Open Access Library Journal
Vol.06 No.03(2019), Article ID:91276,14 pages
10.4236/oalib.1105245
Fixed Point Results for K-Iteration Using Non-Linear Type Mappings
Anju Panwar1, Ravi Parkash Bhokal2*
1Department of Mathematics, M. D. U. Rohtak, Haryana, India
2Government College, Dujana, Jhajjar (Haryana), India
Copyright © 2019 by author(s) and Open Access Library Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Received: February 13, 2019; Accepted: March 17, 2019; Published: March 20, 2019
ABSTRACT
In this paper we establish convergence and stability results using general contractive condition, quasi-nonexpansive mapping and mean non expansive mapping for K-iteration process. We shall also generalize the K-iteration process for a pair of distinct mappings and with the help of example we claim that the generalized iteration process has better convergence rate than the K-iteration process for single mapping and some of the existing iteration processes. Suitable examples are given in the support of main results.
Subject Areas:
Mathematical Analysis
Keywords:
K-Iteration Process, Opial’s Condition, Mean Non-Expansive Mapping, Quasi Non-Expansive Mapping
1. Introduction and Preliminary Definitions
Let be a metric space and be a self map defined on X. Let denote the set of fixed point of T. For , the sequence defined by
(1.1)
is called the Picard iteration.
For , the sequence defined by
*Corrosponding author.
(1.2)
where is a sequence in such that is called the Mann iteration process [1] .
In 2013, Khan [2] produced a new type of iteration process by introducing the concept of the following Picard-Mann hybrid iterative process for a single mapping T. For the initial value , the sequence defined by
,
(1.3)
where is a sequence in .
Khan [2] showed that the rate of convergence of Picard-Mann hybrid iterative process is more than the Picard iteration scheme, Mann iteration scheme [1] and Ishikawa iterative schemes [3] .
In this direction Gursoy and Karakaya [4] , gave new iteration process as follows:
For the initial value , the sequence defined by
(1.4)
where , is a sequence in is known as Picard-S iterative process. By giving appropriate example, Gursoy and Karakaya [4] proved that their iterative process has better convergence rate than Picard, Mann, Ishikawa, Noor and Normal-S iterative processes.
Karakaya et al. in their paper [5] , introduced a new hybrid iterative process as
(1.5)
where , is a sequence in .
With the help of suitable example it was claimed by Karakaya et al. [5] , that their iteration process converges faster than the iteration process of Gursoy and Karakaya [4] .
In 2016, Thakur et al. [6] introduced a new iteration scheme called Thakur New Iteration Scheme as for the initial value , the sequence defined by
(1.6)
where , is a sequence in .
In [6] it was claimed that the Thakur New Iteration Scheme has higher convergence rate than the iteration process of Karakaya et al. [7] .
In the recent work of Hussain et al. [8] , a new iteration scheme has been developed and it is claimed that it has better convergence rate than the iterative process Thakur et al. [6] . This iteration process is called K-iteration process and is given as:
For the initial value , the sequence defined by
(1.7)
where , is a sequence in .
In the present work we shall generalize some convergence and stability results for K-iteration process. We shall also prove convergence and stability results for more general form of K-iteration process and K-iteration process for a pair of two distinct mappings.
Definition 1.1 [3] : Let X be a real Banach space. The mapping is said to be asymptotically quasi-nonexpansive if and there exists a sequence with as such that
(1.8)
for all and .
Definition 1.2 [9] : Let X be a real Banach space. The mapping is said to be mean non-expansive if there exists two non negative real numbers such that and for all ,
Definition 1.3 [10] : Let be any sequence in X. Then the iterative process which converges to a fixed point q, is said to be stable with respect to the mapping T if for , we have if and only if .
Definition 1.4 [7] : A space X is said to satisfy Opial’s condition if for each sequence in X such that converges weakly to x we have for all , following holds:
1) ,
2) .
Lemma 1.5 [11] : Let and be non-negative real sequences satisfying the inequality:
,
where , for all , and as , then .
Lemma 1.6 [12] : Let be a real number such that , and be a sequence of positive numbers such that . Then for any sequence of positive numbers satisfying , we have .
Lemma 1.7 [13] : Let X be a real Banach space and be any sequence in X such that for all . Let and be non-negative real sequences satisfying , and holds for some . Then .
2. Main Results
Theorem 2.1: Let X be a Banach space and be a mapping satisfying the condition
(2.1)
where and . Let be the sequence defined by the K-iterative process given by (1.7). Then the sequence converges strongly to .
Proof: From (1.7) and (2.1) we have,
(2.2)
And
(2.3)
Again using (1.7) and (2.1) we get,
(2.4)
Using (2.4) in (2.3) we get,
(2.5)
Using (2.5) in (2.2) we get,
Since and . Hence by using lemma (1.6), we have
Hence the sequence converges strongly to q.
Corollary 2.2: (Akewe and Okeke [14] ) Let X be a Banach space and be a mapping satisfying the condition
where and . Let be the sequence defined by the Picard-Mann hybrid iterative process given by (1.3). Then the sequence converges strongly to q.
Remark 2.3: Theorem 2.1 gives generalization to many results in the literature by considering a wider class of contractive type operators and more general iterative process, including the results of Chidume [15] , Bosede and Rhoades [16] and Akewe and Okeke [14] .
Theorem 2.4: Let X be a Banach space and be a mapping satisfying the condition
where and . Let be the sequence defined by the K-iterative process given by (1.7). Then the iteration process (1.7) is T-stable.
Proof: By theorem 2.1, the sequence converges strongly to q. Let , and be real sequences in X.
Let , where
,
,
,
and let .
We shall prove that .
Now,
(2.6)
(2.7)
Again using (1.7) and (2.1) we get,
(2.8)
Using (2.8) in (2.7) we get,
(2.9)
Using (2.9) in (2.6) we get,
(2.10)
Since and since we have by lemma (1.6)
Conversely let . We shall show that .
Now
(2.11)
Substituting (2.9) in (2.11),
(2.12)
Since , we have from (2.12) . Hence the K-iteration scheme is T-stable.
From theorem 2.4, we have the following corollary.
Corollary 2.5: Let X be a Banach space and be a mapping satisfying the condition
,
where and . Let be the sequence defined by the Picard-Mann hybrid iterative process given by (1.3). Then the iteration process (1.3) is T-stable.
Example 2.6: Let and consider the mapping . The clearly the mapping T satisfies the inequality (2.1). Now . Now we claim that the K-iteration scheme (1.7) is T-stable. Let us take and consider the sequences . Then clearly .
Now
(2.13)
Taking limit in (2.13), we have . Hence the K-iteration process is T-stable.
Now we shall prove the convergence and stability results for asymptotically quasi-nonexpansive mapping by considering the more general form of K-iteration process as:
,
,
, where , (2.14)
Theorem 2.7: Let H be a non-empty closed convex subset of a Banach space X and be asymptotically quasi-nonexpansive mapping with real sequence . Let be the sequence defined by the K-iterative process given by (2.14) and satisfies the assumption that . Then the sequence converges strongly to some fixed point q of the mapping T.
Proof: From the iterative process (2.14) we have,
(2.15)
and
(2.16)
Again using (2.14) we have,
(2.17)
By repeating the above process, we have the following inequalities
So we can write,
Since for all . Now , so we can write,
(2.18)
Taking limit in (2.18), we have , that is the sequence converges strongly to fixed point q of the mapping T.
Theorem 2.8: Let H be a non-empty closed convex subset of a Banach space X and be asymptotically quasi-nonexpansive mapping with real sequence . Let be the sequence defined by the K-iterative process given by (2.14) and satisfies the assumption that . Then the iterative process (2.14) is T-stable.
Proof: Let be any arbitrary sequence. Let the sequence generated by the iterative process (2.14) is converging to the fixed point q.
Let
We shall prove that if and only if .
First suppose . Now we have
(2.19)
where , and .
Now using (2.19) together with lemma (1.5), we have that is .
Conversely let . we have
Taking limit both sides of (6) we have . Hence (2.14) is T-stable.
Now we shall prove the convergence results for mean non-expansive mapping by modifying the K-iteration process for two mappings as:
,
,
, where , (2.20)
Lemma 2.9: Let H be a non-empty closed convex subset of a Banach space X and be two mean non-expansive mapping such that . Let be the sequence defined by the K-iterative process given by (2.20). Then exists for some .
Proof: We have
(2.21)
Again using (2.20) and (2.21)
(2.22)
Again using (2.20) and (2.22)
(2.23)
This shows that is non-increasing and bounded sequence for . Hence exists.
Lemma 2.10: Let
Proof: Let . In lemma (2.9) we have proved the existence of
. Let . (2.24)
W.L.O.G. let .
Now from (2.20) and (2.24) we have,
(2.25)
Now
Implies that (2.26)
Now
and hence
which implies that (2.27)
Taking limit inferior in (2.27) we obtain
(2.28)
From (2.20) and (2.28) we have
(2.29)
Now from (2.24), (2.26), (2.29) and lemma (1.7), we have .
Now,
(2.30)
Using the conditions of the lemma in (2.30), we can write
(2.31)
Using (2.24), (2.30), (2.31) along with the lemma (1.7), we have
Theorem 2.11: Let H be a non-empty closed convex subset of a Banach space X satisfying Opial’s condition and S, T and be same as defined in the lemma (2.10) .Then the sequence converges weakly to some .
Proof: From lemma (2.10) we have, .
Since X is uniformly convex and hence it is reflexive so there exists a subsequence of such that converges weakly to some . Since H is closed so . Now we claim the weak convergence of to . Let it is not true, then there exists a subsequence of of which converges weakly to and let . Also . Now from lemma (2.9) and both exist. Using Opial’s condition we have,
This is a contradiction, so we must have . Thus the sequence converges weakly to some .
Theorem 2.12: Let H be a non-empty closed compact subset of a Banach space X and S, T and be same as defined in the lemma (2.10). Then the sequence converges strongly to some .
Proof: Since H is compact and hence it is sequentially compact. So there exists a subsequence of which converges to .
Now
(2.32)
Taking limit in (2.32) we have, that is . We have earlier proved that exists for . Hence the sequence converges strongly to some .
In [8] it is proves that the K-iteration process converges faster than Picard-S, Thakur-New and Vatan two-step iterative process. Now we shall compare the rate of convergence the K-iteration process defined in [8] and our new modified K-iteration process for two mappings.
Table 1. Iterative values of K-iteration process and Modified K-iteration process.
Example 2.13: Let be two mappings defined by and . Let be the sequences defined by . Let the initial approximation be . Clearly S, T has
unique common fixed point 2. The convergence pattern of K-iteration process and modified K-iteration process is shown in Table 1.
Clearly we can conclude from Table 1, that the modified K-iteration process has better rate of convergence than the k-iteration process.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this paper.
Cite this paper
Panwar, A. and Bhokal, R.P. (2019) Fixed Point Results for K-Iteration Using Non-Linear Type Mappings. Open Access Library Journal, 6: e5245. https://doi.org/10.4236/oalib.1105245
References
- 1. Mann, W.R. (1953) Mean Value Methods in Iteration. Proceedings of the American Mathematical Society, 4, 506-510.
https://doi.org/10.1090/S0002-9939-1953-0054846-3 - 2. Khan, S.H. (2013) A Picard-Mann Hybrid Iterative Process. Fixed Point Theory and Applications, 2013, 69. https://doi.org/10.1186/1687-1812-2013-69
- 3. Ishikawa, S. (1974) Fixed Points by a New Iteration Method. Proceedings of the American Mathematical Society, 44, 147-150.
https://doi.org/10.1090/S0002-9939-1974-0336469-5 - 4. Gursoy, F. and Karakaya, V. (2014) A Picard-S Hybrid Type Iteration Method for Solving a Differential Equation with Retared Arguments. 1-16.
- 5. Karakaya, V., Bouzara, N.E.H., Dogan, K. and Atalan, Y. (2015) On Different Results for a New Two Step Iteration Method under Weak Contraction Mapping in Banach Spaces. 1-10. arXiv:1507.00200v1
- 6. Thakur, B.S., Thakur, D. and Postolache, M. (2016) A New Iterative Scheme for Numerical Reckoning Fixed Points of Suzuki’s Generalized Non-Expansive Mappings. Applied Mathematics and Computation, 275, 147-155.
https://doi.org/10.1016/j.amc.2015.11.065 - 7. Opial, Z. (1967) Weak Convergence of the Sequence of Successive Approximations for Non-Expansive Mappings. Bulletin of the American Mathematical Society, 73, 595-597. https://doi.org/10.1090/S0002-9904-1967-11761-0
- 8. Hussain, N., Ullah, K. and Arshad, M. (2018) Fixed Point Approximation of Suzuki Generalized Non-Expansive Mapping via New Faster Iterative Process. arxiv 1802.09888v
- 9. Zhang, S.S. (1975) About Fixed Point Theory for Mean Non Expansive Mappings in Banach Spaces. Journal of Sichuan University, 2, 67-78.
- 10. Harder, A.M. (1987) Fixed Point Theory and Stability Results for Fixed Point Iteration Procedure. University of Missouri-Rolla, Missouri.
- 11. Weng, X. (1991) Fixed Point Iteration for Local Strictly Pseudo-Contractive Mapping. Proceedings of the American Mathematical Society, 113, 727-731.
https://doi.org/10.1090/S0002-9939-1991-1086345-8 - 12. Berinde, V. (2002) On the Stability of Some Fixed Point Procedures. Buletinul ?tiin?ific al Univer-sitatii Baia Mare, Seria B, Fascicola matematic?-informatic?, 18, 7-14.
- 13. Sahu, J. (1991) Weak and Strong Convergence to Fixed Points of Asymptotically Non-Expansive Mappings. Bulletin of the Australian Mathematical Society, 43, 153-159. https://doi.org/10.1017/S0004972700028884
- 14. Akewe, H. and Okeke, G.A. (2015) Convergence and Stability Theorems for the Picard-Mann Hybrid Iterative Scheme for a General Class of Contractive-Like Operators. Fixed Point Theory and Applications, 2015, 66.
- 15. Chidume, C.E. (2014) Strong Convergence and Stability of Picard Iteration Sequence for General Class of Contractive-Type Mappings. Fixed Point Theory and Applications, 2014, 233. https://doi.org/10.1186/1687-1812-2014-233
- 16. Bosede, A.O. and Rhoades, B.E. (2010) Stability of Picard and Mann Iteration for a General Class of Functions. Journal of Advanced Mathematical Studies, 3, 23.