Abstract
This article is concerned with coupled coincidence points and common fixed points for two mappings in metric spaces and cone metric spaces. We first establish a coupled coincidence point theorem for two mappings and a common fixed point theorem for two wcompatible mappings in metric spaces. Then, by using a scalarization method, we extend our main theorems to cone metric spaces. Our results generalize and complement several earlier results in the literature. Especially, our main results complement a very recent result due to Abbas et al.
1 Introduction
Throughout this article, unless otherwise specified, we always suppose that ℕ is the set of positive integers and X is a nonempty set. In addition, for convenience, we denote gx = g(x) for each x ∈ X and each mapping g : X → X.
Recently, Abbas et al. [1] introduced the following concept of wcompatible mappings:
Definition 1.1. The mappings g : X → X and F : X × X → X are called wcompatible if g(F(x, y)) = F(gx, gy) whenever gx = F(x, y) and gy = F(y, x).
Moreover, they established several coupled coincidence point theorems and common fixed point theorems for such mappings. The problem investigated in [1] is interesting. In fact, recently, the existence of coupled fixed points, coupled coincidence points, coupled common fixed points, and common fixed points for nonlinear mappings with two variables has attracted more and more attention. For example, Bhashkar and Lakshmikantham [2] investigated some coupled fixed point theorems in partially ordered sets, and they also discussed an application of their result by investigating the existence and uniqueness of the solution for a periodic boundary value problem; Sabetghadam et al. [3] extended some results in [2] to cone metric spaces; Lakshmikantham and Ćirić [4] proved several coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces; Karapinar [5] extended some results of [4] to cone metric spaces; Zoran and Mitrović [6] considered this topic in normed spaces and established a coupled best approximation theorem; Ding et al. [7] established some coupled coincidence and coupled common fixed point theorems in partially ordered metric spaces under some generalized contraction conditions; etc.
The aim of this article is to make further studies on such problems, and to generalize and complement some known results. Next, let us recall some related definitions:
Definition 1.2. [1]Let g : X → X, F : X × X → X be two mappings.
(I) (x, y) ∈ X × X is called a coupled coincidence point of F and g if gx = F(x, y) and gy = F(y, x).
(II) (x, y) ∈ X × X is called a coupled fixed point of F if x = F(x, y) and y = F(y, x).
(III) x ∈ X is called a common fixed point of F and g if x = gx = F(x, x).
2 Metric spaces
Now, let us present one of our main results.
Theorem 2.1. Let (X, d) be a complete metric space. Assume that g : X → X and F : X × X → X are two mappings satisfying
(H1) there exists a nondecreasing function ϕ : [0,+∞) → [0,+∞) such that
for all x, y, u, v ∈ X, where
(H2) F(X × X) ⊆ g(X), and g(X) is a closed subset of X.
Then F and g have a coupled coincidence point in X.
Proof. First, let us present some properties about ϕ which will be used in the sequel. We claim that ϕ(t) <t for each t > 0. In fact, if ϕ(t_{0}) ≥ t_{0 }for some t_{0 }> 0, then, since ϕ is nondecreasing, ϕ^{n}(t_{0}) ≥ t_{0 }for all n ∈ ℕ, which contradicts the condition
Moreover, it is easy to see that ϕ(0) = 0, and thus ϕ(t) ≤ t for all t ≥ 0.
Take x_{0}, y_{0 }∈ X. Since F(X × X) ⊆ g(X), one can construct two sequences {x_{n}}, {y_{n}} in X such that
For any fixed n ∈ ℕ, by (H1), we have
and
where
Since
and
we have
Now, let us prove that for each n ∈ ℕ,
We consider the following three cases:
Case I. If M_{n }= 0 or M_{n }= max{d(gx_{n}, gx_{n1}), d (gy_{n}, gy_{n1})}, then (2.3) obviously holds.
Case II. M_{n }= d(gx_{n}, gx_{n+}_{1}) > 0.
Then, by (2.1),
which is a contradiction.
Case III. M_{n }= d(gy_{n}, gy_{n+1}) > 0.
Similar to Case II, by (2.2), we get a contradiction.
Thus, in all cases, (2.3) holds for each n ∈ ℕ. In addition, combining (2.1) and (2.2), we get that for all n ∈ ℕ:
Let ε > 0 be fixed. Since
Throughout the rest of this article, we denote
for each p ∈ ℕ and each n ∈ ℕ.
Let n >N be fixed. Let us show that for all p ∈ ℕ:
By (2.6), we have
By (2.5) and (2.6), we get
Next, let us show that
where
If
then by (2.5) and (2.8),
which yields
i.e.,
If
In general, in order to prove that
Now, we have proved that (2.7) holds for all p ∈ ℕ, which means that {gx_{n}} and {gy_{n}} are Cauchy sequences. Then, by the completeness of g(X), there exist x, y ∈ X such that
By (H1) we have
and
where
Now, we claim that gx = F(x, y) and gy = F(y, x). In fact, if this is not true, then
which, together with (2.9), yield that c_{n }= max{d(gx, F(x, y)), d(gy, F(y, x))} when n is sufficiently large. Letting n → ∞ in (2.10) and (2.11), it follows that
and
This is a contradiction. Thus, gx = F(x, y) and gy = F(y, x), i.e., (x, y) is a coupled coincidence point of F and g.
Example 2.2. Let X = [2,+∞), d(x, y) = xy, F(x, y) = x + y, g(x) = x^{2}, and
If F and g are wcompatible, we have the following result:
Theorem 2.3. Suppose that all of the assumptions of Theorem 2.1 are satisfied, and F and g are wcompatible. Then F and g have a unique common fixed point.
Proof. We give the proof in 3 steps.
Step 1. We claim that if
then gx_{1 }= gx_{2 }= gy_{1 }= gy_{2}. In fact, by (H1), we have
and
where
which gives that ω = 0, i.e., gx_{1 }= gx_{2 }and gy_{1 }= gy_{2}.
By a similar argument, in the case of
one can also show that gx_{1 }= gy_{2 }and gy_{1 }= gx_{2}. Then, it follows that
Step 2. By Theorem 2.1, (x, y) is a coupled coincidence point of F and g, i.e., gx = F(x, y) and gy = F(y, x). Then, by Step 1, we have gx = gy. Let u = gx = gy. Since F and g are wcompatible, we have
Again by Step 1, one obtains gu = gx. Thus u = gx = gu = F(u, u), i.e., u is a common fixed point of F and g.
Step 3. Let v = gv = F(v, v). By Step 1, one can deduce that gv = gu. So u = gu = gv = v, which means that u is the unique common fixed point of F and g.
3 Applications to cone metric spaces
In this section, by a scalarization method used in [7], we apply our main results in metric spaces to cone metric spaces, and obtain some new theorems.
In the following, we always suppose that E is a Banach space, P is a convex cone in E with
In addition, x ≫ y stands for x  y ∈ intP.
First, let us recall some definitions about cone metric space.
Definition 3.1. [8]Let X be a nonempty set and P be a cone in a Banach space E. Suppose that a mapping d : X × X → E satisfies:
(d1) θ ≼ ρ(x, y) for all x,y ∈ X and ρ(x, y) = θ if and only if x = y, where θ is the zero element of P;
(d2) ρ(x, y) = ρ(y, x) for all x, y ∈ X;
(d3) ρ(x, y) ≼ ρ(x, z) + ρ(z, y) for all x, y, z ∈ X.
Then ρ is called a cone metric on X and (X, ρ) is called a cone metric space.
Definition 3.2. Let (X, ρ) be a cone metric space. Let {x_{n}} be a sequence in X and x ∈ X. If ∀c ≫ θ, there exists N ∈ ℕ such that for all n >N, ρ(x_{n}, x) ≪ c, then we say that {x_{n}} converges to x, and we denote it by
Recall that it has been of great interest for many authors to study fixed point theorems in cone metric spaces, and there is a large literature on this topic. We refer the reader to [1,3,5,7,928] and the references therein for some recent developments on this topic.
Next, let us recall some notations and basic results about the scalarization function ξ_{e}.
Lemma 3.3. [[7], Lemma 1.1] The following statements are true:
(i) ξ_{e}(·) is positively homogeneous and continuous on E;
(ii) y, z ∈ E with y ≼ z implies ξ_{e }(y) ≤ ξ_{e }(z);
(ii) ξ_{e }(y + z) ≤ ξ_{e }(y) + ξ_{e }(z) for all y, z ∈ E.
Combining Theorems 2.1 and 2.2 of [7] and, we have the following results:
Theorem 3.4. Let (X, ρ) be a cone metric space with underlying cone P. Then, ξ_{e }○ ρ is a metric on X. Moreover, if (X, ρ) is complete, then (X, ξ_{e }○ ρ) is a complete metric space.
By using Theorems 2.1 and 2.3, one can deduce many results on cone metric spaces. For example, we have the following theorem:
Theorem 3.5. Let (X, ρ) be a cone metric space with underlying cone P. Assume that g:X → X and F:X × X → X are two mappings satisfying that F(X × X) ⊆ g(X), g(X) is a complete cone metric space, and there exists a constant λ ∈ (0,1) such that for each x, y, u, v ∈ X, there is a
where
and co denotes the convex hull. Then F and g have a coupled coincidence point in X. Moreover, if F and g are wcompatible, then F and g have a unique common fixed point.
Proof. Let d = ξ_{e }○ ρ. By Theorem 3.4, d is a metric on X and (g(X), d) is a complete metric space. Then, by Lemma 3.3, we have
where
it is easy to see that all of the assumptions of Theorem 2.1 are satisfied. Thus F and g have a coupled coincidence point in X. In addition, if F and g are wcompatible, by Theorem 2.3, F and g have a unique common fixed point.
Remark 3.6. Theorem 3.5 is a complement of [[1], Theorem 2.4]. Moreover, Theorem 3.5 extends some existing results. For example, one can deduce [[3], Theorem 2.2] from Theorem 3.5. In addition, note that Theorems 3.4 and 3.5 are true and in the context of tvscone metric spaces (for details see [23,28]).
Remark 3.7. It is needed to note that one can also get Theorem 3.5 by using the method of Minkowski functional, which is introduced in [22].
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Acknowledgements
The authors thank the referees for their valuable comments that helped to improve the text. Wei Long acknowledges support from the NSF of China (11101192), the Key Project of Chinese Ministry of Education (211090), the NSF of Jiangxi Province of China (20114BAB211002), and the Foundation of Jiangxi Provincial Education Department (GJJ12205). Third author is thankful to the Ministry of Science and Technological Development of Serbia.
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