Strong Convergence of a New Three Step Iterative Scheme in Banach Spaces

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 iterative 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.


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 : T X X      F T p  X Tp p 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 approximated over the years by various authors.
In a complete metric space, the Picard iterative scheme defined by has been employed to approximate the fixed points of mappings satisfying the inequality for all , x y X  and 2) is called the Banach's contraction condition.Any operator satisfying (1.2) is called a strict contraction.
In 1953, W. R. Mann defined the Mann iterative scheme [1] as where   n  is a sequence of positive numbers in [0,1].
In 1974, S. Ishikawa defined the Ishikawa iterative scheme [2] as In 2000, M. A. Noor defined the Noor iterative scheme [4] as   Recently, Phuengrattana and Suantai defined the SP iteration scheme [5] as where  
3) In addition, when , then the SP iterative scheme (1.7) reduces to the Mann iterative scheme (1.3).
Theorem 1.1.Let (X, d) be a complete metric space and a mapping for which there exists real : T X X  numbers a, b and c satisfying   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 , x y X   and some .

 
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 operators 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 {s n } be defined through the Ishikawa iteration (1.4) and 0


. Then {s n } converges strongly to the fixed point of T.
Fixed point iterative procedures are designed to be applied 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, commonly 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 Ishikawa iterations through examples.S. L. Singh [19] extended the work of Rhoades.Very recently, Phuengrattana 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 : T X X  and 0 x X  .Define the sequence by  The purpose of this paper is to show the convergence of the CR iterative scheme and to prove equivalence between 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 mentioned 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.

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 Tz Tu Tu u Again, from (1.10), we have Substituting (3.4) in (3.1), we obtain where 1 (using and this implies that n x p  ove that as Conversely, we pr , we have (3.10) Substituting (3.9) and (3.10) in (3.8) and rearranging the terms, we have as n   .Hence, using Lemma 1.

Results on Fastness of CR Iterative Scheme for Qu s
In [21] Berinde showed that Picard it is faster than Mann iteration for quasi-contractive operators satisfying g and Rhoades by taking example owed that Ishikawa iteration is faster than Mann iteration 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 iterative schemes.
2) By using definition (1.1), we show that CR iterative scheme is faster than Picard iteration.
1) Example 4.1.Let T: 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    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 16 n  and p 0 = x 0 .Then, from [23], for Noor iteration (1.6), we have .
Also, for CR iteration (1 i .10),we have It is easy to see that  .Hence, we have Therefore, by definition 1.2, CR iterative scheme converges 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 .
It is easy to see that 16 16 1 8 Hence, we have 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 where In order to compare CR and Picard iterations, we must compare the coefficients of the inequalities (4.4) and (4.3). Obviously

Applications
In this section, with the help of computer programs in compare the rate of convergence of Picard, Mann , for all

Example of Decreasin
Let f: [0,1]→[0,1] be defined by Then f is a decreasing function.By taking m = 7, the comparison of convergence of above mentio tion e defined as ned iterative schemes to the exact fixed point p = 0.188348 is listed in the Table 1.

Example of Increasing Func
to find fixed point of the function is an increasing function.The comparison of convergence of above mentioned iterative schemes to the exact fixed point p = 1 is listed in the Table 2.

Example of Cubic Equation
To find solution of cubic 0 means − 1 = 0 can be rewritten as    is listed in the Table 3.

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.
x  a n d w e are looking for the solution x, with 1 2 x   produce a sequence that .If we rearrange the equation, we can will converge to the solution: 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 recorded (discussed in the next section).

Decreasing Function (1 − x) m
1) For m = 8 and x o = 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 iterations, 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 x o = 0.8, the Picard scheme never ) For x o = 0.8, the Picard scheme conve es to a fixed nt in 8 iterations, the Mann scheme converges in 69 iterations, the Ishikawa scheme converges in 34 iterations, the Noor scheme converges in 24 iterations, Agarwal et al. scheme converges in 7 iterations, the SP scheme converges in 6 iterations and the CR scheme converges in 5 iterations.
2) Taking initial guess x o = 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, , x o = 0.8, we obtain that the Mann sc Ishikawa scheme con heme converges in 23 iterations, the verges in 12 iterations, Noor scheal.scheme converges in 6 iterations, t verges in 9 iterations, the Ishika me converges in 9 iterations, Agarwal et he SP scheme converges in 5 iterations and the CR scheme converges in 4 iterations.

Cubic Equation
1) For x o = 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- 2) Taking initial guess x o = 0.1 (away from the solution 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.


and   n  are sequences of positive numbers in [0,1].n T u (1.3)

1 Definition 1 . 1 .
the following lemma and definitions.Lemma 1 [20].If  is a real number such that 0 Suppose that {a n } and {b n } are two real convergent sequences with limits a and b respectively.Then {a n } is said to converge faster than {b n } if lim 0

Theorem 2 . 1 .
Let K be a nonempty closed conve of an arbitrary Banach space X and :

2 ,
CR iterative scheme converge er than Agarwal et al. iterative scheme to the fixed point 0 of T.
, Ishikawa, Noor, Agarwal et al., SP and CR iteration procedures, through examples.The ou ome is listed in the form of Tables 1-4, by taking initial approation x 0 = 0.8 d as well as iric et al.'s results [23], w heme is faster than other it comparison of convergence of above mentioned iterative schemes to the exact fixed point p = 0.754878 of
Mann scheme converges in 29 iterations, the kawa scheme converges in 18 iterations, Noor scheonverges 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. sche es do ot converge 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 away from the solution, the number of iterations increases in each iterative scheme.3) If we increase the value of n  and n  , the solution is obtained in less number of iterations for Noor and Ishikawa schemes while solution is obtained in more s value f number of iterations for Mann, SP and CR chemes.Agarwal et al. iteration converges for increased o If we i crease 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 iterative schem s is M nn, Ishikawa, N or, Ag wal et al. and CR sche e whil R an P sch es c n  and n  .If we increase the value of n n  In this case, increasing order of rate of conv equivalence.