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An approximation of a common fixed point of nonexpansive mappings on convex metric spaces

Abstract

Sokhuma and Kaewkhao (2011) introduced an iteration scheme to compute a common fixed point of a single-valued nonexpansive mapping and a multivalued nonexpansive mapping on a uniformly convex Banach space. In this paper, we extend the above result of Sokhuma and Kaewkhao from a single-valued mapping to a countable number of mappings and, at the same time, we extend the underlying spaces to strictly convex Banach spaces. The corresponding results are also obtained for the CAT(0) space setting.

MSC:47H09, 47H10.

1 Introduction

Let X be a complete metric space, and E a nonempty subset of X. We will denote by 2 E the family of nonempty subsets of E and by FB(E) the family of nonempty bounded closed subsets of E. Let H(,) be theHausdorff distance on FB(X), that is,

H ( A , B ) = max { sup a A dist ( a , B ) , sup b B dist ( b , A ) } , A , B F B ( X ) ,

where dist(a,B)=inf{d(a,b):bB} is the distance from the point a to the subset B.

A mapping t:EE and a multivalued mapping T:EFB(X) are said to be nonexpansive if for each x,yE,

d ( t x , t y ) d ( x , y ) , and H ( T x , T y ) d ( x , y ) ,

respectively. If tx=x, we call x a fixed point of a single-valued mapping t. Moreover, if xTx, we call x a fixed point of a multivalued mapping T. We use the notation Fix(S) to stand for the set of fixed points of a mapping S. Thus Fix(t)Fix(T) is the set of common fixed points of t and T, i.e., xFix(t)Fix(T) if and only if x=txTx.

Following [8], a bounded closed and convex subset E of a Banach space X has the fixed point property for nonexpansive mappings (FPP) (respectively, for multivalued nonexpansive mappings (MFPP)) if every nonexpansive mapping of E into E has a fixed point (respectively, every nonexpansive mapping of E into 2 E with compact convex values has a fixed point). For a bounded closed and convex subset E of a Banach space X, a mapping t:EX is said to satisfy the conditional fixed point property (CFP) if either t has no fixed points, or t has a fixed point in each nonempty bounded closed convex set that leaves t invariant. A set E is said to have the conditional fixed point property for nonexpansive mappings (CFPP) if every nonexpansive t:EE satisfies (CFP). For commuting family of nonexpansive mappings, the following is a remarkable common fixed point property due to Bruck [6].

Theorem 1.1 ([6])

Let X be a Banach space and E a nonempty closed convex subset of X. If E has both the (FPP) and the (CFPP) for nonexpansive mappings, then for any commuting familySof nonexpansive mappings of E into E, there is a common fixed point forS.

Theorem 1.1 was proved by Belluce and Kirk [1] when S is finite and E is weakly compact and has a normal structure; by Belluce and Kirk [2] when E is weakly compact and has a complete normal structure; by Browder [4] when X is uniformly convex and E is bounded; by Lau and Holmes [11] when S is left reversible and E is compact; and finally, by Lim [14] when S is left reversible and E is weakly compact and has a normal structure.

Open Problem (Bruck [6]). Can commutativity of S be replaced by left reversibility?

The answer to this Problem is not known even when the semigroup is left amenable (see [13] for more details).

In 2011, Sokhuma and Kaewkhao [17] introduced a new iteration method for approximating a common fixed point of a pair of a single-valued and a multivalued nonexpansive mappings and proved the following strong convergence theorem:

Theorem 1.2 ([17], Theorem 3.5])

Let E be a nonempty compact convex subset of a uniformly convex Banach space X, and lett:EEandT:EFB(E)be a single-valued and a multivalued nonexpansive mappings respectively, andFix(t)Fix(T)satisfyingTw={w}for allwFix(t)Fix(T). Let{ x n }be the sequence of the modified Ishikawa iteration defined by

y n = ( 1 β n ) x n + β n z n , x n + 1 = ( 1 α n ) x n + α n t y n ,

where x 1 E, z n T x n and0<a α n , β n b<1. Then{ x n }converges strongly to a common fixed point of t and T.

For a single-valued nonexpansive mapping t:EE with Fix(t), where E is a convex nonexpansive retract of a real uniformly smooth Banach space, Reich and Shemen [15], Theorem 3.4] obtained a strong convergence to a fixed point of t of a sequence { x n } of the form

y n = R E [ ( 1 β n ) x n ] , x n + 1 = ( 1 α n ) x n + α n t y n ,

where R E is a retraction on the subset E and the sequences { α n }{ β n } satisfy conditions: (i) 0< lim inf n α n lim sup n α n <1, (ii) lim n β n =0 and n = 1 β n =. Clearly, conditions (i) and (ii) on the sequences { α n }{ β n } are different from the ones in Theorem 1.2.

In 2003, Suzuki [18] proved the following result.

Theorem 1.3 ([18], Theorem 2])

Let E be a compact convex subset of a strictly convex Banach space X. Let{ t n :nN}be a sequence of nonexpansive mappings on E with n = 1 Fix( t n ). Let{ γ n }be a sequence of positive numbers such that n = 1 γ n <1, and let{ I n }be a sequence of subsets ofNsatisfying I n I n + 1 fornNand n = 1 I n =N. Define a sequence{ x n }in E by x 1 Cand

x n + 1 = ( 1 i I n γ i ) x n + i I n γ i t i x n

fornN. Then{ x n }converges strongly to a common fixed point of{ t n :nN}.

The purpose of this paper is to extend Theorem 1.2 to countably many numbers of single-valued nonexpansive mappings on strictly convex Banach spaces, thereby the result in Theorem 1.3 is covered. The results for CAT(0) spaces are also derived. Our main discoveries are Theorem 3.2 and Theorem 3.6.

2 Preliminaries

We recall that the graph G(U) of a multivalued mapping U:E 2 X is G(U)={(x,y)X×X;xE,yUx}. The following theorem is essentially proved by Dozo [10].

Theorem 2.1 ([10], Theorem 3.1])

Let X be a Banach space which satisfies Opial’s condition, E be a weakly compact convex subset of X. LetT:EK(X), whereK(X)is a family of nonempty compact subsets of X. Then the graph ofU=ITis closed in(X,σ(X, X ))×(X,), where I denotes the identity on X, σ(X, X )the weak topology andthe norm (or strong) topology.

We will use the theorem in the following form: If { x n } is a sequence in E such that { x n } converges weakly to some zE and {dist( x n ,T x n )} converges to 0, then zTz.

Let { t n :nN} be a family of nonexpansive mappings from E to E. The following lemma proved by Bruck [5] plays a very important role to our proof of the main result.

Lemma 2.2 ([5], Lemma 3])

Let E be a nonempty closed convex subset of a strictly convex Banach space X, let{ t n :nN}be a family of single-valued nonexpansive mappings on E. Suppose n = 1 Fix( t n )is nonempty. Given{ λ n }a sequence of positive numbers with n = 1 λ n =1. Then a mapping t on E defined by

tx= n = 1 λ n t n x

for allxEis well defined, nonexpansive andFix(t)= n = 1 Fix( t n ).

The following results show examples when the required condition on the nonemptiness of the common fixed point set always satisfies:

Theorem 2.3 ([8], Theorem 3.1])

Let E be a weakly compact convex subset of a Banach space X. Suppose E has (MFPP) and (CFPP). LetSbe any commuting family of nonexpansive self-mappings of E. IfT:EKC(E)is a multivalued nonexpansive mapping which commutes with every member ofS, whereKC(E)is the family of nonempty compact convex subsets of E. ThenF(S)Fix(T)whereF(S)= t S Fix(t).

Theorem 2.4 ([8], Theorem 3.2])

Let X be a Banach space satisfying the Kirk-Massa condition, i.e., the asymptotic center of each bounded sequence of X in each bounded closed and convex subset is nonempty and compact. Let E be a weakly compact convex subset of X and letSbe any commuting family of nonexpansive self-mappings of E. SupposeT:EKC(E)is a multivalued mapping satisfying condition( C λ )for someλ(0,1)which commutes with every member ofS. If T is upper semi-continuous, thenF(S)Fix(T).

Note that strictly convex Banach spaces satisfy the condition in the above theorems.

Remark 2.5 In our main theorems (Theorem 3.2 and Theorem 3.6), we assume the following conditions:

F ( S ) Fix ( T ) and T w = { w } for all w F ( S ) Fix ( T ) .
(2.1)

It is an open problem to find a sufficient condition to assure that the condition (2.1) is satisfied.

Let (X,d) be a metric space. A geodesic joining xX to yX is a mapping c from a closed interval [0,l]R to X such that c(0)=xc(l)=y and d(c(t),c( t ))=|t t | for all t, t [0,l]. Thus c is an isometry and d(x,y)=l. The image of c is called a geodesic (or metric) segment joining x and y. We denote [x,y] for this geodesic if it is unique. Write c(α0+(1α)l)=αx(1α)y for α(0,1). The space X is said to be a geodesic space if every two points of X are joined by a geodesic. It is said to be of hyperbolic type [12] if it satisfies:

d ( p , α x ( 1 α ) y ) αd(p,x)+(1α)d(p,y)
(2.2)

for all pX. Let { v 1 , v 2 ,, v n }X and { λ 1 , λ 2 ,, λ n }(0,1) with i = 1 n λ i =1. It had been defined, by induction, in [7] that

i = 1 n λ i v i :=(1 λ n ) ( λ 1 1 λ n v 1 λ 2 1 λ n v 2 λ n 1 1 λ n v n 1 ) λ n v n .
(2.3)

The definition of in (2.3) is an ordered one in the sense that it depends on the order of points v 1 ,, v n . Under (2.2) we can see that

d ( i = 1 n λ i v i , x ) n = 1 n λ i d( v i ,x)
(2.4)

for each xX.

Following [3], a metric space X is said to be a CAT(0)space if it is geodesically connected and if every geodesic triangle in X is at least as thin as its comparison triangle in the Euclidean plane E 2 . In fact (cf. [3] p.163), the following are equivalent for a geodesic space X:

  1. (i)

    X is a CAT(0) space.

  2. (ii)

    X satisfies the (CN) inequality: If x 0 , x 1 X and x 0 x 1 2 is the midpoint of x 0 and x 1 , then

    d 2 ( y , x 0 x 1 2 ) 1 2 d 2 (y, x 0 )+ 1 2 d 2 (y, x 1 ) 1 4 d 2 ( x 0 , x 1 ),for allyX.

Lemma 2.6 ([3], Proposition 2.2]) Let X be aCAT(0)space. Then for eachp,q,r,sXandα[0,1]

d ( α p ( 1 α ) q , α r ( 1 α ) s ) α d ( p , r ) + ( 1 α ) d ( q , s ) .
(2.5)

In particular, (2.2) holds inCAT(0)spaces.

In [9] the element x= n = 1 λ n v n has been defined. Let { λ n } be a given sequence in (0,1) such that n = 1 λ n =1, let { v n } be a bounded sequence in X, and let v 0 be an arbitrary point in X. Let λ n = i = n + 1 λ i and assume that i = n λ i 0 as n. Set

s n := λ 1 v 1 λ 2 v 2 λ n v n λ n v 0 .

Thus, by (2.3),

s n = ( i = 1 n λ i ) w n λ n v 0 ,
(2.6)

where w 1 = v 1 and for each n2

w n = λ 1 i = 1 n λ i v 1 λ 2 i = 1 n λ i v 2 λ n i = 1 n λ i v n .

We know that { s n } is a Cauchy sequence (see [9]). Thus s n x as n for some xX. Write

x= n = 1 λ n v n .

By (2.6), d( s n , w n ) λ n d( w n , v 0 ), it is seen that lim n s n = lim n w n . Thus the limit x is independent of the choice of v 0 .

Lemma 2.7 ([9], Lemma 3.8])

Let C be a nonempty closed convex subset of a completeCAT(0)space X, let{ t n :nN}be a family of single-valued nonexpansive mappings on C. Suppose n = 1 Fix( t n )is nonempty. Definet:CCbyt(x)= n = 1 λ n t n (x)for allxCwhere{ λ n }(0,1)with n = 1 λ n =1and i = n λ i 0asn . Then t is nonexpansive andFix(t)= n = 1 Fix( t n ).

3 Main results

3.1 Strictly convex Banach spaces

The following result is a generalization of the result of [16], Lemma 1.3].

Lemma 3.1 Let E be a compact subset of a strictly convex Banach space X, let{ α n }be a sequence of real numbers such that0<a α n b<1for allnN, and let{ u n }, { v n }be sequences of E satisfying, for somec0,

  1. (i)

    lim sup n u n c,

  2. (ii)

    lim sup n v n c and

  3. (iii)

    lim n α n u n +(1 α n ) v n =c.

Then, lim n u n v n =0.

Proof We suppose on the contrary that lim sup n u n v n 0. Since E and [a,b] are compact, there exist subsequences { u n k } of { u n }, { v n k } of { v n } and { α n k } of { α n } such that lim k u n k =u, lim k v n k =v, lim k α n k =α for some u,vE with uv and for some α[0,1]. From (i) and (ii) we have u= lim k u n k c and v= lim k v n k c. Using the strict convexity of X and (iii), we have c= lim k α n k u n k +(1 α n k ) v n k =αu+(1α)v<αu+(1α)vc, a contradiction. Hence lim n u n v n =0. □

Now we introduce a new iteration method for a family of single-valued nonexpansive mappings and a multivalued nonexpansive mapping. Let E be a nonempty bounded closed convex subset of a Banach space X, let { t n :nN} be a family of single-valued nonexpansive mappings on E, and let T:EFB(E) be a multivalued nonexpansive mapping. Given a sequence of positive numbers { γ n } with n = 1 γ n <1. The sequence { x n } of the modified Ishikawa iteration is defined by x 1 E, and

y n = ( 1 β n ) x n + β n z n , x n + 1 = ( 1 i = 1 n γ i ) x n + i = 1 n γ i t i y n ,
(3.1)

where z n T x n , and 0<a β n b<1. Put F:=( n Fix( t n ))Fix(T).

Theorem 3.2 Let E be a nonempty compact convex subset of a strictly convex Banach space X, let{ t n :nN}be a family of single-valued nonexpansive mappings on E, and letT:EFB(E)be a multivalued nonexpansive mapping. SupposeFandTw={w}for allwF. Given a sequence of positive numbers{ γ n }with n = 1 γ n <1and{ β n }with0<a β n b<1. Then the sequence{ x n }defined by (3.1) converges strongly to somevF.

Proof We follow the proof of [17], Theorem 3.6] and split the proof into five steps.

Step 1. lim n x n w exists for all wF:

We first note that, since Tw={w},

z n w=dist( z n ,Tw)H(T x n ,Tw) x n w.

Consider the following estimates:

x n + 1 w ( 1 i = 1 n γ i ) x n w + i = 1 n γ i t i y n w ( 1 i = 1 n γ i ) x n w + i = 1 n γ i y n w ( 1 i = 1 n γ i ) x n w + ( i = 1 n γ i ) ( ( 1 β n ) x n w + β n z n w ) x n w .

Therefore, { x n w} is a bounded decreasing sequence in R, and hence lim n x n w exists.

Step 2. lim n x n i = 1 n γ i t i y n i = 1 n γ i =0:

From Step 1, suppose lim n x n w=c. We have

i = 1 n γ i t i y n i = 1 n γ i w 1 i = 1 n γ i i = 1 n γ i t i y n i = 1 n γ i w y n w x n w .

Thus

lim sup n i = 1 n γ i t i y n i = 1 n γ i w lim sup n y n w lim sup n x n w=c.
(3.2)

We also have

c = lim n x n + 1 w = lim n ( 1 i = 1 n γ i ) x n + i = 1 n γ i t i y n w = lim n ( 1 i = 1 n γ i ) ( x n w ) + i = 1 n γ i ( i = 1 n γ i t i t n i = 1 n γ i w ) .

By Lemma 3.1, since 0< γ 1 < i = 1 n γ i i = 1 γ i <1, .

Step 3. lim n x n z n =0:

From (3.1), we can see that

x n + 1 w ( 1 i = 1 n γ i ) x n w+ i = 1 n γ i y n w,

and hence x n + 1 w x n w i = 1 n γ i ( y n w x n w). Therefore, ( x n + 1 w x n w i = 1 n γ i )+ x n w y n w and by (3.2) we obtain

c = lim inf n { ( x n + 1 w x n w i = 1 n γ i ) + x n w } lim inf n y n w lim sup n y n w c .

Thus c= lim n y n w= lim n (1 β n )( x n w)+ β n ( z n w). By Lemma 3.1, since 0<a β n b<1, lim n x n z n =0.

Step 4. lim n x n i = 1 γ i t i x n i = 1 γ i =0:

We note from Step 3 that

i = 1 n γ i t i x n i = 1 n γ i i = 1 n γ i t i y n i = 1 n γ i x n y n = β n x n z n 0asn
(3.3)

and

t i x n t i x n w + w x n w + w x 1 w + w : = M

for all iN. Therefore,

x n i = 1 γ i t i x n i = 1 γ i x n i = 1 n γ i t i x n i = 1 n γ i + i = 1 n γ i t i x n i = 1 n γ i i = 1 γ i t i x n i = 1 γ i x n i = 1 n γ i t i x n i = 1 n γ i + i = 1 n γ i t i x n i = 1 n γ i i = 1 n γ i t i x n i = 1 γ i + 1 i = 1 γ i i = n + 1 γ i t i x n x n i = 1 n γ i t i x n i = 1 n γ i + i = n + 1 γ i ( i = 1 n γ i ) ( i = 1 γ i ) i = 1 n γ i M + i = n + 1 γ i i = 1 γ i M = x n i = 1 n γ i t i x n i = 1 n γ i + 2 i = n + 1 γ i i = 1 γ i M x n i = 1 n γ i t i y n i = 1 n γ i + i = 1 n γ i t i y n i = 1 n γ i i = 1 n γ i t i x n i = 1 n γ i + 2 i = n + 1 γ i i = 1 γ i M .

From Step 2 and (3.3), we obtain lim n x n i = 1 γ i t i x n i = 1 γ i =0.

Step 5. lim n x n =vF:

Define a mapping t:EE by

tx= n = 1 γ n t n x n = 1 γ n

for any xE. By Lemma 2.2, t is well defined, nonexpansive and Fix(t)= n = 1 Fix( t n ). Since E is compact, there exists a subsequence { x n k } of { x n } which converges to v for some vE. Using Step 3 and Step 4, we have

t v v lim k ( t v t x n k + t x n k x n k + x n k v ) lim k ( t x n k x n k + 2 x n k v ) = 0

and

dist ( v , T v ) v x n k + dist ( x n k , T x n k ) + H ( T x n k , T v ) v x n k + x n k z n k + x n k v 0 as k .

It follows that vFix(T)Fix(t)=F. Since lim n x n v exists by Step 1, lim n x n v= lim k x n k v=0. □

The following example shows that the condition ‘Tw={w} for all wF’ in Theorem 3.2 is necessary.

Example 3.3 We consider the space X of Example 3.9 in [8]. Let X be the Hilbert space R 2 with the usual norm, and let f:[0,1][0,1] be a continuous strictly concave function such that f(0)= 1 2 f(1)=1 and f (x)1 for all x[0,1]. Let ε n = i = 1 n ( 1 2 ) i + 1 T: [ 0 , 1 ] 2 FB( [ 0 , 1 ] 2 ) be defined by T(a,b)=[0,1]×[f(a),1] and t n : [ 0 , 1 ] 2 [ 0 , 1 ] 2 be defined by

t n (a,b)= { ( a , ε n ) , b < ε n , ( a , b ) , otherwise.

It is straightforward showing that T and each t n are nonexpansive. Set x 1 =(1,0) [ 0 , 1 ] 2 and for a subsequence { γ n } in (0,1) with n = 1 γ n <1. Let { x n =( a n , b n )} be a sequence in [ 0 , 1 ] 2 defined as

y n = 1 2 x n + 1 2 z n , x n + 1 = ( 1 i = 1 n γ i ) x n + i = 1 n γ i t i y n ,
(3.4)

where

z n = { ( 0 , f ( a n ) ) , n is odd , ( 1 , f ( a n ) ) , n is even .

We will show that { x n } does not converge to a common fixed point of T and { t n }.

Proof Clearly, { z n } is a divergent sequence. We note that ε n 1 2 and for each y=(a,b) [ 0 , 1 ] 2 with b 1 2 , we have t i y=y for all i. If we put y n =( c n , d n ), then d n 1 2 for all n. Since n = 1 γ n <1, we must have d( x n , z n )0 as n. Suppose { x n } converges to z for some zF={(a,b) [ 0 , 1 ] 2 :bf(a)}. Thus { z n } also converges to z, a contradiction. □

It is noticed that F is not convex. Thus it is not a nonexpansive retract of any convex set. It can be also observed that if we redefine the mapping T as T(a,b)={a}×[ 1 + b 2 ,1] we can easily verify that T is nonexpansive and the condition (2.1) is satisfied.

Remark 3.4 With the same proof, Theorem 3.2 is valid when { x n } is of the following form: For a permutation π on N, define { x n } in E by x 1 E and

y n = ( 1 β n ) x n + β n z n , x n + 1 = ( 1 i = 1 n γ π ( i ) ) x n + i = 1 n γ π ( i ) t π ( i ) y n ,

z n T x n , and 0<a β n b<1.

Note also that the above result is equivalent to:

Let { I n } be a sequence of subsets of N satisfying I n I n + 1 for nN and n = 1 I n =N. Define { x n } in E by x 1 E and

y n = ( 1 β n ) x n + β n z n , x n + 1 = ( 1 i I n γ i ) x n + i I n γ i t i y n ,

z n T x n , and 0<a β n b<1. Then the sequence { x n } converges strongly to some vF.

Thus Theorem 3.2 contains Theorem 1.3.

With the application of the demiclosedness principle (Theorem 2.1), a weak convergence version of Theorem 3.2 also holds:

Theorem 3.5 Let X be a strictly convex Banach space satisfying the Opial’s condition, E be a weakly compact convex subset of X, let{ t n :nN}be a family of single-valued nonexpansive mappings on E, and letT:EFB(E)be a multivalued nonexpansive mapping. SupposeFandTw={w}for allwF. Given a sequence of positive numbers{ γ n }with0< n = 1 γ n <1and{ β n }with0<a β n b<1. Then the sequence{ x n }defined by (3.1) converges weakly to somevF.

Proof In the proof of Theorem 3.2, by applying the Opial’s condition, it follows from a standard argument that { x n } converges weakly to some vE. Then Theorem 2.1 implies that v is a point in F. □

3.2 CAT(0) spaces

Let E be a nonempty bounded closed convex subset of a complete CAT(0) space X, let { t n :nN} be a family of single-valued nonexpansive mappings on E, and T:EFB(E) be a multivalued nonexpansive mapping. Given { γ n } a sequence of positive numbers with n = 1 γ n <1 and i = n γ i 0 as n where γ n = i = n + 1 γ i . The sequence { x n } of the modified Ishikawa iteration is defined by

y n = ( 1 β n ) x n β n z n , x n + 1 = ( 1 i = 1 n γ i ) x n ( i = 1 n γ i ) i = 1 n γ i i = 1 n γ i t i y n ,
(3.5)

where x 1 E, z n T x n , and 0<a β n b<1. Put F:= n = 1 Fix( t n )Fix(T).

Theorem 3.6 Let E be a compact convex subset of a completeCAT(0)space X. Let{ t n :nN}be a family of single-valued nonexpansive mappings on E, and letT:EFB(E)be a multivalued nonexpansive mapping. SupposeFandTw={w}for allwF. Given{ γ n }a sequence of positive numbers with n = 1 γ n <1and i = n γ i 0asnwhere γ n = i = n + 1 γ i . If0<a β n b<1, then the sequence{ x n }defined by (3.5) converges strongly to somevF.

Proof The proof follows along the lines with the proof of Theorem 3.2. Recall that w 1 x= t 1 x and w n x= i = 1 n γ i i = 1 n γ i t i x for all n2. Thus, by (3.5),

x n + 1 = ( 1 i = 1 n γ i ) x n ( i = 1 n γ i ) w n y n .

As before, we consider the proof in 5 steps. Because of the same details in some cases, we only present proofs for Step 2 to Step 4.

Step 2. lim n d( x n , w n y n )=0:

Let wF, we have w n w=w for all n. Using the nonexpansiveness of w n , we see that

d ( w n y n , w ) d ( y n , w ) ( 1 β n ) d ( x n , w ) + β n d ( z n , w ) d ( x n , w ) .
(3.6)

By (3.6) and using (CN) inequality,

d 2 ( x n + 1 , w ) ( 1 i = 1 n γ i ) d 2 ( x n , w ) + ( i = 1 n γ i ) d 2 ( w n y n , w ) i = 1 n γ i ( 1 i = 1 n γ i ) d 2 ( x n , w n y n ) d 2 ( x n , w ) i = 1 n γ i ( 1 i = 1 n γ i ) d 2 ( x n , w n y n ) .

Let γ= i = 1 γ i . Since 0< γ 1 i = 1 n γ i γ<1,

γ 1 (1γ) d 2 ( x n , w n y n ) i = 1 n γ i ( 1 i = 1 n γ i ) d 2 ( x n , w n y n ) d 2 ( x n ,w) d 2 ( x n + 1 ,w).

This implies that

n = 1 [ γ 1 ( 1 γ ) d 2 ( x n , w n y n ) ] d 2 ( x 1 ,w)<,

and hence lim n d( x n , w n y n )=0.

Step 3. lim n d( x n , z n )=0:

Using (3.6) and (CN) inequality, we have

d 2 ( w n y n , w ) d 2 ( y n , w ) ( 1 β n ) d 2 ( x n , w ) + β n d 2 ( z n , w ) β n ( 1 β n ) d 2 ( x n , z n ) d 2 ( x n , w ) β n ( 1 β n ) d 2 ( x n , z n ) ,

and thus

d 2 ( x n + 1 , u ) ( 1 i = 1 n γ i ) d 2 ( x n , w ) + i = 1 n γ i d 2 ( w n y n , w ) d 2 ( x n , w ) β n ( i = 1 n γ i ) ( 1 β n ) d 2 ( x n , z n ) .

As before,

a γ 1 (1b) d 2 ( x n , z n ) β n ( i = 1 n γ i ) (1 β n ) d 2 ( x n , z n ) d 2 ( x n ,w) d 2 ( x n + 1 ,w).

This also implies that lim n d( x n , z n )=0.

Step 4. lim n d( x n ,t x n )=0, where t= i = 1 γ i i = 1 γ i t i :

Since E is compact, there exists a subsequence { y n } of { y n } such that y n y as n for some yE. Using the nonexpansiveness of w n and t, we have

d ( w n y n , t y n ) d ( w n y n , w n y ) + d ( w n y , t y ) + d ( t y , t y n ) 2 d ( y n , y ) + d ( w n y , t y ) 0 as n .

Therefore, lim n d( w n y n ,t y n )=0. From Step 2 and Step 3 we have

d ( x n , t x n ) d ( x n , t y n ) + d ( t y n , t x n ) d ( x n , t y n ) + d ( y n , x n ) d ( x n , w n y n ) + d ( w n y n , t y n ) + β n d ( x n , z n ) 0 as n .

 □

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Acknowledgements

The authors are grateful to the referees for their valuable comments and suggestions. They also would like to thank the Junior Science Talent Project (JSTP) under Thailand’s National Science and Technology Development Agency for financial support.

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Anakkamatee, W., Dhompongsa, S. An approximation of a common fixed point of nonexpansive mappings on convex metric spaces. Fixed Point Theory Appl 2012, 112 (2012). https://doi.org/10.1186/1687-1812-2012-112

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Keywords

  • common fixed point
  • nonexpansive mapping
  • strictly convex Banach space
  • CAT(0) space