In this article, we establish a common fixed point theorem satisfying integral type contractive condition for two pairs of weakly compatible mappings with E. A. property and also generalize Theorem (2) of B.E. Rhoades [1] in dislocated metric space.
In 1986, S. G. Matthews [
The study of fixed point theorems of mappings satisfying contractive conditions of integral type has been a very interesting field of research activity after the establishment of a theorem by A. Branciari [
We start with the following definitions, lemmas and theorems.
Definition 1 [
1.
2.
3.
Then d is called dislocated metric (or d-metric) on X and the pair
Definition 2 [
Definition 3 [
Definition 4 [
Lemma 1 [
Definition 5 Let A and S be two self mappings on a set X. If
Definition 6 [
Definition 7 [
for some
Now we establish a common fixed point theorem for two pairs of weakly compatible mappings using E. A. pro- perty.
Theorem 1 Let (X, d) be a dislocated metric space. Let
where
is a Lebesgue integrable mapping which is summable, non-negative and such that
1. The pairs
2. The pairs
if T(X) is closed then
1) the maps A and T have a coincidence point.
2 the maps B and S have a coincidence point.
3) the maps A, B, S and T have an unique common fixed point.
Proof. Assume that the pair
for some
From condition (2) we have
where
Taking limit as
Since
Hence we have
which is a contradiction, since
Assume
where
Since
So, taking limit as
which is a contradiction. Hence
This proves that v is the coincidence point of
Again, since
Now we claim that
where
Since
So if
or
Hence,
Therefore,
This represents that w is the coincidence point of the maps B and S.
Hence,
Since the pairs
We claim
where
Since
So if
or
Hence,
Therefore,
Uniqueness:
If possible, let
where
Since
So if
or
or
Hence,
Now we have the following corollaries:
If we take T = S in Theorem (1) the we obtain the following corollary
Corollary 1 Let (X,d) be a dislocated metric space. Let
where
is a Lebesgue integrable mapping which is summable, non-negative and such that
1. The pairs
2. The pairs
if S(X) is closed then
1) the maps A and S have a coincidence point
2) the maps B and S have a coincidence point
3) the maps A, B and S have an unique common fixed point.
If we take B = A in Theorem (1) we obtain the following corollary.
Corollary 2 Let (X, d) be a dislocated metric space. Let
where
is a Lebesgue integrable mapping which is summable, non-negative and such that
1. The pairs
2. The pairs
if T(X) is closed then
1) the maps A and T have a coincidence point.
2) the maps A and S have a coincidence point.
3) the maps A, S and T have an unique common fixed point.
If we take T = S and B = A in Theorem (1) then we obtain the following corollary
Corollary 3 Let (X, d) be a dislocated metric space. Let
where
is a Lebesgue integrable mapping which is summable, non-negative and such that
1. The pairs
2. The pair
if S(X) is closed then maps A and S have a unique common fixed point.
If we put S = T = I (Identity map) then we obtain the following corollary.
Corollary 4 Let (X, d) be a dislocated metric space. Let
where
is a Lebesgue integrable mapping which is summable, non-negative and such that
if the pair (A, B) satisfy E.A. property and are weakly compatible then the maps A and B have an unique common fixed point.
Remarks: Our result extends the result of [
Now we establish a fixed point theorem which generalize Theorem (2) of B. E. Rhoades [
Theorem 2 Let (X, d) be a complete dislocated metric space,
that for each
where
and
is a lebesgue integrable mapping which is summable , non negative and such that
for each
Proof. Let
now by (19)
But,
and similarly we can obtain,
Hence
Therefore by (21)
Similarly we can obtain,
Hence
Now taking limit as
by (20)
Now we claim that
If possible let
Using (19) we have,
Now using (22)
Since by triangle inequality and (23)
Hence
and
Similarly
Hence, from (20), (23), (24), (25), (26), (27) and (28)
which is a contradiction. Hence
From the condition (18)
Now taking limit as
which implies
So from the relation (20) we obtain
Uniqueness:
Let z and w two fixed point fixed points of the function f.
Applying condition (19) we obtain
If maximum of the given expression in the set is
which is a contradiction, since
P. M. Geethu Krishnan,A. Sobha,Mini P. Balakrishnan,R. Sumangala,Dinesh Panthi,Panda Sumati Kumari, (2016) Some Integral Type Fixed Point Theorems in Dislocated Metric Space. American Journal of Computational Mathematics,06,88-97. doi: 10.4236/ajcm.2016.62010