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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
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