Abstract
In this work, we propose the three-step Thakur iterative process associated with two mappings in the setting of Banach space. Using this Thakur iteration, we approximate a common fixed point for a pair of noncyclic, relatively nonexpansive mappings. And we support our main result with a numerical example. Also, we give a stronger version of our main result by using von Neumann sequences. Finally, we provide some corollaries on the convergence of common best proximity points in uniformly convex Banach space.
1. Introduction and Preliminaries
In recent years, the convergence of iterative processes for fixed points and common fixed points has become an attractive problem in the theory of nonlinear analysis. In the literature, there are many articles that provide different kinds of iterative processes and their convergence results. At this point, Picard and Mann’s iterative processes are well-known iterative procedures that often help to find fixed points of a mapping of the form , where is Banach space. Here, we recall the following:(i)Picard iteration: let . Then iteration is defined by(ii)Mann iteration: let . Then iteration is defined by
The Picard iteration is a basic tool to find fixed points, and it was an important starting point for the improvement of other iterative processes. At the same time, the Picard iteration fails to converge a fixed point for the class of nonexpansive mappings ( see [1]).
Later on, Ishikawa [2] iteration, a two-step iteration process helps to approximate fixed points of nonexpansive mappings. For a starting point , this iterative scheme is defined bywhere and are sequences in .
Agarwal et al. [3] introduced two-step iteration process in 2007 for an arbitrary , it is defined aswhere and are sequences in .
In 2000, Noor [4] introduced the following iteration scheme: starting with , we define iteratively bywhere and are sequences in .
In the sequel, the following iterative process is defined by Thakur et al. in [5]: for an arbitrarily chosen element , the sequence is generated bywhere , and are sequences in .
Using the Mann iteration process, Eldred et al. [6] proved the convergence result of the fixed point for noncyclic, relatively nonexpansive mappings in the uniformly convex Banach space. One can note that the relatively nonexpansive mappings need not be continuous. Also, Gabeleh et al. [7] proved strong and weak convergence of the Ishikawa iterative scheme for noncyclic relatively nonexpansive mappings in uniformly convex Banach spaces. Gabriela et al. [8] proved the convergence of Thakur iteration for Suzuki-type nonexpansive mappings. This class of mappings properly contains the class of nonexpansive mappings. Recently, Abdeljawad et al. [9] approximated fixed points and best proximity points for relatively nonexpansive mappings through the Agarwal iterative process.
On the other hand, while solving systems of equations of the form ( and are selfmappings), one needs to study common fixed points and their convergence theorems. So, the researchers are showing an interest in finding common fixed points for different kinds of mappings through the well-known iterative processes. In the literature, Rashwan [10] proved the convergence of Mann iteration to a common fixed point for a pair of mappings defined on Banach space. Later on, Ćirić et al. [11] proved the convergence of Ishikawa iteration to common fixed points of two self-mappings in complete convex metric space.
In the sequel, the researchers showed interest in approximating common fixed points for a pair of nonexpansive mappings in the setting of Banach space. For example, Maingé [12] approximated common fixed points for nonexpansive mappings in Hilbert space. Also, Song et al. [13] provided a strong convergence result of common fixed points for a family of nonexpansive mappings in the setting of reflexive Banach spaces. Later on, Gu et al. [14] proved the convergence of Ishikawa iterations associated with two mappings to the common fixed point in uniformly convex Banach space. Also, Gopi et al. [15] found a common fixed point for a pair of relatively nonexpansive mappings and found a common best proximity point for a pair of non-self relatively nonexpansive mappings via the Ishikawa iterative process. And, Pragadeeswarar et al. [16] proved the convergence of a common best proximity point for a pair of mean nonexpansive mappings.
In the light of the above literature survey, one can think of how the three-step Thakur iterative process will approach the common fixed point for a pair of noncyclic, relatively nonexpansive mappings. So, we want to approximate such a common fixed point using Thakur’s iterative process.
The purpose of this paper is to present convergence results of the Thakur iterative process for common fixed points of a pair of noncyclic, relatively nonexpansive mappings in uniformly convex Banach space. Using the von Neumann sequence, we prove the strong convergence result of the Thakur iterative process. To support our main result, we provide a numerical example and we compare the Thakur iteration is how faster than other known iterative processes. Finally, we use projective operators to find the best common proximity point.
The following notations are used subsequently: let and be nonempty subsets of a Banach space .
If is convex, a closed subset of a reflexive and strictly convex space, then contains one element and if and are convex, closed subsets of a reflexive space, with either or is bounded, then .
First, we reconstruct the Thakur iteration associated with two noncyclic mappings , with is convex, as follows: for an arbitrary chosen element , the sequence is generated bywhere , and are sequences in satisfying the following condition: (R) .
Also, we provide the Abbas, Noor, Agarwal, and Ishikawa iterations associated with two noncyclic mappings , with is convex, as follows:(i)Abbas: let . Then the iteration is defined by(ii)Noor: let . Then the iteration is defined by(iii)Agarwal: let . Then the iteration is defined by(iv)Ishikawa: let . Then the iteration is defined by
The following definitions and theorems are very useful to our results:
Definition 1. Let and be nonempty subsets of a metric space . An element is said to be the best proximity points of the nonself mapping if it satisfies the condition that
Definition 2 (see [6]). Let and be nonempty subsets of a Banach space . A mapping is relatively nonexpansive, if
Definition 3 (see [17]). Let be a Banach space. For every , we define the modulus of convexity of bywhere is the unit ball of Banach space .
The norm is called uniformly convex if for all . The space is then called uniformly convex space.
Definition 4 (see [18]). Let be a Banach space. The pair of mappings is said to be mean nonexpansive iffor all and .
Remark 1. In Definition 4, for , then the pair of mappings is said to be nonexpansive.
Lemma 1 (see [19]). Suppose be a uniformly convex Banach space. Suppose , and is a sequence in . Suppose are sequences in such that for all . We define in by . If , then .
Lemma 2 (see [5]). Suppose is a uniformly convex Banach space and for all . Let and be two sequences of such that , and hold for some . Then .
Proposition 1 (see [20]). If is a uniformly convex space and and , then for any , if are such that , then there exists , such that .
Here, we prove a result that shows Thakur iteration converges to the common fixed point of a pair of nonexpansive self-mappings. This result helps to prove our main theorem.
Theorem 1. Let be a nonempty bounded closed convex subset of a uniformly convex Banach space and suppose is a pair of nonexpansive mappings with a nonempty common fixed point set. For an arbitrary chosen , let the sequence be generated by (7) where , and . Then and . Moreover, if lies in a compact set, then converges to a common fixed point of and .
Proof. By assumption, there exists such that . Now, from (7), we haveIn the same way, we can obtainNow, using inequality equation (17), one getsTherefore, by equations (17) and (19), we obtainThis implies that the sequence is nonincreasing and bounded below by 0. Hence there exists , such that .
Case 1. Suppose . FirstAs , we get .From the Thakur iteration, we obtainNow, by equation (17), we have.As , we obtain .Also, by equations (17) and (19), we obtainAs , we obtain . So, we get .
Since is contained in a compact set, has a subsequence that converges to a point . Also and converge to . This implies that converges to . From the Thakur iteration, we can deduce that , implies . Then . And also . Since and are continuous, it implies that . Therefore , which completes the proof.
Case 2. If . Suppose there exists a subsequence of and an such that for all .
Since the modulus of convexity of of is a continuous and increasing function, we choose as small that , where .
Now we choose , such that . Now we haveNow, by Proposition 1, we can obtainAlso, using equation (26), we getTherefore, the equation (25) becomesSince there exists such that ,Suppose that we choose very small , we have , which is a contradiction. This implies that .
Now we prove that . For, we define , and . Now, using equation (17), we get.also, by (19), we obtainTherefore and also . From Thakur’s iteration, we obtain . Dividing by , we getThen . Now we prove that . Now,By Lemma 1, . Therefore .
Since is contained in a compact set, has a subsequence that converges to a point . Also converges to . Now, . As , we obtain . Since is continuous, . So, we have . As , we get . Therefore, . Since is a common fixed point, implies exists. Therefore, . So, , which completes the proof.
Let be a convex closed subset of a Hilbert Space . Then for , we know that is the nearest point to and unique point of . And also is nonexpansive and distinguished by Kolmogorov’s criterion as , for all and .
Let and be two convex closed subsets of . We define.Then and . When and are closed, the convergence of these sequences in norm was proved by von Neumann [21]. The sequences and are called von Neumann sequences or alternating projection algorithms for two sets.
Definition 5 (see [22]). Let and be nonempty closed convex subsets of a Hilbert space . We say that is boundedly regular if for each bounded subset of and for each there exists such that.where is the displacement vector from to . ( is the unique vector satisfying ).
Theorem 2 (see [22]). If is boundedly regular, then the von Neumann sequences converge in the norm.
Theorem 3 (see [22]). If or is boundedly compact, then is boundedly regular.
Lemma 3 (see [19]). Let be a nonempty closed and convex subset and be a nonempty closed subset of a uniformly convex Banach space. Let and be sequences in and be a sequence in satisfying the following:(1),(2).Then converges to zero.
Corollary 1 (see [19]). Let be a nonempty closed convex subset and be a nonempty closed subset of uniformly convex Banach space. Let be a sequence in and such that . Then converges to .
Proposition 2 (see [23]). Let and be two closed and convex subsets of a Hilbert space . Then , and for and .
Lemma 4. Let and be two closed and convex subsets of a Hilbert space . For each .
Lemma 5 (see [24]). Let be a nonempty, bounded, closed, and convex pair in a reflexive and strictly convex Banach space . We define asThen the following statements hold:(1) for any and .(2) is an isometry, that is, for all .(3) is affine.
Definition 6 (see [25]). If then the pair is said to have -property if for any and
Lemma 6 (see [26]). Every, nonempty, bounded, closed and convex pair in a uniformly convex Banach space has the -property.
Lemma 7 (see [27]). Let be a nonempty, closed, and convex pair in a uniformly convex Banach space . Then for the projection mapping defined in equation (17) we have both and are continuous.
2. Main Results
Theorem 4. Let and be nonempty bounded closed convex subsets of a uniformly convex Banach space and suppose satisfy(1) and ;(2) for ; and(3) for ,with a nonempty common fixed-point set. For an arbitrary chosen , let the sequence be generated by (7) where , where and Suppose , then and . Moreover, if lies in a compact set, then and converge to a common fixed point of and .
Proof. If , then and by Theorem 1 we can prove the result from the truth that is nonexpansive. Therefore, let us take that . For a common fixed point of and , we getIn the same way, we can obtain.Now, using inequality equation (39), one getsTherefore, by equations (39) and (41), we obtainThis implies that the sequence is nonincreasing. Then we can find such that .
Suppose there exists a subsequence of and an such that for all . Since the modulus of convexity of of is continuous and increasing function, we choose as small that , where . Now we choose , such that . Now we haveNow, by Proposition 1, we can obtainAlso, using equation (44), we getTherefore, the equation (43) becomesSince there exists such that ,Suppose that we choose very small , we have , which is a contradiction. This implies that . From the Thakur iteration, we have , which implies .
Now we prove that . For, we define , and . Now, using equation (17), we getalso, by equation (19), we obtainTherefore and also . From Thakur’s iteration, we obtain . Dividing by , we getThen . Now we prove that . Now,By Lemma 1, . Therefore .
Since , and from equations (39) and (41), we can obtainandAlso, we have .
Taking lim sup on both sides, we obtainNowas , by , we getSo, from equations (54) and (56), we obtain . On the other hand, we haveand this yields thatSo, by equations (53) and (58), we deduceUsing Lemma 2, we get , as . From the Thakur iteration, we have , which implies .
Since is contained in a compact set, has a subsequence that converges to a point . Also and converge to . Since , there exists such that . Therefore, , which gives that . Let and choose such that . So, we have and . So . By strict convexity of the norm, . From , follows . Then .
On the other hand, , and . So . By strict convexity of the norm, . From , follows . Then . Therefore, . Let . Then we haveTherefore, . By Lemma 6, we get . In particular, . In the same way, we can prove that . So and . Since , we can obtain exists. Therefore, This implies . Also, .
Corollary 2. Let and be nonempty bounded closed convex subsets of a uniformly convex Banach space and suppose satisfy(1) and ;(2)for ; and(3) for ,with a nonempty common fixed-point set. For an arbitrary chosen , let the sequence be generated by (7) where , where and , then , and . Moreover, if lies in a compact set, then , and converge to a common fixed point of and .
Corollary 3. Let and be nonempty bounded closed convex subsets of a uniformly convex Banach space and suppose satisfy(1) and ;(2) for ; and(3) for ,with a nonempty common fixed-point set. Let , and define where , where and then , and . Moreover, if lies in a compact set, then , and converge to a common fixed point of and .
Proof. One can note that . By Theorem 4, the result follows.
We illustrate the above theorem through the following example.
Example 1. Let with . Let and , then . And we define a pair of mappings by and . For , we haveFor , we haveClearly, the set is common fixed points of and Fix . Let , then the Thakur iteration becomesUsing MATLAB coding, we give the following Table 1 to show that the iteration , and , converge to a common fixed point of for an initial point .
In the same way, for the above example, the iterations (9), (10), (11), and (12) become(i)Abbas: for an initial point ,(ii)Agarwal: for an initial point ,(iii)Noor: for an initial point ,(iv)Ishikawa: for an initial point ,Using MATLAB coding, we give the following Table 2, which compares Thakur iteration with Abbas, Agarwal, Noor, and Ishikawa iterations.
Using MATLAB coding, we give the following Figure 1, which compares convergence of Thakur iteration with Abbas, Agarwal, Noor, and Ishikawa iterations by the plot.
Now we omit the assumptions on constants , and in the above theorem and we provide the following theorem by using the condition (R) on constants , and .
Lemma 8 (see [7]). A Banach space is uniformly convex if and only if for each fixed number , there exists a continuous strictly increasing function if and only if , such thatfor all and all such that and .
Lemma 9 (see [7]). We consider a strictly increasing function with . If a sequence in satisfies , then .
Lemma 10 (see [7]). Let be a nonempty and closed pair in a uniformly convex Banach space such that is convex. Let and be sequences in and be a sequence in such that and , then we have .
Theorem 5. Let and be nonempty bounded closed convex subsets of a uniformly convex Banach space and suppose satisfy(1) and ;(2) for ; and(3) for ,with a nonempty common fixed-point set. For an arbitrary chosen , let the sequence be generated by (7) where satisfy (R) and . Then and . Moreover, if lies in a compact set, then , and converge to a common fixed point of and .
Proof. Let be a common fixed point of and . Then from Lemma 8, there exists continuous strictly increasing function such thatTherefore, we can deduce the following inequality:Now, we proceed with the following:
Suppose , and satisfy (R). From equation (73), we getAs , we get . In view of the fact that , implies , so .
As in Theorem 4, we can prove , and . Now, since lies in a compact subset then has a convergent subsequence , converging to some point . Also, we have .
Now . So . From , follows . Then, .
On the other hand, . So . From , follows . Then . Therefore, .
Let . Then we haveTherefore . By Lemma 6, we get . In particular . In the same way, we can prove that .
Since and , we get that exists. Sowhich gives .
In the next result, we provide a stronger version to approximate the common fixed point via von Neumann sequences.
Theorem 6. Let and be nonempty bounded closed convex subsets of a Hilbert space and suppose satisfy(1) and ;(2) for ; and(3) for ,with a nonempty common fixed-point set. Let , and define where , where and , then . Moreover, if lies in a compact set and then converges to a common fixed point of .
Proof. If , then and is a pair of nonexpansive with , the usual Thakur iteration. So, let us take that . Let be a common fixed point of and . Now, by equations (39) and (41), we obtainThis implies that the sequence is nonincreasing. Then we can find such that .
Suppose there exists a subsequence of and an such that for all .
Since the modulus of convexity of of is continuous and increasing function we choose as small that , where .
Now we choose , such that . Now we haveNowAlso, using equation (79), we getTherefore, the equation (78) becomesSince there exists such that ,Suppose that we choose very small , we have , which is a contradiction. This implies that . Now we prove that . From the Thakur iteration, we get . Since we obtain .
Since is contained in a compact set, has a subsequence that converges to a point . Also converges to . From the given sequence, we obtainSince implies . Then . Therefore, , which implies that . Also, we have as .
Now, , which implies that.
. Hence, .
Similarly, , which implies that.
. Hence, .
Also, .
So .
And also .
So .
Now . Thus .
For any , we have and .
Similarly, . Thus .
For any , we have and . By Theorem 2, for each the sequence converges to some . Now,So .
Therefore and similarly, we get .
In the same way, we prove that and .
Now we define by .
Since , then we conclude that is continuous. Therefore is continuous and converges pointwise to zero. Since , by Lemma 4, we obtain . Therefore converges uniformly on the compact set.Therefore,Since , we get , which gives that . Therefore and , which completes the proof.
Suppose is a Hilbert space and let and be nonempty bounded closed convex subsets of and suppose satisfy(1) and ;(2) for ; and(3) for .We consider and . From Proposition 2, for and and for and , by Theorem 4 and Theorem 6, we give the following results on the convergence of best proximity points.
Corollary 4. Let and be nonempty bounded closed convex subsets of a Hilbert space and suppose satisfy(1) and ;(2) for ; and(3) for .If is mapped into a compact subset of , then for any the sequence is defined by , where , converges to in such that .
Corollary 5. Let and be nonempty bounded closed convex subsets of a Hilbert space and suppose satisfy(1) and ;(2) for ; and(3) for .If is mapped into a compact subset of , then for any the sequence defined by , where converges to in such that , provided .
Corollary 6. Let and be nonempty bounded closed convex subsets of a Hilbert space and suppose satisfy(1) and ;(2) for ; and(3)for .If is mapped into a compact subset of , then for any the sequence defined by , where converges to in such that .
Proof.. The result follows from Corollary 4.
Corollary 7. Let and be nonempty bounded closed convex subsets of a Hilbert space and suppose satisfy(1) and ;(2) for ; and(3) for .Let , and define , where , where and . If is mapped into a compact subset of and , then converges to in such that .
Proof. The result follows from Theorem 6.
3. Conclusions
The fixed-point theorems provide sufficient conditions to ensure the existence of fixed points in different domains. Briefly, the fixed-point theorem possesses the solution of equations of the form , where is self-mapping. On the other hand, researchers want to find numerically such a fixed point by using different types of iterative processes for selfcontractive type operators in metric spaces, Hilbert spaces, or several classes of Banach spaces. One of the most famous iterative schemes is Picard’s iterative process. Many research papers were presented for approaching the fixed point through Picard’s iterative process. Later, for fast convergence, many iterative processes were found to approximate fixed points numerically. In this article, we consider the Thakur iterative process for fast convergence of common fixed points for relatively nonexpansive mappings in uniformly convex Banach spaces. Also, we approximate the common fixed point via the von Neumann iterative process in Hilbert space settings. We provide an example to illustrate our main result. As a consequence of our main results, we find common best proximity points for cyclic relatively nonexpansive mappings in Hilbert space.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest with this study.
Acknowledgments
This work has been partially funded by the Basque Government through Grant IT1207-19.