IJANAL International Journal of Analysis 2314-4998 2314-498X Hindawi 10.1155/2017/6173468 6173468 Research Article Best Proximity Point Results for Some Contractive Mappings in Uniform Spaces http://orcid.org/0000-0002-2602-7573 Olisama Victoria 1 Olaleru Johnson 2 Akewe Hudson 2 Sritharan Sivaguru 1 Department of Mathematics Adeniran Ogunsanya College of Education Otto/Ijanikin Lagos Nigeria aocoed.edu.ng 2 Department of Mathematics University of Lagos Lagos Nigeria unilag.edu.ng 2017 2642017 2017 20 06 2016 22 12 2016 26 02 2017 2642017 2017 Copyright © 2017 Victoria Olisama et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We introduce the concept of Jav-distance (an analogue of b-metric), ϕp-proximal contraction, and ϕp-proximal cyclic contraction for non-self-mappings in Hausdorff uniform spaces. We investigate the existence and uniqueness of best proximity points for these modified contractive mappings. The results obtained extended and generalised some fixed and best proximity points results in literature. Examples are given to validate the main results.

1. Introduction

The importance of fixed point theory emerges from the fact that it furnishes a unified approach and constitutes an important tool in solving equations which are not necessarily linear. A large number of problems can be formulated as nonlinear equations of the form T(x)=x, where T is a self-mapping in some framework; see  and other references therein. Nevertheless, an equation of the type T(x)=x does not necessarily possess a solution if T happens to be a non-self-mapping. In this case, one seeks an appropriate solution that is optimal in the sense that d(x,T(x)) is minimum. That is, we resolve a problem of finding an element x such that x is in best proximity to T(x) in some sense.

Best proximity point theorem analyzes the condition under which the optimisation problem, namely, infxAd(x,Tx), has a solution. The point x is called the best proximity point of T:AB, if d(x,Tx)=d(A,B), where d(A,B)=inf{d(x,y):xA;yB}. Note that the best proximity point reduces to a fixed point if T is a self-mapping.

A best proximity point problem is a problem of achieving the minimum distance between two sets through a function defined on one of the sets to the other.

The very popular best approximation theorem is due to Fan . If A is a nonempty compact subset of a Hausdorff locally convex topological vector space B and T:AB is a continuous mapping, then there exists an element xA such that d(x,Tx)=d(A,Tx). Fan’s results are not without shortcomings; the best approximation theorem only ensures the existence of approximate solutions, without necessarily yielding an optimal solution. But the best proximity point theorem provides sufficient conditions that ensure the existence of approximate solutions which are also optimal.

Afterwards many authors such as Eldred and Veeramani  have derived extensions of Fan’s Theorem and the best approximation theorems in many directions. Significant best proximity point results are in  and other references therein.

In fixed point theory, other spaces of study other than metric spaces have been used by different authors. Pseudometric spaces interestingly generalise metric spaces. One of the spaces in literature that generalises the metric and pseudometric spaces is the uniform space.

Weil  was the first to characterise uniform spaces in terms of a family of pseudometrics and Bourbaki  provided the definition of a uniform structure in terms of entourages. Aamri and El Moutawakil  gave some results on common fixed point for some contractive and expansive maps in uniform spaces and provided the definition of A-distance and E-distance.

Also, Olatinwo  established some common fixed point theorems for self-mappings in uniform spaces by using the A- and E-distances. Dhagat et al.  proved some common fixed point theorems for pairs of weakly and semicompatible mappings using E-distances in uniform spaces. Hussain et al.  apply the concept of cyclic (ψ) contractions to establish certain fixed and common point theorems on a Hausdorff uniform space.

In another development, Geraghty  introduced the generalised contraction self-map using comparison functions.

Another useful result is by Karapinar and Erhan  who gave the definition of a k-contractive map for non-self-mappings and Karapinar  who established some results on best proximity points of ψ-Geraghty contractive non-self-mappings.

Also, Basha  gave some necessary and sufficient conditions to claim the existence of unique best proximity points for proximal contractions in metric spaces. Mongkolkeha et al.  introduced proximal cyclic contractions in metric spaces which are more general than the class of proximal contractions given by Basha .

Motivated by the results above, we develop the concept of ϕp-proximal contraction and ϕp-proximal cyclic contractions in uniform spaces and obtain the existence and uniqueness of best proximity points of these non-self-contractive mappings using Jav-distance function.

2. Preliminaries

The following definitions are fundamental to our work.

Definition 1 (see [<xref ref-type="bibr" rid="B7">13</xref>]).

A uniform space (X,Γ) is a nonempty set X equipped with a uniform structure which is a family Γ of subsets of Cartesian product X×X which satisfy the following conditions:

If UΓ, then U contains the diagonal Δ={(x,x):xX}.

If UΓ, then U-1={(y,x):(x,y)U} is also in Γ.

If U,VΓ, then UVΓ.

If UΓ, and VX×X which contains U, then VΓ.

If UΓ, then there exists VΓ such that whenever (x,y) and (y,z) are in V, then (x,z) is in U.

Γ is called the uniform structure or uniformity of X and its elements are called entourages, neighborhoods, surroundings, or vicinities.

Definition 2 (see [<xref ref-type="bibr" rid="B2">14</xref>]).

Let (X,Γ) be a uniform space. A function p:X×XR+ is said to be an

A-distance if, for any VΓ, there exists δ>0 such that if p(z,x)δ and p(z,y)δ for some zX, then (x,y)V;

E-distance if p is an A-distance and p(x,y)p(x,z)+p(z,y),x,y,zX.

Another extension of a metric space is the b-metric space.

Definition 3 (see [<xref ref-type="bibr" rid="B5">2</xref>]).

Let X be a nonempty set and s1 be a given real number. A map d:X×XR is said to be a b-metric if and only if, for all x,y,zX, the following conditions are satisfied:

d(x,y)>0 with xy and d(x,y)=0 if and only if x=y.

d(x,y)=d(y,x).

(1)dx,ysdx,z+dz,y.

The pair (X,d) is called a b-metric space. If s=1, it becomes a metric space.

Examples in literature to show that b-metric is a generalisation of a metric space are in [16, 21].

Now, we introduce the concept of Jav-distance.

Definition 4.

Let (X,Γ) be a uniform space. A function p:X×XR+ is said to be a Jav-distance if

p is an A-distance,

p(x,y)s[p(x,z)+p(z,y)],x,y,zX,s1.

Note that the function p reduces to an E-distance if the constant s is taken as 1.

Example 5.

Let (X,Γ) be a uniform space and let d be a b metric on X. It is clear that (X,Γd) is a uniform space where Γd is the set of all subsets of X×X satisfying Bϵ={(x,y)X2:d(x,y)<ϵ} for some ϵ>0. Moreover, if ΓΓd, then d is an Jav-distance on (X,Γ).

Also, the following definition is required.

Definition 6 (see [<xref ref-type="bibr" rid="B7">13</xref>]).

Let (X,Γ) be a uniform space and p an A-distance on X

If VΓ,(x,y)V, and (y,x)V,x and y are said to be V-close. A sequence (xn) is a Cauchy sequence for Γ if, for any VΓ, there exists N1 such that xn and xm are V-close for n,mN. The sequence (xn)X is a p-Cauchy sequence if for every ϵ>0 there exists n0N such that p(xn,xm)<ϵ for all n,mN.

X is S-complete if for any p-Cauchy sequence {xn}, there exists xX such that limnp(xn,x)=0.

f:XX is p-continuous if limnp(xn,x)=0 implies limnp(f(xn),f(x))=0.

X is said to be p-bounded if δp(X)=sup{p(x,y):x,yX}<.

To guarantee the uniqueness of the limit of the Cauchy sequence for Γ, the uniform space (X,Γ) needs to be Hausdorff.

Definition 7 (see [<xref ref-type="bibr" rid="B7">13</xref>]).

A uniform space (X,Γ) is said to be Hausdorff if and only if the intersection of all the VΓ reduces to the diagonal Δ of X, Δ={(x,x),xX}. In other words, (x,y)V for all VΓ implies x=y.

A uniform structure Γ defines a unique topology τ(Γ) on X for which the neighborhoods of xX are the sets V(x)={yX:(x,y)V},VΓ.

f : X X is τ(Γ) continuous if limnxn=x with respect to τ(Γ) implies limnf(xn)=f(x) with respect to τ(Γ).

Observe that all the above maps are self-mappings.

A large number of articles investigate non-self-contractive mappings on metric spaces. Some of these are given below.

Definition 8 (see [<xref ref-type="bibr" rid="B15">19</xref>]).

Let (X,d) be a metric space and A and B be nonempty subsets of X. A mapping T:AB is said to be a k-contraction if there exists k[0,1) such that (2)dTx,Tykdx,y,x,yA.

Definition 9 (see [<xref ref-type="bibr" rid="B1">23</xref>]).

Let A and B be nonempty subsets of a metric space (X,d) and let T:ABAB such that T:ABAB,

T is cyclic if T(A)B and T(B)A.

T is called a cyclic contraction if (3)dTx,Tykdx,y+1-kdA,B,xA,  yB,for some k[0,1).

T:ABAB is called a cyclic ϕ-contraction if ϕ:[0,)[0,) is a strictly increasing map (4)dTx,Tydx,y-ϕdx,y+ϕdA,B,xA,  yB.

Note that (4) becomes (3) with ϕ(q)=(1-k)q for all q0. But the converse is not true in general (see ).

Among the generalisations of the Banach contraction is the proximal contraction given by Basha in  and the proximal cyclic contraction in .

Definition 10 (see [<xref ref-type="bibr" rid="B6">21</xref>]).

Let (A,B) be a nonempty subset of a complete metric space (X,d). A mapping T:AB is said to be a proximal contraction if there exists a nonnegative real number α<1 such that (5)du,Tx=dA,Bdv,Ty=dA,Bdu,vαdx,y,for all u,x,v,yA.

A0={xA:p(x,y)=p(A,B)  for  some  yB},

B0={yB:p(x,y)=p(A,B)  for  some  xA}.

Basha  proved the following theorem.

Theorem 11 (see [<xref ref-type="bibr" rid="B6">21</xref>]).

Let A,B be two nonempty subsets of a complete metric space (X,d). Suppose that T(A0) is nonempty and closed. Let T:AB satisfy the following conditions:

T is a proximal contraction,

T(A0)B0.

Then there exists a point xA such that d(x,T(x))=d(A,B). Moreover, if T is injective on A, then the point x such that d(x,T(x))=d(A,B) is unique.

Definition 12 (see [<xref ref-type="bibr" rid="B18">22</xref>]).

Let S:AB and T:BA. The pair (S,T) is called a proximal cyclic contraction pair if there exists α[0,1) such that (6)da,Sx=dA,Bdb,Ty=dA,Bda,bαdx,y+1-αdA,B,for all a,xA,b,yB.

Given nonempty subsets A and B of a uniform space (X,Γ), we adopt the following notations and definitions used for metric spaces to the context of uniform spaces.

Definition 13.

Let S:AB and g:AA be an isometry. The mapping S is said to preserve the isometric distance with respect to g if (7)pSgx,Sgy=pgx,gy,x,yA.

Definition 14.

An element x is called a best proximity point of a mapping T:AB if it satisfies the condition that p(x,T(x))=p(A,B)=inf{p(x,y):xA;yB}.

Now, we give the definition of ϕp-proximal contraction and ϕp-proximal cyclic contraction for non-self-mapping in uniform spaces.

Definition 15.

Let (A,B) be a pair of nonempty subsets of an S-complete Hausdorff uniform space (X,Γ) such that p is an Jav-distance on X. A mapping T:AB is said to be a ϕp-proximal contraction if there exists a nondecreasing continuous weak comparison function ϕ:R+R+ satisfying the following.

For each t(0,),0<ϕ(t) and ϕ(0)=0,

limnϕn(t)=0,t(0,),

ϕ(t)<tt(0,),

n=0ϕn(t) converges for any t, such that j,l,k,mA, such that (8)pj,Tk=pA,Bpl,Tm=pA,Bpj,lϕpk,m.

Definition 16.

Let (A,B) be a pair of nonempty subsets of S-complete Hausdorff uniform space (X,Γ) such that p is an Jav-distance on X. Suppose S:AB and T:BA are mappings. The pair (S,T) is said to be a ϕp-proximal cyclic contraction if there exists a nondecreasing continuous weak comparison function ϕ:R+R+ satisfying μ1μ4 above, such that (9)pj,Sk=pA,Bpl,Tm=pA,Bpj,lϕpk,m+pA,B-ϕpA,B,for all j,kA and l,mB.

It is easy to see that a self-mapping that is a ϕp-proximal contraction is a contraction. But a non-self ϕp-proximal contraction is not necessarily a contraction map. If ϕ(w)=αw and Jav-distance p is replaced with a metric d, (9) reduces to (6). Similarly, (8) reduces to (5). Also, (9) and (8) reduce to (2) if A=B,S=T, ϕ(w)=kw and if the Jav-distance p is replaced with a metric d, in the sense that Γ={(x,y)X2:d(x,y)<sϵ},s0.

The following example shows that E-distance function p is different from the metric distance function d. In fact, the E-distance function p reduces to the metric distance function d when X is a metric space.

Example 17.

Let A=(-,0] and B=[2,+) be nonempty closed subsets of X=R with the usual metric. Let H:AB be a mapping given by H(x)=-4/x and u=-1,v=0,x=-4,y=-2 and let ϕ(x)=x/3. It is easy to see that d(-1,H(-4))=d(A,B)=d(0,H(-2))=2.

Clearly, H:AB is not a ϕ-proximal contraction; that is, d(-1,0)>ϕ(d(-4,-2)).

H has no best proximity point since there is no xA such that d(x,H(x))=2.

Now, taking u=x-2,x=u+2,x<u. And consider p defined as p(x,y)=2x.

Clearly, p(u,v)ϕ(p(x,y)) for all u,v,x,yA.H is a ϕp-proximal contraction and −1 is the unique best proximity point of H.

The following Lemma, which is true for self-mappings (see Lemma 2.4 ) can be proved for non-self-mappings.

Lemma 18 (see [<xref ref-type="bibr" rid="B2">14</xref>]).

Let (X,Γ) be a Hausdorff uniform space and p be an A-distance on X. Let {xn}n=0,{yn}n=0 be arbitrary sequences in X and {αn}n=0,{βn}n=0 be sequences in R+ converging to 0. Then, for x,y,zX, the following holds:

If p(xn,y)αn and p(xn,z)βnnN, then y=z. In particular, p(x,y)=0 and p(x,z)=0, and then y=z.

If p(xn,yn)=p(A,B) and p(xn,zn)=p(A,B), then yn=zn.

If p(xn,yn)αn and p(xn,z)βnnN, then, (yn)n=0 converges to z.

If p(xn,xm)αnm>n, then {xn}n=0 is a p-Cauchy sequence in (X,Γ).

The major aim of this paper is to prove results similar to Theorem 11 above in uniform spaces and give the modification of results on proximal contractions in  in uniform spaces.

3. Main Results

We give the first theorem.

Theorem 19.

Let (A,B) be a pair of nonempty subset X of an S-complete Hausdorff uniform space (X,Γ) such that p is an Jav-distance on X and is A0. Suppose a map F:AB is such that F(A0)B0 is a ϕp-proximal contraction. Then there exists a unique point xA0 such that p(x,F(x))=p(A,B).

Proof.

Let x0A0, since A0 and F(A0)B0. There exists x1A such that p(x1,F(x0))=p(A,B). Also, since F(x1)B0, there exists x2A0 such that p(x2,F(x1))=p(A,B). Furthermore, we obtain the sequences {xn} and {xn+1} subsets of A0 such that (10)pxn,Fxn-1=pA,B,(11)pxn+1,Fxn=pA,B,nN.We show that {xn} is a complete p-Cauchy sequence whose limit is the unique best proximity point of F. Since F is a ϕp-proximal contraction, from (10) and (11) we have (12)pxn,xn+1ϕpxn-1,xn.Thus by induction, (13)pxn,xn+1ϕnpx0,x1for any n=1,2,.

Since p is an Jav-distance, we have p(xn,xm)s[p(xn,xn+1)++p(xm-1,xm)],s1. Now for r1, (14)pxn,xn+rsϕnpx0,x1++ϕn+r-1px0,x1.Let Jn=st=0nϕt(p(x0,x1)),n0. Then (15)pxn,xn+rJn+r-1-Jn-1.Suppose p(x0,x1)>0, and since ϕ is a weak comparison function, by Definition 15(μ4), it follows that (16)t=0ϕtpx0,x1<.So there exists a J[0,) such that limnJn=J. Then by (15), (17)limnpxn,xn+r=0.Repeating the same argument, we obtain limnp(xn+r,xn)=0.

Therefore, the sequence {xn} is a p-Cauchy in the S-complete space (X,Γ). Hence there exists xA0 such that (18)limnpxn,x=0,since A0 is closed. We prove that x is the best proximity point of F; that is, p(x,F(x))=p(A,B).

Also, since F(x0)B0 and F(x)B0, there exists an element qA0 such that (19)pq,Fx=pA,B.Using (19) and (11) and since F is a ϕp-proximal contraction, (20)pq,xn+1ϕpx,xn.As n,p(q,xn+1)0 since p(x,xn)0. Therefore, xnq and thus q=x. So from (19), (21)px,Fx=pA,B.To guarantee the uniqueness of x, we show that (X,Γ) is Hausdorff. Suppose there exists y such that (22)py,Fy=pA,B.By the ϕp-proximal contraction F, (23)px,yϕpx,y<px,y,which implies p(x,y)=0. Similarly, p(y,x)=0. But by the second property of Jav-distance, (24)px,xspx,y+py,x.Hence, (25)px,x=0.We conclude that x=y.

Corollary 20.

Let (X,d) be a complete metric space. Suppose f:AA satisfies d(f(x),f(y))kd(x,y),k(0,1); then f has a unique fixed point.

Proof.

Set ϕ(t)=kt, A=B, and Γ={(x,y)X2:d(x,y)<ϵ} in Theorem 19, to obtain the result.

Corollary 21 (see [<xref ref-type="bibr" rid="B15">19</xref>]).

Let A and B be two nonempty subsets of a complete metric space (X,d). Suppose f:AB satisfies d(f(x),f(y))kd(x,y),k(0,1). Then f has a unique best proximity point.

Proof.

Set ϕ(t)=kt,j=f(m),l=f(k) and Γ={(x,y)X2:d(x,y)<ϵ} in Theorem 19, to obtain the corollary.

Corollary 22 (see [<xref ref-type="bibr" rid="B6">21</xref>]).

Let A and B be two nonempty subsets of a complete metric space (X,d). Suppose A0 is nonempty and closed and T:AB satisfies the following conditions:

T is a proximal contraction,

T(A0)B0.

Then there exists a unique point xA such that d(x,T(x))=d(A,B). Moreover, xA, and there exists a sequence {xn}A such that d(xn+1,T(xn))=d(A,B) for every nN{0} and xnx.

Proof.

Set ϕ(t)=kt and Γ={(x,y)X2:d(x,y)<ϵ} in Theorem 19.

Now, we establish some results of best proximity point for ϕp-proximal cyclic contractions in uniform spaces.

Theorem 23.

Let (A,B) be a pair of nonempty closed subset X of a p-bounded and S-complete Hausdorff uniform space (X,Γ) such that A0,B0 and p is an Jav-distance on X. Let F:AB, G:BA, and h:ABAB satisfy the following conditions:

the pair (F,G) is a ϕp-proximal cyclic contraction,

F(A0)B0, G(B0)A0,

A0h(A0) and B0h(B0),

h is isometry.

Then there exist unique points xA and yB such that (26)phx,Fx=phy,Gy=px,y=pA,B.Further, if x0 is any fixed element in A0 and y0 is any fixed element in B0, the sequences {xn} and {yn}, defined by (27)phxn+1,Fxn=pA,B,phyn+1,Fyn=pA,B,converge to the best proximity points x and y, respectively.

Proof.

Let x0 be fixed element in A0. Since F(A0)B0 and A0h(A0), it follows that there exists an element x1A0 such that (28)phx1,Fx0=pA,B.Again, since F(A0)B0 and A0h(A0), there exists an element x2A0 such that (29)phx2,Fx1=pA,B.Following the steps in the proof of Theorem 19, we can find xnA0 such that (30)phxn,Fxn-1=pA,B.By induction, one can determine an element xn+1A0 such that(31)phxn+1,Fxn=pA,B.Also, since h is an isometry and by the ϕp-proximity cyclic contraction using (30) and (31), it follows that, for each n1,(32)phxn,hxn+1=pxn,xn+1ϕpxn-1,xn+pA,B-ϕpA,B(33)<ϕ2pxn-2,xn-1<ϕnpx0,x1.Since p is an Jav-distance, we have p(xn,xm)s[p(xn,xn+1)++p(xm-1,xm)]. Now for q1, (34)pxn,xn+qsϕnpx0,x1++sϕn+q-1px0,x1.Let Jn=st=0nϕt(p(x0,x1)),n0, and then (35)pxn,xn+qJn+q-1-Jn-1.Next we show that {xn} is p-Cauchy in the S-complete space X; that is, (36)limnpxn,xn+q=0,limnpxn+q,xn=0,for any q1.

Recall that (37)pxn+1,Fxn=pA,B,if there exists n0N such that xn0+1=xn0, we are done, and xn0 is the required best proximity point of F. Thus we assume that xn+1xn.

Suppose p(x0,x1)>0. Now using Definition 15(μ4), we have (38)t=0ϕtpx0,x1<,so there exists a J[0,) such that limnJn=J.

Then by (35), (39)limnpxn,xn+q=0.Repeating the same argument, we obtain (40)limnpxn+q,xn=0.So the sequence {xn} is p-Cauchy in the S-complete space (X,Γ).

Hence, {xn} converges to some element xA. Similarly, since F(B0)A0 and A0h(A0), there exists a sequence {yn} such that it converges to some element yB and from (31), (41)phyn+1,Gyn=pA,B.Since the pair (F,G) is a p-proximal cyclic contraction and h is isometry, using (31) and (41), we have (42)phxn+1,hyn+1=pxn+1,yn+1ϕpxn,yn+pA,B-ϕpA,B.By (33), on taking limit as n, we have (43)px,yϕpx,y+pA,B-ϕpA,B.We show that p(x,y)=p(A,B). Assume p(x,y)p(A,B), from (43), p(x,y)<p(x,y), a contradiction. Hence, (44)px,y=pA,B.Thus, xA0 and yB0. Since F(A0)B0 and G(B0)A0, there exist τA and ηB such that (45)pτ,Fx=pA,B,pη,Gy=pA,B.Now, we show that τ=h(x) and η=h(y).

Since (F,G) is a ϕp-proximal cyclic contraction, using (44) and (31) we have (46)pτ,hxn+1ϕpx,xn+pA,B-ϕpA,B.Letting n in (46), p(τ,h(x))<p(x,x), and since p is an Jav-distance, (47)pτ,hxspx,xn+pxn,x.Again letting n, we get p(τ,h(x))0 and so, τ=h(x). Therefore we have (48)phx,Fx=pA,B.Similarly, we can obtain η=h(y) and so, (49)phy,Gy=pA,B.Thus, from (44), (48), and (49), we get(50)px,y=phx,Fx=phy,Gy=pA,B.

Next we prove the uniqueness of x and y. Suppose that there exist xaA and yaB with xxa and yya such that (51)phxa,Fxa=pA,B,(52)phya,Gya=pA,B.Since h is an isometry, and F is a p-proximal cyclic contraction, using (48) and (51), we have (53)phx,hxa=px,xaϕpx,xa+pA,B-ϕpA,B.

p ( x , x a ) < p ( x , x a ) , a contradiction. Hence, p(x,xa)=0. Similarly, we show that p(xa,x)=0. But since p is a Jav-distance, we have (54)pxa,xaspxa,x+px,xa.Therefore, (55)pxa,xa=0.Now we have p(xa,xa)=0 and p(x,xa)=0. By Lemma 18(a), we conclude that xa=x. Similarly, ya=y.

Corollary 24 (see [<xref ref-type="bibr" rid="B12">17</xref>]).

Let (X,Γ) be a S-complete Hausdorff uniform space and p an E-distance on X. Suppose T:XX is a cyclic ψ-contraction such that p(f(x),f(y))ψ(p(x,y)), for all x,yX, where ψ is a weak comparison function. Then f has a unique fixed point.

Proof.

The proof follows from Theorem 23 if F=G, A=B, and Jav-distance is reduced to E-distance function.

Corollary 25 (see [<xref ref-type="bibr" rid="B11">18</xref>]).

Let (X,d) be a complete metric space and T:XX be a Geraghty contraction satisfying d(Tx,Ty)β(d(x,y))d(x,y) for each x,yX, where βS. Then T has a unique fixed point.

Proof.

The proof follows from Theorem 23 if A=B,F=G,ϕ(t)=β(t)(t) and p is a metric distance.

We give the following example to show that (9) generalises (6).

Example 26.

Let A,BX=R+ such that A=[1/4,1/2],B=[3/4,1]. Clearly, d(A,B)=1/4. Suppose u=1/4,v=3/4,x=1/2, and y=1.

Let S(x),T(x), and p(x,y) be defined by(56)Sx=3x2,Tx=x4,xA;x2,otherwise,px,y=y3,xA,  yB;2y3,otherwise.Now, for F(A0)B0 we obtain

A0={xA:p(x,y)=p(A,B)=1/4  for  some  yB}={1/2},

B0={yB:p(x,y)=p(A,B)=1/4  for  some  xA}={3/4}.

Now, (9) generalises (6) in the sense that

(1) (57)d14,S12=dA,Bd34,T1=dA,Bd14,34kd12,1+1-k14,for all u,xA;v,yB.

1 / 2 k ( 1 / 2 ) + ( 1 - k ) 1 / 4 = 1 / 4 ( k + 1 ) , k [ 0,1 ) a contradiction.

Hence, (6) fails. (S,T) is not a proximal cyclic contraction. We see that (S,T) has no unique best proximity point since there is no xA such that d(u,S(u))=d(v,T(v))=1/4.

But taking u=x-1,x=u+1,x<u,

(2) (58)pu,vϕpx,y-ϕpA,B+pA,B,u,xA,  v,yBbecomes pu,v<px,y.  (S,T) is a ϕp-proximal contraction. Clearly, p(x,S(x))=p(y,T(y))=1/4 and 1/2 is the unique best proximity point of the pair S, while 3/4 is the unique best proximity point of the pair T. Hence, (9) is different from (6).

We give the following examples to show that the ϕp-proximal cyclic contraction is different from the Geraghty contraction.

Example 27.

Consider the usual metric (X,d),X=[0,1] and d(x,y)=x-y and let A=[0,1/10] and B=[1/5,1]. Obviously, d(A,B)=1/10, A0=1/10 and B0=1/5, F(A0)B0. Let S:AB,T:BA, be defined as S(k)=5k+1/5,T(m)=m/10 taking j=1/50,k=1/25,i=2/5 and m=1. Also, consider β(x)=1-x. And p and ψ are defined as follows:(59)ψx=2x,x0,12;15x,x12,1.px,y=y4,y0,12,  y>x;2y,y12,1;1,otherwise.We show that S is not a Geraghty contraction ψdSx,Syβ(ψ(d(j,k)))(ψ(d(j,k)))d(2/5,8/15)>4/50×1/50, a contradiction. S is not a Geraghty contraction.

We see that S has no best proximity points since there is no kA such that d(k,S(k))=1/25.

But (S,T) is a ϕp-proximal cyclic contraction. Clearly taking m=2l, p(j,S(k))=p(A,B)=p(l,T(m)) implies p(j,l)ϕ(p(k,m))+p(A,B)-ϕ(p(A,B)),j,kA and l,mB.

( S , T ) is a ϕp-proximal cyclic contraction and 1/25 is the unique best proximity point of S while 2/5 is the unique best proximity point of T.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

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

Abbas M. Khan A. R. Nazir T. Coupled common fixed point results in two generalized metric spaces Applied Mathematics and Computation 2011 217 13 6328 6336 10.1016/j.amc.2011.01.006 MR2773377 2-s2.0-79952003710 Bakhtin I. A. The contraction mapping principle in almost metric spaces Journal of Functional Analysis 1989 30 26 37 Olaleru J. O. Okeke G. A. Akewe H. Coupled fixed point theorems for generalised ϕ-mappings satisfying contractive condition of integral type on cone metric spaces International Journal of Mathematical Modeling & Computations 2012 2 2 87 98 Olaleru J. O. Olaoluwa H. O. Common fixed points of four mappings satisfying weakly contractive-like condition in cone metric spaces Applied Mathematical Sciences 2013 7 57–60 2897 2908 10.12988/ams.2013.13257 MR3065193 2-s2.0-84878627215 Fan K. Extensions of two fixed point theorems of F. E. Browder Mathematische Zeitschrift 1969 112 234 240 10.1007/BF01110225 MR0251603 ZBL0185.39503 2-s2.0-0000310867 Eldred A. A. Veeramani P. Existence and convergence of best proximity points Journal of Mathematical Analysis and Applications 2006 323 2 1001 1006 10.1016/j.jmaa.2005.10.081 MR2260159 2-s2.0-33750609842 Abkar A. Gabeleh M. A note on some best proximity point theorems proved under P-property Abstract and Applied Analysis 2013 2013 3 189567 10.1155/2013/189567 Mihaela A. P. Best proximity point theorems for weak cyclic Kannan contractions Filomat 2011 25 1 145 154 10.2298/fil1101145p MR2932779 2-s2.0-79960776428 Olaleru J. O. Olisama O. V. Coupled best proximity points of generalised Hardy-Rogers type cyclic (ω)-contraction mappings International Journal of Mathematical Analysis and Optimization 2015 1 33 54 Olisama V. O. Olaleru J. O. Olaoluwa H. Quadruple best proximity points of Q-cyclic contraction pair in abstract metric spaces Asian Journal of Mathematics and Applications 2015 2015 21 ama0206 Raj V. S. A best proximity point theorem for weakly contractive non-self-mappings Nonlinear Analysis: Theory, Methods & Applications 2011 74 14 4804 4808 10.1016/j.na.2011.04.052 MR2810719 2-s2.0-79957926969 Weil A. Sur les Espaces a Structure Uniforme et sur la Topologie Generale 1937 551 Paris, France Hermann Actualités Scientifiques et Industrielles Bourbaki N. Topologie Generale, Chapitre 1: Structures Topologiques, Chapitre 2: Structures Uniformes. Quatrieme Edition 1965 Paris, France Hermann Actualites Scientifiques et Industrielles, no. 1142 Aamri M. El Moutawakil D. Common fixed point theorems for E-contractive or E-expansive maps in uniform spaces Acta Mathematica Academiae Paedagogicae Nyí Regyháziensis (New Series) 2004 20 1 83 89 MR2129672 Olatinwo M. O. Some common fixed point theorems for self mappings in uniform space Acta Mathematica Academiae Paedagogiace Nyíregyháziensis 2007 23 1 47 54 Dhagat V. B. Singh V. Nath S. Fixed point theorems in uniform space International Journal of Mathematical Analysis 2009 3 4 197 202 MR2517578 2-s2.0-70349433864 Hussain N. Karapinar E. Sedghi S. Shobkolaei N. Firouzian S. Cyclic ϕ-contractions in uniform spaces and related fixed point results Abstract and Applied Analysis 2014 2014 7 976859 10.1155/2014/976859 MR3182315 2-s2.0-84897545831 Geraghty M. A. On contractive mappings Proceedings of the American Mathematical Society 1973 40 604 608 10.2307/2039421 MR0334176 Karapinar E. Erhan M. Best proximity point on different type contractions Applied Mathematics & Information Sciences 2011 5 3 558 569 Karapinar E. On best proximity point of ψ-Geraghty contractions Fixed Point Theory and Applications 2013 2013, article 200 10.1186/1687-1812-2013-200 MR3095321 2-s2.0-84899892844 Sadiq Basha S. Best proximity points: optimal solutions Journal of Optimization Theory and Applications 2011 151 1 210 216 10.1007/s10957-011-9869-4 MR2836473 Mongkolkeha C. Cho Y. J. Kumam P. Best proximity points for Geraghty's proximal contraction mappings Fixed Point Theory and Applications 2013 2013, article 180 10.1186/1687-1812-2013-180 MR3089869 2-s2.0-84893841471 Al-Thagafi M. A. Shahzad N. Convergence and existence results for best proximity points Nonlinear Analysis: Theory, Methods & Applications 2009 70 10 3665 3671 10.1016/j.na.2008.07.022 MR2504453 2-s2.0-61749086943 Kirk W. A. Reich S. Veeramani P. Proximinal retracts and best proximity pair theorems Numerical Functional Analysis and Optimization 2003 24 7-8 851 862 10.1081/nfa-120026380 MR2011594