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A further remark to paper ‘Convergence theorems for the common solution for a finite family of ϕ-strongly accretive operator equations’

Journal of Inequalities and Applications20132013:35

https://doi.org/10.1186/1029-242X-2013-35

Received: 2 September 2012

Accepted: 6 January 2013

Published: 30 January 2013

Abstract

In this note, we point out several gaps in Gurudwan and Sharma (Appl. Math. Comput. 217(15):6748-6754, 2011) and Yang (Appl. Math. Comput. 218(21):10367-10369, 2012) and give the main results under weaker conditions.

MSC:47H10, 47H09, 46B20.

Keywords

uniformly continuousΦ-strongly accretivemulti-step iteration with errorsBanach space

1 Introduction

Recently, Gurudwan, Sharma [1] and Yang [2] studied the strong convergence of the sequence, respectively, which was defined by

for approximation of a common solution of a finite family of uniformly continuous Φ-strongly accretive operator equations. Their results are as follows.

Theorem GS [[1], Theorem 3.1]

Let E be an arbitrary real Banach space and let { A i } i = 1 N : E E be uniformly continuous ϕ-strongly accretive operators and each range of either A i or ( I A i ) be bounded. Let, for i = 1 , , N , { u n i } n = 1 be sequences in E and { a n i } n = 1 , { b n i } n = 1 , { c n i } n = 1 be real sequences in [ 0 , 1 ] satisfying
  1. (i)

    a n i + b n i + c n i = 1 ,

     
  2. (ii)

    n = 0 b n N = ,

     
  3. (iii)

    n = 0 c n < ,

     
  4. (iv)

    lim n b n i = lim n c n i = lim n c n i b n i + c n i = 0 , i = 1 , , N , n 1 .

     

For any given f E , define { S i } i = 1 N : E E by S i x = x A i x + f , i = 1 , , N , x E . Then the multi-step iterative sequence with errors { x n } n = 1 defined by the above converges strongly to the unique solution of the operator equations { A i x } i = 1 N = f .

On the basis of the above result, Yang [2] proved the following convergence theorem.

Thoerem Yang [[2], Theorem 2]

Let E be an arbitrary real Banach space and let { A i } i = 1 N : E E be uniformly continuous ϕ-strongly accretive operators and each range of either A i or ( I A i ) is bounded. Let for i = 1 , , N , { u n i } n = 1 be bounded sequences in E and { a n i } n = 1 , { b n i } n = 1 , { c n i } n = 1 be real sequences in [ 0 , 1 ] satisfying
  1. (i)

    a n i + b n i + c n i = 1 ,

     
  2. (ii)

    n = 0 b n N = ,

     
  3. (iii)

    lim n b n i = lim n c n i = lim n c n i b n i + c n i = 0 , i = 1 , , N , n 1 .

     

For any given f E , define { S i } i = 1 N : E E by S i x = ( I A i ) x + f , i = 1 , , N , x E . Then the multi-step iterative sequence with errors { x n } n = 1 defined by the above converges strongly to the unique solution of the operator equations { A i x } i = 1 N = f .

However, after careful reading of their works, we discovered that there exist some problems in references [1] and [2] as follows.

Problem 1 In the proof course of Theorem 3.1 of Gurudwan and Sharma [1], which happens in line 11 of page 6751. Here, it is defective that they obtained x y ϕ i 1 ( A i x A i y ) , that is, A i x A i y , j ( x y ) ϕ ( x y ) x y ϕ ( x y ) A i x A i y , but we cannot deduce x y ϕ i 1 ( A i x A i y ) . The reason is that it is possible A i x A i y does not belong to R ( ϕ ) (range of ϕ). A counterexample is as follows. Let us define ϕ : [ 0 , + ) [ 0 , + ) by ϕ ( α ) = 2 α 1 2 α + 1 ; then it can be easily seen that ϕ is increasing with ϕ ( 0 ) = 0 , but lim α + ϕ ( α ) = 1 and ϕ 1 ( 2 ) makes no sense (see [3]).

Problem 2 In the paper of Yang [2], he referred to the mistakes of ‘ x n m + j i q < ϵ for j 1 to deduce x n q 0 ( n )’ in [1] and cited an example, i.e.,

Now, we want to clarify the fact. Let { γ n } be a real sequence, { γ n m } be some infinite subsequence of { γ n } and { n m } be neither odd nor even sequence, then the conclusions are as follows:

(C-i) lim n γ n = 0 ϵ > 0 , nonnegative integer n 0 such that | γ n m + j | < ϵ for n m n 0 , j 1 .

(C-ii) lim n γ n = 0 lim m γ n m = 0 and lim m γ n m + j = 0 for j 1 .

Indeed, the above example () does not satisfy the conclusion (C-i), it just illustrates the result (C-ii). Therefore, the note given by Yang [2] confused the conclusions (C-i) and (C-ii).

The aim of this paper is to generalize the results of papers [1] and [2]. For this, we need the following knowledge.

2 Preliminary

Let E be a real Banach space and E be its dual space. The normalized duality mapping J : E 2 E is defined by
J ( x ) = { f E : x , f = x 2 = f 2 } , x E ,

where , denotes the generalized duality pairing. The single-valued normalized duality mapping is denoted by j.

An operator T : E E is said to be strongly accretive if there exists a constant k > 0 , and for x , y E , j ( x y ) J ( x y ) such that
T x T y , j ( x y ) k x y 2 ,
without loss of generality, we assume that k ( 0 , 1 ) . The operator T is called ϕ-strongly accretive if for any x , y E , there exist j ( x y ) J ( x y ) and a strictly increasing continuous function ϕ : [ 0 , + ) [ 0 , + ) with ϕ ( 0 ) = 0 such that
T x T y , j ( x y ) ϕ ( x y ) x y .

It is obvious that a strongly accretive operator must be the ϕ-strongly accretive in the special case in which ϕ ( t ) = k t , but the converse is not true in general. That is, the class of strongly accretive operators is a proper subclass of the class of ϕ-strongly accretive operators.

In order to obtain the main conclusion of this paper, we need the following lemmas.

Lemma 2.1 [1]

Suppose that E is an arbitrary Banach space and A : E E is a continuous ϕ-strongly accretive operator. Then the equation A x = f has a unique solution for any f E .

Lemma 2.2 [4]

Let E be a real Banach space and let J : E 2 E be a normalized duality mapping. Then
x + y 2 x 2 + 2 y , j ( x + y ) ,
(2.1)

for all x , y E and j ( x + y ) J ( x + y ) .

Lemma 2.3 [5]

Let { δ n } n = 0 , { λ n } n = 0 and { γ n } n = 0 be three nonnegative real sequences and ϕ : [ 0 , + ) [ 0 , + ) be a strictly increasing and continuous function with ϕ ( 0 ) = 0 satisfying the following inequality:
δ n + 1 2 δ n 2 λ n ϕ ( δ n + 1 ) + γ n , n 0 ,
(2.2)

where λ n [ 0 , 1 ] with n = 0 λ n = , γ n = o ( λ n ) . Then δ n 0 as n .

3 Main results

Theorem 3.1 Let E be an arbitrary real Banach space and { A i } i = 1 N : E E be N uniformly continuous ϕ-strongly accretive operators. For i = 1 , 2 , , N , let { u n i } n = 1 be bounded sequences in E and { a n i } n = 1 , { b n i } n = 1 , { c n i } n = 1 be real sequences in [ 0 , 1 ] satisfying
  1. (i)

    a n i + b n i + c n i = 1 , i = 1 , 2 , , N ;

     
  2. (ii)

    n = 1 b n N = + ;

     
  3. (iii)

    lim n b n i = lim n c n i = 0 , i = 1 , 2 , , N ;

     
  4. (iv)

    c n N = o ( b n N ) .

     
For any given f E , define { S i } i = 1 N : E E with i = 1 N F ( S i ) by S i x = x A i x + f , i = 1 , 2 , , N , x E , where F ( S i ) = { x E : S i x = x } . Then, for some x 0 E , the multi-step iterative sequence with errors { x n } n = 1 defined by

converges strongly to the unique solution of the operator equations { A i x } i = 1 N = f .

Proof Since { A i } i = 1 N : E E is ϕ-strongly accretive operator, we obtain that each equation A i x = f has the unique solution by Lemma 2.1, denote q i , i.e., q i is the unique fixed point of S i by S i x = x A i x + f . Since i = 1 N F ( S i ) , then i = 1 N F ( S i ) is a single set, let q. Meanwhile, there exists a strictly increasing continuous function ϕ : [ 0 , + ) [ 0 , + ) with ϕ ( 0 ) = 0 such that
A i x A i q , j ( x q ) ϕ ( x q ) ,
for x E , q F ( T ) , that is,
Choose some x 0 E and x 0 S i x 0 such that r 0 R ( Φ ) , where
r 0 = max { x 0 S 1 x 0 x 0 q , x 0 S 2 x 0 x 0 q , , x 0 S N x 0 x 0 q } ,

R ( Φ ) is the range of Φ. Indeed, if Φ ( r ) + as r + , then r 0 R ( Φ ) ; if sup { Φ ( r ) : r [ 0 , + ) } = r 1 < + with r 1 < r 0 , then for q E , there exists a sequence { w n } in E such that w n q as n with w n q . Since A i is uniformly continuous, so is S i . Furthermore, we obtain that S i w n S i q as n , then { w n S i w n } is the bounded sequence for i = 1 , 2 , , N . Hence, there exists the common natural number n 0 such that w n S i w n w n q < r 1 2 for n n 0 and i = 1 , 2 , , N , then we redefine x 0 = w n 0 and x 0 S i x 0 x 0 q < r 1 2 . Thus, max 1 i N { x 0 S i x 0 x 0 q } R ( ϕ ) . It is to ensure that Φ 1 ( r 0 ) is defined well.

Step I. We show that { x n } is a bounded sequence.

Set R = Φ 1 ( r 0 ) , then from the above formula (@), we obtain that x 0 q R . Denote
B 1 = { x E : x q R } , B 2 = { x E : x q 2 R } .
Since S i is uniformly continuous, then S i is bounded. We let
M = max 1 i N { sup x B 2 { S i x q + 1 } } + max 1 i N { sup n { u n i q } } .
Next, we want to prove that x n B 1 . If n = 0 , then x 0 B 1 . Now, assume that it holds for some n, i.e., x n B 1 . We prove that x n + 1 B 1 . Suppose it is not the case, then x n + 1 q > R > R 2 . Since S i is uniformly continuous for i = 1 , 2 , , N , then for ϵ 0 = Φ ( R 2 ) 8 R , there exists common δ > 0 such that S i x S i y < ϵ 0 when x y < δ . Denote
τ 0 = min { 1 , R M , Φ ( R 2 ) 8 R ( M + 2 R ) , δ 2 M + 5 R } .
Since b n i , c n i 0 as n for i = 1 , 2 , , p . Without loss of generality, we let 0 b n i , c n i τ 0 for any n 0 and i = 1 , 2 , , N . Since c n N = o ( b n N ) , let c n N < b n N τ 0 . Now, estimate x n i q for i = 1 , 2 , , N . From the multi-step iteration, we have
(3.1)
then x n 1 B 2 . Similarly, we have
(3.2)
then x n 2 B 2 .  , we have
(3.3)
then x n N 1 B 2 . Therefore, we get
(3.4)
And we also have
(3.5)
and
(3.6)
By the uniform continuity of S N , we have
S N x n + 1 S N x n N 1 < Φ ( R 2 ) 8 R .
Using Lemma 2.2 and the above formulas, we have
(3.7)

which is a contradiction. So, x n + 1 B 1 , i.e., { x n } is a bounded sequence, from which it follows that { x n 1 } , { x n 2 } , , { x n N 1 } are all bounded sequences as well.

Step II. We want to prove x n q 0 as n .

Since b n i , c n i 0 as n for i = 1 , 2 , , N and { x n } , { x n N 1 } are bounded. From (3.5) and (3.6), we obtain
lim n x n + 1 x n = 0 , lim n x n + 1 x n N 1 = 0 , lim n S N x n + 1 S N x n N 1 = 0 .
By (3.7), we have
(3.8)
where

By Lemma 2.3, we obtain lim n x n q = 0 . This completes the proof. □

Remark 3.2 Theorem 3.1 generalizes Theorem 3.1 of [1] and Theorem 2 of [2] in the following cases:
  1. (a)

    It is not necessary for each range of A i or I A i to be bounded in [1] and [2].

     
  2. (b)

    The condition of { c n i } is weakened to c n N = o ( b n N ) from lim n c n i b n i + c n i = 0 ( i = 1 , 2 , , N ).

     
  3. (c)

    The proof method of our theorem differs from that of [1] and [2].

     
Theorem 3.3 Let E, { u n i } , { a n i } , { b n i } , { c n i } ( i = 1 , 2 , , N ) be as in Theorem  3.1 and let { T i } i = 1 N : E E be N uniformly continuous ϕ-strongly pseudocontractive mappings. Then, for some x 0 E , the multi-step iterative sequence with errors { x n } n = 1 defined by

converges strongly to the unique common fixed point of { T i } i = 1 N .

Proof See [1]. □

Declarations

Acknowledgements

The authors are very grateful to Professor Yeol-Je Cho for good suggestions which helped to improve the manuscript. This work is supported by the Hebei Natural Science Foundation No. A2011210033.

Authors’ Affiliations

(1)
Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang, China
(2)
Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang, China

References

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Copyright

© Xue and Zhou; licensee Springer 2013

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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