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

Authors: Ke Ding; Jong Kyu Kim; Qiang Lu; Bin Du
Journal: Discrete Dynamics in Nature and Society
 (2017)

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## 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.


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## Body

## 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., [1–30]).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

LetC be a nonempty convex subset of a normed linear space E. Let T be a contraction mapping of C into itself with the contraction constant μ; that is,(1)Tx-Ty≤μx-y,0<μ<1,for any x, y∈C. The set of fixed points of T is denoted by FT=x∈C:Tx=x. The set of natural numbers is denoted by N. αn and βn are two sequences of real numbers such that 0≤αn and 0≤βn for all n∈N. Consider the following scheme:(2)x1∈C,xn+1=αnxn+βnTxn,n∈N.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)
Ifαn=0 and βn=1, scheme (2) reduces to Picard iteration.(ii)
If0<αn<1 and βn=1-αn, scheme (2) reduces to Mann iteration.(iii)
If0<αn<1, βn=1-αn, and Txn=T~((1-γn)xn+γnT~xn), where T~ is a contraction mapping of C into itself and 0<γn<1, 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):(3)y1∈C,yn+1=αnyn+βnTyn+knxn-yn,n∈N,where kn is a scheme of feedback parameters which can be determined later. Let en=xn-yn for n∈N. Then we can construct the following scheme from schemes (2) and (3):(4)e1=x1-y1∈C,en+1=αn-knen+βnTxn-Tyn,n∈N.From [4, 31, 32], the fact limn→∞⁡en=0 will ensure limn→∞⁡xn-yn=0. The main purpose of this paper is to find the conditions to guarantee limn→∞⁡en=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 T.


## 3. Main Results

### 3.1. Convergence Results

Now, we give some convergence results for iteration (3).Theorem 2.
LetC be a nonempty convex subset of a normed linear space E. Let T be a contraction mapping of C into itself and F(T)≠ϕ. If (5)α-+β-μ-1<k_≤ki≤k-≤α_,where α-=max⁡αi, α_=min⁡αi, β-=max⁡βi, k-=max⁡ki, and k_=min⁡ki, i=1,2,…,n, then limn→∞⁡xn+1-yn+1=0, which also implies that scheme (3) and scheme (2) converge to the same fixed point of T.Proof.
From (5), we have (6)0<αi-ki,0<αi-ki+βiμfor i=1,2,…,n. Then,(7)en+1=αn-knen+βnTxn-Tyn≤αn-knen+βnTxn-Tyn≤αn-knen+βnμxn-yn≤αn-kn+βnμen⋮≤∏i=1nαi-ki+βiμei.It follows from (5) that 0<α-+β-μ-k_<1, where α-=max⁡{αi}, β-=max⁡βi, and k_=min⁡{ki}, i=1,2,…,n. Thus,(8)en+1≤∏i=1nαi-ki+βiμei≤α-+β-μ-k_ne1.It is easy to see that en+1→0, as n→∞; that is, limn→∞⁡xn+1-yn+1=0. This completes the proof.Remark 3.
It follows from Theorem2 that |ki|<1 for i=1,2,…,n.Theorem2 can be applied to approximating the fixed point of a nonexpansive mapping where the contraction constant μ=1. If μ=1, Theorem 2 reduces to the following result.Corollary 4.
LetC be a nonempty convex subset of a normed linear space E. Let T be a nonexpansive mapping of C into itself and F(T)≠ϕ. If (9)α-+β--1<k_≤ki≤k-≤α_,where α-=max⁡αi, α_=min⁡αi, β-=max⁡βi, k-=max⁡ki, and k_=min⁡ki, i=1,2,…,n, then limn→∞⁡xn+1-yn+1=0.


### 3.2. Three Special Cases

Now, we use Theorem2 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(10)x^1∈C,x^n+1=Tx^n,n∈N.We define the iteration scheme associated with Picard iteration scheme (10):(11)y^1∈C,y^n+1=Ty^n+knx^n-y^n,n∈N.Let e^n=x^n-y^n for n∈N. Schemes (2) and (3) give the following scheme:(12)e^1=x^1-y^1∈C,e^n+1=-kne^n+Tx^n-Ty^n,n∈N.Then, by the similar proof of Theorem2, we have the following convergence theorem.Theorem 5.
LetC be a nonempty convex subset of a normed linear space E. Let T be a contraction mapping of C into itself and F(T)≠ϕ. If max⁡{|ki|}+μ<1,i=1,2,…,n, then limn→∞⁡x^n+1-y^n+1=0.Second, we consider the Mann iteration scheme. The Mann iteration scheme is defined by(13)xˇ1∈C,xˇn+1=1-βnxˇn+βnTx˘n,n∈N.We construct the following iteration scheme associated with Mann iteration scheme (13): (14)yˇ1∈C,yˇn+1=1-βnyˇn+βnTyˇn+knxˇn-yˇn,n∈N.By defining an error variable eˇi=xˇi-yˇi for i=1,2,…,n, we obtain the following iteration scheme:(15)eˇ1=xˇ1-yˇ1∈C,eˇn+1=1-βn-kneˇn+βnTxˇn-Tyˇn,n∈N.Then, from the similar proof for Theorem2, we derive the following convergence result.Theorem 6.
LetC be a nonempty convex subset of a normed linear space E. Let T be a contraction mapping of C into itself and F(T)≠ϕ. If β-μ-β_<k_i≤ki≤k-<1-β-, where β_=min⁡{βi}, β-=max⁡{βi}, k-=max⁡{ki}, and k_=min⁡{ki}, i=1,2,…,n, then limn→∞⁡xˇn+1-yˇn+1=0.Third, we consider the Ishikawa iteration scheme. The Ishikawa iteration scheme is defined by(16)x~1∈C,x~n+1=1-βnx~n+βnT~1-γnx~n+γnT~x~n,n∈N,where T~ is a contraction mapping of C into itself with the contraction constant μ. We generate the following iteration scheme associated with Ishikawa iteration scheme (16):(17)y~1∈C,y~n+1=1-βny~n+βnT~1-γny~n+γnT~y~n+knx~n-y~n,n∈N.After defining an error variable e~i=x~i-y~i for i=1,2,…,n, we obtain the error scheme: (18)e~1=x~1-y~1∈C,e~n+1=1-βn-kne~n+βnT~1-γnx~n+γnT~x~n-T~1-γny~n+γnT~y~n,n∈N.Then, from the similar proof for Theorem2, we achieve the following convergence theorem.Theorem 7.
LetC be a nonempty convex subset of a normed linear space E. Let T be a contraction mapping of C into itself and F(T)≠ϕ. If β-μ(1-γ_+γ-μ)-β_<k_≤ki≤k-<1-β-, where β_=min{βi}, β-=max⁡{βi}, γ_=min⁡γi, γ-=max⁡γi, k-=max⁡{ki}, and k_=min⁡{ki}, i=1,2,…,n, then limn→∞⁡x~n+1-y~n+1=0.


### 3.3. Impact ofki to the Convergent Rate

Next, we analyze the influence of sizeki to the convergence rate of (3). We first give another iteration scheme associated with iteration scheme (2): (19)z1∈C,zn+1=αnzn+βnTzn+k˘nxn-zn,n∈N,where k˘n>kn. 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{an} and {bn} be two sequences of real numbers which converges to a and b, respectively. We say that the sequence {an} converges faster than {bn} if limn→∞⁡|an-a|/|bn-b|=0.Definition 9.
Let{xn}, {yn}, {zn} be three iterative schemes which satisfy zn→xn,yn→xn as n→∞. Let {an} and {bn} be two sequences of real numbers which converge to 0. We say that the scheme {zn} converges faster than {yn} to {xn}, if (20)zn-xn≤an,for  alln∈N,yn-xn≤bn,for  alln∈N,and an converges faster than bn.By the similar method for Theorem2, we have (21)zn+1-xn+1≤α-+β-μ-k_˘nz1-x1,where k_˘=min⁡{k˘i},i=1,2,…,n, and 0<α-+β-μ-k_˘<1. It follows from k_<k_˘ that 0<α-+β-μ-k_˘<α-+β-μ-k_<1, which implies (22)limn→∞⁡zn+1-xn+1yn+1-xn+1=limn→∞⁡α-+β-μ-k_˘α-+β-μ-k_nz1-x1y1-x1=0.Hence, from the above mentions, we have the following comparison result for the convergence rate according to the size ki.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 ki increases which means that the convergence rate of iteration scheme defined by (3) can be controlled by the adjustment of size ki.Remark 12.
IfT is a nonexpansive mapping, i.e., μ=1, limitation (22) reduces to (23)limn→∞⁡zn+1-xn+1yn+1-xn+1=limn→∞⁡α-+β--k_˘α-+β--k_nz1-x1y1-x1=0,which means that Theorem 10 is still valid for the nonexpansive mapping.


## 3.1. Convergence Results

Now, we give some convergence results for iteration (3).Theorem 2.
LetC be a nonempty convex subset of a normed linear space E. Let T be a contraction mapping of C into itself and F(T)≠ϕ. If (5)α-+β-μ-1<k_≤ki≤k-≤α_,where α-=max⁡αi, α_=min⁡αi, β-=max⁡βi, k-=max⁡ki, and k_=min⁡ki, i=1,2,…,n, then limn→∞⁡xn+1-yn+1=0, which also implies that scheme (3) and scheme (2) converge to the same fixed point of T.Proof.
From (5), we have (6)0<αi-ki,0<αi-ki+βiμfor i=1,2,…,n. Then,(7)en+1=αn-knen+βnTxn-Tyn≤αn-knen+βnTxn-Tyn≤αn-knen+βnμxn-yn≤αn-kn+βnμen⋮≤∏i=1nαi-ki+βiμei.It follows from (5) that 0<α-+β-μ-k_<1, where α-=max⁡{αi}, β-=max⁡βi, and k_=min⁡{ki}, i=1,2,…,n. Thus,(8)en+1≤∏i=1nαi-ki+βiμei≤α-+β-μ-k_ne1.It is easy to see that en+1→0, as n→∞; that is, limn→∞⁡xn+1-yn+1=0. This completes the proof.Remark 3.
It follows from Theorem2 that |ki|<1 for i=1,2,…,n.Theorem2 can be applied to approximating the fixed point of a nonexpansive mapping where the contraction constant μ=1. If μ=1, Theorem 2 reduces to the following result.Corollary 4.
LetC be a nonempty convex subset of a normed linear space E. Let T be a nonexpansive mapping of C into itself and F(T)≠ϕ. If (9)α-+β--1<k_≤ki≤k-≤α_,where α-=max⁡αi, α_=min⁡αi, β-=max⁡βi, k-=max⁡ki, and k_=min⁡ki, i=1,2,…,n, then limn→∞⁡xn+1-yn+1=0.


## 3.2. Three Special Cases

Now, we use Theorem2 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(10)x^1∈C,x^n+1=Tx^n,n∈N.We define the iteration scheme associated with Picard iteration scheme (10):(11)y^1∈C,y^n+1=Ty^n+knx^n-y^n,n∈N.Let e^n=x^n-y^n for n∈N. Schemes (2) and (3) give the following scheme:(12)e^1=x^1-y^1∈C,e^n+1=-kne^n+Tx^n-Ty^n,n∈N.Then, by the similar proof of Theorem2, we have the following convergence theorem.Theorem 5.
LetC be a nonempty convex subset of a normed linear space E. Let T be a contraction mapping of C into itself and F(T)≠ϕ. If max⁡{|ki|}+μ<1,i=1,2,…,n, then limn→∞⁡x^n+1-y^n+1=0.Second, we consider the Mann iteration scheme. The Mann iteration scheme is defined by(13)xˇ1∈C,xˇn+1=1-βnxˇn+βnTx˘n,n∈N.We construct the following iteration scheme associated with Mann iteration scheme (13): (14)yˇ1∈C,yˇn+1=1-βnyˇn+βnTyˇn+knxˇn-yˇn,n∈N.By defining an error variable eˇi=xˇi-yˇi for i=1,2,…,n, we obtain the following iteration scheme:(15)eˇ1=xˇ1-yˇ1∈C,eˇn+1=1-βn-kneˇn+βnTxˇn-Tyˇn,n∈N.Then, from the similar proof for Theorem2, we derive the following convergence result.Theorem 6.
LetC be a nonempty convex subset of a normed linear space E. Let T be a contraction mapping of C into itself and F(T)≠ϕ. If β-μ-β_<k_i≤ki≤k-<1-β-, where β_=min⁡{βi}, β-=max⁡{βi}, k-=max⁡{ki}, and k_=min⁡{ki}, i=1,2,…,n, then limn→∞⁡xˇn+1-yˇn+1=0.Third, we consider the Ishikawa iteration scheme. The Ishikawa iteration scheme is defined by(16)x~1∈C,x~n+1=1-βnx~n+βnT~1-γnx~n+γnT~x~n,n∈N,where T~ is a contraction mapping of C into itself with the contraction constant μ. We generate the following iteration scheme associated with Ishikawa iteration scheme (16):(17)y~1∈C,y~n+1=1-βny~n+βnT~1-γny~n+γnT~y~n+knx~n-y~n,n∈N.After defining an error variable e~i=x~i-y~i for i=1,2,…,n, we obtain the error scheme: (18)e~1=x~1-y~1∈C,e~n+1=1-βn-kne~n+βnT~1-γnx~n+γnT~x~n-T~1-γny~n+γnT~y~n,n∈N.Then, from the similar proof for Theorem2, we achieve the following convergence theorem.Theorem 7.
LetC be a nonempty convex subset of a normed linear space E. Let T be a contraction mapping of C into itself and F(T)≠ϕ. If β-μ(1-γ_+γ-μ)-β_<k_≤ki≤k-<1-β-, where β_=min{βi}, β-=max⁡{βi}, γ_=min⁡γi, γ-=max⁡γi, k-=max⁡{ki}, and k_=min⁡{ki}, i=1,2,…,n, then limn→∞⁡x~n+1-y~n+1=0.


## 3.3. Impact ofki to the Convergent Rate

Next, we analyze the influence of sizeki to the convergence rate of (3). We first give another iteration scheme associated with iteration scheme (2): (19)z1∈C,zn+1=αnzn+βnTzn+k˘nxn-zn,n∈N,where k˘n>kn. 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{an} and {bn} be two sequences of real numbers which converges to a and b, respectively. We say that the sequence {an} converges faster than {bn} if limn→∞⁡|an-a|/|bn-b|=0.Definition 9.
Let{xn}, {yn}, {zn} be three iterative schemes which satisfy zn→xn,yn→xn as n→∞. Let {an} and {bn} be two sequences of real numbers which converge to 0. We say that the scheme {zn} converges faster than {yn} to {xn}, if (20)zn-xn≤an,for  alln∈N,yn-xn≤bn,for  alln∈N,and an converges faster than bn.By the similar method for Theorem2, we have (21)zn+1-xn+1≤α-+β-μ-k_˘nz1-x1,where k_˘=min⁡{k˘i},i=1,2,…,n, and 0<α-+β-μ-k_˘<1. It follows from k_<k_˘ that 0<α-+β-μ-k_˘<α-+β-μ-k_<1, which implies (22)limn→∞⁡zn+1-xn+1yn+1-xn+1=limn→∞⁡α-+β-μ-k_˘α-+β-μ-k_nz1-x1y1-x1=0.Hence, from the above mentions, we have the following comparison result for the convergence rate according to the size ki.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 ki increases which means that the convergence rate of iteration scheme defined by (3) can be controlled by the adjustment of size ki.Remark 12.
IfT is a nonexpansive mapping, i.e., μ=1, limitation (22) reduces to (23)limn→∞⁡zn+1-xn+1yn+1-xn+1=limn→∞⁡α-+β--k_˘α-+β--k_nz1-x1y1-x1=0,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.IfTxn=-βnxn2 and αn=βn=r, then scheme (2) reduces to the following logistic map (24)xn+1=rxn-rxn2,n∈N,where x1∈C. If 0<r<0.5 and 0<x1<1, then we can derive 0<xn<1, which implies that (25)Txn-Tyn=rxn2-yn2=rxn+ynxn-yn<xn-ynfor any 0<xn,yn<1.Here, we consider another logistic map(26)yn+1=ryn-ryn2,n∈N,where y1∈C. Defining en=xn-yn, we can have the following scheme: (27)en+1=r1-xn+ynen,n∈N,where e1=x1-y1∈C.Definition 13.
Iflimn→∞⁡xn-yn=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 Theorem2 with ki=0 and r1-xn+yn<0.5, we can derive the following result.Theorem 14.
If0<r<0.5 and 0<x1,y1<1, 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.
ConsiderTx=x2-8x+40 on [4.1,10] with the contraction constant μ=0.78. The fixed point of T is x=5. We first construct the iteration scheme (11) associated with Picard iteration (10). From Theorem 5, we know that max⁡{|ki|}+μ<1,i=1,2,…,n, which implies |ki|<0.22 for i=1,2,…,n. We choose ki=0.21, for i=1,2,…,n, and x1=6 and y^1=8. Figure 1 gives a demonstration of trajectory of e^n+1 of (11). From Figure 1, we can see that limn→∞⁡e^n+1=0; that is, limn→∞⁡xn+1-y^n+1=0, which indicates the effectiveness of Theorem 5.
Then, we construct the iteration scheme (14) associated with Mann iteration (13) with βi=0.75 for i=1,2,…,n. From Theorem 6, we can get β-μ-β_<k_≤ki≤k-<1-β-, which indicates -0.165<ki<0.25 for i=1,2,…,n. We choose ki=0.24, for i=1,2,…,n, and x1=6 and yˇ1=8. Figure 2 provides the trajectory of eˇn+1 of (14). From Figure 2, we can observe that limn→∞⁡eˇn+1=0; that is, limn→∞⁡xn+1-yˇn+1=0, which indicates the effectiveness of Theorem 6.
Now, we construct the iteration scheme (17) associated with Ishikawa iteration (16) with βi=0.5 and γi=0.75 for i=1,2,…,n. From Theorem 7, we can have β-μ(1-γ_+γ-μ)-β_<k_≤ki≤k-<1-β-, which implies -0.1744<ki<0.5 for i=1,2,…,n. We choose ki=0.49, for i=1,2,…,n, and x1=6 and y~1=8. Figure 3 reveals the trajectory of e~n+1 of (17). It follows from Figure 3 that limn→∞⁡e~n+1=0; that is, limn→∞⁡xn+1-y~n+1=0, which indicates the effectiveness of Theorem 7.
Finally, we compare the convergence rates of (3) with different ki. We choose scheme (14) with βi=0.6. From Theorem 6, we can have -0.132<ki<0.4 for i=1,2,…,n. Let x1=6 and y1=z1=9. We choose ki as 0.39, 0.01, and -0.1 to approximate the fixed point, respectively. Table 1 shows that the convergence rate of (3) increases as ki increases, which also illustrates the effectiveness of Theorem 10.Table 1
Comparison for convergent rates of iteration scheme (3) with different ki.
n
=
1
n
=
2
n
=
3
⋯
n
=
17
n
=
18
n
=
19
n
=
20
y
n as ki=0.39
9.00
6.63
5.58
⋯
5.00
5.00
5.00
5.00
y
n as ki=0.01
9.00
7.77
6.80
⋯
5.02
5.01
5.01
5.00
y
n as ki=-0.1
9.00
8.10
7.30
⋯
5.05
5.03
5.02
5.01Figure 1
The trajectory ofe^n+1 with μ=0.78 and ki=0.21.Figure 2
The trajectory ofeˇn+1 with μ=0.78 and ki=0.24.Figure 3
The trajectory ofe~n+1 with μ=0.78 and ki=0.49.Example 16.
Consider the logistic maps described by (24) and (26) with r=0.4 and x1=0.9 and y1=0.6. Figure 4 reveals the global synchronization of the logistic maps described by (24) and (26) which illustrates the effectiveness of Theorem 14.Figure 4
The trajectories ofxn, yn, en.


## 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.



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