In [10] it is proved that with the classic metric given by.
A resource unit is allocated to the user with the highest QoS metric given by (12).
On the set of natural numbers construct the partial metric given by (1.2).
Obviously, this space with the metric given by (3.2). is a complete metric space.
Note that (({mathcal{B}}({mathcal{Z}}),d)) is a complete metric space, where d is the metric given by (21).
This set with the metric given by d ( u, v ) = max t ∈ I | u ( t ) − v ( t ) |, ∀ u, v ∈ C ( I, R ), is a complete metric space.
We consider the space X = C [0,T],ℝ) of continuous functions defined on [0,T] endowed with the (G-complete) G-metric given by G ( u, v, w ) = max t ∈ [ 0, T ] u ( t ) - v ( t ) + max t ∈ [ 0, T ] u ( t ) - w ( t ) + max t ∈ [ 0, T ] v ( t ) - w ( t ) for all u, v, w ∈ X.
We consider the space X = C ( [ 0, T ], ℜ ) of continuous functions defined on [ 0, T ] endowed with the b-metric given by d ( u, v ) = max t ∈ [ 0, T ] | u ( t ) − v ( t ) | p. for all u, v ∈ X, where s = 2 p − 1 and p ≥ 1.
Let d ( u, v ) = max t ∈ [ a, b ] | u ( t ) − v ( t ) |. Equip X with the G-metric given by G ( u, v, w ) = max { d ( u, v ), d ( v, w ), d ( w, u ) }. for all u, v, w ∈ X. Evidently, ( X, G ) is a complete G-metric space.
where τ : W × D → W, f : W × D → R, K : W × D × R → R. Let B ( W ) denote the space of all bounded real-valued functions on W. Clearly, this space endowed with the G-metric given by. is a G-complete metric space.
Concretely, the extended dual complexity space is formed by the pair ((mathcal {C},e_{mathcal{C}})), where mathcal{C}=Biggl{ finmathcal{TC}: sum_{n=1}^{infty }2^{-n}f(n)< inftyBiggr} and (e_{mathcal{C}}) is the extended quasi-metric given by e_{mathcal{C}}(f,g)= left { textstylebegin{array}l@{quad}l} sum_{n=1}^{infty}2^{-n} (g(n -f(n -f & mbox{if }fpreceq_{mathcal{C}} g, infty& mbox{if otherwise}fpreceq_{mathcal{
