 Hassen Aydi^{1}Email author,
 Bessem Samet^{2} and
 Calogero Vetro^{3}
https://doi.org/10.1186/16871812201127
© Aydi et al; licensee Springer. 2011
Received: 5 February 2011
Accepted: 8 August 2011
Published: 8 August 2011
Abstract
In this paper, we introduce the concepts of compatible mappings, bcoupled coincidence point and bcommon coupled fixed point for mappings F, G : X × X → X, where (X, d) is a cone metric space. We establish bcoupled coincidence and bcommon coupled fixed point theorems in such spaces. The presented theorems generalize and extend several wellknown comparable results in the literature, in particular the recent results of Abbas et al. [Appl. Math. Comput. 217, 195202 (2010)]. Some examples are given to illustrate our obtained results. An application to the study of existence of solutions for a system of nonlinear integral equations is also considered.
2010 Mathematics Subject Classifications: 54H25; 47H10.
Keywords
1 Introduction
Ordered normed spaces and cones have applications in applied mathematics, for instance, in using Newton's approximation method [1–4] and in optimization theory [5]. Kmetric and Knormed spaces were introduced in the mid20th century ([2]; see also [3, 4, 6]) by using an ordered Banach space instead of the set of real numbers, as the codomain for a metric. Huang and Zhang [7] reintroduced such spaces under the name of cone metric spaces, and went further, defining convergent and Cauchy sequences in the terms of interior points of the underlying cone. Afterwards, many papers about fixed point theory in cone metric spaces were appeared (see, for example, [8–15]).
The following definitions and results will be needed in the sequel.
 (a)
P is closed, nonempty and P ≠ {0_{ E }},
 (b)
a, b ∈ ℝ, a, b ≥ 0, x, y ∈ P imply that ax + by ∈ P,
 (c)
P ∩ (P) = {0_{ E }},
where 0 _{ E } is the zero vector of E.
Given a cone define a partial ordering ≼ with respect to P by x ≼ y if and only if y  x ∈ P. We shall write x ≪ y for y  x ∈ IntP, where IntP stands for interior of P. Also, we will use x ≺ y to indicate that x ≼ y and x ≠ y. The cone P in a normed space (E, ·) is called normal whenever there is a number k ≥ 1 such that for all x, y ∈ E, 0 _{ E } ≼ x ≼ y implies x ≤ ky. The least positive number satisfying this norm inequality is called the normal constant of P.
Definition 2. [7]. Let X be a nonempty set. Suppose that d : X × X → E satisfies:
(d1) 0 _{ E } ≼ d(x, y) for all x, y ∈ X and d(x, y) = 0 _{ E } if and only if x = y,
(d2) d(x, y) = d(y, x) for all x, y ∈ X,
(d3) d(x, y) ≼ d(x, z) + d(z, y) 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 3. [7]. Let (X, d) be a cone metric space, {x_{ n } } a sequence in X and x ∈ X. For every c ∈ E with c ≫ 0_{ E }, we say that {x_{ n } } is
(C1) a Cauchy sequence if there is some k ∈ ℕ such that, for all n, m ≥ k, d(x_{ n } , x_{ m } ) ≪ c,
(C2) a convergent sequence if there is some k ∈ ℕ such that, for all n ≥ k, d(x_{ n } , x) ≪ c. Then x is called limit of the sequence {x_{ n } }.
Note that every convergent sequence in a cone metric space X is a Cauchy sequence. A cone metric space X is said to be complete if every Cauchy sequence in X is convergent in X.
Recently, Abbas et al. [8] introduced the concept of wcompatible mappings and established coupled coincidence point and coupled point of coincidence theorems for mappings satisfying a contractive condition in cone metric spaces.
In this paper, we introduce the concepts of compatible mappings, bcoupled coincidence point and bcommon coupled fixed point for mappings F, G : X × X → X, where (X, d) is a cone metric space. We establish bcoupled coincidence and bcommon coupled fixed point theorems in such spaces. The presented theorems generalize and extend several wellknown comparable results in the literature, in particular the recent results of Abbas et al. [8] and the result of Olaleru [13]. Some examples and an application to nonlinear integral equations are also considered.
2 Main results
We start by recalling some definitions.
Definition 4. [16]. An element (x, y) ∈ X × X is called a coupled fixed point of mapping F : X × X → X if x = F(x, y) and y = F(y, x).
 (i)
a coupled coincidence point of mappings F : X × X → X and g : X → X if gx = F(x, y) and gy = F(y, x), and (gx, gy) is called coupled point of coincidence,
 (ii)
a common coupled fixed point of mappings F : X × X → X and g : X → X if x = gx = F(x, y) and y = gy = F(y, x).
Note that if g is the identity mapping, then Definition 5 reduces to Definition 4.
Definition 6. [8]. The mappings F : X × X → X and g : X → X are called wcompatible if g(F(x, y)) = F(gx, gy) whenever gx = F(x, y) and gy = F(y, x).
Now, we introduce the following definitions.
 (i)
a bcoupled coincidence point of mappings F, G : X × X → X if G(x, y) = F(x, y) and G(y, x) = F(y, x), and (G(x, y), G(y, x)) is called bcoupled point of coincidence,
 (ii)
a bcommon coupled fixed point of mappings F, G : X × X → X if x = G(x, y) = F(x, y) and y = G(y, x) = F(y, x).
for all x, y ∈ X. Then, (π/2, 0) is a bcoupled coincidence point of F and G, and (1, 0) is a bcoupled point of coincidence.
for all x, y ∈ X. Then, (1, 2) is a bcommon coupled fixed point of F and G.
whenever F(x, y) = G(x, y) and F(y, x) = G(y, x).
for all x, y ∈ X. One can show easily that (x, y) is a bcoupled coincidence point of F and G if and only if x = y. Moreover, we have F(G(x, x), G(x, x)) = G(F(x, x), F(x, x)) for all x ∈ X. Then, F and G are compatible.
Now, we prove our first result.
Theorem 1. Let (X, d) be a cone metric space with a cone P having nonempty interior. Let F, G : X × X → X be mappings satisfying
(h1) for any (x, y) ∈ X × X, there exists (u, v) ∈ X × X such that F(x, y) = G(u, v) and F(y, x) = G(v, u),
(h2) {(G(x, y), G(y, x)): x, y ∈ X} is a complete subspace of (X × X, ν),
where a_{ i } , i = 1, ..., 10 are nonnegative real numbers such that . Then F and G have a bcoupled coincidence point (x, y) ∈ X × X, that is, F(x, y) = G(x, y) and F(y, x) = G(y, x).
meaning that (x_{1}, y_{1}) is a bcoupled coincidence point of F and G.
It follows that d(F(x, y), G(x, y)) = d(F(y, x), G(y, x)) = 0_{ E }, that is, F(x, y) = G(x, y) and F(y, x) = G(y, x). Then, we proved that (x, y) is a bcoupled coincidence point of the mappings F and G. □
As consequences of Theorem 1, we give the following corollaries.
Corollary 1. Let (X, d) be a cone metric space with a cone P having nonempty interior. Let F, G : X × X → X be mappings satisfying
(h1) for any (x, y) ∈ X × X, there exists (u, v) ∈ X × X such that F(x, y) = G(u, v) and F(y, x) = G(v, u),
(h2) {(G(x, y), G(y, x)): x, y ∈ X} is a complete subspace of (X × X, ν),
where α_{ i } , i = 1, ..., 5 are nonnegative real numbers such that . Then F and G have a bcoupled coincidence point (x, y) ∈ X × X, that is, F(x, y) = G(x, y) and F(y, x) = G(y, x).
for all x, y, u, v ∈ X, where a_{ i } , i = 1, ..., 10 are nonnegative real numbers such that . If F(X × X) ⊆ g(X) and g(X) is a complete subset of X, then F and g have a coupled coincidence point in X, that is, there exists (x, y) ∈ X × X such that gx = F(x, y) and gy = F(y, x).
We will check that all the hypotheses of Theorem 1 are satisfied.

Hypothesis (h1):
Let (x, y) ∈ X × X. Since F(X × X) ⊆ g(X), there exists u ∈ X such that F(x, y) = gu = G(u, v) for any v ∈ X. Then, (h1) is satisfied.

Hypothesis (h2):
Then, {(G(x, y), G(y, x)): x, y ∈ X} is a complete subspace of (X × X, ν), and so the hypothesis (h2) is satisfied.

Hypothesis (h3):
The hypothesis (h3) follows immediately from (19).
Now, all the hypotheses of Theorem 1 are satisfied. Then, F and G have a bcoupled coincidence point (x, y) ∈ X × X, that is, F(x, y) = G(x, y) = gx and F(y, x) = G(y, x) = gy. Thus, (x, y) is a coupled coincidence point of F and g □
for all x, y, u, v ∈ X, where α_{ i } , i = 1, ..., 5 are nonnegative real numbers such that . If F(X × X) ⊆ g(X) and g(X) is a complete subset of X, then F and g have a coupled coincidence point in X, that is, there exists (x, y) ∈ X × X such that gx = F(x, y) and gy = F(y, x).
Remark 1.
Now, we are ready to state and prove a result of bcommon coupled fixed point.
Theorem 2. Let F, G : X × X → X be two mappings which satisfy all the conditions of Theorem 1. If F and G are compatible, then F and G have a unique bcommon coupled fixed point. Moreover, the bcommon coupled fixed point of F and G is of the form (u, u) for some u ∈ X.
meaning the uniqueness of the bcoupled point of coincidence of F and G, that is, (G(x, y), G(y, x)).
This means that the unique bcoupled point of coincidence of F and G is (G(x, y), G(x, y)).
Hence, (u, u) is the unique bcommon coupled fixed point of F and G. This makes end to the proof. □
Corollary 4. Let F : X × X → X and g : X → X be two mappings which satisfy all the conditions of Corollary 2. If F and g are wcompatible, then F and g have a unique common coupled fixed point. Moreover, the common fixed point of F and g is of the form (u, u) for some u ∈ X.
Thus, we proved that F and G are compatible mappings, and the desired result follows immediately from Theorem 2. □
Remark 2. Corollary 4 generalizes Theorem 2.11 of [8].
for all x, u ∈ X, where α_{ i } ∈ [0, 1), i = 1, ..., 5 and . Suppose that f and g are weakly compatible, f(X) ⊆ g(X) and g(X) is a complete subspace of X. Then the mappings f and g have a unique common fixed point.
Then, F and G satisfy the hypothesis (h 3) of Theorem 1. Clearly, hypothesis (h 1) of Theorem 1 is satisfied since f(X) ⊆ g(X). The hypothesis (h 2) is also satisfied since g(X) is a complete subspace of X.
Thus, we proved that F and G are compatible mappings. Therefore, from Theorem 2, F and G have a unique bcommon coupled fixed point (u, u) ∈ X × X such that u = F(u, u) = G(u, u), that is, u = fu = gu. This makes end to the proof. □
Now, we give an example to illustrate our obtained results.
We will check that all the hypotheses of Theorem 1 are satisfied.

Hypothesis (h 1):
Let (x, y) ∈ X × X be fixed. We consider the following cases.
Case1: x = y.
In this case, F(x, y) = 0 = G(x, y) and F(y, x) = 0 = G(y, x).
Case2: x > y.
Case3: x < y.
Thus, we proved that (h 1) is satisfied.

Hypothesis (h 2):
Since φ is continuous and [0, 1] is compact, then Λ = φ([0, 1] × [0, 1]) is compact. On the other hand, ([0, 1] × [0, 1], ν) is complete. Then, we deduce that Λ is complete.

Hypothesis (h 3):
Then, (h 3) is satisfied with a_{1} = a_{2} = ⋯ = a_{8} = a_{10} = 0 and a_{9} = 1/3.
All the required hypotheses of Theorem 1 are satisfied. Consequently, F and G have a bcoupled coincidence point.
This implies that F and G are compatible. Applying our Theorem 2, we obtain the existence and uniqueness of bcommon coupled fixed point of F and G. In this example, (0, 0) is the unique bcommon coupled fixed point.
3 Application
In this section, we study the existence of solutions of a system of nonlinear integral equations using the results proved in Section 2.
where t ∈ [0, T], T > 0.
 (a)
k : [0, T] × [0, T] → ℝ is a continuous function,
 (b)
a ∈ C([0, T], ℝ),
 (c)
f : [0, T] × ℝ × ℝ → ℝ is a continuous function,
 (d)
G : C([0, T], ℝ) × C([0, T], ℝ) → C([0, T], ℝ) is a mapping satisfying:
for all t ∈ [0, T],
 (e)
 (f)
Now, we formulate our result.
Theorem 3. Under hypotheses (a)  (f), system (25)(26) has at least one solution in C([0, T], ℝ).
It is easy to show that (x, y) is a solution to (25)(26) if and only if (x, y) is a bcoupled coincidence point of F and G. To establish the existence of such a point, we will use our Theorem 1. Then, we have to check that all the hypotheses of Theorem 1 are satisfied.
for all x, y, u, v ∈ X. Then, hypothesis (h3) is satisfied with a_{9} = MT < 1 (from condition (f)) and a_{1} = a_{2} = ⋯ = a_{8} = a_{10} = 0.
Now, applying Theorem 2, we obtain the existence of a solution to system (25)(26). □
Declarations
Acknowledgements
Calogero Vetro was supported by Università degli Studi di Palermo, Local University Project R. S. ex 60%.
Authors’ Affiliations
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