#### Abstract

This paper deals with designing a new iteration scheme associated with a given scheme for contraction mappings. This new scheme has a similar structure to that of the given scheme, in which those two iterative schemes converge to the same fixed point of the given contraction mapping. The positive influence of feedback parameters on the convergence rate of this new scheme is investigated. Moreover, the derived convergence and comparison results can be extended to nonexpansive mappings. As an application, the derived results are utilized to study the synchronization of logistic maps. Two illustrated examples are used to reveal the effectiveness of our results.

#### 1. Introduction

Fixed point theory has achieved great progress since the last two decades. Various schemes have been constructed to approximate the fixed point of a contraction mapping (see, e.g., [130]).

For a contraction mapping, we can define an iteration scheme which converges to the fixed point of that mapping. Here is a question whether we can design another iteration scheme with a similar structure to that of given scheme to approximate the fixed point. Motivated by this question, we design a new iteration scheme which is associated with the given iteration scheme.

This new scheme has a similar structure to that of the given scheme. Those two schemes converge to the same fixed point of the given contraction mapping. The convergence rate of this new scheme can be accelerated by the increase of the feedback parameters. Those convergence and comparison criteria can be applied to nonexpansive mappings. Moreover, the derived results are utilized to study the synchronization of logistics maps. Two examples are used to reveal the effectiveness of our results.

#### 2. Preliminaries

Let be a nonempty convex subset of a normed linear space . Let be a contraction mapping of into itself with the contraction constant ; that is,for any , . The set of fixed points of is denoted by . The set of natural numbers is denoted by . and are two sequences of real numbers such that and for all . Consider the following scheme:

Remark 1. It should be pointed out that scheme (2) is a general framework which includes the following well-known schemes as special cases.(i)If and , scheme (2) reduces to Picard iteration.(ii)If and , scheme (2) reduces to Mann iteration.(iii)If , , and , where is a contraction mapping of into itself and , scheme (2) reduces to Ishikawa iteration.

For the fixed point scheme described by (2), a question naturally arises whether we can design another iteration scheme with a similar structure to scheme (2) to approximate the fixed point. Moreover, this new scheme has a similar structure to that of the given scheme. Those two schemes converge to the same fixed point of the given contraction mapping.

Motivated by this question, we define the following scheme associated with scheme (2):where is a scheme of feedback parameters which can be determined later. Let for . Then we can construct the following scheme from schemes (2) and (3):From [4, 31, 32], the fact will ensure . The main purpose of this paper is to find the conditions to guarantee , which means that scheme (3) has a similar structure to scheme (2). Schemes (2) and (3) converge to the same fixed point of the given contraction mapping .

#### 3. Main Results

##### 3.1. Convergence Results

Now, we give some convergence results for iteration (3).

Theorem 2. Let be a nonempty convex subset of a normed linear space . Let be a contraction mapping of into itself and If where , , , , and , , then , which also implies that scheme (3) and scheme (2) converge to the same fixed point of .

Proof. From (5), we have for . Then,It follows from (5) that , where , , and , . Thus,It is easy to see that , as ; that is, . This completes the proof.

Remark 3. It follows from Theorem 2 that for .

Theorem 2 can be applied to approximating the fixed point of a nonexpansive mapping where the contraction constant . If , Theorem 2 reduces to the following result.

Corollary 4. Let be a nonempty convex subset of a normed linear space . Let be a nonexpansive mapping of into itself and . If where , , , , and , , then

##### 3.2. Three Special Cases

Now, we use Theorem 2 to construct the associated schemes for Picard iteration scheme, Mann iteration scheme, and Ishikawa iteration scheme for contraction mappings and derive the convergence theorems for those schemes, respectively. First, we consider the Picard iteration scheme. The Picard iteration scheme is defined byWe define the iteration scheme associated with Picard iteration scheme (10):Let for . Schemes (2) and (3) give the following scheme:

Then, by the similar proof of Theorem 2, we have the following convergence theorem.

Theorem 5. Let be a nonempty convex subset of a normed linear space . Let be a contraction mapping of into itself and If , then

Second, we consider the Mann iteration scheme. The Mann iteration scheme is defined by We construct the following iteration scheme associated with Mann iteration scheme (13): By defining an error variable for , we obtain the following iteration scheme:

Then, from the similar proof for Theorem 2, we derive the following convergence result.

Theorem 6. Let be a nonempty convex subset of a normed linear space . Let be a contraction mapping of into itself and If , where , , , and , , then

Third, we consider the Ishikawa iteration scheme. The Ishikawa iteration scheme is defined by where is a contraction mapping of into itself with the contraction constant . We generate the following iteration scheme associated with Ishikawa iteration scheme (16):After defining an error variable for , we obtain the error scheme:

Then, from the similar proof for Theorem 2, we achieve the following convergence theorem.

Theorem 7. Let be a nonempty convex subset of a normed linear space . Let be a contraction mapping of into itself and If , where , , , , , and , , then

##### 3.3. Impact of to the Convergent Rate

Next, we analyze the influence of size to the convergence rate of (3). We first give another iteration scheme associated with iteration scheme (2): where . In order to compare the convergence rate of (3) with that of (19), we give the following definitions for the convergent rates of two different iteration schemes.

Definition 8 (see [4]). Let and be two sequences of real numbers which converges to and , respectively. We say that the sequence converges faster than if

Definition 9. Let , , be three iterative schemes which satisfy as . Let and be two sequences of real numbers which converge to . We say that the scheme converges faster than to , if and converges faster than .

By the similar method for Theorem 2, we have where , and . It follows from that , which implies Hence, from the above mentions, we have the following comparison result for the convergence rate according to the size .

Theorem 10. The iteration scheme defined by (19) converges faster than the iteration scheme defined by (3).

Remark 11. The convergence rate of iteration scheme defined by (3) increases as increases which means that the convergence rate of iteration scheme defined by (3) can be controlled by the adjustment of size .

Remark 12. If is a nonexpansive mapping, i.e., , limitation (22) reduces to which means that Theorem 10 is still valid for the nonexpansive mapping.

#### 4. An Application to Synchronization of Logistic Maps

Logistic maps are classical discrete systems which can generate bifurcation and chaos. Synchronization of two logistic maps, which means the state variable of one logistic map is eventually equal to the counterpart of another logistic map, has been widely used in secure communication, image encryption, and signal transmission [22, 31]. Our results can be applied to studying the synchronization of logistic maps.

If and , then scheme (2) reduces to the following logistic map where . If and , then we can derive , which implies that for any .

Here, we consider another logistic map where . Defining , we can have the following scheme: where .

Definition 13. If , the logistic map described by (24) is said to achieve the global synchronization with the logistic map described by (26).

By using the similar proof method of Theorem 2 with and , we can derive the following result.

Theorem 14. If and , the logistic map described by (24) achieves the global synchronization with the logistic map described by (26).

#### 5. Two Illustrated Examples

Example 15. Now we give an example for the main theorems with numerical analysis.
Consider on with the contraction constant . The fixed point of is . We first construct the iteration scheme (11) associated with Picard iteration (10). From Theorem 5, we know that , which implies for . We choose , for , and and . Figure 1 gives a demonstration of trajectory of of (11). From Figure 1, we can see that ; that is, , which indicates the effectiveness of Theorem 5.
Then, we construct the iteration scheme (14) associated with Mann iteration (13) with for . From Theorem 6, we can get , which indicates for . We choose , for , and and . Figure 2 provides the trajectory of of (14). From Figure 2, we can observe that ; that is, , which indicates the effectiveness of Theorem 6.
Now, we construct the iteration scheme (17) associated with Ishikawa iteration (16) with and for . From Theorem 7, we can have , which implies for . We choose , for , and and . Figure 3 reveals the trajectory of of (17). It follows from Figure 3 that ; that is, , which indicates the effectiveness of Theorem 7.
Finally, we compare the convergence rates of (3) with different . We choose scheme (14) with . From Theorem 6, we can have for . Let and . We choose as , , and to approximate the fixed point, respectively. Table 1 shows that the convergence rate of (3) increases as increases, which also illustrates the effectiveness of Theorem 10.

Example 16. Consider the logistic maps described by (24) and (26) with and and . Figure 4 reveals the global synchronization of the logistic maps described by (24) and (26) which illustrates the effectiveness of Theorem 14.

#### 6. Conclusions and Future Works

For a given convergent scheme to approximate the fixed point of a contraction mapping, we have provided an associated scheme which had a similar structure to that of the given scheme. We have derived conditions to ensure this new scheme and the given scheme to converge to the same fixed point. We have used our derived results to construct the associated schemes for Picard, Mann, and Ishikawa iterative schemes for contraction mappings and derived the convergence theorems for those schemes, respectively. Moreover, we can accelerate the convergence rate of this new scheme by controlling the feedback parameter. We have extended those convergence and comparison results to nonexpansive mappings. In addition, we have utilized those derived results to investigate the synchronization of logistic maps. We have used two examples to illustrate the effectiveness of our derived results. In this paper, we only consider the linear feedback in the scheme. Our future research focus is to design a faster scheme by using the nonlinear feedback.

#### Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

#### Acknowledgments

The authors are most grateful to the suggestions of colleagues of Zhejiang Open Foundation of the Most Important Subjects for this paper. This paper is partially supported by the National Natural Science Foundation of China under Grants 61561023 and 71461011, the Zhejiang Open Foundation of the Most Important Subjects, the Key Project of Youth Science Fund of Jiangxi China under Grant 20133ACB21009, the project of Science and Technology Fund of Jiangxi Education Department of China under Grant GJJ160429, the project of Jiangxi E-Commerce High Level Engineering Technology Research Centre, and the Basic Science Research Program through the National Research Foundation (NRF) grant funded by Ministry of Education of the Republic of Korea (2015R1D1A1A09058177).