Abstract
Motivated by Abdeljawad (Fixed Point Theory Appl. 2013:19, 2013), we establish some common fixed point theorems for three and four self-mappings satisfying generalized Meir-Keeler α-contraction in metric spaces. As a consequence, the results of Rao and Rao (Indian J. Pure Appl. Math. 16(1):1249-1262, 1985), Jungck (Int. J. Math. Math. Sci. 9(4):771-779, 1986), and Abdeljawad itself are generalized, extended and improved. Sufficient examples are given to support our main results.
MSC:47H10, 54H25.
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1 Introduction and preliminaries
The Meir-Keeler contractive condition [1] is one of the interesting aspects to study metrical fixed point theory, that is, for given , there exists a such that
This contraction has further been generalized and studied by various authors (see [2–15]). Very recently, Abdeljawad [16] (see also [17]) established some fixed point results for α-contractive-type maps (due to Samet et al. [18]) to Meir-Keeler versions for single and a pair of maps. In this article, we prove some common fixed point theorems for three and four self-mappings satisfying generalized Meir-Keeler α-contractions. Thus, we provide an affirmative answer to the question of Abdeljawad (see [16], Remark 17).
Let us recall some definitions, which we will use in our main results.
Let be self-mappings of a set X, and let be a mapping, then the mapping f is called α-admissible if
and the pair is called α-admissible if
Let f and g () be two self-mappings defined on a metric space , then f is called g-absorbing if there exists some real number such that for all x in X. Analogously, g will be called f-absorbing if there exists some real number such that for all x in X. The pair of self-maps will be called absorbing if it is both g-absorbing as well as f-absorbing. In particular, if we take g to be the identity map on X, then f is trivially I-absorbing. Similarly, I is also f-absorbing in respect to f.
Definition 1.3 (cf. [21])
Two self-mappings f and g of a metric space are called reciprocally continuous if and only if and whenever is a sequence in X such that for some .
2 Main results
We begin with the following definitions.
Definition 2.1 Let be three self-mappings of a non-empty set X, and let be a mapping, then the pair is called α-admissible with respect to T (in short, is -admissible) if for all ,
Definition 2.2 Let be four self-mappings of a non-empty set X, and let be a mapping, then the pair is called α-admissible with respect to S and T (in short, is -admissible) if for all ,
Clearly, if (identity map), then the definitions above imply Definition 1.1.
In order to extend and improve the result contained in [16] for three self-mappings, we now introduce the concept of generalized Meir-Keeler -contractive mappings as follows.
Definition 2.3 Let be a metric space, and are self-mappings. Then we say that the pair is a generalized Meir-Keeler -contractive pair of type (, respectively) if given an , there exists a such that
where
and
Definition 2.4 Let f, g, and T be three self-mappings on a metric space such that . If for a point , there exists a sequence such that , , , then is called the orbit for at . The space is called -orbitally complete at iff every Cauchy sequence in converges to a point in X. X is called -orbitally complete if it is so at every .
Our first result is the following.
Theorem 2.1 Let be an -orbitally complete metric space. Suppose that is generalized Meir-Keeler -contractive pair of type and satisfies the following conditions:
-
(i)
is -admissible;
-
(ii)
there exists such that ;
-
(iii)
on the -orbit of , we have for all n even and odd.
Then is a Cauchy sequence. Moreover, if
-
(iv)
for all n, and implies that for all n;
-
(v)
one of the pairs and is absorbing as well as reciprocal continuous.
Then f, g, and T have a common fixed point.
Proof Let such that . Define the sequences and in X given by the rule
Since is -admissible, we have
which gives
Again by (i), we have
which gives
Inductively, we have
The fact that is generalized Meir-Keeler -contractive implies that
Now, to obtain a common fixed point of f, g, and T, we take the following steps.
Step 1: We show that there exists a point such that as . For this, first, we claim that is a Cauchy sequence. Two cases arise: either for some n or for each n.
Case I: Suppose that for some n. We first assume that n is even, i.e., but , then by (6),
which is a contradiction. Hence . By proceeding in this way, we obtain for all . Similar is the case when n is odd. Thus, we conclude that is a Cauchy sequence.
Case II: Suppose that for all integers n. Applying (6), we have
Similarly, it can be shown that
Thus, is strictly decreasing sequence of positive real numbers, and, therefore, converges to a limit . If possible, suppose that . Then given , there exists a positive integer such that
where . So by Eqs. (5) and (6), we have
that is, , which is a contradiction. Hence
We now show that is a Cauchy sequence.
Suppose that it is not. Then there exists an such that for each positive integer m, n with , we have . Choose a number δ, for which contractive condition (4) is satisfied. Since , there exists integer such that for all . With this choice of N, pick m, n with such that
in which it is clear that . Otherwise, we have
which contradicts (9). Also from (9), it follows that
Without loss of generality, we may assume that n is even. Suppose that
then
which is a contradiction to (9). So we have
Similarly, suppose that
then
which is a contradiction to (9). So we have
Thus, there exists the smallest odd integer such that
and hence,
Now,
Thus, there exists an odd integer such that
Since we have
So, using (4) and assumption (iii), we get
that is, . But then
which contradicts (11). Therefore, is a Cauchy sequence. Since X is -orbitally complete, so there exists a point such that as . Consequently, and .
Step 2: We show that z is common fixed point of . In view of assumption (v), without loss of generality, let the pair be absorbing and reciprocal continuous. Then the reciprocal continuity of f and T implies that
Since T is f-absorbing, so there exists an such that
Letting , we get . Similarly, since f is T-absorbing, so we have
letting , we get . By the uniqueness of the limit, we have .
Now, suppose that , then by assumption (iv) and Eq. (6), we have
Letting , we get , which implies that . Thus, z is a common fixed point of f, g, and T. This completes the proof of the theorem. □
By putting and (identity map) in Theorem 2.1, we get the following result as a corollary.
Corollary 2.1 Let be an f-orbitally complete metric space, where f is a self-mapping on X. Also, let be a mapping. Assume the following:
-
(i)
f is α-admissible;
-
(ii)
there exists an such that ;
-
(iii)
for given , there exists a such that
-
(iv)
on the f-orbit of , we have for all n even and odd.
Then, f has a fixed point in the f-orbit of , or f has a fixed point z and .
Example 2.1 Let be endowed with the standard metric for all . Define by
Then f is not a Meir-Keeler contraction. To see this consider , , and , then for any , we have , but . However, f is a generalized Meir-Keeler α-contraction, where is defined by
Clearly, f has two fixed points, namely and . Notice that .
For the uniqueness of the fixed point of a generalized Meir-Keeler α-contractive mapping, we will consider the following hypothesis.
-
(H)
For all fixed points x and y of , we have .
Theorem 2.2 Adding condition (H) to the hypotheses of Theorem 2.1 (resp., Corollary 2.1), we obtain the uniqueness of the common fixed point of f, g, and T.
Proof Let z be the common fixed point obtained as and u is another common fixed point. Then, (6) and condition (H) yield to
Thus, we reach , and hence . □
The following example illustrates Theorem 2.2.
Example 2.2 Let and d be the usual metric on X. Define as follows:
In this example the mappings f, g, and T do not satisfy the general Meir-Keeler contractive condition. To see this, consider , and , then for any , we have , but . However, f, g, and T satisfy the generalized Meir-Keeler α-contractive condition (4) with the mapping defined by
Also, all the hypotheses of Theorem 2.1 with condition (H) are satisfied, and clearly is our unique common fixed point. Indeed, hypothesis (ii) is satisfied with , and here is a sequence, for which hypotheses (iii) and (iv) are satisfied. Also in view of the sequence , here both pairs and are reciprocal continuous as well as absorbing. Notice that is the point of discontinuity of the mappings g and T.
Theorem 2.3 The conclusion of Theorem 2.1 remains true if the assumption (v) of Theorem 2.1 is replaced by one of the following conditions:
-
(a)
for all with right-hand side positive.
-
(b)
for all with right-hand side positive.
Proof In view of Theorem 2.1, we have that is a Cauchy sequence, and as , and, consequently, and also converge to z as .
Clearly, for infinitely many n. We can as well assume that for all n.
If (a) holds, then
Letting , we get , i.e., . If (b) holds, then also .
Now, suppose that . Since , so by assumption (iv) and Eq. (6), we have
letting , we get , which implies that .
Now, let , then again by the process above, we have
letting , we get , which implies that . Thus, z is the common fixed point of f, g, and T. □
The following example demonstrates Theorem 2.3.
Example 2.3 Let and d be the usual metric on X. Define as follows:
Here the mappings f, g, and T satisfy all the conditions of Theorem 2.3 with the mapping defined by
Clearly, none of the pairs and are reciprocal continuous. To see this consider the sequence , then , but . Therefore, is not reciprocal continuous. To see that is not reciprocal continuous, one can consider the sequence . Here, the involved mappings satisfy condition (a) of Theorem 2.3, and they have the unique common fixed .
Remark 2.1 Theorem 2.3 generalizes and extends Theorem 1.2 of Rao and Rao [22].
Theorem 2.4 Theorem 2.1 remains true if we replace by and condition (iv) by the following (iv′):
(iv′) for all n and implies that for all n, where .
Proof The proof of follows from Theorem 2.1. Now, suppose that , then by the help of condition (iv’), we have
By letting , we conclude that , and hence . Thus, z is a common fixed point of f, g, and T. □
Example 2.2 above also satisfies Theorem 2.4.
Remark 2.2 Theorem 2.4 generalizes and extends Theorem 1.3 of Rao and Rao [22].
By taking (identity map) in Theorem 2.4, we derive the following result as a corollary.
Corollary 2.2 Let be an -orbitally complete metric space, where f, g are self-mappings of X. Also, let be a mapping. Assume the following:
-
(i)
is α-admissible, and there exists an such that ;
-
(ii)
for given , there exists a such that
where
-
(iii)
on the -orbit of , we have for all n even and odd;
-
(iv)
for n, and implies that for all n, where .
Then, the pair has a common fixed point provided it is absorbing as well as reciprocal continuous.
Remark 2.3 Corollary 2.2 improves Theorem 8 contained in [16].
The next result is a common fixed point theorem for four self-mappings.
Theorem 2.5 Let f, g, S, and T be four self-mappings on a complete metric space such that and , and they satisfy the following conditions:
-
(i)
the pair is -admissible;
-
(ii)
there exists a point such that ;
-
(iii)
for given , there exists a such that
(12)
where
-
(iv)
there exists a sequence in X such that for all n even and odd;
Then f, g, S, and T have a common fixed point provided both the pair and are absorbing as well as reciprocal continuous.
Proof Let such that . Define sequences and in X as
This can be done since and .
Since is -admissible, we have
which gives
Again by (i), we have
which gives
Inductively, we obtain
that is, , when n is odd and when n is even.
By assumption (iii), we have
Now, we claim that is a Cauchy sequence.
Case I: If for some n. We first assume that n is odd, i.e., and suppose that , then by applying (13) and (14), we get
a contradiction. Hence . By proceeding in this manner, we obtain for all . Similarly, when we assume n as even, then we obtain for all , and so is a Cauchy sequence.
Case II: If for each n. Applying (13) and (14), we get
Similarly, we obtain . Thus, is a strictly decreasing sequence of positive numbers, and, therefore, tends to a limit . If possible, suppose that . Then given , there exists a positive integer N such that for each , we have
where . Then by applying (14), we have
that is, , which is a contradiction, and hence,
Now, we show that is a Cauchy sequence. Suppose that it is not, then there exists an such that for each integer N, there exist integers such that . Choose a number δ, , for which contractive condition (12) is satisfied. By virtue of (16), there exists an integer N such that for all . With this choice of N, pick integers such that
in which it is clear that . Also from (17), it follows that .
If not, then
which is a contradiction. Without loss of generality, we can assume that n is even. From (17), there exists the smallest odd integer such that
and hence . So we have
Thus, there exists an odd integer such that
Therefore, we have
so that by (12) and assumption (iv), we get
i.e., . But then
which contradicts (19). Therefore, is a Cauchy sequence. By the completeness of X, there exists a such that as and, consequentially, , , and as .
Since the pair is reciprocal continuous and absorbing, so by reciprocal continuity, we have and as . By absorbing property, there is an such that and , which letting gives and . Thus, we have . Similarly, the absorbing and reciprocal continuity of the pair provides us . Thus, z is a common fixed point of f, g, S, and T. □
Theorem 2.6 Adding the condition (H-2): For all common fixed points x and y of f, g, S, and T, , to the hypotheses of Theorem 2.5, the uniqueness of the fixed point is obtained.
Remark 2.4 Theorem 2.6 generalizes, extends and improves the results of Jungck (Theorem 3.1, [8]), Cho et al. (Theorem 3.2, [4]) and Rao and Rao [22].
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Acknowledgements
The authors would like to express their thanks to Prof. M. A Khamsi and the referees for their helpful comments and suggestions. The first author is thankful to S. V. National Institute of Technology, Surat, India for awarding Senior Research Fellow. The third author gratefully acknowledges the support from the CSIR, Govt. of India, Grant No.-25(0215)/13/EMR-II.
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Patel, D.K., Abdeljawad, T. & Gopal, D. Common fixed points of generalized Meir-Keeler α-contractions. Fixed Point Theory Appl 2013, 260 (2013). https://doi.org/10.1186/1687-1812-2013-260
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DOI: https://doi.org/10.1186/1687-1812-2013-260