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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 658971, 7 pages
http://dx.doi.org/10.1155/2012/658971
Research Article

On T-Stability of the Picard Iteration for Generalized φ-Contraction Mappings

1Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran
2Department of Mathematics, University Politehnica of Bucharest, 060042 Bucharest, Romania
3Department of Mathematics, Shahid Madani University of Azabaidjan, Azabaidjan, Tabriz, Iran

Received 20 May 2012; Accepted 6 June 2012

Academic Editor: Yonghong Yao

Copyright © 2012 R. H. Haghi 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.

Abstract

We introduce some results on T-stability of the Picard iteration for φ-contraction and generalized φ-contraction mappings on metric spaces.

1. Introduction

It is known that iteration methods are numerical procedures which compute a sequence of gradually accurate iterates to approximate the solution of a class of problems. Such methods are useful tools of applied mathematics for solving real life problems ranging from economics and finance or biology to transportation, network analysis, or optimization. An iteration method is considered to be sound if possesses some qualitative properties such as convergence and stability. That is why several scientists paid and still pay attention to the qualitative study of iteration methods; please, see [17].

There are some papers about the stability or different iteration methods. In [3], Harder and Hicks studied the stability of Picard iteration for several contractivity conditions [7], while in [6] Rhoades introduced a contractivity condition independent of that in [7] to obtain stability results for Mann, Kirk, or Massa iteration processes. Meantime, Bosede and Rhoades [2] introduced stability results of Picard and Mann iteration for a general class of functions; also, see [4], while Rezapour et al. [5] studied the almost stability of Mann iteration for 𝜑-contraction mappings and the stability of Picard iteration for mappings satisfying a contractive condition of integral type. In the present paper, we introduce our new results on stability of Picard iteration for 𝜑-contraction and generalized 𝜑-contraction mappings on metric spaces.

2. Previous Notation and Definitions

Let (𝑋,𝑑) be a complete metric space, 𝑇𝑋𝑋 a map and 𝑥𝑛+1=𝑓(𝑇,𝑥𝑛) an iteration procedure. Suppose that 𝑇 has at least one fixed point and that sequence {𝑥𝑛} converges to a fixed point 𝑥𝑋. We denote the set of fixed points of mapping 𝑇 by 𝐹𝑇. Let {𝑦𝑛} be an arbitrary sequence in 𝑋 and 𝜖𝑛=𝑑(𝑦𝑛+1,𝑓(𝑇,𝑦𝑛)).

If lim𝑛𝜖𝑛=0 implies that lim𝑛𝑦𝑛=𝑥, then the iteration procedure 𝑥𝑛+1=𝑓(𝑇,𝑥𝑛) is said to be T-stable (e.g., [1, 6]).

If {𝑦𝑛} is a bounded sequence and lim𝑛𝜖𝑛=0 implies that lim𝑛𝑦𝑛=𝑥, then the iteration procedure 𝑥𝑛+1=𝑓(𝑇,𝑥𝑛) is said to be boundedly T-stable.

In most papers on T-stability, some authors consider the notion of boundedly T-stability instead of T-stability. Here, we mention the Picard iteration methods. Let 𝑥0𝑋. The Picard iteration is given by 𝑥𝑛+1=𝑇𝑥𝑛.

The following example illustrates that the notion of T-Stability is different from the notion of boundedly T-stability.

Example 2.1. Consider mapping 𝑇[0,)[0,) given by 𝑇𝑥=(1/2)(𝑥+1) whenever 𝑥[0,1] and 𝑇𝑥=𝑥+1 whenever 𝑥>1. Put 𝑦𝑛=𝑛+(1/𝑛) for all 𝑛1. Note that {𝑦𝑛} is unbounded, while lim𝑛|𝑦𝑛+1𝑇𝑦𝑛|=0.

3. Main Results

Now, we are ready to state and prove our main results.

Definition 3.1 (see [1]). A function 𝜑[0,)[0,) is said to be comparison if 𝜑 is increasing and 𝜑𝑛(𝑡) converges to 0 for all 𝑡0.

Note that if 𝜑 is comparison, then 𝜑(𝑡)<𝑡 for all 𝑡>0 and 𝜑(0)=0.

Definition 3.2 (see [1]). Let (𝑋,𝑑) be a metric space, and let 𝜑[0,)[0,) be a comparison function. A mapping 𝑇𝑋𝑋 is called 𝜑-contraction whenever 𝑑(𝑇𝑥,𝑇𝑦)𝜑(𝑑(𝑥,𝑦)),(3.1) for all 𝑥,𝑦𝑋.

We say that 𝜑[0,)[0,) is a subadditive comparison function whenever 𝜑 is comparison and 𝜑(𝑡+𝑠)𝜑(𝑡)+𝜑(𝑠) for all 𝑡,𝑠[0,).

There are many subadditive comparison mappings.

For example, if we consider 𝜆<1 and 𝑔[0,)[0,𝜆) is a decreasing function, then 𝜑(𝑡)=𝑡0𝑔(𝑥)𝑑𝑥 is a comparison function. In fact, 𝜑 is increasing because 𝑔>0. Also, 𝜑(𝑡)<min{𝑡,𝜆}. Hence, 𝜑𝑛(𝑡) converges to 0 for all 𝑡0. Since 𝑔 is decreasing, we have 𝜑(𝑢+𝑣)=0𝑢+𝑣𝑔(𝑥)𝑑𝑥=𝑢0𝑔(𝑥)𝑑𝑥+𝑢𝑢+𝑣𝑔(𝑥)𝑑𝑥𝑢0𝑔(𝑥)𝑑𝑥+𝑣0𝑔(𝑥)𝑑𝑥=𝜑(𝑢)+𝜑(𝑣),(3.2) for all 𝑢,𝑣0.

In particular, if we consider 𝑔(𝑡)=𝜆𝑒𝑡, it follows that 𝜑(𝑡)=𝑡0𝑔(𝑥)𝑑𝑥 is a subadditive comparison function.

Theorem 3.3. Let (𝑋,𝑑) be a complete metric space, and, 𝜑[0,)[0,) a subadditive comparison function. If 𝑇𝑋𝑋 is a 𝜑-contraction, then the Picard iteration is T-stable.

Proof. By using Theorem 2.7 in [1], we conclude that 𝑇 has a unique fixed point 𝑞.
Let {𝑦𝑛} be a sequence in 𝑋 with lim𝑛𝑑(𝑦𝑛+1,𝑇𝑦𝑛)=0.
First, we show that {𝑦𝑛} is bounded. If {𝑦𝑛} is not bounded, then there exist subsequence {𝑧𝑛} of {𝑦𝑛} for which 𝑑(𝑧𝑛,𝑞)𝑛. Since lim𝑛𝑑(𝑦𝑛+1,𝑇𝑦𝑛)=0, we can take a subsequence {𝑥𝑛} of {𝑧𝑛} such that 𝑑(𝑥𝑛+1,𝑇𝑥𝑛)1/𝑛2. Now, we have 𝑑𝑇𝑥𝑛𝑑𝑥,𝑞𝜑𝑛𝑑𝑥,𝑞𝜑𝑛,𝑇𝑥𝑛1+𝜑2𝑑𝑥𝑛11,𝑞1+221++𝑛2+𝜑𝑛𝑑𝑥1,𝑞𝑖=11𝑖2𝑥+𝑑1.,𝑞(3.3) Thus, {𝑇𝑥𝑛} is bounded and so is {𝑥𝑛}. This is a contradiction. Therefore {𝑦𝑛} is bounded.
Now, choose 𝑀>0 such that 𝑑(𝑦𝑛,𝑞)<𝑀 for all 𝑛1. For each 𝜀>0 there exist natural numbers 𝑝0 and 𝑁 such that 𝜑𝑝0𝑦(𝑀)<𝜀,𝑑𝑛+1,𝑇𝑦𝑛<𝜀,(3.4) for all 𝑛𝑁. But we have 𝑑𝑦𝑛+2𝑦,𝑞𝑑𝑛+2,𝑇𝑦𝑛+1+𝑑𝑇𝑦𝑛+1𝑦,𝑞𝑑𝑛+2,𝑇𝑦𝑛+1𝑑𝑦+𝜑𝑛+1,𝑑𝑦,𝑞𝑛+3𝑦,𝑞𝑑𝑛+3,𝑇𝑦𝑛+2+𝑑𝑇𝑦𝑛+2𝑦,𝑞𝑑𝑛+3,𝑇𝑦𝑛+2𝑑𝑦+𝜑𝑛+2𝑦,𝑞𝑑𝑛+3,𝑇𝑦𝑛+2𝑑𝑦+𝜑𝑛+2,𝑇𝑦𝑛+1+𝜑2𝑑𝑦𝑛+1.,𝑞(3.5) By continuing this process, we obtain 𝑑𝑦𝑛+𝑝0+1𝑦,𝑞𝑑𝑛+𝑝0+1,𝑇𝑦𝑛+𝑝0𝑑𝑦+𝜑𝑛+𝑝0,𝑇𝑦𝑛+𝑝01++𝜑𝑝01𝑑𝑦𝑛+1,𝑇𝑦𝑛+𝜑𝑝0𝑑𝑦𝑛𝑦,𝑞<𝑑𝑛+𝑝0+1,𝑇𝑦𝑛+𝑝0𝑑𝑦+𝜑𝑛+𝑝0,𝑇𝑦𝑛+𝑝01++𝜑𝑝01𝑑𝑦𝑛+1,𝑇𝑦𝑛+𝜀.(3.6) Hence, lim𝑛sup𝑑(𝑦𝑛,𝑞)𝜀. Since 𝜀>0 was arbitrary, lim𝑛𝑑(𝑦𝑛,𝑞)=0.

Definition 3.4 (see [1]). A function 𝜑5++ is called (5-dimensional) comparison function whenever 𝜑(𝑢)𝜑(𝑣), for each 𝑢,𝑣5+ with 𝑢𝑣, and the function 𝜓++,𝜓(𝑡)=𝜑(𝑡,𝑡,𝑡,𝑡,𝑡)(3.7) satisfies lim𝑛𝜓𝑛(𝑡)=0, for all 𝑡>0.
Note that 𝜓 is a comparison function, while the following are 5-dimensional comparison functions:(i)𝜑(𝑡)=𝑎max{𝑡1,𝑡2,𝑡3,𝑡4,𝑡5} for each 𝑡=(𝑡1,𝑡2,𝑡3,𝑡4,𝑡5), where 𝑎[0,1),(ii)𝜑(𝑡)=𝑎max{𝑡1,𝑡2,𝑡3,𝑡4,(1/2)(𝑡4+𝑡5)}, 𝑎[0,1),(iii)𝜑(𝑡)=𝑎𝑡1+𝑏(𝑡2+𝑡3), 𝑎,𝑏+ with 𝑎+2𝑏<1,(iv)𝜑(𝑡)=𝑎max{𝑡2,𝑡3}, 𝑎(0,1).
In the previous four examples, function 𝜓 given by (3.7) is a subadditive comparison function.

Definition 3.5. Let (𝑋,𝑑) be a metric space, and, 𝜑5++, a 5-dimensional comparison function. A mapping 𝑇𝑋𝑋 is called generalized 𝜑-contraction whenever 𝑑(𝑇𝑥,𝑇𝑦)𝜑(𝑑(𝑥,𝑦),𝑑(𝑥,𝑇𝑥),𝑑(𝑦,𝑇𝑦),𝑑(𝑥,𝑇𝑦),𝑑(𝑦,𝑇𝑥)),(3.8) for all 𝑥,𝑦 in 𝑋.

In the sequel, we will use functions 𝜑 such that 𝜓 is subadditive.

Lemma 3.6. Let (𝑋,𝑑) be a metric space, and let 𝑇𝑋𝑋 be a generalized 𝜑-contraction map. Suppose {𝑦𝑛} is a bounded sequence in 𝑋 such that lim𝑛𝑑(𝑦𝑛+1,𝑇𝑦𝑛)=0. Let 𝑝𝑛 be the diameter of the set 𝐴𝑛={𝑦𝑖}𝑖𝑛{𝑇𝑦𝑖}𝑖𝑛. Then, lim𝑛𝑝𝑛=0. In particular, lim𝑛𝑑(𝑦𝑛,𝑇𝑦𝑛)=0.

Proof. By using definition of 𝑇, for each 𝑛 and 𝑖,𝑗𝑛 we have 𝑑𝑇𝑦𝑖,𝑇𝑦𝑗𝑑𝑦𝜑𝑖,𝑦𝑗𝑦,𝑑𝑖,𝑇𝑦𝑖𝑦,𝑑𝑗,𝑇𝑦𝑗𝑦,𝑑𝑖,𝑇𝑦𝑗𝑦,𝑑𝑗,𝑇𝑦𝑖𝑝𝜓𝑛.(3.9) Let 𝜖𝑖=𝑑(𝑦𝑖+1,𝑇𝑦𝑖). Then 𝑑𝑦𝑖,𝑦𝑗𝑦𝑑𝑖,𝑇𝑦𝑖1+𝑑𝑇𝑦𝑖1,𝑇𝑦𝑗1+𝑑𝑇𝑦𝑗1,𝑦𝑗𝜖𝑖1𝑝+𝜓𝑛1+𝜖𝑗1,𝑑𝑦𝑖,𝑇𝑦𝑗𝑦𝑑𝑖,𝑇𝑦𝑖1+𝑑𝑇𝑦𝑖1,𝑇𝑦𝑗𝜖𝑖1𝑝+𝜓𝑛1.(3.10)
Let 𝑎𝑛=sup𝑖𝑛2𝜖𝑖. It is easy to see that lim𝑛𝑎𝑛=0, and we have 𝑝𝑛𝑎𝑛𝑝+𝜓𝑛1.(3.11)
By using (3.11), we observe that 𝜓𝑝𝑛𝑎𝜓𝑛+𝜓2𝑝𝑛1.(3.12) Since {𝑦𝑛} is bounded, {𝑇𝑦𝑛} so is. Choose 𝑀>0 such that 𝑝𝑛𝑀 for all 𝑛1. Since 𝜓 is comparison, for each 𝜀>0 there exists a natural number 𝑘0 such that 𝜓𝑘0(𝑀)<𝜀/2. But, for each 𝑛1 we obtain 𝜓𝑝𝑛+1𝑎𝜓𝑛+1+𝜓2𝑝𝑛𝑎𝜓𝑛+1+𝜓2𝑎𝑛+𝜓3𝑝𝑛1.(3.13) Hence, 𝜓𝑝𝑛+2𝑎𝜓𝑛+2+𝜓2𝑎𝑛+1+𝜓3𝑎𝑛+𝜓4𝑝𝑛1.(3.14) Since 𝜓(𝑡)<𝑡, for all 𝑡>0, and 𝜓 is increasing, then 𝑘+1𝑖=1𝜓𝑖(𝑎𝑛𝑖+3)0, for all natural numbers 𝑘. Thus by continuing these relations, for each 𝑘𝑘0 we have 𝜓𝑝𝑛+𝑘𝑘+1𝑖=1𝜓𝑖𝑎𝑛𝑖+3+𝜓𝑘+2𝑝𝑛1𝑘+1𝑖=1𝜓𝑖𝑎𝑛𝑖+3+𝜀.(3.15) It implies that lim𝑛sup𝜓(𝑝𝑛)𝜀. Since 𝜀>0 was arbitrary, lim𝑛sup𝜓(𝑝𝑛)=0. Therefore by using (3.11), lim𝑛𝑝𝑛=0.

Theorem 3.7. Let (𝑋,𝑑) be a metric space, let 𝑇𝑋𝑋 be a generalized 𝜑-contraction map, and let 𝐹𝑇={𝑞}. Then the Picard iteration is boundedly T-stable.

Proof. Let {𝑦𝑛} be a bounded sequence in 𝑋 such that lim𝑛𝑑(𝑦𝑛+1,𝑇𝑦𝑛)=0. Choose 𝑀>0 such that 𝑑(𝑦𝑛,𝑞)<𝑀 for all 𝑛1. Observe that 𝑑𝑇𝑦𝑛𝑑𝑦,𝑞𝜑𝑛𝑦,𝑞,𝑑𝑛,𝑇𝑦𝑛𝑦,0,𝑑𝑛,𝑞,𝑑𝑞,𝑇𝑦𝑛𝑑𝑦𝜓max𝑛𝑦,𝑞,𝑑𝑛,𝑇𝑦𝑛,𝑑𝑞,𝑇𝑦𝑛.(3.16) If max{𝑑(𝑦𝑛,𝑞),𝑑(𝑦𝑛,𝑇𝑦𝑛),𝑑(𝑞,𝑇𝑦𝑛)}=𝑑(𝑇𝑦𝑛,𝑞), then 𝑑(𝑇𝑦𝑛,𝑞)=0.
Without loss of generality, suppose that the last equality does not hold. Therefore, we get 𝑑𝑇𝑦𝑛𝑑𝑦,𝑞𝜓max𝑛𝑦,𝑞,𝑑𝑛,𝑇𝑦𝑛𝑑𝑦𝜓𝑛𝑦,𝑞+𝑑𝑛,𝑇𝑦𝑛𝑑𝑦𝜓𝑛𝑑𝑦,𝑞+𝜓𝑛,𝑇𝑦𝑛.(3.17)
For any given 𝜀>0, choose 𝑝0 such that 𝜓𝑝0(𝑀)<𝜀.(3.18)
Now, for each 𝑛1 we have 𝑑𝑇𝑦𝑛+1𝑑𝑦,𝑞𝜓𝑛+1,𝑇𝑦𝑛+1𝑑𝑦+𝜓𝑛+1𝑑𝑦,𝑞𝜓𝑛+1,𝑇𝑦𝑛+1𝑑𝑦+𝜓𝑛+1,𝑇𝑦𝑛+𝑑𝑇𝑦𝑛𝑑𝑦,𝑞𝜓𝑛+1,𝑇𝑦𝑛+1𝑑𝑦+𝜓𝑛+1,𝑇𝑦𝑛+𝜓2𝑑𝑦𝑛,𝑇𝑦𝑛+𝜓2𝑑𝑦𝑛.,𝑞(3.19)
Similarly 𝑑𝑇𝑦𝑛+2𝑑𝑦,𝑞𝜓𝑛+2,𝑇𝑦𝑛+2𝑑𝑦+𝜓𝑛+2𝑑𝑦,𝑞𝜓𝑛+2,𝑇𝑦𝑛+2𝑑𝑦+𝜓𝑛+2,𝑇𝑦𝑛+1+𝑑𝑇𝑦𝑛+1𝑑𝑦,𝑞𝜓𝑛+2,𝑇𝑦𝑛+2𝑑𝑦+𝜓𝑛+2,𝑇𝑦𝑛+1+𝜓2𝑑𝑦𝑛+1,𝑇𝑦𝑛+1+𝜓2𝑑𝑦𝑛+1,𝑇𝑦𝑛+𝜓3𝑑𝑦𝑛,𝑇𝑦𝑛+𝜓3𝑑𝑦𝑛.,𝑞(3.20)
Now for each 𝑝𝑝0 we obtain 𝑑𝑇𝑦𝑛+𝑝,𝑞𝑝𝑖=1𝜓𝑝𝑖+1𝑑𝑦𝑛+𝑖,𝑇𝑦𝑛+𝑖+𝑝𝑖=1𝜓𝑝𝑖+1𝑑𝑦𝑛+𝑖,𝑇𝑦𝑛+𝑖1+𝜓𝑝+1𝑑𝑦𝑛,𝑞+𝜓𝑝+1𝑑𝑦𝑛,𝑇𝑦𝑛𝑝𝑖=1𝜓𝑝𝑖+1𝑑𝑦𝑛+𝑖,𝑇𝑦𝑛+𝑖+𝑝𝑖=1𝜓𝑝𝑖+1𝑑𝑦𝑛+𝑖,𝑇𝑦𝑛+𝑖1+𝜓𝑝+1𝑑𝑦𝑛,𝑇𝑦𝑛+𝜀.(3.21)
If 𝑛, then by a similar method in Lemma 3.6, lim𝑛sup𝑑(𝑇𝑦𝑛,𝑞)𝜀. Since 𝜀>0 is arbitrary, lim𝑛𝑑(𝑇𝑦𝑛,𝑞)=0.
Finally note that the inequality 𝑑𝑦𝑛𝑦,𝑞𝑑𝑛,𝑇𝑦𝑛+𝑑𝑇𝑦𝑛,𝑞(3.22) implies that lim𝑛𝑑(𝑦𝑛,𝑞)=0.
The proof is complete.

Remark 3.8. Let (𝑋,𝑑) be a complete metric space, and let 𝑇𝑋𝑋 be a mapping for which there exists [0,1) satisfying 𝑑(𝑇𝑥,𝑇𝑦)max{𝑑(𝑥,𝑦),𝑑(𝑥,𝑇𝑥),𝑑(𝑦,𝑇𝑦),𝑑(𝑥,𝑇𝑦),𝑑(𝑦,𝑇𝑥)},(3.23) for all 𝑥,𝑦 in 𝑋. If we define 𝜑(𝑡)=max{𝑡1,𝑡2,𝑡3,𝑡4,𝑡5}, then by using Theorem 3.7, the Picard iteration is boundedly T-stable. Consider that some contractive conditions are special cases of (3.8), and, for each of those, the Picard iteration is boundedly T-stable. For example, Theorem 1 in [6] and Theorems 1 and 2 in [3] are special cases of Theorem 3.7.

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