 Research
 Open Access
 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
References
 Kantorovich LV: The majorant principle and Newton's method. Dokl Akad Nauk SSSR (NS) 1951, 76: 17–20.Google Scholar
 Kantorovich LV: On some further applications of the Newton approximation method. Vestn Leningr Univ Ser Mat Mekh Astron 1957, 12: 68–103.Google Scholar
 Vandergraft JS: Newton's method for convex operators in partially ordered spaces. SIAM J Numer Anal 1967, 4: 406–432. 10.1137/0704037MathSciNetView ArticleGoogle Scholar
 Zabreǐko PP: Kmetric and Knormed spaces: survey. Collect Math 1997, 48: 825–859.MathSciNetGoogle Scholar
 Deimling K: Nonlinear Functional Analysis. Springer; 1985.View ArticleGoogle Scholar
 Aliprantis CD, Tourky R: Cones and duality. Graduate Studies in Mathematics, American Mathematical Society, Providence, RI 2007., 84:Google Scholar
 Huang LG, Zhang X: Cone metric spaces and fixed point theorems of contractive mappings. J Math Anal Appl 2007, 332: 1467–1475.View ArticleGoogle Scholar
 Abbas M, Ali Khan M, Radenović S: Common coupled fixed point theorems in cone metric spaces for w compatible mappings. Appl Math Comput 2010, 217: 195–202. 10.1016/j.amc.2010.05.042MathSciNetView ArticleGoogle Scholar
 Altun I, Damjanović B, Djorić D: Fixed point and common fixed point theorems on ordered cone metric spaces. Appl Math Lett 2010, 23: 310–316. 10.1016/j.aml.2009.09.016MathSciNetView ArticleGoogle Scholar
 Beg I, Azam A, Arshad M: Common fixed points for maps on topological vector space valued cone metric spaces. Int J Math Math Sci 2009, 2009: 8. Article ID560264MathSciNetView ArticleGoogle Scholar
 Di Bari C, Vetro P: φ pairs and common fixed points in cone metric spaces. Rend Cir Mat Palermo 2008, 57: 279–285. 10.1007/s1221500800209MathSciNetView ArticleGoogle Scholar
 Di Bari C, Vetro P: Weakly φ pairs and common fixed points in cone metric spaces. Rend Cir Mat Palermo 2009, 58: 125–132. 10.1007/s1221500900124MathSciNetView ArticleGoogle Scholar
 Olaleru JO: Some generalizations of fixed point theorems in cone metric spaces. Fixed Point Theory Appl 2009, 2009: 10. Article ID 657914MathSciNetView ArticleGoogle Scholar
 Rezapour Sh, Hamlbarani R: Some notes on the paper "Cone metric spaces and fixed point theorems of contractive mappings". J Math Anal Appl 2008, 345: 719–724. 10.1016/j.jmaa.2008.04.049MathSciNetView ArticleGoogle Scholar
 Vetro P: Common fixed points in cone metric spaces. Rend Circ Mat Palermo 2007, 56: 464–468. 10.1007/BF03032097MathSciNetView ArticleGoogle Scholar
 Bhashkar TG, Lakshmikantham V: Fixed point theorems in partially ordered cone metric spaces and applications. Nonlinear Anal 2006, 65: 825–832. 10.1016/j.na.2005.10.015MathSciNetView ArticleGoogle Scholar
 Lakshmikantham V, Ćirić L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal 2009, 70: 4341–4349. 10.1016/j.na.2008.09.020MathSciNetView ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.