eu ON THE RATE OF CONVERGENCE AND DATA DEPENDENCE OF JUNGCK MULTISTEP ITERATIVE SCHEMES

Abstract: In this paper we prove data dependence of Jungck-Multistep iterative schemes using quasi contractive operators, that is, by using approximate quasi -contractive operators we estimate the point of coincidence of the given operators. Also following Biazar and Amriteimoori [5] we modify JungckMultistep iterative schemes and with the help of numerical example compare their rate of convergence. Our results generalize some of the result in literature of fixed point theory.

Let X be a Banach space, Y an arbitrary set and S, T : Y → X such that T (Y ) ⊆ S(Y ).For x 0 ∈ Y , the iterative scheme: was introduced by Jungck [21] in 1976 and it becomes the Picard iterative scheme when S = I d (identity mapping) and Y = X.Singh et al. [38] defined the Jungck-Mann iterative scheme as where {α n } is a sequences of positive numbers in [0, 1].In [28], Olatinwo defined the Jungck-Ishikawa and Jungck-Noor [24] iterative schemes as and respectively, where {α n }, {β n } and {γ n } are sequences of positive numbers in [0, 1].Recently, Chugh and Kumar defined the Jungck-SP [11] iterative schemes as Olaleru and Akewe [29] defined the Jungck multistep Noor iteration as Recently Hudson Akewe [2] defined the Jungck multi step SP iteration as Bizare and Amriteimoori [7] improved the picard iteration under following conditions: (i) Initial approximation is chosen in the interval [a, b], where function is defined.
(ii) Function has continuous derivative on (a, b).
).Let {x n } converges to α.If there exists an integer constant q and a real +ve constant C such that lim q is called order and C is called constant of convergence.
. ., q −1 and f q (x) = 0 then sequence x n is of order q.
To improve the order of convergence of fixed iterative schemes, such that f ′ (α), f ′′ (α), . . ., f k (α) = 0. We determine λ i (i = 1, 2, . . ., k) from the following equation (α) = 0 yields to a system of linear equations which after solving [7] converted into upper triangular matrix which have nonzero diagonal entries.It means determinant is nonzero.So we determine λ i (i = 1, 2, . . ., k) uniquely.Now the new Picard iteration becomes where Bhagwati Parsad and Ritu Shani [27] modify Ishikawa and Jungck-Ishikawa iteration as: and Then new modified Jungck-SP, Jungck-Noor, Jungck-Multistep iterative schemes are New Jungck modified multistep Noor iteration scheme New modified Jungck modified multistep SP iteration scheme Where {α n } and {β i n } k i=1 are real sequences in [0, 1].The iterative scheme (1.1) was used by Jungck [21] to prove some of the common fixed point's results using the following Jungck-contraction Olatinwo [28] used the following more general contractive definitions than (1.12) to prove the stability and strong convergence results for the Jungck-Ishikawa iteration process: (a) There exists a real number a ∈ [0, 1) and a monotone increasing function φ : R + → R + such that φ(0) = 0 and ∀ x, y ∈ Y , we have (b) There exists real numbers M ≥ 0, a ∈ [0, 1) and a monotone increasing function φ : R + → R + such that φ(0) = 0 and ∀ x, y ∈ Y , we have Olatinwo [25] used the convergence of Jungck-Noor iterative scheme (1.4) to approximate the coincidence points of some pairs of generalized contractiveoperators satisfying ∀ x, y ∈ Y , where δ is a real number ∈ [0, 1).
In this paper we prove data dependence results for more generalized Jungck multistep SP and Noor iterative schemes using quasi contractive operators satisfying (1.13) and show the comparison between rate of convergence of Jungck multistep SP, Noor and modified Jungck multistep SP, Noor with the help of a numerical example.
To prove data dependence result we will use following definition and results.

Definition 1.3 ([4]
).Let T 1 , T 2 be two operators.We say T 2 is approximate operator of T 1 if for all x ∈ Xand for a fixed ǫ > 0, we have T 1 x − T 2 x ≤ ǫ.Lemma 1.4 ([37]).Let {α n } ∞ n=0 be a nonnegative sequence for which there exists n 0 ∈ I, such that for all n ≥ n 0 it satisfies the following inequality: ).Let (X, ) be an arbitrary Banach space and S, T : Y → X are nonself operators on an arbitrary set Y such that T (Y ) ⊆ S(Y ), where S(Y ) is complete subspace of X and S is injective operator.Let z be coincidence point of S and T , i.e., Sz = T z = p (say).Suppose S and T satisfy condition (1.13).For y 0 ∈ Y , let {Sx n } ∞ n=0 be Jungck multistep SP iterative scheme defined by (1.7), where {α n } and Then the Jungck multistep SP iterative scheme {Sx n } converges strongly to p.

Theorem 1.6 ([29]). Let (X,
) be an arbitrary Banach space and S, T : Y → Xare nonself operators on arbitrary set Y such that T (Y ) ⊆ S(Y ), where S(Y ) is complete subspace of X and S is injective operator.Let z be coincidence point of S and T , i.e., Sz = T z = p (say).Suppose S and T satisfy condition (1.13).For y 0 ∈ Y , let {Sx n } ∞ n=0 be Jungck multistep Noor iterative scheme defined by (1.6), where {α n } and Noor iterative scheme {Sx n } converges strongly to p.
Theorem 2.2.Let T 1 , S 1 : Y → E be mappings satisfying (1.13).Let T 2 , S 2 be approximate operators of T 1 , S 1 , respectively, as in Definition 1. 4 Let p = T 1 q 1 = S 1 p 1 and q = S 2 q 2 = T 2 p 2 , then we have the following estimate: Proof.For a given x 0 ∈ E and u 0 ∈ E we consider the following iterative schemes for T 1 and T and then using (1.13), (2.13) and (2.14), yield the following estimates: and Combining (2.15), (2.16) and (2.17), we have Thus inductively, we get Now by combining (2.19) and (2.20) Using (1.13), (2.13) and (2.14) which further implies where Since Hence using (2.23) and (2.24) in (2.22), we get Let us denote Now from Theorem 1.5, we have Since ϕ is continuous, we have Hence using Lemma 1, (2.25) yields Remark 2.2.Since the iteration (1.5) is special case of iterative scheme (1.7), so.Theorem 1.6 generalizes existing result for (1.5).By taking k = 3 and using Remark 1.1 in Theorem 2.2, data dependence results for the iterative schemes (1.5) and for SP iterative scheme can be obtained easily.
To find the fixed point we write p 1 (x) and p 2 (x) as for Jungck modified multi step SP and Noor iteration as both p 1 (x) and p 2 (x) has unique root in the interval (0, 1) so we convert this in the fixed point form and take α = 0.5 and α = 0.35 respectively.Now we solve it by For respective value of α, λ 1 , λ 2 , λ 3 . . .λ k can be determined uniquely from system of linear equations as in [3] for α = .5 and T 2 (x) = e (1−x) 2 − 1 we have

Observations on the Basis of Graphs and Tables
1. Simple multi step SP iteration for p 1 (x) converge faster than modified multi step SP iteration and speed of convergence increases as value of a and b increases between [0, 1] the result holds same for p 2 (x) but in this case simple multi step SP iteration does not converges for the value a = 0.9 and b = 0.9 but modified multi step SP iteration converges for all value of a and b.

Conclusion
On the analysis of table and graph of multi step and modified multi step SP, Noor iterations for p 1 (x) but p 2 (x) we conclude that modified multi step Noor  a = 0.9, b = 0.9 a = 0.9, b = 0.9 a = 0.9, b = 0.9 a = 0.9, b = 0.9  Where initial approximation is x 0 = 0.6, a = 0.9 and b = 0.9.

Figure 5 :
Figure 5: Graphical observations of Jungck multi step SP and Noor for p 1 (x).Where initial approximation is x 0 = 0.6, a = 0.9 and b = 0.9.Here MSP 0 , MN 0 show the graph for Table1.5.The merging point with value 3.46761 is common fixed point for p 1 (x).While Jungck multi step SP and Noor do not converge for p 2 (x) and {S 1 x n } ∞ n=0 , {S 2 u n } ∞ n=0 be two Jungck multi step SP iterative schemes defined by (1.7) associated to T 1 , S 1 and T 2 , S 2 , respectively, where {α n } ∞ n=0 , {β n } ∞ n=0 and {γ n } ∞ n=0 are real sequences in [0, 1) satisfying 1 , MN 1 show the graph for Table 1.3.The merging point with value 3.46761 and 1.41239 is common fixed point for p 1 (x) and p 2 (x)

Table 5 :
1 , MMN 1 show the graph for Table 1.4.The merging point with value 0.339981 and 0.41239 is common fixed point for p 1 (x) and p 2 (x) Jungck multi step SP and Noor for p 1 (x)

Table 6 :
Jungck modified multi step SP and Noor for p 1 (x) and p 2 (x)