# Fixed Point Theorems and Error Bounds in Partial Metric Spaces

Authors: S. A. M. Mohsenalhosseini
Journal: ISRN Computational Mathematics
 (2013)

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## Abstract

This paper is introduced as a survey of result on some generalization of Banach’s fixed point and their approximations to the fixed point and error bounds, and it contains some new fixed point theorems and applications on dualistic partial metric spaces.


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## Body

## 1. Introduction

The partial metric spaces were introduced in [1] as a part of the study of denotational semantics of dataflow networks. He established the precise relationship between partial metric spaces and the weightable quasi-metric spaces, and proved a partial metric generalization of Banach contraction mapping theorem.A partial metric [1] on a set X is a function p:X×X→[0,∞) such that for all x,y,z∈X(1)
x=y⇔p(x,x)=p(x,y)=p(y,y);(2)
p(x,x)≤p(x,y);(3)
p(x,y)=p(y,x);(4)
p(x,z)≤p(x,y)+p(y,z)-p(y,y).A partial metric space is a pair(X,p), where p is a partial metric on X. If p is a partial metric on X, then the function ps:X×X→[0,∞) given by ps(x,y)=2p(x,y)-p(x,x)-p(y,y) is a (usual) metric on X.Each partial metricp on X induces a T0 topology τp on X which has as a basis 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 all x∈X and ϵ>0. Similarly, closed p-ball is defined as Bp(x,ϵ)={y∈X:p(x,y)≤p(x,x)+ϵ}.A sequence{xn}n∈N in a partial metric space (X,p) is called a Cauchy sequence if there exists (and is finite) limn,mp(xn,xm) [1].Note that{xn}n∈N is a Cauchy sequence in (X,p) if and only if it is a Cauchy sequence in the metric space (X,ps) [1].A partial metric space(X,p) is said to be complete if every Cauchy sequence {xn}n∈N in X converges, with respect to τp to a point x∈X such that p(x,x)=limn,mp(xn,xm) [1].A mappingT:X→X is said to be continuous at x0∈X, if for ϵ>0, there exists δ>0 such that T(Bp(x0,δ))⊂Bp(T(x0),ϵ) [2].Definition 1 (see [1]).
An open ball for a partial metricp:X×X→[0,∞) is a set of the form Bϵp(x):={y∈X:p(x,y)<ϵ} for each ϵ>0 and x∈X.In [3], O’Neill proposed one significant change to Matthews definition of the partial metrics, and that was to extend their range from R+ to R. In the following, partial metrics in the O’Neill sense will be called dualistic partial metrics and a pair (X,p) such that X is a nonempty set and p is a dualistic partial metric on X will be called a dualistic partial metric space.A dualistic partial metric on a setX is a partial metric p:X×X→R. A dualistic partial metric space is a pair (X,p), where p is a dualistic partial metric on X.A quasi-metric on a setX is a nonnegative real-valued function d on X×X such that for all x, y, z∈X(i)
d(x,y)=d(y,x)=0⇔x=y,(ii)
d(x,y)≤d(x,z)+d(z,y).Lemma 2 (see [1]).
If(X,p) is a dualistic partial metric space, then the function dp:X×X→R+ defined by dp(x,y)=p(x,y)-p(x,x), is a quasi-metric on X such that τ(p)=τ(dp).Lemma 3 (see [1]).
A dualistic partial metric space(X,p) is complete if and only if the metric space (X,(dp)s) is complete. Furthermore limn→∞(dp)s(a,xn)=0 if and only if p(a,a)=limn→∞p(a,xn)=limn,m→∞p(xn,xm).Before stating our main results, we establish some (essentially known) correspondences between dualistic partial metrics and quasi-metric spaces. Also refer to definition ofϵ-Fixed point and the existence of ϵ-Fixed point for ϵ>0. Our basic references for quasi-metric spaces are [4, 5] and for ϵ-Fixed point is [6].Ifd is a quasi-metric on X, then the function ds defined on X×X by ds(x,y)=max{d(x,y),d(y,x)}, is a metric on X.Definition 4 (see [6]).
Let(X,p) be a dualistic partial metric space and T:X→X be a map. Then x0∈X is ϵ-fixed point for T if
(1)dp(Tx0,x0)≤ϵ.
We say T has the ϵ-fixed point property if for every ϵ>0, AF(T)≠∅ where
(2)AF(T)={x0∈X:d(Tx0,x0)≤ϵ}.Theorem 5 (see [6]).
Let(X,p) be a dualistic partial metric space and T:X→X be a map, x0∈X and ϵ>0. If dp(Tn(x0), Tn+k(x0))→0 as n→∞ for some k>0, then Tk has an ϵ-fixed point.Definition 6 (see [7]).
LetT:A∪B→A∪B, be continues map such that T(A)⊆B, T(B)⊆A and ϵ>0. We define diameter PTa(A,B) by
(3)diam(PTa(A,B))=sup{d(x,y):x,y∈PTa(A,B)}.Theorem 7 (see [1]).
The partial metric contraction mapping theorem. Let(X,p) be a complete partial metric space and T:X→X be a map such that for all x,y∈X(4)p(Tx,Ty)≤Lp(x,y):0≤L<1,
then T has a unique fixed point u, and Tn(x)→u as n→∞ for each x∈X.


## 2. Some Result Fixed Point on Partial Metric

In this section, we give some result on fixed point andϵ-fixed point in dualistic partial metric space and its diameter.Definition 8.
An open ball for a dualistic partial metricp:X×X→R is a set of the form Bϵp(x):={y∈X:p(x,y)<ϵ} for each ϵ>0 and x∈X.Definition 9.
LetT be a mapping of a complete dualistic partial metric X into itself [T:X→X], then T is called a partial metric contraction mapping if there exists a constant L, 0≤L<1, such that p(Tx,Ty)≤Lp(x,y) for all x,y∈X.Theorem 10.
Every contraction mappingT defined on a complete dualistic partial metric X into itself has a unique fixed point u∈X. Moreover, if x0 is any point in X and the sequence xn is defined by
(5)x1=T(x0),x2=T(x1),…,xn=T(xn-1),
then limn→∞xn=u and
(6)p(xn,u)≤Ln1-Lp(x1,x0):0≤L<1.Proof.
Existence of a fixed point. Letx0 be an arbitrary point in X, and we defined by x1=T(x0), x2=T(x1),…, xn=T(xn-1). Then,
(7)x2=T(x1)=T(T(x0))=T2(x0)x3=T(x2)=T(T(x1))=T(T(T(x0)))=T3(x0)dd⋮xn=Tn(x0).
Ifm>n, say m=n+α, α=1,2,3,….
Then(8)p(xn+α,xn)=p(Tn+α(x0),Tn(x0))=p(T(Tn+α-1(x0)),T(Tn-1(x0)))≤Lp(Tn+α-1(x0),Tn-1(x0)).
Continuing this processn-1 times, we have(9)p(xn+α,xn)≤Lnp(Tα(x0),x0)
for n=0,1,2,…, and all α≥1.
However,(10)p(Tα(x0),x0)≤p(Tα(x0),Tα-1(x0))+p(Tα-1(x0),Tα-2(x0))+⋯+p(T(x0),x0).
Therefore, we see that(11)p(xn+α,xn)≤Ln[Lα-1p(x1,x0)+Lα-2p(x1,x0)+⋯+p(x1,x0)]≤Lnp(x1,x0)[1+L+L2+⋯+Lα-1+Lα].
Hence,(12)p(xn+α,xn)≤Lnp(x1,x0)11-L.
Asn, m=n+α→∞, from (12), we see that p(xn+α,xn)→0; that is, {xn} is a Cauchy sequence in the metric space X. Hence, {xn} must be convergent, say limn→∞xn=u.
SinceT is continuous, we have
(13)Tu=T(limn→∞xn)=limn→∞T(xn)=limn→∞xn+1
or Tu=u. Thus, u is a fixed point of T.
Uniqueness of the fixed pointT.
Letv be another fixed point of T. Then Tv=v. We also have p(T(u),T(v))≤Lp(u,v). But p(T(u),T(v))=p(u,v) which implies that p(u,v)≤Lp(u,v) where 0≤L<1. This is possible only when p(u,v)=0, that is, u=v. This proves that the fixed point of T is unique.Corollary 11.
Let(X,p) be a complete dualistic partial metric space and Brp(x0):={y∈X:p(x0,y)<r}. Let T:Brp→X be a partial metric contraction mapping If p(Ty0,y0)<(1-L)r, then T has a fixed point.Proof.
Chooser<ϵ so that p(Ty0,y0)≤(1-L)r<(1-L)ϵ. We show that T maps the closed ball K={y:p(y,y0)≤ϵ} into itself; for if y∈K, then
(14)p(T(y),y0)≤p(T(y),T(y0))+p(T(y0),y0)≤Lp(y,y0)+(1-L)ϵ≤Lϵ+ϵ-Lϵ=ϵ.
Since K is complete and T:K→K satisfy in (5) thus by Theorem 10, T has a fixed point.Theorem 12.
IfX is a complete dualistic partial metric space, and T:X→X is such that Tr is contraction for some integer r>0, then Tr has an unique fixed point.Proof.
SinceTr, where r is a positive integer, is a contraction mapping by Theorem 10, there exists an unique fixed point u of Tr, that is, Tr(u)=u. We want to show that u ia a fixed point of T, that is, T(u)=u. Let S=Tr; therefore, S(u)=u. This implies that
(15)S2(u)=S(S(u))=S(u)=u,S3(u)=S2(S(u))=S2(u)h=u,Sn(u)=Sn-1(S(u))=Sn-1(u)=⋯=u,
and so T(u)=T(Sn(u))=sn(T(u))=sny say
(16)p(Sn(y),u)=p(Sn(y),Sn(u))≤Lp(Sn-1(y),Sn-1(u))≤L2p(Sn-2(y),Sn-2(u))⋮≤Lnp(y,u),limn→∞p(Sn(y),u)=0,
as Ln→0. Thus, limn→∞Sn(y)=u and we have T(u)=u.Theorem 13.
Let(X,p) is a complete dualistic partial metric space, and let T:X→X and S:X→X be two maps contraction. If for every x∈X,
(17)p(Tx,Sy)≤λ,λ>0, chosen suitably. Then for every x∈X,
(18)p(Tm(x),Sm(x))≤λ1-Lm1-L:0≤L<1,m=1,2,….Proof.
The relation is true form=1. We use the principle of induction in order to prove this relation. Let it be true for all m≥1. Then
(19)p(Tn+1(x)-Sn+1(x))=p(TTm(x),SSm(x))≤p(TTm(x),TSm(x))+p(TSm(x),SSm(x))≤Lp(Tm(x),Sm(x))+λ≤Lλ1-Lm1-L+λ=Lλ-Lm+1λ+λ-Lλ1-L=λ1-Lm+11-L.
Thus, the relation is true for m+1.Corollary 14.
Let(X,p) is a complete dualistic partial metric space and let T:X→X be a partial contractive map on X. Moreover, the iteration sequence x1=T(x0), x2=T(x1)=T2x0,…, xn=Tnx0,… with arbitrary x0∈X converges to the unique fixed point u of T. Error estimate is the following estimate (prior estimate):
(20)p(xm,u)≤Lm1-Lp(x0,x1),
and the posterior estimate
(21)p(xm,u)≤L1-Lp(xm-1,xm).Proof.
The First statement is obvious by(22)p(xm,xn)≤p(xm,xm+1)+p(xm+1,xm+2)+⋯+p(xn-1,xn)≤(Lm+Lm+1+⋯+Ln-1)p(x0,x1).
Thus
(23)p(xm,xn)≤Lm1-Lp(x0,x1)(n>m).
Now, inequality (20) follows from (23) by letting n→∞. We have
(24)p(xm,u)≤Lm1-Lp(x0,x1).
Now, for inequality (21) taking m=1 and writing y0 for x0 and y1 for x1, we have from (23)
(25)p(y1,u)≤L1-Lp(y0,y1).
Setting y0=xm-1, we have y1=Ty0=xm and obtain (21).Corollary 15.
Let(X,p) be a complete dualistic partial metric space and let T:X→X be a contraction on a closed ball B=B(x0,r)={x:p(x,x0)≤r}. Moreover, assume that p(x0,Tx0)<(1-L)r. Then, prior error estimate is the following estimate:
(26)p(xm,u)≤Lmr,
and the posterior estimate
(27)p(xm,u)≤Lr.Proof.
By Corollary11 the iteration sequence as (5) is converges to the unique fixed point u of T; hence, by Corollary 14, we have
(28)p(xm,u)≤Lm1-Lp(x0,x1),
since x1=Tx0 and p(x0,Tx0)<(1-L)r, we have
(29)p(xm,u)≤Lm1-Lp(x0,x1)=Lm1-Lp(x0,Tx0)<Lm1-L·(1-L)r=Lm·r.
Therefore p(xm,u)<Lmr. Also, by Corollary 14 we have
(30)p(xm,u)≤L1-Lp(xm-1,xm),
since xm-1=Tm-1x0 and p(x0,Tx0)<(1-L)r, we have
(31)p(xm,u)≤L1-Lp(xm-1,xm)=L1-Lp(Tm-1x0,Tmx0)=L1-Lp(Tm-1x0,T(Tm-1x0))<L1-L(1-L)r<Lr.
Therefore p(xm,u)<Lr.


## 3. Applications of Banach Contraction Principle on Complete Dualistic Partial Metric Space

In this section, we apply Theorem10 to prove existence of the solutions a system of n linear algebraic equations with n unknowns, and we show that applied of Corollaries 14 and 15 in numerical analysis.


### 3.1. Application 3.1

Suppose we want to find the solution of a system ofn linear algebraic equations with n unknowns, then
(32)a11x1+a12x2+⋯+a1nxn=b1a21x1+a22x2+⋯+a2nxn=b2⋮an1x1+an2x2+⋯+annxn=bn.This system can be written as(33)x1=(1-a11)x1-a12x2-a13x3-⋯-a1nxn+b1x2=-a21x1+(1-a22)x2-a23x3-⋯-a2nxn+b2x3=-a31x1-a32x3+(1-a33)x3-⋯-a3nxn+b3⋮xn=-an1x1-an2x2-an3x3-⋯+(1-ann)xn+bn.
By assuming αij=-aij+δij, where
(34)δij={0i≠j1i=j.
Equation (33) can be written in the following equivalent form:
(35)xi=∑j=1n‍αijxj+bi,i=1,2,3,…,n.Ifx=(x1,x2,…,xn)∈Rn then (35) can be written in the form Tx=x, where T is defined by
(36)Tx=y,wherey=(y1,y2,…,yn),(37)yi=∑j=1n‍αijxj+bi,i=1,2,3,…,n,T:Rn⟶Rn,(αij)isan×nmatrix.Finding solutions of the system described by (32) or (35) is thus equivalent to finding the fixed point of the operator equation, (36). In order to find a unique solution of T, that is, a unique solution of (32), we apply Theorem 10. In fact, we prove the following result. Equation (32) has a unique solution, if
(38)∑j=1n‍|αij|=∑j=1n‍|-aij+δij|≤L<1,i=1,…,n.
For x=(x1,…,xn) and x′=(x1′,x2′,…,xn′), we have
(39)p(Tx,Tx′)=p(y,y′),
where
(40)y=(y1,y2,…,yn)∈Rn,y′=(y1′,y2′,…,yn′)∈Rn,yi=∑j=1n‍αijxj+bi,yi′=∑j=1n‍αijxj′+bi,i=1,2,…,n.
If y=(y1,y2,…,yn)∈Rn, then p(y,y)=sup1≤i≤n|yi|. Therefore,
(41)p(Tx,Tx′)=sup1≤i≤n|yi-yi′|=sup1≤i≤n|∑j=1nαijxj+bi‍-∑j=1nαijxj′-bi‍|=sup1≤i≤n|∑j=1nαij(xj-xj′)‍|≤sup1≤i≤n∑j=1n|αij||xj-xj′|‍≤sup1≤j≤n|xj-xj′|sup1≤i≤n∑j=1n‍|αij|≤Lsup1≤j≤n|xj-xj′|.
Since p(x,x′)=sup1≤j≤n|xj-xj′|, we have p(Tx,Tx′)≤Lp(x,x′), 0≤L<1, that is, T is a contraction mapping of the complete dualistic partial metric space Rn into itself. Hence, by Theorem 10, there exists a unique fixed point u of T in Rn, that is, u is a unique solution of (32).Theorem 16.
Iff(t) is a nonlinear integral equation as the following:
(42)f(t)=∫0t‍e-vtcos(αf(v))dv,0≤t≤1;0<α<1,
then it has a unique solution.Proof.
We apply Theorem10 and we can prove that this equation has a unique continuous real-valued solution f(t). Let X=C[0,1] and the mapping T:X→X, defined by T(f)=f for f∈X, where X is a complete dualistic partial metric space with sup p(x,y), is a contraction mapping:
(43)cos(αa)-cos(αb)=α(b-a)sinβ,
where β lies between αa and αb. Therefore, |cos(αa)-cos(αb)|≤α|b-a|. For functions a(t) and b(t); we get
(44)|cosαa(t)-cosαb(t)|≤sup0≤t≤1|a(t)-b(t)|=p(a,b).
For f=Tf and g=Tg, we have
(45)|Tf-Tg|=|∫0t‍e-vt|cos(αf(v))-cos(αg(v))|dv|≤∫0t‍e-vt|cos(αf(v))-cos(αg(v))|dv≤αp(f,g)∫0t‍e-vtdv≤αp(f,g).
Taking sup over 0≤t≤1, we get
(46)supt|Tf(t)-Tg(t)|≤αp(f,g)
or
(47)p(T(f),T(g))≤αp(f,g).Theorem 17.
Letx0 be an initial value and the iterative sequence {xn} as the following:
(48)xn=g(xn-1)n=1,2,….
If g is continuously differentiable on some interval K=[x0-r,x0+r] and satisfies |g′(x)|≤L<1 on K as well as
(49)|g(x0)-x0|<(1-L)r,
then x=g(x) has a unique solution u on K, the iterative sequence {xm} converges to that solution, and one has the error estimates
(50)|x-xm|<Lmr,|x-xm|<Lr.Proof.
Suppose thatp(x,g(x))=|x-g(x)| for x∈K. By the mean-value theorem and the given condition, g(x) is a contraction mapping of the complete dualistic partial metric space K into itself. Hence, by Corollary 11, there exists a unique fixed point u of g in K, that is, u is a unique solution of x=g(x). Also, the iteration sequence {xm} converges to u. Moreover, by p(x,g(x))=|x-g(x)| and Corollary 15, it has the prior error estimate
(51)|x-xm|≤Lmr,
and the posterior estimate
(52)|x-xm|≤Lr.


## 3.1. Application 3.1

Suppose we want to find the solution of a system ofn linear algebraic equations with n unknowns, then
(32)a11x1+a12x2+⋯+a1nxn=b1a21x1+a22x2+⋯+a2nxn=b2⋮an1x1+an2x2+⋯+annxn=bn.This system can be written as(33)x1=(1-a11)x1-a12x2-a13x3-⋯-a1nxn+b1x2=-a21x1+(1-a22)x2-a23x3-⋯-a2nxn+b2x3=-a31x1-a32x3+(1-a33)x3-⋯-a3nxn+b3⋮xn=-an1x1-an2x2-an3x3-⋯+(1-ann)xn+bn.
By assuming αij=-aij+δij, where
(34)δij={0i≠j1i=j.
Equation (33) can be written in the following equivalent form:
(35)xi=∑j=1n‍αijxj+bi,i=1,2,3,…,n.Ifx=(x1,x2,…,xn)∈Rn then (35) can be written in the form Tx=x, where T is defined by
(36)Tx=y,wherey=(y1,y2,…,yn),(37)yi=∑j=1n‍αijxj+bi,i=1,2,3,…,n,T:Rn⟶Rn,(αij)isan×nmatrix.Finding solutions of the system described by (32) or (35) is thus equivalent to finding the fixed point of the operator equation, (36). In order to find a unique solution of T, that is, a unique solution of (32), we apply Theorem 10. In fact, we prove the following result. Equation (32) has a unique solution, if
(38)∑j=1n‍|αij|=∑j=1n‍|-aij+δij|≤L<1,i=1,…,n.
For x=(x1,…,xn) and x′=(x1′,x2′,…,xn′), we have
(39)p(Tx,Tx′)=p(y,y′),
where
(40)y=(y1,y2,…,yn)∈Rn,y′=(y1′,y2′,…,yn′)∈Rn,yi=∑j=1n‍αijxj+bi,yi′=∑j=1n‍αijxj′+bi,i=1,2,…,n.
If y=(y1,y2,…,yn)∈Rn, then p(y,y)=sup1≤i≤n|yi|. Therefore,
(41)p(Tx,Tx′)=sup1≤i≤n|yi-yi′|=sup1≤i≤n|∑j=1nαijxj+bi‍-∑j=1nαijxj′-bi‍|=sup1≤i≤n|∑j=1nαij(xj-xj′)‍|≤sup1≤i≤n∑j=1n|αij||xj-xj′|‍≤sup1≤j≤n|xj-xj′|sup1≤i≤n∑j=1n‍|αij|≤Lsup1≤j≤n|xj-xj′|.
Since p(x,x′)=sup1≤j≤n|xj-xj′|, we have p(Tx,Tx′)≤Lp(x,x′), 0≤L<1, that is, T is a contraction mapping of the complete dualistic partial metric space Rn into itself. Hence, by Theorem 10, there exists a unique fixed point u of T in Rn, that is, u is a unique solution of (32).Theorem 16.
Iff(t) is a nonlinear integral equation as the following:
(42)f(t)=∫0t‍e-vtcos(αf(v))dv,0≤t≤1;0<α<1,
then it has a unique solution.Proof.
We apply Theorem10 and we can prove that this equation has a unique continuous real-valued solution f(t). Let X=C[0,1] and the mapping T:X→X, defined by T(f)=f for f∈X, where X is a complete dualistic partial metric space with sup p(x,y), is a contraction mapping:
(43)cos(αa)-cos(αb)=α(b-a)sinβ,
where β lies between αa and αb. Therefore, |cos(αa)-cos(αb)|≤α|b-a|. For functions a(t) and b(t); we get
(44)|cosαa(t)-cosαb(t)|≤sup0≤t≤1|a(t)-b(t)|=p(a,b).
For f=Tf and g=Tg, we have
(45)|Tf-Tg|=|∫0t‍e-vt|cos(αf(v))-cos(αg(v))|dv|≤∫0t‍e-vt|cos(αf(v))-cos(αg(v))|dv≤αp(f,g)∫0t‍e-vtdv≤αp(f,g).
Taking sup over 0≤t≤1, we get
(46)supt|Tf(t)-Tg(t)|≤αp(f,g)
or
(47)p(T(f),T(g))≤αp(f,g).Theorem 17.
Letx0 be an initial value and the iterative sequence {xn} as the following:
(48)xn=g(xn-1)n=1,2,….
If g is continuously differentiable on some interval K=[x0-r,x0+r] and satisfies |g′(x)|≤L<1 on K as well as
(49)|g(x0)-x0|<(1-L)r,
then x=g(x) has a unique solution u on K, the iterative sequence {xm} converges to that solution, and one has the error estimates
(50)|x-xm|<Lmr,|x-xm|<Lr.Proof.
Suppose thatp(x,g(x))=|x-g(x)| for x∈K. By the mean-value theorem and the given condition, g(x) is a contraction mapping of the complete dualistic partial metric space K into itself. Hence, by Corollary 11, there exists a unique fixed point u of g in K, that is, u is a unique solution of x=g(x). Also, the iteration sequence {xm} converges to u. Moreover, by p(x,g(x))=|x-g(x)| and Corollary 15, it has the prior error estimate
(51)|x-xm|≤Lmr,
and the posterior estimate
(52)|x-xm|≤Lr.



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