Approximation of Fixed Points and Best Proximity Points of Relatively Nonexpansive Mappings

Abdeljawad, Thabet; Ullah, Kifayat; Ahmad, Junaid; De La Sen, Manuel; Ulhaq, Azhar

doi:https://doi.org/10.1155/2020/8821553

Journal of Mathematics

On this page

Abstract Introduction Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2020 | Article ID 8821553 | https://doi.org/10.1155/2020/8821553

Approximation of Fixed Points and Best Proximity Points of Relatively Nonexpansive Mappings

Thabet Abdeljawad,^1,2,3Kifayat Ullah,⁴Junaid Ahmad,⁴Manuel De La Sen,⁵and Azhar Ulhaq⁴

Academic Editor: Hijaz Ahmad

Received31 Aug 2020

Revised09 Sept 2020

Accepted16 Sept 2020

Published29 Oct 2020

Abstract

In this article, we study the Agarwal iterative process for finding fixed points and best proximity points of relatively nonexpansive mappings. Using the Von Neumann sequence, we establish the convergence result in a Hilbert space framework. We present a new example of relatively nonexpansive mapping and prove that its Agarwal iterative process is more efficient than the Mann and Ishikawa iterative processes.

1. Introduction

Let be a nonempty subset of a Banach space . A self-map of is said to be nonexpansive mapping if

The class of nonexpansive mappings is important as an application point of view. One of the celebrated result of Kirk [1] states that any self nonexpansive mapping of closed bounded convex subset of a reflexive Banach space has a fixed point provided that has normal structure. This result was also independently proved in the same year by Browder [2] and Gohde [3] in uniformly convex Banach space (in short UCBS). After this celebrated result, many generalizations of nonexpansive mappings have been published [4–14]. Among the other things, one of the natural generalization of nonexpansive mappings was given by Eldred et al. [15] as follows. Let and be two nonempty subsets of a Banach space . A self-map of is said to be relatively nonexpansive if

Iterative methods played a very important role in variational inequalities and many other areas of applied sciences (e.g., see [16–27] and others). One of the earlier iterative scheme is the Picard iteration process, , which converges very well for Banach contraction mappings. However, this scheme is not suitable for finding fixed points of nonexpansive mappings and hence for the generalized nonexpansive mappings. Let E be a nonempty subset set of a Banach space X. In [30], Eldred and Praveen studied Mann [29] iterative process for finding fixed points and best proximity points of relatively nonexpansive mappings. In [30], Gopi and Pragadeeswara studied Ishikawa [31] iterative process for finding fixed points and best proximity points of relatively nonexpansive mappings.

Motivated by the above work, we study the Agarwal [32] iterative process for finding fixed points and best proximity points of relatively nonexpansive mappings. We present a new example of relatively nonexansive mapping and prove that its Agarwal iterative process is more efficient than the Mann [29] and Ishikawa [31] iterative processes.

Now, we present some notations which will be used in the sequel:

Notice that is singleton, provided that is closed convex in a reflexive and strictly convex space. Moreover, if and are a closed convex in a reflexive space, such that one of the and is bounded, then .

A handful of definitions and theorems given below correspond to our results.

Definition 1. Suppose that and be two nonempty subsets of a metric space. A point is said to be a best proximity point of the nonself-map provided that

Theorem 1 (see [15]). Suppose and be two nonempty bounded closed convex subsets of a UCBS. Assume that satisfies(i)(ii)Then, there exist .

Theorem 2 (see [15]). Suppose and be two nonempty closed bounded convex subsets of a UCBS. If satisfies the following:(i)(ii)Then, there exist .

Theorem 3 (see [1–3]). Assume that be a self-map on a closed convex bounded subset of a UCBS. If is nonexpansive, then has a fixed point.

Proposition 1 (see [33]). Suppose that is a UCBS, and , then for each and be such that , then there exists some such that .

Lemma 1 (see [34]). Let for every . Assume that and are sequence in a UCBS such that . Define in X by . If , then .

Now, we are going to show that, under some appropriate assumptions, Agarwal’s [32] iteration converges to a fixed point of a given nonexpansive mapping. This result is useful for the upcoming main results.

Theorem 4. Suppose be a nonempty bounded closed convex subset of a UCBS , and assume that be a self-map nonexpansive map of . Choose and set , where and and . Then, . Moreover, if lies in a compact set, then converges to a fixed point of .

Proof. By Theorem 3, there exists some such that . Now,It follows that the sequence is nonincreasing and bounded below by 0. Hence, we have .

Case 1. If , thenIf , then . Now,If , then .

Case 2. If , we need to show that . Suppose not, then one have a subsequence of and a positive real number such that for all .
Since the modulus of convexity of is continuous as well as increasing function, one can select some as small such that , where .
Now, we select , such that . By Proposition 1, we haveSince there exist , such that ,Select very small , we have which is contradiction. This implies that the .
Now, we prove that We have .
Now, we define , , and . One can note that . Now,Therefore, . From Agarwal’s iteration, we obtain . Dividing by , we obtainHence, . Now, we show that . Now,By Lemma 1, . This implies that . Therefore, .
Since is contained in a compact set, has a subsequence that converges to point . Also, and converge to b. This implies that converge to b. Then, . In particular, . Since is continuous, implies that . Therefore, .

Theorem 5 (see [28]). Let and be nonempty closed bounded convex subset of a UCBS. Let satisfy(1)(2)

Let and define , where and . Moreover, if lies in a compact set, converges to a fixed point of T.

Assume that be a convex closed subset of a Hilbert space . Then, for , is the nearest to u and element of . Furthermore, is nonexpansive and distinguished by Kolmogorove’s criterion:

Assume that and are two convex closed subsets of . Set

Then, the sequences and . When and are closed, the convergence of these sequences were established by Von Neumann in [35]. The sequences and are called Von Neumann sequences (sometimes called alternating projection algorithm for two sets).

Definition 2. (see [36]). Suppose and are two nonempty convex closed subsets of a Hilbert space . Then, is called boundedly regular provided that, for every bounded subset of and for every one can select a such thatwhere is the displacement vector from the set to set ( is the unique vector such that).

Theorem 6 (see [36]). Suppose is boundedly regular; then, the Von Neumann sequence converges in norm.

Theorem 7 (see [36]). Assume that one of the and is boundedly compact; then, is boundedly regular.

Lemma 2 (see [37]). Suppose that be a nonempty convex closed subset and be a nonempty closed subset of a UCBS. Assume that and be sequences in and be a sequence in such that(i)(ii)Then, converges to 0.

Corollary 1 (see [37]). Suppose be a nonempty closed convex subset and be a nonempty closed subset of a UCBS. Assume that be a sequence in and such that . Then, converges to .

Proposition 2 (see [15]). Suppose and be two closed and convex subset of a Hilbert space . Then, , , and for each and .

Lemma 3. Suppose and be two closed and convex subset of a Hilbert space . Then, for each , we have

2. Main Results

Theorem 8. Suppose and be nonempty bounded closed convex subsets of a UCBS and assume that such that(i)(ii)Select and set , where and and . Suppose , . Moreover, if lies in a compact set, then converges to a fixed point of .

Proof. If , then , and by Theorem 4, we can establish the theorem from the fact that is nonexpansive. Let . By Theorem 2, there exists such that . Now,which implies thatHence, the sequence is nonincreasing. So, one can select a such that . Assume that there is a subsequence of and an such that for every .
However, the modulus of convexity of is continuous and increasing function, and we choose as small that , where .
Now, we choose k, such that . By Proposition 1, we haveSince there exist such that ,Suppose choosing very small , we have , which is a contradiction. This implies that .
Now, we prove that . We have . Now, we define , and . One can note that . Now,Therefore, . From Agarwal’s iteration, we obtain . Dividing by , we obtainThen, . Now, we prove that . Now,By Lemma 1, . This shows that . Therefore, .
Since is contained in a compact set, has a subsequence that converges to point . Also, and converge to .
Since , there exists , such that . Therefore, , which gives that .
Let and choose such that .
We have , and . So, . By strict convexity of the norm, . It follows that .

Corollary 2. Suppose and are two nonempty bounded closed convex subsets of a UCBS , and assume that is such that(1)(2)Choose and set , where and and . Then, . Moreover, if contained in a compact set, then converges to a fixed point of .

Corollary 3. Suppose and are two nonempty bounded closed convex subsets of a Hilbert space , and assume that be a relatively nonexpansive mapping such that(1)(2)Choose and set , where and and . Then, . Moreover, if is mapped into a compact subset of , then converges to a fixed point of .

Proof. One can note that , and by Theorem 8, the result follows.
Now, we present a new example of relatively nonexpansive mappings and prove that its Agarwal [32] iterative process is better than the Mann [29] and Ishikawa [31] iterative processes.

Example 1. Take ,DefineLet and . Then,From the above process, we get is relatively nonexpansive mapping. The iterative values for are shown below in Table 1 and Figure 1.

Remark 1. From Table 1 and Figure 1, we see that Agarwal iterates converges faster to than the Ishikawa and Mann iterates for the class of relatively nonexpansive mappings.
The stronger version for the approximation of fixed point by using Von Neumann sequences are follows.

Theorem 9. Let and be nonempty bounded closed convex subsets of a Hilbert space and suppose is such that(1)(2)Let and define , where with restriction and and . Then, . Moreover, if is lies in a compact set , then converges to a fixed point of .

Proof. If , then and is nonexpansive with , the usual Agarwal’s iteration. So, let us take that . By Theorem 2, there exist such that . Now,which implies thatHence, the e sequence is nonincreasing. So, one can choose some with . Assume that one can find a subsequence, namely, of and some positive such that for every . Since the modulus of convexity of of is continuous as well as increasing function, one may choose as small such that , where .
Now, we select , such that . By Proposition 1, we haveSince there exist such that :If we select a small , then which is clearly a contradiction. This implies that
Since the set is contained in a compact set, so the sequence has a subsequence such that it converges to a some point . Also, converge to . From the given sequence, one hasSince which implies that . Therefore, , which implies that . Also, we have as .
Now, , which gives that. Therefore, .
Also, . So,
Now, . Thus, .
For any n, and . By Theorem 6, for each , the sequence converges to some . Now,So, .
Therefore, , and similarly .
Now, we define by .
Since , then we conclude that is continuous. Therefore, is continuous and converges pointwise to zero. Since , by Lemma 3, we obtain . Therefore, converges uniformly on the compact set:Therefore,Since , we get , which gives that . Therefore, , which completes the proof.
Suppose is a Hilbert space and assume that be as in Theorem 1. Consider and .
From Proposition 2, for and , by Theorems 8 and 9, we give the following results on convergence of best proximity points.

Corollary 4. Suppose and are two nonempty bounded closed convex subsets of a Hilbert space . Assume that be as in Theorem 1. If is mapped into a compact subset of , then for every the sequence generated by by converges to in such that

Corollary 5. Suppose and are two nonempty bounded closed convex subsets of a Hilbert space . Assume that be as in Theorem 1. If is mapped into a compact subset of , then for any the sequence defined by converges to in such that provided

Corollary 6. Suppose and are two nonempty bounded closed convex subsets of a Hilbert space . Assume that be as in Theorem 1. If is mapped into a compact subset of , then for every the sequence generated by converges to in such that

Proof. The result follows by Corollary 4.

Corollary 7. Suppose and are two nonempty bounded closed convex subsets of a Hilbert space . Assume that be as in Theorem 1. Choose and set , where and and . If is mapped into a compact subset of and , then converges to in such that .

Proof. The result follows by Theorem 9.

3. Conclusions

In this article, we have used the Agarwal iterative process for finding fixed points and best proximity points of relatively nonexpansive mappings. Using the Von Neumann sequence, we have established the convergence result in a Hilbert space framework. We have offered a new example of relatively nonexpansive mapping and proved that its Agarwal iterative process is more efficient than the Mann and Ishikawa iterative processes.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

Acknowledgments

The authors are grateful to the Spanish Government for Grant RTI2018-094336-B-I00 (MCIU/AEI/FEDER, UE) and Basque Government for Grant IT1207-19.

References

W. A. Kirk, “A fixed point theorem for mappings which do not Increase Distances,” The American Mathematical Monthly, vol. 72, no. 9, pp. 1004–1006, 1965.
View at: Publisher Site | Google Scholar
F. E. Browder, “Nonexpansive nonlinear operators in a Banach space,” Proceedings of the National Academy of Sciences, vol. 54, no. 4, pp. 1041–1044, 1965.
View at: Publisher Site | Google Scholar
D. Göhde, “Zum Prinzip der Kontraktiven Abbildung,” Mathematische Nachrichten, vol. 30, no. 3-4, pp. 251–258, 1965.
View at: Publisher Site | Google Scholar
K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 35, no. 1, p. 171, 1972.
View at: Publisher Site | Google Scholar
K. Aoyama and F. Kohsaka, “Fixed point theorem for -nonexpansive mappings in Banach spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 13, pp. 4387–4391, 2011.
View at: Publisher Site | Google Scholar
J. S. Bae, “Fixed point theorems of generalized nonexpansive mappings,” Journal of the Korean Mathematical Society, vol. 21, no. 2, pp. 233–248, 1984.
View at: Google Scholar
J. Bogin, “A generalization of a fixed point theorem of Goebel, Kirk and Shimi,” Canadian Mathematical Bulletin, vol. 19, no. 1, pp. 7–12, 1976.
View at: Publisher Site | Google Scholar
K. Goebel, W. A. Kirk, and T. N. Shimi, “A fixed point theorem in uniformly convex spaces,” Bollettino Della Unione Matematica Italiana, vol. 7, pp. 67–75, 1973.
View at: Google Scholar
E. Llorens Fuster and E. Moreno Gálvez, “The Fixed Point Theory for some generalized nonexpansive mappings,” Abstract and Applied Analysis, vol. 2011, Article ID 435686, 15 pages, 2011.
View at: Publisher Site | Google Scholar
T. Suzuki, “Fixed point theorems and convergence theorems for some generalized nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 340, no. 2, pp. 1088–1095, 2008.
View at: Publisher Site | Google Scholar
R. Pant and R. Shukla, “Approximating fixed points of generalized α-nonexpansive mappings in Banach spaces,” Numerical Functional Analysis and Optimization, vol. 38, no. 2, pp. 248–266, 2017.
View at: Publisher Site | Google Scholar
J. García-Falset, E. Llorens-Fuster, and T. Suzuki, “Fixed point theory for a class of generalized nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 375, no. 1, pp. 185–195, 2011.
View at: Publisher Site | Google Scholar
B. Patir, N. Goswami, and V. N. Mishra, “Some results on fixed point theory for a class of generalized nonexpansive mappings,” Fixed Point Theory and Applications, vol. 19, 2018.
View at: Publisher Site | Google Scholar
E. Karapinar and K. Tas, “Generalized (C)-conditions and related fixed point theorems,” Computers and Mathematics with Applications, vol. 61, pp. 3370–3380, 2011.
View at: Google Scholar
A. A. Eldred, W. A. Kirk, and P. Veeramani, “Proximal normal structure and relatively nonexpansive mappings,” Studia Mathematica, vol. 171, no. 3, pp. 283–293, 2005.
View at: Publisher Site | Google Scholar
H. Ahmad, “Variational iteration method with an auxiliary parameter for solving differential equations of the fifth order,” Nonlinear Science Letters A, vol. 9, no. 1, pp. 27–35, 2018.
View at: Google Scholar
H. Ahmad, A. R. Seadawy, and T. A. Khan, “Numerical solution of Korteweg–de Vries-Burgers equation by the modified variational iteration algorithm-II arising in shallow water waves,” vol. 95, no. 4, 2020.
View at: Publisher Site | Google Scholar
H. Ahmad, T. A. Khan, P. S. Stanimirovic, and I. Ahmad, “Modified variational iteration technique for the numerical solution of fifth order KdV type equations,” Journal of Applied and Computational Mechanics, vol. 6, pp. 1220–1227, 2020.
View at: Google Scholar
M. Nadeem, F. Li, and H. Ahmad, “Modified Laplace variational iteration method for solving fourth-order parabolic partial differential equation with variable coefficients,” Computers & Mathematics with Applications, vol. 78, no. 6, pp. 2052–2062, 2019.
View at: Publisher Site | Google Scholar
H. Ahmad, T. A. Khan, and S.-W. Yao, “An efficient approach for the numerical solution of fifth-order KdV equations,” Open Mathematics, vol. 18, no. 1, pp. 738–748, 2020.
View at: Publisher Site | Google Scholar
K. Ullah, F. Ayaz, and J. Ahmad, “Some convergence results of M iterative process in Banach spaces,” Asian-European Journal of Mathematics, Article ID 2150017, 2019.
View at: Google Scholar
K. Ullah, J. Ahmad, and M. D. L. Sen, “On generalized nonexpansive maps in Banach spaces,” Computation, vol. 8, no. 3, p. 61, 2020.
View at: Publisher Site | Google Scholar
T. Abdeljawad, K. Ullah, J. Ahmad, and N. Mlaiki, “Iterative approximations for a class of generalized nonexpansive operators in Banach spaces,” Discrete Dynamics in Nature and Society, vol. 2020, Article ID 4627260, 6 pages, 2020.
View at: Publisher Site | Google Scholar
K. Ullah, M. S. U. Khan, N. Muhammad, and J. Ahmad, “Approximation of endpoints for multivalued nonexpansive mappings in geodesic spaces,” Asian-European Journal of Mathematics, vol. 13, no. 8, Article ID 2050141, 2020.
View at: Publisher Site | Google Scholar
K. Ullah, J. Ahmad, and N. Muhammad, “Approximation of endpoints for multi-valued mappings in metric spaces,” Journal of Linear and Toplogical Algebra, vol. 9, no. 2, pp. 129–137, 2020.
View at: Google Scholar
T. Abdeljawad, K. Ullah, J. Ahmad, and N. Mlaiki, “Iterative approximation of endpoints for multivalued mappings in Banach spaces,” Journal of Function Spaces, vol. 2020, Article ID 2179059, 5 pages, 2020.
View at: Publisher Site | Google Scholar
K. Ullah, J. Ahmad, M. Arshad, M. de la Sen, and M. S. U. Khan, “Approximating stationary points of multi-valued generalized nonexpansive mappings in metric spaces,” Advances in Mathematical Physics, vol. 2020, Article ID 9086078, 6 pages, 2020.
View at: Publisher Site | Google Scholar
A. A. Eldred and A. Praveen, “Convergence of Mann’s iteration for relatively nonexpansive mappings,” Fixed Point Theory, vol. 18, no. 2, pp. 1–9, 2017.
View at: Publisher Site | Google Scholar
W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, no. 3, pp. 506–510, 1953.
View at: Publisher Site | Google Scholar
R. Gopi and V. Pragadeeswarar, “Convergence of the Ishikawa iteration process for relatively non expansive mappings,” 2019, http://arxiv.org/abs/1910.05541.
View at: Google Scholar
S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, no. 1, p. 147, 1974.
View at: Publisher Site | Google Scholar
R. P. Agarwal, D. O’Regan, and D. R. Sahu, “Iterative construction of fixed points of nearly asymptotically nonexpansive mappings,” Search Results, vol. 8, no. 1, pp. 61–79, 2007.
View at: Google Scholar
C. Chidume, Some Geometric Properties of Banach Spaces, Springer, London, UK, 2009.
W. G. Dotson, “On the Mann iterative process,” Transactions of the American Mathematical Society, vol. 149, no. 1, p. 65, 1970.
View at: Publisher Site | Google Scholar
J. Von Neumann, Functional Operators: Measures and Integrals, Princeton University Press, Princeton, NJ, USA, 1950.
H. K. Bauschke and J. M. Borwien, “On the convergence of von Neumann’s alternating projection algorithm for two sets,” Set-Valued Anal., vol. 1, pp. 283–293, 1993.
View at: Publisher Site | Google Scholar
A. A. Eldred and P. Veeramani, “Existence and convergence of best proximity points,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1001–1006, 2006.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2020 Thabet Abdeljawad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

802

Downloads

655

Citations