# Common Fixed Points of a Pair of Hardy Rogers Type Mappings on a Closed Ball in Ordered Dislocated Metric Spaces

Authors: Muhammad Arshad; Abdullah Shoaib; Pasquale Vetro
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

Common fixed point results for mappings satisfying locally contractive conditions on a closed ball in an ordered complete dislocated metric space have been established.
The notion of dominated mappings is applied to approximate the unique solution of nonlinear functional equations. Our results improve several well-known conventional results.


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## Body

## 1. Introduction and Preliminaries

LetT:X→X be a mapping. A point x∈X is called a fixed point of Tif x=Tx. Fixed points results of mappings satisfying certain contractive conditions on the entire domain have been at the centre of rigorous research activity, (e.g., see [1, 2]), and it has a wide range of applications in different areas such as nonlinear and adaptive control systems, parameterize estimation problems, computing magnetostatic fields in a nonlinear medium, and convergence of recurrent networks (see [3–5]).Recently, many results appeared related to fixed point theorems in complete metric spaces endowed with a partial ordering. Ran and Reurings [6] proved an analogue of Banach's fixed point theorem in metric space endowed with a partial order and gave applications to matrix equations. In this way, they weakened the usual contractive condition. Subsequently, Nieto and Rodríguez-López [7] extended this result in [6] for nondecreasing mappings and applied it to obtain a unique solution for a 1st-order ordinary differential equation with periodic boundary conditions. Thereafter, many works related to fixed point problems have also been considered in partially ordered metric spaces (see [8–10]).Hitzler and Seda [11] introduce the concept of a dislocated topology and its corresponding generalized metric named as dislocated metric and have established a fixed point theorem in complete dislocated metric spaces which generalizes the Banach contraction principle. The notion of dislocated topology has useful applications in the context of logic programming semantics (see [12]). Further useful results can be seen in [13–16].From the application point of view the situation is not yet completely satisfactory because it frequently happens that a mappingT is a contraction not on the entire space X but merely on a subset Y of X. However, if Y is closed and a sequence {xn} in X converges to some x in X, then by imposing a subtle restriction on the choice of x0, one may force the sequence to stay eventually in Y. In this case, one can establish the existence of a fixed point of T. Arshad et al. [17] proved a significant result concerning the existence of fixed points of a mapping satisfying contractive conditions on closed ball in a complete dislocated metric space. Other results on closed ball can be seen in [18–20]. The dominated mapping [21]which satisfies the condition fx⪯x occurs very naturally in several practical problems. For example, if x denotes the total quantity of food produced over a certain period of time and f(x) gives the quantity of food consumed over the same period in a certain town, then we must have fx⪯x. In this paper, we will exploit this concept. We have obtained fixed point theorems for contractive dominated self-mappings in an ordered complete dislocated metric space on a closed ball to generalize, extend, and improve a classical fixed point result in [22]. We have used weaker contractive conditions and weaker restrictions to obtain unique fixed point. Our results do not exist even yet in metric spaces. We have given an example which shows how this result can be used when the corresponding results cannot.Consistent with [12, 21, 22], the following definitions and results will be needed in the sequel.Definition 1.
LetX be a nonempty set and let dl:X×X→[0,+∞) be a function, called a dislocated metric (or simply dl-metric) if the following conditions hold for any x,y,z∈X: (i)
ifdl(x,y)=0, then x=y,(ii)
dl(x,y)=dl(y,x),(iii)
dl(x,y)≤dl(x,z)+dl(z,y)-dl(z,z).The pair (X,dl) is then called a dislocated metric space.It is clear that ifdl(x,y)=0, then from (i), x=y. But if x=y, dl(x,y) may not be 0.Remark 2.
From (iii) of Definition1, we deduce
(1)dl(x,y)≤dl(x,y)+dl(z,z)≤dl(x,z)+dl(z,y)
for all x,y,z∈X.Recently Sarma and Kumari [15] proved results that establish existence of a topology induced by a dislocated metric and that this topology is metrizable. This topology has as a base the family of sets {B(x,ε)∪{x}:x∈X,ε>0}, where B(x,ε) is an open ball and B(x,ε)={y∈X:dl(x,y)<ε} for some x∈X and ε>0. Examples of the dl-metric spaces are given below.Example 3.
IfX=[0,+∞), then dl(x,y)=x+y defines a dislocated metric dl on X.Example 4.
IfX=[0,+∞), then dl(x,y)=max{x,y} defines a dislocated metric dl on X.Definition 5 (see [13]).
A sequence{xn} in a dl-metric space (X,dl) is called a Cauchy sequence if given ε>0, there exists n0∈N such that for all n,m≥n0, we have dl(xm,xn)<ε.Definition 6 (see [13]).
A sequence{xn} in a dl-metric space converges with respect to dl if there exists x∈X such that dl(xn,x)→0 as n→+∞. In this case, x is called limit of {xn}, and we write xn→x.Definition 7.
Adl-metric space (X,dl) is called complete if every Cauchy sequence in X converges to a point in X.Definition 8.
LetX be a nonempty set. Then (X,⪯,dl) is called an ordered dislocated metric space if (i)dl is a dislocated metric on X and (ii) ⪯ is a partial order on X.Definition 9.
Let(X,⪯) be a partial ordered set. Then x,y∈X are called comparable if x⪯y or y⪯x holds.Definition 10 (see [21]).
Let(X,⪯) be a partially ordered set. A self-mapping f on X is called dominated if fx⪯x for each x in X.Example 11.
LetX=[0,1] be endowed with the usual ordering and f:X→X defined by fx=xn for some n∈ℕ. Since fx=xn≤x for all x∈X, therefore f is a dominated mapping.Definition 12.
LetX be a nonempty set and T,f:X→X. A point y∈X is called point of coincidence of T and f if there exists a point x∈X such that y=Tx=fx. The mappings T,f are said to be weakly compatible if they commute at their coincidence point (i.e., Tfx=fTx whenever Tx=fx). If ⪯ is a partial order on X, then the mapping T is called f-dominated if Tx⪯fx for all x∈X; see [23].We require the following lemmas for subsequent use.Lemma 13 (see [24]).
LetX be a nonempty set and f:X→X a function. Then there exists a subset E⊂X such that fE=fX and f:E→X is one-to-one.Lemma 14 (see [25]).
LetX be a nonempty set and the mappings S,T,f:X→X have a unique point of coincidence v in X. If (S,f) and (T,f) are weakly compatible, then S,T,f have a unique common fixed point.Theorem 15 (see [22]).
Let(X,d) be a complete metric space, S:X→X a mapping, r>0, and x0 an arbitrary point in X. Suppose that there exists k∈[0,1) with
(2)d(Sx,Sy)≤kd(x,y),∀x,y∈Y=B(x0,r)¯
and d(x0,Sx0)<(1-k)r. Then there exists a unique point x* in B(x0,r)¯ such that x*=Sx*.


## 2. Fixed Points of Hardy-Rogers Mappings

The Banach contraction principle in the setting of dislocated metric spaces was established in [26] (see also [11]).Theorem 16.
Let(X,dl) be a complete dislocated metric space, and let T:X→X be a contraction. Then T has a unique fixed point.The following example shows that there exists a contraction in a dislocated metric space that is not a contraction in a metric space.Example 17 (see [27, Example 1]).
LetX=[0,1] and dl(x,y)=max{x,y}. Fix r∈[0,1), and define T:X→X by
(3)Tx={0ifx=0r2n-r2n-12n(2nx-1)if12n≤x≤12n-1r2n+r2n+12n(2nx-1)if12n+1≤x≤12n.
For allx,y∈X, we have
(4)dl(Tx,Ty)=max{Tx,Ty}≤max{rx,ry}=rmax{x,y}=rdl(x,y),
which ensures that T is a contraction, and so, by Theorem 16, T has a unique fixed point in X.
Now, we show thatT is not a contraction in the complete metric space (X,d), where d(x,y)=|x-y| for all x,y∈X. In fact, if for odd n>1 we choose x=1/(2n-1) and y=1/(n-1), we have
(5)d(Tx,Ty)=rn-1≤d(x,y)=n(n-1)(2n-1)≤35(n-1)
which is not satisfied if r>3/5. Consequently, T is not a contraction if r>3/5.Thus, the above example gives a strong motivation for studying fixed point results in the setting of dislocated metric spaces.For some results of fixed point in the setting of complete dislocated metric spaces for mappings that satisfy a contractive condition of Hardy and Rogers see [28]. The following theorem is one of our main results.Theorem 18.
Let(X,⪯,dl) be an ordered complete dislocated metric space, x0∈X, r>0, and S,T:X→X two dominated mappings. Suppose that there exists a,b,c∈[0,1) with a+2b+2c<1 such that
(6)dl(Sx,Ty)≤adl(x,y)+b[dl(x,Sx)+dl(y,Ty)]+c[dl(y,Sx)+dl(x,Ty)]
for all comparable elements x,y in B(x0,r)¯ and
(7)dl(x0,Sx0)≤(1-λ)r,
where λ=(a+b+c)/(1-b-c). If for a nonincreasing sequence {xn} in B(x0,r)¯, xn→u implies that u⪯xn, then there exists a point x* in B(x0,r)¯ such that dl(x*,x*)=0 and x*=Sx*=Tx*.Proof.
Choose a pointx1 in X such that x1=Sx0. As Sx0⪯x0 so x1⪯x0, and let x2=Tx1. Now Tx1⪯x1 gives x2⪯x1, continuing this process and choosing xn in X such that
(8)x2k+1=Sx2k,x2k+2=Tx2k+1,wherek=0,1,2,….
Clearly, x2k+1=Sx2k⪯x2k=Tx2k-1⪯x2k-1, and this implies that the sequence {xn} is nonincreasing.
Now, we will prove thatxn∈B(x0,r)¯ for all n∈N, by mathematical induction. By using inequality (7), we have
(9)dl(x0,x1)≤r,
and so x1∈B(x0,r)¯. Assume that x2,…,xj∈B(x0,r)¯ for some j∈N. Now, if 2k+1≤j, by using inequality (6) and Remark 2, we obtain
(10)dl(x2k+1,x2k+2)=dl(Sx2k,Tx2k+1)≤a[dl(x2k,x2k+1)]+b[dl(x2k,Sx2k)+dl(x2k+1,Tx2k+1)]+c[dl(x2k,Tx2k+1)+dl(x2k+1,Sx2k)]≤a[dl(x2k,x2k+1)]+b[dl(x2k,x2k+1)+dl(x2k+1,x2k+2)]+c[dl(x2k,x2k+1)+dl(x2k+1,x2k+2)],
and hence
(11)dl(x2k+1,x2k+2)≤λdl(x2k,x2k+1).
Similarly, if 2k≤j, we deduce
(12)dl(x2k+1,x2k)≤λdl(x2k,x2k-1).
By the previous inequalities, we get
(13)dl(x2k+1,x2k+2)≤λdl(x2k,x2k+1)≤⋯≤λ2k+1dl(x0,x1),dl(x2k+1,x2k)≤λdl(x2k,x2k-1)≤⋯≤λ2kdl(x0,x1).
Thus, from inequality (13), we have
(14)dl(xj,xj+1)≤λjdl(x0,x1)forsomej∈N.
Now, using (14) and (7), we get
(15)dl(x0,xj+1)≤dl(x0,x1)+⋯+dl(xj,xj+1)-[dl(x1,x1)+⋯+dl(xj,xj)]≤dl(x0,x1)[1+⋯+λj-1+λj]≤(1-λ)r(1-λj+1)1-λ<r.
Thus xj+1∈B(x0,r)¯; that is, xn∈B(x0,r)¯ for all n∈N. Since xn+1⪯xn for all n∈N, then it follows that
(16)dl(xn,xn+1)≤λndl(x0,x1)∀n∈N.
So we have
(17)dl(xn+i,xn)≤dl(xn+i,xn+i-1)+⋯+dl(xn+1,xn)≤λn+i-1dl(x0,x1)+⋯+λndl(x0,x1)≤λndl(x0,x1)(1-λi)1-λ⟶0asn⟶+∞.
Hence the sequence {xn} is a Cauchy sequence in (B(x0,r)¯,dl). Now, B(x0,r)¯ is complete since it is closed in X. Therefore there exists a point x*∈B(x0,r)¯ with
(18)limn→+∞dl(xn,x*)=0.
By assumptions x*⪯xn as xn→x*, we have
(19)dl(Sx*,x*)≤dl(Sx*,Tx2n+1)+dl(x2n+2,x*)-dl(x2n+2,x2n+2)≤dl(x*,x2n+2)+adl(x*,x2n+1)+b{dl(x*,Sx*)+dl(x2n+1,Tx2n+1)}+c{dl(x*,Tx2n+1)+dl(x2n+1,Sx*)}≤dl(x*,x2n+2)+adl(x*,x2n+1)+b{dl(x*,Sx*)+dl(x2n+1,x2n+2)}+c{dl(x*,x2n+2)+dl(x2n+1,x*)+dl(x*,Sx*)}.
On taking limit as n→+∞ in the previous inequality, by using inequality (16) and (18), we get
(20)(1-b-c)dl(Sx*,x*)≤0,
which implies that x*=Sx*. Similarly, from
(21)dl(x*,Tx*)≤dl(x*,x2n+1)+dl(x2n+1,Tx*)-dl(x2n+1,x2n+1),
we can obtain x*=Tx*. Hence S and T have a common fixed point in B(x0,r)¯. Now,
(22)dl(x*,x*)=dl(Sx*,Tx*)≤adl(x*,x*)+b[dl(x*,Sx*)+dl(x*,Tx*)]+c[dl(x*,Sx*)+dl(x*,Tx*)]=(a+2b+2c)dl(x*,x*)<dl(x*,x*),
and this implies that
(23)dl(x*,x*)=0.Example 19.
LetX=[0,+∞)∩ℚ be endowed with the order x⪯y if dl(x,x)≤dl(y,y), and define dl:X×X→X as dl(x,y)=x+y. Then (X,dl) is an ordered complete dislocated metric space. Let S,T:X→X be defined by
(24)Sx={x7ifx∈[0,1]∩Xx-13ifx∈(1,+∞)∩X,Tx={2x7ifx∈[0,1]∩Xx-14ifx∈(1,+∞)∩X.
Clearly, S and Tare dominated mappings. For x0=1, r=2, a=1/7, and b=c=1/10, then λ=3/7, B(x0,r)¯=[0,1]∩X, and (1-λ)r=8/7=dl(x0,Sx0).
Now, ifx,y∈(1,+∞)∩X, then
(25)x-13+y-14≥17{x+y}+110{2x-13+2y-14}+110{y+x-13+x+y-14},
and so
(26)dl(Sx,Ty)≥adl(x,y)+b[dl(x,Sx)+dl(y,Ty)]+c[dl(y,Sx)+dl(x,Ty)].
Thus, the contractive condition does not hold on X. Now if x,y∈B(x0,r)¯, then
(27)dl(Sx,Ty)=x7+2y7=17{x+2y}≤17{x+y}+110{x+x7+y+2y7}+110{y+x7+x+2y7}=adl(x,y)+b[dl(x,Sx)+dl(y,Ty)]+c[dl(y,Sx)+dl(x,Ty)].
Therefore, all the conditions of Theorem 18 are satisfied. Moreover, 0 is the common fixed point of S and T and dl(0,0)=0.If in Theorem18 we choose T=S, then we obtain the following corollary.Corollary 20.
Let(X,⪯,dl) be an ordered complete dislocated metric space, x0∈X, r>0, and S:X→X a dominated mapping. Suppose that there exist a,b,c∈[0,1) with a+2b+2c<1 such that
(28)dl(Sx,Sy)≤adl(x,y)+b[dl(x,Sx)+dl(y,Sy)]+c[dl(y,Sx)+dl(x,Sy)],
for all comparable elements x,y in B(x0,r)¯ and
(29)dl(x0,Sx0)≤(1-λ)r,
where λ=(a+b+c)/(1-b-c). If for a nonincreasing sequence {xn} in B(x0,r)¯,xn→u implies that u⪯xn, then there exists a point x* in B(x0,r)¯ such that x*=Sx* and dl(x*,x*)=0.In Theorem18, the condition “for a nonincreasing sequence xn→u implies that u⪯xn” is imposed to restrict condition (6) only for comparable elements. However, the following result relaxes this restriction but imposes that condition (6) holds for all elements in B(x0,r)¯. In Theorem 18, the common fixed point of S and T may not be unique, whereas without order we can obtain a unique fixed point of S and T separately, which is proved in the following theorem.Theorem 21.
Let(X,dl) be a complete dislocated metric space, x0∈X, r>0, and S,T:X→X two dominated mappings. Suppose that there exist a,b,c∈[0,1) with a+2b+2c<1 such that
(30)dl(Sx,Ty)≤adl(x,y)+b[dl(x,Sx)+dl(y,Ty)]+c[dl(y,Sx)+dl(x,Ty)],
for all elements x,y in B(x0,r)¯ and
(31)dl(x0,Sx0)≤(1-λ)r,
where λ=(a+b+c)/(1-b-c). Then there exists a unique point x* in B(x0,r)¯ such that x*=Sx*=Tx* and dl(x*,x*)=0. Moreover, S and T have no fixed point other than x*.Proof.
By following similar arguments, as we used to prove Theorem18, we can obtain a point x* in B(x0,r)¯ such that x*=Sx*=Tx* and dl(x*,x*)=0. Let y=Ty. Then, y is a fixed point of T, and it may not be the fixed point of S. Then,
(32)dl(x*,y)=dl(Sx*,Ty)≤adl(x*,y)+b[dl(x*,Sx*)+dl(y,Ty)]+c[dl(y,Sx*)+dl(x*,Ty)]≤adl(x*,y)+b[dl(y,x*)+dl(x*,y)]+2cdl(x*,y)≤(a+2b+2c)dl(x*,y)<dl(x*,y).
This shows that x*=y. Thus T has no fixed point other than x*. Similarly S has no fixed point other than x*.In Theorem18, condition (7) is imposed to restrict condition (6) only for x,y in B(x0,r)¯, and Example 19 explains the utility of this restriction. However, the following result relaxes condition (7) but imposes condition (6) for all comparable elements in the whole space X. Moreover, we introduce a weaker restriction to obtain a unique common fixed point.Theorem 22.
Let(X,⪯,dl) be an ordered complete dislocated metric space, x0∈X, and S,T:X→X two dominated mappings. Suppose that there exist a,b,c∈[0,1) with a+2b+2c<1 such that
(33)dl(Sx,Ty)≤adl(x,y)+b[dl(x,Sx)+dl(y,Ty)]+c[dl(y,Sx)+dl(x,Ty)],
for all comparable elements x,y in X. If for a nonincreasing sequence {xn} in X,xn→u implies that u⪯xn, then there exists a point x* in X such that x*=Sx*=Tx* and dl(x*,x*)=0. Moreover, the point x* is unique if for any two points x,y in X there exists a point z0∈X such that z0⪯x and z0⪯y.Proof.
By following similar arguments of Theorem18, we can obtain a point x* in X such that x*=Sx*=Tx*. By Theorem 21, x* is unique common fixed point for all comparable elements. Now if x* and y are not comparable and y=Sy=Ty, then there exists a point z0∈X such that z0⪯x* and z0⪯y. Choose a point z1 in X such that z1=Tz0. As Tz0⪯z0 so z1⪯z0, and let z2=Sz1. Now Sz1⪯z1 gives z2⪯z1, continuing this process and having chosen zn in X such that
(34)z2i+1=Tz2i,z2i+2=Sz2i+1,z2i+1=Tz2i⪯z2i=Sz2i-1⪯z2i-1,wherei=0,1,2,….
It follows that zn+1⪯zn⪯⋯⪯z0⪯x*. Following similar arguments as we have used to prove inequality (16), we have
(35)dl(zn,zn+1)≤λndl(z0,z1)∀n∈N.
As z0⪯x* and z0⪯y, it follows that zn⪯Tx* and zn⪯Ty for all n∈N. Then, for i∈N,
(36)dl(Sz2i-1,Tx*)≤adl(z2i-1,x*)+b[dl(z2i-1,Sz2i-1)+dl(x*,Tx*)]+c[dl(z2i-1,Tx*)+dl(x*,Sz2i-1)],
and so
(37)dl(z2i,x*)=dl(Sz2i-1,x*)≤δdl(z2i-1,x*)+μdl(z2i-1,z2i),
where δ=(a+c)/(1-c) and μ=b/(1-c).
Similarly, we deduce(38)dl(x*,z2i-1)=dl(Sx*,Tz2i-2)≤δdl(x*,z2i-2)+μdl(z2i-2,z2i-1).
Since δ,μ≤λ, we get
(39)dl(x*,zn)≤δdl(x*,zn-1)+μdl(zn-1,zn)≤λdl(x*,zn-1)+λndl(z0,z1)
and hence
(40)dl(x*,zn)≤λndl(x*,z0)+nλndl(z0,z1).
On taking limit asn→+∞ in (40), we have
(41)dl(x*,zn)=0.
Similarly,
(42)dl(zn,y)⟶0asn⟶+∞.
Now by using inequality (41) and (42), we have
(43)dl(x*,y)≤dl(x*,zn)+dl(zn,y)-dl(zn,zn)⟶0asn⟶+∞.
So, x*=y.Remark 23.
In Theorem18, the common fixed point of S and T may not be unique. However, fixed point is unique in Theorem 18, if for every pair of elements x,y in B(x0,r)¯there exists a point z0∈B(x0,r)¯ such that z0⪯x and z0⪯y and zn∈B(x0,r)¯, where z1=Tz0, z2=Sz1,….Metric version of Theorem18 is given below.Theorem 24.
Let(X,⪯,d) be an ordered complete metric space, x0∈X, r>0, and S,T:X→X two dominated mappings. Suppose that there exist a,b,c∈[0,1) with a+2b+2c<1 such that
(44)d(Sx,Ty)≤ad(x,y)+b[d(x,Sx)+d(y,Ty)]+c[d(y,Sx)+d(x,Ty)],
for all comparable elements x,y in B(x0,r)¯ and
(45)d(x0,Sx0)≤(1-λ)r,
where λ=(a+b+c)/(1-b-c). If for a nonincreasing sequence {xn} in B(x0,r)¯, xn→u implies that u⪯xn, then there exists a point x* in B(x0,r)¯ such that x*=Sx*=Tx*.Now we apply our Theorem22 to obtain a unique common fixed point of three mappings in ordered complete dislocated metric space. To this aim the innovative technique introduced in [29].Theorem 25.
Let(X,⪯,dl) be an ordered dislocated metric space and S,T,f:X→X such that S,T are f-dominated mappings and SX∪TX⊂fX. Assume that the following condition holds for a,b,c∈[0,1) such that a+2b+2c<1:
(46)dl(Sx,Ty)≤adl(fx,fy)+b[dl(fx,Sx)+dl(fy,Ty)]+c[dl(fy,Sx)+dl(fx,Ty)],
for all comparable elements fx,fy∈fX.
Suppose that for a nonincreasing sequence{xn} in fX, xn→u implies that u⪯xn. Also for any two points z and x in fX there exists a point y∈fX such that y⪯z, y⪯x. If fX is a complete subspace of Xand (S,f) and (T,f) are weakly compatible, then S, T and f have a unique common fixed point fz in fX. Moreover dl(fz,fz)=0.Proof.
By Lemma13, there exists E⊂X such that fE=fX and f:E→X is one-to-one. Now since SX∪TX⊂fX, we define two mappings g,h:fE→fE by g(fx)=Sx and h(fx)=Tx, respectively. Since f is one-to-one on E, then g,h are well defined. As Sx⪯fx implies that g(fx)⪯fx and Tx⪯fx implies that h(fx)⪯fx, therefore g and h are dominated mappings. Let y0=fx0; choose a point y1 in fX such that y1=h(y0).As h(y0)⪯y0, so y1⪯y0, and let y2=g(y1). Now g(y1)⪯y1 gives y2⪯y1. Continuing this process and having chosen yn in fX such that
(47)y2i+1=h(y2i),y2i+2=g(y2i+1),wherei=0,1,2,…,
then yn+1⪯yn for all n∈N. Note that for fx,fy∈fX, where fx and fy are comparable and a,b,c∈[0,1) such that a+2b+2c<1, by using inequality (42), we have
(48)dl(g(fx),h(fy))≤adl(fx,fy)+b[dl(fx,g(fx))+dl(fy,h(fy))]+c[dl(fy,g(fx))+dl(fx,h(fy))].
AsfX is a complete space, so that all conditions of Theorem 22 are satisfied, we deduce that there exists a unique common fixed point fz∈fX of g and h. Also dl(fz,fz)=0. Now fz=g(fz)=h(fz) or fz=Sz=Tz=fz. Thus fz is a point of coincidence of S, T and f. Let v∈fX be another point of coincidence of f, S and T; then there exists u∈X such that v=fu=Su=Tu, which implies that fu=g(fu)=h(fu). A contradiction as fz∈fX is a unique common fixed point of g and h. Hence v=fz. Thus S, T and f have a unique point of coincidence fz∈fX. Now since (S,f) and (T,f) are weakly compatible, by Lemma 14fz is a unique common fixed point of S, T and f.Unique common fixed point result of three mappings in complete dislocated metric space is given below, which can be proved with the help of Theorem21.Theorem 26.
Let(X,dl) be a dislocated metric space andS, T and fself-mappings on X such that SX∪TX⊂fX. Assume that the following condition holds:
(49)dl(Sx,Ty)≤adl(fx,fy)+b[dl(fx,Sx)+dl(fy,Ty)]+c[dl(fy,Sx)+dl(fx,Ty)],
for all elements fx,fy∈B(fx0,r)¯ where a,b,c∈[0,1) such that a+2b+2c<1 and for r>0,
(50)dl(fx0,Tx0)≤(1-λ)r,
whereλ=(a+b+c)/(1-b-c). If fX, is a complete subspace of Xand (S,f) and (T,f) are weakly compatible, then S, T and f have a unique common fixed point fz in B(fx0,r)¯. Moreover, dl(fz,fz)=0.In the following theorem we use Theorem21 to establish the existence of a unique common fixed point of four mappings on closed ball in complete dislocated metric space.Theorem 27.
Let(X,dl) be a dislocated metric space andS, T, g, and f self-mappings on X such that SX,TX⊂fX=gX. Assume that the following condition holds:
(51)dl(Sx,Ty)≤adl(fx,gy)+b[dl(fx,Sx)+dl(gy,Ty)]+c[dl(gy,Sx)+dl(fx,Ty)],
for all elements fx,fy∈B(fx0,r)¯⊆fX, such that a,b,c∈[0,1) with a+2b+2c<1 and for r>0,
(52)dl(fx0,Sx0)≤(1-λ)r,
whereλ=(a+b+c)/(1-b-c). If fX is a complete subspace of Xand (S,f) and (T,g) are weakly compatible, then S, T, f, and g have a unique common fixed point fz in B(fx0,r)¯. Moreover, dl(fz,fz)=0.Proof.
We use the technique introduced in [29]. By Lemma 13, there exists E1,E2⊂X such that fE1=fX=gX=gE2, f:E1→X, and g:E2→X are one-to-one. Now define the mappings A,B:fE1→fE1 by A(fx)=Sx and B(gx)=Tx, respectively. Since f,g are one-to-one on E1 and E2, respectively, then the mappings A,B are well defined. As fx0∈B(fx0,r)¯⊆fX, then fx0∈fX. Let y0=fx0, choose a point y1 in fX such that y1=A(y0), and let y2=B(y1); continuing this process and chosen yn in fX such that
(53)y2i+1=A(y2i),y2i+2=B(y2i+1),wherei=0,1,2,….
Following similar arguments of Theorem 18, we deduce that yn∈B(fx0,r)¯. Also by inequality (51), we have
(54)dl(fx0,A(fx0))≤(1-λ)r.
By using inequality (46), for fx,gy∈B(fx0,r)¯ and a+2b+2c<1 we have
(55)dl(A(fx),B(gy))≤adl(fx,gy)+b[dl(fx,A(fx))+dl(gy,B(gy))]+c[dl(gy,A(fx))+dl(fx,B(gy))].
As fX is a complete space, all conditions of Theorem 21 are satisfied; we deduce that there exists a unique common fixed point fz∈B(fx0,r)¯ of A and B. Furthermore A and B have no fixed point other than fz. Also dl(fz,fz)=0. Now fz=A(fz)=B(fz) or fz=Sz=fz. Thus fz is a point of coincidence of f and S. Let w∈B(fx0,r)¯ be another point of coincidence of S and f, then there exists u∈B(fx0,r)¯ such that w=fu=Su, which implies that fu=A(fu). A contradiction as fz∈B(fx0,r)¯ is a unique fixed point of A. Hence w=fz. Thus S and f have a unique point of coincidence fz∈B(fx0,r)¯. Since (S,f) are weakly compatible, by Lemma 14  fz is a unique common fixed point of S and f. As fX=gX then there exists v∈X such that fz=gv. Now as A(fz)=B(fz)=fz implies that A(gv)=B(gv)=gv⇒Tv=gv, thus gv is the point of coincidence of T and g. Now if Tx=gx, then we have B(gx)=gx, a contradiction. This implies that gv=gx. As (T,g) are weakly compatible, we obtain that gv is the unique common fixed point for T and g. But gv=fz. Thus S, T, g, and f have a unique common fixed point fz∈B(fx0,r)¯.Theorem 28.
Let(X,⪯,dl) be an ordered dislocated metric space and S,T,f:X→X such that S,T are f-dominated mappings and SX∪TX⊂fX. Assume that the following condition holds for a,b,c∈[0,1) such that a+2b+2c<1(56)dl(Sx,Ty)≤adl(fx,fy)+b[dl(fx,Sx)+dl(fy,Ty)]+c[dl(fy,Sx)+dl(fx,Ty)],
for all comparable elements fx,fy∈fX. Suppose that for a nonincreasing sequence {xn} in fX, xn→u implies that u⪯xn. Also for any two points z and x in fX there exists a point y∈fX such that y⪯z, y⪯x. If fX is a complete subspace of X, then S, T and f have a unique point of coincidence fz in fX. Moreover dl(fz,fz)=0.Theorem 29.
Let(X,dl) be a dislocated metric space andS, T, g, and f self-mappings on X such that SX,TX⊂fX=gX. Assume that the following condition holds:
(57)dl(Sx,Ty)≤adl(fx,gy)+b[dl(fx,Sx)+dl(gy,Ty)]+c[dl(gy,Sx)+dl(fx,Ty)],
for all elements fx,fy∈B(fx0,r)¯⊆fX, with a,b,c∈[0,1) such that a+2b+2c<1 and for r>0,
(58)dl(fx0,Sx0)≤(1-λ)r,
whereλ=(a+b+c)/(1-b-c). If fX is a complete subspace of X, then S, T, f, and g have a unique point of coincidence fz in B(fx0,r)¯. Moreover, dl(fz,fz)=0.Remark 30.
We can obtain the metric version of all theorems which are still not present in the literature.


## 3. Conclusion

Azam et al. [18] very recently exploited the idea of fixed points and proved a significant result concerning the existence of fixed points for fuzzy mappings on closed ball in a complete metric space. We continue their investigations, and in this paper, some common fixed point theorems for mappings under Hardy Rogers contractive conditions on a closed ball in a ordered complete dislocated metric space (X,⪯,dl) have been discussed. Our analysis is based on the simple observation that fixed point results can be deduced from fixed point theory of mappings on closed balls. Practically speaking, there are many situations in which the mappings are not contractive on the whole space but instead they are contractive on its subsets. We feel that this aspect of finding the fixed points via closed balls was overlooked and our paper will bring a lot of interest into this area. Furthermore, we have applied the concept of dominated mappings in the process of investigating the existence of unique fixed point of contractive mappings on closed balls in the settings of ordered dislocated metric spaces.



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*Source: 638181-2013-10-23.xml*