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

Abstract

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.

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., [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

$$\begin{array}{c}{x}_1,x_2\in X,x_1\le x_2\Rightarrow F\left(x_1,y,z\right)\le F\left(x_2,y,z\right),\\ y_1,y_2\in X,y_1\le y_2\Rightarrow F\left(x,y_1,z\right)\ge F\left(x,y_2,z\right),\\ z_1,z_2\in X,z_1\le z_2\Rightarrow F\left(x,y,z_1\right)\le F\left(x,y,z_2\right).\end{array}$$

Definition 1.2 Let F : X³ → X. An element (x, y, z) is called a tripled fixed point of F if

$$F\left(x,y,z\right)=x,F\left(y,x,y\right)=yandF\left(z,y,x\right)=z.$$

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

$$d\left(F\left(x,y,z\right),F\left(u,v,w\right)\right)\le jd\left(x,u\right)+rd\left(y,v\right)+ld\left(z,w\right),$$

(1)

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

$$F\left(x,y,z\right)=x,F\left(y,x,y\right)=yandF\left(z,y,x\right)=z,$$

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

$$\begin{array}{c}{x}_1,x_2\in X,g{x}_1\le g{x}_2\Rightarrow F\left(x_1,y,z\right)\le F\left(x_2,y,z\right),\\ y_1,y_2\in X,g{y}_1\le g{y}_2\Rightarrow F\left(x,y_1,z\right)\ge F\left(x,y_2,z\right),\\ z_1,z_2\in X,g{z}_1\le g{z}_2\Rightarrow F\left(x,y,z_1\right)\le F\left(x,y,z_2\right).\end{array}$$

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

$$F\left(x,y,z\right)=gx,F\left(y,x,y\right)=gy,andF\left(z,y,x\right)=gz.$$

(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

$$F\left(x,y,z\right)=gx=x,F\left(y,x,y\right)=gy=y,andF\left(z,y,x\right)=gz=z.$$

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

$$g\left(F\left(x,y,z\right)\right)=F\left(gx,gy,gz\right).$$

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

$$F\left(x,y,z,w\right)=x,F\left(y,z,w,x\right)=y,F\left(z,w,x,y\right)=z,andF\left(w,x,y,z\right)=w.$$

Let (X, d) be a metric space. The mapping $\bar{d}:X^4\to X,$ given by

$$\bar{d}\left(\left(x,y,z,w\right),\left(u,v,h,l\right)\right)=d\left(x,y\right)+d\left(y,v\right)+d\left(z,h\right)+d\left(w,l\right),$$

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,

$$\begin{array}{c}{x}_1,x_2\in X,x_1\le x_{2}impliesF\left(x_1,y,z,w\right)\le F\left(x_2,y,z,w\right),\\ y_1,y_2\in X,y_1\le y_{2}impliesF\left(x,y_2,z,w\right)\le F\left(x,y_1,z,w\right),\\ z_1,z_2\in X,z_1\le z_{2}impliesF\left(x,y,z_1,w\right)\le F\left(x,y,z_2,w\right),\end{array}$$

and

$$w_1,w_2\in X,w_1\le w_{2}impliesF\left(x,y,z,w_2\right)\le F\left(x,y,z,w_1\right).$$

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.

2 Main results

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

$$\begin{array}{c}{x}_1,x_2\in X,g{x}_1\le g{x}_2\Rightarrow F\left(x_1,y,z,w\right)\le F\left(x_2,y,z,w\right),\\ y_1,y_2\in X,g{y}_1\le g{y}_2\Rightarrow F\left(x,y_1,z,w\right)\ge F\left(x,y_2,z,w\right),\\ z_1,z_2\in X,g{z}_1\le g{z}_2\Rightarrow F\left(x,y,z_1,w\right)\le F\left(x,y,z_2,w\right)and\\ w_1,w_2\in X,g{w}_1\le g{w}_2\Rightarrow F\left(x,y,z,w_1\right)\ge F\left(x,y,z,w_2\right).\end{array}$$

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

$$F\left(x,y,z,w\right)=gx,F\left(y,z,w,x\right)=gy,F\left(z,w,x,y\right)=gz,andF\left(w,x,y,z\right)=gw.$$

(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

$$\begin{array}{c}F\left(x,y,z,w\right)=gx=x,F\left(y,z,w,x\right)=gy=y,\\ F\left(z,w,x,y\right)=gz=z,andF\left(w,x,y,z\right)=gw=w.\end{array}$$

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

$$g\left(F\left(x,y,z,w\right)\right)=F\left(gx,gy,gz,gw\right).$$

Let Φ be the set of all functions ϕ : [0, ∞) → [0, ∞) such that:

• 1. ϕ(t) < t for all t ∈ (0,+∞).

• 2. $\underset{r\to t^{+}}{lim}\varphi \left(r\right)<t$ for all t ∈ (0,+∞).

For simplicity, we define the following.

$$M\left(x,y,z,w,u,v,h,l\right)=min\left\{\begin{array}{c}\hfill d\left(F\left(x,y,z,w\right),gx\right),d\left(F\left(x,y,z,w\right),gu\right),\hfill \\ \hfill d\left(F\left(u,v,h,l\right),gu\right)\hfill \end{array}\right\}.$$

(2)

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

$$\begin{array}{c}d\left(F\left(x,y,z,w\right),F\left(u,v,,h,l\right)\right)\le \varphi \left(max\left\{d\left(gx,gu\right),d\left(gy,gv\right),d\left(gz,gh\right),d\left(gw,gl\right)\right\}\right)\\ +LM\left(x,y,z,w,u,v,h,l\right)\end{array}$$

(3)

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

$$\begin{array}{c}g{x}_0\le F\left(x_0,y_0,z_0,w_0\right),g{y}_0\ge F\left(y_0,z_0,w_0,x_0\right),\\ g{z}_0\le F\left(z_0,w_0,x_0,y_0\right),andg{w}_0\ge F\left(w_0,x_0,y_0,z_0\right),\end{array}$$

then there exist x, y, z, w ∈ X such that

$$F\left(x,y,z,w\right)=gx,F\left(y,z,w,x\right)=gy,F\left(z,w,x,y\right)=gzandF\left(w,x,y,z\right)=gw$$

that is, F and g have a quadruple coincidence point.

Proof. Let x_0, y_0, z_0, w₀ ∈ X such that

$$\begin{array}{c}g{x}_0\le F\left(x_0,y_0,z_0,w_0\right),g{y}_0\ge F\left(y_0,z_0,w_0,x_0\right),\\ g{z}_0\le F\left(z_0,w_0,x_0,y_0\right)\text{ and }g{w}_0\ge F\left(w_0,x_0,y_0,z_0\right).\end{array}$$

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

$$\begin{array}{c}g{x}_1=F\left(x_0,y_0,z_0,w_0\right),g{y}_1=F\left(y_0,z_0,w_0,x_0\right),\\ g{z}_1=F\left(z_0,w_0,x_0,y_0\right)\text{and}g{w}_1=F\left(w_0,x_0,y_0,z_0\right).\end{array}$$

(4)

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

$$\begin{array}{c}g{x}_{n+1}=F\left(x_n,y_n,z_n,w_n\right),g{y}_{n+1}=F\left(y_n,z_n,w_n,x_n\right),\\ g{z}_{n+1}=F\left(z_n,w_n,x_n,y_n\right),\text{and}g{w}_{n+1}=F\left(w_n,x_n,y_n,z_n\right).\end{array}$$

(5)

We shall show that

$$g{x}_n\le g{x}_{n+1},g{y}_{n+1}\le g{y}_n,g{z}_n\le g{z}_{n+1},\text{ and }g{w}_{n+1}\le g{w}_n\text{ for }n=0,1,2,\dots$$

(6)

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

$$g{x}_0\le g{x}_1,g{y}_1\le g{y}_0,g{z}_0\le g{z}_1,\text{ and }g{w}_1\le g{w}_0$$

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

$$\begin{array}{cc}\hfill g{x}_{n+1}& =F\left(x_n,y_n,z_n,w_n\right)\le F\left(x_{n+1},y_n,z_n,w_n\right)\hfill \\ \le F\left(x_{n+1},y_n,z_{n+1},w_n\right)\le F\left(x_{n+1},y_{n+1},z_{n+1},w_n\right)\hfill \\ \le F\left(x_{n+1},y_{n+1},z_{n+1},w_{n+1}\right)=g{x}_{n+2},\hfill \end{array}$$

$$\begin{array}{cc}\hfill g{y}_{n+2}& =F\left(y_{n+1},z_{n+1},w_{n+1},x_{n+1}\right)\le F\left(y_{n+1},z_n,x_{n+1},w_{n+1}\right)\hfill \\ \le F\left(y_n,z_n,x_{n+1},w_{n+1}\right)\le F\left(y_n,z_n,x_n,w_{n+1}\right)\hfill \\ \le F\left(y_n,z_n,x_n,w_n\right)=g{y}_{n+1},\hfill \end{array}$$

$$\begin{array}{cc}\hfill g{z}_{n+1}& =F\left(z_n,y_n,x_n,w_n\right)\le F\left(z_{n+1},y_n,x_n,w_n\right)\hfill \\ \le F\left(z_{n+1},y_{n+1},x_n,w_n\right)\le F\left(z_{n+1},y_{n+1},x_{n+1},w_n\right)\hfill \\ \le F\left(z_{n+1},y_{n+1},x_{n+1},w_{n+1}\right)=g{z}_{n+2},\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill g{w}_{n+2}& =F\left(w_{n+1},x_{n+1},y_{n+1},z_{n+1}\right)\le F\left(w_{n+1},x_n,y_{n+1},z_{n+1}\right)\hfill \\ \le F\left(w_n,x_n,y_{n+1},z_{n+1}\right)\le F\left(w_n,x_n,y_n,z_{n+1}\right)\hfill \\ \le F\left(w_n,x_n,y_n,z_n\right)=g{w}_{n+1}.\hfill \end{array}$$

Thus, (6) holds for any n ∈ ℕ. Assume for some n ∈ ℕ,

$$g{x}_n=g{x}_{n+1},g{y}_n=g{y}_{n+1},g{z}_n=g{z}_{n+1},\text{and}g{w}_n=g{w}_{n+1}$$

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

$$g{x}_n\ne g{x}_{n+1}\text{or}g{y}_n\ne g{y}_{n+1}\text{or}g{z}_n\ne g{z}_{n+1}\text{ or }g{w}_n\ne g{w}_{n+1}.$$

(7)

By (2) and (5), it is easy that

$$\begin{array}{c}M\left(x_{n-1},y_{n-1},z_{n-1},w_{n-1},x_n,y_n,z_n,w_n\right)=M\left(y_n,z_n,w_n,x_n,y_{n-1},z_{n-1},w_{n-1},x_{n-1}\right)\hfill \\ =M\left(z_{n-1},y_{n-1},x_{n-1},z_n,y_n,x_n\right)\hfill \\ =M\left(w_n,x_n,y_n,z_n,w_{n-1},x_{n-1},y_{n-1},z_{n-1}\right)=0\text{for all}n\ge 1.\hfill \end{array}$$

(8)

Due to (3) and (8), we have

$$\begin{array}{c}d\left(g{x}_n,g{x}_{n+1}\right)=d\left(F\left(x_{n-1},y_{n-1},z_{n-1},w_{n-1}\right),F\left(x_n,y_n,z_n,w_n\right)\right)\hfill \\ \le \varphi \left(max\left\{d\left(g{x}_{n-1},g{x}_n\right),d\left(g{y}_{n-1},g{y}_n\right),d\left(g{z}_{n-1},g{z}_n\right),d\left(g{w}_{n-1},g{w}_n\right)\right\}\right)\hfill \\ +LM\left(x_{n-1},y_{n-1},z_{n-1},w_{n-1},x_n,y_n,z_n,w_n\right)\hfill \\ =\varphi \left(max\left\{d\left(g{x}_{n-1},g{x}_n\right),d\left(g{y}_{n-1},g{y}_n\right),d\left(g{z}_{n-1},g{z}_n\right),d\left(g{w}_{n-1},g{w}_n\right)\right\}\right),\hfill \end{array}$$

(9)

$$\begin{array}{c}d\left(g{y}_n,g{y}_{n+1}\right)=d\left(F\left(y_n,z_n,w_n,x_n\right),y_{n-1},F\left(y_{n-1},z_{n-1},w_{n-1},x_{n-1}\right)\right)\hfill \\ \le \varphi \left(max\left\{d\left(g{y}_{n-1},g{y}_n\right),d\left(g{x}_{n-1},g{x}_n\right),d\left(g{z}_{n-1},g{z}_n\right),d\left(g{w}_{n-1},g{w}_n\right)\right\}\right),\hfill \\ +LM\left(y_n,z_n,w_n,w_n,y_{n-1},z_{n-1},w_{n-1},x_{n-1}\right)\hfill \\ =\varphi \left(max\left\{d\left(g{y}_{n-1},g{y}_n\right),d\left(g{x}_{n-1},g{x}_n\right),d\left(g{z}_{n-1},g{z}_n\right),d\left(g{w}_{n-1},g{w}_n\right)\right\}\right),\hfill \end{array}$$

(10)

$$\begin{array}{c}d\left(g{z}_n,g{z}_{n+1}\right)=d\left(F\left(z_{n-1},w_{n-1},x_{n-1},y_{n-1}\right),F\left(z_n,w_n,x_n,y_n\right)\right)\hfill \\ \le \varphi \left(max\left\{,d\left(g{z}_{n-1},g{z}_n\right),d\left(g{w}_{n-1},g{w}_n\right),d\left(g{x}_{n-1},g{x}_n\right),d\left(g{y}_{n-1},g{y}_n\right)\right\}\right)\hfill \\ +LM\left(z_{n-1},w_{n-1},x_{n-1},y_{n-1},z_n,w_n,x_n,y_n\right)\hfill \\ =\varphi \left(max\left\{d\left(g{z}_{n-1},g{z}_n\right),d\left(g{w}_{n-1},g{w}_n\right),d\left(g{x}_{n-1},g{x}_n\right),d\left(g{y}_{n-1},g{y}_n\right)\right\}\right)\hfill \end{array}$$

(11)

and

$$\begin{array}{c}d\left(g{w}_n,g{w}_{n+1}\right)=d\left(F\left(w_n,x_n,y_n,z_n\right),F\left(w_{n-1},x_{n-1},y_{n-1},z_{n-1}\right)\right)\hfill \\ \le \varphi \left(max\left\{d\left(g{w}_{n-1},g{w}_n\right),d\left(g{x}_{n-1},g{x}_n\right),d\left(g{y}_{n-1},g{y}_n\right),d\left(g{z}_{n-1},g{z}_n\right)\right\}\right),\hfill \\ +LM\left(w_n,x_n,y_n,z_n,w_{n-1},x_{n-1},y_{n-1},z_{n-1}\right)\hfill \\ =\varphi \left(max\left\{d\left(g{w}_{n-1},g{w}_n\right),d\left(g{x}_{n-1},g{x}_n\right),d\left(g{y}_{n-1},g{y}_n\right),d\left(g{z}_{n-1},g{z}_n\right)\right\}\right).\hfill \end{array}$$

(12)

Having in mind that ϕ (t) < t for all t > 0, so from (9)-(12) we obtain that

$$\begin{array}{cc}\hfill 0& <max\left\{d\left(g{x}_n,g{x}_{n+1}\right),d\left(g{y}_n,g{y}_{n+1}\right),d\left(g{z}_n,g{z}_{n+1}\right),d\left(g{w}_n,g{w}_{n+1}\right)\right\}\hfill \\ \le \varphi \left(max\left\{d\left(g{z}_{n-1},g{z}_n\right),d\left(g{y}_{n-1},g{y}_n\right),d\left(g{x}_{n-1},g{x}_n\right),d\left(g{w}_{n-1},g{w}_n\right)\right\}\right)\hfill \\ <max\left\{d\left(g{z}_{n-1},g{z}_n\right),d\left(g{y}_{n-1},g{y}_n\right),d\left(g{x}_{n-1},g{x}_n\right),d\left(g{w}_{n-1},g{w}_n\right)\right\}.\hfill \end{array}$$

(13)

It follows that

$$max\left\{\begin{array}{c}\hfill d\left(g{x}_n,g{x}_{n+1}\right),d(g{y}_n,g{y}_{n+1},\hfill \\ \hfill d\left(g{z}_n,g{z}_{n+1}\right),d\left(g{w}_n,g{w}_{n+1}\right)\hfill \end{array}\right\}<max\left\{\begin{array}{c}\hfill d\left(g{z}_{n-1},g{z}_n\right),d\left(g{y}_{n-1},g{y}_n\right),\hfill \\ \hfill d\left(g{x}_{n-1},g{x}_n\right),d\left(g{w}_{n-1},g{w}_n\right)\hfill \end{array}\right\}.$$

(14)

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

$$\underset{n\to +\infty }{lim}max\left\{d\left(g{x}_n,g{x}_{n+1}\right),d\left(g{y}_n,g{y}_{n+1}\right),d\left(g{z}_n,g{z}_{n+1}\right),d\left(g{w}_n,g{w}_{n+1}\right)\right\}=r.$$

Suppose that r > 0. Letting n → +∞ in (13), we obtain that

$$0<r\le \underset{n\to +\infty }{lim}\varphi \left(max\left\{\begin{array}{c}\hfill d\left(g{z}_{n-1},g{z}_n\right),d\left(g{y}_{n-1},g{y}_n\right),\hfill \\ \hfill d\left(g{x}_{n-1},g{x}_n\right),d\left(g{w}_{n-1},g{w}_n\right)\hfill \end{array}\right\}\right)=\underset{t\to r^{+}}{lim}\varphi \left(t\right)<r.$$

(15)

It is a contradiction. We deduce that

$$\underset{n\to +\infty }{lim}max\left\{d\left(g{x}_n,g{x}_{n+1}\right),d\left(g{y}_n,g{y}_{n+1}\right),d\left(g{z}_n,g{z}_{n+1}\right),d\left(g{w}_n,g{w}_{n+1}\right)\right\}=0.$$

(16)

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,

$$\underset{n,m\to +\infty }{lim}d\left(g{x}_m,g{x}_n\right)\ne 0\text{or}\underset{n,m\to +\infty }{lim}d\left(g{y}_m,g{y}_n\right)\ne 0$$

or

$$\underset{n,m\to +\infty }{lim}d\left(g{z}_m,g{z}_n\right)\ne 0\text{ or }\underset{n,m\to +\infty }{lim}d\left(g{w}_m,g{w}_n\right)\ne 0.$$

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

$$max\left\{d\left(g{x}_{m_k},g{x}_{n_k}\right),d\left(g{y}_{m_k},g{y}_{n_k}\right),d\left(g{z}_{m_k},g{z}_{n_k}\right),d\left(g{w}_{m_k},g{w}_{n_k}\right)\right\}\ge \epsilon .$$

(17)

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

$$max\left\{d\left(g{x}_{m_k},g{x}_{n_k-1}\right),d\left(g{y}_{m_k},g{y}_{n_k-1}\right),d\left(g{z}_{m_k},g{z}_{n_k-1}\right),d\left(g{w}_{m_k},g{w}_{n_k-1}\right)\right\}<\epsilon .$$

(18)

By triangular inequality and (18), we have

$$\begin{array}{c}d\left(g{x}_{m_k},g{x}_{n_k}\right)\le d\left(g{x}_{m_k},g{x}_{n_k-1}\right)+d\left(g{x}_{n_k-1},g{x}_{n_k}\right)\\ <\epsilon +d\left(g{x}_{n_k-1},g{x}_{n_k}\right).\end{array}$$

Thus, by (16) we obtain

$$\underset{k\to +\infty }{lim}d\left(g{x}_{m_k},g{x}_{n_k}\right)\le \underset{k\to +\infty }{lim}d\left(g{x}_{m_k},g{x}_{n_k-1}\right)\le \epsilon .$$

(19)

Similarly, we have

$$\underset{k\to +\infty }{lim}d\left(g{y}_{m_k},g{y}_{n_k}\right)\le \underset{k\to +\infty }{lim}d\left(g{y}_{m_k},g{y}_{n_k-1}\right)\le \epsilon ,$$

(20)

$$\underset{k\to +\infty }{lim}d\left(g{z}_{m_k},g{z}_{n_k}\right)\le \underset{k\to +\infty }{lim}d\left(g{z}_{m_k},g{z}_{n_k-1}\right)\le \epsilon ,$$

(21)

and

$$\underset{k\to +\infty }{lim}d\left(g{w}_{m_k},g{w}_{n_k}\right)\le \underset{k\to +\infty }{lim}d\left(g{w}_{m_k},g{w}_{n_k-1}\right)\le \epsilon .$$

(22)

Again by (18), we have

$$\begin{array}{cc}\hfill d\left(g{x}_{m_k},g{x}_{n_k}\right)& \le d\left(g{x}_{m_k},g{x}_{m_k-1}\right)+d\left(g{x}_{m_k-1},g{x}_{n_k-1}\right)+d\left(g{x}_{n_k-1},g{x}_{n_k}\right)\hfill \\ \le d\left(g{x}_{m_k},g{x}_{m_k-1}\right)+d\left(g{x}_{m_k-1},g{x}_{m_k}\right)\hfill \\ +d\left(g{x}_{m_k},g{x}_{n_k-1}\right)+d\left(g{x}_{n_k-1},g{x}_{n_k}\right)\hfill \\ <d\left(g{x}_{m_k},g{x}_{m_k-1}\right)+d\left(g{x}_{m_k-1},g{x}_{m_k}\right)+\epsilon +d\left(g{x}_{n_k-1},g{x}_{n_k}\right).\hfill \end{array}$$

Letting k → + ∞ and using (16), we get

$$\underset{k\to +\infty }{lim}d\left(g{x}_{m_k},g{x}_{n_k}\right)\le \underset{k\to +\infty }{lim}d\left(g{x}_{m_k-1},g{x}_{n_k-1}\right)\le \epsilon ,$$

(23)

$$\underset{k\to +\infty }{lim}d\left(g{y}_{m_k},g{y}_{n_k}\right)\le \underset{k\to +\infty }{lim}d\left(g{y}_{m_k-1},g{y}_{n_k-1}\right)\le \epsilon ,$$

(24)

$$\underset{k\to +\infty }{lim}d\left(g{z}_{m_k},g{z}_{n_k}\right)\le \underset{k\to +\infty }{lim}d\left(g{z}_{m_k-1},g{z}_{n_k-1}\right)\le \epsilon$$

(25)

and

$$\underset{k\to +\infty }{lim}d\left(g{w}_{m_k},g{w}_{n_k}\right)\le \underset{k\to +\infty }{lim}d\left(g{w}_{m_k-1},g{w}_{n_k-1}\right)\le \epsilon .$$

(26)

Using (17) and (23)-(26), we have

$$\begin{array}{c}\underset{k\to +\infty }{lim}max\left\{d\left(g{x}_{m_k},g{x}_{n_k}\right),d\left(g{y}_{m_k},g{y}_{n_k}\right),d\left(g{z}_{m_k},g{z}_{n_k}\right),d\left(g{w}_{m_k},g{w}_{n_k}\right)\right\}\hfill \\ =\underset{k\to +\infty }{lim}max\left\{d\left(g{x}_{m_k-1},g{x}_{n_k-1}\right),d\left(g{y}_{m_k-1},g{y}_{n_k-1}\right),d\left(g{z}_{m_k-1},g{z}_{n_k-1}\right),d\left(g{w}_{m_k-1},g{w}_{n_k-1}\right)\right\}\hfill \\ =\epsilon .\hfill \end{array}$$

(27)

By (16), it is easy to see that

$$\begin{array}{c}\underset{k\to +\infty }{lim}M\left(x_{m_k-1},y_{m_k-1},z_{m_k-1},w_{m_k-1},x_{n_k-1},y_{n_k-1},z_{n_k-1},w_{n_k-1}\right)\hfill \\ =\underset{k\to +\infty }{lim}M\left(y_{n_k-1},z_{n_k-1},w_{n_k-1},x_{n_k-1},y_{m_k-1},z_{m_k-1},w_{m_k-1},x_{m_k-1}\right)\hfill \\ =\underset{k\to +\infty }{lim}M\left(z_{m_k-1},w_{m_k-1},x_{m_k-1},y_{m_k-1},z_{n_k-1},w_{n_k-1},x_{n_k-1},y_{m_k-1}\right)\hfill \\ =\underset{k\to +\infty }{lim}M\left(w_{n_k-1},x_{n_k-1},y_{m_k-1},z_{n_k-1},w_{m_k-1},x_{m_k-1},y_{m_k-1},z_{m_k-1}\right)=0.\hfill \end{array}$$

(28)

Now, using inequality (3), we obtain

$$\begin{array}{c}d\left(g{x}_{m_k},g{x}_{n_k}\right)=d\left(F\left(x_{m_k-1},y_{m_k-1},z_{m_k-1},w_{m_k-1}\right),F\left(x_{n_k-1},y_{n_k-1},z_{n_k-1},w_{n_k-1}\right)\right)\hfill \\ \le \varphi \left(max\left\{d\left(x_{m_k-1},x_{n_k-1}\right),d\left(y_{m_k-1},y_{n_k-1}\right),d\left(z_{m_k-1},z_{n_k-1}\right),d\left(w_{m_k-1},w_{n_k-1}\right)\right\}\right)\hfill \\ +LM\left(x_{m_k-1},y_{m_k-1},z_{m_k-1},w_{m_k-1},x_{n_k-1},y_{n_k-1},z_{n_k-1},w_{n_k-1}\right),\hfill \end{array}$$

(29)

$$\begin{array}{c}d\left(g{y}_{n_k},g{y}_{m_k}\right)=d\left(F\left(y_{n_k-1},z_{n_k-1},w_{n_k-1},x_{n_k-1}\right),F\left(y_{m_k-1},z_{m_k-1},w_{m_k-1},x_{m_k-1}\right)\right)\hfill \\ \le \varphi (max\{d\left(y_{m_k-1},y_{n_k-1}\right),d\left(z_{m_k-1},z_{n_k-1}\right),d\left(w_{m_k-1},w_{n_k-1},d\left(x_{m_k-1},x_{n_k-1}\right)\right\})\hfill \\ +LM\left(y_{n_k-1},z_{n_k-1},w_{n_k-1},x_{n_k-1},y_{m_k-1},z_{m_k-1},w_{m_k-1},x_{m_k-1}\right),\hfill \end{array}$$

(30)

$$\begin{array}{c}d\left(g{z}_{m_k},g{z}_{n_k}\right)=d\left(F\left(z_{m_k-1},w_{m_k-1},x_{m_k-1},y_{m_k-1}\right),F\left(z_{n_k-1},w_{n_k-1},x_{n_k-1},y_{n_k-1}\right)\right)\hfill \\ \le \varphi \left(max\left\{d\left(z_{m_k-1},z_{n_k-1}\right),d(w_{m_k-1},w_{n_k-1},d\left(x_{m_k-1},x_{n_k-1}\right),d\left(y_{m_k-1},y_{n_k-1}\right)\right\}\right)\hfill \\ +LM\left(z_{m_k-1},w_{m_k-1},x_{m_k-1},y_{m_k-1},z_{n_k-1},w_{n_k-1},x_{n_k-1},y_{m_k-1}\right)\hfill \end{array}$$

(31)

and

$$\begin{array}{c}d(g{w}_{n_k},g{w}_{m_k})=d\left(F\right(w_{n_k-1},x_{n_k-1},y_{n_k-1},z_{n_k-1}),F(w_{m_k-1},x_{m_k-1},y_{m_k-1},z_{m_k-1}\left)\right)\hfill \\ \le \varphi (max\left\{d(w_{m_k-1},w_{n_k-1},d(x_{m_k-1},x_{n_k-1}),d(y_{m_k-1},y_{n_k-1}),d(z_{m_k-1},z_{n_k-1})\right\})\hfill \\ +LM(w_{n_k-1},x_{n_k-1},y_{n_k-1},z_{n_k-1},w_{m_k-1},x_{m_k-1},y_{m_k-1},z_{m_k-1}).\hfill \end{array}$$

(32)

From (29)-(32), we deduce that

$$\begin{array}{c}max\left\{d\left(g{x}_{m_k},g{x}_{n_k}\right),d\left(g{y}_{m_k},g{y}_{n_k}\right),d\left(g{z}_{m_k},g{z}_{n_k}\right),d\left(g{w}_{m_k},g{w}_{n_k}\right)\right\}\hfill \\ \le \varphi \left(max\left\{d\left(x_{m_k-1},x_{n_k-1}\right),d\left(y_{m_k-1},y_{n_k-1}\right),d\left(z_{m_k-1},z_{n_k-1}\right),d\left(g{w}_{m_k},g{w}_{n_k}\right)\right\}\right)\hfill \\ +LM\left(x_{m_k-1},y_{m_k-1},z_{m_k-1},w_{m_k-1},x_{n_k-1},y_{n_k-1},z_{n_k-1},w_{n_k-1}\right)\hfill \\ +LM\left(y_{n_k-1},z_{n_k-1},w_{n_k-1},x_{n_k-1},y_{m_k-1},z_{m_k-1},w_{m_k-1},x_{m_k-1}\right)\hfill \\ +LM\left(z_{m_k-1},w_{m_k-1},x_{m_k-1},y_{m_k-1},z_{n_k-1},w_{n_k-1},x_{n_k-1},y_{m_k-1}\right)\hfill \\ +LM\left(w_{n_k-1},x_{n_k-1},y_{n_k-1},z_{n_k-1},w_{m_k-1},x_{m_k-1},y_{m_k-1},z_{m_k-1}\right).\hfill \end{array}$$

(33)

Letting k → +∞ in (33) and having in mind (27) and (28), we get that

$$0<\epsilon \le \underset{t\to {\epsilon }^{+}}{lim}\varphi \left(t\right)<\epsilon ,$$

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

$$\underset{n\to +\infty }{lim}g{x}_n=x,\underset{n\to +\infty }{lim}g{y}_n=y,\underset{n\to +\infty }{lim}g{y}_n=y,\text{and}\underset{n\to +\infty }{lim}g{w}_n=w.$$

(34)

From (34) and the continuity of g, we have

$$\underset{n\to +\infty }{lim}g\left(g{x}_n\right)=gx,\underset{n\to +\infty }{lim}g\left(g{y}_n\right)=gy,\underset{n\to +\infty }{lim}g\left(g{z}_n\right)=gz,\text{and}\underset{n\to +\infty }{lim}g\left(g{w}_n\right)=gw.$$

(35)

From (5) and the commutativity of F and g, we have

$$g\left(g{x}_{n+1}\right)=g\left(F\left(x_n,y_n,z_n,w_n\right)\right)=F\left(g{x}_n,g{y}_n,g{z}_n,g{w}_n\right),$$

(36)

$$g\left(g{y}_{n+1}\right)=g\left(F\left(y_n,z_n,w_n,x_n\right)\right)=F\left(g{y}_n,g{z}_n,g{w}_n,g{x}_n\right),$$

(37)

$$g\left(g{z}_{n+1}\right)=g\left(F\left(z_n,w_n,x_n,y_n\right)\right)=F\left(g{z}_n,g{w}_n,g{x}_n,y_n\right),$$

(38)

and

$$g\left(g{w}_{n+1}\right)=g\left(F\left(w_n,x_n,y_n,z_n\right)\right)=F\left(g{w}_n,g{x}_n,y_n,g{z}_n\right).$$

(39)

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

$$\begin{array}{cc}\hfill gx& =\underset{n\to +\infty }{lim}g\left(g{x}_{n+1}\right)=\underset{n\to +\infty }{lim}F\left(g{x}_n,g{y}_n,g{z}_n,g{w}_n\right)\hfill \\ =F\left(\underset{n\to +\infty }{lim}g{x}_n,\underset{n\to +\infty }{lim}g{y}_n,\underset{n\to +\infty }{lim}g{z}_n,\underset{n\to +\infty }{lim}g{w}_n\right)\hfill \\ =F\left(x,y,z,w\right),\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill gy& =\underset{n\to +\infty }{lim}g\left(g{y}_{n+1}\right)=\underset{n\to +\infty }{lim}F\left(g{y}_n,g{z}_n,g{w}_n,g{x}_n\right)\hfill \\ =F\left(\underset{n\to +\infty }{lim}g{y}_n,\underset{n\to +\infty }{lim}g{z}_n,\underset{n\to +\infty }{lim}g{w}_n,\underset{n\to +\infty }{lim}g{w}_n\right)\hfill \\ =F\left(y,z,w,x\right),\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill gz& =\underset{n\to +\infty }{lim}g\left(g{z}_{n+1}\right)=\underset{n\to +\infty }{lim}F\left(g{z}_n,g{w}_n,g{x}_n,g{y}_n\right)\hfill \\ =F\left(\underset{n\to +\infty }{lim}g{z}_n,\underset{n\to +\infty }{lim}g{w}_n,\underset{n\to +\infty }{lim}g{x}_n,\underset{n\to +\infty }{lim}g{y}_n\right)\hfill \\ =F\left(z,w,x,y\right),\hfill \end{array}$$

(42)

and

$$\begin{array}{cc}\hfill gw& =\underset{n\to +\infty }{lim}g\left(g{w}_{n+1}\right)=\underset{n\to +\infty }{lim}F\left(g{w}_n,g{x}_n,g{y}_n,g{z}_n\right)\hfill \\ =F\left(\underset{n\to +\infty }{lim}g{w}_n,\underset{n\to +\infty }{lim}g{x}_n,\underset{n\to +\infty }{lim}g{y}_n,\underset{n\to +\infty }{lim}g{z}_n\right)\hfill \\ =F\left(w,x,y,z\right).\hfill \end{array}$$

(43)

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 non-decreasing sequence a _n → a, then a _n ≤ a for all n,

(ii) if non-increasing 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 : X⁴ → X and g : X → X are such that F has the mixed g-monotone property. Assume that there exist ϕ ∈ Φ and L ≥ 0 such that

$$\begin{array}{c}d\left(F\left(x,y,z,w\right),F\left(u,v,,h,l\right)\right)\le \varphi \left(max\left\{d\left(gx,gu\right),d\left(gy,gv\right),d\left(gz,gh\right),d\left(gw,gl\right)\right\}\right)\\ +LM\left(x,y,z,w,u,v,h,l\right)\end{array}$$

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 (X⁴) ⊂ g(X) and (g(X), d) is a complete metric space. If there exist x₀, y₀, z₀, w₀ ∈ X such that 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 there exist x, y, z, w ∈ X such that

$$F\left(x,y,z,w\right)=gx,F\left(y,z,w,x\right)=gy,F\left(z,w,x,y\right)=gzandF\left(w,x,y,z\right)=gw$$

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

$$g{x}_n\to gx,g{y}_n\to gy,g{z}_n\to gz,\text{and}g{w}_n\to gw.$$

(44)

Since {gx _n }, {gz _n } are non-decreasing and {gy _n }, {gw _n } are non-increasing, then since (X, d, ≤) is regular we have

$$g{x}_n\le gx,g{y}_n\ge gy,g{z}_n\le gz,g{w}_n\ge gw$$

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

$$g{x}_n=g{x}_{n+1}=F\left(x_n,y_n,z_n,w_n\right),g{y}_n=g{y}_{n+1}=F\left(y_n,z_n,w_n,x_n\right),$$

and

$$g{z}_n=g{z}_{n+1}=F\left(z_n,w_n,x_n,y_n\right),g{w}_n=g{w}_{n+1}=F\left(w_n,w_n,y_n,z_n\right),$$

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

$$\begin{array}{c}d\left(gx,F\left(x,y,z,w\right)\right)\le d\left(gx,g{x}_{n+1}\right)+d\left(g{x}_{n+1},F\left(x,y,z,w\right)\right)\hfill \\ =d\left(gx,g{x}_{n+1}\right)+d\left(F\left(x_n,y_n,z_n,w_n\right),F\left(x,y,z,w\right)\right)\hfill \\ \le d\left(gx,g{x}_{n+1}\right)+\varphi \left(max\left\{\begin{array}{c}d\left(g{x}_n,gx\right),d\left(g{y}_n,gy\right),\hfill \\ d\left(g{z}_n,gz\right),d\left(g{w}_n,gw\right)\hfill \end{array}\right\}\right)+LM\left(x_n,y_n,z_n,w_n,x,y,z,w\right)\hfill \\ <d\left(gx,g{x}_{n+1}\right)+max\left\{d\left(g{x}_n,gx\right),d\left(g{y}_n,gy\right),d\left(g{z}_n,gz\right),d\left(g{w}_n,gw\right)\right\}+LM\left(x_n,y_n,z_n,w_n,x,y,z,w\right).\hfill \end{array}$$

(45)

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 right-hand 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

$$F\left(x,y,z,w\right)=gy,F\left(z,w,x,y\right)=gz,\text{and}F\left(w,x,y,z\right)=gw.$$

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 : 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 a non-decreasing function and L ≥ 0 such that

$$\begin{array}{cc}d\left(F\left(x,y,z,w\right),F\left(u,v,h,l\right)\right)\hfill & \le \varphi \left(\frac{d\left(gx,gu\right)+d\left(gy,gv\right)+d\left(gz,gh\right)+d\left(gw,gl\right)}{4}\right)\hfill \\ +LM\left(x,y,z,w,u,v,h,l\right),\hfill \end{array}$$

for any x, y, z, w, u, v, h, l,∈ X for which gx ≤ gu, gv ≤ gy, gz ≤ gw, 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 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 there exist x, y, z, w ∈ X such that

$$F\left(x,y,z,w\right)=gx,F\left(y,z,w,x\right)=gy,F\left(z,w,x,y\right)=gz,andF\left(w,x,y,z\right)=gw.$$

Proof. It suffices to remark that

$$\frac{d\left(gx,gu\right)+d\left(gy,gv\right)+d\left(gz,ph\right),d\left(gw,gl\right)}{4}\le max\left\{\begin{array}{c}d\left(gx,gu\right),d\left(gu,gv\right),\\ d\left(gz,gh\right),d\left(gw,gl\right)\end{array}\right\}.$$

Then, we apply Theorem 2.1, since ϕ is assumed to be non-decreasing.

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 : X⁴ → X and g : X → X are such that F has the mixed g-monotone property. Assume also that there exist ϕ ∈ Φ a non-decreasing function and L ≥ 0 such that

$$\begin{array}{cc}d\left(F\left(x,y,z,w\right),F\left(u,v,h,l\right)\right)\hfill & \le \varphi \left(\frac{d\left(gx,gu\right)+d\left(gy,gv\right)+d\left(gz,gh\right)+d\left(gw,gl\right)}{4}\right)\hfill \\ +LM\left(x,y,z,w,u,v,h,l\right),\hfill \end{array}$$

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 (X⁴) ⊂ g(X) and (g(X), d) is a complete metric space.

If there exist x₀, y₀, z₀, w₀ ∈ X such that 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 there exist x, y, z, w ∈ X such that

$$F\left(x,y,z,w\right)=gx,F\left(y,z,w,x\right)=gy,F\left(z,w,x,y\right)=gz,andF\left(w,x,y,z\right)=gw.$$

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 : X⁴ → X and g : X → X are such that F is continuous and has the mixed g-monotone property. Assume that there exist k ∈ [0, 1) and L ≥ 0 such that

$$d\left(F\left(x,y,z,w\right),F\left(u,v,,h,l\right)\right)\le kmax\left\{\begin{array}{c}d\left(gx,gu\right),d\left(gy,gv\right),\\ d\left(gz,gh\right),d\left(gw,gl\right)\end{array}\right\}\text{ + }LM\text{(}x\text{,}y\text{,}z\text{,}w\text{,}u\text{,}v\text{,}h\text{,}l\text{),}$$

for any x, y, z, w, u, v, h, l ∈ X for which^:gx ≤ gu, gv ≤ gy, gz ≤ gw, 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 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 there exist x, y, z, w ∈ X such that

$$F\left(x,y,z,w\right)=gx,f\left(y,z,w,x\right)=gy,f\left(z,w,x,y\right)=gz,andF\left(w,x,y,z\right)=gw.$$

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 : X⁴ → X and g : X → X are such that F has the mixed g-monotone property. Assume that there exist k ∈ [0, 1) and L ≥ 0 such that

$$d\left(F\left(x,y,z,w\right),F\left(u,v,,h,l\right)\right)\le kmax\left\{\begin{array}{c}d\left(gx,gu\right),d\left(gy,gv\right),\\ d\left(gz,gh\right),d\left(gw,gl\right)\end{array}\right\}\text{ + }LM\text{(}x\text{,}y\text{,}z\text{,}w\text{,}u\text{,}v\text{,}h\text{,}l\text{),}$$

for any x, y, z, w, u, v, h, l ∈ X for which gx ≤ gu, gv ≤ gy, gz ≤ gw, and gl ≤ gw. Suppose F (X⁴) ⊂ g(X) and (g(X), d) is a complete metric space.

If there exist x₀, y₀, z₀, w₀ ∈ X such that gx₀ ≤ F (x₀, y₀, z₀, w₀), gy₀ ≥ F (y₀, z₀, w_0, x₀), gz₀ ≤ F (z₀, w₀, x₀, y₀), and gw₀ ≥ F (w₀, x₀, y₀, z₀), then there exist x, y, z, w ∈ X such that

$$F\left(x,y,z,w\right)=gx,F\left(y,z,w,x\right)=gy,F\left(z,w,x,y\right)=gz,andF\left(w,x,y,z\right)=gw.$$

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 : X⁴ → X and g : X → X are such that F is continuous and has the mixed g-monotone property. Assume that there exist k ∈ [0, 1) and L ≥ 0 such that

$$d\left(F\left(x,y,z,w\right),F\left(u,v,h,l\right)\right)\le \frac{k}{4}\left\{\begin{array}{c}d\left(gx,gu\right)+d\left(gy,gv\right)+\\ d\left(gz,gh\right)+d\left(gw,gl\right)\end{array}\right\}+LM\left(x,y,z,w,u,v,h,l\right),$$

for any x, y, z, w, ∈ X for which ^:gx ≤ gu, gv ≤ gy, gz ≤ gw, and gl ≤ gw. Also, suppose F (X⁴) ⊂ g(X) and (g(X), g is continuous and commutes with F.

If there exist x₀, y₀, z₀, w₀ ∈ X such that gx₀ ≤ F (x₀, y₀, z₀, w₀), gy₀ ≥ F (y₀, z₀, w_0, x₀), gz₀ ≤ F (z₀, w₀, x₀, y₀), and gw₀ ≥ F (w₀, x₀, y₀, z₀), then there exist x, y, z, w ∈ X such that

$$F\left(x,y,z,w\right)=gx,F\left(y,z,w,x\right)=gy,F\left(z,w,x,y\right)=gz,andF\left(w,x,y,z\right)=gw.$$

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 : X⁴ → X and g : X → X are such that F has the mixed g-monotone property. Assume that there exist k ∈ [0, 1) and L ≥ 0 such that

$$d\left(F\left(x,y,z,w\right),F\left(u,v,h,l\right)\right)\le \frac{k}{4}\left\{\begin{array}{c}d\left(gx,gu\right)+d\left(gy,gv\right)+\\ d\left(gz,gh\right)+d\left(gw,gl\right)\end{array}\right\}+LM\left(x,y,z,w,u,v,h,l\right),$$

for any x, y, z, w, ∈ X for which gx ≤ gu, gv ≤ gy, gz ≤ gw, and gl ≤ gw. Suppose F (X⁴) ⊂ g(X) and (g(X), d) is a complete metric space.

If there exist x₀, y₀, z₀, w₀ ∈ X such that gx₀ ≤ F (x₀, y₀, z₀, w₀), gy₀ ≥ F (y₀, z₀, w_0, x₀), gz₀ ≤ F (z₀, w₀, x₀, y₀), and gw₀ ≥ F (w₀, x₀, y₀, z₀), then there exist x, y, z, w ∈ X such that

$$F\left(x,y,z,w\right)=gx,F\left(y,z,w,r\right)=gy,F\left(z,w,x,y\right)=gz,andF\left(w,x,y,z\right)=gw.$$

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 X⁴ 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) ∈ X⁴

$$\left(x,y,z,w\right)\le \left(u,v,r,h\right)\iff x\le u,y\ge v,z\le r\text{and }w\ge l$$

(46)

We say that (x, y, z, w) and (u, v, r, l) are comparable if

$$\left(x,y,z,w\right)\le \left(u,v,r,l\right)\text{or}\text{(}u,v,r,l)\le \left(x,y,z,w\right).$$

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) ∈ X⁴, there exists(a, b, c, d) ∈ X⁴ such that

$$\left(F\left(a,b,c,d\right),F\left(b,c,d,a\right),F\left(c,d,a,b\right),F\left(d,a,b,c\right)\right)$$

is comparable to

$$\begin{array}{c}\left(F\left(x,y,z,w\right),F\left(y,z,w,x\right),F\left(z,w,x,y\right),F\left(w,x,y,z\right)\right)and\\ \left(F\left(u,v,r,l\right),F\left(v,r,l,u\right),F\left(r,l,u,v\right),F\left(l,u,v,r\right)\right).\end{array}$$

Then, F and g have a unique quadruple common fixed point (x, y, z, w) such that

$$\begin{array}{c}x=gx=F\left(x,y,z,w\right),y=gy=F\left(y,z,w,x\right),\\ z=gz=F\left(z,w,x,y\right),andw=gw=F\left(w,x,y,z\right).\end{array}$$

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,

$$\begin{array}{c}F\text{(}x,y,z,w\text{)}=gx,F\text{(}u,v,r,l\text{)}=gu,\\ F\text{(}y,z,w,x\text{)}=gy,F\text{(}v,r,l,u\text{)}=gv,\\ F\text{(}z,w,x,y\text{)}=gz,F\text{(}r,l,u,v\text{)}=gr,\\ F\text{(}w,x,y,z\text{)}=gw,F\text{(}l,u,v,r\text{)}=gl.\end{array}$$

(47)

We shall show that (gx, gy, gz, gw) and (gu, gv, gr, gl) are equal. By assumption, there exists (a, b, c, d) ∈ X⁴ 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

a₀ = a, b₀ = b, c₀ = c, d₀ = d and for any n ≥ 1

$$\begin{array}{c}g{a}_n=F\text{(}{a}_{n--1}\text{,}{b}_{n-1}\text{,}{c}_{n-1}\text{,}{d}_{n-1}\text{)},\hfill \\ g{b}_n=F\text{(}{b}_{n-1}\text{,}{c}_{n-1}\text{,}{d}_{n-1}\text{,}{a}_{n-1}\text{)},\hfill \\ g{c}_n=F\text{(}{c}_{n-1}\text{,}{d}_{n-1}\text{,}{a}_{n-1}\text{,}{b}_{n-1}\text{)},\hfill \\ g{d}_n=F\text{(}{d}_{n-1},a_{n-1},b_{n-1},c_{n-1}\text{)},\hfill \end{array}$$

(48)

for all n. Further, set x₀ = x, y₀ = y, z₀ = z, w₀ = w and u₀ = u, v₀ = v, r₀ = r, l₀ = 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

$$\begin{array}{c}g{x}_n=F\left(x,y,z,w\right),g{u}_n=F\left(u,v,r,l\right),\\ g{y}_n=F\left(y,z,w,x\right),g{v}_n=F\left(v,r,l,u\right),\\ g{z}_n=F\left(z,w,x,y\right),g{r}_n=F\left(r,l,u,v\right),\\ g{w}_n=F\left(w,x,y,z\right),g{l}_n=F\left(l,u,v\text{,}r\right)\end{array}$$

(49)

for all n ≥ 1. Since

$$\begin{array}{cc}\hfill \text{(}F\text{(}x\text{,}y\text{,}z\text{,}w\text{),}F\text{(}y\text{,}z\text{,}w\text{,}x\text{),}F\text{(}z\text{,}w\text{,}x\text{,}y\text{),}F\text{(}w\text{,}x\text{,}y\text{,}z\text{))}& =\text{(}g{x}_1\text{,}g{y}_1\text{,}g{z}_1\text{,}g{w}_1)\hfill \\ =(gx\text{,}gy\text{,}gz\text{,}gw\text{)}\hfill \end{array}$$

is comparable to

$$\left(F\left(a,b,c,d\right),F\left(b,c,d,a\right),F\left(c,d,a,b\right),F\left(d,a,b,c\right)\right)=\left(g{a}_1,g{b}_1,g{c}_1,g{d}_1\right),$$

then it is easy to show (gx, gy, gz, gw) ≥ (ga₁, gb₁, gc₁, gd₁). Recursively, we get that

$$\left(g{a}_n,g{b}_n,g{c}_n,g{d}_n\right)\le \left(gx,gy,gz,gw\right)\text{for all }n.$$

(50)

From (2) and (47), it is obvious that

$$\begin{array}{c}M\text{(}{a}_n\text{,}{b}_n\text{,}{c}_n\text{,}{d}_n\text{,}x\text{,}y\text{,}z\text{,}w\text{)}=M\text{(}y\text{,}z\text{,}w\text{,}x,b_n\text{,}{c}_n\text{,}{d}_n\text{,}{a}_n\text{))}\\ =M\text{(}{c}_n\text{,}{d}_n\text{,}{a}_n\text{,}{b}_n\text{,}z\text{,}w\text{,}x\text{,}y\text{)}=M\text{(}w\text{,}x\text{,}y\text{,}z\text{,}{d}_n\text{,}{a}_n\text{,}{b}_n\text{,}{c}_n\text{)}=\text{0}.\end{array}$$

(51)

By (50), (51), and (3), we have

$$\begin{array}{c}d\left(g{a}_{n+1},gx\right)=d\left(F\left(a_n,b_n,c_n,d_n\right),F\left(x,y,z,w\right)\right)\hfill \\ \le \varphi \left(max\left\{d\left(gx,g{a}_n\right),d\left(gy,g{b}_n\right),d\left(gz,g{c}_n\right),d\left(gw,g{d}_n\right)\right\}\right),\hfill \\ +LM\left(a_n,b_n,c_n,d_n,x,y,z,w\right)\hfill \\ =\varphi \left(max\left\{d\left(gx,g{a}_n\right),d\left(gy,g{b}_n\right),d\left(gz,g{c}_n\right),d\left(gw,g{d}_n\right)\right\}\right),\hfill \end{array}$$

(52)

$$\begin{array}{c}d\left(gy,g{b}_{n+1}\right)=d\left(F\left(y,z,w,x\right),F\left(b_n,c_n,d_n,a_n\right)\right)\hfill \\ \le \varphi \left(max\left\{d\left(g{a}_n,gx\right),d\left(g{b}_n,gy\right),d\left(g{c}_n,gz\right),d\left(g{d}_n,gw\right)\right\}\right)\hfill \\ +LM\left(y,z,w,x,b_n,c_n,d_n,a_n\right)\hfill \\ =\varphi \left(max\left\{d\left(g{a}_n,gx\right),d\left(g{b}_n,gy\right),d\left(g{c}_n,gz\right),d\left(g{d}_n,gw\right)\right\}\right),\hfill \end{array}$$

(53)

$$\begin{array}{c}d\left(g{c}_{n+1},gz\right)=d\left(F\left(c_n,d_n,a_n,b_n\right),F\left(z,w,x,y\right)\right)\hfill \\ \le \varphi \left(max\left\{d\left(g{a}_n,gx\right),d\left(g{b}_n,gy\right),d\left(g{c}_n,gz\right),d\left(g{d}_n,gw\right)\right\}\right)\hfill \\ +LM\left(c_n,d_n,a_n,b_n,z,w,x,y\right)\hfill \\ =\varphi \left(max\left\{d\left(g{a}_n,gx\right),d\left(g{b}_n,gy\right),d\left(g{c}_n,gz\right),d\left(g{d}_n,gw\right)\right\}\right)\hfill \end{array}$$

(54)

and

$$\begin{array}{c}d\left(gw,g{d}_{n+1}\right)=d\left(F\left(w,x,y,z\right),F\left(d_n,a_n,b_n,c_n\right)\right)\hfill \\ \le \varphi \left(max\left\{d\left(g{a}_n,gx\right),d\left(g{b}_n,gy\right),d\left(g{c}_n,gz\right),d\left(g{d}_n,gw\right)\right\}\right)\hfill \\ +LM\left(w,x,y,z,d_n,a_n,b_n,c_n\right)\hfill \\ =\varphi \left(max\left\{d\left(g{d}_n,gw\right),d\left(g{a}_n,gx\right),d\left(g{b}_n,gy\right),d\left(g{c}_n,gz\right)\right\}\right).\hfill \end{array}$$

(55)

From (52)-(55), it follows that

$$max\left\{\begin{array}{c}\hfill d\left(gz,g{c}_{n+1}\right),d\left(gy,g{b}_{n+1}\right),\hfill \\ \hfill d\left(gx,g{a}_{n+1}\right),d\left(gw,g{d}_{n+1}\right)\hfill \end{array}\right\}\le \varphi \left(max\left\{\begin{array}{c}\hfill d\left(gz,g{c}_n\right),d\left(gy,g{b}_n\right),\hfill \\ \hfill d\left(gx,g{a}_n\right),d\left(gw,g{d}_n\right)\hfill \end{array}\right\}\right).$$

(56)

Therefore, for each n ≥ 1,

$$max\left\{\begin{array}{c}d\left(gz,g{c}_n\right),d\left(gy,g{b}_n\right),\\ d\left(gx,g{a}_n\right),d\left(gw,g{d}_n\right)\end{array}\right\}\le {\varphi }^n\left(max\left\{\begin{array}{c}d\left(gz,g{c}_0\right),d\left(gy,g{b}_0\right),\\ d\left(gx,g{a}_0\right),d\left(gw,g{d}_0\right)\end{array}\right\}\right).$$

(57)

It is known that ϕ(t) < t and $\underset{r\to t^{+}}{lim}\varphi \left(r\right)<t$ imply $\underset{n\to \infty }{lim}{\varphi }^n\left(t\right)=0$ for each t > 0. Thus, from (57)

$$\underset{n\to \infty }{lim}max\left\{d\left(gz,g{c}_n\right),d\left(gy,g{b}_n\right),d\left(gx,g{a}_n\right),d\left(gw,g{d}_n\right)\right\}=0.$$

This yields that

$$\underset{n\to \infty }{lim}d\left(gx,g{a}_n\right)=0,\underset{n\to \infty }{lim}d\left(gy,g{b}_n\right)=0,\underset{n\to \infty }{lim}d\left(gz,g{c}_n\right)=0\text{and}\underset{n\to \infty }{lim}d\left(gw,g{d}_n\right)=0.$$

(58)

Analogously, we may show that

$$\underset{n\to \infty }{lim}d\left(gu,g{a}_n\right)=0,\underset{n\to \infty }{lim}d\left(gv,g{b}_n\right)=0,\underset{n\to \infty }{lim}d\left(gr,g{c}_n\right)=0\text{ and }\underset{n\to \infty }{lim}d\left(gl,g{d}_n\right)=0.$$

(59)

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

$$\begin{array}{c}g{x}^{\prime }=g\left(gx\right)=g\left(F\left(x,y,z,w\right)\right)=F\left(gx,gy,gz,gw\right),\\ g{y}^{\prime }=g\left(gy\right)=g\left(F\left(y,z,w,x\right)\right)=F\left(gy,gz,gw,gx\right),\\ g{z}^{\prime }=g\left(gz\right)=g\left(F\left(z,w,x,y\right)\right)=F\left(gz,gw,gx,gy\right)\end{array}$$

and

$$g{w}^{\prime }=g\left(gw\right)=g\left(F\left(w,x,y,z\right)\right)=F\left(gw,gx,gy,gz\right)$$

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

$$g{x}^{\prime }=gx=x^{\prime },g{y}^{\prime }=gy=y^{\prime }\text{and}g{z}^{\prime }=gz=z^{\prime },g{w}^{\prime }=gw=w^{\prime }.$$

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: X⁴→X be defined by

$$g\left(x\right)=\frac{3}{4}x,F\left(x,y,z,w\right)=\frac{x-y+z-w}{8},\mathit{\text{for all}}x,y,z,w\in X$$

Take ϕ : [0, ∞) → [0, ∞) be given by $\varphi \left(t\right)=\frac{2}{3}t$ 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

$$\begin{array}{cc}\hfill d\left(F\left(x,y,z,w\right),F\left(u,v,h,l\right)\right)& =\frac{u-x}{8}+\frac{y-v}{8}+\frac{h-z}{8}+\frac{w-l}{8}\hfill \\ \le \frac{1}{2}\left[max\left\{\left(u-x\right),\left(y-v\right),\left(h-z\right),\left(w-l\right)\right\}\right]\hfill \\ =\frac{2}{3}max\left\{d\left(gx,gu\right),d\left(gy,gv\right),d\left(gz,gh\right),d\left(gw,gl\right)\right\}\hfill \\ \le \varphi \left(max\left\{d\left(gx,gu\right),d\left(gy,gv\right),d\left(gz,gh\right),d\left(gw,gl\right)\right\}\right)\hfill \\ +LM\left(x,y,z,w,u,v,h,l\right),\hfill \end{array}$$

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

$$\left\{\begin{array}{c}X=Q+A_1^{*}X{A}_1-B_1^{*}Y{B}_1+A_2^{*}Z{A}_2-B_2^{*}T{B}_2\\ Y=Q+A_1^{*}Y{A}_1-B_1^{*}Z{B}_1+A_2^{*}T{A}_2-B_2^{*}X{B}_2\\ Z=Q+A_1^{*}Z{A}_1-B_1^{*}T{B}_1+A_2^{*}X{A}_2-B_2^{*}Y{B}_2\\ T=Q+A_1^{*}T{A}_1-B_1^{*}X{B}_1+A_2^{*}Y{A}_2-B_2^{*}Z{B}_2,\end{array}\right.$$

(60)

where $A_1,A_2,B_1,B_2\in \mathcal{M}\left(n\right)$: the set of all n × n matrices, $Q\in \mathcal{P}\left(n\right):$ the set of all n × n positive definite matrices, and $\mathcal{H}\left(n\right)$ is the set of all n × n Hermitian matrices.

We endow $\mathcal{H}\left(n\right)$ with the partial order ≼ given by

$$M,N\in \mathcal{H}\left(n\right),M\preccurlyeq N\iff N-M\in \mathcal{P}\left(n\right).$$

For a fixed $P\in \mathcal{P}\left(n\right)$, we consider

$$\left|\right|H|{|}_{1,P}=tr\left(P^{\frac{1}{2}}H{P}^{\frac{1}{2}}\right).$$

for all $H\in \mathcal{H}\left(n\right)$, where tr is the trace operator. The space $\mathcal{H}\left(n\right)$ equipped with the metric induced by $\left|\right|.|{|}_{1,P}$ 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

$$0\le tr\left(AB\right)=tr\left(BA\right)\le \left|\right|A\left|\right|tr\left(B\right),$$

where ||.|| is the spectral norm.

Theorem 3.1 Suppose that there exists $P\in \mathcal{P}\left(n\right)$ such that

$$k=4max\left\{\left|\right|P^{-\frac{1}{2}}{A}_1^{*}P{A}_1{P}^{-\frac{1}{2}}\left|\right|,\left|\right|P^{-\frac{1}{2}}{A}_2^{*}P{A}_2{P}^{-\frac{1}{2}}\left|\right|,\left|\right|P^{-\frac{1}{2}}{B}_1^{*}P{B}_1{P}^{-\frac{1}{2}}\left|\right|,\left|\right|P^{-\frac{1}{2}}{B}_2^{*}P{B}_2{P}^{-\frac{1}{2}}\left|\right|\right\}<1.$$

(61)

Suppose also that

$$0\preccurlyeq \sum _{i=1}^2{A}_i^{*}Q{A}_{i}andQ\preccurlyeq \sum _{i=1}^2{B}_i^{*}Q{B}_i.$$

(62)

Then, the system (60) has one and only one solution $\left(X_1,X_2,X_3,X_4\right)\in {\left(\mathcal{H}\left(n\right)\right)}^4$.

Proof. Consider the mappings $F:{\left(\mathcal{H}\left(n\right)\right)}^4\to \mathcal{H}\left(n\right)$ and $g:\mathcal{H}\left(n\right)\to \mathcal{H}\left(n\right)$ defined by

$$F\left(X_1,X_2,X_3,X_4\right)=Q+A_1^{*}{X}_1{A}_1-B_1^{*}{X}_2{B}_1+A_2^{*}{X}_3{A}_2-B_2^{*}{X}_4{B}_2\text{ and}gX=X,$$

for all $X,X_i\in \mathcal{H}\left(n\right)$ i =1, . . . , 4.

For all $X_i,Y_i\in \mathcal{H}\left(n\right)$ i = 1. . . , 4 with gX₁ ≼ gY₁, gY₂ ≼ gX₂, gX₃ ≼ gY₃ and gY₄ ≼ gX₄, by using Lemma 3.1, we have

$$\begin{array}{c}\left|\right|F\left(Y_1,Y_2,Y_3,Y_4\right)-F\left(X_1,X_2,X_3,X_4\right)|{|}_{1,P}\hfill \\ =\left|\right|A_1^{*}\left(Y_1-X_1\right)A_1-B_1^{*}\left(Y_2-X_2\right)B_1+A_2^{*}\left(Y_3-X_3\right)A_2-B_2^{*}\left(Y_4-X_4\right)B_2|{|}_{1,P}\hfill \\ =tr\left[P^{\frac{1}{2}}\left(A_1^{*}\left(Y_1-X_1\right)A_1-B_1^{*}\left(Y_2-X_2\right)B_1+A_2^{*}\left(Y_3-X_3\right)A_2-B_2^{*}\left(Y_4-X_4\right)B_2\right)P^{\frac{1}{2}}\right]\hfill \\ =tr\left[A_{1}P{A}_1^{*}\left(Y_1-X_1\right)\right]+tr\left[B_{1}P{B}_1^{*}\left(X_2-Y_2\right)\right]+tr\left[A_{2}P{A}_2^{*}\left(Y_3-X_3\right)\right]+tr\left[B_{2}P{B}_2^{*}\left(X_4-Y_4\right)\right]\hfill \\ =tr\left[A_{1}P{A}_1^{*}{P}^{-\frac{1}{2}}{P}^{\frac{1}{2}}\left(Y_1-X_1\right)P^{\frac{1}{2}}{P}^{-\frac{1}{2}}\right]+tr\left[B_{1}P{B}_1^{*}{P}^{-\frac{1}{2}}{P}^{\frac{1}{2}}\left(X_2-Y_2\right)P^{\frac{1}{2}}{P}^{-\frac{1}{2}}\right]\hfill \\ +tr\left[A_{2}P{A}_2^{*}{P}^{-\frac{1}{2}}{P}^{\frac{1}{2}}\left(Y_3-X_3\right)P^{\frac{1}{2}}{P}^{-\frac{1}{2}}\right]+tr\left[B_{2}P{B}_2^{*}{P}^{-\frac{1}{2}}{P}^{\frac{1}{2}}\left(X_4-Y_4\right)P^{\frac{1}{2}}{P}^{-\frac{1}{2}}\right]\hfill \\ \le \left|\right|P^{-\frac{1}{2}}{A}_{1}P{A}_1^{*}{P}^{-\frac{1}{2}}\left|\right|tr\left(P^{\frac{1}{2}}\left(Y_1-X_1\right)P^{\frac{1}{2}}\right)+\left|\right|P^{-\frac{1}{2}}{B}_{1}P{B}_1^{*}{P}^{-\frac{1}{2}}\left|\right|tr\left(P^{\frac{1}{2}}\left(X_2-Y_2\right)P^{\frac{1}{2}}\right)\hfill \\ +\left|\right|P^{-\frac{1}{2}}{A}_{2}P{A}_2^{*}{P}^{-\frac{1}{2}}\left|\right|tr\left(P^{\frac{1}{2}}\left(Y_3-X_3\right)P^{\frac{1}{2}}\right)+\left|\right|P^{-\frac{1}{2}}{B}_{2}P{B}_2^{*}{P}^{-\frac{1}{2}}\left|\right|tr\left(P^{\frac{1}{2}}\left(X_4-Y_4\right)P^{\frac{1}{2}}\right)\hfill \\ =\left|\right|P^{-\frac{1}{2}}{A}_{1}P{A}_1^{*}{P}^{-\frac{1}{2}}\left|\right|\left|\right|Y_1-X_1|{|}_{1,P}+|\left|P^{-\frac{1}{2}}{B}_{1}P{B}_1^{*}{P}^{-\frac{1}{2}}\right|\left|\right||X_2-Y_2|{|}_{1,P}\hfill \\ +\left|\right|P^{-\frac{1}{2}}{A}_{2}P{A}_2^{*}{P}^{-\frac{1}{2}}\left|\right|\left|\right|Y_3-X_3|{|}_{1,P}+|\left|P^{-\frac{1}{2}}{B}_{2}P{B}_2^{*}{P}^{-\frac{1}{2}}\right|\left|\right||X_4-Y_4|{|}_{1,P}\hfill \\ \le \frac{k}{4}\left(\left|\right|g{Y}_1-g{X}_1|{|}_{1,P}+||g{X}_2-g{Y}_2|{|}_{1,P}+\left|\right|g{Y}_3-g{X}_3|{|}_{1,P}+||g{X}_4-g{Y}_4|{|}_{1,P}\right).\hfill \end{array}$$

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 g 0 ≽ 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 $\mathcal{H}\left(n\right)$). Then, there exist $X_1,X_2,X_3,X_4\in \mathcal{H}\left(n\right)$ such that

$$\begin{array}{c}F\left(X_1,X_2,X_3,X_4\right)=X_1,F\left(X_2,X_3,X_4,X_1\right)=X_2,\\ F\left(X_3,X_4,X_1,X_2\right)=X_3\text{ and }F\left(X_4,X_1,X_2,X_4\right)=X_4.\end{array}$$

On the other hand, for all $X,Y\in \mathcal{H}\left(n\right)$ 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).