1. INTRODUCTION:

The Banach contraction principle is one of the pivotal results of analysis. It is widely considered as the source if metric fixed point theory. Also, its significance lies in its vast applicability in a number of branches of mathematics. Generalization of the above principle has been a heavily investigated branch of research.

The fixed points of mappings in ordered metric space are of great use in many mathematical problems in applied and pure mathematics. The first result in this direction was obtained by Ran and Reurings [1], in this study, the authors present some applications of their obtained results of matrix equations. In [2], Nieto and Lopez extended the result of Ran and Reurings [3], for non decreasing mappings and applied their result to get a unique solution for a first order differential equation. While Agrawal et al. [4] and O’Regan and Petrutel [5] studied some results for generalized contractions in ordered metric spaces. Bhaskar and Lakshmikantham [6] introduced the notion of a coupled fixed point of mapping from They established some coupled fixed point results and applied there results to the study of existence and uniqueness of solution for a periodic boundary value problem. Lakshmikantham and Ciric [7] introduced the concept of coupled coincidence point and proved coupled coincidence and coupled common fixed point results for mappings F from and satisfying non linear contraction in ordered metric space.

In this paper, we drive new coupled fixed point theorems for mapping having the mixed monotone property in partially ordered metric space.

2. Preliminaries:

We recall the definitions and results that will be needed in the sequel.

Definition 2.1 A partially ordered set is a set P and a binary relation, denoted by such that for all

Definition 2.2 A sequence in a metric space is said to be convergent to a point denoted by if

Definition 2.3 A sequence in a metric space is said to be Cauchy sequence if if for all

Definition 2.4 A metric space is said to be complete if every Cauchy sequence in X is convergent.

Definition 2.5 Let be a partially ordered set and. The mapping F is said to has the mixed monotone property if is non – decreasing in and is monotone non-increasing in , that is, for any ,

Definition 2.6 An element is called a coupled fixed point of the mapping if

Theorem 2.7:

Let be a partially ordered set and suppose there exists a metric d on X 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 all if there exist two elements with

and

there exist such that and .

3. Main Results:

Theorem 3.1 Let be a partially ordered set and suppose there exists a metric d on X 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

(3.1.1)

For all if there exist two elements with and , there exist such that and .

Proof:

Let with

& (3.1.2)

Define the sequence and in X such that,

& (3.1.3)

For all

We claim that is monotone non decreasing monotone non increasing

and for all (3.1.4)

From we have

And

Thus i.e equation true for some

Now suppose that equation hold for some n.

We shall prove that the equation is true for

Now then by mixed monotone property of F, we have

Thus by the mathematical induction principle equation holds for all n in N.

Since and , from we have,

If we take ,

, which contradiction of the hypothesis,

This implies,

Similarly since and and from we have

By adding we get,

let us denote then

Similarly it can be proved that

Therefore

By repeating we get,

This implies that,

Thus

For each we have

By adding these, we get

This implies that,

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

Thus by taking limit as in (3.1.3) we get,

Therefore

Thus F has a coupled fixed point in X.

Theorem 3.2:

(3.2.1)

For all if there exist two elements with and , there exist such that and .

Proof:

Let with

and (3.2.2)

Define the sequence and in X such that,

and (3.2.3)

For all

We claim that is monotone non decreasing monotone non increasing

and for all (3.2.4)

From we have

and

Thus i.e equation true for some

Now suppose that equation and hold for some n.

We shall prove that the equation is true for

Now then by mixed monotone property of F, we have

Thus by the mathematical induction principle equation holds for all n in N.

Since and , from we have,

This implies,

Similarly since and and from we have

By adding we get,

Let us denote and then

Similarly it can be proved that

Therefore

By repeating we get,

This implies that,

Thus

For each we have

By adding these, we get

This implies that,

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

Thus by taking limit as in (3.1.3) we get,

Therefore

Thus F has a coupled fixed point in X.

ACKNOWLEDGEMENT:

The author is very grateful to the reviewers for their reading the manuscript and valuable comments.

REFERENCES:

1. 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).

2. Nieto, JJ, Lopez, RR: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 22, 223{239 (2005)

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

4. Agarwal, R.P., El-Gebeily, MA, Oregano, D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 87, 1{8 (2008)

5. O'Regan, D, Petrutel, A: Fixed point theorems for generalized contractions in ordered metric spaces. J.Math. Anal. Appl. 341, 241{1252 (2008)

6. Bhaskar. TG, Lakshmikantham, V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65 1379{1393 (2006)

7. Lakshmikantham, V, Ciric, Lj: Coupled ¯fixed point theorems for nonlinear contractions in partiallyordered metric spaces. Nonlinear Anal. 70, 4341{4349 (2009)

8. Abbas, M, Cho, YJ, Nazir, T: Common ¯fixed point theorems for four mappings in TVS-valued cone metric spaces. J. Math. Inequal. 5, 287{299 (2011)

9. Abbas, M, Khan, MA, Radenovic, S: Common coupled ¯fixed point theorem in cone metric space for w-compatible mappings. Appl. Math. Comput. 217 195{202 (2010)

10. Cho, YJ, He, G, Huang, NJ: The existence results of coupled quasi-solutions for a class of operator equations. Bull. Korean Math. Soc. 47, 455{465 (2010)

Received on 27.03.2020 Modified on 12.04.2020

Research J. Science and Tech. 2020; 12(2): 103-109.

DOI: 10.5958/2349-2988.2020.00013.3