We prove the existence and uniqueness of a common fixed point of compatible mappings of integral type in modular metric spaces.
MSC:47H09, 47H10, 46A80.
Hinweise
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
1 Introduction and preliminaries
The metric fixed point theory is very important and useful in mathematics. It can be applied in various branches of mathematics, variational inequalities optimization and approximation theory. In 1976, Jungck [1] proved a common fixed point theorem for commuting maps generalizing the Banach contraction mapping principle. This result was further generalized and extended in various ways by many authors. On the other hand, Sessa [2] defined weak commutativity as follows.
Let be a metric space, the self-mappings f, g are said to be weakly commuting if for all . Further, Jungck [3] introduced more generalized commutativity, the so-called compatibility, which is more general than weak commutativity. Let f, g be self-mappings of a metric space . The mappings f and g are said to be compatible if , whenever is a sequence in X such that for some . Clearly, weakly commuting mappings are compatible, but neither implication is reversible. Let with the usual metric. We define mappings f and g on X by
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Let be a sequence in X with , then and . Thus the pair is compatible on X. In [4] Branciari obtained a fixed point theorem for a single mapping satisfying an analogue of the Banach contraction principle for integral type inequality (see also [5‐7]). Vijayaraju et al. [8] proved the existence of the unique common fixed point theorem for a pair of maps satisfying a general contractive condition of integral type. Recently, Razani and Moradi [9] proved the common fixed point theorem of integral type in modular spaces. The purpose of this paper is to generalize and improve Jungck’s fixed point theorem [3] and Branciari’s result [4] to compatible maps in metric modular spaces. The notions of a metric modular on an arbitrary set and the corresponding modular space, more general than a metric space, were introduced and studied recently by Chistyakov [10]. In the sequel, we recall some basic concepts about modular metric spaces.
Definition 1.1 A function is said to be a metric modular on X if it satisfies the following three axioms:
(i)
given , for all if and only if ;
(ii)
for all and ;
(iii)
for all and .
If, instead of (i), we have only the condition (i)′ for all and , then ω is said to be a (metric) pseudo-modular on X. The main property of a (pseudo)modular ω on a set X is the following: given , the function is non-increasing on . In fact, if , then (iii), (i)′ and (ii) imply
for all . If follows that at each point the right limit and the left limit exist in and the following two inequalities hold:
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for all . We know that if , the set is a metric space, called a modular space, whose metric is given by
for all . We know that (see [10]) if X is a real linear space, and
for all and , then ρ is modular on X if and only if ω is metric modular on X.
Example 1.2 The following indexed objects ω are simple examples of (pseudo)modulars on a set X. Let and , we have:
(a)
if , if ;
and if is a (pseudo)metric space with (pseudo)metric d, then we also have:
(b)
, where is a nondecreasing function;
(c)
if , and if ;
(d)
if , and if .
Definition 1.3 Let be a modular metric space.
(1)
The sequence in is said to be convergent to if as for all .
(2)
The sequence in is said to be Cauchy to if as for all .
(3)
A subset C of is said to be closed if the limit of a convergent sequence of C always belongs to C.
(4)
A subset C of is said to be complete if any Cauchy sequence of C is a convergent sequence and its limit is in C.
(5)
A subset C of is said to be bounded if for all , .
2 A common fixed point theorem for contractive condition maps
Here, the existence of a common fixed point for ω-compatible mappings satisfying a contractive condition of integral type in modular metric spaces is studied. We recall the following definition.
Definition 2.1 Let be a modular metric space induced by metric modular ω. Two self-mappings T, h of are called ω-compatible if , whenever is a sequence in such that and for some point and for .
Theorem 2.2Letbe a complete modular metric space. Suppose that , andare twoω-compatible mappings such thatand
(2.1)
for someand for , whereis a Lebesgue integrable function which is summable, nonnegative and for all ,
(2.2)
If one ofTorhis continuous, then there exists a unique common fixed point ofTandh.
Proof Let x be an arbitrary point of and generate inductively the sequence as follows: for each n and , that is possible as . For each integer and for all , (2.1) shows that
By the principle of mathematical induction, we can easily show that
which, upon taking the limit as , yields
Hence (2.2) implies that
We now show that is Cauchy. So, for all , there exists such that for all with and . Without loss of generality, suppose and . Observe that for , there exists such that
for all . We thus obtain
for all . This implies that is a Cauchy sequence. Since is complete, there exists such that as . If T is continuous, then and . By the ω-compatibility of , we have as for . Moreover, since .
In the sequel, we prove that z is a common fixed point of T and h. By (2.1), we get
Taking the limit as yields
which implies that for . Hence . It follows from that there exists a point such that . By (2.1), we get
Taking the limit as yields
and so
Hence and also (see [11]). In addition, if one considers h to be continuous (instead of T), then by a similar argument (as above), one can prove .
Finally, suppose that z and ω are two arbitrary common fixed points of T and h. Then we have
which implies that for and hence . □
The following theorem is another version of Theorem 2.2 when , by adding the restriction that , where B is a closed and bounded subset of .
Theorem 2.3Letbe a complete modular metric space, and letBbe a closed and bounded subset of . Suppose thatare twoω-compatible mappings such thatand
(2.3)
for alland for , wherewith , andis a Lebesgue integrable function which is summable, nonnegative and for all , . If one ofTorhis continuous, thenTandhhave a unique common fixed point.
Proof Let and . Let be the sequence generated in the proof of Theorem 2.2. Then
for . Since B is bounded,
which implies that . Therefore, is Cauchy. Since is complete and B is closed, there exists such that . If T is continuous, then and . Then, by ω-compatibility of , we have as for . Moreover, . Next, we prove that z is a fixed point of T. It follows from (2.3) that
Taking the limit as yields
So for and hence . Since , there exists a point such that , and
Taking the limit as yields
Hence and also (see [11]). In addition, if one considers h to be continuous (instead of T), then by a similar argument (as above), one can prove .
Let z and ω be two arbitrary common fixed points of T and h. Then
which implies that for and hence . □
3 A common fixed point theorem for quasi-contraction maps
In this section, we prove Theorem 2.2 for a quasi-contraction map of integral type. To this end, we present the following definition.
Definition 3.1 Two self-mappings of a modular metric space are -generalized contractions of integral type if there exist and with such that
(3.1)
where , and is a Lebesgue integrable function which is summable, nonnegative and for all , , and .
Theorem 3.2Letbe a complete modular metric space. Suppose thatTandhare -generalized contractions of integral type self-maps ofand . If one ofTorhis continuous, thenTandhhave a unique common fixed point.
Proof Choose . Let x be an arbitrary point of and generate inductively the sequence as follows: and . We thus obtain
for , where
It follows from that
Moreover,
Then
and
Continuing this process, we get
So as n tends to infinity. Suppose . Since is a decreasing function, one may write , whenever . Taking the limit from both sides of this inequality shows that for and . Thus we have for any . Now, we show that is Cauchy. Since for , for , there exists such that for all with and . Without loss of generality, suppose and . Observe that for , there exists such that
for all . Now we have
for all . This implies is a Cauchy sequence. Since is complete, there exists such that as . Next we prove that z is a fixed point of T. If T is continuous, then and . By the ω-compatibility of , we have as for . Moreover, since . Note that
where
Taking the limit as , we get
and so . Since , there exists a point such that . We have
and
Taking the limit as , we get
It follows that and also (see [11]). Moreover, if h is continuous (instead of T), then by a similar argument (as above), we can prove . Let z and ω be two arbitrary fixed points of T and h. Then
Therefore,
which implies that . □
4 Generalization
Here, we extend the results of the last section. We need a general contractive inequality of integral type. Let be a set of nonnegative real numbers and consider (∗) as a nondecreasing and right-continuous function such that for any .
To prove the next theorem, we need the following lemma [12].
Lemma 4.1Let . if only if , wheredenotes thek-times repeated composition ofϕwith itself.
Next, we prove a modified version of Theorem 2.2.
Theorem 4.2Letbe a complete modular metric space. Suppose that , andare twoω-compatible mappings such thatand
whereϕis a function satisfying the property (∗), andis a Lebesgue integrable mapping which is summable, nonnegative and for all , and . If one ofTorhis continuous, thenTandhhave a unique common fixed point.
Proof Let x be an arbitrary point of and generate inductively the sequence as follows: and , that is possible as ,
for each integer and . By the principle of mathematical induction, we can easily see that
Taking the limit as , we obtain, by Lemma 4.1,
Using the same method as in the proof of Theorem 2.2, we show that T and h have a unique common fixed point. □
Applying the method of proof of Theorem 3.2, we get the following result.
Theorem 4.3Letbe a complete modular metric space. Suppose , andsuch that
where , ϕis a function satisfying the property (∗) and . If one ofTorhis continuous, then there exists a unique common fixed point ofhandT.
Proof The proof is similar to the proof of Theorem 3.2. □
Now we provide examples to validate and illustrate Theorems 2.2 and 3.2.
Example 4.4 Let and for . Define the mapping by
Then all the hypotheses of Theorem 2.2 are satisfied with for and , , .
Example 4.5 Let and for . Define the mapping by
Then all the hypotheses of Theorem 3.2 are satisfied with for and , . See [4] for details.
5 Application
In the section, we assume that , , ℕ denotes the set of all positive integers, ‘opt’ stands for ‘sup’ or ‘inf’, Y is a Banach space and is an ω-complete space. Suppose that , and denotes the complete space of all bounded real-valued functions on S with the norm
and such that φ is Lebesgue integrable, summable on each compact subset of and for each . We prove the solvability of the functional equations
(5.1)
for all in . First, we recall the following lemma [13].
LetEbe a set, and letbe mappings. Ifandare bounded, then
Theorem 5.2Let , , , . Suppose thatuandHare bounded such that
(5.2)
for all , and someand . Then the functional equation (5.1) has a unique solutionandconverges towfor each , where the mappingAis defined by
(5.3)
Proof By boundedness of u and H, there exists such that
(5.4)
It is easy to show that A is a self-mapping in by (5.3), (5.4) and Lemma 5.1. Using [[14], Theorem 12.34] and , we conclude that for each , there exists such that
(5.5)
where denotes the Lebesgue measure of C.
Let , . Suppose that . Clearly, for , (5.3) implies that there exist satisfying
(5.6)
(5.7)
(5.8)
(5.9)
Put , , , .
From (5.6) and (5.9), it follows that
Similarly, from (5.7) and (5.8), we get
So
Then
(5.10)
for each .
Similarly, we infer that (5.10) holds also for . Combining (5.2), (5.5) and (5.10) yields
which means that
for each . Letting in the above inequality, we deduce that
Thus Theorem 5.2 follows from Theorem 2.2. This completes the proof. □
Acknowledgements
The authors are grateful to the two anonymous reviewers for their valuable comments and suggestions.
Open Access
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Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.