Abstract
Two common fixed theorems for weakly compatible mappings satisfying general contractive conditions of integral type in metric spaces are proved and an illustrative example is provided. The results obtained in this paper substantially extend and improve several previous results, particularly Theorem 2.1 of Branciari (Int. J. Math. Math. Sci. 29(9):531-536, 2002), Theorem 2 of Rhoades (Int. J. Math. Math. Sci. 2003(63):4007-4013, 2003) and Theorem 2 of Vijayaraju et al. (Int. J. Math. Math. Sci. 2005(15):2359-2364, 2005). A nontrivial example with uncountably many points is also provided to support the results presented herein.
MSC:54H25.
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1 Introduction and preliminaries
In 2002, Branciari [1] introduced the notion of contractive mappings of integral type in metric spaces and proved the following fixed point theorem for the contractive mapping of integral type, which is a nice generalization of the Banach contraction principle.
Theorem 1.1 ([1])
Let be a complete metric space, , and let be a mapping such that
where is a Lebesgue integrable mapping which is summable on each compact subset of and such that for all ,
Then f has a unique fixed point such that for each , .
Afterward, the researchers [2–17] and others extended the result to more general contractive conditions of integral type. In particular, Rhoades [13] proved the following extension of Theorem 1.1.
Theorem 1.2 ([13])
Let be a complete metric space, , be a mapping such that
where
is a Lebesgue integrable mapping which is summable on each compact subset of and such that for all ,
Then f has a unique fixed point and, for each , .
Vijayaraju et al. [17] extended further Theorems 1.1 and 1.2 from a single mapping to a pair of mappings. Using a rational expression for a contractive condition of integral type, Vetro [16] extended also Theorem 1.1 and proved the following common fixed point theorem for weakly compatible mappings.
Theorem 1.3 ([16])
Let be a metric space and let A, B, S and T be self-mappings of X with and such that
where
, , and is a Lebesgue integrable mapping on each compact subset of and such that for all ,
Suppose that one of , , and is a complete subset of X and the pairs and are weakly compatible. Then A, B, S and T have a unique common fixed point in X.
Motivated and inspired by the results in [1–17], in this paper we introduce more general contractive mappings of integral type, which include the contractive mappings of integral type in [1, 4, 13, 16, 17] as special cases, and we establish the existence and uniqueness of common fixed points for these contractive mappings of integral type with weak compatibility. Our results extend, improve and unify the corresponding results in [1, 4, 13, 16, 17]. A nontrivial example with uncountably many points is also provided to support the results presented herein.
Throughout this paper, we assume that , , , where ℕ denotes the set of all positive integers and
Φ = { satisfies that φ is Lebesgue integrable, summable on each compact subset of and for each },
Ψ = { is upper semi-continuous on , and for each },
= { is nondecreasing on , and for each }.
Recall that a pair of self-mappings f and g in a metric space are said to be weakly compatible if for all the equality implies .
Lemma 1.4 ([9])
Let and be a nonnegative sequence. Then
if and only if .
2 Common fixed point theorems
Now we show two common fixed point theorems for four contractive mappings of integral type in metric spaces.
Theorem 2.1 Let A, B, S and T be self-mappings of a metric space such that
(C1) and ;
(C2) the pairs and are weakly compatible;
(C3) one of , , and is a complete subset of X and
where is in and
Then A, B, S and T have a unique common fixed point in X.
Proof Let . It follows from (C1) that there exist two sequences and in X satisfying
Put for each .
Firstly we show that A, B, S and T have at most a common fixed point in X. Suppose that u and v are two different common fixed points of A, B, S and T in X. It follows from (2.1), (2.2) and that
and
which is a contradiction. Hence A, B, S and T have at most a common fixed point in X.
Secondly we show that A, B, S and T have a common fixed point if there exist satisfying
Assume that (2.4) holds for some . Put . Note that (C2) implies that
Suppose that . In view of (2.1), (2.2), (2.4), (2.5) and , we infer that
and
which is impossible. Consequently, . Similarly we conclude that . That is, c is a common fixed point of A, B, S and T.
Thirdly we show that (2.4) holds for some . In order to prove (2.4), we have to consider three possible cases as follows.
Case 1. There exists satisfying . We claim that . Otherwise . Using (2.1)-(2.3) and , we deduce that
and
which is a contradiction. Hence . It follows that
Put and . It is easy to see that (2.4) holds and is a common fixed point of A, B, S and T.
Case 2. There exists satisfying . As in the proof of Case 1, we infer similarly that (2.4) holds for and , and is a common fixed point of A, B, S and T.
Case 3. for all . Now we claim that for all . Suppose that for some . By virtue of (2.1), (2.2) and , we arrive at
and
which is absurd. Hence for each . As in the proofs of (2.6) and (2.7), we infer similarly that for all . Consequently, is a nonincreasing positive sequence, which means that there exists a constant with
Suppose that . Making use of (2.1), (2.2), (2.6), (2.8) and and Lemma 1.4, we get that
which is a contradiction. Hence . That is,
In order to prove that is a Cauchy sequence, by (2.9) we need only to prove that is a Cauchy sequence. Suppose that is not a Cauchy sequence. It follows that there exists such that for each even integer 2k there are even integers , with and
For every even integer 2k, let be the least even integer exceeding satisfying (2.10). It follows that
Note that
In terms of (2.9)-(2.12), we know that
In light of (2.1), (2.2), (2.9), (2.13), and Lemma 1.4, we deduce that
and
which is a contradiction. Therefore is a Cauchy sequence.
Assume that is complete. Notice that , which implies that converges to a point . Obviously . Put . It follows that . Suppose that . In view of (2.1)-(2.3), , Lemma 1.4 and , we infer that
and
which is a contradiction. Therefore, , which together with (C1) means that . Put , that is, . Suppose that . By means of (2.1), (2.2) and , we get that
and
which is impossible. That is, . Hence (2.4) holds.
Assume that is complete. Notice that , which implies that converges to a point . Obviously . Put . It follows that . Observe that , which implies that there exists with . As in the proof of completeness of , we infer that (2.4) holds. Similarly we conclude that (2.4) holds if one of and is complete. This completes the proof. □
As in the proof of Theorem 2.1 we have the following result and omit its proof.
Theorem 2.2 Let A, B, S and T be self-mappings of a metric space satisfying (C1)-(C3) and
where is in and is defined by (2.2). Then A, B, S and T have a unique common fixed point in X.
Remark 2.3 Theorems 2.1 and 2.2 extend, improve and unify Theorem 2.1 in [1, 4], Theorem 2 in [13, 16, 17] and Corollary 3 in [17]. The following example reveals that Theorem 2.2 is both an indeed generalization of Theorem 2.1 in [1], Theorem 2 in [13, 17] and Corollary 3 in [17], and different from Theorems 3.1-3.3 in [9].
Example 2.4 Let be endowed with the Euclidean metric for all . Let be defined by
Now we claim that Theorem 2 and Corollary 3 in [17] cannot be used to prove the existence of common fixed points of the mappings S and T in X, and Theorem 2 in [13], Theorem 2.1 in [1] and Theorems 3.1-3.3 in [9] are useless in proving the existence of fixed points of the mapping S in X.
Suppose that there exists satisfying the condition of Theorem 2 in [17], that is,
where
Put and . It follows from (2.15), (2.16) and that
and
which is impossible.
Suppose that there exist and satisfying the condition of Corollary 3 in [17], that is,
where
Take and . It follows from (2.17), (2.18), and that
and
which is a contradiction.
Suppose that there exist and satisfying the condition of Theorem 2 in [13], that is,
where
Put and . It follows from (2.19), (2.20), and that
and
which is absurd. Observe that Theorem 2 in [13] generalizes Theorem 2.1 in [1], hence Theorem 2.1 in [1] cannot be used to prove the existence of fixed points of S in X.
Suppose that there exists satisfying the condition of Theorem 3.1 in [9], that is,
where
Put and . It follows from (2.21), (2.22) and that
which is impossible.
Suppose that there exists satisfying the condition of Theorem 3.2 in [9], that is,
where
Put and . It follows from (2.23), (2.24) and that
which is a contradiction.
Suppose that there exists satisfying the condition of Theorem 3.3 in [9], that is,
where
Put and . It follows from (2.25), (2.26) and that
which is impossible.
Next we prove, by using Theorem 2.2, that the mappings A, B, S and T have a unique common fixed point in X, where are defined by
Define two functions by
It is easy to see that (C1), (C2) and (C3) hold. Let . In order to verify (2.14), we have to consider two possible cases as follows.
Case 1. . It is clear that
Case 2. . Note that ψ is nondecreasing on ,
and
Hence (2.14) holds. That is, the conditions of Theorem 2.2 are satisfied. Consequently, Theorem 2.2 implies that A, B, S and T have a unique common fixed point .
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Acknowledgements
This research was supported by the Science Research Foundation of Educational Department of Liaoning Province (L2012380) and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT & Future Planning (2013R1A1A2057665).
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Liu, Z., Han, Y., Kang, S.M. et al. Common fixed point theorems for weakly compatible mappings satisfying contractive conditions of integral type. Fixed Point Theory Appl 2014, 132 (2014). https://doi.org/10.1186/1687-1812-2014-132
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DOI: https://doi.org/10.1186/1687-1812-2014-132