Strong convergence of modified Halpern’s iterations for a k strictly pseudocontractive mapping

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R E S E A R C H

Open Access

Strong convergence of modiﬁed Halpern’s

iterations for a

k

-strictly pseudocontractive

mapping

Suhong Li

1,2*

_{, Lihua Li}

1

_{, Lingmin Zhang}

1

_{and Xiujuan He}

3

*_{Correspondence:}

[email protected]

1_{College of Mathematics and}

Information Technology, Hebei Normal University of Science and Technology, Qinhuangdao, Hebei 066004, China

2_{Institute of Mathematics and}

Systems Science, Hebei Normal University of Science and Technology, Qinhuangdao, Hebei 066004, China

Full list of author information is available at the end of the article

Abstract

In this paper, we discuss three modiﬁed Halpern iterations as follows:

xn+1=

α

nu+ (1 –

α

n)

(

(1 –

δ

)xn+

δ

Txn

)

, (I)

xn+1=

α

n

(

(1 –

δ

)u+

δ

xn

)

+ (1 –

α

n)Txn, (II)

xn+1=

α

nu+

β

nxn+

γ

nTxn, n≥0, (III)

and obtained the strong convergence results of the iterations (I)-(III) for ak-strictly pseudocontractive mapping, where{

α

n}satisﬁes the conditions: (C1) limn→∞

α

n= 0

and (C2)∞_n₌₁

α

n= +∞, respectively. The results presented in this work improve the

corresponding ones announced by many other authors.

1 Introduction

LetHbe a real Hilbert space with the inner product·,·and the norm · and letCbe a nonempty closed convex subset ofH.

Recall that a mappingT with domainD(T) and rangeR(T) in the Hilbert spaceHis called strongly pseudo-contractive if, for allx,y∈D(T), there existsk∈(, ) such that

Tx–Ty,x–y ≤kx–y, (.)

whileTis said to be pseudo-contractive if (.) holds fork= . A mappingTis said to be Lipschitzian if, for allx,y∈D(T), there existsL>  such that

Tx–Ty ≤Lx–y. (.)

A mappingT is called nonexpansive if (.) holds forL=  and, further,T is said to be contractive ifL< .Tis said to be ﬁrmly nonexpansive if for allx,y∈D(T),

Tx–Ty≤ x–y,Tx–Ty.

Firmly nonexpansive mappings could be looked upon as an important subclass of nonex-pansive mappings. A mappingTis calledk-strictly pseudocontractive, if for allx,y∈D(T),

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there existsλ>  such that

Tx–Ty,x–y ≤ x–y–λx–y– (Tx–Ty). (.)

Without loss of generality, we may assume thatλ∈(, ). In Hilbert spacesH, (.) is equivalent to the inequality

Tx–Ty≤ x–y+k(I–T)x– (I–T)y, k= ( – λ) < ,

and we can assume also thatk≥ so thatk∈[, ).

It is obvious that ak-strictly pseudocontractive mapping is Lipschitzian withL=k+_k . The class of nonexpansive mappings is a subclass of strictly pseudocontractive mappings in a Hilbert space, but the converse implication may be false. We remark that the class of strongly pseudo-contractive mappings is independent from the class ofk-strict pseudo-contractions.

In , Halpern [] was the ﬁrst who introduced the following iteration scheme for a nonexpansive mappingTwhich was referred to as Halpern iteration: For any initialization

x∈Cand any anchoru∈C,αn∈[, ],

xn+=αnu+ ( –αn)Txn, ∀n≥. (.)

He proved that the sequence (.) converges weakly to a ﬁxed point ofT, whereαn=n–a,

a∈(, ). In , Lions [] further proved that the sequence (.) converges strongly to a ﬁxed point ofT in a Hilbert space, where{αn}satisﬁes the following conditions:

(C) lim n→∞αn= ;

(C)

∞

n=

αn= +∞;

(C) lim n→∞

|αn+–αn|

α

n+

= .

But, in [, ], the real sequence {αn} excluded the canonical choiceαn= _n_+. In , Wittmann [] proved, still in Hilbert spaces, the strong convergence of the sequence (.) to a ﬁxed point ofT, where{αn}satisﬁes the following conditions:

(C) lim n→∞αn= ;

(C)

∞

n=

αn= +∞;

(C)

∞

n=

|αn+–αn|< +∞.

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Takahashi [] extended Wittmann’s result to Banach spaces. In , Xu [] obtained a strong convergence theorem, where{αn}satisﬁes the following conditions: (C), (C) and (C)limn→∞|αn+αn+–αn|= . In particular, the canonical choice ofαn=



n+ satisﬁes the

con-ditions (C), (C) and (C).

However, is a real sequence {αn}satisfying the conditions (C) and (C) suﬃcient to guarantee the strong convergence of Halpern’s iteration (.) for nonexpansive mappings? It remains an open question, see [].

Some mathematicians considered the open question. In [], Song proved that for a firmly nonexpansive mappingT, an important subclass of nonexpansive mappings, the answer of the Halpern open problem is affirmative. A partial answer to this question was given independently by Chidume and Chidume [] and Suzuki []. They defined the se-quence{xn}by

xn+=αnu+ ( –αn)

( –δ)xn+δTxn

, (.)

whereδ∈[, ],Iis the identity, and obtained the strong convergence of the iteration (.), where{αn}satisﬁes the conditions (C) and (C). Recently, Xu [] gave another partial answer to this question. He obtained the strong convergence of the iterative sequence

xn+=αn

( –δ)u+δxn

+ ( –αn)Txn, (.)

whereδ∈[, ] and{αn}satisﬁes the conditions (C) and (C).

In [], Liang-Gen Hu introduced the modiﬁed Halpern’s iteration: For anyu,x∈C,

the sequence{xn}is deﬁned by

xn+=αnu+βnxn+γnTxn, n≥, (.)

where {αn}, {βn},{γn} are three real sequences in [, ], satisfyingαn+βn+γn= . He showed that the sequence{αn}satisfying the conditions (C) and (C) is suﬃcient to guar-antee the strong convergence of the modiﬁed Halpern’s iterative sequence (.) for non-expansive mappings.

The purpose of this paper is to present a significant answer to the above open question. We will show that the sequence{αn}satisfying the conditions (C) and (C) is sufficient to guarantee the strong convergence of the modified Halpern’s iterative sequences (.)-(.) fork-strictly pseudocontractive mappings, respectively. The results present in this paper improve and develop the corresponding results of [, , , ].

2 Preliminaries

In what follows we will need the following:

Lemma .[] Let H be a real Hilbert space,then the following well-known results hold:

(i) tx+ ( –t)y₌_tx_{+ ( –}_t₎_y_–_t_{( –}_t₎_x_–_y_{for all}_x_,_y_∈_H_{and for all} t∈[, ].

(ii) x+y_≤_x_{+ }_y_,_x₊_y_,_∀_x_,_y_∈_H_.

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its nearest point denoted byPCxinC. ThusPCxis the unique point inCsuch that

x–PCx ≤ x–y, ∀y∈C.

It is known that for eachx∈H,

x–PCx,y–PCx ≤, ∀y∈C. (.)

Lemma .[] Let C be a nonempty closed convex subset of a real Hilbert space H,T :

C→C be a k-strictly pseudocontractive mapping.Then(I–T)is demiclosed at zero.

Lemma .[] Let{an}be a sequence of nonnegative real numbers such that an+≤( –

δn)an+δnξn,∀n≥,where{δn}is a sequence in[, ]and{ξn}is a sequence in R satisfying

the following conditions:

(i) ∞_n_=δn= +∞;

(ii) lim sup_n_→∞ξn≤or ∞

n=δn|ξn|< +∞.

Thenlimn→∞an= .

3 Main results

In this section, proving the following theorems, we show that the conjunction of (C) and (C) is a suﬃcient condition on our iteration (I)-(III), respectively.

Theorem . Let C be a closed and convex subset of a real Hilbert space H,T:C→C be

a k-strictly pseudocontractive mapping such that F(T)=∅.For an arbitrary initial value x∈C and ﬁxed anchor u∈C,deﬁne iteratively a sequence{xn}as follows:

xn+=αnu+βnxn+γnTxn, (I)

where{αn},{βn},{γn}are three real sequences in(, ),satisfyingαn+βn+γn= and <k< βn

βn+γn.Suppose that{αn}satisﬁes the conditions: (C)limn→∞αn= ; (C) ∞

n=αn= +∞.

Then as n→ ∞,{xn}converges strongly to some ﬁxed point x*of T and x*=PF(T)u,where PF(T)is the metric projection from H onto F(T).

Proof Firstly, we show that{xn}is bounded. Rewrite the iterative process (I) as follows:

xn+=αnu+βnxn+γnTxn

=αnu+ ( –αn)

βnxn+γnTxn  –αn

=αnu+ ( –αn)yn, (.)

whereyn=βnxn_–+_αγnTxn_n . Take anyp∈F(T), then, from Lemma . and (.), we estimate as follows:

xn+–p

=αnu–p+ ( –αn)yn–p–αn( –αn)u–yn

=αnu–p+ ( –αn)

βn  –αn

xn–p+ ( –αn)

γn  –αn

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– ( –αn) βn  –αn

γn  –αn

xn–Txn–αn( –αn)u–yn

=αnu–p+βnxn–p+γnTxn–p–αn( –αn)u–yn

– βnγn  –αnxn

–Txn

≤αnu–p+βnxn–p+γn

xn–p+kxn–Txn

– βnγn  –αn

xn–Txn–αn( –αn)u–yn

=αnu–p+ ( –αn)xn–p–

βn  –αn

–k

γnxn–Txn

–αn( –αn)u–yn

≤αnu–p+ ( –αn)xn–p

≤max u–p,xn–p

. (.)

By induction,

xn+–p≤max u–p,x–p

.

This proves the boundedness of the sequence{xn}, which leads to the boundedness of

{Txn}.

Next, we claim that

lim

n→∞xn–Txn= .

In fact, we have from (.) (for some appropriate constantM> ) that

xn+–p≤αnu–p+ ( –αn)xn–p–

βn  –αn

–k

γnxn–Txn

=αn

u–p–xn–p

+xn–p–

βn  –αn

–k

γnxn–Txn

≤αnM+xn–p–

βn  –αn

–k

γnxn–Txn,

which implies that

βn  –αn

–k

γnxn–Txn–αnM≤ xn–p–xn+–p. (.)

If ( βn

–αn–k)γnxn–Txn

_–_α

nM≤, then

βn  –αn

–k

γnxn–Txn≤αnM,

xn–Txn≤

αn

( βn

–αn–k)γn

M,

and hence the desired result is obtained by the condition (C) and  <k< βn

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If ( βn

–αn–k)γnxn–Txn

_–_α

nM> , then following (.), we have

m

n=

βn  –αn

–k

γnxn–Txn–αnM

≤ x–p–xm–p≤ x–p.

Then ∞ n= βn  –αn

–k

γnxn–Txn–αnM

< +∞.

Thus

lim n→∞

βn  –αn

–k

γnxn–Txn–αnM

= ,

and hence

lim

n→∞xn–Txn= . (.)

In order to provexn→x*=PF(T)u, we next show that

lim sup n→∞ PF(T)

u–u,PF(T)u–xn+ ≤.

Indeed, we can take a subsequence{xnk}of{xn}such that

lim sup

n→∞ PF(T)u–u,PF(T)u–xn+=klim→∞PF(T)u–u,PF(T)u–xnk+.

We may assume thatxnkzsinceHis reﬂexive and{xn}is bounded. From (.), it follows from Lemma . thatz∈F(T).

From (.), we conclude

lim sup

n→∞ PF(T)u–u,PF(T)u–xn+=klim→∞PF(T)u–u,PF(T)u–xnk+

=PF(T)u–u,PF(T)u–z ≤. (.)

Finally, we show thatxn→PF(T)u. As a matter of fact, from Lemma . and (.), we obtain

xn+–PF(T)u

≤( –αn)yn–PF(T)u+ αnu–PF(T)u,xn+–PF(T)u

= ( –αn)

βn  –αnxn

–PF(T)u+ γn  –αnTxn

–PF(T)u– βnγn

( –αn)xn–Txn 

+ αnu–PF(T)u,xn+–PF(T)u

≤( –αn)

βn  –αnxn

–PF(T)u+ γn  –αnxn

–PF(T)u+ γnk  –αnxn

–Txn

– βnγn  –αn

xn–Txn

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= ( –αn)xn–PF(T)u–γn(βn–k)xn–Txn+ αnu–PF(T)u,xn+–PF(T)u

≤( –αn)xn–PF(T)u+ αnu–PF(T)u,xn+–PF(T)u.

It follows from the conditions (C), (C) and (.), using Lemma ., that

lim

n→∞xn–PF(T)u= .

This completes the proof of Theorem ..

Theorem . Let C be a closed and convex subset of a real Hilbert space H,T:C→C be

xn+=αn

( –δ)u+δxn

+ ( –αn)Txn, (II)

where{αn} ⊂(, ),  <k<αnδ.Suppose that{αn}satisﬁes the conditions: (C)limn→∞αn= ; (C)∞_n_=αn=∞.Then as n→ ∞,{xn}converges strongly to some ﬁxed point x*of T

and x*₌_P

F(T)u,where PF(T)is the metric projection from H onto F(T).

Proof Firstly, we show that{xn}is bounded. Rewrite the iterative process (II) as follows:

yn= ( –δ)u+δxn,

xn+=αnyn+ ( –αn)Txn.

(.)

Take anyp∈F(T), then, from Lemma . and (.), we estimate as follows:

xn+–p

=αnyn–p+ ( –αn)Txn–p–αn( –αn)yn–Txn

=αn( –δ)u–p+αnδxn–p–αn( –δ)δu–xn

+ ( –αn)Txn–p–αn( –αn)( –δ)u–Txn

–αn( –αn)δxn–Txn+αn( –αn)δ( –δ)u–xn

≤αn( –δ)u–p+αnδxn–p–αn( –δ)δxn–u+ ( –αn)xn–p

+ ( –αn)kxn–Txn–αn( –αn)( –δ)Txn–u

–αn( –αn)δxn–Txn+αn( –αn)δ( –δ)xn–u

≤ –αn( –δ)xn–p+αn( –δ)u–p– ( –αn)(αnδ–k)xn–Txn

≤ –αn( –δ)

xn–p+αn( –δ)u–p

≤max u–p,xn–p

.

By induction,

xn+–p≤max u–p,x–p

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This proves the boundedness of the sequence{xn}, which implies that the sequence{Txn} is bounded also.

Using the same technique as in Theorem ., we can prove

lim

n→∞xn–Txn= 

and

lim sup

n→∞ PF(T)u–u,PF(T)u–xn+ ≤.

Finally, we show thatxn→PF(T)u. Writingzn=αnδxn_{–(–}+(–δ)ααnn)Txn, then

xn+= ( –δ)αnu+

 – ( –δ)αn

zn,

from Lemma . and (.), we obtain

xn+–PF(T)u ≤ – ( –δ)αn

zn–PF(T)u+ ( –δ)αnu–PF(T)u,xn+–PF(T)u

= – ( –δ)αn

αnδ  – ( –δ)αnxn

–PF(T)u+

 –αn  – ( –δ)αnTxn

–PF(T)u

– αn( –αn)δ

[ – ( –δ)αn]xn–Txn 

+ ( –δ)αnu–PF(T)u,xn+–PF(T)u

≤ – ( –δ)αn

αnδ  – ( –δ)αn

xn–PF(T)u+

 –αn  – ( –δ)αn

xn–PF(T)u

+  –αn  – ( –δ)αnkxn

–Txn–

αn( –αn)δ  – ( –δ)αnxn

–Txn

= – ( –δ)αn

xn–PF(T)u–

( –αn)(αnδ–k)  – ( –δ)αn

xn–Txn

≤ – ( –δ)αn

xn–PF(T)u+ ( –δ)αnu–PF(T)u,xn+–PF(T)u.

It follows from the conditions (C), (C) and (.), using Lemma ., that

lim

This completes the proof of Theorem ..

Theorem . Let C be a closed and convex subset of a real Hilbert space H,T:C→C be

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where Tβ=βI+ ( –β)T,{αn} ⊂[, ],β∈(k, ).Suppose that{αn}satisﬁes the conditions: (C)limn→∞αn= ; (C)

∞

n=αn=∞.Then as n→ ∞,{xn}converges strongly to some

ﬁxed point x*of T,and x*=PF(T)u,where PF(T)is the metric projection from H onto F(T).

Proof It is easy to see thatF(Tβ) =F(T)=∅. For anyx,y∈C, we have

Tβx–Tβy

=β(x–y) + ( –β)(Tx–Ty)

=βx–y+ ( –β)Tx–Ty–β( –β)x–Tx– (y–Ty)

≤βx–y+ ( –β)x–y+kx–Tx– (y–Ty)

–β( –β)x–Tx– (y–Ty)

=x–y– ( –β)(β–k)x–Tx– (y–Ty)

=x–y–β–k

 –βx–Tβx– (y–Tβy) 

≤ x–y– (β–k)x–Tβx– (y–Tβy)



.

Thus, for allx∈Cand for allp∈F(Tβ) =F(T), we have

Tβx–p≤ x–p– (β–k)x–Tβx.

This implies thatTβis a quasi-ﬁrmly type nonexpansive mapping (see, for example, []).

Tβ is also a strongly quasi-nonexpansive mapping (see, for example, []). Hence it fol-lows from [, ] (see Theorem . and Remark  of [] or Corollary  of []) that{xn} converges strongly to a pointx*_∈_F₍_T

β) =F(T). Finally, we showx*₌_P

F(T)u. From Lemma . and the iterative process (III), we estimate

as follows:

xn+–PF(T)u

≤( –αn)β(xn–PF(T)u) + ( –β)(Txn–PF(T)u) 

+ αnu–PF(T)u,xn+–PF(T)u

≤( –αn)βxn–PF(T)u+ ( –β)Txn–PF(T)u–β( –β)xn–Txn

≤( –αn)βxn–PF(T)u+ ( –β)xn–PF(T)u+ ( –β)kxn–Txn

–β( –β)xn–Txn

= ( –αn)

xn–PF(T)u– ( –β)(β–k)xn–Txn

+ αnu–PF(T)u,xn+–PF(T)u

≤( –αn)xn–PF(T)u+ αnu–PF(T)u,xn+–PF(T)u.

It follows from the conditions (C), (C) and

lim

n→∞u–PF(T)u,xn+–PF(T)u=

u–PF(T)u,x*–PF(T)u

(10)

using Lemma ., that

lim

This completes the proof of Theorem ..

Remark . Theorems .-. improve the main results of [, , , ] from a

nonexpan-sive mapping to ak-strictly pseudocontractive mapping, respectively. Theorems .-. show that the real sequence{αn}satisfying the two conditions (C) and (C) is suﬃcient for the strong convergence of the iterative sequences (I)-(III) fork-strictly pseudocontrac-tive mappings, respecpseudocontrac-tively. Therefore, our results give a signiﬁcant partial answer to the open question.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

SL and LL carried out the proof of convergence of the theorems. XH and LZ carried out the check of the manuscript. All authors read and approved the ﬁnal manuscript.

Author details

1_{College of Mathematics and Information Technology, Hebei Normal University of Science and Technology,}

Qinhuangdao, Hebei 066004, China. 2_{Institute of Mathematics and Systems Science, Hebei Normal University of Science}

and Technology, Qinhuangdao, Hebei 066004, China.3_{Hebei Business and Trade School, Shijiazhuang, Hebei 050000,}

China.

Acknowledgements

The authors are very grateful to the referees for their careful reading, comments and suggestions, which improved the presentation of this article. The ﬁrst author was supported by the Natural Science Foundational Committee of Hebei Province (Z2011113) and Hebei Normal University of Science and Technology (ZDJS 2009 and CXTD2010-05).

Received: 27 June 2012 Accepted: 20 February 2013 Published: 12 March 2013 References

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doi:10.1186/1029-242X-2013-98