T -Stability of Picard Iteration in Metric Spaces

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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008, Article ID 418971,4pages doi:10.1155/2008/418971

Research Article

T -Stability of Picard Iteration in Metric Spaces

Yuan Qing¹and B. E. Rhoades²

1Department of Mathematics, Beijing University of Aeronautics and Astronautics, Beijing 100083, China

2Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA

Correspondence should be addressed to Yuan Qing,[email protected] Received 10 July 2007; Accepted 11 January 2008

Recommended by H´el`ene Frankowska

We establish a general result for the stability of Picard’s iteration. Several theorems in the literature are obtained as special cases.

Copyrightq2008 Y. Qing and B. E. Rhoades. 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.

LetX, dbe a complete metric space andTa self-map ofX. Letxn1 fT, xnbe some iteration procedure. Suppose thatFT, the fixed point set ofT, is nonempty and thatxnconverges to a pointq ∈ FT. Let{yn} ⊂X and definen dyn1, fT, yn. If limn 0 implies that limyn q, then the iteration procedurexn1 fT, xnis said to beT-stable. Without loss of generality, we may assume that{yn} is bounded, for if{yn}is not bounded, then it cannot possibly converge. If these conditions hold forxn1 Txn, that is, Picard’s iteration, then we will say that Picard’s iteration isT-stable.

We will obtain suﬃcient conditions that Picard’s iteration isT-stable for an arbitrary self-map, and then demonstrate that a number of contractive conditions are PicardT-stable.

We will need the following lemma from1.

Lemma 1. Let{xn},{n}be nonnegative sequences satisfyingxn1≤hxnn,for alln∈N,0≤h <

1, limn0. Then, limxn0.

Theorem 1. LetX, dbe a nonempty complete metric space andTa self-map ofXwithFT/∅. If there exist numbersL≥0,0≤h <1,such that

dTx, q≤Ldx, Tx hdx, q 1

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2 Fixed Point Theory and Applications for eachx∈X, q∈FT, and, in addition,

limd yn, Tyn

0, 2

then Picard’s iteration isT-stable.

Proof. First, we show that the fixed pointqofTis unique. Supposepis another fixed point of T, then

dp, q dTp, q≤Ldp, Tp hdp, q hdp, q. 3

Since 0≤h <1, sodp, q 0, that is,pq.

Let{yn} ⊂X,ndyn1, Tyn, and limn0. We need to show that limynq.

Using1,2, andLemma 1,

d yn1, q

≤d

yn1, Tyn

d

Tyn, q

≤nLd yn, Tyn

hd

yn, q

, 4

and limynq.

Corollary 1. LetX, d be a nonempty complete metric space andT a self-map ofX satisfying the following: there exists 0≤h <1, such that, for eachx, y∈X,

dTx, Ty≤hmax

dx, y, dx, Tx, dy, Ty, dx, Ty, dy, Tx

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Then, Picard’s iteration isT-stable.

Proof. From2, Theorem 11,T has a unique fixed pointq. Also, T satisfies1. It remains to show that2is satisfied.

Definepnto be the diameter of the orbit ofyn; that is,pn δOyn, Tyn, . . .. First, we show thatpnis bounded:

d Tyn, q

≤hmax d

yn, q , d

yn, Tyn

, d yn, Tq

, d q, Tyn

, d q, Tq

≤hmax d

yn, q , d

yn, Tyn

, d yn, q

, d q, Tyn

,0

hmax d

yn, q , d

yn, Tyn

, d yn, q

, d q, Tyn

.

6

Hence,dTyn, q≤hdyn, qordTyn, q≤hdyn, TynordTyn, q≤hdq, Tyn. IfdTyn, q≤hdyn, q, it is clear that

d Tyn, q

≤hd yn, q

≤ h 1−hd

yn, q

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IfdTyn, q≤hdq, Tyn, then

d Tyn, q

0≤ h 1−hd

yn, q

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Y. Qing and B. E. Rhoades 3 IfdTyn, q≤hdyn, Tyn, then

d yn, Tyn

≤d

Tyn, q d

yn, q

≤hd yn, Tyn

d

yn, q

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Hence,dTyn, q≤h/1−hdyn, q. Now it is easy to see that{Tyn}is bounded and so is {pn}, since{yn}is bounded.

For anyi, j≥n, using5, d

Tyi, Tyj

≤hmax

d yi, yj

, d yi, Tyi

, d yj, Tyj

, d yi, Tyj

, d yj, Tyi

≤hpn. 10

Thus,

d yi, Tyj

≤d

yi, Tyi−1 d

Tyi−1, Tyj

≤i−1hpn−1. 11

But d

yi, yj

≤d

yi, Tyi−1 d

Tyi−1, Tyj−1 d

Tyj−1, yj

≤i−1hpn−1i−1, 12

which implies that

pn≤2i−1hpn−1, 13

and limpn0 byLemma 1. Sincedyn, Tyn≤pn, limdyn, Tyn 0.

The conclusion now follows fromTheorem 1.

Corollary 2see3, Theorem 1. LetX, dbe a nonempty complete metric space andTa self-map ofXsatisfying

dTx, Ty≤Ldx, Tx adx, y 14

for allx, y∈X, whereL≥0,0≤a <1. Suppose thatThas a fixed pointp. Then,Tis PicardT-stable.

Proof. SinceT satisfies14for allx, y ∈ X, thenT satisfies inequality 1of our paper. Let {yn} ⊂ Xand definen dyn1, yn. From the proof of Theorem 1 of3, limdyn, Tyn 0.

Therefore, by our theoremTheorem 1,Tis PicardT-stable.

Definition5of this paper is actually Definition24of2. Therefore, many contractive conditions are special cases of 5, and, for each of these, Picard’s iteration isT-stable. For example, Theorems 1 and 2 of4and Theorem 1 of5are special cases ofCorollary 1.

We will not examine the analogues ofTheorem 1for Mann, Ishikawa, Kirk, or any other iteration scheme since, if one obtains convergence to a fixed point for a map using Picard’s iteration, there is no point in considering any other more complicated iteration procedure.

Acknowledgment

This article is partly supported by the National Natural Science Foundation of China no.

10271012.

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4 Fixed Point Theory and Applications References

1Q. Liu, “A convergence theorem of the sequence of Ishikawa iterates for quasi-contractive mappings,”

Journal of Mathematical Analysis and Applications, vol. 146, no. 2, pp. 301–305, 1990.

2B. E. Rhoades, “A comparison of various definitions of contractive mappings,” Transactions of the Amer- ican Mathematical Society, vol. 226, pp. 257–290, 1977.

3M. O. Osilike, “Stability results for fixed point iteration procedures,” Journal of the Nigerian Mathematical Society, vol. 14-15, pp. 17–29, 1995.

4A. M. Harder and T. L. Hicks, “Stability results for fixed point iteration procedures,” Mathematica Japon- ica, vol. 33, no. 5, pp. 693–706, 1988.

5B. E. Rhoades, “Fixed point theorems and stability results for fixed point iteration procedures,” Indian Journal of Pure and Applied Mathematics, vol. 21, no. 1, pp. 1–9, 1990.