Open Access

Viscosity approximation methods for two nonexpansive semigroups in CAT(0) spaces

Journal of Inequalities and Applications20142014:283

https://doi.org/10.1186/1029-242X-2014-283

Received: 6 November 2013

Accepted: 10 July 2014

Published: 15 August 2014

Abstract

The purpose of this paper is by using the viscosity approximation method to study the strong convergence problem for two one-parameter continuous semigroups of nonexpansive mappings in CAT(0) spaces. Under suitable conditions, some strong convergence theorems for the proposed implicit and explicit iterative schemes to converge to a common fixed point of two one-parameter continuous semigroups of nonexpansive mappings are proved, which is also a unique solution of some kind of variational inequalities. The results presented in this paper extend and improve the corresponding results of some others.

MSC:47J05, 47H09, 49J25.

Keywords

viscosity approximation method nonexpansive semigroup implicit and explicit iterative scheme CAT(0) space fixed point

1 Introduction

Throughout this paper, we assume that X is a CAT(0) space, is the set of positive integers, is the set of real numbers, R + is the set of nonnegative real numbers and C is a nonempty closed and convex subset of a complete CAT(0) space X.

A family of mappings T : = { T ( t ) : t R + } : C C is called a one-parameter continuous semigroup of nonexpansive mappings if the following conditions are satisfied:
  1. (i)
    for each t R + , T ( t ) is a nonexpansive mapping on C, i.e.,
    d ( T ( t ) x , T ( t ) y ) d ( x , y ) , x , y C ;
     
  2. (ii)

    T ( s + t ) = T ( t ) T ( s ) for all t , s R + ;

     
  3. (iii)

    for each x X , the mapping T ( ) x from R + into C is continuous.

     
A family of mappings T : = { T ( t ) : t R + } is called a one-parameter strongly continuous semigroup of nonexpansive mappings if conditions (i), (ii), (iii) and the following condition are satisfied:
  1. (iv)

    T ( 0 ) x = x for all x C .

     
In the sequel, we shall denote by the common fixed point set of T , that is,
F : = F ( T ) = { x C : T ( t ) x = x , t R + } = t R + F ( T ( t ) ) .
It is well known that one classical way to study nonexpansive mappings is to use the contractions to approximate nonexpansive mappings. More precisely, take t ( 0 , 1 ) and define a contraction T t : C C by
T t = t u + ( 1 t ) T x , x C ,
(1.1)

where u C is an arbitrary fixed element. In the case of T having a fixed point, Browder [1] proved that x t converged strongly to a fixed point of T that is nearest to u in the framework of Hilbert spaces. Reich [2] extended Browder’s result to the setting of a uniformly smooth Banach space and proved that x t converged strongly to a fixed point of T.

Halpern [3] introduced the following explicit iterative scheme (1.2) for a nonexpansive mapping T on a subset C of a Hilbert space:
x n + 1 = α n u + ( 1 α n ) T x n .
(1.2)
He proved that the sequence { x n } converged to a fixed point of T. In [4], Shioji and Takahashi introduced the following implicit iteration in a Hilbert space:
x n = α n u + ( 1 α n ) 1 t n 0 t n T ( t ) x n d t .
(1.3)

Under suitable conditions, they proved strong convergence of { x n } to a member of .

Later, Suzuki [5] introduced in a Hilbert space the following iteration process:
x n + 1 = α n u + ( 1 α n ) T ( t n ) x n , n 1 ,
(1.4)
where { T ( t ) : t 0 } is a strongly continuous semigroup of nonexpansive mappings on C such that F . Under suitable conditions he proved that { x n } converged strongly to the element of nearest to u. Using Moudafi’s viscosity approximation methods, Song and Xu [6], Cho and Kang [7] introduced the following iteration process:
x n = α n f ( x n ) + ( 1 α n ) T ( t n ) x n , n 1 ,
(1.5)
and
x n + 1 = α n f ( x n ) + ( 1 α n ) T ( t n ) x n , n 1 .
(1.6)

They proved that { x n } defined by (1.5) and (1.6) both converged to the same point of in a reflexive strictly convex Banach space with a uniformly Gâteaux differentiable norm.

In a similar way, Dhompongsa et al. [8] extended Browder’s implicit iteration to a strongly continuous semigroup of nonexpansive mappings { T ( t ) : t 0 } in a complete CAT(0) space X. Under suitable conditions he proved that the sequence converged strongly to the element of nearest to u. Using Moudafi’s viscosity approximation methods, Shi and Chen [9] studied the convergence theorems of the following Moudafi’s viscosity iterations for a nonexpansive mapping T:
x t = t f ( x t ) ( 1 t ) T x t ,
(1.7)
and
x n + 1 = α n f ( x n ) ( 1 α n ) T x n .
(1.8)

They proved that { x t } defined by (1.7) and { x n } defined by (1.8) converged strongly to a fixed point of T in the framework of CAT(0) spaces.

Very recently, Wangkeeree and Preechasilp [10] extended the results of [9] to a one-parameter continuous semigroup of nonexpansive mappings T : = { T ( t ) : t R + } in CAT(0) spaces. Under suitable conditions they proved that the iterative schemes { x n } both converged strongly to the same point x ˜ such that x ˜ = P F f ( x ˜ ) , which is the unique solution of the variational inequality
x ˜ f x ˜ , x x ˜ 0 , x F .
(1.9)

Motivated and inspired by the research going on in this direction, especially inspired by Wangkeeree and Preechasilp [10], in this paper we study the strong convergence theorems of Moudafi’s viscosity approximation methods for two one-parameter continuous semigroups of nonexpansive mappings in CAT(0) spaces. We prove that the implicit and explicit iteration algorithms both converge strongly to the same point x ˜ such that x ˜ = P F f ( x ˜ ) , which is the unique solution of the variational inequality (1.9) where is the set of common fixed points of the two semigroups of nonexpansive mappings.

2 Preliminaries and lemmas

In this paper, we write ( 1 t ) x t y for the unique point z in the geodesic segment joining from x to y such that
d ( x , z ) = t d ( x , y ) , d ( y , z ) = ( 1 t ) d ( x , y ) .
(2.1)

Lemma 2.1 [11]

A geodesic space X is a CAT(0) space if and only if the following inequality
d 2 ( ( 1 t ) x t y , z ) ( 1 t ) d 2 ( x , z ) + t d 2 ( y , z ) t ( 1 t ) d 2 ( x , y )
(2.2)
is satisfied for all x , y , z X and t [ 0 , 1 ] . In particular, if x, y, z are points in a CAT(0) space and t [ 0 , 1 ] , then
d ( ( 1 t ) x t y , z ) ( 1 t ) d ( x , z ) + t d ( y , z ) .
(2.3)

Lemma 2.2 [12]

Let X be a CAT(0) space, p , q , r , s X and λ [ 0 , 1 ] . Then
d ( λ p ( 1 λ ) q , λ r ( 1 λ ) s ) λ d ( p , r ) + ( 1 λ ) d ( q , s ) .
By induction, we write
m = 1 n λ m x m : = ( 1 λ n ) ( λ 1 1 λ n x 1 λ 2 1 λ n x 2 λ n 1 1 λ n x n 1 ) λ n x n .
(2.4)
Lemma 2.3 Let X be a CAT(0) space, then, for any sequence { λ m } m = 1 n in [ 0 , 1 ] satisfying m = 1 n λ m = 1 and for any { x m } m = 1 n X , the following conclusions hold:
d ( m = 1 n λ m x m , x ) m = 1 n λ m d ( x m , x ) , x X ;
(2.5)
and
d 2 ( m = 1 n λ m x m , x ) m = 1 n λ m d 2 ( x m , x ) λ 1 λ 2 d 2 ( x 1 , x 2 ) , x X .
(2.6)
Proof It is obvious that (2.5) holds for n = 2 . Suppose that (2.5) holds for some n 2 . From (2.3) and (2.4) we have
d ( m = 1 n + 1 λ m x m , x ) = d ( ( 1 λ n + 1 ) ( λ 1 1 λ n + 1 x 1 λ 2 1 λ n + 1 x 2 λ n 1 λ n + 1 x n ) λ n + 1 x n + 1 , x ) ( 1 λ n + 1 ) d ( λ 1 1 λ n + 1 x 1 λ 2 1 λ n + 1 x 2 λ n 1 λ n + 1 x n , x ) + λ n + 1 d ( x n + 1 , x ) λ 1 d ( x 1 , x ) + λ 2 d ( x 2 , x ) + + λ n d ( x n , x ) + λ n + 1 d ( x n + 1 , x ) = m = 1 n + 1 λ m d ( x m , x ) .

This implies that (2.5) holds.

Next, we prove that (2.6) holds.

Indeed, it is obvious that (2.6) holds for n = 2 . Suppose that (2.6) holds for some n 2 . Next we prove that (2.6) is also true for n + 1 .

In fact, we have
d 2 ( m = 1 n + 1 λ m x m , x ) = d 2 ( m = 1 n λ m x m λ n + 1 x n + 1 , x ) .
From (2.2) and (2.4) and the assumption of induction, we have
d 2 ( m = 1 n + 1 λ m x m , x ) = d 2 ( m = 1 n λ m x m λ n + 1 x n + 1 , x ) = d 2 ( ( 1 λ n + 1 ) m = 1 n λ m 1 λ n + 1 x m λ n + 1 x n + 1 , x ) ( 1 λ n + 1 ) d 2 ( m = 1 n λ m 1 λ n + 1 x m , x ) + λ n + 1 d 2 ( x n + 1 , x ) ( 1 λ n + 1 ) m = 1 n λ m 1 λ n + 1 d 2 ( x m , x ) λ 1 λ 2 d 2 ( x 1 , x 2 ) + λ n + 1 d 2 ( x n + 1 , x ) = m = 1 n + 1 λ m d 2 ( x m , x ) λ 1 λ 2 d 2 ( x 1 , x 2 ) .

This completes the proof of (2.6). □

The concept of Δ-convergence introduced by Lim [13] in 1976 was shown by Kirk and Panyanak [14] in CAT(0) spaces to be very similar to the weak convergence in the Banach space setting (see also [15]). Now, we give the concept of Δ-convergence.

Let { x n } be a bounded sequence in a CAT(0) space X. For x X , we set
r ( x , { x n } ) = lim sup n d ( x , x n ) .
The asymptotic radius r ( { x n } ) of { x n } is given by
r ( { x n } ) = inf x X { r ( x , { x n } ) } ,
and the asymptotic center A ( { x n } ) of { x n } is the set
A ( { x n } ) = { x X : r ( x , { x n } ) = r ( { x n } ) } .

It is known from Proposition 7 of [16] that in a complete CAT(0) space, A ( { x n } ) consists of exactly one point. A sequence { x n } X is said to Δ-converge to x X if A ( { x n k } ) = { x } for every subsequence { x n k } of { x n } .

The uniqueness of an asymptotic center implies that a CAT(0) space X satisfies Opial’s property, i.e., for given { x n } X such that { x n } Δ-converges to x and given y X with y x ,
lim sup n d ( x n , x ) < lim sup n d ( x n , y ) .

Lemma 2.4 [14]

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

Berg and Nikolaev [17]introduced the concept of quasilinearization as follows. Let us denote a pair ( a , b ) X × X by a b and call it a vector. Then quasilinearization is defined as a map , : ( X × X ) × ( X × X ) R defined by
a b , c d = 1 2 ( d 2 ( a , d ) + d 2 ( b , c ) d 2 ( a , c ) d 2 ( b , d ) ) ( a , b , c , d X ) .
(2.7)
It is easily seen that a b , c d = c d , a b , a b , c d = b a , c d and a x , c d + x b , c d = a b , c d for all a , b , c , d X . We say that X satisfies the Cauchy-Schwarz inequality if
a b , c d d ( a , b ) d ( c , d )
(2.8)

for all a , b , c , d X .

Recently, Dehghan and Rooin [18] presented a characterization of metric projection in CAT(0) spaces as follows.

Lemma 2.5 Let C be a nonempty convex subset of a complete CAT(0) space X, x X and u C . Then u = P C x if and only if
y u , u x 0 , y C .
(2.9)

Lemma 2.6 [19]

Let X be a complete CAT(0) space, { x n } be a sequence in X and x X . Then { x n } Δ-converges to x if and only if lim sup n x x n , x y 0 for all y X .

Lemma 2.7 [20]

Let { a n } be a sequence of nonnegative real numbers satisfying the property a n + 1 ( 1 α n ) a n + α n β n , n 0 , where { α n } ( 0 , 1 ) and { β n } R such that
  1. (i)

    n = 0 α n = ;

     
  2. (ii)

    lim sup n β n 0 or n = 0 | α n β n | < .

     

Then { a n } converges to zero as n .

3 Viscosity approximation iteration algorithms

In this section, we present the strong convergence theorems of Moudafi’s viscosity approximation implicit and explicit iteration algorithms for two one-parameter continuous semigroups of nonexpansive mappings T : = { T ( t ) : t R + } and S : = { S ( s ) : s R + } in CAT(0) spaces.

Before proving main results, we need the following two vital lemmas.

Lemma 3.1 [10, 21]

Let X be a complete CAT(0) space. Then, for all u , x , y X , the following inequality holds:
d 2 ( x , u ) d 2 ( y , u ) + 2 x y , x u .
Lemma 3.2 Let X be a complete CAT(0) space. For any u , v , w X and r , s , t [ 0 , 1 ] , r + s + t = 1 , let z = r u s v t w . Then, for any x , y X , the following inequality holds:
z x , z y r u x , z y + s v x , z y + t w x , z y + r t d 2 ( u , w ) + s t d 2 ( v , w ) .
Proof It follows from (2.1) and (2.6) that
d 2 ( u , z ) = d 2 ( u , r u ( 1 r ) ( s 1 r v t 1 r w ) ) = ( 1 r ) 2 d 2 ( u , s 1 r v t 1 r w ) ( 1 r ) 2 ( s 1 r d 2 ( u , v ) + t 1 r d 2 ( u , w ) s 1 r t 1 r d 2 ( v , w ) ) = ( 1 r ) s d 2 ( u , v ) + ( 1 r ) t d 2 ( u , w ) s t d 2 ( v , w ) .
Similarly, we can obtain d 2 ( v , z ) ( 1 s ) r d 2 ( v , u ) + ( 1 s ) t d 2 ( v , w ) r t d 2 ( u , w ) and d 2 ( w , z ) ( 1 t ) r d 2 ( w , u ) + ( 1 t ) s d 2 ( w , v ) r s d 2 ( u , v ) . Therefore, we have
r d 2 ( u , z ) + s d 2 ( v , z ) + t d 2 ( w , z ) ( 1 r ) r s d 2 ( u , v ) + ( 1 r ) r t d 2 ( u , w ) r s t d 2 ( v , w ) + ( 1 s ) s r d 2 ( v , u ) + ( 1 s ) s t d 2 ( v , w ) r s t d 2 ( u , w ) + ( 1 t ) t r d 2 ( w , u ) + ( 1 t ) t s d 2 ( w , v ) r s t d 2 ( u , v ) = r s d 2 ( u , v ) + r t d 2 ( u , w ) + s t d 2 ( v , w ) .
(3.1)
From (2.6) and (3.1), we have that
2 z x , z y = d 2 ( z , y ) + d 2 ( x , z ) d 2 ( x , y ) r d 2 ( u , y ) + s d 2 ( v , y ) + t d 2 ( w , y ) r s d 2 ( u , v ) + r d 2 ( x , z ) + s d 2 ( x , z ) + t d 2 ( x , z ) r d 2 ( x , y ) s d 2 ( x , y ) t d 2 ( x , y ) = 2 r u x , z y + 2 s v x , z y + 2 t w x , z y r s d 2 ( u , v ) + r d 2 ( u , z ) + s d 2 ( v , z ) + t d 2 ( w , z ) 2 r u x , z y + 2 s v x , z y + 2 t w x , z y + r t d 2 ( u , w ) + s t d 2 ( v , w ) ,

which is the desired result. □

Now we are in a position to state and prove our main results.

Theorem 3.3 Let C be a closed convex subset of a complete CAT(0) space X, and let { T ( t ) } and { S ( s ) } be two one-parameter continuous semigroups of nonexpansive mappings on C satisfying F : = F ( T ) F ( S ) and both uniformly asymptotically regular (in short, u.a.r.) on C, that is, for all h , k 0 and any bounded subset B of C,
lim t sup x B d ( T ( h ) ( T ( t ) x ) , T ( t ) x ) = 0 , lim s sup x B d ( S ( k ) ( S ( s ) x ) , S ( s ) x ) = 0 .
Let f be a contraction on C with coefficient α ( 0 , 1 ) . Suppose that the sequence { x n } is given by
x n = α n f ( x n ) β n T ( t n ) x n γ n S ( s n ) x n
(3.2)
for all n 0 , where { α n } , { β n } , { γ n } ( 0 , 1 ) and t n , s n [ 0 , ) satisfy the following conditions:
  1. (i)

    α n + β n + γ n = 1 ;

     
  2. (ii)

    lim n α n = 0 , γ n = o ( α n ) ;

     
  3. (iii)

    lim n t n = , lim n s n = ;

     
  4. (iv)

    for any bounded subset B of C, lim n sup x B T ( t n ) x , S ( s n ) x = 0 .

     
Then { x n } converges strongly to x ˜ such that x ˜ = P F f ( x ˜ ) , which is equivalent to the following variational inequality:
x ˜ f ( x ˜ ) , x x ˜ 0 , x F .
(3.3)

Proof We shall divide the proof of Theorem 3.3 into five steps.

Step 1. The sequence { x n } defined by (3.2) is well defined for all n 0 .

In fact, let us define mappings G , M : C C by
G n ( x ) : = α n f ( x ) β n T ( t n ) x γ n S ( s n ) x , x C
and
M n ( x ) : = β n 1 α n T ( t n ) x γ n 1 α n S ( s n ) x , x C ,
respectively. For any x , y C , from Lemma 2.2, we have
d ( M n ( x ) , M n ( y ) ) = d ( β n 1 α n T ( t n ) x γ n 1 α n S ( s n ) x , β n 1 α n T ( t n ) y γ n 1 α n S ( s n ) y ) β n 1 α n d ( T ( t n ) x , T ( t n ) y ) + γ n 1 α n d ( S ( s n ) x , S ( s n ) y ) β n 1 α n d ( x , y ) + γ n 1 α n d ( x , y ) = d ( x , y ) .
Therefore we have that
d ( G n ( x ) , G n ( y ) ) = d ( α n f ( x ) ( 1 α n ) M n ( x ) , α n f ( y ) ( 1 α n ) M n ( y ) ) α n d ( f ( x ) , f ( y ) ) + ( 1 α n ) d ( M n ( x ) , M n ( y ) ) α n α d ( x , y ) + ( 1 α n ) d ( x , y ) = ( 1 α n ( 1 α ) ) d ( x , y ) .

This implies that G n is a contraction mapping. Hence, the sequence { x n } is well defined for all n 0 .

Step 2. The sequence { x n } is bounded.

For any p F , from Lemma 2.3, we have that
d ( x n , p ) = d ( α n f ( x n ) β n T ( t n ) x n γ n S ( s n ) x n , p ) α n d ( f ( x n ) , p ) + β n d ( T ( t n ) x n , p ) + γ n d ( S ( s n ) x n , p ) α n d ( f ( x n ) , p ) + β n d ( x n , p ) + γ n d ( x n , p ) = α n d ( f ( x n ) , p ) + ( 1 α n ) d ( x n , p ) .
(3.4)
Then
d ( x n , p ) d ( f ( x n ) , p ) d ( f ( x n ) , f ( p ) ) + d ( f ( p ) , p ) α d ( x n , p ) + d ( f ( p ) , p ) .
This implies that
d ( x n , p ) 1 1 α d ( f ( p ) , p ) .

Hence { x n } is bounded, so are { T ( t n ) x n } , { S ( s n ) x n } and { f ( x n ) } .

Step 3. For any h , k 0 , lim n d ( x n , T ( h ) x n ) = 0 and lim n d ( x n , S ( k ) x n ) = 0 .

From Lemma 2.3 and condition (ii), we have
d ( x n , T ( t n ) x n ) = d ( α n f ( x n ) β n T ( t n ) x n γ n S ( s n ) x n , T ( t n ) x n ) α n d ( f ( x n ) , T ( t n ) x n ) + γ n d ( S ( s n ) x n , T ( t n ) x n ) 0 ( n )
and
d ( x n , S ( s n ) x n ) = d ( α n f ( x n ) β n T ( t n ) x n γ n S ( s n ) x n , S ( s n ) x n ) α n d ( f ( x n ) , S ( s n ) x n ) + β n d ( T ( t n ) x n , S ( s n ) x n ) 0 ( n ) .
Since { T ( t ) } and { S ( s ) } is u.a.r., we obtain that for all h , k > 0 ,
lim n d ( T ( h ) ( T ( t n ) x n ) , T ( t n ) x n ) lim n sup x B d ( T ( h ) ( T ( t n ) x ) , T ( t n ) x ) = 0
and
lim n d ( S ( k ) ( S ( s n ) x n ) , S ( s n ) x n ) lim n sup x B d ( S ( k ) ( S ( s n ) x ) , S ( s n ) x ) = 0 ,
where B is any bounded subset of C containing { x n } . Hence, we have
d ( x n , T ( h ) x n ) d ( x n , T ( t n ) x n ) + d ( T ( t n ) x n , T ( h ) ( T ( t n ) x n ) ) + d ( T ( h ) ( T ( t n ) x n ) , T ( h ) x n ) 2 d ( x n , T ( t n ) x n ) + d ( T ( t n ) x n , T ( h ) ( T ( t n ) x n ) ) 0 ( n )
and
d ( x n , S ( k ) x n ) d ( x n , S ( s n ) x n ) + d ( S ( s n ) x n , S ( k ) ( S ( s n ) x n ) ) + d ( S ( k ) ( S ( s n ) x n ) , S ( k ) x n ) 2 d ( x n , S ( s n ) x n ) + d ( S ( s n ) x n , S ( k ) ( S ( s n ) x n ) ) 0 ( n ) .

Step 4. The sequence { x n } contains a subsequence converging strongly to x ˜ such that x ˜ = P F f ( x ˜ ) , which is equivalent to (3.3).

Since { x n } is bounded, by Lemma 2.4, there exists a subsequence { x n j } of { x n } (without loss of generality, we denote it by { x j } ) which Δ-converges to a point x ˜ .

First we claim that x ˜ F = F ( T ) F ( S ) . Since every CAT(0) space has Opial’s property, for any h 0 , if T ( h ) x ˜ x ˜ , we have
lim sup j d ( x j , T ( h ) x ˜ ) lim sup j ( d ( x j , T ( h ) x j ) + d ( T ( h ) x j , T ( h ) x ˜ ) ) lim sup j ( d ( x j , T ( h ) x j ) + d ( x j , x ˜ ) ) = lim sup j d ( x j , x ˜ ) < lim sup j d ( x j , T ( h ) x ˜ ) .

This is a contraction, and hence x ˜ F ( T ) . Similarly, we can obtain that x ˜ F ( S ) . So we have x ˜ F .

Next we prove that { x j } converges strongly to x ˜ . Indeed, it follows from Lemma 3.2 that
d 2 ( x j , x ˜ ) = x j x ˜ , x j x ˜ α j f ( x j ) x ˜ , x j x ˜ + β j T ( t j ) x j x ˜ , x j x ˜ + γ j S ( s j ) x j x ˜ , x j x ˜ + α j N j α j f ( x j ) x ˜ , x j x ˜ + β j d ( T ( t j ) x j , x ˜ ) d ( x j , x ˜ ) + γ j d ( S ( s j ) x j , x ˜ ) d ( x j , x ˜ ) + α j N j α j f ( x j ) x ˜ , x j x ˜ + ( 1 α j ) d 2 ( x j , x ˜ ) + α j N j ,
where N j : = γ j α j β j d 2 ( T ( t j ) x j , S ( s j ) x j ) + γ j d 2 ( f ( x j ) , S ( s j ) x j ) . It follows that
d 2 ( x j , x ˜ ) f ( x j ) x ˜ , x j x ˜ + N j = f ( x j ) f ( x ˜ ) , x j x ˜ + f ( x ˜ ) x ˜ , x j x ˜ + N j d ( f ( x j ) , f ( x ˜ ) ) d ( x j , x ˜ ) + f ( x ˜ ) x ˜ , x j x ˜ + N j α d 2 ( x j , x ˜ ) + f ( x ˜ ) x ˜ , x j x ˜ + N j ,
and thus
d 2 ( x j , x ˜ ) 1 1 α f ( x ˜ ) x ˜ , x j x ˜ + 1 1 α N j .
(3.5)
Since { x j } Δ-converges to x ˜ , by Lemma 2.6 we have
lim sup n f ( x ˜ ) x ˜ , x j x ˜ 0 .

It follows from (3.5) and lim j N j = 0 that { x j } converges strongly to x ˜ .

Next we show that x ˜ solves the variational inequality (3.3). Applying Lemma 2.3, for any q F , we have
d 2 ( x j , q ) = d 2 ( α j f ( x j ) β j T ( t j ) x j γ j S ( s j ) x j , q ) α j d 2 ( f ( x j ) , q ) + β j d 2 ( T ( t j ) x j , q ) + γ j d 2 ( S ( s j ) x j , q ) α j β j d 2 ( f ( x j ) , T ( t j ) x j ) α j d 2 ( f ( x j ) , q ) + ( 1 α j ) d 2 ( x j , q ) α j β j d 2 ( f ( x j ) , T ( t j ) x j ) .
This implies that
d 2 ( x j , q ) d 2 ( f ( x j ) , q ) β j ( d ( f ( x j ) , x j ) + d ( x j , T ( t j ) x j ) ) 2 .
Taking the limit through j , we can obtain
d 2 ( x ˜ , q ) d 2 ( f ( x ˜ ) , q ) d 2 ( f ( x ˜ ) , x ˜ ) .
(3.6)
On the other hand, from (2.7) we have
x f ( x ˜ ) , q x ˜ = 1 2 [ d 2 ( x ˜ , x ˜ ) + d 2 ( f ( x ˜ ) , q ) d 2 ( x ˜ , q ) d 2 ( f ( x ˜ ) , x ˜ ) ] .
(3.7)
From (3.6) and (3.7) we have
x f ( x ˜ ) , q x ˜ 0 , q F .

That is, x ˜ solves inequality (3.3).

Step 5. The sequence { x n } converges strongly to x ˜ .

Assume that x n i x ˆ as n . By the same argument, we get that x ˆ F and solves the variational inequality (3.3), i.e.,
x ˜ f ( x ˜ ) , x ˜ x ˆ 0
(3.8)
and
x ˆ f ( x ˆ ) , x ˆ x ˜ 0 .
(3.9)
Adding up (3.8) and (3.9), we get that
0 x ˜ f ( x ˜ ) , x ˜ x ˆ x ˆ f ( x ˆ ) , x ˜ x ˆ = x ˜ f ( x ˆ ) , x ˜ x ˆ + f ( x ˆ ) f ( x ˜ ) , x ˜ x ˆ x ˆ x ˜ , x ˜ x ˆ x ˜ f ( x ˆ ) , x ˜ x ˆ = x ˜ x ˆ , x ˜ x ˆ f ( x ˆ ) f ( x ˜ ) , x ˆ x ˜ x ˜ x ˆ , x ˜ x ˆ d ( f ( x ˆ ) , f ( x ˜ ) ) d ( x ˆ , x ˜ ) d 2 ( x ˜ , x ˆ ) α d 2 ( x ˆ , x ˜ ) = ( 1 α ) d 2 ( x ˜ , x ˆ ) .

Since 0 < α < 1 , we have that d ( x ˜ , x ˆ ) = 0 , and so x ˜ = x ˆ . Hence the sequence { x n } converges strongly to x ˜ , which is the unique solution to the variational inequality (3.3).

This completes the proof. □

Theorem 3.4 Let C be a closed convex subset of a complete CAT(0) space X, and let { T ( t ) } and { S ( s ) } be two one-parameter continuous semigroups of nonexpansive mappings on C satisfying F : = F ( T ) F ( S ) and both uniformly asymptotically regular on C. Let f be a contraction on C with coefficient α ( 0 , 1 ) . Suppose that { x n } is given by
x n + 1 = α n f ( x n ) β n T ( t n ) x n γ n S ( s n ) x n
(3.10)
for all n 0 , where { α n } , { β n } , { γ n } ( 0 , 1 ) and t n , s n [ 0 , ) satisfy the following conditions:
  1. (i)

    α n + β n + γ n = 1 ;

     
  2. (ii)

    lim n α n = 0 , n = 0 α n = and γ n = o ( α n ) ;

     
  3. (iii)

    for all n 0 , α n < 1 α ;

     
  4. (iv)

    lim n t n = and lim n s n = ;

     
  5. (iv)

    for any bounded subset B of C, lim n sup x B d ( T ( t n ) x , S ( s n ) x ) = 0 .

     

Then { x n } converges strongly to x ˜ such that x ˜ = P F f ( x ˜ ) , which is equivalent to the variational inequality (3.3).

Proof We first show that the sequence { x n } is bounded. For any p F , we have that
d ( x n + 1 , p ) = d ( α n f ( x n ) β n T ( t n ) x n γ n S ( s n ) x n , p ) α n d ( f ( x n ) , p ) + β n d ( T ( t n ) x n , p ) + γ n d ( S ( s n ) x n , p ) α n ( d ( f ( x n ) , f ( p ) ) + d ( f ( p ) , p ) ) + β n d ( x n , p ) + γ n d ( x n , p ) ( α n α + 1 α n ) d ( x n , p ) + α n d ( f ( p ) , p ) = ( 1 α n ( 1 α ) ) d ( x n , p ) + α n ( 1 α ) 1 1 α d ( f ( p ) , p ) max { d ( x n , p ) , 1 1 α d ( f ( p ) , p ) } .
By induction, we have
d ( x n , p ) max { d ( x 0 , p ) , 1 1 α d ( f ( p ) , p ) }

for all n 0 . Hence { x n } is bounded, so are { T ( t n ) x n } , { S ( s n ) x n } and { f ( x n ) } .

In view of condition (ii), we have
d ( x n + 1 , T ( t n ) x n ) α n d ( f ( x n ) , T ( t n ) x n ) + γ n d ( S ( s n ) x n , T ( t n ) x n ) 0 ( n ) .
Since { T ( t ) } is u.a.r and lim n t n = , then for all h 0 , we obtain that
lim n d ( T ( h ) ( T ( t n ) x n ) , T ( t n ) x n ) lim n sup x B d ( T ( h ) ( T ( t n ) x ) , T ( t n ) x ) = 0 ,
where B is any bounded subset of C containing { x n } . Hence
d ( x n + 1 , T ( h ) x n + 1 ) d ( x n + 1 , T ( t n ) x n ) + d ( T ( t n ) x n , T ( h ) ( T ( t n ) x n ) ) + d ( T ( h ) ( T ( t n ) x n ) , T ( h ) x n + 1 ) 2 d ( x n + 1 , T ( t n ) x n ) + d ( T ( t n ) x n , T ( h ) ( T ( t n ) x n ) ) 0 ( n ) .
(3.11)
Similarly, for all k 0 , we have
lim n d ( x n + 1 , S ( k ) x n + 1 ) = 0 .
(3.12)
Let { z m } be a sequence in C such that
z m = α m f ( z m ) β m T ( t m ) z m γ m S ( s m ) z m .

It follows from Theorem 3.3 that { z m } converges strongly to a fixed point x ˜ F , which solves the variational inequality (3.3).

Now we claim that
lim sup n f ( x ˜ ) x ˜ , x n + 1 x ˜ 0 .
Indeed, it follows from Lemma 3.2 that
d 2 ( z m , x n + 1 ) = z m x n + 1 , z m x n + 1 α m f ( z m ) x n + 1 , z m x n + 1 + β m T ( t m ) z m x n + 1 , z m x n + 1 + γ m S ( s m ) z m x n + 1 , z m x n + 1 + α m N m = α m f ( z m ) f ( x ˜ ) , z m x n + 1 + α m f ( x ˜ ) x ˜ , z m x n + 1 + α m x ˜ z m , z m x n + 1 + α m z m x n + 1 , z m x n + 1 + β m T ( t m ) z m T ( t m ) x n + 1 , z m x n + 1 + β m T ( t m ) x n + 1 x n + 1 , z m x n + 1 + γ m S ( s m ) z m S ( s m ) x n + 1 , z m x n + 1 + γ m S ( s m ) x n + 1 x n + 1 , z m x n + 1 + α m N m α m α d ( z m , x ˜ ) d ( z m , x n + 1 ) + α m f ( x ˜ ) x ˜ , z m x n + 1 + α m d ( x ˜ , z m ) d ( z m , x n + 1 ) + α m d 2 ( z m , x n + 1 ) + β m d 2 ( z m , x n + 1 ) + β m d ( T ( t m ) x n + 1 , x n + 1 ) d ( z m , x n + 1 ) + γ m d 2 ( z m , x n + 1 ) + γ m d ( S ( s m ) x n + 1 , x n + 1 ) d ( z m , x n + 1 ) + α m N m α m α d ( z m , x ˜ ) M + α m f ( x ˜ ) x ˜ , z m x n + 1 + α m d ( x ˜ , z m ) M + d 2 ( z m , x n + 1 ) + β m d ( T ( t m ) x n + 1 , x n + 1 ) M + γ m d ( S ( s m ) x n + 1 , x n + 1 ) M + α m N m ,
where
N m : = γ m α m β m d 2 ( T ( t m ) z m , S ( s m ) z m ) + γ m d 2 ( f ( z m ) , S ( s m ) z m )
and
M sup m , n 1 { d ( z m , x n ) } .
This implies that
f ( x ˜ ) x ˜ , x n + 1 z m ( 1 + α ) M d ( z m , x ˜ ) + d ( T ( t m ) x n + 1 , x n + 1 ) α m M + γ m α m M d ( S ( s m ) x n + 1 , x n + 1 ) + N m .
(3.13)
Taking the upper limit as n first, and then m , from (3.11), (3.12) and lim m N m = 0 , we get
lim sup m lim sup n f ( x ˜ ) x ˜ , x n + 1 z m 0 .
(3.14)
Since
f ( x ˜ ) x ˜ , x n + 1 x ˜ = f ( x ˜ ) x ˜ , x n + 1 z m + f ( x ˜ ) x ˜ , z m x ˜ f ( x ˜ ) x ˜ , x n + 1 z m + d ( f ( x ˜ ) , x ˜ ) d ( z m , x ˜ ) .
Thus, by taking the upper limit as n first, and then m , it follows from z m x ˜ and (3.14) that
lim sup n f ( x ˜ ) x ˜ , x n + 1 x ˜ 0 .
Finally, we prove that x n x ˜ as n . In fact, for any n 0 , letting
y n = α n x ˜ β n T ( t n ) x n γ n S ( s n ) x n ,
from Lemma 3.1 and Lemma 3.2, we have that
d 2 ( x n + 1 , x ˜ ) d 2 ( y n , x ˜ ) + 2 x n + 1 y n , x n + 1 x ˜ ( β n d ( T ( t n ) x n , x ˜ ) + γ n d ( S ( s n ) x n , x ˜ ) ) 2 + 2 [ α n f ( x n ) y n , x n + 1 x ˜ + β n T ( t n ) x n y n , x n + 1 x ˜ + γ n S ( s n ) x n y n , x n + 1 x ˜ ] ( 1 α n ) 2 d 2 ( x n , x ˜ ) + 2 [ α n 2 f ( x n ) x ˜ , x n + 1 x ˜ + α n β n f ( x n ) T ( t n ) x n , x n + 1 x ˜ + α n γ n f ( x n ) S ( s n ) x n , x n + 1 x ˜ + β n α n T ( t n ) x n x ˜ , x n + 1 x ˜ + β n 2 T ( t n ) x n T ( t n ) x n , x n + 1 x ˜ + β n γ n T ( t n ) x n S ( s n ) x n , x n + 1 x ˜ + γ n α n S ( s n ) x n x ˜ , x n + 1 x ˜ + γ n β n S ( s n ) x n T ( t n ) x n , x n + 1 x ˜ + γ n 2 S ( s n ) x n S ( s n ) x n , x n + 1 x ˜ + α n N n ] ( 1 α n ) 2 d 2 ( x n , x ˜ ) + 2 [ α n 2 f ( x n ) x ˜ , x n + 1 x ˜ + α n β n f ( x n ) x ˜ , x n + 1 x ˜ + α n γ n f ( x n ) x ˜ , x n + 1 x ˜ + β n 2 d ( T ( t n ) x n , T ( t n ) x n ) d ( x n + 1 , x ˜ ) + γ n 2 d ( S ( s n ) x n , S ( s n ) x n ) d ( x n + 1 , x ˜ ) + α n N n ] = ( 1 α n ) 2 d 2 ( x n , x ˜ ) + 2 α n ( f ( x n ) x ˜ , x n + 1 x ˜ + N n ) = ( 1 α n ) 2 d 2 ( x n , x ˜ ) + 2 α n f ( x n ) f ( x ˜ ) , x n + 1 x ˜ + 2 α n ( f ( x ˜ ) x ˜ , x n + 1 x ˜ + N n ) ( 1 α n ) 2 d 2 ( x n , x ˜ ) + 2 α n α d ( x n , x ˜ ) d ( x n + 1 , x ˜ ) + 2 α n ( f ( x ˜ ) x ˜ , x n + 1 x ˜ + N n ) ( 1 α n ) 2 d 2 ( x n , x ˜ ) + α n α ( d 2 ( x n , x ˜ ) + d 2 ( x n + 1 , x ˜ ) ) + 2 α n ( f ( x ˜ ) x ˜ , x n + 1 x ˜ + N n ) ,
where N n : = γ n α n β n d 2 ( T ( t n ) x n , S ( s n ) x n ) + γ n d 2 ( x ˜ , S ( s n ) x n ) . This implies that
d 2 ( x n + 1 , x ˜ ) 1 ( 2 α ) α n + α n 2 1 α α n d 2 ( x n , x ˜ ) + 2 α n 1 α α n ( f ( x ˜ ) x ˜ , x n + 1 x ˜ + N n ) = ( 1 α n ( 2 2 α α n ) 1 α α n ) d 2 ( x n , x ˜ ) + 2 α n 1 α α n ( f ( x ˜ ) x ˜ , x n + 1 x ˜ + N n ) .
Then it follows that
d 2 ( x n + 1 , x ˜ ) ( 1 α n ) d 2 ( x n , x ˜ ) + α n β n ,
where
α n = α n ( 2 2 α α n ) 1 α α n , β n = 2 2 2 α α n ( f ( x ˜ ) x ˜ , x n + 1 x ˜ + N n ) .

Applying Lemma 2.7 and lim n N n = 0 , we can conclude that x n x ˜ as n . This completes the proof. □

Declarations

Acknowledgements

The authors would like to express their thanks to the editors and referees for their helpful comments and advice. This work was supported by the Scientific Research Fund of Sichuan Provincial Education Department (13ZA0199) and the Scientific Research Fund of Sichuan Provincial Department of Science and Technology (2012JYZ011) and the National Natural Science Foundation of China (Grant No. 11361070).

Authors’ Affiliations

(1)
Department of Mathematics, Yibin University
(2)
College of Statistics and Mathematics, Yunnan University of Finance and Economics

References

  1. Browder FE: Fixed point theorems for noncompact mappings in Hilbert spaces. Proc. Natl. Acad. Sci. USA 1965, 53: 1272–1276. 10.1073/pnas.53.6.1272MathSciNetView ArticleGoogle Scholar
  2. Reich S: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl. 1980, 75: 287–292. 10.1016/0022-247X(80)90323-6MathSciNetView ArticleGoogle Scholar
  3. Halpern B: Fixed points of nonexpansive maps. Bull. Am. Math. Soc. 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0View ArticleGoogle Scholar
  4. Shioji N, Takahashi W: Strong convergence theorems for asymptotically nonexpansive semi-groups in Hilbert spaces. Nonlinear Anal. 1998, 34: 87–99. 10.1016/S0362-546X(97)00682-2MathSciNetView ArticleGoogle Scholar
  5. Suzuki T: On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces. Proc. Am. Math. Soc. 2003, 131: 2133–2136. 10.1090/S0002-9939-02-06844-2View ArticleGoogle Scholar
  6. Song Y, Xu S: Strong convergence theorems for nonexpansive semigroup in Banach spaces. J. Math. Anal. Appl. 2008, 338: 152–161. 10.1016/j.jmaa.2007.05.021MathSciNetView ArticleGoogle Scholar
  7. Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl. Math. Lett. 2011,24(2):224–228. 10.1016/j.aml.2010.09.008MathSciNetView ArticleGoogle Scholar
  8. Dhompongsa S, Fupinwong W, Kaewkhao A: Common fixed points of a nonexpansive semigroup and a convergence theorem for Mann iterations in geodesic metric spaces. Nonlinear Anal. 2009, 70: 4268–4273. 10.1016/j.na.2008.09.012MathSciNetView ArticleGoogle Scholar
  9. Shi LY, Chen RD: Strong convergence of viscosity approximation methods for nonexpansive mappings in CAT(0) spaces. J. Appl. Math. 2012., 2012: Article ID 421050 10.1155/2012/421050Google Scholar
  10. Wangkeeree R, Preechasilp P: Viscosity approximation methods for nonexpansive semigroups in CAT(0) spaces. Fixed Point Theory Appl. 2013., 3013: Article ID 160 10.1186/1687-1812-2013-160Google Scholar
  11. Dhompongsa S, Panyanak B: On Δ-convergence theorems in CAT(0) spaces. Comput. Math. Appl. 2008,56(10):2572–2579. 10.1016/j.camwa.2008.05.036MathSciNetView ArticleGoogle Scholar
  12. Bridson M, Haefliger A: Metric Spaces of Nonpositive Curvature. Springer, Berlin; 1999.View ArticleGoogle Scholar
  13. Lim TC: Remarks on some fixed point theorems. Proc. Am. Math. Soc. 1976, 60: 179–182. 10.1090/S0002-9939-1976-0423139-XView ArticleGoogle Scholar
  14. Kirk WA, Panyanak B: A concept of convergence in geodesic spaces. Nonlinear Anal. 2008, 68: 3689–3696. 10.1016/j.na.2007.04.011MathSciNetView ArticleGoogle Scholar
  15. Ahmadi Kakavandi B: Weak topologies in complete CAT(0) metric spaces. Proc. Am. Math. Soc. 2013, 141: 1029–1039. S 0002–9939(2012)11743–5MathSciNetView ArticleGoogle Scholar
  16. Dhompongsa S, Kirk WA, Sims B: Fixed points of uniformly Lipschitzian mappings. Nonlinear Anal. 2006, 65: 762–772. 10.1016/j.na.2005.09.044MathSciNetView ArticleGoogle Scholar
  17. Kirk WA, Panyanak B: A concept of convergence in geodesic spaces. Nonlinear Anal., Theory Methods Appl. 2008,68(12):3689–3696. 10.1016/j.na.2007.04.011MathSciNetView ArticleGoogle Scholar
  18. Berg ID, Nikolaev IG: Quasilinearization and curvature of Alexandrov spaces. Geom. Dedic. 2008, 133: 195–218. 10.1007/s10711-008-9243-3MathSciNetView ArticleGoogle Scholar
  19. Dehghan H, Rooin J: A characterization of metric projection in CAT(0) spaces. In International Conference on Functional Equation, Geometric Functions and Applications. Payame Noor University, Tabriz; 2012:41–43. (ICFGA 2012) 10–12th, May 2012Google Scholar
  20. Xu HK: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 2003, 116: 659–678. 10.1023/A:1023073621589MathSciNetView ArticleGoogle Scholar
  21. Liu XD, Chang SS: Viscosity approximation methods for hierarchical optimization problems in CAT(0) spaces. J. Inequal. Appl. 2013., 2013: Article ID 47Google Scholar

Copyright

© Tang and Chang; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.