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A common fixed point theorem for two weakly compatible pairs in G-metric spaces using the property E.A
Fixed Point Theory and Applications volume 2013, Article number: 41 (2013)
Abstract
In view of the fact that the fixed point theory provides an efficient tool in many fields of pure and applied sciences, we use the notion of the property E.A to prove a common fixed point theorem for weakly compatible mappings. The presented results are applied to obtain the solution of an integral equation and the bounded solution of a functional equation arising in dynamic programming.
MSC:47H10, 54H25.
1 Introduction
Inspired by the fact that the metric fixed point theory provides an efficient tool in many fields of pure and applied sciences, many authors investigated the possibility to generalize the notion of a metric space. In this direction, Gahler [1, 2] introduced the notion of a 2-metric space, while Dhage [3] introduced the concept of a D-metric space. Later on, Mustafa and Sims [4] showed that most of the results concerning Dhage’s D-metric spaces are invalid. Therefore, they introduced a new notion of a generalized metric space, called G-metric space. After then, many authors studied fixed and common fixed points in generalized metric spaces; see [4–15].
Here, we give preliminaries and basic definitions which are helpful in the sequel. First, we introduce the concepts of a G-metric and a G-metric space.
Definition 1.1 [4]
Let X be a nonempty set and be a function satisfying the following properties:
-
(G1)
if ;
-
(G2)
for all with ;
-
(G3)
for all with ;
-
(G4)
(symmetry in all three variables);
-
(G5)
for all (rectangle inequality).
Then the function G is called a generalized metric or, more specifically, a G-metric on X, and the pair is called a G-metric space.
Definition 1.2 A G-metric space is said to be symmetric if for all .
Example 1.3 Let and be defined by , , . It is easy to show that the function G satisfies all properties of Definition 1.1, but for all with . Therefore, G is not symmetric.
Definition 1.4 [4]
Let be a G-metric space, and let be a sequence of points of X. A point is said to be the limit of the sequence if , and we say that the sequence is G-convergent to x or G-converges to x.
Thus, in a G-metric space if for any , there exists such that for all .
Proposition 1.5 [4]
Let be a G-metric space. Then the following are equivalent:
-
(1)
is G-convergent to x;
-
(2)
as ;
-
(3)
as .
Definition 1.6 [4]
Let be a G-metric space. A sequence is called G-Cauchy if for every , there is such that for all ; that is, as .
Proposition 1.7 [4]
Let be a G-metric space. Then the following are equivalent:
-
(1)
the sequence is G-Cauchy;
-
(2)
for every , there is such that for all .
Proposition 1.8 [4]
Let be a G-metric space. Then the function is jointly continuous in all three of its variables.
Definition 1.9 [4]
A G-metric space is called G-complete if every G-Cauchy sequence in is G-convergent in .
Proposition 1.10 [4]
Let be a G-metric space. Then, for any , it follows that
-
(i)
if , then ;
-
(ii)
;
-
(iii)
;
-
(iv)
;
-
(v)
;
-
(vi)
.
An interesting observation is that any G-metric space induces a metric on X given by
Moreover, is G-complete if and only if is complete.
It was observed that in the symmetric case ( is symmetric), many fixed point theorems on G-metric spaces are particular cases of the existing fixed point theorems in metric spaces. This allows us to readily transport many results from the metric spaces into the G-metric spaces.
On the other hand, by reasoning on the properties of the mappings, the practice of coining weaker forms of commutativity to ensure the existence of a common fixed point for self-mappings on metric spaces is still on. To read more in this direction, we refer to [16] and the references therein. Here, for our further use, we recall only the two fundamental notions of ‘weakly compatible mappings’ and ‘property E.A’; see also [17, 18].
In 1976, Jungck [19] introduced the notion of weakly compatible mappings as follows.
Definition 1.11 Let S and T be two self-mappings of a metric space . Then the pair is said to be weakly compatible if they commute at their coincidence points, that is, if for some , then .
In 2002, Amari and El Moutawakil [20] introduced a new concept of the property E.A in metric spaces to generalize the concept of noncompatible mappings. Then, they proved some common fixed point theorems.
Definition 1.12 Let S and T be two self-mappings of a metric space . Then the pair is said to satisfy the property E.A if there exists a sequence in X such that for some .
Example 1.13 Let . Define by and for all . Consider the sequence in X. Clearly, , and so S and T satisfy the property E.A.
Example 1.14 Let . Define by and for all . Suppose that the property E.A holds. Then there exists a sequence in X such that for some . It follows that and and so, by Definition 1.12, but . Therefore, S and T do not satisfy the property E.A.
In conclusion of this preliminary section, we consider the following set:
Let Φ denote the set of all the functions such that:
-
(1)
φ is non-decreasing;
-
(2)
for all .
If , then it is an easy matter to show that and for all ; see Matkowski [21].
In this paper, by merging the above notions, we prove a common fixed point theorem for two pairs of weakly compatible mappings in a G-metric space. The presented results are applied to obtain the solution of an integral equation and the bounded solution of a functional equation arising in dynamic programming.
2 Main results
The following is the main result of this section.
Theorem 2.1 Let be a G-metric space and be four self-mappings such that:
-
(i)
and ;
-
(ii)
one of the pairs and satisfies the property E.A;
-
(iii)
for all , , where ;
-
(iv)
one of , , and is a complete subset of X.
Then the pairs and have a coincidence point. Further, if and are weakly compatible, then A, B, S and T have a unique common fixed point in X.
Proof Suppose the pair satisfies the property E.A. Then there exists a sequence in X such that for some . Since , there exists a sequence in X such that . Hence . We will show that . From (iii), we have
Taking the limit as (upper limit) and using the fact that is continuous at , we get
and so . Thus, we have . Suppose that is a complete subset of X. Then for some . Now, we will show that . From (iii), we have
Taking the limit as , by the property of φ, we get
which implies . Therefore, u is a coincidence point of the pair . The weak compatibility of A and S implies that and hence . Since , there exists such that . We claim that . Suppose not, from (iii) and using the fact that , we have
This implies that and hence . It follows that also the pair has a coincidence point. Thus, we have .
Now, if B and T are weakly compatible, then we obtain and show that Au is a common fixed point of A, B, S and T. For and , from (iii) and the property of φ, we get
which implies . Therefore, is a common fixed point of A and S. Similarly, one can prove that Bv is a common fixed point of B and T. Since , we deduce that Au is a common fixed point of A, B, S and T. Now, we have only to show that the common fixed point is unique. Suppose to the contrary that w and z, with , are two common fixed points of A, B, S and T. Then, from (iii) and the property of φ, we have
that is a contradiction and so must be . Therefore, A, B, S and T have a unique common fixed point. Clearly, proceeding on the foregoing lines, one can obtain the same conclusion in case (instead of ) one of , and is a complete subset of X, and in case (instead of ) satisfies the property E.A. □
If we assume in above Theorem 2.1, we deduce the following result involving three self-mappings.
Corollary 2.2 Let be a G-metric space and be three mappings such that:
-
(i)
and ;
-
(ii)
one of the pairs and satisfies the property E.A;
-
(iii)
for all , , where ;
-
(iv)
one of , and is a complete subset of X.
Then the pairs and have a coincidence point. Further, if and are weakly compatible, then A, B and S have a unique common fixed point in X.
Example 2.3 Let and be defined by for all . Define also by , and for all and by for all . Clearly, the hypotheses (i) and (iv) of Corollary 2.2 hold trivially. Moreover, the pair satisfies the property E.A. Here we show only that the hypothesis (iii) holds. In fact, for all , we have , , , , and consequently,
Then, by Corollary 2.2, the pairs and have a coincidence point, that is, . Moreover, since and are weakly compatible, then is the unique common fixed point of A, B and S in X.
3 Applications to integral and functional equations
In this section we illustrate two useful applications of our presented results. Firstly, we show how it is possible to obtain the solution of an integral equation by applying Corollary 2.2. Let and be the space of all the real continuous functions defined on Ω. Obviously, this space endowed with the G-metric given by
is a G-complete metric space.
Let and be two continuous functions. Consider an integral equation of the following type:
We will prove the following theorem.
Theorem 3.1 Suppose there exists such that:
-
(i)
for all ;
-
(ii)
, where .
Then integral equation (3.1) has a solution in .
Proof Define and . Now
Thus, all the hypotheses of Corollary 2.2 are satisfied with , the identity mapping on Ω, and for all and . Therefore, there is a unique solution of integral equation (3.1) in . □
Now, we study the existence and uniqueness of the bounded solution of a functional equation using again Corollary 2.2. Here we assume that U and V are Banach spaces, is a state space and is a decision space.
It is well known that the dynamic programming provides useful tools for mathematical optimization and computer programming as well; see [22–24]. In particular, the problem of dynamic programming related to a multistage process reduces to the problem of solving the functional equation
where , , .
Let denote the space of all bounded real-valued functions on W. Clearly, this space endowed with the G-metric given by
is a G-complete metric space.
We will prove the following theorem.
Theorem 3.2 Let and be two bounded functions and let be defined by
for all and . Assume that the following condition holds:
where , and . Then functional equation (3.2) has a unique bounded solution.
Proof Note that is a complete G-metric space. Let ε be an arbitrary positive number, and , then there exist such that
Then from (3.4) and (3.7), it follows easily that
Hence we get
Similarly, from (3.5) and (3.6), we obtain
Therefore, from (3.8) and (3.9), we have
which implies
Since the above inequality is true for any and is taken arbitrary, then we conclude immediately that
Thus, all the hypotheses of Corollary 2.2 are satisfied with and , the identity mapping on . Therefore, functional equation (3.2) has a unique bounded solution. □
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Manro, S., Bhatia, S.S., Kumar, S. et al. A common fixed point theorem for two weakly compatible pairs in G-metric spaces using the property E.A. Fixed Point Theory Appl 2013, 41 (2013). https://doi.org/10.1186/1687-1812-2013-41
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DOI: https://doi.org/10.1186/1687-1812-2013-41