# On a New Faster  Implicit Fixed Point Iterative Scheme in Convex Metric Spaces

Authors: Renu Chugh; Preety Malik; Vivek Kumar
Journal: Journal of Function Spaces
 (2015)

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## Abstract

The purpose of this paper is to consider a new implicit iteration and study its strong convergence, stability, and data dependence. It is proved through numerical examples that newly introduced iteration has better convergence rate than well known implicit Mann iteration as well as implicit Ishikawa iteration and implicit iterations converge faster as compared to corresponding explicit iterations. Applications of implicit iterations to RNN (Recurrent Neural Networks) analysis are also presented.


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## Body

## 1. Introduction

In recent years, numerous papers have been published on explicit iterations in various spaces [1–6], but there are very few works on implicit iterations (regarding convergence rate and data dependence) [7–13]. Implicit iterations have an advantage over explicit iterations for nonlinear problems as they provide better approximation of fixed points and are widely used in many applications when explicit iterations are inefficient. Approximation of fixed points in computer oriented programs by using implicit iterations can reduce the computational cost of the fixed point problems. The study of stability of iterations enjoys a celebrated place in applied sciences and engineering due to chaotic behavior of functions in discrete dynamics and other numerical computations. Data dependence of fixed points is a related and new issue which has been studied by many authors; see [4, 14] and references therein. In computational mathematics, it is of theoretical and practical importance to compare the convergence rate of iterations and to find out, if possible, which one of them converges more rapidly to the fixed point. Recent works in this direction are [1, 3, 4, 15–17]. In concrete, a fixed point iteration is valuable from a numerical point of view and is useful for applications if it satisfies the following requirements:(a)
it converges to fixed point of an operator;(b)
it isT-stable;(c)
it is faster as compared to other iterations existing in the literature;(d)
it shows data dependence results.Motivated by the fact that three-step iterations give better approximation than one-step and two-step iterations [18], we define a new and more general three-step implicit iteration (IN) which satisfies the above requirements.LetK be a nonempty convex subset of a convex metric space X and let T:K→K be a given mapping. For the real sequences {αn}n=0∞, {βn}n=0∞, and {γn}n=0∞ in [0,1], Noor iteration [19] in convex metric spaces can be written asNxn=Wxn-1,Tyn-1,αn,yn-1=Wxn-1,Tzn-1,βn,zn-1=Wxn-1,Txn-1,γn,n=0,1,2,….Putting γn=1 in (N), we get well known Ishikawa iteration [20, 21] in convex metric spaces:Ixn=Wxn-1,Tyn-1,αn,yn-1=Wxn-1,Txn-1,βn,n=0,1,2….Putting γn=βn=1 in (N), we get well known Mann iteration [21, 22] in convex metric spaces:Mxn=Wxn-1,Txn-1,αn,n=0,1,2….Forx0∈K, we define the following iteration, namely, implicit Noor iteration in convex metric spaces:INxn=Wyn-1,Txn,αn,yn-1=Wzn-1,Tyn-1,βn,zn-1=Wxn-1,Tzn-1,γn,where {αn},{βn} and {γn} are sequences in [0,1].Equivalence form of iteration(IN) in linear space can be written as(1)xn=αnyn-1+1-αnTxn,yn-1=βnzn-1+1-βnTyn-1,zn-1=γnxn-1+1-γnTzn-1.Putting γn=1 in (IN), we get well known implicit Ishikawa iteration [23]:IIxn=Wyn-1,Txn,αn,yn-1=Wxn-1,Tyn-1,βn.Putting γn=βn=1 in (IN), we get well known implicit Mann iteration [2, 6, 11, 24]:IMxn=Wxn-1,Txn,αn=αnxn-1+1-αnTxn.Zamfirescu operators [25] are the most general contractive like operators which have been studied by several authors, satisfying the following condition: for each pair of points x,y in X at least one of the following is true:(2)idTx,Ty≤pdx,yiidTx,Ty≤qdx,Tx+dy,TyiiidTx,Ty≤rdx,Ty+dy,Tx,where p, q, and r are nonnegative constants satisfying 0≤p≤1, 0≤q, and r≤1/2.Z-operators are equivalent to the following contractive condition:(3)dTx,Ty≤cmax⁡⁡dx,y,dx,Tx+dy,Ty2,000000000dx,Ty+dy,Tx20000000000000∀x,y∈X,0<c<1.The contractive condition (3) implies(4)dTx,Ty≤2adx,Tx+adx,y,∀x,y∈X,where a=max⁡⁡{c,c/(2-c)} [see [15]].In [5], Rhoades used the following more general contractive condition than (4): there exists c∈[0,1) such that(5)dTx,Ty≤cmax⁡⁡dx,y,dx,Tx+dy,Ty2,000000000dx,Ty,dy,Txdx,Tx+dy,Ty2∀x,y∈X.In [26], Osilike used a more general contractive definition than those of Rhoades: there exist a∈[0,1), L≥0 such that(6)dTx,Ty≤Ldx,Tx+adx,y∀x,y∈X.We use the contractive condition due to Imoru and Olatinwo [27], which is more general than (6): there exist a∈[0,1) and a monotone increasing function φ:R+→R+ with φ(0)=0, such that(7)dTx,Ty≤ϕdx,Tx+adx,y,00000a∈0,1,∀x,y∈X.Also, we use the following definitions and lemmas to achieve our main results.Definition 1 (see [23]).
A mapW:X2×[0,1]→X is a convex structure on X if (8)du,Wx,y,λ≤λdu,x+1-λdu,yfor all x,y,u∈X and λ∈[0,1]. A metric space (X,d) together with a convex structure W is known as convex metric space and denoted by (X,d,W). A nonempty subset C of a convex metric space is convex if W(x,y,λ)∈C for all x,y∈C and λ∈[0,1].All normed spaces and their subsets are the examples of convex metric spaces. But there are many examples of convex metric spaces which are not embedded in any normed space (see [23, 28]). After that several authors extended this concept in many ways; one such convex structure is hyperbolic space which was introduced by Kohlenbach [29] as follows.Definition 2 (see [29]).
A hyperbolic space(X,d,W) is a metric space (X,d) together with a convexity mapping W:X2×[0,1]→X satisfying(W1)
d(z,W(x,y,λ))≤(1-λ)d(z,x)+λd(z,y),(W2)
d(W(x,y,λ1),W(x,y,λ2))=λ1-λ2d(x,y),(W3)
W(x,y,λ)=W(y,x,1-λ),(W4)
d(W(x,z,λ),W(y,w,λ))≤(1-λ)d(x,y)+λd(z,w) for all x,y,z,w∈X and λ,λ1,λ2∈[0,1].Evidently every hyperbolic space is a convex metric space but the converse may not true. For example, if we take X=R, W(x,y,λ)=λx+(1-λ)y and define d(x,y)=x-y/(1+x-y) for x,y∈R, then (X,d,W) is a convex metric space but not a hyperbolic space.The stability of explicit as well as implicit iterations has extensively been studied by various authors [4, 7, 21, 27, 30–32] due to its increasing importance in computational mathematics, especially due to revolution in computer programming. The concept of T-stability in convex metric space setting was given by Olatinwo [21].Definition 3 (see [21]).
Let(X,d,W) be a convex metric space and let T:X→X a self-mapping.
Let{xn}n=0∞⊂X be the sequence generated by an iterative scheme involving T which is defined by (9)xn+1=fT,αnxn,n=0,1,2…,where x0∈X is the initial approximation and fT,αnxn is some function having convex structure such that αn∈[0,1]. Suppose that {xn} converges to a fixed point p of  T. Let {yn}n=0∞⊂X be an arbitrary sequence and set εn=d(yn+1,fT,αnyn). Then, the iteration (9) is said to be T-stable with respect to T if and only if  limn→∞⁡εn=0 implies limn→∞⁡yn=p.Lemma 4 (see [4, 15]).
Ifδ is a real number such that 0≤δ<1 and {∈n}n=0∞ is a sequence of positive numbers such that limn→∞⁡∈n=0, then for any sequence of positive numbers {un}n=0∞ satisfying(10)un+1≤δun+∈n,n=0,1,2,…,one has limn→∞⁡un=0.Definition 5 (see [15]).
Suppose{an} and {bn} are two real convergent sequences with limits a and b, respectively. Then {an} is said to converge faster than {bn} if (11)limn→∞an-abn-b=0.Definition 6 (see [15]).
Let{un} and {vn} be two fixed point iterations that converge to the same fixed point p on a normed space X such that the error estimates (12)un-p≤an,vn-p≤bnare available, where {an} and {bn} are two sequences of positive numbers (converging to zero). If {an} converge faster than {bn}, then one says that {un} converge faster to p than {vn}.Definition 7 (see [14]).
LetT,T1 be two operators on X. One says T1 is approximate operator of T if, for all x∈X and for a fixed ε>0, one has d(Tx,T1x)≤ε.Lemma 8 (see [4, 14]).
Let{an}n=0∞ be a nonnegative sequence for which there exists n0∈N such that, for all n≥n0, one has the following inequality:(13)an+1≤1-rnan+rntn,where rn∈(0,1), for all n∈N, ∑n=1∞rn=∞, and tn≥0∀n∈N.
Then,0≤limn→∞⁡sup⁡an≤limn→∞⁡sup⁡tn.Having introduced the implicit Noor iteration(IN), we use it to prove the results concerning convergence, stability, and convergence rate for contractive condition (7) in convex metric spaces. Also, data dependence result of the same iteration is proved in hyperbolic spaces. Moreover, applications of implicit iterations in RNN analysis will be discussed in the last section.


## 2. Convergence and Stability Results of New Implicit Iteration in Convex Metric Spaces

Theorem 9.
LetK be a nonempty closed convex subset of a convex metric space X and let T be a quasi-contractive operator satisfying (7) with F(T)≠ϕ. Then, for x0∈C, the sequence {xn} defined by (IN), with ∑(1-αn)=∞, converges to the fixed point of  T.Proof.
Using(IN) and (7), we have, for p∈F(T),(14)dxn,p=dWyn-1,Txn,αn,p≤αndyn-1,p+1-αndTxn,p≤αndyn-1,p+1-αnadxn,p,which further implies(15)1-1-αnadxn,p≤αndyn-1,pdxn,p≤αn1-1-αnadyn-1,p.Again from (IN), we have the following estimates:(16)dyn-1,p=dWzn-1,Tyn-1,βn,p≤βndzn-1,p+1-βndTyn-1,p≤βndzn-1,p+1-βnadyn-1,p,which gives(17)dyn-1,p≤βn1-1-βnadzn-1,p,dzn-1,p=dWxn-1,Tzn-1,γn,p≤γndxn-1,p+1-γndTzn-1,p≤γndxn-1,p+1-γnadzn-1,p,implying that(18)dzn-1,p≤γn1-1-γnadxn-1,p.Using (14)–(18), we arrive at(19)dxn,p≤αn1-1-αnaβn1-1-βna×γn1-1-γnadxn-1,p.If we take αn/(1-1-αna)=An/Bn, then(20)1-AnBn=1-αn1-1-αna=1-1-αna+αn1-1-αna≥1-1-αna+αn,and hence(21)AnBn≤1-αna+αn=1-1-a1-αn.Similarly, with ease we can show that(22)βn1-1-βna≤βn+1-βna=1-1-βn1-a≤1,(23)γn1-1-γna≤γn+1-γna=1-1-γn1-a≤1.Using (21)–(23), (19) becomes(24)dxn,p≤1-1-αn1-adxn-1,p≤∏i=1n1-1-αi1-adx0,p≤e-∑i=1n1-αi1-adx0,p.But ∑i=1n(1-αi)=∞; hence (24) yields limn→∞⁡d(xn,p)=0. Therefore {xn} converges to p.Theorem 10.
LetK be a nonempty closed convex subset of a convex metric space X and let T be a quasi-contractive operator satisfying (7) with F(T)≠ϕ. Then, for x0∈C, the sequence {xn} defined by (IN) with αn≤α<1, ∑(1-αn)=∞, is T-stable.Proof.
Suppose that{pn}n=0∞⊂K is an arbitrary sequence, εn=d(pn,W(qn-1,Tpn,αn)), where qn-1=W(rn-1,Tqn-1,βn), rn-1=W(pn-1,Trn-1,γn) and let limn→∞⁡εn=0.
Then, using (7) we have(25)dpn,p≤dpn,Wqn-1,Tpn,αn+dWqn-1,Tpn,αn,p≤εn+αndqn-1,p+1-αndTpn,p≤εn+αndqn-1,p+1-αnφdTp,p+1-αnadpn,p,which implies (26)1-1-αnadpn,p≤εn+αndqn-1,p,and therefore(27)dpn,p≤αn1-1-αnadqn-1,p+εn1-1-αna.Using (21), (27) becomes(28)dpn,p≤1-1-αn1-adqn-1,p+εn1-1-αna.Now, using (22) and (23), we have the following estimates:(29)dqn-1,p≤βn1-1-βnadrn-1,p≤drn-1,p,drn-1,p≤γn1-1-γnadpn-1,p≤dpn-1,p.Using αn≤α<1and a∈[0,1), we have 1-(1-αn)(1-a)<1. Hence using Lemma 4, together with estimates (29), (28) yields limn→∞⁡pn=p.
Conversely, if we letlimn→∞⁡pn=p, then using contractive condition (7), it is easy to see that limn→∞⁡εn=0.
Therefore, the iteration(IN) is T-stable.Remark 11.
As contractive condition (7) is more general than those of (2)–(6), the convergence and stability results for implicit iteration (IN) using contractive conditions (2)–(6) can be obtained as special cases.Remark 12.
As implicit Mann iteration(IM) and implicit Ishikawa iteration (II) are special cases of implicit iteration (IN), results similar to Theorems 9 and 10 hold for these iterations as well.


## 3. Convergence Rate of Implicit Iterations

Theorem 13.
LetK be a nonempty closed convex subset of a convex metric space X and let T be a quasi-contractive operator satisfying (7) with F(T)≠ϕ. Then, for x0∈C, the sequence {xn} defined by (IN) with ∑(1-αn)=∞, αn≤α<1, converges faster than implicit Mann (IM) and implicit Ishikawa (II) iterations to the fixed point of T. Moreover, implicit iterations converge faster than the corresponding explicit iterations.Proof.
For implicit Mann iteration(IM), we have(30)dxn,p=dWxn-1,Txn,αn,p≤αndxn-1,p+1-αndTxn,p≤αndxn-1,p+1-αnadxn,p,which further yield(31)1-1-αnadxn,p≤αndxn-1,p,and so(32)dxn,p≤αn1-1-αnadxn-1,p.Similarly, for implicit Ishikawa (II) and implicit Noor (IN) iterations, we have the estimates (33) and (34), respectively, as follows:(33)dxn,p≤αn1-1-αnaβn1-1-βnadxn-1,p,(34)dxn,p≤αn1-1-αnaβn1-1-βna×γn1-1-γnadxn-1,p.Also, for explicit Mann iteration (M), we have (35)dxn,p=dWxn-1,Txn-1,αn,p≤αndxn-1,p+1-αndTxn-1,p≤αndxn-1,p+1-αnadxn-1,p≤αn+1-αnadxn-1,p.For explicit Ishikawa iteration (I), we have (36)dxn,p=dWxn-1,Tyn-1,αn,p≤αndxn-1,p+1-αndTyn-1,p≤αndxn-1,p+1-αnadyn-1,p,(37)dyn-1,p=dWxn-1,Txn-1,βn,p≤βn+1-βnadxn-1,p.Using (37), (36) becomes(38)dxn,p≤αn+1-αnaβn+1-βnadxn-1,p.Similarly, for explicit Noor iteration (N), we have(39)dxn,p≤αn+1-αnaβn+1-αn1-βna2000×γn+1-γnadxn-1,p.Now, using (21) and (22), we obtain(40)αn1-1-αnaβn1-1-βna≤αn+1-αnaβn+1-βna=αn+1-αna1-1-βn1-a≤αn+1-αnaβn+1-αn1-βna2.Similarly using (21)–(23), we get(41)αn1-1-αnaβn1-1-βnaγn1-1-γna≤αn+1-αnaβn+1-βnaγn+1-γna≤αn+1-αnaβn+1-αn1-βna2×γn+1-γna.Keeping in mind Berinde’s Definitions 5 and 6, inequalities (21), (32), and (35) yield that implicit Mann iteration (IM) converges faster than explicit Mann iteration (M), inequalities (33), (38), and (40) yield that implicit Ishikawa iteration (II) converges faster than explicit Ishikawa iteration (I), and inequalities (34), (39), and (41) yield that implicit Noor iteration (IN) converges faster than explicit Noor iteration (N).
Moreover, again using Berinde’s Definitions5 and 6 with(42)αn1-1-αnaβn1-1-βna≤αn1-1-αna,αn1-1-αnaβn1-1-βnaγn1-1-γna≤αn1-1-αnaβn1-1-βna,and inequalities (32)–(34), we conclude that decreasing order of convergence speed of implicit iterations is as follows: implicit Noor, implicit Ishikawa, and implicit Mann iteration.Example 14.
LetK=[0,1], T(x)=x/4, x0≠0, αn=βn=γn=1-(4/n), n≥25, and for n=1,2,…,24, αn=βn=γn=0; then for implicit Mann iteration (IM), we have(43)xn=αnxn-1+1-αnTxn=1-4nxn-1+4nxn4,which further implies(44)xn1-1n=1-4nxn-1,and so(45)xn=n-4n-1xn-1=∏i=25ni-4i-1x0.Also, for implicit Ishikawa iteration (II), we have(46)xn=n-4n-1yn-1,with (47)yn-1=n-4n-1xn-1.Using (47), (46) becomes(48)xn=n-4n-12xn-1=∏i=25ni-4i-12x0.Similarly, for implicit Noor iteration (IN), we have(49)xn=∏i=25ni-4i-13x0.Using (48) and (49), we have(50)xnINxnII=∏i=25n‍i-4i-1=∏i=25n‍1-3i-1with(51)0≤limn→∞∏i=25n‍1-3i-1≤limn→∞∏i=25n1-1i=limn→∞2425×2526⋯n-1n=limn→∞24n=0.Hence limn→∞⁡|(xn(IN)-0)/(xn(II)-0)|=0. Therefore using Definition 5, implicit Noor iteration (IN) converges faster than the implicit Ishikawa iteration (II) to the fixed point p=0.
Similarly, using (45) and (48), we have(52)limn→∞xnII-0xnIM-0=limn→∞∏i=25ni-4i-1=0.That is, implicit Ishikawa iteration (II) converges faster than the implicit Mann iteration (IM) to the fixed point p=0.
Now, we compare implicit iterations with their corresponding explicit iterations.
For explicit Mann iteration(M), we have(53)xn=αnxn-1+1-αnTxn-1=1-4nxn-1+4nxn-14=1-3nxn-1=∏i=25n1-3ix0.Similarly, for explicit Ishikawa and explicit Noor iterations, we have following estimates (54) and (55), respectively:(54)xn-1=1-3n-3nxn-1=∏i=25n1-3i-3ix0,(55)xn-1=1-3n-3n-3n3/2xn-1=∏i=25n1-3i-3i-3i3/2x0.Using (45) and (53), we arrive at(56)xnIMxnM=∏i=25ni-4i-1ii-3=∏i=25ni-4ii-4i+3,with(57)0≤limn→∞∏i=25ni-4ii-4i+3≤limn→∞∏i=25n1-3i-4i+3≤limn→∞∏i=25n1-1i=limn→∞2425×2526⋯n-1n=limn→∞24n=0.Hence limn→∞⁡|(xn(IM)-0)/(xn(M)-0)|=0. Therefore using Definition 5, implicit Mann iteration (IM) converges faster than the corresponding explicit Mann iteration (M) to the fixed point p=0.
Similarly, using (48) and (54), we have(58)xnIIxnI=∏i=25ni-4i-12ii-3i-3=∏i=25ni2-8i3/2+16ii2-5i3/2+4i+3i-3,with(59)0≤limn→∞∏i=25ni2-8i3/2+16ii2-5i3/2+4i+3i-3≤limn→∞∏i=25n1-3i3/2-12i+3i-3i2-5i3/2+4i+3i-3≤limn→∞∏i=25n1-1i=limn→∞2425×2526⋯n-1n=limn→∞24n=0,which implies that limn→∞⁡|(xn(II)-0)/(xn(I)-0)|=0. That is, implicit Ishikawa iteration (II) converges faster than the corresponding explicit Ishikawa iteration (I) to the fixed point p=0.
Again, similarly using (49) and (55), we have(60)xnINxnN=∏i=25ni-4i-13i3/2i3/2-3i-3i-3=∏i=25ni3-12i5/2+48i2-64i3/2i3-6i5/2+9i2-4i3/2+3i-6i+3,with(61)0≤limn→∞∏i=25ni3-12i5/2+48i2-64i3/2i3-6i5/2+9i2-4i3/2+3i-6i+3≤limn→∞∏i=25n1-6i5/2-39i2+60i3/2+3i-6i+3i3-6i5/2+9i2-4i3/2+3i-6i+3≤limn→∞∏i=25n1-1i=limn→∞2425×2526⋯n-1n=limn→∞24n=0,which implies that limn→∞⁡|(xn(IN)-0)/(xn(N)-0)|=0. That is, implicit Noor iteration (IN) converges faster than the explicit Noor iteration (N) to the fixed point p=0.
Using computer programs in C++, the convergence speed of various iterations is compared and observations are listed in Table1 by taking initial approximation x0=1, T(x)=x/4 and αn=βn=γn=1-(4/n), n≥25. The table reveals that implicit Noor iteration has better convergence rate as compared to implicit Mann iteration as well as implicit Ishikawa iteration and implicit iterations converge faster than corresponding explicit iterations.Table 1
Convergence rate comparison of various iterations.
Number of iterations(n)
Mann iteration (M)
Ishikawa iteration (I)
Noor iteration (N)
Implicit Mann iteration (IM)
Implicit Ishikawa iteration (II)
Implicit Noor iteration (IN)
25
0.4
0.28
0.256
0.25
0.0625
0.015625
26
0.164661
0.082955
0.070051
0.0670294
0.004493
0.000301
27
0.0695938
0.025844
0.020326
0.0191074
0.000365
6.97593e  − 06
28
0.0301378
0.008423
0.006213
0.00575025
3.306536e  − 05
1.90134e  − 07
29
0.0133485
0.002859
0.00199
0.00181636
3.299167e  − 06
5.992477e  − 09
30
0.00603721
0.001007
0.000665
0.000599294
3.591534e  − 07
2.152385e  − 10
31
0.00278426
0.000367
0.000231
0.000205692
4.230913e  − 08
8.702644e  − 12
32
0.00130768
0.000138
8.287781e  − 05
7.31828e  − 005
5.355715e  − 09
3.91946e  − 13
33
0.000624769
5.337885e  − 05
3.075038e  − 05
2.69091e  − 005
7.240995e  − 10
1.948486e  − 14
34
0.000303328
2.120576e  − 05
1.175085e  − 05
1.01987e  − 005
1.040128e  − 10
1.060791e  − 15
35
0.000149513
8.63484e  − 06
4.614614e  − 06
3.97501e  − 006
1.580069e  − 11
6.280786e  − 17
36
7.47563e  − 005
3.59785e  − 06
1.858664e  − 06
1.59e  − 006
2.52811e  − 12
4.019703e  − 18
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## 4. Data Dependence of Implicit Iterations in Hyperbolic Spaces

Theorem 15.
LetT:K→K be a mapping satisfying (7). Let T1 be an approximate operator of T as in Definition 7, and let {xn}n=0∞, {un}n=0∞ be two implicit iterations associated to T,T1 and defined by (IN)(62)un=Wvn-1,T1un,αn,vn-1=Wwn-1,T1vn-1,βn,wn-1=Wun-1,T1wn-1,γn,respectively, where {αn}n=0∞,{βn}n=0∞, and {γn}n=0∞ are real sequences in [0,1] satisfying ∑n=0∞(1-αn)=∞, αn≤βn, αn≤γn. Let p=Tp and q=T1q, then, for ε>0, we have the following estimate:(63)dp,q≤ε1-a2.Proof.
Using Definition2, iteration (IN), and iteration (62), we have the following estimates:(64)dxn,un=dWyn-1,Txn,αn,Wvn-1,T1un,αn≤αndyn-1,vn-1+1-αndTxn,T1un≤αndyn-1,vn-1+1-αn×dTxn,T1xn+dT1xn,T1un≤αndyn-1,vn-1+1-αn×ε+φdxn,T1xn+adxn,un,which further gives(65)1-1-αnadxn,un≤αndyn-1,vn-1+1-αnε+φdxn,T1xn,dxn,un≤αn1-1-αnadyn-1,vn-1+1-αn1-1-αnaε+φdxn,T1xn,with(66)dyn-1,vn-1=dWzn-1,Tyn-1,βn,Wwn-1,T1vn-1,βn≤βndzn-1,wn-1+1-βndTyn-1,T1vn-1,(67)dTyn-1,T1vn-1≤dTyn-1,T1yn-1dTyn-1,T1vn-1≤+dT1yn-1,T1vn-1dTyn-1,T1vn-1≤ε+φdyn-1,T1yn-1dTyn-1,T1vn-1≤+adyn-1,vn-1.Using (66) and (67), we have(68)dyn-1,vn-1≤βn1-1-βnadzn-1,wn-1+1-βn1-1-βna×ε+φdyn-1,T1yn-1,with(69)dzn-1,wn-1≤γndxn-1,un-1+1-γndTzn-1,T1wn-1,(70)dTzn-1,T1wn-1≤dTzn-1,T1zn-1+dT1zn-1,T1wn-1≤ε+φdzn-1,T1zn-1+adzn-1,wn-1.Using (69) and (70) and simplifying, we get the following estimate:(71)dzn-1,wn-1≤γn1-1-γnadxn-1,un-1+1-γn1-1-γnaε+φdzn-1,T1zn-1.Using estimates (65), (68), and (71), we arrive at(72)dxn,un≤αn1-1-αnaβn1-1-βna×γn1-1-γnadxn-1,un-1+αn1-1-αnaβn1-1-βna×1-γn1-1-γnaε+φdzn-1,T1zn-1+αn1-1-αna1-βn1-1-βna×ε+φdyn-1,T1yn-1+αn1-1-αna1-βn1-1-βna×ε+φdxn,T1xn.
Keeping in mind inequalities (21)–(23), (72) reduces to(73)dxn,un≤1-1-αn1-adxn-1,un-1+1-γn1-1-γnaε+φdzn-1,T1zn-1+1-βn1-1-βnaε+φdyn-1,T1yn-1+1-αn1-1-αnaε+φdxn,T1xn≤1-1-αn1-adxn-1,un-1+1-αn1-a1-a1-aε+φdzn-1,T1zn-1+1-αn1-a1-a1-aε+φdyn-1,T1yn-1+1-αn1-a1-a1-aε+φdxn,T1xn.Putting an=d(xn,un), rn=(1-αn)(1-a), and tn=φd(xn,T1xn)+φd(yn-1,T1yn-1)+φd(zn-1,T1zn-1)+3ε/(1-a)2, the above inequality becomes(74)an≤1-rnan-1+rntn.Now from Theorem 9, we have limn→∞⁡d(xn,p)=0, limn→∞⁡d(un,p)=0 and since φ is continuous, hence limn→∞⁡φd(xn,Txn)=limn→∞⁡φd(yn,Tyn)=limn→∞⁡φd(zn,Tzn)=0.
Therefore, using Lemma8, (74) yields(75)dp,q≤3ε1-a2.Remark 16.
Puttingγn=βn=1 in (IN) and (62), data dependence result of implicit Mann (IM) iteration can be proved easily on the same lines as in Theorem 15.Remark 17.
Puttingβn=1 in (IN) and (62), data dependence result of implicit Ishikawa (II) iteration can be proved easily on the same lines as in Theorem 15.


## 5. Applications

### 5.1. Implicit Iterations in RNN (Recurrent Neural Networks) Analysis

Neutral networks are a class of nonlinear functions approximations and stable state is achieved in recurrent autoassociative neural networks using iterations. Here we analyze the convergence speed of implicit iterations in recurrent network and many important results will be studied for decreasing and increasing functions. The achieved results possess multifaceted real line applications and in particular can be helpful to design the inner product kernel of support vector machine with faster convergence rate. (For details about RNN and SVM please refer to [33].)


#### 5.1.1. Decreasing Function(1-x)2

The functionf:[0,1]→[0,1] defined by f(x)=(1-x)2 is a decreasing function with fixed point p=0.381966. By taking initial approximation x0=0.7, 1-αn=1-βn=1-γn=1/n+4, the comparison of convergence of implicit iterations to the fixed point is listed in Table 2.Table 2
Convergence of implicit iterations to the fixed point of decreasing function.
Number of iterations (n)
Implicit Noor iteration (IN)
Implicit  Ishikawa iteration (II)
Implicit Mann iteration (IM)
1
0.38215
0.384209
0.409165
2
0.381968
0.382091
0.388393
3
0.381966
0.381972
0.38334
4
0.381966
0.381966
0.382236
5
0.381966
0.381966
0.382015
6
0.381966
0.381966
0.381974
7
0.381966
0.381966
0.381967
8
0.381966
0.381966
0.381966
9
0.381966
0.381966
0.381966
10
0.381966
0.381966
0.381966


#### 5.1.2. Increasing Function(x/2)+3

The functionf:[6,8]→[6,8] defined by f(x)=(x/2)+3 is an increasing function with fixed point p=6. By taking initial approximation x0=7, 1-αn=1-βn=1-γn=1/5n+1, the comparison of convergence of implicit iterations to the fixed point is listed in Table 3.Table 3
Convergence of implicit iterations to the fixed point of increasing function.
Number of iterations (n)
Implicit Noor iteration (IN)
Implicit Ishikawa iteration (II)
Implicit Mann iteration (IM)
0
7
7
7
1
6.194906
6.336163
6.579796
2
6.019386
6.072164
6.268634
3
6.001241
6.011546
6.107454
4
6.000057
6.001482
6.038496
5
6.000002
6.000159
6.012624
6
6
6.000015
6.003844
7
6
6.000001
6.001098
8
6
6
6.000297
9
6
6
6.000076
10
6
6
6.000019
11
6
6
6.000004
12
6
6
6.000001
13
6
6
6
14
6
6
6
15
6
6
6
16
6
6
6From computational study of convergence of implicit iterations to the fixed point of decreasing as well as increasing functions, the following observations are made.Decreasing Function. For x0=0.7, αn=βn=γn=1/n+4, implicit Noor iteration converges in 3 iterations, implicit Ishikawa iteration converges in 4 iterations, and implicit Mann iteration converges in 8 iterations.Forx0=0.7, αn=βn=γn=1/n+8, implicit Noor, implicit Ishikawa, and implicit Mann iterations converge in 2, 3, and 7 iterations, respectively.Increasing Function. For x0=7, αn=βn=γn=1/5n+1, implicit Noor iteration converges in 7 iterations, implicit Ishikawa iteration converges in 9 iterations, and implicit Mann iteration converges in 14 iterations.Forx0=7, αn=βn=γn=1/5n+3, implicit Noor, implicit Ishikawa, and implicit Mann iterations converge in 6, 8, and 13 iterations, respectively.


## 5.1. Implicit Iterations in RNN (Recurrent Neural Networks) Analysis

Neutral networks are a class of nonlinear functions approximations and stable state is achieved in recurrent autoassociative neural networks using iterations. Here we analyze the convergence speed of implicit iterations in recurrent network and many important results will be studied for decreasing and increasing functions. The achieved results possess multifaceted real line applications and in particular can be helpful to design the inner product kernel of support vector machine with faster convergence rate. (For details about RNN and SVM please refer to [33].)


### 5.1.1. Decreasing Function(1-x)2

The functionf:[0,1]→[0,1] defined by f(x)=(1-x)2 is a decreasing function with fixed point p=0.381966. By taking initial approximation x0=0.7, 1-αn=1-βn=1-γn=1/n+4, the comparison of convergence of implicit iterations to the fixed point is listed in Table 2.Table 2
Convergence of implicit iterations to the fixed point of decreasing function.
Number of iterations (n)
Implicit Noor iteration (IN)
Implicit  Ishikawa iteration (II)
Implicit Mann iteration (IM)
1
0.38215
0.384209
0.409165
2
0.381968
0.382091
0.388393
3
0.381966
0.381972
0.38334
4
0.381966
0.381966
0.382236
5
0.381966
0.381966
0.382015
6
0.381966
0.381966
0.381974
7
0.381966
0.381966
0.381967
8
0.381966
0.381966
0.381966
9
0.381966
0.381966
0.381966
10
0.381966
0.381966
0.381966


### 5.1.2. Increasing Function(x/2)+3

The functionf:[6,8]→[6,8] defined by f(x)=(x/2)+3 is an increasing function with fixed point p=6. By taking initial approximation x0=7, 1-αn=1-βn=1-γn=1/5n+1, the comparison of convergence of implicit iterations to the fixed point is listed in Table 3.Table 3
Convergence of implicit iterations to the fixed point of increasing function.
Number of iterations (n)
Implicit Noor iteration (IN)
Implicit Ishikawa iteration (II)
Implicit Mann iteration (IM)
0
7
7
7
1
6.194906
6.336163
6.579796
2
6.019386
6.072164
6.268634
3
6.001241
6.011546
6.107454
4
6.000057
6.001482
6.038496
5
6.000002
6.000159
6.012624
6
6
6.000015
6.003844
7
6
6.000001
6.001098
8
6
6
6.000297
9
6
6
6.000076
10
6
6
6.000019
11
6
6
6.000004
12
6
6
6.000001
13
6
6
6
14
6
6
6
15
6
6
6
16
6
6
6From computational study of convergence of implicit iterations to the fixed point of decreasing as well as increasing functions, the following observations are made.Decreasing Function. For x0=0.7, αn=βn=γn=1/n+4, implicit Noor iteration converges in 3 iterations, implicit Ishikawa iteration converges in 4 iterations, and implicit Mann iteration converges in 8 iterations.Forx0=0.7, αn=βn=γn=1/n+8, implicit Noor, implicit Ishikawa, and implicit Mann iterations converge in 2, 3, and 7 iterations, respectively.Increasing Function. For x0=7, αn=βn=γn=1/5n+1, implicit Noor iteration converges in 7 iterations, implicit Ishikawa iteration converges in 9 iterations, and implicit Mann iteration converges in 14 iterations.Forx0=7, αn=βn=γn=1/5n+3, implicit Noor, implicit Ishikawa, and implicit Mann iterations converge in 6, 8, and 13 iterations, respectively.


## 5.1.1. Decreasing Function(1-x)2

The functionf:[0,1]→[0,1] defined by f(x)=(1-x)2 is a decreasing function with fixed point p=0.381966. By taking initial approximation x0=0.7, 1-αn=1-βn=1-γn=1/n+4, the comparison of convergence of implicit iterations to the fixed point is listed in Table 2.Table 2
Convergence of implicit iterations to the fixed point of decreasing function.
Number of iterations (n)
Implicit Noor iteration (IN)
Implicit  Ishikawa iteration (II)
Implicit Mann iteration (IM)
1
0.38215
0.384209
0.409165
2
0.381968
0.382091
0.388393
3
0.381966
0.381972
0.38334
4
0.381966
0.381966
0.382236
5
0.381966
0.381966
0.382015
6
0.381966
0.381966
0.381974
7
0.381966
0.381966
0.381967
8
0.381966
0.381966
0.381966
9
0.381966
0.381966
0.381966
10
0.381966
0.381966
0.381966


## 5.1.2. Increasing Function(x/2)+3

The functionf:[6,8]→[6,8] defined by f(x)=(x/2)+3 is an increasing function with fixed point p=6. By taking initial approximation x0=7, 1-αn=1-βn=1-γn=1/5n+1, the comparison of convergence of implicit iterations to the fixed point is listed in Table 3.Table 3
Convergence of implicit iterations to the fixed point of increasing function.
Number of iterations (n)
Implicit Noor iteration (IN)
Implicit Ishikawa iteration (II)
Implicit Mann iteration (IM)
0
7
7
7
1
6.194906
6.336163
6.579796
2
6.019386
6.072164
6.268634
3
6.001241
6.011546
6.107454
4
6.000057
6.001482
6.038496
5
6.000002
6.000159
6.012624
6
6
6.000015
6.003844
7
6
6.000001
6.001098
8
6
6
6.000297
9
6
6
6.000076
10
6
6
6.000019
11
6
6
6.000004
12
6
6
6.000001
13
6
6
6
14
6
6
6
15
6
6
6
16
6
6
6From computational study of convergence of implicit iterations to the fixed point of decreasing as well as increasing functions, the following observations are made.Decreasing Function. For x0=0.7, αn=βn=γn=1/n+4, implicit Noor iteration converges in 3 iterations, implicit Ishikawa iteration converges in 4 iterations, and implicit Mann iteration converges in 8 iterations.Forx0=0.7, αn=βn=γn=1/n+8, implicit Noor, implicit Ishikawa, and implicit Mann iterations converge in 2, 3, and 7 iterations, respectively.Increasing Function. For x0=7, αn=βn=γn=1/5n+1, implicit Noor iteration converges in 7 iterations, implicit Ishikawa iteration converges in 9 iterations, and implicit Mann iteration converges in 14 iterations.Forx0=7, αn=βn=γn=1/5n+3, implicit Noor, implicit Ishikawa, and implicit Mann iterations converge in 6, 8, and 13 iterations, respectively.


## 6. Conclusions

(i)
The speed of implicit iterations depends on parametersαn, βn, and γn.
(ii)
In every case the newly introduced three-step implicit Noor iteration has better convergence rate as compared to other iterations and hence has a good potential for further applications in various disciplines of science.



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*Source: 905834-2015-05-18.xml*