Back
 JAMP  Vol.7 No.5 , May 2019
N-Order Fixed Point Theory for N-Order Generalized Meir-Keeler Type Contraction in Partially Ordered Metric Spaces
Abstract: This paper concerns N-order fixed point theory in partially ordered metric spaces. For the sake of simplicity, we start our investigations with the tripled case. We define tripled generalized Meir-Keeler type contraction which extends the definition of [Bessem Samet, Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces, Nonlinear Anal. 72 (2010), 4508-4517]. We then discuss the existence and uniqueness of tripled fixed point theorems in partially ordered metric spaces. For general cases, we generalized our results to the N-order case. The results will promote the study of N-order fixed point theory.

1. Introduction and Preliminaries

Banach contraction principle [1] is classical and powerful in fixed point theory. It has been widely generalized (see [2] [3] [4] and others). Recently, fixed point theory in partially ordered metric spaces has been presented by many scholars: Ran and Reurings [5] , Agarwal et al. [6] , Bhsakar and Lakshmikantham [7] , Samet [8] , Berinde and Borcut [9] , Amini-Harandi [10] , etc., considered some coupled and tripled fixed point theorems. For more fixed point theorems in partially ordered metric spaces, one can refer to [11] [12] [13] and others.

This paper focuses on the tripled and N-order fixed point theory. For convenience, we denote N + = { 1,2, , n , } . Let ( X , , d ) denote a partially ordered set ( X , ) endowed a metric d (i.e., ( X , d ) is a metric space). Our work is carried out on the following two preliminaries: a result about fixed point in partially ordered metric space in [6] and a definition of generally Meir-Keeler type function for the case of coupled fixed points in [8] .

Lemma 1.1 ( [6] ). Let ( X , , d ) be a partially ordered metric space and suppose the metric space ( X , d ) is complete. Assume there is a nondecreasing function φ : [ 0, ) [ 0, ) with φ n ( t ) 0 as n for each t > 0 . If f : X X is a nondecreasing mapping with

d ( f ( x ) , f ( y ) ) φ ( d ( x , y ) ) , x y .

Assume that either

1) f is continuous or,

2) If a nondecreasing sequence x n x , then x n x , n N + .

If x 0 X with x 0 f ( x 0 ) then f has a fixed point. If for each x , y X , there exists z X which is comparable to x and y, then the fixed point of f is unique.

Definition 1 ( [8] ) Let ( X , , d ) be a partially ordered metric space and F : X × X X be a mapping. F is called generalized Meir-Keeler type function if for all ε > 0 there exists δ ( ε ) > 0 such that

x u , y v , ε 1 2 ( d ( x , u ) + d ( y , v ) ) < ε + δ ( ε ) d ( F ( x , y ) , F ( u , v ) ) < ε . (1.1)

Let ( X , ) be a partially ordered set with a metric d on X, M = X × X × X and F : M X be a given mapping. Let be the partial order on M : ( x , y , z ) ( u , v , w ) x u , y v , z w . We employ the notion of tripled fixed point introduced by Samet and Vetro which is investigated by Amini-Harandi [10] .

Definition 2 ( [11] ) An element x , y , z X is called a tripled fixed point of F : M X if

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

In this paper, we first define N-order generalized Meir-Keeler type contraction by adding some parameters (see Definition 3 and Definition 5), which is an extension of Definition 1. Then we use a simple approach introduced by [10] to discuss N-order fixed point theorems. We start our discussions with the tripled case. Section 2 devotes to tripled fixed point theorems. Section 3 devotes to N-order fixed point theory. Section 4 gives two examples to illustrate the results obtained in Section 2.

2. Tripled Fixed Point Theory

Recalling that ( X , , d ) is a partially ordered set with a metric d on X and M = X × X × X . Let ρ be the metric and be the partially order on M . For each ( x , y , z ) , ( u , v , w ) M , we define

ρ ( ( x , y , z ) , ( u , v , w ) ) = d ( x , u ) + d ( y , v ) + d ( z , w )

( x , y , z ) ( u , v , w ) x u , y v , z w

and

( x , y , z ) ( u , v , w ) at least one of the inequalities x < u , y < v and z < w hold .

Now, we define tripled generalized Meir-Keeler type contraction which is a useful tool for the following theorems in this section.

Definition 3 Let ( X , , d ) be a partially ordered metric space and F : M X be a mapping. F is called a tripled generalized Meir-Keeler type contraction if for all ε > 0 there exists δ ( ε ) > 0 such that

( x , y , z ) ( u , v , w ) , ε 1 3 [ l d ( x , u ) + k d ( y , v ) + j d ( z , w ) ] < ε + δ ( ε ) d ( F ( x , y , z ) , F ( u , v , w ) ) < ε (2.1)

where l , k , j are constants with 0 < l + k + j < 3 .

Theorem 2.1 Let ( X , , d ) be a partially ordered metric space. Let l , k , j be the given constants with 0 < l + k + j < 3 . If F : M X is a tripled generalized Meir-Keeler contraction mapping, then

d ( F ( x , y , z ) , F ( u , v , w ) ) < 1 3 [ l d ( x , u ) + k d ( y , v ) + j d ( z , w ) ]

for all ( x , y , z ) ( u , v , w ) .

Proof. Let ( x , y , z ) , ( u , v , w ) M such that ( x , y , z ) ( u , v , w ) . Then it follows that

1 3 [ l d ( x , u ) + k d ( y , v ) + j d ( z , w ) ] > 0.

Setting

ε = 1 3 [ l d ( x , u ) + k d ( y , v ) + j d ( z , w ) ] ,

we have

0 < ε = 1 3 [ l d ( x , u ) + k d ( y , v ) + j d ( z , w ) ] < ε + δ ( ε ) .

By F : M X being a tripled generalized Meir-Keeler type contraction, then

d ( F ( x , y , z ) , F ( u , v , w ) ) < ε = 1 3 [ l d ( x , u ) + k d ( y , v ) + j d ( z , w ) ] . □

Let F : M X be a mapping. We say F is nondecreasing in each of its variables if

x 1 , x 2 X , x 1 < x 2 F ( x 1 , y , z ) < F ( x 2 , y , z ) , y , z X ,

y 1 , y 2 X , y 1 < y 2 F ( x , y 1 , z ) < F ( x , y 2 , z ) , x , z X ,

and

z 1 , z 2 X , z 1 < z 2 F ( x , y , z 1 ) < F ( x , y , z 2 ) , x , y X .

By the monotone property of F, we can get

( x , y , z ) , ( u , v , w ) M , ( x , y , z ) ( u , v , w ) F ( x , y , z ) < F ( u , v , w ) . (2.2)

For all n N + , n > 1 , we define:

F n ( x , y , z ) = F ( F n 1 ( x , y , z ) , F n 1 ( y , z , x ) , F n 1 ( z , x , y ) ) (2.3)

with F 1 = F .

In order to investigate the tripled fixed point of F, we introduce a mapping T : M M which is defined by

T ( x , y , z ) = ( F ( x , y , z ) , F ( y , z , x ) , F ( z , x , y ) ) . (2.4)

Obviously, by the definition of ρ , we have

ρ ( T ( x , y , z ) , T ( u , v , w ) ) = d ( F ( x , y , z ) , F ( u , v , w ) ) + d ( F ( y , z , x ) , F ( v , w , u ) ) + d ( F ( z , x , y ) , F ( w , u , v ) ) . (2.5)

Simultaneously, by (2.3) and (2.4), we have

T n ( x , y , z ) = ( F n ( x , y , z ) , F n ( y , z , x ) , F n ( z , x , y ) )

with T 1 = T , and we have

F n ( x , y , z ) = F ( T n 1 ( x , y , z ) ) .

Theorem 2.2 Let ( X , , d ) be a partially ordered metric space and l , k , j be the given constants with 0 < l + k + j < 3 . Let F : M X be nondecreasing in each of its variables and be a tripled generalized Meir-Keeler type contraction. There exist ( x , y , z ) , ( u , v , w ) M with ( x , y , z ) ( u , v , w ) . Then, for n N + , we have

1) T n ( x , y , z ) T n ( u , v , w ) ;

2) ρ ( T n + 1 ( x , y , z ) , T n + 1 ( u , v , w ) ) < ρ ( T n ( x , y , z ) , T n ( u , v , w ) ) ;

3) ρ ( T n ( x , y , z ) , T n ( u , v , w ) ) 0, n .

Proof. We first prove 1). Since ( x , y , z ) ( u , v , w ) , due to the monotone property of F and (2.2), we have F ( x , y , z ) < F ( u , v , w ) , F ( y , z , x ) < F ( v , w , u ) and F ( z , x , y ) < F ( w , u , v ) . By T 1 = T and (2.4), 1) holds for n = 1 . Now we assume 1) holds for n N + , i.e.

( F n ( x , y , z ) , F n ( y , z , x ) , F n ( z , x , y ) ) = T n ( x , y , z ) T n ( u , v , w ) = ( F n ( u , v , w ) , F n ( v , w , u ) , F n ( w , u , v ) ) .

Then, we obtain

F ( F n ( x , y , z ) , F n ( y , z , x ) , F n ( z , x , y ) ) < F ( F n ( u , v , w ) , F n ( v , w , u ) , F n ( w , u , v ) )

which means F n + 1 ( x , y , z ) < F n + 1 ( u , v , w ) . Using the same strategy, we have F n + 1 ( y , z , x ) < F n + 1 ( v , w , u ) and F n + 1 ( z , x , y ) < F n + 1 ( w , u , v ) . Hence we have T n + 1 ( x , y , z ) T n + 1 ( u , v , w ) , that is, 1) holds for n + 1 . Simultaneously, we can also obtain that T n ( y , z , x ) T n ( v , w , u ) and T n ( z , x , y ) T n ( w , u , v ) .

Now, we prove 2). We consider

ρ ( T n + 1 ( x , y , z ) , T n + 1 ( u , v , w ) ) = d ( F n + 1 ( x , y , z ) , F n + 1 ( u , v , w ) ) + d ( F n + 1 ( y , z , x ) , F n + 1 ( v , w , u ) ) + d ( F n + 1 ( z , x , y ) , F n + 1 ( w , u , v ) ) .

It follows from Theorem 2.1 and 1) that

d ( F n + 1 ( x , y , z ) , F n + 1 ( u , v , w ) ) = d ( F ( T n ( x , y , z ) ) , F ( T n ( u , v , w ) ) ) < 1 3 [ l d ( F n ( x , y , z ) , F n ( u , v , w ) ) + k d ( F n ( y , z , x ) , F n ( v , w , u ) ) + j d ( F n ( z , x , y ) , F n ( w , u , v ) ) ] ,

d ( F n + 1 ( y , z , x ) , F n + 1 ( v , w , u ) ) = d ( F ( T n ( y , z , x ) ) , F ( T n ( v , w , u ) ) ) < 1 3 [ l d ( F n ( y , z , x ) , F n ( v , w , u ) ) + k d ( F n ( z , x , y ) , F n ( w , u , v ) ) + j d ( F n ( x , y , z ) , F n ( u , v , w ) ) ]

and

d ( F n + 1 ( z , x , y ) , F n + 1 ( w , u , v ) ) = d ( F ( T n ( z , x , y ) ) , F ( T n ( w , u , v ) ) ) < 1 3 [ l d ( F n ( z , x , y ) , F n ( w , u , v ) ) + k d ( F n ( x , y , z ) , F n ( u , v , w ) ) + j d ( F n ( y , z , x ) , F n ( v , w , u ) ) ] .

Thus,

ρ ( T n + 1 ( x , y , z ) , T n + 1 ( u , v , w ) ) < 1 3 ( l + k + j ) [ d ( F n ( x , y , z ) , F n ( u , v , w ) ) + d ( F n ( y , z , x ) , F n ( v , w , u ) ) + d ( F n ( z , x , y ) , F n ( w , u , v ) ) ] < ρ ( T n ( x , y , z ) , T n ( u , v , w ) ) .

Last, we prove 3). From 2), we know that lim n ρ ( T n ( x , y , z ) , T n ( u , v , w ) ) exists. If lim n ρ ( T n ( x , y , z ) , T n ( u , v , w ) ) 0 , we suppose that

lim n 1 3 ρ ( T n ( x , y , z ) , T n ( u , v , w ) ) = ε > 0. (2.6)

Then it follows that

1 3 ρ ( T n ( x , y , z ) , T n ( u , v , w ) ) ε , n N + .

By (2.6), we have

lim n 1 3 [ l d ( F n ( x , y , z ) , F n ( u , v , w ) ) + k d ( F n ( y , z , x ) , F n ( v , w , u ) ) + j d ( F n ( z , x , y ) , F n ( w , u , v ) ) ] = ε

which implies that there exists m 0 N + such that

ε 1 3 [ l d ( F m 0 ( x , y , z ) , F m 0 ( u , v , w ) ) + k d ( F m 0 ( y , z , x ) , F m 0 ( v , w , u ) ) + j d ( F m 0 ( z , x , y ) , F m 0 ( w , u , v ) ) ] < ε + δ ( ε ) . (2.7)

Since F is a tripled generalized Meir-Keeler type contraction, we get

ε > d ( F ( F m 0 ( x , y , z ) , F m 0 ( y , z , x ) , F m 0 ( z , x , y ) ) , F ( F m 0 ( u , v , w ) , F m 0 ( v , w , u ) , F m 0 ( w , u , v ) ) ) = d ( F m 0 + 1 ( x , y , z ) , F m 0 + 1 ( u , v , w ) ) . (2.8)

By (2.7), we also have

ε 1 3 [ k d ( F m 0 ( y , z , x ) , F m 0 ( v , w , u ) ) + j d ( F m 0 ( z , x , y ) , F m 0 ( w , u , v ) ) + l d ( F m 0 ( x , y , z ) , F m 0 ( u , v , w ) ) ] < ε + δ ( ε ) ,

and

ε 1 3 [ j d ( F m 0 ( z , x , y ) , F m 0 ( w , u , v ) ) + l d ( F m 0 ( x , y , z ) , F m 0 ( u , v , w ) ) + k d ( F m 0 ( y , z , x ) , F m 0 ( v , w , u ) ) ] < ε + δ ( ε ) .

Then, we get

ε > d ( F m 0 + 1 ( y , z , x ) , F m 0 + 1 ( v , w , u ) ) (2.9)

and

ε > d ( F m 0 + 1 ( z , x , y ) , F m 0 + 1 ( w , u , v ) ) . (2.10)

From (2.8)-(2.10), we get

1 3 ρ ( T m 0 + 1 ( x , y , z ) , T m 0 + 1 ( u , v , w ) ) = 1 3 [ d ( F m 0 + 1 ( x , y , z ) , F m 0 + 1 ( u , v , w ) ) + d ( F m 0 + 1 ( y , z , x ) , F m 0 + 1 ( v , w , u ) ) + d ( F m 0 + 1 ( z , x , y ) , F m 0 + 1 ( w , u , v ) ) ] < ε .

This is a contradiction. The proof is completed.

From the definition of T, we observe that the fixed point of T is exactly the tripled fixed point of F, that is,

( x , y , z ) = T ( x , y , z ) x = F ( x , y , z ) , y = F ( y , z , x ) , z = F ( z , x , y ) .

We will obtain the tripled fixed point theorems by investigating the fixed point of T.

Theorem 2.3 Let ( X , , d ) be a partially ordered metric space and ( X , d ) is a complete metric space. Let l , k , j be the given constants with 0 < l + k + j < 3 . Let F : M X be nondecreasing in each of its variables and be a tripled generalized Meir-Keeler contraction. T : M M be a mapping defined as (2.4) satisfying that there exists ( x 0 , y 0 , z 0 ) M with ( x 0 , y 0 , z 0 ) T ( x 0 , y 0 , z 0 ) . Then, there exists ( x * , y * , z * ) M which is a tripled fixed point of F, if either

1) F is continuous or

2) a nondecreasing sequence ( x n , y n , z n ) ( x , y , z ) , then ( x n , y n , z n ) ( x , y , z ) , n N + .

Furthermore, if

3) for ( x , y , z ) , ( u , v , w ) M , there exists ( a , b , c ) M that is comparable to ( x , y , z ) and ( u , v , w ) , we get the uniqueness of tripled fixed point of F and x * = y * = z * .

Proof. Since ( X , d ) is a complete metric space, it is obvious that the metric space ( M , ρ ) is complete. By Theorem 2.2, T is non-decreasing. Meanwhile, by Theorem 2.1 and (2.5), for each ( x , y , z ) , ( u , v , w ) M with ( x , y , z ) ( u , v , w ) , we have

ρ ( T ( x , y , z ) , T ( u , v , w ) ) = d ( F ( x , y , z ) , F ( u , v , w ) ) + d ( F ( y , z , x ) , F ( v , w , u ) ) + d ( F ( z , x , y ) , F ( w , u , v ) ) < 1 3 ( l + k + j ) ρ ( ( x , y , z ) , ( u , v , w ) ) .

By Lemma 1.1, we deduce that T has a unique fixed point denoted by ( x * , y * , z * ) , then ( x * , y * , z * ) is the unique tripled fixed point of F.

However, we can check that ( y * , z * , x * ) is also a tripled fixed point of F. In fact, since ( x * , y * , z * ) is the tripled fixed point of F, i.e., x * = F ( x * , y * , z * ) , y * = F ( y * , z * , x * ) , z * = F ( z * , x * , y * ) , we have

y * = F ( y * , z * , x * ) , z * = F ( z * , x * , y * ) , x * = F ( x * , y * , z * )

which implies that ( y * , z * , x * ) is also a tripled fixed point of F. By the uniqueness, we get x * = y * = z * . □

Corollary 1 Suppose that all the hypotheses of Theorem 2.3 are satisfied, then the tripled fixed point ( x * , y * , z * ) can be deduced by

F n ( x 0 , y 0 , z 0 ) x * , F n ( y 0 , z 0 , x 0 ) y * , F n ( z 0 , x 0 , y 0 ) z * , as n . (2.11)

Proof. By examining the proof of Theorem 2.3, ( x * , y * , z * ) is actually the fixed point of T on M . According to the proof of Lemma 1.1 in [6] , we have

lim n T n ( x 0 , y 0 , z 0 ) = ( x * , y * , z * ) .

By the definition of T n , we can easily get (2.11). □

Theorem 2.4 In addition to the hypotheses of Theorem 2.3 except (3), we have x * = y * = z * by adding the hypotheses (3*): x 0 , y 0 , z 0 in X are comparable.

Proof. Without the restriction of the generality, we assume that x 0 y 0 z 0 . Setting ( x 1 , y 1 , z 1 ) = ( x 0 , y 0 , z 0 ) and ( u 1 , v 1 , w 1 ) = ( y 0 , z 0 , z 0 ) , it’s easy to see that ( x 1 , y 1 , z 1 ) ( u 1 , v 1 , w 1 ) . From Theorem 1.1, we have ρ ( T n ( x 1 , y 1 , z 1 ) , T n ( u 1 , v 1 , w 1 ) ) 0 as n , which implies that

d ( F n ( x 1 , y 1 , z 1 ) , F n ( u 1 , v 1 , w 1 ) ) 0, n ,

i.e.,

d ( F n ( x 0 , y 0 , z 0 ) , F n ( y 0 , z 0 , z 0 ) ) 0, n . (2.12)

By the similar strategy, setting ( x 2 , y 2 , z 2 ) = ( y 0 , z 0 , x 0 ) and ( u 2 , v 2 , w 2 ) = ( y 0 , z 0 , z 0 ) , we can get

d ( F n ( y 0 , z 0 , x 0 ) , F n ( y 0 , z 0 , z 0 ) ) 0, n . (2.13)

It follows from the triangular inequality that

d ( x * , y * ) d ( x * , F n ( x 0 , y 0 , z 0 ) ) + d ( F n ( x 0 , y 0 , z 0 ) , F n ( y 0 , z 0 , z 0 ) ) + d ( F n ( y 0 , z 0 , z 0 ) , F n ( y 0 , z 0 , x 0 ) ) + d ( F n ( y 0 , z 0 , x 0 ) , y * ) .

Taking the limit as n , by (2.11), (2.12) and (2.13), we get x * = y * .

Similarly, by setting

( x 3 , y 3 , z 3 ) = ( y 0 , z 0 , x 0 ) , ( u 3 , v 3 , w 3 ) = ( z 0 , z 0 , y 0 )

and

( x 4 , y 4 , z 4 ) = ( z 0 , x 0 , y 0 ) , ( u 4 , v 4 , w 4 ) = ( z 0 , z 0 , y 0 ) ,

we can get two equalities,

d ( F n ( y 0 , z 0 , x 0 ) , F n ( z 0 , z 0 , y 0 ) ) 0, n (2.14)

and

d ( F n ( z 0 , x 0 , y 0 ) , F n ( z 0 , z 0 , y 0 ) ) 0, n (2.15)

respectively. Then it follows from (2.11), (2.14) and (2.15) that

d ( y * , z * ) d ( y * , F n ( y 0 , z 0 , x 0 ) ) + d ( F n ( y 0 , z 0 , x 0 ) , F n ( z 0 , z 0 , y 0 ) ) + d ( F n ( z 0 , z 0 , y 0 ) , F n ( z 0 , x 0 , y 0 ) ) + d ( F n ( z 0 , x 0 , y 0 ) , z * ) 0.

We get y * = z * . Hence we have x * = y * = z * . □

3. N-Order Fixed Point Theorems

Let ( X , , d ) be a partially ordered set with a metric d on X. Let K = X N , η be the metric on K and be the partially order. For each x = ( x 1 , , x N ) , y = ( y 1 , , y N ) K , we define

η ( x , y ) = d ( x 1 , y 1 ) + + d ( x N , y N )

x y x 1 y 1 , , x N y N

and

x y there exists 1 i N , such that x i < y i .

Definition 4 [11] Let X be a non-empty set and F : K X be a given mapping. An element x K is called a N-order fixed point of F if

x 1 = F ( x 1 , , x N ) , x 2 = F ( x 2 , , x N , x 1 ) , , x N = F ( x N , x 1 , , x N 1 ) .

We introduce generally N-order generalized Meir-Keeler type contraction.

Definition 5 Let ( X , , d ) be a partially ordered metric space and F : K X be a mapping. F is called a N-order generalized Meir-Keeler contraction if for all ε > 0 there exists δ ( ε ) > 0 such that for x , y K

x y , ε k 1 d ( x 1 , y 1 ) + + k N d ( x N , y N ) N < ε + δ ( ε ) d ( F ( x ) , F ( y ) ) < ε (3.16)

where k 1 , , k N are constants with 0 < k 1 + + k N < N .

Substituting the tripled case with N-order case in the discussions of Section 3, by the similar strategy, we can obtain the same results with Theorem 2.1, Theorem 2.2, Theorem 2.3, Corollary 1 and Theorem 2.4.

4. The Examples

This section provides two examples to illustrate Theorem 2.3 and Theorem 2.4.

Example 1 This example is aroused by [13] . Let X = R , d ( x , y ) = | x y | and F : M X , defined by

F = 4 x 4 y + 3 z + 1 15 .

It is easy to check that F satisfies all the hypotheses of Theorem 2.3 with

l = 1 , k = 1 , j = 3 4 , δ ( ε ) = 1 4 ε

and ( x * , y * , z * ) = ( 1 12 , 1 12 , 1 12 ) is the unique tripled fixed point of F.

Example 2 Let

X = { ( 0 , 1 ) , ( 0 , 2 ) , ( 1 , 3 ) , ( 1 , 0 ) , ( 2 , 0 ) , ( 3 , 0 ) } .

For x = ( x 1 , x 2 ) , y = ( y 1 , y 2 ) X , d ( x , y ) = | x 1 y 1 | + | x 2 y 2 | and x y x 1 y 1 , x 2 y 2 . F : M X is defined by

F ( x , y , z ) = { ( 0 , 3 ) , x , y , z { ( 0 , 1 ) , ( 0 , 2 ) , ( 0 , 3 ) } ( 3 , 3 ) , x , y , z { ( 1 , 1 ) , ( 2 , 0 ) , ( 3 , 0 ) } (4.1)

It is easy to check that:

1) F is continues on M ;

2) F is a tripled generally Meir-Keeler type contraction. In fact, we can deduce that

d ( F ( x , y , z ) , F ( u , v , w ) ) = 0 for each ( x , y , z ) ( u , v , w ) ;

3) Setting x 0 = y 0 = z 0 = ( 0 , 1 ) , then we have F ( x 0 , y 0 , z 0 ) = F ( y 0 , z 0 , x 0 ) = F ( z 0 , x 0 , y 0 ) = ( 0 , 3 ) . Clearly, we have ( x 0 , y 0 , z 0 ) T ( x 0 , y 0 , z 0 ) ;

4) Setting ( x , y , z ) = ( 0 , 1 , 0 , 2 , 0 , 3 ) , ( u , v , w ) = ( 1 , 0 , 2 , 0 , 3 , 0 ) , there are no elements in M which are comparable to ( x , y , z ) and ( u , v , w ) .

The above 4) implies that F doesn’t satisfy all the hypotheses of Theorem 2.3. However, the above 1)-3) imply that F satisfies all the hypotheses of Theorem 2.4, then F has the unique tripled fixed point ( x * , y * , z * ) with x * = y * = z * = ( 0 , 3 ) .

5. Conclusion

In this paper, we extend the definition generalized Meir-Keeler type contraction to N-ordered case. And we use it to discuss N-order fixed point theorems. In future work, we will study N-ordered fixed point theory with invariant set.

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 11701390).

Cite this paper: Wang, S. and Zhang, J. (2019) N-Order Fixed Point Theory for N-Order Generalized Meir-Keeler Type Contraction in Partially Ordered Metric Spaces. Journal of Applied Mathematics and Physics, 7, 1174-1184. doi: 10.4236/jamp.2019.75078.
References

[1]   Bananch, S. (1922) Sur les opérations dans les ensembles anstraits et application qux équations intégrales. Fundamenta Mathematicae, 3, 133-181.
https://doi.org/10.4064/fm-3-1-133-181

[2]   Branciari, A. (2022) A Fixed Point Theorem for Mappings Satisfying a General Contractive Condition of Integral Type. International Journal of Mathematics and Mathematical Sciences, 29, 531-536.
https://doi.org/10.1155/S0161171202007524

[3]   Meir, A. and Keeler, E. (1969) A Theorem on Contraction Mappings. Journal of Mathematical Analysis and Applications, 58, 326-329.
https://doi.org/10.1016/0022-247X(69)90031-6

[4]   Suzuki, T. (2007) Meir-Keeler Contrctions of Integral Type Are Still Meir-Keeler Contrctions. International Journal of Mathematics and Mathematical Sciences, 2007, Article ID: 39281.
https://doi.org/10.1155/2007/39281

[5]   Ran, A.C.M. and Reurings, M.C.B. (2004) A Fixed Point Theorem in Partially Ordered Sets and Some Applications to Matrix Equations. Proceedings of the American Mathematical Society, 132, 1435-1443.
https://doi.org/10.1090/S0002-9939-03-07220-4

[6]   Agarwal, R.P., EI-Gebeily, M.A. and O’Regan, D. (2008) Generalized Contractions in Partially Ordered Metric Spaces. Applicable Analysis, 87, 1-8.
https://doi.org/10.1080/00036810701556151

[7]   Gnana Bhaskar, T. and Lakshmikantham, V. (2006) Fixed Point Theorems in Partially Ordered Metric Spaces and Applications. Nonlinear Analysis, 65, 1379-1393.
https://doi.org/10.1016/j.na.2005.10.017

[8]   Samet, B. (2010) Coupled Fixed Point Theorems for a Generalized Meir-Keeler Contraction in Partially Ordered Metric Spaces. Nonlinear Analysis, 72, 4508-4517.
https://doi.org/10.1016/j.na.2010.02.026

[9]   Borcut, M. and Berinde, V. (2011) Tripled Fixed Point Theorems for Contractive Type Mappings in Partially Ordered Metric Spaces. Nonlinear Analysis, 74, 4889-4897.
https://doi.org/10.1016/j.na.2011.03.032

[10]   Amini-Harandi, A. (2013) Coupled and Tripled Fixed Point Theory in Partially Prdered Metric Spaces with Application to Initial Value Problem. Mathematical and Computer Modelling, 57, 2343-2348.
https://doi.org/10.1016/j.mcm.2011.12.006

[11]   Samet, B. and Vetro, C. (2010) Coupled Fixed Point, F-Invariant Set and Fixed Point of N-Order. Annals of Functional Analysis, 1, 46-56.
https://doi.org/10.15352/afa/1399900586

[12]   Sintunavarat, W., Kumam, P. and Cho, Y.J. (2012) Coupled Fixed Point Theorems for Nonlinear Contractions without Mixed Monotone Property. Fixed Point Theory and Applications, 2012, 170.
https://doi.org/10.1186/1687-1812-2012-170

[13]   Berinde, V. and Borcut, M. (2011) Tripled Fixed Point Theorems for Contractive Type Mappings in Partially Ordered Metric Spaces. Nonlinear Analysis, 74, 4889-4897.
https://doi.org/10.1016/j.na.2011.03.032

 
 
Top