Abstract
Two results involving the existence, uniqueness and iterative approximations of fixed points for two contractive mappings of integral type are proved in complete metric spaces. Two nontrivial examples are included.
MSC:54H25.
Similar content being viewed by others
1 Introduction
In recent years, there has been increasing interest in the study of fixed points and common fixed points of mappings satisfying contractive conditions of integral type, see, for example, [1–14] and the references cited therein. Branciari [4] introduced first the contractive mapping of integral type as follows:
where is a constant, = { satisfies that φ is Lebesgue integrable, summable on each compact subset of and for each } and proved the existence of a fixed point for the mapping in complete metric spaces. Rhoades [10] and Liu et al. [8] extended Branciari’s result and obtained a few fixed point theorems for the contractive mappings of integral type below:
and
where is a constant, and is a function with , . Mongkolkeha and Kumam [9] proved fixed point and common fixed point theorems for ρ-compatible mapping satisfying a generalized weak contraction of integral type in modular spaces. Sintunavarat and Kumam [11, 12] gave common fixed point theorems for single-valued and multi-valued mappings satisfying strict general contractive conditions of integral type.
Inspired and motivated by the results in [1–14], in this paper, we introduce two new classes of contractive mappings of integral type in complete metric spaces and study the existence, uniqueness and iterative approximations of fixed points for the mappings. The results obtained in this paper generalize and improve Theorem 2.1 in [4], Theorem 3.1 in [8] and Theorem 2 in [10]. Two nontrivial examples are constructed.
2 Preliminaries
Throughout this paper, we assume that , , ℕ denotes the set of all positive integers, ,
= { is upper semi-continuous on , and , };
= { is right upper semi-continuous on , and , };
= { is continuous, , , and , for each sequence with };
= { is strictly increasing, , continuous at 0 and for each sequence in with };
= { is in and is left continuous on };
-
(a1)
;
-
(a2)
;
-
(a3)
.
Let T be a mapping from a metric space into itself, and let be a function. Put
and denote the right and left limits of the function ψ at , respectively.
The following lemmas play important roles in this paper.
Lemma 2.1 [8]
Let and be a nonnegative sequence with . Then
Lemma 2.2 [8]
Let and be a nonnegative sequence. Then
if and only if .
3 Fixed point theorems and examples
In this section, we prove two fixed point theorems for two classes of contractive mappings of integral type and display two examples as applications of the theorems.
Theorem 3.1 Let be a complete metric space, and let be a mapping satisfying
where φ, ϕ and ψ satisfy (a1) or (a2). Then T has a unique fixed point and for each .
Proof Let be an arbitrary point in X. Put for each . Assume that for some . It is easy to see that is a fixed point of T, and there is nothing to prove. Assume that for all . From (3.1) and one of (a1) and (a2), we obtain that
which implies that
Note that (3.3) yields that the sequence is positive and strictly decreasing. Thus, there exists a constant with
Suppose that . Taking upper limit in (3.2) and using (3.4), Lemma 2.1 and one of (a1) and (a2), we conclude that
which is absurd, and hence , that is,
which together with one of (a1) and (a2) guarantees that
Now, we show that is a Cauchy sequence. Suppose that is not a Cauchy sequence, which means that there is a constant such that for each positive integer k, there are positive integers and with satisfying
For each positive integer k, let denote the least integer exceeding and satisfying the inequality above. It follows that
Note that
Letting in (3.7) and using (3.5) and (3.6), we conclude that
In view of (3.1), we deduce that
Assume that (a1) holds. Taking upper limit in (3.9) and using (3.8) and Lemma 2.1, we get that
which is a contradiction.
Assume that (a2) holds. In view of (3.8), there exists satisfying
It follows from the inequality above and that
which together with (3.9) and gives that
which ensures that
that is,
which together with (3.6) and (3.8) implies that
Taking upper limit in (3.9) and using Lemma 2.1, (a2) and the equations above, we conclude that
which is a contradiction.
Thus, is a Cauchy sequence. Since is a complete metric space, there exists a point such that . By (3.1), Lemma 2.2 and one of and , we arrive at
that is,
which together with Lemma 2.2 means that
Note that (3.10) and one of and ensure that . Consequently, we conclude immediately that
which gives that .
Next, we show that a is a unique fixed point T in X. Suppose that T has another fixed point . It follows from (3.1) and one of (a1) and (a2) that
which is a contradiction. This completes the proof. □
Remark 3.2 Theorem 3.1 generalizes Theorem 2.1 in [4] and Theorem 3.1 in [8]. The example below is an application of Theorem 3.1.
Example 3.3 Let be endowed with the Euclidean metric . Define and by
It is easy to see that (a2) holds. Put with . To verify (3.1), we need to consider four possible cases as follows.
Case 1. Let . It follows that
Case 2. Let and . Note that and . It follows that
Case 3. Let and . Notice that . It follows that
Case 4. Let and . It follows that
That is, (3.1) holds. Thus, Theorem 3.1 implies that T has a unique fixed point and for each .
Theorem 3.4 Let be a complete metric space, and let be a mapping satisfying
where φ, ϕ and ψ satisfy (a1) or (a3). Then T has a unique fixed point and for each .
Proof Let be an arbitrary point in X. Put for each . Assume that for some . It is easy to see that is a fixed point of T, and there is nothing to prove. Assume that for all . From (3.11) and one of (a1) and (a3), we obtain that
where
Suppose that for some . It follows from (3.12) and (3.13) that
which is a contradiction. Consequently, we deduce that
In view of (3.12), (3.14) and one of (a1) and (a3), we get that
which implies that
which means that there exists a constant with .
Now we show that . Otherwise, . Taking upper limit in (3.15) and using Lemma 2.1 and one of (a1) and (a3), we conclude that
which is impossible. Hence , that is,
which together with one of (a1) and (a3) gives that
Next, we claim that is a Cauchy sequence. Suppose that is not a Cauchy sequence, which means that there is a constant such that for each positive integer k, there are positive integers and with such that
For each positive integer k, let denote the least integer exceeding and satisfying the inequality above. Obviously, (3.6)-(3.8) hold. Note that
Combining (3.8), (3.17) and (3.18), we infer that
In light of (3.11), we deduce that
Assume that (a1) holds. Taking upper limit in (3.20) and using (3.8), (3.19) and Lemma 2.1, we get that
which is a contradiction.
Assume that (a3) holds. Note that (3.19) implies that there exists with
By virtue of (a3) and (3.21), we deduce that
In terms of (3.20), (3.22) and (a3), we get that
which yields that
that is,
which together with (3.6), (3.8) and (3.19) implies that
Taking upper limit in (3.20) and using (3.23), (a3) and Lemma 2.1, we conclude that
which is a contradiction.
Thus, is a Cauchy sequence. It follows from completeness of that there exists a point with .
Next, we show that a is a fixed point of T in X. Suppose that . Notice that
which guarantees that there exists satisfying
Let (a1) hold. In light of (3.11), (3.24) and Lemma 2.1, we infer that
which is a contradiction.
Let (a3) hold. In view of (3.11) and (3.24), we deduce that
which yields that
that is,
which together with (3.25) and means that
which is a contradiction.
Hence T has a fixed point . Finally, we show that a is a unique fixed point of T in X. Suppose that T has another fixed point . It follows from (3.11) and one of (a1) and (a3) that
which is a contradiction. This completes the proof. □
Remark 3.5 Theorem 3.4 extends Theorem 2 in [10]. The following example is an application of Theorem 3.4.
Example 3.6 Let be endowed with the Euclidean metric . Define and by
Clearly, (a1) holds, ϕ and ψ are strictly increasing in . Put with . To prove (3.11), we need to consider three possible cases as follows.
Case 1. Let . It follows that
and
Case 2. Let . Notice that
Suppose that . It follows that
Suppose that . It follows that
Case 3. Let and . It follows that
and
that is, (3.11) holds. Thus, Theorem 3.4 implies that T has a unique fixed point and for each .
References
Aliouche A: A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type. J. Math. Anal. Appl. 2006, 322: 796–802. 10.1016/j.jmaa.2005.09.068
Altun I, Türkoǧlu D, Rhoades BE: Fixed points of weakly compatible maps satisfying a general contractive of integral type. Fixed Point Theory Appl. 2007., 2007: Article ID 17301 10.1155/2007/17301
Beygmohammadi M, Razani A: Two fixed-point theorems for mappings satisfying a general contractive condition of integral type in the modular space. Int. J. Math. Math. Sci. 2010., 2010: Article ID 317107 10.1155/2010/317107
Branciari A: A fixed point theorem for mappings satisfying a general contractive condition of integral type. Int. J. Math. Math. Sci. 2002, 29: 531–536. 10.1155/S0161171202007524
Djoudi A, Aliouche A: Common fixed point theorems of Greguš type for weakly compatible mappings satisfying contractive conditions of integral type. J. Math. Anal. Appl. 2007, 329: 31–45. 10.1016/j.jmaa.2006.06.037
Djoudi A, Merghadi F: Common fixed point theorems for maps under a contractive condition of integral type. J. Math. Anal. Appl. 2008, 341: 953–960. 10.1016/j.jmaa.2007.10.064
Jachymski J: Remarks on contractive conditions of integral type. Nonlinear Anal. 2009, 71: 1073–1081. 10.1016/j.na.2008.11.046
Liu Z, Li X, Kang SM, Cho SY: Fixed point theorems for mappings satisfying contractive conditions of integral type and applications. Fixed Point Theory Appl. 2011., 2011: Article ID 64 10.1186/1687-1812-2011-64
Mongkolkeha C, Kumam P: Fixed point and common fixed point theorems for generalized weak contraction mappings of integral type in modular spaces. Int. J. Math. Math. Sci. 2011., 2011: Article ID 705943 10.1155/2011/705943
Rhoades BE: Two fixed point theorems for mappings satisfying a general contractive condition of integral type. Int. J. Math. Math. Sci. 2003, 63: 4007–4013. 10.1155/S0161171203208024
Sintunavarat W, Kumam P: Greguš-type common fixed point theorems for tangential multivalued mappings of integral type in metric spaces. Int. J. Math. Math. Sci. 2011., 2011: Article ID 923458 10.1155/2011/923458
Sintunavarat W, Kumam P: Greguš type fixed points for a tangential multi-valued mappings satisfying contractive conditions of integral type. J. Inequal. Appl. 2011., 2011: Article ID 3 10.1186/1029-242X-2011-3
Suzuki T: Meir-Keeler contractions of integral type are still Meir-Keeler contractions. Int. J. Math. Math. Sci. 2007., 2007: Article ID 39281 10.1155/2007/39281
Vijayaraju P, Rhoades BE, Mohanraj R: A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type. Int. J. Math. Math. Sci. 2005, 15: 2359–2364. 10.1155/IJMMS.2005.2359
Acknowledgements
The authors would like to thank the referees for useful comments and suggestions. This research was supported by the Science Research Foundation of Educational Department of Liaoning Province (L2012380).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Liu, Z., Lu, Y. & Kang, S.M. Fixed point theorems for mappings satisfying contractive conditions of integral type. Fixed Point Theory Appl 2013, 267 (2013). https://doi.org/10.1186/1687-1812-2013-267
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2013-267