Characterizations of quasi-metric and G-metric completeness involving w-distances and fixed points

Example 1

Let q : R × R → R + defined by q ( x , y ) = max { x − y , 0 } . Then, q is a non- T 1 quasi-metric on X , and the topology T q is the so-called lower topology on R . Note also that q s is the usual metric on R .

Proposition 2

Let ( X , q ) be a quasi-metric space. Then, a function F : X × X → R + is a Q-function on ( X , q ) if and only if it is a w-distance on ( X , q ) .

Proof

It was noted in [42, page 128] that every w-distance on ( X , q ) is a Q-function on ( X , q ) .

Now suppose that F is a Q-function on ( X , q ) , which is not a w-distance on ( X , q ) . Then, there is x ∈ X for which the function F ( x , ⋅ ) : X → R + is not T q ∗ -lsc. Therefore, there exists a sequence ( y n ) n ∈ N in X that T q ∗ -converges to some y ∈ X , and an ε > 0 and a subsequence ( y k n ) n ∈ N of ( y n ) n ∈ N such that F ( x , y ) ≥ ε + F ( x , y k n ) , for all n ∈ N . Put M = F ( x , y ) − ε / 2 . Then, M > 0 and F ( x , y k n ) < M for all n ∈ N . Since ( y k n ) n ∈ N is T q ∗ -convergent to y and F is a Q -function, we deduce that F ( x , y ) ≤ M , a contradiction. Hence, F is a w-distance on ( X , q ) .□

Remark 3

Note that although the authors of [42] worked in the realm of T 1 quasi-metric spaces, Proposition 2 remains valid for every quasi-metric space.

Example 4

Let q be the quasi-metric on R given by q ( x , x ) = 0 for all x ∈ R , and q ( x , y ) = ∣ y ∣ otherwise. Since q ( x , 0 ) = 0 for all x ∈ R , we deduce that ( R , q ) is a non- T 1 quasi-metric space. Now, let W : R × R → R + defined by W ( x , y ) = ∣ y ∣ for all x , y ∈ R . Then, W is a Q -function on ( R , q ) [42, Example 1(a)], so it is a w-distance on ( R , q ) by Proposition 2. We shall show this fact directly for the sake of completeness. To this end, it suffices to verify condition (w2). Indeed, fix x ∈ R and let ( y n ) n ∈ N be a non-eventually constant sequence in R that T q ∗ -converges to some y ∈ R . Then, q ( y n , y ) → 0 as n → ∞ , and q ( y n , y ) = ∣ y ∣ eventually, so y = 0 . Hence, W ( x , y ) = 0 , and thus W ( x , ⋅ ) is T q ∗ -lsc. We conclude that W is a w-distance on ( R , q ) .

Example 5

Let q be the quasi-metric on R given by q ( x , y ) = x − y if y ≤ x , and q ( x , y ) = 2 ( y − x ) otherwise. Clearly, ( R , q ) is a T 1 quasi-metric space. Now, let W : R × R → R + be defined by W ( x , y ) = ∣ x − y ∣ for all x , y ∈ R . Then, W is a Q -function on ( R , q ) [42, Example 1(b)], so it is a w-distance on ( R , q ) by Proposition 2. We shall show directly this fact for the sake of completeness. To this end, it suffices to verify condition (w2). Indeed, fix x ∈ R and let ( y n ) n ∈ N be a non-eventually constant sequence in R that T q ∗ -converges to some y ∈ R . Then, q ( y n , y ) → 0 as n → ∞ , so, by the definition of q , ∣ y n − y ∣ → 0 as n → ∞ . Since W ( x , y ) ≤ W ( x , y n ) + ∣ y n − y ∣ , we deduce that W ( x , ⋅ ) is T q ∗ -lsc, so W is a w-distance on ( R , q ) .

Example 6

On the set PF of partial functions, we define a relation ⊑ PF as follows:

It is clear that ⊑ PF  is a partial order on PF (i.e., a reflexive, antisymmetric, and transitive relation).

Now, let q PF be the non- T 1 quasi-metric on PF given by q PF ( f , g ) = 0 if f ⊑ PF g , and q PF ( f , g ) = 1 otherwise. It is well known, and easily checked, that the topology induced by q PF agrees with the famous Alexandroff topology on PF, that is, any topology where the intersection of an arbitrary family of open sets is open. Note that ( q PF ) s is the discrete metric on PF, i.e., ( q PF ) s ( f , g ) = 1 whenever f ≠ g , and hence the quasi-metric space ( PF , q PF ) is ( q PF ) ∗ -half complete because every Cauchy sequence in ( PF , q PF ) is eventually constant.

Let f 0 be the element of PF such that ℓ ( f 0 ) = 1 and f 0 ( 1 ) = 1 .

Define a function W PF : PF × PF → R + as follows:

W PF ( f 0 , f 0 ) = 0 and W PF ( f , g ) = ℓ ( g ) otherwise.

We are going to show that W PF is a w -distance on ( PF , q PF ) , i.e., that it satisfies conditions (w1), (w2) and (w3). Indeed,

For (w1), let f , g , h ∈ PF . Since W PF ( f , g ) = W PF ( h , g ) , we immediately obtain that W PF ( f , g ) ≤ W PF ( f , h ) + W PF ( h , g ) .

For (w2), fix f ∈ PF and let ( g j ) j ∈ N be a sequence in PF that T ( q PF ) ∗ -converges to a g ∈ PF . Then, q PF ( g j , g ) → 0 as j → ∞ , so there is j 0 ∈ N such that q ( g j , g ) = 0 for all j ≥ j 0 . This implies that g j ⊑ PF g , and hence, ℓ ( g j ) = ℓ ( g ) for all j ≥ j 0 . If f = g = f 0 , we have W PF ( f , g ) = 0 . Otherwise, we obtain W PF ( f , g ) = W PF ( f , g j ) for all j ≥ j 0 . Consequently, W PF ( f , ⋅ ) is T ( q PF ) ∗ -lsc.

For (w3), fix ε > 0 . Put δ = min { 1 / 2 , ε } . Let f , g , h ∈ P F such that W PF ( f , g ) ≤ δ and W PF ( f , h ) ≤ δ . Then, f = g = h = f 0 , so q PF ( f , g ) = 0 ≤ ε .

Theorem 7

For a quasi-metric space ( X , q ) , the following statements are equivalent.

1. ( X , q ) is q ∗ -right complete.

2. If T is a self-mapping T of X such that there is a T q ∗ -nearly lsc function ϕ : X → R + fulfilling, for every x ∈ X ,

   q ( x , T x ) ≤ ϕ ( x ) − ϕ ( T x ) ,

   then, there exists u ∈ X satisfying ϕ ( u ) = ϕ ( T u ) .

Remark 8

We recall that, according to [26], given a quasi-metric space ( X , q ) , a function f : X → R is T q -nearly lsc provided that whenever ( x n ) n ∈ N is a sequence of distinct points in X that T q -converges to some x ∈ X , we have f ( x ) ≤ liminf n → ∞ f ( x n ) . Furthermore, the notions of T q -nearly lsc and T q -lsc coincide whenever ( X , q ) is a T 1 quasi-metric space.

Example 9

Let X be the set of all ordinals less than the first uncountable ordinal number ω 1 . Consider the non- T 1 quasi-metric q on X given by q ( x , y ) = 0 if x ≤ y , and q ( x , y ) = 1 otherwise. It is clear that ( X , q ) is q ∗ -right complete because every left Cauchy sequence ( x n ) n ∈ N is T q ∗ -convergent to x ≔ sup { x n : n ∈ N } . Now consider the self-mapping of X given by T x = x + 1 for all x ∈ X . Then, T has no fixed points but one has q ( x , T x ) = 0 = ϕ ( x ) − ϕ ( T x ) , for all x ∈ X , where ϕ ( x ) = 0 for all x ∈ X .

Definition 10

Let ( X , q ) be a quasi-metric space. A self-mapping T of X is called a W -Caristi mapping (on ( X , q ) ) if there exist a w -distance W on ( X , q ) and a T q ∗ -lsc function ϕ : X → R + such that

for all x ∈ X .

Lemma 11

Let X be a (non-empty) set, ℱ : X ↦ 2 X a multivalued mapping, and ϕ a function from X to R + . Then, for each x ∈ X , there is a sequence ( x n ) n ∈ N 0 in X such that x 0 = x , x n + 1 ∈ ℱ x n , and

for all n ∈ N 0 .

If, in addition, there is a function W : X × X → R + satisfying the triangle inequality and verifying

for all n ∈ N 0 , then, for each δ > 0 , there is n δ ∈ N 0 such that

whenever m > n ≥ n δ .

Proof

Let x ∈ X . Put x 0 = x . Since ℱ x 0 ≠ ∅ , there exists x 1 ∈ ℱ x 0 such that ϕ ( x 1 ) < 1 + inf ϕ ( ℱ x 0 ) .

Analogously, there exists x 2 ∈ ℱ x 1 such that ϕ ( x 2 ) < 2 − 1 inf ϕ ( ℱ x 1 ) .

Thus, we inductively deduce the existence of a sequence ( x n ) n ∈ N 0 in X such that x n + 1 ∈ ℱ x n and ϕ ( x n + 1 ) < 2 − n + inf ϕ ( ℱ x n ) , for all n ∈ N 0 .

Now, suppose that there is a function W : X × X → R + satisfying the triangle inequality and verifying W ( x n , x n + 1 ) ≤ ϕ ( x n ) − ϕ ( x n + 1 ) , for all n ∈ N 0 .

Then, ( ϕ ( x n ) ) n ∈ N 0 is a non-increasing sequence in R + , and hence, it is a Cauchy sequence in R + when endowed with the usual metric. Consequently, given δ > 0 , there is n δ ∈ N 0 such that ϕ ( x n ) − ϕ ( x m ) < δ , for all n , m ≥ n δ .

Since W satisfies the triangle inequality, we deduce that

whenever m > n ≥ n δ .□

Theorem 12

Every W-Caristi mapping on a q ∗ -half complete quasi-metric space ( X , q ) has a fixed point.

Proof

Let T be a W -Caristi mapping on a q ∗ -half complete quasi-metric space ( X , q ) . Then, there exist a w -distance W on ( X , q ) and a T q ∗ -lsc function ϕ : X → R + such that

for all x ∈ X .

Define a multivalued mapping ℱ : X ↦ 2 X by

for all x ∈ X .

Note that ℱ is well-defined because T x ∈ ℱ x , and thus ℱ x ∈ 2 X for all x ∈ X .

Fix now an x ∈ X . By the first part of Lemma 11, there is a sequence ( x n ) n ∈ N 0 in X such that x 0 = x , x n + 1 ∈ ℱ x n and ϕ ( x n + 1 ) < 2 − n + inf ϕ ( ℱ x n ) , for all n ∈ N 0 .

Since x n + 1 ∈ ℱ x n , it follows from the definition of the multivalued mapping ℱ that W ( x n , x n + 1 ) ≤ ϕ ( x n ) − ϕ ( x n + 1 ) , for all n ∈ N 0 .

Therefore, by the second part of Lemma 11, we obtain that, for each δ > 0 , there is n δ ∈ N 0 such that W ( x n , x m ) < δ , whenever m > n ≥ n δ .

Choose an arbitrary ε > 0 . Let δ ≔ δ ( ε ) for which condition (w3) is fulfilled. Then, for every j , k > n δ , we have W ( x n δ , x j ) < δ and W ( x n δ , x k ) < δ , so q ( x j , x k ) ≤ ε and q ( x k , x j ) ≤ ε .

This implies that ( x n ) n ∈ N 0 is a Cauchy sequence in the metric space ( X , q s ) . Since ( X , q ) is q ∗ -sequentially complete, there exists u ∈ X such that q ( x n , u ) → 0 as n → ∞ .

Next, we shall prove that w ( u , T u ) = 0 . To this end, we shall show four claims.

Claim 1. W ( x n , u ) → 0 as n → ∞ .

Indeed, given δ > 0 , there is n δ ∈ N 0 such that W ( x n , x m ) < δ , whenever m > n ≥ n δ . Fix n ≥ n δ . By condition (w2), there is m > n such that W ( x n , u ) < δ + W ( x n , x m ) . Hence,

for all n ≥ n δ .

Claim 2. u ∈ ⋂ n ∈ N 0 ℱ x n .

Indeed, fix n ∈ N 0 . Choose an arbitrary δ > 0 . By Claim 1 and the fact that ϕ is T q ∗ -lsc, we deduce the existence of an m > n such that W ( x m , u ) < δ and ϕ ( u ) − ϕ ( x m ) < δ .

Taking into account that x m ∈ ℱ x n , we obtain

Since δ is arbitrary, we conclude that W ( x n , u ) ≤ ϕ ( x n ) − ϕ ( u ) , so u ∈ ℱ x n .

Claim 3. ϕ ( u ) = inf n ∈ N 0 ϕ ( x n ) .

Indeed, by Claim 2, u ∈ ℱ x n for all n ∈ N 0 . So, by the definition of ℱ , ϕ ( u ) ≤ ϕ ( x n ) for all n ∈ N 0 . Thus, ϕ ( u ) ≤ inf n ∈ N 0 ϕ ( x n ) .

On the other hand, we have

for all n ∈ N 0 , and hence inf n ∈ N 0 ϕ ( x n ) ≤ ϕ ( u ) .

Claim 4. ϕ ( T u ) = inf n ∈ N 0 ϕ ( x n ) .

Indeed, since T is W -Caristi mapping, we have T u ∈ ℱ u , and, by Claim 2, we also have that u ∈ ℱ x n for all n ∈ N 0 . Therefore,

for all n ∈ N 0 , so that T u ∈ ⋂ n ∈ N 0 ℱ x n . As in the proof of Claim 2, we obtain that ϕ ( T u ) ≤ inf n ∈ N 0 ϕ ( x n ) , and also

for all n ∈ N 0 , so inf n ∈ N 0 ϕ ( x n ) ≤ ϕ ( T u ) .

From Claims 3 and 4, we have that ϕ ( u ) = ϕ ( T u ) , and, consequently, w ( u , T u ) = 0 .

Finally, we prove that u = T u . Indeed, by Claim 1, the triangle inequality, and the fact that w ( u , T u ) = 0 , we deduce that W ( x n , T u ) → 0 as n → ∞ .

Now, choose an arbitrary ε > 0 . Let δ > 0 for which condition (w3) is fulfilled. Take n ∈ N 0 such that W ( x n , u ) ≤ δ and W ( x n , T u ) ≤ δ . From condition (w3), it follows that q s ( u , T u ) ≤ ε . Hence, u = T u , and thus, u is a fixed point of T . Furthermore, W ( u , u ) = 0 .□

Example 13

Let ( PF , q PF ) be the quasi-metric space of Example 6, and let T be the self-mapping of PF defined as follows:

For each f ∈ P F with ℓ ( f ) ≥ 2 , T f is the only element of PF such that ℓ ( T f ) = ℓ ( f ) − 1 and T f ( n ) = f ( n ) for all n ∈ { 1 , … , ℓ ( T f ) } , and for each f ∈ P F with ℓ ( f ) = 1 , put T f = f 0 .

We are going to prove that T is a W PF -Caristi mapping on ( PF , q PF ) , where W PF is the w -distance constructed in Example 6.

Indeed, define a function ϕ : P F → R + by ϕ ( f ) = ( ℓ ( f ) ) 2 for all f ∈ P F \ { f 0 } , and ϕ ( f 0 ) = 0 .

Clearly, ϕ is T ( q PF ) ∗ -lsc. Indeed, let ( g j ) j ∈ N be a sequence in PF that T ( q PF ) ∗ -converges to a g ∈ PF \ { f 0 } . Then, q PF ( g j , g ) → 0 as j → ∞ , so there is j 0 ∈ N such that q ( g j , g ) = 0 for all j ≥ j 0 . This implies that g j ⊑ PF g , and hence ℓ ( g j ) = ℓ ( g ) for all j ≥ j 0 , so ϕ ( g ) = ϕ ( g j ) , for all j ≥ j 0 .

Since W PF ( f 0 , T f 0 ) = W PF ( f 0 , f 0 ) = 0 , we obtain W PF ( f 0 , T f 0 ) = ϕ ( f 0 ) − ϕ ( T f 0 ) .

Now let f ∈ P F \ { f 0 } . If ℓ ( f ) ≥ 2 , we obtain

and if ℓ ( f ) = 1 , we obtain

We have shown that T is a w PF -Caristi mapping on the ( q PF ) ∗ -half complete quasi-metric space ( PF , q PF ) . Hence, we can apply Theorem 12 to conclude that T has a fixed point. In fact f 0 is the unique fixed point of T .

Finally, we shall show that ( PF , q PF ) is not ( q PF ) ∗ -right complete, and thus we cannot apply Theorem 7.

Indeed, consider the sequence ( g j ) j ∈ N in PF such that ℓ ( g j ) = 1 and g j ( 1 ) = j for all j ∈ N . Since q PF ( g i , g j ) = 0 whenever i ≤ j , we deduce that ( g j ) j ∈ N is a ( q PF ) ∗ -right Cauchy sequence in ( PF , q PF ) . Suppose that there is g ∈ PF such that q PF ( g j , g ) → 0 as j → ∞ . Then, there is j 0 ∈ N such that q PF ( g j , g ) = 0 for all j ≥ j 0 . Hence, ℓ ( g j ) = ℓ ( g ) = 1 and g j ( 1 ) ≤ g ( 1 ) for all j ≥ j 0 , so j ≤ g ( 1 ) for all j ≥ j 0 , a contradiction. Consequently, ( PF , q PF ) is not ( q PF ) ∗ -right complete.