Cho et al. [Comput. Math. Appl. 61(2011), 1254–1260] studied common fixed point
theorems on cone metric spaces by using the concept of c-distance. In this paper, we prove
some coupled fixed point theorems in ordered cone metric spaces by using the concept of
c-distance in cone metric spaces.
1. Introduction
Many fixed point theorems have been proved for mappings on cone metric spaces in the sense of Huang and Zhang [1]. For some more results on fixed point theory and applications in cone metric spaces, we refer the readers to [2–15]. Recently, Bhaskar and Lakshmikantham [16] introduced the concept of a coupled coincidence point of a mapping F from X×X into X and a mapping g from X into X and studied fixed point theorems in partially ordered metric spaces. For some more results on couple fixed point theorems, refer to [17–23].
Recently, Cho et al. [7] introduced a new concept of c-distance in cone metric spaces, which is a cone version of w-distance of Kada et al. [24] (see also [25]) and proved some fixed point theorems for some contractive type mappings in partially ordered cone metric spaces using the c-distance.
In this paper, we prove some coupled fixed point theorems in ordered cone metric spaces by using the concept of c-distance.
2. Preliminaries
In this paper, assume that E is a real Banach space. Let P be a subset of E with int(P)≠∅. Then P is called a cone if the following conditions are satisfied:
P is closed and P≠{θ};
a,b∈R+, x,y∈P implies ax+by∈P;
x∈P∩-P implies x=θ.
For a cone P, define the partial ordering ⪯ with respect to P by x⪯y if and only if y-x∈P. We write x≺y to indicate that x⪯y but x≠y, while x≪y stand for y-x∈intP.
It can be easily shown that λint(P)⊆int(P) for all positive scalars λ.
Definition 2.1 (see [1]).
Let X be a nonempty set. Suppose that the mapping d:X×X→E satisfies the following conditions:
θ⪯d(x,y) for all x,y∈X and d(x,y)=θ if and only if x=y;
d(x,y)=d(y,x) for all x,y∈X;
d(x,y)⪯d(x,z)+d(y,z) for all x,y,z∈X.
Then d is called a cone metric on X, and (X,d) is called a cone metric space.
Definition 2.2 (see [1]).
Let (X,d) be a cone metric space. Let (xn) be a sequence in X and x∈X.
If, for any c∈X with θ≪c, there exists N∈N such that d(xn,x)≪c for all n≥N, then (xn) is said to be convergent to a point x∈X and x is the limit of (xn). We denote this by limn→∞xn=x or xn→x as n→∞.
If, for any c∈E with θ≪c, there exists N∈N such that d(xn,xm)≪c for all n,m≥N, then (xn) is called a Cauchy sequence in X.
The space (X,d) is called a complete cone metric space if every Cauchy sequence is convergent.
Definition 2.3 (see [7]).
Let (X,⊑) be a partially ordered set, and let F:X×X→X be a function. Then the mapping F is said to have the mixed monotone property if F(x,y) is monotone nondecreasing in x and is monotone nonincreasing in y; that is,
x1⊑x2impliesF(x1,y)⊑F(x2,y)
for all y∈X and
y1⊑y2impliesF(x,y2)⊑F(x,y1)
for all x∈X.
Definition 2.4 (see [7]).
An element (x,y)∈X×X is called a coupled fixed point of a mapping F:X×X→X if F(x,y)=x and F(y,x)=y.
Recently, Cho et al. [7] introduced the concept of c-distance on cone metric space (X,d) which is a generalization of w-distance of Kada et al. [24].
Definition 2.5 (see [7]).
Let (X,d) be a cone metric space. Then a function q:X×X→E is called a c-distance on X if the following are satisfied:
θ⪯q(x,y) for all x,y∈X;
q(x,z)⪯q(x,y)+q(y,z) for all x,y,z∈X;
for any x∈X, if there exists u=ux∈P such that q(x,yn)⪯u for each n≥1, then q(x,y)⪯u whenever (yn) is a sequence in X converging to a point y∈X;
for any c∈E with θ≪c, there exists e∈E with 0≤e such that q(z,x)≪e and q(z,y)≪c imply d(x,y)≪c.
Cho et al. [7] noticed the following important remark in the concept of c-distance on cone metric spaces.
Remark 2.6 (see [7]).
Let q be a c-distance on a cone metric space (X,d). Then
q(x,y)=q(y,x) does not necessarily hold for all x,y∈X,
q(x,y)=θ is not necessarily equivalent to x=y for all x,y∈X.
The following lemma is crucial in proving our results.
Lemma 2.7 (see [7]).
Let (X,d) be a cone metric space, and let q be a c-distance on X. Let (xn) and (yn) be sequences in X and x,y,z∈X. Suppose that (un) is a sequence in P converging to θ. Then the following hold:
if q(xn,y)⪯un and q(xn,z)⪯un, then y=z;
if q(xn,yn)⪯un and q(xn,z)⪯un, then (yn) converges to a point z∈X;
if q(xn,xm)⪯un for each m>n, then (xn) is a Cauchy sequence in X;
If q(y,xn)⪯un, then (xn) is a Cauchy sequence in X.
3. Main Results
In this section, we prove some coupled fixed point theorems by using c-distance in partially ordered cone metric spaces.
Theorem 3.1.
Let (X,⊑) be a partially ordered set, and suppose that (X,d) is a complete cone metric space. Let q be a c-distance on X, and let F:X×X→X be a continuous function having the mixed monotone property such that
q(F(x,y),F(x*,y*))⪯k2(q(x,x*)+q(y,y*))
for some k∈[0,1) and all x,y,x*,y*∈X with (x⊑x*)∧(y⊒y*) or (x⊒x*)∧(y⊑y*). If there exist x0,y0∈X such that x0⊑F(x0,y0) and F(y0,x0)⊑y0, then F has a coupled fixed point (u,v). Moreover, one has q(v,v)=θ and q(u,u)=θ.
Proof.
Let x0,y0∈X be such that x0⊑F(x0,y0) and F(y0,x0)⊑y0. Let x1=F(x0,y0) and y1=F(y0,x0). Since F has the mixed monotone property, we have x0⊑x1 and y1⊑y0. Continuing this process, we can construct two sequences (xn) and (yn) in X such that
xn=F(xn-1,yn-1)⊑xn+1=F(xn,yn),yn+1=F(yn,xn)⊑yn=F(yn-1,xn-1).
Let n∈N. Now, by (3.1), we have
q(xn,xn+1)=q(F(xn-1,yn-1),F(xn,yn))⪯k2(q(xn-1,xn)+q(yn-1,yn)),q(xn+1,xn)=q(F(xn,yn),F(xn-1,yn-1))⪯k2(q(xn,xn-1)+q(yn,yn-1)).
From (3.3), it follows that
q(xn,xn+1)+q(xn+1,xn)⪯k2(q(xn-1,xn)+q(yn-1,yn)+q(xn,xn-1)+q(yn,yn-1)).
Similarly, we have
q(yn,yn+1)+q(yn+1,yn)⪯k2(q(xn-1,xn)+q(yn-1,yn)+q(xn,xn-1)+q(yn,yn-1)).
Thus it follows from (3.4) and (3.5) that
q(xn,xn+1)+q(xn+1,xn)+q(yn,yn+1)+q(yn+1,yn)⪯k(q(xn-1,xn)+q(yn-1,yn)+q(xn,xn-1)+q(yn,yn-1)).
Repeating (3.6) n-times, we get
q(xn,xn+1)+q(xn+1,xn)+q(yn,yn+1)+d(yn+1,yn)⪯kn(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)).
Thus we have
q(xn,xn+1)⪯kn(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)),q(yn,yn+1)⪯kn(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)).
Let m,n∈N with m>n. Since
q(xn,xm)⪯∑i=nm-1q(xi,xi+1),q(yn,ym)⪯∑i=nm-1q(yi,yi+1),
and k<1, we have
q(xn,xm)⪯kn1-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)),q(yn,ym)⪯kn1-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)).
From Lemma 2.7 (3), it follows that (xn) and (yn) are Cauchy sequences in (X,d). Since X is complete, there exist u,v∈X such that xn→u and yn→v. Since F is continuous, we have
xn+1=F(xn,yn)⟶F(u,v),yn+1=F(yn,xn)⟶F(v,u).
By the uniqueness of the limits, we get u=f(u,v) and v=F(v,u). Thus (u,v) is a coupled fixed point of F.
Moreover, by (3.1), we have
q(u,u)=q(F(u,v),F(u,v))⪯k2(q(u,u)+q(v,v)),q(v,v)=q(F(v,u),F(v,u))⪯k2(q(v,v)+q(u,u)).
Therefore, we get
q(u,u)+q(v,v)⪯k(q(v,v)+q(u,u)).
Since k<1, we conclude that q(u,u)+q(v,v)=θ, and hence q(u,u)=θ and q(v,v)=θ. This completes the proof.
Theorem 3.2.
In addition to the hypotheses of Theorem 3.1, suppose that any two elements x and y in X are comparable. Then the coupled fixed point has the form (u,u), where u∈X.
Proof.
As in the proof of Theorem 3.1, there exists a coupled fixed point (u,v)∈X×X. Here u=F(u,v) and v=F(v,u). By the additional assumption and (3.1), we have
q(u,v)=q(F(u,v),F(v,u))⪯k2(q(u,v)+q(v,u)),q(v,u)=q(F(v,u),F(u,v))⪯k2(q(v,u)+q(u,v)).
Thus we have
q(u,v)+q(v,u)⪯k(q(v,u)+q(u,v)).
Since k<1, we get q(u,v)+q(v,u)=θ. Hence q(u,v)=θ and q(v,u)=θ. Let un=θ and xn=u. Then
q(xn,u)⪯un,q(xn,v)⪯un.
From Lemma 2.7 (1), we have u=v. Hence the coupled fixed point of F has the form (u,u). This completes the proof.
Theorem 3.3.
Let (X,⊑) be a partially ordered set, and suppose that (X,d) is a complete cone metric space. Let q be a c-distance on X, and let F:X×X→X be a function having the mixed monotone property such that
q(F(x,y),F(x*,y*))⪯k4(q(x,x*)+q(y,y*))
for some k∈(0,1) and all x,y,x*,y*∈X with (x⊑x*)∧(y⊒y*) or (x⊒x*)∧(y⊑y*). Also, suppose that X has the following properties:
if (xn) is a nondecreasing sequence in X with xn→x, then xn⊑x for all n≥1;
if (xn) is a nonincreasing sequence in X with xn→x, then x⊑xn for all n≥1.
Assume there exist x0,y0∈X such that x0⊑F(x0,y0) and F(y0,x0)⊑y0. If y0⊑x0, then F has a coupled fixed point.
Proof.
As in the proof of Theorem 3.1, we can construct two Cauchy sequences (xn) and (yn) in X such that
x0⊑x1⊑⋯⊑xn⊑⋯,y0⊒y1⊒⋯yn⊒⋯.
Moreover, we have that (xn) converges to a point u∈X and (yn) converges to v∈X,
q(xn,xm)⪯kn1-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)),q(yn,ym)⪯kn1-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1))
for each n>m≥1. By (q3), we have
q(xn,u)⪯kn1-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)),q(yn,v)⪯kn1-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)),
and so
q(xn,u)+q(yn,v)⪯2kn1-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)).
By the properties (a) and (b), we have
v⊑yn⊑y0⊑x0⊑xn⊑u.
By (3.17), we have
q(xn,F(u,v))=q(F(xn-1,yn-1),F(u,v))⪯k4(q(xn-1,u)+q(yn-1,v)),q(yn,F(v,u))=q(F(yn-1,xn-1),F(v,u))⪯k4(q(yn-1,v)+q(xn-1,u)).
Thus we have
q(xn,F(u,v))+q(yn,F(v,u))⪯k2(q(xn-1,u)+q(yn-1,v)).
By (3.21), we get
q(xn,F(u,v))+q(yn,F(v,u))⪯k2⋅2kn-11-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1))=kn1-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)).
Therefore, we have
q(xn,F(u,v))⪯kn1-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)),q(yn,F(v,u))⪯kn1-k(q(x1,x0)+q(y1,y0)+q(x0,x1)+q(y0,y1)).
By using (3.20) and (3.26), Lemma 2.7 (1) shows that u=F(u,v) and v=F(v,u). Therefore, (u,v) is a coupled fixed point of F. This completes the proof.
Example 3.4.
Let E=CR1[0,1] with ∥x∥=∥x∥∞+∥x′∥∞ and P={x∈E:x(t)≥0,t∈[0,1]}. Let X=[0,+∞) (with usual order), and let d:X×X→E be defined by d(x,y)(t)=|x-y|et. Then (X,d) is an ordered cone metric space (see [7, Example 2.9]). Further, let q:X×X→E be defined by q(x,y)(t)=yet. It is easy to check that q is a c-distance. Consider now the function F:X×X→X defined by
F(x,y)={18(x-y),x≥y,0,x<y.
Then it is easy to see that
q(F(x,y),F(u,v))⪯16(q(x,u)+q(y,v))
for all x,y,u,v∈X with (x≤u)∧(y≥v) or (x≥u)∧(y≤v). Note that 0≤F(0,1) and 1≥F(1,0). Thus, by Theorem 3.1, it follows that F has a coupled fixed point in E. Here (0,0) is a coupled fixed point of F.
Acknowledgments
The first author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No.: 2011–0021821).
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