An intermixed algorithm for strict pseudo-contractions in Hilbert spaces

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R E S E A R C H

Open Access

An intermixed algorithm for strict

pseudo-contractions in Hilbert spaces

Zhangsong Yao

1

_{, Shin Min Kang}

2*

_{and Hong-Jun Li}

3

*_{Correspondence:}

[email protected]

2_{Department of Mathematics and}

the RINS, Gyeongsang National University, Jinju, 660-701, Korea Full list of author information is available at the end of the article

Abstract

An intermixed algorithm for two strict pseudo-contractions in Hilbert spaces have been presented. It is shown that the suggested algorithms converge strongly to the fixed points of two strict pseudo-contractions, independently. As a special case, we can find the common fixed points of two strict pseudo-contractions in Hilbert spaces.

MSC: 47H09; 47H10

Keywords: intermixed algorithm; strict pseudo-contraction; ﬁxed point; strong convergence

1 Introduction

LetCbe a nonempty closed convex subset of a real Hilbert spaceHwith its inner product

·,·and norm · .

Deﬁnition . A mappingT:C→Cis said to be nonexpansive if

Tx–Ty ≤ x–y

for allx,y∈C.

We useFix(T) to denote the set of ﬁxed points ofT.

Deﬁnition . A mappingT :C→C is said to be strictly pseudo-contractive if there exists a constant ≤λ<  such that

Tx–Ty≤ x–y+λ(I–T)x– (I–T)y, ∀x,y∈C.

Remark . It is well known that the class of strictly pseudo-contractive mappings

prop-erly includes the class of nonexpansive mappings.

Iterative construction of fixed points of nonlinear mappings has a long history and is still an active field in the nonlinear functional analysis. LetCbe a nonempty closed con-vex subset of a real Hilbert space. LetT:C→Cbe a nonlinear mapping. Let{αn}be a real number sequence in (, ). For arbitrarily fixedx∈C, define a sequence{xn}in the

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following manner:

xn+=αnxn+ ( –αn)Txn, n≥. (.)

Iteration (.) is said to be a Mann iteration []; it has been studied extensively in the literature. IfT is a nonexpansive mapping withFix(T)=∅and{αn}satisﬁes the condi-tion∞_n_=αn( –αn) =∞, then the sequence{xn}generated by Mann’s algorithm

con-verges weakly to a fixed point ofT []. Now, it is well known that Mann’s algorithm fails, in general, to converge strongly in the setting of infinite-dimensional Hilbert spaces []. Iterative methods for nonexpansive mappings have been investigated extensively in the literature; see [–] and the references therein. However, iterative methods for strictly pseudo-contractive mappings are far less developed than those for nonexpansive map-pings though Browder and Petryshyn [] initiated their work in . However, strictly pseudo-contractive mappings have more powerful applications than nonexpansive map-pings, for example, to solve inverse problems (see Scherzer []). Therefore it is interesting to develop the algorithms for finding the fixed points of strictly pseudo-contractive map-pings. Now, we know that Mann’s algorithm is not good enough for approximating fixed points of (even if Lipschitz continuous) pseudo-contractions. Thus, we have to find other type of iterative algorithms; see [–]. The first such an attempt was done by Ishikawa [] who introduced the following Ishikawa algorithm:

yn= ( –βn)xn+βnTxn,

xn+= ( –αn)xn+αnTyn,

n≥,

where{αn}and{βn}are sequences in the interval [, ],Tis a (nonlinear) self-mapping of C, and the initial guessx∈Cis selected arbitrarily. (Ishikawa’s algorithm can be viewed

as a double-step (or two-level) Mann’s algorithm.) Ishikawa proved that his algorithm con-verges in norm to a ﬁxed point of a Lipschitz pseudo-contractionTif{αn}and{βn}satisfy certain conditions and ifT is compact.

On the other hand, iterative methods for approximating the common fixed points of a finite (or an infinite) family of nonlinear mappings have been considered by many authors. For the related work, we refer the reader to [–, , ]. Above discussion suggests the following question.

Question . Could we construct an iterative algorithm such that it converges strongly to the ﬁxed points of a ﬁnite family of strict pseudo-contractions?

It is our purpose in this paper to construct redundant intermixed algorithms for two strict pseudo-contractions. It is shown that the suggested algorithms converge strongly to the fixed points of two strict pseudo-contractions, independently. As a special case, we can find the common fixed points of two strict pseudo-contractions in Hilbert spaces.

2 Preliminaries

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with the property:

x–PCx ≤ x–y, y∈C.

Note thatPCis characterized by the inequality:

PCx∈C, x–PCx,y–PCx ≤, y∈C.

Consequently,PCis nonexpansive.

In order to prove our main results, we need the following well-known lemmas.

Lemma .([]) Let C be a nonempty closed convex subset of a real Hilbert space H.Let T:C→C be aλ-strictly pseudo-contractive mapping.Then I–T is demi-closed at,i.e., if xnx∈C and xn–Txn→,then x=Tx.

Lemma .([]) Let{xn}and{yn}be bounded sequences in a Banach space E and{βn}

be a sequence in[, ]with <lim inf_n→∞βn≤lim supn→∞βn< .Suppose that xn+= ( –

βn)xn+βnznfor all n≥andlim supn→∞(zn+–zn–xn+–xn)≤.Thenlimn→∞zn–

xn= .

Lemma .([]) Assume{an}is a sequence of nonnegative real numbers such that an+≤

( –γn)an+γnδn,n≥where{γn}is a sequence in(, )and{δn}is a sequence in R such

that

(i) ∞_n_=γn=∞; (ii) lim sup_n→∞δn≤or

∞

n=|δnγn|<∞.

Thenlimn→∞an= .

3 Main results

LetCbe a nonempty closed convex subset of a real Hilbert spaceH. LetT:C→Cbe aλ-strict pseudo-contraction. Letf :C→H be a ρ-contraction andg:C→H be a

ρ-contraction. (A mappingf :C→His said to be contractive iff(x) –f(y) ≤ρx–y

for someρ∈[, ) and for allx,y∈C.) Letk∈(,  –λ) be a constant.

Now we propose the following redundant intermixed algorithm for two strict pseudo-contractionsSandT.

Algorithm . For arbitrarily givenx∈C,y∈C, let the sequences{xn} and{yn}be

generated iteratively by

xn+= ( –βn)xn+βnPC[αnf(yn) + ( –k–αn)xn+kTxn], n≥,

yn+= ( –βn)yn+βnPC[αng(xn) + ( –k–αn)yn+kSyn], n≥,

(.)

where{αn}and{βn}are two real number sequences in (, ).

Remark . Note that the deﬁnition of the sequence {xn} is involved in the sequence

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Theorem . Suppose thatFix(S)=∅andFix(T)=∅.Assume the following conditions are satisﬁed:

(C) lim_n→∞αn= and ∞

n=αn=∞; (C) βn∈[ξ,ξ]⊂(, )for alln≥.

Then the sequences{xn}and{yn}generated by(.)converge strongly to the ﬁxed points PFix(T)f(y∗)and PFix(S)g(x∗)of T and S,respectively,where x∗∈Fix(T)and y∗∈Fix(S).

Proof First, we give the following propositions.

Proposition . The sequences{xn}and{yn}are bounded.

In order to prove this proposition, we need the following result.

Proposition . The mapping PC[αf+ ( –k–α)I+kT]is contractive for small enoughα.

Proof Letx,y∈C. Then we have

PC

αf(x) + ( –k–α)x+kTx–PC

αf(y) + ( –k–α)y+kTy

≤αf(x) –f(y)+ ( –k–α)(x–y) +k(Tx–Ty)

=αf(x) –f(y)+ ( –α)  –k–α

 –α (x–y) +

k

 –α(Tx–Ty)



≤αf(x) –f(y)+ ( –α) –k–α

 –α (x–y) +

k

 –α(Tx–Ty)



≤αρx–y+

( –k–α)

 –α x–y

₊ k

 –αTx–Ty



+( –k–α)k

 –α Tx–Ty,x–y

≤αρx–y+

( –k–α)

 –α x–y

₊ k

 –α

x–y+λ(I–T)x– (I–T)y

+( –k–α)k

 –α x–y

_– –λ

 (I–T)x– (I–T)y



=αρx–y+

  –α

λk– ( –λ)( –k–α)k(I–T)x– (I–T)y + ( –α)x–y

= k

 –α

k– ( –α)( –λ)(I–T)x– (I–T)y+ – ( –ρ)α

x–y.

Thus, we get

PC

αf(x) + ( –k–α)x+kTx–PC

αf(y) + ( –k–α)y+kTy

≤  –( –ρ)α 

x–y

for allx,y∈Cask≤( –α)( –λ) (that is,α≤ –_–k_λ).

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Proof SinceFix(S)=∅andFix(T)=∅, we can choosex∗∈Fix(T) andy∗∈Fix(S). From (.), we have

xn+–x∗=( –βn)xn+βnPC

αnf(yn) + ( –k–αn)xn+kTxn

–x∗

≤βnPC

αnf(yn) + ( –k–αn)xn+kTxn

–x∗

+ ( –βn)xn–x∗

≤βnαnf(yn) –x∗+βn( –k–αn)

xn–x∗

+kTxn–Tx∗

+ ( –βn)xn–x∗ ≤βnαnf(yn) –f

y∗+βnαnf

y∗–x∗+ ( –βn)xn–x∗

+βn( –αn)xn–x∗ ≤ρβnαnyn–y∗+βnαnf

y∗–x∗+ ( –αnβn)xn–x∗ ≤ρβnαnyn–y∗+βnαnf

y∗–x∗+ ( –αnβn)xn–x∗,

whereρ=max{ρ,ρ}. Similarly, we have yn+–y∗≤ρβnαnxn–x∗+βnαng

x∗–y∗+ ( –αnβn)yn–y∗ ≤ρβnαnxn–x∗+βnαng

x∗–y∗+ ( –αnβn)yn–y∗.

Hence, we obtain

xn+–x∗+yn+–y∗

≤ – ( –ρ)αnβnxn–x∗+yn–y∗+αnβnf

y∗–x∗+gx∗–y∗

≤maxxn–x∗+yn–y∗,

f(y∗) –x∗+g(x∗) –y∗  –ρ

.

By induction, we have

xn–x∗+yn–y∗

≤maxx–x∗+y–y∗,

f(y∗) –x∗+g(x∗) –y∗  –α

.

So,{xn}and{yn}are bounded.

Proposition . xn–Txn →andyn–Syn →.

Proof We ﬁrst estimatexn+–xn. Setun=PC[αnf(yn) + ( –k–αn)xn+kTxn],n≥. It

follows that

un+–un ≤αn+f(yn+) + ( –k–αn+)xn++kTxn+

–αnf(yn) – ( –k–αn)xn+kTxn

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+αn+f(yn+)+xn

+αnf(yn)+xn

≤( –αn+)xn+–xn+αn+f(yn+)+xn

+αnf(yn)+xn

.

Sinceαn→, we deduce that

lim sup

n→∞

un+–un–xn+–xn

≤.

From Lemma ., we get

lim

n→∞un–xn=  and n→∞lim xn+–xn= .

From (.), we derive

xn+–Txn ≤( –βn)xn–Txn+βnαnf(yn) –Txn

+βn( –k–αn)xn–Txn

= – (k+αn)βn

xn–Txn+βnαnf(yn) –Txn.

Thus,

xn–Txn ≤ xn–xn++xn+–Txn ≤ – (k+αn)βn

xn–Txn+βnαnf(yn) –Txn

+xn–xn+.

It follows that

xn–Txn ≤

 (k+αn)βn

xn–xn++βnαnf(yn) –Txn →.

Similarly, we can obtain

lim

n→∞yn–Syn= .

By Proposition ., we know that the mappingPC[αf+ ( –k–α)I+kT] is contractive

for small enoughα. Thus, the equationx=PC[tf(x) + ( –k–t)x+kTx] has a unique ﬁxed

point, denoted byxt, that is,

xt=PC

tf(xt) + ( –k–t)xt+kTxt

(.)

for small enought. In order to prove Theorem ., we need the following lemma.

Lemma . SupposeFix(T)=∅.Then,as t→,the net{xt}deﬁned by(.)converges

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Proof Letx∗∈Fix(T). From (.), we have

_x_t–x∗=PC

tf(xt) + ( –k–t)xt+kTxt

–x∗

≤tf(xt) –x∗+( –k–t)

xt–x∗

+kTxt–x∗

≤tρxt–x∗+tf

x∗–x∗+ ( –t)xt–x∗,

hence,

xt–x∗≤

  –ρ

fx∗–x∗.

Thus,{xt}is bounded. Again, from (.), we get

xt–Txt ≤tf(xt) –Txt+ ( –k–t)xt–Txt.

It follows that

xt–Txt ≤

t

k+tf(xt) –Txt→.

Let {tn} ⊂(, ). Assume that tn→ asn→ ∞. Putxn:=xtn. We have limn→∞xn– Txn= . Setyt=tf(xt) + ( –k–t)xt+kTxt, for allt. Then we havext=PCyt, and for any

x∗∈Fix(T),

xt–x∗=xt–yt+yt–x∗

=xt–yt+t

f(xt) –x∗

+ ( –k–t)xt–x∗

+kTxt–x∗

.

From the property of the metric projection, we deduce

xt–yt,xt–x∗

≤.

So,

xt–x∗ 

=xt–yt,xt–x∗

+( –k–t)xt–x∗

+kTxt–x∗

,xt–x∗

+tf(xt) –x∗,xt–x∗

≤( –k–t)xt–x∗

+kTxt–x∗xt–x∗

+tf(xt) –f

x∗,xt–x∗

+tfx∗–x∗,xt–x∗

≤ – ( –ρ)txt–x∗ 

+tfx∗–x∗,xt–x∗

.

Hence,

xt–x∗ 

≤ 

( –ρ)

fx∗–x∗,xt–x∗

, ∀x∗∈Fix(T).

By similar arguments to [], we ﬁnd that the net{xt}converges strongly tox∗∈Fix(T).

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Remark . From Lemma ., we know that the net{xt}deﬁned byxt=PC[tu+ ( –k–

t)xt+kTxt] whereu∈H, converges toPFix(T)u. Letx∗∈Fix(T) andy∗∈Fix(S). If we take

u=f(y∗), then the net{xt}deﬁned byxt=PC[tf(y∗) + ( –k–t)xt+kTxt], converges to

PFix(T)f(y∗).

Finally, we prove thatxn→PFix(T)f(y∗) andyn→PFix(S)g(x∗), wherex∗∈Fix(T) andy∗∈

Fix(S). We note the following fact. If the sequence{wn}is bounded andwn–Twn →,

we easily deduce that

lim sup

n→∞

fPFix(S)g

x∗–PFix(T)f

y∗,wn–PFix(T)f

y∗≤.

Setvn=PC[αng(xn) + ( –k–αn)yn+kSyn] for alln≥. Thus, we deduce that the sequences {un}and{vn}satisfy: (){un}and{vn}are bounded; ()un–Tun → andvn–Svn →.

Therefore,

lim sup

n→∞

fPFix(S)g

x∗–PFix(T)f

y∗,un–PFix(T)f

y∗≤

and

lim sup

n→∞

gPFix(T)f

y∗–PFix(S)g

x∗,vn–PFix(S)g

x∗≤.

Next, we estimate un–PFix(T)f(y∗). Set u˜n=αnf(yn) + ( –k–αn)xn+kTxn andv˜n=

αng(xn) + ( –k–αn)yn+kSynfor alln. We have _u_n_–_P_Fix₍_T₎_f

y∗

=PC[u˜n] –PFix(T)f

y∗

≤u˜n–PFix(T)f

y∗,un–PFix(T)f

y∗

=αnf(yn) + ( –k–αn)xn+kTxn–PFix(T)f

y∗,un–PFix(T)f

y∗

≤αn

f(yn) –PFix(T)f

y∗,un–PFix(T)f

y∗

+ ( –αn)xn–PFix(T)f

y∗un–PFix(T)f

y∗

≤ –αn

 xn–PFix(T)f

y∗+

un–PFix(T)f

y∗

+αn

f(yn) –f

PFix(S)g

x∗,un–PFix(T)f

y∗

+αn

fPFix(S)g

x∗–PFix(T)f

y∗,un–PFix(T)f

y∗

≤ –αn

 xn–PFix(T)f

y∗+

un–PFix(T)f

y∗

+αnρyn–PFix(S)g

x∗un–PFix(T)f

y∗

+αn

fPFix(S)g

x∗–PFix(T)f

y∗,un–PFix(T)f

y∗

≤ –αn

 xn–PFix(T)f

y∗+

un–PFix(T)f

y∗

+αnρ

 yn–PFix(S)g

x∗+un–PFix(T)f

y∗

+αn

fPFix(S)g

x∗–PFix(T)f

y∗,un–PFix(T)f

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It follows that

un–PFix(T)f

y∗

≤  –αn

 –αnρ

xn–PFix(T)f

y∗+ αnρ  –αnρ

yn–PFix(S)g

x∗

+ αn  –αnρ

fPFix(S)g

x∗–PFix(T)f

y∗,un–PFix(T)f

y∗.

Thus,

xn+–PFix(T)f

y∗

≤( –βn)xn–PFix(T)f

y∗+βnun–PFix(T)f

y∗

≤

 –  –ρ  –αnρ

αnβn

xn–PFix(T)f

y∗+ αnβnρ  –αnρ

yn–PFix(S)g

x∗

+ αnβn  –αnρ

fPFix(S)g

x∗–PFix(T)f

y∗,un–PFix(T)f

y∗.

Similarly, we also have

yn+–PFix(S)g

x∗

≤

 –  –ρ  –αnρ

αnβn

yn–PFix(S)g

x∗+ αnβnρ  –αnρ

xn–PFix(T)f

y∗

+ αnβn  –αnρ

gPFix(T)f

y∗–PFix(S)g

x∗,vn–PFix(S)g

x∗.

Therefore,

xn+–PFix(T)f

y∗+yn+–PFix(S)g

x∗

≤

 –  – ρ  –αnρ

αnβn

xn–PFix(T)f

y∗+yn–PFix(S)g

x∗

+ αnβn  –αnρ

fPFix(S)g

x∗–PFix(T)f

y∗,un–PFix(T)f

y∗

+ αnβn  –αnρ

gPFix(T)f

y∗–PFix(S)g

x∗,vn–PFix(S)g

x∗.

We can check that all assumptions of Lemma . are satisﬁed. Therefore, xn →

PFix(T)f(y∗) andyn→PFix(S)g(x∗). This completes the proof.

Algorithm . For arbitrarily givenx∈C, let the sequence{xn}be generated iteratively

by

xn+= ( –βn)xn+βnPC

( –k–αn)xn+kTxn

, n≥, (.)

where{αn}and{βn}are two real number sequences in (, ).

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(C) lim_n→∞αn= and ∞

n=αn=∞; (C) βn∈[ξ,ξ]⊂(, )for alln≥.

Then the sequence{xn}generated by(.)converge strongly to the ﬁxed points PFix(T)(),

which is the minimum norm element inFix(T).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved the ﬁnal manuscript.

Author details

1_{School of Information Engineering, Nanjing Xiaozhuang University, Nanjing, 211171, China.}2_{Department of}

Mathematics and the RINS, Gyeongsang National University, Jinju, 660-701, Korea. 3_{Department of Mathematics, Tianjin} Polytechnic University, Tianjin, 300387, China.

Acknowledgements

The authors are grateful to the three reviewers for their valuable comments and suggestions. Zhangsong Yao was supported by the Scientiﬁc Research Project of Nanjing Xiaozhuang University (2015NXY46).

Received: 29 April 2015 Accepted: 4 November 2015 References

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