Research Article | Open Access
R. H. Haghi, M. Postolache, Sh. Rezapour, "On T-Stability of the Picard Iteration for Generalized φ-Contraction Mappings", Abstract and Applied Analysis, vol. 2012, Article ID 658971, 7 pages, 2012. https://doi.org/10.1155/2012/658971
On T-Stability of the Picard Iteration for Generalized φ-Contraction Mappings
We introduce some results on T-stability of the Picard iteration for φ-contraction and generalized φ-contraction mappings on metric spaces.
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 [1–7].
There are some papers about the stability or different iteration methods. In , Harder and Hicks studied the stability of Picard iteration for several contractivity conditions , while in  Rhoades introduced a contractivity condition independent of that in  to obtain stability results for Mann, Kirk, or Massa iteration processes. Meantime, Bosede and Rhoades  introduced stability results of Picard and Mann iteration for a general class of functions; also, see , while Rezapour et al.  studied the almost stability of Mann iteration for -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 -contraction and generalized -contraction mappings on metric spaces.
2. Previous Notation and Definitions
Let be a complete metric space, a map and an iteration procedure. Suppose that has at least one fixed point and that sequence converges to a fixed point . We denote the set of fixed points of mapping by . Let be an arbitrary sequence in and .
If is a bounded sequence and implies that , then the iteration procedure 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 . The Picard iteration is given by .
The following example illustrates that the notion of T-Stability is different from the notion of boundedly T-stability.
Example 2.1. Consider mapping given by whenever and whenever . Put for all . Note that is unbounded, while
3. Main Results
Now, we are ready to state and prove our main results.
Definition 3.1 (see ). A function is said to be comparison if is increasing and converges to 0 for all .
Note that if is comparison, then for all and .
Definition 3.2 (see ). Let be a metric space, and let be a comparison function. A mapping is called -contraction whenever for all .
We say that is a subadditive comparison function whenever is comparison and for all .
There are many subadditive comparison mappings.
For example, if we consider and is a decreasing function, then is a comparison function. In fact, is increasing because . Also, . Hence, converges to 0 for all . Since is decreasing, we have for all .
In particular, if we consider , it follows that is a subadditive comparison function.
Theorem 3.3. Let be a complete metric space, and, a subadditive comparison function. If is a -contraction, then the Picard iteration is T-stable.
Proof. By using Theorem 2.7 in , we conclude that has a unique fixed point .
Let be a sequence in with
First, we show that is bounded. If is not bounded, then there exist subsequence of for which . Since we can take a subsequence of such that . Now, we have Thus, is bounded and so is . This is a contradiction. Therefore is bounded.
Now, choose such that for all . For each there exist natural numbers and such that for all . But we have By continuing this process, we obtain Hence, . Since was arbitrary, .
Definition 3.4 (see ). A function is called (5-dimensional) comparison function whenever , for each with , and the function
satisfies , for all .
Note that is a comparison function, while the following are 5-dimensional comparison functions:(i) for each , where ,(ii), ,(iii), with ,(iv), .
In the previous four examples, function given by (3.7) is a subadditive comparison function.
Definition 3.5. Let be a metric space, and, , a 5-dimensional comparison function. A mapping is called generalized -contraction whenever for all in .
In the sequel, we will use functions such that is subadditive.
Lemma 3.6. Let be a metric space, and let be a generalized -contraction map. Suppose is a bounded sequence in such that . Let be the diameter of the set . Then, In particular,
Proof. By using definition of , for each and we have
Let . Then
Let . It is easy to see that , and we have
By using (3.11), we observe that Since is bounded, so is. Choose such that for all . Since is comparison, for each there exists a natural number such that . But, for each we obtain Hence, Since , for all , and is increasing, then , for all natural numbers . Thus by continuing these relations, for each we have It implies that . Since was arbitrary, . Therefore by using (3.11), .
Theorem 3.7. Let be a metric space, let be a generalized -contraction map, and let . Then the Picard iteration is boundedly T-stable.
Proof. Let be a bounded sequence in such that Choose such that for all . Observe that
If , then .
Without loss of generality, suppose that the last equality does not hold. Therefore, we get
For any given , choose such that
Now, for each we have
Now for each we obtain
If , then by a similar method in Lemma 3.6, Since is arbitrary, .
Finally note that the inequality implies that
The proof is complete.
Remark 3.8. Let be a complete metric space, and let be a mapping for which there exists satisfying for all in . If we define , 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  and Theorems 1 and 2 in  are special cases of Theorem 3.7.
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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.