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Explicit averaging cyclic algorithm for common fixed points of a finite family of asymptotically strictly pseudocontractive mappings in q-uniformly smooth Banach spaces

Fixed Point Theory and Applications20122012:167

https://doi.org/10.1186/1687-1812-2012-167

  • Received: 9 June 2012
  • Accepted: 13 September 2012
  • Published:

Abstract

Let E be a real q-uniformly smooth Banach space which is also uniformly convex and K be a nonempty, closed and convex subset of E. We obtain a weak convergence theorem of the explicit averaging cyclic algorithm for a finite family of asymptotically strictly pseudocontractive mappings of K under suitable control conditions, and elicit a necessary and sufficient condition that guarantees strong convergence of an explicit averaging cyclic process to a common fixed point of a finite family of asymptotically strictly pseudocontractive mappings in q-uniformly smooth Banach spaces. The results of this paper are interesting extensions of those known results.

MSC:47H09, 47H10.

Keywords

  • asymptotically strictly pseudocontractive mappings
  • weak and strong convergence
  • explicit averaging cyclic algorithm
  • fixed points
  • q-uniformly smooth Banach spaces

1 Introduction

Let E and E be a real Banach space and the dual space of E, respectively. Let J q ( q > 1 ) denote the generalized duality mapping from E into 2 E given by J q ( x ) = { f E : x , f = x q  and  f = x q 1 } for all x E , where , denotes the generalized duality pairing between E and E . In particular, J 2 is called the normalized duality mapping and it is usually denoted by J. If E is smooth or E is strictly convex, then J q is single-valued. In the sequel, we will denote the single-valued generalized duality mapping by j q .

Let K be a nonempty subset of E. A mapping T : K K is called asymptotically κ-strictly pseudocontractive with sequence { κ n } n = 1 [ 1 , ) such that lim n κ n = 1 (see, e.g., [13]) if for all x , y K , there exist a constant κ [ 0 , 1 ) and j q ( x y ) J q ( x y ) such that
T n x T n y , j q ( x y ) κ n x y q κ x y ( T n x T n y ) q , n 1 .
(1)
If I denotes the identity operator, then (1) can be written in the form
(2)
The class of asymptotically κ-strictly pseudocontractive mappings was first introduced in Hilbert spaces by Qihou [3]. In Hilbert spaces, j q is the identity, and it is shown by Osilike et al. [2] that (1) (and hence (2)) is equivalent to the inequality
T n x T n y 2 λ n x y 2 + λ x y ( T n x T n y ) 2 ,

where lim n λ n = lim n [ 1 + 2 ( κ n 1 ) ] = 1 , λ = ( 1 2 κ ) [ 0 , 1 ) .

A mapping T with domain D ( T ) and range R ( T ) in E is called strictly pseudocontractive of Browder-Petryshyn type [4] if for all x , y D ( T ) , there exist κ [ 0 , 1 ) and j q ( x y ) J q ( x y ) such that
T x T y , j q ( x y ) x y q κ x y ( T x T y ) q .
(3)
If I denotes the identity operator, then (3) can be written in the form
( I T ) x ( I T ) y , j q ( x y ) κ ( I T ) x ( I T ) y q .
(4)
In Hilbert spaces, (3) (and hence (4)) is equivalent to the inequality
T x T y 2 x y 2 + k x y ( T x T y ) 2 , k = ( 1 2 κ ) < 1 .

It is shown in [5] that the class of asymptotically κ-strictly pseudocontractive mappings and the class of κ-strictly pseudocontractive mappings are independent.

A mapping T is said to be uniformly L-Lipschitzian if there exists a constant L > 0 such that, for all x , y K ,
T n x T n y L x y , n 1 .

Let { T j } j = 0 N 1 be N asymptotically strictly pseudocontractive self-mappings of K, and denote the common fixed points set of { T j } j = 0 N 1 by F : = j = 0 N 1 F ( T j ) , where F ( T j ) : = { x K : T j x = x } . We consider the following explicit averaging cyclic algorithm.

For a given x 0 K , and a real sequence { α n } n = 0 ( 0 , 1 ) , the sequence { x n } n = 0 is generated as follows:
x 1 = α 0 x 0 + ( 1 α 0 ) T 0 x 0 , x 2 = α 1 x 1 + ( 1 α 1 ) T 1 x 1 , x N = α N 1 x N 1 + ( 1 α N 1 ) T N 1 x N 1 , x N + 1 = α N x N + ( 1 α N ) T 0 2 x N , x N + 2 = α N + 1 x N + 1 + ( 1 α N + 1 ) T 1 2 x N + 1 , x 2 N = α 2 N 1 x 2 N 1 + ( 1 α 2 N 1 ) T N 1 2 x 2 N 1 , x 2 N + 1 = α 2 N x 2 N + ( 1 α 2 N ) T 0 3 x 2 N , x 2 N + 2 = α 2 N + 1 x 2 N + 1 + ( 1 α 2 N + 1 ) T 1 3 x 2 N + 1 ,
The algorithm can be expressed in a compact form as
x n + 1 = α n x n + ( 1 α n ) T i ( n ) k ( n ) x n , n 0 ,
(5)

where n = ( k 1 ) N + i with i = i ( n ) I = { 0 , 1 , 2 , , N 1 } , k = k ( n ) 1 a positive integer and lim n k ( n ) = . The cyclic algorithm was first studied by Acedo and Xu [6] for the iterative approximation of common fixed points of a finite family of strictly pseudocontractive mappings in Hilbert spaces, and it is better than implicit iteration methods.

In [7] Xiaolong Qin et al. proved the following theorem in a Hilbert space.

Theorem QCKS Let K be a closed and convex subset of a Hilbert space H and N 1 be an integer. Let, for each 1 i N , T i : K K be an asymptotically κ i -strictly pseudo-contractive mapping for some 0 κ i < 1 and a sequence { k n , i } such that n = 0 ( k n , i 1 ) < . Let κ = max { κ i : 1 i N } and κ n = max { κ n , i : 1 i N } . Assume that F . For any x 0 K , let { x n } be the sequence generated by the cyclic algorithm (5). Assume that the control sequence { α n } is chosen such that κ + ϵ α n 1 ϵ for all n 0 and a small enough constant ϵ ( 0 , 1 ) . Then { x n } converges weakly to a common fixed point of the family { T i } i = 1 N .

Osilike and Shehu [8] extended the result of Theorem QCKS from a Hilbert space to 2-uniformly smooth Banach spaces which are also uniformly convex. They proved the following theorem.

Theorem OS Let E be a real 2-uniformly smooth Banach space which is also uniformly convex, and K be a nonempty, closed and convex subset of E. Let { T j } j = 0 N 1 be N asymptotically λ j -strictly pseudocontractive self-mappings of K for some 0 λ j < 1 with a sequence { κ n ( j ) } n = 0 [ 1 , ) such that n = 0 ( κ n ( j ) 1 ) < , j J = { 0 , 1 , 2 , , N 1 } , and F . Let { α n } satisfy the conditions
( i ) 0 α n < 1 , n 0 , ( i i ) 0 < a 1 α n b < 2 λ C 2 ,

where λ = min j J { λ j } and C 2 is the constant appearing in the inequality (7) with q = 2 . Let { x n } be the sequence generated by the cyclic algorithm (5). Then { x n } converges weakly to a common fixed point of the family { T j } j = 0 N 1 .

We would like to point out that the condition ( ii ) in Theorem OS excludes the natural choice 1 1 n for α n . This is overcome by this paper. Moreover, we improve and extend the result of Theorem OS from 2-uniformly smooth Banach spaces to q-uniformly smooth Banach spaces which are also uniformly convex. We prove that if { α n } satisfies the conditions
( i ) μ α n < 1 , n 0 , ( i i ) n = 0 ( 1 α n ) [ q λ C q ( 1 α n ) q 1 ] = ,
(6)

where μ = max { 0 , 1 ( q λ C q ) 1 q 1 } , λ = min j J { λ j } , then the iterative sequence (5) converges weakly to a common fixed point of the family { T j } j = 0 N 1 .

Furthermore, we elicit a necessary and sufficient condition that guarantees strong convergence of the iterative sequence (5) to a common fixed point of the family { T j } j = 0 N 1 in q-uniformly smooth Banach spaces.

We will use the notation:
  1. 1.

    for weak convergence.

     
  2. 2.

    ω W ( x n ) = { x : x n j x } denotes the weak ω-limit set of { x n } .

     

2 Preliminaries

Let E be a real Banach space. The modulus of smoothness of E is the function ρ E : [ 0 , ) [ 0 , ) defined by
ρ E ( τ ) = sup { 1 2 ( x + y + x y ) 1 : x 1 , y τ } .

E is uniformly smooth if and only if lim τ 0 [ ρ E ( τ ) / τ ] = 0 .

Let q > 1 . E is said to be q-uniformly smooth (or to have a modulus of smoothness of power type q > 1 ) if there exists a constant c > 0 such that ρ E ( τ ) c τ q . Hilbert spaces, L p (or l p ) spaces ( 1 < p < ) and the Sobolev spaces W m p ( 1 < p < ) are q-uniformly smooth. Hilbert spaces are 2-uniformly smooth while
L p ( or  l p ) or W m p is { p -uniformly smooth if  1 < p 2 , 2 -uniformly smooth if  p 2 .

Theorem HKX ([[9], p.1130])

Let q > 1 and let E be a real q-uniformly smooth Banach space. Then there exists a constant C q > 0 such that, for all x , y E ,
x + y q x q + q y , j q ( x ) + C q y q .
(7)
E is said to have a Fréchet differentiable norm if, for all x U = { x E : x = 1 } ,
lim t 0 x + t y x t
exists and is attained uniformly in y U . In this case, there exists an increasing function b : [ 0 , ) [ 0 , ) with lim t 0 + [ b ( t ) / t ] = 0 such that, for all x , h E ,
1 2 x 2 + h , j ( x ) 1 2 x + h 2 1 2 x 2 + h , j ( x ) + b ( h ) .
(8)

It is well known (see, for example, [[10], p.107]) that a q-uniformly smooth Banach space has a Fréchet differentiable norm.

Lemma 2.1 ([[5], p.1338])

Let E be a real q-uniformly smooth Banach space which is also uniformly convex. Let K be a nonempty, closed and convex subset of E and T : K K be an asymptotically κ-strictly pseudocontractive mapping with a nonempty fixed point set. Then ( I T ) is demiclosed at zero, that is, if whenever { x n } D ( T ) such that { x n } converges weakly to x D ( T ) and { ( I T ) x n } converges strongly to 0, then T x = x .

Lemma 2.2 ([[2], p.80])

Let { a n } n = 0 , { b n } n = 0 , { δ n } n = 0 be sequences of nonnegative real numbers satisfying the following inequality:
a n + 1 ( 1 + δ n ) a n + b n , n 0 .

If n = 0 δ n < and n = 0 b n < , then lim n a n exists. If, in addition, { a n } n = 0 has a subsequence which converges strongly to zero, then lim n a n = 0 .

Lemma 2.3 ([[2], p.78])

Let E be a real Banach space, K be a nonempty subset of E and T : K K be an asymptotically κ-strictly pseudocontractive mapping. Then T is uniformly L-Lipschitzian.

Lemma 2.4 Let E be a real q-uniformly smooth Banach space which is also uniformly convex, and let K be a nonempty, closed and convex subset of E. Let, for each 0 j N 1 , T j : K K be an asymptotically λ j -strictly pseudocontractive mapping with F . Let { x n } n = 0 be the sequence satisfying the following conditions:
  1. (a)

    lim n x n p exists for every p F ;

     
  2. (b)

    lim n x n T j x n = 0 , for each 0 j N 1 ;

     
  3. (c)

    lim n t x n + ( 1 t ) p 1 p 2 exists for all t [ 0 , 1 ] and for all p 1 , p 2 F .

     

Then the sequence { x n } converges weakly to a common fixed point of the family { T j } j = 0 N 1 .

Proof Since lim n x n p exists, then { x n } is bounded. By (b) and Lemma 2.1, we have ω W ( x n ) F . Assume that p 1 , p 2 ω W ( x n ) and that { x n i } and { x m j } are subsequences of { x n } such that x n i p 1 and x m j p 2 , respectively. Since E is a real q-uniformly smooth Banach space, which is also uniformly convex, then E has a Fréchet differentiable norm. Set x = p 1 p 2 , h = t ( x n p 1 ) in (8), we obtain
1 2 p 1 p 2 2 + t x n p 1 , j ( p 1 p 2 ) 1 2 t x n + ( 1 t ) p 1 p 2 2 1 2 p 1 p 2 2 + b ( t x n p 1 ) + t x n p 1 , j ( p 1 p 2 ) ,
where b is an increasing function. Since x n p 1 M , n 0 , for some M > 0 , then
1 2 p 1 p 2 2 + t x n p 1 , j ( p 1 p 2 ) 1 2 t x n + ( 1 t ) p 1 p 2 2 1 2 p 1 p 2 2 + b ( t M ) + t x n p 1 , j ( p 1 p 2 ) .
Therefore,
1 2 p 1 p 2 2 + t lim sup n x n p 1 , j ( p 1 p 2 ) 1 2 lim n t x n + ( 1 t ) p 1 p 2 2 1 2 p 1 p 2 2 + b ( t M ) + t lim inf n x n p 1 , j ( p 1 p 2 ) .

Hence, lim sup n x n p 1 , j ( p 1 p 2 ) lim inf n x n p 1 , j ( p 1 p 2 ) + b ( t M ) / t . Since lim t 0 + [ b ( t M ) / t ] = 0 , then lim n x n p 1 , j ( p 1 p 2 ) exists. Since lim n x n p 1 , j ( p 1 p 2 ) = p p 1 , j ( p 1 p 2 ) , for all p ω W ( x n ) . Set p = p 2 . We have p 2 p 1 , j ( p 1 p 2 ) = 0 , that is, p 2 = p 1 . Hence, ω W ( x n ) is a singleton, so that { x n } converges weakly to a common fixed point of the family { T j } j = 0 N 1 . □

3 Main results

Theorem 3.1 Let E be a real q-uniformly smooth Banach space which is also uniformly convex and K be a nonempty, closed and convex subset of E. Let N 1 be an integer and J = { 0 , 1 , 2 , , N 1 } . Let, for each j J , T j : K K be an asymptotically λ j -strictly pseudocontractive mapping for some 0 λ j < 1 with sequences { κ n , j } n = 0 [ 1 , ) such that n = 0 ( κ n 1 ) < , where κ n = max j J { κ n , j } , and F : = j = 0 N 1 F ( T j ) . Let λ = min j J { λ j } . Let { α n } satisfy the conditions (6) and { x n } be the sequence generated by the cyclic algorithm (5). Then { x n } converges weakly to a common fixed point of the family { T j } j = 0 N 1 .

Proof Pick a p F . We firstly show that lim n x n p exists. To see this, using (2) and (7), we obtain
x n + 1 p q = x n p ( 1 α n ) [ x n p ( T i ( n ) k ( n ) x n p ) ] q x n p q + C q ( 1 α n ) q x n p ( T i ( n ) k ( n ) x n p ) q q ( 1 α n ) x n p ( T i ( n ) k ( n ) x n p ) , j q ( x n p ) x n p q + C q ( 1 α n ) q x n p ( T i ( n ) k ( n ) x n p ) q q ( 1 α n ) { λ i ( n ) x n p ( T i ( n ) k ( n ) x n p ) q ( κ k ( n ) , i ( n ) 1 ) x n p q } = [ 1 + q ( 1 α n ) ( κ k ( n ) , i ( n ) 1 ) ] x n p q ( 1 α n ) [ q λ i ( n ) C q ( 1 α n ) q 1 ] x n T i ( n ) k ( n ) x n q [ 1 + q ( 1 α n ) ( κ k ( n ) 1 ) ] x n p q ( 1 α n ) [ q λ C q ( 1 α n ) q 1 ] x n T i ( n ) k ( n ) x n q ,
(9)
where κ k ( n ) = max i J { κ k ( n ) , i ( n ) } . Since μ α n < 1 for all n, where μ = max { 0 , 1 ( q λ C q ) 1 q 1 } , we get ( 1 α n ) [ q λ C q ( 1 α n ) q 1 ] 0 . Therefore, (9) implies
x n + 1 p q [ 1 + q ( 1 α n ) ( κ k ( n ) 1 ) ] x n p q .
(10)
Let δ n = 1 + q ( 1 α n ) ( κ k ( n ) 1 ) . Since n = 0 ( κ n 1 ) < , we have
n = 0 ( δ n 1 ) = q n = 0 ( 1 α n ) ( κ k ( n ) 1 ) q N n = 1 ( κ n 1 ) < ,

then (10) implies lim n x n p exists by Lemma 2.2 (and hence the sequence { x n p } is bounded, that is, there exists a constant M > 0 such that x n p < M ).

Then we prove lim n x n T j x n = 0 , j J . In fact, it follows from (9) that
( 1 α n ) [ q λ C q ( 1 α n ) q 1 ] x n T i ( n ) k ( n ) x n q x n p q x n + 1 p q + q ( 1 α n ) ( κ k ( n ) 1 ) x n p q .
Then
n = 0 ( 1 α n ) [ q λ C q ( 1 α n ) q 1 ] x n T i ( n ) k ( n ) x n q < x 0 p q + M q n = 0 ( δ k ( n ) 1 ) < .
(11)

Since n = 0 ( 1 α n ) [ q λ C q ( 1 α n ) q 1 ] = , then (11) implies that lim inf n x n T i ( n ) k ( n ) x n = 0 . Thus lim n x n T i ( n ) k ( n ) x n = 0 .

For all n > N , we have k ( n ) 1 = k ( n N ) and i ( n ) = i ( n N ) . By Lemma 2.3, we know that T j is uniformly L j -Lipschitzian, then there exists a constant L = max j J { L j } , such that
T j n x T j n y L x y , n 0 , x , y K  and  j J .
Thus
x n T i ( n ) x n x n T i ( n ) k ( n ) x n + T i ( n ) k ( n ) x n T i ( n ) x n x n T i ( n ) k ( n ) x n + L T i ( n ) k ( n ) 1 x n x n x n T i ( n ) k ( n ) x n + L T i ( n ) k ( n ) 1 x n T i ( n N ) k ( n ) 1 x n N + L T i ( n N ) k ( n ) 1 x n N x n N 1 + L x n N 1 x n x n T i ( n ) k ( n ) x n + L 2 x n x n N + L T i ( n N ) k ( n N ) x n N x n N 1 + L x n N 1 x n .
Observe that
x n x n + 1 = ( 1 α n ) x n T i ( n ) k ( n ) x n 0 as  n .
Consequently,
x n x n + l 0 as  n ,  for all integer  l .
Observe also that
x n 1 T i ( n ) k ( n ) x n x n x n 1 + x n T i ( n ) k ( n ) x n 0 as  n .
Hence,
lim n x n T i ( n ) x n = 0 .
Consequently, for all j J , we have
x n T n + j x n x n x n + j + x n + j T n + j x n + j + L x n x n + j 0 as  n .
Thus,
lim n x n T j x n = 0 , j J .
Now we prove that for all p 1 , p 2 F , lim n t x n + ( 1 t ) p 1 p 2 exists for all t [ 0 , 1 ] . Let a n ( t ) = t x n + ( 1 t ) p 1 p 2 . It is obvious that lim n a n ( 0 ) = p 1 p 2 and lim n a n ( 1 ) = lim n x n p 2 exist. So, we only need to consider the case of t ( 0 , 1 ) . Define A n : K K by
A n x = α n x + ( 1 α n ) T i ( n ) k ( n ) x , x K .
Then for all x , y K
A n x A n y q x y q q ( 1 α n ) ( I T i ( n ) k ( n ) ) x ( I T i ( n ) k ( n ) ) y , j q ( x y ) + C q ( 1 α n ) q x y ( T i ( n ) k ( n ) x T i ( n ) k ( n ) y ) q [ 1 + q ( 1 α n ) ( κ k ( n ) 1 ) ] x y q ( 1 α n ) [ q λ C q ( 1 α n ) q 1 ] x y ( T i ( n ) k ( n ) x T i ( n ) k ( n ) y ) q .

By the choice of α n , we have ( 1 α n ) [ q λ C q ( 1 α n ) q 1 ] 0 , so it follows that A n x A n y q [ 1 + q ( 1 α n ) ( κ k ( n ) 1 ) ] x y q = δ n x y q . For the convenience of the following discussion, set η n = ( δ n ) 1 q , then A n x A n y η n x y .

Set S n , m = A n + m 1 A n + m 2 A n , m 1 . We have
S n , m x S n , m y ( j = n n + m 1 η j ) x y for all  x , y K ,
and
S n , m x n = x n + m , S n , m p = p for all  p F .
Set b n , m = S n , m ( t x n + ( 1 t ) p 1 ) t S n , m x n ( 1 t ) S n , m p 1 . If x n p 1 = 0 for some n 0 , then x n = p 1 for any n n 0 so that lim n x n p 1 = 0 , in fact, { x n } converges strongly to p 1 F . Thus we may assume x n p 1 > 0 for any n 0 . Let δ denote the modulus of convexity of E. It is well known (see, for example, [[11], p.108]) that
t x + ( 1 t ) y 1 2 min { t , ( 1 t ) } δ ( x y ) 1 2 t ( 1 t ) δ ( x y )
(12)
for all t [ 0 , 1 ] and for all x , y E such that x 1 , y 1 . Set
w n , m = S n , m p 1 S n , m ( t x n + ( 1 t ) p 1 ) t ( j = n n + m 1 η j ) x n p 1 , z n , m = S n , m ( t x n + ( 1 t ) p 1 ) S n , m x n ( 1 t ) ( j = n n + m 1 η j ) x n p 1 .
Then w n , m 1 and z n , m 1 so that it follows from (12) that
2 t ( 1 t ) δ ( w n , m z n , m ) 1 t w n , m + ( 1 t ) z n , m .
(13)
Observe that
w n , m z n , m = b n , m t ( 1 t ) ( j = n n + m 1 η j ) x n p 1
and
t w n , m + ( 1 t ) z n , m = S n , m x n S n , m p 1 ( j = n n + m 1 η j ) x n p 1 ,
it follows from (13) that
(14)
Since E is uniformly convex, then δ ( s ) / s is nondecreasing, and since ( j = n n + m 1 η j ) x n p 1 ( j = n n + m 1 η j ) η n 1 x n 1 p 1 ( j = n n + m 1 η j ) ( j = 0 n 1 η j ) x 0 p 1 = ( j = 0 n + m 1 η j ) x 0 p 1 , hence it follows from (14) that
( j = 0 n + m 1 η j ) x 0 p 1 2 δ ( 4 ( j = 0 n + m 1 η j ) x 0 p 1 b n , m ) ( j = n n + m 1 η j ) x n p 1 x n + m p 1 ( since  t ( 1 t ) 1 4  for all  t [ 0 , 1 ] ) .
Since lim n j = 0 n + m 1 η j exits and lim n j = 0 n + m 1 η j 0 . Also since lim n j = n n + m 1 η j = 1 and lim n x n p 1 exists, then the continuity of δ and δ ( 0 ) = 0 yield lim n b n , m = 0 uniformly for all m 1 . Observe that
a n + m ( t ) t x n + m + ( 1 t ) p 1 p 2 + ( S n , m ( t x n + ( 1 t ) p 1 ) t S n , m x n ( 1 t ) S n , m p 1 ) + S n , m ( t x n + ( 1 t ) p 1 ) t S n , m x n ( 1 t ) S n , m p 1 = S n , m ( t x n + ( 1 t ) p 1 ) S n , m p 2 + b n , m ( j = n n + m 1 η j ) t x n + ( 1 t ) p 1 p 2 + b n , m = ( j = n n + m 1 η j ) a n ( t ) + b n , m .

Hence lim sup n a n ( t ) lim inf n a n ( t ) , this ensures that lim n a n ( t ) exists for all t ( 0 , 1 ) .

Now apply Lemma 2.4 to conclude that { x n } converges weakly to a common fixed point of the family { T j } j = 0 N 1 . □

Theorem 3.2 Let E be a real q-uniformly smooth Banach space, and let K be a nonempty, closed and convex subset of E. Let N 1 be an integer and J = { 0 , 1 , 2 , , N 1 } . Let, for each j J , T j : K K be an asymptotically λ j -strictly pseudocontractive mapping for some 0 λ j < 1 with sequences { κ n , j } n = 0 [ 1 , ) such that n = 0 ( κ n 1 ) < , where κ n = max j J { κ n , j } , and F : = j = 0 N 1 F ( T j ) . Let λ = min j J { λ j } . Let { α n } satisfy the conditions (6) and { x n } be the sequence generated by the cyclic algorithm (5). Then { x n } converges strongly to a common fixed point of the family { T j } j = 0 N 1 if and only if
lim inf n d ( x n , F ) = 0 ,

where d ( x n , F ) = inf p F x n p .

Proof It follows from (10) that
x n + 1 p q δ n x n p q .

Thus [ d ( x n + 1 p ) ] q δ n [ d ( x n p ) ] q , and it follows from Lemma 2.2 that lim n d ( x n , F ) exists.

Now if { x n } converges strongly to a common fixed point p of the family { T j } j = 0 N 1 , then lim n x n p = 0 . Since
0 d ( x n , F ) x n p ,

we have lim inf n d ( x n , F ) = 0 .

Conversely, suppose lim inf n d ( x n , F ) = 0 , then the existence of lim n d ( x n , F ) implies that lim n d ( x n , F ) = 0 . Thus, for arbitrary ϵ > 0 , there exists a positive integer n 0 such that d ( x n , F ) < ϵ 2 for any n n 0 .

From (10), we have
x n + 1 p q x n p q + M q ( δ n 1 ) , n 0 ,
and for some M > 0 , x n p < M . Now, an induction yields
x n p q x n 1 p q + M q ( δ n 1 1 ) x n 2 p q + M q ( δ n 2 1 ) + M q ( δ n 1 1 ) x l p q + M q j = l n 1 ( δ j 1 ) , n 1 l 0 .
Since n = 0 ( δ n 1 ) < , then there exists a positive integer n 1 such that j = n ( δ j 1 ) < ( ϵ 2 M ) q , n n 1 . Choose N = max { n 0 , n 1 } , then for all n , m N + 1 and for all p F , we have
x n x m x n p + x m p [ x N p q + M q j = N n 1 ( δ j 1 ) ] 1 q + [ x N p q + M q j = N m 1 ( δ j 1 ) ] 1 q [ x N p q + M q j = N ( δ j 1 ) ] 1 q + [ x N p q + M q j = N ( δ j 1 ) ] 1 q = 2 [ x N p q + M q j = N ( δ j 1 ) ] 1 q .
Taking infimum over all p F , we obtain
x n x m 2 { [ d ( x N , F ) ] q + M q j = N ( δ j 1 ) } 1 q < 2 [ ( ϵ 2 ) q + M q ( ϵ 2 M ) q ] 1 q < 2 ϵ .
Thus { x n } n = 0 is Cauchy. Suppose lim n x n = u . Then for all j J we have
0 u T j u u x n + x n T j x n + L x n u 0 as  n .

Thus u F ( T j ) , j J , and hence u F . □

Declarations

Authors’ Affiliations

(1)
School of Mathematics and Physics, North China Electric Power University, Baoding, Hebei, 071003, P.R. China
(2)
School of Economics, Renmin University of China, Beijing, 100872, P.R. China
(3)
Easyway Company Limited, Beijing, 100872, P.R. China

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© Zhang and Xie; licensee Springer 2012

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 cited.

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