On -Stability of Picard Iteration in Cone Metric Spaces

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Volume 2009, Article ID 751090,6pages doi:10.1155/2009/751090

Research Article

On

T

-Stability of Picard Iteration in

Cone Metric Spaces

M. Asadi,

1

_{H. Soleimani,}

1

_{S. M. Vaezpour,}

2, 3

_{and B. E. Rhoades}

4

1_{Department of Mathematics, Science and Research Branch, Islamic Azad University (IAU),}

Tehran 14778 93855, Iran

2_{Department of Mathematics, Amirkabir University of Technology, Tehran 15916 34311, Iran} 3_{Department of Mathematics, Newcastle University, Newcastle, NSW 2308, Australia} 4_{Department of Mathematics, Indiana University, Bloomington, IN 46205, USA}

Correspondence should be addressed to S. M. Vaezpour,vaez@aut.ac.ir

Received 28 March 2009; Revised 28 September 2009; Accepted 19 October 2009

Recommended by Brailey Sims

The aim of this work is to investigate theT-stability of Picard’s iteration procedures in cone metric spaces and give an application.

Copyrightq2009 M. Asadi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction and Preliminary

LetEbe a real Banach space. A nonempty convex closed subsetP ⊂Eis called a cone inEif it satisfies the following:

iPis closed, nonempty, andP /{0},

iia, b∈R, a, b≥0, andx, y∈Pimply thataxby∈P,

iiix∈Pand−x∈P imply thatx0.

The spaceEcan be partially ordered by the coneP ⊂ E; by defining,x ≤ y if and only if

y−x∈P. Also, we writexyify−x∈intP, where intPdenotes the interior ofP. A conePis called normal if there exists a constantK >0 such that 0≤x≤yimplies

x ≤Ky.

In the following we always suppose thatEis a real Banach space,Pis a cone inE, and

≤is the partial ordering with respect toP.

Definition 1.1see1. LetXbe a nonempty set. Assume that the mappingd:X×X → E

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i0≤dx, yfor allx, y∈Xanddx, y 0 if and only ifxy,

iidx, y dy, xfor allx, y∈X,

iiidx, y≤dx, z dz, yfor allx, y, z∈X.

Thendis called a cone metric onX, andX, dis called a cone metric space.

Definition 1.2. LetT :X → X be a map for which there exist real numbersa, b, csatisfying 0 < a < 1,0 < b <1/2,0 < c < 1/2. ThenT is called a Zamfirescu operatorif, for each pair

x, y∈X,Tsatisfies at least one of the following conditions:

1dTx, Ty≤adx, y,

2dTx, Ty≤bdx, Tx dy, Ty,

3dTx, Ty≤cdx, Ty dy, Tx.

Every Zamfirescu operatorTsatisfies the inequality:

dTx, Ty≤δdx, y2δdx, Tx 1.1

for allx, y∈X, whereδmax{a, b/1−b, c/1−c}, with 0< δ <1. For normed spaces see

2.

Lemma 1.3 see 3. Let {an} and {bn} be nonnegative real sequences satisfying the following

inequality:

an1≤1−λnanbn, 1.2

whereλn∈0,1,for alln≥n0,

_∞

n1λn∞,andbn/λn → 0asn → ∞.Thenlimn→ ∞an0.

Remark 1.4. Let {an} and {bn} be nonnegative real sequences satisfying the following

inequality:

an1≤λan−mbn, 1.3

whereλ∈0,1,for alln≥n0and for some positive integer numberm. Ifbn → 0 asn → ∞.

Then limn→ ∞an 0.

Lemma 1.5. Let P be a normal cone with constant K, and let{an} and {bn} be sequences in E

satisfying the following inequality:

an1≤hanbn, 1.4

whereh∈0,1andbn → 0asn → ∞.Thenlimn→ ∞an0.

Proof. Letmbe a positive integer such thathm_{K <}₁_._{By recursion we have}

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so

an1 ≤Kbnhbn−1· · ·hmbn−mhm1Kan−m, 1.6

and then byRemark 1.4 an → 0.Thereforean → 0.

2. T

-Stability in Cone Metric Spaces

LetX, dbe a cone metric space, andT a self-map ofX. Letx0be a point ofX, and assume

thatxn1 fT, xnis an iteration procedure, involvingT, which yields a sequence{xn}of

points fromX.

Definition 2.1see 4. The iteration procedurexn1 fT, xnis said to be T-stable with

respect toT if{xn}converges to a fixed pointqofT and whenever{yn}is a sequence inX

with limn→ ∞dyn1, fT, yn 0 we have limn→ ∞ynq.

In practice, such a sequence{yn}could arise in the following way. Letx0be a point in X. Setxn1 fT, xn. Lety0 x0. Nowx1 fT, x0. Because of rounding or discretization

in the functionT, a new valuey1approximately equal tox1might be obtained instead of the

true value offT, x0. Then to approximatey2, the value fT, y1is computed to yieldy2,

an approximation offT, y1. This computation is continued to obtain{yn}an approximate

sequence of{xn}.

One of the most popular iteration procedures for approximating a fixed point ofT is Picard’s iteration defined byxn1Txn. If the conditions ofDefinition 2.1hold forxn1Txn,

then we will say that Picard’s iteration isT-stable.

Recently Qing and Rhoades 5 established a result for the T-stability of Picard’s iteration in metric spaces. Here we are going to generalize their result to cone metric spaces and present an application.

Theorem 2.2. LetX, dbe cone metric space,Pa normal cone, andT :X → X withFT/∅.If there exist numbersa≥0and0≤b <1,such that

dTx, q≤adx, Tx bdx, q 2.1

for eachx ∈ X, q ∈ FTand in addition, whenever {yn} is a sequence withdyn, Tyn → 0as

n → ∞, then Picard’s iteration isT-stable.

Proof. Suppose{yn} ⊆X, cndyn1, Tynandcn → 0.We shall show thatyn → q.Since

dyn1, q

≤dyn1, Tyn

dTyn, q

≤cnad

yn, Tyn

bdyn, q

, 2.2

if we putan:dTyn, qandbn:cnadyn, TyninLemma 1.5, then we haveyn → q.

Note that the fixed pointqofTis unique. Because ifpis another fixed point ofT, then

dp, qdTp, q≤adp, Tpbdp, qbdp, q, 2.3

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Corollary 2.3. LetX, dbe a cone metric space,P a normal cone, andT :X → Xwithq∈FT.

If there exists a numberλ∈0,1,such thatdTx, Ty≤λdx, y,for eachx, y∈X,then Picard’s iteration isT-stable.

Corollary 2.4. LetX, dbe a cone metric space,P a normal cone, andT :X → Xis a Zamfirescu operator withFT/∅ and whenever{yn}is a sequence withdyn, Tyn → 0as n → ∞, then

Picard’s iteration isT-stable.

Definition 2.5 see6. LetX, d be a cone metric space. A map T : X → X is called a quasicontraction if for some constant λ ∈ 0,1 and for every x, y ∈ X, there exists u ∈ CT;x, y≡ {dx, y, dx, Tx, dy, Ty, dx, Ty, dy, Tx},such thatdTx, Ty≤λu.

Lemma 2.6. IfT is a quasicontraction with 0 < λ < 1/2, thenT is a Zamfirescu operator and so satisfies2.1.

Proof. Letλ∈0,1/2for everyx, y∈Xwe havedTx, Ty≤λufor someu∈CT;x, y.In the case thatudx, Tywe have

dTx, Ty≤λdx, Ty≤λdx, Tx λdTx, Ty. 2.4

So

dTx, Ty≤ λ

1−λdx, Tx≤2 λ

1−λdx, Tx λ

1−λd

x, y. 2.5

Put δ : λ/1−λ so 0 < δ < 1. The other cases are similarly proved. Therefore T is a Zamfirescu operator.

Theorem 2.7. LetX, dbe a nonempty complete cone metric space,P be a normal cone, andT a quasicontraction and self map ofXwith some0< λ <1/2.Then Picard’s iteration isT-stable.

Proof. By 6, Theorem 2.1,T has a unique fixed point q ∈ X. AlsoT satisfies2.1. So by

Theorem 2.2it is enough to show thatdyn, Tyn → 0.We have

dyn, Tyn

≤dyn, Tyn−1

dTyn−1, Tyn

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Putbn :dyn, Tyn, cn :dyn1, Tynanddn:dTyn−1, Tyn.Thereforecn → 0 asn → ∞

and

bn≤cn−1dn≤cn−1λun, 2.7

where

un∈C

T, yn−1, yn

dyn−1, yn

, dyn−1, Tyn−1

, dyn, Tyn

, dyn−1, Tyn

, dyn, Tyn−1

.

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Hence we haveun bn orun ≤ sbn−1lcn−1 wheres 0,1 or 1/1−λandl 1 or 1λ.

Therefore by2.7,bn ≤ λl1cn−1 λsbn−1 by 0 ≤ λs < 1.Now byLemma 1.5we have bn → 0.

3. An Application

Theorem 3.1. LetX : C0,1,Rwithf_∞:sup₀_≤_x_≤₁|fx|forf∈Xand letTbe a self map ofXdefined byTfx ₀1Fx, ftdtwhere

aF:0,1×R → Ris a continuous function,

bthe partial derivativeFyofFwith respect toyexists and|Fyx, y| ≤Lfor someL∈0,1, cfor every real number0≤a <1one hasax≤Fx, ayfor everyx, y∈0,1.

LetP :{x, y∈R2_|_{x, y}_≥₀_}_{be a normal cone and}_{X, d}_{the complete cone metric space defined} bydf, g f−g_∞, αf−g_∞whereα≥0.Then,

iPicard’s iteration isT-stable if0≤L <1/2,

iiPicard’s iteration fails to beT-stable if1/2≤L <1and1₀Fx, tdt /x.

Proof. iWe haveT being a continuous quasicontraction map with 0 ≤ λ: L <1/2; so by

Theorem 2.7, Picard’s iteration isT-stable.

ii Put ynx : nx/n1 so yn ∈ X and dyn, h → 0, where hx x.Also

dyn1, Tyn → 0,since

yn1−Tyn_∞ sup

0≤x≤1

n1

n2x− ₁

0 F

x, nt n1

dt

≤ sup

0≤x≤1

n_n1₂x− nx n1

−→0,

3.1

asn → ∞.Butyn → hand his not a fixed point forT.Therefore Picard’s iteration is not

T-stable.

Example 3.2. LetF1x, y:xy/4 andF2x, y:xy/2.ThereforeF1andF2satisfy the

hypothesis ofTheorem 3.1whereF1has propertyiandF2has propertyii. So the self maps T1, T2ofXdefined byT1fx x 1/4

₁

0ftdtandT2fx x 1/2

₁

0ftdthave unique

fixed points but Picard’s iteration isT-stable forT1but notT-stable forT2.

References

1 L.-G. Huang and X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings,”

Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1468–1476, 2007.

2 X. Zhiqun, “Remarks of equivalence among Picard, Mann, and Ishikawa iterations in normed spaces,”

Fixed Point Theory and Applications, vol. 2007, Article ID 61434, 5 pages, 2007.

3 F. P. Vasilev,Numerical Methodes for Solving Extremal Problems, Nauka, Moscow, Russian, 2nd edition, 1988.

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5 Y. Qing and B. E. Rhoades, “T-stability of Picard iteration in metric spaces,”Fixed Point Theory and Applications, vol. 2008, Article ID 418971, 4 pages, 2008.