1. Introduction and Preliminaries

The Banach contraction mapping principle has been generalized in several directions. One of these generalizations, known as the Meir-Keeler fixed point theorem ^[11], has been obtained by the following more general assumption: for all ε>0 there exists δ(ε) > 0 such that

(1)

Bhaskar and Lakshmikantham ^[3] introduced the notion of coupled fixed point, mixed monotone mappings in the setting of single-valued mappings and established some coupled fixed point theorems for a mapping with the mixed monotone property in the setting of partially ordered metric spaces.

In ^[3], Bhaskar and Lakshmikantham introduced the following.

Definition 1. Let be a partially ordered set and endow the product space with the following partial order:

(2)

Definition 2. An element is called a coupled fixed point of the mapping if

(3)

Definition 3. Let be a partially ordered set. Suppose be a given mapping. We say that F has the mixed monotone property if for all , we have

(4)

and

(5)

Lakshmikantham and Ciric ^[10] extended the notion of mixed monotone property to mixed g-monotone property and established coupled coincidence point results using a pair of commutative mappings, which generalized the results of Bhaskar and Lakshmikantham ^[3].

In ^[10], Lakshmikantham and Ciric introduced the following:

Definition 4. An element is called a coupled coincidence point of the mappings and if

(6)

Definition 5. an element is called a common coupled fixed point of the mappings and if

(7)

Definition 6. An element x 2 X is called a common fixed point of the mappings and if

(8)

Definition 7. The mappings and are said to be commutative if

(9)

Definition 8. Let be a partially ordered set. Suppose and are given mappings. We say that F has the mixed g-monotone property if for all ; we have

(10)

and

(11)

If g is the identity mapping on X; then F satisfies the mixed monotone property.

These results used to study the existence and uniqueness of solution for periodic boundary value problems. Hussain et al. ^[9] introduced a new concept of generalized compatibility of a pair of mappings defined on a product space and proved some coupled coincidence point results.

In ^[9], Hussain et al. introduced the following:

Definition 9. An element is called a coupled coincidence point of mappings if

(12)

Example 10. Let be defined by F(x,y) = xy and G(x,y) = 2/3 (x + y) for all . Note that (0,0), (1,2) and (2,1) are coupled coincidence points of F and G.

Definition 11. Let be two mappings. We say that the pair {F,G} is commuting if

(13)

Definition 12. Let . We say that the pair {F,G} is generalized compatible if

whenever (x_n) and (y_n) are sequences in X such that

Obviously, a commuting pair is a generalized compatible but not conversely in general.

Coupled fixed point theory have developed literature, some of the instances of these works are [1,2,4,5,6,7,8,11,12,13,15]. Recently Samet et al. ^[14] claimed that most of the coupled fixed point theorems in the setting of single valued mappings on ordered metric spaces are consequences of well-known fixed point theorems.

In ^[13], Samet established the coupled fixed points of mixed strict monotone generalized Meir-Keeler operators and also established the existence and uniqueness results for coupled fixed point. Berinde and Pecurar ^[2] obtained more general coupled fixed point theorems for mixed monotone operators satisfying a generalized symmetric Meir-Keeler contractive condition.

In this paper, we introduce the concept of generalized weakly compatibility for the pair {F,G} of mappings and also introduce the concept of common fixed point of the mappings . We establish a common fixed point theorem for generalized weakly compatible pair of mappings without mixed monotone property of any mapping under generalized symmetric Meir-Keeler contraction on a non complete metric space, which is not partially ordered. We also give an example to support our result presented here. We extend and generalize the results of Berinde and Pecurar ^[2], Bhaskar and Lakshmikantham ^[3], Meir and Keeler ^[11], Samet ^[13] and many other results in the existing literature.

2. Main Results

First, we introduce the following:

Definition 13. An element is called a common fixed point of the mappings if

Definition 14. Let X be a non-empty set. The mappings are called generalized weakly compatible mappings if F(x, y) = G(x, y), F(y, x) = G(y, x) implies that G(F(x, y), F(y, x)) = F(G(x, y), G(y, x)), G(F(y, x), F(x, y)) = F(G(y, x), G(x, y)), for all . Obviously, a generalized compatible pair is generalized weakly compatible but converse is not true in general.

Example 15. Let (X, d) be a usual metric space where . Define by

Let . Then, we have

So F and G are not generalized compatible. From F(x, y) = G(x, y), F(y, x) =G(y, x), we can get (x, y) = (0, 0) and we have G(F(0, 0), F(0, 0)) = F(G(0,0), G(0, 0)), G(F(0, 0), F(0, 0)) = F(G(0, 0), G(0, 0)), which implies that F and G are generalized weakly compatible.

Theorem 16. Let (X, d) be a metric space. Assume be two generalized weakly compatible mappings and for each ε> 0, there exists δ(ε) > 0 such that

implies

(14)

for all . Suppose that for any , there exist such that

(15)

Suppose that is complete. Then there exists a unique such that x = G(x,x) = F(x,x).

Proof. Let x₀, y₀ be two arbitrary points in X. From (15); we can choose such that

Continuing this process, we can construct sequences {x_n} and {y_n} in X such that

(16)

The proof is divided into 4 steps.

Step 1. Prove that {G(x_n,y_n)} and {G(y_n,x_n)} are Cauchy sequences.

Now, by (14), for each ε> 0, there exists δ(ε) > 0 such that

implies

(17)

Condition (17) implies the strict contractive condition

(18)

for . Thus, by (18), we have

which shows that the sequence of nonnegative numbers given by

(19)

is non-increasing, Therefore, there exists some ε≥0 such that

We shall prove that ε= 0. Suppose, to the contrary, that ε> 0. Then there exists a positive integer p such that

which, by (17); implies

it follows, by (16) and (19); that

which is a contradiction. Thus ε = 0 and hence

(20)

Let now ε> 0 be arbitrary and δ(ε) the corresponding value from the hypothesis of our theorem. By (20), there exists a positive integer k such that

(21)

For this fixed number k, consider now the set A_k = {(G(x, y), G(y, x)): G(x_k, y_k) ≤ G(x, y), G(y, x) ≥ G(y_k, x_k), ½ [d(G(x_k, y_k), G(x, y))+d(G(y_k, x_k), G(y,x))] < ε + δ(ε). By (21), A_k ≠ø. We claim that

(22)

Let . Then

(23)

which, by (14), implies

(24)

Now, by (21) and (24), we have

Thus . Again

Thus and by induction,

This implies that for all n, m > k, we have

This shows that and are Cauchy sequences in X.

Step 2. Prove that G and F have a coupled coincidence point.

Since is complete, then there exist and such that

(25)

Now, by (18), we have

Taking limit as n→ 1 in the above inequality and using (25), we have

which implies that

Since F and G are generalized weakly compatible, we get that

which implies that

that is, (x, y) is a coupled coincidence point of F and G.

Step 3. Prove that G(x, y) = y and G(y, x) = x.

If, not. Then by (18), we have

Letting n→∞ in the above inequality and using (25), we have

which is a contradiction. Thus we must have G(x, y) = y and G(y, x) = x.

Step 4. Prove that x = y.

If, not. Then by (18), we have

Letting n→∞ in the above inequality and using (25), we get

which is a contradiction. Thus x = y.

Example 17. Suppose that , equipped with the usual metric . Let be defined as

From F(x, y) = G(x, y), F(y, x) = G(y, x), we can get (x, y) = (0, 0) and we have G(F(0, 0), F(0, 0)) = F(G(0, 0), G(0, 0)), G(F(0, 0), F(0, 0)) = F(G(0, 0), G(0, 0)), which implies that F and G are generalized weakly compatible.

Now, we prove that for any , there exist such that

Let be fixed. We consider the following cases:

Case 1: If x = y, then we have F(x, y) = 0 = G(x, y) and F(y, x) = 0 =G(y, x).

Case 2: If x > y, then we have and .

Case 3: If x < y, then we have and . Now, we shall show that the mappings F and G satisfy the condition (14): For each , we have

Then

Thus the contractive condition (14) is satisfied for all . In addition, all the other conditions of Theorem 16 are satisfied and 0 is a unique common fixed point of F and G.

Corollary 18. Let (X, d) be a metric space. Assume be two generalized compatible mappings satisfying (14), (15) and is complete. Then there exists a unique such that x = G(x, x) = F(x, x).

Corollary 19. Let (X, d) be a metric space. Assume be two commuting mappings satisfying (14), (15) and is complete. Then there exists a unique such that x = G(x, x) = F(x, x).

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