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
1) F is (a)-admissible.
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.