# Fixed Point Theorems and Asymptotically Regular Mappings in Partial Metric Spaces

Authors: Satish Shukla; Ishak Altun; Ravindra Sen
Journal: ISRN Computational Mathematics
 (2013)

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## Abstract

The notion of asymptotically regular mapping in partial metric spaces is introduced, and a fixed point result for the mappings of this class is proved. Examples show that there are cases when
new results can be applied, while old ones (in metric space) cannot. Some common fixed point theorems for sequence of mappings in partial metric spaces are also proved which generalize and improve some known results
in partial metric spaces.


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## Body

## 1. Introduction

Matthews [1] introduced partial metric spaces as a part of the study of denotational semantics of data flow networks. In partial metric space, the usual metric was replaced by partial metric, with a property that the self-distance of any point may not be zero. In fact, it is widely recognized that partial metric spaces play an important role in constructing models in the theory of computation. Partial metric has applications in the branches of science where the size of data point is represented by its self-distance. The fixed point of a contraction mapping in partial metric space has zero self-distance; that is, fixed point is a total object. Every metric space is a partial metric space with zero self-distance that is, partial metric spaces are the generalization of metric spaces.O’Neill [2] generalized the concept of partial metric space a bit further by admitting negative distances. The partial metric defined by O’Neill is called dualistic partial metric. Heckmann [3] generalized it by omitting small self-distance axiom. The partial metric defined by Heckmann is called weak partial metric.Banach contraction principle ensures the existence and uniqueness of a fixed point of a contractive self-map of metric space and has many applications in applied sciences. The fixed point result of Matthews is the generalization of the following Banach contraction principle.Let(X,d) be a complete metric space and let f be a self-map on X. If there exists λ∈[0,1) such that d(fx,fy)≤λd(x,y) for all x,y∈X, then f has a unique fixed point in X.The fixed point result of Matthews is generalized by several authors for single self map in partial metric spaces (see, e.g., [4–6]). Almost all contractive conditions in these papers imply the asymptotic regularity of the mapping under consideration.The purpose of this paper is to prove some common fixed point theorems for a sequence of self maps on partial metric spaces and generalize the result of Matthews. The notion of asymptotically regular mapping in partial metric spaces is introduced and a fixed point result for the mappings of this class is also proved.


## 2. Definitions and Preliminaries

First, we recall some definitions and properties of partial metric spaces.Definition 1 (see [1]).
A partial metric on a nonempty setX is a function p:X×X→ℝ+ (ℝ+ stands for nonnegative reals) such that for all x,y,z∈X(p1)
x=y⇔p(x,x)=p(x,y)=p(y,y),(p2)
p(x,x)≤p(x,y),(p3)
p(x,y)=p(y,x),(p4)
p(x,y)≤p(x,z)+p(z,y)-p(z,z). A partial metric space is a pair (X,p) such that X is a nonempty set and p is a partial metric on X.It is clear that ifp(x,y)=0, then from (p1) and (p2), x=y. But if x=y, p(x,y) may not be 0.Example 2.
Letp:ℝ+×ℝ+→ℝ+ be defined by p(x,y)=max{x,y}, for all x,y∈ℝ+, and then (ℝ+,p) is a partial metric space.Each partial metric onX generates a T0 topology τp on X which has a base the family of open p-balls {Bp(x,ϵ):x∈X,ϵ>0}, where Bp(x,ϵ)={y∈X:p(x,y)<p(x,x)+ϵ} for every x∈X and ϵ>0.Theorem 3 (see [1]).
For each partial metricp:X×X→ℝ+ the pair (X,d), where d(x,y)=2p(x,y)-p(x,x)-p(y,y) for all x,y∈X is a metric space.Here,(X,d) is called induced metric space, and d is metric induced by p.Throughout this paper, we suppose that(X,p) is induced metric space and d is metric induced by p.Let(X,p) be partial metric space. Then, (1)
a sequence{xn} in (X,p) converges to a point x∈X if and only if p(x,x)=limn→∞p(xn,x),(2)
a sequence{xn} in (X,p) is called Cauchy sequence if there exists limn,m→∞p(xn,xm) and is finite,(3)
(X,p) is said to be complete if every Cauchy sequence {xn} in X converges, with respect to τp, to a point x∈X such that p(x,x)=limn,m→∞p(xn,xm).Lemma 4 (see [1, 7]).
Let(X,p) be partial metric space, and then(a)
{xn} is a Cauchy sequence in (X,p) if and only if it is a Cauchy sequence in metric space (X,d).(b)
(X,p) is complete if and only if the metric space (X,d) is complete. Furthermore, limn→∞d(xn,x)=0 if and only if
(1)p(x,x)=limn→∞p(xn,x)=limn,m→∞p(xn,xm).Definition 5.
A self mapf on a partial metric space (X,p) is said to be asymptotically regular at a point x in X, if
(2)limn→∞p(fnx,fn+1x)=0,
where fn denotes the nth iterate of f at x.Note that Banach type contractions, generalized contractions, and so forth are asymptotically regular at every point of space.Example 6.
Letp:ℝ+×ℝ+→ℝ+ be defined by p(x,y)=max{x,y}, for all x,y∈ℝ+, and then (ℝ+,p) is a partial metric space. The mapping f:X→X is defined by fx=x/2, for every x∈ℝ+. Then f is asymptotically regular at every point of ℝ+.


## 3. Fixed Point Theorems

First, we prove the fixed point result for asymptotically regular mappings.Theorem 7.
Let(X,p) be a complete partial metric space and let f be a self map on X, satisfying the following condition:
(3)p(fx,fy)≤a1ϕ[min{p(x,fx),p(y,fy)}]+a2ψ[p(x,fx)·p(y,fy)]+a3p(x,y)+a4[p(x,fx)+p(y,fy)]+a5[p(x,fy)+p(y,fx)]
for all x,y∈X, where ai=ai(x,y)(i=1,2,3,4,5) are nonnegative functions such that for arbitrarily fixed k>0,0<λ<1 and for all x,y∈X(4)a1(x,y),a2(x,y)≤k,(5)a3(x,y)+a4(x,y)+2a5(x,y)≤λ,
where ϕ,ψ:ℝ+→ℝ+ are functions such that ϕ(0)=ψ(0)=0 and ϕ,ψ are continuous at 0. If f is asymptotically regular at some x0∈X, and then f has a unique fixed point u in X with p(u,u)=0.Proof.
Letxn+1=fxn for every n≥0. If there exists n such that xn=xn+1, then xn is a fixed point of f. Suppose that xn+1≠xn for every n≥0. We show that {xn} is a Cauchy sequence.
Letm>n denote pn=p(xn+1,xn) and then from (3) we obtain
(6)p(xn,xm)≤p(xn,xn+1)+p(xn+1,xm+1)+p(xm+1,xm)-p(xn+1,xn+1)-p(xm+1,xm+1)≤pn+pm+p(fxn,fxm).
Using (3), we obtain
(7)p(xn,xm)≤pn+pm+a1ϕ[min{pn,pm}]+a2ψ[pn·pm]+a3p(xn,xm)+a4[pn+pm]+a5[p(xn,xm+1)+p(xm,xn+1)],
where ai=ai(xn,xm). As
(8)p(xn,xm+1)+p(xm,xn+1)≤p(xn,xm)+pm-p(xm,xm)+p(xm,xn)+pn-p(xn,xn)≤2p(xn,xm)+pn+pm.
Therefore from (7), it follows that
(9)p(xn,xm)≤pn+pm+a1ϕ[min{pn,pm}]+a2ψ[pn·pm]+a3p(xn,xm)+a4[pn+pm]+a5[2p(xn,xm)+pn+pm]=(1+a4+a5)(pn+pm)+(a3+2a5)p(xn,xm)+a1ϕ[min{pn,pm}]+a2ψ[pn·pm].
Using (4) and (5), we obtain
(10)p(xn,xm)≤(1+λ)(pn+pm)+λp(xn,xm)+kϕ[min{pn,pm}]+kψ[pn·pm](1-λ)p(xn,xm)≤(1+λ)(pn+pm)+kϕ[min{pn,pm}]+kψ[pn·pm].
Since f is asymptotically regular at x0 and ϕ and ψ are continuous at zero, therefore from the above inequality we obtain
(11)limn,m→∞p(xn,xm)=0.
Hence, {xn} is a Cauchy sequence in (X,p), and by Lemma 4, it is a Cauchy sequence in (X,d). Using completeness of (X,p) and Lemma 4, it follows that (X,d) is complete and {xn} converges in (X,d). Thus, limn→∞d(xn,u)=0 for some u∈X.
Again, from Lemma4, and (11), we have
(12)p(u,u)=limn→∞p(xn,u)=limn,m→∞p(xn,xm)=0.
To prove that fu=u, let us consider the following inequalities:
(13)p(u,fu)≤p(u,xn+1)+p(xn+1,fu)-p(xn+1,xn+1)≤p(u,xn+1)+p(fxn,fu),p(u,fu)≤p(u,xn+1)+a1ϕ[min{pn,p(u,fu)}]+a2ψ[pn·p(u,fu)]+a3p(xn,u)+a4[pn+p(u,fu)]+a5[p(xn,fu)+p(u,fxn)],
where ai=ai(xn,u). As
(14)p(xn,fu)+p(u,fxn)≤p(xn,u)+p(u,fu)-p(u,u)+p(u,xn+1)=p(xn,u)+p(u,fu)+p(u,xn+1).
Therefore, (13) gives
(15)p(u,fu)≤(1+a5)p(u,xn+1)+(a3+a5)p(xn,u)+a4pn+(a4+a5)p(u,fu)+a1ϕ[min{pn,p(u,fu)}]+a2ψ[pn·p(u,fu)].
Using (4) and (5), we obtain
(16)p(u,fu)≤(1+λ)p(u,xn+1)+λp(xn,u)+λpn+λp(u,fu)+kϕ[min{pn,p(u,fu)}]+kψ[pn·p(u,fu)].
From (12) and the above inequality, it follows that
(17)p(u,fu)≤λp(u,fu)<p(u,fu).
This contradiction shows that p(u,fu)=0, that is, fu=u. Thus, u is a fixed point of f.
Letv be another fixed point of f and p(u,v)>0. From (3), we obtain
(18)p(u,v)=p(fu,fv)≤a1ϕ[min{p(u,fu),p(v,fv)}]+a2ψ[p(u,fu)·p(v,fv)]+a3p(u,v)+a4[p(u,fu)+p(v,fv)]+a5[p(u,fv)+p(v,fu)]=a1ϕ[min{p(u,u),p(v,v)}]+a2ψ[p(u,u)·p(v,v)]+a3p(u,v)+a4[p(u,u)+p(v,v)]+a5[p(u,v)+p(v,u)]=a1ϕ[0]+a2ϕ[0]+a3p(u,v)+a4p(v,v)+a5[p(u,v)+p(v,u)]≤(a3+a4+2a5)p(u,v),
where ai=ai(u,v). The above inequality with (5) gives
(19)p(u,v)≤λp(u,v)<p(u,v).
This contradiction proves uniqueness.Takinga1(x,y)=a2(x,y)=1/(1+p(x,y)), ϕ(t)=ψ(t)=α·t, 0<α<1, a3(x,y)=β, a4(x,y)=a5(x,y)=0 in the above theorem, we obtain following corollary.Corollary 8.
Let(X,p) be a complete partial metric space and let f be a self map on X, satisfying the following condition:
(20)p(fx,fy)≤αmin{p(x,fx),p(y,fy)}+p(x,fx)·p(y,fy)1+p(x,y)+βp(x,y)
for all x,y∈X, where α,β are nonnegative reals, such that α<1,β<1. If f is asymptotically regular at some x0∈X, then f has a unique fixed point u in X with p(u,u)=0.Again takinga1(x,y)=a2(x,y)=0 and a3(x,y)=α,a4(x,y)=β,a5(x,y)=γ in Theorem 7, we obtain following corollary.Corollary 9.
Let(X,p) be a complete partial metric space and f be a self map on X, satisfying following condition:
(21)p(fx,fy)≤αp(x,y)+β[p(x,fx)+p(y,fy)]+γ[p(x,fy)+p(y,fx)]
for all x,y∈X, where α,β,γ are nonnegative reals, such that α+β+2γ<1. If f is asymptotically regular at some x0∈X, then f has a unique fixed point u in X with p(u,u)=0.Takingα=γ=0 in above the corollary, we obtain the following result.Corollary 10.
Let(X,p) be a complete partial metric space and let f be a self map on X, satisfying the following condition:
(22)p(fx,fy)≤β[p(x,fx)+p(y,fy)]
for all x,y∈X, where 0≤β<1. If f is asymptotically regular at some x0∈X, then f has a unique fixed point u in X with p(u,u)=0.The following example shows that the assumption of asymptotic regularity in above theorems cannot be dropped.Example 11.
LetX=[0,1], then (X,p), where p(x,y)=max{x,y} is a complete partial metric space. Define a self map f on X, as follows:
(23)fx={1ifx∈[0,12];x2ifx∈(12,1].
Take β=7/10 then f satisfies the contractive condition of Corollary 10, but f is not asymptotically regular at any point of X, and has no fixed point in X.
Results similar to the above corollaries are available in usual metric spaces (see, e.g., [8]). In the following we illustrate the existence of self map which satisfies contractive condition of Corollary 10, in partial metric space but not in usual metric space.Example 12.
LetX=[0,1], then (X,p), where p(x,y)=max{x,y} is a complete partial metric space. Define a self map f on X, as follows:
(24)fx={x2ifx∈[0,12);14ifx∈[12,1].
Note that f satisfies all the conditions of Corollary 10, with β=9/10, and has a unique fixed point u=0. But f does not satisfy the contractive condition d(fx,fy)≤β[d(x,fx)+d(y,fy)] for all x,y∈X, in usual metric space (X,d) with d(x,y)=|x-y| (which is also the induced metric); for example, if we take x=1/2,y=0, then there is no β such that 0≤β<1 and d(fx,fy)≤β[d(x,fx)+d(y,fy)]. Therefore, results of usual metric spaces cannot be applied.Now, we will prove some common fixed point theorems.Theorem 13.
Let(X,p) be a complete partial metric space and {fn} be a sequence of self maps on X satisfying the following.
There exista1,a2,a3,a4,a5∈[0,1) with a1+a2+a3+a4+a5<1 such that for all x,y∈X(25)p(fix,fjy)≤a1p(x,y)+a2p(x,fix)+a3p(y,fjy)+a4p(y,fix)+a5p(x,fjy).
Then, all the mappings of sequence {fn} have a unique common fixed point u in X with p(u,u)=0.Proof.
Letx0∈X and xn+1=fn+1xn for every n≥0. We show that {xn} is a Cauchy sequence.
If there existsk≥0, such that xk=xk+1, then from (25) we obtain
(26)p(xk+2,xk+1)=p(fk+2xk+1,fk+1xk)≤a1p(xk+1,xk)+a2p(xk+1,fk+2xk+1)+a3p(xk,fk+1xk)+a4p(xk,fk+2xk+1)+a5p(xk+1,fk+1xk)=a1p(xk+1,xk)+a2p(xk+1,xk+2)+a3p(xk,xk+1)+a4p(xk,xk+2)+a5p(xk+1,xk+1)=a1p(xk+1,xk+1)+a2p(xk+1,xk+2)+a3p(xk+1,xk+1)+a4p(xk+1,xk+2)+a5p(xk+1,xk+1).
As p(xk+1,xk+1)≤p(xk+2,xk+1) we have
(27)p(xk+2,xk+1)≤(a1+a2+a3+a4+a5)p(xk+2,xk+1),
as a1+a2+a3+a4+a5<1, it follows that p(xk+2,xk+1)=0,thatis,xk+2=xk+1.
Similarly, it can be seen thatxk=xk+r for every r>0. Thus, {xn} is a Cauchy sequence.
Assume thatxn≠xn+1 for every n≥0. Denote pn=p(xn+1,xn), then from (25) it follows that
(28)pn=p(fn+1xn,fnxn-1)≤a1p(xn,xn-1)+a2p(xn,fn+1xn)+a3p(xn-1,fnxn-1)+a4p(xn-1,fn+1xn)+a5p(xn,fnxn-1)=a1p(xn,xn-1)+a2p(xn,xn+1)+a3p(xn-1,xn)+a4p(xn-1,xn+1)+a5p(xn,xn)≤a1pn-1+a2pn+a3pn-1+a4pn-1+a4pn-a4p(xn,xn)+a5p(xn,xn)
and so
(29)(1-a2-a4)pn≤(a1+a3+a4)pn-1+(a5-a4)p(xn,xn).
Using, symmetry of p, we obtain
(30)(1-a3-a5)pn≤(a1+a2+a5)pn-1+(a4-a5)p(xn,xn).
It follows from (29) and (30) that
(31)(2-a2-a3-a4-a5)pn≤(2a1+a2+a3+a4+a5)pn-1pn≤2a1+a2+a3+a4+a52-a2-a3-a4-a5pn-1pn≤λpn-1,
where λ=(2a1+a2+a3+a4+a5)/(2-a2-a3-a4-a5)<1, which implies pn≤λpn-1≤λ2pn-2≤⋯≤λnp0, where p0=p(x1,x0).
Form>n, we obtain
(32)p(xn,xm)≤p(xn,xn+1)+p(xn+1,xn+2)+⋯+p(xm-1,xm)-[p(xn+1,xn+1)+p(xn+2,xn+2)+⋯+p(xm-1,xm-1)]≤pn+pn+1+⋯+pm-1≤λnp0+λn+1p0+⋯+λm-1p0=λnp0(1+λ+λ2+⋯λm-1)≤λn1-λp0.
As λ<1, it follows that λn→0 as n→∞, which implies
(33)limn,m→∞p(xn,xm)=0.
Hence {xn} is a Cauchy sequence in (X,p), and by Lemma 4, it is a Cauchy sequence in (X,d). Using completeness of (X,p) and Lemma 4, it follows that (X,d) is complete and {xn} converges in (X,d). Thus limn→∞d(xn,u)=0 for some u∈X.
Again from Lemma4, and (33) we have
(34)p(u,u)=limn→∞p(xn,u)=limn,m→∞p(xn,xm)=0.
To prove that fiu=u for any arbitrary fixed i∈ℕ, let us consider the following inequalities:
(35)p(fiu,u)≤p(fiu,fnxn-1)+p(fnxn-1,u)-p(fnxn-1,fnxn-1)≤p(fiu,fnxn-1)+p(xn,u)≤a1p(u,xn-1)+a2p(u,fiu)+a3p(xn-1,fnxn-1)+a4p(xn-1,fiu)+a5p(u,fnxn-1)+p(xn,u)=a1p(u,xn-1)+a2p(u,fiu)+a3p(xn-1,xn)+a4p(xn-1,fiu)+a5p(u,xn)+p(xn,u)≤(a1+a3+a4)p(xn-1,u)+(a3+a5+1)p(xn,u)+(a2+a4)p(u,fiu)-(a3+a4)p(u,u),p(u,fiu)≤a1+a3+a41-a2-a4p(xn-1,u)+a3+a5+11-a2-a4p(xn,u).
Using (34), it follows that p(fiu,u)=0;thatis,fiu=u. Thus, u is a common fixed point of all the maps of sequence {fn}.
Letv be another common fixed point of all the maps of sequence {fn} and p(u,v)>0.
For anyn∈ℕ, from (25), we obtain
(36)p(u,v)=p(fnu,fnv)≤a1p(u,v)+a2p(u,fnu)+a3p(v,fnv)+a4p(v,fnu)+a5p(u,fnv)=a1p(u,v)+a2p(u,u)+a3p(v,v)+a4p(v,u)+a5p(u,v).
Using (p2), we have
(37)p(u,v)≤(a1+a2+a3+a4+a5)p(u,v)<p(u,v).
This contradiction proves uniqueness.Takingfn=f for every n∈ℕ in above theorem we obtain following Hardy-Rogers-type result and an improvement to Theorem 2, of Altun et al. [5].Corollary 14.
Let(X,p) be a complete partial metric space. Let f be a self map on X, satisfying the following.
There exista1,a2,a3,a4,a5∈[0,1) with a1+a2+a3+a4+a5<1 such that for all x,y∈X(38)p(fx,fy)≤a1p(x,y)+a2p(x,fx)+a3p(y,fy)+a4p(y,fx)+a5p(x,fy).
Then, the f has a unique fixed point u in X with p(u,u)=0.Theorem 15.
Let(X,p) be a complete partial metric space. Let {fn} be a sequence of self maps on X satisfying the following.
There existsa1,a2,a3,a4,a5∈[0,1) with a1+a2+a3+a4+a5<1 and positive integer mi for each i∈ℕ, such that, for all x,y∈X(39)p(fimix,fjmjy)≤a1p(x,y)+a2p(x,fimix)+a3p(y,fjmjy)+a4p(y,fimix)+a5p(x,fjmjy).
Then all the maps of sequence {fn} have a unique common fixed point u in X with p(u,u)=0.Proof.
Note that the sequence{fnmn} satisfies all the conditions of Theorem 13, therefore, all the maps of sequence {fnmn} have a unique common fixed point u in X with p(u,u)=0, that is, fimiu=u for every i∈ℕ. Taking x=fiu,y=u in (39) with the fact that fimifiu=fifimiu=fiu for every i∈ℕ, we obtain fiu=u. Thus, u is a common fixed point of all the maps of sequence {fn}. Its uniqueness follows from the fact that fiu=u implies fimiu=u for every i∈ℕ.



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