- Research
- Open access
- Published:
Generalized -expansive mappings and related fixed-point theorems
Journal of Inequalities and Applications volume 2014, Article number: 22 (2014)
Abstract
In this paper, we introduce a new class of expansive mappings called generalized -expansive mappings and investigate the existence of a fixed point for the mappings in this class. We conclude that several fixed-point theorems can be considered as a consequence of main results. Moreover, some examples are given to illustrate the usability of the obtained results.
MSC:46T99, 54H25, 47H10, 54E50.
1 Introduction
Fixed-point theory has attracted many mathematicians since it provides a simple proof for the existence and uniqueness of the solutions to various mathematical models (integral and partial differential equations, variational inequalities etc.). After the celebrated results of Banach [1], fixed-point theory became one of the most interesting topics in nonlinear analysis. Consequently, a number of the papers have appeared since then; see e.g. [2–10] and references therein. Among them, we mention the α-ψ-contractive mapping, which was introduced by Samet et al. [9] via α-admissible mappings. In this paper, the authors established various fixed-point theorems for such mappings in complete metric spaces. Furthermore, Samet et al. [9] stated that several existing results can be concluded from their main results. For the sake of completeness, we recall some basic definitions and fundamental results.
Definition 1.1 [9]
Let φ denote the family of all functions which satisfy:
-
(i)
for each , where is the n th iterate of ψ.
-
(ii)
ψ is non-decreasing.
Definition 1.2 [9]
Let be a metric space and be a given self mapping. T is said to be an α-ψ-contractive mapping if there exist two functions and such that
for all .
Definition 1.3 [9]
Let and . T is said to be α-admissible if
Now, we give some examples of α-admissible mappings.
Example 1.4 Let X be the set of all non-negative real numbers. Let us define the mapping by
and define the mapping by for all . Then T is α-admissible.
In what follows, we present the main results of Samet et al. [9].
Theorem 1.1 [9]
Let be a complete metric space and be an α-ψ-contractive mapping satisfying the following conditions:
-
(i)
T is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
T is continuous.
Then T has a fixed point, that is, there exists such that .
Theorem 1.2 [9]
Let be a complete metric space and be an α-ψ-contractive mapping satisfying the following conditions:
-
(i)
T is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
if is a sequence in X such that for all n and as , then for all n.
Then T has a fixed point.
Samet et al. [9] added the following condition (H) to the hypotheses of Theorem 1.1 and Theorem 1.2 to assure the uniqueness of the fixed point:
-
(H)
For all , there exists such that and .
Afterwards, Karapınar and Samet [10] generalized these notions to obtain further fixed-point results in the setting of complete metric space.
Definition 1.5 [10]
Let be a metric space and be a given mapping. We say that T is a generalized α-ψ-contractive mapping if there exist two functions and such that for all , we have
where .
Theorem 1.3 [10]
Let be a complete metric space. Suppose that is a generalized α-ψ-contractive mapping and satisfies the following conditions:
-
(i)
T is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
T is continuous.
Then there exists such that .
Theorem 1.4 [10]
Let be a complete metric space. Suppose that is a generalized α-ψ-contractive mapping and the following conditions hold:
-
(i)
T is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
if is a sequence in X such that for all n and as , then there exists a subsequence of such that for all k.
Then there exists such that .
Theorem 1.5 [10]
Adding condition (H) to the hypotheses of Theorem 1.3 (resp. Theorem 1.4), we find that u is the unique fixed point of T.
On the other hand, Shahi et al. [11] introduced a new category of expansive mappings called -expansive mappings as a complement of the concept of α-ψ-contractive type mappings. The authors in [11] also studied several fixed-point theorems for these mappings in the context of complete metric spaces.
We recollect the notion of -expansive mappings as follows. Let χ denote all functions which satisfy the following properties:
() ξ is non-decreasing,
() for each , where is the n th iterate of ξ,
() , .
Lemma 1.6 [9]
If is a non-decreasing function, then for each , implies .
Definition 1.6 [11]
Let be a metric space and be a given mapping. We say that T is an -expansive mapping if there exist two functions and such that
for all .
Remark 1.1 If is an expansion mapping, then T is an -expansive mapping, where for all and for all and some .
The main result of Shahi et al. [11] is the following.
Theorem 1.7 [11]
Let be a complete metric space and be a bijective, -expansive mapping satisfying the following conditions:
-
(i)
is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
T is continuous.
Then T has a fixed point, that is, there exists such that .
Motivated by the above ideas, we aim to give a new concept of generalized -expansive mappings. The results proved in this paper extend and generalize many existing results in the literature. We also illustrate some examples to support our statements.
2 Main results
We begin this section with the following definition.
Definition 2.1 Let be a metric space and be a given mapping. We say that T is a generalized -expansive mapping if there exists two functions and such that for all , we have
where .
Theorem 2.1 Let be a complete metric space and be a bijective, generalized -expansive mapping satisfying the following conditions:
-
(i)
is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
T is continuous.
Then T has a fixed point, that is, there exists such that .
Proof Let be such that . Let us define the sequence in X by
Now, if for any , one sees that is a fixed point of T from the definition. Without loss of generality, we can suppose for each . Since is an α-admissible mapping and , we deduce that . Continuing this process, we get
for all . Applying inequality (3) with , , we obtain
Owing to the fact that for all n, we have
Now, if for some , then
which is a contradiction. Hence, for all , we obtain
By induction, we have
For any , we infer that
From (), it follows that is a Cauchy sequence in the complete metric space . So, there exists such that as . From the continuity of T, it follows that as . Owing to the uniqueness of the limit, we get , that is, u is a fixed point of T. This completes the proof. □
In the sequel, we prove that Theorem 2.1 still holds for T not necessarily continuous, assuming the following condition:
-
(P)
If is a sequence in X such that for all n and as , then
(5)
for all n.
Theorem 2.2 If in Theorem 2.1 we replace the continuity of T by the condition (P), then the result holds true.
Proof Following the proof of Theorem 2.1, we see that is a sequence in X such that for all n and as . Now, in view of condition (P), we infer that
for all . Owing to inequalities (3) and (4), we get
Letting in the above inequality and due to the continuity of ξ at , we obtain
That is, either or . This implies that u is a fixed point of T. □
We shall present some examples to illustrate the validity of our results.
Example 2.2 Let endowed with the metric
Define the mappings and by for all . Consider the mapping by
Clearly, T is continuous and generalized -expansive mapping with for all . In fact, for all , we have
Moreover, there exists such that . Now, we proceed to show that is α-admissible. Let such that implying that . Now, by the definition of and α, we obtain . Thus, , that is, is α-admissible. Now, all the conditions of Theorem 2.1 are satisfied. Consequently, T has a fixed point. In this example, 0 and 1 are two fixed points of T.
Now, we give an example involving a function T that is not continuous.
Example 2.3 Let endowed with the standard metric for all . Define the mappings and by
and
Owing to the discontinuity of T at 1, Theorem 2.1 is not applicable in this case. Clearly, T is a generalized -expansive mapping with for all . In fact, for all , we infer that
Moreover, there exists such that . Now, we need to show that is α-admissible. Let such that implying that and . By the definition of and α, we get . Thus, is α-admissible.
Now, let be a sequence in X such that for all n and as . Since for all n, in view of definition of α, we get for all n and . Thus, .
Therefore, all the conditions of Theorem 2.2 are satisfied, and so T has a fixed point. Here, 0 and 1 are two fixed points of T.
If we take in Theorem 2.1, we get the following result.
Corollary 2.3 Let be a complete metric space and be a bijective map. Suppose that T satisfies the following condition:
where and . Suppose also that
-
(i)
there exists such that ;
-
(ii)
T is continuous.
Then T has a fixed point, that is, there exists such that .
Remark 2.1 Let be a metric space and be a map. Then the following inequality is evidently satisfied:
where .
As a consequence of the observation above, one can deduce the following results from Theorem 2.1.
Corollary 2.4 Let be a complete metric space and be a bijective. Suppose that T satisfies the following condition:
where and . Suppose also that
-
(i)
is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
T is continuous.
Then T has a fixed point, that is, there exists such that .
Corollary 2.5 Let be a complete metric space and be a bijective map. Suppose that T satisfies for
We suppose also that
-
(i)
is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
T is continuous.
Then T has a fixed point, that is, there exists such that .
Corollary 2.6 Let be a complete metric space and be a bijective map. Suppose that T satisfies for
Suppose also that
-
(i)
is α-admissible;
-
(ii)
there exists such that ;
-
(iii)
T is continuous.
Then T has a fixed point, that is, there exists such that .
Corollary 2.7 Let be a complete metric space and be a bijective map. Suppose that T is continuous, satisfying the following condition:
where . Then T has a fixed point, that is, there exists such that .
Proof By taking for all in Corollary 2.6, we get the proof of this corollary. □
Corollary 2.8 Let be a complete metric space and be a bijective map. Suppose that T is continuous, satisfying the following condition:
where . Then T has a fixed point, that is, there exists such that .
Proof By taking , where and in Corollary 2.7, we get the proof of this corollary. □
Remark 2.2 If we replace the continuity assumption of T by the condition (P) in Corollary 2.5, Corollary 2.4, Corollary 2.6, Corollary 2.7,Corollary 2.8, then the result holds true.
3 Consequences
3.1 Fixed-point theorems on metric spaces endowed with a partial order
Recently, there have been tremendous growth in the study of fixed-point problems of contractive mappings in metric spaces endowed with a partial order. The first result in this direction was given by Turinici [12], where he generalized the Banach contraction principle in partially ordered sets. Some applications of Turinici’s theorem to matrix equations were demonstrated by Ran and Reurings [13]. Later, numerous important results had been obtained concerning the existence of a fixed point for contraction type mappings in partially ordered metric spaces by Bhaskar and Lakshmikantham [4], Nieto and Lopez [7, 14], Agarwal et al. [15], Lakshmikantham and Ćirić [6] and Samet [16] etc. In this section, we will deduce some fixed-point results on a metric space endowed with a partial order. For this, we require the following concepts.
Definition 3.1 Let be a partially ordered set and be a given mapping. We say that T is non-decreasing with respect to ⪯ if
Definition 3.2 Let be a partially ordered set. A sequence is said to be non-decreasing with respect to ⪯ if for all n.
Definition 3.3 Let be a partially ordered set and d be a metric on X. We say that is regular if for every non-decreasing sequence such that as , there exists a subsequence of such that for all k.
Now, we have the following result.
Corollary 3.1 Let be a partially ordered set and d be a metric on X such that is complete. Let be a bijective mapping such that is a non-decreasing mapping with respect to ⪯ satisfying the following condition for all with :
where and
Suppose also that
-
(i)
there exists such that ;
-
(ii)
T is continuous or is regular.
Then T has a fixed point, that is, there exists such that .
Proof Let us define the mapping by
Clearly, T is a generalized -expansive mapping, that is,
for all . In view of condition (i), we have . Owing to the monotonicity of , we get
This shows that is α-admissible. Now, if T is continuous, the existence of a fixed point follows from Theorem 2.1. Suppose now that is regular. Assume that is a sequence in X such that for all n and as . Due to the regularity hypotheses, there exists a subsequence of such that for all k. Now, in view of the definition of α, we obtain for all k. Thus, we get the existence of a fixed point in this case from Theorem 2.2. □
Regarding Lemma 2.1, the following is a natural consequence of the above corollary.
Corollary 3.2 Let be a partially ordered set and d be a metric on X such that is complete. Let be a bijective mapping such that is a non-decreasing mapping with respect to ⪯ satisfying the following condition for all with :
where . Suppose also that
-
(i)
there exists such that ;
-
(ii)
T is continuous or is regular.
Then T has a fixed point, that is, there exists such that .
Corollary 3.3 Let be a partially ordered set and d be a metric on X such that is complete. Let be a bijective mapping such that be a non-decreasing mapping with respect to ⪯ satisfying the following condition for all with :
where . Suppose also that
-
(i)
there exists such that ;
-
(ii)
T is continuous or is regular.
Then T has a fixed point, that is, there exists such that .
Proof If we take in Corollary 3.1, then we get the proof of this corollary. □
Corollary 3.4 Let be a partially ordered set and d be a metric on X such that is complete. Let be a bijective mapping such that is a non-decreasing mapping with respect to ⪯ satisfying the following condition for all with :
where . Suppose also that
-
(i)
there exists such that ;
-
(ii)
T is continuous or is regular.
Then T has a fixed point, that is, there exists such that .
Proof If we take , where and in Corollary 3.3, then we get the proof of this corollary. □
3.2 Fixed-point theorems for cyclic contractive mappings
Kirk et al. [17] in 2003 generalized the Banach contraction mapping principle by introducing cyclic representations and cyclic contractions. A mapping is called cyclic if and , where A, B are nonempty subsets of a metric space . Moreover, T is called a cyclic contraction if there exists such that for all and . It is to be noted that although a contraction is continuous, cyclic contractions need not be. This is one of the important benefits of this theorem. In the last decade, various authors have used the cyclic representations and cyclic contractions to derive various fixed-point results. See for example [18–23].
Corollary 3.5 Let be nonempty closed subsets of a complete metric space and be a given bijective mapping, where . Suppose that the following conditions hold:
-
(I)
and ;
-
(II)
there exists a function such that
(19)
Then T has a unique fixed point that belongs to .
Proof As and are closed subsets of the complete metric space , then is complete. Let us define the mapping
In view of (II) and the definition of α, we infer that
for all . Thus, T is a generalized -expansive mapping.
Let such that . If , from (I), , which implies that . Therefore, in all cases, we have . This implies that is α-admissible. Also, in view of (I), for any , we get , thereby implying that .
Now, let be a sequence in X such that for all n and as . From the definition of α, we have
As is a closed set with respect to the Euclidean metric, we infer that
which implies that . So, we obtain from the definition of α the result that for all n. □
From Lemma 2.1, one get deduce the following result.
Corollary 3.6 Let be nonempty closed subsets of a complete metric space and be a given bijective mapping, where . Suppose that the following conditions hold:
-
(I)
and ;
-
(II)
there exist constants a, b, c such that
(23)
Then T has a unique fixed point that belongs to .
Corollary 3.7 Let be nonempty closed subsets of a complete metric space and be a given bijective mapping, where . Suppose that the following conditions hold:
-
(I)
and ;
-
(II)
there exists a function such that
(24)
Then T has a unique fixed point that belongs to .
Proof If we take in Corollary 3.5, then we get the proof of this corollary. □
Corollary 3.8 Let be nonempty closed subsets of a complete metric space and be a given bijective mapping, where . Suppose that the following conditions hold:
-
(I)
and ;
-
(II)
there exists a constant such that
(25)
Then T has a unique fixed point that belongs to .
Proof By taking , where and in Corollary 3.7, we get the proof of this corollary. □
References
Banach S: Sur les operations dans les ensembles abstraits et leur application aux equations itegrales. Fundam. Math. 1922, 3: 133-181.
Caccioppoli R: Un teorema generale sull-esistenza di elementi uniti in una trasformazione funzionale. Atti Accad. Naz. Lincei, Rend. Cl. Sci. Fis. Mat. Nat. 1930, 11: 794-799. (in Italian)
Kannan R: Some results on fixed points. Bull. Calcutta Math. Soc. 1968, 10: 71-76.
Bhaskar TG, Lakshmikantham V: Fixed point theory in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 1379-1393. 10.1016/j.na.2005.10.017
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
Lakshmikantham V, Ćirić L: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 2009, 70: 4341-4349. 10.1016/j.na.2008.09.020
Nieto JJ, Lopez RR: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223-239. 10.1007/s11083-005-9018-5
Saadati R, Vaezpour SM: Monotone generalized weak contractions in partially ordered metric spaces. Fixed Point Theory 2010, 11: 375-382.
Samet B, Vetro C, Vetro P: Fixed point theorem for α - ψ contractive type mappings. Nonlinear Anal. 2012, 75: 2154-2165. 10.1016/j.na.2011.10.014
Karapınar E, Samet B: Generalized α - ψ -contractive type mappings and related fixed point theorems with applications. Abstr. Appl. Anal. 2012., 2012: Article ID 793486 10.1155/2012/793486
Shahi P, Kaur J, Bhatia SS:Fixed point theorems for-expansive mappings in complete metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 157
Turinici M: Abstract comparison principles and multivariable Gronwall-Bellman inequalities. J. Math. Anal. Appl. 1986, 117: 100-127. 10.1016/0022-247X(86)90251-9
Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc. Am. Math. Soc. 2004, 132: 1435-1443. 10.1090/S0002-9939-03-07220-4
Nieto JJ, Lopez RR: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations. Acta Math. Sin. Engl. Ser. 2007, 23: 2205-2212. 10.1007/s10114-005-0769-0
Agarwal RP, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 1-8. 10.1080/00036810701714164
Samet B: Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces. Nonlinear Anal. TMA 2010. 10.1016/j.na.2010.02.026
Kirk WA, Srinivasan PS, Veeramani P: Fixed points for mappings satisfying cyclical contractive conditions. Fixed Point Theory 2003, 4: 79-89.
Agarwal RP, Alghamdi MA, Shahzad N: Fixed point theory for cyclic generalized contractions in partial metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 40
Karapınar E: Fixed point theory for cyclic weak ϕ -contraction. Appl. Math. Lett. 2011, 24: 822-825. 10.1016/j.aml.2010.12.016
Karapınar E, Sadaranagni K:Fixed point theory for cyclic-contractions. Fixed Point Theory Appl. 2011., 2011: Article ID 69
Pacurar M, Rus IA: Fixed point theory for cyclic φ -contractions. Nonlinear Anal. 2010, 72: 1181-1187. 10.1016/j.na.2009.08.002
Petric MA: Some results concerning cyclic contractive mappings. Gen. Math. 2010, 18: 213-226.
Rus IA: Cyclic representations and fixed points. Ann. “Tiberiu Popoviciu” Sem. Funct. Equ. Approx. Convexity 2005, 3: 171-178.
Acknowledgements
The authors are grateful to the reviewers for their careful reviews and useful comments.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally in writing this article. 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
Karapınar, E., Shahi, P., Kaur, J. et al. Generalized -expansive mappings and related fixed-point theorems. J Inequal Appl 2014, 22 (2014). https://doi.org/10.1186/1029-242X-2014-22
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-22