# Best Proximity Points for Generalized Proximal Weak Contractions Satisfying Rational Expression on Ordered Metric Spaces

Authors: V. Pragadeeswarar; M. Marudai
Journal: Abstract and Applied Analysis
 (2015)

---

## Abstract

We introduce a generalized proximal weak contraction of rational type for the non-self-map and proved results to ensure the existence and uniqueness of best proximity point for such mappings in the setting of partially ordered metric spaces. Further, our results provides an extension of a result due to  Luong and Thuan (2011) and also it provides an extension of Harjani (2010) to the case of self-mappings.


---

## Body

## 1. Introduction and Preliminaries

The fixed point theory of partially ordered metric space was introduced by Ran and Reurings [1], where they extended the Banach contraction principle in partially ordered sets with some applications to linear and nonlinear matrix equations. Subsequently, Nieto and Rodríguez-López [2] extended the result of Ran and Reurings and apply their results to obtain a unique solution for a first order ordinary differential equation with periodic boundary conditions. The following notion of an altering distance function was introduced by Khan et al. in [3].Definition 1.
A functionψ:[0,∞)→[0,∞) is said to be an altering distance function if it satisfies the following conditions.(i)
ψis continuous and nondecreasing.(ii)
ψ(t)=0 if and only if t=0.In [3], Khan et al. proved the fixed point theorems by using altering distance function together with contractive type condition.Motivated by the interesting paper of Jaggi [4], in [5] Harjani et al. proved the following fixed point theorem in partially ordered metric spaces.Theorem 2 (see [5]).
Let(X,≤) be an ordered set and suppose that there exists a metric d in X such that (X,d) is a complete metric space.
LetT:X→X be a nondecreasing mapping such that (1)dTx,Ty≤αdx,Txdy,Tydx,y+βdx,yhhhhhhhhhhforx,y∈X,x≥y,x≠y.Also, assume either T is continuous or X has the property that(2)xnisanondecreasingsequenceinXkkkksuchthatxn⟶x,thenx=sup⁡{xn}.If there exists x0∈X such that x0≤Tx0, then T has a fixed point.In [6] Luong and Thuan proved the following theorem.Theorem 3 (see [6]).
Let(X,≤) be an ordered set and suppose that there exists a metric d in X such that (X,d) is a complete metric space. Let T:X→X be a nondecreasing mapping such that(3)dTx,Ty≤mx,y-ϕmx,yddddforx,y∈X,x≥y,x≠y,where ϕ:[0,∞)→[0,∞) is a lower semicontinuous function with ϕ(t)=0 if and only if t=0, and m(x,y)=max⁡dx,Txdy,Ty/dx,y,dx,y.
Also, assume eitherT is continuous or X has the property (2).
If there existsx0∈X such that x0≤Tx0, then T has a fixed point.In this article, we attempt to give a generalization of Theorem3 by considering a non-self-map T. Before getting into the details of our main theorem, let us give a brief discussion of best proximity point results.


### 1.1. Best Proximity Point

LetA≠∅ be a subset of a metric space (X,d). A mapping T:A→X has a fixed point in A if the fixed point equation Tx=x has at least one solution. That is, x∈A is a fixed point of T if dx,Tx=0. If the fixed point equation Tx=x does not possess a solution, then dx,Tx>0 for all x∈A. In such a situation, it is our aim to find an element x∈A such that dx,Tx is minimum in some sense. The best approximation theory and best proximity pair theorems are studied in this direction. Here we state the following well-known best approximation theorem due to Fan [7].Theorem 4 (see [7]).
LetA be a nonempty compact convex subset of a normed linear space X and T:A→X be a continuous function. Then there exists x∈A such that x-Tx=dTx,A:=inf⁡∥Tx-a∥:a∈A.Such an elementx∈A in Theorem 4 is called a best approximant of T in A. Note that if x∈A is a best approximant, then x-Tx need not be the optimum. Best proximity point theorems have been explored to find sufficient conditions so that the minimization problem min⁡x∈A∥x-Tx∥ has at least one solution. To have a concrete lower bound, let us consider two nonempty subsets A,B of a metric space X and a mapping T:A→B. The natural question is whether one can find an element x0∈A such that dx0,Tx0=min⁡dx,Tx:x∈A. Since dx,Tx≥dA,B, the optimal solution to the problem of minimizing the real valued function x→dx,Tx over the domain A of the mapping T will be the one for which the value d(A,B) is attained. A point x0∈A is called a best proximity point of T if dx0,Tx0=dA,B. Note that if dA,B=0, then the best proximity point is nothing but a fixed point of T.The existence and convergence of best proximity points is an interesting topic of optimization theory which recently attracted the attention of many authors [8–16]. Also one can find the existence of best proximity point in the setting of partially ordered metric space in [17–24].The purpose of this article is to present best proximity point theorems for non-self-mappings in the setting of partially ordered metric spaces, thereby producing optimal approximate solutions forTx=x, where T is a non-self-mapping. When the map T is considered to be a self-map and ψ is defined as identity function, then our result reduces to the fixed point theorem of Luong and Thuan [6].Given nonempty subsetsA and B of a metric space X, the following notions are used subsequently: (4)d(A,B):=inf⁡dx,y:x∈A,y∈B,A0=x∈A:dx,y=dA,Bforsomey∈B,B0=y∈B:dx,y=dA,Bforsomex∈A.In [14], the authors discussed sufficient conditions which guarantee the nonemptiness of A0 and B0. Moreover, in [12], the authors proved that A0 is contained in the boundary of A in the setting of normed linear spaces.Definition 5 (see [17]).
A mappingT:A→B is said to be proximally increasing if it satisfies the condition that (5)kkkkkkkkkky1≤y2d(x1,Ty1)=d(A,B)d(x2,Ty2)=d(A,B)⟹x1≤x2,where x1,x2,y1,y2∈A.One can see that, for a self-mapping, the notion of proximally increasing mapping reduces to that of increasing mapping.Definition 6.
A mappingT:A→B is said to be generalized proximal weak contraction of rational type if it satisfies the condition that (6)kkkkx≤y,x≠ydu1,Tx=dA,Bdu2,Ty=dA,B⟹ψdu1,u2≤ψmx,y-ϕmx,y,where u1,u2,x,y∈A, ψ is an altering distance function, ϕ:[0,∞)→[0,∞) is a nondecreasing function with ϕ(t)=0 if and only if t=0, and m(x,y)=max⁡dx,u1dy,u2/dx,y,dx,y.One can see that, for a self-mapping, the notion of generalized proximal weak contraction of rational type reduces to generalized weak contraction of rational type.


## 1.1. Best Proximity Point

LetA≠∅ be a subset of a metric space (X,d). A mapping T:A→X has a fixed point in A if the fixed point equation Tx=x has at least one solution. That is, x∈A is a fixed point of T if dx,Tx=0. If the fixed point equation Tx=x does not possess a solution, then dx,Tx>0 for all x∈A. In such a situation, it is our aim to find an element x∈A such that dx,Tx is minimum in some sense. The best approximation theory and best proximity pair theorems are studied in this direction. Here we state the following well-known best approximation theorem due to Fan [7].Theorem 4 (see [7]).
LetA be a nonempty compact convex subset of a normed linear space X and T:A→X be a continuous function. Then there exists x∈A such that x-Tx=dTx,A:=inf⁡∥Tx-a∥:a∈A.Such an elementx∈A in Theorem 4 is called a best approximant of T in A. Note that if x∈A is a best approximant, then x-Tx need not be the optimum. Best proximity point theorems have been explored to find sufficient conditions so that the minimization problem min⁡x∈A∥x-Tx∥ has at least one solution. To have a concrete lower bound, let us consider two nonempty subsets A,B of a metric space X and a mapping T:A→B. The natural question is whether one can find an element x0∈A such that dx0,Tx0=min⁡dx,Tx:x∈A. Since dx,Tx≥dA,B, the optimal solution to the problem of minimizing the real valued function x→dx,Tx over the domain A of the mapping T will be the one for which the value d(A,B) is attained. A point x0∈A is called a best proximity point of T if dx0,Tx0=dA,B. Note that if dA,B=0, then the best proximity point is nothing but a fixed point of T.The existence and convergence of best proximity points is an interesting topic of optimization theory which recently attracted the attention of many authors [8–16]. Also one can find the existence of best proximity point in the setting of partially ordered metric space in [17–24].The purpose of this article is to present best proximity point theorems for non-self-mappings in the setting of partially ordered metric spaces, thereby producing optimal approximate solutions forTx=x, where T is a non-self-mapping. When the map T is considered to be a self-map and ψ is defined as identity function, then our result reduces to the fixed point theorem of Luong and Thuan [6].Given nonempty subsetsA and B of a metric space X, the following notions are used subsequently: (4)d(A,B):=inf⁡dx,y:x∈A,y∈B,A0=x∈A:dx,y=dA,Bforsomey∈B,B0=y∈B:dx,y=dA,Bforsomex∈A.In [14], the authors discussed sufficient conditions which guarantee the nonemptiness of A0 and B0. Moreover, in [12], the authors proved that A0 is contained in the boundary of A in the setting of normed linear spaces.Definition 5 (see [17]).
A mappingT:A→B is said to be proximally increasing if it satisfies the condition that (5)kkkkkkkkkky1≤y2d(x1,Ty1)=d(A,B)d(x2,Ty2)=d(A,B)⟹x1≤x2,where x1,x2,y1,y2∈A.One can see that, for a self-mapping, the notion of proximally increasing mapping reduces to that of increasing mapping.Definition 6.
A mappingT:A→B is said to be generalized proximal weak contraction of rational type if it satisfies the condition that (6)kkkkx≤y,x≠ydu1,Tx=dA,Bdu2,Ty=dA,B⟹ψdu1,u2≤ψmx,y-ϕmx,y,where u1,u2,x,y∈A, ψ is an altering distance function, ϕ:[0,∞)→[0,∞) is a nondecreasing function with ϕ(t)=0 if and only if t=0, and m(x,y)=max⁡dx,u1dy,u2/dx,y,dx,y.One can see that, for a self-mapping, the notion of generalized proximal weak contraction of rational type reduces to generalized weak contraction of rational type.


## 2. Main Results

Now, let us state our main result.Theorem 7.
LetX be a nonempty set such that (X,≤) is a partially ordered set and (X,d) is a complete metric space. Let A and B be nonempty closed subsets of the metric space (X,d) such that A0≠∅. Let T:A→B satisfy the following conditions.(i)
T is continuous, proximally increasing, and generalized proximal weak contraction of rational type such that T(A0)⊆B0.(ii)
There existx0 and x1 in A0 such that (7)dx1,Tx0=dA,B,x0≤x1.Then, there exists an element x in A such that (8)dx,Tx=dA,B.Further, the sequence {xn}, defined by (9)dxn+1,Txn=dA,Bforn≥1,converges to the element x.Proof.
By hypothesis there exist elementsx0 and x1 in A0 such that (10)dx1,Tx0=dA,B,x0≤x1.Because of the fact that T(A0)⊆B0, there exists an element x2 in A0 such that (11)dx2,Tx1=dA,B.Since T is proximally increasing, we get x1≤x2.
Continuing this process, we can construct a sequence(xn) in A0 such that(12)dxn+1,Txn=dA,B∀n∈Nwithx0≤x1≤x2≤⋯xn≤xn+1⋯.If there exist n0 such that xn0=xn0+1, then d(xn0+1,Txn0)=d(xn0,Txn0)=d(A,B). This means that xn0 is a best proximity point of T and the proof is finished. Thus, we can suppose that xn≠xn+1 for all n.
Sincexn-1<xn, we get(13)ψdxn,xn+1≤ψmax⁡dxn-1,xndxn,xn+1dxn-1,xn,dxn-1,xn-ϕmax⁡dxn-1,xndxn,xn+1dxn-1,xn,dxn-1,xn=ψmax⁡dxn,xn+1,dxn-1,xn-ϕmax⁡dxn,xn+1,dxn-1,xn.Suppose that there exists m0 such that d(xm0,xm0+1)>d(xm0-1,xm0), and from (13), we have (14)ψdxm0,xm0+1≤ψdxm0,xm0+1-ϕdxm0,xm0+1<ψdxm0,xm0+1.
Hence, the sequence{d(xn,xn+1)} is monotone, nonincreasing and bounded. Thus, there exists r≥0 such that(15)limn→∞dxn,xn+1=r≥0.Since {d(xn,xn+1)} is a nonincreasing sequence, from (13), we get(16)ψdxn,xn+1≤ψdxn-1,xn-ϕdxn-1,xn,kkkkkkkkkkkkkkkkkkkkkkkkk∀xn-1<xn,n≥1.Suppose that limn→∞d(xn,xn+1)=r>0. Then the inequality (16) (17)ψdxn,xn+1≤ψdxn-1,xn-ϕdxn-1,xn≤ψd(xn-1,xn)implies that(18)limn→∞ϕdxn,xn+1=0.But, as 0<r≤d(xn,xn+1) and ϕ is nondecreasing function, (19)0<ϕr≤ϕdxn,xn+1,and this gives us limn→∞ϕ(d(xn,xn+1))≥ϕ(r)>0 which contradicts (18). Hence,(20)limn→∞dxn,xn+1=0.Now to prove that {xn} is a Cauchy sequence, suppose {xn} is not a Cauchy sequence. Then there exist ϵ>0 for which we can find subsequences {xm(k)} and {xn(k)} of {xn} such that n(k) is smallest index for which(21)nk>mk>k,dxmk,xnk≥ϵ.This means that(22)dxmk,xnk-1<ϵ,ϵ≤dxmk,xnkk≤dxmk,xnk-1+dxn(k)-1,xn(k)k<ϵ+dxnk-1,xnk.Letting k→∞ and using (20) we can conclude that(23)limk→∞dxmk,xnk=ϵ.By triangle inequality (24)dxnk,xmk≤dxnk,xnk-1+dxnk-1,xmk-1+dxmk-1,xmk,dxnk-1,xmk-1≤dxnk-1,xnk+dxnk,xmk+dxmk,xmk-1.Letting k→∞ in the above two inequalities, using (20) and (23), we get(25)limk→∞dxnk-1,xmk-1=ϵ.Since mk<nk, xm(k)-1≤xn(k)-1, from (16), we have (26)0<ψϵ≤ψdxmk,xnkk≤ψdxmk-1,xnk-1-ϕdxmk-1,xnk-1k≤ψdxmk-1,xnk-1.Using (25) and continuity of ψ in the above inequality we can obtain(27)limk→∞ϕdxmk-1,xnk-1=0.But, from limk→∞d(xn(k)-1,xm(k)-1)=ϵ we can find k0∈N such that for any k≥k0(28)ϵ2≤dxnk-1,xmk-1and consequently, (29)0<ϕϵ2≤ϕdxnk-1,xmk-1fork≥k0.Therefore, 0<ϕϵ/2≤ϕdxnk-1,xmk-1 and this contradicts (27). Thus, xn is a Cauchy sequence in A and hence converges to some element x in A. Since T is continuous, we have Txn→Tx.
Hence the continuity of the metric functiond implies that d(xn+1,Txn)→d(x,Tx). But (12) shows that the sequence d(xn+1,Txn) is a constant sequence with the value d(A,B). Therefore, d(x,Tx)=d(A,B). This completes the proof.Corollary 8.
LetX be a nonempty set such that (X,≤) is a partially ordered set and (X,d) is a complete metric space. Let A be a nonempty closed subset of the metric space (X,d). Let T:A→A satisfy the following conditions.(i)
T is continuous, proximally increasing, and generalized proximal weak contraction of rational type.(ii)
There exist elementsx0 and x1 in A such that dx1,Tx0=0 with x0≤x1.Then, there exist an element x in A such that d(x,Tx)=0.In what follows we prove that Theorem7 is still valid for T which is not necessarily continuous, assuming the following hypothesis in A. A has the property that(30)xnisanondecreasingsequenceinAkkksuchthatxn⟶x,thenx=sup⁡xn.Theorem 9.
Assume the conditions (30) and A0 is closed in X instead of continuity of T in Theorem 7; then the conclution of Theorem 7 holds.Proof.
Following the proof of Theorem7, there exists a sequence {xn} in A satisfying the following condition:(31)dxn+1,Txn=dA,B∀n∈Nwithx0≤x1≤x2≤⋯xn≤xn+1⋯.and xn converges to x in A. Note that the sequence {xn} in A0 and A0 is closed. Therefore, x∈A0. Since T(A0)⊆B0, we get Tx∈B0.
SinceTx∈B0, there exist y1∈A such that(32)dy1,Tx=dA,B.Since {xn} is a nondecreasing sequence and xn→x, then x=sup⁡{xn}. Particularly, xn≤x for all n. Since T is proximally increasing and from (31) and (32), we obtain xn+1≤y1. But x=sup⁡{xn} which implies x≤y1. Therefore, we get that there exist elements x and y1 in A0 such that(33)dy1,Tx=dA,B,x≤y1.Consider the sequence {yn} that is constructed as follows:(34)dyn+1,Tyn=dA,B∀n∈N.withx=y0≤y1≤y2≤⋯yn≤yn+1⋯.Arguing like above Theorem 7, we obtain that {yn} is a nondecreasing sequence and yn→y for certain y∈A. From (30), we have y=sup⁡{yn}. Since xn<x=y0≤y1≤yn≤y for all n, suppose that x≠y, T is generalized proximal weak contraction of rational type; from (31) and (34) we have (35)ψdxn+1,yn+1≤ψmax⁡dxn,xn+1dyn,yn+1dxn,yn,dxn,yn-ϕmax⁡dxn,xn+1dyn,yn+1dxn,yn,dxn,yn.Taking limit as n→∞ in the above inequality, we have (36)ψdx,y≤ψmax⁡0,dx,y-ϕmax⁡0,dx,y<ψdx,ywhich is a contradiction. Hence, x=y. We have x=y0≤y1≤yn=x, and therefore yn=x, for all n. From (34), we obtain x is a best proximity point for T. The proof is complete.Corollary 10.
Assume the condition (30) instead of continuity of T in the Corollary 8; then the conclusion of Corollary 8 holds.Now, we present an example where it can be appreciated that hypotheses in Theorems7 and 9 do not guarantee uniqueness of the best proximity point.Example 11.
LetX=0,1,1,0,-1,0,0,-1⊂R2 and consider the usual order (x,y)⪯(z,t)⇔x≤z and y≤t.Thus,X,⪯ is a partially ordered set. Besides, X,d2 is a complete metric space considering d2 the euclidean metric. Let A={(0,1),(1,0)} and B=0,-1,-1,0 be a closed subset of X. Then, d(A,B)=2, A=A0 and B=B0. Let T:A→B be defined as T(x,y)=-y,-x. Then, it can be seen that T is continuous, proximally increasing mappings such that T(A0)⊆B0. The only comparable pairs of elements in A are x⪯x for x∈A and there are no elements such that x≺y for x,y∈A. Hence, T is generalized proximal weak contraction of rational type. It can be shown that the other hypotheses of the Theorems 7 and 9 are also satisfied.However,T has two best proximity points (0,1) and (1,0).Theorem 12.
In addition to the hypotheses of Theorem7 (resp., Theorem 9), suppose that(37)foreveryx,y∈A0,thereexistz∈A0ggggggthatiscomparabletoxandyand then T has a unique best proximity point.Proof.
From Theorem7 (resp., Theorem 9), the set of best proximity points of T is nonempty. Suppose that there exist elements x,y in A which are best proximity points. We distinguish two cases:
Case 1. If x and y are comparable.
Sinced(x,Tx)=d(A,B) and d(y,Ty)=d(A,B).
SinceT is a generalized proximal weak contraction of rational type, we get (38)ψdx,y≤ψmax⁡⁡dx,xdy,ydx,y,dx,y-ϕmax⁡dx,xdy,ydx,y,dx,y=ψ(d(x,y))-ϕdx,ywhich implies ϕ(d(x,y))=0, and by our assumption about ϕ, we get d(x,y)=0 or x=y.
Case 2. If x is not comparable to y.
By the condition (37) there exist z0∈A0 comparable to x and y. We define a sequence {zn} as d(zn+1,Tzn)=d(A,B). Since z0 is comparable with x, we may assume that z0≤x. Now using T is proximally increasing, it is easy to show that zn≤x for all n.
Suppose that there existn0>1 such that x=zn0, and again by using T which is proximally increasing, we get x≤zn0+1. But, zn≤x for all n. Therefore, x=zn0+1. Arguing like above, we obtain x=zn for all n≥n0. Hence, zn→x as n→∞.
On the other hand, ifzn-1≠x for all n, now using T is a generalized proximal weak contraction of rational type, we have (39)ψ(d(zn,x))≤ψ(max⁡{d(zn-1,zn)d(x,x)d(zn-1,x),d(zn-1,x)})-ϕ(max⁡{d(zn-1,zn)d(x,x)d(zn-1,x),d(zn-1,x)})=ψ(d(zn-1,x))-ϕ(d(zn-1,x))<ψ(d(zn-1,x)).Since ψ is nondecreasing, we get d(zn,x)≤d(zn-1,x) Hence, the sequence {d(zn,x)} is monotone, nonincreasing and bounded. Thus, there exist r≥0 such that(40)limn→∞dzn,x=r≥0.Suppose that limn→∞d(zn,x)=r>0. Then the inequality (41)ψdzn,x≤ψdzn-1,x-ϕdzn-1,x≤ψdzn-1,ximplies that(42)limn→∞ϕdzn-1,x=0.But, as 0<r≤d(zn-1,x) and ϕ is nondecreasing function, 0<ϕ(r)≤ϕ(d(zn-1,x)), and this gives limn→∞ϕ(d(zn-1,x))≥ϕ(r)>0 which contradicts (42). Hence, limn→∞d(zn,x)=0. Analogously, it can be proved that limn→∞d(zn,y)=0. Finally, the uniqueness of the limit gives us x=y.Let us illustrate the above theorem with the following example.Example 13.
LetX=R2 and consider the order (x,y)⪯(z,t)⇔x≤z and y≤t, where ≤ is usual order in R.
Thus,(X,⪯) is a partially ordered set. Besides, (X,d1) is a complete metric space where the metric is defined as d1((x1,y1),(x2,y2))=x1-x2+y1-y2. Let A={(0,x):x∈[0,∞)} and B={(1,x):x∈[0,∞)} be a closed subset of X. Then, d(A,B)=1,A=A0 and B=B0. Let T:A→B be defined as T(0,x)=1,x/2. Then, it can be seen that T is continuous, proximally increasing mappings and proximally weak increasing such that T(A0)⊆B0. Now, we have to prove T is a generalized proximal weak contraction of rational type. That is to prove (43)kkkkkkx≤y,x≠yd0,x2,T0,x=1d0,y2,T0,y=1⟹ψd0,x2,0,y2≤ψm0,x,0,y-ϕm0,x,0,y,where m((0,x),(0,y))=max⁡xy/4y-x,y-x.
Note thatd((0,x/2),(0,y/2))=(1/2)(y-x) and (44)m0,x,0,y=xy4y-xif9xy≥x2+y2,y-xif9xy≤x2+y2.For ψ(t)=2t and ϕ(t)=t, we have ψd0,x/2,0,y/2=y-x and (45)ψm0,x,0,y-ϕm0,x,0,y=3xy8y-xif9xy≥x2+y2,y-xif9xy≤x2+y2.Now, we easily conclude that the mapping T is a generalized proximal weak contraction of rational type. Hence all the hypotheses of the Theorem 12 are satisfied. Also, it can be observed that (0,0) is the unique best proximity point of the mapping T.The following result, due to Luong and Thuan [6], is a corollary from the above Theorem 12, by taking A=B.Corollary 14.
In addition to the hypothesis of Corollary8 (resp., Corollary 10), suppose that(46)foreveryx,y∈A,thereexistz∈Akkkkkthatiscomparabletoxandyand then T has a unique fixed point.



---

*Source: 361657-2015-03-22.xml*