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.
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