Abstract
We put forward a new general iterative process. We prove a convergence result as well as a stability result regarding this new iterative process for weak contraction operators.
1. Introduction and Preliminaries
Throughout this paper, by , we denote the set of all positive integers. In this paper, we obtain results on the stability and strong convergence for a new iteration process (3) in an arbitrary Banach space by using weak contraction operator in the sense of Berinde [1]. Also, we obtain that the iteration procedure (3) can be useful method for solution of delay differential equations. To obtain solution of delay differential equation by using fixed point theory, some authors have done different studies. One can find these works in [2, 3]. Many results of stability have been established by some authors using different contractive mappings. The first study on the stability of the Picard iteration under Banach contraction condition was done by Ostrowski [4]. Some other remarkable results on the concept of stability can be found in works of the following authors involving Harder and Hicks [5, 6], Rhoades [7, 8], Osilike [9], Osilike and Udomene [10], and Singh and Prasad [11]. In 1988, Harder and Hicks [5] established applications of stability results to first order differential equations. Osilike and Udomene [10] developed a short proof of stability results for various fixed point iteration processes. Afterward, in following studies, same technique given in [10] has been used, by Berinde [12], Olatinwo [13], Imoru and Olatinwo [14], Karakaya et al. [15], and some authors.
Let be complete metric space and a self-map on ; and the set of fixed points of in is defined by . Let be the sequence generated by an iteration involving which is defined by where is the initial point and is a proper function. Suppose that sequence converges to a fixed point of . Let and set
Then, the iteration procedure (1) is said to be stable or stable with respect to if and only if implies .
Now, let be a convex subset of a normed space and a self-map on . We introduce a new two-step iteration process which is a generalization of Ishikawa iteration process as follows: for , where , , and satisfy the following conditions,,.
In the following remark, we show that the new iteration process is more general than the Ishikawa and Mann iteration processes.
Remark 1. (1)If , then (3) reduces to the Ishikawa iteration process in [16].(2)If , then (3) reduces to the Mann iteration process in [17].
Lemma 2 (see [18]). If is a real number such that and is a sequence of positive real numbers such that , then for any sequence of positive numbers satisfying one has
Lemma 3 (see [2]). Let be a sequence of positive real numbers including zero satisfying
If and , then .
A mapping is said to be contraction if there is a fixed real number such that for all .
This contraction condition has been generalized by many authors. For example, Kannan [19] shows that there exists such that, for all ,
Chatterjea [20] shows that there exists such that, for all ,
In 1972, Zamfirescu [21] obtained the following theorem.
Theorem 4 (see [21]). Let be a complete metric space and a mapping for which there exist real numbers , and satisfying , such that, for each pair , at least one of the following conditions is performed:(i),(ii),(iii).
Then has a unique fixed point and the Picard iteration defined by
converges to for any arbitrary but fixed .
In 2004, Berinde introduced the definition which is a generalization of the above operators.
Definition 5 (see [1]). A mapping is said to be a weak contraction operator, if there exist and such that for all .
Theorem 6 (see [1]). Let be Banach space. Assume that is a nonempty closed convex subset and is a mapping satisfying (11). Then .
Definition 7 (see [22]). Let and be two iteration processes and let both and be converging to the same fixed point of a self-mapping . Assume that Then, it is said that converges faster than to fixed point of .
The rate of convergence of the Picard and Mann iteration processes in terms of Zamfirescu operators in arbitrary Banach setting was compared by Berinde [22]. Using this class of operator, the Mann iteration method converges faster than the Ishikawa iteration method that was shown by Babu and Vara Prasad [23]. After a short time, Qing and Rhoades [24] showed that the claim of Babu and Vara Prasad [23] is false. There are many studies which have been made on the rate of convergence as given in [15, 25, 26] which are just a few of them.
2. Main Results
Theorem 8. Let be a nonempty closed convex subset of an arbitrary Banach space and let be a mapping satisfying (11). Let be defined through the new iteration (3) and , where with satisfying , . Then converges strongly to fixed point of .
Proof. From Theorems 4 and 6, it is clear that has a unique fixed point in and .
From (3), we have
In addition,
Substituting (14) in (13), we have the following estimates:
Since , we have
for all .
Since , , and , we have
So yields . This completes the proof of theorem.
Theorem 9. Let be Banach space and a self-mapping with fixed point with respect to weak contraction condition in the sense of Berinde (11). Let be iteration process (3) converging to fixed point of , where and such that for all . Then two-step iteration process is stable.
Proof. Let be iteration process (3) converging to . Assume that is an arbitrary sequence in . Set
where . Suppose that . Then, we shall prove that . Using contraction condition (11), we have
We estimate in (19) as follows:
Substituting (20) in (19), we have
Since , we have
Since and using Lemma 2, we obtain .
Conversely, letting , we show that as follows:
Since , it follows that . Therefore the iteration scheme is stable.
Example 10 (see [24]). Let , , , , and , , , for all . It is easy to show that is a weak contraction operator satisfying (11) with a unique fixed point . Furthermore, for all , , and . Then the new iterative process is faster than the Ishikawa iterative process. Assume that is initial point for the new and Ishikawa iterative processes, respectively. Firstly, we consider the new iterative process, and we have
Secondly, we consider the Ishikawa iterative process, and we have
Now, taking the above two equalities, we obtain
It is clear that
Therefore, the proof is completed.
Now, we can give Table 1 and Figures 1 and 2 to support and reinforce our claim in the Example 10.
Finally, we check that this iteration procedure can be applied to find the solution of delay differential equations.
2.1. An Application
Throughout the rest of this paper, the space equipped with Chebyshev norm denotes the space of all continuous functions. It is well known that is a real Banach space with respect to norm; more details can be found in [2, 27].
Now, we will consider a delay differential equation such that and an assumed solution
Assume that the following conditions are satisfied:, ,,,there exists the following inequality: for all and such that ,, and according to a solution of problem (28)-(29) we infer the function . The problem can be reconstituted as follows: Also, the map is defined by the following form:
Using weak-contraction mapping, we obtain the following.
Theorem 11. We suppose that conditions are performed. Then the problem (28)-(29) has a unique solution in .
Proof. We consider iterative process (3) for the mapping . The fixed point of is shown via such that .
For the first part, that is, for , it is clear that . Therefore, letting , we obtain
Hence, we obtain
By continuing this way, we have
Hence, we obtain
Substituting (36) into (34), we obtain
Since , we have
We take and , and then the conditions of Lemma 3 immediately imply .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The authors would like to thank Yıldız Technical University Scientific Research Projects Coordination Department under Project no. BAPK 2014-07-03-DOP02 for financial support during the preparation of this paper.