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Coupled fixed point results in cone metric spaces for compatible mappings
Fixed Point Theory and Applications volume 2011, Article number: 27 (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.
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.
Definition 1. [4, 7]. Let E be a real Banach space. A subset P of E is called a cone if and only if:

(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).
Definition 5. [17]. An element (x, y) ∈ X × X is called

(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.
Definition 7. An element (x, y) ∈ X × X is called

(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).
Example 1. Let × = ℝ and F, G : X × X → X the mappings defined by
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.
Example 2. Let X = ℝ and F, G : X × X → X the mappings defined by
for all x, y ∈ X. Then, (1, 2) is a bcommon coupled fixed point of F and G.
Definition 8. The mappings F, G : X × X → X are called compatible if
whenever F(x, y) = G(x, y) and F(y, x) = G(y, x).
Example 3. Let X = ℝ and F, G : X × X → X the mappings defined by
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 arecompatible.
If (X, d) is a cone metric space, we endow the product set X × X by the cone metric ν defined by
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),
(h3) for any x, y, u, v ∈ X,
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).
Proof. Let x_{0} and y_{0} be two arbitrary points in X. By (h1), there exists (x_{1}, y_{1}) such that
Continuing this process, we can construct two sequences {x_{ n } } and {y_{ n } } in X such that
For any n ∈ ℕ, let z_{ n } ∈ X and t_{ n } ∈ X as follows
Now, taking (x, y) = (x_{ n } , y_{ n } ) and (u, v) = (x_{n+1}, y_{n+1}) in the considered contractive condition and using (2), we have
Then, using the triangular inequality, one can write for any n ∈ ℕ*
Therefore,
Similarly, taking (x, y) = (y_{ n } , x_{ n } ) and (u, v) = (y_{n+1}, x_{n+1}) and reasoning as above, we obtain
Adding (4) to (5), we have
Let us denote
then, we deduce that
On the other hand, we have
from which by the triangular inequality, it follows that
Therefore,
Similarly, we find
Summing (8) to (9) and referring to (6), we get
Finally, from (7) and (10), we have for any n ∈ ℕ*
that is
where
Consequently, we have
If δ_{0} = 0_{ E }, we get d(z_{0}, z_{1}) + d(t_{0}, t_{1}) = 0_{ E }, that is, z_{0} = z_{1} and t_{0} = t_{1}. Therefore, from (2) and (6), we have
and
meaning that (x_{1}, y_{1}) is a bcoupled coincidence point of F and G.
Now, assume that δ_{0} ≻ 0_{ E }. If m > n, we have
Summing the two above inequalities, we obtain using also (13) and (6)
As , we have 0 ≤ α < 1. Hence, for any c ∈ E with c ≫ 0_{ E }, there exists N ∈ ℕ such that for any n ≥ N, we have . Furthermore, for any m > n ≥ N, we get
Thus, we proved that for any c ≫ 0_{ E }, there exists n ∈ ℕ such that
This implies that {(z_{ n } , t_{ n } )} is a Cauchy sequence in the cone metric space (X × X, ν). On the other hand, we have (z_{ n } , t_{ n } ) = (G(x_{n+1}, y_{n+1}), G(y_{n+1}, x_{n+1})) ∈ {(G(x, y), G(y, x)): x, y ∈ X} that is a complete subspace of (X × X, ν) (from (h2)). Hence, there exists (z, t) ∈ {(G(x, y), G(y, x)): x, y ∈ X} such that for all c ≫ 0_{ E }, there exists such that
This implies that there exist x, y ∈ X such that z = G(x, y) and t = G(y, x) with
and
Now, we prove that F(x, y) = G(x, y) and F(y, x) = G(y, x), that is, (x, y) is a bcoupled coincidence point of F and G. First, by the triangular inequality, we have
On the other hand, applying the contractive condition in (h3), we get
Combining the above inequality with (16), and using again the triangular inequality, we get
Therefore, we have
Similarly, one can find
Summing (17) and (18), we get
Therefore, we have
where
From (13), (14) and (15), for any c ≫ 0_{ E }, there exists N ∈ ℕ such that
for all n ≥ N. Thus, for all n ≥ N, we have
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),
(h3) for any x, y, u, v ∈ X,
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).
Corollary 2. Let (X, d) be a cone metric space with a cone P having nonempty interior, F : X × X → X and g : X → X be mappings satisfying
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).
Proof. Consider the mapping G : X × X → X defined by
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):
Let {x_{ n } } and {y_{ n } } be two sequences in X such that {(G(x_{ n } , y_{ n } ), G(y_{ n } , x_{ n } ))} is a Cauchy sequence in (X × X, ν). Then, for every c ≫ 0_{ E }, there exists N ∈ ℕ such that
that is,
This implies that {gx_{ n } } and {gy_{ n } } are Cauchy sequences in (g(X), d). Since g(X) is complete, there exist x, y ∈ X such that
that is,
Therefore,
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 □
Corollary 3. Let (X, d) be a cone metric space with a cone P having nonempty interior, F : X × X → X and g : X → X be mappings satisfying
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.

Putting a_{2} = a_{4} = a_{6} = a_{8} = 0 in Corollary 2, we obtain Theorem 2.4 of Abbas et al. [8];

Putting α_{2} = α_{4} = 0 in Corollary 3, we obtain Corollary 2.5 of [8].
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.
Proof. First, we'll show that the bcoupled point of coincidence is unique. Suppose that (x, y) and (x*, y*) ∈ X × X with G(x, y) = F(x, y), G(y, x) = F(y, x), F(x*, y*) = G(x*, y*) and F(y*, x*) = G(y*, x*). Using (h3), we get
Similarly, we obtain
Therefore, summing the two previous inequalities, we get
Since a_{5} + a_{6} + a_{7} + a_{8} + a_{9} + a_{10} < 1, we obtain
which implies that
meaning the uniqueness of the bcoupled point of coincidence of F and G, that is, (G(x, y), G(y, x)).
Again, using (h3), we have
Similarly,
A summation gives
The fact that a_{5} + a_{6} + a_{7} + a_{8} + a_{9} + a_{10} < 1 yields that
In view of (20) and (21), one can assert that
This means that the unique bcoupled point of coincidence of F and G is (G(x, y), G(x, y)).
Now, let u = G(x, y), then we have u = G(x, y) = F(x, y) = G(y, x) = F(y, x). Since F and G are compatible, we have
that is, thanks to (22)
Consequently, (u, u) is a bcoupled coincidence point of F and G, and so (G(u, u), G(u, u)) is a bcoupled point of coincidence of F and G, and by its uniqueness, we get G(u, u) = G(x, y). Thus, we obtain
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.
Proof. From the proof of Corollary 2 and the result given by Theorem 2, we have only to show that F and G are compatible, where G : X × X → X is defined by G(x, y) = gx for all x, y ∈ X. Let (x, y) ∈ X × X such that F(x, y) = G(x, y) and F(y, x) = G(y, x). From the definition of G, we get F(x, y) = gx and F(y, x) = gy. Since F and g are wcompatible, this implies that
Using (23), we have
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].
Corollary 5. [13]. Let (X, d) be a cone metric space and f, g : X → X be mappings such that
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.
Proof. Consider the mappings F, G : X × X → X defined by F(x, y) = fx and G(x, y) = gx for all x, y ∈ X. Then, the contractive condition (24) implies that
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.
Now, we will show that F and G are compatible mappings. Let (x, y) ∈ X × X such that F(x, y) = G(x, y) and F(y, x) = G(y, x). This implies that fx = gx. Since f and g are weakly compatible, we have f(gx) = g(fx). Then, we have
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.
Example 4. Let X = [0, 1] endowed with the standard metric d(x, y) = x  y for all x, y ∈ X. Define the mappings G, F : X × X → X by
We will check that all the hypotheses of Theorem 1 are satisfied.

Hypothesis (h 1):
Let us prove that for any x, y ∈ X, there exist u, v ∈ X such that
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.
In this case, we have
Case3: x < y.
In this case, we have
Thus, we proved that (h 1) is satisfied.

Hypothesis (h 2):
Let us prove that Λ := {(G(x, y), G(y, x)): x, y ∈ [0, 1]} is a complete subspace of ([0, 1] × [0, 1], ν). Define the function φ : [0, 1] × [0, 1] → ℝ^{2}by
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):
For all x, y, u, v ∈ X, we have
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.
In this case, for any x, y ∈ [0, 1], (x, y) is a bcoupled coincidence point if and only if x = y. Moreover, we have
This implies that F and G arecompatible. 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.
Consider the following system of integral equations:
where t ∈ [0, T], T > 0.
Let X = C([0, T], ℝ) be the set of continuous functions defined on [0, T] endowed with the metric given by
We consider the following assumptions:

(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:
(d.1) For all x, y ∈ C([0, T], ℝ), there exist u, v ∈ C([0, T], ℝ) such that
for all t ∈ [0, T],
(d.2) The set {(G(x, y), G(y, x)): x, y ∈ C([0, T], ℝ)} is closed,

(e)
For all t ∈ [0, T], for all x, y, u, v ∈ X, we have

(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).
Proof. We consider the operator F : X × X → X defined by
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.

Hypotheses (h1)(h2) follow immediately from assumption (d).

Hypothesis (h3): Let x, y, u, v ∈ X. For all t ∈ [0, T], we have
Using condition (e), we get
Using condition (f), we obtain
This implies that
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). □
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Acknowledgements
Calogero Vetro was supported by Università degli Studi di Palermo, Local University Project R. S. ex 60%.
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Aydi, H., Samet, B. & Vetro, C. Coupled fixed point results in cone metric spaces for compatible mappings. Fixed Point Theory Appl 2011, 27 (2011). https://doi.org/10.1186/16871812201127
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DOI: https://doi.org/10.1186/16871812201127