Extensions of Banach contraction principle to partial cone metric spaces over a non-normal solid cone

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R E S E A R C H

Open Access

Extensions of Banach contraction principle to

partial cone metric spaces over a non-normal

solid cone

Shujun Jiang

1

_{and Zhilong Li}

2*

*_{Correspondence:}

[email protected] 2_{School of Statistics, Jiangxi}

University of Finance and Economics, Nanchang, 330013, China

Full list of author information is available at the end of the article

Abstract

In this paper, we present some extensions of Banach contraction principle to partial cone metric spaces over a non-normal solid cone, which improve many recent ﬁxed point results in cone metric spaces and partial cone metric spaces. An example is given to support the usability of our results.

MSC: 06A07; 47H10

Keywords: Banach contraction principle; partial cone metric space; non-normal cone; solid cone

1 Introduction

The Banach contraction principle is the most celebrated fixed point theorem, which has been extended in various directions. In , Huang and Zhang [] introduced cone met-ric spaces and extended the Banach contraction principle to cone metmet-ric spaces over a normal solid cone, being unaware that cone metric spaces already existed under the name ofK-metric spaces andK-normed spaces that were introduced and used in the middle of the th century in [–]. Furthermore, Huang and Zhang defined the convergence via interior points of the cone. Such an approach allows the investigation of the case that the cone is not necessarily normal, for example, the authors in [–] established many fixed point results and common fixed point results in cone metric spaces over a non-normal cone. In , based on the definition of cone metric spaces and partial metric spaces, which were introduced by Matthews [], Sonmez [, ] defined a partial cone met-ric space and considered the extensions of Banach contraction principle to partial cone metric spaces.

It is worth mentioning that in most of the preceding references concerned with ﬁxed point results of contractions in cone metric spaces and partial cone metric spaces, the contractions are always assumed to be restricted with a constant. In [], Agarwal consid-ered a contraction restricted with a positive linear mapping and proved the following ﬁxed point theorem in cone metric spaces.

Theorem (See []) Let(X,d)be a complete cone metric space overRn

+and T:X→X.If

there exists a linear bounded mapping L:Rn

+→Rn+with the spectral radius r(L) < such

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that

d(Tx,Ty)Ld(x,y), ∀x,y∈X. ()

Then T has a unique ﬁxed point x∗∈X.

It is clear thatRn

+is a normal solid cone ofRnendowed with the usual norm. Motivated by [–, , ], we in this paper shall extend Theorem  to partial cone metric spaces over a non-normal solid cone of an abstract normed vector space.

2 Preliminaries

LetEbe a topological vector space. A cone ofEis a nonempty closed subsetPofEsuch that

(i) ax+by∈Pfor eachx,y∈Pand eacha,b≥, and (ii) P∩(–P) ={θ}, whereθis the zero element ofE.

Each cone PofEdetermines a partial order onE byxy⇐⇒y–x∈P for each

x,y∈X.

A conePof a topological vector spaceE, is solid [] ifintP=∅, whereintPis the interior ofP. For eachx,y∈Ewithy–x∈intP, we writexy. A conePof a normed vector space (E, · ), is normal [] if there existsN>  such thatxyimplies thatx ≤Nyfor eachx,y∈P, and the minimalNis called a normal constant ofP.

Lemma  Let P be a solid cone of a normed vector space(E,·),and let{un}be a sequence

in E.Then un→·θimplies that for each∈intP,there exists a positive integer nsuch that ±un∈intP,i.e.,unfor all n≥n.

Proof For each∈intP, there exists someε>  such thatx<εimplies that±x∈intP

for eachx∈E. Ifun→·θ, then for thisε, there exists a positive integernsuch thatun<ε for eachn≥n, and hence±un∈intPfor eachn≥n,i.e., –un for eachn≥n.

The proof is complete.

Remark  The converse of Lemma  is true provided thatPis normal. In fact, for each ε> , there exists some∈intPsuch that<__Nε_+, whereNdenotes the normal con-stant ofP. Note that for this, there exists a positive integernsuch that –unfor

eachn≥n, and soθ un+≤. Thenun ≤ un++ ≤(N+ )<εfor

eachn≥nby the normality ofP. This forces thatun ·

→θ.

The following example shows that the converse of Lemma  may not be true ifPis non-normal.

Example  LetE=C_R[, ] with the normu=u∞+u∞andP={u∈E:u(t)≥ ,t∈[, ]}, which is a non-normal solid cone []. Letun(t) =sin_nnt. Clearly,un= , and

soun

·

θ. On the other hand, letvn(t)≡ _n, thenvn∈P,vn

·

→θ and –vnunvn. By

Lemma , for each∈intP, there exists a positive integernsuch thatθvnfor all n≥n, and hence –unfor alln≥n.

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(d) d(x,y) =θ⇐⇒x=y; (d) d(x,y) =d(y,x);

(d) d(x,y)d(x,z) +d(z,y).

The pair (X,d) is called a cone metric space overP. A partial cone metric [, ] onXis a mappingp:X×X→Psuch that for eachx,y,z∈X,

(p) p(x,y) =p(x,x) =p(y,y)⇐⇒x=y; (p) p(x,y) =p(y,x);

(p) p(x,x)p(x,y);

(p) p(x,y)p(x,z) +p(z,y) –p(z,z).

The pair (X,p) is called a partial cone metric space overP.

Each cone metric is certainly a partial cone metric. The following example shows that there does exist some partial cone metric which is not a cone metric.

Example  LetE=C

R[, ] with the normu=u∞+u∞, and X=P={u∈E:

u(t)≥,t∈[, ]}. Deﬁne a mappingp:X×X→Pby

p(x,y) = ⎧ ⎨ ⎩

x, x=y,

x+y, otherwise.

For each x,y∈X, p(x,y) =p(y,x) =x andp(x,x) =p(x,y) =x when x=y, andp(x,y) =

p(y,x) =x+yandp(x,x) =xx+y=p(x,y) when x=y, i.e., (p) and (p) are satis-ﬁed. For each x,y∈X, p(x,y) =p(x,x) =p(y,y) =xwheneverx=y, andx=ywhenever

p(x,x) =p(y,y),i.e., (p) is satisﬁed. For eachx,y,z∈X,

p(x,y) =x=p(x,z) +p(y,z) –p(z,z), whenx=y=z,

p(x,y) =xx+y+z=p(x,z) +p(y,z) –p(z,z), whenx=y,y=z,

p(x,y) =x+y=p(x,z) +p(y,z) –p(z,z), whenx=y,y=z,

p(x,y) =x+y=p(x,z) +p(y,z) –p(z,z), whenx=y,x=z,

p(x,y) =x+yx+y+z=p(x,z) +p(y,z) –p(z,z), whenx=y,y=z,x=z,

i.e., (p) is satisﬁed for eachx,y,z∈X. Hencepis partial cone metric, but not a cone metric, sincep(x,x)=θfor eachx∈Xwithx=θ.

Each partial cone metricponXover a solid cone generates a topologyτponX, which

has a base of the family of openp-balls{Bp(x,) :x∈X,θ}, whereBp(x,) ={y∈X:

p(x,y)p(x,x) +}for eachx∈Xand each∈intP.

Deﬁnition  Let (X,p) be a partial cone metric space over a solid conePof a topological vector spaceE.

(i) A sequence{xn}inXconverges [] tox∈X(denote byxn τp

→x), if for each

∈intP, there exists a positive integernsuch thatp(xn,x)p(x,x) +for each n≥n. A sequence{xn}inXstrongly converges [] tox∈X(denote byxn

s-τp

→x), if

limn→∞p(xn,x) =limn→∞p(xn,xn) =p(x,x).

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θ-complete, if eachθ-Cauchy sequence{xn}ofXconverges to a pointx∈Xsuch thatp(x,x) =θ.

It follows from Lemma  and Remark  that each strongly convergent sequence{xn}of a partial cone metric spaceX is convergent wheneverEis a normed vector space, and the converse is true provided thatP is a normal. The following example will show that there exists some sequence of a partial cone metric, which is convergent but not strongly convergent ifPis non-normal.

Example  Let (X,p),EandPbe the same ones as those in Example , and letun(t) = +sinnt

n+ . Thenun

τp

→θ, butuns-τpθ. In fact, it is clear thatun∈P,p(un,θ) =un,unvand

vn→· θ, wherev(t)≡ _n_+. Then by Lemma , for each ∈intP, there exists a positive

integernsuch thatθ p(un,θ)vnfor alln≥n,i.e.,un

τp

→θ. On the other hand,

p(un,θ) –p(θ,θ)=un= , and henceuns-τpθ.

Deﬁnition  Let (X,p) be a partial cone metric space over a solid conePof a normed

vector space (E, · ). A sequence{xn}inXis Cauchy [, ], if there existsu∈Pwith

u<∞such thatlim_m_,_n_→∞p(xn,xm) =u. The partial cone metric space (X,p) is complete

[, ], if each Cauchy sequence{xn}ofXstrongly converges to a pointx∈Xsuch that

p(x,x) =u.

IfPis a normal solid cone of a normed vector space (E, · ), then each complete partial cone metric space isθ-complete by Lemma  and Remark . But the converse is not true, the following example shows that a partial cone metric space which isθ-complete, is not necessarily complete.

Example  LetX={(x,x, . . . ,xk) :xi≥,xi∈Q,i= , , . . . ,k}, E=Rk with the norm

x=k_i_=x

i,P=Rk+, whereQdenotes the set of rational numbers. Deﬁne a mapping

p:X×X→Pas follows:

p(x,y) = (x∨y,x∨y, . . . ,xk∨yk), ∀x,y∈X.

Clearly, (X,p) is a partial cone metric space,p(x,x) =xfor eachx∈X,p(x,θ) =θ⇐⇒x=θ,

Pis normal.

Let{yn}be a sequence in (X,p), whereyn= (yn,yn, . . . ,ynk). If{yn}isθ-Cauchy, then by Remark  and the normality ofP,limm,n→∞p(yn,ym) =θ, and so for eachε> , there

exists n such that p(yn,ym)<ε for each m,n≥n. Thus,yni∨ymi=p(yni,ymi) <ε for each m,n≥n and each ≤i≤k. This means limn→∞yni=  for each ≤i≤k, i.e.,limn→∞yn=θ. Therefore,limn→∞p(yn,θ) =limn→∞yn=θ =p(θ,θ),i.e.,yn→τp θ, and hence (X,p) isθ-complete sinceθ∈X.

Letyni=i( + _n)nfor each nand each ≤i≤k, ande= (e, e, . . . ,ke). It is clear that

limm,n→∞p(yn,ym) =e, and hence {yn} is a Cauchy sequence in (X,p). If there exists x∈X such that p(x,x) =e, thenx=e, which contradicts to the fact thate∈/ X since

e∈/ Q. This meansp(x,x)=efor eachx∈X, and so there does not existx∈Xsuch that

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3 Extensions of Banach contraction principle

In this section, we present some extensions of Banach contraction principle in the setting of partial cone metric spaces over a non-normal solid cone of an abstract normed vector space.

Theorem  Let(X,p)be aθ-complete partial cone metric space over a solid cone P of a normed vector space(E, · )and T :X→X.If there exists a linear bounded mapping L:P→P with the spectral radius r(L) < such that

p(Tx,Ty)Lp(x,y), ∀x,y∈X. ()

Then T has a unique ﬁxed point x∗∈X.In addition,for each x∈X,let

xn=Txn–=Tnx, ∀n, ()

then there exists a positive integer nsuch that yn τp

→x∗,where{yn}is a subsequence of{xn} deﬁned by yn=Tnn_x

.

Proof Byr(L) <  and Gelfand’s formula, there exists  <β<  such that

lim

n→∞ n

Ln₌_r₍_L₎≤_β,

which implies that there exists a positive integernsuch that

_Ln_≤_βn_, _∀n_≥_n

. ()

Clearly,

yn=Tnn_x

=TnT(n–)nx=Tnyn–, ∀n. ()

By (), () andL(P)⊂P,

p(yn,yn+) =pTnx(n–)n,T

n_xnn 

Ln_p₍_yn

–,yn) · · · Lnnp(y,y), ∀n,

and so by (p),

p(yn,ym)p(yn,yn+) +p(yn+,yn+) +· · ·+p(ym–,ym)

m–

i=n Lin_p₍_y

,y),

∀m>n. ()

By (),

m–

i=n

Likp(y,y)

≤p(y,y)

m–

i=n

Lni≤_p₍_y ,y)

m–

i=n

βin

=p(y,y)(β

nn_–_β(m+)n₎

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which implies thatm_i₌–_n Lik_p₍_y

,y) ·

→θ(n→ ∞) byβ< . Then by () and Lemma , for each∈intP, there exists a positive integernsuch that

p(yn,ym)

m–

i=n

Likp(y,y), ∀m,n≥n,

which implies that{yn}is aθ-Cauchy sequence in (X,p). Moreover by theθ-completeness of (X,p), there exists somex∗∈Xsuch thatyn→τp x∗andp(x∗,x∗) =θ, and so there exists a positive integern≥nsuch that

p yn,x∗

, ∀n≥n. ()

Sincer(L) < , then (I–L)(P)⊂P. Thus, by (p), (), () and (),

p Tn_x∗_,_x∗_p _yn

+,Tnx∗

+pyn+,x∗

Ln_p _yn_,_x∗₊_p_yn +,x∗

p yn,x∗

+pyn+,x∗

, ∀n≥n,

which together with the arbitrary property ofimplies thatp(Tn_x∗_,_x∗_{) =}_θ_{, and hence}

Tn_x∗₌_x∗ _{by (p) and (p),}_i.e._,_x∗ _{is a ﬁxed point of}_Tn_{. Let}_x∈_X_{be a ﬁxed point of}

Tn_,_i.e._,_Tn_x₌_x_{. Note that}Ln_{<  by (), then the inverse of}_I_–_Ln_{exists, denote it by} (I–Ln₎–_{. Moreover by Neumann’s formula, (}_I_–_Ln₎–₍_P₎⊂_P_{. By (), we have}_p₍_x_,_x∗_{) =}

p(Tn_x_,_Tn_x∗₎_Ln_p₍_x_,_x∗_{), and hence (}_I_–_Ln₎_p₍_x_,_x∗₎_θ_{. Act it with (}_I_–_Ln₎–_{, then}

p(x,x∗)θ. This implies thatp(x,x∗) =θ, and hencex=x∗by (p) and (p). Hencex∗is the unique ﬁxed point ofTn_.

Note thatx∗is a fixed point ofTn_{, then}_Tn₍_Tx∗_{) =}_T₍_Tn_x∗_{) =}_Tx∗_,_i.e._,_Tx∗ _{is also a} fixed point ofTn_{. By the uniqueness of fixed point of}_Tn_{, we have}_Tx∗₌_x∗_,_i.e._,_x∗_{is also} a fixed point ofT. Letybe a fixed point ofT. It is clear thatyis also a fixed point ofTn_, and soy=x∗by the uniqueness of fixed point ofTn_{. Hence}_x∗_{is the unique fixed point}

ofT. The proof is complete.

Remark  It is clear that Theorem  is exactly a special case of Theorem  withE=Rn_and P=Rn

+. LetLu=cufor some constantc∈[, ), thenr(L) =c< , and so Theorem  of [], Theorem  of [] and Theorem  of [] directly follow from Theorem . In addition, the normality ofPnecessarily assumed in [, , , ] has been removed in Theorem . Therefore, Theorem  indeed improves the corresponding results in [, , , ].

The following example shows the usability of Theorem .

Example  Let (X,p), E andP be the same ones as those in Example . Let (Tx)(t) = (Lx)(t)_tx(s)dsfor eachx∈X, wheret∈[,a],a> . Clearly,θis the unique ﬁxed point ofT.

For eachx,y∈X,p(Tx,Ty) =_tx(s)ds=Lp(x,y) wheneverx=y, andp(Tx,Ty) =_t[x(s) +

y(s)]ds=Lp(x,y) wheneverx= y,i.e., () is satisﬁed. It is clear that (Ln_x₎₍_t₎_≤tn

n!x∞for eacht∈[,a], and henceLn_x

∞≤a

n

n!x∞. Note that (Lnx)(t) = (Ln–x)(t), then

Lnx=Lnx_∞+ Lnx_∞≤

an n! +

an

(n– )!

x∞≤

an n! +

an

(n– )!

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which implies that Ln_≤ an n! +

an

(n–)!. Therefore by Gelfand’s formula, r(L) =

limn→∞ n √

Ln_{=  since}_lim n→∞ √n

n! = , and henceT has a unique ﬁxed point inXby Theorem .

However, the existence of ﬁxed point ofTcannot derive from the ﬁxed point results in [–, ], sincePis non-normal, andpis not a cone metric by Example  and Example , and there does not exist a constantc∈[, ) such thatp(Tx,Ty)cp(x,y).

Theorem  Let(X,p)be aθ-complete partial cone metric space over a solid cone P of a normed vector space(E, · )and T:X→X.If there exist four nonnegative constants c,

c,cand cwith c+c+c+ c< such that

p(Tx,Ty)cp(x,y) +cp(x,Tx) +cp(y,Ty) +c

p(x,Ty) +p(y,Tx), ∀x,y∈X. ()

Then T has a unique ﬁxed point x∗∈X,and for each x∈X,xn

τp

→x∗,where xnis deﬁned by().

Proof By (), () and (p),

p(xn,xn+)

=p(Txn–,Txn)

cp(xn–,xn) +cp(xn–,xn) +cp(xn,xn+) +c

p(xn–,xn+) +p(xn,xn)

cp(xn–,xn) +cp(xn–,xn) +cp(xn,xn+) +c

p(xn–,xn) +p(xn,xn+)

, ∀n,

and so

p(xn,xn+)cp(xn–,xn), ∀n,

wherec=c+c+c

–c–c <  byc+c+c+ c< . Moreover by (p),

p(xn,xm)

m–

i=n

p(xi,xi+)

m–

i=n

cip(x,x)

cn_p₍_x

,x)

 –c , ∀m>n. ()

Sincec< , then cnp(x,x) –c

·

→θ, and hence by Lemma , for each∈intP, there exists a positive integernsuch thatc

n_p₍_x

,x)

–c for alln≥n. Thus, by (),

p(xn,xm)c

n_p₍_x

,x)

 –c , ∀m,n≥n,

i.e.,{xn}is aθ-Cauchy sequence. Therefore, by theθ-completeness of (X,p), there exists

x∗∈Xsuch thatxn τp

→x∗andp(x∗,x∗) =θ, and so there exists a positive integern≥n such that

p xn,x∗ ( –c–c)

( +c+ c)

, ∀n≥n, ()

and

p(xn,xn+)

( –c–c) c

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By (p) and (),

p Tx∗,x∗p Tx∗,xn+

+p xn+,x∗

cp x∗,xn

+cp x∗,Tx∗

+cp(xn,xn+)

+c

px∗,xn+

+p xn,Tx∗

+p x∗,xn+

cp x∗,xn

+cp x∗,Tx∗

+cp(xn,xn+)

+c

px∗,xn+

+p xn,x∗+px∗,Tx∗+px∗,xn+

= (c+c)px∗,xn

+ (c+c)px∗,Tx∗

+cp(xn,xn+) + ( +c)p x∗,xn+

, ∀n,

and so

p Tx∗,x∗(c+c)p(x

∗_,_xn_{) +}_c

p(xn,xn+) + ( +c)p(x∗,xn+)  –c–c

, ∀n.

Then by () and (),

p Tx∗,x∗, ∀n≥n,

which together with the arbitrary property ofimplies thatp(Tx∗,x∗) =θ, and soTx∗=x∗

by (p) and (p). Letxbe a ﬁxed point ofT. Then by () and (p),

p x∗,x=p Tx∗,Tx

cpx∗,x

+cp x∗,Tx∗

+cp(x,Tx) +c

p x∗,Tx+p x,Tx∗

= (c+ c)p x∗,x

+cp(x,x)(c+c+ c)p x∗,x

.

This forces thatp(x∗,x) =θ sincec+c+ c< , and sox=x∗by (p) and (p). Hencex∗ is the unique ﬁxed point ofT. The proof is complete.

Remark  Theorem  and Theorem  of [] are special cases of Theorem  in the setting of cone metric spaces withc=c= ,c=candc=c=c= , respectively, and Theo-rem  of [] and TheoTheo-rem  of [] are special cases of TheoTheo-rem  withc=c= ,c=c. In addition,Pis not necessarily normal in Theorem . Compared with the corresponding results of [, ], the partial cone metric spaceXis only assumed to beθ-complete, but not complete in Theorem  and Theorem .

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors have contributed in obtaining the new results presented in this article. All authors read and approved the ﬁnal manuscript.

Author details

1_{Department of Mathematics, Jiangxi University of Finance and Economics, Nanchang, 330013, China.}2_{School of}

(9)

Acknowledgements

The work was supported by the Natural Science Foundation of China (11161022), the Natural Science Foundation of Jiangxi Province (20114BAB211006, 20122BAB201015), the Educational Department of Jiangxi Province (GJJ12280, GJJ13297) and Program for Excellent Youth Talents of JXUFE (201201).

Received: 24 June 2013 Accepted: 9 September 2013 Published:07 Nov 2013

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