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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’

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.

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 :EE 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, n1.

For any given fE, define { S i } i = 1 N :EE by S i x=x A i x+f, i=1,,N, xE. 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 :EE 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, n1.

For any given fE, define { S i } i = 1 N :EE by S i x=(I A i )x+f, i=1,,N, xE. 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 xy ϕ i 1 ( A i x A i y), that is, A i x A i y,j(xy)ϕ(xy)xyϕ(xy) A i x A i y, but we cannot deduce xy 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 j1 to deduce x n q0 (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 =0ϵ>0, nonnegative integer n 0 such that | γ n m + j |<ϵ for n m n 0 , j1.

(C-ii) lim n γ n =0 lim m γ n m =0 and lim m γ n m + j =0 for j1.

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 } ,xE,

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

An operator T:EE is said to be strongly accretive if there exists a constant k>0, and for x,yE, j(xy)J(xy) 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,yE, there exist j(xy)J(xy) 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 ) xy.

It is obvious that a strongly accretive operator must be the ϕ-strongly accretive in the special case in which ϕ(t)=kt, 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:EE is a continuous ϕ-strongly accretive operator. Then the equation Ax=f has a unique solution for any fE.

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,yE 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 ,n0,
(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 :EE 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 fE, define { S i } i = 1 N :EE with i = 1 N F( S i ) by S i x=x A i x+f, i=1,2,,N, xE, where F( S i )={xE: 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 :EE 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 xE, qF(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 qE, 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 qR. 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 xy<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 n0 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 q0 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 :EE 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]. □

References

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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.

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Correspondence to Zhiqun Xue.

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Xue, Z., Zhou, H. A further remark to paper ‘Convergence theorems for the common solution for a finite family of ϕ-strongly accretive operator equations’. J Inequal Appl 2013, 35 (2013). https://doi.org/10.1186/1029-242X-2013-35

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