DOI QR코드

DOI QR Code

HUGE CONTRACTION ON PARTIALLY ORDERED METRIC SPACES

  • DESHPANDE, BHAVANA (DEPARTMENT OF MATHEMATICS, GOVT. ARTS & SCIENCE P. G. COLLEGE) ;
  • HANDA, AMRISH (DEPARTMENT OF MATHEMATICS, GOVT. P. G. ARTS AND SCIENCE COLLEGE) ;
  • KOTHARI, CHETNA (DEPARTMENT OF MATHEMATICS, GOVT. P. G. COLLEGE)
  • Received : 2015.11.15
  • Accepted : 2016.02.02
  • Published : 2016.02.28

Abstract

We establish coincidence point theorem for g-nondecreasing mappings satisfying generalized nonlinear contraction on partially ordered metric spaces. We also obtain the coupled coincidence point theorem for generalized compatible pair of mappings F, G : X2 → X by using obtained coincidence point results. Furthermore, an example is also given to demonstrate the degree of validity of our hypothesis. Our results generalize, modify, improve and sharpen several well-known results.

Keywords

1. INTRODUCTION AND PRELIMINARIES

In the sequel, we denote by X a non-empty set and ≤ will represent a partial order on X. Given n ∈ ℕ with n ≥ 2, let Xn be the nth Cartesian product X × X ×...× X (n times). For simplicity, if x ∈ X, we denote g(x) by gx.

The idea of the coupled fixed point was initiated by Guo and Lakshmikantham [9] in 1987.

Definition 1 ([9]). Let F : X2 → X be a given mapping. An element (x, y) ∈ X2 is called a coupled fixed point of F if

Following this paper, Bhaskar and Lakshmikantham [2] where the authors introduced the notion of mixed monotone property for F : X2 → X (wherein X is an ordered metric space) and utilized the same to prove some theorems on the existence and uniqueness of coupled fixed points.

Definition 2 ([2]). Let (X, ≤) be a partially ordered set. Suppose F : X2 → X be a given mapping. We say that F has the mixed monotone property if for all x, y ∈ X, we have

and

In 2009, Lakshmikantham and Ciric [15] generalized these results for nonlinear contraction mappings by introducing the notions of coupled coincidence point and mixed g-monotone property.

Definition 3 ([15]). Let F : X2 → X and g : X → X be given mappings. An element (x, y) ∈ X2 is called a coupled coincidence point of the mappings F and g if

Definition 4 ([15]). Let F : X2 → X and g : X → X be given mappings. An element (x, y) ∈ X2 is called a common coupled fixed point of the mappings F and g if

Definition 5 ([15]). The mappings F : X2 → X and g : X → X are said to be commutative if

Definition 6 ([15]). Let (X, ≤) be a partially ordered set. Suppose F : X2 → X and g : X → X are given mappings. We say that F has the mixed g-monotone property if for all x, y ∈ X, we have

and

If g is the identity mapping on X, then F satisfies the mixed monotone property.

Subsequently, Choudhury and Kundu [3] introduced the notion of compatibility and by using this notion to improve the results of Lakshmikantham and Ciric [15], thenafter several authors established coupled fixed/coincidence point theorems by using this notion.

Definition 7 ([3]). The mappings F : X2 → X and g : X → X are said to be compatible if

whenever {xn} and {yn} are sequences in X such that

A great deal of these studies investigate contractions on partially ordered metric spaces because of their applicability to initial value problems defined by differential or integral equations.

Hussain et al. [11] introduced the notion of generalized compatibility of a pair {F, G}, of mappings F, G : X × X → X, then the authors employed this notion to obtained coupled coincidence point results for such a pair of mappings involving (φ, ψ)-contractive condition without mixed G-monotone property of F.

Definition 8 ([11]). Suppose that F, G : X2 → X are two mappings. The mapping F is said to be G−increasing with respect to ≤ if for all x, y, u, v ∈ X with G(x, y) ≤ G(u, v) we have F(x, y) ≤ F(u, v).

Definition 9 ([11]). Let F, G : X2 → X be two mappings. We say that the pair {F, G} is commuting if

Definition 10 ([11]). Suppose that F, G : X2 → X are two mappings. An element (x, y) ∈ X2 is called a coupled coincidence point of mappings F and G if

Definition 11 ([11]). Let (X, ≤) be a partially ordered set, F : X2 → X and g : X → X are two mappings. We say that F is g-increasing with respect to ≤ if for any x, y ∈ X,

and

Definition 12 ([11]). Let (X, ≤) be a partially ordered set, F : X2 → X be a mapping. We say that F is increasing with respect to ≤ if for any x, y ∈ X,

and

Definition 13 ([11]). Let F, G : X2 → X are two mappings. We say that the pair {F, G} is generalized compatible if

whenever (xn) and (yn) are sequences in X such that

Obviously, a commuting pair is a generalized compatible but not conversely in general.

Erhan et al. [7], announced that the results established in Hussain et al. [11] can be easily derived from the coincidence point results in the literature.

In [7], Erhan et al. recalled the following basic definitions:

Definition 14 ([1, 8]). A coincidence point of two mappings T, g : X → X is a point x ∈ X such that Tx = gx.

Definition 15 ([7]). An ordered metric space (X, d, ≤) is a metric space (X, d) provided with a partial order ≤ .

Definition 16 ([2, 11]). An ordered metric space (X, d, ≤) is said to be non-decreasing-regular (respectively, non-increasing-regular) if for every sequence {xn} ⊆ X such that {xn} → x and xn ≤ xn+1 (respectively, xn ≥ xn+1) for all n, we have that xn ≤ x (respectively, xn ≥ x) for all n. (X, d, ≤) is said to be regular if it is both non-decreasing-regular and non-increasing-regular.

Definition 17 ([7]). Let(X, ≤) be a partially ordered set and let T, g : X → X be two mappings. We say that T is (g, ≤)-non-decreasing if Tx ≤ Ty for all x, y ∈ X such that gx ≤ gy. If g is the identity mapping on X, we say that T is ≤-non-decreasing.

Remark 18 ([7]). If T is (g, ≤)-non-decreasing and gx = gy, then Tx = Ty. It follows that

Definition 19 ([18]). Let (X, ≤) be a partially ordered set and endow the product space X2 with the following partial order:

Definition 20 ([3, 10, 17, 18]). Let (X, d, ≤) be an ordered metric space. Two mappings T, g : X → X are said to be O-compatible if

provided that {xn} is a sequence in X such that {gxn} is ≤-monotone, that is, it is either non-increasing or non-decreasing with respect to ≤ and

Samet et al. [20] declared that most of the coupled fixed point theorems for single-valued mappings on ordered metric spaces can be derived from well-known fixed point theorems.

On the other hand, Ding et al. [6] proved coupled coincidence and common coupled fixed point theorems for generalized nonlinear contraction on partially ordered metric spaces which generalize the results of Lakshmikantham and Ciric [15]. Our fundamental sources are [4-7, 11-14, 16, 18-20].

In this paper, we obtain a coincidence point theorem for g-non-decreasing mappings satisfying generalized nonlinear contraction on partially ordered metric spaces. With the help of our result, we derive a coupled coincidence point theorem of generalized compatible pair of mappings F, G : X2 → X. We also give an example and an application to integral equation to support our results. Our results generalize, extend, modify, improve and sharpen the results of Bhaskar and Lakshmikantham [2], Ding et al. [6] and Lakshmikantham and Ciric [15].

 

2. MAIN RESULTS

Lemma 21. Let (X, d) be a metric space. Suppose Y = X2 and define δ : Y × Y → [0, +∞) by

Then δ is metric on Y and (X, d) is complete if and only if (Y, δ) is complete.

Let Φ denote the set of all functions φ : [0, +∞) → [0, +∞) satisfying

(iφ) φ is non-decreasing, (iiφ) limn→∞ φn (t) = 0 for all t > 0, where φn+1(t) = φn (φ(t)).

It is clear that φ(t) < t for each t > 0. In fact, if φ(t0) ≥ t0 for some t0 > 0, then, since φ is non-decreasing, φn (t0) ≥ t0 for all n ∈ ℕ, which contradicts with limn→∞ φn (t0) = 0. In addition, it is easy to see that φ(0) = 0.

Theorem 22. Let (X, d, ≤) be a partially ordered metric space and let T, g : X → X be two mappings such that the following properties are fulfilled:

(i) T(X) ⊆ g(X),

(ii) T is (g, ≤)-non-decreasing,

(iii) there exists x0 ∈ X such that gx0 ≤ Tx0,

(iv) there exists φ ∈ Φ such that

d(Tx, Ty) ≤ φ (M(x, y)),

where

for all x, y ∈ X such that gx ≤ gy. Also assume that, at least, one of the following conditions holds:

(a) (X, d) is complete, T and g are continuous and the pair (T, g) is O-compatible,

(b) (X, d) is complete, T and g are continuous and commuting,

(c) (g(X), d) is complete and (X, d, ≤) is non-decreasing-regular,

(d) (X, d) is complete, g(X) is closed and (X, d, ≤) is non-decreasing-regular,

(e) (X, d) is complete, g is continuous, the pair (T, g) is O-compatible and (X, d, ≤) is non-decreasing-regular.

Then T and g have, at least, a coincidence point.

Proof. We divide the proof into four steps.

Step 1. We claim that there exists a sequence {xn} ⊆ X such that {gxn} is ≤-non-decreasing and gxn+1 = Txn, for all n ≥ 0. Let x0 ∈ X be arbitrary. Since Tx0 ∈ T(X) ⊆ g(X), therefore there exists x1 ∈ X such that Tx0 = gx1. Then gx0 ≤ Tx0 = gx1. Since T is (g, ≤)-non-decreasing, therefore Tx0 ≤ Tx1. Again, since Tx1 ∈ T(X) ⊆ g(X), therefore there exists x2 ∈ X such that Tx1 = gx2. Then gx1 = Tx0 ≤ Tx1 = gx2. Since T is (g, ≤)-non-decreasing, therefore Tx1 ≤ Tx2. Repeating this argument, there exists a sequence such that {gxn} is ≤-non-decreasing, gxn+1 = Txn ≤ Txn+1 = gxn+2 and

Step 2. We claim that is a Cauchy sequence in X. Now, by contractive condition (iv), we have

where

If d(gxn+1, gxn+2) ≥ d(gxn, gxn+1). Then

From (23), (24) and by the fact that φ(t) < t for all t > 0, we get

d(gxn+1, gxn+2) ≤ φ (d(gxn+1, gxn+2)) < d(gxn+1, gxn+2),

which is a contradiction. Hence, d(gxn, gxn+1) ≥ d(gxn+1, gxn+2). Then

Thus, by (23) and (25), we have for all n ∈ ℕ,

where

δ = d(gx0, gx1).

Without loss of generality, we can assume that d(gx0, gx1) ≠ 0. In fact, if this is not true, then gx0 = gx1 = Tx0, that is, x0 is a coincidence point of g and T.

Thus, for m, n ∈ ℕ with m > n, by triangle inequality and (26), we get

which implies, by (iiφ), that {gxn} is a Cauchy sequence in X.

Step 3. We claim that T and g have a coincidence point distinguishing between cases (a) − (e).

Suppose now that (a) holds, that is, (X, d) is complete, T and g are continuous and the pair (T, g) is O-compatible. Since (X, d) is complete, therefore there exists z ∈ X such that {gxn} → z and {Txn} → z. Since T and g are continuous, therefore {Tgxn} → Tz and {ggxn} → gz. Since the pair (T, g) is O-compatible, therefore limn→∞ d(gTxn, Tgxn) = 0. Thus, we conclude that

that is, z is a coincidence point of T and g.

Suppose now that (b) holds, that is, (X, d) is complete, T and g are continuous and commuting. It is evident that (b) implies (a).

Suppose now that (c) holds, that is, (g(X), d) is complete and (X, d, ≤) is non-decreasing-regular. As {gxn} is a Cauchy sequence in the complete space (g(X), d), so there exists y ∈ g(X) such that {gxn} → y. Let z ∈ X be any point such that y = gz, then {gxn} → gz. Indeed, as (X, d, ≤) is non-decreasing-regular and {gxn} is ≤-non-decreasing and converging to gz, we deduce that gxn ≤ gz for all n ≥ 0. Applying the contractive condition (iv), we get

where

Since {gxn} → gz, therefore there exists n0 ∈ ℕ such that for all n > n0,

By (27) and (28), we get

d(gxn+1, Tz) ≤ φ (d(gz, Tz).

Now, we claim that d(gz, Tz) = 0. If this is not true, then d(gz, Tz) > 0, which, by the fact that φ(t) < t for all t > 0, implies

d(gxn+1, Tz) < d(gz, Tz).

Letting n → ∞ in the above inequality and using limn→∞ gxn = gz, we get

d(gz, Tz) < d(gz, Tz).

which is a contradiction. Hence we must have d(gz, Tz) = 0, that is, z is a coincidence point of T and g.

Suppose now that (d) holds, that is, (X, d) is complete, g(X) is closed and (X, d, ≤) is non-decreasing-regular. It follows from the fact that a closed subset of a complete metric space is also complete. Then, (g(X), d) is complete and (X, d, ≤) is non-decreasing-regular. Thus (d) implies (c).

Suppose now that (e) holds, that is, (X, d) is complete, g is continuous, the pair (T, g) is O-compatible and (X, d, ≤) is non-decreasing-regular. As (X, d) is complete, so there exists z ∈ X such that {gxn} → z. Since Txn = gxn+1 for all n, we also have that {Txn} → z. As g is continuous, then {ggxn} → gz. Furthermore, since the pair (T, g) is O-compatible, we have limn→∞ d(ggxn+1, Tgxn) = limn→∞ d(gTxn, Tgxn) = 0. As {ggxn} → gz the previous property means that {Tgxn} → gz.

Indeed, as (X, d, ≤) is non-decreasing-regular and {gxn} is ≤-non-decreasing and converging to z, we deduce that gxn ≤ z for all n ≥ 0. Applying the contractive condition (iv), we get

where

Since {ggxn} → gz, therefore there exists n0 ∈ ℕ such that for all n > n0,

By (29) and (30), we get

d(Tgxn, Tz) ≤ φ (d(gz, Tz)),

Now, we claim that d(gz, Tz) = 0. If this is not true, then d(gz, Tz) > 0, which, by the fact that φ(t) < t for all t > 0, implies

d(Tgxn, Tz) < d(gz, Tz).

Letting n → ∞ in the above inequality and using {Tgxn} → gz, we get

d(gz, Tz) < d(gz, Tz),

which is a contradiction. Hence we must have d(gz, Tz) = 0, that is, z is a coincidence point of T and g.                                        ⧠

Next, we derive the two dimensional version of Theorem 22. For the ordered metric space (X, d, ≤), let us consider the ordered metric space (X2 , δ, ⊆), where δ was defined in Lemma 21 and ⊆ was introduced in (19). Define the mappings TF, TG : X2 → X2 , for all (x, y) ∈ X2 , by,

Under these conditions, the following properties hold:

Lemma 23. Let (X, d, ≤) be a partially ordered metric space and let F, G : X2 → X be two mappings. Then

(1) (X, d) is complete if and only if (X2 , δ) is complete.

(2) If (X, d, ≤) is regular, then (X2 , δ, ⊆) is also regular.

(3) If F is d-continuous, then TF is δ-continuous.

(4) If F is G-increasing with respect to ≤, then TF is (TG, ⊆)-nondecreasing.

(5) If there exist two elements x0, y0 ∈ X with G(x0, y0) ≤ F(x0, y0) and G(y0, x0) ≥ F(y0, x0), then there exists a point (x0, y0) ∈ X2 such that TG(x0, y0) ⊆ TF(x0, y0).

(6) For any x, y ∈ X, there exist u, v ∈ X such that F(x, y) = G(u, v) and F(y, x) = G(v, u), then TF (X2 ) ⊆ TG(X2).

(7) Assume there exists φ ∈ Φ such that

where

for all x, y, u, v ∈ X, where G(x, y) ≤ G(u, v) and G(y, x) ≥ G(v, u), then

δ(TF(x, y), TF(u, v)) ≤ φ(Mδ((x, y), (u, v))),

where

for all (x, y), (u, v) ∈ X2, where TG(x, y) ⊆ TG(u, v).

(8) If the pair {F, G} is generalized compatible, then the mappings TF and TG are O-compatible in (X2 , δ, ⊆).

(9) A point (x, y) ∈ X2 is a coupled coincidence point of F and G if and only if it is a coincidence point of TF and TG.

Proof. Statement (1) follows from Lemma 21 and (2), (3), (5), (6) and (9) are obvious.

(4) Assume that F is G-increasing with respect to ≤ and let (x, y), (u, v) ∈ X2 be such that TG(x, y) ⊆ TG(u, v). Then G(x, y) ≤ G(u, v) and G(y, x) ≥ G(v, u). Since F is G-increasing with respect to ≤, we have that F(x, y) ≤ F(u, v) and F(y, x) ≥ F(v, u). Therefore TF(x, y) ⊆ TF(u, v) which shows that TF is (TG, ⊆)-non-decreasing.

(7) Let (x, y), (u, v) ∈ X2 be such that TG(x, y) ⊆ TG(u, v). Therefore G(x, y) ≤ G(u, v) and G(y, x) ≥ G(v, u). From (32), we have

Furthermore G(y, x) ≥ G(v, u) and G(x, y) ≤ G(u, v), the contractive condition (32) implies that

Combining (33) and (34), we get

It follows from (35) that

(8) Let {(xn, yn)} ⊆ X2 be any sequence such that and (Note that it is not require to suppose that {TG(xn, yn)} is ⊆-monotone). Thus

and

Therefore

Since the pair {F, G} is generalized compatible, therefore

In particular,

Hence, the mappings TF and TG are O-compatible in (X2 , δ, ⊆).                                         ⧠

Theorem 24. Let (X, ≤) be a partially ordered set such that there exists a complete metric d on X. Assume F, G : X2 → X be two generalized compatible mappings such that F is G-increasing with respect to ≤, G is continuous and there exist two elements x0, y0 ∈ X with

G(x0, y0) ≤ F(x0, y0) and G(y0, x0) ≥ F(y0, x0).

Suppose that there exists φ ∈ Φ satisfying (32) and for any x, y ∈ X, there exist u, v ∈ X such that

Also suppose that either

(a) F is continuous or

(b) (X, d, ≤) is regular.

Then F and G have a coupled coincidence point.

Proof. It is only require to use Theorem 22 to the mappings T = TF and g = TG in the ordered metric space (X2 , δ, ⊆) with Lemma 23.                                         ⧠

Corollary 25. Let (X, ≤) be a partially ordered set such that there exists a complete metric d on X. Assume F, G : X2 → X be two commuting mappings satisfying (32) and (36) such that F is G-increasing with respect to ≤, G is continuous and there exist two elements x0, y0 ∈ X with

G(x0, y0) ≤ F(x0, y0) and G(y0, x0) ≥ F(y0, x0).

Also suppose that either

(a) F is continuous or

(b) (X, d, ≤) is regular.

Then F and G have a coupled coincidence point.

Next, we deduce results without g-mixed monotone property of F.

Corollary 26. Let (X, ≤) be a partially ordered set such that there exists a complete metric d on X, F : X × X → X and g : X → X be two compatible mappings such that F is g-increasing with respect to ≤. Assume there exists φ ∈ Φ such that

where

for all x, y, u, v ∈ X, where gx ≤ gu and gy ≥ gv. Furthermore F(X × X) ⊆ g(X), g is continuous and monotone increasing with respect to ≤ . Also suppose that either

(a) F is continuous or

(b) (X, d, ≤) is regular.

If there exist two elements x0, y0 ∈ X with

gx0 ≤ F(x0, y0) and gy0 ≥ F(y0, x0).

Then F and g have a coupled coincidence point.

Corollary 27. Let (X, ≤) be a partially ordered set such that there exists a complete metric d on X. Assume F : X × X → X and g : X → X be two commuting mappings satisfying (37) such that F is g-increasing with respect to ≤ . Furthermore F(X × X) ⊆ g(X), g is continuous and monotone increasing with respect to ≤ . Also suppose that either

(a) F is continuous or

(b) (X, d, ≤) is regular.

If there exist two elements x0, y0 ∈ X with

gx0 ≤ F(x0, y0) and gy0 ≥ F(y0, x0).

Then F and g have a coupled coincidence point.

Now, we deduce result without mixed monotone property of F.

Corollary 28. Let (X, ≤) be a partially ordered set such that there exists a complete metric d on X. Assume F : X × X → X be an increasing mapping with respect to ≤ and there exists φ ∈ Φ such that

d(F(x, y), F(u, v)) ≤ φ(m(x, y, u, v)),

where

for all x, y, u, v ∈ X, where x ≤ u and y ≥ v. Also suppose that either

(a) F is continuous or

(b) (X, d, ≤) is regular.

If there exist two elements x0, y0 ∈ X with

x0 ≤ F(x0, y0) and y0 ≥ F(y0, x0).

Then F has a coupled fixed point.

Example 29. Suppose that X = [0, 1], equipped with the usual metric d : X × X → [0, +∞) with the natural ordering of real numbers ≤ . Let F, G : X × X → X be defined as

Define φ : [0, +∞) → [0, +∞) as follows

First, we shall show that the contractive condition (32) holds for the mappings F and G. Let x, y, u, v ∈ X such that G(x, y) ≤ G(u, v) and G(y, x) ≥ G(v, u), we have

Thus the contractive condition (32) holds for all x, y, u, v ∈ X. In addition, like in [11], all the other conditions of Theorem 24 are satisfied and z = (0, 0) is a coincidence point of F and G.

Remark 30. Using the same technique that can be used in [12 − 14, 18, 19, 20] it is possible to derive tripled, quadruple and in general, multidimensional coincidence point theorems from Theorem 22.

References

  1. R.P. Agarwal, R.K. Bisht & N. Shahzad: A comparison of various noncommuting conditions in metric fixed point theory and their applications. Fixed Point Theory Appl. 2014, Article ID 38.
  2. T.G. Bhaskar & V. Lakshmikantham: Fixed point theorems in partially ordered metric spaces and applications. Nonlinear Anal. 65 (2006), no. 7, 1379-1393. https://doi.org/10.1016/j.na.2005.10.017
  3. B.S. Choudhury & A. Kundu: A coupled coincidence point results in partially ordered metric spaces for compatible mappings. Nonlinear Anal. 73 (2010), 2524-2531. https://doi.org/10.1016/j.na.2010.06.025
  4. B. Deshpande & A. Handa. Nonlinear mixed monotone-generalized contractions on partially ordered modified intuitionistic fuzzy metric spaces with application to integral equations. Afr. Mat. 26 (2015), no. 3-4, 317-343. https://doi.org/10.1007/s13370-013-0204-0
  5. B. Deshpande & A. Handa, Application of coupled fixed point technique in solving integral equations on modified intuitionistic fuzzy metric spaces. Adv. Fuzzy Syst. 2014, Article ID 348069.
  6. H.S. Ding, L. Li & S. Radenovic: Coupled coincidence point theorems for generalized nonlinear contraction in partially ordered metric spaces. Fixed Point Theory Appl. 2012:96.
  7. I.M. Erhan, E. Karapınar, A. Roldan & N. Shahzad: Remarks on coupled coincidence point results for a generalized compatible pair with applications. Fixed Point Theory Appl. 2014:207.
  8. K. Goebel: A coincidence theorem. Bull. Acad. Pol. Sci., Ser. Sci. Math. Astron. Phys. 16 (1968), 733-735.
  9. D. Guo & V. Lakshmikantham: Coupled fixed points of nonlinear operators with applications. Nonlinear Anal. 11 (1987), no. 5, 623-632. https://doi.org/10.1016/0362-546X(87)90077-0
  10. N.M. Hung, E. Karapınar & N.V. Luong: Coupled coincidence point theorem for O-compatible mappings via implicit relation. Abstr. Appl. Anal. 2012, Article ID 796964.
  11. N. Hussain, M. Abbas, A. Azam & J. Ahmad: Coupled coincidence point results for a generalized compatible pair with applications. Fixed Point Theory Appl. 2014:62.
  12. E. Karapınar & A. Roldan: A note on n-Tuplet fixed point theorems for contractive type mappings in partially ordered metric spaces. J. Inequal. Appl. 2013, Article ID 567.
  13. E. Karapınar, A. Roldan, C. Roldan & J. Martinez-Moreno: A note on N-Fixed point theorems for nonlinear contractions in partially ordered metric spaces. Fixed Point Theory Appl. 2013, Article ID 310.
  14. E. Karapinar, A. Roldan, N. Shahzad & W. Sintunavarat: Discussion on coupled and tripled coincidence point theorems for ϕ−contractive mappings without the mixed g-monotone property. Fixed Point Theory Appl. 2014, Article ID 92.
  15. V. Lakshmikantham & L. Ciric: Coupled fixed point theorems for nonlinear contractions in partially ordered metric spaces. Nonlinear Anal. 70 (2009), no. 12, 4341-4349. https://doi.org/10.1016/j.na.2008.09.020
  16. N.V. Luong & N.X. Thuan: Coupled fixed points in partially ordered metric spaces and application. Nonlinear Anal. 74 (2011), 983-992. https://doi.org/10.1016/j.na.2010.09.055
  17. N.V. Luong & N.X. Thuan: Coupled points in ordered generalized metric spaces and application to integro-differential equations. Comput. Math. Appl. 62 (2011), no. 11, 4238-4248. https://doi.org/10.1016/j.camwa.2011.10.011
  18. S.A. Al-Mezel, H. Alsulami, E. Karapinar & A. Roldan: Discussion on multidimensional coincidence points via recent publications. Abstr. Appl. Anal. 2014, Article ID 287492.
  19. A. Roldan, J. Martinez-Moreno, C. Roldan & E. Karapinar: Some remarks on multi-dimensional fixed point theorems. Fixed Point Theory Appl. 2013, Article ID 158.
  20. B. Samet, E. Karapinar, H. Aydi & V.C. Rajic: Discussion on some coupled fixed point theorems. Fixed Point Theory Appl. 2013:50.