# 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 : X4X and g : XX 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 , where he extended the Banach contraction principle in partially ordered sets. In 2004, Ran and Reurings  presented some applications of Turinici's theorem to matrix equations. Following these initial articles, some remarkable results were reported see, e.g., .

Gnana Bhashkar and Lakshmikantham in  introduced the concept of a coupled fixed point of a mapping F : X × XX and investigated some coupled fixed point theorems in partially ordered complete metric spaces. Later, Lakshmikantham and Ćirić  proved coupled coincidence and coupled common fixed point theorems for nonlinear mappings F : X × XX and g : XX 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  introduced the concept of tripled fixed point in ordered sets.

For simplicity, we denote by Xk where k . Let us recall some basic definitions.

Definition 1.1 (See ) 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

$x 1 , x 2 ∈ X , x 1 ≤ x 2 ⇒ F ( x 1 , y , z ) ≤ F ( x 2 , y , z ) , y 1 , y 2 ∈ X , y 1 ≤ y 2 ⇒ F ( x , y 1 , z ) ≥ F ( x , y 2 , z ) , z 1 , z 2 ∈ X , z 1 ≤ z 2 ⇒ F ( x , y , z 1 ) ≤ F ( x , y , z 2 ) .$

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

$F ( x , y , z ) = x , F ( y , x , y ) = y a n d F ( z , y , x ) = z .$

Also, Berinde and Borcut  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

$d ( F ( x , y , z ) , F ( u , v , w ) ) ≤jd ( x , u ) +rd ( y , v ) +ld ( z , w ) ,$
(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 x0, y0, z0 X such that x0 ≤ F (x0, y0, z0), y0 ≥ F (y0, x0, z0) and z0F (z0, y0, x0), then there exist x, y, z X such that

$F ( x , y , z ) =x,F ( y , x , y ) =yandF ( z , y , x ) =z,$

that is, F has a tripled fixed point.

Recently, Aydi et al.  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

$x 1 , x 2 ∈ X , g x 1 ≤ g x 2 ⇒ F ( x 1 , y , z ) ≤ F ( x 2 , y , z ) , y 1 , y 2 ∈ X , g y 1 ≤ g y 2 ⇒ F ( x , y 1 , z ) ≥ F ( x , y 2 , z ) , z 1 , z 2 ∈ X , g z 1 ≤ g z 2 ⇒ F ( x , y , z 1 ) ≤ F ( x , y , z 2 ) .$

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

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

(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

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

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

$g ( F ( x , y , z ) ) =F ( g x , g y , g z ) .$

The notion of fixed point of order N ≥ 3 was first introduced by Samet and Vetro . Very recently, Karapinar used the concept of quadruple fixed point and proved some fixed point theorems on the topic . Following this study, quadruple fixed point is developed and some related fixed point theorems are obtained in .

Definition 1.7  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

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

Let (X, d) be a metric space. The mapping $d ̄ : X 4 →X,$ given by

$d ̄ ( ( x , y , z , w ) , ( u , v , h , l ) ) =d ( x , y ) +d ( y , v ) +d ( z , h ) +d ( w , l ) ,$

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

Definition 1.8  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,

$x 1 , x 2 ∈ X , x 1 ≤ x 2 i m p l i e s F ( x 1 , y , z , w ) ≤ F ( x 2 , y , z , w ) , y 1 , y 2 ∈ X , y 1 ≤ y 2 i m p l i e s F ( x , y 2 , z , w ) ≤ F ( x , y 1 , z , w ) , z 1 , z 2 ∈ X , z 1 ≤ z 2 i m p l i e s F ( x , y , z 1 , w ) ≤ F ( x , y , z 2 , w ) ,$

and

$w 1 , w 2 ∈X, w 1 ≤ w 2 impliesF ( x , y , z , w 2 ) ≤F ( x , y , z , w 1 ) .$

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

$x 1 , x 2 ∈ X , g x 1 ≤ g x 2 ⇒ F ( x 1 , y , z , w ) ≤ F ( x 2 , y , z , w ) , y 1 , y 2 ∈ X , g y 1 ≤ g y 2 ⇒ F ( x , y 1 , z , w ) ≥ F ( x , y 2 , z , w ) , z 1 , z 2 ∈ X , g z 1 ≤ g z 2 ⇒ F ( x , y , z 1 , w ) ≤ F ( x , y , z 2 , w ) a n d w 1 , w 2 ∈ X , g w 1 ≤ g w 2 ⇒ F ( x , y , z , w 1 ) ≥ F ( x , y , z , w 2 ) .$

Definition 2.2 Let F : X4X and g : X → X. An element (x, y, z, w) is called a quadruple coincidence point of F and g if

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

(gx, gy, gz, gw) is said a quadruple point of coincidence of F and g.

Definition 2.3 Let F : X4X and g : X → X. An element (x, y, z, w) is called a quadruple common fixed point of F and g if

$F ( x , y , z , w ) = g x = x , F ( y , z , w , x ) = g y = y , F ( z , w , x , y ) = g z = z , a n d F ( w , x , y , z ) = g w = w .$

Definition 2.4 Let X be a non-empty set. Then we say that the mappings F : X4X and g : X → × are commutative if for all x, y, z, w X

$g ( F ( x , y , z , w ) ) = F ( g x , g y , g z , g w ) .$

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

1. 1.

ϕ(t) < t for all t (0,+∞).

2. 2.

$lim r → t + ϕ ( r ) for all t (0,+∞).

For simplicity, we define the following.

$M ( x , y , z , w , u , v , h , l ) =min d ( F ( x , y , z , w ) , g x ) , d ( F ( x , y , z , w ) , g u ) , d ( F ( u , v , h , l ) , g u ) .$
(2)

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

$d ( F ( x , y , z , w ) , F ( u , v , , h , l ) ) ≤ ϕ ( max { d ( g x , g u ) , d ( g y , g v ) , d ( g z , g h ) , d ( g w , g l ) } ) + L M ( x , y , z , w , u , v , h , l )$
(3)

for any x, y, z, w, u, v, h, l X for which gxgu, gvgy, gzgh and glgw. Suppose F (X4) g(X), g is continuous and commutes with F. If there exist x0, y0, z0, w0 X such that

$g x 0 ≤ F ( x 0 , y 0 , z 0 , w 0 ) , g y 0 ≥ F ( y 0 , z 0 , w 0 , x 0 ) , g z 0 ≤ F ( z 0 , w 0 , x 0 , y 0 ) , a n d g w 0 ≥ F ( w 0 , x 0 , y 0 , z 0 ) ,$

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

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

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

Proof. Let x0, y0, z0, w0 X such that

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

$g x 1 = F ( x 0 , y 0 , z 0 , w 0 ) , g y 1 = F ( y 0 , z 0 , w 0 , x 0 ) , g z 1 = F ( z 0 , w 0 , x 0 , y 0 ) and g w 1 = F ( w 0 , x 0 , y 0 , z 0 ) .$
(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

$g x n + 1 = F ( x n , y n , z n , w n ) , g y n + 1 = F ( y n , z n , w n , x n ) , g z n + 1 = F ( z n , w n , x n , y n ) , and g w n + 1 = F ( w n , x n , y n , z n ) .$
(5)

We shall show that

(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

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 ≤ gxn+1, gyn+1gy n , gz n ≤ gzn+1and gwn+1gw n , we obtain

$g x n + 1 = F ( x n , y n , z n , w n ) ≤ F ( x n + 1 , y n , z n , w n ) ≤ F ( x n + 1 , y n , z n + 1 , w n ) ≤ F ( x n + 1 , y n + 1 , z n + 1 , w n ) ≤ F ( x n + 1 , y n + 1 , z n + 1 , w n + 1 ) = g x n + 2 ,$
$g y n + 2 = F ( y n + 1 , z n + 1 , w n + 1 , x n + 1 ) ≤ F ( y n + 1 , z n , x n + 1 , w n + 1 ) ≤ F ( y n , z n , x n + 1 , w n + 1 ) ≤ F ( y n , z n , x n , w n + 1 ) ≤ F ( y n , z n , x n , w n ) = g y n + 1 ,$
$g z n + 1 = F ( z n , y n , x n , w n ) ≤ F ( z n + 1 , y n , x n , w n ) ≤ F ( z n + 1 , y n + 1 , x n , w n ) ≤ F ( z n + 1 , y n + 1 , x n + 1 , w n ) ≤ F ( z n + 1 , y n + 1 , x n + 1 , w n + 1 ) = g z n + 2 ,$

and

$g w n + 2 = F ( w n + 1 , x n + 1 , y n + 1 , z n + 1 ) ≤ F ( w n + 1 , x n , y n + 1 , z n + 1 ) ≤ F ( w n , x n , y n + 1 , z n + 1 ) ≤ F ( w n , x n , y n , z n + 1 ) ≤ F ( w n , x n , y n , z n ) = g w n + 1 .$

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

$g x n = g x n + 1 , g y n = g y n + 1 , g z n = g z n + 1 , and g w n = g w n + 1$

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 that at least

(7)

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

(8)

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

$d ( g x n , g x n + 1 ) = d ( F ( x n - 1 , y n - 1 , z n - 1 , w n - 1 ) , F ( x n , y n , z n , w n ) ) ≤ ϕ ( max { d ( g x n - 1 , g x n ) , d ( g y n - 1 , g y n ) , d ( g z n - 1 , g z n ) , d ( g w n - 1 , g w n ) } ) + L M ( x n - 1 , y n - 1 , z n - 1 , w n - 1 , x n , y n , z n , w n ) = ϕ ( max { d ( g x n - 1 , g x n ) , d ( g y n - 1 , g y n ) , d ( g z n - 1 , g z n ) , d ( g w n - 1 , g w n ) } ) ,$
(9)
$d ( g y n , g y n + 1 ) = d ( F ( y n , z n , w n , x n ) , y n - 1 , F ( y n - 1 , z n - 1 , w n - 1 , x n - 1 ) ) ≤ ϕ ( max { d ( g y n - 1 , g y n ) , d ( g x n - 1 , g x n ) , d ( g z n - 1 , g z n ) , d ( g w n - 1 , g w n ) } ) , + L M ( y n , z n , w n , w n , y n - 1 , z n - 1 , w n - 1 , x n - 1 ) = ϕ ( max { d ( g y n - 1 , g y n ) , d ( g x n - 1 , g x n ) , d ( g z n - 1 , g z n ) , d ( g w n - 1 , g w n ) } ) ,$
(10)
$d ( g z n , g z n + 1 ) = d ( F ( z n - 1 , w n - 1 , x n - 1 , y n - 1 ) , F ( z n , w n , x n , y n ) ) ≤ ϕ ( max { , d ( g z n - 1 , g z n ) , d ( g w n - 1 , g w n ) , d ( g x n - 1 , g x n ) , d ( g y n - 1 , g y n ) } ) + L M ( z n - 1 , w n - 1 , x n - 1 , y n - 1 , z n , w n , x n , y n ) = ϕ ( max { d ( g z n - 1 , g z n ) , d ( g w n - 1 , g w n ) , d ( g x n - 1 , g x n ) , d ( g y n - 1 , g y n ) } )$
(11)

and

$d ( g w n , g w n + 1 ) = d ( F ( w n , x n , y n , z n ) , F ( w n - 1 , x n - 1 , y n - 1 , z n - 1 ) ) ≤ ϕ ( max { d ( g w n - 1 , g w n ) , d ( g x n - 1 , g x n ) , d ( g y n - 1 , g y n ) , d ( g z n - 1 , g z n ) } ) , + L M ( w n , x n , y n , z n , w n - 1 , x n - 1 , y n - 1 , z n - 1 ) = ϕ ( max { d ( g w n - 1 , g w n ) , d ( g x n - 1 , g x n ) , d ( g y n - 1 , g y n ) , d ( g z n - 1 , g z n ) } ) .$
(12)

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

$0 < max { d ( g x n , g x n + 1 ) , d ( g y n , g y n + 1 ) , d ( g z n , g z n + 1 ) , d ( g w n , g w n + 1 ) } ≤ ϕ ( max { d ( g z n - 1 , g z n ) , d ( g y n - 1 , g y n ) , d ( g x n - 1 , g x n ) , d ( g w n - 1 , g w n ) } ) < max { d ( g z n - 1 , g z n ) , d ( g y n - 1 , g y n ) , d ( g x n - 1 , g x n ) , d ( g w n - 1 , g w n ) } .$
(13)

It follows that

$max d ( g x n , g x n + 1 ) , d ( g y n , g y n + 1 , d ( g z n , g z n + 1 ) , d ( g w n , g w n + 1 ) < max d ( g z n - 1 , g z n ) , d ( g y n - 1 , g y n ) , d ( g x n - 1 , g x n ) , d ( g w n - 1 , g w n ) .$
(14)

Thus, {max{d(gx n , gxn+1), d(gy n , gyn+1), d(gz n , gzn+1), d(gw n , gwn+1)}} is a positive decreasing sequence. Hence, there exists r ≥ 0 such that

$lim n → + ∞ max { d ( g x n , g x n + 1 ) , d ( g y n , g y n + 1 ) , d ( g z n , g z n + 1 ) , d ( g w n , g w n + 1 ) } =r.$

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

$0
(15)

It is a contradiction. We deduce that

$lim n → + ∞ max { d ( g x n , g x n + 1 ) , d ( g y n , g y n + 1 ) , d ( g z n , g z n + 1 ) , d ( g w n , g w n + 1 ) } =0.$
(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,

$lim n , m → + ∞ d ( g x m , g x n ) ≠ 0 or lim n , m → + ∞ d ( g y m , g y n ) ≠ 0$

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

$max { d ( g x m k , g x n k ) , d ( g y m k , g y n k ) , d ( g z m k , g z n k ) , d ( g w m k , g w n k ) } ≥ε.$
(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

$max { d ( g x m k , g x n k - 1 ) , d ( g y m k , g y n k - 1 ) , d ( g z m k , g z n k - 1 ) , d ( g w m k , g w n k - 1 ) } <ε.$
(18)

By triangular inequality and (18), we have

$d ( g x m k , g x n k ) ≤ d ( g x m k , g x n k - 1 ) + d ( g x n k - 1 , g x n k ) < ε + d ( g x n k - 1 , g x n k ) .$

Thus, by (16) we obtain

$lim k → + ∞ d ( g x m k , g x n k ) ≤ lim k → + ∞ d ( g x m k , g x n k - 1 ) ≤ε.$
(19)

Similarly, we have

$lim k → + ∞ d ( g y m k , g y n k ) ≤ lim k → + ∞ d ( g y m k , g y n k - 1 ) ≤ε,$
(20)
$lim k → + ∞ d ( g z m k , g z n k ) ≤ lim k → + ∞ d ( g z m k , g z n k - 1 ) ≤ε,$
(21)

and

$lim k → + ∞ d ( g w m k , g w n k ) ≤ lim k → + ∞ d ( g w m k , g w n k - 1 ) ≤ε.$
(22)

Again by (18), we have

$d ( g x m k , g x n k ) ≤ d ( g x m k , g x m k - 1 ) + d ( g x m k - 1 , g x n k - 1 ) + d ( g x n k - 1 , g x n k ) ≤ d ( g x m k , g x m k - 1 ) + d ( g x m k - 1 , g x m k ) + d ( g x m k , g x n k - 1 ) + d ( g x n k - 1 , g x n k ) < d ( g x m k , g x m k - 1 ) + d ( g x m k - 1 , g x m k ) + ε + d ( g x n k - 1 , g x n k ) .$

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

$lim k → + ∞ d ( g x m k , g x n k ) ≤ lim k → + ∞ d ( g x m k - 1 , g x n k - 1 ) ≤ε,$
(23)
$lim k → + ∞ d ( g y m k , g y n k ) ≤ lim k → + ∞ d ( g y m k - 1 , g y n k - 1 ) ≤ε,$
(24)
$lim k → + ∞ d ( g z m k , g z n k ) ≤ lim k → + ∞ d ( g z m k - 1 , g z n k - 1 ) ≤ε$
(25)

and

$lim k → + ∞ d ( g w m k , g w n k ) ≤ lim k → + ∞ d ( g w m k - 1 , g w n k - 1 ) ≤ε.$
(26)

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

$lim k → + ∞ max { d ( g x m k , g x n k ) , d ( g y m k , g y n k ) , d ( g z m k , g z n k ) , d ( g w m k , g w n k ) } = lim k → + ∞ max { d ( g x m k - 1 , g x n k - 1 ) , d ( g y m k - 1 , g y n k - 1 ) , d ( g z m k - 1 , g z n k - 1 ) , d ( g w m k - 1 , g w n k - 1 ) } = ε .$
(27)

By (16), it is easy to see that

$lim k → + ∞ M ( x m k - 1 , y m k - 1 , z m k - 1 , w m k - 1 , x n k - 1 , y n k - 1 , z n k - 1 , w n k - 1 ) = lim k → + ∞ M ( y n k - 1 , z n k - 1 , w n k - 1 , x n k - 1 , y m k - 1 , z m k - 1 , w m k - 1 , x m k - 1 ) = lim k → + ∞ M ( z m k - 1 , w m k - 1 , x m k - 1 , y m k - 1 , z n k - 1 , w n k - 1 , x n k - 1 , y m k - 1 ) = lim k → + ∞ M ( w n k - 1 , x n k - 1 , y m k - 1 , z n k - 1 , w m k - 1 , x m k - 1 , y m k - 1 , z m k - 1 ) = 0 .$
(28)

Now, using inequality (3), we obtain

$d ( g x m k , g x n k ) = d ( F ( x m k - 1 , y m k - 1 , z m k - 1 , w m k - 1 ) , F ( x n k - 1 , y n k - 1 , z n k - 1 , w n k - 1 ) ) ≤ ϕ ( max { d ( x m k - 1 , x n k - 1 ) , d ( y m k - 1 , y n k - 1 ) , d ( z m k - 1 , z n k - 1 ) , d ( w m k - 1 , w n k - 1 ) } ) + L M ( x m k - 1 , y m k - 1 , z m k - 1 , w m k - 1 , x n k - 1 , y n k - 1 , z n k - 1 , w n k - 1 ) ,$
(29)
$d ( g y n k , g y m k ) = d ( F ( y n k - 1 , z n k - 1 , w n k - 1 , x n k - 1 ) , F ( y m k - 1 , z m k - 1 , w m k - 1 , x m k - 1 ) ) ≤ ϕ ( max { d ( y m k - 1 , y n k - 1 ) , d ( z m k - 1 , z n k - 1 ) , d ( w m k - 1 , w n k - 1 , d ( x m k - 1 , x n k - 1 ) } ) + L M ( y n k - 1 , z n k - 1 , w n k - 1 , x n k - 1 , y m k - 1 , z m k - 1 , w m k - 1 , x m k - 1 ) ,$
(30)
$d ( g z m k , g z n k ) = d ( F ( z m k - 1 , w m k - 1 , x m k - 1 , y m k - 1 ) , F ( z n k - 1 , w n k - 1 , x n k - 1 , y n k - 1 ) ) ≤ ϕ ( max { d ( z m k - 1 , z n k - 1 ) , d ( w m k - 1 , w n k - 1 , d ( x m k - 1 , x n k - 1 ) , d ( y m k - 1 , y n k - 1 ) } ) + L M ( z m k - 1 , w m k - 1 , x m k - 1 , y m k - 1 , z n k - 1 , w n k - 1 , x n k - 1 , y m k - 1 )$
(31)

and

$d ( g w n k , g w m k ) = d ( F ( w n k - 1 , x n k - 1 , y n k - 1 , z n k - 1 ) , F ( w m k - 1 , x m k - 1 , y m k - 1 , z m k - 1 ) ) ≤ ϕ ( max { d ( w m k - 1 , w n k - 1 , d ( x m k - 1 , x n k - 1 ) , d ( y m k - 1 , y n k - 1 ) , d ( z m k - 1 , z n k - 1 ) } ) + L M ( w n k - 1 , x n k - 1 , y n k - 1 , z n k - 1 , w m k - 1 , x m k - 1 , y m k - 1 , z m k - 1 ) .$
(32)

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

$max { d ( g x m k , g x n k ) , d ( g y m k , g y n k ) , d ( g z m k , g z n k ) , d ( g w m k , g w n k ) } ≤ ϕ ( max { d ( x m k - 1 , x n k - 1 ) , d ( y m k - 1 , y n k - 1 ) , d ( z m k - 1 , z n k - 1 ) , d ( g w m k , g w n k ) } ) + L M ( x m k - 1 , y m k - 1 , z m k - 1 , w m k - 1 , x n k - 1 , y n k - 1 , z n k - 1 , w n k - 1 ) + L M ( y n k - 1 , z n k - 1 , w n k - 1 , x n k - 1 , y m k - 1 , z m k - 1 , w m k - 1 , x m k - 1 ) + L M ( z m k - 1 , w m k - 1 , x m k - 1 , y m k - 1 , z n k - 1 , w n k - 1 , x n k - 1 , y m k - 1 ) + L M ( w n k - 1 , x n k - 1 , y n k - 1 , z n k - 1 , w m k - 1 , x m k - 1 , y m k - 1 , z m k - 1 ) .$
(33)

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

$0<ε≤ lim t → ε + ϕ ( t ) <ε,$

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

$lim n → + ∞ g x n =x, lim n → + ∞ g y n =y, lim n → + ∞ g y n =y, and lim n → + ∞ g w n =w.$
(34)

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

$lim n → + ∞ g ( g x n ) = g x , lim n → + ∞ g ( g y n ) = g y , lim n → + ∞ g ( g z n ) = g z , and lim n → + ∞ g ( g w n ) = g w .$
(35)

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

$g ( g x n + 1 ) =g ( F ( x n , y n , z n , w n ) ) =F ( g x n , g y n , g z n , g w n ) ,$
(36)
$g ( g y n + 1 ) =g ( F ( y n , z n , w n , x n ) ) =F ( g y n , g z n , g w n , g x n ) ,$
(37)
$g ( g z n + 1 ) = g ( F ( z n , w n , x n , y n ) ) = F ( g z n , g w n , g x n , y n ) ,$
(38)

and

$g ( g w n + 1 ) =g ( F ( w n , x n , y n , z n ) ) =F ( g w n , g x n , y n , g z n ) .$
(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

$g x = lim n → + ∞ g ( g x n + 1 ) = lim n → + ∞ F ( g x n , g y n , g z n , g w n ) = F ( lim n → + ∞ g x n , lim n → + ∞ g y n , lim n → + ∞ g z n , lim n → + ∞ g w n ) = F ( x , y , z , w ) ,$
(40)
$g y = lim n → + ∞ g ( g y n + 1 ) = lim n → + ∞ F ( g y n , g z n , g w n , g x n ) = F ( lim n → + ∞ g y n , lim n → + ∞ g z n , lim n → + ∞ g w n , lim n → + ∞ g w n ) = F ( y , z , w , x ) ,$
(41)
$g z = lim n → + ∞ g ( g z n + 1 ) = lim n → + ∞ F ( g z n , g w n , g x n , g y n ) = F ( lim n → + ∞ g z n , lim n → + ∞ g w n , lim n → + ∞ g x n , lim n → + ∞ g y n ) = F ( z , w , x , y ) ,$
(42)

and

$g w = lim n → + ∞ g ( g w n + 1 ) = lim n → + ∞ F ( g w n , g x n , g y n , g z n ) = F ( lim n → + ∞ g w n , lim n → + ∞ g x n , lim n → + ∞ g y n , lim n → + ∞ g z n ) = F ( w , x , y , z ) .$
(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 bb 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 : X4X and g : XX are such that F has the mixed g-monotone property. Assume that there exist ϕ Φ and L ≥ 0 such that

$d ( F ( x , y , z , w ) , F ( u , v , , h , l ) ) ≤ ϕ ( max { d ( g x , g u ) , d ( g y , g v ) , d ( g z , g h ) , d ( g w , g l ) } ) + L M ( x , y , z , w , u , v , h , l )$

for any x, y, z, w, u, v, h, l X for which gxgu, gvgy, gzgh, and glgw. 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 gx0F (x0, y0, z0, w0), gy0F (y0, z0, w0, x0), gz0F (z0, w0, x0, y0) and gw0F (w0, x0, y0, z0), then there exist x, y, z, w X such that

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

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

$g x n →gx,g y n →gy,g z n →gz, and g w n →gw.$
(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

$g x n ≤gx,g y n ≥gy,g z n ≤gz,g w n ≥gw$

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 gxn+1gx = gx n , gygyn+1gy n = gy, gz = gz n gzn+1gz = gz n , and gwgwn+1gw n = gw, which implies that

$g x n =g x n + 1 =F ( x n , y n , z n , w n ) ,g y n =g y n + 1 =F ( y n , z n , w n , x n ) ,$

and

$g z n =g z n + 1 =F ( z n , w n , x n , y n ) ,g w n =g w n + 1 =F ( w n , w n , y n , z n ) ,$

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

$d ( g x , F ( x , y , z , w ) ) ≤ d ( g x , g x n + 1 ) + d ( g x n + 1 , F ( x , y , z , w ) ) = d ( g x , g x n + 1 ) + d ( F ( x n , y n , z n , w n ) , F ( x , y , z , w ) ) ≤ d ( g x , g x n + 1 ) + ϕ max d ( g x n , g x ) , d ( g y n , g y ) , d ( g z n , g z ) , d ( g w n , g w ) + L M ( x n , y n , z n , w n , x , y , z , w ) < d ( g x , g x n + 1 ) + max { d ( g x n , g x ) , d ( g y n , g y ) , d ( g z n , g z ) , d ( g w n , g w ) } + L M ( x n , y n , z n , w n , x , y , z , w ) .$
(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

$F ( x , y , z , w ) =gy,F ( z , w , x , y ) =gz, and F ( w , x , y , z ) =gw.$

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 : X4X and g : XX 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

$d ( F ( x , y , z , w ) , F ( u , v , h , l ) ) ≤ ϕ d ( g x , g u ) + d ( g y , g v ) + d ( g z , g h ) + d ( g w , g l ) 4 + L M ( x , y , z , w , u , v , h , l ) ,$

for any x, y, z, w, u, v, h, l, X for which gxgu, gvgy, gzgw, and glgw. Suppose F (X4) g(X), g is continuous and commutes with F .

If there exist x0, y0, z0, w0 X such that gx0F (x0, y0, z0, w0), gy0F (y0, z0, w0, x0), gz0F (z0, w0, x0, y0), and gw0F (w0, x0, y0, z0), then there exist x, y, z, w X such that

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

Proof. It suffices to remark that

$d ( g x , g u ) + d ( g y , g v ) + d ( g z , p h ) , d ( g w , g l ) 4 ≤ max d ( g x , g u ) , d ( g u , g v ) , d ( g z , g h ) , d ( g w , g l ) .$

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 : X4X and g : XX 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

$d ( F ( x , y , z , w ) , F ( u , v , h , l ) ) ≤ ϕ d ( g x , g u ) + d ( g y , g v ) + d ( g z , g h ) + d ( g w , g l ) 4 + L M ( x , y , z , w , u , v , h , l ) ,$

for any x, y, z, w, u, v, h, l X for which gxgu, gvgy, gzgw, and glgw. 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 gx0F (x0, y0, z0, w0), gy0F (y0, z0, w0, x0), gz0F (z0, w0, x0, y0), and gw0F (w0, x0, y0, z0), then there exist x, y, z, w X such that

$F ( x , y , z , w ) =gx,F ( y , z ,$