Coupled Fixed Point for (α, Ψ)-Contractive in Partially Ordered Metric Spaces Using Compatible Mappings
Preeti, Sanjay Kumar

Abstract

In this paper, first we introduce notions of (α, Ψ)-contractive and (α)-admissible for a pair of map and prove a coupled coincidence point theorem for compatible mappings using these notions. Our work extends and generalizes the results of Mursaleen et al. [1]. At the end, we will provide an example in support of our result.

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Preeti, &. and Kumar, S. (2015) Coupled Fixed Point for (α, Ψ)-Contractive in Partially Ordered Metric Spaces Using Compatible Mappings. Applied Mathematics, 6, 1380-1388. doi: 10.4236/am.2015.68130.

1. Introduction

Fixed point theorems give the conditions under which maps have solutions.

Fixed point theory is a beautiful mixture of Analysis, Topology and Geometry. Fixed points Theory has been playing a vital role in the study of nonlinear phenomena. In particular, fixed point techniques have been applied in diverse fields as Biology, Chemistry, and Economics, Engineering, Game theory and Physics. The usefulness of the concrete applications has increased enormously due to the development of accurate techniques for computing fixed points.

The fixed point theory has many important applications in numerical methods like Newton-Raphson Method and establishing Picard’s Existence Theorem regarding existence and uniqueness of solution of first order differential equation, existence of solution of integral equations and a system of linear equations. The credit of making the concept of fixed point theory useful and popular goes to polish mathematician Stefan Banach. In 1922, Banachproved a fixed point theorem, which ensures the existence and uniqueness of a fixed point under appropriate conditions. This result of Banach is known as Banach fixed point theoremor contraction mapping principle, “Let x be any non empty set and be a completemetric space If T is mapping of X into itself satisfying for each where, then T has a unique fixed point in X”. This principle provides a technique for solving a variety of applied problems in Mathematical sciences and Engineering and guarantees the existence and uniqueness of fixed points of certain self maps of metric spaces and provides a constructive method to find out fixed points. Now the question arise what type of problems have the fixed point. The fixed point problems can be elaborated in the following manner:

1) What functions/maps have a fixed point?

2) How do we determine the fixed point?

3) Is the fixed point unique?

Currently, fixed point theory has been receiving much attention on in partially ordered metric spaces; that is, metric spaces endowed with a partial ordering. Turinici [2] extending the Banach contraction principle in the setting of partially ordered sets and laid the foundation a new trend in fixed point theory. Ran and Reurings [3] developed some applications of Turinici’s theorem to matrix equations and established some results in this direction. The results were further extended by Nieto and Rodŕguez-Ĺpez [4] [5] for non-decreasing mappings. Bhaskar and Lakshmikantham [6] [7] introduced the new notion of coupled fixed points for the mappings satisfying the mixed monotone property in partially ordered spaces and discussed the existence and uniqueness of a solution for a periodic boundary value problem. Later on, Lakshmikantham and Ciríc [8] proved coupled coincidence and coupled common fixed point theorems for nonlinear contractive mappings in partially ordered complete metric spaces.

Choudhury and Kundu [9] , proved the coupled coincidence result for compatible mappings in the settings of partially ordered metric space. Recently, Samet et al. [10] [11] have introduced the notion of α-ψ-contractive and α-admissible mapping and proved fixed point theorems for such mappings in complete metric spaces. For more results regarding coupled fixed points in various metric spaces one can refer to [12] -[23] .

In this paper, we will generalize the results of Mursaleen et al. [1] for α-ψ-contractive and α-admissible mappings using compatible mappings under α-ψ-contractions and α-admissible conditions.

2. Mathematical Preliminaries

In order to obtain our results we need to consider the followings.

Definition 2.1. [6] . Let be a partially ordered set and be a mapping. Then a map F is said to have the mixed monotone property if is monotone non-decreasing in x and is monotone non-increasing in y; that is, for any,

implies and

implies.

Definition 2.2. [6] . An element is said to be a coupled fixed point of the mapping if

and.

Definition 2.3. [8] . Let be a partially ordered set and and be two mappings. We say F has the mixed g-monotone 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 any

; implies and

; implies.

Definition 2.4. [8] . An element is called a coupled coincidence point of mappings and if

and.

Choudhury et al. [9] introduced the notion of compatible maps in partially ordered metric spaces as follows:

Definition 2.5. [9] . The mappings F and g where and be are said to be compatible if

and

whenever and are sequences in X, such that

and for all are satisfied.

In order to obtain our results we need to consider the followings.

Definition 2.6. [1] . Denote by Ψ the family of non-decreasing functions such that

for all, where ψn is the nth iterate of ψ satisfying

1).

2) for all and

3) for all.

Lemma 2.7. [1] . If is non-decreasing and right continuous, then

as for all if and only if for all.

Definition 2.8. [1] . Let be a partially ordered metric space and then F is said to be α-contractive if there exist two functions

and such that

with and.

Definition 2.9. [1] . Let and be two mappings. Then F is said to be (α)-admissible if

implies, for all.

Now, we will introduce our notions:

Definition 2.10. Let be a partially ordered metric space and and be two mappings. Then the maps F and g are said to be (α, ψ)-contractive if there exist two functions

and such that

for

with and.

Definition 2.11. Let, and be mappings.

Then F and g are said to be (α)-admissible if

implies, for all

.

3. Main Results

Recently, Mursaleen et al. [1] proved the following coupled fixed point theorem with α-ψ-contractive conditions in partial ordered metric spaces:

Theorem 3.1 [1] Let be a partially ordered set and there exists a metric d on X such that is a complete metric space. Let be mapping and suppose F has mixed monotone property. Suppose there exists and

Such that for, the following holds:

, with and.

Suppose also that

2) There exists such that

and

3) F is continuous.

If there exists such that and.

Then F has a coupled fixed point, that is, there exist, such that.

Now we are ready to prove our results for compatible mappings.

Theorem 3.2 Let be a partially ordered set and there exists a metric d on X such that is a complete metric space. Let be mapping and be another mapping. Suppose F has g-mixed monotone property and there exists and

(3.3)

For all with and.

Suppose also that

1) F and g are (a)-admissible.

2) There exists such that

and

3), g is continuous and F and g are compatible in X.

4) F is continuous.

If there exists such that and.

Then F and g has coupled coincidence point that is there exist, such that

.

Proof: Let be such that

and

and and.

Let be such that, and.

Continuing this process, we can construct two sequences and in X as follows:

and for all,.

Now we will show that

and for all,. (3.4)

For, since, and

and as and

We have, and.

Thus (3.4) holds for.

Now suppose that (3.4) holds for some fixed.

Then, since and.

Therefore, by g-mixed monotone property of F, we have

and

.

From above, we conclude that

.

Thus, by mathematical induction, we conclude that (3.4) holds for all.

If following holds for some,

Then obviously, and, i.e., F has coupled coincidence point.

Now, we assume that for all,.

Since, F and g a-admissible, we have

, ,

implies,.

Thus by mathematical induction, we have

(3.5)

Similarly, we have

for all,. (3.6)

From (3.3) and conditions 1) and 2) of hypothesis, we get

(3.7)

Similarly, we have

(3.8)

On adding (3.7) and (3.8), we get

Repeating the above process, we get

For there exists such that

Let be such that, then by using the triangle inequality, we have

that is;

Since, and

.

Hence, and are Cauchy sequences in.

Since, is complete, therefore, and are convergent in.

There exists, such that

and

Since, F and g are compatible mappings; therefore, we have

(3.9)

(3.10)

Next we will show that and.

For all, we have

Taking limit in the above inequality by continuity of F and g and from (3.9) we get

.

Similarly, we have.

Thus

and.

Hence, we have proved that F and g has coupled coincidence point.

Now, we will replace continuity of F in the theorem 3.2 by a condition on sequences.

Theorem 3.3. Let be a partially ordered set and there exists a metric d on such that is a complete metric space. Let and be maps and F has g-mixed monotone property. Suppose there exists such that for, the following holds:

1) Inequality (3.3) and conditions 1), 2) and 3) hold.

2) if and are sequences in X such that

and

for all n and and, for all, then

and.

If there exists such that and.

Then F and g has coupled coincidence point, that is, there exist, such that

and.

Proof. Proceeding along the same lines as in the proof of Theorem 3.2, we know that and

are Cauchy sequences in the complete metric space. Then there exists such that and

(3.11)

Similarly, (3.12)

Using the triangle inequality, (3.11) and the property of for all t > 0, we get

Similarly, on using (3.12), we have

Proceeding limit in above two inequalities, we get

and.

Thus, and.

Remark. On putting, identity map, we get the required result of Mursaleen et al. [14] .

Example 3.4. Let. Then is a partially ordered set with the natural ordering of real numbers. Let

for.

Then is a complete metric space.

Let be defined as, for all.

Let be defined as

Let be defined as for.

Let, and be two sequences in X such that,

, , ,.

Then obviously, and

Now, for all, ,

and

Then it follows that,

and

Hence, the mappings F and g are compatible in X.

Consider a mapping be such that

Thus (3.3) holds for for all, and we also see that and F satisfies g-

mixed monotone property. Let and. Then and

. Thus, all the conditions of theorem 3.2 are satisfied. Here

is a coupled coincidence point of g and F in X.

Conflicts of Interest

The authors declare no conflicts of interest.

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