R E S E A R C H
Open Access
Fixed point theorems of ordered contractive
mappings on cone metric spaces over Banach
algebras
Jiandong Yin
*, Tao Wang and Qi Yan
*Correspondence: [email protected] Department of Mathematics, Nanchang University, Nanchang, 330031, China
Abstract
In this paper, the concept of ordered contractive mappings is introduced on cone metric spaces over Banach algebras, and some existence theorems of fixed points for such mappings are obtained. As an application of a result, an example is given at the end of the paper.
MSC: 54H25; 47H10
Keywords: fixed point theorems; ordered contractive mappings; cone metric spaces over Banach algebras; spectral radius
1 Introduction
The study of fixed points for nonlinear mappings is an important subject of nonlinear function analysis, and the theory of fixed points is frequently applied to nonlinear integral equations and differential equations (see [–] and the references given there). Besides, the work of Caristi [], in which a partial ordering was introduced in metric spaces by a function and a fixed point theorem was proved, is worthy of attention. In , in the work of Huang and Zhang [], the concept of cone metric spaces as a generalization of general metric spaces was introduced, in which the distanced(x,y) ofxandyis defined to be a vector in an ordered Banach spaceE. It was also proved that the Banach contrac-tion principle remains true in the setting of cone metric spaces. After that, based on the work of Huang and Zhang [], the fixed point results of some mappings with certain con-tractive property on cone metric spaces appeared like mushrooms after rain (see [–] and the references therein and [–]). Among those works, the results of [, ] attract much attention since they give an answer to the natural problem that whether cone met-ric spaces are equivalent to metmet-ric spaces in terms of the existence of fixed points of the involved mappings. Concretely, the authors showed that any cone metric space (X,d) is equivalent to a usual metric space (X,d∗) if the real-valued metric functiond∗is defined by a nonlinear scalarization functionξe(see []) or by a Minkowski functionalqe(see []).
In , in order to generalize the Banach contraction principle to a more general form, Liu and Xu [, ] introduced the concept of cone metric spaces over Banach algebras by replacing Banach spaces with Banach algebras and proved some fixed point theorems of generalized Lipschitz mappings with weaker and natural conditions on the generalized
Lipschitz constantkby means of spectral radius and pointed out that it is significant to in-troduce the concept of cone metric spaces over Banach algebras because it can be proved that cone metric spaces over Banach algebras are not equivalent to metric spaces in terms of the existence of fixed points of the generalized Lipschitz mappings (see []). Liu and Xu [] showed that their results could not be reduced to a consequence of correspond-ing results in metric spaces by means of the methods in the literature. In , Xu and Radenović [] keenly discovered that the proofs of the main results of [] strongly de-pend on the condition that the underlying solid cone is normal. Naturally, they considered that whether the conclusions of [] remain valid if ‘normal’ is deleted from the hypothe-ses. By means of some properties of spectral radius, they proved that the main results of [] still hold without the assumption of normality of the cone involved. Hence Xu and Radenović [] improved the results of [].
In this paper, on the basis of [, ], we introduce a partial ordering given by a continu-ous function in cone metric spaces over Banach algebras as well as the concept of ordered contractive mappings that differ from the known contractive mappings, and present sev-eral fixed point results of such mappings under some natural conditions. Finally, as an application of one of our results, we give a concrete example.
2 Preliminaries
Consistent with Huang and Zhang [] and Liu and Xu [], the following definitions and results are needed in the sequel.
LetAalways be a real Banach algebra, that is,Ais a real Banach space in which an operation of multiplication is defined, subject to the following properties (for allx,y,z∈A, α∈R):
() (xy)z=x(yz);
() x(y+z) =xy+xzand(x+y)z=xz+yz; () α(xy) = (αx)y=x(αy);
() xy ≤ xy.
Here and subsequently, we assume that a Banach algebra has a unit (i.e., a multiplicative identity)esuch thatex=xe=xfor allx∈A.x∈Ais said to be invertible if there is an inverse elementy∈Asuch thatxy=yx=e. The inverse ofxis denoted byx–. We refer
the reader to [] for more details.
A non-empty closed convex subsetPof a Banach algebraAis called a cone if
(i) {θ,e} ⊂P; (ii) αP+βP⊂P; (iii) P=PP⊂P;
(iv) P∩(–P) ={θ},
whereθ denotes the null of the Banach algebraA.
Fix a coneP⊂A, a partial ordering ‘’ with respect toPcan be defined byxyif and only ify–x∈P.x≺ystands forxyandx=y.xystands fory–x∈int(P), hereint(P) denotes the interior ofP.Pis called a solid cone ifint(P)=∅.Pis called normal if there exists a positive constantNsuch that for allx,y∈A,θxy⇒ x ≤Ny.
Definition .([]) LetXbe a non-empty set andAbe a real Banach algebra. Suppose
that the mappingd:X×X→Asatisfies:
() d(x,y) =d(y,x)for allx,y∈X;
() d(x,y)d(x,z) +d(z,y)for allx,y,z∈X.
Thendis called a cone metric onXand (X,d) is called a cone metric space over the Banach algebraA.
See [] for some examples of cone metric spaces over Banach algebras.
Definition .([]) Let (X,d) be a cone metric space over the Banach algebraA,x∈X and let{xn}be a sequence inX. Then:
() {xn}converges toxif for eachc∈Awithθc, there is a natural numberNsuch thatd(xn,x)cfor alln>N. We denote this bylimn→∞xn=xorxn→x.
() {xn}is a Cauchy sequence if for eachc∈Awithθc, there is a natural numberN
such thatd(xn,xm)cfor alln,m>N.
() (X,d)is a complete cone metric space if every Cauchy sequence is convergent.
Definition . Let (X,d) be a cone metric space over the Banach algebraAandϕ:X→A
be a mapping. A relation ‘≤’ (for the sake of differing from the partial ordering ‘’ inA, we denote it by ‘≤’) inXis defined as follows:
x,y∈X, x≤y if and only if d(x,y)ϕ(x) –ϕ(y).
Clearly ‘≤’ is a partial ordering inXandx≤yimpliesϕ(x)ϕ(y). Here we call it the partial ordering induced byϕ. Meanwhile, (X,d) is called a partial ordering cone metric space over the Banach algebraA.
Definition . Let (X,d) be a partial ordering cone metric space over the Banach
alge-braA. We say thatx,y∈Xare comparable ifx≤yory≤xholds.
Ifx≤yandx=y, we writex<y. We writex=x∨yify≤xand writey=x∨yifx≤y. The same notions as those in Definition . can be defined inAas follows: For anyx,y∈A, if xyoryxholds, we say thatxandyare comparable. Letu,v∈A. Similarly, we write v=u∨vifuvand writeu=v∨uifvu. From Definition ., it is evident thatϕ(x) andϕ(y) are comparable inAifx,y∈Xare comparable.
Remark . Suppose thatAis a real Banach algebra,u,v,w∈A. The following results
are clear.
(i) Ifuandvare comparable, thenu–vandv–uare comparable and
θ(u–v)∨(v–u).
(ii) Ifuandv,uandwtogether withvandware comparable, then
(u–v)∨(v–u)(u–w)∨(w–u)+(w–v)∨(v–w).
Definition . Let (X,d) be a cone metric space over the Banach algebraAandA:X→X
be a mapping. We say thatAis continuous if for any{xn} ⊂X,xn→ximpliesAxn→Ax
(n→ ∞).
byϕ, whereϕ:X→Ais continuous,Pis a solid cone ofAwhich gives the partial ordering ‘’ inA.
Definition . A mappingA:X→Xis said to beϕ-ordered contractive if there exists
k∈Pwith ≤r(k) < such that for anyx,y∈X, ifxandyare comparable, thenAxand Ayare comparable and
ϕ(Ax) –ϕ(Ay)∨ϕ(Ay) –ϕ(Ax)kϕ(x) –ϕ(y)∨ϕ(y) –ϕ(x). (.)
Remark . See Example . for a support example of Definition ., Definition .,
Def-inition . and Definition ..
Lemma .([]) Let{xn}and{yn}be two sequences in X and xn→x,yn→yas n→ ∞.
Then d(xn,yn)→d(x,y) (n→ ∞).
Lemma . Let u,vn∈X(n= , , . . .).If for any natural number n,u and vnare
compa-rable and vn→v(n→ ∞),then u and vare comparable.
Proof Since for any natural numbern,uandvnare comparable, there exists a subsequence {vnk}of {vn}such that for any k, u≤vnk or vnk ≤u. Without loss of generality, we as-sume that for anyk,u≤vnk. Asvn→vwhenn→ ∞, we havevnk→vask→ ∞. By Lemma . and the continuity ofϕand notingu≤vnkfor allk≥, we have
d(u,v) = lim
k→∞d(u,vnk)k→∞lim
ϕ(u) –ϕ(vnk)
=ϕ(u) –ϕ(v).
Sou≤v. That isuandvare comparable.
Lemma . Let un,vn∈X(n= , , , . . .).If for any natural number n,unand vnare
com-parable and un→u,vn→v(n→ ∞),then uand vare comparable.In particular,if
for any natural number n,un≤vnand un→u,vn→v(n→ ∞),then u≤v.
Proof Because the proof is similar to that of Lemma ., we omit it.
Lemma .([]) Let x,y be vectors inA.If x and y commute,then the spectral radius r satisfies the following properties:
(i) r(xy)≤r(x)r(y); (ii) r(x+y)≤r(x) +r(y); (iii) |r(x) –r(y)| ≤r(x–y).
Lemma .([]) Let k∈A. If≤r(k) < ,then e–k is invertible and r((e–k)–)≤
( –r(k))–.
3 Main results
Theorem . If a continuous mapping A:X→X isϕ-ordered contractive and there exists x∈X such that xand Axare comparable,then A has a fixed point x∗ in X,that is,
Ax∗=x∗.Furthermore,the iterative sequence xn=Axn–(n= , , . . .)converges to x∗and
dx∗,x
(e–k)–ϕ(Ax) –ϕ(x)
∨ϕ(x) –ϕ(Ax)
,
where k is the element in P satisfying(.).
Proof Put
x=Ax, . . . ,xn=Axn–, . . . , n= , , . . . . (.)
SincexandAxare comparable andAisϕ-ordered contractive,x=Axandx=Ax
are comparable. By induction, it is not difficult to prove that for any natural numbern,xn
andxn+are comparable. AsAisϕ-ordered contractive, there existsk∈Pwith ≤r(k) <
such that for any natural numbern,
ϕ(xn+) –ϕ(xn)
∨ϕ(xn) –ϕ(xn+)
=ϕ(Axn) –ϕ(Axn–)
∨ϕ(Axn–) –ϕ(Axn)
kϕ(xn) –ϕ(xn–)
∨ϕ(xn–) –ϕ(xn)
· · ·
knϕ(Ax) –ϕ(x)
∨ϕ(x) –ϕ(Ax)
.
By the definition of ‘’ and the ordered contractive property ofA, we get that
d(xn+,xn)
ϕ(xn+) –ϕ(xn)
∨ϕ(xn) –ϕ(xn+)
· · ·
knϕ(Ax) –ϕ(x)
∨ϕ(x) –ϕ(Ax)
.
For any pair of natural numbersn,mwithm>n, we have
d(xn,xm)d(xn,xn+) +d(xn+,xn+) +· · ·+d(xm–,xm)
kn+kn++· · ·+km–ϕ(Ax) –ϕ(x)
∨ϕ(x) –ϕ(Ax)
∞
i=
ki
knϕ(Ax) –ϕ(x)
∨ϕ(x) –ϕ(Ax)
= (e–k)–knϕ(Ax) –ϕ(x)
∨ϕ(x) –ϕ(Ax)
.
Then by the same argument as that used in Theorem . of [], we can prove that{xn}is a Cauchy sequence. SinceXis complete, there existsx∗∈Xsuch thatxn→x∗(n→ ∞).
The continuity ofAimplies thatxn+=Axn→Ax∗(n→ ∞). SoAx∗=x∗, that is,x∗ is a
fixed point ofA. Furthermore, by Lemma ., we have
dx∗,x
= lim
n→∞d(xn,x)
lim n→∞
n
i=
∞
i=
ki–ϕ(Ax) –ϕ(x)
∨ϕ(x) –ϕ(Ax)
= (e–k)–ϕ(Ax) –ϕ(x)
∨ϕ(x) –ϕ(Ax)
.
Theorem . If A:X→X isϕ-ordered contractive and there exists x∈X such that for
any natural number n,xand Anxare comparable,then the same conclusions as those of
Theorem.hold.
Proof Define{xn}byxn=Axn–(n= , , . . .). By a proof similar to that of Theorem .,
one can prove easily that{xn}is a Cauchy sequence. The completeness ofXimplies that there isx∗∈Xsuch thatxn→x∗(n→ ∞). Next, we show thatx∗is a fixed point ofA.
For any pair of natural numbersn,mwithn<m,xandxm–n=Am–nxare
compara-ble according to the given conditions. AsAisϕ-ordered contractive,AxandAxm–nare
comparable. Continue this progress, we get thatxn=Anxandxm=Anxm–n=Amxare
comparable. Letm→ ∞, Lemma . shows that for any natural numbern,xnandx∗are
comparable. BecauseAisϕ-ordered contractive,AxnandAx∗are comparable and there
existsk∈Pwith ≤r(k) < such that
ϕ(Axn) –ϕ
Ax∗∨ϕAx∗–ϕ(Axn)
kϕ(xn) –ϕ
x∗∨ϕx∗–ϕ(xn)
. (.)
Noting the ordered contractive property ofA, we have
dxn+,Ax∗
=dAxn,Ax∗
ϕ(Axn) –ϕ
Ax∗∨ϕAx∗–ϕ(Axn)
kϕ(xn) –ϕ
x∗∨ϕx∗–ϕ(xn)
.
Letun=d(xn+,Ax∗),vn=k[(ϕ(xn) –ϕ(x∗))∨(ϕ(x∗) –ϕ(xn))],n= , , , . . . . Then, for any
natural numbern,un≤vn. Notice thatun=d(xn+,Ax∗)→d(x∗,Ax∗) and the continuity
ofϕ, by Lemma ., we haved(x∗,Ax∗)k[ϕ(x∗) –ϕ(x∗)] =θ, soAx∗=x∗. Moreover, we get that
d(xn+,xn)
ϕ(xn+) –ϕ(xn)
∨ϕ(xn) –ϕ(xn+)
=ϕ(Axn) –ϕ(Axn–)
∨ϕ(Axn–) –ϕ(Axn)
kϕ(xn) –ϕ(xn–)
∨ϕ(xn–) –ϕ(xn)
· · ·
knϕ(Ax) –ϕ(x)
∨ϕ(x) –ϕ(Ax)
.
Hence Lemma . gives that
dx∗,x
= lim
n→∞d(xn,x)
lim n→∞
n
i=
lim n→∞
n
i=
ki–ϕ(Ax) –ϕ(x)
∨ϕ(x) –ϕ(Ax)
= (e–k)–ϕ(Ax) –ϕ(x)
∨ϕ(x) –ϕ(Ax)
.
Theorem . Let A:X→X be a continuous mapping satisfying the following conditions:
(i) There existλ,λ∈Pwith≤r(λ) +r(λ) < such that for any comparable pair
x,y∈X,AxandAyare comparable.Moreover,ifxandAx,yandAyare comparable,
then
ϕ(Ax) –ϕ(Ay)∨ϕ(Ay) –ϕ(Ax)
λ
ϕ(Ax) –ϕ(x)∨ϕ(x) –ϕ(Ax)
+λ
ϕ(Ay) –ϕ(y)∨ϕ(y) –ϕ(Ay).
(ii) There existsx∈Xsuch thatxandAxare comparable.
Then A has a fixed point x∗ in X. Furthermore,the iterative sequence xn=Axn– (n=
, , . . .)converges to x∗and
dx∗,x
e– (e–λ)–λ
–
ϕ(Ax) –ϕ(x)
∨ϕ(x) –ϕ(Ax)
.
Proof Letxn=Axn–(n= , , . . .). SincexandAxare comparable, according to the given
condition (i) and by induction, it is easy to verify that for any natural numbern,xnand
xn+=Axnare comparable. Therefore
d(xn,xn+)
ϕ(xn) –ϕ(xn+)
∨ϕ(xn+) –ϕ(xn)
=ϕ(Axn–) –ϕ(Axn)
∨ϕ(Axn) –ϕ(Axn–)
λ
ϕ(Axn–) –ϕ(xn–)
∨ϕ(xn–) –ϕ(Axn–)
+λ
ϕ(Axn) –ϕ(xn)
∨ϕ(xn) –ϕ(Axn)
=λ
ϕ(xn) –ϕ(xn–)
∨ϕ(xn–) –ϕ(xn)
+λ
ϕ(xn+) –ϕ(xn)
∨ϕ(xn) –ϕ(xn+)
,
which yields that
d(xn,xn+)
ϕ(xn) –ϕ(xn+)
∨ϕ(xn+) –ϕ(xn)
(e–λ)–λ
ϕ(xn–) –ϕ(xn)
∨ϕ(xn) –ϕ(xn–)
· · ·
(e–λ)–λ
n
ϕ(x) –ϕ(x)
∨ϕ(x) –ϕ(x)
.
Lemma . together with Lemma . shows thatr((e–λ)–λ)≤r((e–λ)–)r(λ)≤( –
r(λ))–r(λ). Asr(λ) +r(λ) < ,r((e–λ)–λ)≤( –r(λ))–r(λ) < . Thus by the same
Ax∗(n→ ∞). SoAx∗=x∗. Furthermore, by Lemma ., we have
dx∗,x
= lim
n→∞d(xn,x)
lim n→∞
n
i=
d(xi,xi–)
lim n→∞
n
i=
(e–λ)–λ
i–
ϕ(Ax) –ϕ(x)
∨ϕ(x) –ϕ(Ax)
=e– (e–λ)–λ
–
ϕ(Ax) –ϕ(x)
∨ϕ(x) –ϕ(Ax)
.
Theorem . Suppose that A:X→X is a mapping satisfying the following conditions:
(i) There existλ,λ∈Pwith≤r(λ) +r(λ) < such that for any comparable pair
x,y∈X,AxandAyare comparable.Moreover,ifxandAx,yandAyare comparable,
then
ϕ(Ax) –ϕ(Ay)∨ϕ(Ay) –ϕ(Ax)
λ
ϕ(Ax) –ϕ(x)∨ϕ(x) –ϕ(Ax)
+λ
ϕ(Ay) –ϕ(y)∨ϕ(y) –ϕ(Ay).
(ii) There existsx∈Xsuch that for any natural numbern,xandAnxare comparable.
Then the same conclusions as those in Theorem.hold.
Proof Setxn=Axn–(n= , , . . .). It follows by the same method as in Theorem . that
{xn}is a Cauchy sequence and there existsx∗inXsuch thatxn→x∗(n→ ∞). Next we
show thatx∗is a fixed point ofA.
Also, by a proof similar to that of Theorem . and by Lemma ., we get that for any natural numbern,xnandxn+=Axnare comparable,xnandx∗are also comparable.
Ac-cording to the given condition (i), for any natural numbern,AxnandAx∗are comparable.
Letn→ ∞, by Lemma .,x∗andAx∗are comparable. So
ϕ(Axn) –ϕ
Ax∗∨ϕAx∗–ϕ(Axn)
λ
ϕ(Axn) –ϕ(xn)
∨ϕ(xn) –ϕ(Axn)
+λ
ϕAx∗–ϕx∗∨ϕx∗–ϕAx∗.
Letn→ ∞, the continuity ofϕtogether with Lemma . implies that
θ ϕx∗–ϕAx∗∨ϕAx∗–ϕx∗
λ
ϕAx∗–ϕx∗∨ϕx∗–ϕAx∗. (.)
Note that ≤r(λ) < , by Proposition .(ii) of [], we have
The partial ordering inXgives that
dx∗,Ax∗ϕx∗–ϕAx∗∨ϕAx∗–ϕx∗. (.)
It is evident from (.) and (.) thatAx∗=x∗, sox∗is a fixed point ofA.
By the same argument as in Theorem . and the partial ordering inX, we have
d(xn,xn+)
ϕ(xn) –ϕ(xn+)
∨ϕ(xn+) –ϕ(xn)
(e–λ)–λ
ϕ(xn) –ϕ(xn–)
∨ϕ(xn–) –ϕ(xn)
· · ·
(e–λ)–λ
n
ϕ(x) –ϕ(x)
∨ϕ(x) –ϕ(x)
.
From Lemma ., we have
dx∗,x
= lim
n→∞d(xn,x)
lim n→∞
n
i=
d(xi,xi–)
lim n→∞
n
i=
(e–λ)–λ
i–
ϕ(Ax) –ϕ(x)
∨ϕ(x) –ϕ(Ax)
=e– (e–λ)–λ
–
ϕ(Ax) –ϕ(x)
∨ϕ(x) –ϕ(Ax)
.
Theorem . Assume that A:X→X is continuous and satisfies the following:
(i) There existλ,λ∈Pwith≤r(λ) < such that for any comparable pairx,y∈X,
AxandAyare comparable.Moreover,ifxandAx,yandAyare comparable,then
ϕ(Ax) –ϕ(Ay)∨ϕ(Ay) –ϕ(Ax)
λ
ϕ(Ax) –ϕ(y)∨ϕ(y) –ϕ(Ax)
+λ
ϕ(Ay) –ϕ(x)∨ϕ(x) –ϕ(Ay).
(ii) There existsx∈Xsuch thatxandAx,xandAxare comparable.
Then A has a fixed point x∗ in X. Furthermore,the iterative sequence xn=Axn– (n=
, , . . .)converges to x∗and
dx∗,x
e–λ(e–λ)–
–
ϕ(Ax) –ϕ(x)
∨ϕ(x) –ϕ(Ax)
.
Proof Pickxn=Axn–(n= , , . . .). SincexandAx,xandAxare comparable, for any
natural numbern,xnandxn+as well asxnandxn+are comparable. Thereforeϕ(xn) and
ϕ(xn+),ϕ(xn) andϕ(xn+) are comparable. It follows from Remark . that
θ ϕ(xn) –ϕ(xn+)
∨ϕ(xn+) –ϕ(xn)
=ϕ(Axn–) –ϕ(Axn)
∨ϕ(Axn) –ϕ(Axn–)
λ
ϕ(Axn–) –ϕ(xn)
∨ϕ(xn) –ϕ(Axn–)
+λ
ϕ(Axn) –ϕ(xn–)
∨ϕ(xn–) –ϕ(Axn)
=λ
ϕ(xn+) –ϕ(xn–)
∨ϕ(xn–) –ϕ(xn+)
λ
ϕ(xn+) –ϕ(xn)
∨ϕ(xn) –ϕ(xn+)
+ϕ(xn) –ϕ(xn–)
∨ϕ(xn–) –ϕ(xn)
.
Therefore
θ ϕ(xn) –ϕ(xn+)
∨ϕ(xn+) –ϕ(xn)
(e–λ)–λ
ϕ(xn) –ϕ(xn–)
∨ϕ(xn–) –ϕ(xn)
· · ·
(e–λ)–λ
n
ϕ(x) –ϕ(x)
∨ϕ(x) –ϕ(x)
.
The definition of the partial ordering inXgives that
d(xn,xn+)
ϕ(xn) –ϕ(xn+)
∨ϕ(xn+) –ϕ(xn)
(e–λ)–λ
ϕ(xn) –ϕ(xn–)
∨ϕ(xn–) –ϕ(xn)
(e–λ)–λ
n
ϕ(x) –ϕ(x)
∨ϕ(x) –ϕ(x)
.
Since ≤r(λ) <,{xn}is a Cauchy sequence. Hence there existsx∗∈Xsuch thatxn→x∗
(n→ ∞). The continuity ofAshows that
Ax∗= lim
n→∞Axn=n→∞limxn+=x
∗, (.)
that is,x∗is a fixed point ofA. From Lemma ., we have
dx∗,x
= lim
n→∞d(xn,x)
lim n→∞
n
i=
d(xi,xi–)
lim n→∞ n i=
(e–λ)–λ
i–
ϕ(Ax) –ϕ(x)
∨ϕ(x) –ϕ(Ax)
=e– (e–λ)–λ
–
ϕ(Ax) –ϕ(x)
∨ϕ(x) –ϕ(Ax)
.
Theorem . Let A:X→X be a mapping and satisfy the following conditions:
(i) There existλ,λ∈Pwith≤r(λ) < such that for any comparable pairx,y∈X,
AxandAyare comparable.Moreover,ifxandAx,yandAyare also comparable,
then
ϕ(Ax) –ϕ(Ay)∨ϕ(Ay) –ϕ(Ax)
λ
ϕ(Ax) –ϕ(y)∨ϕ(y) –ϕ(Ax)
+λ
(ii) There existsx∈Xsuch that for any natural numbern,xandAnxare comparable.
Then the same conclusions as those in Theorem.hold.
Proof Define{xn}byxn=Axn–(n= , , . . .). As in the proof of Theorem ., there exists
x∗∈Xsuch thatxn→x∗(n→ ∞) and
ϕ(Axn) –ϕ
Ax∗∨ϕAx∗–ϕ(Axn)
λ
ϕ(Axn) –ϕ
x∗∨ϕx∗–ϕ(Axn)
+λ
ϕAx∗–ϕ(xn)
∨ϕ(xn) –ϕ
Ax∗.
Letn→ ∞, the continuity ofϕimplies that
θ ϕx∗–ϕAx∗∨ϕAx∗–ϕx∗
λ
ϕAx∗–ϕx∗∨ϕx∗–ϕAx∗.
Note that ≤r(λ) < , by Proposition .(ii) of [], we have
ϕx∗=ϕAx∗. (.)
Sincex∗andAx∗are comparable,
dx∗,Ax∗ϕx∗–ϕAx∗∨ϕAx∗–ϕx∗. (.)
From (.) and (.), we haveAx∗=x∗, sox∗is a fixed point ofA. By the same method as in Theorem ., we have
d(xn,xn+)
ϕ(xn) –ϕ(xn+)
∨ϕ(xn+) –ϕ(xn)
(e–λ)–λ
ϕ(xn) –ϕ(xn–)
∨ϕ(xn–) –ϕ(xn)
(e–λ)–λ
n
ϕ(x) –ϕ(x)
∨ϕ(x) –ϕ(x)
.
Lemma . shows that
dx∗,x
= lim
n→∞d(xn,x)
lim n→∞
n
i=
d(xi,xi–)
lim n→∞
n
i=
(e–λ)–λ
i–
ϕ(Ax) –ϕ(x)
∨ϕ(x) –ϕ(Ax)
=e– (e–λ)–λ
–
ϕ(Ax) –ϕ(x)
∨ϕ(x) –ϕ(Ax)
.
Example . LetA=R. For eachx= (x
,x)∈A, letx=|x|+|x|. The multiplication
is defined by
ThenAis a Banach algebra with unite= (, ). LetP={(x,x)∈R|x≥,x≥}and
X=R. A metricdonXis defined by
d(x,y) =d(x,x), (y,y)
=|x–y|,|x–y|
∈P.
Then (X,d) is a complete cone metric space over the Banach algebraA. Now define mappingT:X→Xby
T(x,y) = lnex–+ ,tan
πarctan(y+ )
.
ThenT has a fixed point inX.
In fact, letk= (e,π)∈Pand defineϕ:X→Abyϕ(x,y) = (–e
x–x, –arctan(y+ ) –y).
Thenk∈Pwith <r(k) < and it is not difficult to verify thatϕis continuous onXand the partial ordering inXcan be induced byϕ. Clearly,Tis continuous and for any com-parable pairx= (x,x),y= (y,y)∈X,TxandTyare comparable. Moreover, by simple
calculations, we can obtain that
ϕ(Tx) –ϕ(Ty)∨ϕ(Ty) –ϕ(Tx)≤kϕ(x) –ϕ(y)∨ϕ(y) –ϕ(x).
Thus T is ϕ-ordered contractive. Takex= (–, ), thenx= (–, )≤Tx= (ln(e–+
),tan()), that is,xandTxare comparable. Hence by Theorem .,Thas a fixed point
inX.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Received: 5 January 2015 Accepted: 23 March 2015
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