Abstract
Using the notion of compatible mappings in the setting of a partially ordered metric space, we prove the existence and uniqueness of coupled coincidence points involving a (ϕ, ψ)contractive condition for a mappings having the mixed gmonotone property. We illustrate our results with the help of an example.
Keywords:
coupled coincidence point; partially ordered metric space; mixed gmonotone property1 Introduction
The Banach contraction principle is the most celebrated fixed point theorem. Afterward many authors obtained many important extensions of this principle (cf. [116]). Recently Bhaskar and Lakshmikantham [5], Nieto and Lopez [12,13], Ran and Reurings [14] and Agarwal et al. [3] presented some new results for contractions in partially ordered metric spaces. Bhaskar and Lakshmikantham [5] noted that their theorem can be used to investigate a large class of problems and have discussed the existence and uniqueness of solution for a periodic boundary value problem.
Recently, Luong and Thuan [11] presented some coupled fixed point theorems for a mixed monotone mapping in a partially ordered metric space which are generalizations of the results of Bhaskar and Lakshmikantham [5]. In this paper, we establish the existence and uniqueness of coupled coincidence point involving a (ϕ,ψ)contractive condition for mappings having the mixed gmonotone property. We also illustrate our results with the help of an example.
2 Preliminaries
A partial order is a binary relation ≼ over a set X which is reflexive, antisymmetric, and transitive. Now, let us recall the definition of the monotonic function f : X → X in the partially order set (X, ≼). We say that f is nondecreasing if for x, y ∈ X, x ≼ y, we have fx ≼ fy. Similarly, we say that f is nonincreasing if for x, y ∈ X, x ≼ y, we have fx ≽ fy. Any one could read on [9] for more details on fixed point theory.
Definition 2.1 [10](Mixed gMonotone Property)
Let (X, ≼) be a partially ordered set and F : X × X → X. We say that the mapping F has the mixed gmonotone property if F is monotone gnondecreasing in its first argument and is monotone gnonincreasing in its second argument. That is, for any x, y ∈ X,
and
Definition 2.2 [10](Coupled Coincidence Point)
Let (x, y) ∈ X × X, F : X × X → X and g : X → X. We say that (x, y) is a coupled coincidence point of F and g if F(x, y) = gx and F(y, x) = gy for x, y ∈ X.
Definition 2.3 [10]Let X be a nonempty set and let F : X × X → X and g : X → X. We say F and g are commutative if, for all x, y ∈ X,
Definition 2.4 [6]The mapping F and g where F : X × X → X and g : X → X, are said to be compatible if
and
whenever {x_{n}} and {y_{n}} are sequences in X, such that lim_{n}_{→∞} F (x_{n}, y_{n}) = lim_{n}_{→∞} gx_{n }= x and lim_{n}_{→∞} F (y_{n}, x_{n}) = lim_{n}_{→∞} gy_{n }= y, for all x, y ∈ X are satisfied.
3 Existence of coupled coincidence points
As in [11], let ϕ denote all functions ϕ : [0, ∞) → [0, ∞) which satisfy
1. ϕ is continuous and nondecreasing,
2. ϕ (t) = 0 if and only if t = 0,
3. ϕ (t + s) ≤ ϕ (t) + ϕ (s), ∀t, s ∈ [0, ∞)
and let ψ denote all the functions ψ : [0, ∞) → (0, ∞) which satisfy lim_{t}_{→}_{r }ψ (t) > 0 for all r > 0 and
For example [11], functions ϕ_{1}(t) = kt where k > 0,
are in Ψ,
Now, let us start proving our main results.
Theorem 3.1 Let (X, ≼) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete metric space. Let F : X × X → X be a mapping having the mixed gmonotone property on X such that there exist two elements x_{0}, y_{0 }∈ X with
Suppose there exist ϕ ∈ Φ and ψ ∈ Ψ such that
for all x, y, u, v ∈ X with gx ≽ gu and gy ≼ gv. Suppose F(X × X) ⊆ g(X), g is continuous and compatible with F and also suppose either
(a) F is continuous or
(b) X has the following property:
(i) if a nondecreasing sequence {x_{n}} → x, then x_{n }≼ x, for all n,
(ii) if a nonincreasing sequence {y_{n}} → y, then y ≼ y_{n}, for all n.
Then there exists x, y ∈ X such that
i.e., F and g have a coupled coincidence point in X.
Proof. Let x_{0}, y_{0 }∈ X be such that gx_{0 }≼ F (x_{0}, y_{0}) and gy_{0 }≽ F (y_{0}, x_{0}).
Using F(X × X) ⊆ g(X), we construct sequences {x_{n}} and {y_{n}} in X as
We are going to prove that
and
To prove these, we are going to use the mathematical induction.
Let n = 0. Since gx_{0 }≼ F(x_{0}, y_{0}) and gy_{0 }≽ F(y_{0}, x_{0}) and as gx_{1 }= F(x_{0}, y_{0}) and gy_{1 }= F (y_{0}, x_{0}), we have gx_{0 }≼ gx_{1 }and gy_{0 }≽ gy_{1}. Thus (5) and (6) hold for n = 0.
Suppose now that (5) and (6) hold for some fixed n ≥ 0, Then, since gx_{n }≼ gx_{n}_{+1 }and gy_{n }≽ gy_{n}_{+1 }, and by mixed gmonotone property of F, we have
and
Using (7) and (8), we get
Hence by the mathematical induction we conclude that (5) and (6) hold for all n ≥ 0. Therefore,
and
Since gx_{n }≽ gx_{n  }_{1 }and gy_{n }≼ gy_{n  }_{1 }, using (3) and (4), we have
Similarly, since gy_{n  }_{1 }≽ gy_{n }and gx_{n  }_{1 }≼ gx_{n}, using (3) and (4), we also have
Using (11) and (12), we have
By property (iii) of ϕ, we have
Using (13) and (14), we have
which implies, since ψ is a nonnegative function,
Using the fact that ϕ is nondecreasing, we get
Set
Now we would like to show that δ_{n }→ 0 as n → ∞. It is clear that the sequence {δ_{n}} is decreasing. Therefore, there is some δ ≥ 0 such that
We shall show that δ = 0. Suppose, to the contrary, that δ > 0. Then taking the limit as n → ∞ (equivalently, δ_{n }→ δ) of both sides of (15) and remembering lim_{t}_{→}_{r }ψ(t) > 0 for all r > 0 and ϕ is continuous, we have
a contradiction. Thus δ = 0, that is
Now, we will prove that {gx_{n}} and {gy_{n}} are Cauchy sequences. Suppose, to the contrary, that at least one of {gx_{n}} or {gy_{n}} is not Cauchy sequence. Then there exists an ε > 0 for which we can find subsequences {gx_{n}(_{k})}, {gx_{m}(_{k})} of {gx_{n}} and {gy_{n}(_{k})}, {gy_{m}(_{k})} of {gy_{n}} with n(k) > m(k) ≥ k such that
Further, corresponding to m(k), we can choose n(k) in such a way that it is the smallest integer with n(k) > m(k) and satisfying (18). Then
Using (18), (19) and the triangle inequality, we have
Letting k → ∞ and using (17), we get
By the triangle inequality
Using the property of ϕ, we have
Since n(k) > m(k), hence gx_{n}(_{k}) ≽ gx_{m}(_{k}) and gy_{n}(_{k}) ≽ gy_{m}(_{k}). Using (3) and
(4), we get
By the same way, we also have
Inserting (22) and (23) in (21), we have
Letting k → ∞ and using (17) and (20), we get
a contradiction. This shows that {gx_{n}} and {gy_{n}} are Cauchy sequences.
Since X is a complete metric space, there exist x, y ∈ X such that
Since F and g are compatible mappings, we have
and
We now show that gx = F(x, y) and gy = F(y, x). Suppose that the assumption (a) holds. For all n ≥ 0, we have,
Taking the limit as n → ∞, using (4), (24), (25) and the fact that F and g
are continuous, we have d(gx, F(x, y)) = 0.
Similarly, using (4), (24), (26) and the fact that F and g are continuous, we have d(gy, F(y, x)) = 0.
Combining the above two results we get
Finally, suppose that (b) holds. By (5), (6) and (24), we have {gx_{n}} is a nondecreasing sequence, gx_{n }→ x and {gy_{n}} is a nonincreasing sequence, gy_{n }→ y as n → ∞. Hence, by assumption (b), we have for all n ≥ 0,
Since F and g are compatible mappings and g is continuous, by (25) and (26)
we have
and,
Now we have
Taking n → ∞ in the above inequality, using (4) and (21) we have,
Using the property of ϕ, we get
Since the mapping g is monotone increasing, using (3), (27) and (30), we have for all n ≥ 0,
Using the above inequality, using (24) and the property of ψ, we get ϕ(d(gx, F(x, y))) = 0, thus d(gx, F(x, y)) = 0. Hence gx = F(x, y).
Similarly, we can show that gy = F(y, x). Thus we proved that F and g have a coupled coincidence point.
Corollary 3.1 [11]Let (X, ≼) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete metric space. Let F : X × X → X be a mapping having the mixed monotone property on X such that there exist two elements x_{0}, y_{0 }∈ X with
Suppose there exist ϕ ∈ Φ and ψ ∈ Ψ such that
for all x, y, u, v ∈ X with x ≥ u and y ≤ v. Suppose either
(a) F is continuous or
(b) X has the following property.
(i) if a nondecreasing sequence {x_{n}} → x, then x_{n }≼ x, for all n,
(ii) if a nonincreasing sequence {y_{n}} → y, then y ≼ y_{n}, for all n,
then there exist x, y ∈ X such that
that is, F has a coupled fixed point in X.
Corollary 3.2 [11] Let (X, ≼) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete metric space. Let F : X × X → X be a mapping having the mixed monotone property on X such that there exist two elements x_{0}, y_{0 }∈ X with
Suppose there exists ψ ∈ Ψ such that
for all x, y, u, v ∈ X with x ≥ u and y ≤ v. Suppose either
(a) F is continuous or
(b) X has the following property:
(i) if a nondecreasing sequence {x_{n}} → x, then x_{n }≼ x, for all n,
(ii) if a nonincreasing sequence {y_{n}} → y, then y ≼ y_{n}, for all n,
then there exist x, y ∈ X such that
that is, F has a coupled fixed point in X.
Proof. Take ϕ(t) = t in Corollary 3.1, we get Corollary 3.2.
Corollary 3.3 [5] eses of Corollary 3.1, suppose that for Let (X, ≼) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete metric space. Let F : X × X → X be a mapping having the mixed monotone property on X such that there exist two elements x_{0}, y_{0 }∈ X with
Suppose there exists a real number k ∈ [0, 1) such that
for all x, y, u, v ∈ X with x ≥ u and y ≥ v. Suppose either
(a) F is continuous or
(b) X has the following property.
(i) if a nondecreasing sequence {x_{n}} → x, then x_{n }≼ x, for all n,
(ii) if a nonincreasing sequence {y_{n}} → y, then y ≼ y_{n}, for all n,
then there exist x, y ∈ X such that
that is, F has a coupled fixed point in X.
Proof. Taking
4 Uniqueness of coupled coincidence point
In this section, we will prove the uniqueness of the coupled coincidence point. Note that if (X, ≼) is a partially ordered set, then we endow the product X × X with the following partial order relation, for all (x, y), (u, v) ∈ X × X,
Theorem 4.1 In addition to hypotheses of Theorem 3.1, suppose that for every (x, y), (z, t) in X × X, if there exists a (u, v) in X ×X that is comparable to (x, y) and (z, t), then F has a unique coupled coincidence point.
Proof. From Theorem 3.1, the set of coupled coincidence points of F and g is nonempty. Suppose (x, y) and (z, t) are coupled coincidence points of F and g, that is gx = F(x, y), gy = F(y, x), gz = F(z, t) and gt = F(t, z). We are going to show that gx = gz and gy = gt. By assumption, there exists (u, v) ⊂ X × X that is comparable to (x, y) and (z, t). We define sequences {gu_{n}}, {gv_{n}} as follows
Since (u, v) is comparable with (x, y), we may assume that (x, y) ≽ (u, v) = (u_{0}, v_{0}). Using the mathematical induction, it is easy to prove that
Using (3) and (31), we have
Similarly
Using (32), (33) and the property of φ, we have
which implies, using the property of ψ,
Thus, using the property of ϕ,
That is the sequence {d(gx, gu_{n})+ d(gy, gv_{n})} is decreasing. Therefore, there exists α ≥ 0 such that
We will show that α = 0. Suppose, to the contrary, that α > 0. Taking the limit as n → ∞ in (34), we have, using the property of ψ,
a contradiction. Thus. α = 0, that is,
It implies
Similarly, we show that
Using (36) and (37) we have gx = gz and gy = gt.
Corollary 4.1 [11]In addition to hypotheses of Corollary 3.1, suppose that for every (x, y), (z, t) in X × X, if there exists a (u, v) in X × X that is comparable to (x, y) and (z, t), then F has a unique coupled fixed point.
5 Example
Example 5.1 Let X = [0, 1]. Then (X, ≤) is a partially ordered set with the natural ordering of real numbers. Let
Then (X, d) is a complete metric space.
Let g : X → X be defined as
and let F : X × X → X be defined as
F obeys the mixed gmonotone property.
Let ϕ : [0, ∞) → [0, ∞) be defined as
and let ψ : [0, ∞) → [0, ∞) be defined as
Let {x_{n}} and {y_{n}} be two sequences in X such that lim_{n}_{→∞} F (x_{n}, y_{n}) = a, lim_{n}_{→∞} gx_{n }= a, lim_{n}_{→∞} F (y_{n}, x_{n}) = b and lim_{n}_{→∞} gy_{n }= b Then obviously, a = 0 and b = 0. Now, for all n ≥ 0,
and
Then it follows that,
and
Hence, the mappings F and g are compatible in X. Also, x_{0 }= 0 and y_{0 }= c(> 0) are two points in X such that
and
We next verify the contraction (3). We take x, y, u, v, ∈ X, such that gx ≥ gu and gy ≤ gv, that is, x^{2 }≥ u^{2 }and y^{2 }≤ v^{2}.
We consider the following cases:
Case 1. x ≥ y, u ≥ v. Then,
Case 2. x ≥ y, u < v.Then
Case 3. x < y and u ≥ v. Then
Case 4. x < y and u < v with x^{2 }≤ u^{2 }and y^{2 }≥ v^{2}. Then, F(x, y) = 0 and F(u, v) = 0, that is,
Therefore all conditions of Theorem 3.1 are satisfied. Thus the conclusion follows.
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
The authors have been working together on each step of the paper such as the literature review, results and examples.
Acknowledgements
The authors would like to thank the referees for the invaluable comments that improved this paper.
References

Abbas, M, Ali Khan, M, Radenovic, S: Common coupled fixed point theorems in cone metric spaces for wcompatible mappings. Appl Math Comput. 217, 195–201 (2010). Publisher Full Text

Abbas, M, Ali Khan, M, Nazir, T: Coupled common fixed point results in two generalized metric spaces. Appl Math Comput. 217, 6328–6336 (2011). Publisher Full Text

Agarwal, RP, ElGebeily, MA, O'Regan, D: Generalized contractions in partially ordered metric spaces. Appl Anal. 87, 1–8 (2008). Publisher Full Text

Altun, I, Erduran, A: Fixed point theorems for monotone mappings on partial metric spaces. Fixed Point Theory Appl. 2011, Article ID 508730 (2011)

Bhaskar, TG, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal: Theorey Methods Appl. 65, 1379–1393 (2006). Publisher Full Text

Choudhury, BS, Kundu, A: A coupled coincidence point result in partially ordered metric spaces for compatible mappings. Nonlinear Anal. 73, 2524–2531 (2010). Publisher Full Text

Hussain, N, Shah, MH, Kutbi, MA: Coupled coincidence point theorems for nonlinear contractions in partially ordered quasimetric spaces with a Qfunction. Fixed Point Theory Appl. 2011, 21 Article ID 703938 (2011)
Article ID 703938
BioMed Central Full Text 
Khamsi, MA, Hussain, N: KKM mappings in metric type spaces. Nonlinear Anal: Theory Methods Appl. 73, 3123–3129 (2010). Publisher Full Text

Khamsi, MA, Kirk, W: An Introduction to Metric Spaces and Fixed Point Theory, Pure and Applied Mathematics. WileyInterscience, New York (2001)

Lakshmikantham, V, Ćirifić, L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal: Theorey Methods Appl. 70(12), 4341–4349 (2009). Publisher Full Text

Luong, NV, Thuan, NX: Coupled fixed point in partially ordered metric spaces and applications. Nonlinear Anal: Theorey Methods Appl. 74, 983–992 (2011). Publisher Full Text

Nieto, JJ, RodriguezLopez, R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order. 22, 223–239 (2005). Publisher Full Text

Nieto, JJ, Lopez, RR: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math Sinica Engl Ser. 23(12), 2205–2212 (2007). Publisher Full Text

Ran, ACM, Reurings, MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc Am Math Soc. 132, 1435–1443 (2004). Publisher Full Text

Sabetghadam, F, Masiha, HP, Sanatpour, AH: Some coupled fixed point theorems in cone metric spaces. Fixed Point Theory Appl. 2009, Article ID 125426 (2009)

Ray, BK: On Ćirić's fixed point theorem. Fund Math. XCIV, 221–229 (1977)