Keywords:
quadruple coincidence point; quadruple common fixed point; ordered set; metric space; generalized contractionAbstract
In this manuscript, we prove some quadruple coincidence and common fixed point theorems for F : X^{4 }→ X and g : X → X satisfying generalized contractions in partially ordered metric spaces. Our results unify, generalize and complement various known results from the current literature. Also, an application to matrix equations is given.
2000 Mathematics subject Classifications: 46T99; 54H25; 47H10; 54E50.
1 Introduction and preliminaries
Existence of fixed points in partially ordered metric spaces was first investigated by Turinici [1], where he extended the Banach contraction principle in partially ordered sets. In 2004, Ran and Reurings [2] presented some applications of Turinici's theorem to matrix equations. Following these initial articles, some remarkable results were reported see, e.g., [313].
Gnana Bhashkar and Lakshmikantham in [14] introduced the concept of a coupled fixed point of a mapping F : X × X → X and investigated some coupled fixed point theorems in partially ordered complete metric spaces. Later, Lakshmikantham and Ćirić [15] proved coupled coincidence and coupled common fixed point theorems for nonlinear mappings F : X × X → X and g : X → X in partially ordered complete metric spaces. Various results on coupled fixed point have been obtained, since then see, e.g., [6,9,1633]. Recently, Berinde and Borcut [34] introduced the concept of tripled fixed point in ordered sets.
For simplicity, we denote by X^{k }where k ∈ ℕ. Let us recall some basic definitions.
Definition 1.1 (See [34]) Let (X, ≤) be a partially ordered set and F: X^{3 }→ X. The mapping F is said to has the mixed monotone property if for any x, y, z ∈ X
Definition 1.2 Let F : X^{3 }→ X. An element (x, y, z) is called a tripled fixed point of F if
Also, Berinde and Borcut [34] proved the following theorem:
Theorem 1.1 Let (X,≤, d) be a partially ordered set and suppose there is a metric d on X such that (X, d) is a complete metric space. Let F : X^{3 }→ X having the mixed monotone property. Suppose there exist j, r, l ≥ 0 with j + r + l < 1 such that
for any x, y, z ∈ X for which × ≤ u, v ≤ y and z ≤ w. Suppose either F is continuous or X has the following properties:
1. if a nondecreasing sequence x_{n }→ x, then x_{n }≤ x for all n,
2. if a nonincreasing sequence y_{n }→ y, then y ≤ y_{n }for all n.
If there exist x_{0, }y_{0, }z_{0 }∈ X such that x_{0 }≤ F (x_{0, }y_{0, }z_{0}), y_{0 }≥ F (y_{0, }x_{0, }z_{0}) and z_{0 }≤ F (z_{0, }y_{0, }x_{0}), then there exist x, y, z ∈ X such that
that is, F has a tripled fixed point.
Recently, Aydi et al. [35] introduced the following concepts.
Definition 1.3 Let (X, ≤) be a partially ordered set. Let F : X^{3 }→ X and g : X → X. The mapping F is said to has the mixed gmonotone property if for any x, y, z ∈ X
Definition 1.4 Let F : X^{3 }→ X and g : X → X. An element (x, y, z) is called a tripled coincidence point of F and g if
(gx, gy, gz) is said a tripled point of coincidence of F and g.
Definition 1.5 Let F : X^{3 }→ X and g : X → X. An element (x, y, z) is called a tripled common fixed point of F and g if
Definition 1.6 Let X be a nonempty set. Then we say that the mappings F : X^{3 }→ X and
g : X → X are commutative if for all x, y, z ∈ X
The notion of fixed point of order N ≥ 3 was first introduced by Samet and Vetro [36]. Very recently, Karapinar used the concept of quadruple fixed point and proved some fixed point theorems on the topic [37]. Following this study, quadruple fixed point is developed and some related fixed point theorems are obtained in [3841].
Definition 1.7 [38]Let X be a nonempty set and F : X^{4 }→ X be a given mapping. An element (x, y, z, w) ∈ X × X × X × X is called a quadruple fixed point of F if
Let (X, d) be a metric space. The mapping given by
defines a metric on X^{4}, which will be denoted for convenience by d.
Definition 1.8 [38]Let (X, ≤) be a partially ordered set and F : X^{4 }→ X be a mapping. We say that F has the mixed monotone property if F (x, y, z, w) is monotone nondecreasing in x and z and is monotone nonincreasing in y and w; that is, for any x, y, z, w ∈ X,
and
In this article, we establish some quadruple coincidence and common fixed point theorems for F : X^{4 }→ X and g : X → X satisfying nonlinear contractions in partially ordered metric spaces. Also, some interesting corollaries are derived and an application to matrix equations is given.
2 Main results
We start this section with the following definitions.
Definition 2.1 Let (X, ≤) be a partially ordered set. Let F : X^{4 }→ X and g : X → X. The mapping F is said to has the mixed gmonotone property if for any x, y, z, w ∈ X
Definition 2.2 Let F : X^{4 }→ X and g : X → X. An element (x, y, z, w) is called a quadruple coincidence point of F and g if
(gx, gy, gz, gw) is said a quadruple point of coincidence of F and g.
Definition 2.3 Let F : X^{4 }→ X and g : X → X. An element (x, y, z, w) is called a quadruple common fixed point of F and g if
Definition 2.4 Let X be a nonempty set. Then we say that the mappings F : X^{4 }→ X and g : X → × are commutative if for all x, y, z, w ∈ X
Let Φ be the set of all functions ϕ : [0, ∞) → [0, ∞) such that:
1. ϕ(t) < t for all t ∈ (0,+∞).
For simplicity, we define the following.
Now, we state the first main result of this article.
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. Suppose F : X^{4 }→ X and g : X → X are such that F is continuous and has the mixed gmonotone property. Assume also that there exist ϕ ∈ Φ and L ≥ 0 such that
for any x, y, z, w, u, v, h, l ∈ X for which gx ≤ gu, gv ≤ gy, gz ≤ gh and gl ≤ gw. Suppose F (X^{4}) ⊂ g(X), g is continuous and commutes with F. If there exist x_{0}, y_{0}, z_{0}, w_{0 }∈ X such that
then there exist x, y, z, w ∈ X such that
that is, F and g have a quadruple coincidence point.
Proof. Let x_{0, }y_{0, }z_{0, }w_{0 }∈ X such that
Since F (X^{4}) ⊂ g(X), then we can choose x_{1}, y_{1}, z_{1}, w_{1 }∈ X such that
Taking into account F (X^{4}) ⊂ g(X), by continuing this process, we can construct sequences {x_{n}}, {y_{n}}, {z_{n}}, and {w_{n}} in X such that
We shall show that
For this purpose, we use the mathematical induction. Since, gx_{0 }≤ F (x_{0}, y_{0}, z_{0}, w_{0}), gy_{0 }≥ F (y_{0}, z_{0}, w_{0}, x_{0}), gz_{0 }≤ F (z_{0}, w_{0}, x_{0}, y_{0}), and gw_{0 }≥ F (w_{0}, x_{0}, y_{0}, z_{0}), then by (4), we get
that is, (6) holds for n = 0.
We presume that (6) holds for some n > 0. As F has the mixed gmonotone property and gx_{n }≤ gx_{n+1}, gy_{n+1 }≤ gy_{n}, gz_{n }≤ gz_{n+1 }and gw_{n+1 }≤ gw_{n}, we obtain
and
Thus, (6) holds for any n ∈ ℕ. Assume for some n ∈ ℕ,
then, by (5), (x_{n}, y_{n}, z_{n}, w_{n}) is a quadruple coincidence point of F and g. From now on, assume for any n ∈ ℕ that at least
By (2) and (5), it is easy that
Due to (3) and (8), we have
and
Having in mind that ϕ (t) < t for all t > 0, so from (9)(12) we obtain that
It follows that
Thus, {max{d(gx_{n}, gx_{n+1}), d(gy_{n}, gy_{n+1}), d(gz_{n}, gz_{n+1}), d(gw_{n}, gw_{n+1})}} is a positive decreasing sequence. Hence, there exists r ≥ 0 such that
Suppose that r > 0. Letting n → +∞ in (13), we obtain that
It is a contradiction. We deduce that
We shall show that {gx_{n}}, {gy_{n}}, {gz_{n}}, and {gw_{n}} are Cauchy sequences in the metric space (X, d). Assume the contrary, that is, one of the sequence {gx_{n}}, {gy_{n}}, {gz_{n}} or {gw_{n}} is not a Cauchy, that is,
or
This means that there exists ε > 0, for which we can find subsequences of integers (m_{k}) and (n_{k}) with n_{k }> m_{k }> k such that
Further, corresponding to m_{k }we can choose n_{k }in such a way that it is the smallest integer with n_{k }> m_{k }and satisfying (17). Then
By triangular inequality and (18), we have
Thus, by (16) we obtain
Similarly, we have
and
Again by (18), we have
Letting k → + ∞ and using (16), we get
and
Using (17) and (23)(26), we have
By (16), it is easy to see that
Now, using inequality (3), we obtain
and
From (29)(32), we deduce that
Letting k → +∞ in (33) and having in mind (27) and (28), we get that
it is a contradiction. Thus, {gx_{n}}, {gy_{n}}, {gz_{n}}, and {gw_{n}} are Cauchy sequences in (X, d).
Since (X, d) is complete, there exist x, y, z, w ∈ X such that
From (34) and the continuity of g, we have
From (5) and the commutativity of F and g, we have
and
Now we shall show that gx = F (x, y, z, w), gy = F (y, z, w, x), gz = F (z, w, x, y), and gw = F (w, x, y, z).
By letting n → +∞ in (36)  (39), by (34), (35) and the continuity of F , we obtain
and
We have proved that F and g have a quadruple coincidence point. This completes the proof of Theorem 2.1.
In the following theorem, we omit the continuity hypothesis of F. We need the following definition.
Definition 2.5 Let (X, ≤) be a partially ordered metric set and d be a metric on X. We say that (X, d, ≤) is regular if the following conditions hold:
(i) if nondecreasing sequence a_{n }→ a, then a_{n }≤ a for all n,
(ii) if nonincreasing sequence b_{n }→ b, then b ≤ b_{n }for all n.
Theorem 2.2 Let (X, ≤) be a partially ordered set and d be a metric on X such that (X, d, ≤) is regular. Suppose F : X^{4 }→ X and g : X → X are such that F has the mixed gmonotone property. Assume that there exist ϕ ∈ Φ and L ≥ 0 such that
for any x, y, z, w, u, v, h, l ∈ X for which gx ≤ gu, gv ≤ gy, gz ≤ gh, and gl ≤ gw. Also, suppose F (X^{4}) ⊂ g(X) and (g(X), d) is a complete metric space. If there exist x_{0}, y_{0}, z_{0}, w_{0 }∈ X such that gx_{0 }≤ F (x_{0}, y_{0}, z_{0}, w_{0}), gy_{0 }≥ F (y_{0}, z_{0}, w_{0}, x_{0}), gz_{0 }≤ F (z_{0}, w_{0}, x_{0}, y_{0}) and gw_{0 }≥ F (w_{0}, x_{0}, y_{0}, z_{0}), then there exist x, y, z, w ∈ X such that
that is, F and g have a quadruple coincidence point.
Proof. Proceeding exactly as in Theorem 2.1, we have that {gx_{n}}, {gy_{n}}, {gz_{n}}, and {gw_{n}} are Cauchy sequences in the complete metric space (g(X), d). Then, there exist x, y, z, w ∈ X such that
Since {gx_{n}}, {gz_{n}} are nondecreasing and {gy_{n}}, {gw_{n}} are nonincreasing, then since (X, d, ≤) is regular we have
for all n. If gx_{n }= gx, gy_{n }= gy, gz_{n }= gz, and gw_{n }= gw for some n ≥ 0, then gx = gx_{n }≤ gx_{n+1 }≤ gx = gx_{n}, gy ≤ gy_{n+1 }≤ gy_{n }= gy, gz = gz_{n }≤ gz_{n+1 }≤ gz = gz_{n}, and gw ≤ gw_{n+1 }≤ gw_{n }= gw, which implies that
and
that is, (x_{n}, y_{n}, z_{n}, w_{n}) is a quadruple coincidence point of F and g. Then, we suppose that (gx_{n}, gy_{n}, gz_{n}, gw_{n}) ≠ (gx, gy, gz, gw) for all n ≥ 0. By (3), consider now
Taking n → ∞ and using (44), the quantity M(x_{n}, y_{n}, z_{n}, w_{n}, x, y, z, w) tends to 0 and so the righthand side of (45) tends to 0, hence we get that d(gx, F (x, y, z, w)) = 0. Thus, gx = F (x, y, z, w). Analogously, one finds
Thus, we proved that F and g have a quartet coincidence point. This completes the proof of Theorem 2.2.
Corollary 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. Suppose F : X^{4 }→ X and g : X → X are such that F is continuous and has the mixed gmonotone property. Assume also that there exist ϕ ∈ Φ a nondecreasing function and L ≥ 0 such that
for any x, y, z, w, u, v, h, l,∈ X for which gx ≤ gu, gv ≤ gy, gz ≤ gw, and gl ≤ gw. Suppose F (X^{4}) ⊂ g(X), g is continuous and commutes with F .
If there exist x_{0}, y_{0}, z_{0}, w_{0 }∈ X such that gx_{0 }≤ F (x_{0}, y_{0}, z_{0}, w_{0}), gy_{0 }≥ F (y_{0}, z_{0}, w_{0}, x_{0}), gz_{0 }≤ F (z_{0}, w_{0}, x_{0}, y_{0}), and gw_{0 }≥ F (w_{0}, x_{0}, y_{0}, z_{0}), then there exist x, y, z, w ∈ X such that
Proof. It suffices to remark that
Then, we apply Theorem 2.1, since ϕ is assumed to be nondecreasing.
Similarly, as an easy consequence of Theorem 2.2 we have the following corollary.
Corollary 2.2 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^{4 }→ X and g : X → X are such that F has the mixed gmonotone property. Assume also that there exist ϕ ∈ Φ a nondecreasing function and L ≥ 0 such that
for any x, y, z, w, u, v, h, l ∈ X for which gx ≤ gu, gv ≤ gy, gz ≤ gw, and gl ≤ gw. Also, suppose F (X^{4}) ⊂ g(X) and (g(X), d) is a complete metric space.
If there exist x_{0}, y_{0}, z_{0}, w_{0 }∈ X such that gx_{0 }≤ F (x_{0}, y_{0}, z_{0}, w_{0}), gy_{0 }≥ F (y_{0}, z_{0}, w_{0}, x_{0}), gz_{0 }≤ F (z_{0}, w_{0}, x_{0}, y_{0}), and gw_{0 }≥ F (w_{0}, x_{0}, y_{0}, z_{0}), then there exist x, y, z, w ∈ X such that
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^{4 }→ X and g : X → X are such that F is continuous and has the mixed gmonotone property. Assume that there exist k ∈ [0, 1) and L ≥ 0 such that
for any x, y, z, w, u, v, h, l ∈ X for which^{:}gx ≤ gu, gv ≤ gy, gz ≤ gw, and gl ≤ gw. Suppose F (X^{4}) ⊂ g(X), g is continuous and commutes with F.
If there exist x_{0}, y_{0}, z_{0}, w_{0 }∈ X such that gx_{0 }≤ F (x_{0}, y_{0}, z_{0}, w_{0}), gy_{0 }≥ F (y_{0}, z_{0}, w_{0}, x_{0}), gz_{0 }≤ F (z_{0}, w_{0}, x_{0}, y_{0}), and gw_{0 }≥ F (w_{0}, x_{0}, y_{0}, z_{0}), then there exist x, y, z, w ∈ X such that
Proof. It suffices to take ϕ (t) = kt in Theorem 2.1.
Corollary 2.4 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^{4 }→ X and g : X → X are such that F has the mixed gmonotone property. Assume that there exist k ∈ [0, 1) and L ≥ 0 such that
for any x, y, z, w, u, v, h, l ∈ X for which gx ≤ gu, gv ≤ gy, gz ≤ gw, and gl ≤ gw. Suppose F (X^{4}) ⊂ g(X) and (g(X), d) is a complete metric space.
If there exist x_{0}, y_{0}, z_{0}, w_{0 }∈ X such that gx_{0 }≤ F (x_{0}, y_{0}, z_{0}, w_{0}), gy_{0 }≥ F (y_{0}, z_{0}, w_{0, }x_{0}), gz_{0 }≤ F (z_{0}, w_{0}, x_{0}, y_{0}), and gw_{0 }≥ F (w_{0}, x_{0}, y_{0}, z_{0}), then there exist x, y, z, w ∈ X such that
Proof. It suffices to take ϕ (t) = kt in Theorem 2.2.
Corollary 2.5 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^{4 }→ X and g : X → X are such that F is continuous and has the mixed gmonotone property. Assume that there exist k ∈ [0, 1) and L ≥ 0 such that
for any x, y, z, w, ∈ X for which ^{:}gx ≤ gu, gv ≤ gy, gz ≤ gw, and gl ≤ gw. Also, suppose F (X^{4}) ⊂ g(X) and (g(X), g is continuous and commutes with F.
If there exist x_{0}, y_{0}, z_{0}, w_{0 }∈ X such that gx_{0 }≤ F (x_{0}, y_{0}, z_{0}, w_{0}), gy_{0 }≥ F (y_{0}, z_{0}, w_{0, }x_{0}), gz_{0 }≤ F (z_{0}, w_{0}, x_{0}, y_{0}), and gw_{0 }≥ F (w_{0}, x_{0}, y_{0}, z_{0}), then there exist x, y, z, w ∈ X such that
Proof. It suffices to take ϕ (t) = kt in Corollary 2.1.
Corollary 2.6 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^{4 }→ X and g : X → X are such that F has the mixed gmonotone property. Assume that there exist k ∈ [0, 1) and L ≥ 0 such that
for any x, y, z, w, ∈ X for which gx ≤ gu, gv ≤ gy, gz ≤ gw, and gl ≤ gw. Suppose F (X^{4}) ⊂ g(X) and (g(X), d) is a complete metric space.
If there exist x_{0}, y_{0}, z_{0}, w_{0 }∈ X such that gx_{0 }≤ F (x_{0}, y_{0}, z_{0}, w_{0}), gy_{0 }≥ F (y_{0}, z_{0}, w_{0, }x_{0}), gz_{0 }≤ F (z_{0}, w_{0}, x_{0}, y_{0}), and gw_{0 }≥ F (w_{0}, x_{0}, y_{0}, z_{0}), then there exist x, y, z, w ∈ X such that
Proof. It suffices to take ϕ (t) = kt in Corollary 2.2.
Remark 1 • Corollary 2.4 of Karapinar [39]is a particular case of Corollary 2.5 by taking L = 0 and g = I_{X }the identity on X.
• Corollary 2.4 of Karapinar [39]is a particular case of Corollary 2.6 by taking L = 0 and g = I_{X }.
• Theorem 2.6 of Berinde and Karapinar [40]is a particular case of Corollary 2.1 by taking L = 0.
• Theorem 2.6 of Berinde and Karapinar [40]is a particular case of Corollary 2.1 by taking L = 0.
Now, we shall prove the existence and uniqueness of quadruple common fixed point. For a product X^{4 }of a partial ordered set (X, ≤), we define a partial ordering in the following way: For all (x, y, z, w), (u, v, r, h) ∈ X^{4}
We say that (x, y, z, w) and (u, v, r, l) are comparable if
Also, we say that (x, y, z, w) is equal to (u, v, r, l) if and only if x = u, y = v, z = r and w = l.
Theorem 2.3 In addition to hypotheses of Theorem 2.1, suppose that for all (x, y, z, w), (u, v, r, l) ∈ X^{4}, there exists(a, b, c, d) ∈ X^{4 }such that
is comparable to
Then, F and g have a unique quadruple common fixed point (x, y, z, w) such that
Proof. The set of quadruple coincidence points of F and g is not empty due to Theorem 2.1. Assume, now, (x, y, z, w) and (u, v, r, l) are two quadruple coincidence points of F and g, that is,
We shall show that (gx, gy, gz, gw) and (gu, gv, gr, gl) are equal. By assumption, there exists (a, b, c, d) ∈ X^{4 }such that (F (a, b, c, d), F (b, c, d, a), F (c, d, a, b), F (d, a, b, c)) is comparable to (F (x, y, z, w), F (y, z, w, x), F (z, w, x, y), F (w, x, y, z)) and (F (u, v, r, l), F (v, r, l, u), F (r, l, u, v), F (l, u, v, r)).
Define sequences {ga_{n}}, {gb_{n}}, {gc_{n}}, and {gd_{n}} such that
a_{0 }= a, b_{0 }= b, c_{0 }= c, d_{0 }= d and for any n ≥ 1
for all n. Further, set x_{0 }= x, y_{0 }= y, z_{0 }= z, w_{0 }= w and u_{0 }= u, v_{0 }= v, r_{0 }= r, l_{0 }= l and on the same way define the sequences {gx_{n}}, {gy_{n}}, {gz_{n}}, {gw_{n}} and {gu_{n}}, {gv_{n}}, {gr_{n}}, {gl_{n}}. Then, it is easy that
for all n ≥ 1. Since
is comparable to
then it is easy to show (gx, gy, gz, gw) ≥ (ga_{1}, gb_{1}, gc_{1}, gd_{1}). Recursively, we get that
From (2) and (47), it is obvious that
By (50), (51), and (3), we have
and
From (52)(55), it follows that
Therefore, for each n ≥ 1,
It is known that ϕ(t) < t and imply for each t > 0. Thus, from (57)
This yields that
Analogously, we may show that
Combining (58) and (59) yields that (gx, gy, gz, gw) and (gu, gv, gr, gl) are equal.
Since gx = F(x, y, z, w), gy = F(y, z, w, x), gz = F(z, w, x. y), and gz = F(z, w, x, y), by commutativity of F and g we have
and
where gx = x', gy = y', gz = z', and gw = w'. Thus, (x', y', z', w') is a quadruple coincidence point of F and g. Consequently, (gx', gy', gz', gz') and (gx, gy, gz, gw) are equal. We deduce
Therefore, (x', y', z', w') is a quadruple common fixed of F and g. Its uniqueness follows easily from (3).
Example 2.1 Let X = ℝ be endowed with the usual ordering and the usual metric, which is complete.
Let g: X → X and F: X^{4}→X be defined by
Take ϕ : [0, ∞) → [0, ∞) be given by for all t ∈ [0, ∞).
We will check that the contraction (3) is satisfied for all x, y, z, w, u, v, h, l ∈ X satisfying gx ≤ gu, gv ≤ gy, gz ≤ gh, and gl ≤ gw. In this case, we have
for arbitrary L ≥ 0.
It is obvious that the other hypotheses of Theorem 2.3 are satisfied. We deduce that (0, 0, 0, 0) is the unique quadruple common fixed point of F and g.
3 Application to matrix equations
In this section, we study the existence and uniqueness of solutions (X, Y, Z, T) to the system of matrix equations
where : the set of all n × n matrices, the set of all n × n positive definite matrices, and is the set of all n × n Hermitian matrices.
We endow with the partial order ≼ given by
for all , where tr is the trace operator. The space equipped with the metric induced by is a complete metric space for any positive definite matrix P (see [42]).
The following lemma will be useful for our application.
Lemma 3.1 Let A ≽ 0 and B ≽ 0 be n × n matrices. Then, we have
where . is the spectral norm.
Theorem 3.1 Suppose that there exists such that
Suppose also that
Then, the system (60) has one and only one solution .
Proof. Consider the mappings and defined by
For all i = 1. . . , 4 with gX_{1 }≼ gY_{1}, gY_{2 }≼ gX_{2}, gX_{3 }≼ gY_{3 }and gY_{4 }≼ gX_{4}, by using Lemma 3.1, we have
Thus, we proved that the contractive condition given in Corollary 2.5 is satisfied for all L ≥ 0. Moreover, from (62), we have letting gQ ≼ F (Q, 0, Q, 0) and g0 ≽ F (0, Q, 0, Q). Applying Corollary 2.5, F and g have a coupled coincidence point (and so a quadrupled fixed point since g is the identity on ). Then, there exist such that
On the other hand, for all there is a greatest lower bound and a least upper bound, hence it is obvious that the hypotheses of Theorem 2.3 hold, so the uniqueness of that quadrupled fixed point of F, which is also the unique solution of the system (60).
Competing interests
The authors declare that they have no competing interests.
Authors' contributions
All authors have contributed in obtaining the new results presented in this article. All authors read and approve the final manuscript.
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