Mixed g-monotone property and quadruple fixed point theorems in partially ordered metric spaces

In this manuscript, we prove some quadruple coincidence and common fixed point theorems for F : X⁴ → 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.

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., [3–13].

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, 16–33]. Recently, Berinde and Borcut [34] introduced the concept of tripled fixed point in ordered sets.

For simplicity, we denote\underset{k\text{ times}}{\underbrace{X\times X\cdots X\times X}} 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: X³ → X. The mapping F is said to has the mixed monotone property if for any x, y, z ∈ X

Definition 1.2 Let F : X³ → 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 : X³ → 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 non-decreasing sequence x _n → x, then x _n ≤ x for all n,

2. if a non-increasing sequence y _n → y, then y ≤ y _n for all n.

If there exist x_0, y_0, z₀ ∈ X such that x₀ ≤ F (x_0, y_0, z₀), y₀ ≥ F (y_0, x_0, z₀) and z₀ ≤ F (z_0, y_0, x₀), 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 : X³ → X and g : X → X. The mapping F is said to has the mixed g-monotone property if for any x, y, z ∈ X

Definition 1.4 Let F : X³ → 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 : X³ → 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 non-empty set. Then we say that the mappings F : X³ → 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 [38–41].

Definition 1.7 [38] Let X be a nonempty set and F : X⁴ → 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 \bar{d}:X^4\to X, given by

defines a metric on X⁴, which will be denoted for convenience by d.

Definition 1.8 [38] Let (X, ≤) be a partially ordered set and F : X⁴ → X be a mapping. We say that F has the mixed monotone property if F (x, y, z, w) is monotone non-decreasing in x and z and is monotone non-increasing 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 : X⁴ → 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.

We start this section with the following definitions.

Definition 2.1 Let (X, ≤) be a partially ordered set. Let F : X⁴ → X and g : X → X. The mapping F is said to has the mixed g-monotone property if for any x, y, z, w ∈ X

Definition 2.2 Let F : X⁴ → 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 : X⁴ → 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 non-empty set. Then we say that the mappings F : X⁴ → 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. 1.

   ϕ(t) < t for all t ∈ (0,+∞).

2. 2.

   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 : X⁴ → X and g : X → X are such that F is continuous and has the mixed g-monotone 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 (X⁴) ⊂ g(X), g is continuous and commutes with F. If there exist x₀, y₀, z₀, w₀ ∈ 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, w₀ ∈ X such that

Since F (X⁴) ⊂ g(X), then we can choose x₁, y₁, z₁, w₁ ∈ X such that

Taking into account F (X⁴) ⊂ 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, gx₀ ≤ F (x₀, y₀, z₀, w₀), gy₀ ≥ F (y₀, z₀, w₀, x₀), gz₀ ≤ F (z₀, w₀, x₀, y₀), and gw₀ ≥ F (w₀, x₀, y₀, z₀), 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 g-monotone property and gx _n ≤ gx_n+1, gy_n+1≤ gy _n , gz _n ≤ gz_n+1and 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