# Best Proximity Point Theorems for a Berinde MT-Cyclic Contraction on a Semisharp Proximal Pair

Authors: Chalongchai Klanarong; Tanadon Chaobankoh
Journal: International Journal of Mathematics and Mathematical Sciences
 (2018)

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## Abstract

In this paper, a new type of non-self-mapping, called Berinde MT-cyclic contractions, is introduced and studied. Best proximity point theorems for this type of mappings in a metric space are presented. Some examples illustrating our main results are also given. Our results generalize and improve some known results in the literature.


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## Body

## 1. Introduction and Preliminaries

Several problems in a real world can be modeled in the form of operator equations. An equationSx=x, which is called the fixed point equation, is one of the important means for solving some problems among them. Fixed point theory is then considered as an important tool for solving such problems. The well-known fixed point theorem for contraction mappings was given by Banach [1]. It is known as the Banach contraction principle. This principle guarantees that each contraction mapping from a complete metric space X into itself always has a unique fixed point.In 2004, Berinde [2] introduced and studied the concept of weak contraction mappings in the context of a complete metric space. Let (X,d) be a metric space. A mapping S:X→X is called aweak contraction if there exist a∈(0,1) and L≥0 such that(1)dSx,Sy≤adx,y+Ldx,Sy,for  allx,y∈X.A fixed point theorem of this type of mapping was proved in [2]. It extended and generalized that of the Banach contraction principle and others; see [2] and references therein.On the other hand, if the fixed point equationSx=x does not have a solution, then d(x,Sx)>0 for all x∈X. In this situation, it is natural to ask whether we can find an approximate solution x such that the error is (2)dx,Sx=miny∈X⁡dy,Sy.In order to have a concrete lower bound, let us consider two nonempty subsets W and V of a metric space X and a mapping S:W→V. It is observed that d(x,Sx)≥d(W,V) for all x∈W, where d(W,V)=infdx,y:x∈Wandy∈V. So we are interested to find a point x0∈W such that(3)dx0,Sx0=dW,V.Such point x0 is called abest proximity point of the mapping S, and d(x0,Sx0) is called theglobal minimum value of d(W,V).The best proximity point theorem was first studied by Fan [3], in 1969. He proved that if W is a nonempty compact convex subset in a normed space X and S:W→X is a continuous mapping, then there exists u∈W such that u-Su=d(Su,W) where d(Su,W)≔min⁡Su-a:a∈W. Especially, if S(W)⊆W, then we get that u is a fixed point of S.Several years later, the above result has been studied and generalized by many researchers, such as Reich [4], Sehgal and Singh [5], Vetrivel et al. [6], Anuradha and Veeramani [7], Basha [8, 9], Kirk et al. [10], Raj [11], Gabeleh [12], Abkar and Gabeleh [13], Eldred and Veeramani [14], and Du and Lakzian [15] and references therein. Some recent research papers worth mentioning are [16–19].Throughout this paper, we denote byW and V nonempty subsets of a metric space (X,d). We also require the following notions:(4)dW,V≔inf⁡dx,y:x∈Wandy∈V,W0≔x∈W:dx,y=dW,Vfor  somey∈V,V0≔y∈V:dx,y=dW,Vfor  somex∈W.A mappingS:W∪V→W∪V is called acyclic mapping if S(W)⊆V and S(V)⊆W. And a point x∈W∪V is said to be abest proximity point of S if d(x,Sx)=d(W,V). In 2006, Eldred and Veeramani [14] introduced the concept of cyclic contraction and proved the existence of a best proximity point for this type of mapping on a complete metric space.Definition 1 (see [14]).
A mappingS:W∪V→W∪V is called acyclic contraction if the following conditions hold: (i)
S is a cyclic mapping;(ii)
there existsk∈(0,1) such that d(Sx,Sy)≤kd(x,y)+(1-k)d(W,V) for all x∈W,y∈V.The concept ofMT-function was used by Reich [20] and Mizoguchi and Takahashi [21] to define a class of multivalued mappings which is more general than that of contraction mappings. After that Du [22, 23] studied the class of multivalued mappings generated by Mizoguchi and Takahashi functions (or MT-functions) and gave characterizations of MT-functions.Definition 2 (see [22]).
A functionφ:[0,∞)→[0,1) is said to be anMT-function (orR-function) if lim⁡sups→t+⁡φ(s)<1 for all t∈[0,∞).Theorem 3 (see [23]).
Letφ:[0,∞)→[0,1) be a function. Then the following statements are equivalent. (a)
φ is an MT-function; i.e., lim⁡sups→t+⁡φ(s)<1 for all t∈[0,∞).(b)
For eacht∈[0,∞), there exist rt∈[0,1) and εt>0 such that φ(s)≤rt  for all s∈(t,t+εt).(c)
For any nonincreasing sequence{xn}n∈N in [0,∞), 0≤supn∈N⁡φ(xn)<1.It is clear that ifφ is a nondecreasing function or a nonincreasing function, then φ is an MT-function. For more examples and details, see [15, 22, 23].Consequently, Du and Lakzian [15] introduced MT-cyclic contractions with respect to φ and proved the existence and convergence theorems for this type of non-self-mapping in metric spaces.Definition 4 (see [15]).
If a mapS:W∪V→W∪V satisfies the following: then S is called anMT-cyclic contraction withrespect toφ on W∪V.(i)
S is a cyclic mapping;(ii)
there exists an MT-functionφ:[0,∞)→[0,1) such that d(Sx,Sy) ≤ φ(d(x,y))d(x,y) + (1-φ(d(x,y)))d(W,V) for all x∈W and y∈V.It is obvious that ifφ(t)=k with k∈[0,1), then S is a cyclic contraction, and, hence, an MT-cyclic contraction with respect to φ which is more general than that of cyclic contraction. For example of an MT-cyclic contraction with respect to φ, but is not a cyclic contraction, see [15].In 2009, Suzuki et al. [24] introduced the concept of the property UC of two nonempty subsets of a metric space as follows.Definition 5 (see [24]).
Let(W,V) be a pair of nonempty subsets of a metric space (X,d). The pair (W,V) is said to satisfy theproperty UC if {xn} and {zn} are sequences in W and {yn} is a sequence in V such that(5)limn→∞⁡dxn,yn=dW,V=limn→∞⁡dzn,yn,and then limn→∞⁡d(xn,zn)=0.Later, in 2011, Kosuru and Veeramani [25] introduced the concept of semisharp proximal pair of two nonempty subsets of a metric space. This concept is again more general than that of the property UC.Definition 6 (see [25]).
Let(W,V) be a pair of nonempty subsets of a metric space (X,d). The pair (W,V) is said to be asemisharp proximal pair if for each w∈W and u∈V there exist at most one x∈V and y∈W such that d(w,x)=d(W,V)=d(y,u).Example 7.
Consider the spaceX of all real valued continuous functions on [0,1] with the supremum norm; i.e., X=(C[0,1],∥·∥∞). Set(6)W=ft:t∈0,1andV=gt:t∈0,1,where(7)ftx=t+x,ifx∈0,12;t-x+1,ifx∈12,1,and (8)gtx=t+x+1,ifx∈0,12;t-x+2,ifx∈12,1.It is easy to show that ft-gt=1 and ft-gk>1 for all t≠k∈[0,1]. Hence d(W,V)=1 and (W,V) is a semisharp proximal pair.Note that the property UC implies semisharp proximality. In 2015, R. Espinola, et al. [26] introduced the concept of a proximally complete pair (W,V) of subsets of a metric space. They proved existences and convergence theorems of best proximity points for cyclic contraction mappings. They obtained a useful theorem presented as follows.Theorem 8 (Espinola et al. [26, Theorem 3.3]).
Let(W,V) be a pair of complete subsets of a metric space satisfying the UC property. If xn is a sequence in W∪V with x2n∈W and x2n+1∈V, for all n∈N, then the sequences x2n and x2n+1 have convergent subsequences in W and V, respectively.By those works mentioned above, we aim to introduce a new type of single-valued, non-self-mapping which is more general than that of contractions, cyclic contractions, andMT-cyclic contractions. The best proximity point theorems for this type of mappings in metric spaces will be investigated. Our main results extend and generalize those of Du and Lakzian [15], Eldred and Veeramani [14], and others.


## 2. Main Results

By using ideas of cyclic contractions, MT-functions, and weak contractions, we shall first introduce Berinde MT-cyclic contractions with respect toφ and prove the existence and convergence theorems for this type of non-self-mapping in metric spaces.Definition 9.
LetS:W∪V→W∪V be a cyclic mapping. The mapping S is said to be aBerinde MT-cyclic contraction with respect toφ if there exists an MT-function φ:[0,∞)→[0,1) and L≥0 such that(9)dSx,Sy≤φdx,ydx,y+1-φdx,ydW,V+Lmin⁡dy,Sx,dx,Sy,for all x∈W and y∈V.It is easy to see that a Berinde MT-cyclic contraction with respect toφ can be reduced to an MT-cyclic contraction with respect to φ.Remark 10.
IfS:W∪V→W∪V is a Berinde MT-cyclic contraction with respect to φ, then S satisfies the following condition:(10)dSx,Sy≤dx,y+Lmin⁡dy,Sx,dx,Sy,for all x∈W and y∈V. To see this, we can write (9) in the form d(Sx,Sy)-d(W,V)≤φ(d(x,y))[d(x,y)-d(W,V)]+Lmindy,Sx,dx,Sy, for all x∈W and y∈V. Because of φ(d(x,y))<1, it follows that(11)dSx,Sy-dW,V≤dx,y-dW,V+Lmin⁡dy,Sx,dx,Sy,and, for all x∈W and y∈V, hence (10) is satisfied.Example 11.
Letl∞ be the metric space consisting of all bounded real sequences with supremum metric d∞ and let {en} be the canonical basis of c0, where c0 is the space of all null sequences. Let {τn} be a sequence of positive real numbers satisfying τ1=τ2 and 0<a≤τn+1<τn for n≥2 and for some positive real number a. Thus {τn} is convergent. Put xn=τnen for n∈N and let X=x1,x2,x3,…. Then X is a bounded and complete subset of l∞, and hence (X,d∞) is a complete metric space with d∞(xn,xm)=τn if m>n.
LetW=x1,x3,x5,…, V=x1,x2,x4,… and let S:W∪V→W∪V be defined by (12)Sxn=x1ifn=1,2;xn+3ifn≥3,and define φ:[0,∞)→[0,1) by (13)φt=τn+3τnift=τnfor  somen∈Nwithn≥3;0otherwise.Then d(W,V)=0 and S(W)⊆V,S(V)⊆W. Since lim⁡sups→t+⁡φ(s)=0<1 for all t∈[0,∞), we have that φ is an MT-function. Next, we show that S is a Berinde MT-cyclic contraction with respect to φ. Obviously, d(Sx1,Sxi) for i=1,2,3,… satisfy (9) with L≥1. We will consider three cases as follows.
Case 1. For i≥3 and xi∈W, we have (14)dSx2,Sxi=dx1,xi+3=τ1≤Lτ2=φdx2,xidx2,xi+1-φdx2,xidW,V+Lmin⁡dx2,xi+3,dxi,x1,forL≥1.
Case 2. For i≥4 and xi∈V, we get (15)dSx3,Sxi=dx6,xi+3=τ6≤τ6+Lmin⁡dx3,xi+3,dxi,x6=φdx3,xidx3,xi+1-φdx3,xidW,V+Lmin⁡dx3,xi+3,dxi,x6,forL≥0.
Case 3. For j>i≥4 and xi, xj∈W∪V, we have (16)dSxi,Sxj=dxi+3,xj+3=τi+3≤τi+3+Lmin⁡dxi,xj+3,dxj,xi+3=φdxi,xjdxi,xj+1-φdxi,xjdW,V+Lmin⁡dxi,xj+3,dxj,xi+3,forL≥0.From all of the above cases, we can now conclude that S is a Berinde MT-cyclic contraction with respect to φ and L≥1. We note from Case 1 that S is not an MT-cyclic contraction with respect to φ.Proposition 12.
LetW and V be nonempty subsets of a metric space (X,d) and S:W∪V→W∪V be a Berinde MT-cyclic contraction with respect to φ. Then starting with any given x0∈W∪V, define a sequence {xn} in W∪V by xn+1=Sxn for all n≥0; we have d(xn,xn+1)→d(W,V) as n→∞.Proof.
Letx0∈W∪V be given. Define a sequence {xn} in W∪V by (17)xn+1=Sxn,for  eachn≥0.Suppose that x0∈W (when x0∈V is similar); then x1=Sx0∈V, and so, x2=Sx1∈W. Since S is a Berinde MT-cyclic contraction with respect to φ, we have (18)dx1,x2=dSx0,Sx1≤φdx0,x1dx0,x1+1-φdx0,x1dW,V+Lmin⁡dx1,Sx0,dx0,Sx1=φdx0,x1dx0,x1+1-φdx0,x1dW,V+Lmin⁡dx1,x1,dx0,x2=φdx0,x1dx0,x1+1-φdx0,x1dW,V.Again, since x2∈W and S is a cyclic mapping, we get x3=Sx2∈V. By the Berinde MT-cyclic contraction with respect to φ of S, we have (19)dx2,x3=dSx1,Sx2≤φdx1,x2dx1,x2+1-φdx1,x2dW,V+Lmin⁡dx1,Sx2,dx2,Sx1=φdx1,x2dx1,x2+1-φdx1,x2dW,V+Lmin⁡dx1,x3,dx2,x2=φdx1,x2dx1,x2+1-φdx1,x2dW,V.By induction, we can show that, for each n≥1,(20)dxn,xn+1≤φdxn-1,xndxn-1,xn+1-φdxn-1,xndW,V.By Remark 10, we have (21)dxn,xn+1=dSxn-1,Sxn≤dxn-1,xn+Lmin⁡dxn-1,Sxn,dxn,Sxn-1=dxn-1,xn,for all n≥1. It means that {d(xn,xn+1)} is a nonincreasing sequence. By Theorem 3, we get 0≤supn∈N⁡φ(d(xn,xn+1))<1.
Putk≔supn∈N⁡φdxn,xn+1. Thus 0≤φ(d(xn,xn+1))≤k<1, for all n≥1. It follows from (20) that (22)dxn,xn+1≤kdxn-1,xn+1-kdW,V,for  alln≥1.Hence for each n≥1, we have (23)dxn,xn+1≤kdxn-1,xn+1-kdW,V≤kkdxn-2,xn-1+1-kdW,V+1-kdW,V=k2dxn-2,xn-1+1-k2dW,V⋮≤kndx0,x1+1-kndW,V.Since d(W,V)≤d(xn,xn+1) and kn→0 as n→∞, by taking n→∞ in the above inequality, we obtain that (24)dxn,xn+1→dW,V,asn→∞.The proof is now completed.Example 13.
Letτn=1/3+1/n for all n∈N. Then limn→∞⁡τn=1/3. Put xn=τnen for n∈N and let X=x1,x2,x3,… be a bounded and complete subset of l∞. Then (X,d∞) be a complete metric space with d∞(xn,xm)=τn if m>n. Set W=x1,x3,x5,… and V=x2,x4,x6,…. So limn→∞⁡d(xn,xn+1)=limn→∞⁡τn=1/3=d(W,V). Let S:W∩V→W∩V be defined by(25)Sxn=xn+1for  alln∈N.
It is easy to see thatS(W)=V and S(V)⊂W and so S is a cyclic mapping. Define φ:[0,∞)→[0,1) as (26)φt=τn+1τnift=τnfor  somen∈N;0otherwise.Then φ is an MT-function. Now, we will show that S is a Berinde MT-cyclic contraction with respect to φ. For m,n∈N with m>n, (27)dSxn,Sxm=τn+1<τn+1+131-τn+1τn+Lmin⁡dxn,Sxm,dxm,Sxn=φdxn,xmdxn,xm+1-φdxn,xmdW,V+Lmin⁡dxn,Sxm,dxm,Sxn.Hence S is a Berinde MT-cyclic contraction with respect to φ. Therefore, all the assumptions of Proposition 12 hold.The following result is obtained immediately from Proposition12 because every nondecreasing function or nonincreasing function is an MT-function.Corollary 14.
LetW and V be nonempty subsets of a metric space (X,d) and S:W∪V→W∪V be a cyclic mapping. Let x0∈W be given and define a sequence {xn} in W∪V by xn+1=Sxn, for all n≥0. Suppose that there exists a nondecreasing (or nonincreasing) function η:[0,∞)→[0,1) such that for all x∈W and y∈V(28)dSx,Sy≤ηdx,ydx,y+1-ηdx,ydW,V+Lmin⁡dy,Sx,dx,Sy.Then limn→∞d(xn,xn+1)=d(W,V).By Proposition12, if S is a cyclic contraction or an MT-cyclic contraction with respect to φ, then we obtain directly the following results which were proved by Eldred and Veeramani [14] and Du and Lakzian [15], respectively.Corollary 15 (Eldred and Veeramani [14, Proposition 3.1]).
LetW and V be nonempty subsets of a metric space (X,d) and S:W∪V→W∪V be a cyclic contraction. Then starting with any x0∈W and defining a sequence {xn} in W∪V by xn+1=Sxn, for all n≥0, then limn→∞⁡d(xn,xn+1)=d(W,V).Corollary 16 (Du and Lakzian [15, Theorem 2.1]).
LetW and V be nonempty subsets of a metric space (X,d) and S:W∪V→W∪V be an MT-cyclic contraction with respect to φ. Then starting with any x0∈W∪V, define a sequence {xn} in W∪V by xn+1=Sxn for all n≥0; we have d(xn,xn+1)→d(W,V) as n→∞.Observe that ifW and V are nonempty subsets of a metric space (X,d) and S:W∪V→W∪V is a cyclic mapping with x0∈W, define a sequence {xn} in W∪V by xn+1=Sxn, for all n≥0; then {x2n} and {x2n+1} are subsequences of {xn} in W and V, respectively. Similarly, if x0∈V, then {x2n} and {x2n+1} are subsequences of {xn} in V and W, respectively. Moreover, d(W,V)≤d(xn,xn+1), for all n≥0.Theorem 17.
LetW and V be nonempty subsets of a metric space (X,d) and S:W∪V→W∪V be a cyclic mapping. Let x0∈W be given and define a sequence {xn} in W∪V by xn+1=Sxn, for all n≥0. Suppose that the following conditions hold: (i)
d(Sx,Sy)≤d(x,y)+Lmindx,Sy,dy,Sx for all x∈W and y∈V with L≥0;(ii)
{x2n} has a convergent subsequence in W;(iii)
limn→∞⁡d(xn,xn+1)=d(W,V).
Then there existsx∈W such that d(x,Sx)=d(W,V).Proof.
Letx0∈W and {x2nk} be a subsequence of {x2n} such that (29)limk→∞⁡x2nk=x,for  somex∈W.In fact, for each k∈N,(30)dW,V≤dx,x2nk-1≤dx,x2nk+dx2nk,x2nk-1.It follows by (iii) that (31)limk→∞⁡dx,x2nk-1=dW,V,ask→∞.From (i), for each k∈N, we have (32)dW,V≤dSx,x2nk=dSx,Sx2nk-1≤dx,x2nk-1+Lmin⁡dx,x2nk,dx2nk-1,Sx≤dx,x2nk-1+Ldx,x2nk.Taking k→∞ in the above inequality, we obtain that(33)dx,Sx=limk→∞⁡dx2nk,Sx=dW,V.The proof is completed.Using the same proof as Theorem17, we obtain a similar result.Theorem 18.
LetW and V be nonempty subsets of a metric space (X,d) and S:W∪V→W∪V be a cyclic mapping. Let x0∈W be given and define a sequence {xn} in W∪V by xn+1=Sxn, for all n≥0. Suppose that the following conditions hold: (i)
d(Sx,Sy)≤d(x,y)+Lmin{d(x,Sy),d(y,Sx)} for all x∈W and y∈V with L≥0;(ii)
{x2n+1} has a convergent subsequence in V;(iii)
limn→∞⁡d(xn,xn+1)=d(W,V).
Then there existsy∈V such that d(y,Sy)=d(W,V).Applying Proposition12 and Theorems 17 and 18, we establish the following new best proximity point theorems for a Berinde MT-cyclic contraction with respect to φ.Theorem 19.
LetW and V be nonempty subsets of a metric space (X,d) and S:W∪V→W∪V be a Berinde MT-cyclic contraction with respect to φ. Let x0∈W be given and define a sequence {xn} in W∪V by xn+1=Sxn, for all n≥0. If {x2n} has a convergent subsequence in W, then there exists x∈W such that d(x,Sx)=d(W,V).Theorem 20.
LetW and V be nonempty subsets of a metric space (X,d) and S:W∪V→W∪V be a Berinde MT-cyclic contraction with respect to φ. Let x0∈W be given and define a sequence {xn} in W∪V by xn+1=Sxn, for all n≥0. If {x2n+1} has a convergent subsequence in V, then there exists y∈V such that d(y,Sy)=d(W,V).By Theorems19 and 20, we obtain the next result.Corollary 21.
LetW and V be nonempty subsets of a metric space (X,d) and S:W∪V→W∪V be a Berinde MT-cyclic contraction with respect to φ. Let x0∈W be given and define a sequence {xn} in W∪V by xn+1=Sxn, for all n≥0. Suppose that W or V is compact. Then there exists x∈W∪V such that d(x,Sx)=d(W,V).Theorem 22.
LetW and V be nonempty subsets of a metric space (X,d) such that (W,V) is a semisharp proximal pair and S:W∪V→W∪V is a Berinde MT-cyclic contraction with respect to φ. Let x0∈W be given and define a sequence {xn} in W∪V by xn+1=Sxn, for all n≥0. If {x2n} has a convergent subsequence in W, then the following hold: (i)
there existsx∈W such that d(x,Sx)=d(W,V);(ii)
x and Sx are fixed point of S2 in W and V, respectively.Proof.
By Theorem19, there exists x∈W such that d(x,Sx)=d(W,V) and it follows that (34)dW,V≤dSx,S2x≤φdx,Sxdx,Sx+1-φdx,SxdW,V+Lmin⁡dx,S2x,dSx,Sx=φdx,Sxdx,Sx+1-φdx,SxdW,V=dW,V.Hence d(Sx,S2x)=d(W,V). In the semisharp proximality of (W,V), we have S2x=x. Consider(35)dx,S2Sx=dx,SS2x=dx,Sx=dW,V,which implies that S2(Sx)=Sx. Therefore, x and Sx are fixed points of S2 in W and V, respectively.Using the proof of Theorem22, we obtain the following result.Theorem 23.
LetW and V be nonempty subsets of a metric space (X,d) such that (W,V) is a semisharp proximal pair, and S:W∪V→W∪V is a Berinde MT-cyclic contraction with respect to φ. Let x0∈W be given and define a sequence {xn} in W∪V by xn+1=Sxn, for all n≥0. If {x2n+1} has a convergent subsequence in V, then the following hold: (i)
there existsx∈V such that d(x,Sx)=d(W,V);(ii)
x and Sx are fixed point of S2 in V and W, respectively.By using Theorem8, we have the following corollary.Corollary 24.
LetW and V be nonempty complete subsets of a metric space (X,d) such that (W,V) satisfies the property UC, and S:W∪V→W∪V is a Berinde MT-cyclic contraction with respect to φ. Let x0∈W be given and define a sequence {xn} in W∪V by xn+1=Sxn, for all n≥0. Then the following hold: (i)
there existsx∈W such that d(x,Sx)=d(W,V);(ii)
x and Sx are fixed points of S2 in W and V, respectively.We have discussed that, under some specific conditions, a Berinde MT-cyclic contraction with respect toφ can be reduced to a cyclic contraction or an MT-cyclic contraction with respect to φ. Thus, Theorems 17 and 19 are generalizations of the results proved by Eldred and Veeramani [14] and Du and Lakzian [15], respectively. Hence, the three following corollaries are obtained directly from those theorems.Corollary 25 (Eldred and Veeramani [14, Proposition 3.2]).
LetW and V be nonempty closed subsets of a complete metric space X. Let S:W∪V→W∪V be a cyclic contraction, x0∈W, and define xn+1=Sxn for all n≥0. Suppose that {x2n} has a convergent subsequence in W. Then there exists x∈W such that d(x,Sx)=d(A,B).Corollary 26 (Du and Lakzian [15, Theorem 2.3]).
LetW and V be nonempty subsets of a metric space (X,d) and S:W∪V→W∪V be a cyclic mapping. Let x0∈W be given and define a sequence {xn} in W∪V by xn+1=Sxn, for all n≥0. Suppose that the following conditions hold: (i)
d(Sx,Sy)≤d(x,y) for all x∈W and y∈V;(ii)
{x2n} has a convergent subsequence in W;(iii)
limn→∞⁡d(xn,xn+1)=d(W,V).
Then there existsx∈W such that d(x,Sx)=d(W,V).Corollary 27 (Du and Lakzian [15, Theorem 2.4]).
LetW and V be nonempty subsets of a metric space (X,d) and S:W∪V→W∪V be a MT-cyclic contraction with respect to φ. Let x0∈W be given and define a sequence {xn} in W∪V by xn+1=Sxn, for all n≥0. Suppose that {x2n} has a convergent subsequence in W. Then there exists x∈W such that d(x,Sx)=d(W,V).



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