Open Access

The mean ergodic theorem for nonexpansive mappings in multi-Banach spaces

Journal of Inequalities and Applications20142014:259

https://doi.org/10.1186/1029-242X-2014-259

Received: 11 April 2014

Accepted: 13 June 2014

Published: 22 July 2014

Abstract

In this paper, we prove a mean ergodic theorem for nonexpansive mappings in multi-Banach spaces.

MSC:39A10, 39B72, 47H10, 46B03.

Keywords

ergodic theoremnonexpansive mappingmulti-Banach space

1 Introduction

Let X be a Banach space and C be a closed convex subset of X. For each j 1 , a mapping T j : C C is said to be nonexpansive on C if
T j x T j y x y

for all x , y C . For each j 1 , let F ( T j ) be the set of fixed points of T j . If X is a strictly convex Banach space, then F ( T j ) is closed and convex.

In [1], Baillon proved the first nonlinear ergodic theorem such that, if X is a real Hilbert space and F ( T j ) for each j 1 , then, for each x C , the sequence { S n , j x } defined by
S n , j x = 1 n ( x + T j x + + T j n 1 x )

converges weakly to a fixed point of T j . It was also shown by Pazy [2] that, if X is a real Hilbert space and S n , j x converges weakly to y C , then y F ( T ) . These results were extended by Baillon [3], Bruck [4] and Reich [5, 6] and [7].

2 Multi-Banach spaces

The notion of a multi-normed space was introduced by Dales and Polyakov in [8]. This concept is somewhat similar to an operator sequence space and has some connections with the operator spaces and Banach lattices. Observations on multi-normed spaces and examples are given in [810].

Let ( E , ) be a complex normed space and let k N . We denote by E k the linear space E E consisting of k-tuples ( x 1 , , x k ) , where x 1 , , x k E . The linear operations on E k are defined coordinate-wise. The zero element of either E or E k is denoted by 0. We denote by N k the set { 1 , 2 , , k } and by Σ k the group of permutations on k symbols.

Definition 2.1 A multi-norm on { E k : k N } is a sequence { k } k N such that k is a norm on E k for each k N with k 2 satisfying the following conditions:

(A1) ( x σ ( 1 ) , , x σ ( k ) ) k = ( x 1 , , x k ) k ( σ Σ k , x 1 , , x k E );

(A2) ( α 1 x 1 , , α k x k ) k ( max i N k | α i | ) ( x 1 , , x k ) k ( α 1 , , α k C , x 1 , , x k E );

(A3) ( x 1 , , x k 1 , 0 ) k = ( x 1 , , x k 1 ) k 1 ( x 1 , , x k 1 E );

(A4) ( x 1 , , x k 1 , x k 1 ) k = ( x 1 , , x k 1 ) k 1 ( x 1 , , x k 1 E ).

In this case, we say that { ( E k , k ) } k N is a multi-normed space.

Lemma 2.2 ([10])

Suppose that { ( E k , k ) } k N is a multi-normed space and take k N . Then we have the following:
  1. (1)

    ( x , , x ) k = x ( x E );

     
  2. (2)

    max i N k x i x 1 , , x k k i = 1 k x i k max i N k x i ( x 1 , , x k E ).

     

It follows from (2) that, if ( E , ) is a Banach space, then ( E k , k ) is a Banach space for each k N . In this case { ( E k , k ) } k N is a multi-Banach space.

Now, we give two important examples of multi-norms for an arbitrary normed space E [8].

Example 2.3 The sequence { k } k N on { E k : k N } defined by
( x 1 , , x k ) k : = max i N k x i ( x 1 , , x k E )

is a multi-norm, which is called the minimum multi-norm. The terminology ‘minimum’ is justified by the property (2).

Example 2.4 Let { ( k α : k N ) : α A } be the (nonempty) family of all multi-norms on { E k : k N } . For each k N , set
( x 1 , , x k ) k : = sup α A ( x 1 , , x k ) k α ( x 1 , , x k E ) .

Then { k } k N is a multi-norm on { E k : k N } , which is called the maximum multi-norm.

We need the following observation, which can easily be deduced from the triangle inequality for the norm k and the property (2) of multi-norms.

Lemma 2.5 Suppose that k N and ( x 1 , , x k ) E k . For each j { 1 , , k } , let { x n j } n 1 be a sequence in E such that lim n x n j = x j . Then, for each ( y 1 , , y k ) E k , we have
lim n ( x n 1 y 1 , , x n k y k ) = ( x 1 y 1 , , x k y k ) .
Definition 2.6 Let { ( E k , k ) } k N be a multi-normed space. A sequence { x n } n 1 in E is called a multi-null sequence if, for any ε > 0 , there exists n 0 N such that
sup k N ( x n , , x n + k 1 ) k < ε ( n n 0 ) .
Let x E . We say that the sequence { x n } n 1 is multi-convergent to a point x E and write
lim n x n = x

if { x n x } n is a multi-null sequence.

3 Main results

To prove the main results in this paper, first, we introduce some lemmas.

Lemma 3.1 ([11])

Let { ( X j , j ) } j N be a uniformly convex multi-Banach space with modulus of the convexity δ. Let x j , y j X . If ( x 1 , , x j ) j r , ( y 1 , , y j ) j r , r R and ( x 1 y 1 , , x j y j ) j ϵ > 0 , then
( λ x 1 + ( 1 λ ) y 1 , , λ x j + ( 1 λ ) y j ) j r ( 1 2 λ ( 1 λ ) δ R ( ϵ ) )

for all λ [ 0 , 1 ] , where δ R ( ϵ ) = δ ( ϵ R ) .

To proceed, let { ( X j , j ) } j N denote a uniformly convex multi-Banach space with modulus of the convexity δ.

Lemma 3.2 Let C be a closed convex subset of X and for each j 1 , T j : C C be a nonexpansive mapping. Let x C , f j F ( T j ) for each j 1 and 0 < α β < 1 . Then, for any ϵ > 0 , there exists N > 0 such that, for all n N ,
( T 1 k ( λ T 1 n x + ( 1 λ ) f 1 ) ( λ T 1 n + k x + ( 1 λ ) f 1 ) , , T j k ( λ T j n x + ( 1 λ ) f j ) ( λ T j n + k x + ( 1 λ ) f j ) ) j < ϵ

for all k > 0 and λ [ α , β ] .

Proof Put
r = lim n ( T 1 n x f 1 , , T j n x f j ) j , R = ( x f 1 , , x f j ) j , c = min { 2 λ ( 1 λ ) : α λ β } .
For given ϵ > 0 , choose d > 0 such that r r + d > 1 c δ R ( ϵ ) . Then there exists N > 0 such that, for all n N ,
( T 1 n x f 1 , , T j n x f j ) j < r + d .
For each n N , k > 0 and α λ β , we put
u j = ( 1 λ ) ( T j k z f j ) , v j = λ ( T j n + k x T j k z ) ,
where z j = λ T j n x + ( 1 λ ) f j . Then we have
( u 1 , , u j ) j λ ( 1 λ ) ( T 1 n x f 1 , , T j n x f j ) j
and
( v 1 , , v j ) j λ ( 1 λ ) ( T 1 n x f 1 , , T j n x f j ) j .
Suppose that
( u 1 v 1 , , u j v j ) j = ( T 1 k z ( λ T 1 n + k x + ( 1 λ ) f 1 ) , , T 1 k z ( λ T j n + k x + ( 1 λ ) f j ) ) j ϵ .
Then, by Lemma 3.1, we have
( λ u 1 + ( 1 λ ) v 1 , , λ u j + ( 1 λ ) v j ) j = λ ( 1 λ ) ( T 1 n + k x f 1 , , T j n + k x f j ) j λ ( 1 λ ) ( T 1 n x f 1 , , T j n x f j ) j ( 1 2 λ ( 1 λ ) δ R ( ϵ ) ) λ ( 1 λ ) ( T 1 n x f 1 , , T j n x f j ) j ( 1 c δ R ( ϵ ) ) .
Hence we have
( r + d ) ( 1 c δ R ( ϵ ) ) < r ( r + d ) ( 1 C δ R ( ϵ ) ) ,

which is a contradiction. This completes the proof. □

Lemma 3.3 (Browder [12])

Let C be a closed convex subset of X and T j : C C be a nonexpansive mapping. If { u i } is a weakly convergent sequence in C with the weak limit u 0 and lim i u i T j u i = 0 , then u 0 is a fixed point of T j .

Lemma 3.4 Let C be a closed convex subset of X and, for each j 1 , T j : C C be a nonexpansive mapping. Then, for all x C and n > 0 ,
lim i sup j ( T 1 k S n , 1 T 1 i x S n , 1 T 1 k T 1 i x , , T j k S n , j T j i x S n , j T j k T j i x ) j = 0
(1)

uniformly for each k 1 .

Proof By induction on n, we prove this lemma. First, we prove the conclusion in the case n = 2 . Put
r = lim n sup j 1 ( T 1 n + 1 x T 1 n x , , T j n + 1 x T j n x ) j , R = ( x T 1 x , , x T j x ) j , x i , j = T j i x

for each i 1 .

If r 0 , then, for any ϵ > 0 , choose c > 0 such that r r + c > 1 δ R ( ϵ ) / 2 . Then there exists N > 0 such that, for all i N ,
( T 1 k x i , 1 T 1 k + 1 x i , 1 , , T j k x i , j T j k + 1 x i , j ) j r + c
for each k 1 . If we put
u j = 1 2 ( T j k z T j k x i , j ) , v j = 1 2 ( T j k + 1 x i , j T j k z j ) ,
where i N , k > 0 and z j = 1 2 ( x i , j + T j x i , j ) , then we have
( u 1 , , u j ) j 1 2 ( z 1 x i , 1 z j x i , j ) j = 1 4 ( T 1 x i , 1 x i , 1 , , T j x i , j x i , j ) j 1 4 ( r + c ) .
Similarly, we have ( v 1 , , v j ) j 1 4 ( r + c ) . Suppose that
( u 1 v 1 , , u j v j ) j = ( T 1 k z 1 1 2 ( T 1 k + 1 x i , 1 + T 1 k x i , 1 ) , , T j k z j 1 2 ( T j k + 1 x i , j + T j k x i , j ) ) j ϵ .
Then, by Lemma 3.1, we have
1 2 ( u 1 + v 1 , , u j + v j ) j = 1 4 ( T 1 k + 1 x i , 1 T 1 k x i , 1 , , T j k + 1 x i , j T j k x i , j ) j 1 4 ( r + c ) ( 1 1 2 δ R ( ϵ ) ) ,

which contradicts r > ( r + c ) ( 1 1 2 δ R ( ϵ ) ) .

If r = 0 , then, for any ϵ > 0 , choose i > 0 so large that sup j ( u 1 , , u j ) j < ϵ 2 . Hence we have
sup j 1 ( T 1 k z 1 1 2 ( T 1 k + 1 x i , 1 + T 1 k x i , 1 ) , , T j k z j 1 2 ( T j k + 1 x i , j + T j k x i , j ) ) j = sup j 1 ( u 1 v 1 , , u j v j ) j sup j 1 ( u 1 , , u j ) j + sup j 1 ( v 1 , , v j ) j < ϵ .

This completes the proof of the case n = 2 .

Now, suppose that
lim i sup j 1 ( T 1 k S n 1 , 1 x i , 1 S n 1 , 1 T 1 k x i , 1 , , T j k S n 1 , j x i , j S n 1 , j T j k x i , j ) j = 0
uniformly for each k 1 . We claim that
lim i sup j 1 ( S n 1 , 1 T 1 x i , 1 x i , 1 , , S n 1 , j T j x i , j x i , j ) j
exists. Put
r = lim inf i sup j 1 ( S n 1 , 1 T 1 x i , 1 x i , 1 , , S n 1 , j T j x i , j x i , j ) j .
For any ϵ > 0 , choose i > 0 such that
sup j 1 ( S n 1 , 1 T 1 x i , 1 x i , 1 , , S n 1 , j T j x i , j x i , j ) j < r + ϵ 2
and
sup j 1 ( S n 1 , 1 T 1 k x i + 1 , 1 T 1 k S n 1 , 1 x i + 1 , 1 , , S n 1 , j T j k x i + 1 , j T j k S n 1 , j x i + 1 , j ) j < ϵ 2 .
Then we have
sup j 1 ( S n 1 , 1 T 1 x i + k , 1 x i + k , 1 , , S n 1 , j T j x i + k , j x i + k , j ) j sup j 1 ( S n 1 , 1 T 1 k x i + 1 , 1 T 1 k S n 1 , 1 x i + 1 , 1 , , S n 1 , j T j k x i + 1 , j T j k S n 1 , j x i + 1 , j ) j + sup j 1 ( T 1 k S n 1 , 1 x i + 1 , 1 T 1 k x i , 1 , , T j k S n 1 , j x i + 1 , j T j k x i , j ) j < ϵ 2 + r + ϵ 2 = r + ϵ
for all k 1 . Therefore, we have
lim sup i sup j 1 ( S n 1 , 1 T 1 x i , 1 x i , 1 , , S n 1 , j T j x i , j x i , j ) j = lim sup k sup j 1 ( S n 1 , 1 T 1 x i + k , 1 x i + k , 1 , , S n 1 , j T j x i + k , j x i + k , j ) j < r + ϵ .
Since ϵ > 0 is arbitrary, we have
lim sup i sup j 1 ( S n 1 , 1 T 1 x i , 1 x i , 1 , , S n 1 , j T j x i , 1 x i , 1 ) j lim inf i sup j 1 ( S n 1 , 1 T 1 x i , 1 x i , 1 , , S n 1 , j T j x i , j x i , j ) j ,

i.e., lim i sup j 1 ( S n 1 , 1 T 1 x i , 1 x i , 1 , , S n 1 , j T j x i , j x i , j ) j exists.

Now, we put
r = lim i sup j 1 ( S n 1 , 1 T 1 x i , 1 x i , 1 , , S n 1 , j T j x i , j x i , j ) j .
If r 0 , then, for any ϵ, choose c > 0 such that
r c r + 2 c > 1 ( 2 ( n 1 ) n 2 ) δ 3 r ( ϵ ) .
Then there exists N > 0 such that, if, for all i N , we put
u j = n ( n 1 ) ( T j k S n , j x i , j T j k x i , j ) , v j = n ( S n 1 , j T j k x i + 1 , j T j k S n , j x i , j ) ,
so
( u 1 , , u j ) j ( S n 1 , 1 T 1 x i , 1 x i , 1 , , S n 1 , j T j x i , j x i , j ) j r + c , ( v 1 , , v 2 ) j n ( S n 1 , 1 T 1 k x i + 1 , 1 T 1 k S n 1 , 1 x i + 1 , 1 , , S n 1 , j T j k x i + 1 , j T j k S n 1 , j x i + 1 , j ) j + ( S n 1 , 1 T 1 x i , 1 x i , 1 , , S n 1 , j T j x i , j x i , j ) j r + 2 c
and
( u 1 v 1 , , u j v j ) j = n n 1 ( T 1 k S n , 1 x i , 1 S n , 1 T 1 k x i , 1 , , T j k S n , j x i , j S n , j T j k x i , j ) j .
Hence, by the method in the proof of the case n = 2 , we have
sup j 1 ( T 1 k S n , 1 x i , 1 S n , 1 T 1 k x i , 1 , , T j k S n , j x i , j S n , j T j k x i , j ) j < ϵ

for all k 1 and i N .

If r = 0 , then, as in the proof of the case n = 2 , there exists N such that, for each i N ,
sup j 1 ( u 1 , , u j ) j < ϵ 2 , sup j 1 ( v 1 , , v j ) j < ϵ 2 .
Therefore, we have
sup j 1 ( T 1 k S n , 1 x i , 1 S n , 1 T 1 k x i , 1 , , T j k S n , j x i , j S n , j T j k x i , j ) j < ϵ .

This completes the proof. □

Now, assume that the norm of X is Frechet differentiable and then we have the following.

Proposition 3.5 ([4, 6, 13])

Let C be a closed convex subset of X and, for each j 1 , T j : C C be a nonexpansive mapping. If we put W j ( x ) = m c o ¯ { T j k x : k m } for all x C , then W j ( x ) F ( T j ) is at most one point.

In this paper, we give a new proof of the following theorem, which is due to Reich [6].

Theorem 3.6 Let { ( X j , j ) } j N be a uniformly convex multi-Banach space which has the Fréchet differentiable norm. Let C be a closed convex subset of X and, for each j 1 , T j : C C be a nonexpansive mapping. Then the following statements are equivalent:
  1. (1)

    F ( T j ) .

     
  2. (2)

    { T j n x } is bounded for all x C .

     
  3. (3)

    For all x C , { S n T j i x } converges weakly to a point ( y 1 , , y j ) C j uniformly for each i 1 .

     

Proof (1)  (2) is well known in [12].

(3)  (2) Suppose that, for some x C , there exists an unbounded subsequence { T j n i x } of { T j n x } . For each j 1 , since T j is a nonexpansive mapping, it follows that, for each m > 0 , the sequence { S m < j T j n i x } is also unbounded, which contradicts the condition (3).

(2)  (3) Since { T j n x } is bounded and
( T 1 S n , 1 T 1 i x S n , 1 T 1 i x , , T j S n , j T j i x S n , j T j i x ) j ( T 1 S n , 1 T 1 i x S n , 1 T 1 T 1 i x , , T j S n , j T j i x S n , j T j T j i x ) j + ( S n , 1 T 1 T 1 i x S n , 1 T 1 i x , , S n , j T j T j i x S n , j T j i x ) j ( T 1 S n , 1 T 1 i x S n , 1 T 1 T 1 i x , , T j S n , j T j i x S n , j T j T j i x ) j + 1 n ( T 1 i + 1 + n x T 1 i x , , T j i + 1 + n x T j i x ) j ,
there exists a sequence { S n , j T j i n x } such that
lim n sup j 1 ( T 1 S n , 1 T 1 i n x S n , 1 T 1 i n x , , T j S n , j T j i n x S n , j T j i n x ) j = 0 .
Then, by Lemma 3.3 and Proposition 3.5, it follows that any weakly multi-convergent subsequence of { S n , j T j i n x } multi-converges weakly to a point y j , i.e., S n , j T j i n x y j , where y j = W j ( x ) F ( T j ) . Also, by Lemma 3.4, it follows that
lim n sup j 1 ( T 1 S n , 1 T 1 i n + k n + i x S n , 1 T 1 i n + k n + i x , , T j S n , j T j i n + k n + i x S n , j T j i n + k n + i x ) j = 0

for all i , k 1 . Therefore, S n , j T j i n + k n x i y j uniformly for each k 1 .

On the other hand, for each n 1 with m i n , we have
S m , j T j i x = 1 m k = 0 m 1 T j k x i = 1 m ( k = i n + t n m 1 T j k x i + n ( k = 0 t S n T j i n + k n x i ) + k = 0 i n T j k x i ) ,

where m = t n + i n + r , r < n . Since { S n , j T j i n + k n x i } multi-converges to y j uniformly for each k 1 , it follows that { S m , j T j i x } converges weakly to y j uniformly for each i 1 . This completes the proof. □

Declarations

Acknowledgements

The third author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2011-0021821).

Authors’ Affiliations

(1)
Department of Mathematics, Science and Research Branch, Islamic Azad University
(2)
Department of Mathematics and Computer Science, Iran University of Science and Technology
(3)
Department of Mathematics Education and the RINS, Gyeongsang National University

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© Kenari et al.; licensee Springer. 2014

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