Data dependence results of new multi-step and S-iterative schemes for contractive-like operators

R E S E A R C H

Open Access

Data dependence results of new multi-step

and S-iterative schemes for contractive-like

operators

Faik Gürsoy

_{, Vatan Karakaya}

_{and Billy E Rhoades}

*_{Correspondence:}

[email protected]; [email protected]

2_{Department of Mathematical}

Engineering, Faculty of Chemical and Metallurgical Engineering, Yildiz Technical University, Davutpasa Campus, Esenler, Istanbul, 34210, Turkey

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

Abstract

In this paper, we prove that the convergence of a new iteration and S-iteration can be used to approximate the fixed points of contractive-like operators. We also prove some data dependence results for these new iteration and S-iteration schemes for contractive-like operators. Our results extend and improve some known results in the literature.

MSC: Primary 47H10

Keywords: new multi-step iteration; S-iteration; data dependence; contractive-like operator

1 Introduction

Contractive mappings and iteration procedures are some of the main tools in the study of fixed point theory. There are many contractive mappings and iteration schemes that have been introduced and developed by several authors to serve various purposes in the literature of this highly active research area,viz., [–] among others.

Whether an iteration method used in any investigation converges to a fixed point of a contractive type mapping corresponding to a particular iteration process is of utmost im-portance. Therefore it is natural to see many works related to the convergence of iteration methods such as [–].

Fixed point theory is concerned with investigating a wide variety of issues such as the existence (and uniqueness) of fixed points, the construction of fixed points,etc.One of these themes is data dependency of fixed points. Data dependency of fixed points has been the subject of research in fixed point theory for some time now, and data dependence research is an important theme in its own right.

Several authors who have made contributions to the study of data dependence of fixed points are Rus and Muresan [], Ruset al.[, ], Berinde [], Espínola and Petruşel [], Markin [], Chifu and Petruşel [], Olantiwo [, ], Şoltuz [, ], Şoltuz and Grosan [], Chugh and Kumar [] and the references therein.

points of contractive-like operators. Motivated by the works of Şoltuz [, ], Şoltuz and Grosan [], and Chugh and Kumar [], we prove two data dependence results for the new multi-step iteration and S-iteration schemes by employing contractive-like operators. As a background of our exposition, we now mention some contractive mappings and iteration schemes.

In [] Zamfirescu established an important generalization of the Banach fixed point theorem using the following contractive condition. For a mappingT:E→E, there exist real numbersa,b,csatisfying  <a< ,  <b,c< / such that, for each pairx,y∈X, at least one of the following is true:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

(z) Tx–Ty ≤ax–y,

(z) Tx–Ty ≤b(x–Tx+y–Ty),

(z) Tx–Ty ≤c(x–Ty+y–Tx).

(.)

A mappingT satisfying the contractive conditions (z), (z) and (z) in (.) is called a

Zamfirescu operator. An operator satisfying condition (z) is called aKannan operator,

while the mapping satisfying condition (z) is called aChatterjea operator. As shown in

[], the contractive condition (.) leads to ⎧

⎨ ⎩

(b) Tx–Ty ≤δx–y+ δx–Txif one uses (z), and

(b) Tx–Ty ≤δx–y+ δx–Tyif one uses (z),

(.)

for allx,y∈E, whereδ:=max{a, b

–b, c

–c},δ∈[, ), and it was shown that this class of op-erators is wider than the class of Zamfirescu opop-erators. Any mapping satisfying condition (b) or (b) is called a quasi-contractive operator.

Extending the above definition, Osilike and Udomene [] considered operatorsT for which there exist real numbersL≥ andδ∈[, ) such that for allx,y∈E,

Tx–Ty ≤δx–y+Lx–Tx. (.)

Imoru and Olantiwo [] gave a more general definition: An operator T is called a contractive-like operator if there exists a constantδ∈[, ) and a strictly increasing and continuous functionϕ: [,∞)→[,∞), withϕ() = , such that for eachx,y∈E,

Tx–Ty ≤δx–y+ϕx–Tx. (.)

A map satisfying (.) need not have a fixed point, even ifEis complete. For example, let

E= [,∞) and defineTby

Tx= ⎧ ⎨ ⎩

., ≤x≤., ., . <x.

If ≤x≤. <y, thenTx–Ty= ..

Defineϕbyϕ(t) =Ltfor anyL≥. Thenϕis increasing, continuous, andϕ() = . Also,

x–Tx=  –xso thatϕ(x–Tx) =L( –x)≥.L≥.. Therefore

. =Tx–Ty ≤Lx–Tx ≤δx–y+Lx–Tx

for any ≤δ< , and (.) is satisfied for ≤x≤. <y. ButThas no fixed point. However, using (.) it is obvious that ifThas a fixed point, then it is unique.

From now on, we demand thatNdenotes the set of all nonnegative integers. LetXbe a Banach space, letE⊂Xbe a nonempty closed, convex subset ofX, and letT be a self-map on E. DefineFT:={p∈X:p=Tp}to be the set of fixed points ofT. Let{αn}∞n=, {βn}∞n=,{γn}∞n=and{βni}∞n=,i= ,k– ,k≥ be real sequences in [, ) satisfying certain

conditions.

In [] Rhoades and Şoltuz introduced a multi-step iterative procedure given by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈E,

yk–

n = ( –βnk–)xn+βnk–Txn, k≥,

yi

n= ( –βni)xn+βniTyin+, i= ,k– ,

xn+= ( –αn)xn+αnTyn, n∈N.

(.)

The sequence{xn}∞n=defined by

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x∈E,

xn+= ( –αn)Txn+αnTyn,

yn= ( –βn)xn+βnTxn, n∈N,

(.)

is known as the S-iteration process (see [, , ]). Thianwan [] defined a two-step iteration{xn}∞n=by

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x∈E,

xn+= ( –αn)yn+αnTyn,

yn= ( –βn)xn+βnTxn, n∈N.

(.)

Recently Phuengrattana and Suantai [] introduced an SP iteration method defined by

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈E,

xn+= ( –αn)yn+αnTyn,

yn= ( –βn)zn+βnTzn,

zn= ( –γn)xn+γnTxn, n∈N.

We shall employ the following iterative process. For an arbitrary fixed orderk≥, ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈E,

xn+= ( –αn)yn+αnTyn,

y

n= ( –βn)yn+βnTyn,

y

n= ( –βn)yn+βnTyn,

· · ·,

yk–

n = ( –βnk–)ynk–+βnk–Tykn–,

yk–

n = ( –βnk–)xn+βnk–Txn, n∈N,

(.)

or, in short, ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈E,

xn+= ( –αn)yn+αnTyn,

yi

n= ( –βni)yin++βniTyin+, i= ,k– ,

yk_n–= ( –β_nk–)xn+βnk–Txn, n∈N,

(.)

where

{αn}∞n=⊂[, ), ∞

n=

αn=∞, (.)

and

β_ni∞_n=⊂[, ), i= ,k– . (.)

Remark  If eachγn= , then SP iteration (.) reduces to two-step iteration (.). By takingk=  andk=  in (.), we obtain iterations (.) and (.), respectively.

We shall need the following definition and lemma in the sequel.

Definition [] LetT,T :X→Xbe two operators. We say thatT is an approximate

operator forTif, for someε> , we have

Tx–Tx ≤ε

for allx∈X.

Lemma [] Let{an}∞n= be a nonnegative sequence for which one assumes that there

exists an n∈Nsuch that for all n≥n,

an+≤( –μn)an+μnηn

is satisfied,whereμn∈(, )for all n∈N, ∞

n=μn=∞andηn≥,∀n∈N.Then the

following holds: ≤ lim

2 Main results

For simplicity we use the following notation throughout this section.

For any iterative process,{xn}∞n=and{un}∞n=denote iterative sequences associated toT

andT, respectively.

Theorem  Let T:E→E be a map satisfying(.)with FT=∅,and let{xn}∞n=be a

se-quence defined by(.),then the sequence{xn}∞n=converges to the unique fixed point of T.

Proof The proof can be easily obtained by using the argument in the proof of ([],

The-orem .).

This result allows us to give the next theorem.

Theorem  Let T :E→E be a map satisfying(.) with FT=∅, and letT be an

ap-proximate operator of T as in Definition.Let{xn}∞n=,{un}∞n=be two iterative sequences

defined by(.)with real sequences{αn}∞n=,{βni}∞n=⊂[, )satisfying(i) ≤βni <αn≤,

i= ,k– , (ii)αn=∞.If p=Tp and q=Tq,then we have

p–q ≤ kε

 –δ.

Proof For givenx∈Eandu∈E, we consider the following multi-step iteration forT

andT: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

x∈E,

xn+= ( –αn)yn+αnTyn,

yi

n= ( –βni)yin++βniTyin+, i= ,k– ,

yk–

n = ( –βnk–)xn+βnk–Txn, k≥,n∈N,

(.)

and ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

u∈E,

un+= ( –αn)vn+αnTvn,

vi

n= ( –βni)vin++βniTvin+, i= ,k– ,

vk–

n = ( –βnk–)un+βnk–Tun, k≥,n∈N.

(.)

Thus, from (.), (.) and (.), we have the following inequalities.

xn+–un+=( –αn)

y_n–v_n+αn

Ty_n–Tv_n

≤( –αn)y_n–v_n+αnTyn–Tvn

= ( –αn)y_n–v_n+αnTyn–Tvn+Tvn–Tvn

≤( –αn)y_n–v_n+αnTyn–Tvn+αnTvn–Tvn

≤( –αn)yn–vn+αnδyn–vn+αnϕyn–Tyn+αnε

y_n–v_n= –β_ny_n–v_n+β_nTy_n–Tv_n

≤ –β_ny_n–v_n+β_nTy_n–Tv_n

≤ –β_ny_n–v_n+β_nTy_n–Tv_n+β_nTv_n–Tv_n

≤ –β_ny_n–v_n+β_nδy_n–v_n+β_nϕy_n–Ty_n+β_nε

= –β_n( –δ)y_n–v_n+β_nϕy_n–Ty_n+β_nε, (.) _y

n–vn= –βn

y_n–v_n+β_nTy_n–Tv_n

≤ –β_ny_n–v_n+β_nTy_n–Tv_n

≤ –β_ny_n–v_n+β_nTy_n–Tv_n+β_nTv_n–Tv_n

≤ –β_ny_n–v_n+β_nδy_n–v_n+β_nϕy_n–Ty_n+β_nε

= –β_n( –δ)y_n–v_n+β_nϕy_n–Ty_n+β_nε. (.)

Combining (.), (.) and (.), we obtain

xn+–un+ ≤

 –αn( –δ)

 –β_n( –δ) –β_n( –δ)y_n–v_n

+ –αn( –δ) –β_n( –δ)β_nϕy_n–Ty_n

+ –αn( –δ) –β_n( –δ)β_nε

+ –αn( –δ)β_nϕy_n–Ty_n

+ –αn( –δ)

β_nε+αnϕyn–Tyn+αnε. (.)

Thus, by induction, we get

xn+–un+ ≤

 –αn( –δ)

 –β_n( –δ)· · · –β_nk–( –δ)yk_n––vk_n–

+ –αn( –δ)

 –β_n( –δ)· · · –β_nk–( –δ)β_nk–ϕy_nk––Tyk_n–

+· · ·+ –αn( –δ)β_nϕy_n–Ty_n+αnϕyn–Tyn + –αn( –δ) –β_n( –δ)· · · –β_nk–( –δ)β_nk–ε

+· · ·+ –αn( –δ)β_nε+αnε. (.)

Again, using (.), (.) and (.), we get

yk_n––v_nk–= –β_nk–(xn–un) +βnk–(Txn–Tun)

≤ –β_nk–xn–un+βnk–Txn–Tun

≤ –β_nk–xn–un+βnk–Txn–Tun+βnk–Tun–Tun

≤ –β_nk–( –δ)xn–un+βnk–ϕ

xn–Txn

Substituting (.) in (.), we have

xn+–un+ ≤

 –αn( –δ)

 –β_n( –δ)· · · –β_nk–( –δ)xn–un + –αn( –δ)

 –β_n( –δ)· · · –β_nk–( –δ)β_nk–ϕxn–Txn

+ –αn( –δ)

 –β_n( –δ)· · · –β_nk–( –δ)β_nk–ϕy_nk––Tyk_n–

+· · ·+ –αn( –δ)

β_nϕy_n–Ty_n+αnϕyn–Tyn + –αn( –δ)

 –β_n( –δ)· · · –β_nk–( –δ)β_nk–ε

+· · ·+ –αn( –δ)β_nε+αnε. (.)

Sinceδ∈[, ) and{αn}∞n=,{βni}∞n=⊂[, ), fori= ,k– , we have

 –αn( –δ)

 –β_n( –δ)· · · –β_ni( –δ)≤ –αn( –δ)

. (.)

From inequality (.) and assumption (i) in (.), it follows

xn+–un+ ≤

 –αn( –δ)xn–un +αnϕ

xn–Txn

+αnϕykn––Tykn– +· · ·+αnϕyn–Tyn+αnϕyn–Tyn +αnε+αnε+· · ·+αnε+αnε

= –αn( –δ)xn–un +αn( –δ)

ϕ(xn–Txn) +ϕ(ykn––Tykn–)  –δ

+· · ·+ϕ(y



n–Tyn) +kε  –δ

. (.)

Define

an:=xn–un,

μn:=αn( –δ)∈(.),

ηn:=

ϕ(xn–Txn) +ϕ(ykn––Tykn–)  –δ +· · ·+

ϕ(y

n–Tyn) +kε  –δ

.

From Theorem  it follows thatlimn→∞xn–p= . SinceT satisfies condition (.) and

Tp=p∈FT,

≤ xn–Txn

≤ xn–p+δp–xn+ϕ

p–Tp

= ( +δ)xn–p → asn→ ∞. (.)

Sinceβ_ni∈[, ),∀n∈N,i= ,k– , using (.) and (.), we have

≤y_n–Ty_n=y_n–p+p–Ty_n

≤y_n–p+Tp–Ty_n

≤y_n–p+δp–y_n+ϕp–Tp

= ( +δ)y_n–p

= ( +δ) –β_ny_n+β_nTy_n–p –β_n+β_n ≤( +δ)  –β_ny_n–p+β_nTy_n–Tp

≤( +δ)  –β_ny_n–p+β_nδy_n–p

= ( +δ) –β_n( –δ)y_n–p

≤ · · ·

≤( +δ) –β_n( –δ)· · · –β_nk–( –δ)yk_n––p

≤( +δ) –β_n( –δ)· · · –β_nk–( –δ)xn–p

≤( +δ)xn–p → asn→ ∞. (.)

It is easy to see from (.) that this result is also valid forTy

n–yn, . . . ,Tynk––ykn–. Sinceϕis continuous, we have

lim

n→∞ϕ

xn–Txn

= lim

n→∞ϕy



n–Tyn=· · ·=_nlim_→∞ϕykn––Tykn–= . (.) Hence an application of Lemma  to (.) leads to

p–q ≤ kε

 –δ. (.)

As shown by Hussainet al.([], Theorem ), in an arbitrary Banach spaceX, the S-iteration{xn}∞n= given by (.) converges to the fixed point ofT, whereT :E→E is a

mapping satisfying condition (.).

Theorem  Let T:E→E be a map satisfying(.)with FT =∅,and let{xn}∞n=be

de-fined by(.)with real sequences{βn}∞n=,{αn}∞n=⊂[, )satisfying

∞

n=αn=∞.Then the

sequence{xn}∞n=converges to the unique fixed point of T.

Proof The argument is similar to the proof of Theorem  of [], and is thus omitted.

Theorem  Let T,T be two operators as in Theorem.Let{xn}∞n=,{un}∞n=be S-iterations

defined by(.)with real sequences{βn}∞n=,{αn}n∞=⊂[, )satisfying(i)  ≤αn,∀n∈N,

and(ii)∞_n=αn=∞.If p=Tp and q=Tq,then we have

p–q ≤ ε

 –δ.

Proof For a givenx∈Eandu∈E, we consider the following iteration forT andT:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

x∈E,

xn+= ( –αn)Txn+αnTyn,

yn= ( –βn)xn+βnTxn, n∈N

(.)

and ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

u∈E,

un+= ( –αn)Tun+αnTvn,

vn= ( –βn)un+βnTun, n∈N.

(.)

Using (.), (.) and (.), we obtain the following estimates:

xn+–un+=( –αn)(Txn–Tun) +αn(Tyn–Tvn)

≤( –αn)Txn–Tun+αnTyn–Tvn

= ( –αn)Txn–Tun+Tun–Tun+αnTyn–Tvn+Tvn–Tvn

≤( –αn) Txn–Tun+Tun–Tun

+αn Tyn–Tvn+Tvn–Tvn

≤( –αn) δxn–un+ϕ

xn–Txn

+ε

+αn δyn–vn+ϕ

yn–Tyn

+ε, (.)

yn–vn=( –βn)(xn–un) +βn(Txn–Tun)

≤( –βn)xn–un+βnTxn–Tun

= ( –βn)xn–un+βnTxn–Tun+Tun–Tun

≤( –βn)xn–un+βn Txn–Tun+Tun–Tun

≤( –βn)xn–un+βn δxn–un+ϕ

xn–Txn

+ε

= –βn( –δ)xn–un+βnϕ

xn–Txn

+βnε. (.)

Combining (.) and (.), we get

xn+–un+ ≤ ( –αn)δ+αnδ

 –βn( –δ)

xn–un +{ –αn+αnδβn}ϕ

xn–Txn

+αnϕ

yn–Tyn

For{αn}∞n=,{βn}∞n=⊂[, ) andδ∈[, ),

( –αn)δ<  –αn,  –βn( –δ) < , αnδβn<αn. (.)

It follows from assumption (i) that

 –αn<αn, ∀n∈N. (.)

Therefore, combining (.) and (.) to (.) gives

xn+–un+ ≤

 –αn( –δ)xn–un + αnϕ

xn–Txn

+αnϕ

yn–Tyn

+αnε+αnε+αnε, (.)

or, equivalently,

xn+–un+ ≤

 –αn( –δ)xn–un

+αn( –δ){ϕ(xn–Txn) +ϕ(yn–Tyn) + ε}

 –δ . (.)

Now define

an:=xn–un,

μn:=αn( –δ)∈(, ),

ηn:=

ϕ(xn–Txn) +ϕ(yn–Tyn) + ε

 –δ .

From Theorem , we havelim_n→∞xn–p= . SinceTsatisfies condition (.), andTp=

p∈FT, using an argument similar to that in the proof of Theorem ,

lim

n→∞xn–Txn=nlim→∞yn–Tyn= . (.)

Using the fact thatϕis continuous, we have

lim

n→∞ϕ

xn–Txn

= lim

n→∞ϕ

yn–Tyn

= . (.)

An application of Lemma  to (.) leads to

p–q ≤ ε

 –δ. (.)

3 Conclusion

Since the iterative schemes (.) and (.) are special cases of the iterative process (.), Theorem  generalizes Theorem . of [] and Theorem . of []. By takingk=  and

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics, Faculty of Science and Letters, Yildiz Technical University, Davutpasa Campus, Esenler,}

Istanbul, 34220, Turkey.2_{Department of Mathematical Engineering, Faculty of Chemical and Metallurgical Engineering,}

Yildiz Technical University, Davutpasa Campus, Esenler, Istanbul, 34210, Turkey.3_{Department of Mathematics, Indiana}

University, Bloomington, IN 47405-7106, USA.

Acknowledgements

The first two authors would like to thank Yıldız Technical University Scientific Research Projects Coordination Unit under project number BAPK 2012-07-03-DOP02 for financial support during the preparation of this manuscript.

Received: 5 December 2012 Accepted: 10 March 2013 Published: 28 March 2013 References

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doi:10.1186/1687-1812-2013-76