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.
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
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 [
Choudhury and Kundu [
In this paper, we will generalize the results of Mursaleen et al. [
In order to obtain our results we need to consider the followings.
Definition 2.1. [
Definition 2.2. [
Definition 2.3. [
Definition 2.4. [
Choudhury et al. [
Definition 2.5. [
whenever
In order to obtain our results we need to consider the followings.
Definition 2.6. [
1)
2)
3)
Lemma 2.7. [
Definition 2.8. [
Definition 2.9. [
Now, we will introduce our notions:
Definition 2.10. Let
with
Definition 2.11. Let
Then F and g are said to be (α)-admissible if
Recently, Mursaleen et al. [
Theorem 3.1 [
Such that for
Suppose also that
1) F is (a)-admissible.
2) There exists
3) F is continuous.
If there exists
Then F has a coupled fixed point, that is, there exist,
Now we are ready to prove our results for compatible mappings.
Theorem 3.2 Let
For all
Suppose also that
1) F and g are (a)-admissible.
2) There exists
3)
4) F is continuous.
If there exists
Then F and g has coupled coincidence point that is there exist,
Proof: Let
and
Let
Continuing this process, we can construct two sequences
Now we will show that
For
and as
We have,
Thus (3.4) holds for
Now suppose that (3.4) holds for some fixed
Then, since
Therefore, by g-mixed monotone property of F, we have
From above, we conclude that
Thus, by mathematical induction, we conclude that (3.4) holds for all
If following holds for some
Then obviously,
Now, we assume that
Since, F and g a-admissible, we have
implies,
Thus by mathematical induction, we have
Similarly, we have
From (3.3) and conditions 1) and 2) of hypothesis, we get
Similarly, we have
On adding (3.7) and (3.8), we get
Repeating the above process, we get
For
Let
that is;
Since,
Hence,
Since,
There exists,
Since, F and g are compatible mappings; therefore, we have
Next we will show that
For all
Taking limit
Similarly, we have
Thus
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
1) Inequality (3.3) and conditions 1), 2) and 3) hold.
2) if
for all n and
If there exists
Then F and g has coupled coincidence point, that is, there exist,
Proof. Proceeding along the same lines as in the proof of Theorem 3.2, we know that
are Cauchy sequences in the complete metric space
Similarly,
Using the triangle inequality, (3.11) and the property of
Similarly, on using (3.12), we have
Proceeding limit
Thus,
Remark. On putting
Example 3.4. Let
Then
Let
Let
Let
Let,
Then obviously,
Now, for all
Then it follows that,
Hence, the mappings F and g are compatible in X.
Consider a mapping
Thus (3.3) holds for
mixed monotone property. Let
is a coupled coincidence point of g and F in X.
 Preeti,SanjayKumar, (2015) Coupled Fixed Point for (α, Ψ)-Contractive in Partially Ordered Metric Spaces Using Compatible Mappings. Applied Mathematics,06,1380-1388. doi: 10.4236/am.2015.68130