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Applied Mathematics and Computation 231 (2014) 521–535
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Analytical and numerical treatment of Jungck-type iterative schemes Abdul Rahim Khan a,⇑, Vivek Kumar b, Nawab Hussain c a
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia Department of Mathematics, KLP College, Rewari, India c Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia b
a r t i c l e
i n f o
Keywords: Jungck-type iterative schemes Data dependence Convergence rate Strong convergence Stability
a b s t r a c t In this paper, we introduce a new and general Jungck-type iterative scheme for a pair of nonself mappings and study its strong convergence, stability and data dependence. It is exhibited that our iterative scheme has much better convergence rate than those of Jungck–Mann, Jungck–Ishikawa, Jungck–Noor and Jungck–CR iterative schemes. Numerical examples in support of validity and applications of our results are provided. Our results are extension, improvement and generalization of many known results in the literature of fixed point theory. 2014 Elsevier Inc. All rights reserved.
1. Introduction Fixed point theory deals with finding fixed point of mappings and proving their uniqueness and iterative convergence. Data dependence of fixed points is a related and new issue which has become an important subject for research. The data dependence of various iterative schemes has been studied by many authors; see [1–3]. These authors have studied data dependence of fixed points in the context of self operators. Inspired by the work of Takahashi et al. [17] and Khan [6], here we continue study of convergence rate, stability and data dependence of fixed points through Jungck-type iterative schemes of nonself maps. Let X be a Banach space, Y an arbitrary set and S, T : Y ? X such that T(Y) # S(Y). For an 2 ½0; 1, Singh et al. [7] defined the Jungck–Mann iterative scheme as
Sxnþ1 ¼ ð1 an ÞSxn þ an Txn :
ð1:1Þ
For an ; bn ; an 2 ½0; 1, Olatinwo defined the Jungck–Ishikawa [8] and Jungck–Noor [9] iterative schemes as
Sxnþ1 ¼ ð1 an ÞSxn þ an Tyn ; Syn ¼ ð1 bn ÞSxn þ bn Txn ;
ð1:2Þ
and
Sxnþ1 ¼ ð1 an ÞSxn þ an Tyn ; Syn ¼ ð1 bn ÞSxn þ bn Tzn ; Szn ¼ ð1 an ÞSxn þ an Txn ; ⇑ Corresponding author. E-mail addresses: [email protected] (A.R. Khan), [email protected] (V. Kumar), [email protected] (N. Hussain). 0096-3003/$ - see front matter 2014 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.12.150
ð1:3Þ
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respectively, to approximate 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. Jungck [11] used the iterative scheme Sxnþ1 ¼ Txn n = 0, 1, . . ., to approximate common fixed points of the mappings S and T satisfying the following Jungck-contraction
dðTx; TyÞ 6 adðSx; SyÞ;
0 6 a < 1:
ð1:4Þ
Olatinwo and Imoru [10], studied the generalized Zamfirescu operators for the pair (S, T), satisfying the following condition: for each pair of points x, y in Y at least one of the following is true:
ðiÞ dðTx; TyÞ 6 adðSx; SyÞ ðiiÞ dðTx; TyÞ 6 b½dðSx; TxÞ þ dðSy; TyÞ ðiiiÞ dðTx; TyÞ 6 c½dðSx; TyÞ þ dðSy; TxÞ;
ð1:5Þ
where a, b, c are nonnegative constants satisfying 0 6 a 6 1, 0 6 b, c 6 The contractive condition (1.5) implies
dðTx; TyÞ 6 2ddðSx; TxÞ þ ddðSx; SyÞ;
8x;
y 2 Y;
1 . 2
ð1:6Þ
b c where 0 6 d < 1; d = max fa; 1b ; 1c g (see [4,5]). Singh et al. [7] established some stability results for Jungck and Jungck–Mann iterative schemes for both contractive conditions (1.4) and (1.7): for some q 2 ½0; 1Þ and 0 6 L
dðTx; TyÞ 6 qdðSx; SyÞ þ LdðSx; TxÞ:
ð1:7Þ
Obviously, (1.7) is more general than (1.4)–(1.6). Recently, Hussain et al. [12] used the following more general contractive condition than (1.7) to prove stability and strong convergence results for the Jungck-type iterative schemes: there exists a real number q e [0, 1) and a monotone increasing function /: R+ ? R+ such that /(0) = 0 and " x, y e Y, we have
kTx Tyk 6 /ðkSx TxkÞ þ qkSx Syk:
ð1:8Þ
Inspired by the above mentioned contributions, for an þ bn ; bn þ cn ; an 2 ½0; 1, we define Jungck–Khan iterative schemes for the pairs (T, S) and (T1, S1), respectively as follows:
Sxnþ1 ¼ ð1 an bn ÞSxn þ an Tyn þ bn Txn ; Syn ¼ ð1 bn cn ÞSxn þ bn Tzn þ cn Txn ; Szn ¼ ð1 an ÞSxn þ an Txn ;
ð1:9Þ
S1 unþ1 ¼ ð1 an bn ÞS1 un þ an T 1 v n þ bn T 1 un ; S1 v n ¼ ð1 bn cn ÞS1 un þ bn T 1 wn þ cn T 1 un ; S1 wn ¼ ð1 an ÞS1 un þ an T 1 un :
ð1:10Þ
Remark 1.1. Putting bn ¼ 0; cn ¼ 0 in Jungck–Khan iterative scheme (1.9), we get Jungck–Noor iterative scheme. Similarly, putting bn ¼ 0; cn ¼ 0; an ¼ 0 and bn ¼ 0; cn ¼ 0; an ¼ 0; bn ¼ 0; in (1.9), respectively, we get Jungck–Ishikawa and Jungck– Mann iterative schemes. Remark 1.2. If X = Y and S = Id (identity mapping), then Jungck–Noor (1.3), Jungck–Ishikawa (1.2) and the Jungck–Mann (1.1) iterative schemes become Noor [13], Ishikawa [14] and the Mann [15] iterative schemes, respectively. Definition 1.3 [16]. Let f and g be two selfmaps 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. Definition 1.4 [7]. Let S, T : Y ! X be nonself operator for an arbitrary set Y such that T(Y) # S(Y) and p a point of coincidence of S and T. Let fSxn g1 n¼0 X, be the sequence generated by an iterative procedure
Sxnþ1 ¼ f ðT; xn Þ; n ¼ 0; 1; . . . ;
ð1:11Þ
A.R. Khan et al. / Applied Mathematics and Computation 231 (2014) 521–535
523
1 where x0 2 X is the initial approximation and f is some function. Suppose fSxn g1 n¼0 converges to p. Let fSyn gn¼0 X be an arbitrary sequence and set en = d(Syn, f(T, yn)), n = 0, 1, . . . Then, the iterative procedure (1.11) is said to be (S, T)-stable or stable if and only if lim en ¼ 0 implies lim Syn = p. n!1
n!1
Definition 1.5 (5). 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
an a ¼ 0: lim n!1 bn b
Definition 1.6 ([1,5]). Let T1, T2 be two operators defined on a normed space (X, ||.||). We say T2 is approximate operator of T1 if for all x 2 X and for a fixed e > 0, we have kT 1 x T 2 xk 6 e. Lemma 1.7 [1]. Let fan g1 n¼0 be a nonnegative sequence for which there exist n0 2 N such that for all n P n0 , one has the following inequality:
anþ1 6 ð1 r n Þan þ rn tn ; where r n 2 (0, 1), for all n 2 N;
P1
n¼1 r n
¼ 1 and tn P 0 8n 2 N. Then, 0 6 lim sup an 6 lim sup tn . n!1
n!1
Lemma 1.8 [5]. If d is a real number such that 0 6 d < 1 and f2n g1 n¼0 is a sequence of positive numbers such that lim 2n ¼ 0, then n!1 for any sequence of positive numbers fun g1 n¼0 satisfying
unþ1 6 dun þ 2n ; n ¼ 0; 1; 2; . . . ; we have lim un ¼ 0. n!1
We establish here that Jungck–Khan scheme (1.9) introduced in this paper has much better convergence rate and it is more general as compared to Jungck–Mann, Jungck–Ishikawa and Jungck–Noor iterative schemes. As Jungck–Khan is independant of recently defined Jungck–CR scheme, so we shall discuss data dependence problem of both of these schemes. Numerical examples in support of validity and applications of our results are also provided. 2. Convergence and stability of Jungck–Khan iterative scheme Theorem 2.1. Let (X, ||.||) be an arbitrary Banach space and S, T: Y ? X be nonself operators satisfying contractive condition (1.8), on an arbitrary set Y with TðYÞ # SðYÞ, S(Y) is a complete subspace of X. Let Sz = Tz = p(say) and for x0 Y, let fSxn g1 n¼0 be the P P1 1 Jungck–Khan iterative scheme defined by (1.9) with 1 a ¼ 1 or b ¼ 1. Then the iterative scheme fSx g converges n n n n¼1 n¼1 n¼0 strongly to p. Also, p will be the unique common fixed point of S, T provided Y = X, and S and T are weakly compatible. Proof. It follows from (1.9) that
kSxnþ1 pk 6 ð1 an bn ÞkSxn pk þ an kTyn pk þ bn kTxn pk:
ð2:1Þ
Now, we have the following estimates:
kTxn pk 6 qkSxn pk þ /ðkSz TzkÞ ¼ qkSxn pk;
ð2:2Þ
kSzn pk 6 ð1 an ÞkSxn pk þ an kTxn pk 6 ð1 an ð1 qÞÞkSxn pk;
ð2:3Þ
and
kTyn pk 6 qkSyn pk 6 qð1 bn cn ÞkSxn pk þ qbn kTzn pk þ qcn kTxn pk 6 qð1 bn cn ÞkSxn pk þ bn q2 kSzn pk þ cn q2 kSxn pk 6 ½q bn qð1 qÞ cn qð1 qÞ an bn q2 ð1 qÞkSxn pk:
ð2:4Þ
Using estimates (2.2)–(2.4), (2.1) yields
kSxnþ1 pk 6 ½1 an ð1 qÞ bn ð1 qÞ an bn qð1 qÞ an cn qð1 qÞ an an bn q2 ð1 qÞkSxn pk;
ð2:5Þ
6 ½1 an ð1 qÞkSxn pk; ð1qÞ
6 e
1 X
ak
k¼0
kSx0 pk:
ð2:6Þ
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Pn P1 ð1qÞ a k¼0 k ? 0 as n ? 1. Hence, it follows from (2.6) that Since 0 6 q < 1, ak e [0, 1] and lim n¼0 an ¼ 1, so e n!1 kSxnþ1 pk = 0, which implies that the iterative scheme fSxn g1 n¼0 converges strongly to p. Now, we prove p is unique common fixed point of S and T, when Y = X. Let there exists another point of coincidence say p⁄. Then, there exists z⁄ e X such that Sz⁄ = Tz⁄ = p⁄. But from (1.8) we have
0 6 kp p k ¼ kTz Tz k 6 qkSz Sz k þ /ðkSz TzkÞ ¼ qkp p k; which implies p ¼ p as 0 6 q < 1. h Now, as S and T are weakly compatible and p = Tz = Sz, so Tp = TTz = TSz = STz and hence Tp = Sp. Therefore Tp is a point of coincidence of S, T and since the point of coincidence is unique so p = Tp. Thus Tp = Sp = p and therefore p is unique common fixed point of S and T. The main result in [10, Theorem 3.1] now becomes corollary of our Theorem 2.1 by employing the condition of weak compatibility in place of injectivity of S. Theorem 2.2. Let (X, ||.||) be a normed space and S, T: Y ? X are nonself operators on an arbitrary set Y such that TðYÞ # SðYÞ, where S(Y) is a complete subspace of X. Let p be a point of coincidence of S and T, i.e., Sz = Tz = p (say). Suppose that S and T satisfy condition (1.8). For x0 e Y, let fSxn g1 n¼0 be the Jungck–Khan iterative scheme converging to p, with 0 < a 6 an or 0 < b 6 bn , for all n. Then, the iterative scheme fSxn g1 n¼0 is (S, T)-stable. Proof. Let fSpn g1 n¼0 X be an arbitrary sequence, en ¼ kSpnþ1 ð1 an bn ÞSpn an Tqn bn Tpn k, n = 0, 1, 2, 3, . . ., where Sqn ¼ ð1 bn cn ÞSpn þ bn Tr n þ cn Tpn ; Sr n ¼ ð1 an ÞSpn þ an Tpn and let lim en ¼ 0. n!1 Then, using iterative scheme (1.9), we have
kSpnþ1 pk 6 kSpnþ1 ð1 an bn ÞSpn an Tqn bn Tpn k þ kð1 an bn ÞSpn þ an Tqn þ bn Tpn ð1 an bn þ bn þ an Þpk 6 en þ ð1 an bn ÞkSpn pk þ an kTqn pk þ bn kTpn pk 6 en þ ð1 an bn ÞkSpn pk þ an qkSqn pk þ bn qkSpn pk ¼ en þ ½1 an bn ð1 qÞkSpn pk þ an qkSqn pk: ð2:7Þ Now, we have the following estimates:
kSqn pk ¼ kð1 bn cn ÞSpn þ bn Tr n þ cn Tpn ð1 bn cn þ bn þ cn Þpk 6 ð1 bn cn ÞkSpn pk þ bn kTrn pk þ cn kTpn pk 6 ½1 bn cn ð1 qÞkSpn pk þ bn qkSr n pk;
ð2:8Þ
and
kSr n pk ¼ kð1 an ÞSpn þ an Tpn ð1 an þ an Þpk ¼ ½1 an ð1 qÞkSpn pk:
ð2:9Þ
It follows from (2.7), (2.8), and (2.9) that
kSpnþ1 pk 6 ½1 an ð1 qÞ bn ð1 qÞkSpn pk þ en 6 1 an ð1 qÞkSpn pk þ en :
ð2:10Þ
Using 0 < a 6 an and q 2 [0, 1) we have ½1 an ð1 qÞ < 1; 8n 2 N. Hence using Lemma (1.8), (2.10) yields lim Spnþ1 ¼ p. n!1 Conversely, let lim Spnþ1 ¼ p. Then using contractive condition (1.8) and the triangle inequality, we have n!1
en ¼ kSpnþ1 ð1 an bn ÞSpn an Tqn bn Tpn k 6 kSpnþ1 pk þ kð1 an bn ÞSpn þ an Tqn þ bn Tpn ð1 an bn þ bn þ an Þpk 6 kSpnþ1 pk þ kð1 an bn Þ kSpn pk þ an kTqn pk þ bn kTpn pk 6 kSpnþ1 pk þ kð1 an bn Þ kSpn pk þ an qkSqn pk þ bn qkSpn pk ¼ kSpnþ1 pk þ ½1 an ð1 qÞ bn ð1 qÞkSpn pkðusing ð2:7Þ and ð2:8ÞÞ ¼ ½1 an ð1 qÞkSpn pk þ kSpnþ1 pk: ð2:11Þ and hence lim
n!1
en ¼ 0: Therefore Jungck–Khan iterative scheme (1.9) is (S, T)-stable. h
3. Direct comparison of convergence rate In computational mathematics, convergence speed of an iterative scheme plays a vital role and it has been studied by many authors [4,5,12]. For recent work in this direction, we refer the reader to [12] and references therein. Theorem 3.1. Let (X, ||.||) be an arbitrary Banach space and S, T : Y ? X be nonself operators on an arbitrary set Y satisfying contractive condition (1.8). Assume that TðYÞ # SðYÞ, S(Y) is a complete subspace of X and Sz = Tz = p (say). For x0 e Y, let Jungck– Mann (JM), Jungck–Ishikawa (JI), Jungck–Noor (JN) and Jungck–Khan (JK) iterative scheme be defined by (1.1)–(1.3), (1.9), P 1 ; lim an ¼ 0; 1 respectively, with 0 6 an < 1þq n¼0 an ¼ 1 and 0 < l 6 bn ; n 2 N. Then Jungck–Khan iterative scheme converges n!1
faster than Jungck–Mann, Jungck–Ishikawa and Jungck–Noor iteratives schemes.
A.R. Khan et al. / Applied Mathematics and Computation 231 (2014) 521–535
525
Proof. For Jungck–Mann iterative scheme (1.1) we have
kSxnþ1 pk P ð1 an ÞkSxn pk an kTxn pk P ð1 an ÞkSxn pk an qkSxn pk ¼ ½1 an ð1 þ qÞkSxn pk P
n Y
½1 ai ð1 þ qÞkSx0 pk:
ð3:1Þ
i¼1
For Jungck–Ishikawa iterative scheme (1.2) we have
kSxnþ1 pk P ð1 an ÞkSxn pk an qkSyn pk P ½1 an an qð1 bn ð1 qÞkSxn pk P ½1 an ð1 þ qÞkSxn pk P
n Y
½1 ai ð1 þ qÞkSx0 pk:
ð3:2Þ
i¼1
Similarly, for Jungck–Noor iterative scheme (1.3), it is easy to see that
kSxnþ1 pk P
n Y
½1 ai ð1 þ qÞkSx0 pk:
ð3:3Þ
i¼1
Also, for Jungck–Khan iterative scheme (1.9), using (2.5) we have n
kSxnþ1 pk 6 ½1 bn ð1 qÞkSxn pk 6 ½1 lð1 qÞkSxn pk 6 ½1 lð1 qÞ kSx0 pk:
ð3:4Þ
Using (3.3) and (3.4), we have n
kSxnþ1 ðJKÞ pk ½1 lð1 qÞ kSx0 pk 6 n : Y kSxnþ1 ðJNÞ pk ½1 ai ð1 þ qÞkSx0 pk i¼1 ½1lð1qÞn
Now, let pn ¼ Qn
i¼1
Then
pnþ1 pn
½1ai ð1þqÞ
.
½1lð1qÞ ¼ ½1 anþ1 ð1þqÞ. pnþ1 n!1 pn
Using lim an ¼ 0, we get lim n!1
¼ 1 lð1 qÞ < 1.
P1
kSxnþ1 ðJKÞpk lim pn ¼ 0. Hence lim kSx ¼ 0, i.e. Jungck–Khan Therefore by ratio test n¼0 pn < 1 ; which further yields nþ1 ðJNÞpk n!1 n!1 iterative scheme (1.9) converges faster than Jungck–Noor iterative scheme (1.3) to p. Similar argument can be applied to show that
lim
n!1
kSxnþ1 ðJKÞ pk ¼ 0 and kSxnþ1 ðJIÞ pk
lim
kSxnþ1 ðJKÞ pk ¼ 0; pk
n!1 kSxnþ1 ðJMÞ
which imply that the new iterative scheme (1.9) converges faster than Jungck–Ishikawa (1.2) and Jungck–Mann (1.1) iterative schemes to p.The following example shows the validity of Theorem 3.1. h Example 3.2. Let T, S: [1, 4] ? [1, 16] be defined as Tx = 2x + 3, Sx = x2. It is easy to see that (T, S) is a quasi-contractive operator pair satisfying (1.8) with point of coincidence 9. By taking initial approximation x0 = 2, /ðtÞ ¼ 2qt and 1 n an ¼ an ¼ bn ¼ pffiffiffiffiffiffiffiffi ; bn ¼ cn ¼ pffiffiffiffiffiffiffiffi ; n P 1; the obtained results are listed in Table 1, showing convergence of different 2nþ1 2 n þ1
Jungck type schemes to p = 9 = T3 = S3. Remark 3.3. Table 1 shows that Jungck–Khan iterative scheme (1.9) has much better convergence rate than Jungck–Noor, P 1 Jungck–Mann and Jungck–Ishikawa iterative schemes in the context of conditions 0 6 an < 1þq ; lim an ¼ 0; 1 n¼0 an ¼ 1 and n!1 0 < l 6 bn ; n 2 N; used in Theorem 3.1. Moreover, the iterative scheme (1.9) can have better convergence rate than the above P1 P1 mentioned iterative schemes in the context of condition n¼0 an ¼ 1 or n¼0 bn ¼ 1 used in Theorem 2.1, as shown in Tables 2–5. The following Jungck–CR iterative schemes are recently defined and studied in [12]:
Sxnþ1 ¼ ð1 an ÞSyn þ an Tyn ; Syn ¼ ð1 bn ÞTxn þ bn Tzn ; Szn ¼ ð1 an ÞSxn þ an Txn ; S1 unþ1 ¼ ð1 an ÞS1 v n þ an T 1 v n ;
ð3:5Þ
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A.R. Khan et al. / Applied Mathematics and Computation 231 (2014) 521–535
Table 1 1 n Convergence of different Jungck type iterative schemes to coincidence point p = 9 of Tx = 2x + 3 and Sx = x2. (x0 = 2, an ¼ an ¼ bn ¼ pffiffiffiffiffiffiffiffi ; n P 1). ; bn ¼ cn ¼ pffiffiffiffiffiffiffiffi 2nþ1 2 n þ1
n
1 2 3 4 5 6 7 8 9 10 11 – 199 200 201 202 203 204 205 – 226 227 228 229
Jungck–Khan iterative scheme
Jungck–Noor iterative scheme
Jungck–Ishikawa iterative scheme
Jungck–Mann iterative scheme
Jungck–CR iterative scheme
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
7 8.93074 8.99917 8.99996 9 9
4 8.79343 8.9975 8.99988 8.99999 9
2.96537 2.99958 2.99998 3 3 3
–
–
–
7 8.01791 8.37444 8.55903 8.67111 8.74553 8.79787 8.83624 8.86522 8.88766 8.90536 – 9 9 9 9
4 6.29486 7.22116 7.72571 8.04039 8.25277 8.40384 8.51541 8.60021 8.66612 8.71831 – 8.99999 8.99999 9 9
2.50896 2.68722 2.77952 2.83556 2.87276 2.89894 2.91812 2.93261 2.94383 2.95268 2.95977 – 3 3 3 3
–
–
–
7 7.78834 8.15816 8.37441 8.51562 8.68633 8.7408 8.78303 8.81643 8.84328 8.68633 – 9 9 9 9 9 9 9 – 9 9 9 9
4 5.73205 6.65165 7.22106 7.60551 8.08358 8.23921 8.36086 8.45772 8.53599 8.08358 – 8.99999 8.99999 8.99999 8.99999 8.99999 8.99999 8.99999 – 8.99999 8.99999 9 9
2.39417 2.57908 2.6872 2.75781 2.80712 2.8704 2.89152 2.90822 2.92164 2.93259 2.8704 – 3 3 3 3 3 3 3 – 3 3 3 3
2.83474 2.96628 2.99238 2.99817 2.99954 2.99988 2.99997 2.99999 3 3 3 –
–
2.48741 2.66946 2.76556 2.82449 2.86384 2.91207 2.92754 2.93954 2.94902 2.95663 2.91207 – 3 3 3 3 3 3 3 –
4 8.03578 8.79879 8.95435 8.98903 8.99725 8.99929 8.99981 8.99995 8.99999 9 –
–
4 6.1872 7.12601 7.64835 7.97774 8.36157 8.48015 8.57051 8.6409 8.69673 8.36157 – 8.99999 8.99999 8.99999 8.99999 8.99999 9 9 –
7 8.66949 8.93255 8.98476 8.99634 8.99908 8.99976 8.99994 8.99998 9 9 –
–
7 7.97482 8.33892 8.53113 8.64898 8.78328 8.82414 8.85509 8.87908 8.89804 8.78328 – 9 9 9 9 9 9 9 –
–
–
–
Table 2 1 Convergence of different Jungck type iterative schemes to coincidence point p = 9 of Tx = 2x + 3 and Sx = x2. (x0 = 1.4, an ¼ an ¼ bn ¼ pffiffiffiffiffiffiffiffi ¼ bn ¼ cn ; n P 0). 2nþ1 n
Jungck–Khan iterative scheme
Jungck–CR iterative scheme
Txn
Sxn
xn+1
0 1 – 2 6 7 8 9 – 19 20 – 28 29 30 31 – 131 132 133 134 135 – 161 162 – 210 211 212
5.9 10.3826 – 9.06366 9.00235 9.0014 9.00087 9.00057 – 9.00002 9.00002 – 9 9 9 9 –
2.1025 13.6255 – 9.192 9.00706 9.0042 9.00262 9.0017 – 9.00007 9.00005 – 9.00001 9.00001 9.00001 9 –
3.69128 3.03183 – 3.00981 3.0007 3.00044 3.00028 3.00019 – 3.00001 3.00001 – 3 3 3 3 –
–
–
–
–
–
–
5.9 8.86734 – 8.9788 8.99994 8.99998 9 9 –
–
2.1025 8.60642 – 8.9365 8.99983 8.99995 8.99999 9 –
–
Jungck–Noor iterative scheme
2.93367 2.9894 – 2.99785 2.99999 3 3 3 –
–
Jungck–Ishikawa iterative scheme
Jungck–Mann iterative scheme
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
5.9 8.86734 – 8.93037 8.98124 8.98506 8.98788 8.99001 – 8.99771 8.99797 – 8.99905 8.99914 8.99922 8.99929 – 9 9 9 9 9 –
2.1025 8.60642 – 8.79233 8.94381 8.95524 8.96366 8.97005 – 8.99314 8.99391 – 8.99715 8.99741 8.99765 8.99787 – 8.99999 8.99999 8.99999 9 9 –
2.93367 2.96519 – 2.97737 2.99253 2.99394 2.995 2.99583 – 2.99899 2.9991 – 2.99957 2.99961 2.99964 2.99968 – 3 3 3 3 3 –
–
–
2.1025 7.85798 – 8.3779 8.82725 8.86209 8.88783 8.90741 – 8.97865 8.98105 – 8.9911 8.99193 8.99268 8.99334 – 8.99998 8.99998 8.99998 8.99998 8.99999 – 8.99999 9 –
2.80321 2.89446 – 2.93086 2.97693 2.98125 2.98453 2.98708 – 2.99684 2.99719 – 2.99865 2.99878 2.99889 2.99899 – 3 3 3 3 3 – 3 3 –
5.9 7.85798 – 8.303 8.77562 8.81736 8.84896 8.87349 – 8.96802 8.97144 – 8.98608 8.98733 8.98845 8.98946 – 8.99997 8.99997 8.99997 8.99997 8.99997 – 8.99999 8.99999 – 9 9 9
2.1025 5.9 – 7.03044 8.33946 8.46043 8.55259 8.62447 – 8.90433 8.91453 – 8.95829 8.96204 8.96539 8.96839 – 8.9999 8.9999 8.99991 8.99991 8.99992 – 8.99997 8.99997 – 8.99999 8.99999 9
2.42899 2.6515 – 2.75673 2.90868 2.92448 2.93674 2.94646 – 2.98572 2.98721
–
5.9 8.60642 – 8.78892 8.94214 8.95385 8.96249 8.96906 – 8.99288 8.99368 – 8.99703 8.99731 8.99756 8.99778 – 8.99999 8.99999 8.99999 8.99999 9 – 9 9 –
2.99367 2.99423 2.99473 2.99518 – 2.99998 2.99998 2.99999 2.99999 2.99999 – 3 3 – 3 3 3
527
A.R. Khan et al. / Applied Mathematics and Computation 231 (2014) 521–535
Table 3 1 Convergence of different Jungck type iterative schemes to coincidence point p⁄ = 2 of T1 x = 2x and S1 x = x2 + 1. (x0 = 1.45, an ¼ an ¼ bn ¼ pffiffiffiffiffiffiffiffi ¼ bn ¼ cn ; n P 0). 2nþ1 n
0 1 2 3 4 5 – 239 240 241 242 243 244 –
Jungck–Khan iterative scheme
Jungck–CR iterative scheme
Jungck–Noor iterative scheme
Jungck–Ishikawa iterative scheme
Jungck–Mann iterative scheme
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
2.9 2.27239 2.24124 2.22399 2.21203 2.20288 – 2.06767 2.06756 2.06746 2.06735 2.06725 2.06715 –
3.1025 2.29094 2.25579 2.23653 2.22327 2.21317 – 2.06882 2.06871 2.0686 2.06849 2.06838 2.06827 –
1.1362 1.12062 1.11199 1.10601 1.10144 1.09774 – 1.03378 1.03373 1.03368 1.03363 1.03357 1.03352 –
2.9 2.56974 2.46094 2.39454 2.34758 2.3119
3.1025 2.65089 2.51405 2.43346 2.37778 2.33622
1.28487 1.23047 1.19727 1.17379 1.15595 1.14179
2.01514 2.01508 2.01502 2.01497 2.01491 2.01485
2.0152 2.01514 2.01508 2.01502 2.01496 2.0149
1.00754 1.00751 1.00748 1.00745 1.00742 1.0074
2.9 2.56974 2.5085 2.47398 2.44982 2.43119 – 2.14368 2.14345 2.14322 2.143 2.14277 2.14254 –
3.1025 2.65089 2.57314 2.53014 2.5004 2.47768 – 2.14884 2.1486 2.14835 2.14811 2.14787 2.14762 –
1.28487 1.25425 1.23699 1.22491 1.2156 1.20802 – 1.07173 1.07161 1.0715 1.07138 1.07127 1.07116 –
2.9 2.65089 2.58363 2.54358 2.51491 2.49256 – 2.152 2.15175 2.15149 2.15124 2.15099 2.15074 –
3.1025 2.75681 2.66879 2.61744 2.58119 2.55322 – 2.15778 2.15751 2.15723 2.15696 2.15669 2.15642 –
1.32545 1.29182 1.27179 1.25745 1.24628 1.23713 – 1.07587 1.07575 1.07562 1.0755 1.07537 1.07525 –
2.9 2.75681 2.69617 2.65567 2.6249 2.59998 – 2.17362 2.1733 2.17298 2.17267 2.17236 2.17205 –
3.1025 2.9 2.81733 2.76314 2.72252 2.68998 – 2.18115 2.18081 2.18046 2.18012 2.17979 2.17945 –
1.3784 1.34808 1.32783 1.31245 1.29999 1.28951 – 1.08665 1.08649 1.08634 1.08618 1.08602 1.08587 –
S1 v n ¼ ð1 bn ÞT 1 un þ bn T 1 wn ; S1 wn ¼ ð1 an ÞS1 un þ an T 1 un ;
ð3:6Þ
where an ; bn ; an 2 ½0; 1 Remark 3.4. Putting an ¼ 0; bn ¼ 1 and an ¼ 0 in the iterative scheme (3.5), we get Jungck–S [12] and Jungck–Agarwal et al. [12] iterative schemes, respectively. Remark 3.5. It is not possible to compare directly Jungck–Khan (1.9) and Jungck–CR iterative scheme (3.5). However, Table 2 and Table 5 show that Jungck–Khan iterative scheme can have better convergence rate than Jungck–CR iterative scheme.
4. Data dependence In this section we establish data dependence of Jungck–Khan and Jungck–CR iterative schemes and deduce as corollaries certain recent data dependence results for well-known iterative schemes like Jungck–Noor, Jungck–Mann, Jungck–Ishikawa, Jungck–Agarwal et al. and Jungck–S. Theorem 4.1. Let (X, ||.||) be an arbitrary Banach space and (S, T), (S1, T1) : Y ? X be nonself operator pairs on an arbitrary set Y with (S, T) satisfying contractive condition (1.8) such that kTx T 1 xk 6 e; kSx S1 xk 6 e1 . Assume that TðYÞ # SðYÞ, T 1 ðYÞ # S1 ðYÞ, 1 where S(Y) and S1 ðYÞ are complete subspaces of X with Sz = Tz = p and S1z⁄ = T1 z⁄ = p⁄. Suppose that fSxn g1 n¼0 , fSun gn¼0 are ⁄ Jungck–Khan iterative schemes (1.9) and (1.10) associated with (S, T) and (S1, T1), which converge to p and p , respectively. Then we have
kp p k 6 where
10e2 ; 1q
e2 = max {e, e1 }, provided bn 6 an ; 8n 2 N and
P1
n¼1
an ¼ 1.
Proof. It follows from (1.9) and (1.10) that
kSxnþ1 S1 unþ1 k 6 ð1 an bn ÞkSxn S1 un k þ an kTyn T 1 v n k þ bn kTxn T 1 un k:
ð4:1Þ
Now we have the following estimates:
kTyn T 1 v n k 6 kTyn T v n k þ kT v n T 1 v n k 6 qkSyn Sv n k þ /ðkSyn Tyn kÞ þ e;
ð4:2Þ
kSyn Sv n k 6 kSyn S1 v n k þ kS1 v n Sv n k 6 kSyn S1 v n k þ e1 ;
ð4:3Þ
kSyn S1 v n k 6 ð1 bn cn ÞkSxn S1 un k þ bn kTzn T 1 wn k þ cn kTxn T 1 un k;
ð4:4Þ
kTzn T 1 wn k 6 kTzn Twn k þ kTwn T 1 wn k 6 qkSzn Swn k þ /ðkSzn Tzn kÞ þ e;
ð4:5Þ
kSzn Swn k 6 kSzn S1 wn k þ kS1 wn Swn k ¼ kSzn S1 wn k þ e1 ;
ð4:6Þ
528
Table 4 1 Convergence of different Jungck type iterative schemes to common fixed point p = 0.5 of Tx = 12 ð12 þ xÞ and Sx = 1 x. (x0 = 0.6, an ¼ an ¼ bn ¼ pffiffiffiffiffiffiffiffi ¼ bn ¼ cn ; n P 0). 2nþ1 n
Txn
Sxn
xn+1
0.55 0.491406 0.500002 0.5
0.4 0.517188 0.499996 0.5
0.482812 0.500004 0.5 0.5
– 0.5 0.5 0.5 – 0.5 0.5 0.5 0.5 0.5
– 0.5 0.5 0.5 – 0.5 0.5 0.5 0.5 0.5
– 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
Jungck–CR iterative scheme
0.55 0.496094 0.500568 0.499892 0.500024 0.499994 0.500002 0.5 0.5
0.4 0.507813 0.498864 0.500217 0.499952 0.500012 0.499997 0.500001 0.5
0.492188 0.501136 0.499783 0.500048 0.499988 0.500003 0.499999 0.5 0.5
Jungck–Noor iterative scheme
Jungck–Ishikawa iterative scheme
Jungck–Mann iterative scheme
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
0.55 0.519531 0.509514 0.505203 0.503063 0.501901 0.501227 0.500817 0.500558 – 0.500002 0.500002 0.500001 – 0.5 0.5 0.5
0.4 0.460938 0.480972 0.489595 0.493873 0.496199 0.497546 0.498366 0.498883 – 0.499996 0.499997 0.499997 – 0.499999 0.499999 0.5
0.539063 0.519028 0.510405 0.506127 0.503801 0.502454 0.501634 0.501117 0.500779 – 0.500003 0.500003 0.500002 – 0.500001 0.5 0.5
0.55 0.521875 0.511214 0.506318 0.503795 0.502389 0.501559 0.501048 0.500721 – 0.500003 0.500002 0.500002 – 0.5 0.5 0.5 0.5 0.5
0.4 0.45625 0.477573 0.487364 0.49241 0.495222 0.496882 0.497905 0.498558 – 0.499995 0.499996 0.499996 – 0.499999 0.499999 0.499999 0.499999 0.5
0.54375 0.522427 0.512636 0.50759 0.504778 0.503118 0.502095 0.501442 0.501012 – 0.500004 0.500004 0.500003 – 0.500001 0.500001 0.500001 0.5 0.5
0.55 0.5125 0.504845 0.502276 0.501196 0.500678 0.500406 0.500254 0.500164 – 0.5 0.5 0.5 – 0.5 0.5 0.5 0.5 0.5
0.4 0.475 0.490309 0.495449 0.497608 0.498643 0.499187 0.499492 0.499672 – 0.499999 0.499999 0.5 – 0.5 0.5 0.5 0.5 0.5
0.525 0.509691 0.504551 0.502392 0.501357 0.500813 0.500508 0.500328 0.500218 – 0.500001 0.5 0.5 – 0.5 0.5 0.5 0.5 0.5
A.R. Khan et al. / Applied Mathematics and Computation 231 (2014) 521–535
0 1 2 3 4 5 6 7 8 – 30 31 32 – 40 41 42 43 44
Jungck–Khan iterative scheme
Table 5 1 Convergence of different Jungck type iterative schemes to common fixed point p⁄ = 1 of T1 = 1þx and S1x = x2. (x0 = 0.6, an ¼ an ¼ bn ¼ pffiffiffiffiffiffiffiffi ¼ bn ¼ cn ; n P 0). 2 2nþ1 n
Jungck–CR iterative scheme
Jungck–Noor iterative scheme
Jungck–Ishikawa iterative scheme
Jungck–Mann iterative scheme
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
0.8 0.968824 0.990049 –
0.36 0.879183 0.960594 –
0.937647 0.980099 0.991652 –
0.8 0.974178 0.996068 – 0.999999 1 1
0.36 0.89938 0.984332 – 0.999997 0.999999 1
0.948356 0.992135 0.99869 – 1 1 1
– 1 1 1 – 1 1
– 0.999999 0.999999 1 – 1 1
– 1 1 1 – 1 1
0.8 0.895889 0.933565 – 0.983801 0.986873 0.989223 – 0.999473 0.999522 0.999566 – 1 1 1
0.36 0.626913 0.751914 – 0.936254 0.948183 0.957358 – 0.997891 0.998089 0.998266 – 0.999999 0.999999 1
0.791778 0.86713 0.90729 – 0.973747 0.978447 0.982111 – 0.999044 0.999132 0.999211 – 1 1 1
– –
– –
– –
–
–
–
–
–
–
0.8 0.893826 0.931824 – 0.98323 0.986401 0.988828 – 0.99945 0.999502 0.999548 – 1 1 1 1 –
0.36 0.620394 0.745889 – 0.934045 0.946342 0.955813 – 0.997803 0.998009 0.998193 – 0.999999 0.999999 0.999999 1 –
0.787651 0.863648 0.904551 – 0.972801 0.977657 0.981445 – 0.999004 0.999096 0.999178 – 1 1 1 1 –
0.8 0.880789 0.919165 – 0.977739 0.981719 0.984813 – 0.999155 0.999232 0.999301 – 1 1 1 1 – 1 1 1
0.36 0.58 0.702796 – 0.912937 0.928211 0.940176 – 0.996623 0.99693 0.997205 – 0.999999 0.999999 0.999999 0.999999 – 0.999999 0.999999 1
0.761577 0.83833 0.882776 – 0.963437 0.969627 0.974522 – 0.998464 0.998601 0.998725 – 1 1 1 1 – 1 1 1
A.R. Khan et al. / Applied Mathematics and Computation 231 (2014) 521–535
0 1 2 – 7 8 9 – 33 34 35 – 174 175 176 177 – 190 191 192
Jungck–Khan iterative scheme
529
530
A.R. Khan et al. / Applied Mathematics and Computation 231 (2014) 521–535
and
kSzn S1 wn k 6 ð1 an ÞkSxn S1 un k þ an kTxn T 1 un k 6 ð1 an ÞkSxn S1 un k þ an kTxn Tun k þ an kTun T 1 un k 6 ð1 an ÞkSxn S1 un k þ an qkSxn Sun k þ an /ðkSxn Txn kÞ þ an e 6 ð1 an ÞkSxn S1 un k þ an qkSxn S1 un k þ an qkS1 un Sun k þ an /ðkSxn Txn kÞ þ an e 6 ½1 an ð1 qÞkSxn S1 un k þ an /ðkSxn Txn kÞ þ an qe1 þ an e:
ð4:7Þ
Using estimates (4.5)–(4.7), we find
kTzn T 1 wn k 6 q½1 an ð1 qÞkSxn S1 un k þ /ðkSzn Tzn kÞ þ qan /ðkSxn Txn kÞ þ qe1 þ e þ an q2 e1 þ an qe:
ð4:8Þ
It follows from (4.4) and (4.8) that
kSyn S1 v n k 6 ð1 bn cn ÞkSxn S1 un k þ bn q½1 an ð1 qÞÞkSxn S1 un k þ an bn q/ðkSxn Txn kÞ þ bn /ðkSzn Tzn kÞ þ cn kTxn T 1 un k þ bn e þ an bn e þ an bn q2 e1 þ bn qe1 ; 6 ½1 bn ð1 qÞ cn kSxn S1 un k þ an bn q/ðkSxn Txn kÞ þ bn /ðkSzn Tzn kÞ þ cn kTxn T 1 un k þ bn e þ an bn e þ an bn q2 e1 þ bn qe1 :
ð4:9Þ
Also, (4.2) and (4.9) together yield
kTyn T 1 v n k 6 q½1 bn ð1 qÞ cn kSxn S1 un k þ an bn q2 /ðkSxn Txn kÞ þ bn q/ðkSzn Tzn kÞ þ qcn kTxn T 1 un k þ /ðkSyn Tyn kÞ þ bn qe þ bn q2 e1 þ e þ an bn qe þ an bn q3 e1 :
ð4:10Þ
Using (4.10), (4.1) yields
kSxnþ1 S1 unþ1 k 6 ð1 an bn ÞkSxn S1 un k þ an q½1 bn ð1 qÞ cn kSxn S1 un k þ an an bn q2 /ðkSxn Txn kÞ þ an bn q/ðkSzn Tzn kÞ þ an qcn kTxn T 1 un k þ an bn qe þ an bn q2 e1 þ an qe1 þ an /ðkSyn Tyn kÞ þ an e þ bn kTxn T 1 un k þ an an bn qe þ an an bn q3 e1 6 ð1 an ð1 qÞ an qcn bn ÞkSxn S1 un k þ an an bn q2 /ðkSxn Txn kÞ þ an bn q/ðkSzn Tzn kÞ þ an /ðkSyn Tyn kÞ þ ðan qcn þ bn ÞkTxn T 1 un k þ an qe1 ð1 þ bn qÞ þ an eð1 þ bn qÞ þ an an bn qe þ an an bn q3 e1 :
ð4:11Þ
Now
kSxn Txn k 6 kSxn pk þ kp Txn k 6 kSxn pk þ qkSxn pk ¼ ð1 þ qÞkSxn pk; which yields
lim kSxn Txn k ¼ 0: ðusing lim Sxn ¼ pÞ
n!1
n!1
ð4:12Þ
Also,
kSyn Tyn k 6 kSyn pk þ kp Tyn k 6 kSyn pk þ qkSyn pk ¼ ð1 þ qÞkSyn pk; with
kSyn pk 6 ð1 bn cn ÞkSxn pk þ bn kTzn pk þ cn kTxn pk 6 ð1 bn cn ÞkSxn pk þ bn qkSzn pk þ cn qkSxn pk 6 ð1 bn cn ÞkSxn pk þ þbn q½ð1 an ÞkSxn pk þ an kTxn pk þ cn qkSxn pk 6 ð1 bn cn ÞkSxn pk þ þbn qð1 an ÞkSxn pk þ bn an q2 kSxn pk þ cn qkSxn pk; implies lim kSyn Tyn k ¼ 0: n!1
ð4:13Þ
Similarly
kSzn Tzn k 6 kSzn pk þ kp Tzn k 6 kSzn pk þ qkSzn pk ¼ ð1 þ qÞkSzn pk; with
kSzn pk 6 ð1 an ÞkSxn pk þ an qkSxn pk ¼ ½1 an ð1 qÞkSxn pk; implies
lim kSzn Tzn k ¼ 0:
n!1
ð4:14Þ
A.R. Khan et al. / Applied Mathematics and Computation 231 (2014) 521–535
531
Moreover,
kTxn T 1 un k 6 kTxn Tun k þ kTun T 1 un k 6 qkSxn Sun k þ /ðkSxn Txn kÞ þ e 6 qkSxn S1 un k þ qkS1 un Sun k þ /ðkSxn Txn kÞ þ e 6 qkSxn S1 un k þ qe1 þ /ðkSxn Txn kÞ þ e:
ð4:15Þ
Using estimate (4.15), (4.11) becomes
kSxnþ1 S1 unþ1 k 6 ½1 an ð1 qÞ bn ð1 qÞ an qcn ð1 an cn qÞkSxn S1 un k þ an an bn q2 /ðkSxn Txn kÞ þ an qcn /ðkSxn Txn kÞ þ bn /ðkSxn Txn kÞ þ an bn q/ðkSzn Tzn kÞ þ an /ðkSyn Tyn kÞ þ ðan qe1 þ an eÞð1 þ bn q þ an bn q þ cn qÞ þ qe1 bn þ bn e ¼ ð1 r n ÞkSxn S1 un k þ rn t n ; ðusing bn 6 an Þ;
ð4:16Þ
2 a b q2 /ðkSx Tx kÞ þ qc /ðkSx Tx kÞ þ /ðkSx Tx kÞ 3 n n n n n n n n n 7 6 þbn q/ðkSzn Tzn kÞ þ /ðkSyn Tyn kÞ 7 6 7 6 þðqe þ eÞð1 þ b q þ a b q þ c qÞ þ qe þ e 7 6 1 n n n n 1 where rn ¼ an ð1 qÞ and t n ¼ 6 7: 7 6 1q 7 6 5 4 Lemma 1.7 and estimates (4.12)–(4.14), (4.16) yield
kp p k 6 where
10e2 ; 1q
e2 = max{e, e1 }. Hence the result. h
Corollary 4.2. Let (X, ||.||) be an arbitrary Banach space and (S, T), (S1, T1): Y ? X be nonself operator pairs on an arbitrary set Y with (S, T) satisfying contractive condition (1.8) such that kTx T 1 xk 6 e; kSx S1 xk 6 e1 . Assume that TðYÞ # SðYÞ, 1 T 1 ðYÞ # S1 ðYÞ, where S(Y) and S1 ðYÞ are complete subspaces of X with Sz = Tz = p and S1z⁄ = T1z⁄ = p⁄. Let fSxn g1 n¼0 , fSun gn¼0 be ⁄ the Jungck–Noor iterative schemes associated with (S, T) and (S1, T1), and converging to p and p , respectively. Then we have
kp p k 6 where
6e2 ; 1q
e2 = max{e, e1 }, e; e1 > 0; provided
P1
n¼1
an ¼ 1.
Proof. Putting bn ¼ cn ¼ 0 in (1.9) we get the desired result. h Remark 4.3. As Jungck–Ishikawa [12] and the Jungck–Mann (1.1) iterative schemes are special cases of Jungck–Khan iterative scheme (1.9), results similar to Corollary 4.2 hold for these iterative schemes. Also putting S = Id (identity mapping), Y = X, in (1.9), data dependence results of Ishikawa iterative scheme [1, Theorem 3.2] and Noor iterative scheme [2, Theorem 3.1] can be obtained as corollaries. Now we establish data dependence result for Jungck–CR scheme. Theorem 4.4. Let (X, ||.||) be an arbitrary Banach space and (S, T), (S1, T1): Y ? X be nonself operator pairs on an arbitrary set Y with (S, T) satisfying contractive condition (1.8) such that kTx T 1 xk 6 e; kSx S1 xk 6 e1 . Assume that TðYÞ # SðYÞ, T 1 ðYÞ # S1 ðYÞ, where S(Y) and S1 ðYÞ are complete subspaces of X with Sz = Tz = p and S1z⁄ = T1 z⁄ = p⁄. Suppose that fSxn g1 n¼0 , ⁄ fSun g1 n¼0 are the Jungck–CR iterative schemes (3.5) and (3.6) associated with (S, T) and (S1, T1), which converge to p and p , respectively. Then we have
kp p k 6 where
6e2 ; 1q
e2 = max{e, e1 }, provided an ð1 qÞ P 12 ; 8n 2 N and
P1
n¼1
an ¼ 1.
Proof. It follows from (3.5) and (3.6) that
kSxnþ1 S1 unþ1 k 6 ð1 an ÞkSyn S1 v n k þ an kTyn T 1 v n k:
ð4:17Þ
Now we have the following estimates:
kTyn T 1 v n k 6 kTyn T v n k þ kT v n T 1 v n k 6 qkSyn Sv n k þ /ðkSyn Tyn kÞ þ e;
ð4:18Þ
532
A.R. Khan et al. / Applied Mathematics and Computation 231 (2014) 521–535
kSyn Sv n k 6 kSyn S1 v n k þ kS1 v n Sv n k 6 kSyn S1 v n k þ e1 :
ð4:19Þ
Using (4.17), (4.18), and (4.19) becomes
kSxnþ1 S1 unþ1 k 6 ½1 an ð1 qÞkSyn S1 v n k þ an /ðkSyn Tyn kÞ þ an e þ an qe1 :
ð4:20Þ
Now we have the following estimates:
kSyn S1 v n k 6 ð1 bn ÞkTxn T 1 un k þ bn kTzn T 1 wn k 6 ð1 bn ÞkTxn Tun k þ ð1 bn ÞkTun T 1 un k þ bn kTzn Twn k þ bn kTwn T 1 wn k 6 ð1 bn ÞqkSxn Sun k þ ð1 bn Þ/ðkSxn Txn kÞ þ ð1 bn Þe þ bn qkSzn Swn k þ bn /ðkSzn Tzn kÞ þ bn e ¼ ð1 bn ÞqkSxn Sun k þ ð1 bn Þ/ðkSxn Txn kÞ þ bn /ðkSzn Tzn kÞ þ bn qkSzn Swn k þ e; kSxn Sun k 6 kSxn S1 un k þ kS1 un Sun k 6 kSxn S1 un k þ e1 ;
ð4:21Þ ð4:22Þ
kSzn Swn k 6 kSzn S1 wn k þ kS1 wn Swn k 6 ð1 an ÞkSxn S1 un k þ an kTxn T 1 un k þ e1 6 ð1 an ÞkSxn S1 un k þ an kTxn Tun k þ an kTun T 1 un k þ e1 6 ð1 an ÞkSxn S1 un k þ an qkSxn Sun k þ an /ðkSxn Txn kÞ þ an eþ e1 6 ½1 an ð1 qÞkSxn S1 un k þ an /ðkSxn Txn kÞ þ an qe1 þ an e þ e1 :
ð4:23Þ
Using estimates (4.21)–(4.23) yields
kSyn S1 v n k 6 ð1 bn ÞqkSxn S1 un k þ ð1 bn Þqe1 þ ð1 bn Þ/ðkSxn Txn kÞ þ bn /ðkSzn Tzn kÞ þ bn q½1 an ð1 qÞkSxn S1 un k þ an bn q/ðkSxn Txn kÞ þ an bn q2 e1 þ an bn qe þ bn qe1 þ e 6 ½ð1 bn Þq þ bn qð1 an ð1 qÞÞkSxn S1 un k þ ð1 bn Þ/ðkSxn Txn kÞ þ bn /ðkSzn Tzn kÞ þ an bn q/ðkSxn Txn kÞ þ eð an bn q þ 1Þ þ qe1 ð an bn q þ 1Þ:
ð4:24Þ
It follows from (4.20) and (4.24) that
kSxnþ1 S1 unþ1 k 6 ½1 an ð1 qÞkSxn S1 un k þ ½1 an ð1 qÞ/ð kSxn Txn kÞ þ ½1 an ð1 qÞ/ðkSzn Tzn kÞ þ ½1 an ð1 qÞan bn q/ðkSxn Txn kÞ þ ½1 an ð1 qÞðe þ qe1 Þðan bn q þ 1Þ þ an ðe þ qe1 Þ þ an /ð kSyn Tyn kÞ:
ð4:25Þ
Now an ð1 qÞ P 12 implies1 an ð1 qÞ 6 an ð1 qÞ 6 an . Hence inequality (4.25) implies
kSxnþ1 S1 unþ1 k 6 ½1 an ð1 qÞkSxn S1 un k þ an /ðkSxn Txn kÞ þ an /ðkSzn Tzn kÞ þ an an bn q/ðkSxn Txn kÞ þ an ðe þ qe1 Þð an bn q þ 1Þ þ an ðe þ qe1 Þ þ an /ðkSyn Tyn kÞ; ¼ ð1 r n ÞkSxn S1 un k þ rn t n ;
ð4:26Þ
n kÞþðeþqe1 Þðan bn qþ1Þþðeþqe1 Þþ/ð kSyn Tyn kÞ ð/ðkSxn Txn kÞþ/ðkSzn Tzn kÞþan bn q/ðkSxn Tx1q Þ.
where rn = an ð1 qÞ and tn = As in Theorem 4.1, it is easy to see that lim kSzn Tzn k ¼ lim kSxn Txn k ¼ lim kSyn Tyn k ¼ 0. n!1
n!1
n!1
Hence Lemma 1.7 and (4.26) yield
kp p k 6 where
6e2 ; 1q
e2 = max{e, e1 }. Hence the result. h
As Jungck–Agarwal et al. iterative scheme [12] is a special case of Jungck–CR iterative scheme, we have the following result: Corollary 4.5. Let (X, ||.||) be an arbitrary Banach space and (S, T), (S1, T1): Y ? X be nonself operator pairs on an arbitrary set Y with (S, T) satisfying contractive condition (1.8) such that kTx T 1 xk 6 e; kSx S1 xk 6 e1 . Assume that TðYÞ # SðYÞ, T 1 ðYÞ # S1 ðYÞ, where S(Y) and S1 ðYÞ are complete subspaces of X with Sz = Tz = p and S1z⁄ = T1z⁄ = p⁄. Suppose that fSxn g1 n¼0 , ⁄ fSun g1 n¼0 are the Jungck–Agarwal et al. iterative schemes associated with (S, T) and (S1, T1), which converge to p and p , respectively. Then we have
kp past k 6 where
4e2 ; 1q
e2 = max {e, e1 }, provided an ð1 qÞ P 12 ; 8n 2 N and
P1
n¼1
an ¼ 1.
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A.R. Khan et al. / Applied Mathematics and Computation 231 (2014) 521–535
Proof. Putting an ¼ 0; bn ¼ an in the iterative scheme (3.5), we get the desired result. h Remark 4.6. Set an ¼ 0; bn ¼ 1; an ¼ an in (3.5) to obtain data dependence result [3, Theorem 4] for Jungck–S iterative scheme studied in [12]. The following two examples reveal the validity of our Theorem 4.1. Example 4.7. Let T, S, T1, and S1: [1, 4] ? [1, 17] be defined as Tx = 2x + 3, Sx = x2, T1x = 2x and S1x = x2 + 1. It is easy to see that (S, T) and (S1, T1) are approximate operator pairs with (S, T) satisfying (1.8) and /ðtÞ ¼ 2qt. Also, here e = 3, e1 ¼ 1 and so 1 ffi e2 = 3. Taking initial approximation x0 = 1.45 and an ¼ bn ¼ an ¼ bn ¼ cn ¼ pffiffiffiffiffiffiffiffi ; n P 0; results obtained are listed in Table 2 2nþ1 and Table 3 which show convergence of different Jungck type schemes associated with (S, T) and (S1, T1), respectively, to p = 9 = T3 = S3 and p⁄ = 2 = T11 = S11. Example 4.8. Let T, S, T1, and S1: [0, 1] ? [0, 1] be defined by T(x) = 12 ð12 þ xÞ, S(x) = 1 x; T1x = 1þx and S1x = x2. It is easy to see 2 that (S, T) and (S1, T1) are approximate operator pairs with (S, T) satisfying (1.8) and /ðtÞ ¼ 2qt. Also, here e = 0.25, e1 ¼ 1 and 1 ; n P 0; results obtained are listed in so e2 = 1. By taking initial approximation x0 = 0.6 and an ¼ bn ¼ an ¼ bn ¼ cn ¼ pffiffiffiffiffiffiffiffi 2nþ4 Table 4 and Table 5 which show convergence of different Jungck type schemes associated to (S, T) and (S1, T1), respectively, to p = 0.5 = T 0.5 = S 0.5 and p⁄ = 1 = T11 = S11. The following example reveals the validity of Theorem 4.4. Example 4.9. Let T, S, T1, and S1: [1, 4] ? [1, 17] be defined by Tx = 2x + 3, Sx = x2, T1x = 2x and S1x = x2 + 1. It is easy to see that (S, T) and (S1, T1) are approximate operator pairs with (T, S) satisfying (1.8) and /ðtÞ ¼ 2qt. Also, here e = 3, e1 ¼ 1 and so e2 = 3. n 1 ffi ; bn ¼ an ¼ bn ¼ cn ¼ pffiffiffiffiffiffiffiffi ; n P 1; the results obtained are listed in By taking initial approximation x0 = 2 and an ¼ nþ0:2 2nþ1 Table 6 which shows convergence of Jungck–CR iterative schemes associated with (S, T) and (S1, T1), respectively, to p = 9 = T3 = S3 and p⁄ = 2 = T11 = S11. 5. Applications In this section, with the help of computer programs in C++, we apply Jungck–Khan iterative scheme (1.9) to solve various type of problems, some of which are not easy to solve by other methods. Solution of Legendre polynomial 63 x5 35 x3 þ 15 x¼0 8 4 8 To solve this polynomial we rewrite it in the form Sx = Tx, where Tx = 63x5 þ 15x and Sx ¼ 70x3 : Here the operators S, T : 1 ; n P 0; the ½0:2; 0:9 ! ½0:56; 51:03 are such that T(x) 2 SðxÞ; 8x 2 ½0:2; 0:9: Taking x0 ¼ 0:2; an ¼ bn ¼ an ¼ bn ¼ cn ¼ pffiffiffiffiffiffiffiffi 2nþ4 convergence of Jungck–Khan iterative scheme to the coincedence point 0.538471 of S and T is listed in the Table 7. Solution of equation ex ðx2 þ 5x þ 2Þ þ 1 ¼ 0
Table 6 Convergence of Jungck–CR iterative scheme to coincidence point p = 9 of Tx = 2x + 3, and Sx = x2 as well as to coincidence point p⁄ = 2 of T1 x = 2x and S1x = x2 + 1. n 1 (x0 = 2, an ¼ nþ0:2 ; bn ¼ an ¼ bn ¼ cn ¼ pffiffiffiffiffiffiffiffi ; n P 1). 2nþ1 n
Txn
Sxn
xn+1
T1xn
S1xn
xn+1
1 2 3 4 5 6 7 8 9 10 11 12 13 – 246 247 248 249 –
7 8.46082 8.8424 8.9523 8.98526 8.99538 8.99854 8.99953 8.99985 8.99995 8.99998 8.99999 9 – 9 9 9 9 –
4 7.45515 8.53341 8.85746 8.95583 8.98615 8.99562 8.9986 8.99955 8.99986 8.99995 8.99998 9 – 9 9 9 9 –
2.73041 2.9212 2.97615 2.99263 2.99769 2.99927 2.99977 2.99993 2.99998 2.99999 3 3 3 – 3 3 3 3 –
4 3.12411 2.76002 2.56778 2.45078 2.37269 2.31711 2.27565 2.24359 2.2181 2.19736 2.18017 2.1657 – 2.0082 2.00816 2.00813 2.0081 –
5 3.44001 2.90442 2.64838 2.50158 2.40741 2.34225 2.29465 2.25843 2.22999 2.2071 2.18829 2.17256 – 2.00821 2.00818 2.00815 2.00811 –
1.56205 1.38001 1.28389 1.22539 1.18634 1.15855 1.13783 1.1218 1.10905 1.09868 1.09009 1.08285 1.07667 – 1.00408 1.00407 1.00405 1.00403 –
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We rewrite above equation in the form Sx = Tx, where the operators T, S: ½2; 0:5 ! ½10; 2:5 are defined by 1 ; n P 0, Tx = ex x2 2 and Sx = 5x, respectively. With initial approximation x0 = 1 and an ¼ bn ¼ an ¼ bn ¼ cn ¼ pffiffiffiffiffiffiffiffi 2nþ4 the convergence of Jungck–Khan iterative scheme to the coincedence point 0.579167 of S and T (and solution of equation) is listed in the Table 8. Solution of quadratic equation x2 10x þ 9 ¼ 0 In order to solve this equation by Jungck–Khan iterative scheme (1.9), we write it in the form Sx = Tx, where the operators T, S: ½1; 4 ! ½10; 40 are defined by Tx = x2 þ 9 and Sx = 10x, respectively. With initial approximation x0 = 2 and an ¼ bn ¼ an ¼ bn ¼ cn ¼ n15 ; n P 1; the convergence of Jungck–Khan iterative scheme to the coincedence point 1 of S and T (and solution of quadratic equation) is listed in the Table 9. For detailed study, these programs are executed after changing the parameters and some of the observations are given below:
5.1. Observations 5.1.1. Solution of Legendre polynomial 1. Taking initial guess xo = 0.5 (near coincidence point), Jungck–Khan iterative scheme (1.9) converges in 24 iterations. 1 2. Taking an ¼ bn ¼ an ¼ bn ¼ cn ¼ p4ffiffiffiffiffiffiffiffi ; n P 0; and xo = 0.2, we observe that Jungck–Khan iterative scheme converges in 21 2nþ4 iterations. Table 7 Convergence of Jungck–Khan iterative scheme to coincidence point p = 0.538471 of Tx = 63x5 þ 15x and Sx ¼ 70x3 : 1 (x0 ¼ 0:2, an ¼ bn ¼ an ¼ bn ¼ cn ¼ pffiffiffiffiffiffiffiffi ; n P 0Þ. 2nþ4 n
Txn
Sxn
xn+1
0 1 2 3 4 5 – 23 24 25 26 27
3.02016 6.84522 8.68351 9.63872 10.1666 10.47 – 10.929 10.929 10.929 10.9291 10.9291
3.02017 6.84524 8.68353 9.63874 10.1666 10.47 – 10.929 10.929 10.9291 10.9291 10.9291
0.408546 0.476129 0.504846 0.519245 0.527105 0.531609 – 0.538469 0.53847 0.53847 0.538471 0.538471
Table 8 Convergence of Jungck–Khan iterative scheme to coincidence point p = 0.579167 of Tx = ex x2 2 and Sx = 5x. 1 (x0 ¼ 1; an ¼ bn ¼ an ¼ bn ¼ cn ¼ pffiffiffiffiffiffiffiffi ; n P 0Þ. 2nþ4 n
Txn
Sxn
xn+1
0 1 2 3 4 5 6
3.36792 2.92951 2.89837 2.89604 2.89585 2.89583 2.89583
3.36792 2.92951 2.89837 2.89604 2.89585 2.89583 2.89583
0.629975 0.583365 0.579509 0.579196 0.579169 0.579167 0.579167
Table 9 Convergence of Jungck–Khan iterative scheme to coincidence point p = 1 of Tx = x2 þ 9 and Sx = 10x. (x0 ¼ 2; an ¼ bn ¼ an ¼ bn ¼ cn ¼ n15 ; n P 1Þ. n
Txn
Sxn
xn+1
1 2 3 4 5 6 7
13 10.0288 10.0056 10.0011 10.0002 10 10
13 10.0288 10.0056 10.0011 10.0002 10 10
1.01428 1.0028 1.00056 1.00011 1.00002 1 1
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535
5.1.2. Solution of equation ex ðx2 þ 5x þ 2Þ þ 1 ¼ 0 1. Taking initial guess x0 = 2 (away from coincidence point), Jungck–Khan iterative scheme converges in 7 iterations. 1 2. Taking an ¼ bn ¼ an ¼ bn ¼ cn ¼ p8ffiffiffiffiffiffiffiffi ; n P 0; and x0 = 1, we observe that Jungck–Khan iterative scheme converges in 5 2nþ4 iterations. 5.1.3. Solution of quadratic equation 1. Taking initial guess x0 = 4 (away from coincidence point), Jungck–Khan iterative scheme converges in 9 iterations. 1 2. Taking an ¼ bn ¼ an ¼ bn ¼ cn ¼ n0:1 ; n P 1 and x0 = 2, we observe that Jungck–Khan iterative scheme converges in 5 iterations. 6. Conclusions Keeping in mind the results of Sections 2-4, Tables 1–9 and observations made in Section 5, we make the following remarks: The newly introduced Jungck-type iterative scheme is more general as well as has much better convergence rate as compared to Jungck–Mann, Jungck–Ishikawa and Jungck–Noor iterative schemes and hence has a good potential for further applications. Data dependency of fixed points is performed for nonself operators for a more general iterative scheme with a higher convergence rate. Many typical problems can be solved using Jungck-type iterative schemes. The speed of iterative schemes depend on parameters an , bn etc. For initial guess, near the solution, there is a decrease in number of iterations to converge to the solution of desired problem.
Acknowledgments The authors A.R. Khan and N. Hussain acknowledge gratefully the support of KACST, Riyad, Saudi Arabia for supporting Project J-P-11-0623. The authors would like to thank the referee for his/her valuable suggestions to improve presentation of the paper. References [1] S.M. Soltuz, T. Grosan, Data dependence for Ishikawa iteration when dealing with contractive like operators, Fixed Point Theory Appl. 2008 (2008) 7. 242916. [2] R. Chugh, V. Kumar, Data dependence of Noor and SP iterative schemes when dealing with quasi-contractive operators, Int. J. Comput. Appl. 40 (2011) 41–46. [3] F. Gursoy, V. Karakaya, B. Rhoades, Data dependence results of new multi-step and S-iterative schemes for contractive-like operators, Fixed Point Theory Appl. 76 (2013) 12. [4] V. Berinde, Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators, Fixed Point Theory Appl. 2004 (2004) 716359, http://dx.doi.org/10.1155/S1687182004311058. [5] V. Berinde, Iterative Approximation of Fixed Points, Springer, 2007. [6] A.R. Khan, On modified Noor iterations for asymptotically nonexpansive mappings, Bull. Belg. Math. Soc. Simon Stevin 17 (2010) 127–140. [7] S.L. Singh, C. Bhatnagar, S.N. Mishra, Stability of Jungck-type iterative procedures, Int. J. Math. Math. Sci. 193035–3043 (2005). [8] M.O. Olatinwo, Some stability and strong convergence results for the Jungck–Ishikawa iteration process, Creating Math. Inf. 17 (2008) 33–42. [9] M.O. Olatinwo, A generalization of some convergence results using the Jungck–Noor three-step iteration process in an arbitrary Banach space, Fasciculi Math. 40 (2008) 37–43. [10] M.O. Olatinwo, C.O. Imoru, Some convergence results for the Jungck–Mann and the Jungck–Ishikawa iteration processes in the class of generalized Zamfirescu operators, Acta Math. Univ. Comenianae LXXVII, 2 (2008) 299–304. [11] G. Jungck, Commuting mappings and fixed points, Am. Math. Monthly 83 (1976) 261–263. [12] N. Hussain, V. Kumar, M.A. Kutbi, On the rate of convergence of Jungck-type iterative schemes, Abstract Appl. Anal. 2013 (2013) 15. Article ID 132626. [13] M.A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000) 217–229. [14] S. Ishikawa, Fixed points by a new iteration method, Proc. Am. Math. Soc. 44 (1974) 147–150. [15] W.R. Mann, Mean value methods in iteration, Proc. Am. Math. Soc. 4 (1953) 506–510. [16] G. Jungck, N. Hussain, Compatible maps and invariant approximations, J. Math. Anal. Appl. 325 (2007) 1003–1012. [17] W. Takahashi, S. Matsushita, Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions, Nonlinear Anal. 68 (2008) 412–419.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Analytical and numerical treatment of Jungck-type iterative schemes Abdul Rahim Khan a,⇑, Vivek Kumar b, Nawab Hussain c a
Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia Department of Mathematics, KLP College, Rewari, India c Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia b
a r t i c l e
i n f o
Keywords: Jungck-type iterative schemes Data dependence Convergence rate Strong convergence Stability
a b s t r a c t In this paper, we introduce a new and general Jungck-type iterative scheme for a pair of nonself mappings and study its strong convergence, stability and data dependence. It is exhibited that our iterative scheme has much better convergence rate than those of Jungck–Mann, Jungck–Ishikawa, Jungck–Noor and Jungck–CR iterative schemes. Numerical examples in support of validity and applications of our results are provided. Our results are extension, improvement and generalization of many known results in the literature of fixed point theory. 2014 Elsevier Inc. All rights reserved.
1. Introduction Fixed point theory deals with finding fixed point of mappings and proving their uniqueness and iterative convergence. Data dependence of fixed points is a related and new issue which has become an important subject for research. The data dependence of various iterative schemes has been studied by many authors; see [1–3]. These authors have studied data dependence of fixed points in the context of self operators. Inspired by the work of Takahashi et al. [17] and Khan [6], here we continue study of convergence rate, stability and data dependence of fixed points through Jungck-type iterative schemes of nonself maps. Let X be a Banach space, Y an arbitrary set and S, T : Y ? X such that T(Y) # S(Y). For an 2 ½0; 1, Singh et al. [7] defined the Jungck–Mann iterative scheme as
Sxnþ1 ¼ ð1 an ÞSxn þ an Txn :
ð1:1Þ
For an ; bn ; an 2 ½0; 1, Olatinwo defined the Jungck–Ishikawa [8] and Jungck–Noor [9] iterative schemes as
Sxnþ1 ¼ ð1 an ÞSxn þ an Tyn ; Syn ¼ ð1 bn ÞSxn þ bn Txn ;
ð1:2Þ
and
Sxnþ1 ¼ ð1 an ÞSxn þ an Tyn ; Syn ¼ ð1 bn ÞSxn þ bn Tzn ; Szn ¼ ð1 an ÞSxn þ an Txn ; ⇑ Corresponding author. E-mail addresses: [email protected] (A.R. Khan), [email protected] (V. Kumar), [email protected] (N. Hussain). 0096-3003/$ - see front matter 2014 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.amc.2013.12.150
ð1:3Þ
522
A.R. Khan et al. / Applied Mathematics and Computation 231 (2014) 521–535
respectively, to approximate 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. Jungck [11] used the iterative scheme Sxnþ1 ¼ Txn n = 0, 1, . . ., to approximate common fixed points of the mappings S and T satisfying the following Jungck-contraction
dðTx; TyÞ 6 adðSx; SyÞ;
0 6 a < 1:
ð1:4Þ
Olatinwo and Imoru [10], studied the generalized Zamfirescu operators for the pair (S, T), satisfying the following condition: for each pair of points x, y in Y at least one of the following is true:
ðiÞ dðTx; TyÞ 6 adðSx; SyÞ ðiiÞ dðTx; TyÞ 6 b½dðSx; TxÞ þ dðSy; TyÞ ðiiiÞ dðTx; TyÞ 6 c½dðSx; TyÞ þ dðSy; TxÞ;
ð1:5Þ
where a, b, c are nonnegative constants satisfying 0 6 a 6 1, 0 6 b, c 6 The contractive condition (1.5) implies
dðTx; TyÞ 6 2ddðSx; TxÞ þ ddðSx; SyÞ;
8x;
y 2 Y;
1 . 2
ð1:6Þ
b c where 0 6 d < 1; d = max fa; 1b ; 1c g (see [4,5]). Singh et al. [7] established some stability results for Jungck and Jungck–Mann iterative schemes for both contractive conditions (1.4) and (1.7): for some q 2 ½0; 1Þ and 0 6 L
dðTx; TyÞ 6 qdðSx; SyÞ þ LdðSx; TxÞ:
ð1:7Þ
Obviously, (1.7) is more general than (1.4)–(1.6). Recently, Hussain et al. [12] used the following more general contractive condition than (1.7) to prove stability and strong convergence results for the Jungck-type iterative schemes: there exists a real number q e [0, 1) and a monotone increasing function /: R+ ? R+ such that /(0) = 0 and " x, y e Y, we have
kTx Tyk 6 /ðkSx TxkÞ þ qkSx Syk:
ð1:8Þ
Inspired by the above mentioned contributions, for an þ bn ; bn þ cn ; an 2 ½0; 1, we define Jungck–Khan iterative schemes for the pairs (T, S) and (T1, S1), respectively as follows:
Sxnþ1 ¼ ð1 an bn ÞSxn þ an Tyn þ bn Txn ; Syn ¼ ð1 bn cn ÞSxn þ bn Tzn þ cn Txn ; Szn ¼ ð1 an ÞSxn þ an Txn ;
ð1:9Þ
S1 unþ1 ¼ ð1 an bn ÞS1 un þ an T 1 v n þ bn T 1 un ; S1 v n ¼ ð1 bn cn ÞS1 un þ bn T 1 wn þ cn T 1 un ; S1 wn ¼ ð1 an ÞS1 un þ an T 1 un :
ð1:10Þ
Remark 1.1. Putting bn ¼ 0; cn ¼ 0 in Jungck–Khan iterative scheme (1.9), we get Jungck–Noor iterative scheme. Similarly, putting bn ¼ 0; cn ¼ 0; an ¼ 0 and bn ¼ 0; cn ¼ 0; an ¼ 0; bn ¼ 0; in (1.9), respectively, we get Jungck–Ishikawa and Jungck– Mann iterative schemes. Remark 1.2. If X = Y and S = Id (identity mapping), then Jungck–Noor (1.3), Jungck–Ishikawa (1.2) and the Jungck–Mann (1.1) iterative schemes become Noor [13], Ishikawa [14] and the Mann [15] iterative schemes, respectively. Definition 1.3 [16]. Let f and g be two selfmaps 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. Definition 1.4 [7]. Let S, T : Y ! X be nonself operator for an arbitrary set Y such that T(Y) # S(Y) and p a point of coincidence of S and T. Let fSxn g1 n¼0 X, be the sequence generated by an iterative procedure
Sxnþ1 ¼ f ðT; xn Þ; n ¼ 0; 1; . . . ;
ð1:11Þ
A.R. Khan et al. / Applied Mathematics and Computation 231 (2014) 521–535
523
1 where x0 2 X is the initial approximation and f is some function. Suppose fSxn g1 n¼0 converges to p. Let fSyn gn¼0 X be an arbitrary sequence and set en = d(Syn, f(T, yn)), n = 0, 1, . . . Then, the iterative procedure (1.11) is said to be (S, T)-stable or stable if and only if lim en ¼ 0 implies lim Syn = p. n!1
n!1
Definition 1.5 (5). 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
an a ¼ 0: lim n!1 bn b
Definition 1.6 ([1,5]). Let T1, T2 be two operators defined on a normed space (X, ||.||). We say T2 is approximate operator of T1 if for all x 2 X and for a fixed e > 0, we have kT 1 x T 2 xk 6 e. Lemma 1.7 [1]. Let fan g1 n¼0 be a nonnegative sequence for which there exist n0 2 N such that for all n P n0 , one has the following inequality:
anþ1 6 ð1 r n Þan þ rn tn ; where r n 2 (0, 1), for all n 2 N;
P1
n¼1 r n
¼ 1 and tn P 0 8n 2 N. Then, 0 6 lim sup an 6 lim sup tn . n!1
n!1
Lemma 1.8 [5]. If d is a real number such that 0 6 d < 1 and f2n g1 n¼0 is a sequence of positive numbers such that lim 2n ¼ 0, then n!1 for any sequence of positive numbers fun g1 n¼0 satisfying
unþ1 6 dun þ 2n ; n ¼ 0; 1; 2; . . . ; we have lim un ¼ 0. n!1
We establish here that Jungck–Khan scheme (1.9) introduced in this paper has much better convergence rate and it is more general as compared to Jungck–Mann, Jungck–Ishikawa and Jungck–Noor iterative schemes. As Jungck–Khan is independant of recently defined Jungck–CR scheme, so we shall discuss data dependence problem of both of these schemes. Numerical examples in support of validity and applications of our results are also provided. 2. Convergence and stability of Jungck–Khan iterative scheme Theorem 2.1. Let (X, ||.||) be an arbitrary Banach space and S, T: Y ? X be nonself operators satisfying contractive condition (1.8), on an arbitrary set Y with TðYÞ # SðYÞ, S(Y) is a complete subspace of X. Let Sz = Tz = p(say) and for x0 Y, let fSxn g1 n¼0 be the P P1 1 Jungck–Khan iterative scheme defined by (1.9) with 1 a ¼ 1 or b ¼ 1. Then the iterative scheme fSx g converges n n n n¼1 n¼1 n¼0 strongly to p. Also, p will be the unique common fixed point of S, T provided Y = X, and S and T are weakly compatible. Proof. It follows from (1.9) that
kSxnþ1 pk 6 ð1 an bn ÞkSxn pk þ an kTyn pk þ bn kTxn pk:
ð2:1Þ
Now, we have the following estimates:
kTxn pk 6 qkSxn pk þ /ðkSz TzkÞ ¼ qkSxn pk;
ð2:2Þ
kSzn pk 6 ð1 an ÞkSxn pk þ an kTxn pk 6 ð1 an ð1 qÞÞkSxn pk;
ð2:3Þ
and
kTyn pk 6 qkSyn pk 6 qð1 bn cn ÞkSxn pk þ qbn kTzn pk þ qcn kTxn pk 6 qð1 bn cn ÞkSxn pk þ bn q2 kSzn pk þ cn q2 kSxn pk 6 ½q bn qð1 qÞ cn qð1 qÞ an bn q2 ð1 qÞkSxn pk:
ð2:4Þ
Using estimates (2.2)–(2.4), (2.1) yields
kSxnþ1 pk 6 ½1 an ð1 qÞ bn ð1 qÞ an bn qð1 qÞ an cn qð1 qÞ an an bn q2 ð1 qÞkSxn pk;
ð2:5Þ
6 ½1 an ð1 qÞkSxn pk; ð1qÞ
6 e
1 X
ak
k¼0
kSx0 pk:
ð2:6Þ
524
A.R. Khan et al. / Applied Mathematics and Computation 231 (2014) 521–535
Pn P1 ð1qÞ a k¼0 k ? 0 as n ? 1. Hence, it follows from (2.6) that Since 0 6 q < 1, ak e [0, 1] and lim n¼0 an ¼ 1, so e n!1 kSxnþ1 pk = 0, which implies that the iterative scheme fSxn g1 n¼0 converges strongly to p. Now, we prove p is unique common fixed point of S and T, when Y = X. Let there exists another point of coincidence say p⁄. Then, there exists z⁄ e X such that Sz⁄ = Tz⁄ = p⁄. But from (1.8) we have
0 6 kp p k ¼ kTz Tz k 6 qkSz Sz k þ /ðkSz TzkÞ ¼ qkp p k; which implies p ¼ p as 0 6 q < 1. h Now, as S and T are weakly compatible and p = Tz = Sz, so Tp = TTz = TSz = STz and hence Tp = Sp. Therefore Tp is a point of coincidence of S, T and since the point of coincidence is unique so p = Tp. Thus Tp = Sp = p and therefore p is unique common fixed point of S and T. The main result in [10, Theorem 3.1] now becomes corollary of our Theorem 2.1 by employing the condition of weak compatibility in place of injectivity of S. Theorem 2.2. Let (X, ||.||) be a normed space and S, T: Y ? X are nonself operators on an arbitrary set Y such that TðYÞ # SðYÞ, where S(Y) is a complete subspace of X. Let p be a point of coincidence of S and T, i.e., Sz = Tz = p (say). Suppose that S and T satisfy condition (1.8). For x0 e Y, let fSxn g1 n¼0 be the Jungck–Khan iterative scheme converging to p, with 0 < a 6 an or 0 < b 6 bn , for all n. Then, the iterative scheme fSxn g1 n¼0 is (S, T)-stable. Proof. Let fSpn g1 n¼0 X be an arbitrary sequence, en ¼ kSpnþ1 ð1 an bn ÞSpn an Tqn bn Tpn k, n = 0, 1, 2, 3, . . ., where Sqn ¼ ð1 bn cn ÞSpn þ bn Tr n þ cn Tpn ; Sr n ¼ ð1 an ÞSpn þ an Tpn and let lim en ¼ 0. n!1 Then, using iterative scheme (1.9), we have
kSpnþ1 pk 6 kSpnþ1 ð1 an bn ÞSpn an Tqn bn Tpn k þ kð1 an bn ÞSpn þ an Tqn þ bn Tpn ð1 an bn þ bn þ an Þpk 6 en þ ð1 an bn ÞkSpn pk þ an kTqn pk þ bn kTpn pk 6 en þ ð1 an bn ÞkSpn pk þ an qkSqn pk þ bn qkSpn pk ¼ en þ ½1 an bn ð1 qÞkSpn pk þ an qkSqn pk: ð2:7Þ Now, we have the following estimates:
kSqn pk ¼ kð1 bn cn ÞSpn þ bn Tr n þ cn Tpn ð1 bn cn þ bn þ cn Þpk 6 ð1 bn cn ÞkSpn pk þ bn kTrn pk þ cn kTpn pk 6 ½1 bn cn ð1 qÞkSpn pk þ bn qkSr n pk;
ð2:8Þ
and
kSr n pk ¼ kð1 an ÞSpn þ an Tpn ð1 an þ an Þpk ¼ ½1 an ð1 qÞkSpn pk:
ð2:9Þ
It follows from (2.7), (2.8), and (2.9) that
kSpnþ1 pk 6 ½1 an ð1 qÞ bn ð1 qÞkSpn pk þ en 6 1 an ð1 qÞkSpn pk þ en :
ð2:10Þ
Using 0 < a 6 an and q 2 [0, 1) we have ½1 an ð1 qÞ < 1; 8n 2 N. Hence using Lemma (1.8), (2.10) yields lim Spnþ1 ¼ p. n!1 Conversely, let lim Spnþ1 ¼ p. Then using contractive condition (1.8) and the triangle inequality, we have n!1
en ¼ kSpnþ1 ð1 an bn ÞSpn an Tqn bn Tpn k 6 kSpnþ1 pk þ kð1 an bn ÞSpn þ an Tqn þ bn Tpn ð1 an bn þ bn þ an Þpk 6 kSpnþ1 pk þ kð1 an bn Þ kSpn pk þ an kTqn pk þ bn kTpn pk 6 kSpnþ1 pk þ kð1 an bn Þ kSpn pk þ an qkSqn pk þ bn qkSpn pk ¼ kSpnþ1 pk þ ½1 an ð1 qÞ bn ð1 qÞkSpn pkðusing ð2:7Þ and ð2:8ÞÞ ¼ ½1 an ð1 qÞkSpn pk þ kSpnþ1 pk: ð2:11Þ and hence lim
n!1
en ¼ 0: Therefore Jungck–Khan iterative scheme (1.9) is (S, T)-stable. h
3. Direct comparison of convergence rate In computational mathematics, convergence speed of an iterative scheme plays a vital role and it has been studied by many authors [4,5,12]. For recent work in this direction, we refer the reader to [12] and references therein. Theorem 3.1. Let (X, ||.||) be an arbitrary Banach space and S, T : Y ? X be nonself operators on an arbitrary set Y satisfying contractive condition (1.8). Assume that TðYÞ # SðYÞ, S(Y) is a complete subspace of X and Sz = Tz = p (say). For x0 e Y, let Jungck– Mann (JM), Jungck–Ishikawa (JI), Jungck–Noor (JN) and Jungck–Khan (JK) iterative scheme be defined by (1.1)–(1.3), (1.9), P 1 ; lim an ¼ 0; 1 respectively, with 0 6 an < 1þq n¼0 an ¼ 1 and 0 < l 6 bn ; n 2 N. Then Jungck–Khan iterative scheme converges n!1
faster than Jungck–Mann, Jungck–Ishikawa and Jungck–Noor iteratives schemes.
A.R. Khan et al. / Applied Mathematics and Computation 231 (2014) 521–535
525
Proof. For Jungck–Mann iterative scheme (1.1) we have
kSxnþ1 pk P ð1 an ÞkSxn pk an kTxn pk P ð1 an ÞkSxn pk an qkSxn pk ¼ ½1 an ð1 þ qÞkSxn pk P
n Y
½1 ai ð1 þ qÞkSx0 pk:
ð3:1Þ
i¼1
For Jungck–Ishikawa iterative scheme (1.2) we have
kSxnþ1 pk P ð1 an ÞkSxn pk an qkSyn pk P ½1 an an qð1 bn ð1 qÞkSxn pk P ½1 an ð1 þ qÞkSxn pk P
n Y
½1 ai ð1 þ qÞkSx0 pk:
ð3:2Þ
i¼1
Similarly, for Jungck–Noor iterative scheme (1.3), it is easy to see that
kSxnþ1 pk P
n Y
½1 ai ð1 þ qÞkSx0 pk:
ð3:3Þ
i¼1
Also, for Jungck–Khan iterative scheme (1.9), using (2.5) we have n
kSxnþ1 pk 6 ½1 bn ð1 qÞkSxn pk 6 ½1 lð1 qÞkSxn pk 6 ½1 lð1 qÞ kSx0 pk:
ð3:4Þ
Using (3.3) and (3.4), we have n
kSxnþ1 ðJKÞ pk ½1 lð1 qÞ kSx0 pk 6 n : Y kSxnþ1 ðJNÞ pk ½1 ai ð1 þ qÞkSx0 pk i¼1 ½1lð1qÞn
Now, let pn ¼ Qn
i¼1
Then
pnþ1 pn
½1ai ð1þqÞ
.
½1lð1qÞ ¼ ½1 anþ1 ð1þqÞ. pnþ1 n!1 pn
Using lim an ¼ 0, we get lim n!1
¼ 1 lð1 qÞ < 1.
P1
kSxnþ1 ðJKÞpk lim pn ¼ 0. Hence lim kSx ¼ 0, i.e. Jungck–Khan Therefore by ratio test n¼0 pn < 1 ; which further yields nþ1 ðJNÞpk n!1 n!1 iterative scheme (1.9) converges faster than Jungck–Noor iterative scheme (1.3) to p. Similar argument can be applied to show that
lim
n!1
kSxnþ1 ðJKÞ pk ¼ 0 and kSxnþ1 ðJIÞ pk
lim
kSxnþ1 ðJKÞ pk ¼ 0; pk
n!1 kSxnþ1 ðJMÞ
which imply that the new iterative scheme (1.9) converges faster than Jungck–Ishikawa (1.2) and Jungck–Mann (1.1) iterative schemes to p.The following example shows the validity of Theorem 3.1. h Example 3.2. Let T, S: [1, 4] ? [1, 16] be defined as Tx = 2x + 3, Sx = x2. It is easy to see that (T, S) is a quasi-contractive operator pair satisfying (1.8) with point of coincidence 9. By taking initial approximation x0 = 2, /ðtÞ ¼ 2qt and 1 n an ¼ an ¼ bn ¼ pffiffiffiffiffiffiffiffi ; bn ¼ cn ¼ pffiffiffiffiffiffiffiffi ; n P 1; the obtained results are listed in Table 1, showing convergence of different 2nþ1 2 n þ1
Jungck type schemes to p = 9 = T3 = S3. Remark 3.3. Table 1 shows that Jungck–Khan iterative scheme (1.9) has much better convergence rate than Jungck–Noor, P 1 Jungck–Mann and Jungck–Ishikawa iterative schemes in the context of conditions 0 6 an < 1þq ; lim an ¼ 0; 1 n¼0 an ¼ 1 and n!1 0 < l 6 bn ; n 2 N; used in Theorem 3.1. Moreover, the iterative scheme (1.9) can have better convergence rate than the above P1 P1 mentioned iterative schemes in the context of condition n¼0 an ¼ 1 or n¼0 bn ¼ 1 used in Theorem 2.1, as shown in Tables 2–5. The following Jungck–CR iterative schemes are recently defined and studied in [12]:
Sxnþ1 ¼ ð1 an ÞSyn þ an Tyn ; Syn ¼ ð1 bn ÞTxn þ bn Tzn ; Szn ¼ ð1 an ÞSxn þ an Txn ; S1 unþ1 ¼ ð1 an ÞS1 v n þ an T 1 v n ;
ð3:5Þ
526
A.R. Khan et al. / Applied Mathematics and Computation 231 (2014) 521–535
Table 1 1 n Convergence of different Jungck type iterative schemes to coincidence point p = 9 of Tx = 2x + 3 and Sx = x2. (x0 = 2, an ¼ an ¼ bn ¼ pffiffiffiffiffiffiffiffi ; n P 1). ; bn ¼ cn ¼ pffiffiffiffiffiffiffiffi 2nþ1 2 n þ1
n
1 2 3 4 5 6 7 8 9 10 11 – 199 200 201 202 203 204 205 – 226 227 228 229
Jungck–Khan iterative scheme
Jungck–Noor iterative scheme
Jungck–Ishikawa iterative scheme
Jungck–Mann iterative scheme
Jungck–CR iterative scheme
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
7 8.93074 8.99917 8.99996 9 9
4 8.79343 8.9975 8.99988 8.99999 9
2.96537 2.99958 2.99998 3 3 3
–
–
–
7 8.01791 8.37444 8.55903 8.67111 8.74553 8.79787 8.83624 8.86522 8.88766 8.90536 – 9 9 9 9
4 6.29486 7.22116 7.72571 8.04039 8.25277 8.40384 8.51541 8.60021 8.66612 8.71831 – 8.99999 8.99999 9 9
2.50896 2.68722 2.77952 2.83556 2.87276 2.89894 2.91812 2.93261 2.94383 2.95268 2.95977 – 3 3 3 3
–
–
–
7 7.78834 8.15816 8.37441 8.51562 8.68633 8.7408 8.78303 8.81643 8.84328 8.68633 – 9 9 9 9 9 9 9 – 9 9 9 9
4 5.73205 6.65165 7.22106 7.60551 8.08358 8.23921 8.36086 8.45772 8.53599 8.08358 – 8.99999 8.99999 8.99999 8.99999 8.99999 8.99999 8.99999 – 8.99999 8.99999 9 9
2.39417 2.57908 2.6872 2.75781 2.80712 2.8704 2.89152 2.90822 2.92164 2.93259 2.8704 – 3 3 3 3 3 3 3 – 3 3 3 3
2.83474 2.96628 2.99238 2.99817 2.99954 2.99988 2.99997 2.99999 3 3 3 –
–
2.48741 2.66946 2.76556 2.82449 2.86384 2.91207 2.92754 2.93954 2.94902 2.95663 2.91207 – 3 3 3 3 3 3 3 –
4 8.03578 8.79879 8.95435 8.98903 8.99725 8.99929 8.99981 8.99995 8.99999 9 –
–
4 6.1872 7.12601 7.64835 7.97774 8.36157 8.48015 8.57051 8.6409 8.69673 8.36157 – 8.99999 8.99999 8.99999 8.99999 8.99999 9 9 –
7 8.66949 8.93255 8.98476 8.99634 8.99908 8.99976 8.99994 8.99998 9 9 –
–
7 7.97482 8.33892 8.53113 8.64898 8.78328 8.82414 8.85509 8.87908 8.89804 8.78328 – 9 9 9 9 9 9 9 –
–
–
–
Table 2 1 Convergence of different Jungck type iterative schemes to coincidence point p = 9 of Tx = 2x + 3 and Sx = x2. (x0 = 1.4, an ¼ an ¼ bn ¼ pffiffiffiffiffiffiffiffi ¼ bn ¼ cn ; n P 0). 2nþ1 n
Jungck–Khan iterative scheme
Jungck–CR iterative scheme
Txn
Sxn
xn+1
0 1 – 2 6 7 8 9 – 19 20 – 28 29 30 31 – 131 132 133 134 135 – 161 162 – 210 211 212
5.9 10.3826 – 9.06366 9.00235 9.0014 9.00087 9.00057 – 9.00002 9.00002 – 9 9 9 9 –
2.1025 13.6255 – 9.192 9.00706 9.0042 9.00262 9.0017 – 9.00007 9.00005 – 9.00001 9.00001 9.00001 9 –
3.69128 3.03183 – 3.00981 3.0007 3.00044 3.00028 3.00019 – 3.00001 3.00001 – 3 3 3 3 –
–
–
–
–
–
–
5.9 8.86734 – 8.9788 8.99994 8.99998 9 9 –
–
2.1025 8.60642 – 8.9365 8.99983 8.99995 8.99999 9 –
–
Jungck–Noor iterative scheme
2.93367 2.9894 – 2.99785 2.99999 3 3 3 –
–
Jungck–Ishikawa iterative scheme
Jungck–Mann iterative scheme
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
5.9 8.86734 – 8.93037 8.98124 8.98506 8.98788 8.99001 – 8.99771 8.99797 – 8.99905 8.99914 8.99922 8.99929 – 9 9 9 9 9 –
2.1025 8.60642 – 8.79233 8.94381 8.95524 8.96366 8.97005 – 8.99314 8.99391 – 8.99715 8.99741 8.99765 8.99787 – 8.99999 8.99999 8.99999 9 9 –
2.93367 2.96519 – 2.97737 2.99253 2.99394 2.995 2.99583 – 2.99899 2.9991 – 2.99957 2.99961 2.99964 2.99968 – 3 3 3 3 3 –
–
–
2.1025 7.85798 – 8.3779 8.82725 8.86209 8.88783 8.90741 – 8.97865 8.98105 – 8.9911 8.99193 8.99268 8.99334 – 8.99998 8.99998 8.99998 8.99998 8.99999 – 8.99999 9 –
2.80321 2.89446 – 2.93086 2.97693 2.98125 2.98453 2.98708 – 2.99684 2.99719 – 2.99865 2.99878 2.99889 2.99899 – 3 3 3 3 3 – 3 3 –
5.9 7.85798 – 8.303 8.77562 8.81736 8.84896 8.87349 – 8.96802 8.97144 – 8.98608 8.98733 8.98845 8.98946 – 8.99997 8.99997 8.99997 8.99997 8.99997 – 8.99999 8.99999 – 9 9 9
2.1025 5.9 – 7.03044 8.33946 8.46043 8.55259 8.62447 – 8.90433 8.91453 – 8.95829 8.96204 8.96539 8.96839 – 8.9999 8.9999 8.99991 8.99991 8.99992 – 8.99997 8.99997 – 8.99999 8.99999 9
2.42899 2.6515 – 2.75673 2.90868 2.92448 2.93674 2.94646 – 2.98572 2.98721
–
5.9 8.60642 – 8.78892 8.94214 8.95385 8.96249 8.96906 – 8.99288 8.99368 – 8.99703 8.99731 8.99756 8.99778 – 8.99999 8.99999 8.99999 8.99999 9 – 9 9 –
2.99367 2.99423 2.99473 2.99518 – 2.99998 2.99998 2.99999 2.99999 2.99999 – 3 3 – 3 3 3
527
A.R. Khan et al. / Applied Mathematics and Computation 231 (2014) 521–535
Table 3 1 Convergence of different Jungck type iterative schemes to coincidence point p⁄ = 2 of T1 x = 2x and S1 x = x2 + 1. (x0 = 1.45, an ¼ an ¼ bn ¼ pffiffiffiffiffiffiffiffi ¼ bn ¼ cn ; n P 0). 2nþ1 n
0 1 2 3 4 5 – 239 240 241 242 243 244 –
Jungck–Khan iterative scheme
Jungck–CR iterative scheme
Jungck–Noor iterative scheme
Jungck–Ishikawa iterative scheme
Jungck–Mann iterative scheme
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
2.9 2.27239 2.24124 2.22399 2.21203 2.20288 – 2.06767 2.06756 2.06746 2.06735 2.06725 2.06715 –
3.1025 2.29094 2.25579 2.23653 2.22327 2.21317 – 2.06882 2.06871 2.0686 2.06849 2.06838 2.06827 –
1.1362 1.12062 1.11199 1.10601 1.10144 1.09774 – 1.03378 1.03373 1.03368 1.03363 1.03357 1.03352 –
2.9 2.56974 2.46094 2.39454 2.34758 2.3119
3.1025 2.65089 2.51405 2.43346 2.37778 2.33622
1.28487 1.23047 1.19727 1.17379 1.15595 1.14179
2.01514 2.01508 2.01502 2.01497 2.01491 2.01485
2.0152 2.01514 2.01508 2.01502 2.01496 2.0149
1.00754 1.00751 1.00748 1.00745 1.00742 1.0074
2.9 2.56974 2.5085 2.47398 2.44982 2.43119 – 2.14368 2.14345 2.14322 2.143 2.14277 2.14254 –
3.1025 2.65089 2.57314 2.53014 2.5004 2.47768 – 2.14884 2.1486 2.14835 2.14811 2.14787 2.14762 –
1.28487 1.25425 1.23699 1.22491 1.2156 1.20802 – 1.07173 1.07161 1.0715 1.07138 1.07127 1.07116 –
2.9 2.65089 2.58363 2.54358 2.51491 2.49256 – 2.152 2.15175 2.15149 2.15124 2.15099 2.15074 –
3.1025 2.75681 2.66879 2.61744 2.58119 2.55322 – 2.15778 2.15751 2.15723 2.15696 2.15669 2.15642 –
1.32545 1.29182 1.27179 1.25745 1.24628 1.23713 – 1.07587 1.07575 1.07562 1.0755 1.07537 1.07525 –
2.9 2.75681 2.69617 2.65567 2.6249 2.59998 – 2.17362 2.1733 2.17298 2.17267 2.17236 2.17205 –
3.1025 2.9 2.81733 2.76314 2.72252 2.68998 – 2.18115 2.18081 2.18046 2.18012 2.17979 2.17945 –
1.3784 1.34808 1.32783 1.31245 1.29999 1.28951 – 1.08665 1.08649 1.08634 1.08618 1.08602 1.08587 –
S1 v n ¼ ð1 bn ÞT 1 un þ bn T 1 wn ; S1 wn ¼ ð1 an ÞS1 un þ an T 1 un ;
ð3:6Þ
where an ; bn ; an 2 ½0; 1 Remark 3.4. Putting an ¼ 0; bn ¼ 1 and an ¼ 0 in the iterative scheme (3.5), we get Jungck–S [12] and Jungck–Agarwal et al. [12] iterative schemes, respectively. Remark 3.5. It is not possible to compare directly Jungck–Khan (1.9) and Jungck–CR iterative scheme (3.5). However, Table 2 and Table 5 show that Jungck–Khan iterative scheme can have better convergence rate than Jungck–CR iterative scheme.
4. Data dependence In this section we establish data dependence of Jungck–Khan and Jungck–CR iterative schemes and deduce as corollaries certain recent data dependence results for well-known iterative schemes like Jungck–Noor, Jungck–Mann, Jungck–Ishikawa, Jungck–Agarwal et al. and Jungck–S. Theorem 4.1. Let (X, ||.||) be an arbitrary Banach space and (S, T), (S1, T1) : Y ? X be nonself operator pairs on an arbitrary set Y with (S, T) satisfying contractive condition (1.8) such that kTx T 1 xk 6 e; kSx S1 xk 6 e1 . Assume that TðYÞ # SðYÞ, T 1 ðYÞ # S1 ðYÞ, 1 where S(Y) and S1 ðYÞ are complete subspaces of X with Sz = Tz = p and S1z⁄ = T1 z⁄ = p⁄. Suppose that fSxn g1 n¼0 , fSun gn¼0 are ⁄ Jungck–Khan iterative schemes (1.9) and (1.10) associated with (S, T) and (S1, T1), which converge to p and p , respectively. Then we have
kp p k 6 where
10e2 ; 1q
e2 = max {e, e1 }, provided bn 6 an ; 8n 2 N and
P1
n¼1
an ¼ 1.
Proof. It follows from (1.9) and (1.10) that
kSxnþ1 S1 unþ1 k 6 ð1 an bn ÞkSxn S1 un k þ an kTyn T 1 v n k þ bn kTxn T 1 un k:
ð4:1Þ
Now we have the following estimates:
kTyn T 1 v n k 6 kTyn T v n k þ kT v n T 1 v n k 6 qkSyn Sv n k þ /ðkSyn Tyn kÞ þ e;
ð4:2Þ
kSyn Sv n k 6 kSyn S1 v n k þ kS1 v n Sv n k 6 kSyn S1 v n k þ e1 ;
ð4:3Þ
kSyn S1 v n k 6 ð1 bn cn ÞkSxn S1 un k þ bn kTzn T 1 wn k þ cn kTxn T 1 un k;
ð4:4Þ
kTzn T 1 wn k 6 kTzn Twn k þ kTwn T 1 wn k 6 qkSzn Swn k þ /ðkSzn Tzn kÞ þ e;
ð4:5Þ
kSzn Swn k 6 kSzn S1 wn k þ kS1 wn Swn k ¼ kSzn S1 wn k þ e1 ;
ð4:6Þ
528
Table 4 1 Convergence of different Jungck type iterative schemes to common fixed point p = 0.5 of Tx = 12 ð12 þ xÞ and Sx = 1 x. (x0 = 0.6, an ¼ an ¼ bn ¼ pffiffiffiffiffiffiffiffi ¼ bn ¼ cn ; n P 0). 2nþ1 n
Txn
Sxn
xn+1
0.55 0.491406 0.500002 0.5
0.4 0.517188 0.499996 0.5
0.482812 0.500004 0.5 0.5
– 0.5 0.5 0.5 – 0.5 0.5 0.5 0.5 0.5
– 0.5 0.5 0.5 – 0.5 0.5 0.5 0.5 0.5
– 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5
Jungck–CR iterative scheme
0.55 0.496094 0.500568 0.499892 0.500024 0.499994 0.500002 0.5 0.5
0.4 0.507813 0.498864 0.500217 0.499952 0.500012 0.499997 0.500001 0.5
0.492188 0.501136 0.499783 0.500048 0.499988 0.500003 0.499999 0.5 0.5
Jungck–Noor iterative scheme
Jungck–Ishikawa iterative scheme
Jungck–Mann iterative scheme
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
0.55 0.519531 0.509514 0.505203 0.503063 0.501901 0.501227 0.500817 0.500558 – 0.500002 0.500002 0.500001 – 0.5 0.5 0.5
0.4 0.460938 0.480972 0.489595 0.493873 0.496199 0.497546 0.498366 0.498883 – 0.499996 0.499997 0.499997 – 0.499999 0.499999 0.5
0.539063 0.519028 0.510405 0.506127 0.503801 0.502454 0.501634 0.501117 0.500779 – 0.500003 0.500003 0.500002 – 0.500001 0.5 0.5
0.55 0.521875 0.511214 0.506318 0.503795 0.502389 0.501559 0.501048 0.500721 – 0.500003 0.500002 0.500002 – 0.5 0.5 0.5 0.5 0.5
0.4 0.45625 0.477573 0.487364 0.49241 0.495222 0.496882 0.497905 0.498558 – 0.499995 0.499996 0.499996 – 0.499999 0.499999 0.499999 0.499999 0.5
0.54375 0.522427 0.512636 0.50759 0.504778 0.503118 0.502095 0.501442 0.501012 – 0.500004 0.500004 0.500003 – 0.500001 0.500001 0.500001 0.5 0.5
0.55 0.5125 0.504845 0.502276 0.501196 0.500678 0.500406 0.500254 0.500164 – 0.5 0.5 0.5 – 0.5 0.5 0.5 0.5 0.5
0.4 0.475 0.490309 0.495449 0.497608 0.498643 0.499187 0.499492 0.499672 – 0.499999 0.499999 0.5 – 0.5 0.5 0.5 0.5 0.5
0.525 0.509691 0.504551 0.502392 0.501357 0.500813 0.500508 0.500328 0.500218 – 0.500001 0.5 0.5 – 0.5 0.5 0.5 0.5 0.5
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0 1 2 3 4 5 6 7 8 – 30 31 32 – 40 41 42 43 44
Jungck–Khan iterative scheme
Table 5 1 Convergence of different Jungck type iterative schemes to common fixed point p⁄ = 1 of T1 = 1þx and S1x = x2. (x0 = 0.6, an ¼ an ¼ bn ¼ pffiffiffiffiffiffiffiffi ¼ bn ¼ cn ; n P 0). 2 2nþ1 n
Jungck–CR iterative scheme
Jungck–Noor iterative scheme
Jungck–Ishikawa iterative scheme
Jungck–Mann iterative scheme
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
Txn
Sxn
xn+1
0.8 0.968824 0.990049 –
0.36 0.879183 0.960594 –
0.937647 0.980099 0.991652 –
0.8 0.974178 0.996068 – 0.999999 1 1
0.36 0.89938 0.984332 – 0.999997 0.999999 1
0.948356 0.992135 0.99869 – 1 1 1
– 1 1 1 – 1 1
– 0.999999 0.999999 1 – 1 1
– 1 1 1 – 1 1
0.8 0.895889 0.933565 – 0.983801 0.986873 0.989223 – 0.999473 0.999522 0.999566 – 1 1 1
0.36 0.626913 0.751914 – 0.936254 0.948183 0.957358 – 0.997891 0.998089 0.998266 – 0.999999 0.999999 1
0.791778 0.86713 0.90729 – 0.973747 0.978447 0.982111 – 0.999044 0.999132 0.999211 – 1 1 1
– –
– –
– –
–
–
–
–
–
–
0.8 0.893826 0.931824 – 0.98323 0.986401 0.988828 – 0.99945 0.999502 0.999548 – 1 1 1 1 –
0.36 0.620394 0.745889 – 0.934045 0.946342 0.955813 – 0.997803 0.998009 0.998193 – 0.999999 0.999999 0.999999 1 –
0.787651 0.863648 0.904551 – 0.972801 0.977657 0.981445 – 0.999004 0.999096 0.999178 – 1 1 1 1 –
0.8 0.880789 0.919165 – 0.977739 0.981719 0.984813 – 0.999155 0.999232 0.999301 – 1 1 1 1 – 1 1 1
0.36 0.58 0.702796 – 0.912937 0.928211 0.940176 – 0.996623 0.99693 0.997205 – 0.999999 0.999999 0.999999 0.999999 – 0.999999 0.999999 1
0.761577 0.83833 0.882776 – 0.963437 0.969627 0.974522 – 0.998464 0.998601 0.998725 – 1 1 1 1 – 1 1 1
A.R. Khan et al. / Applied Mathematics and Computation 231 (2014) 521–535
0 1 2 – 7 8 9 – 33 34 35 – 174 175 176 177 – 190 191 192
Jungck–Khan iterative scheme
529
530
A.R. Khan et al. / Applied Mathematics and Computation 231 (2014) 521–535
and
kSzn S1 wn k 6 ð1 an ÞkSxn S1 un k þ an kTxn T 1 un k 6 ð1 an ÞkSxn S1 un k þ an kTxn Tun k þ an kTun T 1 un k 6 ð1 an ÞkSxn S1 un k þ an qkSxn Sun k þ an /ðkSxn Txn kÞ þ an e 6 ð1 an ÞkSxn S1 un k þ an qkSxn S1 un k þ an qkS1 un Sun k þ an /ðkSxn Txn kÞ þ an e 6 ½1 an ð1 qÞkSxn S1 un k þ an /ðkSxn Txn kÞ þ an qe1 þ an e:
ð4:7Þ
Using estimates (4.5)–(4.7), we find
kTzn T 1 wn k 6 q½1 an ð1 qÞkSxn S1 un k þ /ðkSzn Tzn kÞ þ qan /ðkSxn Txn kÞ þ qe1 þ e þ an q2 e1 þ an qe:
ð4:8Þ
It follows from (4.4) and (4.8) that
kSyn S1 v n k 6 ð1 bn cn ÞkSxn S1 un k þ bn q½1 an ð1 qÞÞkSxn S1 un k þ an bn q/ðkSxn Txn kÞ þ bn /ðkSzn Tzn kÞ þ cn kTxn T 1 un k þ bn e þ an bn e þ an bn q2 e1 þ bn qe1 ; 6 ½1 bn ð1 qÞ cn kSxn S1 un k þ an bn q/ðkSxn Txn kÞ þ bn /ðkSzn Tzn kÞ þ cn kTxn T 1 un k þ bn e þ an bn e þ an bn q2 e1 þ bn qe1 :
ð4:9Þ
Also, (4.2) and (4.9) together yield
kTyn T 1 v n k 6 q½1 bn ð1 qÞ cn kSxn S1 un k þ an bn q2 /ðkSxn Txn kÞ þ bn q/ðkSzn Tzn kÞ þ qcn kTxn T 1 un k þ /ðkSyn Tyn kÞ þ bn qe þ bn q2 e1 þ e þ an bn qe þ an bn q3 e1 :
ð4:10Þ
Using (4.10), (4.1) yields
kSxnþ1 S1 unþ1 k 6 ð1 an bn ÞkSxn S1 un k þ an q½1 bn ð1 qÞ cn kSxn S1 un k þ an an bn q2 /ðkSxn Txn kÞ þ an bn q/ðkSzn Tzn kÞ þ an qcn kTxn T 1 un k þ an bn qe þ an bn q2 e1 þ an qe1 þ an /ðkSyn Tyn kÞ þ an e þ bn kTxn T 1 un k þ an an bn qe þ an an bn q3 e1 6 ð1 an ð1 qÞ an qcn bn ÞkSxn S1 un k þ an an bn q2 /ðkSxn Txn kÞ þ an bn q/ðkSzn Tzn kÞ þ an /ðkSyn Tyn kÞ þ ðan qcn þ bn ÞkTxn T 1 un k þ an qe1 ð1 þ bn qÞ þ an eð1 þ bn qÞ þ an an bn qe þ an an bn q3 e1 :
ð4:11Þ
Now
kSxn Txn k 6 kSxn pk þ kp Txn k 6 kSxn pk þ qkSxn pk ¼ ð1 þ qÞkSxn pk; which yields
lim kSxn Txn k ¼ 0: ðusing lim Sxn ¼ pÞ
n!1
n!1
ð4:12Þ
Also,
kSyn Tyn k 6 kSyn pk þ kp Tyn k 6 kSyn pk þ qkSyn pk ¼ ð1 þ qÞkSyn pk; with
kSyn pk 6 ð1 bn cn ÞkSxn pk þ bn kTzn pk þ cn kTxn pk 6 ð1 bn cn ÞkSxn pk þ bn qkSzn pk þ cn qkSxn pk 6 ð1 bn cn ÞkSxn pk þ þbn q½ð1 an ÞkSxn pk þ an kTxn pk þ cn qkSxn pk 6 ð1 bn cn ÞkSxn pk þ þbn qð1 an ÞkSxn pk þ bn an q2 kSxn pk þ cn qkSxn pk; implies lim kSyn Tyn k ¼ 0: n!1
ð4:13Þ
Similarly
kSzn Tzn k 6 kSzn pk þ kp Tzn k 6 kSzn pk þ qkSzn pk ¼ ð1 þ qÞkSzn pk; with
kSzn pk 6 ð1 an ÞkSxn pk þ an qkSxn pk ¼ ½1 an ð1 qÞkSxn pk; implies
lim kSzn Tzn k ¼ 0:
n!1
ð4:14Þ
A.R. Khan et al. / Applied Mathematics and Computation 231 (2014) 521–535
531
Moreover,
kTxn T 1 un k 6 kTxn Tun k þ kTun T 1 un k 6 qkSxn Sun k þ /ðkSxn Txn kÞ þ e 6 qkSxn S1 un k þ qkS1 un Sun k þ /ðkSxn Txn kÞ þ e 6 qkSxn S1 un k þ qe1 þ /ðkSxn Txn kÞ þ e:
ð4:15Þ
Using estimate (4.15), (4.11) becomes
kSxnþ1 S1 unþ1 k 6 ½1 an ð1 qÞ bn ð1 qÞ an qcn ð1 an cn qÞkSxn S1 un k þ an an bn q2 /ðkSxn Txn kÞ þ an qcn /ðkSxn Txn kÞ þ bn /ðkSxn Txn kÞ þ an bn q/ðkSzn Tzn kÞ þ an /ðkSyn Tyn kÞ þ ðan qe1 þ an eÞð1 þ bn q þ an bn q þ cn qÞ þ qe1 bn þ bn e ¼ ð1 r n ÞkSxn S1 un k þ rn t n ; ðusing bn 6 an Þ;
ð4:16Þ
2 a b q2 /ðkSx Tx kÞ þ qc /ðkSx Tx kÞ þ /ðkSx Tx kÞ 3 n n n n n n n n n 7 6 þbn q/ðkSzn Tzn kÞ þ /ðkSyn Tyn kÞ 7 6 7 6 þðqe þ eÞð1 þ b q þ a b q þ c qÞ þ qe þ e 7 6 1 n n n n 1 where rn ¼ an ð1 qÞ and t n ¼ 6 7: 7 6 1q 7 6 5 4 Lemma 1.7 and estimates (4.12)–(4.14), (4.16) yield
kp p k 6 where
10e2 ; 1q
e2 = max{e, e1 }. Hence the result. h
Corollary 4.2. Let (X, ||.||) be an arbitrary Banach space and (S, T), (S1, T1): Y ? X be nonself operator pairs on an arbitrary set Y with (S, T) satisfying contractive condition (1.8) such that kTx T 1 xk 6 e; kSx S1 xk 6 e1 . Assume that TðYÞ # SðYÞ, 1 T 1 ðYÞ # S1 ðYÞ, where S(Y) and S1 ðYÞ are complete subspaces of X with Sz = Tz = p and S1z⁄ = T1z⁄ = p⁄. Let fSxn g1 n¼0 , fSun gn¼0 be ⁄ the Jungck–Noor iterative schemes associated with (S, T) and (S1, T1), and converging to p and p , respectively. Then we have
kp p k 6 where
6e2 ; 1q
e2 = max{e, e1 }, e; e1 > 0; provided
P1
n¼1
an ¼ 1.
Proof. Putting bn ¼ cn ¼ 0 in (1.9) we get the desired result. h Remark 4.3. As Jungck–Ishikawa [12] and the Jungck–Mann (1.1) iterative schemes are special cases of Jungck–Khan iterative scheme (1.9), results similar to Corollary 4.2 hold for these iterative schemes. Also putting S = Id (identity mapping), Y = X, in (1.9), data dependence results of Ishikawa iterative scheme [1, Theorem 3.2] and Noor iterative scheme [2, Theorem 3.1] can be obtained as corollaries. Now we establish data dependence result for Jungck–CR scheme. Theorem 4.4. Let (X, ||.||) be an arbitrary Banach space and (S, T), (S1, T1): Y ? X be nonself operator pairs on an arbitrary set Y with (S, T) satisfying contractive condition (1.8) such that kTx T 1 xk 6 e; kSx S1 xk 6 e1 . Assume that TðYÞ # SðYÞ, T 1 ðYÞ # S1 ðYÞ, where S(Y) and S1 ðYÞ are complete subspaces of X with Sz = Tz = p and S1z⁄ = T1 z⁄ = p⁄. Suppose that fSxn g1 n¼0 , ⁄ fSun g1 n¼0 are the Jungck–CR iterative schemes (3.5) and (3.6) associated with (S, T) and (S1, T1), which converge to p and p , respectively. Then we have
kp p k 6 where
6e2 ; 1q
e2 = max{e, e1 }, provided an ð1 qÞ P 12 ; 8n 2 N and
P1
n¼1
an ¼ 1.
Proof. It follows from (3.5) and (3.6) that
kSxnþ1 S1 unþ1 k 6 ð1 an ÞkSyn S1 v n k þ an kTyn T 1 v n k:
ð4:17Þ
Now we have the following estimates:
kTyn T 1 v n k 6 kTyn T v n k þ kT v n T 1 v n k 6 qkSyn Sv n k þ /ðkSyn Tyn kÞ þ e;
ð4:18Þ
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A.R. Khan et al. / Applied Mathematics and Computation 231 (2014) 521–535
kSyn Sv n k 6 kSyn S1 v n k þ kS1 v n Sv n k 6 kSyn S1 v n k þ e1 :
ð4:19Þ
Using (4.17), (4.18), and (4.19) becomes
kSxnþ1 S1 unþ1 k 6 ½1 an ð1 qÞkSyn S1 v n k þ an /ðkSyn Tyn kÞ þ an e þ an qe1 :
ð4:20Þ
Now we have the following estimates:
kSyn S1 v n k 6 ð1 bn ÞkTxn T 1 un k þ bn kTzn T 1 wn k 6 ð1 bn ÞkTxn Tun k þ ð1 bn ÞkTun T 1 un k þ bn kTzn Twn k þ bn kTwn T 1 wn k 6 ð1 bn ÞqkSxn Sun k þ ð1 bn Þ/ðkSxn Txn kÞ þ ð1 bn Þe þ bn qkSzn Swn k þ bn /ðkSzn Tzn kÞ þ bn e ¼ ð1 bn ÞqkSxn Sun k þ ð1 bn Þ/ðkSxn Txn kÞ þ bn /ðkSzn Tzn kÞ þ bn qkSzn Swn k þ e; kSxn Sun k 6 kSxn S1 un k þ kS1 un Sun k 6 kSxn S1 un k þ e1 ;
ð4:21Þ ð4:22Þ
kSzn Swn k 6 kSzn S1 wn k þ kS1 wn Swn k 6 ð1 an ÞkSxn S1 un k þ an kTxn T 1 un k þ e1 6 ð1 an ÞkSxn S1 un k þ an kTxn Tun k þ an kTun T 1 un k þ e1 6 ð1 an ÞkSxn S1 un k þ an qkSxn Sun k þ an /ðkSxn Txn kÞ þ an eþ e1 6 ½1 an ð1 qÞkSxn S1 un k þ an /ðkSxn Txn kÞ þ an qe1 þ an e þ e1 :
ð4:23Þ
Using estimates (4.21)–(4.23) yields
kSyn S1 v n k 6 ð1 bn ÞqkSxn S1 un k þ ð1 bn Þqe1 þ ð1 bn Þ/ðkSxn Txn kÞ þ bn /ðkSzn Tzn kÞ þ bn q½1 an ð1 qÞkSxn S1 un k þ an bn q/ðkSxn Txn kÞ þ an bn q2 e1 þ an bn qe þ bn qe1 þ e 6 ½ð1 bn Þq þ bn qð1 an ð1 qÞÞkSxn S1 un k þ ð1 bn Þ/ðkSxn Txn kÞ þ bn /ðkSzn Tzn kÞ þ an bn q/ðkSxn Txn kÞ þ eð an bn q þ 1Þ þ qe1 ð an bn q þ 1Þ:
ð4:24Þ
It follows from (4.20) and (4.24) that
kSxnþ1 S1 unþ1 k 6 ½1 an ð1 qÞkSxn S1 un k þ ½1 an ð1 qÞ/ð kSxn Txn kÞ þ ½1 an ð1 qÞ/ðkSzn Tzn kÞ þ ½1 an ð1 qÞan bn q/ðkSxn Txn kÞ þ ½1 an ð1 qÞðe þ qe1 Þðan bn q þ 1Þ þ an ðe þ qe1 Þ þ an /ð kSyn Tyn kÞ:
ð4:25Þ
Now an ð1 qÞ P 12 implies1 an ð1 qÞ 6 an ð1 qÞ 6 an . Hence inequality (4.25) implies
kSxnþ1 S1 unþ1 k 6 ½1 an ð1 qÞkSxn S1 un k þ an /ðkSxn Txn kÞ þ an /ðkSzn Tzn kÞ þ an an bn q/ðkSxn Txn kÞ þ an ðe þ qe1 Þð an bn q þ 1Þ þ an ðe þ qe1 Þ þ an /ðkSyn Tyn kÞ; ¼ ð1 r n ÞkSxn S1 un k þ rn t n ;
ð4:26Þ
n kÞþðeþqe1 Þðan bn qþ1Þþðeþqe1 Þþ/ð kSyn Tyn kÞ ð/ðkSxn Txn kÞþ/ðkSzn Tzn kÞþan bn q/ðkSxn Tx1q Þ.
where rn = an ð1 qÞ and tn = As in Theorem 4.1, it is easy to see that lim kSzn Tzn k ¼ lim kSxn Txn k ¼ lim kSyn Tyn k ¼ 0. n!1
n!1
n!1
Hence Lemma 1.7 and (4.26) yield
kp p k 6 where
6e2 ; 1q
e2 = max{e, e1 }. Hence the result. h
As Jungck–Agarwal et al. iterative scheme [12] is a special case of Jungck–CR iterative scheme, we have the following result: Corollary 4.5. Let (X, ||.||) be an arbitrary Banach space and (S, T), (S1, T1): Y ? X be nonself operator pairs on an arbitrary set Y with (S, T) satisfying contractive condition (1.8) such that kTx T 1 xk 6 e; kSx S1 xk 6 e1 . Assume that TðYÞ # SðYÞ, T 1 ðYÞ # S1 ðYÞ, where S(Y) and S1 ðYÞ are complete subspaces of X with Sz = Tz = p and S1z⁄ = T1z⁄ = p⁄. Suppose that fSxn g1 n¼0 , ⁄ fSun g1 n¼0 are the Jungck–Agarwal et al. iterative schemes associated with (S, T) and (S1, T1), which converge to p and p , respectively. Then we have
kp past k 6 where
4e2 ; 1q
e2 = max {e, e1 }, provided an ð1 qÞ P 12 ; 8n 2 N and
P1
n¼1
an ¼ 1.
533
A.R. Khan et al. / Applied Mathematics and Computation 231 (2014) 521–535
Proof. Putting an ¼ 0; bn ¼ an in the iterative scheme (3.5), we get the desired result. h Remark 4.6. Set an ¼ 0; bn ¼ 1; an ¼ an in (3.5) to obtain data dependence result [3, Theorem 4] for Jungck–S iterative scheme studied in [12]. The following two examples reveal the validity of our Theorem 4.1. Example 4.7. Let T, S, T1, and S1: [1, 4] ? [1, 17] be defined as Tx = 2x + 3, Sx = x2, T1x = 2x and S1x = x2 + 1. It is easy to see that (S, T) and (S1, T1) are approximate operator pairs with (S, T) satisfying (1.8) and /ðtÞ ¼ 2qt. Also, here e = 3, e1 ¼ 1 and so 1 ffi e2 = 3. Taking initial approximation x0 = 1.45 and an ¼ bn ¼ an ¼ bn ¼ cn ¼ pffiffiffiffiffiffiffiffi ; n P 0; results obtained are listed in Table 2 2nþ1 and Table 3 which show convergence of different Jungck type schemes associated with (S, T) and (S1, T1), respectively, to p = 9 = T3 = S3 and p⁄ = 2 = T11 = S11. Example 4.8. Let T, S, T1, and S1: [0, 1] ? [0, 1] be defined by T(x) = 12 ð12 þ xÞ, S(x) = 1 x; T1x = 1þx and S1x = x2. It is easy to see 2 that (S, T) and (S1, T1) are approximate operator pairs with (S, T) satisfying (1.8) and /ðtÞ ¼ 2qt. Also, here e = 0.25, e1 ¼ 1 and 1 ; n P 0; results obtained are listed in so e2 = 1. By taking initial approximation x0 = 0.6 and an ¼ bn ¼ an ¼ bn ¼ cn ¼ pffiffiffiffiffiffiffiffi 2nþ4 Table 4 and Table 5 which show convergence of different Jungck type schemes associated to (S, T) and (S1, T1), respectively, to p = 0.5 = T 0.5 = S 0.5 and p⁄ = 1 = T11 = S11. The following example reveals the validity of Theorem 4.4. Example 4.9. Let T, S, T1, and S1: [1, 4] ? [1, 17] be defined by Tx = 2x + 3, Sx = x2, T1x = 2x and S1x = x2 + 1. It is easy to see that (S, T) and (S1, T1) are approximate operator pairs with (T, S) satisfying (1.8) and /ðtÞ ¼ 2qt. Also, here e = 3, e1 ¼ 1 and so e2 = 3. n 1 ffi ; bn ¼ an ¼ bn ¼ cn ¼ pffiffiffiffiffiffiffiffi ; n P 1; the results obtained are listed in By taking initial approximation x0 = 2 and an ¼ nþ0:2 2nþ1 Table 6 which shows convergence of Jungck–CR iterative schemes associated with (S, T) and (S1, T1), respectively, to p = 9 = T3 = S3 and p⁄ = 2 = T11 = S11. 5. Applications In this section, with the help of computer programs in C++, we apply Jungck–Khan iterative scheme (1.9) to solve various type of problems, some of which are not easy to solve by other methods. Solution of Legendre polynomial 63 x5 35 x3 þ 15 x¼0 8 4 8 To solve this polynomial we rewrite it in the form Sx = Tx, where Tx = 63x5 þ 15x and Sx ¼ 70x3 : Here the operators S, T : 1 ; n P 0; the ½0:2; 0:9 ! ½0:56; 51:03 are such that T(x) 2 SðxÞ; 8x 2 ½0:2; 0:9: Taking x0 ¼ 0:2; an ¼ bn ¼ an ¼ bn ¼ cn ¼ pffiffiffiffiffiffiffiffi 2nþ4 convergence of Jungck–Khan iterative scheme to the coincedence point 0.538471 of S and T is listed in the Table 7. Solution of equation ex ðx2 þ 5x þ 2Þ þ 1 ¼ 0
Table 6 Convergence of Jungck–CR iterative scheme to coincidence point p = 9 of Tx = 2x + 3, and Sx = x2 as well as to coincidence point p⁄ = 2 of T1 x = 2x and S1x = x2 + 1. n 1 (x0 = 2, an ¼ nþ0:2 ; bn ¼ an ¼ bn ¼ cn ¼ pffiffiffiffiffiffiffiffi ; n P 1). 2nþ1 n
Txn
Sxn
xn+1
T1xn
S1xn
xn+1
1 2 3 4 5 6 7 8 9 10 11 12 13 – 246 247 248 249 –
7 8.46082 8.8424 8.9523 8.98526 8.99538 8.99854 8.99953 8.99985 8.99995 8.99998 8.99999 9 – 9 9 9 9 –
4 7.45515 8.53341 8.85746 8.95583 8.98615 8.99562 8.9986 8.99955 8.99986 8.99995 8.99998 9 – 9 9 9 9 –
2.73041 2.9212 2.97615 2.99263 2.99769 2.99927 2.99977 2.99993 2.99998 2.99999 3 3 3 – 3 3 3 3 –
4 3.12411 2.76002 2.56778 2.45078 2.37269 2.31711 2.27565 2.24359 2.2181 2.19736 2.18017 2.1657 – 2.0082 2.00816 2.00813 2.0081 –
5 3.44001 2.90442 2.64838 2.50158 2.40741 2.34225 2.29465 2.25843 2.22999 2.2071 2.18829 2.17256 – 2.00821 2.00818 2.00815 2.00811 –
1.56205 1.38001 1.28389 1.22539 1.18634 1.15855 1.13783 1.1218 1.10905 1.09868 1.09009 1.08285 1.07667 – 1.00408 1.00407 1.00405 1.00403 –
534
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We rewrite above equation in the form Sx = Tx, where the operators T, S: ½2; 0:5 ! ½10; 2:5 are defined by 1 ; n P 0, Tx = ex x2 2 and Sx = 5x, respectively. With initial approximation x0 = 1 and an ¼ bn ¼ an ¼ bn ¼ cn ¼ pffiffiffiffiffiffiffiffi 2nþ4 the convergence of Jungck–Khan iterative scheme to the coincedence point 0.579167 of S and T (and solution of equation) is listed in the Table 8. Solution of quadratic equation x2 10x þ 9 ¼ 0 In order to solve this equation by Jungck–Khan iterative scheme (1.9), we write it in the form Sx = Tx, where the operators T, S: ½1; 4 ! ½10; 40 are defined by Tx = x2 þ 9 and Sx = 10x, respectively. With initial approximation x0 = 2 and an ¼ bn ¼ an ¼ bn ¼ cn ¼ n15 ; n P 1; the convergence of Jungck–Khan iterative scheme to the coincedence point 1 of S and T (and solution of quadratic equation) is listed in the Table 9. For detailed study, these programs are executed after changing the parameters and some of the observations are given below:
5.1. Observations 5.1.1. Solution of Legendre polynomial 1. Taking initial guess xo = 0.5 (near coincidence point), Jungck–Khan iterative scheme (1.9) converges in 24 iterations. 1 2. Taking an ¼ bn ¼ an ¼ bn ¼ cn ¼ p4ffiffiffiffiffiffiffiffi ; n P 0; and xo = 0.2, we observe that Jungck–Khan iterative scheme converges in 21 2nþ4 iterations. Table 7 Convergence of Jungck–Khan iterative scheme to coincidence point p = 0.538471 of Tx = 63x5 þ 15x and Sx ¼ 70x3 : 1 (x0 ¼ 0:2, an ¼ bn ¼ an ¼ bn ¼ cn ¼ pffiffiffiffiffiffiffiffi ; n P 0Þ. 2nþ4 n
Txn
Sxn
xn+1
0 1 2 3 4 5 – 23 24 25 26 27
3.02016 6.84522 8.68351 9.63872 10.1666 10.47 – 10.929 10.929 10.929 10.9291 10.9291
3.02017 6.84524 8.68353 9.63874 10.1666 10.47 – 10.929 10.929 10.9291 10.9291 10.9291
0.408546 0.476129 0.504846 0.519245 0.527105 0.531609 – 0.538469 0.53847 0.53847 0.538471 0.538471
Table 8 Convergence of Jungck–Khan iterative scheme to coincidence point p = 0.579167 of Tx = ex x2 2 and Sx = 5x. 1 (x0 ¼ 1; an ¼ bn ¼ an ¼ bn ¼ cn ¼ pffiffiffiffiffiffiffiffi ; n P 0Þ. 2nþ4 n
Txn
Sxn
xn+1
0 1 2 3 4 5 6
3.36792 2.92951 2.89837 2.89604 2.89585 2.89583 2.89583
3.36792 2.92951 2.89837 2.89604 2.89585 2.89583 2.89583
0.629975 0.583365 0.579509 0.579196 0.579169 0.579167 0.579167
Table 9 Convergence of Jungck–Khan iterative scheme to coincidence point p = 1 of Tx = x2 þ 9 and Sx = 10x. (x0 ¼ 2; an ¼ bn ¼ an ¼ bn ¼ cn ¼ n15 ; n P 1Þ. n
Txn
Sxn
xn+1
1 2 3 4 5 6 7
13 10.0288 10.0056 10.0011 10.0002 10 10
13 10.0288 10.0056 10.0011 10.0002 10 10
1.01428 1.0028 1.00056 1.00011 1.00002 1 1
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5.1.2. Solution of equation ex ðx2 þ 5x þ 2Þ þ 1 ¼ 0 1. Taking initial guess x0 = 2 (away from coincidence point), Jungck–Khan iterative scheme converges in 7 iterations. 1 2. Taking an ¼ bn ¼ an ¼ bn ¼ cn ¼ p8ffiffiffiffiffiffiffiffi ; n P 0; and x0 = 1, we observe that Jungck–Khan iterative scheme converges in 5 2nþ4 iterations. 5.1.3. Solution of quadratic equation 1. Taking initial guess x0 = 4 (away from coincidence point), Jungck–Khan iterative scheme converges in 9 iterations. 1 2. Taking an ¼ bn ¼ an ¼ bn ¼ cn ¼ n0:1 ; n P 1 and x0 = 2, we observe that Jungck–Khan iterative scheme converges in 5 iterations. 6. Conclusions Keeping in mind the results of Sections 2-4, Tables 1–9 and observations made in Section 5, we make the following remarks: The newly introduced Jungck-type iterative scheme is more general as well as has much better convergence rate as compared to Jungck–Mann, Jungck–Ishikawa and Jungck–Noor iterative schemes and hence has a good potential for further applications. Data dependency of fixed points is performed for nonself operators for a more general iterative scheme with a higher convergence rate. Many typical problems can be solved using Jungck-type iterative schemes. The speed of iterative schemes depend on parameters an , bn etc. For initial guess, near the solution, there is a decrease in number of iterations to converge to the solution of desired problem.
Acknowledgments The authors A.R. Khan and N. Hussain acknowledge gratefully the support of KACST, Riyad, Saudi Arabia for supporting Project J-P-11-0623. The authors would like to thank the referee for his/her valuable suggestions to improve presentation of the paper. References [1] S.M. Soltuz, T. Grosan, Data dependence for Ishikawa iteration when dealing with contractive like operators, Fixed Point Theory Appl. 2008 (2008) 7. 242916. [2] R. Chugh, V. Kumar, Data dependence of Noor and SP iterative schemes when dealing with quasi-contractive operators, Int. J. Comput. Appl. 40 (2011) 41–46. [3] F. Gursoy, V. Karakaya, B. Rhoades, Data dependence results of new multi-step and S-iterative schemes for contractive-like operators, Fixed Point Theory Appl. 76 (2013) 12. [4] V. Berinde, Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators, Fixed Point Theory Appl. 2004 (2004) 716359, http://dx.doi.org/10.1155/S1687182004311058. [5] V. Berinde, Iterative Approximation of Fixed Points, Springer, 2007. [6] A.R. Khan, On modified Noor iterations for asymptotically nonexpansive mappings, Bull. Belg. Math. Soc. Simon Stevin 17 (2010) 127–140. [7] S.L. Singh, C. Bhatnagar, S.N. Mishra, Stability of Jungck-type iterative procedures, Int. J. Math. Math. Sci. 193035–3043 (2005). [8] M.O. Olatinwo, Some stability and strong convergence results for the Jungck–Ishikawa iteration process, Creating Math. Inf. 17 (2008) 33–42. [9] M.O. Olatinwo, A generalization of some convergence results using the Jungck–Noor three-step iteration process in an arbitrary Banach space, Fasciculi Math. 40 (2008) 37–43. [10] M.O. Olatinwo, C.O. Imoru, Some convergence results for the Jungck–Mann and the Jungck–Ishikawa iteration processes in the class of generalized Zamfirescu operators, Acta Math. Univ. Comenianae LXXVII, 2 (2008) 299–304. [11] G. Jungck, Commuting mappings and fixed points, Am. Math. Monthly 83 (1976) 261–263. [12] N. Hussain, V. Kumar, M.A. Kutbi, On the rate of convergence of Jungck-type iterative schemes, Abstract Appl. Anal. 2013 (2013) 15. Article ID 132626. [13] M.A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000) 217–229. [14] S. Ishikawa, Fixed points by a new iteration method, Proc. Am. Math. Soc. 44 (1974) 147–150. [15] W.R. Mann, Mean value methods in iteration, Proc. Am. Math. Soc. 4 (1953) 506–510. [16] G. Jungck, N. Hussain, Compatible maps and invariant approximations, J. Math. Anal. Appl. 325 (2007) 1003–1012. [17] W. Takahashi, S. Matsushita, Strong convergence theorems for nonexpansive nonself-mappings without boundary conditions, Nonlinear Anal. 68 (2008) 412–419.
