Mathematica Moravica Vol . 18-2 ( 2014 ) , 45 – 62 Coupled Common Fixed Point Theorems in Partially Ordered G-metric Spaces for Nonlinear Contractions

The aim of this paper is to prove coupled coincidence and coupled common fixed point theorems for a mixed g-monotone mapping satisfying nonlinear contractive conditions in the setting of partially ordered G-metric spaces. Present theorems are true generalizations of the recent results of Choudhury and Maity [Math Comput. Modelling 54 (2011), 73–79], and Luong and Thuan [Math. Comput. Modelling 55 (2012), 1601–1609].


Introduction and Preliminaries
In [12], Mustafa together with Sims introduced the generalized structure of metric spaces, called G-metric spaces.Afterwards, numerous fixed point theorems in this generalized structure were proved by different authors.Works noted in [1,6,9,11,13,14,17,21] are some examples in this direction.Bhaskar and Lakshmikantham [3] introduced the notion of coupled fixed point and proved some coupled fixed point theorems for a mapping satisfying mixed monotone property in partially ordered metric spaces.As an application, they discussed the existence and uniqueness of solution for a periodic boundary value problem.Lakshmikantham and Ćirić [8] extended the notion of mixed monotone property to mixed g-monotone property and generalized the results of Bhaskar and Lakshmikantham [3] by establishing the existence of coupled coincidence point results using a pair of commutative mappings.These results have been extended and generalized by several authors.References [7,15,16] are some examples of these works.Now-a-days authors have keen interest in proving fixed point theorems in partially ordered metric spaces subjected to nonlinear contractive conditions, see [2,4,10,[18][19][20]. Our paper deals with the establishment of some coupled coincidence and coupled common fixed point results for a mixed g-monotone mapping satisfying nonlinear contractive conditions in partially ordered G-metric spaces.Our results generalize the recent results of Choudhury and Maity [5] and Luong and Thuan [9].We give also an example to illustrate our results.We now recall some definitions and properties in G-metric spaces (see [12]).Definition 1.1.Let X be a nonempty set.Suppose that G : X × X × X → [0, +∞) is a function satisfying the following conditions: (G1) G(x, y, z) = 0 if and only if x = y = z; (G2) 0 < G(x, x, y) for all x, y ∈ X with x = y; (G3) G(x, x, y) ≤ G(x, y, z) for all x, y, z ∈ X with y = z; (G4) G(x, y, z) = G(x, z, y) = G(y, z, x) = . . .(symmetry in all three variables); (G5) G(x, y, z) ≤ G(x, a, a) + G(a, y, z) for all x, y, z, a ∈ X (rectangle inequality).Then G is called a G-metric on X and (X, G) is called a G-metric space.Definition 1.2.Let (X, G) be a G-metric space.We say that {x n } is: (i) a G-Cauchy sequence if, for any ε > 0, there is an N ∈ N (the set of all positive integers) such that for all n, m, l Proposition 1.1.Let (X, G) be a G-metric space.The following are equivalent: Proposition 1.2.Let (X, G) be a G-metric space.Then the following are equivalent: Proposition 1.4.Let (X, G) be a G-metric space.Then the function G(x, y, z) is jointly continuous in all three of its variables.

Definition 1.3 ([5]
).Let (X, G) be a G-metric space.A mapping F : X × X → X is said to be continuous if for any two G-convergent sequences {x n } and {y n } converging to x and y respectively, An interesting observation is that any G-metric space (X, G) induces a metric d G on X given by d G (x, y) = G(x, y, y) + G(y, x, x), for all x, y ∈ X.
Now, we recall some definitions introduced in [3,8].Let (X, ) be a partially ordered set and g : X → X be a mapping.The mapping g is said to be non-decreasing if for all x, y ∈ X, x y implies g(x) g(y).Similarly, g is said to be non-increasing, if for all x, y ∈ X, x y implies g(x) g(y).
Bhaskar and Lakshmikantham [3] introduced the following notions of mixed monotone mapping and coupled fixed point.Definition 1.4.Let (X, ) be a partially ordered set and F : X × X → X.The mapping F is said to have the mixed monotone property if F is monotone non-decreasing in its first argument and is monotone nonincreasing in its second argument, that is, for all x 1 , x 2 ∈ X, x 1 x 2 implies F (x 1 , y) F (x 2 , y), for any y ∈ X and for all y 1 , y 2 ∈ X, y 1 y 2 implies F (x, y 1 ) F (x, y 2 ), for any x ∈ X.
The concept of the mixed monotone property was generalized by Lakshmikantham and Ćirić [8] as follows.
Definition 1.5 ([8]).Let (X, ) be a partially ordered set and F : X × X → X and g : X → X.The mapping F is said to have the mixed gmonotone property if F is monotone g-non-decreasing in its first argument and is monotone g-non-increasing in its second argument, that is, for all x 1 , x 2 ∈ X, g(x 1 ) g(x 2 ) implies F (x 1 , y) F (x 2 , y), for any y ∈ X and for all y 1 , y 2 ∈ X, g(y 1 ) g(y 2 ) implies F (x, y 1 ) F (x, y 2 ), for any x ∈ X.

Definition 1.10 ([9]). Let Θ denote all functions
Let (X, ) be a partially ordered set, F : X × X → X be a mapping having the mixed monotone property and suppose there exists a G-metric such that (X, G) is a G-metric space.Choudhury and Maity [5] established some fixed point results for the mapping F under the following contractive condition for w u x and y v z, where k ∈ [0, 1).
Recently, Luong and Thuan [9] generalized the results of Choudhury and Maity [5] by proving some coupled fixed point theorems in partially ordered G-metric spaces under a nonlinear contractive condition of the form for w u x and y v z, where θ ∈ Θ.

Main Results
Our first result is the following coupled coincidence point theorem.
Theorem 2.1.Let (X, ) be a partially ordered set and suppose there is a G-metric G on X such that (X, G) is a complete G-metric space.Let F : X × X → X and g : X → X be mappings such that F has the mixed g-monotone property on X and there exist two elements x 0 , y 0 ∈ X with g(x 0 ) F (x 0 , y 0 ) and g(y 0 ) F (y 0 , x 0 ).Suppose there exist φ ∈ Φ and θ ∈ Θ such that for all (x, y), (u, v), (w, z) ∈ X × X with g(w) g(u) g(x) and g(y) g(v) g(z).Further suppose that F is continuous, F (X × X) ⊆ g(X), g is continuous and commutes with F .Then, there exist x, y ∈ X such that F (x, y) = g(x) and F (y, x) = g(y), that is, F and g have a coupled coincidence point (x, y) ∈ X × X.
Continuing this process, we can construct two sequences {x n } and Now we prove that for all n ≥ 0, ) and g(y n ) g(y n+1 ).
Thus by the mathematical induction, we conclude that (3) holds for all n ≥ 0.
By (11), using the properties of φ, we get ( 12) Since n(i) > m(i), g(x n(i) ) g(x m(i) ) and g(y m(i) ) g(y n(i) ), by (1) we deduce (13) φ(G(g(x Similarly, we have ( 14) φ(G(g(y n(i)+1 ), g(y n(i)+1 ), g(y Inserting ( 13) and ( 14) in ( 12), we get ( 15) By (8), the sequences have subsequences converging to ε 1 and ε 2 (say) respectively and ε 1 + ε 2 = ε > 0. By passing to subsequences, we may assume that Taking i → +∞ in (15) and using ( 7), ( 8), the properties of φ and θ, we have G(g(y n(i) ), g(y n(i) ), g(y m(i) )) which is a contradiction.Thus, {g(x n )} and {g(y n )} are Cauchy sequences.Since the G-metric space (X, G) is complete, there exist x, y ∈ X such that {g(x n )} and {g(y n )} are convergent to x and y respectively, that is from Proposition 1.1, we have Using the continuity of g and Proposition 1.3, we get (16) Since, g(x n+1 ) = F (x n , y n ) and g(y n+1 ) = F (y n , x n ), hence the commutativity of F and g yields that Next we show that F (x, y) = g(x) and F (y, x) = g(y).The mapping F is continuous and since the sequences {g(x n )} and {g(y n )} are respectively G-convergent to x and y, hence using Definition 1.3 the sequence {F (g(x n ), g(y n ))} is G-convergent to F (x, y).Therefore, by (17), {g(g(x n+1 ))} is G-convergent to F (x, y).By uniqueness of limit and using ( 16), we have F (x, y) = g(x).Similarly, we can show that F (y, x) = g(y).Hence (x, y) is a coupled coincidence point of F and g.
Theorem 2.2.Let (X, ) be a partially ordered set and G be a G-metric on X.Let F : X × X → X and g : X → X be mappings such that F has the mixed g-monotone property on X and there exist two elements x 0 , y 0 ∈ X with g(x 0 ) F (x 0 , y 0 ) and g(y 0 ) F (y 0 , x 0 ).Suppose there exist φ ∈ Φ and θ ∈ Θ such that (18) φ(G(F (x, y), F (u, v), F (w, z)) − θ(G(g(x), g(u), g(w)), G(g(y), g(v), g(z))) for all (x, y), (u, v), (w, z) ∈ X × X with g(w) g(u) g(x) and g(y) g(v) g(z).Further suppose that (g(X), G) or (F (X × X), G) is complete, F (X × X) ⊆ g(X) and the following conditions hold: (i) if a non-decreasing sequence {x n } in X converges to x ∈ X, then x n x for all n, (ii) if a non-increasing sequence {y n } in X converges to y ∈ X, then y n y for all n.Then, there exist x, y ∈ X such that F (x, y) = g(x) and F (y, x) = g(y), that is, F and g have a coupled coincidence point (x, y) ∈ X × X.
Proof.Following the proof of Theorem 2.1, it follows that {g(x n )} and {g(y n )} are Cauchy sequences.Now, we distinguish the following two cases.Case 1.If (g(X), G) is complete, then there exist x, y ∈ X such that g(x n ) → g(x) and g(y n ) → g(y) as n → +∞.Since {g(x n )} is nondecreasing and {g(y n )} is non-increasing, by given hypotheses, we have g(x n ) g(x) and g(y) g(y n ) for all n ≥ 0. Then using (18) and the properties of φ and θ, we have Letting n → +∞ in the last inequality and using the properties of φ, we obtain φ(G(F (x, y), g(x), g(x))) ≤ 0, which implies that G(F (x, y), g(x), g(x)) = 0, that is, F (x, y) = g(x).Similarly, it can be shown that , so there exist x, y ∈ X such that p = g(x) and q = g(y) and from here onwards the proof follows as in Case 1.
If g = I, the identity mapping in Theorem 2.1, then we deduce the following result of coupled fixed point.
Corollary 2.1.Let (X, ) be a partially ordered set and suppose there is a G-metric G on X such that (X, G) is a complete G-metric space.Let F : X × X → X be a continuous mapping such that F has the mixed monotone property on X and there exist two elements x 0 , y 0 ∈ X with x 0 F (x 0 , y 0 ) and y 0 F (y 0 , x 0 ).Also suppose there exist φ ∈ Φ and θ ∈ Θ such that for all (x, y), (u, v), (w, z) ∈ X × X with w u x and y v z Then, there exist x, y ∈ X such that F (x, y) = x and F (y, x) = y, that is, F has a coupled fixed point (x, y) ∈ X × X.
If φ = g = I, the identity mapping in Theorem 2.1, then we obtain the result of Luong and Thuan [9] in the form of following corollary.

Corollary 2.2 ([9]
).Let (X, ) be a partially ordered set and suppose there is a G-metric G on X such that (X, G) is a complete G-metric space.Let F : X × X → X be a mapping such that F is continuous and has the mixed monotone property on X and there exist two elements x 0 , y 0 ∈ X with x 0 F (x 0 , y 0 ) and y 0 F (y 0 , x 0 ).Also suppose there exists θ ∈ Θ such that for all (x, y), (u, v), (w, z) ∈ X × X with w u x and y v z.Then, there exist x, y ∈ X such that F (x, y) = x and F (y, x) = y, that is, F has a coupled fixed point (x, y) ∈ X × X.
Corollary 2.3.Let (X, ) be a partially ordered set and suppose there is a G-metric G on X such that (X, G) is a complete G-metric space.Let F : X × X → X and g : X × X be mappings such that F is continuous, F has the g-mixed monotone property on X and there exist two elements x 0 , y 0 ∈ X with g(x 0 ) F (x 0 , y 0 ) and g(y 0 ) F (y 0 , x 0 ).Also suppose there exist φ ∈ Φ and ψ ∈ Ψ such that φ(G(F (x, y), F (u, v), F (w, z)) − ψ(max{G(g(x), g(u), g(w)), G(g(y), g(v), g(z))}) for all (x, y), (u, v), (w, z) ∈ X × X with g(w) g(u) g(x) and g(y) g(v) g(z).Then, there exist x, y ∈ X such that F (x, y) = g(x) and F (y, x) = g(y), that is, F and g have a coupled coincidence point (x, y) ∈ X × X.
Corollary 2.4.Let (X, ) be a partially ordered set and suppose there is a G-metric G on X such that (X, G) is a complete G-metric space.Let F : X × X → X be a mapping such that F is continuous and has the mixed monotone property on X and there exist two elements x 0 , y 0 ∈ X with x 0 F (x 0 , y 0 ) and y 0 F (y 0 , x 0 ).Also suppose there exists k ∈ [0, 1) such that for all (x, y), (u, v), (w, z) ∈ X × X with w u x and y v z.Then, there exist x, y ∈ X such that F (x, y) = x and F (y, x) = y, that is, F has a coupled fixed point (x, y) ∈ X × X.Now, we give sufficient conditions for uniqueness of the coupled fixed point.If (X, ) is a partially ordered set, then we endow the product space X × X with the following partial order: for (x, y), (u, v) ∈ X × X, (u, v) (x, y) ⇔ x u, y v. Theorem 2.3.In addition to the hypotheses of Theorem 2.1, suppose that for every (x, y), (x * , y * ) ∈ X × X there exists (u, v) ∈ X × X such that (F (u, v), F (v, u)) is comparable to (F (x, y), F (y, x)) and (F (x * , y * ), F (y * , x * )).Then F and g have a unique coupled common fixed point, that is, there exists a unique (x, y) ∈ X × X such that x = g(x) = F (x, y) and y = g(y) = F (y, x).
Coupled Common Fixed Point Theorems . . .Thus condition (19) holds in all the cases.Hence by Corollary 2.1, F has a coupled fixed point (0, 0) ∈ X × X.Note that Corollary 2.1 is not applicable in respect of the usual order of real numbers because condition (19) does not hold.In fact, in this case, from condition (19)