Abstract
The main purpose of this paper is to give some fixed point results for mappings involving generalized -contractions in partially ordered metric spaces. Our results generalize, extend, and unify several well-known comparable results in the literature (Jaggi in Indian J. Pure Appl. Math. 8(2):223-230, 1977, Harjani et al. in Nonlinear Anal. 71:3403-3410, 2009, Luong and Thuan in Fixed Point Theory Appl. 2011:46, 2011). The presented results are supported by three illustrative examples.
MSC: 46N40, 47H10, 54H25, 46T99.
Similar content being viewed by others
1 Introduction and preliminaries
The Banach contraction mapping principle [1] is one of the pivotal results of analysis. It is widely considered as the source of metric fixed point theory. Also, its significance lies in its application in a vast number of branches of mathematics. Generalizations of this principle have been investigated heavily (see Jaggi [2], Harjani et al. [3], Luong and Thuan [4]). In particular, in 1977, Jaggi [2] proved the following theorem satisfying a contractive condition of a rational type.
Theorem 1 Let be a complete metric space. Let be a continuous mapping such that
for all distinct points where with . Then T has a unique fixed point.
Existence of fixed point in partially ordered sets has been recently studied in [3–53].
Recently, Harjani et al. [3] proved the ordered version of Theorem 1. Very recently, Luong and Thuan [4] generalized the results of [3] and proved the following.
Theorem 2 Let be a partially ordered set. Suppose there exists a metric d such that is a metric space. Let be a non-decreasing mapping such that
for all distinct points with where is a lower semi-continuous function with the property that if and only if , and
Also, assume either
-
(i)
T is continuous or
-
(ii)
if is a non-decreasing sequence in X such that , then .
If there exists such that , then T has a fixed point.
Set and . For some work on the class of Φ or the class of Ψ, we refer the reader to [21, 51, 54].
In 2004, Berinde [55] introduced an almost contraction, a new class of contractive type mappings which exhibits totally different features more than the one of the particular results incorporated [1, 16, 39, 50], i.e., an almost contraction generally does not have a unique fixed point; see Example 1 in [55]. Thereafter, many authors presented several interesting and useful facts about almost contractions; see [42, 56–59].
The purpose of this article is to generalize the above results for a mapping involving a generalized -almost contraction. Some examples are also presented to show that our results are effective.
2 Main result
Our essential result is given as follows.
Theorem 3 Let be a partially ordered set. Suppose there exists a metric d such that is a complete metric space. Let be a non-decreasing mapping which satisfies the inequality
for all distinct points with where , , and
Also, assume either
-
(i)
T is continuous or
-
(ii)
if is a non-decreasing sequence in X such that , then .
If there exists such that , then T has a fixed point.
Proof Let such that . We define a sequence in X as follows:
Since T is a non-decreasing mapping together with (2.2), we have . Inductively, we obtain
Assume that there exists such that . Since , then T has a fixed point. Suppose that for all . Thus, by (2.3) we have
Regarding (2.4), the condition (2.1) implies that
where
Suppose that for some . Then the inequality (2.5) turns into
Regarding (2.4) and the property of ψ, this is a contradiction. Thus, for all . Therefore, the inequality (2.5) yields
Since ϕ is non-decreasing, we have . Consequently, is a decreasing sequence of positive real numbers which is bounded below. So, there exists such that . We claim that . Suppose, to the contrary, that . By taking the limit of the supremum in the relation , as , we get
which is a contradiction. Hence, we conclude that , that is,
We prove that the sequence is Cauchy in X. Suppose, to the contrary, that is not a Cauchy sequence. So, there exists such that
where and are subsequences of with
Moreover, is chosen to be the smallest integer satisfying (2.8). Thus, we have
By the triangle inequality, we get
Keeping (2.7) in mind and letting in the above inequality, we get
Due to the triangle inequality, we have
and
By using (2.7), (2.11), and letting in (2.12) and (2.13), we get
Analogously, we derive
Since we have . By (2.1) we have
where
Letting in (2.16) (and hence in (2.17)), and taking (2.7), (2.11), (2.14), and (2.15) into account, we obtain
which is a contradiction. Thus, is a Cauchy sequence in X. Since X is a complete metric space, there exists such that .
We will show that z is a fixed point of T. Assume that (i) holds. Then by the continuity of T, we have
Suppose that (ii) holds. Since is a non-decreasing sequence and then . Hence, for all . Since T is a non-decreasing mapping, we conclude that , or equivalently,
Then , and we get .
To this end, we construct a new sequence as follows:
Since , we have . Hence we find that is a non-decreasing sequence. By repeating the discussion above, one can conclude that is Cauchy. Thus there exists such that . By (ii), we have and so we have . From (2.19), we get
If then the proof is finished. Suppose that . On account of (2.20), the expression (2.1) implies that
where
Letting in (2.21) (and hence (2.22)), we obtain
which is a contradiction. So and we have , then . □
If we take in Theorem 3 we get the following result.
Theorem 4 Let be a partially ordered set. Suppose there exists a metric d such that is a complete metric space. Let be a non-decreasing mapping which satisfies the inequality
for all distinct with where , and
Also, assume either
-
(i)
T is continuous or
-
(ii)
if is a non-decreasing sequence in X such that , then .
If there exists such that , then T has a fixed point.
Other corollaries could be derived.
Corollary 5 Let be a partially ordered set. Suppose there exists a metric d such that is a complete metric space. Let be a non-decreasing mapping such that
for all distinct with where , and
Also, assume either
-
(i)
T is continuous or
-
(ii)
if is a non-decreasing sequence in X such that , then .
If there exists such that , then T has a fixed point.
Proof Take in Theorem 3. □
Corollary 6 Let be a partially ordered set. Suppose there exists a metric d X such that is a complete metric space. Let be a non-decreasing mapping such that
for all distinct with where and
Also, assume either
-
(i)
T is continuous or
-
(ii)
if is a non-decreasing sequence in X such that , then .
If there exists such that , then T has a fixed point.
Proof Take for all in Corollary 5. □
Corollary 7 Let be a partially ordered set. Suppose there exists a metric d such that is a complete metric space. Let be a non-decreasing mapping such that
for all distinct with where with . Also, assume either
-
(i)
T is continuous or
-
(ii)
if is a non-decreasing sequence in X such that , then .
If there exists such that , then T has a fixed point.
Proof Take and for all in Corollary 6. Indeed,
□
Theorem 8 In addition to the hypotheses of Theorem 3, assume that
then T has a unique fixed point.
Proof Suppose, to the contrary, that x and y are fixed points of T where . By (2.28), there exists a point which is comparable with x and y. Without loss of generality, we choose . We construct a sequence as follows:
Since T is non-decreasing, implies . By induction, we get .
If for some then for all . So . Analogously, we get , which completes the proof.
Consider the other case, that is, for all . Then, by (2.1), we observe that
for all distinct with where , and
Thus,
which is a contradiction. This ends the proof. □
Remark
-
Corollary 5 is a generalization of Theorem 2.1 of Luong and Thuan [4].
-
Corollary 7 (with ) corresponds to Theorem 2.2 and Theorem 2.3 of Harjani, López and Sadarangani [3].
-
Theorem 2.28 generalizes Theorem 2.4 of Luong and Thuan [4].
Now, we give some examples illustrating our results.
Example 9 Let be endowed with the usual metric for all , and . Consider the mapping
We define the functions by and . Now, we will check that all the hypotheses required by Theorem 4 (Theorem 3 with ) are satisfied.
First, X has the property: if is a non-decreasing sequence in X such that , then . Indeed, let be a non-decreasing sequence in X with respect to ⪯ such that as . We have for all .
-
If , then . From the definition of ⪯, we have . By induction, we get for all and . Then for all and .
-
If , then . From the definition of ⪯, we have . By induction, we get for all and . Then for all and .
-
If , then . From the definition of ⪯, we have . By induction, we get for all . Suppose that there exists such that . From the definition of ⪯, we get for all . Thus, we have and for all . Now, suppose that for all . In this case, we get and for all and .
Thus, we proved that in all cases, we have .
Let such that and , so we have only and . In particular
so (2.23) holds easily. On the other hand, it is obvious that T is a non-decreasing mapping with respect to ⪯ and there exists such that . All the hypotheses of Theorem 4 are verified and is a fixed point of T.
Note that Theorem 1 is not applicable. Indeed, taking and
for any such that . Also, we could not apply Theorem 2 in this example. Indeed, for and (that is, and ), we have
Example 10 Let be endowed with the Euclidean metric and the order ⪯ given as follows:
Define by if and if . Define the functions by and .
Take and . It means that . In particular, and . This implies that (2.23) holds. It is easy that X satisfies the property: if is a non-decreasing sequence in X such that , then for all . Also, the other conditions of Theorem 4 are satisfied and is a fixed point of T.
Notice that we cannot apply Theorem 1 (since T is not continuous) nor Theorem 2 to this example. Indeed, letting and (that is, ), we have
Example 11 Let with the Euclidean distance . is, obviously, a complete metric space. Moreover, we consider the order ≤ in X given by . We also consider given by , and . Take and . Obviously, T is a continuous and non-decreasing mapping since and . Let and , then necessarily and . Then
so (2.23) holds. Also, , therefore all conditions in Theorem 4 hold and there are two fixed points which are and . The non-uniqueness follows from the fact that the partial order ≤ is not total.
Note that Theorem 1 is not applicable. Indeed, taking and
for any such that . Also, we could not apply Theorem 2 in this example. Indeed, for and we have
References
Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 1922, 3: 133–181.
Jaggi DJ: Some unique fixed point theorems. Indian J. Pure Appl. Math. 1977,8(2):223–230.
Harjani J, López B, Sadarangani K: A fixed point theorem for mappings satisfying a contractive condition of rational type on a partially ordered metric space. Abstr. Appl. Anal. 2010., 2010: Article ID 190701
Luong NV, Thuan NX: Fixed point theorem for generalized weak contractions satisfying rational expressions in ordered metric spaces. Fixed Point Theory Appl. 2011., 2011: Article ID 46
Agarwal RA, El-Gebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 109–116. 10.1080/00036810701556151
Amini-Harandi A, Emami H: A fixed point theorem for contraction type maps in partially ordered metric spaces and application to ordinary differential equations. Nonlinear Anal. 2010, 72: 2238–2242. 10.1016/j.na.2009.10.023
Altun I, Simsek H: Some fixed point theorems on ordered metric spaces and applications. Fixed Point Theory Appl. 2010., 2010: Article ID 621469
Arshad M, Karapınar E, Jamshaid A: Some unique fixed point theorems for rational contractions in partially ordered metric spaces. J. Inequal. Appl. 2013., 2013: Article ID 248
Aydi H: Fixed point results for weakly contractive mappings in ordered partial metric spaces. J. Adv. Math. Stud. 2011,4(2):1–12.
Aydi H: Common fixed point results for mappings satisfying -weak contractions in ordered partial metric spaces. Int. J. Math. Stat. 2012,12(2):63–64.
Aydi H, Nashine HK, Samet B, Yazidi H: Coincidence and common fixed point results in partially ordered cone metric spaces and applications to integral equations. Nonlinear Anal. 2011,74(17):6814–6825. 10.1016/j.na.2011.07.006
Aydi H, Shatanawi W, Postolache M, Mustafa Z, Tahat N: Theorems for Boyd-Wong type contractions in ordered metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 359054
Aydi H, Karapınar E, Postolache M: Tripled coincidence point theorems for weak f -contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 44
Aydi H, Postolache M, Shatanawi W: Coupled fixed point results for -weakly contractive mappings in ordered G -metric spaces. Comput. Math. Appl. 2012,63(1):298–309. 10.1016/j.camwa.2011.11.022
Chandok S, Karapınar E: Common fixed point of generalized rational type contraction mappings in partially ordered metric spaces. Thai J. Math. 2013,11(2):251–260.
Chandok S, Postolache M: Fixed point theorem for weakly Chatterjea-type cyclic contractions. Fixed Point Theory Appl. 2013., 2013: Article ID 28
Chandok S, Mustafa Z, Postolache M: Coupled common fixed point theorems for mixed g -monotone mappings in partially ordered G -metric spaces. Sci. Bull. ‘Politeh.’ Univ. Buchar., Ser. A, Appl. Math. Phys. 2013,75(4):11–24.
Choudhury BS, Metiya N: Coincidence point and fixed point theorems in ordered cone metric spaces. J. Adv. Math. Stud. 2012,5(2):20–31.
Choudhury BS, Metiya N, Postolache M: A generalized weak contraction principle with applications to coupled coincidence point problems. Fixed Point Theory Appl. 2013., 2013: Article ID 152
Dey D, Ganguly A, Saha M: Fixed point theorems for mappings under general contractive condition of integral type. Bull. Math. Anal. Appl. 2011, 3: 27–34.
Ðorić D: Common fixed point for generalized -weak contractions. Appl. Math. Lett. 2009, 22: 1896–1900. 10.1016/j.aml.2009.08.001
Gnana Bhaskar T, Lakshmikantham V: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 2006, 65: 1379–1393. 10.1016/j.na.2005.10.017
Harjani J, Sadarangani K: Fixed point theorems for weakly contractive mappings in partially ordered sets. Nonlinear Anal. 2009, 71: 3403–3410. 10.1016/j.na.2009.01.240
Haghi RH, Postolache M, Rezapour S: On T -stability of the Picard iteration for generalized f -contraction mappings. Abstr. Appl. Anal. 2012., 2012: Article ID 658971
Ding H-S, Li L, Radenović S: Coupled coincidence point theorems for generalized nonlinear contraction in partially ordered metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 96
Karapınar E: Couple fixed point theorems for nonlinear contractions in cone metric spaces. Comput. Math. Appl. 2010, 59: 3656–3668. 10.1016/j.camwa.2010.03.062
Karapınar E: Weak φ -contraction on partial metric spaces and existence of fixed points in partially ordered sets. Math. Æterna 2011, 1: 237–244.
Karapınar E, Marudai M, Pragadeeswarar AV: Fixed point theorems for generalized weak contractions satisfying rational expression on a ordered partial metric space. Lobachevskii J. Math. 2013,34(1):116–123. 10.1134/S1995080213010083
Nieto JJ, Rodríguez-López R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239. 10.1007/s11083-005-9018-5
Miandaragh MA, Postolache M, Rezapour S: Some approximate fixed point results for generalized α -contractive mappings. Sci. Bull. ‘Politeh.’ Univ. Buchar., Ser. A, Appl. Math. Phys. 2013,75(2):3–10.
Miandaragh MA, Postolache M, Rezapour S: Approximate fixed points of generalized convex contractions. Fixed Point Theory Appl. 2013., 2013: Article ID 255
Abbas M, Nazir T, Radenović S: Common fixed points of four maps in partially ordered metric spaces. Appl. Math. Lett. 2011, 24: 1520–1526. 10.1016/j.aml.2011.03.038
Abbas M, Nazir T, Radenović S: Common coupled fixed points of generalized contractive mappings in partially ordered metric spaces. Positivity 2013. 10.1007/s11117-012-0219-z
Nieto JJ, Rodríguez-López R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equation. Acta Math. Sin. Engl. Ser. 2007,23(12):2205–2212. 10.1007/s10114-005-0769-0
O’Regan D, Petrusel A: Fixed point theorems for generalized contractions in ordered metric spaces. J. Math. Anal. Appl. 2008, 341: 1241–1252. 10.1016/j.jmaa.2007.11.026
Olatinwo MO, Postolache M: Stability results for Jungck-type iterative processes in convex metric spaces. Appl. Math. Comput. 2012,218(12):6727–6732. 10.1016/j.amc.2011.12.038
Petrusel A, Rus IA: Fixed point theorems in ordered L -spaces. Proc. Am. Math. Soc. 2006, 134: 411–418.
Ran ACM, Reurings MVB: 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
Rhoades BE: A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc. 1977, 226: 257–290.
Radenović S, Kadelburg Z, Jandrlić A: Some results on weakly contractive maps. Bull. Iran. Math. Soc. 2012,38(3):625–645.
Radenović S, Kadelburg Z: Generalized weak contractions in partially ordered metric spaces. Comput. Math. Appl. 2010, 60: 1776–1783. 10.1016/j.camwa.2010.07.008
Shobkolaei N, Sedghi S, Roshan JR, Altun I:Common fixed point of mappings satisfying almost generalized -contractive condition in partially ordered partial metric spaces. Appl. Math. Comput. 2012, 219: 443–452. 10.1016/j.amc.2012.06.063
Shatanawi W, Postolache M: Common fixed point results of mappings for nonlinear contractions of cyclic form in ordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 60
Shatanawi W, Postolache M: Some fixed point results for a G -weak contraction in G -metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 815870
Shatanawi W, Postolache M: Common fixed point theorems for dominating and weak annihilator mappings in ordered metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 271
Shatanawi W, Postolache M: Coincidence and fixed point results for generalized weak contractions in the sense of Berinde on partial metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 54
Shatanawi W, Pitea A: Fixed and coupled fixed point theorems of omega-distance for nonlinear contraction. Fixed Point Theory Appl. 2013., 2013: Article ID 275
Shatanawi W, Pitea A: ω -distance and coupled fixed point in G -metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 208
Shatanawi W, Pitea A: Some coupled fixed point theorems in quasi-partial metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 153
Zamfirescu T: Fix point theorems in metric spaces. Arch. Math. 1972, 23: 292–298. 10.1007/BF01304884
Zhang Q, Song Y: Fixed point theory for generalized ϕ -weak contraction. Appl. Math. Lett. 2009, 22: 75–78. 10.1016/j.aml.2008.02.007
Golubović Z, Kadelburg Z, Radenović S: Common fixed points of ordered g -quasicontractions and weak contractions in ordered metric spaces. Fixed Point Theory Appl. 2012., 2012: Article ID 20
Kadelburg Z, Radenović S, Pavlović M: Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces. Comput. Math. Appl. 2010, 59: 3148–3159. 10.1016/j.camwa.2010.02.039
Khan KS, Swaleh M, Sessa S: Fixed point theorems for altering distances between the points. Bull. Aust. Math. Soc. 1984,30(1):1–9. 10.1017/S0004972700001659
Berinde V: Approximation fixed points of weak contractions using the Picard iteration. Nonlinear Anal. Forum 2004, 9: 43–53.
Abbas M, Vetro P, Khan SH: On fixed points of Berinde’s contractive mappings in cone metric spaces. Carpath. J. Math. 2010,26(2):121–133.
Aghajani A, Radenović S, Roshan JR:Common fixed point results for four mappings satisfying almost generalized -contractive condition in partially ordered metric spaces. Appl. Math. Comput. 2012, 218: 5665–5670. 10.1016/j.amc.2011.11.061
Babu GVR, Sandhya ML, Kameswari MVR: A note on a fixed point theorem of Berinde on weak contraction. Carpath. J. Math. 2008,24(1):8–12.
Berinde V: General constructive fixed point theorem for Ćirić-type almost contractions in metric spaces. Carpath. J. Math. 2008,24(2):10–19.
Acknowledgements
The authors express their gratitude to the referees for constructive and useful remarks and suggestions.
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 and significantly 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
Mustafa, Z., Karapınar, E. & Aydi, H. A discussion on generalized almost contractions via rational expressions in partially ordered metric spaces. J Inequal Appl 2014, 219 (2014). https://doi.org/10.1186/1029-242X-2014-219
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1029-242X-2014-219