# Fixed Point and Common Fixed Point Theorems on Ordered Cone b-Metric Spaces

Authors: Sahar Mohammad Abusalim; Mohd Salmi Md Noorani
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

The concept of a cone b-metric space has been introduced recently as a generalization of a b-metric space and a cone metric space in 2011. The aim of this paper is to establish some fixed point and common fixed point theorems on ordered cone b-metric spaces. The proposed theorems expand and generalize several well-known comparable results in the literature to ordered cone b-metric spaces. Some supporting examples are given.


---

## Body

## 1. Introduction

Fixed point theory has attracted many researchers since 1922 with the admired Banach fixed point theorem. This theorem supplies a method for solving a variety of applied dilemma in mathematical sciences and engineering. A large literature on this subject exists, and this is a very active area of research at present. Banach contraction principle has been generalized in dissimilar directions in different spaces by mathematicians over the years; for more details on this and related topics, we refer to [1–6] and references therein.In contemporary time, fixed point theory has evolved speedily in partially ordered cone metric spaces; that is, cone metric spaces equipped with a partial ordering, for some new results in ordered metric spaces see [7]. A coming early result in this bearing was constituted by Altun and Durmaz [8] under the condition of normality for cones. Then, Altun et al. [9] generalized the results of Altun and Durmaz [8] by omitting the assumption of normality condition for cones. Afterward, several authors have studied fixed point and common fixed point problems in ordered cone metric spaces; for more details see [10–17].In 2011, Hussain and Shah [18] presented cone b-metric spaces as a generalization of b-metric spaces and cone metric spaces; for some new results in b-metric spaces see [19]. They not only constructed some topological properties in such spaces but also ameliorated some current results about KKM mappings in the setting of a cone b-metric space. After some time, many authors have been motivated to demonstrate fixed point theorems as well as common fixed point theorems for two or more mappings on cone b-metric spaces by the incipient work of Hussain and Shah [18] (see [20–23] and the references therein).In [8], Altun and Durmaz proved the following results under the condition of normality for cones.Theorem 1 (see [8]).
Let(X,⊑) be a partially ordered set, suppose that there exists a cone metric d in X such that the cone metric space (X,d) is complete, and let P be a normal cone with normal constant K. Let f:X→X be a continuous and nondecreasing mapping with respect to ⊑. Suppose that the following three assertions hold: (i)
there existsk∈[0,1) such that d(fx,fy)⪯kd(x,y) for all x,y∈X with y⊑x;(ii)
there existsx0∈X such that x0⊑fx0.Then f has a fixed point in X.In [9], Altun et al. generalized the above theorem and proved it without normality condition for cones.Theorem 2 (see [9]).
Let(X,⊑) be a partially ordered set and suppose that there exists a cone metric d in X such that the cone metric space (X,d) is complete over a solid cone P. Let f:X→X be a continuous and nondecreasing mapping with respect to ⊑. Suppose that the following two assertions hold: (i)
there existk,l,r∈[0,1) with k+2l+2r<1 such that
(1)d(fx,fy)⪯kd(x,y)+l(d(fx,x)+d(fy,y))+r(d(fx,y)+d(fy,x))
for all x,y∈X with y⊑x;(ii)
there existsx0∈X such that x0⊑fx0.Then f has a fixed point in X.Theorem 3 (see [9]).
Let(X,⊑) be a partially ordered set and suppose that there exists a cone metric d in X such that the cone metric space (X,d) is complete over a solid cone P. Let f:X→X be a nondecreasing mapping with respect to ⊑. Suppose that the following three assertions hold: (i)
there existk,l,r∈[0,1) with k+2l+2r<1 such that
(2)d(fx,fy)⪯kd(x,y)+l(d(fx,x)+d(fy,y))+r(d(fx,y)+d(fy,x))
for all x,y∈X with y⊑x;(ii)
there existsx0∈X such that x0⊑fx0;(iii)
if an increasing sequence{xn} converges to x in X, then xn⊑x for all n.Then f has a fixed point in X.In the same paper, they also presented the following two common fixed point results in ordered cone metric spaces.Theorem 4 (see [9]).
Let(X,⊑) be a partially ordered set and suppose that there exists a cone metric d in X such that the cone metric space (X,d) is complete over a solid cone P. Let f,g:X→X be two weakly increasing mappings with respect to ⊑. Suppose that the following three assertions hold:(i)
there existk,l,r∈[0,1) with k+2l+2r<1 such that
(3)d(fx,gy)⪯kd(x,y)+l(d(x,fx)+d(y,gy))+r(d(y,fx)+d(x,gy))
for all comparative x,y∈X;(ii)
f or g is continuous.Then f and g have a common fixed point x*∈X.Theorem 5 (see [9]).
Let(X,⊑) be a partially ordered set and suppose that there exists a cone metric d in X such that the cone metric space (X,d) is complete over a solid cone P. Let f,g:X→X be two weakly increasing mappings with respect to⊑. Suppose that the following three assertions hold: (i)
there existk,l,r∈[0,1) with k+2l+2r<1 such that
(4)d(fx,gy)⪯kd(x,y)+l(d(x,fx)+d(y,gy))+r(d(y,fx)+d(x,gy))
for all comparative x,y∈X;(ii)
if an increasing sequence{xn} converges to x in X, then xn⊑x for all n.Then f and g have a common fixed point x*∈X.In this paper, we prove some fixed point and common fixed point theorems on ordered cone b-metric spaces. Our results extend and generalize several well-known comparable results in the literature to ordered cone b-metric spaces. Throughout this paper, we do not impose the normality condition for the cones, but the only assumption is that the coneP is solid, that is, int P≠∅.The following definitions and results shall be needed in the sequel.LetE be a real Banach space and θ denotes the zero element in E. A cone P is a subset of E such that(1)
P is nonempty closed set and P≠{θ};(2)
ifa, b are nonnegative real numbers and x,y∈P, then ax+by∈P;(3)
x∈P and -x∈P imply x=θ.For any cone P⊂E, the partial ordering ⪯ with respect to P is defined by x⪯y if and only if y-x∈P. The notation of ≺ stands for x⪯y but x≠y. Also, we use x≪y to indicate that y-x∈int P, where int P denotes the interior of P. A cone P is called normal if there exists the number K such that
(5)θ⪯x⪯y⟹∥x∥≤K∥y∥,
for all x,y∈E. The least positive number K satisfying the above condition is called the normal constant of P.Definition 6 (see [18]).
LetX be a nonempty set and E a real Banach space equipped with the partial ordering ⪯ with respect to the cone P. A vector-valued function d:X×X→E is said to be a cone b-metric function on X with the constant s≥1 if the following conditions are satisfied:(1)
θ⪯d(x,y) for all x,y∈X and d(x,y)=θ if and only if x=y;(2)
d(x,y)=d(y,x) for all x,y∈X;(3)
d(x,y)⪯s(d(x,y)+d(y,z)) for all x,y,z∈X.Then pair (X,d) is called a cone b-metric space (or a cone metric type space); we shall use the first mentioned term.Observe that ifs=1, then the ordinary triangle inequality in a cone metric space is satisfied; however, it does not hold true when s>1. Thus the class of cone b-metric spaces is effectively larger than that of the ordinary cone metric spaces. That is, every cone metric space is a cone b-metric space, but the converse need not be true. The following examples show the above remarks.Example 7.
LetX={-1,0,1},E=ℝ2,andP={(x,y):x≥0,y≥0}. Define d:X×X→P by d(x,y)=d(y,x) for all x,y∈X,d(x,x)=θ,x∈X, and d(-1,0)=(3,3),d(-1,1)=d(0,1)=(1,1). Then (X,d) is a complete cone b-metric space but the triangle inequality is not satisfied. Indeed, we have that d(-1,1)+d(1,0)=(1,1)+(1,1)=(2,2)≺(3,3)=d(-1,0). It is not hard to verify that s=3/2.Example 8.
LetX=ℝ, E=ℝ2, andP={(x,y)∈E:x≥0,y≥0}. Define d:X×X→E by d(x,y)=(|x-y|2,|x-y|2). Then, it is easy to see that (X,d) is a cone b-metric space with the coefficient s=2. But it is not a cone metric spaces since the triangle inequality is not satisfied.Definition 9 (see [18]).
Let(X,d) be a cone b-metric space, {xn} a sequence in X and x∈X. (1)
For allc∈E with θ≪c, if there exists a positive integer N such that d(xn,x)≪c for all n>N, then xn is said to be convergent and x is the limit of {xn}. One denotes this by xn→x.(2)
For allc∈E with θ≪c, if there exists a positive integer N such that d(xn,xm)≪c for all n,m>N, then {xn} is called a Cauchy sequence in X.(3)
A cone metric space(X,d) is called complete if every Cauchy sequence in X is convergent.The following lemma is useful in our work.Lemma 10 (see [24]).
(
1
)
IfE is a real Banach space with a cone P and a⪯λa where a∈P and 0≤λ<1, then a=θ.
(
2
)
Ifc∈intP, θ⪯an, and an→θ, then there exists a positive integer N such that an≪c for all n≥N.
(
3
)
Ifa⪯b and b≪c, then a≪c.
(
4
)
Ifθ⪯u≪c for each θ≪c, then u=θ.


## 2. Fixed Point Results

In this section, we prove some fixed point theorems on ordered cone b-metric space. We begin with a simple but a useful lemma.Lemma 11.
Let{xn} be a sequence in a cone b-metric space (X,d) with the coefficient s≥1 relative to a solid cone P such that
(6)d(xn,xn+1)⪯hd(xn-1,xn),
where h∈[0,1/s) and n=1,2,…. Then {xn} is a Cauchy sequence in (X,d).Proof.
Letm>n≥1. It follows that
(7)d(xn,xm)⪯sd(xn,xn+1)+s2d(xn+1,xn+2)+⋯+sm-nd(xm-1,xm).
Now, (6) and sh<1 imply that
(8)d(xn,xm)⪯sd(xn,xn+1)+s2d(xn+1,xn+2)+⋯+sm-nd(xm-1,xm)⪯shnd(x0,x1)+s2hn+1d(x0,x1)+⋯+sm-nhm-1d(x0,x1)=(shn+s2hn+1+⋯+sm-nhm-1)d(x0,x1)=shn(1+sh+(sh)2+⋯+(sh)m-n-1)d(x0,x1)⪯shn1-shd(x0,x1)⟶θasn⟶∞.
According to Lemma 10(2), and for any c∈E with c≫θ, there exists N0∈ℕ such that for any n>N0, (shn/(1-sh))d(x0,x1)≪c. Furthermore, from (8) and for any m>n>N0, Lemma 10(3) shows that
(9)d(xn,xm)≪c.
Hence, by Definition 9(2){xn} is a Cauchy sequence in X.Theorem 12.
Let(X,⊑) be a partially ordered set and suppose that there exists a cone b-metric d in X such that the cone b-metric space (X,d) is complete with the coefficient s≥1 relative to a solid cone P. Let f:X→X be a continuous and nondecreasing mapping with respect to ⊑. Suppose that the following three assertions hold: (i)
there existai,i=1,…,5, such that 2sa1+(s+1)(a2+a3)+(s2+s)(a4+a5)<2 with ∑i=15‍ai<1,
(10)d(fx,fy)⪯a1d(x,y)+a2d(fx,x)+a3d(fy,y)+a4d(fx,y)+a5d(fy,x)
for all x,y∈X with y⊑x;(ii)
there existsx0∈X such that x0⊑fx0.Then f has a fixed point x*∈X.Proof.
Ifx0=fx0, then the proof is finished. Suppose that x0≠fx0. Since x0⊑fx0 and f is nondecreasing with respect to⊑, we obtain by induction that x0⊑fx0=x1⊑f1x0=x2⊑⋯⊑fn-1x0=xn⊑fnx0=xn+1⊑⋯. Then we have,
(11)d(xn+1,xn)=d(fnx0,fn-1x0)=d(f(fn-1x0),f(fn-2x0))⪯a1d(fn-1x0,fn-2x0)+a2d(fn-1x0,fn-2x0)+a3d(fnx0,fn-1x0)+a4d(fnx0,fn-2x0)+a5d(fn-1x0,fn-1x0)=a1d(xn,xn-1)+a2d(xn+1,xn)+a3d(xn,xn-1)+a4d(xn+1,xn-1)+a5d(xn,xn)⪯a1d(xn,xn-1)+a2d(xn+1,xn)+a3d(xn,xn-1)+sa4(d(xn+1,xn)+d(xn,xn-1)).
Then, one can assert that
(12)d(xn+1,xn)⪯(a1+a3+sa4)d(xn,xn-1)+(a2+sa4)d(xn+1,xn).
On the other hand, we have
(13)d(xn,xn+1)=d(fn-1x0,fnx0)=d(f(fn-2x0),f(fn-1x0))⪯a1d(fn-2x0,fn-1x0)+a2d(fn-2x0,fn-1x0)+a3d(fn-1x0,fnx0)+a4d(fn-1x0,fn-1x0)+a5d(fnx0,fn-2x0)=a1d(xn,xn-1)+a2d(xn,xn-1)+a3d(xn+1,xn)+a4d(xn,xn)+a5d(xn+1,xn-1)⪯a1d(xn,xn-1)+a2d(xn,xn-1)+a3d(xn+1,xn)+sa5(d(xn+1,xn)+d(xn,xn-1)).
Then, one can assert that
(14)d(xn+1,xn)⪯(a1+a2+sa5)d(xn,xn-1)+(a3+sa5)d(xn+1,xn).
Adding (12) and (14), we get
(15)d(xn+1,xn)⪯2a1+a2+a3+sa4+sa52-(a2+a3+sa4+sa5)d(xn,xn-1)=λd(xn,xn-1),
where λ=(2a1+a2+a3+sa4+sa5)/(2-(a2+a3+sa4+sa5))<1/s. According to Lemma 11, we have {xn} is a Cauchy sequence in X. Since X is complete, there exists x*∈X such that xn→x*. Since f is continuous, then x*=limxn+1=limfnx0=limf(fn-1x0)=f(limfn-1x0)=f(limxn)=f(x*). Therefore, x* is a fixed point of f.If we use condition (iii) instead of the continuity off in Theorem 12, we have the following result.Theorem 13.
Let(X,⊑) be a partially ordered set and suppose that there exists a cone b-metric d in X such that the cone b-metric space (X,d) is complete with the coefficient s≥1 relative to a solid cone P. Let f:X→X be a nondecreasing mapping with respect to⊑. Suppose that the following three assertions hold: (i)
there existai,i=1,…,5, such that 2sa1+(s+1)(a2+a3)+(s2+s)(a4+a5)<2 with ∑i=15‍ai<1,
(16)d(fx,fy)⪯a1d(x,y)+a2d(fx,x)+a3d(fy,y)+a4d(fx,y)+a5d(fy,x)
for all x,y∈X with y⊑x;(ii)
there existsx0∈X such that x0⊑fx0;(iii)
if an increasing sequence{xn} converges to x in X, then xn⊑x for all n.Then f has a fixed point x*∈X.Proof.
As in the Theorem12, we can construct an increasing sequence {xn} and prove that there exists x*∈X such that xn→x*. Now, condition (iii) implies xn⊑x* for all n. Therefore, we can use condition (i) and so
(17)d(fxn,fx*)⪯a1d(xn,x*)+a2d(fxn,xn)+a3d(fx*,x*)+a4d(fxn,x*)+a5d(fx*,xn).
Taking n→∞, we have d(x*,fx*)⪯(a3+a5)d(x*,fx*)d(x*,fx*). Since (a3+a5)<1, Lemma 10(1) shows that d(x*,fx*)=θ; that is, x*=fx*. Therefore x* is a fixed point of f.


## 3. Common Fixed Point Results

Now, we give two common fixed point theorems on ordered cone b-metric spaces. We need the following definition.Definition 14 (see [9]).
Let(X,⊑) be a partially ordered set. Two mappings f,g:X→X are said to be weakly increasing if fx⊑gfx and gx⊑fgx hold for all x∈X.Theorem 15.
Let(X,⊑) be a partially ordered set and suppose that there exists a cone b-metric d in X such that the cone b-metric space (X,d) is complete with the coefficient s≥1 relative to a solid cone P. Let f,g:X→X be two weakly increasing mappings with respect to⊑. Suppose that the following three assertions hold:(i)
there existai,i=1,…,5, such that 2sa1+(s+1)(a2+a3)+(s2+s)(a4+a5)<2 with ∑i=15‍ai<1,
(18)d(fx,gy)⪯a1d(x,y)+a2d(x,fx)+a3d(y,gy)+a4d(y,fx)+a5d(x,gy)
for all comparative x,y∈X;(ii)
f or g is continuous.Then f and g have a common fixed point x*∈X.Proof.
Letx0 be an arbitrary point of X and define a sequence {xn} in X as follows: x2n+1=fx2n and x2n+2=gx2n+1 for all n>0. Note that, since f and g are weakly increasing, we have x1=fx0⊑gfx0=gx1=x2 and x2=gx1⊑fgx1=fx2=x3, and continuing this process we have x1⊑x2⊑⋯⊑xn⊑xn+1⊑⋯. That is, the sequence {xn} is nondecreasing. Now, since x2n and x2n+1 are comparative, we can use the inequality (18), and then we have
(19)d(x2n+1,x2n+2)=d(fx2n,gx2n+1)⪯a1d(x2n,x2n+1)+a2d(x2n,fx2n)+a3d(x2n+1,gx2n+1)+a4d(x2n+1,fx2n)+a5d(x2n,gx2n+1)⪯a1d(x2n,x2n+1)+a2d(x2n,x2n+1)+a3d(x2n+1,x2n+2)+a4d(x2n+1,x2n+1)+a5d(x2n,x2n+2)⪯(a1+a2)d(x2n,x2n+1)+a3d(x2n+1,x2n+2)+sa5(d(x2n,x2n+1)+d(x2n+1,x2n+2))=(a1+a2+sa5)d(x2n,x2n+1)+(a3+sa5)d(x2n+1,x2n+2).
Hence,
(20)(1-(a3+sa5))d(x2n+1,x2n+2)⪯(a1+a2+sa5)d(x2n,x2n+1).
On the other hand and by symmetry we have(21)d(x2n+2,x2n+1)=d(gx2n+1,fx2n)⪯a1d(x2n+1,x2n)+a2d(x2n+1,gx2n+1)+a3d(x2n,fx2n)+a4d(x2n,gx2n+1)+a5d(x2n+1,fx2n)⪯a1d(x2n+1,x2n)+a2d(x2n+1,x2n+2)+a3d(x2n,x2n+1)+a4d(x2n,x2n+2)+a5d(x2n+1,x2n+1)⪯(a1+a3)d(x2n,x2n+1)+a2d(x2n+1,x2n+2)+sa4(d(x2n,x2n+1)+d(x2n+1,x2n+2))=(a1+a3+sa4)d(x2n,x2n+1)+(a2+sa4)d(x2n+1,x2n+2).
Hence,
(22)(1-(a2+sa4))d(x2n+2,x2n+1)⪯(a1+a3+sa4)d(x2n,x2n+1).
Adding inequalities (20) and (22), we get
(23)d(x2n+1,x2n+2)⪯(2a1+a2+a3+sa4+sa5)2-(a2+a3+sa4+sa5)d(x2n,x2n+1)=λd(x2n,x2n+1),
where λ=(2a1+a2+a3+sa4+sa5)/(2-(a2+a3+sa4+sa5))<1/s. Similarly, it can be shown that
(24)d(x2n+3,x2n+2)⪯λd(x2n+,x2n+2).
Therefore,
(25)d(xn+1,xn+2)⪯λd(xn,xn+1).
According to Lemma 11, we have {xn} is a Cauchy sequence in X. Since X is complete, there exists x*∈X such that xn→x*. Suppose that f is continuous. Then, x*=limxn+1=limfnx0=limf(fn-1x0)=f(limfn-1x0)=f(limxn)=f(x*). Therefore, x* is a fixed point of f. Now, we need to show that x* is a fixed point of g. Since x*⊑x*, we can use the inequality (18) for x=y=x*. Then we have
(26)d(fx*,gx*)⪯a1d(x*,x*)+a2d(x*,fx*)+a3d(x*,gx*)+a4d(x*,fx*)+a5d(x*,gx*)=a1d(x*,x*)+a2d(x*,x*)+a3d(x*,gx*)+a4d(x*,x*)+a5d(x*,gx*)=a3d(x*,gx*)+a5d(x*,gx*)=(a3+a5)d(x*,gx*).
Hence,
(27)d(x*,gx*)⪯(a3+a5)d(x*,gx*).
Since (a3+a5)<1, Lemma 10(1) shows that d(x*,gx*)=θ; that is, x*=gx*. Therefore x* is a fixed point of g. Therefore, f and g have a common fixed point. The proof is similar when g is a continuous mapping.Theorem 16.
Let(X,⊑) be a partially ordered set and suppose that there exists a cone b-metric d in X such that the cone b-metric space (X,d) is complete with the coefficient s≥1 relative to a solid cone P. Let f,g:X→X be two weakly increasing mappings with respect to⊑. Suppose that the following three assertions hold: (i)
there existai,i=1,…,5 such that 2sa1+(s+1)(a2+a3)+(s2+s)(a4+a5)<2 with ∑i=15‍ai<1,
(28)d(fx,gy)⪯a1d(x,y)+a2d(x,fx)+a3d(y,gy)+a4d(y,fx)+a5d(x,gy),
for all comparative x,y∈X;(ii)
if an increasing sequence{xn} converges to x in X, then xn⊑x for all n.Then f and g have a common fixed point x*∈X.Proof.
As in Theorem15, we can construct an increasing sequence {xn} and prove that there exists x*∈X such that xn→x*, also; by the construction of xn, gxn→x*. Now, condition (iii) implies xn⊑x* for all n. Putting x=x* and y=xn in (28), we get
(29)d(fx*,gxn)⪯a1d(x*,xn)+a2d(x*,fx*)+a3d(xn,gxn)+a4d(xn,fx*)+a5d(x*,gxn)=a1d(xn,x*)+a2d(fx*,x*)+a3d(xn,gxn)+a4d(xn,fx*)+a5d(gxn,x*)⪯a1d(xn,x*)+a2(d(fx*,gxn)+d(gxn,x*))+a3(d(xn,x*)+d(x*,gxn))+a4(+d(gxn,fx*)d(xn,x*)+d(x*,gxn)+d(gxn,fx*))+a5d(gxn,x*)=(a1+a3+a4)d(xn,x*)+(a2+a3+a4+a5)d(gxn,x*)+(a2+a4)d(fx*,gxn).
Hence,
(30)d(fx*,gxn)⪯a1+a3+a41-(a2+a4)d(xn,x*)+a2+a3+a4+a51-(a2+a4)d(gxn,x*).
Since xn→x* and gxn→x*, then by Definition 9(1) and for c≫θ there exists N0∈ℕ such that for all n>N0, d(xn,x*)≪c(1-(a2+a4))/2(a1+a3+a4), and d(gxn,x*)≪c(1-(a2+a4))/2(a2+a3+a4+a5). Then we have
(31)d(gxn,fx*)=d(fx*,gxn)⪯a1+a3+a41-(a2+a4)d(xn,x*)+a2+a3+a4+a51-(a2+a4)d(gxn,x*)≪a1+a3+a41-(a2+a4)c(1-(a2+a4))2(a1+a3+a4)+a2+a3+a4+a51-(a2+a4)c(1-(a2+a4))2(a2+a3+a4+a5)=c2+c2=c.
Now again, according to Definition 9(1) it follows that gxn→fx*. It follows that fx*=x*. In a similar way and using that x*⊑x*, we can prove that gx*=x*. Therefore, f and g have a common fixed point.Now, we present two examples to illustrate our results. In the first example (the case of a normal cone), the conditions of Theorem12 are fulfilled, but Theorem 2 of Altun et al. [9, Theorem 12] cannot be applied. In the second example (the case of a nonnormal cone), the conditions of Theorem 12 are fulfilled, but Theorem 3 of Altun et al. [9, Theorem 13] cannot be applied.Example 17.
LetX=[0,1] endowed with the standard order and E=ℝ2 and let P={(x,y):x,y≥0}. Define d:X×X→E as in Example 8. Define f:X→X by f(x)=x2/6. Then f is a continuous and nondecreasing mapping with respect to ⊑. Then we have
(32)d(fx,fy)=d(x26,y26)=(|x26-y26|2,|x26-y26|2)=136|x+y|2(|x-y|2,|x-y|2)⪯436(|x-y|2,|x-y|2)⪯436d(x,y),
where a1=4/36, a2=a3=a4=a5=0. It is clear that the conditions of Theorem 12 are satisfied. Therefore, f has a fixed point x=0.Example 18.
LetX=[0,∞) endowed with the standard order and E=Cℝ1[0,1] with ∥u∥=∥u∥∞+∥u′∥∞,u∈E and let P={u∈E:u(t)≥0on[0,1]}. It is well known that this cone is solid, but it is not normal. Define a cone metric d:X×X→E by d(x,y)(t)=|x-y|2et. Then (X,d) is a complete cone b-metric space with the coefficient s=2. Let us define f:X→X by f(x)=x/2. Then f is a continuous and nondecreasing mapping with respect to ⊑. Then we have f is an increasing mapping; also we have
(33)d(fx,fy)(t)=|12x-12y|2et=14|x-y|2et⪯14|x-y|2et+15|x2|2et⪯14d(x,y)(t)+15d(fx,x)(t),
where a1=1/4, a2=1/5, a3=a4=a5=0. It is clear that the conditions of Theorem 12 are satisfied. Therefore, f has a fixed point x=0.



---

*Source: 815289-2013-05-13.xml*