 Research
 Open Access
Fixed point results for contractions involving generalized altering distances in ordered metric spaces
 Hemant Kumar Nashine^{1},
 Bessem Samet^{2} and
 Jong Kyu Kim^{3}Email author
https://doi.org/10.1186/1687181220115
© Nashine et al; licensee Springer. 2011
 Received: 30 December 2010
 Accepted: 21 June 2011
 Published: 21 June 2011
Abstract
In this article, we establish coincidence point and common fixed point theorems for mappings satisfying a contractive inequality which involves two generalized altering distance functions in ordered complete metric spaces. As application, we study the existence of a common solution to a system of integral equations.
2000 Mathematics subject classification. Primary 47H10, Secondary 54H25
Keywords
 Coincidence point
 Common fixed point
 Complete metric space
 Generalized altering distance function
 Weakly contractive condition
 Weakly increasing
 Partially ordered set
Introduction and Preliminaries
There are a lot of generalizations of the Banach contractionmapping principle in the literature (see [1–31] and others).
A new category of contractive fixed point problems was addressed by Khan et al. [1]. In this study, they introduced the notion of an altering distance function which is a control function that alters distance between two points in a metric space.
 (i)
φ is continuous.
 (ii)
φ is nondecreasing.
 (iii)
φ (t) = 0 ⇔ t = 0.
Khan et al. [1] proved the following result:
for all x, y ∈ X and for some 0 < c < 1. Then, T has a unique fixed point.
Letting φ(t) = t in Theorem 1.2, we retrieve immediately the Banach contraction principle.
In 1997, Alber and GuerreDelabriere [2] introduced the concept of weak contractions in Hilbert spaces. This concept was extended to metric spaces in [3].
where φ: [0, +∞) → [0, +∞) is an altering distance function.
Theorem 1.4. [3]Let (X, d) be a complete metric space and T : X → X be a weakly contractive map. Then, T has a unique fixed point.
Weak inequalities of the above type have been used to establish fixed point results in a number of subsequent studies, some of which are noted in [4–7]. In [5], Choudhury introduced the concept of a generalized altering distance function.
 (i)
φ is continuous.
 (ii)
φ is nondecreasing in all the three variables.
 (iii)
φ (x, y, z) = 0 ⇔ x = y = z = 0.
In [5], Choudhury proved the following common fixed point theorem:
for all x, y ∈ X, where ψ_{1}and ψ_{2}are generalised altering distance functions, and Φ_{1}(x) = ψ_{1}(x, x, x). Then, S and T have a common fixed point.
Recently, there have been so many exciting developments in the field of existence of fixed point in partially ordered sets (see [8–27] and the references cited therein). The first result in this direction was given by Turinici [27], where he extended the Banach contraction principle in partially ordered sets. Ran and Reurings [24] presented some applications of Turinici's theorem to matrix equations. The obtained result by Turinici was further extended and refined in [20–23].
In this article, we obtain coincidence point and common fixed point theorems in complete ordered metric spaces for mappings, satisfying a contractive condition which involves two generalized altering distance functions. Presented theorems are the extensions of Theorem 1.6 of Choudhury [5]. In addition, as an application, we study the existence of a common solution for a system of integral equations.
Main Results
At first, we introduce some notations and definitions that will be used later. The following definition was introduced by Jungck [28].
Definition 2.1. [28] Let (X, d) be a metric space and f, g : X → X. If w = fx = gx, for some x ∈ X, then x is called a coincidence point of f and g, and w is called a point of coincidence of f and g. The pair {f, g} is said to be compatible if and only if , whenever {x_{ n } } is a sequence in X such that for some t ∈ X.
In [19], Nashine and Samet introduced the following concept:
Remark 2.3. If R : X → X is the identity mapping (Rx = x for all x ∈ X), then S and T are weakly increasing with respect to R implies that S and T are weakly increasing mappings. It is noted that the notion of weakly increasing mappings was introduced in [9] (also see [16, 29]).
Then, we will show that the mappings S and T are weakly increasing with respect to R.
Let x ∈ X. We distinguish the following two cases.

First case: x = 0 or x ≥ 1.
 (i)
Let y ∈ R ^{1}(Tx), that is, Ry = Tx. By the definition of T, we have Tx = 0 and then Ry = 0. By the definition of R, we have y = 0 or y ≥ 1. By the definition of S, in both cases, we have Sy = 0. Then, Tx = 0 = Sy.
 (ii)
Let y ∈ R ^{1}(Sx), that is, Ry = Sx. By the definition of S, we have Sx = 0, and then Ry = 0. By the definition of R, we have y = 0 or y ≥ 1. By the definition of T, in both cases, we have Ty = 0. Then, Sx = 0 = Ty.

Second case: 0 < x < 1.
 (i)
 (ii)
Thus, we proved that S and T are weakly increasing with respect to R.
We will show that the mappings S and T are weakly increasing with respect to R.
Let x, y ∈ X such that y ∈ R^{1}(Tx). By the definition of S, we have Sy = 1. On the other hand, Tx ∈ {1, 3} and (1, 1), (3, 1) ∈≤. Thus, we have Tx ≤ Sy for all y ∈ R^{1}(Tx).
Let x, y ∈ X such that y ∈ R^{1}(Sx). By the definitions of S and R, we have R^{1}(Sx) = R^{1}(1) = {1}. Then, we have y = 1. On the other hand, 1 = Sx ≤ Ty = T 1 = 1. Then, Sx ≤ Ty for all y ∈ R^{1}(Sx). Thus, we proved that S and T are weakly increasing with respect to R.
Our first result is as follows.
where ψ_{1}and ψ_{2}are generalized altering distance functions, and Φ_{1}(x) = ψ_{1}(x, x, x).
 (i)
T, S, and R are continuous.
 (ii)
TX ⊆ RX, SX ⊆ RX.
 (iii)
T and S are weakly increasing with respect to R.
 (iv)
the pairs {T, R} and {S, R} are compatible.
Then, T, S, and R have a coincidence point, that is, there exists u ∈ X such that Ru = Tu = Su.
Proof. Let x_{0} ∈ X be an arbitrary point. Since TX ⊆ RX, there exists x_{1} ∈ X such that Rx_{1} = Tx_{0}. Since SX ⊆ RX, there exists x_{2} ∈ X such that Rx_{2} = Sx_{1}.
Hence, by induction, (2.3) holds.
Now, we will prove our result on three steps.
which implies that r = 0. Hence, (2.5) is proved.
Step II. We claim that {Rx_{ n } } is a Cauchy sequence.
which implies that ψ_{2}(ε, 0, 0) = 0, that is a contradiction since ε > 0. We deduce that {Rx_{ n } } is a Cauchy sequence.
Step III. Existence of a coincidence point.
that is, u is a coincidence point of T, S, and R. This completes the proof.
In the next theorem, we omit the continuity hypotheses on T, S, and R.
Definition 2.7. Let (X,≤, d) be a partially ordered metric space. We say that X is regular if the following hypothesis holds: if {z_{ n } } is a nondecreasing sequence in X with respect to ≤ such that z_{ n } → z ∈ X as n → +∞, then z_{ n } ≤ z for all .
Now, our second result is the following.
 (i)
X is regular.
 (ii)
T and S are weakly increasing with respect to R.
 (iii)
RX is a closed subset of (X, d).
 (iv)
TX ⊆ RX, SX ⊆ R X.
Then, T, S, and R have a coincidence point.
Hence, v is a coincidence point of T, S, and R. This completes the proof.
Now, we present an example to illustrate the obtained result given by the previous theorem. Moreover, in this example, we will show that Theorem 1.6 of Choudhury cannot be applied.
We will show that T and S are weakly increasing with respect to R. In the case under study, we have to check that Tx ≤ T(Tx) for all x ∈ X.
Thus, we have proved that T and S are weakly increasing with respect to R.
Now, we will show that (X, ≤, d) is regular.
Let {z_{ n } } be a nondecreasing sequence in X with respect to ≤ such that z_{ n }→ z ∈ X as n → +∞. Then, we have z_{ n }≤ z_{n+1}, for all .

If z_{0} = 4, then z_{0} = 4 ≤ z_{1}. From the definition of ≤, we have z_{1} = 4. By induction, we get z_{ n }= 4 for all and z = 4. Then, z_{ n }≤ z for all .

If z_{0} = 5, then z_{0} = 5 ≤ z_{1}. From the definition of ≤, we have z_{1} = 5. By induction, we get z_{ n }= 5 for all and z = 5. Then, z_{ n }≤ z for all .

If z_{0} = 6, then z_{0} = 6 ≤ z_{1}. From the definition of ≤, we have z_{1} ∈ {6, 4}. By induction, we get z_{ n }∈ {6, 4} for all . Suppose that there exists p ≥ 1 such that z_{ p }= 4. From the definition of ≤, we get z_{ n }= z_{ p }= 4 for all n ≥ p. Thus, we have z = 4 and z_{ n }≤ z for all . Now, suppose that z_{ n }= 6 for all . In this case, we get z = 6, and z_{ n }≤ z for all . Thus, we proved that in all the cases considered, we have z_{ n }≤ z for all . Then, (X, ≤, d) is regular.
Clearly, ψ_{1} and ψ_{2} are the generalized altering distance functions, and for every x, y ∈ X such that Rx ≤ Ry, inequality (2.1) is satisfied.
Now, we can apply Theorem 2.8 to deduce that T, S, and R have a coincidence point u = 4. Note that u is also a fixed point of T since S = T, and R is the identity mapping.
Thus, Inequality (1.2) is not satisfied for x = 4 and y = 5. Then, Theorem 1.6 of Choudhury [5] cannot be applied in this case.
If R : X → X is the identity mapping, we can deduce easily the following common fixed point results.
The next result is an immediate consequence of Theorem 2.6.
 (i)
T and S are continuous.
 (ii)
T and S are weakly increasing.
Then, T and S have a common fixed point, that is, there exists u ∈ X such that u = Tu = Su.
The following result is an immediate consequence of Theorem 2.8.
 (i)
X is regular.
 (ii)
T and S are weakly increasing.
Then, T and S have a common fixed point.
where s > 0 and 0 < k = k_{1} + k_{2} + k_{3}< 1, then we obtain the following results.
The next result is an immediate consequence of Corollary 2.10.
 (i)
T and S are continuous.
 (ii)
T and S are weakly increasing.
Then, T and S have a common fixed point, that is, there exists u ∈ X such that u = Tu = Su.
The next result is an immediate consequence of Corollary 2.11.
 (i)
X is regular.
 (ii)
T and S are weakly increasing.
Then, T and S have a common fixed point.
Remark 2.14. Other fixed point results may also be obtained under specific choices of ψ_{1} and ψ_{2}.
Application
where T > 0.
The purpose of this section is to give an existence theorem for common solution of (3.1) using Corollary 2.13. This application is inspired in [9].
Moreover, in [20], it is proved that (C(I), ≤) is regular.
Now, we will prove the following result.
 (i)
K _{1}, K _{2} : I × I × ℝ → ℝ, and g : ℝ → ℝ are continuous;
 (ii)
 (iii)
Then, the integral equations (3.1) have a solution u* ∈ C(I).
Then, we have Tx ≤ STx and Sx ≤ TSx for all x ∈ C(I). This implies that T and S are weakly increasing.
Hence, the contractive condition required by Corollary 2.13 is satisfied with s = 1, k_{1} = α, and k_{2} = k_{3} = 0.
Now, all the required hypotheses of Corollary 2.13 are satisfied. Then, there exists u* ∈ C(I), a common fixed point of T and S, that is, u* is a solution to (3.1).
Declarations
Acknowledgements
The authors thank the referees for their valuable comments and suggestions. This work was supported by the Kyungnam University Research Fund, 2010.
Authors’ Affiliations
References
 Khan MS, Swaleh M, Sessa S: Fixed point theorems by altering distances between the points. Bull Aust Math Soc 1984, 30: 1–9. 10.1017/S0004972700001659MathSciNetView ArticleGoogle Scholar
 Alber YaI, GuerreDelabriere S: Principles of weakly contractive maps in Hilbert spaces. Oper Theory Adv Appl 1997, 98: 7–22.MathSciNetGoogle Scholar
 Rhoades BE: Some theorems on weakly contractive maps. Nonlinear Anal 2001, 47: 2683–2693. 10.1016/S0362546X(01)003881MathSciNetView ArticleGoogle Scholar
 Abbas M, Ali Khan M: Common fixed point theorem of two mappings satisfying a generalized weak contractive condition. Int J Math Math Sci 2009, 2009: 9. Article ID 131068View ArticleGoogle Scholar
 Choudhury BS: A common unique fixed point result in metric spaces involving generalized altering distances. Math Commun 2005, 10: 105–110.MathSciNetGoogle Scholar
 Dutta PN, Choudhury BS: A generalisation of contraction principle in metric spaces. Fixed Point Theory Appl 2008, 2008: 8. Article ID 406368MathSciNetView ArticleGoogle Scholar
 Zhang Q, Song Y: Fixed point theory for generalized φweakly contraction. Appl Math Lett 2009, 22: 75–78. 10.1016/j.aml.2008.02.007MathSciNetView ArticleGoogle Scholar
 Agarwal RP, ElGebeily MA, O'Regan D: Generalized contractions in partially ordered metric spaces. Appl Anal 2008, 87: 109–116. 10.1080/00036810701556151MathSciNetView ArticleGoogle Scholar
 Altun I, Simsek H: Some fixed point theorems on ordered metric spaces and application. Fixed Point Theory Appl 2010, 2010: 17. Article ID 621492MathSciNetGoogle 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
 Beg I, Butt AR: Fixed point for setvalued mappings satisfying an implicit relation in partially ordered metric spaces. Nonlinear Anal 2009, 71: 3699–3704. 10.1016/j.na.2009.02.027MathSciNetView ArticleGoogle Scholar
 Beg I, Butt AR: Fixed points for weakly compatible mappings satisfying an implicit relation in partially ordered metric spaces. Carpathian J Math 2009, 25: 1–12.MathSciNetGoogle Scholar
 Bhaskar TG, 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
 Ćirić N, Cakić Lj, Rajović M, Ume JS: Monotone generalized nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl 2008, 2008: 11. Article ID 131294Google Scholar
 Lakshmikantham V, Ćirić Lj: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal 2009, 70: 4341–4349. 10.1016/j.na.2008.09.020MathSciNetView ArticleGoogle Scholar
 Dhage BC, O'Regan D, Agrawal RP: Common fixed point theorems for a pair of countably condensing mappings in ordered Banach spaces. J Appl Math Stoch Anal 2003, 16: 243–248. 10.1155/S1048953303000182View 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
 Harjani J, Sadarangani K: Generalized contractions in partially ordered metric spaces and applications to ordianry differential equations. Nonlinear Anal 2010, 72: 1188–1197. 10.1016/j.na.2009.08.003MathSciNetView ArticleGoogle Scholar
 Nashine HK, Samet B: Fixed point results for mappings satisfying ( ψ , φ )weakly contractive condition in partially ordered metric spaces. Nonlinear Anal 2011, 74: 2201–2209. 10.1016/j.na.2010.11.024MathSciNetView ArticleGoogle Scholar
 Nieto JJ, RodríguezLópez R: Contractive mapping theorems in partially ordered sets and applications to ordianry differential equations. Order 2005, 22: 223–239. 10.1007/s1108300590185MathSciNetView ArticleGoogle Scholar
 Nieto JJ, RodríguezLópez R: Existence and uniqueness of fixed point in partially ordered sets and applications to ordianry differential equations. Acta Math Sinica (English Series) 2007,23(12):2205–2212. 10.1007/s1011400507690View ArticleGoogle Scholar
 Nieto JJ, Pouso RL, RodríguezLópez R: Fixed point theorems in ordered abstract spaces. Proc Am Math Soc 2007, 135: 2505–2517. 10.1090/S0002993907087291View 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
 Ran ACM, Reurings MCB: A fixed point theorem in partially ordered sets and some applications to matrix equations. Proc Am Math Soc 2004,132(5):1435–1443. 10.1090/S0002993903072204MathSciNetView ArticleGoogle Scholar
 Samet B: Coupled fixed point theorems for a generalized MeirKeeler contraction in partially ordered metric spaces. Nonlinear Anal 2010, 72: 4508–4517. 10.1016/j.na.2010.02.026MathSciNetView ArticleGoogle Scholar
 Samet B, Yazidi H: Coupled fixed point theorems in partially ordered ε chainable metric spaces. J Math Comput Sci 2010,1(3):142–151.Google Scholar
 Turinici M: Abstract comparison principles and multivariable GronwallBellman inequalities. J Math Anal Appl 1986, 117: 100–127. 10.1016/0022247X(86)902519MathSciNetView ArticleGoogle Scholar
 Jungck G: Compatible mappings and common fixed points. Int J Math Math Sci 1986, 9: 771–779. 10.1155/S0161171286000935MathSciNetView ArticleGoogle Scholar
 Dhage BC: Condensing mappings and applications to existence theorems for common solution of differential equations. Bull Korean Math Soc 1999,36(3):565–578.MathSciNetGoogle Scholar
 Boyd DW, Wong JSW: On nonlinear contractions. Proc Am Math Soc 1969, 20: 458–464. 10.1090/S00029939196902395599MathSciNetView ArticleGoogle Scholar
 Reich S: Some fixed point problems. Atti Acad Naz Linei 1974, 57: 194–198.Google 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 cited.