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In this paper, we propose a new perspective to discuss the N-order fixed point theory of set-valued and single-valued mappings. There are two aspects in our work: we first define a product metric space with a graph for the single-valued mapping whose conversion makes the results and proofs concise and straightforward, and then we propose an
*SG*-contraction definition for set-valued mapping which is more general than some recent contraction’s definition. The results obtained in this paper extend and unify some recent results of other authors. Our method to discuss the N-order fixed point unifies
*N*-order fixed point theory of set-valued and single-valued mappings.

Since Banach Contraction Principle was first shown in 1922 [

The definition of coupled fixed point was given by Bhaskar and Lakshmikantham [

Based on the results obtained in [

d ( F ( x , y ) , F ( u , v ) ) ≤ k ( d ( x , u ) + d ( y , v ) 2 ) (1.1)

The inequalities similar with (1.1) appeared in many literatures about fixed point theories such as [

We focus on two aspects: one is to construct a concise method to study the fixed point theory; the other is to construct the N-order fixed point theories of single-valued and set-valued mappings. First, we define ( X N , D , G ) which is induced by ( X , d , G ) . We transfer the N-order fixed point of F : X N → X N (resp., C B ( X N ) ), to the fixed point of F : X N → X N (resp., C B ( X N ) ). Next, we construct the N-order fixed point theories of single-valued and set-valued mappings.

We first introduce a terminology of graph theory. Consider a complete metric space ( X , d ) endowed with a directed graph G, denoted by ( X , d , G ) . The vertices set V ( G ) coincides X and the edges set E ( G ) contains the diagonal elements of X × X . Assume that G has no parallel edges. The metric between two vertices can be treated as the weight so that G is a weighted graph. Moreover, let G − 1 denote the conversion of the graph G, i.e., E ( G − 1 ) = { ( y , x ) : ( x , y ) ∈ E ( G ) } . Simultaneously, let G ¯ denote the undirected graph obtained from G, that is E ( G ¯ ) = E ( G ) ∪ E ( G − 1 ) . We say that G is weakly connected if is G ¯ connected.

Let X N denote the product space of N X^{’}s and X = X N . We say that ( X , D , G ) is induced by ( X , d , G ) if V ( G ) = X , and ∀ x = ( ξ 1 , ξ 2 , ⋯ , ξ n ) , y = ( η 1 , η 2 , ⋯ , η n ) ∈ X , E ( G ) and D are defined respectively by:

( x , y ) ∈ E ( G ) ⇔ ( ξ i , η i ) ∈ E ( G ) , i = 1 , 2 , ⋯ , N , (2.2)

and

D ( x , y ) = d ( ξ 1 , η 1 ) + ⋯ + d ( ξ N , η N ) (2.3)

In addition, we use Ɗ to denote the Pompeiu-Hausdorff distance between two sets.

In this paper, we mainly consider the following two properties of ( X , d , G ) and the mapping f : X → Y where Y is a metric space with a graph.

( P 1 ) : For any sequence ( x n ) n ∈ N in X, if x n → x and ( x n , x n + 1 ) ∈ E ( G ) for any n ∈ N , then ( x n , x ) ∈ E ( G ) .

( P 2 ) : for given x and a sequence { x n } in X with ( x n , x n + 1 ) ∈ E ( G ) and lim n → ∞ x n = x , we have lim n → ∞ f ( x n ) = f ( x ) .

In this section, We make use of f, F, F to denote the following mappings, respectively, f : X → X , F : X → X and F : X → X .

For simplicity of description, we denote X f = { x ∈ X : ( x , f ( x ) ) ∈ E ( G ) } .

Definition 1. [

ξ 1 = F ( ξ 1 , ξ 2 , ⋯ , ξ N ) , ξ 2 = F ( ξ 2 , ⋯ , ξ N , ξ 1 ) , ⋯ , ξ N = F ( ξ N , ξ 1 , ⋯ , ξ N − 1 ) .

When N = 1 , 2 , 3 , we call it a fixed point, coupled fixed point and tripled fixed point respectively.

Definition 2. Let F : X → X be a mapping. We say that F is NG-contractive if it satisfies the following two conditions:

1) NG-monotone property, i.e., for all x , y ∈ X with ( x , y ) ∈ E ( G ) , it holds that ( F ( x ) , F ( y ) ) ∈ E ( G ) ; (3.1)

2) There exists k ∈ ( 0 , 1 ) such that d ( F ( x ) , F ( y ) ) ≤ k D ( x , y ) for all x , y ∈ X N with ( x , y ) ∈ E ( G ) .

Particularly, when F : X → X satisfies (1) and (2) for N = 1 , it is called a G-contraction.

The definition “NG-contractive” was first introduced in [

We consider the definition of N-order fixed point of the mapping F : X → X and find that the N-order fixed point of the mapping F is just the fixed point of the mapping F : X → X defined by

∀ x ∈ X , F ( x ) = ( F ( ξ 1 , ξ 2 , ⋯ , ξ N ) , F ( ξ 2 , ξ 3 , ⋯ , ξ 1 ) ⋯ , F ( ξ N , ξ 1 , ⋯ , ξ N − 1 ) ) , (3.2)

which implies that the study of the N-order fixed point can be converted to that of fixed point of F . Simultaneously, F has many properties relative to F. According to Definition 2 and by (2.2), (2.3) and (3.2), if F is G-contractive, then we have:

1) for x , y ∈ X with ( x , y ) ∈ E ( G ) ⇒ ( F ( x ) , F ( y ) ) ∈ E ( G ) ⇒ ( F ( x ) , F ( y ) ) ∈ E ( G ) ;

2) there exists α ∈ ( 0 , 1 ) such that d ( F ( x ) , F ( y ) ) ≤ α D ( x , y ) for all x , y ∈ X with ( x , y ) ∈ E ( G ) , and then d ( F ( x ) , F ( y ) ) ≤ α D ( x , y ) .

Thus, if F is G-contractive, F must be NG-contractive. However, if F is NG-contractive, F is not necessarily G-contractive. We can show it by the following example.

Example 3.1. Let X = R be endowed with the Euclidean distance | ⋅ | . For any ξ , η ∈ X , we say that ( ξ , η ) ∈ E ( G ) if ξ ≤ η . ( R 3 , D , G ) is induced by ( R , | ⋅ | , G ) . For x , y ∈ R 3 with ( x , y ) ∈ E ( G ) , by (2.3), we have D ( x , y ) = | ξ 1 − η 1 | + | ξ 2 − η 2 | + | ξ 3 − η 3 | . Let F : R 3 → R be defined by

F ( x ) = 1 2 ( | ξ 1 | + | ξ 2 | + | ξ 3 | ) .

It is obvious that

d ( F ( x ) , F ( y ) ) = | F ( x ) − F ( y ) | ≤ 1 2 ( | ξ 1 − η 1 | + | ξ 2 − η 2 | + | ξ 3 − η 3 | ) = 1 2 D ( x , y )

which implies that F is NG-contractive. However,

D ( F ( x ) , F ( y ) ) = d ( F ( x ) , F ( y ) ) + d ( F ( ξ 2 , ξ 3 , ξ 1 ) , F ( η 2 , η 3 , η 1 ) ) + d ( F ( ξ 3 , ξ 1 , ξ 2 ) , F ( η 3 , η 1 , η 2 ) ) = | F ( x ) − F ( y ) | + | F ( ξ 2 , ξ 3 , ξ 1 ) − F ( η 2 , η 3 , η 1 ) | + | F ( ξ 3 , ξ 1 , ξ 2 ) − F ( η 3 , η 1 , η 2 ) | ≤ 3 2 ( x , y ) .

Notice that when x = ( 1 , 1 , 1 ) and y = ( 2 , 2 , 2 ) , then D ( F ( x ) , F ( y ) ) = 3 2 D ( x , y ) which means F is not G-contractive.

Lemma 3.1. [

1) for any x ∈ X f , f | [ x ] G has a unique fixed point;

2) if G is weakly connected, then f has a unique fixed point in G;

3) if X ′ = ∪ { [ x ] G ¯ : x ∈ X f } , then f | X ′ has a fixed point in X;

4) if f ( X ) ⊂ E ( G ) then f has a fixed point;

5) fix f ≠ ∅ if and only if X f ≠ ∅ .

Lemma 3.2. [

In the sequel, we study the N-order fixed point results of F : X → X by finding the fixed point of F : X → X and we give a concise proof. We try to ensure the G-contractive property of F in Theorem 4.3 and Corollary 1.

Theorem 3.1. Suppose that ( X , D , G ) is induced by ( X , d , G ) and ( X , D , G ) satisfies ( P 1 ) . Suppose that F : X → X satisfies the NG-monotone property (3.4) and there are some constants k 1 , ⋯ , k N with 0 < k 1 + ⋯ + k N < 1 such that

d ( F ( x ) , F ( y ) ) ≤ k 1 d ( ξ 1 , η 1 ) + ⋯ + k N d ( ξ N , η N ) (3.3)

with ( ξ i , η i ) ∈ E ( G ) for i = 1 , ⋯ , N

Suppose that there is a point x 0 = ( ξ 1 0 , ξ 2 0 , ⋯ , ξ N 0 ) ∈ X such that

( ξ i 0 , F ( ξ i 0 , ξ i + 1 0 , ⋯ , ξ N 0 , ξ 1 0 , ⋯ , ξ i − 1 0 ) ) ∈ E ( G ) , ∀ i = 1 , 2 , ⋯ , N . (3.4)

Define F : X → X by (3.2). Then,

1) for any x ∈ X F , F | [ x ] G ¯ has a unique N-order fixed point;

2) if G is weakly connected, then F has a unique N-order fixed point in G ;

3) if X ′ = ∪ { [ x ] G ¯ : x ∈ X F } , then F | X ′ has an N-order fixed point in X ;

4) if F ( X ) ⊂ E ( G ) then F has an N-order fixed point;

5) fix F = ∅ if and only if X F ≠ ∅ .

Moreover, if x ∗ = ( ξ 1 ∗ , ξ 2 ∗ , ⋯ , ξ N ∗ ) is an N-order fixed point of F with ξ 1 ∗ = ξ 2 ∗ = ⋯ = ξ N ∗ .

Proof. (1) Since ( X , D , G ) is induced by ( X , d , G ) and ( X , d ) is complete, we know that ( X , D ) is complete, and for all x , y ∈ X with x = ( ξ 1 , ⋯ , ξ N ) , y = ( η 1 , ⋯ , η N ) , G is defined as (2.2) and D is defined as (2.3). Then, for x , y ∈ E ( G ) , we obtain that

( ξ 2 , ⋯ , ξ N , ξ 1 ; η 2 , ⋯ , η N , η 1 ) ∈ E ( G ) , ⋯ , ( ξ N , ⋯ , ξ N − 1 ; η N , ⋯ , η N − 1 ) ∈ E ( G ) . (3.5)

Considering (2.2), (2.3), (3.1), we have ( F ( x ) , F ( y ) ) ∈ E ( G ) and

D ( F ( x ) , F ( y ) ) = d ( F ( x ) , F ( y ) ) + d ( F ( ξ 2 , ⋯ , ξ N , ξ 1 ) , F ( η 2 , ⋯ , η N , η 1 ) ) + ⋯ + d ( F ( ξ N , ⋯ , ξ N − 1 ) , F ( η N , ⋯ , η N − 1 ) ) ≤ ( k 1 d ( ξ 1 , η 1 ) + ⋯ k N d ( ξ N , η N ) ) + ( k 1 d ( ξ 2 , η 2 ) + ⋯ k N d ( ξ 1 , η 1 ) ) + ⋯ + ( k 1 d ( ξ N , η N ) + ⋯ k N d ( ξ N − 1 , η N − 1 ) ) = ( k 1 + ⋯ k N ) ( d ( ξ 1 , η 1 ) + ⋯ + k N d ( ξ N , η N ) ) = ( k 1 + ⋯ k N ) D ( x , y ) .

Thus F is a G-contraction on X satisfying ( x 0 , F ( x 0 ) ) ∈ E ( G ) by (4.13). Then, by Lemma 3.1, we obtain that F has a unique fixed point x ∗ = ( ξ 1 ∗ , ξ 2 ∗ , ⋯ , ξ N ∗ ) in [ x 0 ] G which is just the N-order fixed point of F.

At the same time, we have

ξ 1 ∗ = F ( ξ 1 ∗ , ξ 2 ∗ , ⋯ , ξ N ∗ ) , ξ 2 ∗ = F ( ξ 2 ∗ , ⋯ , ξ N ∗ , ξ 1 ∗ ) , ⋯ , ξ N ∗ = F ( ξ N ∗ , ξ 1 ∗ , ⋯ , ξ N − 1 ∗ )

which implies that ( ξ 2 ∗ , ⋯ , ξ N ∗ , ξ 1 ∗ ) , ⋯ , ( ξ N ∗ , ξ 1 ∗ , ⋯ , ξ N − 1 ∗ ) are all N-order fixed points of F. By the uniqueness, we can get ξ 1 ∗ = ξ 2 ∗ = ⋯ = ξ N ∗ .

(2)-(5) can be proved by Lemma 3.1 similarly. We omit them.

Theorem 3.2. If we replace the condition “ ( X , D , G ) satisfies ( P 1 ) ” by “ F : X → X satisfies ( P 2 ) ” and the other conditions in Theorem 3.1 are satisfied, we can get the same results with Theorem 3.1.

Corollary 1. Suppose that F : X → X satisfies ( P 2 ) and F : X → X satisfies the NG-monotone property (3.1). And

d ( F ( x ) , F ( y ) ) ≤ k N D ( x , y ) , ∀ x , y ∈ X with ( x , y ) ∈ E ( G ) (3.6)

where 0 < k < 1 . If there exists x 0 = ( ξ 1 0 , ξ 2 0 , ⋯ , ξ N 0 ) ∈ X such that

∀ i , ( ξ i 0 , F ( ξ i 0 , ξ i + 1 0 , ⋯ , ξ N 0 , ξ 1 0 , ⋯ , ξ i − 1 0 ) ) ∈ E ( G ) .

Then there is an N-order fixed point x ∗ = ( ξ 1 ∗ , ξ 2 ∗ , ⋯ , ξ N ∗ ) of F. If G is weakly connected, x ∗ is the unique N-order fixed point with ξ 1 ∗ = ξ 2 ∗ = ⋯ = ξ N ∗ .

Proof. We can easily prove that F defined as (3.2) is G-contractive. By Theorem 3.2, we can get the proof.

Corollary 2. Suppose that a mapping F : X → X has an N-order fixed point x ∗ = ( ξ 1 ∗ , ξ 2 ∗ , ⋯ , ξ N ∗ ) . Then x ∗ is the unique N-order fixed point and satisfies ξ 1 ∗ = ξ 2 ∗ = ⋯ = ξ N ∗ if

( P 3 ) : for each x , y ∈ X , there exists z ∈ X such that ( x , z ) ∈ E ( G ¯ ) and ( x , y ) ∈ E ( G ¯ )

Proof. It is clear that G ¯ is connected. By theorem 3.2, we can obtain the result.

In partially ordered set, the assumption ( P 3 ) implies the comparable property which is widely used in the uniqueness of fixed point theorems.

Remark The conditions (3.5) appeared in many documents with N = 1 , 2 , 3 , such as, Bhaskar and Lakshmikantham [

The followings are excerpted from [

Corollary 3. [

1) F has mixed monotone property: for any ξ 1 , ξ 2 , η 1 , η 2 , ξ , η ∈ X

ξ 1 ≤ ξ 2 ⇒ F ( ξ 1 , η ) ≤ F ( ξ 2 , η ) , η 1 ≤ η 2 ⇒ F ( ξ , η 2 ) ≤ F ( ξ , η 1 ) ;

2) there is a k ∈ [ 0 , 1 ) such that for all ξ 1 , ξ 2 , η 1 , η 2 ∈ X with ξ 1 ≤ ξ 2 and η 2 ≤ η 1 ,

d ( F ( ξ 1 , ξ 2 ) , F ( η 1 , η 2 ) ) ≤ k ( d ( ξ 1 , η 1 ) + d ( ξ 2 , η 2 ) 2 )

If there exists ( ξ 1 0 , ξ 2 0 ) with ξ 1 0 ≤ F ( ξ 1 0 , ξ 2 0 ) and F ( ξ 2 0 , ξ 1 0 ) ≤ ξ 2 0 , then F has a coupled fixed point. Assume that the comparable property holds, we can obtain the uniqueness of the coupled fixed point.

Proof. Set a graph G on X defined as: for x = ( ξ 1 , ξ 2 ) , y = ( η 1 , η 2 ) ∈ X 2 , we say

( x , y ) ∈ E ( G ) ⇔ x ≼ y ⇔ ξ 1 ≤ η 1 and η 2 ≤ ξ 2 .

Then by mixed monotone property, we know

( x , y ) ∈ E ( G ) ⇒ F ( x ) = F ( ξ 1 , ξ 2 ) ≤ F ( η 1 , ξ 2 ) ≤ F ( η 1 , η 2 ) = F ( y ) ⇒ ( F ( x ) , F ( y ) ) ∈ E ( G ) .

By Theorem 3.2 and Corollary 2, we can get the proof.

Corollary 4 [

d ( F ( ξ 1 , ξ 2 ) , F ( η 1 , η 2 ) ) ≤ k ( d ( ξ 1 , η 1 ) + d ( ξ 2 , η 2 ) 2 )

for ( ξ 1 , η 1 ) , ( η 2 , ξ 2 ) ∈ E ( G ) . If there exists ( ξ 1 0 , ξ 2 0 ) with and ( ξ 1 0 , F ( ξ 1 0 , ξ 2 0 ) ) ∈ E ( G ) , then F has a coupled fixed point.

Proof. We can get the proof by Theorem 3.2.

Let ( X , d , g ) and ( X , D , G ) be defined in Section 2 and

C B ( X ) = { S : S isanonemptyclosedandboundedsubsetof X } .

We make use of h, H, H to denote the following different set-valued mappings, i.e.,

h : X → C B ( X ) , H : X → C B ( X ) and H : X → C B ( X ) .

For simplicity of description, we denote

X h = { x ∈ X : ( x , u ) ∈ E ( G ) forsome u ∈ h ( x ) } .

Let D denote the Pompeiu-Hausdorff metric between two sets on C B ( X ) , i.e., for A , B ∈ C B ( X ) ,

D ( A , B ) = max { sup a ∈ A d ( a , B ) , sup b ∈ B d ( b , A ) }

where d ( a , B ) = i n f b ∈ B d ( a , b ) .

Definition 3. Let X be a non-empty set and h : X → C B ( X ) be a set-valued mapping. An element x ∈ X is called a fixed point of h provided x ∈ h ( x ) .

In order to discuss the fixed point theory of set-valued mappings, many contraction’s notations have been introduced such as [

Definition 4. [

Definition 5. [

Definition 6. [

For all x , y ∈ X , we have h ( x ) , h ( y ) ∈ C B ( X ) and, for a given number α > 0 , for any u ∈ h ( x ) , there is a point v ∈ h ( y ) such that d ( u , v ) ≤ D ( h ( x ) , h ( y ) ) + α ( [

Definition 7. Let ( X , d ) be a non-empty metric space with a graph G. Let h : X → C B ( X ) be a set-valued mapping. There exists k ∈ ( 0 , 1 ) such that, for ∀ x , y ∈ X with ( x , y ) ∈ E ( G ) and any u ∈ h ( x ) , for any α > 0 , there exists v ∈ h ( y ) such that ( u , v ) ∈ E ( G ) and d ( u , v ) ≤ k d ( x , y ) + α . We say that the mapping h : X → C B ( X ) is SG-contractive.

If h is a single-valued mapping, then SG-contractive property of h implies that, ∀ x , y ∈ X with ( x , y ) ∈ E ( G ) , it holds that h ( x ) , h ( y ) ∈ E ( G ) and d ( h ( x ) , h ( y ) ) ≤ k d ( x , y ) which is called G-contractive (see Definition 2).

Let’s investigate the above definitions. “Set-valued contractive” in Definition 4 imlies “SG-contractive” in Definition 7. In Definition 5, the condition “if u ∈ h ( x ) and v ∈ h ( y ) with d ( u , v ) ≤ k d ( x , y ) + α for each α > 0 , then ( u , v ) ∈ E ( G ) “ is very hard. It is obvious that “G-contractive” in Definition 5 implies “SG-contractive” in Definition 7. And, “monotone increasing G-contractive” in Definition 6 implies “SG-contractive” in Definition 7. Since, for u ∈ h ( x ) , there may not be a point v ∈ h ( y ) such that d ( u , v ) ≤ D ( h ( x ) , h ( y ) ) (see [

Definition 8. The point x = ( ξ 1 , ξ 2 , ⋯ , ξ N ) ∈ X is said to be an N-order fixed point of H : X → C B ( X ) if

ξ i ∈ H ( ξ i , ξ i + 1 , ⋯ , ξ N , ξ 1 , ⋯ , ξ i − 1 ) , i = 1 , 2 , ⋯ , N .

For a set-valued mapping H : X → C B ( X ) , we define H : X → C B ( X ) as: for x = ( ξ 1 , ξ 2 , ⋯ , ξ N ) ∈ X ,

H ( x ) = { y = ( η 1 , η 2 , ⋯ , η N ) : η i ∈ F ( ξ i , ξ i + 1 , ⋯ , ξ N , ξ 1 , ⋯ , ξ i − 1 ) } . (4.1)

We can see that:

x is an N-order fixed point of H

⇔ ξ i ∈ H ( ξ i , ξ i + 1 , ⋯ , ξ N , ξ 1 , ⋯ , ξ i − 1 ) , i = 1 , 2 , ⋯ , N

⇔ x ∈ H ( x )

⇔ x is a fixed point of H

Thus, we can research the N-order fixed point theory by studying the fixed point theory.

Theorem 4.1. Suppose that ( X , d , G ) has the property ( P 1 ) . Let h : X → C B ( X ) be a set-valued SG-contraction. If X h ≠ ∅ , then the following statements hold:

1) for any x ∈ X h , h | [ x ] G ¯ has a fixed point;

2) if G is weakly connected, then h has a fixed point in G;

3) if X ′ = ∪ { [ x ] G ¯ : x ∈ X h } , then h | X ′ has a fixed point in X;

4) if h ( x ) ⊂ E ( G ) then h has a fixed point;

5) fix h ≠ ∅ if and only if X h ≠ ∅ .

Proof. (1) Considering X h ≠ ∅ , we take a point x 0 ∈ X h . Then there is a point x 1 ∈ h ( x 0 ) such that ( x 0 , x 1 ) ∈ E ( G ) . Noting that h : X → C B ( X ) is an SG-contractive mapping, for k ∈ [ 0 , 1 ) , there is a point x 2 ∈ h ( x 1 ) such that

( x 1 , x 2 ) ∈ E ( G ) and d ( x 1 , x 2 ) ≤ k d ( x 0 , x 1 ) + k

By the similar deduction, there is a point x 3 such that

( x 2 , x 3 ) ∈ E ( G ) and d ( x 2 , x 3 ) ≤ k d ( x 1 , x 2 ) + k 2 ≤ k 2 d ( x 1 , x 2 ) + 2 k 2 .

⋯

We can construct a sequence { x n } such that for n ≥ 1

x n + 1 ∈ h ( x n ) , ( x n , x n + 1 ) ∈ E ( G )

and

d ( x n , x n + 1 ) ≤ k d ( x n − 1 , x n ) + k n ≤ k n d ( x 1 , x 2 ) + n k n (4.2)

Since ∑ n = 0 ∞ d ( x n , x n + 1 ) < ∞ , we get that { x n } is a Cauchy sequence and converges a point x ∗ ∈ X .

Now, We claim that x ∗ ∈ h ( x ∗ ) under the condition ( P 1 ) . By ( P 1 ) , we have ( x n , x ∗ ) ∈ E ( G ) . For x n + 1 ∈ h ( x n ) , by Definition 7, for α = 1 n , there exists a point y n ∈ h ( x ∗ ) such that ( x n + 1 , y n ) ∈ E ( G ) and d ( x n + 1 , y n ) ≤ k d ( x n , x ) + 1 n . Thus, for any n ≥ 1 ,

d ( x , y n ) ≤ d ( x , x n + 1 ) + d ( x n + 1 , y n ) ≤ d ( x , x n + 1 ) + k d ( x n , x ) + 1 n

which implies lim n → ∞ = x ∗ . Noting that h ( x ∗ ) ∈ C B ( X ) , we get x ∗ ∈ h ( x ∗ ) .

We can get the proof of (2)-(5) by the similar deduction of Theorem 3.1 in [

Theorem 4.2. Let ( X , d , G ) be the complete metric space endowed with a graph G. Suppose that h has the property ( P 2 ) . Let h : X → C B ( X ) be a set-valued SG-contraction. Suppose that h has the property ( P 2 ) . If X h ≠ ∅ , then conclusions obtained in Theorem 4.1 remain true.

By the Similar deduction in Section 3.1, in order to discuss the N-order fixed point of H, we try to ensure the SG-contraction of H .

Theorem 4.3. Suppose that ( X , d , G ) has the property ( P 1 ) . Let H : X → C B ( X ) be a set-valued mapping. Suppose that there are some constants k 1 , ⋯ , k N with 0 < k 1 + ⋯ + k N < 1 such that for any u ∈ H ( x ) , for any α > 0 , there is a point v ∈ H ( y ) such that

d ( u , v ) ≤ k 1 d ( ξ 1 , η 1 ) + ⋯ + k N ( ξ N , η N ) + α (4.3)

for ( x , y ) ∈ E ( G ) . Suppose that there is a point x 0 = ( ξ 1 0 , ξ 2 0 , ⋯ , ξ N 0 ) ∈ X such that there is a point u 0 = ( u 1 0 , u 2 0 , ⋯ , u N 0 ) ∈ X with ( ξ i 0 , u i 0 ) ∈ E ( G ) and

u i 0 ∈ H ( ξ i 0 , ξ i + 1 0 , ⋯ , ξ N 0 , ξ 1 0 , ⋯ , ξ i − 1 0 ) , ∀ i = 1 , 2 , ⋯ , N . (4.4)

Then

1) for any x ∈ X H , H | [ x ] G ¯ has an N-order fixed point;

2) if G is weakly connected, then H has an N-order fixed point in G ;

3) if X ′ = ∪ { [ x ] G ¯ : x ∈ X H } , then H | X ′ has an N-order fixed point in X ;

4) if H ( X ⊂ ) E ( G ) then H has an N-order fixed point;

5) fix H ≠ ∅ if and only if X H ≠ ∅ .

Proof. (1) Since ( X , d ) is complete, we know that ( X , D ) is complete. Let H : X → C B ( X ) be defined by (4.10).

We first show that H is SG-contractive. For x = ( ξ 1 , ⋯ , ξ N ) , y = ( η 1 , ⋯ , η N ) with ( x , y ) ∈ E ( G ) , we obtain that

( ξ 2 , ⋯ , ξ N , ξ 1 ; η 2 , ⋯ , η N , η 1 ) ∈ E ( G ) , ⋯ , ( ξ N , ⋯ , ξ N − 1 ; η N , ⋯ , η N − 1 ) ∈ E ( G ) (4.5)

Let u = ( u 1 , ⋯ , u N ) ∈ H ( x ) and α > 0 be arbitrary. Considering (4.3), for u 1 ∈ H ( x ) , there is a point v 1 ∈ H ( y ) such that

d ( u 1 , v 1 ) ≤ k 1 d ( ξ 1 , η 1 ) + ⋯ + k N d ( ξ N , η N ) + α N

⋯

For u n ∈ H ( ξ N , ξ 1 , ⋯ , ξ N − 1 ) , there is a point v n ∈ H ( η N , η 1 , ⋯ , η N − 1 ) such that

d ( u n , v n ) ≤ k N d ( ξ N , η N ) + k 1 d ( ξ 1 , η 1 ) + ⋯ + k N − 1 d ( ξ N − 1 , η N − 1 ) + α N

Then, the point v = ( v 1 , v 2 , ⋯ , v N ) belongs to H ( y ) and

d ( u , v ) = d ( u 1 , v 1 ) + d ( u 2 , v 2 ) + ⋯ + d ( u N , v N ) ≤ ( k 1 d ( ξ 1 , η 1 ) + ⋯ + k N d ( ξ N , η N ) + α N ) + ( k 1 d ( ξ 2 , η 2 ) + ⋯ + k N d ( ξ 1 , η 1 ) + α N ) + ⋯

+ ( k 1 d ( ξ N , η N ) + ⋯ + k N d ( ξ N − 1 , η N − 1 ) + α N ) = ( k 1 + ⋯ + k N ) ( d ( ξ 1 , η 1 ) + ⋯ + k N d ( ξ N , η N ) ) + α = ( k 1 + ⋯ + k N ) D ( x , y ) + α

Thus H is an SG-contraction on X .

Next ( X , D , G ) has the property ( P 1 ) . In fact, for any sequence ( x n ) n ∈ N in X , if ( ξ 1 n , ⋯ , ξ N n ) = x n → x = ( ξ 1 , ⋯ , ξ N ) and ( x n , x n + 1 ) ∈ E ( G ) for any n ∈ N , then

ξ i n → ξ i , ( ξ i n , ξ i n + 1 ) ∈ E ( G ) , i = 1 , ⋯ , N .

By the property ( P 1 ) of ( X , d , G ) , we get

( ξ i n , ξ ) ∈ E ( G ) ,

then ( x n , x ) ∈ E ( G ) .

Last, we can see that ( x 0 , u 0 ) ∈ X H which implies that X H ≠ ∅ . By Theorem 4.3, we get the proof.

Theorem 4.4. Let H : X → C B ( X ) be a set-valued mapping and H satisfy the property ( P 2 ) .

Suppose that there are some constants k 1 , ⋯ , k N with 0 < k 1 + ⋯ + k N < 1 such that for any u ∈ H ( x ) , for any α > 0 , there is a point v ∈ H ( y ) such that (4.12) holds for ( x , y ) ∈ E ( G ) . Suppose that there is a point x 0 = ( ξ 1 0 , ξ 2 0 , ⋯ , ξ N 0 ) ∈ X N such that (4.13) holds for some u 0 = ( u 1 0 , u 2 0 , ⋯ , u N 0 ) with ( ξ i 0 , u i 0 ) ∈ E ( G ) . Then the conclusions in Theorem 4.3 hold.

Corollary 5. [

Proof. The “the mixed G-monotone property” and the “(MBL) condition” on the page 9 in [

Let f : I × R × R × R → R . We discuss the differential equations

{ x ˙ ( t ) = f ( t , x ( t ) , y ( t ) , z ( t ) ) y ˙ ( t ) = f ( t , y ( t ) , z ( t ) , x ( t ) ) , t ∈ I = [ 0 , 1 ] z ˙ ( t ) = f ( t , z ( t ) , x ( t ) , y ( t ) ) (5.1)

with the initial condition

x ( 0 ) = y ( 0 ) = z ( 0 ) = x 0 (5.2)

Suppose that there are three continuous mapping α ( t ) , β ( t ) and γ ( t ) with the initial condition α ( 0 ) = β ( 0 ) = γ ( 0 ) = x 0 . If there is a constant l > 0 such that

max t ∈ I | α ( t ) − ( x 0 + ∫ 0 t f ( t , α ( s ) , β ( s ) , γ ( s ) ) d s ) | ≤ l

max t ∈ I | β ( t ) − ( x 0 + ∫ 0 t f ( t , β ( s ) , γ ( s ) , α ( s ) ) d s ) | ≤ l

max t ∈ I | γ ( t ) − ( x 0 + ∫ 0 t f ( t , γ ( s ) , α ( s ) , β ( s ) ) d s ) | ≤ l

then ( α ( t ) , β ( t ) , γ ( t ) ) is called an l−lower solution of differential Equations (5.1) with the initial condition (5.2). Although we use the lower solution definition, we can’t judge ( α ( t ) , β ( t ) , γ ( t ) ) is bigger or smaller than the solution of (5.1) since l is arbitrary bigger than 0. This notation of l-lower solution only shows the relative distance between ( α ( t ) , β ( t ) , γ ( t ) ) and the solution of (5.1).

Theorem 5.1. Let f ∈ C ( I × R × R × R , R ) . Consider the differential Equations (5.1) with the initial condition (5.2) and a constant l > 0 . Suppose that the following conditions hold:

1) for x , y , z , u , v , w ∈ X with | x − u | ≤ l , | y − v | ≤ l , | z − w | ≤ l , we have

| f ( t , x , y , z ) − f ( t , u , v , w ) | ≤ k 1 | x − u | + k 2 | y − v | + k 3 | z − w |

where 0 ≤ k 1 , k 2 , k 3 < 1 with k 1 + k 2 + k 3 < 1 ;

2) there is an l-lower solution of (5.1).

Then the differential Equations (5.1) with initial condition (5.2) has a unique solution ( x ∗ , x ∗ , x ∗ ) in C ( I , R ) × C ( I , R ) × C ( I , R ) . That is, x ∗ is the solution of the differential

x ˙ ( t ) = f ( t , x ( t ) , x ( t ) , x ( t ) ) , t ∈ [ 0 , 1 ] (5.3)

with the initial condition x ( 0 ) = x 0 .

Proof Let X = C ( I , R ) . It is clear that ( X , d ) is a complete metric space with the metric d defined as

∀ x , y ∈ X , d ( x , y ) = max t ∈ I | x ( t ) − y ( t ) | .

Denote

B l = { x ∈ X : max t ∈ I | x ( t ) | ≤ l } .

We know that B is a closed subset of X. Let G be a graph on X with V ( G ) = X and

∀ x , y ∈ X , ( x , y ) ∈ E ( G ) ⇔ x − y ∈ B l (5.4)

Clearly, E ( G ) is an equivalence relation and ( x , x ) ∈ E ( G ) , ∀ x ∈ X . And besides, G is weakly connected.

Define F : X 3 → X by

F ( x , y , z ) ( t ) = x 0 + ∫ 0 t f ( t , x ( s ) , y ( s ) , z ( s ) ) d s (5.5)

By f’s continuity and (5.4), we know that F is continuous. Then the solution of the differential

Equations (5.1) with initial condition (5.2) are just the tripled fixed point of F. In the following, we discuss the tripled fixed point of F. We will testify that F satisfies the all conditions in Theorem 4.3.

For x , y , z , u , v , w ∈ X , define a graph G and a metric D on X 3 by

( ( x , y , z ) , ( u , v , w ) ) ∈ E ( G ) ⇔ x − u ∈ B l , y − v ∈ B l , z − w ∈ B l

and

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

Obviously, ( X 3 , D , G ) is induced by ( X , d , G ) . We now prove ( F ( x , y , z ) , F ( u , v , w ) ) ∈ E ( G ) if ( ( x , y , z ) , ( u , v , w ) ) ∈ E ( G ) . Since

( ( x , y , z ) , ( u , v , w ) ) ∈ E ( G )

⇔ x − u ∈ B l , y − v ∈ B l , z − w ∈ B l

⇔ max t ∈ I | x ( t ) − u ( t ) | ≤ l , max t ∈ I | y ( t ) − v ( t ) | ≤ l , max t ∈ I | z ( t ) − w ( t ) | ≤ l

⇔ d ( x , u ) ≤ l , d ( y , v ) ≤ l , d ( z , w ) ≤ l ,

by (i), we have for t ∈ I ,

| F ( x , y , z ) ( t ) − F ( u , v , w ) ( t ) | = | ∫ 0 t f ( t , x ( s ) , y ( s ) , z ( s ) ) d s − ∫ 0 t f ( t , u ( s ) , v ( s ) , w ( s ) ) d s | ≤ ∫ 0 t | f ( t , x ( s ) , y ( s ) , z ( s ) ) − f ( t , u ( s ) , v ( s ) , w ( s ) ) | d s ≤ ∫ 0 t k 1 | x ( s ) − u ( s ) | + k 2 | y ( s ) − v ( s ) | + k 3 | z ( s ) − w ( s ) | d s ≤ ∫ 0 t k 1 d ( x , u ) + k 2 d ( y , u ) + k 3 d ( z , w ) d s

≤ k 1 d ( x , u ) + k 2 d ( y , u ) + k 3 d ( z , w ) (which is a constant)

≤ ( k 1 + k 2 + k 3 ) max { d ( x , u ) , d ( y , v ) , d ( z , w ) } ≤ l

which implies F ( x , y , z ) − F ( u , v , w ) ∈ B l and then ( F ( x , y , z ) , F ( u , v , w ) ) ∈ E ( G ) .

Moreover, by the above deduction, we have

d ( F ( x , y , z ) , F ( u , v , w ) ) = max t ∈ I | F ( x , y , z ) ( t ) − F ( u , v , w ) ( t ) | ≤ k 1 d ( x , u ) + k 2 d ( y , v ) + k 3 d ( z , w )

By (2), the differential Equations (5.1) has an l-lower solution denoted ( α ( t ) , β ( t ) , γ ( t ) ) , we know

( α , F ( α , β , γ ) ) ∈ E ( G ) , ( β , F ( β , γ , α ) ) ∈ E ( G ) , ( γ , F ( γ , α , β ) ) ∈ E ( G ) .

Thus by Theorem 4.3, F has a unique fixed point ( x ∗ , y ∗ , z ∗ ) with x ∗ = y ∗ = z ∗ .

Thus, the different Equations (5.1) with initial condition (5.2) has a unique solution ( x ∗ , x ∗ , x ∗ ) and which means

x ∗ ( t ) = F ( x ∗ , x ∗ , x ∗ ) ( t ) = x 0 + ∫ 0 t f ( t , x ∗ ( s ) , x ∗ ( s ) , x ∗ ( s ) ) d s

which is equivalent to the different Equations (5.3) with the initial condition x ( 0 ) = x 0 .

Theorem 5.2. Let f : I × R × R × R → R satisfy: forgiven x , y , z ∈ R and a sequence { x n , y n , z n } with | x n − x n + 1 | ≤ l , | y n − y n + 1 | ≤ l , | z n − z n + 1 | ≤ l and lim n → ∞ ( x n , y n , z n ) = ( x , y , z ) , we have lim n → ∞ f ( t , x n , y n , z n ) = f ( t , x , y , z ) . Consider the differential Equations (5.1) with the initial condition (5.2) and a constant l > 0 . Suppose that the conditions (1) and (2) in Theorem 5.1 are satisfied. Then we can get the same result with Theorem 5.1.

Proof. Define F as (5.4), then F satisfies ( P 2 ) with N = 3 . By the similar induction of Theorem 4.3, we can get the proof.

We establish N-order fixed point theorems of set-valued and single-valued mapping on product metric space with a graph. We build a unified method to study the N-order fixed point theory.

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

The authors declare no conflicts of interest regarding the publication of this paper.

Liang, X.N. and Wang, S.Y. (2020) A General Method of Researching the N-Ordered Fixed Point on a Metric Space with a Graph. Journal of Applied Mathematics and Physics, 8, 2846-2860. https://doi.org/10.4236/jamp.2020.812210