We first show that the metrics introduced the Lemma 1.3 and the Theorem 1.5 are equivalent. Then, we provide some examples involving the metric defined in Lemma 1.3.

**Theorem 2.1**
*For every cone metric D* : *X × X* → *E there exists a metric*
*which is equivalent to D on X*.

**Proof**. Define

*d*(

*x, y*) = inf {||

*u*||:

*D*(

*x, y*) ≼

*u*}. By the Lemma 1.3

*d* is a metric. We shall now show that each sequence {

*x*
_{
n
}} ⊆

*X* which converges to a point

*x* ∈

*X* in the (

*X, d*) metric also converges to

*x* in the (

*X, D*) metric, and conversely. We have

Put *v*
_{
n
} := *u*
_{
nn
} then
and *D*(*x*
_{
n
}
*, x*) ≼ *v*
_{
n
}. Now if *x*
_{
n
} → *x* in (*X, d*) then *d*(*x*
_{
n
}
*, x*) → 0 and so *v*
_{
n
} → 0 too, therefore for all *c* ≻≻ 0 there exists
such that *v*
_{
n
} ≺≺ *c* for all *n ≥ N*. This implies that *D*(*x*
_{
n
}
*, x*) ≺≺ *c* for all *n ≥ N*. Namely *x*
_{
n
} → *x* in (*X, D*).

Conversely, for every real *ε >* 0, choose *c* ∈ *E* with *c* ≻≻ 0 and ||*c*|| *< ε*. Then there exists
such that *D*(*x*
_{
n
}
*, x*) ≺≺ *c* for all *n ≥ N*. This means that for all *ε >* 0 there exists
such that *d*(*x*
_{
n
}
*, x*) *≤* ||*c*|| *< ε* for all *n ≥ N*. Therefore *d*(*x*
_{
n
}
*, x*) → 0 as *n* → *∞* so *x*
_{
n
} → *x* in (*X, d*).

□

**Theorem 2.2**
*If d*
_{1}(*x, y*) = inf {||*u*||: *D*(*x, y*) ≼ *u*} *and d*
_{2}(*x, y*) = *ξ*
_{
e
}(*D*(*x, y*)) *where D is a cone metric on X. Then d*
_{1}
*is equivalent with d*
_{2}.

**Proof**. Let

then

so by Theorem 2.1 in

so

and or *εe − D*(*x*
_{
n
}
*, x*) ∈ *K°* for all *n ≥ N*. So *D*(*x*
_{
n
}
*, x*) ∈ *e* - *K°* for *n ≥ N*. Now by [[1], Lemma 1.1 (iv)] *ξ*
_{
e
}(*D*(*x*
_{
n
}
*, x*)) *< ε* for all *n ≥ N*. Namely *d*
_{2}(*x*
_{
n
}
*, x*) *< ε* for all *n ≥ N* therefore
or
.

Conversely,

hence

so

, therefore

So *D*(*x*
_{
n
}
*, x*) ∈ *εe−K°* for *n ≥ N* by [[1], Lemma 1.1 (iv)]. Hence, *D*(*x*
_{
n
}
*, x*) = *εe−k* for some *k* ∈ *K°*, so *D*(*x*
_{
n
}
*, x*) ≺≺ *εe* for *n ≥ N* this implies that
and again by Theorem 2.1
. □

In the following examples, we use the metric of Lemma 1.3.

**Example 2.3**
*Let*
*with* ||

*a*|| = 1

*and for every*
*define*
*Then D is a cone metric on*
*and its equivalent metric d is*
*which is a discrete metric*.

**Example 2.4**
*Let a, b ≥* 0

*and consider the cone metric*
*with*
*where d*
_{1},

*d*
_{2}
*are metrics on*
.

*Then its equivalent metric is*
*In particular if d*
_{1}(

*x, y*):=

*|x − y| and d*
_{2}(

*x, y*):=

*α|x − y|, where α ≥* 0

*then D is the same famous cone metric which has been introduced in [*[

2]

*, Example 1] and its equivalent metric is*
**Example 2.5**
*For q >* 0,

*b >* 1,

*E* =

*l*
^{
q
}
*, P* = {{

*x*
_{
n
}}

_{
n
}
_{≥1} :

*x*
_{
n
}
*≥* 0,

*for all n*}

*and* (

*X, ρ*)

*a metric space, define D* :

*X × X* →

*E which is the same cone metric as [*[

4]

*, Example 1.3] by*
*Then its equivalent metric on × is*