An Iteration Scheme for Contraction Mappings with an Application to Synchronization of Discrete Logistic Maps

1School of Automation, Hangzhou Dianzi University, Zhejiang 310018, China 2Jiangxi E-Commerce High Level Engineering Technology Research Centre, Jiangxi University of Finance and Economics, Nanchang 330013, China 3Department of Mathematics Education, Kyungnam University, Changwon, Gyeongnam 631-701, Republic of Korea 4School of Information Technology, Jiangxi University of Finance and Economics, Nanchang 330013, China


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., ).
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.

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,      −      ≤       −      , 0 <  < 1, for any ,  ∈ .The set of fixed points of  is denoted by () = { ∈  :  = }.The set of natural numbers is denoted by N. {  } and {  } are two sequences of real numbers such that 0 ≤   and 0 ≤   for all  ∈ N. 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.
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  ∈ N. Then we can construct the following scheme from schemes (2) and (3): From [4,31,32], the fact lim →∞ ‖  ‖ = 0 will ensure lim →∞ ‖  −   ‖ = 0.The main purpose of this paper is to find the conditions to guarantee lim →∞ ‖  ‖ = 0, 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 .

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 () ̸ = .
Theorem 2 can be applied to approximating the fixed point of a nonexpansive mapping where the contraction constant  = 1.If  = 1, 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 () ̸ = .

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 by We define the iteration scheme associated with Picard iteration scheme (10): Let ê = x − ŷ for  ∈ N. Schemes ( 2) and (3) give the following scheme: Then, by the similar proof of Theorem 2, we have the following convergence theorem.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 ě  = x  − y  for  = 1, 2, . . ., , we obtain the following iteration scheme: Then, from the similar proof for Theorem 2, we derive the following convergence result.Third, we consider the Ishikawa iteration scheme.The Ishikawa iteration scheme is defined by where T 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 ẽ = x − ỹ for  = 1, 2, . . ., , we obtain the error scheme: Then, from the similar proof for Theorem 2, we achieve the following convergence theorem.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 k  >   .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 and {  } converges faster than {  }.
which means that Theorem 10 is still valid for the nonexpansive mapping.

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.
Definition 13.If lim →∞ |  −   | = 0, 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   = 0 and |(1 − (  +   ))| < 0.5, we can derive the following result.

Two Illustrated Examples
Example 15.Now we give an example for the main theorems with numerical analysis.

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.
−   )   +   (  −   ) Hence, from the above mentions, we have the following comparison result for the convergence rate according to the size   .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   .

Table 1 :
Comparison for convergent rates of iteration scheme (3) with different   .