On T -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

1Department of Mathematics, Science and Research Branch, Islamic Azad University (IAU), Tehran 14778 93855, Iran

2Department of Mathematics, Amirkabir University of Technology, Tehran 15916 34311, Iran

3Department of Mathematics, Newcastle University, Newcastle, NSW 2308, Australia

4Department of Mathematics, Indiana University, Bloomington, IN 46205, USA

Correspondence should be addressed to S. M. Vaezpour,[email protected] 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 satisfies the following:

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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 operator if, 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:

d

Tx, Ty

≤δd x, y

2δdx, 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:

a_n1≤1−λ_nanb_n, 1.2

whereλ_n∈0,1,for alln≥n₀,_∞

n1λ_n∞,andb_n/λ_n → 0 asn → ∞.Then lim_{n→ ∞}a_n0.

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

a_n1≤λa_n−mb_n, 1.3

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

Then limn→ ∞an 0.

Lemma 1.5. LetP be a normal cone with constant K, and let{an} and {bn} be sequences in E satisfying the following inequality:

a_n1≤hanbn, 1.4

whereh∈0,1andbn → 0 asn → ∞.Then lim_n_{→ ∞}an0.

Proof. Letmbe a positive integer such thath^mK <1.By recursion we have

a_n1≤bnhb_n−1· · ·h^mb_n−mh^m1a_n−m, 1.5

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so

an1 ≤Kbnhb_n−1· · ·h^mb_n−mh^m1Kan−m, 1.6 and then byRemark 1.4 an → 0.Thereforea_n → 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 thatx_n1 fT, xnis an iteration procedure, involvingT, which yields a sequence{xn}of points fromX.

Definition 2.1see 4. The iteration procedurex_n1 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→ ∞dy_n1, fT, yn 0 we have limn→ ∞ynq.

In practice, such a sequence{yn}could arise in the following way. Letx₀be a point in X. Setx_n1 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 byx_n1Txn. If the conditions ofDefinition 2.1hold forx_n1Txn, 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≥0 and 0≤b <1,such that

d Tx, q

≤adx, Tx bd x, q

2.1 for eachx ∈ X, q ∈ FTand in addition, whenever {yn} is a sequence withdyn, Ty_n → 0 as n → ∞, then Picard’s iteration isT-stable.

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

y_n1, q

≤d

y_n1, Ty_n d

Ty_n, q

≤c_nad

y_n, Ty_n bd

y_n, q

, 2.2

if we puta_n:dTyn, qandb_n:c_nadyn, Ty_ninLemma 1.5, then we havey_n → q.

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

p, q d

Tp, q

≤ad p, Tp

bd p, q

, 2.3

which impliespq.

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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, Ty_n → 0 asn → ∞, 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

d

Tx, Ty

≤λd x, Ty

≤λdx, Tx λd

Tx, Ty

. 2.4

So

d

Tx, 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 some 0< 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, Ty_n → 0.We have

d

y_n, Ty_n

≤d

y_n, Ty_n−1 d

Ty_n−1, Ty_n

. 2.6

Putb_n :dyn, Ty_n, cn :dyn1, Ty_nandd_n:dTyn−1, Ty_n.Thereforec_n → 0 asn → ∞ and

bn≤c_n−1dn≤c_n−1λun, 2.7

where

u_n∈C

T, y_n−1, y_n

d

y_n−1, y_n , d

y_n−1, Ty_n−1 , d

y_n, Ty_n , d

y_n−1, Ty_n , d

y_n, Ty_n−1 . 2.8

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

Therefore by2.7,b_n ≤ λl1c_n−1 λsb_n−1 by 0 ≤ λs < 1.Now byLemma 1.5we have b_n → 0.

3. An Application

Theorem 3.1. LetX : C0,1,Rwithf_∞:sup_0≤x≤1|fx|forf∈Xand letTbe a self map ofXdefined byTfx ₁

0Fx, ftdtwhere

aF:0,1×R → Ris a continuous function,

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

LetP :{x, y∈R²|x, y≥0}be a normal cone andX, dthe complete cone metric space defined bydf, g f−g_∞, αf−g_∞whereα≥0.Then,

iPicard’s iteration isT-stable if 0≤L <1/2,

iiPicard’s iteration fails to beT-stable if 1/2≤L <1 and₁

0Fx, 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 y_nx : nx/n1 so y_n ∈ X and dyn, h → 0, where hx x.Also dyn1, Tyn → 0,since

y_n1−Ty_n

∞ sup

0≤x≤1

n1 n2x−

₁

0

F

x, nt n1

dt

≤ sup

0≤x≤1

n1

n2x− nx n1

−→0,

3.1

asn → ∞.Buty_n → 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 1/4₁

0ftdtandT2fx x 1/2₁

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.

4 A. M. Harder and T. L. Hicks, “Stability results for fixed point iteration procedures,” Mathematica Japonica, vol. 33, no. 5, pp. 693–706, 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.

6 D. Ilić and V. Rakoćević, “Quasi-contraction on a cone metric space,” Applied Mathematics Letters, vol.

22, no. 5, pp. 728–731, 2009.