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.
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., [
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.
Let
It should be pointed out that scheme (
If
If
If
For the fixed point scheme described by (
Motivated by this question, we define the following scheme associated with scheme (
Now, we give some convergence results for iteration (
Let
From (
It follows from Theorem
Theorem
Let
Now, we use Theorem
Then, by the similar proof of Theorem
Let
Second, we consider the Mann iteration scheme. The Mann iteration scheme is defined by
Then, from the similar proof for Theorem
Let
Third, we consider the Ishikawa iteration scheme. The Ishikawa iteration scheme is defined by
Then, from the similar proof for Theorem
Let
Next, we analyze the influence of size
Let
Let
By the similar method for Theorem
The iteration scheme defined by (
The convergence rate of iteration scheme defined by (
If
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 [
If
Here, we consider another logistic map
If
By using the similar proof method of Theorem
If
Now we give an example for the main theorems with numerical analysis.
Consider
Then, we construct the iteration scheme (
Now, we construct the iteration scheme (
Finally, we compare the convergence rates of (
Comparison for convergent rates of iteration scheme (











9.00  6.63  5.58 

5.00  5.00  5.00  5.00 

9.00  7.77  6.80 

5.02  5.01  5.01  5.00 

9.00  8.10  7.30 

5.05  5.03  5.02  5.01 
The trajectory of
The trajectory of
The trajectory of
Consider the logistic maps described by (
The trajectories of
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.
The authors declare that there are no conflicts of interest regarding the publication of this paper.
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 ECommerce 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).