txn,xn)=0.

Proof

Consider,

d(txn,xn)=d(xn,txn)d(xn,tnxn)+d(tnxn,txn)d(xn,tnxn)+k1[d(tn1xn,tn1xn1)+d(tn1xn1,xn)]d(xn,tnxn)+k1kn1d(xn,xn1)+k1d(tn1xn1,xn)d(xn,tnxn)+k1kn1αn1d(tn1yn1,xn1)+k1(1αn1)d(xn1,tn1xn1)+k1kn1αn1d(yn1,xn1)d(xn,tnxn)+k1kn1αn1d(tn1yn1,xn1)+k1(1αn1)d(xn1,tn1xn1)+k1kn1αn1βn1d(zn1,xn1).

It follows from Lemmas 3.2 – 3.4 that,

limnd(txn,xn)=0.

Theorem 3.7

Let K be a nonempty bounded closed convex subset of a complete CAT(0) space X, t : KK and T : KPB(K) a single-valued asymptotically nonexpansive mapping and a multivalued nonexpansive mapping respectively with Fix(t) ∩ Fix(T) ≠ ∅︀ such that PT is nonexpansive and n=1(kn1)<. Let {xn} be the sequence of Ishikawa iterates defined by (1.2). Then {xn} is Δ–convergent to y implies yFix(t) ∩ Fix(T).

Proof

Since {xn} is Δ–convergent to y. From Lemma 3.6,

limnd(txn,xn)=0.

By Corollary 1.6, yK and ty = y, it follows that y ∈ Fix(t). It follows from Lemma 2.9 that,

dist(y,PT(y))d(y,xn)+dist(xn,PT(xn))+H(PT(xn),PT(y))d(y,xn)+d(xn,zn)+d(xn,y)0 as n.

It follows that y ∈ Fix(PT) then y ∈ Fix(T). Therefore y ∈ Fix(t) ∩ Fix(T) as desired.

Theorem 3.8

Let K be a nonempty bounded closed convex subset of a complete CAT(0) space X, t : KK and T : KPB(K) a single-valued asymptotically nonexpansive mapping and a multivalued nonexpansive mapping respectively with Fix(t) ∩ Fix(T) ≠ ∅︀ such that PT is nonexpansive and n=1(kn1)<. Let {xn} be the sequence of Ishikawa iterates defined by (1.2). Then {xn} is Δ–convergent to a common fixed point of t and T.

Proof

Since Lemma 3.6 guarantees that {xn} is bounded and limnd(txn,xn)=0. So, let ωw(xn) := ∪A({un}) where the union is taken over all subsequences {un} of {xn}. If ωw(xn) ⊂ Fix(t) ∩ Fix(T), then there exists a subsequence {un} of {xn} such that A({un}) = {u}. By Lemmas 1.2 and 1.3 there exists a subsequence {vn} of {un} such that Δlimnvn=vK. Since

limnd(tvn,vn)=0, it follows that v ∈ Fix(t). Since,

dist(v,PT(v))dist(v,PT(vn))+H(PT(vn),PT(v))d(v,zn)+d(vn,v)d(v,vn)+d(vn,zn)+d(vn,v)0 as n.

It follows that v ∈ Fix(PT) and v ∈ Fix(T) by Lemma 3.1. Therefore v ∈ Fix(t) ∩ Fix(T) as desired. Suppose that uv, since t is a single-valued asymptotically nonexpansive mapping and v ∈ Fix(t) ∩ Fix(T),limnd(xn,v) exists by Lemma 3.2. Then by the uniqueness of asymptotic centres,

limsupnd(vn,v)<limsupnd(vn,u)limsupnd(un,u)<limsupnd(un,v)=limsupnd(xn,v)=limsupnd(vn,v)

a contradiction, and hence u = v ∈ Fix(t) ∩ Fix(T).

To show that {xn} is Δ–convergent to a common fixed point of t and T, it suffices to show that ωw(xn) consists of exactly one point. Let {un} be a subsequence of {xn}. By Lemmas 1.2 and 1.3 there exists a subsequence {vn} of {un} such that Δlimnvn=vK. Let A({un}) = {u} and A({xn}) = {x}. It has seen that u = v and v ∈ Fix(t) ∩ Fix(T).

It can complete the proof by showing that x = v. Suppose not, since limnd(xn,v) exists, then by the uniqueness of asymptotic centres,

limsupnd(vn,v)<limsupnd(vn,x)limsupnd(xn,x)<limsupnd(xn,v)=limsupnd(vn,v)

a contradiction, and hence the conclusion follows.

Acknowledgements

I would like to thank the Institute for Research and Development, Phranakhon Rajabhat University, for financial support.

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