# Thakur’s Iterative Scheme for Approximating Common Fixed Points to a Pair of Relatively Nonexpansive Mappings

Authors: R. Gopi; V. Pragadeeswarar; M. De La Sen
Journal: Journal of Mathematics
 (2022)

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## 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.


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## Body

## 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 formF:X⟶X, where X is Banach space. Here, we recall the following:(i)
Picard iteration: letw0∈X. Then iteration is defined by(1)wn+1=Fwn.(ii)
Mann iteration: letw0∈X. Then iteration is defined by(2)wn+1=1−ηnwn+ηnFwn,ηn∈0,1.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 w0∈X, this iterative scheme is defined by(3)wn+1=1−ηnwn+ηnFun,un=1−γnwn+γnFwn,where ηn and γn are sequences in 0,1.Agarwal et al. [3] introduced two-step iteration process in 2007 for an arbitrary w0∈X, it is defined as(4)wn+1=1−ηnFwn+ηnFun,un=1−γnwn+γnFwn,where ηn and γn are sequences in 0,1.In 2000, Noor [4] introduced the following iteration scheme: starting with w0∈X, we define wn iteratively by(5)wn+1=1−ηnwn+ηnFvn,vn=1−δnwn+δnFun,un=1−γnwn+γnFwn,where ηn,δn and γn are sequences in 0,1.In the sequel, the following iterative process is defined by Thakur et al. in [5]: for an arbitrarily chosen element w0∈X, the sequence wn is generated by(6)wn+1=1−ηnFun+ηnFvn,vn=1−δnun+δnFun,un=1−γnwn+γnFwn,where ηn,δn, and γn are sequences in 0,1.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 formFx=x,Gx=x (F and G 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: letM and N be nonempty subsets of a Banach space X.(7)dw,N=infw−z:z∈N,dM,N=infw−z:w∈M,z∈N,PMw=z∈M:w−z=dw,M,M0=w∈M:w−z′=dM,Nfor somez′∈N,N0=z∈N:w′−z=dM,Nfor somew′∈M.IfM is convex, a closed subset of a reflexive and strictly convex space, then PMw contains one element and if M and N are convex, closed subsets of a reflexive space, with either M or N is bounded, then M0≠∅.First, we reconstruct the Thakur iteration associated with two noncyclic mappingsF,G:M∪N⟶M∪N, with M is convex, as follows: for an arbitrary chosen element w0∈M, the sequence wn is generated by(8)wn+1=1−ηnGun+ηnFvn,vn=1−δnun+δnGun,un=1−γnwn+γnFwn,where ηn,δn, and γn are sequences in 0,1 satisfying the following condition: (R) 0<ε≤γn1−γn.Also, we provide the Abbas, Noor, Agarwal, and Ishikawa iterations associated with two noncyclic mappingsF,G:M∪N⟶M∪N, with M is convex, as follows:(i)
Abbas: letw0∈M. Then the iteration is defined by(9)wn+1=1−ηnFun+ηnGvn,vn=1−δnFwn+δnGun,un=1−γnwn+γnFwn,ηn,δn,γn∈0,1.(ii)
Noor: letw0∈M. Then the iteration is defined by(10)wn+1=1−ηnwn+ηnFvn,vn=1−δnwn+δnGun,un=1−γnwn+γnFwn,ηn,δn,γn∈0,1.(iii)
Agarwal: letw0∈M. Then the iteration is defined by(11)wn+1=1−ηnFwn+ηnGun,un=1−γnwn+γnFwn,ηn,γn∈0,1.(iv)
Ishikawa: letw0∈M. Then the iteration is defined by(12)wn+1=1−ηnwn+ηnGun,un=1−γnwn+γnFwn,ηn,γn∈0,1.The following definitions and theorems are very useful to our results:Definition 1.
LetM and N be nonempty subsets of a metric space X,d. An element w∈M is said to be the best proximity points of the nonself mapping F:M⟶N if it satisfies the condition that(13)dw,Fw=dM,N.Definition 2 (see [6]).
LetM and N be nonempty subsets of a Banach space X. A mapping F:M∪N⟶M∪N is relatively nonexpansive, if(14)Fw−Fz≤w−z,for allw∈M,z∈N.Definition 3 (see [17]).
LetX,⋅ be a Banach space. For every ε∈0,2, we define the modulus of convexity of ⋅ by(15)δXε=inf1−x+y2:x,y∈BX,x−y≥ε,where BX is the unit ball of Banach space X.
The norm is called uniformly convex ifδXε>0 for all ϵ∈0,2. The space X,⋅ is then called uniformly convex space.Definition 4 (see [18]).
LetX be a Banach space. The pair of mappings F,G:X⟶X is said to be mean nonexpansive if(16)Fu−Gv≤au−v+bu−Fu+v−Gv+cu−Gv+v−Fu,for all u,v∈X,a,b,c∈0,1 and a+2b+2c≤1.Remark 1.
In Definition4, for a=1,b=c=0, then the pair of mappings F,G:X⟶X is said to be nonexpansive.Lemma 1 (see [19]).
SupposeX be a uniformly convex Banach space. Suppose 0<a<b<1, and tn is a sequence in a,b. Suppose wn,vn are sequences in X such that wn≤1,vn≤1 for all n. We define zn in X by zn=1−tnwn+tnvn. If limn⟶∞zn=1, then limn⟶∞wn−vn=0.Lemma 2 (see [5]).
SupposeE is a uniformly convex Banach space and 0<p≤tn≤q<1 for all n∈ℕ. Let xn and yn be two sequences of E such that limsupn⟶∞xn≤r,limsupn⟶∞yn≤r, and limsupn⟶∞tnxn+1−tnyn=r hold for some r≥0. Then limn⟶∞xn−yn=0.Proposition 1 (see [20]).
IfX is a uniformly convex space and η∈0,1 and ε>0, then for any d>0, if w,z∈X are such that w≤d,z≤d,w−z≥ε, then there exists δ=δε/d>0, such that ηw+1−ηz≤1−2δε/dminη,1−ηd.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.
LetK be a nonempty bounded closed convex subset of a uniformly convex Banach space X and suppose G,F:K⟶K is a pair of nonexpansive mappings with a nonempty common fixed point set. For an arbitrary chosen w0∈K, let the sequence wn be generated by (7) where ηn,δn,γn∈ε,1−ε, and ε∈0,1/2. Then limn⟶∞wn−Fwn=0 and limn⟶∞Gun−Fvn=0. Moreover, if FK lies in a compact set, then wn converges to a common fixed point of G and F.Proof.
By assumption, there existsz∈K such that Gz=Fz=z. Now, from (7), we have(17)un−z=1−γnwn+γnFwn−z=1−γnwn−z+γnFwn−z≤1−γnwn−z+γnFwn−Gz≤1−γnwn−z+γnwn−z=wn−z.
In the same way, we can obtain(18)vn−z=1−δnun+δnGun−z=1−δnun−z+δnGun−z≤1−δnun−z+δnGun−Fz≤1−δnun−z+δnun−z=un−z.
Now, using inequality equation (17), one gets(19)vn−z≤wn−z.
Therefore, by equations (17) and (19), we obtain(20)wn+1−z=1−ηnGun+ηnFvn−z=1−ηnGun−z+ηnFvn−z≤1−ηnGun−Fz+ηnFvn−Gz≤1−ηnun−z+ηnvn−z≤1−ηnwn−z+ηnwn−z=wn−z.
This implies that the sequencewn−z is nonincreasing and bounded below by 0. Hence there exists d≥0, such that wn−z⟶d.Case 1.
Supposed=0. First(21)wn−Fwn≤wn−z+z−Fwn≤wn−z+Gz−Fwn≤wn−z+z−wn.As n⟶∞, we get wn−Fwn⟶0.From the Thakur iteration, we obtain(22)wn+1−wn=1−ηnGun+ηnFvn−wn=Gun−wn+ηnFvn−Gun≤Gun−wn+ηnFvn−Gun.
Now, by equation (17), we have.(23)Gun−wn≤Gun−z+z−wn=Gun−Fz+z−wn≤un−z+z−wn≤2wn−z.As n⟶∞, we obtain Gun−wn⟶0.Also, by equations (17) and (19), we obtain(24)Fvn−Gun≤vn−un=vn−z+z−un≤2wn−z.As n⟶∞, we obtain Fvn−Gun⟶0. So, we get wn+1−wn⟶0.
SinceFK is contained in a compact set, Fwn has a subsequence Fwnk that converges to a point z∈K. Also wnk and wnk+1 converge to z. This implies that wn converges to z. From the Thakur iteration, we can deduce that un−wn=γnFwn−wn, implies un−wn⟶0. Then un⟶z. And also Gun⟶z,Fwn⟶z. Since F and G are continuous, it implies that Gun⟶Gz,Fwn⟶Fz. Therefore Fz=Gz=z, which completes the proof.Case 2.
Ifwn−z⟶d>0. Suppose there exists a subsequence wnk of wn and an ϵ>0 such that wnk−Fwnk≥ε>0 for all k.
Since the modulus of convexity ofδ of X is a continuous and increasing function, we choose ξ>0 as small that 1−cδε/d+ξd+ξ<d, where c>0.
Now we choosek, such that wnk−z≤d+ξ. Now we have(25)z−wnk+1=z−1−ηnkGunk+ηnkFvnk=1−ηnkz+ηnkz−1−ηnkG1−γnkwnk+γnkFwnk+ηnkFvnk≤1−ηnkz−G1−γnkwnk+γnkFwnk+ηnkz−Fvnk=1−ηnkFz−G1−γnkwnk+γnkFwnk+ηnkGz−Fvnk≤1−ηnkz−1−γnkwnk+γnkFwnk+ηnkz−vnk.
Now, by Proposition1, we can obtain(26)z−1−γnkwnk+γnkFwnk=1−γnkz−wnk+γnkz−Fwnk≤1−2δεd+ξminγnk,1−γnkd+ξ.Also, using equation (26), we get(27)z−vnk=z−1−δnkunk+δnkGunk=1−δnkz−unk+δnkz−Gunk≤1−δnkz−unk+δnkz−Gunk=1−δnkz−unk+δnkFz−Gunk≤1−δnkz−unk+δnkz−unk=z−unk≤1−2δεd+ξminγnk,1−γnkd+ξ.
Therefore, the equation (25) becomes(28)z−wnk+1≤1−2δεd+ξminγnk,1−γnkd+ξ.
Since there existsl>0 such that 2minγnk,1−γnk≥l,(29)1−2δεd+ξminγnk,1−γnkd+ξ≤1−lδεd+ξd+ξ.
Suppose that we choose very smallξ>0, we have 1−lδε/d+ξd+ξ<d, which is a contradiction. This implies that limn⟶∞wn−Fwn=0.
Now we prove thatGun−Fvn⟶0. For, we define an=wn+1−z/wn−z, bn=Gun−z/wn−z and cn=Fvn−z/wn−z. Now, using equation (17), we get.(30)Gun−z=Gun−Fz≤un−z≤wn−z,also, by (19), we obtain(31)Fvn−z=Fvn−Gz≤vn−z≤wn−z.
Thereforebn=Gun−z/wn−z≤wn−z/wn−z=1 and also cn=Fvn−z/wn−z≤wn−z/wn−z=1. From Thakur’s iteration, we obtain wn+1−z=1−ηnGun−z+ηnFvn−z. Dividing by wn−z, we get(32)wn+1−zwn−z=1−ηnGun−zwn−z+ηnFvn−zwn−z.
Thenan=1−ηnbn+ηncn. Now we prove that an⟶1. Now,(33)limn⟶∞an=limn⟶∞wn+1−zwn−z=dd=1.
By Lemma1, bn−cn⟶0. Therefore Gun−Fvn⟶0.
SinceFK is contained in a compact set, Fwn has a subsequence Fwnk that converges to a point a∈K. Also wnk converges to a. Now, Fwnk−Ga≤wnk−a. As k⟶∞, we obtain a=Ga. Since G is continuous, Gwnk⟶Ga. So, we have Gwnk−Fa≤wnk−a. As k⟶∞, we get Fa=Ga. Therefore, Fa=Ga=a. Since a is a common fixed point, implies limn⟶∞wn−a exists. Therefore, limn⟶∞wn−a=limk⟶∞wnk−a=0. So, wn⟶a, which completes the proof.
LetM be a convex closed subset of a Hilbert Space X. Then for w∈X, we know that PMw is the nearest point to w and unique point of M. And also PM is nonexpansive and distinguished by Kolmogorov’s criterion as w−PMw,PMw−a≥0, for all w∈X and a∈M.
LetM and N be two convex closed subsets of X. We define.(34)Pw=PMPNwfor eachw∈X.
ThenPnw⊂M and PNPnw⊂N. When M and N are closed, the convergence of these sequences in norm was proved by von Neumann [21]. The sequences Pnw and PNPnw are called von Neumann sequences or alternating projection algorithms for two sets.Definition 5 (see [22]).
LetM and N be nonempty closed convex subsets of a Hilbert space X. We say that M,N is boundedly regular if for each bounded subset S of X and for each ε>0 there exists δ>0 such that.(35)maxdw,M,dw,N−v≤δdw,N≤ε,∀w∈X,where v=PN−M0 is the displacement vector from M to N. (v is the unique vector satisfying v=dM,N).Theorem 2 (see [22]).
IfM,N is boundedly regular, then the von Neumann sequences converge in the norm.Theorem 3 (see [22]).
IfM or N is boundedly compact, then M,N is boundedly regular.Lemma 3 (see [19]).
LetM be a nonempty closed and convex subset and N be a nonempty closed subset of a uniformly convex Banach space. Let wn and an be sequences in M and zn be a sequence in N satisfying the following:(1)
wn−zn⟶dM,N,(2)
an−zn⟶dM,N.
Thenwn−an converges to zero.Corollary 1 (see [19]).
LetM be a nonempty closed convex subset and N be a nonempty closed subset of uniformly convex Banach space. Let wn be a sequence in M and z0∈N such that wn−z0⟶dM,N. Then wn converges to PMz0.Proposition 2 (see [23]).
LetM and N be two closed and convex subsets of a Hilbert space X. Then PNM⊆N,PMN⊆M, and PNw−PMz≤w−z for w∈M and z∈N.Lemma 4.
LetM and N be two closed and convex subsets of a Hilbert space X. For each w∈X.(36)Pn+1w−a≤Pnw−a,for eacha∈M0∪N0.Lemma 5 (see [24]).
LetM,N be a nonempty, bounded, closed, and convex pair in a reflexive and strictly convex Banach space X. We define P:M0∪N0⟶M0∪N0 as(37)Px=PM0x,ifx∈N0,PN0x,ifx∈M0,
Then the following statements hold:(1
)x−Px=dM,N for any x∈M0∪N0 and PM0⊆N0,PN0⊆M0.(2
)P is an isometry, that is, Px−Py=x−y for all x,y∈M0×N0.(3
)P is affine.Definition 6 (see [25]).
IfM0≠∅ then the pair M,N is said to have P-property if for any u1,u2∈M0 and v1,v2∈N0(38)du1,v1=dM,Ndu2,v2=dM,Ndu1,u2=dv1,v2.Lemma 6 (see [26]).
Every, nonempty, bounded, closed and convex pair in a uniformly convex Banach spaceX has the P-property.Lemma 7 (see [27]).
LetM,N be a nonempty, closed, and convex pair in a uniformly convex Banach space X. Then for the projection mapping P:M0∪N0⟶M0∪N0 defined in equation (17) we have both P|M0 and P|N0 are continuous.


## 2. Main Results

Theorem 4.
LetM and N be nonempty bounded closed convex subsets of a uniformly convex Banach space and suppose F,G:M∪N⟶M∪N satisfy(1)
GM⊆M,GN⊆N,FM⊆M and FN⊆N;(2)
Fu−Gv≤u−v for u∈M,v∈N; and(3)
Fu−Gv≤u−v for u∈N,v∈M,
with a nonempty common fixed-point set. For an arbitrary chosenw0∈M, let the sequence wn be generated by (7) where ηn,δn,γn∈ε,1−ε, where ε∈0,1/2 and n=0,1,2,… Suppose dwn,M0⟶0, then limn⟶∞Gun−Fvn=0,limn⟶∞un−Gun=0 and limn⟶∞wn−Fwn=0. Moreover, if FM lies in a compact set, then wn,vn and un converge to a common fixed point of G and F.Proof.
IfdM,N=0, then M0=N0=M∩N and by Theorem 1 we can prove the result from the truth that F,G:M∩N⟶M∩N is nonexpansive. Therefore, let us take that dM,N>0. For a common fixed point z∈N of F and G, we get(39)un−z=1−γnwn+γnFwn−z=1−γnwn−z+γnFwn−z≤1−γnwn−z+γnFwn−z=1−γnwn−z+γnFwn−Gz≤1−γnwn−z+γnwn−z=wn−z.
In the same way, we can obtain.(40)vn−z=1−δnun+δnGun−z=1−δnun−z+δnGun−z≤1−δnun−z+δnGun−z=1−δnun−z+δnGun−Fz≤1−δnun−z+δnun−z=un−z.
Now, using inequality equation (39), one gets(41)vn−z≤wn−z.
Therefore, by equations (39) and (41), we obtain(42)wn+1−z=1−ηnGun+ηnFvn−z=1−ηnGun−z+ηnFvn−z≤1−ηnGun−Fz+ηnFvn−Gz≤1−ηnun−z+ηnvn−z≤1−ηnwn−z+ηnwn−z=wn−z.
This implies that the sequencewn−z is nonincreasing. Then we can find d>0 such that limn⟶∞wn−z=d.
Suppose there exists a subsequencewnk of wn and an ε>0 such that wnk−Fwnk≥ε>0 for all k. Since the modulus of convexity of δ of X is continuous and increasing function, we choose ξ>0 as small that 1−cδε/d+ξd+ξ<d, where c>0. Now we choose k, such that wnk−z≤d+ξ. Now we have(43)z−wnk+1=z−1−ηnkGunk+ηnkFvnk=1−ηnkz+ηnkz−1−ηnkG1−γnkwnk+γnkFwnk+ηnkFvnk≤1−ηnkz−G1−γnkwnk+γnkFwnk+ηnkz−Fvnk=1−ηnkFz−G1−γnkwnk+γnkFwnk+ηnkGz−Fvnk≤1−ηnkz−1−γnkwnk+γnkFwnk+ηnkz−vnk.
Now, by Proposition1, we can obtain(44)z−1−γnkwnk+γnkFwnk=1−γnkz−wnk+γnkz−Fwnk≤1−2δεd+ξminγnk,1−γnkd+ξ.Also, using equation (44), we get(45)z−vnk=z−1−δnkunk+δnkGunk=1−δnkz−unk+δnkz−Gunk≤1−δnkz−unk+δnkz−Gunk=1−δnkz−unk+δnkFz−Gunk≤1−δnkz−unk+δnkz−unk=z−unk≤1−2δεd+ξminγnk,1−γnkd+ξ.
Therefore, the equation (43) becomes(46)z−wnk+1≤1−2δεd+ξminγnk,1−γnkd+ξ.
Since there existsl>0 such that 2minγnk,1−γnk≥l,(47)1−2δεd+ξminγnk,1−γnkd+ξ≤1−lδεd+ξd+ξ.
Suppose that we choose very smallξ>0, we have 1−lδε/d+ξd+ξ<d, which is a contradiction. This implies that limn⟶∞wn−Fwn=0. From the Thakur iteration, we have un−wn=γnwn−Fwn, which implies un−wn⟶0.
Now we prove thatGun−Fvn⟶0. For, we define an=wn+1−z/wn−z, bn=Gun−z/wn−z and cn=Fvn−z/wn−z. Now, using equation (17), we get(48)Gun−z=Gun−Fz≤un−z≤wn−z,also, by equation (19), we obtain(49)Fvn−z=Fvn−Gz≤vn−z≤wn−z.
Thereforebn=Gun−z/wn−z≤wn−z/wn−z=1 and also cn=Fvn−z/wn−z≤wn−z/wn−z=1. From Thakur’s iteration, we obtain wn+1−z=1−ηnGun−z+ηnFvn−z. Dividing by wn−z, we get(50)wn+1−zwn−z=1−ηnGun−zwn−z+ηnFvn−zwn−z.
Thenan=1−ηnbn+ηncn. Now we prove that an⟶1. Now,(51)limn⟶∞an=limn⟶∞wn+1−zwn−z=dd=1.
By Lemma1, bn−cn⟶0. Therefore Gun−Fvn⟶0.
Sincelimn⟶∞wn−z=d, and from equations (39) and (41), we can obtain(52)limsupn⟶∞un−z≤d,and(53)limsupn⟶∞vn−z≤d.
Also, we haveGun−z=Gun−Fz≤un−z.
Taking lim sup on both sides, we obtain(54)limsupn⟶∞Gun−z≤d.
Now(55)wn+1−z=1−ηnGun+ηnFvn−z=Gun−z+ηnFvn−Gun,as n⟶∞, by Fvn−Gun⟶0, we get(56)d≤liminfn⟶∞Gun−z.
So, from equations (54) and (56), we obtain limn⟶∞Gun−z=d. On the other hand, we have(57)Gun−z≤Gun−Fvn+Fvn−z=Gun−Fvn+Fvn−Gz=Gun−Fvn+vn−z,and this yields that(58)d≤liminfn⟶∞vn−z.
So, by equations (53) and (58), we deduce(59)limn⟶∞vn−z=d.
Using Lemma2, we get un−Gun⟶0, as n⟶∞. From the Thakur iteration, we have vn−un=δnun−Gun, which implies vn−un⟶0.
SinceFM is contained in a compact set, Fwn has a subsequence Fwnk that converges to a point a∈M. Also wnk and unk converge to a. Since dwn,M0⟶0, there exists an⊆M0 such that wn−an⟶0. Therefore, ank⟶a, which gives that a∈M0. Let D=dM,N and choose b∈N0 such that a−b=D. So, we have wnk−b⟶a−b=D and wnk−b≥Fwnk−Gb⟶a−Gb. So a−Gb=D. By strict convexity of the norm, Gb=b. From Fa−Gb≤a−b, follows Fa−Gb=D. Then Fa=a.
On the other hand,unk−b⟶a−b=D, and unk−b≥Gunk−Fb⟶a−Fb. So a−Fb=D. By strict convexity of the norm, Fb=b. From Ga−Fb≤a−b, follows Ga−Fb=D. Then Ga=a. Therefore, Fa=Ga=a. Let x∈M0. Then we have(60)Fx−GPx≤x−Px=dM,N.
Therefore,Fx−GPx=dM,N=Fx−PFx. By Lemma 6, we get GPx=PFx. In particular, GPa=PFa. In the same way, we can prove that FPa=PGa. So FPa=Pa and GPa=Pa. Since Pa∈N0, we can obtain limn⟶∞wn−Pa exists. Therefore, (61)limn⟶∞wn−Pa=limk⟶∞wnk−Pa=a−Pa=dM,N.
This implieswn⟶a. Also, vn⟶a,un⟶a.Corollary 2.
LetM and N be nonempty bounded closed convex subsets of a uniformly convex Banach space and suppose F,G:M∪N⟶M∪N satisfy(1)
GM⊆M,GN⊆N,FM⊆M and FN⊆N;(2)
foru∈M,v∈N; and(3)
Fu−Gv≤u−v for u∈N,v∈M,
with a nonempty common fixed-point set. For an arbitrary chosenw0∈M0, let the sequence wn be generated by (7) where ηn,δn,γn∈ε,1−ε, where ε∈0,1/2 and n=0,1,2,…, then limn⟶∞Gun−Fvn=0,limn⟶∞un−Gun=0, and limn⟶∞wn−Fwn=0. Moreover, if FM lies in a compact set, then wn,vn, and un converge to a common fixed point of G and F.Corollary 3.
LetM and N be nonempty bounded closed convex subsets of a uniformly convex Banach space and suppose F,G:M∪N⟶M∪N satisfy(1)
GM⊆M,GN⊆N,FM⊆M and FN⊆N;(2)
Fu−Gv≤u−v for u∈M,v∈N; and(3)
Fu−Gv≤u−v for u∈N,v∈M,
with a nonempty common fixed-point set. Letw0∈M0, and define wn+1=Pn1−ηnGun+ηnFvn where vn=1−δnun+δnGun,un=1−γnwn+γnFwn,ηn,δn,γn∈ε,1−ε, where ε∈0,1/2 and n=0,1,2,… then limn⟶∞Gun−Fvn=0,limn⟶∞un−Gun=0, and limn⟶∞wn−Fwn=0. Moreover, if FM lies in a compact set, then wn,vn, and un converge to a common fixed point of G and F.Proof.
One can note thatPn1−ηnGun+ηnFvn=1−ηnGun+ηnFvn. By Theorem 4, the result follows.
We illustrate the above theorem through the following example.Example 1.
Letℝ2,⋅ with u1,u2−v1,v2=u1−v12+u2−v22. Let M=0,u∈ℝ2:u∈0,1 and N=1,u∈ℝ2:u∈2,3, then dM,N=2. And we define a pair of mappings F,G:M∪N⟶M∪N by F0,u=0,1,F1,u=1,u and G0,u=0,u,G1,u=1,2. For 0,u∈M,1,v∈N, we have(62)G0,u−F1,v=0,u−1,v.
For0,u∈M,1,v∈N, we have(63)F0,u−G1,v=0,1−1,2=2≤0,u−1,v.
Clearly, the set0,1,1,2 is common fixed points of F and G Fix ηn=3/4,δn=3/4,γn=3/4∀n. Let 0,w0∈M, then the Thakur iteration becomes(64)0,un=1−340,wn+34F0,wn=140,wn+340,1=0,wn4+0,34=0,wn+340,vn=1−340,un+34G0,un=140,wn+34+34G0,wn+34=140,wn+34+340,wn+34=0,wn+340,wn+1=1−34G0,un+34F0,vn=14G0,wn+34+340,1=140,wn+34+0,34=0,wn+316+0,34=0,wn+1516.
Using MATLAB coding, we give the following Table1 to show that the iteration 0,wn+1,0,un, and 0,vn, converge to a common fixed point of F,G for an initial point 0,w0=0,0.1∈M.
In the same way, for the above example, the iterations (9), (10), (11), and (12) become(i)
Abbas: for an initial point0,w0∈M,(65)0,wn+1=0,9wn+5564,(ii)
Agarwal: for an initial point0,w0∈M,(66)0,wn+1=0,3wn+1316,(iii)
Noor: for an initial point0,w0∈M,(67)0,wn+1=0,wn+34,(iv)
Ishikawa: for an initial point0,w0∈M,(68)0,wn+1=0,7wn+916.
Using MATLAB coding, we give the following Table2, which compares Thakur iteration with Abbas, Agarwal, Noor, and Ishikawa iterations.
Using MATLAB coding, we give the following Figure1, which compares convergence of Thakur iteration with Abbas, Agarwal, Noor, and Ishikawa iterations by the plot.
Now we omit the assumptions on constantsηn,δn,γn, and dwn,M0⟶0 in the above theorem and we provide the following theorem by using the condition (R) on constants ηn,δn, and γn.Table 1
Thakur iteration
n0,vn=0,vn0,wn+109(0,0.999996566772461)(0,0.999999999986903)10(0,0.999999141693115)(0,0.999999999999181)11(0,0.999999785423279)(0,0.999999999999949)12(0,0.999999946355820)(0,0.999999999999997)13(0, 0.999999986588955)(0,1.000000000000000)⋮⋮⋮24(0,0.999999999999997)25(0,0.999999999999999)26(0,1.000000000000000)Table 2
Comparative results.
nIshikawaNoorAgarwalAbbasThakur10(0,0.999768781231887)(0,0.999999141693115)(0,0.999999951665723)(0,0.999999997282962)(0,0.999999999999181)11(0,0.999898841788951)(0,0.999999785423279)(0,0.999999990937323)(0,0.999999999617984)(0,0.999999999999949)12(0, 0.999955743282666)(0,0.999999946355820)(0,0.999999998300748)(0,0.999999999946289)(0,0.999999999999997)13(0,0.999980637686166)(0,0.999999986588955)(0,0.999999999681390)(0,0.999999999992448)(0,1.000000000000000)14(0,0.999991528987698)(0,0.999999996647239)(0,0.999999999940261)(0,0.999999999998938)(0,1.000000000000000)15(0, 0.999996293932118)(0,0.999999999161810)(0,0.999999999988799)(0,0.999999999999851)(0,1.000000000000000)16(0,0.999998378595301)(0,0.999999999790452)(0,0.999999999997900)(0,0.999999999999979)(0,1.000000000000000)17(0, 0.999999290635444)(0,0.999999999947613)(0,0.999999999999606)(0,0.999999999999997)(0,1.000000000000000)18(0,0.999999689653007)(0,0.999999999986903)(0,0.999999999999926)(0,1.000000000000000)(0,1.000000000000000)19(0,0.999999864223191)(0,0.999999999996726)(0,0.999999999999986)(0,1.000000000000000)(0,1.000000000000000)20(0,0.999999940597646)(0,0.999999999999181)(0,0.999999999999997)(0,1.000000000000000)(0,1.000000000000000)21(0, 0.999999974011470)(0,0.999999999999795)(0,1.000000000000000)(0,1.000000000000000)(0,1.000000000000000)22(0,0.999999988630018)(0,0.999999999999949)(0,1.000000000000000)(0,1.000000000000000)(0,1.000000000000000)23(0,0.999999995025633)(0,0.999999999999987)(0,1.000000000000000)(0,1.000000000000000)(0,1.000000000000000)24(0,0.999999997823714)(0,0.999999999999997)(0,1.000000000000000)(0,1.000000000000000)(0,1.000000000000000)25(0, 0.999999999047875)(0,0.999999999999999)(0,1.000000000000000)(0,1.000000000000000)(0,1.000000000000000)26(0,0.999999999583445)(0,1.000000000000000)(0,1.000000000000000)(0,1.000000000000000)(0,1.000000000000000)⋮⋮⋮⋮⋮⋮41(0,0.999999999999998)42(0,0.999999999999999)43(0,1.000000000000000)Figure 1
Convergence results.Lemma 8 (see [7]).
A Banach spaceX is uniformly convex if and only if for each fixed number r>0, there exists a continuous strictly increasing function ϕ:[0,∞)⟶[0,∞),ϕt=0 if and only if t=0, such that(69)λx+1−λy2≤λx2+1−λy2−λ1−λϕx−y,for all λ∈0,1 and all x,y∈X such that x≤r and y≤r.Lemma 9 (see [7]).
We consider a strictly increasing functionϕ:[0,∞)⟶[0,∞) with ϕ0=0. If a sequence rn in 0,∞ satisfies limn⟶∞ϕrn=0, then limn⟶∞rn=0.Lemma 10 (see [7]).
LetA,B be a nonempty and closed pair in a uniformly convex Banach space X such that A is convex. Let xn and zn be sequences in A and yn be a sequence in B such that limn⟶∞xn−yn=dA,B and limn⟶∞zn−yn=dA,B, then we have limn⟶∞xn−zn=0.Theorem 5.
LetM and N be nonempty bounded closed convex subsets of a uniformly convex Banach space and suppose F,G:M∪N⟶M∪N satisfy(1)
GM⊆M,GN⊆N,FM⊆M and FN⊆N;(2)
Fu−Gv≤u−v for u∈M,v∈N; and(3)
Fu−Gv≤u−v for u∈N,v∈M,
with a nonempty common fixed-point set. For an arbitrary chosenw0∈M0, let the sequence wn be generated by (7) where ηn,δn,γn satisfy (R) and n=0,1,2,…. Then limn⟶∞Gun−Fvn=0,limn⟶∞un−Gun=0 and limn⟶∞wn−Fwn=0. Moreover, if FM lies in a compact set, then wn,vn, and un converge to a common fixed point of G and F.Proof.
Letz∈N0 be a common fixed point of F and G. Then from Lemma 8, there exists continuous strictly increasing function ϕ:[0,∞)⟶[0,∞) such that(70)wn+1−z2=1−ηnGun+ηnFvn−z2=ηnFvn−z+1−ηnGun−z2≤ηnFvn−z2+1−ηnGun−z2−ηn1−ηnϕFvn−Gun=ηnFvn−Gz2+1−ηnGun−Fz2−ηn1−ηnϕFvn−Gun≤ηnvn−z2+1−ηnun−z2=ηn1−δnun+δnGun−z2+1−ηn1−γnwn+γnFwn−z2=ηnδnGun−z+1−δnun−z2+1−ηnγnFwn−z+1−γnwn−z2≤ηnδnGun−z2+ηn1−δnun−z2−ηnδn1−δnϕGun−un+1−ηnγnFwn−z2+1−ηn1−γnwn−z2−1−ηnγn1−γnϕFwn−wn≤ηnδnun−z2+ηn1−δnun−z2+1−ηnγnwn−z2+1−ηn1−γnwn−z2−1−ηnγn1−γnϕFwn−wn≤ηnun−z2+1−ηnwn−z2−1−ηnγn1−γnϕFwn−wn=ηn1−γnwn+γnFwn−z2+1−ηnwn−z2−1−ηnγn1−γnϕFwn−wn=ηnγnFwn−z+1−γnwn−z2+1−ηnwn−z2−1−ηnγn1−γnϕFwn−wn≤ηnγnFwn−z2+ηn1−γnwn−z2−ηnγn1−γnϕFwn−wn+1−ηnwn−z2−1−ηnγn1−γnϕFwn−wn≤wn−z2−γn1−γnϕFwn−wn.
Therefore, we can deduce the following inequality:(71)γn1−γnϕFwn−wn≤wn−z2−wn+1−z2.
Now, we proceed with the following:
Supposeηn,δn, and γn satisfy (R). From equation (73), we get(72)∑n=1mγn1−γnϕFwn−wn≤w1−z2−wm+1−z2.
Asm⟶∞, we get ∑∞∞γn1−γnϕFwn−wn<∞. In view of the fact that γn1−γn≥ε, implies ϕFwn−wn⟶0, so Fwn−wn⟶0.
As in Theorem4, we can prove un−wn⟶0,Gun−Fvn⟶0,Gun−un⟶0, and vn−un⟶0. Now, since FM lies in a compact subset then Fwn has a convergent subsequence Fwnk, converging to some point u∈M0. Also, we have wnk⟶u,unk⟶u,Gunk⟶u.
Nowwnk−Pu≥Fwnk−GPu⟶u−GPu. So u−GPu=D. From Fu−GPu≤u−Pu, follows Fu−GPu=D. Then, Fu=u.
On the other hand,unk−Pu≥Gunk−FPu⟶u−FPu. So u−FPu=D. From Gu−FPu≤u−Pu, follows Gu−FPu=D. Then Gu=u. Therefore, Fu=Gu=u.
Letx∈M0. Then we have(73)Fx−GPx≤x−Px=dM,N.
ThereforeFx−GPx=dM,N=Fx−PFx. By Lemma 6, we get GPx=PFx. In particular GPu=PFu. In the same way, we can prove that FPu=PGu.
SinceFPu=PGu=Pu and GPu=PFu=Pu, we get that limn⟶∞wn−Pu exists. So(74)limn⟶∞wn−Pu=limk⟶∞wnk−Pu=u−Pu=dM,N,which gives wn⟶u.
In the next result, we provide a stronger version to approximate the common fixed point via von Neumann sequences.Theorem 6.
LetM and N be nonempty bounded closed convex subsets of a Hilbert space and suppose F,G:M∪N⟶M∪N satisfy(1)
GM⊆M,GN⊆N,FM⊆M and FN⊆N;(2)
Fu−Gv≤u−v for u∈M,v∈N; and(3)
Fu−Gv≤u−v for u∈N,v∈M,
with a nonempty common fixed-point set. Letw0∈M, and define wn+1=Pn1−ηnGun+ηnFvn where vn=1−δnun+δnGun,un=1−γnwn+γnFwn,ηn,δn,γn∈ε,1−ε, where ε∈0,1/2 and n=0,1,2,…, then limn⟶∞wn−Fwn=0. Moreover, if FM lies in a compact set and un−Gun⟶0,vn−Fvn⟶0 then wn converges to a common fixed point of F,G.Proof.
IfdM,N=0, then M0=N0=M∩N and F,G:M∩N⟶M∩N is a pair of nonexpansive with wn+1=Pn1−ηnGun+ηnFvn=1−ηnGun+ηnFvn, the usual Thakur iteration. So, let us take that dM,N>0. Let z∈N0 be a common fixed point of F and G. Now, by equations (39) and (41), we obtain(75)wn+1−z=Pn1−ηnGun+ηnFvn−z≤1−ηnGun+ηnFvn−z=1−ηnGun−z+ηnFvn−z≤1−ηnun−z+ηnvn−z≤1−ηnwn−z+ηnwn−z=wn−z.
This implies that the sequencewn−z is nonincreasing. Then we can find d>0 such that limn⟶∞wn−z=d.
Suppose there exists a subsequencewnk of wn and an ϵ>0 such that wnk−Fwnk≥ε>0 for all k.
Since the modulus of convexity ofδ of X is continuous and increasing function we choose ξ>0 as small that 1−cδε/d+ξd+ξ<d, where c>0.
Now we choosek, such that wnk−z≤d+ξ. Now we have(76)z−wnk+1=z−Pnk1−ηnkGunk+ηnkFvnk≤z−1−ηnkGunk+ηnkFvnk=1−ηnkz+ηnkz−1−ηnkG1−γnkwnk+γnkFwnk+ηnkFvnk≤1−ηnkz−G1−γnkwnk+γnkFwnk+ηnkz−Fvnk≤1−ηnkz−1−γnkwnk+γnkFwnk+ηnkz−vnk.
Now(77)z−1−γnkwnk+γnkFwnk=1−γnkz−wnk+γnkz−Fwnk≤1−2δεd+ξminγnk,1−γnkd+ξ.Also, using equation (79), we get(78)z−vnk=z−1−δnkunk+δnkGunk=1−δnkz−unk+δnkz−Gunk≤1−δnkz−unk+δnkz−Gunk≤1−δnkz−unk+δnkz−unk=z−unk≤1−2δεd+ξminγnk,1−γnkd+ξ.
Therefore, the equation (78) becomes(79)z−wnk+1≤1−2δεd+ξminγnk,1−γnkd+ξ.
Since there existsl>0 such that 2minγnk,1−γnk≥l,(80)1−2δεd+ξminγnk,1−γnkd+ξ≤1−lδεd+ξd+ξ.
Suppose that we choose very smallξ>0, we have 1−lδε/d+ξd+ξ<d, which is a contradiction. This implies that limn⟶∞wn−Fwn=0. Now we prove that wn+1−wn⟶0. From the Thakur iteration, we get un−wn=γnFwn−wn. Since limn⟶∞wn−Fwn=0 we obtain un−wn⟶0.
SinceFM is contained in a compact set, Fwn has a subsequence Fwnk that converges to a point v0∈M. Also wnk converges to v0. From the given sequence, we obtain(81)wnk+1−wnk=Pnk1−ηnkGunk+ηnkFvnk−wnk≤1−ηnkGunk+ηnkFvnk−wnk=Gunk−wnk+ηnkGunk−Fvnk≤Gunk−unk+unk−wnk+ηnkGunk−unk+unk−vnk+vnk−Fvnk.
SinceGunk−unk⟶0 implies unk−vnk⟶0. Then wnk+1−wnk⟶0. Therefore, wnk+1⟶v0, which implies that wn⟶v0. Also, we have un⟶v0,Gun⟶v0,Fwn⟶v0 as n⟶∞.
Now,Fwn−GPNv0≤wn−PNv0, which implies that.
v0−GPNv0≤v0−PNv0. Hence, GPNv0=PNv0.
Similarly,Gun−FPNv0≤un−PNv0, which implies that.
v0−FPNv0≤v0−PNv0. Hence, FPNv0=PNv0.
Also,GPv0−PNv0=GPv0−FPNv0≤Pv0−PNv0.
SoGPv0=Pv0.
And alsoFPv0−PNv0=FPv0−GPNv0≤Pv0−PNv0.
SoFPv0=Pv0.
NowGPNPv0−Pv0=GPNPv0−FPv0≤PNPv0−Pv0. Thus GPNPv0=PNPv0.
For anyn, we have FPnv0=Pnv0 and GPNPnv0=PNPnv0.
Similarly,FPNPv0−Pv0=FPNPv0−GPv0≤PNPv0−Pv0. Thus FPNPv0=PNPv0.
For anyn, we have GPnv0=Pnv0 and FPNPnv0=PNPnv0. By Theorem 2, for each u∈M the sequence Pnu converges to some ru∈M0. Now,(82)Grv0−PNrv0≤limn⟶∞Grv0−PNPnv0=limn⟶∞Grv0−FPNPnv0≤limn⟶∞rv0−PNPnv0=rv0−PNrv0.
SoGrv0−PNrv0≤rv0−PNrv0.
ThereforeGrv0=rv0 and similarly, we get GPNrv0=PNrv0.
In the same way, we prove thatFrv0=rv0 and FPNrv0=PNrv0.
Now we definegn:M⟶ℝ by gnu=Pnu−ru.
Sinceru−rv=limn⟶∞Pnu−Pnv≤u−v, then we conclude that r is continuous. Therefore gnu is continuous and converges pointwise to zero. Since ru∈M0, by Lemma 4, we obtain gn+1≤gn. Therefore gn converges uniformly on the compact set.(83)F=1−ηnkGunk+ηnkFvnk∪v0.
Therefore,(84)limk⟶∞Pnk1−ηnkGunk+ηnkFvnk−r1−ηnkGunk+ηnkFvnk=0.
Sincer1−ηnkGunk+ηnkFvnk⟶rv0, we get wnk+1⟶rv0, which gives that rv0=v0. Therefore Gv0=Grv0=rv0=v0 and Fv0=Frv0=rv0=v0, which completes the proof.
SupposeX is a Hilbert space and let M and N be nonempty bounded closed convex subsets of X and suppose F,G:M∪N⟶M∪N satisfy(1)
GM⊆N,GN⊆M,FM⊆N and FN⊆M;(2)
Fu−Gv≤u−v for u∈M,v∈N; and(3)
Fu−Gv≤u−v for u∈N,v∈M.
We considerPMG:M⟶M,PNF:N⟶N,PNG:N⟶N and PMF:M⟶M. From Proposition 2, PMFu−PNGv≤u−v for u∈M and v∈N and PNFu−PMGv≤u−v for u∈N and v∈M, by Theorem 4 and Theorem 6, we give the following results on the convergence of best proximity points.Corollary 4.
LetM and N be nonempty bounded closed convex subsets of a Hilbert space and suppose F,G:M∪N⟶M∪N satisfy(1)
GM⊆N,GN⊆M,FM⊆N and FN⊆M;(2)
Fu−Gv≤u−v for u∈M,v∈N; and(3)
Fu−Gv≤u−v for u∈N,v∈M.
IfM is mapped into a compact subset of N, then for any w0∈M0 the sequence is defined by wn+1=1−ηnPMGun+ηnPMFvn, where vn=1−δnun+δnPMGun,un=1−γnwn+γnPMFwn, converges to w in M0 such that w−Fw=w−Gw=dM,N.Corollary 5.
LetM and N be nonempty bounded closed convex subsets of a Hilbert space and suppose F,G:M∪N⟶M∪N satisfy(1)
GM⊆N,GN⊆M,FM⊆N and FN⊆M;(2)
Fu−Gv≤u−v for u∈M,v∈N; and(3)
Fu−Gv≤u−v for u∈N,v∈M.
IfM is mapped into a compact subset of N, then for any w0∈M the sequence defined by wn+1=1−ηnPMGun+ηnPMFvn, where vn=1−δnun+δnPMGun,un=1−γnwn+γnPMFwn converges to w in M0 such that w−Fw=w−Gw=dM,N, provided dwn,M0⟶0.Corollary 6.
LetM and N be nonempty bounded closed convex subsets of a Hilbert space and suppose F,G:M∪N⟶M∪N satisfy(1)
GM⊆N,GN⊆M,FM⊆N and FN⊆M;(2)
Fu−Gv≤u−v for u∈M,v∈N; and(3)
foru∈N,v∈M.
IfM is mapped into a compact subset of N, then for any w0∈M0 the sequence defined by wn+1=Pn1−ηnPMGun+ηnPMFvn, where vn=1−δnun+δnPMGun,un=1−γnwn+γnPMFwn converges to w in M0 such that w−Fw=w−Gw=dM,N.Proof..
The result follows from Corollary4.Corollary 7.
LetM and N be nonempty bounded closed convex subsets of a Hilbert space and suppose F,G:M∪N⟶M∪N satisfy(1)
GM⊆N,GN⊆M,FM⊆N and FN⊆M;(2)
Fu−Gv≤u−v for u∈M,v∈N; and(3)
Fu−Gv≤u−v for u∈N,v∈M.
Letw0∈M, and define wn+1=Pn1−ηnPMGun+ηnPMFvn, where vn=1−δnun+δnPMGun,un=1−δnwn+δnPMFwn,ηn,δn∈ε,1−ε, where ε∈0,1/2 and n=0,1,2,…. If M is mapped into a compact subset of N and un−PMGun⟶0,vn−PMFvn⟶0, then wn converges to w in M0 such that w−Fw=w−Gw=dM,N.Proof.
The result follows from Theorem6.


## 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 formFx=x, where F 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.



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