 Research
 Open Access
A discussion on generalized almost contractions via rational expressions in partially ordered metric spaces
 Zead Mustafa^{1},
 Erdal Karapınar^{2, 3}Email author and
 Hassen Aydi^{4, 5}
https://doi.org/10.1186/1029242X2014219
© Mustafa et al.; licensee Springer. 2014
 Received: 9 October 2013
 Accepted: 15 May 2014
 Published: 2 June 2014
Abstract
The main purpose of this paper is to give some fixed point results for mappings involving generalized $(\varphi ,\psi )$contractions in partially ordered metric spaces. Our results generalize, extend, and unify several wellknown comparable results in the literature (Jaggi in Indian J. Pure Appl. Math. 8(2):223230, 1977, Harjani et al. in Nonlinear Anal. 71:34033410, 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.
Keywords
 ordered set
 metric space
 fixed point
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.
for all distinct points $x,y\in X$ where $\alpha ,\beta \in [0,1)$ with $\alpha +\beta <1$. 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.
 (i)
T is continuous or
 (ii)
if $\{{x}_{n}\}$ is a nondecreasing sequence in X such that ${x}_{n}\to x$, then $x=sup\{{x}_{n}\}$.
If there exists ${x}_{0}\in X$ such that ${x}_{0}\le T{x}_{0}$, then T has a fixed point.
Set $\mathrm{\Phi}=\{\varphi \mid \varphi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})\text{is continuous and nondecreasing with}\varphi (t)=0\text{if and}\text{only if}t=0\}$ and $\mathrm{\Psi}=\{\psi \mid \psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})\text{is lower semi continuous},\psi (t)0\text{for all}t0,\text{and}\psi (0)=0\}$. 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 $T:X\to X$ involving a generalized $(\varphi ,\psi )$almost contraction. Some examples are also presented to show that our results are effective.
2 Main result
Our essential result is given as follows.
 (i)
T is continuous or
 (ii)
if $\{{x}_{n}\}$ is a nondecreasing sequence in X such that ${x}_{n}\to x$, then $x=sup\{{x}_{n}\}$.
If there exists ${x}_{0}\in X$ such that ${x}_{0}\le T{x}_{0}$, then T has a fixed point.
which is a contradiction. Thus, $\{{x}_{n}\}$ is a Cauchy sequence in X. Since X is a complete metric space, there exists $z\in X$ such that ${lim}_{n\to \mathrm{\infty}}{x}_{n}=z$.
Then $z=sup\{{x}_{n}\}$, and we get $z\le Tz$.
which is a contradiction. So $y=z$ and we have $z\le Tz\le z$, then $Tz=z$. □
If we take $L=0$ in Theorem 3 we get the following result.
 (i)
T is continuous or
 (ii)
if $\{{x}_{n}\}$ is a nondecreasing sequence in X such that ${x}_{n}\to x$, then $x=sup\{{x}_{n}\}$.
If there exists ${x}_{0}\in X$ such that ${x}_{0}\le T{x}_{0}$, then T has a fixed point.
Other corollaries could be derived.
 (i)
T is continuous or
 (ii)
if $\{{x}_{n}\}$ is a nondecreasing sequence in X such that ${x}_{n}\to x$, then $x=sup\{{x}_{n}\}$.
If there exists ${x}_{0}\in X$ such that ${x}_{0}\le T{x}_{0}$, then T has a fixed point.
Proof Take $\varphi (t)=t$ in Theorem 3. □
 (i)
T is continuous or
 (ii)
if $\{{x}_{n}\}$ is a nondecreasing sequence in X such that ${x}_{n}\to x$, then $x=sup\{{x}_{n}\}$.
If there exists ${x}_{0}\in X$ such that ${x}_{0}\le T{x}_{0}$, then T has a fixed point.
Proof Take $\psi (t)=(1k)\psi (t)$ for all $t\in [0,\mathrm{\infty})$ in Corollary 5. □
 (i)
T is continuous or
 (ii)
if $\{{x}_{n}\}$ is a nondecreasing sequence in X such that ${x}_{n}\to x$, then $x=sup\{{x}_{n}\}$.
If there exists ${x}_{0}\in X$ such that ${x}_{0}\le T{x}_{0}$, then T has a fixed point.
□
then T has a unique fixed point.
Since T is nondecreasing, $z\le x$ implies $Tz\le Tx=x$. By induction, we get ${z}_{n}\le x$.
If $x={z}_{{N}_{0}}$ for some ${N}_{0}\ge 1$ then ${z}_{n}=T{z}_{n1}=Tx=x$ for all $n\ge {N}_{0}1$. So ${lim}_{n\to \mathrm{\infty}}{z}_{n}=x$. Analogously, we get ${lim}_{n\to \mathrm{\infty}}{z}_{n}=y$, which completes the proof.
which is a contradiction. This ends the proof. □
Remark
Now, we give some examples illustrating our results.
We define the functions $\varphi ,\psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ by $\varphi (t)=2t$ and $\psi (t)=\frac{3}{2}t$. Now, we will check that all the hypotheses required by Theorem 4 (Theorem 3 with $L=0$) are satisfied.
First, X has the property: if $\{{x}_{n}\}$ is a nondecreasing sequence in X such that ${x}_{n}\to x$, then $x=sup\{{x}_{n}\}$. Indeed, let $\{{z}_{n}\}$ be a nondecreasing sequence in X with respect to ⪯ such that ${z}_{n}\to z\in X$ as $n\to +\mathrm{\infty}$. We have ${z}_{n}\u2aaf{z}_{n+1}$ for all $n\in \mathbb{N}$.

If ${z}_{0}=4$, then ${z}_{0}=4\u2aaf{z}_{1}$. From the definition of ⪯, we have ${z}_{1}=4$. By induction, we get ${z}_{n}=4$ for all $n\in \mathbb{N}$ and $z=4$. Then ${z}_{n}\u2aafz$ for all $n\in \mathbb{N}$ and $z=sup\{{z}_{n}\}$.

If ${z}_{0}=5$, then ${z}_{0}=5\u2aaf{z}_{1}$. From the definition of ⪯, we have ${z}_{1}=5$. By induction, we get ${z}_{n}=5$ for all $n\in \mathbb{N}$ and $z=5$. Then ${z}_{n}\u2aafz$ for all $n\in \mathbb{N}$ and $z=sup\{{z}_{n}\}$.

If ${z}_{0}=6$, then ${z}_{0}=6\u2aaf{z}_{1}$. From the definition of ⪯, we have ${z}_{1}\in \{6,4\}$. By induction, we get ${z}_{n}\in \{6,4\}$ for all $n\in \mathbb{N}$. Suppose that there exists $p\ge 1$ such that ${z}_{p}=4$. From the definition of ⪯, we get ${z}_{n}={z}_{p}=4$ for all $n\ge p$. Thus, we have $z=4$ and ${z}_{n}\u2aafz$ for all $n\in \mathbb{N}$. Now, suppose that ${z}_{n}=6$ for all $n\in \mathbb{N}$. In this case, we get $z=6$ and ${z}_{n}\u2aafz$ for all $n\in \mathbb{N}$ and $z=sup\{{z}_{n}\}$.
Thus, we proved that in all cases, we have $z=sup\{{z}_{n}\}$.
so (2.23) holds easily. On the other hand, it is obvious that T is a nondecreasing mapping with respect to ⪯ and there exists ${x}_{0}=6$ such that ${x}_{0}\u2aafT{x}_{0}$. All the hypotheses of Theorem 4 are verified and $u=4$ is a fixed point of T.
Define $T:X\to X$ by $Tx=x$ if $0\le x<1$ and $Tx=0$ if $x\ge 1$. Define the functions $\varphi ,\psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ by $\varphi (t)=4t$ and $\psi (t)=3t$.
Take $x\u2aafy$ and $x\ne y$. It means that $1\le x<y$. In particular, $d(Tx,Ty)=0$ and $M(x,y)=yx$. This implies that (2.23) holds. It is easy that X satisfies the property: if $\{{x}_{n}\}$ is a nondecreasing sequence in X such that ${x}_{n}\to x$, then ${x}_{n}\u2aafx$ for all $n\in \mathbb{N}$. Also, the other conditions of Theorem 4 are satisfied and $u=0$ is a fixed point of T.
so (2.23) holds. Also, $(0,1)\le T((0,1))$, therefore all conditions in Theorem 4 hold and there are two fixed points which are $(0,1)$ and $(1,0)$. The nonuniqueness follows from the fact that the partial order ≤ is not total.
Notes
Declarations
Acknowledgements
The authors express their gratitude to the referees for constructive and useful remarks and suggestions.
Authors’ Affiliations
References
 Banach S: Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fundam. Math. 1922, 3: 133–181.Google Scholar
 Jaggi DJ: Some unique fixed point theorems. Indian J. Pure Appl. Math. 1977,8(2):223–230.MathSciNetGoogle Scholar
 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 190701Google Scholar
 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 46Google Scholar
 Agarwal RA, ElGebeily MA, O’Regan D: Generalized contractions in partially ordered metric spaces. Appl. Anal. 2008, 87: 109–116. 10.1080/00036810701556151MathSciNetView ArticleGoogle Scholar
 AminiHarandi 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.2009.10.023MathSciNetView ArticleGoogle Scholar
 Altun I, Simsek H: Some fixed point theorems on ordered metric spaces and applications. Fixed Point Theory Appl. 2010., 2010: Article ID 621469Google Scholar
 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 248Google Scholar
 Aydi H: Fixed point results for weakly contractive mappings in ordered partial metric spaces. J. Adv. Math. Stud. 2011,4(2):1–12.MathSciNetGoogle Scholar
 Aydi H: Common fixed point results for mappings satisfying $(\psi ,\varphi )$weak contractions in ordered partial metric spaces. Int. J. Math. Stat. 2012,12(2):63–64.MathSciNetGoogle Scholar
 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.2011.07.006MathSciNetView ArticleGoogle Scholar
 Aydi H, Shatanawi W, Postolache M, Mustafa Z, Tahat N: Theorems for BoydWong type contractions in ordered metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 359054Google Scholar
 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 44Google Scholar
 Aydi H, Postolache M, Shatanawi W: Coupled fixed point results for $(\psi ,\varphi )$ weakly contractive mappings in ordered G metric spaces. Comput. Math. Appl. 2012,63(1):298–309. 10.1016/j.camwa.2011.11.022MathSciNetView ArticleGoogle Scholar
 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.MathSciNetGoogle Scholar
 Chandok S, Postolache M: Fixed point theorem for weakly Chatterjeatype cyclic contractions. Fixed Point Theory Appl. 2013., 2013: Article ID 28Google Scholar
 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.MathSciNetGoogle Scholar
 Choudhury BS, Metiya N: Coincidence point and fixed point theorems in ordered cone metric spaces. J. Adv. Math. Stud. 2012,5(2):20–31.MathSciNetGoogle Scholar
 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 152Google Scholar
 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.MathSciNetGoogle Scholar
 Ðorić D: Common fixed point for generalized $(\psi ,\varphi )$weak contractions. Appl. Math. Lett. 2009, 22: 1896–1900. 10.1016/j.aml.2009.08.001MathSciNetView ArticleGoogle Scholar
 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.2005.10.017MathSciNetView ArticleGoogle Scholar
 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.2009.01.240MathSciNetView ArticleGoogle Scholar
 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 658971Google Scholar
 Ding HS, 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 96Google Scholar
 Karapınar E: Couple fixed point theorems for nonlinear contractions in cone metric spaces. Comput. Math. Appl. 2010, 59: 3656–3668. 10.1016/j.camwa.2010.03.062MathSciNetView ArticleGoogle Scholar
 Karapınar E: Weak φ contraction on partial metric spaces and existence of fixed points in partially ordered sets. Math. Æterna 2011, 1: 237–244.Google Scholar
 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/S1995080213010083MathSciNetView ArticleGoogle Scholar
 Nieto JJ, RodríguezLópez R: Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations. Order 2005, 22: 223–239. 10.1007/s1108300590185MathSciNetView ArticleGoogle Scholar
 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.MathSciNetGoogle Scholar
 Miandaragh MA, Postolache M, Rezapour S: Approximate fixed points of generalized convex contractions. Fixed Point Theory Appl. 2013., 2013: Article ID 255Google Scholar
 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.2011.03.038MathSciNetView ArticleGoogle Scholar
 Abbas M, Nazir T, Radenović S: Common coupled fixed points of generalized contractive mappings in partially ordered metric spaces. Positivity 2013. 10.1007/s111170120219zGoogle Scholar
 Nieto JJ, RodríguezLópez 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/s1011400507690MathSciNetView ArticleGoogle Scholar
 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.2007.11.026MathSciNetView ArticleGoogle Scholar
 Olatinwo MO, Postolache M: Stability results for Jungcktype iterative processes in convex metric spaces. Appl. Math. Comput. 2012,218(12):6727–6732. 10.1016/j.amc.2011.12.038MathSciNetView ArticleGoogle Scholar
 Petrusel A, Rus IA: Fixed point theorems in ordered L spaces. Proc. Am. Math. Soc. 2006, 134: 411–418.MathSciNetView ArticleGoogle Scholar
 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/S0002993903072204MathSciNetView ArticleGoogle Scholar
 Rhoades BE: A comparison of various definitions of contractive mappings. Trans. Am. Math. Soc. 1977, 226: 257–290.MathSciNetView ArticleGoogle Scholar
 Radenović S, Kadelburg Z, Jandrlić A: Some results on weakly contractive maps. Bull. Iran. Math. Soc. 2012,38(3):625–645.Google Scholar
 Radenović S, Kadelburg Z: Generalized weak contractions in partially ordered metric spaces. Comput. Math. Appl. 2010, 60: 1776–1783. 10.1016/j.camwa.2010.07.008MathSciNetView ArticleGoogle Scholar
 Shobkolaei N, Sedghi S, Roshan JR, Altun I:Common fixed point of mappings satisfying almost generalized $(S,T)$contractive condition in partially ordered partial metric spaces. Appl. Math. Comput. 2012, 219: 443–452. 10.1016/j.amc.2012.06.063MathSciNetView ArticleGoogle Scholar
 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 60Google Scholar
 Shatanawi W, Postolache M: Some fixed point results for a G weak contraction in G metric spaces. Abstr. Appl. Anal. 2012., 2012: Article ID 815870Google Scholar
 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 271Google Scholar
 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 54Google Scholar
 Shatanawi W, Pitea A: Fixed and coupled fixed point theorems of omegadistance for nonlinear contraction. Fixed Point Theory Appl. 2013., 2013: Article ID 275Google Scholar
 Shatanawi W, Pitea A: ω distance and coupled fixed point in G metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 208Google Scholar
 Shatanawi W, Pitea A: Some coupled fixed point theorems in quasipartial metric spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 153Google Scholar
 Zamfirescu T: Fix point theorems in metric spaces. Arch. Math. 1972, 23: 292–298. 10.1007/BF01304884MathSciNetView ArticleGoogle Scholar
 Zhang Q, Song Y: Fixed point theory for generalized ϕ weak contraction. Appl. Math. Lett. 2009, 22: 75–78. 10.1016/j.aml.2008.02.007MathSciNetView ArticleGoogle Scholar
 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 20Google Scholar
 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.2010.02.039MathSciNetView ArticleGoogle Scholar
 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/S0004972700001659MathSciNetView ArticleGoogle Scholar
 Berinde V: Approximation fixed points of weak contractions using the Picard iteration. Nonlinear Anal. Forum 2004, 9: 43–53.MathSciNetGoogle Scholar
 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.MathSciNetGoogle Scholar
 Aghajani A, Radenović S, Roshan JR:Common fixed point results for four mappings satisfying almost generalized $(S,T)$contractive condition in partially ordered metric spaces. Appl. Math. Comput. 2012, 218: 5665–5670. 10.1016/j.amc.2011.11.061MathSciNetView ArticleGoogle Scholar
 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.MathSciNetGoogle Scholar
 Berinde V: General constructive fixed point theorem for Ćirićtype almost contractions in metric spaces. Carpath. J. Math. 2008,24(2):10–19.MathSciNetGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.