Abstract
In this article, we present tripled coincidence point theorems for F: X^{3 }→ X and g: X → X satisfying weak φcontractions in partially ordered metric spaces. We also provide nontrivial examples to illustrate our results and new concepts presented herein. Our results unify, generalize and complement various known comparable results from the current literature, Berinde and Borcut and Abbas et al.
1 Introduction
Fixed point theory has fascinated hundreds of researchers since 1922 with the celebrated Banach's fixed point theorem. This theorem provides a technique for solving a variety of applied problems in mathematical sciences and engineering. There exists a last literature on the topic and this is a very active field of research at present. There are great number of generalizations of the Banach contraction principle. Bhaskar and Lakshmikantham [1] introduced the notion of coupled fixed point and proved some coupled fixed point results under certain conditions, in a complete metric space endowed with a partial order. Later, Lakshmikantham and Ćirić [2] extended these results by defining the mixed gmonotone property. More accurately, they proved coupled coincidence and coupled common fixed point theorems for a mixed gmonotone mapping in a complete metric space endowed with a partial order. Karapınar [3,4] generalized these results on a complete cone metric space endowed with a partial order. For other results on coupled fixed point theory, we address the readers to [513].
To make our exposition self contained, in this section we recall some previous notations and known results.
For simplicity, we denote from now on by X^{k}, where k ∈ ℕ and X be a nonempty set.
Let (X, ≤) be a partially ordered set. According to [1], the mapping F: X^{2 }→ X is said to have mixed monotone property if F(x, y) is monotone nondecreasing in x and is monotone nonincreasing in y, that is, for any x, y ∈ X,
An element (x, y) ∈ X^{2 }is said to be a coupled fixed point of the mapping F: X^{2 }→ X if
Theorem 1.1. ([1]) Let (X, ≤) be an ordered set such that there exists a metric d on X such that (X, d) is complete. Let F: X^{2 }→ X be a continuous mapping having the mixed monotone property on X. Assume that there exists k ∈ [0, 1) with
If there exist x_{0}, y_{0 }∈ X such that x_{0 }≤ F(x_{0}, y_{0}) and F(y_{0}, x_{0}) ≤ y_{0}, then, there exist x, y ∈ X such that x = F(x, y) and y = F(y, x).
Recently, Samet and Vetro [14] introduced the notion of fixed point of Norder as natural extension of that of coupled fixed point and established some new coupled fixed point theorems in complete metric spaces, using a new concept of Finvariant set. Later, Berinde and Borcut [15] obtained existence and uniqueness of triple fixed point results in a complete metric space, endowed with a partial order.
Again, let (X, ≤) be a partially ordered set. In accordance with [15], the mapping F: X^{3 }→ X is said to have the mixed monotone property if for any x, y, z ∈ X
An element (x, y, z) ∈ X^{3 }is called a tripled fixed point of F if
Berinde and Borcut [15] proved the following theorem.
Theorem 1.2. ([15]) Let (X, ≤) be a partially ordered set and (X, d) be a complete metric space. Let F: X^{3 }→ X be a mapping having the mixed monotone property on X. Assume that there exist constants a, b, c ∈ [0, 1) such that a + b + c < 1 for which
for all x ≥ u, y ≤ υ, z ≥ w. Assume either
(I) F is continuous, or
(II) X has the following properties:
(i) if nondecreasing sequence x_{n }→ x, then x_{n }≤ x for all n,
(ii) if nonincreasing sequence y_{n }→ y, then y_{n }≥ y for all n.
If there exist x_{0}, y_{0}, z_{0 }∈ X such that
then there exist x, y, z ∈ X such that
In this article, we establish tripled coincidence point theorems for F: X^{3 }→ X and g: X → X satisfying nonlinear contractive conditions, in partially ordered metric spaces. The presented theorems extend and improve some results in litterature.
2 Main results
We shall start this section by recalling the following basic notions, introduced by [Abbas, Aydi and Karapınar, Tripled common fixed point in partially ordered metric spaces, submitted]. In this respect, let us consider (X, ≤) a partially ordered set, F: X^{3 }→ X and g: X → X two mappings. The mapping F is said to have the mixed gmonotone property if for any x, y, z ∈ X
An element (x, y, z) is called a tripled coincidence point of F and g if
while (gx, gy, gz) is said a tripled point of coincidence of mappings F and g. Moreover, (x, y, z) is called a tripled common fixed point of F and g if
At last, mappings F and g are called commutative if
In the same paper, they proved the following result.
Theorem 2.1. Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete metric space. Assume there is a function φ: [0, +∞) → [0, +∞) such that φ(t) < t for each t > 0. Also suppose F: X^{3 }→ X and g: X → X are such that F has the mixed gmonotone property and suppose there exist p, q, r ∈ [0, 1) with p + 2q + r < 1 such that
for any x, y, z ∈ X for which gx > gu, gυ ≥ gy and gz ≥ gw.
Suppose F(X^{3}) ⊂ g(X), g is continuous and commutes with F. Suppose either
(a) F is continuous, or
(b) X has the following properties:
(i) if nondecreasing sequence gx_{n }→ x (respectively, gz_{n }→ z), then gx_{n }≤ x (respectively, gz_{n }≤ z) for all n,
(ii) if nonincreasing sequence gy_{n }→ y, then gy_{n }≥ y for all n.
If there exist x_{0}, y_{0}, z_{0 }∈ X such that gx_{0 }≤ F(x_{0}, y_{0}, z_{0}), gy_{0 }≥ F(y_{0}, x_{0}, y_{0}) and gz_{0 }≤ F(z_{0}, y_{0}, x_{0}), then there exist x, y, z ∈ X such that
that is, F and g have a tripled coincidence point.
Before starting to introduce our results, let us consider the set of functions
Our first main result is the following:
Theorem 2.2. Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that (X,d) is a complete metric space. Suppose F: X^{3 }→ X and g: X → X are such that F has the mixed gmonotone property and F(X^{3}) ⊂ g(X). Assume there is a function φ ∈ Φ such that
for any x, y, z, u, υ, w ∈ X for which gx ≥ gu, gυ ≥ gy and gz ≥ gw. Assume that F is continuous, g is continuous and commutes with F. If there exist x_{0}, y_{0}, z_{0 }∈ X such that
then there exist x, y, z ∈ X such that
that is, F and g have a tripled coincidence point.
Proof. Let x_{0}, y_{0}, z_{0 }∈ X be such that gx_{0 }≤ F(x_{0}, y_{0}, z_{0}), gy_{0 }≥ F(y_{0}, x_{0}, y_{0}) and gz_{0 }≤ F(z_{0}, y_{0}, x_{0}). We can choose x_{1}, y_{1}, z_{1 }∈ X such that
This can be done because F(X^{3}) ⊂ g(X). Continuing this process, we construct sequences {x_{n}}, {y_{n}}, and {z_{n}} in X such that
By induction, we will prove that
Since gx_{0 }≤ F(x_{0}, y_{0}, z_{0}), gy_{0 }≥ F(y_{0}, x_{0}, y_{0}), and gz_{0 }≤ F(z_{0}, y_{0}, x_{0}), therefore by (2.4) we have
Thus (2.6) is true for n = 0. We suppose that (2.6) is true for some n > 0. Since F has the mixed gmonotone property, by gx_{n }≤ gx_{n}_{+1}, gy_{n}_{+1 }≤ gy_{n}, and gz_{n }≤ gz_{n}_{+1}, we have that
and
That is, (2.6) is true for any n ∈ ℕ. If for some k ∈ ℕ
then, by (2.5), (x_{k}, y_{k}, z_{k}) is a tripled coincidence point of F and g. From now on, we assume that at least
for any n ∈ ℕ. From (2.6) and the inequality (2.2)
For each n ≥ 1, take
One can write
By (2.7), we have δ_{n }> 0. Having in mind φ(t) < t for each t > 0, so we have φ(δ_{n}) < δ_{n}. From (2.9), we get
that is, the sequence {δ_{n}} is nonnegative and decreasing. Therefore, there exists some δ ≥ 0 such that
We shall prove that δ = 0. Assume, on the contrary, that δ > 0. Then by letting n → +∞ in (2.9) we have
which is a contradiction. Thus, δ = 0, and by (2.10), we get
We now prove that {gx_{n}}, {gy_{n}}, and {gz_{n}} are Cauchy sequences in (X,d).
Suppose, on the contrary, that at least one of {gx_{n}}, {gy_{n}}, and {gz_{n}} is not a Cauchy sequence. So, there exists ε > 0 for which we can find subsequences {gx_{n}_{(k)}}, {gx_{m}_{(k)}} of {gx_{n}}, {gy_{n}_{(k)}}, {gy_{m}_{(k)}} of {gy_{n}}, and {gz_{n}_{(k)}}, {gz_{m}_{(k)}} of {gz_{n}} with n(k) > m(k) ≥ k such that
Additionally, corresponding to m(k), we may choose n(k) such that it is the smallest integer satisfying (2.12) and n(k) > m(k) ≥ k. Thus,
By using triangle inequality and having in mind (2.12) and (2.13)
Letting k → ∞ in (2.14) and using (2.11)
Again by triangle inequality,
Since n(k) > m(k), then
Take (2.17) in (2.2) to get
Combining this in (2.16), we obtain that
Letting k → ∞ and having in mind (2.11) and (2.15), we get
which is a contradiction. This shows that {gx_{n}}, {gy_{n}}, and {gz_{n}} are Cauchy sequences in (X, d).
Since X is complete, there exist x, y, z ∈ X such that
From (2.18) and the continuity of g.
From the commutativity of F and g, we have
Now we shall show that gx = F(x, y, z), gy = F(y, x, y), and gz = F(z, y, x).
Suppose that F is continuous. Letting n → +∞ in (2.20), therefore by (2.18) and (2.19), we obtain
and
We have proved that F and g have a tripled coincidence point.
Corollary 2.3. Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that (X,d) is a complete metric space. Suppose F: X^{3 }→ X and g: X → X are such that F has the mixed gmonotone property and F(X^{3}) ⊂ g(X). Assume there exists α ∈ [0,1) such that
for any x, y, z, u, υ, w ∈ X for which gx ≥ gu, gυ ≥ gy, and gz ≥ gw. Assume that F is continuous, g is continuous and commutes with F. If there exist x_{0}, y_{0}, z_{0 }∈ X such that
then there exist x, y, z ∈ X such that
that is, F and g have a tripled coincidence point.
Proof. It follows by taking φ(t) = αt in Theorem 2.2.
In the following theorem, we omit the continuity hypothesis of F. We need the following definition.
Definition 2.1. Let (X, ≤) be a partially ordered set and d be a metric on X. We say that (X, d,≤) is regular if the following conditions hold:
(i) if a nondecreasing sequence (x_{n}) is such that x_{n }→ x, then x_{n }≤ x for all n,
(ii) if a nonincreasing sequence (y_{n}) is such that y_{n }→ y, then y ≤ y_{n }for all n.
Theorem 2.4. Let (X, ≤) be a partially ordered set and d be a metric on X such that (X, d, ≤) is regular. Suppose that there exist φ ∈ Φ and mappings F: X^{3 }→ X and g: X → X such that (2.2) holds for any x, y, z, u, υ, w ∈ X for which gx ≥ gu, gυ ≥ gy and gz ≥ gw. Suppose also that (g(X), d) is complete, F has the mixed gmonotone property and F(X^{3}) ⊂ g(X). If there exist x_{0}, y_{0}, z_{0 }∈ X such that gx_{0 }≤ F(x_{0}, y_{0}, z_{0}), gy_{0 }≥ F(y_{0}, x_{0}, y_{0}), and gz_{0 }≤ F(z_{0}, y_{0}, x_{0}), then there exist x, y, z ∈ X such that
that is, F and g have a tripled coincidence point.
Proof. Proceeding exactly as in Theorem 2.2, we have that (gx_{n}), (gy_{n}), and (gz_{n}) are Cauchy sequences in the complete metric space (g(X), d). Then, there exist x, y, z ∈ X such that gx_{n }→ gx, gy_{n }→ gy, and gz_{n }→ gz. Since (gx_{n}) and (gz_{n}) are nondecreasing and (gy_{n}) is nonincreasing, using the regularity of (X, d, ≤), we have gx_{n }≤ gx, gz_{n }≤ gz, and gy ≤ gy_{n }for all n ≥ 0. If gx_{n }= gx, gy_{n }= gy, and gz_{n }= gz for some n ≥ 0, then gx = gx_{n }≤ gx_{n}_{+1 }≤ gx = gx_{n}, gz = gz_{n }≤ gz_{n}_{+1 }≤ gz = gz_{n}, and gy ≤ gy_{n}_{+1 }≤ gy_{n }= gy, which implies that gx_{n }= gx_{n}_{+1 }= F(x_{n}, y_{n}, z_{n}), gy_{n }= gy_{n}_{+1 }= F(y_{n}, x_{n}, y_{n}), and gz_{n }= gz_{n}_{+1 }= F(z_{n}, y_{n}, x_{n}), that is, (x_{n}, y_{n}, z_{n}) is a tripled coincidence point of F and g. Then, we suppose that (gx_{n}, gy_{n}, gz_{n}) ≠ (gx, gy, gz) for all n ≥ 0. Using the triangle inequality, (2.2) and the property φ(t) < t for all t > 0,
Taking n → ∞ in the above inequality we obtain that d(gx,F(x, y, z)) = 0, so gx = F(x, y, z).
Analogously, we find that
thus, we have proved that F and g have a tripled coincidence point.
Corollary 2.5. Let (X, ≤) be a partially ordered set and suppose there is a metric d on X such that (X, ≤,d) is regular. Suppose F: X^{3 }→ X and g: X → X are such that F has the mixed gmonotone property and F(X^{3}) ⊂ g(X). Assume there exists α ∈ [0,1) such that
for any x, y, z, u, υ, w ∈ X for which gx ≥ gu, gυ ≥ gy, and gz ≥ gw. Suppose also that (g(X), d) is complete. If there exist x_{0}, y_{0}, z_{0 }∈ X such that
then there exist x, y, z ∈ X such that
that is, F and g have a tripled coincidence point.
Proof. It follows by taking φ(t) = αt in Theorem 2.4.
Now, we shall prove the existence and the uniqueness of a tripled common fixed point theorem. For a product X^{3 }= X × X × X of a partial ordered set (X, ≤), we define a partial ordering in the following way: For all (x, y, z), (u, υ, r) ∈ X^{3}
We say that (x, y, z) and (u, υ, w) are comparable if
Also, we say that (x, y, z) is equal to (u, υ, r) if and only if x = u, y = υ and z = r.
Theorem 2.6. In addition to hypothesis of Theorem 2.2, suppose that for all (x, y, z) and (u, υ, r) in X^{3}, there exists (a, b, c) in X^{3 }such that (F(a, b, c), F(b, a, b), F(c, b, a)) is comparable to (F(x, y, z), F(y, x, y), F(z, y, x)) and (F(u, υ, r), F(υ, u, υ), F(r, υ, u)). Also, assume that φ is nondecreasing. Then, F and g have a unique tripled common fixed point (x, y, z), that is
Proof. Due to Theorem 2.2, the set of tripled coincidence points of F and g is not empty. Assume now, that (x, y, z) and (u,υ,r) are two tripled coincidence points of F and g, that is,
We shall show that (gx, gy, gz) and (gu, gυ, gr) are equal.
By assumption, there is (a, b, c) in X^{3 }such that (F(a, b, c), F(b, a, b), F(c, b, a)) is comparable to (F(x, y, z), F(y, x, y), F(z, y, x)) and (F(u, υ, r), F(υ, u, v), F(r, υ, u)).
Define the sequences {ga_{n}},{gb_{n}}, and {gc_{n}} such that a = a_{0}, b = b_{0}, c = c_{0 }and
for all n. Further, set x_{0 }= x, y_{0 }= y, z_{0 }= z and u_{0 }= u, υ_{0 }= υ, r_{0 }= r, and similar define the sequences {gx_{n}},{gy_{n}}, {gz_{n}} and {gu_{n}},{gυ_{n}}, {gr_{n}}. Then,
for all n ≥ 1. Since (F(x, y, z), F(y, x, y), F(z, y, x)) = (gx_{1}, gy_{1}, gz_{1}) = (gx, gy, gz) is comparable to (F(a, b, c), F(b, a, b), F(c, b, a)) = (ga_{1}, gb_{1}, gc_{1}), then it is easy to show that (gx, gy, gz) ≥ (ga_{1}, gb_{1}, gc_{1}). Recursively, we get that
By (2.24) and (2.2), we have
Set
From (2.25), we deduce that γ_{n}_{+1 }≤ φ(γ_{n}). Since φ is nondecreasing, it follows
From the definition of Φ, we get . Then, we have . Thus,
By analogy, we show that
Combining (2.26) and (2.27) yields that (gx, gy, gz) and (gu, gυ, gr) are equal.
Since gx = F(x, y, z), gy = F(y, x, y), and gz = F(z, y, x), by the commutativity of F and g, we have
Denote gx = x', gy = y', and gz = z'. From the precedent identities,
that is, (x', y', z') is a tripled coincidence point of F and g. Consequently, (gx', gy', gz') and (gx, gy, gz) are equal, that is, gx = gx', gy = gy', and gz = gz'.
We deduce gx' = gx = x', gy' = gy = y', and gz' = gz = z'. Therefore, (x', y', z') is a tripled common fixed of F and g. Its uniqueness follows from Theorem 2.2.
3 Examples
Remark that Theorem 2.2 is more general than Theorem 2.1, since the contractive condition (2.2) is weaker than (2.1), a fact which is clearly illustrated by the following example.
Example 3.1. Let X = ℝ with d(x, y) = x  y and natural ordering and let g: X → X, F: X^{3 }→ X be given by
It is clear that F is continuous and has the mixed gmonotone property. We now take . We shall show that (2.2) holds for all gx ≥ gu, gy ≤ gυ, and gz ≤ gw.
Let x, y, z, u, υ, and w such that gx ≥ gu, gy ≤ gυ, and gz ≤ gw, and by definition of g, it means that x ≥ u, y ≤ υ and z ≤ w, so we have
which is the contractive condition (2.2). On the other hand, x_{0 }= 0, y_{0 }= 0, z_{0 }= 0 satisfy (2.3). All the hypotheses of Theorem 2.2 are verified, and (0,0,0) is a tripled coincidence point of F and g.
On the other hand, assume that (2.1) holds. Then, there exist p,q,r ≥ 0 such that p + 2q + r < 1 and φ satisfying φ(t) < t for each t > 0. If x > u, z = w and y = υ, we have
which implies for any n ≥ 1, and letting n → +∞, we get p ≥ 1, that is a contradiction. Thus, Theorem 2.1 is not applicable in this case.
Following example shows that Theorem 2.2 is more general than Theorem 1.2.
Example 3.2. Let X = ℝ be endowed with the usual ordering and the usual metric. Consider g: X → X and F: X^{3 }→ X be given by the formulas
Take φ: [0, ∞) → [0, ∞) be given by for all t ∈ [0, ∞).
It is clear that all conditions of Theorem 2.2 are satisfied. Moreover, (0,0,0) is a tripled coincidence point (also a common fixed point) of F and g.
Now, for x = u, z = w and υ > y, we have
for any k ∈ [0,1), that is the result of Berinde and Borcut [15]given by Theorem 1.2 is not applicable .
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors contributed equally and significantly in writing this article. All authors read and approve the final manuscript.
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