INTRODUCTION:

The Banach contraction principle is the most celebrated fixed point theorem. Boyd and Wong [4] extended the Banach contradiction principle to the case of non linear contraction mappings. Afterword many authors obtain important fixed point theorems. Recently Bhaskar and Lakshmikantham [2] presented some new results for contractions in partially ordered metric spaces, and 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 valued problem.

After some time, Lakshmikantham and Circ [6] introduced the concept of mixed monotone mapping and generalized the results of Bhaskar and Lakshmikantham [2]. In the present work, we prove some more results for coupled fixed point theorems by using the concept of g-monotone mappings.

Recall that if is partially ordered set and such that for implies then a mapping is said to be non decreasing. similarly, mapping is defined. Bhaskar and Lakshmikantham introduced the following notions of mixed monotone mapping and a coupled fixed point.

Definition- 1:

(Bhaskar and Lakshmikantham [2]) Let be an ordered set and . The mapping F is said to has the mixed monotone property if F is monotone non-decreasing in its first argument and is monotone non-increasing in its second argument, that is, for any

And

Definition – 2:

(Bhaskar and Lakshmikantham [2]) An element is called a coupled fixed point of the mapping if,

The main theoretical results of the Bhaskar and Lakshmikantham in [2] are the following two coupled fixed point theorems.

Theorem – 3:

(Bhaskar and Lakshmikantham [2] Theorem 2.1) Let be partially ordered set and suppose there is a metric d on such that is a complete metric space. Let be a continuous mapping having the mixed monotone property on X. Assume that there exists a with,

for each

If there exist such that,

Then there exists, such that,

Theorem – 4:

(Bhaskar and Lakshmikantham [2] Theorem 2.2) be partially ordered set and suppose there is a metric such that is a complete metric space. Assume that X has the following property,

1. If a non decreasing sequence for all n,

2. If a non increasing sequence , for all n,

Let F : X \times X \rightarrow X be a mapping having the mixed monotone property on X. Assume that there exists a with,

for each,

If there exist such that,

Then there exists, such that,

We note that Bhaskar and Lakshmikantham [2] have discussed the problems of a uniqueness of a coupled fixed point and applied their theorems to problems of the existence and uniqueness of solution for a periodic boundary valued problem.

Main Results:

Analogous with Definition 2 Lakshmikantham and Ciric [6] introduced the following concept of a mixed g- monotone mapping.

Definition – 5:

Let be an ordered set and The mapping F is said to has the mixed g- monotone property if F is g- monotone non-decreasing in its first argument and is g- monotone non-increasing in its second argument, that is, for any

And

Note that if g is the identity mapping, then Definition – 5 , reduces to Definition – 2.

Definition – 6:

(Lakshmikantham and Ciric [6]) An element is called a coupled coincidence point of the mapping if,

Definition – 7:

(Lakshmikantham and Ciric [6])Let X be non empty set and one says F and g are commutative if,

for all .

Now we prove our main result.

Theorem – 8:

Let be a partially ordered set and suppose there is a metric d on X such that is a complete metric space. Suppose are such that F has the mixed g- monotone property and

(1)

for all such that and . Suppose that g is continuous and commute with F and also suppose either

1. If a non decreasing sequence , for all n,

2. If a non increasing sequence , for all n.

If there exist such that

Then there exist such that Since

Proof:

Let be such that Since we can choose such that and Again from we can choose such that Continuing the process we can construct sequence such that

(2)

for all .

We shall show that

for all (3)

And

for all (4)

For this we shall use the mathematical induction. Let n = 0. Since and and as and we have and Thus (3) and (4) hold for n = 0.

Suppose now (3) and (4) holds for some fixed . Then, since and and as F has the mixed g- monotone property, from (3.4) and (2.1).

(5)

And

(6)

and from (3.4) and (2.2),

(7)

(8)

Now from (5) – (8) we get

(9)

And

(10)

Thus by the mathematical induction we conclude that (5) – (8) holds for all . Therefore,

And

Since,

by using, (1) and (2) we have,

This gives,

(11)

Similarly, from (1) and (2), as

(12)

Let us denote and,

then by adding (11) and (12), we get

which implies that,

Thus,

For each we have,

And

by adding the both of above, we get

which implies,

Therefore, are Cauchy sequence in X. since X is complete metric space, there exist such that and

Thus by taking limit as in (2), we get

Therefore, F and g have a coupled fixed point.

Now, we present coupled coincidence and coupled common fixed point results for mappings satisfying contractions of integral type. Denote by the set of functions satisfying the following hypotheses:

1. is a Lebsegue mapping on each compact subet of

2. for any , we have .

Finally we have the following results.

Theorem – 9:

Let be a partially ordered set and suppose there is a metric d on X such that (X,d) is a complete metric space. Suppose are such that F has the mixed g- monotone property and assume that there exist such that,

for all such that with and . Suppose that g is continuous and commute with F. Then there exist such that Since

In the view of Theorem – 8 , we have,

By using, (3.14) we have,

It can be written as,

Similarly we can show that,

Processing the same way it is easy to see that,

And

From the Theorem – 8 , the result is follows and nothing to prove.

Corollary – 10:

Let be a partially ordered set and suppose there is a metric d on X such that is a complete metric space. Suppose are such that F has the mixed g- monotone property such that,

for all such that with and . Suppose that g is continuous and commute with F. Then there exist such that Since

Proof:

In Theorem – 9 , if we take then result is follows.

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Received on 27.03.2020 Modified on 14.04.2020

Research J. Science and Tech. 2020; 12(2): 123-130.

DOI: 10.5958/2349-2988.2020.00015.7