Open Access

General composite implicit iteration process for a finite family of asymptotically pseudo-contractive mappings

  • Balwant Singh Thakur1,
  • Rajshree Dewangan1 and
  • Mihai Postolache2Email author
Fixed Point Theory and Applications20142014:90

https://doi.org/10.1186/1687-1812-2014-90

Received: 3 January 2014

Accepted: 25 March 2014

Published: 7 April 2014

Abstract

In this paper, a modified general composite implicit iteration process is used to study the convergence of a finite family of asymptotically nonexpansive mappings. Weak and strong convergence theorems have been proved, in the framework of a Banach space.

MSC:47H09, 47H10.

Keywords

implicit iteration processasymptotically pseudo-contractive mappingcommon fixed points

1 Introduction

Let K be a nonempty subset of a real Banach space E and let J : E 2 E is the normalized duality mapping defined by
J ( x ) = { f E : x , f = x f ; x = f } , x E ,

where E denotes the dual space of E and , denotes the generalized duality pairing.

It is well known that if E is strictly convex, then J is single valued.

In the sequel, we shall denote the single valued normalized duality mapping by j.

Let K be a nonempty subset of E. A mapping T : K K is said to be L-Lipschitzian if there exists a constant L > 0 such that for all x , y K , we have T x T y L x y . It is said to be nonexpansive if T x T y x y , for all x , y K . T is called asymptotically nonexpansive [1] if there exists a sequence { h n } [ 1 , ) with lim n h n = 1 such that T n x T n y h n x y , for all integers n 1 and all x , y K .

A mapping T is said to be pseudo-contractive [2, 3], if there exists j ( x y ) J ( x y ) such that T x T y , j ( x y ) x y 2 , for all x , y K . T is called strongly pseudo-contractive, if there exists a constant β ( 0 , 1 ) , j ( x y ) J ( x y ) such that T x T y , j ( x y ) β x y 2 , for all x , y K . It is said to be asymptotically pseudo-contractive [4] if there exists a sequence { h n } [ 1 , ) with lim n h n = 1 and j ( x y ) J ( x y ) such that
T n x T n y , j ( x y ) h n x y 2 , x , y K , n 1 .
(1.1)
It follows from Kato [5] that
x y x y + r [ ( h n I T n ) x ( h n I T n ) y ] , x , y K , n 1 , r > 0 .
(1.2)

We use F ( T ) to denote the set of fixed points of T; that is, F ( T ) = { x K : x = T x } .

It follows from the definition that if T is asymptotically nonexpansive, then for all j ( x y ) J ( x y ) ,
T n x T n y , j ( x y ) = x y T n x T n y h n x y 2 .

Hence every asymptotically nonexpansive mapping is asymptotically pseudo-contractive.

It can be observed from the definition that an asymptotically nonexpansive mapping is uniformly L-Lipschitzian, where L = sup n 1 { h n } .

Now consider an example of non-Lipschitzian mapping due to Rhoades [6]. Define a mapping T : [ 0 , 1 ] [ 0 , 1 ] by the formula T x = { 1 x 2 3 } 3 2 , for x [ 0 , 1 ] . Schu [4] used this example to show that the class of asymptotically nonexpansive mappings is a subclass of the class of pseudo-contractive mappings. Since T is not Lipschitzian, it cannot be asymptotically nonexpansive. Also T 2 is the identity mapping and T is monotonically decreasing, and it follows that
| x y | | T n x T n y | = | x y | 2 for all  n = 2 m , m N
and
( x y ) ( T n x T n y ) = ( x y ) ( T x T y ) 0 | x y | 2 for all  n = 2 m 1 , m N .

Hence T is asymptotically pseudo-contractive mapping with constant sequence { 1 } .

The iterative approximation problems for a nonexpansive mapping, an asymptotically nonexpansive mapping, and an asymptotically pseudo-contractive mapping were studied extensively by Browder [7], Kirk [8], Goebel and Kirk [1], Schu [4], Xu [9, 10], Liu [11] in the setting of Hilbert space or uniformly convex Banach space.

In 2001, Xu and Ori [12] introduced the following implicit iteration process for a finite family of nonexpansive self-mappings in Hilbert space:
{ x 0 K arbitrary , x n = α n x n 1 + ( 1 α n ) T n x n , n 1 ,
(1.3)

where { α n } be a sequence in ( 0 , 1 ) and T n = T n mod N . They proved in [12] that the sequence { x n } converges weakly to a common fixed point of T n , n = 1 , 2 , , N .

Later on Osilike and Akuchu [13], and Chen et al. [14] extended the iteration process (1.3) to a finite family of asymptotically pseudo-contractive mapping and a finite family of continuous pseudo-contractive self-mapping, respectively. Zhou and Chang [15] studied the convergence of a modified implicit iteration process to the common fixed point of a finite family of asymptotically nonexpansive mappings. Then Su and Li [16], and Su and Qin [17] introduced the composite implicit iteration process and the general iteration algorithm, respectively, which properly include the implicit iteration process. Recently, Beg and Thakur [18] introduced a modified general composite implicit iteration process for a finite family of random asymptotically nonexpansive mapping and proved strong convergence theorems.

The purpose of this paper is to consider a finite family { T i } i = 1 N of asymptotically pseudo-contractive mappings and to establish convergence results in Banach spaces based on the modified general composite implicit iteration:

For x 0 K , construct a sequence { x n } by
x n = α n x n 1 + ( 1 α n ) T i ( n ) k ( n ) y n , y n = r n x n + s n x n 1 + t n T i ( n ) k ( n ) x n + w n T i ( n ) k ( n ) x n 1
(1.4)

for each n 1 , which can be written as n = ( k ( n ) 1 ) N + i ( n ) , where i ( n ) = 1 , 2 , , N and k ( n ) 1 is a positive integer, with k ( n ) as n . The sequences { α n } , { r n } , { s n } , { t n } and { w n } are in ( 0 , 1 ) such that r n + s n + t n + w n = 1 for all n 1 .

2 Preliminaries

In what follows we shall use the following results.

Lemma 2.1 [19]

Let E be a Banach space, K be a nonempty closed convex subset of E, and T : K K be a continuous and strong pseudo-contraction. Then T has a unique fixed point.

Lemma 2.2 [20]

Let { a n } , { b n } , and { c n } be three nonnegative sequences satisfying the following condition:
a n + 1 ( 1 + b n ) a n + c n for all  n n 0 ,

where n 0 is some nonnegative integer, n = 0 b n < and n = 0 c n < .

Then
  1. (i)

    lim n a n exists;

     
  2. (ii)

    if, in addition, there exists a subsequence { a n i } { a n } such that a n i 0 , then a n 0 as n .

     

Lemma 2.3 [21]

Let E be a uniformly convex Banach space and let a, b be two constants with 0 < a < b < 1 . Suppose that { t n } [ a , b ] is a real sequence and { x n } , { y n } are two sequences in E. Then the conditions
lim sup n x n d , lim sup n y n d and lim n t n x n + ( 1 t n ) y n = d

imply that lim n x n y n = 0 , where d 0 is some constant.

Lemma 2.4 [22]

Let E be a reflexive smooth Banach space with a weakly sequential continuous duality mapping J. Let K be a nonempty bounded and closed convex subset of E and T : K K be a uniformly L-Lipschitzian and asymptotical pseudo-contraction. Then I T is demiclosed at zero, where I is the identical mapping.

We shall denote weak convergence by and strong convergence by →.

A Banach space E is said to satisfy Opial’s condition if for any sequence { x n } E , x n x as n implies
lim sup n x n x < lim sup n x n y , y E  with  x y .

We know that a Banach space with a sequentially continuous duality mapping satisfies Opial’s condition (for details, see [23]).

3 The main results

Throughout this section, E is a uniformly convex Banach space, K a nonempty closed convex subset of E. denotes the set of natural numbers and I = { 1 , 2 , , N } , the set of the first N natural numbers. T i ( i I ) are N uniformly Lipschitzian asymptotically pseudo-contractive self-mappings on K. Let F = i I F ( T i ) .

Since T i ( i I ) are uniformly Lipschitzian, there exist constants L i > 0 such that T i n x T i n y L i x y , for all x , y K , n N and i I . Also, since T i ( i I ) are asymptotically pseudo-contractive; therefore there exist sequences { h n ( i ) } such that T i n x T i n y , j ( x y ) h n ( i ) x y 2 for all x , y K and i I .

Take L = max i I ( L i ) and h n = max i I ( h n ( i ) ) .

Before presenting the main results, we first show that the proposed iteration (1.4) is well defined.

Let T be uniformly Lipschitzian asymptotically pseudo-contractive mapping. For every fixed u K and α ( L + L 2 L + L 2 + 1 , 1 ) , define a mapping S n : K K by the formula
S n x = α u + ( 1 α ) T n a , a = r x + s u + t T n x + w T n u  for all  x K ,
(3.1)

where α , r , s , t , w ( 0 , 1 ) , with ( 1 α ) ( L + L 2 ) < 1 .

Then, for all x , y K , j ( x y ) J ( x y ) , we have
S n y = α u + ( 1 α ) T n b , b = r y + s u + t T n y + w T n u  for all  x K .
(3.2)
Now
T n a T n b , j ( x y ) = T n a T n b x y L a b x y = L r ( x y ) + t ( T n x T n y ) x y L ( r x y + t L x y ) x y = ( L r + t L 2 ) x y 2 ( L + L 2 ) x y 2 ,
so
S n x S n y , j ( x y ) = ( 1 α ) T n a T n b , j ( x y ) ( 1 α ) ( L + L 2 ) x y 2 .
Since ( 1 α ) ( L + L 2 ) ( 0 , 1 ) , S n is strongly pseudo-contractive, which is also continuous, by Lemma 2.1, S n has a unique fixed point x K , i.e.
S n x = α u + ( 1 α ) T n a , a = r x + s u + t T n x + w T n u  for all  x K .
(3.3)

Thus the implicit iteration (1.4) is defined in K for a finite family { T i } of uniformly Lipschitzian asymptotically pseudo-contractive self-mappings on K, provided α n ( α , 1 ) , where α = L + L 2 L + L 2 + 1 , for all n N , L = max i I ( L i ) .

Lemma 3.1 Let E, K, and T i ( i I ) be as defined above and let { x n } be the sequence defined by (1.4), where { α n } is a sequence of real numbers such that 0 < α < α n β < 1 for α = L + L 2 L + L 2 + 1 and β is some constant and satisfying the conditions n = 1 ( 1 α n ) < and lim n h n 1 1 α n = 0 . Let b > 0 be a real number such that t n + w n b / L < 1 . Then
  1. (i)

    lim n x n p exists, for all p F ,

     
  2. (ii)

    lim n d ( x n , F ) exists, where d ( x n , F ) = inf p F x n p ,

     
  3. (iii)

    lim n x n T l x n = 0 , l I .

     
Proof Let p F . Using (1.4), we have
x n p 2 = x n p , j ( x n p ) α n x n 1 p , j ( x n p ) + ( 1 α n ) T i ( n ) k ( n ) y n T i ( n ) k ( n ) x n , j ( x n p ) + ( 1 α n ) h k ( n ) x n p 2 = α n x n 1 p x n p + ( 1 α n ) L y n x n x n p + ( 1 α n ) h k ( n ) x n p 2 .
(3.4)
Using (1.4), we obtain
y n x n = s n ( x n 1 x n ) + t n ( T i ( n ) k ( n ) x n x n ) + w n ( T i ( n ) k ( n ) x n 1 x n ) s n x n 1 p + s n x n p + t n L x n p + t n x n p + w n L x n 1 p + w n x n p .
(3.5)
Substituting (3.5) in (3.4), we get
x n p 2 ( α n + ( 1 α n ) L ( s n + w n L ) ) x n 1 p x n p + ( 1 α n ) [ ( s n + t n + w n + t n L ) L + h k ( n ) ] x n p 2 ( α n + ( 1 α n ) ( 1 + L ) L ) x n 1 p x n p + ( 1 α n ) [ ( 1 + L ) L + h k ( n ) ] x n p 2 ( α n + ( 1 α n ) ( 1 + L ) L ) x n 1 p x n p + [ ( 1 α n ) ( 1 + L ) L + ( 1 α n + μ k ( n ) ) ] x n p 2 ,
(3.6)

where μ k ( n ) = h k ( n ) 1 for all n 1 , by condition n = 1 ( h k ( n ) 1 ) < , we have n = 1 μ k ( n ) < .

Therefore, we have
x n p ( α n + ( 1 α n ) ( 1 + L ) L ) α n μ k ( n ) ( 1 α n ) ( 1 + L ) L x n 1 p [ 1 + μ k ( n ) + 2 ( 1 α n ) ( 1 + L ) L α n μ k ( n ) ( 1 α n ) ( 1 + L ) L ] x n 1 p [ 1 + μ k ( n ) + 2 ( 1 α n ) ( 1 + L ) L 1 ( 1 α n + μ k ( n ) + ( 1 α n ) ( 1 + L ) L ) ] x n 1 p .
(3.7)

Since lim n h k ( n ) 1 1 α n = lim n μ k ( n ) 1 α n = 0 , there exists a M such that μ k ( n ) 1 α n < M .

Now, we consider the second term on the right side of (3.7). We have
( 1 α n + μ k ( n ) + ( 1 α n ) ( 1 + L ) L ) ( 1 α n ) [ 1 + M + ( 1 + L ) L ] .
By condition n = 1 ( 1 α n ) < , we have lim n ( 1 α n ) = 0 , then there exists a natural number N 1 such that if n > N 1 , then
1 ( 1 α n + μ k ( n ) + ( 1 α n ) ( 1 + L ) L ) 1 2 .
Therefore, it follows from (3.7) that
x n p [ 1 + 2 { μ k ( n ) + 2 ( 1 α n ) ( 1 + L ) L } ] x n 1 p = ( 1 + σ n ) x n 1 p ,
(3.8)

where σ n = 2 { μ k ( n ) + 2 ( 1 α n ) ( 1 + L ) L } .

Taking the infimum over p F , we have
d ( x n , F ) ( 1 + σ n ) d ( x n 1 , F ) .
(3.9)
Since n = 1 μ k ( n ) < and n = 1 ( 1 α n ) < , we have
n = 1 σ n < .

Thus, by Lemma 2.2, lim n x n p and lim n d ( x n , F ) exist.

Without loss of generality, we assume
lim n x n p = d 1 .
(3.10)
Set v k ( n ) = h k ( n ) 1 h k ( n ) , and from (1.2), we have
x n p x n p + 1 α n 2 α n h k ( n ) [ ( h k ( n ) I T i ( n ) k ( n ) ) x n ( h k ( n ) I T i ( n ) k ( n ) ) p ] x n p + 1 α n 2 α n [ α n ( x n 1 T i ( n ) k ( n ) x n ) + ( 1 α n ) ( T i ( n ) k ( n ) y n T i ( n ) k ( n ) x n ) ] + ( 1 α n 2 α n ) ( h k ( n ) 1 h k ( n ) ) T i ( n ) k ( n ) x n p = x n p + 1 α n 2 ( x n 1 T i ( n ) k ( n ) x n ) + ( 1 α n ) 2 2 α n ( T i ( n ) k ( n ) y n T i ( n ) k ( n ) x n ) + ( 1 α n 2 α n ) v k ( n ) T i ( n ) k ( n ) x n p x n p + 1 2 ( x n 1 x n ) + ( 1 α n 2 α n ) v k ( n ) T i ( n ) k ( n ) x n p + ( 1 α n ) 2 2 α n L y n x n 1 2 ( x n p ) + 1 2 ( x n 1 p ) + ( 1 α n 2 α n ) v k ( n ) T i ( n ) k ( n ) x n p + ( 1 α n ) 2 2 α n L y n x n .
Thus
lim inf n x n p lim inf n 1 2 ( x n p ) + 1 2 ( x n 1 p ) + lim inf n ( 1 α n 2 α n ) v k ( n ) T i ( n ) k ( n ) x n p + lim inf n ( 1 α n ) 2 2 α n L y n x n .
Since v k ( n ) = h k ( n ) 1 h k ( n ) ( 0 , 1 ) , we have lim n v k ( n ) = 0 and from n = 1 ( 1 α n ) < , we have lim n ( 1 α n ) = 0 and using (3.10), we have
lim inf n 1 2 ( x n p ) + 1 2 ( x n 1 p ) d 1 .
(3.11)
On the other hand, we obtain
lim sup n 1 2 ( x n p ) + 1 2 ( x n 1 p ) lim sup n [ 1 2 x n p + 1 2 x n 1 p ] = d 1 ,
(3.12)
from (3.11) and (3.12), we have
lim n 1 2 ( x n p ) + 1 2 ( x n 1 p ) = d 1 .
It follows from Lemma 2.3 that
lim n x n x n 1 = 0 .
(3.13)
Thus, for any i I , we have
lim n x n x n + i = 0 .
(3.14)
Since 0 < α < α n β < 1 and from (1.4) and (3.13), we get
lim n x n T i ( n ) k ( n ) y n = lim n α n 1 α n x n x n 1 1 1 β lim n x n x n 1 = 0 .
(3.15)
On the other hand, from (3.13) and (3.15)
lim n x n 1 T i ( n ) k ( n ) y n lim n x n 1 x n + lim n x n T i ( n ) k ( n ) y n = 0 .
(3.16)
Now,
T i ( n ) k ( n ) x n x n x n x n 1 + T i ( n ) k ( n ) y n x n 1 + T i ( n ) k ( n ) y n T i ( n ) k ( n ) x n ( 1 + L ) x n x n 1 + T i ( n ) k ( n ) y n x n 1 + L y n x n 1 .
(3.17)
Again, by using (1.4), we obtain
y n x n 1 r n x n + s n x n 1 + t n T i ( n ) k ( n ) x n + w n T i ( n ) k ( n ) x n 1 x n 1 t n T i ( n ) k ( n ) x n x n + w n T i ( n ) k ( n ) x n 1 x n + ( r n + t n + w n ) x n x n 1 ( t n + w n ) T i ( n ) k ( n ) x n x n + ( r n + t n + w n + w n L ) x n x n 1 .
(3.18)
Substituting (3.18) into (3.17), we get
T i ( n ) k ( n ) x n x n ( 1 + L ) x n x n 1 + T i ( n ) k ( n ) y n x n 1 + L ( t n + w n ) T i ( n ) k ( n ) x n x n + L ( r n + t n + w n + w n L ) x n x n 1 .
Since t n + w n b / L < 1 , the above inequality gives
( 1 b ) T i ( n ) k ( n ) x n x n [ 1 + L ( 1 + r n + t n + w n + w n L ) ] x n x n 1 + T i ( n ) k ( n ) y n x n 1 .
Then from (3.13), (3.16), and the above inequality, we have
lim n T i ( n ) k ( n ) x n x n = 0 .
(3.19)
From (3.13), (3.18), and (3.19), we get
lim n y n x n 1 = 0 .
(3.20)
On the other hand, from (3.13) and (3.20) we have
lim n y n x n lim n y n x n 1 + lim n x n 1 x n = 0 .
(3.21)

Since for any positive integer n > N , we can write n = ( k ( n ) 1 ) N + i ( n ) , i ( n ) I .

Let A n = T i ( n ) k ( n ) y n x n 1 , then from (3.16), we have A n 0 . Also,
x n 1 T n x n x n 1 T i ( n ) k ( n ) y n + T i ( n ) k ( n ) y n T n x n = A n + T i ( n ) k ( n ) y n T i ( n ) x n A n + L T i ( n ) k ( n ) 1 y n x n A n + L { T i ( n ) k ( n ) 1 y n T i ( n N ) k ( n ) 1 x n N + T i ( n N ) k ( n ) 1 x n N T i ( n N ) k ( n ) 1 y n N + T i ( n N ) k ( n ) 1 y n N x ( n N ) 1 + x ( n N ) 1 x n } .
(3.22)
Since for each n > N , n = ( n N ) ( mod N ) and n = ( k ( n ) 1 ) N + i ( n ) , n N = ( ( k ( n ) 1 ) 1 ) N + i ( n ) = ( k ( n N ) 1 ) N + i ( n N ) , i.e.
k ( n N ) = k ( n ) 1 and i ( n N ) = i ( n ) .
Therefore from (3.22), we have
x n 1 T n x n A n + L { T i ( n ) k ( n ) 1 y n T i ( n ) k ( n ) 1 x n N + T i ( n N ) k ( n N ) x n N T i ( n N ) k ( n N ) y n N + T i ( n N ) k ( n N ) y n N x ( n N ) 1 + x ( n N ) 1 x n } A n + L { L y n x n N + L x n N y n N + A n N + x ( n N ) 1 x n } A n + L 2 ( y n x n + x n x n N + x n N y n N ) + L ( A n N + x ( n N ) 1 x n ) .
(3.23)
From (3.14), (3.21), and A n 0 , we have
lim n x n 1 T n x n = 0 .
(3.24)
It follows from (3.13) and (3.24) that
lim n x n T n x n lim n { x n x n 1 + x n 1 T n x n } = 0 .
(3.25)
Consequently, for any i I , from (3.14), (3.25), we obtain
x n T n + i x n x n x n + i + x n + i T n + i x n + i + T n + i x n + i T n + i x n ( 1 + L ) x n x n + i + x n + i T n + i x n + i 0 ,
as n . This implies that the sequence
i = 1 N { x n T n + i x n } n = 1 0 , as  n .
Since for each l = 1 , 2 , , N , { x n T l x n } is a subsequence of i = 1 N { x n T n + i x n } , therefore, we have
lim n x n T l x n = 0 , l I .
(3.26)

This completes the proof. □

3.1 Strong convergence theorems

First, we prove necessary and sufficient conditions for the strong convergence of the modified general composite implicit iteration process to a common fixed point of a finite family of asymptotically pseudo-contractive mappings.

Theorem 3.1 Let E, K, and T i ( i I ) be as defined above and { α n } be a sequence of real numbers as in Lemma  3.1. Then the sequence { x n } generated by (1.4) converges strongly to a member of if and only if lim inf n d ( x n , F ) = 0 .

Proof The necessity of the condition is obvious. Thus, we will only prove the sufficiency.

Let lim inf n d ( x n , F ) = 0 . Then from (ii) in Lemma 3.1, we have lim n d ( x n , F ) = 0 .

Next, we show that { x n } is a Cauchy sequence in K. For any given ε > 0 , since lim n d ( x n , F ) = 0 , there exists a natural number n 1 such that d ( x n , F ) < ε / 4 when n n 1 .

Since lim n x n p exists for all p F , we have x n p < M , for all n 1 and some positive number M .

Furthermore n = 1 σ n < implies that there exists a positive integer n 2 such that j = n σ j < ε / 4 M for all n n 2 . Let N = max { n 1 , n 2 } . It follows from (3.8) that
x n p x n 1 p + M σ n .
Now, for all n , m N and for all p F , we have
x n x m x n p + x m p x N p + M j = N + 1 n σ j + x N p + M j = N + 1 m σ j 2 x N p + 2 M j = N σ j .
Taking the infimum over all p F , we obtain
x n x m 2 d ( x N , F ) + 2 M j = N σ j < ε .

This implies that { x n } is a Cauchy sequence. Since E is complete, therefore { x n } is convergent.

Suppose lim n x n = q .

Since K is closed, we get q K , then { x n } converges strongly to q.

It remains to show that q F .

Notice that
| d ( q , F ) d ( x n , F ) | q x n , n N ,

since lim n x n = q and lim n d ( x n , F ) = 0 , we obtain q F .

This completes the proof. □

Corollary 3.1 Suppose that the conditions are the same as in Theorem  3.1. Then the sequence { x n } generated by (1.4) converges strongly to u F if and only if { x n } has a subsequence { x n j } which converges strongly to u F .

A mapping T : K K with F ( T ) is said to satisfy condition (A) [24] on K if there exists a nondecreasing function f : [ 0 , ) [ 0 , ) , with f ( 0 ) = 0 and f ( r ) > r , for all r ( 0 , ) , such that for all x K ,
x T x f ( d ( x , F ( T ) ) ) .
A family { T i } i = 1 N of N self-mappings of K with F = i I F ( T i ) is said to satisfy
  1. (1)
    condition (B) on K [25] if there is a nondecreasing function f : [ 0 , ) [ 0 , ) with f ( 0 ) = 0 and f ( r ) > r for all r ( 0 , ) such that for all x K such that
    max 1 l N { x T l x } f ( d ( x , F ) ) ;
     
  2. (2)
    condition ( C ¯ ) on K [26] if there is a nondecreasing function f : [ 0 , ) [ 0 , ) with f ( 0 ) = 0 and f ( r ) > r for all r ( 0 , ) such that for all x K such that
    { x T l x } f ( d ( x , F ) )
     

for at least one T l , l = 1 , 2 , , N or, in other words, at least one of the T l ’s satisfies condition (A).

Condition (B) reduces to condition (A) when all but one of the T l ’s are identities. Also condition (B) and condition ( C ¯ ) are equivalent (see [26]).

Senter and Dotson [24] established a relation between condition (A) and demicompactness that the condition (A) is weaker than demicompactness for a nonexpansive mapping T defined on a bounded set. Every compact operator is demicompact. Since every completely continuous mapping T : K K is continuous and demicompact, it satisfies condition (A).

Therefore in the next result, instead of complete continuity of mappings T 1 , T 2 , , T N , we use condition ( C ¯ ).

Theorem 3.2 Let E and K be as defined above, T i ( i I ) be N asymptotically pseudo-contractive mappings as defined above and satisfying condition ( C ¯ ) and { α n } be a sequence of real numbers as in Lemma  3.1. Then the sequence { x n } generated by (1.4) converges strongly to a member of .

Proof By Lemma 3.1, we see that lim n x n p and lim n d ( x n , F ) exist.

Let one of the T i ’s, say T l , l I , satisfy condition (A).

By Lemma 3.1, we have lim n x n T l x n = 0 . Therefore we have lim n f ( d ( x n , F ) ) = 0 . By the nature of f and the fact that lim n d ( x n , F ) exists, we have lim n d ( x n , F ) = 0 . By Theorem 3.1, we find that { x n } converges strongly to a common fixed point in .

This completes the proof. □

A mapping T : K K is said to be semicompact, if the sequence { x n } in K such that x n T x n 0 , as n , has a convergent subsequence.

Theorem 3.3 Let E and K be as defined above, and let T i ( i I ) be N asymptotically pseudo-contractive mappings as defined above such that one of the mappings in { T i } i = 1 N is semicompact, and let { α n } be a sequence of real numbers as in Lemma  3.1. Then the sequence { x n } generated by (1.4) converges strongly to a member of .

Proof Without loss of generality, we may assume that T s is semicompact for some fixed s { 1 , 2 , , N } . Then by Lemma 3.1, we have lim n x n T s x n = 0 . So by definition of semicompactness, there exists a subsequence { x n j } of { x n } such that { x n j } converges strongly to x K . Now again by Lemma 3.1, we have
lim n j x n j T l x n j = 0

for all l I . By continuity of T l , we have T l x n j T l x for all l I .

Thus lim j x n j T l x n j = x T l x = 0 for all l I . This implies that x F . Also, lim inf n d ( x n , F ) = 0 . By Theorem 3.1, we find that { x n } converges strongly to a common fixed point in . □

3.2 Weak convergence theorem

Theorem 3.4 Let E be a uniformly convex and smooth Banach space which admits a weakly sequentially continuous duality mapping, and let K and T i ( i I ) be as defined above and { α n } be a sequence of real numbers as in Lemma  3.1. Then the sequence { x n } generated by (1.4) converges weakly to a member of .

Proof Since { x n } is a bounded sequence in K, there exists a subsequence { x n k } { x n } such that { x n k } converges weakly to q K . Hence from Lemma 3.1, we have
lim n x n k T l x n k = 0 , l I .

By Lemma 2.4, we find that ( I T l ) is demiclosed at zero, i.e. ( I T l ) q = 0 , so that q F ( T l ) . By the arbitrariness of l I , we know that q F = l I F ( T l ) .

Next we prove that { x n } converges weakly to q.

If { x n } has another subsequence { x n j } which converges weakly to q 1 q , then we must have q 1 F , and since lim n x n q 1 exists and since the Banach space E has a weakly sequentially duality mapping, it satisfies Opial’s condition, and it follows from a standard argument that q 1 = q . Thus { x n } converges weakly to q F . □

Remark 3.1 Our results improve and generalize the corresponding results of Browder [7], Kirk [8], Goebel and Kirk [1], Schu [4], Xu [9, 10], Liu [11], Zhou and Chang [15], Osilike [27], Osilike and Akuchu [13], Su and Li [16], Su and Qin [17], and many others.

Let K be a nonempty subset of a real Banach space E. Let D be a nonempty bounded subset of K. The set-measure of noncompactness of D, γ ( D ) , is defined as
γ ( D ) = inf { d > 0 : D  can be covered by a finite number of sets of diameter d } .
The ball-measure of compactness of D, χ ( D ) , is defined as
χ ( D ) = inf { r > 0 : D  can be covered by a finite family of balls with centers in  E and radius  r } .
A bounded continuous mapping T : K E is called
  1. (1)

    k-set-contractive if γ ( T ( D ) ) k γ ( D ) , for each bounded subset D of K and some constant k 0 ;

     
  2. (2)

    k-set-condensing if γ ( T ( D ) ) < γ ( D ) , for each bounded subset D of K with γ ( D ) > 0 ;

     
  3. (3)

    k-ball-contractive if χ ( T ( D ) ) k χ ( D ) , for each bounded subset D of K and some constant k 0 ;

     
  4. (4)

    k-ball-condensing if χ ( T ( D ) ) < χ ( D ) , for each bounded subset D of K with χ ( D ) > 0 .

     
A mapping T : K E is called
  1. (5)

    compact if cl ( T ( A ) ) is compact whenever A K is bounded;

     
  2. (6)

    completely continuous if it maps weakly convergence sequences into strongly convergent sequences;

     
  3. (7)

    a generalized contraction if for each x K there exists k ( x ) < 1 such that T x T y k ( x ) x y for all y K ;

     
  4. (8)

    a mapping T : E E is called uniformly strictly contractive (relative to E) if for each x E there exists k ( x ) < 1 such that T x T y k ( x ) x y for all y K . Every k-set-contractive mapping with k < 1 is set-condensing and also every compact mapping is set-condensing.

     
Let K be a nonempty closed bounded subset of E and T : K E a continuous mapping. Then
  1. (a)

    T is strictly semicontractive if there exists a continuous mapping V : E × E E with T ( x ) = V ( x , x ) for x E such that for each x E , V ( , x ) is a k-contraction with k < 1 and V ( x , ) is compact;

     
  2. (b)

    T is of strictly semicontractive type if there exists a continuous mapping V : K × K E with T ( x ) = V ( x , x ) , for x K such that for each x K , V ( , x ) is a k-contraction with some k < 1 independent of x and x V ( , x ) is compact from K into the space of continuous mapping of K into E with the uniform metric;

     
  3. (c)

    T is of strongly semicontractive type relative to X if there exists a mapping V : E × K E with T ( x ) = V ( x , x ) , for x K such that x K , V ( , x ) is uniformly strictly contractive on K relative to E and V ( x , ) is a completely continuous from K to E, uniformly for x K .

     

For details refer to [2830].

Let K be a nonempty closed convex bounded subset of a uniformly convex Banach space E. Suppose T : K K . Then T is semicompact if T satisfies any one of the following conditions [[25], Proposition 3.4]:
  1. (i)

    T is either set-condensing or ball-condensing (or compact);

     
  2. (ii)

    T is a generalized contraction;

     
  3. (iii)

    T is uniformly strictly contractive;

     
  4. (iv)

    T is strictly semicontractive;

     
  5. (v)

    T is of strictly semicontractive type;

     
  6. (vi)

    T is of strongly semicontractive type.

     

Remark 3.2 In view of the above, it is possible to replace the semicompactness assumption in Theorem 3.3 with any of the contractive assumptions (i)-(vi).

We now give an example of asymptotically pseudo-contractive mapping with nonempty fixed point set.

Example 3.1 [31]

Let E = R = ( , ) with usual norm and K = [ 0 , 1 ] and define T : K K by
T x = { 0 if  x = 0 , 1 9 if  x = 1 , x 1 3 n + 1 if  1 3 n + 1 x < 1 3 ( 1 3 n + 1 + 1 3 n ) , 1 3 n x if  1 3 ( 1 3 n + 1 + 1 3 n ) x < 1 3 n
for all n 0 . Then F ( T ) = { 0 } and for any x K , there exists j ( x 0 ) J ( x 0 ) satisfying
T n x T n 0 , j ( x 0 ) = T n x x 1 3 x 2 < x 2

for all n 1 . That is, T is an asymptotically pseudo-contractive mapping with sequence { k n } = 1 .

Declarations

Authors’ Affiliations

(1)
School of Studies in Mathematics, Pt. Ravishankar Shukla University
(2)
Department of Mathematics and Informatics, University Politehnica of Bucharest

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© Thakur et al.; licensee Springer. 2014

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