Periodic and fixed points for mappings in extended b-gauge spaces equipped with a graph

Definition 2.1

A map q : U × U → [ 0 , ∞ ) is a pseudo-metric, if for all e , f , g ∈ U , it satisfies

1. q ( e , e ) = 0 ;

2. q ( e , f ) = q ( f , e ) ; and

3. q ( e , g ) ≤ q ( e , f ) + q ( f , g ) .

Definition 2.2

Each family Q = { q β : β ∈ Ω } of pseudo-metrics q β : U × U → [ 0 , ∞ ) for β ∈ Ω is said to be gauge on U .

Definition 2.3

The family Q = { q β : β ∈ Ω } of pseudo-metrics q β : U × U → [ 0 , ∞ ) for β ∈ Ω is called to be separating if for each pair ( e , f ) where e ≠ f , there is q β ∈ Q such that q β ( e , f ) > 0 .

Definition 2.4

Let Q = { q β : β ∈ Ω } be a family of pseudo-metrics on U . The topology T ( Q ) on U whose subbase is defined by the family ℬ ( Q ) = { B ( e , ε β ) : e ∈ U , ε β > 0 , β ∈ Ω } of all balls B ( e , ε β ) = { f ∈ U : q β ( e , f ) < ε β } is called the induced topology.

Definition 2.5

A topological space ( U , T ) is called a gauge space if there exists gauge Q on U with T = T ( Q ) . The pair ( U , T ( Q ) ) denotes the gauge space and is Hausdorff if Q is separating.

Example 2.6

Let U = R 2 and let q 1 , q 2 : U × U → [ 0 , ∞ ) be defined for all ( e 1 , f 1 ) , ( e 2 , f 2 ) ∈ R 2 by

Then, q 1 and q 2 are pseudo-metrics on U .

Note that q 1 ( ( 2 , 3 ) , ( 2 , 5 ) ) = ∣ 2 − 2 ∣ = 0 , but ( 2 , 3 ) and ( 2 , 5 ) are distinct points. Also, q 2 ( ( 3 , 6 ) , ( 5 , 6 ) ) = ∣ 6 − 6 ∣ = 0 , but ( 3 , 6 ) and ( 5 , 6 ) are the distinct points. Therefore, q 1 and q 2 are not metrics on U .

Let the family Q = { q 1 , q 2 } be a gauge on U . We now look for the topology T ( Q ) induced by gauge Q in the following manner.

First, finding balls B ( e , ε 1 ) for q 1 , where e = ( e 1 , f 1 ) ∈ U and ε 1 > 0 :

Thus, B ( ( e 1 , f 1 ) , ε 1 ) contains all verticle strips in the plane.

Similarly,

Thus, B ( ( e 1 , f 1 ) , ε 2 ) contains all horizontal strips in the plane.

The subbase ℬ ( Q ) for induced topology T ( Q ) is the collection of all vertical and horizontal infinite open strips. Their intersections are open rectangles that form the base of induced topology. The induced topology is thus the usual topology on R 2 . Therefore, ( U , Q ) is a gauge space.

Definition 2.7

A map q : U × U → R + is said to be a b -metric, if for each e , f , g ∈ U , there exists s ≥ 1 such that it satisfies

1. q ( e , f ) = 0 ⇔ e = f ;

2. q ( e , f ) = q ( f , e ) ;

3. q ( e , g ) ≤ s { q ( e , f ) + q ( f , g ) } .

Definition 2.8

A map q : U × U → [ 0 , ∞ ) is a b -pseudo-metric, if there is s ≥ 1 satisfying for all e , f , g ∈ U the following conditions:

1. q ( e , e ) = 0 ;

2. q ( e , f ) = q ( f , e ) ; and

3. q ( e , g ) ≤ s { q ( e , f ) + q ( f , g ) } .

Definition 2.9

Each family Q = { q β : β ∈ Ω } of b -pseudo-metrics q β : U × U → [ 0 , ∞ ) , is called the b -gauge on U .

Definition 2.10

The family Q = { q β : β ∈ Ω } is separating if for each pair ( e , f ) , where e ≠ f , there is q β ∈ Q such that q β ( e , f ) > 0 .

Definition 2.11

Let Q = { q β : β ∈ Ω } be the family of b -pseudo-metrics on U . The topology T ( Q ) on U whose subbase is defined by the family ℬ ( Q ) = { B ( e , ε β ) : e ∈ U , ε β > 0 , β ∈ Ω } , where B ( e , ε β ) = { f ∈ U : q β ( e , f ) < ε β } is called the topology induced by Q . The pair ( U , T ( Q ) ) is called to be a b -gauge space and is Hausdorff if Q is separating.

Example 2.12

[32] Suppose U = C ( [ 0 , ∞ ) , R ) and describe q : U × U → [ 0 , ∞ ) by

Then, d is a b -pseudo-metric, but not a metric, pseudo-metric or b -metric.

Definition 2.13

Let φ : U × U → [ 1 , ∞ ) . A map q : U × U → R + is said to be an extended b -metric, if for all e , f , g ∈ U , it satisfies

1. q ( e , f ) = 0 ⇔ e = f ;

2. q ( e , f ) = q ( f , e ) ;

3. q ( e , g ) ≤ φ ( e , g ) { q ( e , f ) + q ( f , g ) } .

Definition 3.1

[37] A map q : U × U → R + is an extended pseudo- b -metric, if for all e , f , g ∈ U , there exists φ : U × U → [ 1 , ∞ ) satisfying the following conditions:

1. q ( e , e ) = 0 ;

2. q ( e , f ) = q ( f , e ) ; and

3. q ( e , g ) ≤ φ ( e , g ) { q ( e , f ) + q ( f , g ) } .

Example 3.2

Suppose U = [ 0 , 1 ] . Define q : U × U → R + and φ : U × U → [ 1 , ∞ ) by

and

for all e , f ∈ U .

Then, q is an extended pseudo- b -metric on U . Indeed, q ( e , e ) = 0 and q ( e , f ) = q ( f , e ) for all e , f ∈ U . Furthermore, for all e , f , g ∈ U , q ( e , g ) ≤ φ ( e , g ) { q ( e , f ) + q ( f , g ) } holds.

Example 3.3

Let U = { e , f , g } . Define q : U × U → R + and φ : U × U → [ 1 , ∞ ) for all e , f , g ∈ U by

and

Furthermore, q ( e , g ) ≤ φ ( e , g ) { q ( e , f ) + q ( f , g ) } is satisfied. Indeed, q is an extended pseudo- b -metric on U . Note that q is not a pseudo-metric on U as 3 2 = q ( e , f ) + q ( f , g ) < q ( e , g ) = 2 .

This example illustrates that an extended pseudo- b -metric serves as a generalization or extension of a pseudo-metric.

Definition 3.4

The family Q φ ; Ω = { q β : β ∈ Ω } of extended pseudo- b -metrics q β : U × U → [ 0 , ∞ ) , β ∈ Ω , is said to be an extended b -gauge on U ( Ω -index set).

Definition 3.5

The family Q φ ; Ω = { q β : β ∈ Ω } of extended pseudo- b -metrics q β : U × U → [ 0 , ∞ ) , β ∈ Ω is called to be separating if for every pair ( e , f ) , where e ≠ f , there exists q β ∈ Q φ ; Ω such that q β ( e , f ) ≠ 0 .

Definition 3.6

Let the family Q φ ; Ω = { q β : β ∈ Ω } be an extended b -gauge on U . The topology T ( Q φ ; Ω ) on U whose subbase is defined by the family ℬ ( Q φ ; Ω ) = { B ( e , ε β ) : e ∈ U , ε β > 0 , β ∈ Ω } of all balls B ( e , ε β ) = { f ∈ U : q β ( e , f ) < ε β } is called the topology induced by Q φ ; Ω . The topological space ( U , T ( Q φ ; Ω ) ) is an extended b -gauge space, denoted by ( U , Q φ ; Ω ) . We note that ( U , Q φ ; Ω ) is Hausdorff if Q φ ; Ω is separating.

Remark 3.7

For s β = 1 , for each β ∈ Ω , each gauge space is a b s -gauge space, and for φ β ( u , v ) = s , for each β ∈ Ω , where s ≥ 1 , each b -gauge space is an extended b -gauge space. Hence, extended b -gauge space is the largest general space.

Definition 3.8

The family J φ ; Ω = { J β : β ∈ Ω } , where J β : U × U → [ 0 , ∞ ) , β ∈ Ω , is called an extended J φ ; Ω -family of generalized extended pseudo- b -distances on ( U , Q φ ; Ω ) if the following statements hold for all u , v , w ∈ U and for all β ∈ Ω :

1. J β ( u , w ) ≤ φ β ( u , w ) { J β ( u , v ) + J β ( v , w ) } ;

2. for each sequences ( u m : m ∈ N ) and ( v m : m ∈ N ) in U fulfilling

   (3.1) lim m → ∞ sup n > m J β ( u m , u n ) = 0

   and

   (3.2) lim m → ∞ J β ( v m , u m ) = 0 ,

   the following holds:

   (3.3) lim m → ∞ q β ( v m , u m ) = 0 .

Also, we denote

and

Thus, U = U J φ ; Ω 0 ∪ U J φ ; Ω + .

Example 3.9

Let U contain at least two distinct elements, and suppose Q φ ; Ω = { q β : β ∈ Ω } is an extended b -gauge on U . Thus, ( U , Q φ ; Ω ) is an extended b -gauge space.

Let there be at least two distinct but arbitrary and fixed elements in a set F ⊂ U . Let a β ∈ ( 0 , ∞ ) satisfy δ β ( F ) < a β , where δ β ( F ) = sup { q β ( e , f ) : e , f ∈ F } , for all β ∈ Ω . Let J β : U × U → [ 0 , ∞ ) for all e , f ∈ U be defined as

Then, J φ ; Ω = { J β : β ∈ Ω } ∈ J ( U , Q ) .

Note that J β ( e , g ) ≤ φ β ( e , g ) { J β ( e , f ) + J β ( f , g ) } , for all e , f , g ∈ U ; thus, ( J 1 ) is satisfied. Indeed, ( J 1 ) will not satisfied only when there is some e , f , g ∈ U such that J β ( e , g ) = a β , J β ( e , f ) = q β ( e , f ) , J β ( f , g ) = q β ( f , g ) , and φ β ( e , g ) { q β ( e , f ) + q β ( f , g ) } ≤ a β . However, then this gives rise to an element h ∈ { e , g } with h ∉ F , and on other hand, e , f , g ∈ F , which is impossible.

Next, assume that (3.1) and (3.2) hold by the sequences ( u m ) and ( v m ) in U . Then, (3.2) yields that for all 0 < ε < a β , there exists m 1 = m 1 ( β ) ∈ N such that

By (3.5) and (3.4), denoting m 2 = min { m 1 ( β ) : β ∈ Ω } , we have

and

Thus, (3.3) is satisfied. Therefore, J φ ; Ω is a J φ ; Ω -family on U .

Remark 3.10

Let ( U , Q φ ; Ω ) be an extended b -gauge space. Then, the following hold:

1. Q φ ; Ω ∈ J ( U , Q φ ; Ω ) .

2. Let J φ ; Ω ∈ J ( U , Q φ ; Ω ) . If for all β ∈ Ω and for all u , v ∈ U \ J β ( v , v ) = 0 and J β ( u , v ) = J β ( v , u ) , then for each β ∈ Ω , J β is an extended pseudo- b -metric.

3. Several examples of J φ ; Ω ∈ J ( U , Q φ ; Ω ) , which show that the maps J β , β ∈ Ω are not the extended pseudo- b -metrics can be found in the literature (see [37], Example 3).

Proposition 3.11

Let ( U , Q φ ; Ω ) be a Hausdorff extended b-gauge space and the family J φ ; Ω = { J β : β ∈ Ω } be an extended J φ ; Ω -family on U. Then, there exists β ∈ Ω such that

for each e , f ∈ U .

Proof

Let that there be e ≠ f , e , f ∈ U such that J β ( e , f ) = 0 = J β ( f , e ) for all β ∈ Ω . Then, J β ( e , e ) = 0 , for all β ∈ Ω ; using property ( J 1 ) in Definition 3.8, it follows that J β ( e , e ) ≤ φ β ( e , e ) { J β ( e , f ) + J β ( f , e ) } = 0 , for all β ∈ Ω .

Defining sequences ( u m ) and ( v m ) in U by u m = f and v m = e , we see that Conditions (3.1) and (3.2) of property ( J 2 ) are satisfied, and therefore, Condition (3.3) holds, which implies that q β ( e , f ) = 0 , for all β ∈ Ω . But this denies the fact that ( U , Q φ ; Ω ) is a Hausdorff extended b -gauge space. Therefore, our supposition is wrong, and there exists β ∈ Ω such that for all e , f ∈ U ,

Definition 3.12

Let J φ ; Ω = { J β : β ∈ Ω } be an extended J φ ; Ω -family on the extended b -gauge space ( U , Q φ ; Ω ) .

1. A sequence ( v m : m ∈ N ) is an extended J φ ; Ω -Cauchy sequence in U if

   lim m → ∞ sup n > m J β ( v m , v n ) = 0 , for all β ∈ Ω .

2. The sequence ( v m : m ∈ N ) is extended J φ ; Ω -convergent to v ∈ U if lim m → ∞ J φ ; Ω v m = v , where

   lim m → ∞ J φ ; Ω v m = v ⇔ lim m → ∞ J β ( v , v m ) = 0 = lim m → ∞ J β ( v m , v ) , for all β ∈ Ω .

3. If S ( v m : m ∈ N ) J φ ; Ω ≠ ∅ , where

   S ( v m : m ∈ N ) J φ ; Ω = { v ∈ U : lim m → ∞ J φ ; Ω v m = v } ,

   then ( v m : m ∈ N ) in U is an extended J φ ; Ω -convergent sequence in U .

4. The space ( U , Q φ ; Ω ) is called extended J φ ; Ω -sequentially complete, if every extended J φ ; Ω -Cauchy sequence in U , extended J φ ; Ω -converges in U .

Remark 3.13

There exist examples of extended b -gauge space ( U , Q φ ; Ω ) and J φ ; Ω -family on U with J φ ; Ω ≠ Q φ ; Ω such that ( U , Q φ ; Ω ) is J φ ; Ω -sequential complete but not Q φ ; Ω -sequential complete (see [38], Example 3.10).

Definition 3.14

Let T : U → 2 U be a set-valued map. The map T [ k ] is said to be an extended Q φ ; Ω -closed map on U , for some k ∈ N , if for each sequence ( w m : m ∈ N ) in T [ k ] ( U ) , which is extended Q φ ; Ω -convergent in U , one has S ( w m : m ∈ N ) Q φ ; Ω ≠ ∅ and its subsequences ( f m ) and ( z m ) satisfy

f m ∈ T [ k ] ( z m ) , for all m ∈ N

has the property that there exists w ∈ S ( w m : m ∈ N ) Q φ ; Ω such that w ∈ T [ k ] ( w ) .

Definition 3.15

Let J φ ; Ω = { J β : β ∈ Ω } be an extended J φ ; Ω -family on extended b -gauge space ( U , Q φ ; Ω ) . A set Y ∈ 2 U is J φ ; Ω -closed in U if Y = c l U J φ ; Ω ( Y ) , where c l U J φ ; Ω ( Y ) is the J φ ; Ω -closure on Y in U , denotes the collection of all u ∈ U for which there is a sequence ( u m : m ∈ N ) in Y such that it is J φ ; Ω -converges to u .

Define C l J φ ; Ω ( U ) = { Y ∈ 2 U : Y = c l U J φ ; Ω ( Y ) } . Thus, C l J φ ; Ω ( U ) indicates the set of all J φ ; Ω -closed subsets of U .

Definition 3.16

Let J φ ; Ω = { J β : β ∈ Ω } be an extended J φ ; Ω -family on extended b -gauge space U , and let for all e ∈ U , for all F ∈ 2 U , and for all β ∈ Ω ,

Define on C l J φ ; Ω ( U ) the distance D β J φ ; Ω of Hausdorff type for all β ∈ Ω and for all E , F ∈ C l J φ ; Ω ( U ) , where D β J φ ; Ω : C l J φ ; Ω ( U ) × C l J φ ; Ω ( U ) → [ 0 , ∞ ) , β ∈ Ω is defined as follows:

Lemma 3.17

Let ( U , Q φ ; Ω ) be an extended b-gauge space, and let J φ ; Ω = { J β : β ∈ Ω } , where J β : U × U → [ 0 , ∞ ) , be an extended J φ ; Ω -family on U. Then,

for all u , v ∈ U , for all β ∈ Ω , and for all A ⊂ U , where

Proof

From axioms of definition, we can write for all β ∈ Ω ,

By taking infimum of both sides over A , we obtain for all β ∈ Ω ,