# On Rate of Convergence of Jungck-Type Iterative Schemes

Authors: Nawab Hussain; Vivek Kumar; Marwan A. Kutbi
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We introduce a new iterative scheme called Jungck-CR iterative scheme and study the stability and strong convergence of this iterative scheme for a pair of nonself-mappings using a certain contractive condition. Also, convergence speed comparison and applications of Jungck-type iterative schemes will be shown through examples.


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## Body

## 1. Introduction and Preliminaries

LetX be a Banach space, Y an arbitrary set, and S,T:Y→X such that T(Y)⊆S(Y). For x0∈Y, consider the following iterative scheme:
(1)Sxn+1=Txn,n=0,1,….
This scheme is called Jungck iterative scheme and was essentially introduced by Jungck [1] in 1976 and it becomes the Picard iterative scheme when S=Id (identity mapping) and Y=X.Forαn∈[0,1], Singh et al. [2] defined the Jungck-Mann iterative scheme as
(2)Sxn+1=(1-αn)Sxn+αnTxn.
For αn,βn,γn∈[0,1], Olatinwo defined the Jungck-Ishikawa [3] (see also [4, 5]) and Jungck-Noor [6] iterative schemes as
(3)Sxn+1=(1-αn)Sxn+αnTyn,Syn=(1-βn)Sxn+βnTxn,(4)Sxn+1=(1-αn)Sxn+αnTyn,Syn=(1-βn)Sxn+βnTzn,Szn=(1-γn)Sxn+γnTxn,
respectively.Chugh and Kumar [7] defined the Jungck-SP iterative scheme as
(5)Sxn+1=(1-αn)Syn+αnTyn,Syn=(1-βn)Szn+βnTzn,Szn=(1-γn)Sxn+γnTxn,
where {αn}, {βn}, and {γn} are sequences of positive numbers in [0,1].Remark 1.
IfX=Yand S=Id (identity mapping), then the Jungck-SP (5), Jungck-Noor (4), Jungck-Ishikawa (3), and the Jungck-Mann (2) iterative schemes, respectively, become the SP [8], Noor [9], Ishikawa [10] and the Mann [11] iterative schemes.Jungck [1] used the iterative scheme (1) to approximate the common fixed points of the mappings S and T satisfying the following Jungck contraction:
(6)d(Tx,Ty)≤αd(Sx,Sy),0≤α<1.Olatinwo [3] used the following more general contractive definition than (6) to prove the stability and strong convergence results for the Jungck-Ishikawa iteration process: there exists a real number a∈[0,1) and a monotone increasing function ϕ: R+→R+ such that ϕ(0)=0 and for all x,y∈Y, we have
(7)∥Tx-Ty∥≤ϕ(∥Sx-Tx∥)+a∥Sx-Sy∥.
Olatinwo [6] used the convergences of Jungck-Noor iterative scheme (4) to approximate the coincidence points (not common fixed points) of some pairs of generalized contractive-like operators with the assumption that one of each of the pairs of maps is injective.Motivated by the above facts, forαn, βn, and γn∈[0,1], we introduce the following iterative scheme:
(JCR)Sxn+1=(1-αn)Syn+αnTyn,Syn=(1-βn)Txn+βnTzn,Szn=(1-γn)Sxn+γnTxn
and call it Jungck-CR iterative scheme.Remark 2.
Puttingαn=0 and αn=0, βn=1 in Jungck-CR iterative scheme, we get Jungck versions of Agarwal et al. [12] and Sahu and Petruşel [13] iterative schemes, respectively, as defined below:
(JA)Sxn+1=(1-βn)Txn+βnTyn,Syn=(1-γn)Sxn+γnTxn,(JS)Sxn+1=Tyn,Syn=(1-γn)Sxn+γnTxn.We will need the following definitions and lemma.Definition 3 (see [14]).
Let{un} and {vn} be two fixed-point iteration procedures that converge to the same fixed point p on  a normed space X such that the error estimates
(8)∥un-p∥≤an,∥vn-p∥≤bn
are available, where {an} and {bn} are two sequences of positive numbers (converging to zero). If {an} converge faster than {bn}, then we say that {un} converges faster to p than {vn}.Definition 4 (see [15]).
Suppose that{an} and {bn} are two real convergent sequences with limits a and b, respectively. Then, {an} is said to converge faster than {bn} if
(9)limn→∞|an-abn-b|=0.Definition 5 (see [16, 17]).
Letf and g be two self-maps on X. A point x in X is called (1) a fixed point of f if  f(x)=x; (2) coincidence point of a pair (f,g) if  fx=gx; (3) common fixed point of a pair (f,g) if x=fx=gx. If w=fx=gx  for some x in X, then w is called a point of coincidence of  f and g. A pair (f,g) is said to be weakly compatible if f and g commute at their coincidence points.Lemma 6 (see [18]).
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 7 (see [2]).
LetS, T:Y→X be non-self-operators for an arbitrary set Y such that  T(Y)⊆S(Y) and p  a point of coincidence of S and T. Let {Sxn}n=0∞⊂X, be the sequence generated by an iterative procedure
(11)Sxn+1=f(T,xn),n=0,1…,
where  x0∈X  is the initial approximation and  f  is some function. Suppose that {Sxn}n=0∞ converges to p. Let {Syn}n=0∞⊂X  be an arbitrary sequence and set  εn=d(Syn,f(T,yn)),n=0,1,…. Then,  the iterative procedure (11) is said to  be  (S,T)-stable or stable  if and only if limn→∞∈n=0  implies limn→∞Syn=p.The purpose  of this paper is to study the stability and strong convergence of Jungck-CR(JCR) iterative scheme for nonself-mappings in an arbitrary Banach space by employing the contractive conditions (7) and then to compare convergence rates of Jungck-type iterative schemes. Moreover, applications of Jungck-type iterative schemes in recurrent neural networks (RNN) analysis will be discussed.


## 2. Strong Convergence in an Arbitrary Banach Space

Theorem 8.
Let(X,∥·∥) be an arbitrary Banach space, and let   S,T:Y→X   be  nonself-operators on an arbitrary set Y satisfying contractive condition (7). Assume that  T(Y)⊆S(Y),  S(Y)  is a complete subspace of X and Sz=Tz=p (say). For x0∈Y, let {Sxn}n=0∞ be the Jungck-CR  iterative scheme defined by (JCR), where  {αn}, {βn},{γn} are  sequences of positive  numbers in  [0,1] with {αn} satisfying  ∑n=0∞αn=∞.Then, the Jungck-CR iterative  scheme {Sxn}n=0∞ converges strongly to p. Also, p will be the unique common fixed point of S,T  provided that  Y=X, and  S and  T are weakly compatible.Proof.
First, we prove that Jungck-CR iterative  scheme{Sxn}n=0∞ converges strongly to p.
It  follows  from(JCR) and (7) that
(12)∥Sxn+1-p∥=∥(1-αn)Syn+αnTyn-(1-αn+αn)p∥≤(1-αn)∥Syn-p∥+αn∥Tyn-p∥=(1-αn)∥Syn-p∥+αn∥Tz-Tyn∥≤(1-αn)∥Syn-p∥+αn{ϕ(∥Sz-Tz∥)+a∥Sz-Syn∥}=(1-αn)∥Syn-p∥+aαn∥Syn-p∥=[1-αn(1-a)]∥Syn-p∥.
Now, we have the following estimates:
(13)∥Syn-p∥=∥(1-βn)Txn+βnTzn-(1-βn+βn)p∥≤(1-βn)∥Txn-p∥+βn∥Tzn-p∥≤(1-βn)∥Txn-Tz∥+βn∥Tzn-Tz∥≤(1-βn)(ϕ(∥Sz-Tz∥)+a∥Sxn-Sz∥)+βn{ϕ(∥Sz-Tz∥)+a∥Szn-Sz∥}≤(1-βn)a∥Sxn-p∥+βna∥Szn-p∥,∥Szn-p∥=∥(1-γn)Sxn+γnTxn-(1-γn+γn)p∥≤(1-γn)∥Sxn-p∥+γn∥Txn-Tz∥≤(1-γn)∥Sxn-p∥+γn{ϕ(∥Sz-Tz∥)+a∥Sxn-Sz∥}=(1-γn(1-a))∥Sxn-p∥.
It follows from (13) that
(14)∥Syn-p∥≤(1-βn)a∥Sxn-p∥+βna(1-γn(1-a))∥Sxn-p∥.
Using  (1-βn)a≤(1-βn)  and βna(1-γn(1-a))≤βna, inequality  (14) yields
(15)∥Syn-p∥≤(1-βn(1-a))∥Sxn-p∥.
It follows from (15) and (12) that
(16)∥Sxn+1-p∥≤[1-αn(1-a)][1-βn(1-a)]∥Sxn-p∥≤[1-αn(1-a)]∥Sxn-p∥≤∏k=0n[1-αk(1-a)]∥Sx0-p∥≤e-(1-a)∑k=0∞αk∥Sx0-p∥.
Since 0≤a<1,αk∈[0,1]  and  ∑n=0∞αn=∞,  so e-(1-a)∑k=0nαk→0  as  n→∞.
Hence, it follows from (16) that limn→∞∥Sxn+1-p∥=0. Therefore, {Sxn}n=0∞ converges strongly to p.
Now, we prove thatp is unique common fixed point of  S  and  T.
Let there exist another point of coincidence sayp*. Then, there exists  q*∈X such that  Sq*=Tq*=p*. But from (7),  we have
(17)0≤∥p-p*∥=∥Tq-Tq*∥≤ϕ(∥Sq-Tq∥)+a∥Sq-Sq*∥=a∥p-p*∥,
which implies that  p=p* as  0≤a<1.
Now, asS and T are weakly compatible and p=Tq=Sq, so  Tp=TTq=TSq=STq and hence Tp=Sp. Therefore, Tp is a point of coincidence of S, T and since the point of coincidence  is unique then p=Tp. Thus, Tp=Sp=p, and therefore p is unique common fixed point of S and T.Corollary 9.
Let(X,∥·∥) be an arbitrary Banach space, and,  S,T:Y→X  be  nonself-operators on an arbitrary set Y satisfying contractive condition (7). Assume that T(Y)⊆S(Y),   S(Y)  is a complete subspace of X and Sz=Tz=p (say).  For x0∈Y,  let {Sxn}n=0∞ be the  iterative scheme defined by (JA), where  {αn},{βn} are  sequences of positive  numbers in  [0,1] with {αn} satisfying ∑n=0∞αn=∞. Then the Jungck-Agarwal iterative  scheme {Sxn}n=0∞ converges strongly to p. Also, p will be the unique common fixed point of  S,T  provided that Y=X, and  S and T are weakly compatible.Proof.
Puttingαn=0 and   βn=αn, in  iterative scheme (JCR), convergence of iterative scheme (JA) can be proved on the same lines as in Theorem 8.Corollary 10.
Let(X,∥·∥)  be an arbitrary Banach space  and   S, and let T:Y→X  be  nonself-operators on an arbitrary set Y satisfying contractive condition (7). Assume that  T(Y)⊆S(Y),  S(Y)  is a complete subspace of  X and Sz=Tz=p (say). For x0∈Y,  let {Sxn}n=0∞ be the  Jungck-S iterative scheme defined by (JS), where {αn},{βn} are  sequences of positive  numbers in  [0,1] with {αn} satisfying ∑n=0∞αn=∞. Then the Jungck-S iterative  scheme {Sxn}n=0∞  converges strongly to p. Also, p will be the unique common fixed point of S,T  provided   that  Y=X, and  S and T are weakly compatible.Proof.
Puttingαn=0 and  γn=αn,βn=1 in iterative scheme (JCR), convergence of iterative scheme (JS) can be proved on the same lines as in the Theorem 8.The following examples reveal the validity of our results.Example 11.
LetX=Y=[0,1].   Define  T  and  S  by
(18)T(x)={0,x∈[0,1)12,x=1},Sx=x2,αn=βn=γn=12n+4,ϕ(t)=2at.
It is clear that T and S are quasicontractive operators satisfying (7) but do not satisfy contractive  condition (6), with a unique common fixed point 0.Using computer programming in C++ with initial approximationx0=1, convergence of Jungck-CR iterative scheme to the common fixed point 0 is shown in Table 1.Table 1
Number of iterations (n)
Jungck-CR iterative scheme (Sxn+1)
0
1
1
0.5
2
0.125
3
0
4
0Example 12.
LetY=X=[0,1].  Define T and S by T(x)=(1/2)(1/2+x), S(x)=1-x,αn=βn=γn=1/2n+4, and ϕ(t)=2at. It is clear that T and S are weakly compatible quasicontractive operators satisfying (7) with a unique common fixed point 0.5.Using computer programming in C++ with initial approximationx0=0.8, convergence of Jungck-CR iterative scheme to the common fixed point 0.5 is shown in Table 2.Table 2
Number of iterations (n)
Jungck-CR iterative scheme (Sxn+1)
0
0.2
1
0.523438
2
0.496593
3
0.50065
4
0.499855
5
0.500036
6
0.49999
7
0.500003
8
0.499999
9
0.5
10
0.5Theorem 13.
Let(X,∥·∥) be an arbitrary Banach space  and  S, and let T:Y→X  be  nonself operators on an arbitrary set Y satisfying contractive condition (7). Assume that T(Y)⊆S(Y),  S(Y)  is a complete subspace of X, and Sz=Tz=p (say). For x0∈Y and α∈(0,1), let {Sxn}n=0∞ be the Jungck-CR iterative scheme (JCR) converging to p, where {αn},{βn},{γn} are sequences  in  [0,1] with  {αn} satisfying  α≤αn  for all n. Then, the Jungck-CR iterative  scheme is (S,T)-stable.Proof.
Suppose that{Syn}n=0∞⊂X  be an arbitrary sequence, εn=∥Syn+1-(1-αn)Sbn-αnTbn∥, n=0,1,2,3…, where Sbn=(1-βn)Tyn+βnTcn,Scn=(1-γn)Syn+γnTyn and let limn→∞∈n=0.
Then, for Jungck-CR iterative  scheme(JCR), we have
(19)∥Syn+1-p∥≤∥Syn+1-(1-αn)Sbn-αnTbn∥+∥(1-αn)Sbn+αnTbn-(1-αn+αn)p∥≤εn+(1-αn)∥Sbn-p∥+αn∥Tbn-p∥=εn+(1-αn)∥Sbn-p∥+αn∥Tz-Tbn∥≤εn+(1-αn)∥Sbn-p∥+αn{ϕ(∥Sz-Tz∥)+a∥Sz-Sbn∥}=εn+(1-αn)∥Sbn-p∥+αn{ϕ(∥0∥)+a∥Sz-Sbn∥}=[1-αn(1-a)]∥Sbn-p∥+εn.
Now, we have the following estimates:
(20)∥Sbn-p∥=∥(1-βn)Tyn+βnTcn-(1-βn+βn)p∥≤(1-βn)∥Tyn-p∥+βn∥Tcn-p∥=(1-βn)∥Tyn-Tz∥+βn∥Tz-Tcn∥≤(1-βn){ϕ(∥Sz-Tz∥)+a∥Sz-Syn∥}+βn{ϕ(∥Sz-Tz∥)+a∥Sz-Scn∥}≤(1-βn)a∥p-Syn∥+βna∥p-Scn∥,||Scn-p||=∥(1-γn)Syn+γnTyn-(1-γn+γn)p∥≤(1-γn)∥Syn-p∥+γn∥Tyn-p∥=(1-γn)∥Syn-p∥+γn∥Tz-Tyn∥≤(1-γn)∥Syn-Tz∥+γn{ϕ(∥Sz-Tz∥)+a∥Sz-Syn∥}=(1-γn(1-a))∥Syn-p∥.
It follows from (19), (20) that
(21)∥Syn+1-p∥≤[1-αn(1-a)]∥Syn-p∥+εn.
Using  0<α≤αn and  a∈[0,1), we have   [1-αn(1-a)]<1.
Hence using Lemma6, (21) yields limn→∞Syn+1=p.
Conversely, letlimn→∞Syn+1=p. Then, using contractive condition (7) and the triangle inequality, we have
(22)εn=∥Syn+1-(1-αn)Sbn-αnTbn∥≤∥Syn+1-p∥+∥(1-αn+αn)p-(1-αn)Sbn-αnTbn∥≤∥Syn+1-p∥+(1-αn)∥p-Sbn∥+αn∥p-Tbn∥=∥Syn+1-p∥+(1-αn)∥Sbn-p∥+αn∥Tz-Tbn∥≤∥Syn+1-p∥+(1-αn)∥Sbn-p∥+aαn∥Sz-Sbn∥=∥Syn+1-p∥+[1-αn(1-a)]∥Sbn-p∥.
By using estimates (20), (22), yields
(23)εn≤[1-αn(1-a)]∥Syn-p∥+∥Syn+1-p∥.
Hence, limn→∞εn=0.
Therefore, the JCR iterative scheme  is(S,T) stable.


## 3. Results on Direct Comparison of Jungck-Type Iterative Schemes

Various authors [7, 13–15, 19–22] have worked on convergence speed of iterative schemes. In [14], Berinde showed  that Picard iteration is faster than Mann iteration for quasicontractive operators. In [15], Qing and Rhoades by taking  an example showed that Ishikawa iteration is faster than Mann iteration for a certain  quasicontractive operator.  In [20], Hussain et al. provided an example of a quasicontractive operator for which the   iterative scheme due to Agarwal et al. is faster than Mann and Ishikawa iterative  schemes. Recently, Chugh and Kumar [19] showed that SP iterative scheme  with error terms converges faster than Ishikawa and Noor iterative schemes for accretive-type mappings. For recent work in this direction, we refer the reader to [23–27] and references therein.Theorem 14.
Let(X,∥·∥) be an arbitrary Banach space,  and let S,T:Y→X  be  nonself-operators on an arbitrary set Y satisfying contractive condition (7). Assume that  T(Y)⊆S(Y),  S(Y)  is a complete subspace of X and Sz=Tz=p (say). For x0∈Y,  let Jungck-Mann  iterative scheme be defined by (JM) and Jungck-Ishikawa iterative scheme be defined by (JI), with αn∈[0,1/(1+(1+2/m)a)], βn≤1-αn(1-a), for some m>0 and n∈N satisfying ∑n=0∞αn=∞.
Then, the Jungck-Ishikawa iterative scheme converges faster than Jungck-Mann iterative scheme top.Proof.
For Jungck-Mann iterative scheme,  we have(24)∥Sxn+1-p∥≥(1-αn)∥Sxn-p∥-αn∥Txn-p∥≥(1-αn)∥Sxn-p∥-αna∥Sxn-p∥≥[1-αn(1+a)]∥Sun-p∥.
Also, for Jungck-Ishikawa  iterative scheme,  we have
(25)∥Sxn+1-p∥≤(1-αn)∥Sxn-p∥+a∥Tyn-p∥≤(1-αn)∥Sxn-p∥+αna∥Syn-p∥.
But
(26)∥Sxn+1-p∥≤(1-βn)∥Sxn-p∥+βn∥Txn-p∥≤(1-βn(1-a))∥Sxn-p∥.
Hence,
(27)∥Sxn+1-p∥≤(1-αn(1-a)-αnβna(1-a))∥Txn-p∥.
Using (24) and (27), we have
(28)∥JIn+1JMn+1∥≤∏i=0n[(1-αi(1-a)-αiβia(1-a))(1-αi(1+a))].
But we observe that
(29)1-αi(1-a)1-αi(1+a)≤1+m∀i=0,1,2,….
Using (29) together with  βn≤1-αn(1-a), we have
(30)∏i=0n[(1-αi(1-a)-αiβia(1-a))(1-αi(1+a))]≤(1+m)(1-αi(1-a))≤(1+m)e-αi(1-a).
As  ∑n=0∞αn=∞,  so (28)  yields  limn→∞∥(JIn+1-p)/(JMn+1-p)∥=0.
Therefore, by Definition4, Jungck-Ishikawa iterative scheme converges faster than Jungck-Mann iterative scheme to p.Theorem 15.
Let(X,∥·∥) be an arbitrary Banach space,  and let S,T:Y→X  be  nonself-operators on an arbitrary set Y satisfying contractive condition (7). Assume that T(Y)⊆S(Y),  S(Y)  is a complete subspace of X, and Sz=Tz=p (say). For x0∈Y, let Jungck-Noor  iterative scheme be defined by (JN) and Jungck-Ishikawa   iterative scheme defined by (JI), with αn∈[0,1/(1+(1+2/m)a)], βn≤1-αn(1-a),for some m>0 and n∈N satisfying ∑n=0∞αn=∞. Then, the Jungck-Noor iterative  scheme  converges faster than Jungck-Ishikawa iterative scheme  to p.Proof.
For Jungck-Ishikawa iterative scheme,  we have(31)∥Sxn+1-p∥≥[1-αn(1+a)]∥Sun-p∥.
Also, for Jungck-Noor  iterative scheme,  we have
(32)∥Sxn+1-p∥≤(1-αn(1-a)-αnβna(1-a)-αnβnγna2(1-a))×∥Txn-p∥.
Using (31) and (32), we have(33)∥JNn+1JIn+1∥≤∏i=0n[(1-αi(1-a)-αiβia(1-a)-αnβnγna2(1-a))(1-αi(1+a))]≤∏i=0n[1-αi(1-a)-αiβia(1-a)(1-αi(1+a))].Making the same calculations as in Theorem 14, (33) yields
(34)limn→∞∥JNn+1-pJIn+1-p∥=0.
By Definition 4, Jungck-Noor iterative scheme converges faster than Jungck-Ishikawa iterative scheme to p.Theorem 16.
Let(X,∥·∥) be an arbitrary Banach space  and  S,T:Y→X  be  nonself operators on an arbitrary set Y satisfying contractive condition (7). Assume that T(Y)⊆S(Y),  S(Y)  is a complete subspace of X and Sz=Tz=p (say). For  x0∈Y, let Jungck-Noor  iterative scheme be defined by (JN) and Jungck-SP iterative scheme defined by (JSP), with αn∈[0,1/(1+(1+2/m)a)], for some m>0 satisfying ∑n=0∞βn=∞. Then, the Jungck-SP iterative  scheme  converges faster than Jungck-Noor iterative scheme  to p.Proof.
For Jungck-Noor iterative scheme,  we have(35)∥Sxn+1-p∥≥[1-αn(1+a)]∥Sun-p∥.
Also, for Jungck-SP iterative scheme, we have
(36)∥Sxn+1-p∥≤(1-αn(1-a))(1-βn(1-a))×(1-γn(1-a))∥Txn-p∥.
Using (35) and (36), we have
(37)∥JSPn+1-pJNn+1-p∥≤∏i=0n[(1-αn(1-a))(1-βn(1-a))(1-γn(1-a))(1-αi(1+a))]≤∏i=0n[(1-αn(1-a))(1-βn(1-a))(1-αi(1+a))].
We observe that
(38)1-αi(1-a)1-αi(1+a)≤1+m∀i=0,1,2,….
Using (38) together with  ∑n=0∞βn=∞,  (37) yields
(39)limn→∞∥JSPn+1-pJNn+1-p∥=0.
Therefore, by Definition 4, Jungck-SP iterative scheme converges faster than Jungck-Noor iterative scheme p.Theorem 17.
Let(X,∥·∥) be an arbitrary Banach space,  and let S,T:Y→X  be  nonself operators on an arbitrary set Y satisfying contractive condition (7). Assume that T(Y)⊆S(Y),  S(Y)  is a complete subspace of X and Sz=Tz=p (say). For x0∈Y, let Jungck-Agarwal’s et al.  iterative scheme be defined by (JA) and Jungck-SP   iterative scheme be defined by (JSP) with (i)∑n=0∞αn=∞, (ii)limn→∞αn=0, and (iii)limn→∞βn=0. Then, the Jungck-Agarwal iterative  scheme  converges faster than Jungck-SP iterative scheme  to p.Proof.
For Jungck-SP iterative scheme,  we have(40)∥Sxn+1-p∥≥[1-αn(1+a)]∥Sxn-p∥.
Also, for Jungck-Agarwal   iterative scheme,  we have
(41)∥Sxn+1-p∥≤a(1-αnβn(1-a))∥Txn-p∥.
Using (40) and (41), we have
(42)∥JSPn+1-pJAn+1-p∥≤an∏i=0n[(1-αnβn(1-a))(1-αi(1+a))].
Since a∈[0,1) and limn→∞αn=0,limn→∞βn=0.
Hence from (42), we have
(43)limn→∞∥JSPn+1-pJAn+1-p∥=0.
Therefore, by Definition 4, Jungck-SP  iterative scheme converges faster than Jungck-Agarwal et al.’s  iterative scheme to p.Theorem 18.
Let(X,∥·∥) be an arbitrary Banach space,  and let  S, T:Y→X  be  nonself-operators on an arbitrary set Y satisfying contractive condition (7). Assume that T(Y)⊆S(Y),  S(Y)  is a complete subspace of X and Sz=Tz=p (say). For x0∈Y, let Jungck-S  iterative scheme be defined by (JS) and Jungck-Agarwal  iterative scheme defined by (JA). Then, the Jungck-S iterative  scheme  converges faster than Jungck-Agarwal iterative scheme  to p.Proof.
For Jungck-S iterative scheme,  we have(44)∥Sxn+1-p∥≤a(1-αn(1-a))∥Sxn-p∥.
Also,  for Jungck-Agarwal   iterative scheme,  we have
(45)∥Sxn+1-p∥≤a(1-αnβn(1-a))∥Sxn-p∥.
It is obvious that
(46)a(1-αn(1-a))≤a(1-αnβn(1-a))∀n.
Hence by Definition 3, Jungck-S iterative scheme converges faster than  Jungck-Agarwal   iterative scheme.Theorem 19.
Let(X,∥·∥) be an arbitrary Banach space,  and let S,T:Y→X  be  nonself operators on an arbitrary set Y satisfying contractive condition (7). Assume that T(Y)⊆S(Y),  S(Y)  is a complete subspace of X and Sz=Tz=p (say). For x0∈Y, let Jungck-S  iterative scheme be defined by (JS) and Jungck-CR  iterative scheme be defined by (JCR). Then, the Jungck-CR iterative  scheme  converges faster than Jungck-S iterative scheme  to p.Proof.
For Jungck-S iterative scheme,  we have(47)∥Sxn+1-p∥≤a(1-αn(1-a))∥Sxn-p∥.
Also,  for Jungck-CR   iterative scheme,  we have
(48)∥Sxn+1-p∥≤a(1-αn(1-a))(1-βnγn(1-a))∥Sxn-p∥.
It is obvious that
(49)a(1-αn(1-a))(1-βnγn(1-a))≤a(1-αn(1-a))∀n.
Hence by Definition 3, Jungck-CR iterative scheme converges faster than  Jungck-S   iterative scheme.The following example supports  the above results.Example 20.
LetY=[0,1], X=[0,1/2], S:Y→X=x/2, T:Y→X=x/4,  αn=βn=γn=0, n=1,2…n0-1 for some  n0∈N,  and αn=βn=γn=4/n, n≥n0. It is clear that T and S are quasicontractive operators satisfying (7) with the unique common fixed point 0. Also, it is easy to see that Example 20 satisfies all the conditions of Theorem 8 and Theorems 14–19.Proof .
For JM, JI, JN, JA, JS, JSP, and JCR   iterative schemes with initial approximationx0≠0, we have
(50)JMn=∏i=n0n(12-1i)x0,JIn=∏i=n0n(12-1i-2i)x0,JNn=∏i=n0n(1-2i-4i-8i3/2)x0,JSPn=∏i=n0n(12-3i+6i-4i3/2)x0,JAn=∏i=n0n(14-2i)x0,JSn=∏i=n0n(14-2i)x0,JCRn=∏i=n0n(14-12i-2i+4i3/2)x0.
Now, for  n0=16,  consider
(51)|JIn+1JMn+1|=|∏i=16n(1-2/i-4/i)x0∏i=16n(1-2/i)x0|=|∏i=16n[1-4/i(1-2/i)]|=|∏i=16n[1-4(i-2i)]|.
It is easy to see that
(52)0≤limn→∞∏i=16n[1-4(i-2i)]≤limn→∞∏i=16n(1-1i)=limn→∞15n=0.
Hence,   limn→∞|JIn+1/JMn+1|=0.
Therefore, by Definition4, Jungck-Ishikawa iterative scheme converges faster than Jungck-Mann iterative scheme to the common fixed point 0 of T and S.
Similarly, forn0=16,
(53)|JNnJIn|=|∏i=16n(1-2/i-4/i-8/i3/2)x0∏i=16n(1-2/i-4/i)x0|=|∏i=16n[1-8/i3/21-2/i-4/i]|=|∏i=16n[1-8(i3/2-2i-4i)]|
with
(54)0≤limn→∞∏i=16n[1-8(i3/2-2i-4i)]≤limn→∞∏i=16n(1-1i)=limn→∞15n=0
implies
(55)limn→∞|JNnJIn|=0.
Therefore, by Definition 4, JN iterative scheme converges faster than JI iterative scheme to the common fixed point 0 of T and S.
Again, similarly, forn0=100,
(56)|JSPnJNn|=|∏i=100n(1-6/i+12/i-8/i3/2)x0∏i=100n(1-2/i-4/i-8/i3/2)x0|=|∏i=100n[1-(4/i-16/i)1-2/i-4/i-8/i3/2]|=|∏i=100n[1-(4i-16i)(i3/2-2i-4i-8)]|
with
(57)0≤limn→∞∏i=100n[1-(4i-16i)(i3/2-2i-4i-8)]≤limn→∞∏i=100n(1-1i)=limn→∞99n=0
implies
(58)limn→∞|JSPnJNn|=0.
Therefore, by Definition 4, JSP iterative scheme converges faster than JN iterative scheme to the common fixed point 0 of T and  S.
Again, similarly, forn0=100,
(59)|JAnJSPn|=|∏i=100n(1/2-4/i)x0∏i=100n(1-6/i+12/i-8/i3/2)x0|=|∏i=100n[1-(1/2-6/i+16/i-8/i3/2)1-6/i+12/i-8/i3/2]|=|∏i=100n[1-(i3/2-12i+32i-16)(2i3/2-12i+24i-16)]|
with
(60)0≤limn→∞∏i=100n[1-(i3/2-12i+32i-16)(2i3/2-12i+24i-16)]≤limn→∞∏i=100n(1-1i)=limn→∞99n=0
implies
(61)limn→∞|JAnJSPn|=0.
Therefore, by Definition 4, JA iterative scheme converges faster than JSP iterative scheme to the common fixed point 0 of T and S.
Again, forn0=16,
(62)|JSnJAn|=|∏i=16n(1/2-1/i)x0∏i=16n(1/2-4/i)x0|=|∏i=16n[1-(1/i-4/i)1/2-4/i]|=|∏i=16n[1-(2i-8)i-8]|
with
(63)0≤limn→∞∏i=16n[1-(2i-8)i-8]≤limn→∞∏i=16n(1-1i)=limn→∞15n=0
implies
(64)limn→∞|JSnJAn|=0.
Therefore, by Definition 4, JS iterative scheme converges faster than JA iterative scheme to the common fixed point 0 of T and S.
Similarly, again, forn0=16,
(65)|JCRnJSn|=|∏i=16n(1/2-1/i-4/i+8/i3/2)x0∏i=16n(1/2-1/i)x0|=|∏i=16n[1-(4/i-8/i3/2)1/2-1/i]|=|∏i=16n[1-(8i-16)i3/2-2i]|
with
(66)0≤limn→∞∏i=16n[1-(8i-16)i3/2-2i]≤limn→∞∏i=16n(1-1i)=limn→∞15n=0
implies
(67)limn→∞|JCRnJSn|=0.
Therefore, by Definition 4, JCR iterative scheme converges faster than JS iterative scheme to the common fixed point 0 of T and S.
From Example20, we observe that the decreasing order of Jungck-type iterative schemes is as follows:
JCR, JS, JA, JSP, JN, JI, and JM.


## 4. Applications

### 4.1. Jungck-Type Iterative Schemes in RNN Analysis

Recurrent neural networks (RNNs) are a class of densely connected single-layer nonlinear networks of perceptrons. RNNs not only operate on an input space but also on an internal statespace. This is equivalent to a with-memory Iterated Function System [28]. The state space enables the representation (and learning) of temporally/sequentially extended dependencies over unspecified (and potentially infinite) intervals according to
(68)y(t)=G(s(t))s(t)=F(s(t-1),x(t)).
Because of the network’s nonlinearity, a number of undesirable local energy minima emerge from the learning procedure. This has been shown to significantly affect the network’s performance. The iterative schemes like Mann, Ishikawa and J-iteration may be used to estimate the number of iterations required to achieve a stable state in recurrent autoassociative neural networks.


#### 4.1.1. Decreasing Function(1-x)9

In order to solve this function by Jungck-type iterative schemes, we write it in the formSx=Tx, where the functions T, S:[0,1]→[0,2] are defined as T(x)=(1-x)9 and Sx=x, respectively. By taking initial approximation x0=0.8 and αn=βn=γn=1/n+14, the obtained results are listed in Table 3 showing convergence of different Jungck-type schemes to p=0.175699=T0.175699=S0.175699.Table 3
Decreasing function.
n
JN
JCR (JSP)
JA (JI)
JS
JM
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
0
5.12
e
-
007
5.12
e
-
007
0.337789
5.12
e
-
007
5.12
e
-
007
0.337789
5.12
e
-
007
5.12
e
-
007
0.0244889
5.12
e
-
007
5.12
e
-
007
0.0244889
5.12
e
-
007
5.12
e
-
007
0.8
1
0.0244889
0.0244889
0.215511
0.0244889
0.0244889
0.187422
0.8
0.8
0.191339
0.8
0.8
0.337789
5.12
e
-
007
5.12
e
-
007
0.8
2
0.112533
0.112533
0.187233
0.154449
0.154449
0.176002
0.147875
0.147875
0.173023
0.0244889
0.0244889
0.113605
5.12
e
-
007
5.12
e
-
007
0.8
3
0.154772
0.154772
0.179346
0.175121
0.175121
0.175701
0.180901
0.180901
0.175892
0.337789
0.337789
0.214684
5.12
e
-
007
5.12
e
-
007
0.8
4
0.168827
0.168827
0.176923
0.175696
0.175696
0.175699
0.17533
0.17533
0.175696
0.113605
0.113605
0.15714
5.12
e
-
007
5.12
e
-
007
0.8
5
0.173366
0.173366
0.176129
0.175699
0.175699
0.175699
0.175706
0.175706
0.175699
0.214684
0.214684
0.185861
5.12
e
-
007
5.12
e
-
007
0.8
6
0.174877
0.174877
0.175856
0.175699
0.175699
0.175699
0.1757
0.1757
0.175699
0.15714
0.15714
0.170534
5.12
e
-
007
5.12
e
-
007
0.8
7
0.1754
0.1754
0.175758
0.175699
0.175699
0.175699
0.185861
0.185861
0.178428
5.12
e
-
007
5.12
e
-
007
0.8
8
0.175587
0.175587
0.175722
0.175699
0.175699
0.175699
0.170534
0.170534
0.174287
5.12
e
-
007
5.12
e
-
007
0.8
9
0.175656
0.175656
0.175708
0.178428
0.178428
0.176438
5.12
e
-
007
5.12
e
-
007
0.8
10
0.175682
0.175682
0.175703
0.174287
0.174287
0.175315
5.12
e
-
007
5.12
e
-
007
0.8
11
0.175692
0.175692
0.175701
0.176438
0.176438
0.1759
5.12
e
-
007
5.12
e
-
007
0.8
12
0.175697
0.175697
0.1757
0.175315
0.175315
0.175595
5.12
e
-
007
5.12
e
-
007
0.8
13
0.175698
0.175698
0.1757
0.1759
0.1759
0.175754
5.12
e
-
007
5.12
e
-
007
0.8
14
0.175699
0.175699
0.1757
0.175595
0.175595
0.175671
5.12
e
-
007
5.12
e
-
007
0.8
15
0.175699
0.175699
0.175699
0.175754
0.175754
0.175714
5.12
e
-
007
5.12
e
-
007
0.8
16
0.175699
0.175699
0.175699
0.175671
0.175671
0.175692
5.12
e
-
007
5.12
e
-
007
0.8
17
0.175714
0.175714
0.175703
18
0.175692
0.175692
0.175697
19
0.175703
0.175703
0.175701
20
0.175697
0.175697
0.175699
21
0.175701
0.175701
0.1757
22
0.175699
0.175699
0.175699
23
0.1757
0.1757
0.1757
24
0.175699
0.175699
0.175699
25
0.1757
0.1757
0.175699
26
0.175699
0.175699
0.175699
27
0.175699
0.175699
0.175699


#### 4.1.2. Increasing Functionx2-2x-3

In order to solve this function by Jungck-type iterative schemes, we write it in the formSx=Tx, where the functions T, S:[3,4]→[9,16] are defined as Tx=2x+3 and Sx=x2, respectively. By taking initial approximation x0=4 and αn=βn=γn=1/n+14, the obtained results are listed in Table 4 showing convergence of different Jungck-type schemes to p=9=T3=S3.Table 4
Increasing function.
n
JN
JCR (JSP)
JA (JI)
JS
JM
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
0
11
11
3.103748
11
11
3.034385
11
11
3.103748
11
11
3.10375
11
11
3.316625
1
9.207495
9.207495
3.01144
9.06877
9.068771
3.002208
9.207495
9.2075
3.015102
9.207495
9.2075
3.01144
9.63325
9.63325
3.103748
2
9.02288
9.02288
3.00127
9.00442
9.004417
3.000179
9.030205
9.03021
3.002482
9.02288
9.0229
3.00127
9.207495
9.207495
3.034385
3
9.00254
9.00254
3.000141
9.00036
9.000358
3.000017
9.004964
9.00496
3.000437
9.00254
9.0025
3.00014
9.068771
9.068771
3.01144
4
9.000282
9.000282
3.000016
9.00003
9.000033
3.000002
9.000875
9.00088
3.000081
9.000282
9.0003
3.00002
9.02288
9.02288
3.003811
5
9.000031
9.000031
3.000002
9
9.000003
3
9.000162
9.00016
3.000015
9.000031
9
3
9.007622
9.007622
3.00127
6
9.000003
9.000003
3
9
9
3
9.000031
9.00003
3.000003
9.000003
9
3
9.00254
9.00254
3.000423
7
9
9
3
9
9
3
9.000006
9.00001
3.000001
9
9
3
9.000847
9.000847
3.000141
8
9
9
3
9.000001
9
3
9
9
3
9.000282
9.000282
3.000047
9
9
9
3
9.000094
9.000094
3.000016
10
9
9
3
9.000031
9.000031
3.000005
11
9.00001
9.00001
3.000002
12
9.000003
9.000003
3.000001
13
9.000001
9.000001
3
14
9
9
3
15
9
9
3


#### 4.1.3. Oscillating Function1/x

In order to solve this function by Jungck-type iterative schemes, we write it in the formSx=Tx, where the functions T, S:[0.5,2]→[0.25,4] are defined as Tx=1/x and Sx=x2, respectively. By taking initial approximation x0=2 and αn=βn=γn=1/n+14, the obtained results are listed in Table 5 showing convergence of different Jungck type schemes to p=1=T1=S1.Table 5
Oscillatory function.
n
JN
JCR (JSP)
JA (JI)
JS
JM
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
0
0.5
0.5
0.917004
0.5
0.5
0.917004
0.5
0.5
1.18921
0.5
0.5
1.18921
0.5
0.5
0.707107
1
1.09051
1.09051
1.01167
1.09051
1.09051
1.00285
0.840896
0.840896
1.02508
0.84096
0.84096
1.04427
0.41421
0.41421
1.18921
2
0.988463
0.988463
0.9983
0.997163
0.997163
0.999972
0.975531
0.975531
1.0018
0.957603
0.957603
1.01089
0.840896
0.840896
0.917004
3
1.0017
1.0017
1.00027
1.00003
1.00003
1
0.998208
0.998208
1.00005
0.989228
0.989228
1.00271
1.09051
1.09051
1.04427
4
0.999733
0.999733
0.999956
1
1
1
0.999945
0.999945
1
0.997296
0.997296
1.00068
0.957603
0.957603
0.978572
5
1.00004
1.00004
1.00001
1
1
1
1
1
1
0.999323
0.999323
1.00017
1.0219
1.0219
1.01089
6
0.999992
0.999992
0.999999
1
1
1
0.999831
0.999831
1.00004
0.989228
0.989228
0.994599
7
1
1
1
0.999958
0.999958
1.00001
1.00543
1.00543
1.00271
8
1
1
1
0.999989
0.999989
1
0.997296
0.997296
0.999662
9
0.999997
0.999997
1
1.00135
1.00135
1.00017
10
0.999999
0.999999
1
0.999323
0.999323
0.999915
11
1
1
1
1.00034
1.00034
1.00004
12
1
1
1
0.999831
0.999831
0.999979
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
18
0.999997
0.999997
0.999999
19
0.999999
0.999999
1
20
1
1
1
21
1
1
1


#### 4.1.4.  Biquadratic Equationx4-36x2-52x+87=0

In order to solve this equation, we rewrite it in the formSx=Tx, where the functions T, S:[0.5,1.5]→[9,81] are defined as Tx=x4-52x+87 and Sx=36x2, respectively. Taking initial approximation x0=0.5 and αn=βn=γn=1/n+12, the obtained results are listed in Table 6 showing convergence of different Jungck-type schemes to p=36=T1=S1.Table 6
Biquadratic equation.
n
JN
JCR (JSP)
JA (JI)
JS
JM
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
0
61.25
61.25
1.1545
61.25
61.25
1.1545
61.25
61.25
0.761468
61.25
61.25
0.761468
61.25
61.25
1.30437
1
28.2988
28.2988
0.955826
28.2988
28.2988
0.995288
47.9835
47.9835
0.972744
47.9835
47.9835
0.88661
20.874
20.874
0.761468
2
38.2107
38.2107
1.01255
36.2356
36.2356
1
37.3636
37.3636
1.00048
41.6824
41.6824
0.945813
47.9835
47.9835
1.1545
3
35.3724
35.3724
0.996092
35.9999
35.9999
1
35.9758
35.9758
0.999949
38.7123
38.7123
0.973995
28.2988
28.2988
0.88661
4
36.1954
36.1954
1.00129
36
36
1
36.0026
36.0026
1.00001
37.3009
37.3009
0.98749
41.6824
41.6824
1.07603
5
35.9353
35.9353
0.999546
36
36
1
35.9996
35.9996
0.999998
36.6256
36.6256
0.993975
32.2042
32.2042
0.945813
6
36.0227
36.0227
1.00017
36.0001
36.0001
1
36.3013
36.3013
0.997096
38.7123
38.7123
1.03699
7
35.9917
35.9917
0.999937
36
36
1
36.1452
36.1452
0.9986
34.152
34.152
0.973995
8
36.0032
36.0032
1.00002
36
36
1
36.07
36.07
0.999325
37.3009
37.3009
1.01791
9
35.9988
35.9988
0.99999
36.0338
36.0338
0.999674
35.1049
35.1049
0.98749
10
36.0005
36.0005
1
36.0163
36.0163
0.999843
36.6256
36.6256
1.00865
11
35.9998
35.9998
0.999998
36.0078
36.0078
0.999924
35.5675
35.5675
0.993975
12
36.0001
36.0001
1
36.0038
36.0038
0.999963
36.3013
36.3013
1.00418
13
36
36
1
36.0018
36.0018
0.999982
35.7912
35.7912
0.997096
14
36
36
1
36.0009
36.0009
0.999992
36.1452
36.1452
1.00201
15
36.0004
36.0004
0.999996
35.8993
35.8993
0.9986
16
36.0002
36.0002
0.999998
36.07
36.07
1.00097
17
36.0001
36.0001
0.999999
35.9514
35.9514
0.999325
18
36
36
1
36.0338
36.0338
1.00047
19
36
36
1
35.9766
35.9766
0.999674
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
34
36.0001
36.0001
1
35
35.9999
35.9999
0.999999
36
36
36
1
37
36
36
1For detailed study, these programs are again executed after changing the parameters, and some observations are given as below.Decreasing Function
(1)
Taking initial guessxo=0.3 (near common fixed point), Jungck-Noor iterative scheme converges in 14 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 8 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 5 iterations, and Jungck-S iterative scheme converges in 25 iterations while Jungck-Mann iterative scheme shows strange constant behavior.
(2)
Takingαn=βn=γn=1/(1+n)1/6 and xo=0.8, we observe that Jungck-Noor iterative scheme converges in 13 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 11 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 8 iterations, and Jungck-S iterative scheme converges in 27 iterations while Jungck-Mann iterative scheme shows strange constant behavior.Increasing Functions
(1)
Taking initial guessxo=3.2 (near coincidence point), Jungck-Noor iterative scheme converges in 7 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 8  iterations, Jungck-CR  and the Jungck-SP iterative schemes converge in a similar manner in 6 iterations, and Jungck-S iterative scheme converges in 7 iterations while  Jungck-Mann iterative scheme converges in 13 iterations.
(2)
Takingαn=βn=γn=1/(1+n)1/6 and xo=4, we observe  that Jungck-Noor iterative scheme converges in 7 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 8 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 6 iterations, and Jungck-S iterative scheme converges in 7 iterations while Jungck-Mann iterative scheme converges in 14 iterations.Oscillatory Function
(1)
Taking initial guessxo=1.3 (near common fixed point), Jungck-Noor iterative scheme converges in 8 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 6 terations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 5 iterations, Jungck-S iterative scheme converges in 11 iterations while Jungck-Mann iterative scheme converges in 19 iterations.
(2)
Takingαn=βn=γn=1/(1+n)1/6 and xo=2, we observe that Jungck-Noor iterative scheme converges in 8 iterations, Jungck-Ishikawa and Jungck-Agarwal  iterative schemes converge in a similar manner in 9  iterations, Jungck-CR  and the Jungck-SP iterative schemes converge in a similar manner in 6 iterations, Jungck-S iterative scheme converges in 12 iterations while Jungck-Mann iterative scheme converges in 21 iterations.Biquadratic Equation
(1)
Taking initial guessxo=0.8 (near coincidence point), Jungck-Noor iterative scheme converges in 11  iterations,  Jungck-Ishikawa and Jungck-Agarwal  iterative schemes converge in a similar manner in 7  iterations, Jungck-CR  and the Jungck-SP iterative schemes converge in a similar manner in 4 iterations, and Jungck-S iterative scheme converges in 18 iterations while  Jungck-Mann iterative scheme converges in 35 iterations.
(2)
Takingαn=βn=γn=1/(1+n)1/4 and  xo=0.5, we observe  that Jungck-Noor iterative scheme converges in 12 iterations, Jungck-Ishikawa and Jungck-Agarwal  iterative schemes converge in a similar manner in 8 iterations, Jungck-CR  and the Jungck-SP iterative schemes converge in a similar manner in 6 iterations, and Jungck-S iterative scheme converges in 19 iterations while Jungck-Mann iterative scheme converges in 37 iterations.


## 4.1. Jungck-Type Iterative Schemes in RNN Analysis

Recurrent neural networks (RNNs) are a class of densely connected single-layer nonlinear networks of perceptrons. RNNs not only operate on an input space but also on an internal statespace. This is equivalent to a with-memory Iterated Function System [28]. The state space enables the representation (and learning) of temporally/sequentially extended dependencies over unspecified (and potentially infinite) intervals according to
(68)y(t)=G(s(t))s(t)=F(s(t-1),x(t)).
Because of the network’s nonlinearity, a number of undesirable local energy minima emerge from the learning procedure. This has been shown to significantly affect the network’s performance. The iterative schemes like Mann, Ishikawa and J-iteration may be used to estimate the number of iterations required to achieve a stable state in recurrent autoassociative neural networks.


### 4.1.1. Decreasing Function(1-x)9

In order to solve this function by Jungck-type iterative schemes, we write it in the formSx=Tx, where the functions T, S:[0,1]→[0,2] are defined as T(x)=(1-x)9 and Sx=x, respectively. By taking initial approximation x0=0.8 and αn=βn=γn=1/n+14, the obtained results are listed in Table 3 showing convergence of different Jungck-type schemes to p=0.175699=T0.175699=S0.175699.Table 3
Decreasing function.
n
JN
JCR (JSP)
JA (JI)
JS
JM
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
0
5.12
e
-
007
5.12
e
-
007
0.337789
5.12
e
-
007
5.12
e
-
007
0.337789
5.12
e
-
007
5.12
e
-
007
0.0244889
5.12
e
-
007
5.12
e
-
007
0.0244889
5.12
e
-
007
5.12
e
-
007
0.8
1
0.0244889
0.0244889
0.215511
0.0244889
0.0244889
0.187422
0.8
0.8
0.191339
0.8
0.8
0.337789
5.12
e
-
007
5.12
e
-
007
0.8
2
0.112533
0.112533
0.187233
0.154449
0.154449
0.176002
0.147875
0.147875
0.173023
0.0244889
0.0244889
0.113605
5.12
e
-
007
5.12
e
-
007
0.8
3
0.154772
0.154772
0.179346
0.175121
0.175121
0.175701
0.180901
0.180901
0.175892
0.337789
0.337789
0.214684
5.12
e
-
007
5.12
e
-
007
0.8
4
0.168827
0.168827
0.176923
0.175696
0.175696
0.175699
0.17533
0.17533
0.175696
0.113605
0.113605
0.15714
5.12
e
-
007
5.12
e
-
007
0.8
5
0.173366
0.173366
0.176129
0.175699
0.175699
0.175699
0.175706
0.175706
0.175699
0.214684
0.214684
0.185861
5.12
e
-
007
5.12
e
-
007
0.8
6
0.174877
0.174877
0.175856
0.175699
0.175699
0.175699
0.1757
0.1757
0.175699
0.15714
0.15714
0.170534
5.12
e
-
007
5.12
e
-
007
0.8
7
0.1754
0.1754
0.175758
0.175699
0.175699
0.175699
0.185861
0.185861
0.178428
5.12
e
-
007
5.12
e
-
007
0.8
8
0.175587
0.175587
0.175722
0.175699
0.175699
0.175699
0.170534
0.170534
0.174287
5.12
e
-
007
5.12
e
-
007
0.8
9
0.175656
0.175656
0.175708
0.178428
0.178428
0.176438
5.12
e
-
007
5.12
e
-
007
0.8
10
0.175682
0.175682
0.175703
0.174287
0.174287
0.175315
5.12
e
-
007
5.12
e
-
007
0.8
11
0.175692
0.175692
0.175701
0.176438
0.176438
0.1759
5.12
e
-
007
5.12
e
-
007
0.8
12
0.175697
0.175697
0.1757
0.175315
0.175315
0.175595
5.12
e
-
007
5.12
e
-
007
0.8
13
0.175698
0.175698
0.1757
0.1759
0.1759
0.175754
5.12
e
-
007
5.12
e
-
007
0.8
14
0.175699
0.175699
0.1757
0.175595
0.175595
0.175671
5.12
e
-
007
5.12
e
-
007
0.8
15
0.175699
0.175699
0.175699
0.175754
0.175754
0.175714
5.12
e
-
007
5.12
e
-
007
0.8
16
0.175699
0.175699
0.175699
0.175671
0.175671
0.175692
5.12
e
-
007
5.12
e
-
007
0.8
17
0.175714
0.175714
0.175703
18
0.175692
0.175692
0.175697
19
0.175703
0.175703
0.175701
20
0.175697
0.175697
0.175699
21
0.175701
0.175701
0.1757
22
0.175699
0.175699
0.175699
23
0.1757
0.1757
0.1757
24
0.175699
0.175699
0.175699
25
0.1757
0.1757
0.175699
26
0.175699
0.175699
0.175699
27
0.175699
0.175699
0.175699


### 4.1.2. Increasing Functionx2-2x-3

In order to solve this function by Jungck-type iterative schemes, we write it in the formSx=Tx, where the functions T, S:[3,4]→[9,16] are defined as Tx=2x+3 and Sx=x2, respectively. By taking initial approximation x0=4 and αn=βn=γn=1/n+14, the obtained results are listed in Table 4 showing convergence of different Jungck-type schemes to p=9=T3=S3.Table 4
Increasing function.
n
JN
JCR (JSP)
JA (JI)
JS
JM
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
0
11
11
3.103748
11
11
3.034385
11
11
3.103748
11
11
3.10375
11
11
3.316625
1
9.207495
9.207495
3.01144
9.06877
9.068771
3.002208
9.207495
9.2075
3.015102
9.207495
9.2075
3.01144
9.63325
9.63325
3.103748
2
9.02288
9.02288
3.00127
9.00442
9.004417
3.000179
9.030205
9.03021
3.002482
9.02288
9.0229
3.00127
9.207495
9.207495
3.034385
3
9.00254
9.00254
3.000141
9.00036
9.000358
3.000017
9.004964
9.00496
3.000437
9.00254
9.0025
3.00014
9.068771
9.068771
3.01144
4
9.000282
9.000282
3.000016
9.00003
9.000033
3.000002
9.000875
9.00088
3.000081
9.000282
9.0003
3.00002
9.02288
9.02288
3.003811
5
9.000031
9.000031
3.000002
9
9.000003
3
9.000162
9.00016
3.000015
9.000031
9
3
9.007622
9.007622
3.00127
6
9.000003
9.000003
3
9
9
3
9.000031
9.00003
3.000003
9.000003
9
3
9.00254
9.00254
3.000423
7
9
9
3
9
9
3
9.000006
9.00001
3.000001
9
9
3
9.000847
9.000847
3.000141
8
9
9
3
9.000001
9
3
9
9
3
9.000282
9.000282
3.000047
9
9
9
3
9.000094
9.000094
3.000016
10
9
9
3
9.000031
9.000031
3.000005
11
9.00001
9.00001
3.000002
12
9.000003
9.000003
3.000001
13
9.000001
9.000001
3
14
9
9
3
15
9
9
3


### 4.1.3. Oscillating Function1/x

In order to solve this function by Jungck-type iterative schemes, we write it in the formSx=Tx, where the functions T, S:[0.5,2]→[0.25,4] are defined as Tx=1/x and Sx=x2, respectively. By taking initial approximation x0=2 and αn=βn=γn=1/n+14, the obtained results are listed in Table 5 showing convergence of different Jungck type schemes to p=1=T1=S1.Table 5
Oscillatory function.
n
JN
JCR (JSP)
JA (JI)
JS
JM
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
0
0.5
0.5
0.917004
0.5
0.5
0.917004
0.5
0.5
1.18921
0.5
0.5
1.18921
0.5
0.5
0.707107
1
1.09051
1.09051
1.01167
1.09051
1.09051
1.00285
0.840896
0.840896
1.02508
0.84096
0.84096
1.04427
0.41421
0.41421
1.18921
2
0.988463
0.988463
0.9983
0.997163
0.997163
0.999972
0.975531
0.975531
1.0018
0.957603
0.957603
1.01089
0.840896
0.840896
0.917004
3
1.0017
1.0017
1.00027
1.00003
1.00003
1
0.998208
0.998208
1.00005
0.989228
0.989228
1.00271
1.09051
1.09051
1.04427
4
0.999733
0.999733
0.999956
1
1
1
0.999945
0.999945
1
0.997296
0.997296
1.00068
0.957603
0.957603
0.978572
5
1.00004
1.00004
1.00001
1
1
1
1
1
1
0.999323
0.999323
1.00017
1.0219
1.0219
1.01089
6
0.999992
0.999992
0.999999
1
1
1
0.999831
0.999831
1.00004
0.989228
0.989228
0.994599
7
1
1
1
0.999958
0.999958
1.00001
1.00543
1.00543
1.00271
8
1
1
1
0.999989
0.999989
1
0.997296
0.997296
0.999662
9
0.999997
0.999997
1
1.00135
1.00135
1.00017
10
0.999999
0.999999
1
0.999323
0.999323
0.999915
11
1
1
1
1.00034
1.00034
1.00004
12
1
1
1
0.999831
0.999831
0.999979
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
18
0.999997
0.999997
0.999999
19
0.999999
0.999999
1
20
1
1
1
21
1
1
1


### 4.1.4.  Biquadratic Equationx4-36x2-52x+87=0

In order to solve this equation, we rewrite it in the formSx=Tx, where the functions T, S:[0.5,1.5]→[9,81] are defined as Tx=x4-52x+87 and Sx=36x2, respectively. Taking initial approximation x0=0.5 and αn=βn=γn=1/n+12, the obtained results are listed in Table 6 showing convergence of different Jungck-type schemes to p=36=T1=S1.Table 6
Biquadratic equation.
n
JN
JCR (JSP)
JA (JI)
JS
JM
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
0
61.25
61.25
1.1545
61.25
61.25
1.1545
61.25
61.25
0.761468
61.25
61.25
0.761468
61.25
61.25
1.30437
1
28.2988
28.2988
0.955826
28.2988
28.2988
0.995288
47.9835
47.9835
0.972744
47.9835
47.9835
0.88661
20.874
20.874
0.761468
2
38.2107
38.2107
1.01255
36.2356
36.2356
1
37.3636
37.3636
1.00048
41.6824
41.6824
0.945813
47.9835
47.9835
1.1545
3
35.3724
35.3724
0.996092
35.9999
35.9999
1
35.9758
35.9758
0.999949
38.7123
38.7123
0.973995
28.2988
28.2988
0.88661
4
36.1954
36.1954
1.00129
36
36
1
36.0026
36.0026
1.00001
37.3009
37.3009
0.98749
41.6824
41.6824
1.07603
5
35.9353
35.9353
0.999546
36
36
1
35.9996
35.9996
0.999998
36.6256
36.6256
0.993975
32.2042
32.2042
0.945813
6
36.0227
36.0227
1.00017
36.0001
36.0001
1
36.3013
36.3013
0.997096
38.7123
38.7123
1.03699
7
35.9917
35.9917
0.999937
36
36
1
36.1452
36.1452
0.9986
34.152
34.152
0.973995
8
36.0032
36.0032
1.00002
36
36
1
36.07
36.07
0.999325
37.3009
37.3009
1.01791
9
35.9988
35.9988
0.99999
36.0338
36.0338
0.999674
35.1049
35.1049
0.98749
10
36.0005
36.0005
1
36.0163
36.0163
0.999843
36.6256
36.6256
1.00865
11
35.9998
35.9998
0.999998
36.0078
36.0078
0.999924
35.5675
35.5675
0.993975
12
36.0001
36.0001
1
36.0038
36.0038
0.999963
36.3013
36.3013
1.00418
13
36
36
1
36.0018
36.0018
0.999982
35.7912
35.7912
0.997096
14
36
36
1
36.0009
36.0009
0.999992
36.1452
36.1452
1.00201
15
36.0004
36.0004
0.999996
35.8993
35.8993
0.9986
16
36.0002
36.0002
0.999998
36.07
36.07
1.00097
17
36.0001
36.0001
0.999999
35.9514
35.9514
0.999325
18
36
36
1
36.0338
36.0338
1.00047
19
36
36
1
35.9766
35.9766
0.999674
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
34
36.0001
36.0001
1
35
35.9999
35.9999
0.999999
36
36
36
1
37
36
36
1For detailed study, these programs are again executed after changing the parameters, and some observations are given as below.Decreasing Function
(1)
Taking initial guessxo=0.3 (near common fixed point), Jungck-Noor iterative scheme converges in 14 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 8 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 5 iterations, and Jungck-S iterative scheme converges in 25 iterations while Jungck-Mann iterative scheme shows strange constant behavior.
(2)
Takingαn=βn=γn=1/(1+n)1/6 and xo=0.8, we observe that Jungck-Noor iterative scheme converges in 13 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 11 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 8 iterations, and Jungck-S iterative scheme converges in 27 iterations while Jungck-Mann iterative scheme shows strange constant behavior.Increasing Functions
(1)
Taking initial guessxo=3.2 (near coincidence point), Jungck-Noor iterative scheme converges in 7 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 8  iterations, Jungck-CR  and the Jungck-SP iterative schemes converge in a similar manner in 6 iterations, and Jungck-S iterative scheme converges in 7 iterations while  Jungck-Mann iterative scheme converges in 13 iterations.
(2)
Takingαn=βn=γn=1/(1+n)1/6 and xo=4, we observe  that Jungck-Noor iterative scheme converges in 7 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 8 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 6 iterations, and Jungck-S iterative scheme converges in 7 iterations while Jungck-Mann iterative scheme converges in 14 iterations.Oscillatory Function
(1)
Taking initial guessxo=1.3 (near common fixed point), Jungck-Noor iterative scheme converges in 8 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 6 terations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 5 iterations, Jungck-S iterative scheme converges in 11 iterations while Jungck-Mann iterative scheme converges in 19 iterations.
(2)
Takingαn=βn=γn=1/(1+n)1/6 and xo=2, we observe that Jungck-Noor iterative scheme converges in 8 iterations, Jungck-Ishikawa and Jungck-Agarwal  iterative schemes converge in a similar manner in 9  iterations, Jungck-CR  and the Jungck-SP iterative schemes converge in a similar manner in 6 iterations, Jungck-S iterative scheme converges in 12 iterations while Jungck-Mann iterative scheme converges in 21 iterations.Biquadratic Equation
(1)
Taking initial guessxo=0.8 (near coincidence point), Jungck-Noor iterative scheme converges in 11  iterations,  Jungck-Ishikawa and Jungck-Agarwal  iterative schemes converge in a similar manner in 7  iterations, Jungck-CR  and the Jungck-SP iterative schemes converge in a similar manner in 4 iterations, and Jungck-S iterative scheme converges in 18 iterations while  Jungck-Mann iterative scheme converges in 35 iterations.
(2)
Takingαn=βn=γn=1/(1+n)1/4 and  xo=0.5, we observe  that Jungck-Noor iterative scheme converges in 12 iterations, Jungck-Ishikawa and Jungck-Agarwal  iterative schemes converge in a similar manner in 8 iterations, Jungck-CR  and the Jungck-SP iterative schemes converge in a similar manner in 6 iterations, and Jungck-S iterative scheme converges in 19 iterations while Jungck-Mann iterative scheme converges in 37 iterations.


## 4.1.1. Decreasing Function(1-x)9

In order to solve this function by Jungck-type iterative schemes, we write it in the formSx=Tx, where the functions T, S:[0,1]→[0,2] are defined as T(x)=(1-x)9 and Sx=x, respectively. By taking initial approximation x0=0.8 and αn=βn=γn=1/n+14, the obtained results are listed in Table 3 showing convergence of different Jungck-type schemes to p=0.175699=T0.175699=S0.175699.Table 3
Decreasing function.
n
JN
JCR (JSP)
JA (JI)
JS
JM
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
0
5.12
e
-
007
5.12
e
-
007
0.337789
5.12
e
-
007
5.12
e
-
007
0.337789
5.12
e
-
007
5.12
e
-
007
0.0244889
5.12
e
-
007
5.12
e
-
007
0.0244889
5.12
e
-
007
5.12
e
-
007
0.8
1
0.0244889
0.0244889
0.215511
0.0244889
0.0244889
0.187422
0.8
0.8
0.191339
0.8
0.8
0.337789
5.12
e
-
007
5.12
e
-
007
0.8
2
0.112533
0.112533
0.187233
0.154449
0.154449
0.176002
0.147875
0.147875
0.173023
0.0244889
0.0244889
0.113605
5.12
e
-
007
5.12
e
-
007
0.8
3
0.154772
0.154772
0.179346
0.175121
0.175121
0.175701
0.180901
0.180901
0.175892
0.337789
0.337789
0.214684
5.12
e
-
007
5.12
e
-
007
0.8
4
0.168827
0.168827
0.176923
0.175696
0.175696
0.175699
0.17533
0.17533
0.175696
0.113605
0.113605
0.15714
5.12
e
-
007
5.12
e
-
007
0.8
5
0.173366
0.173366
0.176129
0.175699
0.175699
0.175699
0.175706
0.175706
0.175699
0.214684
0.214684
0.185861
5.12
e
-
007
5.12
e
-
007
0.8
6
0.174877
0.174877
0.175856
0.175699
0.175699
0.175699
0.1757
0.1757
0.175699
0.15714
0.15714
0.170534
5.12
e
-
007
5.12
e
-
007
0.8
7
0.1754
0.1754
0.175758
0.175699
0.175699
0.175699
0.185861
0.185861
0.178428
5.12
e
-
007
5.12
e
-
007
0.8
8
0.175587
0.175587
0.175722
0.175699
0.175699
0.175699
0.170534
0.170534
0.174287
5.12
e
-
007
5.12
e
-
007
0.8
9
0.175656
0.175656
0.175708
0.178428
0.178428
0.176438
5.12
e
-
007
5.12
e
-
007
0.8
10
0.175682
0.175682
0.175703
0.174287
0.174287
0.175315
5.12
e
-
007
5.12
e
-
007
0.8
11
0.175692
0.175692
0.175701
0.176438
0.176438
0.1759
5.12
e
-
007
5.12
e
-
007
0.8
12
0.175697
0.175697
0.1757
0.175315
0.175315
0.175595
5.12
e
-
007
5.12
e
-
007
0.8
13
0.175698
0.175698
0.1757
0.1759
0.1759
0.175754
5.12
e
-
007
5.12
e
-
007
0.8
14
0.175699
0.175699
0.1757
0.175595
0.175595
0.175671
5.12
e
-
007
5.12
e
-
007
0.8
15
0.175699
0.175699
0.175699
0.175754
0.175754
0.175714
5.12
e
-
007
5.12
e
-
007
0.8
16
0.175699
0.175699
0.175699
0.175671
0.175671
0.175692
5.12
e
-
007
5.12
e
-
007
0.8
17
0.175714
0.175714
0.175703
18
0.175692
0.175692
0.175697
19
0.175703
0.175703
0.175701
20
0.175697
0.175697
0.175699
21
0.175701
0.175701
0.1757
22
0.175699
0.175699
0.175699
23
0.1757
0.1757
0.1757
24
0.175699
0.175699
0.175699
25
0.1757
0.1757
0.175699
26
0.175699
0.175699
0.175699
27
0.175699
0.175699
0.175699


## 4.1.2. Increasing Functionx2-2x-3

In order to solve this function by Jungck-type iterative schemes, we write it in the formSx=Tx, where the functions T, S:[3,4]→[9,16] are defined as Tx=2x+3 and Sx=x2, respectively. By taking initial approximation x0=4 and αn=βn=γn=1/n+14, the obtained results are listed in Table 4 showing convergence of different Jungck-type schemes to p=9=T3=S3.Table 4
Increasing function.
n
JN
JCR (JSP)
JA (JI)
JS
JM
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
0
11
11
3.103748
11
11
3.034385
11
11
3.103748
11
11
3.10375
11
11
3.316625
1
9.207495
9.207495
3.01144
9.06877
9.068771
3.002208
9.207495
9.2075
3.015102
9.207495
9.2075
3.01144
9.63325
9.63325
3.103748
2
9.02288
9.02288
3.00127
9.00442
9.004417
3.000179
9.030205
9.03021
3.002482
9.02288
9.0229
3.00127
9.207495
9.207495
3.034385
3
9.00254
9.00254
3.000141
9.00036
9.000358
3.000017
9.004964
9.00496
3.000437
9.00254
9.0025
3.00014
9.068771
9.068771
3.01144
4
9.000282
9.000282
3.000016
9.00003
9.000033
3.000002
9.000875
9.00088
3.000081
9.000282
9.0003
3.00002
9.02288
9.02288
3.003811
5
9.000031
9.000031
3.000002
9
9.000003
3
9.000162
9.00016
3.000015
9.000031
9
3
9.007622
9.007622
3.00127
6
9.000003
9.000003
3
9
9
3
9.000031
9.00003
3.000003
9.000003
9
3
9.00254
9.00254
3.000423
7
9
9
3
9
9
3
9.000006
9.00001
3.000001
9
9
3
9.000847
9.000847
3.000141
8
9
9
3
9.000001
9
3
9
9
3
9.000282
9.000282
3.000047
9
9
9
3
9.000094
9.000094
3.000016
10
9
9
3
9.000031
9.000031
3.000005
11
9.00001
9.00001
3.000002
12
9.000003
9.000003
3.000001
13
9.000001
9.000001
3
14
9
9
3
15
9
9
3


## 4.1.3. Oscillating Function1/x

In order to solve this function by Jungck-type iterative schemes, we write it in the formSx=Tx, where the functions T, S:[0.5,2]→[0.25,4] are defined as Tx=1/x and Sx=x2, respectively. By taking initial approximation x0=2 and αn=βn=γn=1/n+14, the obtained results are listed in Table 5 showing convergence of different Jungck type schemes to p=1=T1=S1.Table 5
Oscillatory function.
n
JN
JCR (JSP)
JA (JI)
JS
JM
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
0
0.5
0.5
0.917004
0.5
0.5
0.917004
0.5
0.5
1.18921
0.5
0.5
1.18921
0.5
0.5
0.707107
1
1.09051
1.09051
1.01167
1.09051
1.09051
1.00285
0.840896
0.840896
1.02508
0.84096
0.84096
1.04427
0.41421
0.41421
1.18921
2
0.988463
0.988463
0.9983
0.997163
0.997163
0.999972
0.975531
0.975531
1.0018
0.957603
0.957603
1.01089
0.840896
0.840896
0.917004
3
1.0017
1.0017
1.00027
1.00003
1.00003
1
0.998208
0.998208
1.00005
0.989228
0.989228
1.00271
1.09051
1.09051
1.04427
4
0.999733
0.999733
0.999956
1
1
1
0.999945
0.999945
1
0.997296
0.997296
1.00068
0.957603
0.957603
0.978572
5
1.00004
1.00004
1.00001
1
1
1
1
1
1
0.999323
0.999323
1.00017
1.0219
1.0219
1.01089
6
0.999992
0.999992
0.999999
1
1
1
0.999831
0.999831
1.00004
0.989228
0.989228
0.994599
7
1
1
1
0.999958
0.999958
1.00001
1.00543
1.00543
1.00271
8
1
1
1
0.999989
0.999989
1
0.997296
0.997296
0.999662
9
0.999997
0.999997
1
1.00135
1.00135
1.00017
10
0.999999
0.999999
1
0.999323
0.999323
0.999915
11
1
1
1
1.00034
1.00034
1.00004
12
1
1
1
0.999831
0.999831
0.999979
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
18
0.999997
0.999997
0.999999
19
0.999999
0.999999
1
20
1
1
1
21
1
1
1


## 4.1.4.  Biquadratic Equationx4-36x2-52x+87=0

In order to solve this equation, we rewrite it in the formSx=Tx, where the functions T, S:[0.5,1.5]→[9,81] are defined as Tx=x4-52x+87 and Sx=36x2, respectively. Taking initial approximation x0=0.5 and αn=βn=γn=1/n+12, the obtained results are listed in Table 6 showing convergence of different Jungck-type schemes to p=36=T1=S1.Table 6
Biquadratic equation.
n
JN
JCR (JSP)
JA (JI)
JS
JM
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
T
x
n
S
x
n
x
n
+
1
0
61.25
61.25
1.1545
61.25
61.25
1.1545
61.25
61.25
0.761468
61.25
61.25
0.761468
61.25
61.25
1.30437
1
28.2988
28.2988
0.955826
28.2988
28.2988
0.995288
47.9835
47.9835
0.972744
47.9835
47.9835
0.88661
20.874
20.874
0.761468
2
38.2107
38.2107
1.01255
36.2356
36.2356
1
37.3636
37.3636
1.00048
41.6824
41.6824
0.945813
47.9835
47.9835
1.1545
3
35.3724
35.3724
0.996092
35.9999
35.9999
1
35.9758
35.9758
0.999949
38.7123
38.7123
0.973995
28.2988
28.2988
0.88661
4
36.1954
36.1954
1.00129
36
36
1
36.0026
36.0026
1.00001
37.3009
37.3009
0.98749
41.6824
41.6824
1.07603
5
35.9353
35.9353
0.999546
36
36
1
35.9996
35.9996
0.999998
36.6256
36.6256
0.993975
32.2042
32.2042
0.945813
6
36.0227
36.0227
1.00017
36.0001
36.0001
1
36.3013
36.3013
0.997096
38.7123
38.7123
1.03699
7
35.9917
35.9917
0.999937
36
36
1
36.1452
36.1452
0.9986
34.152
34.152
0.973995
8
36.0032
36.0032
1.00002
36
36
1
36.07
36.07
0.999325
37.3009
37.3009
1.01791
9
35.9988
35.9988
0.99999
36.0338
36.0338
0.999674
35.1049
35.1049
0.98749
10
36.0005
36.0005
1
36.0163
36.0163
0.999843
36.6256
36.6256
1.00865
11
35.9998
35.9998
0.999998
36.0078
36.0078
0.999924
35.5675
35.5675
0.993975
12
36.0001
36.0001
1
36.0038
36.0038
0.999963
36.3013
36.3013
1.00418
13
36
36
1
36.0018
36.0018
0.999982
35.7912
35.7912
0.997096
14
36
36
1
36.0009
36.0009
0.999992
36.1452
36.1452
1.00201
15
36.0004
36.0004
0.999996
35.8993
35.8993
0.9986
16
36.0002
36.0002
0.999998
36.07
36.07
1.00097
17
36.0001
36.0001
0.999999
35.9514
35.9514
0.999325
18
36
36
1
36.0338
36.0338
1.00047
19
36
36
1
35.9766
35.9766
0.999674
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
34
36.0001
36.0001
1
35
35.9999
35.9999
0.999999
36
36
36
1
37
36
36
1For detailed study, these programs are again executed after changing the parameters, and some observations are given as below.Decreasing Function
(1)
Taking initial guessxo=0.3 (near common fixed point), Jungck-Noor iterative scheme converges in 14 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 8 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 5 iterations, and Jungck-S iterative scheme converges in 25 iterations while Jungck-Mann iterative scheme shows strange constant behavior.
(2)
Takingαn=βn=γn=1/(1+n)1/6 and xo=0.8, we observe that Jungck-Noor iterative scheme converges in 13 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 11 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 8 iterations, and Jungck-S iterative scheme converges in 27 iterations while Jungck-Mann iterative scheme shows strange constant behavior.Increasing Functions
(1)
Taking initial guessxo=3.2 (near coincidence point), Jungck-Noor iterative scheme converges in 7 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 8  iterations, Jungck-CR  and the Jungck-SP iterative schemes converge in a similar manner in 6 iterations, and Jungck-S iterative scheme converges in 7 iterations while  Jungck-Mann iterative scheme converges in 13 iterations.
(2)
Takingαn=βn=γn=1/(1+n)1/6 and xo=4, we observe  that Jungck-Noor iterative scheme converges in 7 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 8 iterations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 6 iterations, and Jungck-S iterative scheme converges in 7 iterations while Jungck-Mann iterative scheme converges in 14 iterations.Oscillatory Function
(1)
Taking initial guessxo=1.3 (near common fixed point), Jungck-Noor iterative scheme converges in 8 iterations, Jungck-Ishikawa and Jungck-Agarwal iterative schemes converge in a similar manner in 6 terations, Jungck-CR and the Jungck-SP iterative schemes converge in a similar manner in 5 iterations, Jungck-S iterative scheme converges in 11 iterations while Jungck-Mann iterative scheme converges in 19 iterations.
(2)
Takingαn=βn=γn=1/(1+n)1/6 and xo=2, we observe that Jungck-Noor iterative scheme converges in 8 iterations, Jungck-Ishikawa and Jungck-Agarwal  iterative schemes converge in a similar manner in 9  iterations, Jungck-CR  and the Jungck-SP iterative schemes converge in a similar manner in 6 iterations, Jungck-S iterative scheme converges in 12 iterations while Jungck-Mann iterative scheme converges in 21 iterations.Biquadratic Equation
(1)
Taking initial guessxo=0.8 (near coincidence point), Jungck-Noor iterative scheme converges in 11  iterations,  Jungck-Ishikawa and Jungck-Agarwal  iterative schemes converge in a similar manner in 7  iterations, Jungck-CR  and the Jungck-SP iterative schemes converge in a similar manner in 4 iterations, and Jungck-S iterative scheme converges in 18 iterations while  Jungck-Mann iterative scheme converges in 35 iterations.
(2)
Takingαn=βn=γn=1/(1+n)1/4 and  xo=0.5, we observe  that Jungck-Noor iterative scheme converges in 12 iterations, Jungck-Ishikawa and Jungck-Agarwal  iterative schemes converge in a similar manner in 8 iterations, Jungck-CR  and the Jungck-SP iterative schemes converge in a similar manner in 6 iterations, and Jungck-S iterative scheme converges in 19 iterations while Jungck-Mann iterative scheme converges in 37 iterations.


## 5. Conclusions

The speed of iterative schemes depends onαn, βn, and γn. From Tables 3–6 and observations made above, we make the following conjectures.


### 5.1. Decreasing Function

(1)
Decreasing order of rate of convergence of Jungck type iterative schemes is as follows: Jungck-CR (Jungck-SP), Jungck-Agarwal (Jungck-Ishikawa), Jungck-Noor, and Jungck-S iterative scheme.
(2)
For initial guess near to common fixed point, Jungck-CR (Jungck-SP), Jungck-Noor, and Jungck-S iterative schemes show a decrease while Jungck-Agarwal (Jungck-Ishikawa) iterative scheme shows no change in the number of iterations to converge.


### 5.2. Increasing Functions

(1)
Decreasing order of rate of convergence of Jungck-type iterative schemes is as follows: Jungck-CR (Jungck-SP), Jungck-S (Jungck-Noor), Jungck-Agarwal (Jungck-Ishikawa), and Jungck Mann iterative scheme.
(2)
For initial guess near to the coincidence point, all Jungck-type iterative schemes  show a  decrease in the number of iterations to converge.


### 5.3. Oscillatory Functions

(1)
Decreasing order of rate of convergence of Jungck-type iterative schemes is as follows: Jungck-CR (Jungck-SP), Jungck-Agarwal (Jungck-Ishikawa), Jungck-Noor, Jungck-S, and Jungck-Mann iterative scheme.
(2)
For initial guess near to the common fixed point, Jungck-Mann and Jungck-S  iterative schemes  show a  decr ease  while Jungck-CR (Jungck-SP), Jungck-Agarwal (Jungck-Ishikawa), and Jungck-Noor iterative schemes show  no change  in the number of iterations to converge.


### 5.4. Biquadratic Equation

(1)
Decreasing order of rate of convergence of Jungck type iterative schemes is as follows: Jungck-CR (Jungck-SP), Jungck-Agarwal (Jungck-Ishikawa), Jungck-Noor, Jungck-S, and Jungck-Mann iterative scheme.
(2)
For initial guess near to the coincidence point, all Jungck-type iterative schemes show a  decrease in the number of iterations to converge.Remark 21.
In each case mentioned above, Jungck-CR and Jungck-SP iterative schemes have better convergence rate as compared to other iterative schemes and hence have a good potential for further applications.


## 5.1. Decreasing Function

(1)
Decreasing order of rate of convergence of Jungck type iterative schemes is as follows: Jungck-CR (Jungck-SP), Jungck-Agarwal (Jungck-Ishikawa), Jungck-Noor, and Jungck-S iterative scheme.
(2)
For initial guess near to common fixed point, Jungck-CR (Jungck-SP), Jungck-Noor, and Jungck-S iterative schemes show a decrease while Jungck-Agarwal (Jungck-Ishikawa) iterative scheme shows no change in the number of iterations to converge.


## 5.2. Increasing Functions

(1)
Decreasing order of rate of convergence of Jungck-type iterative schemes is as follows: Jungck-CR (Jungck-SP), Jungck-S (Jungck-Noor), Jungck-Agarwal (Jungck-Ishikawa), and Jungck Mann iterative scheme.
(2)
For initial guess near to the coincidence point, all Jungck-type iterative schemes  show a  decrease in the number of iterations to converge.


## 5.3. Oscillatory Functions

(1)
Decreasing order of rate of convergence of Jungck-type iterative schemes is as follows: Jungck-CR (Jungck-SP), Jungck-Agarwal (Jungck-Ishikawa), Jungck-Noor, Jungck-S, and Jungck-Mann iterative scheme.
(2)
For initial guess near to the common fixed point, Jungck-Mann and Jungck-S  iterative schemes  show a  decr ease  while Jungck-CR (Jungck-SP), Jungck-Agarwal (Jungck-Ishikawa), and Jungck-Noor iterative schemes show  no change  in the number of iterations to converge.


## 5.4. Biquadratic Equation

(1)
Decreasing order of rate of convergence of Jungck type iterative schemes is as follows: Jungck-CR (Jungck-SP), Jungck-Agarwal (Jungck-Ishikawa), Jungck-Noor, Jungck-S, and Jungck-Mann iterative scheme.
(2)
For initial guess near to the coincidence point, all Jungck-type iterative schemes show a  decrease in the number of iterations to converge.Remark 21.
In each case mentioned above, Jungck-CR and Jungck-SP iterative schemes have better convergence rate as compared to other iterative schemes and hence have a good potential for further applications.



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*Source: 132626-2013-05-28.xml*