### Paper Menu >>

### Journal Menu >>

American Journal of Computational Mathematics, 2012, 2, 345-357 http://dx.doi.org/10.4236/ajcm.2012.24048 Published Online December 2012 (http://www.SciRP.org/journal/ajcm) Strong Convergence of a New Three Step Iterative Scheme in Banach Spaces Renu Chugh, Vivek Kumar*, Sanjay Kumar Department of Mathematics, Maharshi Dayanand University, Rohtak, India Email: *ratheevivek15@yahoo.com Received August 11, 2012; revised October 12, 2012; accepted November 3, 2012 ABSTRACT In this paper, we suggest a new type of three step iterative scheme called the CR iterative scheme and study the strong convergence of this iterative scheme for a certain class of quasi-contractive operators in Banach spaces. We show that for the aforementioned class of operators, the CR iterative scheme is equivalent to and faster than Picard, Mann, Ishikawa, Agarwal et al., Noor and SP iterative schemes. Moreover, we also present various numerical examples using computer programming in C++ for the CR iterative scheme to compare it with the other above mentioned iterative schemes. Our results show that as far as the rate of convergence is concerned 1) for increasing functions the CR itera- tive scheme is best, while for decreasing functions the SP iterative scheme is best; 2) CR iterative scheme is best for a certain class of quasi-contractive operators. Keywords: Fixed Point; Various Iterative Schemes; Rate of Convergence 1. Introduction There is a close relationship between the problem of solving a nonlinear equation and that of approximating fixed points of a corresponding contractive type operator. Consequently, there is theoretical and practical interest in approximating fixed points of various contractive type operators. Let , X d , be a complete metric space and a self map for X. Suppose that :TX X F TpXTpp is the set of fixed points of T. There are several iterative processes in the literature for which the fixed points of operators have been approxi- mated over the years by various authors. In a complete metric space, the Picard iterative scheme defined by 0, nn x 1,0,1, nn xTxn (1.1) has been employed to approximate the fixed points of mappings satisfying the inequality ,dTxTy dxy , (1.2) for all , x yX and 0,1 . Condition (1.2) is called the Banach’s contraction con- dition. Any operator satisfying (1.2) is called a strict con- traction. In 1953, W. R. Mann defined the Mann iterative sche- me [1] as 11, nnnn uu where n is a sequence of positive numbers in [0,1]. In 1974, S. Ishikawa defined the Ishikawa iterative scheme [2] as 11 nnnnn s sT t 1 nnnn ts , n Ts (1.4) where n and n are sequences of positive num- bers in [0,1]. In 2007, Agarwal et al. defined the Agarwal et al. ite- rative scheme [3] as 11 nnnnn s Ts Tt 1 nnnn ts , n Ts (1.5) where n and n are sequences of positive num- bers in [0,1]. In 2000, M. A. Noor defined the Noor iterative scheme [4] as 11 nnnn pp n Tq 1 nnnn qp n Tr 1 nnnn rp , n Tp (1.6) where n , n and n are sequences of posi- tive numbers in [0,1]. n Tu (1.3) Recently, Phuengrattana and Suantai defined the SP iteration scheme [5] as *Corresponding author. C opyright © 2012 SciRes. AJCM R. CHUGH ET AL. 346 11 nnnnn x yT y 1 nnnn yz n Tz 1 nnnn zx , n Tx (1.7) where n , n and n are sequences of posi- tive numbers in [0,1]. Remarks: 1) If , then the Noor iterative scheme (1.6) re- duces to the Ishikawa iterative scheme (1.4). 0 n 2) If , then the Noor iterative scheme (1.6) reduces to the Mann iterative scheme (1.3). 0 nn 3) In addition, when , then the SP iterative scheme (1.7) reduces to the Mann iterative scheme (1.3). 0 nn In 1972, Zamfirescu [6] obtained the following inter- esting fixed point theorem. Theorem 1.1. Let (X, d) be a complete metric space and a mapping for which there exists real :TX X numbers a, b and c satisfying 1 0,1,,0, 2 abc such that for each pair , x yX, at least one of the fol- lowing conditions hold 1) ,,dTxTy adxy 2) ,,,dTxTybdxTx dyTy 3),,, .dTxTycdxTy dyTx (1.8) Then T has a fixed point p and the Picard iteration defined by 0 nn x 1, 0,1, nn xTxn converges to p for any arbitrary but fixed 0 x X. The operators satisfying the condition (1.8) are called Zamfirescu operators. Berinde [7] introduced a new class of operators on an arbitrary Banach space satisfying ,2,,dTxTydxTx dxy , (1.9) , x yX and some . 0, 1 He proved that this class is wider than the class of Zamfirescu operators and used the Ishikawa iteration process to approximate fixed points of this class of ope- rators in an arbitrary Banach space given in the form of following: Theorem 1.2 [7]. Let K be a nonempty closed convex subset of an arbitrary Banach space X and be a mapping satisfying (1.9). Let {sn} be defined through the Ishikawa iteration (1.4) and 0 :TK K s K where n , n are sequences of positive numbers in [0,1] with n satisfying 0 n n . Then {sn} converges strongly to the fixed point of T. Several authors [5,8-17] have studied the equivalence between various iterative schemes. S. M. Solutz [15,16] proved that for quasi-contractive operators the itérative processes Picard, Mann, Ishikawa and Noor are équi- valent. Recently, Renu Chugh and Vivek Kumar [17] proved that for quasi-contractive operators satisfying (1.9) Picard, Mann, Ishikawa, Noor and SP iterative schemes are equivalent. Fixed point iterative procedures are designed to be ap- plied in solving equations arising in physical formulation but there is no systematic study of numerical aspects of these iterative procedures. In computational mathematics, it is of vital interest to know which of the given iterative procedures converge faster to a desired solution, com- monly known as rate of convergence. B. E. Rhoades [18] compared the Mann and Ishikawa iterative procedures concerning their rate of convergence. He illustrated the difference in the rate of convergence for increasing and decreasing functions. Indeed he used computer programs, perhaps for the first time to compare the Mann and Ishi- kawa iterations through examples. S. L. Singh [19] ex- tended the work of Rhoades. Very recently, Phuengrat- tana and Suantai [5] proved that SP iterative scheme is equivalent to and faster than Mann, Ishikawa and Noor iterative schemes for increasing functions. Now, we introduce the following CR iterative process: Let X be a Banach space, a self map of X :TX X and 0 x X . Define the sequence by 0 nn x 11 1 1, nnnn nnnn nnnnn n n x yT yTx zxT y Tz x (1.10) where n , n and n are sequences of posi- tive numbers in [0,1] with n satisfying 0 n n . We shall need the following lemma and definitions. Lemma 1 [20]. If is a real number such that 01 and 0 nn is a sequence of positive num- bers such that lim 0 n n , then for any sequence of posi- tive numbers 0 nn satisfying 1,0,1,2, nnn uu n , we have lim 0 n nu . Definition 1.1. Suppose that {an} and {bn} are two real convergent sequences with limits a and b respec- tively. Then {an} is said to converge faster than {bn} if lim 0 n nn aa bb . Copyright © 2012 SciRes. AJCM R. CHUGH ET AL. 347 Definition 1.2 [21]. Suppose that {un} and {vn} are two fixed pointeration procedures both converging to the sam t i e fixed point p with the error estimates ,0,1, nn upan ,0,1, nn vpbn where {an} and {bn} are two real convergent sequences converging to 0. If {an} converges faster than {bn}, then we say {} convefaster than v to p. rwal et al., SP an x subset a map- unrges n The purpose of this paper is to show the convergence of the CR iterative scheme and to prove equivalence be- tween Picard, Mann, Ishikawa, Noor, Aga d CR iterative schemes for quasi-contractive operators satisfying (1.9). We provide an example for which the CR iterative scheme is faster than the other above men- tioned iterative schemes for the aforementioned class of operators. Also, by using computer programs in C++, we compare the above mentioned iterative schemes through examples of increasing and decreasing functions. 2. Result on Strong Convergence Theorem 2.1. Let K be a nonempty closed conve of an arbitrary Banach space X and :TK K ping satisfying (1.9). Let 0 nn xbe defined through the CR iteration (1.10) and 0 x X , where n , n are sequences of positivers in [0,1] with numbe n Satisfying n . Then convetro 0n to the fixed point of T . rges sngly heorem 1hat T K p 0 nn x Proof. T.1 shows t has a unique fixed point in , say . From (1.10), we have 1 11 1 , n nn nn nnn nn n nn xp yp Ty p Tx p Tz p Ty p (2.1) Using (1.9), (2.1) yields 1 1 1 1 n nn nnn nn xp 1 n x p zp yp (2.2) Using (1.9) and (1.10), (2.2) yields 1 2 2 2 3 11 11 1 11 11 1 1 1 11 11 1 n nnn nnnn nnn n nn n nnn nnnn nnnn nnn nnn n nnnn nnnn nnn n xp xp xp Tx p Tz p xp xp xp xp xp xp xp 1 nn n Tx p (2.3) 0 0 0 1 0 11 ek k n n nk k n x pxp xp (2.4) Since 0 < 1, k [0,1] and 0 n n t follow , so as Hence, i 0 1 e0 k k n (2.4) that n. s from . Therefore 0 nn x 1 lim 0 n nxp con- verges strongly to p. 3.ann, Ishikawa, Noor, Agarwal et al., SP and CR Iterative Schemes . Let K be a nononvex X and a mapping itial poe, Equivalence between Picard, M Theorem 3.1empty closed csubset of a Banach space satisfying (1.9). If the in :TK K int is the sam 0, n A nN , then the followings are equiv 1) The Mann iteration (1.3) converges to p. 2) The Agarwal et al. iteration (1.5) converges to p. 3) The CR iteration (1.10 alent: ) converges to p. shall show Using (1.3) and ( Proof. First we prove that 1) 3). Let the Mann it- eration (1.3) converge to p. We that the CR iteration (1.10) also converges to p. 1.10), we have 11nn 1nnn nnn yuTy Tu 12 11 2 . nnnnnnnnn yu yu uTu yu uTu (3.1) xu nn nnnn Copyright © 2012 SciRes. AJCM R. CHUGH ET AL. 348 Using as and Lemma 1.1, (3.7) yields From (1.10), we have 1 11 121 2 121 , nn nnnn nn nnnn nn nnn nnnnnn n nnn nnn nnn nnnn n nn n Tx uTz u Tx TuTu u Tz TuTu u 1nn nn yu x uTuu z u Tu uTu u xu uTu zu (3.2) Again, from (1.10), we have Tu u 1 1 11 21 nn nnn nnn nnn nnnnn nnn nn zu xu Txu . n x uTxTuTuu x uTu u (3.3) Substituting (3.3) in (3.2), we have 1yu x 11 12 1 . nn nnn nn nn nnn n u xu Tx u (3.4) Substituting (3.4) in (3.1), we obtain 11 11 1 nn 11 121 2. nn nnn nnn nn n xu x u uTu (3.5) Also, uTuu pp 1. nnn n n Tu up (3.6) Substituting (3.6) in (3.5), we have 11 11 1 11 1212 nn nnnnn nnn n n xu xu up hx uup 1 nn n (3.7) where 1(using 11 1 nnn h 0, n A nN ) and 11212l . n up n 0 nn xu as In addition n. nnnn x pxu up and this implies that n x p ove that as Conversely, we pr n. n x p impn up. lies .3), (1.9) and (1.1Using (10 ), we have 11 1 2 112 . nn nnn nnn n n nn n nnnnn xu y uTy Tu yu yTy y yTy , we have 1nn nn yu (3.8) n u From (1.10) 2 11 1 11 11 1 11 nn n nnn nn nn nn n nn nn nnn n nn nnn n nnn nn n nn nn yu Tzxxu Txpxpx Tzpxpxu xuxpzp xu xp xp xp xu x 1nn nnn Tx u Tz 11 nn n nnn Tx xx u nn u nn n u . np (3.9) Also, 1 11 1 11 1 11 1, nn nn n nn nn nn nnnn nn n yTy ypTyp yp x pzp x p xp (3.10) Substituting (3.9) and (3.10) in (3.8) and rearranging the terms, we have Copyright © 2012 SciRes. AJCM R. CHUGH ET AL. 349 11 2 2 11 11 1(1 2111 11 121 . nn nnn nnn nnn nnn n xu xu n x p xu xp (3.11) Since 0, n A nN 1 1,.nN , so Also 01n n x p as n. Hence, using Lemma 1.1, (3.11) yields 0 nn xu as n. In addition nnnn up xuxp and this im- plies that as Hence the result. Next, that 1. Let as . Using (1.3) and (1.5), we have n up we show n. ) 2) n up n 11 1 12 1 . nn nnnnnn nnnnnnnn n nnn nnnn nnn su Ts uTt Tu Ts ut uu Tu Ts T Tu win 1n u 11 12 nn nn Tu su u u (3.12) imates: Now, we have the follog est 1 nn nn u pTu n uTu p up (3.13) 2 21. nn nnnn nnn Ts Tus uu Tu s uu p (3.14) It follows using (3.12), (3.13) and (3.14) that 11 (1 11 21 11 21 11 . 141 nn nnnnnn nn n nn nnn n n su 11 1 nn n s u up up su up (3.15) Since and as , hence us- in up 0, 1 n g Lemma 1.1, (3.15) yields up n 0 nn su as n. . nnnn s psu up (3.16) and this implies that n s p ove that as Conversely, we pr n. n s p implies Using (1.3), (1.5) and (1.9 n up. ), we have 11 1 1 2. nn nnnnnn nnnnn nn nnnn su Ts uTt Tu Tsss u tu tTt (3.17) e have twing etes: Now, whe follostima In addition 1 nn n s Ts s p ) (3.18 1 111 nn n nn tTtt p s p (3.19) 1 nn nnnnn tusTsu 1nn n n 1. n n n nn n nn nn u s u Tsu su su sp (3.20) It follows from (3.17), (3.18), (3.19) and (3.20 ) that 11 11 21 11 113 nn nnn n nnn n su sp 11 nn n su 1nn . s us (3.21) If p 0, n A nN , then 01 11 n ad . Also, n s p as n Hence, using Lemma 1.1, . (3.21) yields 0us nn as n. In addition nnnn upussp and this implies that as Hence the result. Keeping in mind Soltuz’s results [15,16] as well as Chugh and Kumar’s result [17], Theorem 3.1 leads to the following corollary: Corollary 3.2. Let K be a nonempty closed con subset of a Banach space X and T: n up n. vex K K oint is a ma satisfying (1.9). If the initial p the pping same, 0, n A nN he Picae he followi 1) Trd it ( , then tuivalent: ration1.1) coo the fixed po ) The Mann iteration (1.3) converges to p, awa iteration (1.4) converges to p, ngs are eq nverges t int p of T, 2 3) The Ishik Copyright © 2012 SciRes. AJCM R. CHUGH ET AL. 350 4) The Noor iteration (1.6) converges to , ) conve .7) conv to p, ai-Contractive Operators eration (1.9). In [22], Qin sh ertain quasi-contractive operator. Ciric, Lee anowed that Nd Ishi- kaactive ope- ra an exor which the ite faster than Mann and Ishikawa iterative schemes. ter than Pi - p 5) The Agarwal et al. iteration (1.5rges to p, 6) The SP iteration (1erges 7) The CR iteration (1.10) converges to p. 4. Results on Fastness of CR Iterative Scheme for Qus In [21] Berinde showed that Picard itis faster than Mann iteration for quasi-contractive operators satisfying g and Rhoades by taking example owed that Ishikawa iteration is faster than Mann ite- ration for a c d Rafiq [23], by providing an example, sh oor iterative scheme can be faster than Mann an wa iterative schemes for some quasi-contr tor. Recently, Nawab Hussian et al. [24] provide ample of a quasi-contractive operator f rative scheme (1.5) due to Agarwal et al. is Now, we show that the CR iteration is fas card, Agarwal et al., Noor and SP iterations for quasi contractive operators satisfying (1.9) as follows: 1) By providing an example 4.1 of a quasi-contractive operator satisfying (1.9), we show that CR iterative scheme is faster than Agarwal et al., Noor and SP ite- rative schemes. 2) By using definition (1.1), we show that CR iterative scheme is faster than Picard iteration. 1) Example 4.1. Let T: :0,1 0,1T be defined by 2 x Tx, 0 nnn , 1, 2,,15n, 4,16 nnn n n . It is clear that T is a quasi- contractive operator satisfying (1.9) with a unique fixed point 0. Also, it is easy to see that T, ,and nn n satisfies all the conditions of Theorem 2.1. We show that C e scheme. R iterative scheme is faster than Agarwal et al., Noor and SP iterative schemes. Proof. First of all we show that CR iterative scheme is faster than Noor iterativ Let 16n and p0 = x0. Then, from [23], for Noor it- eration (1.6), we have 1 16 248 1. n ni0 p p iii (4.1) Also, for CR iteration (1 i .10), we have 10 16 114 8. 2 n ni x x i iii (4.2) So, 0 16 114 8 2 n i x x 1 124 8 1 n n n p 0 3 2 3 16 2 23 2 1. 28 416 n i i ii i p ii iii 16ii ii i It is easy to see that 3 2 3 16 2 16 232 0lim1 28 416 115 lim 1lim0. n ni n nn i ii iii in 1 1 lim 0 n nn x p . Hence, we have Therefore, by definition 1.2, CR iterative scheme con- verges faster than Noor iterative scheme to the fixed point 0 of T. that CR iterative scheme is faster than SP iterative scheme: For SP iteration (1.7) we have Secondly, we show 10 16 612 8 1. n ni x x i iii So, 16 1 1SP n x i 16 3 2 3 16 2 114 8 CR 2 612 8 1 2 10 3232 1. 2241216 n i n n i n i xi iii iii i iii i iii ii i iii 16 161 2 8 1 i 151616 n It is easy to see that Copyright © 2012 SciRes. AJCM R. CHUGH ET AL. 351 3 2 3 16 2 16 10 3232 0lim1 2241216 1 1015 lim 1lim0. 1000 n ni n nn i ii i iii in Hence, we have 1 1 CR lim 0 SP n nn x x . Therefore, by definition 1.2, CR iterative scheme converges faster than SP iterative scheme to the fixed point 0 of T. Next, we show that CR iterative scheme is faster than Agarwal et al. iterative scheme. For Agarwal et al. iteration (1.5), we have 1?0n 16 14 . 2 n i s s i So, 0 16 1 1 0 16 16 114 8 2 14 2 18 1 14 . n i nn n i n i 2 x xi iii ss i iii i It is easy to see that 16 16 18 nn 1 0lim1lim1 14 2 15 lim 0. nn ii n iii i i n Hence, we have 1 1 lim 0 n nn x s . Therefore, again by definition 1.2, CR iterative scheme convergeer than Agarwal et al. iterative scheme to the fixed point 0 of T. 2) Here we show that CR iteration is faster than Picard iteration. Using Picard iteration (1.1) and condition (1.9) we have s fast 1 10 . n n x 1 11 0 11 11 1 , nn n nnnn nn n xp xp qx pqx p nn (4.4) where 11 11 11 nn nnn qnn . (if 0,anN). n In order to compare CR and Picard iterations, we must compare the coefficients of the inequalities (4.4) and (4.3). Obviously 11nn q 10 n as n and hence us- ing definition (1.1), we can say that CR iteration is faster than Picard iteration. Keeping in mind results of example 4.1 Ce conclude that CR iterative scerative schemes for a certain class of quasi-contractive operators. 5. Applications In this section, with the help of computer programs in compare the rate of convergence of Picard, Mann, Ishikawa, Noor, Agarwal et al., SP and CR ite- ration procedures, through examples. The ouome is listed in the form of Tables 1-4, by taking initial appro- ation x0 = 0.8d as well as iric et al.’s results [23], w heme is faster than other it C++, we tc xim an 1 2 1n iterative schemes. 1 nn n ab g , for all 5.1. Example of Decreasin Let f: [0,1]→[0,1] be defined by g Function m 1 f xx , 7, 8,m . Then f is a decreasing function. By taking m = 7, the comparison of convergence of above men- tio tion e defined as ned iterative schemes to the exact fixed point p = 0.188348 is listed in the Table 1. 5.2. Example of Increasing Func pxp (4.3) Also, from (2.3) we have Let f : [0,8]→[0,8] b 29x fx 10 . Then f equation x3 + x2 − 1 = to find fixed point of the function is an increasing function. The comparison of conver- gence of above mentioned iterative schemes to the exact fixed point p = 1 is listed in the Table 2. 5.3. Example of Cubic Equation To find solution of cubic 0 means 12 3 1x as x3 + x2 Copyright © 2012 SciRes. AJCM R. CHUGH ET AL. Copyright © 2012 SciRes. AJCM 352 Table 1. Decreasing function. R. CHUGH ET AL. 353 Table 2. Increasing function. CR iteration SP iteration Noor iteration Picard iteration Mann iteration Ishikawa iteration Agarwal et al. iteration n fxn x n+1 fxn x n+1 fxn x n+1 n fxn x n+1 fxn x n+1 n xn+1 fxn 0 0.964 0.998591 0.964 0.998591 0.964 0.9985910.9 0.9 0.964 0.964 0.964 0.99293 0.964 0.99293 - - - - - - - - - - - - - - - 4 1 1 0.999999 0.999999 0.999974 0.9999220.999850.999850.9947140.9828280.999968 0.999937 0.9999960.999997 5 1 1 1 1 0.999984 0.999950.999970.999970.9965950.9884490.999987 0.999973 0.9999990.999999 6 1 1 1 1 0.99999 0.9999660.9999940.9999940.9977030.9919470.999968 0.999937 1 1 7 1 1 1 1 0.999993 0.9999760.9999990.9999990.9983960.9942270.999995 0.999988 1 1 8 1 1 1 1 0.999995 0.9999831 1 0.9988490.9957670.999998 0.999994 1 1 - - - - - - - - - - - - - - - 23 1 1 1 1 1 1 1 1 0.9999640.999850.999999 0.999997 1 1 24 1 1 1 1 1 1 1 1 0.999970.9998740.999999 0.999998 1 1 - - - - - - - - - - - - - - - 33 1 1 1 1 1 1 1 1 0.9999940.9999741 1 1 1 - - - - - - - - - - - - - - - 68 1 1 1 1 1 1 1 1 1 1 1 1 1 1 69 1 1 1 1 1 1 1 1 1 1 1 1 1 1 − 1 = 0 can be rewritten as 12 3 1 x x. The com- parison of convergence of above mentioned iterative schemes to the exact fixed point p = 0.754878 of 12 3 1x is listed in the Table 3. 5.4. Example of Goat Problem A farmer has a fenced circular pasture of radius a and wants to tie a goat to the fence with a rope of length b so as to allow the goat to graze half the pasture. How long should the rope be to accomplish this? The length of the rope “b” must be longer than “a” and shorter than 2a, i.e. 2ab a. If we let b xa , we get the simplified equation 21 22 42 sin4ππ 2 x xxx x and we are looking for the solution x, with 12x produce a sequence that . If we rear- range the equation, we can will converge to the solution: 222 ππ 442sin 2 1 x xxx x or 221 π442sin 2 π x xx x x . 11x , Let 221 1 π442sin 2 π n nnn n x xx x x 221 π442sin 2. π n nnn x xx x fx and The comparison of convergence of above mentioned iterative schemes to the exact fixed point 1.15863 of the function f(x) is listed in the Table 4. So the rope length “b” should be approximately 1.15863 a. For detailed study, these programs are again executed after changing the parameters and the readings are re- corded (discussed in the next section). 6. Observations 6.1. Decreasing Function (1 − x)m 1) For m = 8 and xo = 0.8, the Picard scheme never converges (oscillates between 0 and 1), the Mann scheme converges in 9 iterations, the Ishikawa scheme converges in 35 iterations, the Noor scheme converges in 10 itera- tions, Agarwal et al. iteration does not converges, the CR scheme converges in 9 iterations and the SP scheme converges in 7 iterations. 2) For m = 30 and xo = 0.8, the Picard scheme never Copyright © 2012 SciRes. AJCM R. CHUGH ET AL. 354 n n Table 3. Cubic equation. CR iteration SP iteratioNoor iteratioPicard iterationMann iteration Ishikawa iteration Agarwal et al. iteration n fxn x n+1 fxn x 1 fxn x 1 fxn x 1 n+n+n+ fxn x n+1fxn x n+1fxn x n+1 0 0.69857 0.68185 0.698555857577 7 0.68184 0.6987 0.6818450.690.6980.69850.698570.698570.8118480.698570.811848 - - - - - - - - - 0.754709 04839 0.754872 04878 04878 0487055 0.754878 04878 04878 04870.755475 0.755263 - - - 0.754878 04878 04878 04870.37240.7240.7545920.75466 0.754878 04878 04878 0.4878 0.4877 0.48780.92170.92170.78 0.7548780.754097 0.755268 0.755124 0.755071 - - - - - - 0.754878 04878 04878 04878 0.4878 0.48780.22410.2410.754691 0.754722 13 0.754878 0.4878 04878 0.4878 0.4878 0.48780.22590.22590.78 0.7548780.754789 0.754925 0.755054 0.755027 - - - - - - - 0.754878 0.78 0.78 0.78 0.78 0.781 1 0.754878 0.7548780.754877 0.754878 0.759426 0.759083 0.754878 04878 04878 04878 488 0.48780.01130.1130.8780.7548780.754878 0.754878 0.750074 0.750431 0.754878 0.754878 0.754878 0.754878 0.754878 0.7548781 1 0.754878 0.7548780.754878 0.754878 0.759867 0.759516 -- - - - - 3 .7583 0.754992 0.754878 0.754385 0.7549630.82640.8264390.761920.7552480.723666 0.769363 0.7566140.75569 4 .75.75.758 0.754782 0.7549 20.65990.6599550.7544580.7548950.737969 0.762659 0.7539570.75435 5 .75.75.758 0.75485 0.7548860.8441340.8441340.7548580.754880.745922 0.759043 - - - - - - - - - - - - 8 .75.75.758 0.754876 0.754878595930.754878 0.7548780.753476 0.755566 9 .75.7575757588887548 - - -- - - - - 12 .75.75.757575484820.754878 0.7548780.754728 0.754956 75 .7575757594947548 - - - - - - - 27 75487548754875487548 28 .75.75.750.757500000754 29 converges (oscillates between 0 and 1), the Mann scheme converges in 11 iterations, the Ishikawa scheme con- ver 37 iterations, the Noor scheme converges in 12 , Agarwal et al. scheme never c ges in iterations onverges, the s and the SP scheme tions, Noor sche- m ergeR scheme converges in 10 itera- tions a SP sch 4) Taking CR scheme converges in 13 iteration converges in 9 iterations. 3) Taking initial guess xo = 0.2 (nearer to the fixed point), Picard scheme never converges (oscillates be- tween 0 and 1), Mann scheme converges in 10 iterations, Ishikawa scheme converges in 40 itera e converges in 10 iterations, Agarwal et al. scheme does not conv, the C nd theeme converges in 8 iterations. 1 4 1 1 nnnn and xo = 0.8, we obtain that the Mann scheme converges in 23 iterations, 56 iterat , the CR scheme converges in 11 iterations and the SP scheme converges in tions. 6.2. Increasing Function the Ishikawa scheme converges inion, the Noor scheme converges in 21 iterations, Agarwal et al. scheme converges in 12 iterations 15 itera- 2910x 1rg poi ) For xo = 0.8, the Picard scheme convees to a fixed nt in 8 iterations, the Mann scheme converges in 69 iterations, the Ishikawa scheme converges in 34 itera- tions, the Noor scheme converges in 24 iterations, Agar- wal et al. scheme converges in 7 iterations, the SP scheme converges in 6 iterations and the CR scheme con- verges in 5 iterations. 2) Taking initial guess xo = 0.6 (away from the fixed ), the Picard scheme converges to a fixed point in 8 ite point rations, the Mann scheme converges in 75 iterations, the Ishikawa scheme converges in 38 iterations, the Noor scheme converges in 27 iterations, Agarwal et al. scheme converges in 6 iterations, the SP scheme converges in 7 iterations and the CR scheme converges in 5 iterations. 3) Taking, 1 4 1 1 nnn n , xo = 0.8, we ob- tain that the Mann sc Ishikawa scheme con heme converges in 23 iterations, the verges in 12 iterations, Noor sche- al. scheme con- verges in 6 iterations, t verges in 9 iterations, the Ishi- ka me converges in 9 iterations, Agarwal et he SP scheme converges in 5 it- erations and the CR scheme converges in 4 iterations. 6.3. Cubic Equation x3 + x2 –1 = 0 1) For xo = 0.8, the Picard scheme never converges to the solution of cubic equation (oscillates between 0 and 1), the Mann scheme con wa scheme converges in 29 iterations, the Noor sche- Copyright © 2012 SciRes. AJCM R. CHUGH ET AL. 355 Table 4. Goat problem. CR iteration SP iteration Noor iteration Picard iteration Mann iteration Ishikawa iteration Agarwal et al. iteration n fxn x n+1 fxn x n+1 fxn x n+1 fxn x n+1 fxn x n+1 fxn x n+1 fxn x n+1 0 1.103589 1.150601 1.103589 1.150601 1.103589 1.1506011.1035891.1035891.1035891.1035891.103589 1.137884 1.1035891.137884 1 1.155492 1.1584 1.103589 1.150601 1.155492 1.1552751.1378841.1378841.1378841.1378841.150601 1.149315 1.1506011.15304 1.15863 1.15863 1.158626 1.158625 1.158512 1.1583991.1586011.1586011.1568521.1550111.158273 1.157929 1.1586271.158627 1.15863 1.15863 1.15863 1.15863 1.158574 1.1585151.1586291.1586291.1576931.1566611.158458 1.158278 1.158631.15863 1.15863 1.15863 1.15863 1.15863 1.1586 1.1585671.158631.158631.1580921.1574741.158538 1.158436 1.158631.15863 1.158631.3 1.158631.15863 15863 1.15863 1.15863 1.15863 1.158616 1.1585991.158631.158631.1583491.1580131.158585 1.158533 1.158631.15863 3 3 - 63 63 3 73 - - - - - - - - - - - - - - - 6 1.158629 1.158629 1.15861 1.158608 1.158414 1.1582291.158441.158441.1556931.1528471.157988 1.157426 1.1586031.158608 7 1.15863 1.15863 1.158622 1.15862 1.158472 1.1583281.1585561.1585561.1563661.1540911.158157 1.15772 1.158621.158621 8 9 1.15863 1.15863 1.158628 1.158628 1.158539 1.1584491.1586191.1586191.1572111.1557071.158355 1.15808 1.1586291.158629 10 1.15863 1.15863 1.15863 1.15863 1.158559 1.1584871.1586261.1586261.1574841.1562431.158414 1.158193 1.158631.15863 11 12 1.15863 1.15863 1.15863 1.15863 1.158585 1.1585371.158631.158631.1578571.1569931.158492 1.158344 1.158631.15863 13 1.15863 1.15863 1.15863 1.15863 1.158594 1.1585541.158631.158631.1579871.1572591.158518 1.158395 1.158631.15863 14 15 1.15863 1.15863 1.15863 1.15863 1.158606 1.15857815861.1581761.1576491.158554 1.158468 1.158631.15863 16 1.15863 1.15863 1.15863 1.15863 1.15861 1.1585861.158631.158631.1582451.1577941.158567 1.158494 1.158631.15863 17 1.15863 1.15863 1.15863 1.15863 1.158613 1.1585931.1583021.1579141.158577 1.158516 1.158631.15863 18 1. - - - - - - - - 47 1.15863 1.15863 1.15863 1.15863 1.15863 1.158631.1586 48 1.15863 1.15863 1.15863 1.15863 1.15863 1.158631.1586 - - - - - - - - 6 - - - - - - - 1.158631.158621.1586071.158629 1.158627 1.158631.15863 1.158631.1586211.1586091.158629 1.158628 1.158631.15863 - -- - - - - 1.158631.1586271.1586231.15863 1.15863 1.158631.15863 - - - - - - - 1.158631.158631.158631.15863 1.15863 1.158631.15863 1.158631.158631.158631.15863 1.15863 1.158631.15863 0 1.15863 1.15863 1.15863 1.15863 1.15863 1.158631.158 - - - - - - - - 89 1.15863 1.15863 1.15863 1.15863 1.15863 1.158631.158 90 1.15863 1.15863 1.15863 1.15863 1.15863 1.158631.1586 me converges in 13 iterations, Agarwal et al. scheme nees, the CR scheme co tions and the SP scheme converges in 5 iterations. ges in 10 iterations and the SP scheme converges ver convergnverges in 6 itera- 2) Taking initial guess xo = 0.1 (away from the solu- tion of cubic equation), Picard scheme never converges (oscillates between 0 and 1), the Mann scheme converges in 10 iterations, the Ishikawa scheme converges in 30 iterations, Noor scheme converges in 21 iterations, Agarwal et al. scheme never converges, the CR scheme conver in 5 iterations. 3) Taking 1 4 1 nnn n and xo = 0.8, we 1 ob Mann scheme converg the Ishikawa scheme converges in 27 iterations, the Noor s to a fixed converges in 90 iterations, t tain that thees in 14 iterations, scheme converges in 9 iterations, Agarwal et al. scheme converges in 10 iterations, the CR scheme converges in 8 iterations and the SP scheme converges in 8 iterations. 6.4. Goat Problem 1) For xo = 0.8, the Picard scheme converge point in 13 iterations, the Mann scheme he Ishikawa scheme converges in 61 itera- tions, the Noor scheme converges in 48 iterations, Agar- wal et al. scheme converges in 11 iterations, the SP scheme converges in 11 iterations and the CR scheme Copyright © 2012 SciRes. AJCM R. CHUGH ET AL. 356 converges in 8 iterations. 2) Taking initial guess x = 0.6 (away from the fixed poinhemes to iterations, the nn sce converges i 104 iterations the Ishi se conerge71 iterations, tNoor s con 12 iterations and the CR scheme converges in 9 itera- tions. o e convergt), the Picard sc a fixed point in 14 Ma chem hem v n, kawa s in he cheme converges in 57 iterations, Agarwal et al. scheme verges in 12 iterations, the SP scheme converges in 3) Taking 1 4 1 1n nnn o w o Ishi me converges in 14 iterations, Agarwal et al. scheme c i 7. Conclusions For d m n f and x = 0.8,e btain that Mann scheme converges in 29 iterations, the kawa scheme converges in 18 iterations, Noor sche- onverges in 10 iterations ,the SP scheme converges in 7 terations and the CR scheme converges in 7 iterations. ecreasing functions, we conclude the followings: 1) Picard and Agarwal et al. schees do ot converge or 1 2 n 1, ra 1 ne i M faster t iteration ) Oincreing the valuef m, Mann, Iikawa Noo r iterations t c e schemeoor e shows anncreaswhile wa schemwin iterations co she. T speedf iterative schees depds on te of convergence of the SP schme s better than other iterative schemes, while CR and ann schem han Is es show hikawa s equivalence. A . lso, Noor scheme is 2n as osh, r, SP and CR schemes require more numbe of o For in onverg itial gu e. ess near3)er to th fixed point, Mann , Nschem ie, Ishika e sho erge. Of s an dec course, th rease e CR the num scheme ber of ows no to nv chang 4)he omen n and n . If we increase the value of n and n , the f fo i nnumr a sa xed ll point is obtained i more ber of iterations chemes. Agarwal et al. scheme converges for incre sed value of n i.e for 1 2 1 nn . In this case, increasing order of rate of conve 1 rgence fo tions increases in each iterative scheme. Hence, clhe initial guess to the fi the result is achieved. 3) If we increase the value of r iterative schemes is Ishikawa, Mann, Noor, Agarwal et al., SP and CR scheme. For increasing functions, we conclude the followings: 1) Increasing order of rate of convergence for iterative schemes is Mann, Ishikawa, Noor, Picard, Agarwal et al., SP and CR scheme. 2) For initial guess away from the fixed point, the number of itera oser txed point, quicker , the fixed btained in less number of iterations for all schemes. Except CR iterative scheme which remn- d. olof cubic euation, we code the followings: n point is o ains u affecte For sution qnclu 1) Picard and Agarwal et al. iterative schemes do not converge for 1 2 1 1 nn f s. and rate of convergence o SP iterative scheme is better than Ishikawa, Noor, Mann and CR sc 2) Fo heme r initial guess away from the solution, the number of iterations increases in each iterative scheme. 3) If we increase the value of n and n , the solu- tion is obtained in less number of iterations for Noor and Ishikawa schemes while solution is obtained in more s valuef number of iterations for Mann, SP and CR chemes. Agarwal et al. iteration converges for increased o n In this case, increasing order of rate of conv equivalence. s is Mann o- e fix pointhe number of iterations increases in each iterative scheme. c,r 3) If we icrease the value of ergence for iterative schemes is Ishikawa, Mann, Agarwal et al., Noor and CR scheme while CR and SP schemes show For the goat problem we conclude the followings: 1) Increasing order of rate of convergence for iterative scheme scheme , Ishika and Ag wa, No arwal eor, Picar t al. sche d, SP an mes sh d CR w equi while SP valence. 2) For initial guess away from thedt, Hence, loser the initial guess to the fixed point quicke the result is achieved. nn and n , the xed poi is oined less mb iterations foall - verence fo iterative schems is Mnn, Ishikawa, Nor, Agwal et al. and CR schee whilR anP sches c REFERENCES fi r ntbtain nuer fo schemes. In this case increasing order of rate of con g ar r e m a e C o emd S show equivalene. [1] W. R. Mann, “Mean Value Methods in Iteration,” Pro- ceedings of the American Mathematical Society, Vol. 4, No. 3, 1953, pp. 506-510. doi:10.1090/S0002-9939-1953-0054846-3 [2] S. Ishikawa, “Fixed Points by a New Iteration Method,” Proceedings of the American Mathematical Society, Vol. 50. -9939-1974-0336469-5 44, 1974, pp. 147-1 doi:10.1090/S0002 [3] R. P. Agarwal, D. O’Regan and D. R. Sahu, “Iterative Construction of Fixed Points of Nearly Asymptotically Nonexpasive Mappings,” Journal of Nonlinear and Con- vex Analysis, Vol. 8, No. 1, 2007, pp. 61-79. [4] M. A. Noor, “New Approximation Schemes for General Variational Inequalities,” Journal of Mathematical Analysis and Applications, Vol. 251, No. 1, 2000, pp. 217-229. and n Copyright © 2012 SciRes. AJCM R. CHUGH ET AL. Copyright © 2012 SciRes. AJCM 357 .7042doi:10.1006/jmaa.2000 [5] W. Pheungrattana and S. Suantai, “On the Rate of Con- vergence of Mann, Ishikawa, Noor and SP Iterations for Continuous on an Arbitrary Interval,” Journal of Compu- tational and Applied Mathematics, Vol. 235, No. 9, 2011, pp. 3006-3914. doi:10.1016/j.cam.2010.12.022 [6] T. Zamfirescu, “Fixed Point Theorems in Metric Spaces,” Archiv der Mathematik, Vol. 23, No. 1, 1972, pp. 292- 298. doi:10.1007/BF01304884 [7] V. Berinde, “On the Convergence of the Ishikawa Itera- ass of Quasi Contrac 2004, pp. 119-126. f ematics and Mathematical Sciences, Vol. 451-459. tion in the Cltive Operators,” Acta Mathematica Universitatis Comenianae, Vol. 73, No. 1, [8] Z. Q. Xue, “Remarks on Equivalence among Picard, Mann and Ishikawa Iterations in Normed Spaces. Fixed Point Theory and Applications, Vol. 2007, 2007, Article ID: 61434. [9] B. E. Rhoades and S. M. Soltuz, “On the Equivalence o Mann and Ishikawa Iteration Methods,” International Journal of Math 2003, 2003, pp. doi:10.1155/S0161171203110198 [1 Mann and Ishikawa Iteration for Non-Lipschitzian Op- erators,” In Mathematical Sciences, Vol. 42, 2003, pp. 2645-2652. 0] B. E. Rhoades, and S. M. Soltuz, “The Equivalence of the ternational Journal of Mathematics and doi:10.1155/S0161171203211418 [11] B. E. Rhoades and S. M. Soltuz, “The Equivalence be- tween the Convergences of Ishikawa and Mann Iterations for Asymptotically Pseudo-Contrave Map,” Journal of cti Mathematical Analysis and Applications, Vol. 283, 2003, pp. 681-688. doi:10.1016/S0022-247X(03)00338-X [12] B. E. Rhoades and S. M. Soltuz, “The Equivalence of Mann and Ishikawa Iteration for a Lipschitzian Psi-Uni- formly Pseudocontractive and Psi-Uniformly Accretive Maps, Tamkang Journal of Mathematics, Vol. 35, 2004, pp. 235-245. [13] B. E. Rhoades and S. M. Soltuz, “The Equivalence be- tween the Convergences of Ishikawa and Mann Iterations for Asymptotically Nonexpansive Maps in the Intermedi- ate Sense and Strongly Successively Pseudocontractive Maps,” Journal of Mathematical Analysis and Applica- tions, Vol. 289, No. 1, 2004, pp. 266-278. doi:10.1016/j.jmaa.2003.09.057 [14] B. E. Rhoades and S. M. Soltuz, “The Equivalence be- tween Mann-Ishikawa Iterations and Multistep Iteration,” Nonlinear Analysis, Vol. 58, No. 1-2, 2004, pp. 219-228. doi:10.1016/j.na.2003.11.013 S. M. Soltuz, “The Equivalence of Picard, Mann and [15] Iterations Deal erators,” Mathematical Communications, Vol. 10, 2005, tra pe Ishikawa ing with Quasi-Contractive Op- pp. 81-89. [16] S. M. Soltuz, “The Equivalence between Krasnoselskij, Mann, Ishikawa, Noor and Multistep Iterations,” Mathe- matical Communications, Vol. 12, 2007, pp. 53-61. [17] R. Chugh and V. Kumar, “Strong Convergence of SP Iterative Scheme for Quasi-Conctive Orators,” In- ternational Journal of Computer Applications, Vol. 31, No. 5, 2011, pp. 21-27. [18] B. E. Rhoades, “Comments on Two Fixed Point Iteration Methods,” Journal of Mathematical Analysis and Appli- cations, Vol. 56, 1976, pp. 741-750. doi:10.1016/0022-247X(76)90038-X [19] S. L. Singh, “A New Approach in Numerical Praxis,” Progress of Mathematics (Varanasi), Vol. 32, No. 2, 1998, pp. 75-89. [2 rs,” Fixed Point Theory and Applications, Vol. 2, 0] V. Berinde, “Iterative Approximation of Fixed Points,” Springer-Verlag, Berlin, 2007. [21] V. Berinde, “Picard Iteration Converges Faster than Mann Iteration Iteration for a Class of Quasi-Contractive Op- erato 2004, pp. 97-105. doi:10.1155/S1687182004311058 [22] Y. Qing and B. E. Rhoades, “Comments on the Rate of Convergence between Mann and Ishikawa Iterations Ap- plied to Zamfirescu Operators,” Fixed Point Theory and rti 87 lications, Vol. 45, 2011, pp. 1-6. Applications, Vol. 2008, 2008, Acle ID: 3504. [23] L. B. Ciric, B. S. Lee and A. Rafiq, “Faster Noor Itera- tions,” Indian Journal of Mathematics, Vol. 52, No. 3, 2010, pp. 429-436. [24] N. Hussian, A. Rafiq, D. Bosko and L. Rade, “On Rate of Convergence of Various Iterative Schemes,” Fixed Point Theory and App |