@ Appl. Gen. Topol. 20, no. 1 (2019), 265-279doi:10.4995/agt.2019.10930
c© AGT, UPV, 2019

A quantitative version of the Arzelà-Ascoli

theorem based on the degree of

nondensifiability and applications

G. Garćıa

Departamento de Matemáticas, UNED, 03202 Elche, Alicante (Spain)

(gonzalogarciamacias@gmail.com)

Communicated by D. Werner

Abstract

We present a novel result that, in a certain sense, generalizes the Arzelà-

Ascoli theorem. Our main tool will be the so called degree of nonden-

sifiability, which is not a measure of noncompactness but can be used

as an alternative tool in certain fixed problems where such measures

do not work out. To justify our results, we analyze the existence of

continuous solutions of certain Volterra integral equations defined by

vector valued functions.

2010 MSC: Primary 46B50; Secondary 47H08; 45D05.

Keywords: Arzelà-Ascoli theorem; degree of nondensifiability; α-dense

curves; measures of noncompactness; Volterra integral equa-

tions.

1. Introduction

To set the notation, let I := [0,1], (X,‖ · ‖) a Banach space and C(I,X) the
Banach space of continuous maps x : I −→ X, endowed the norm ‖x‖∞ :=
max{‖x(t)‖ : t ∈ I}, for each x ∈ C(I,X). Also, BX denotes the class of
non-empty and bounded subsets of X, Conv(B) and B̄, as usual, denote the
convex hull and closure of a subset B of X, respectively.

It is well known, that one of the most useful tools to analyse the compactness
of a subset of C(I,X) is the celebrated Arzelà-Ascoli theorem (see, for instance,

Received 07 November 2018 – Accepted 24 January 2019

http://dx.doi.org/10.4995/agt.2019.10930


G. Garćıa

[29]). So, it is not surprising that this result has been extended and generalized
in many ways; see, for instance, [5, 7, 25] and references therein. Our main
goal will be to state a novel generalization of such a result.

In our context, the Arzelà-Ascoli theorem can be stated as follows: a set
B ∈ BC(I,X) is precompact (i.e., its closure is compact) if, and only if, the
following conditions hold:

(1) For each t ∈ I, the set B(t) := {x(t) : x ∈ B} is precompact.
(2) B is equicontinuous, that is, given ε > 0 there is δ > 0 such that ‖x(t) −

x(t′)‖ ≤ ε for every x ∈ B, whenever |t − t′| ≤ δ.
For a better comprehension of the manuscript, we recall the concept of mea-

sure of noncompactness. As the definition of such measures may vary according
to the author (see, for instance, [1, 3]), here we consider the following one given
in [13]:

Definition 1.1. A map µ : BX −→ R+ := [0,∞) is said to be a measure of
noncompactness, MNC, if it satisfies the following properties:

(i) Regularity: µ(B) = 0 if and only if B ∈ BX is a precompact set.
(ii) Invariant under closure: µ(B) = µ(B̄), for all B ∈ BX.
(iii) Monotony: µ(B1 ∪ B2) = max{µ(B1),µ(B2)}, for all B1, B2 ∈ BX.
(iv) Semi-homogeneity: µ(λB) = |λ|µ(B), for all λ ∈ R and S ∈ BX.
(v) Invariance under translations: µ(x + B) = µ(B), for all x ∈ X and

B ∈ BX.

Two widely studied MNCs are these of Hausdorff and Kuratowski (again,
[1, 3]), denoted by χ and κ respectively and defined as

χ(B) := inf
{

ε > 0 : B can be covered by a finite number of balls

of radius ≤ ε
}

and

κ(B) := inf
{

ε > 0 : B can be covered by a finite number of sets

of diameter ≤ ε
}

,

for every B ∈ BX. For instance, if UX is the closed unit ball of X, we have that
χ(UX) = 1, κ(UX) = 2 if X has infinite dimension and χ(UX) = κ(UX) = 0 if
X has finite dimension.

There are some Arzelà-Ascoli results type for vector-valued functions based
on the MNCs (see, for instance, [6, 14, 23] or [28, Theorem 12.5, p. 255]), the
first of them, due to Ambrosetti [2], was proved in 1967:

Theorem 1.2. Let B ∈ BC(I,X) be equicontinuous. Then,

κ(B) = sup
{

κ
(

B(t)
)

: t ∈ I
}

,

where B(t) := {x(t) : x ∈ B}, for each t ∈ I.

Roughly speaking, the above result “quantifies” the compactness of an equicon-
tinuous B ∈ BC(I,X). Of course, if X := R then the above result holds trivially

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A quantitative version of the Arzelà-Ascoli theorem

from the equicontinuity of B and the Arzelà-Ascoli theorem. Therefore, Theo-
rem 1.2 can be useful in infinite dimensional Banach spaces where, unlike the
finite dimensional case, a bounded set is not necessarily precompact.

On the other hand, our main goal is to prove a more general result, not based
on MNCs, than Theorem 1.2 using the so called degree of nondensifiability φd,
explained in detail in Section 2. We prove this result in Theorem 3.2, and as a
consequence we derive some bounds of the MNCs of arc-connected subsets of
BC(I,X).

As it is shown in [10, 11, 12], we can use φd as an alternative to the MNCs in
certain fixed point problems, because in some cases it seems to work out under
more general conditions than the MNCs. So, with a suitable combination of
a Sadovskĭı fixed point theorem type for φd (see Theorem 2.11) and Theorem
3.2, we will prove in Section 4 a result regarding the existence of solutions of
certain Volterra integral equations.

2. The degree of nondensifiability

In 1997 the concepts of α-dense curve and densifiable set were introduced
by Cherruault and Mora [18] in a metric space (Y,d):

Definition 2.1. Let α ≥ 0 and B ∈ BY , the class of non-empty and bounded
subsets of Y . A continuous map γ : I −→ Y is said to be an α-dense curve in
B if the following conditions hold:

(i) γ(I) ⊂ B.
(ii) For any x ∈ B, there is y ∈ γ(I) such that d(x,y) ≤ α.
B is said to be densifiable if for every α > 0 there is an α-dense curve in B.

Let us note that, given B ∈ BY , the class of α-dense curves in B is non-
empty for any α ≥ Diam(B), the diameter of B. Indeed, fixed x0 ∈ B, the
map given by γ(t) := x0, for each t ∈ I, is an α-dense curve in B, for any
α ≥ Diam(B). Also, note that for B := In, n > 1, a 0-dense curve in B is,
precisely, a space-filling curve in B (see [26]). So, the α-dense curves are a
generalization of these curves.

It can be proved that the class of densifiable sets is strictly between the class
of Peano Continua (i.e. those sets that are the continuous image of I) and the
class of connected and precompact sets. For a detailed exposition of the above
concepts, see [8, 17, 18, 19, 20] and references therein.

On the other hand, from the α-dense curves we can define the following (see
[13, 19]):

Definition 2.2. The degree of nondensifiability φd : BY −→ R+ is defined as
φd(B) := inf

{

α ≥ 0 : Γα,B 6= ∅
}

,

for each B ∈ BY , Γα,B being the class of α-dense curves in B.
Note that φd is well defined, as we have pointed out above, Γα,B 6= ∅

for every α ≥ Diam(B) for each B ∈ BY . So, φd(B) ≤ Diam(B) for each
B ∈ BY . For instance, if Y is a Banach space, we have that φd(UY ) = 0 if Y is

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G. Garćıa

finite dimensional (see Proposition 2.4 below) and φd(UY ) = 1 if Y is infinite
dimensional (see [19]), UY being the closed unit ball of Y .

Example 2.3. Let L1 be the Banach space of absolute value Lebesgue inte-
grable functions defined on I, endowed its usual norm, and define the set

D :=
{

f ∈ L1 : f ≥ 0 and
∫ 1

0

f(x)dx = 1
}

.

Then, one can check that φd(D) = 2 (see [13]). So, the inequalities

1 = φd(UL1) = φd(UL1 ∪ D) < max{φd(D),φd(UL1)} = 2,
hold, where UL1 is the closed unit ball of L

1.

By the above example follows that the φd is not a MNC, as the monotony
condition (iii) of Definition 1.1 is not satisfied. However, the degree of nonden-
sifiability shares some properties with the MNCs, as we show in the next result
(see [13]).

Proposition 2.4. In a complete metric space (Y,d), the degree of nondensifi-
ability φd satisfies:

(i) It is regular on the subclass Ba,Y of bounded and arc-connected sets of
the class BY of bounded sets of Y : for a given B ∈ Ba,Y , φd(B) = 0 if,
and only if, B is precompact.

(ii) It is invariant under closure: φd(B) = φd(B), for any B ∈ BY .
(iii) It is semi-additive on disjoint sets of Ba,Y , that is to say

φd(B1 ∪ B2) = max{φd(B1),φd(B2)}.
whenever B1∩ B2 6= ∅.

Furthermore, if Y is a Banach space, then φd also satisfies:

(I) φd
(

Conv(B1)
)

≤ φd(B1), φd
(

Conv(B1 ∪ B2)
)

≤ max{φd(B1),φd(B2)},
for every B1,B2 ∈ BY .

(II) It is semi-homogeneous, that is, φd(λB) = |λ|φd(B), for λ ∈ R and
B ∈ BY .

(III) It is invariant under translations, that is, φd(y + B) = φd(B), for any
y ∈ Y and B ∈ BY .

The degree of nondensifiability φd and the Hausdorff MNC χ are related by
the following result (see [13]):

Proposition 2.5. For each arc-connected B ∈ BX we have
χ(B) ≤ φd(B) ≤ 2χ(B),

and these inequalities are the best possible if X is infinite dimensional.

For the Kuratowski MNC κ, we have the following result (again, [13]):

Proposition 2.6. For each arc-connected B ∈ BX we have
1

2
κ(B) ≤ φd(B) ≤ κ(B).

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A quantitative version of the Arzelà-Ascoli theorem

Another interesting property of the degree of nondensifiability φd is that it
is an “universal” upper bound for any MNC. Formally (see [13]):

Proposition 2.7. µ : BX −→ R+ a MNC. Then,
µ(B) ≤ µ(UX)φd(B),

for each B ∈ BX, UX being the closed unit ball of X.
Next, we recall some well known definitions regarding with the MNCs (see,

for instance, [1, 3]):

Definition 2.8. Let T : C ⊆ X −→ X continuous and µ a MNC. Given k ≥ 0,
we say that T is k-µ-contractive if

µ
(

T(B)
)

≤ kµ(B),
for each B ∈ BX ∩ C. If for every non-precompact B ∈ BX ∩ C, µ

(

T(B)
)

<
µ(B) we say that T is µ-condensing.

Now, we can state the following fixed point result (again, [1, 3]):

Theorem 2.9 (Sadovskĭı fixed point theorem). Let C ∈ BX be closed and
convex, and T : C −→ C µ-condensing, for certain MNC µ invariant under
the convex hull. Then, T has some fixed point.

Before to state a Sadovskĭı fixed point theorem for the degree of nondensi-
fiability φd, is convenient to formalize the concepts of Definition 2.8 for φd.

Definition 2.10. Let T : C ⊆ X −→ X continuous and k ≥ 0. We will say
that T is k-φd-contractive if

φd
(

T(B)
)

≤ kφd(B),
for each convex B ∈ BX ∩ C. If for every convex and non-precompact B ∈
BX ∩ C, φd

(

T(B)
)

< φd(B), then T is said to be φd-condensing.

Now, we have the following result (we skip the proof because it follows in
the same way than the given in [3], noticing Proposition 2.4, or in [10, Theorem
3.2]):

Theorem 2.11. Let C ∈ BX be closed and convex, and T : C −→ C φd-
condensing. Then, T has some fixed point.

As we have pointed out above, the degree of nondensifiability φd and the
MNCs (and, in particular, the Hausdorff and Kuratowski MNCs) are essentially
different. Then, as expected, Definitions 2.8 and 2.10 are essentially different.
This fact is evidenced in the following examples.

Example 2.12. Let ℓ2 be the Banach space of the real sequences of summable
square, endowed with the usual norm ‖ · ‖. Fixed 2−1/2 < β < 1, let (ξn)n≥1
be a dense sequence in βUℓ2, Uℓ2 being the closed unit ball of ℓ2, and define
T : ℓ2 −→ ℓ2 as

T(x) :=
∑

n≥1

max
{

0,1 − ‖x − en‖
1 − β

}

ξn, ∀x := (xn)n≥1 ∈ ℓ2,

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G. Garćıa

where {en : n ≥ 1} is the standard basis of ℓ2. Clearly, T is continuous and
T(Uℓ2) ⊂ βUℓ2.

Then, in [9, Remark 3.10] it is proved that
(2.1)

κ({en : n ≥ 1}) =
√
2 > 2β = κ(T({en : n ≥ 1})) χ(T(B)) ≤ β ≤ χ(B),

for every non-precompact B ∈ Bℓ2. That is, T is 1-χ-contractive but not
1-κ-contractive.

Now, let B be a non-empty, convex and non-precompact subset of Uℓ2. By
Proposition 2.4, we can assume without loss of generality that θ ∈ B, θ being
the null vector of ℓ2. Then, from the definition of T , T(θ) = θ and ‖T(x)‖ ≤ β
for every x ∈ Uℓ2. Consequently, the identically null map γ : I −→ ℓ2 is a
β-dense curve in T(Uℓ2). Therefore,

(2.2) φd(T(B)) ≤ β.

Finally, in view of (2.1), (2.2) and Proposition 2.5 we conclude that

φd(T(B)) ≤ β ≤ χ(B) ≤ φd(B),

that is, T is a 1-φd-contractive map.

Example 2.13. Let the closed and convex set C := {x ∈ C(I,R) : 0 = x(0) ≤
x(t) ≤ 1 = x(1), t ∈ I} and T : C −→ C given by

T(x)(t) :=















1

2
x(2t), for 0 ≤ t ≤ 1

2

1

2
x(2t − 1) + 1

2
, for 1

2
< t ≤ 1

for each x ∈ C. Then, T is continuous (in fact, affine) and T(C) ⊂ C. One
can check (see [3, Example 2, p. 169]) that T is 1

2
-κ-contractive and 1-χ-

contractive, because of χ(T(C)) = χ(C) = 1/2, whereas in [11] we show that
T is 1

2
-φd-contractive.

To end this section, note that the above examples show that φd and the
MNCs χ and κ are essentially different. Moreover, if we consider the “inner”
Hausdorff MNC (see, for instnace, [1]) defined as

χi(B) := inf
{

ε > 0 : B can be covered by a finite number of balls

centered at points of B and radius ≤ ε
}

,

for each B ∈ BX, in general, χi 6= φd. Indeed, for the set

B :=
{

(x,sin(1/x)) : x ∈ [−1,0] ∪ (0,1]
}

∪
{

[0,y] : y ∈ [−1,1]} ⊂ R2,

we find χi(B) = 0 < 1 = φd(B). Another example that shows that φd and χi
are essentially different can be found in [12, Example 3.4]. Despite its name,
χi is not a MNC (see, for instance, [1, p. 9]).

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A quantitative version of the Arzelà-Ascoli theorem

3. The main result

Before to prove our main result, we need to recall the following definition
(see, for instance, [28, p. 253]):

Definition 3.1. Given B ∈ BC(I,X), the uniform modulus of equicontinuity
of B is defined as

ω(B) := inf
δ>0

sup
|t−s|<δ

sup
x∈B

‖x(t) − x(s)‖.

Is clear that ω(B) is well defined, as ω(B) < ∞ for every B ∈ BC(I,X). Also,
let us note that B is equicontinuous if, and only if, ω(B) = 0.

Now, we are ready to prove our main result:

Theorem 3.2. Let C ∈ BC(I,X) be arc-connected. Then,

(3.1) max
{

Φd(C),
1

2
ω(C)

}

≤ φd(C) ≤ Φd(C) + 2ω(C).

where C(t) := {x(t) : x ∈ C} and Φd(C) := sup{φd
(

C(t)
)

: t ∈ I}, for each
t ∈ I. If C(t) is precompact for every t ∈ I, then

(3.2)
1

2
ω(C) ≤ φd(C) ≤ ω(C).

Proof. The inequality

(3.3) Φd(C) ≤ φd(C).
has been proved in [10, Lemma 3.2]. Let α := φd(C), ε > 0 and γ an (α + ε)-
dense curve in C. Then, given x ∈ C there is y ∈ γ(I) such that
(3.4) ‖x − y‖∞ ≤ α + ε.

Therefore, fixed δ > 0, in view of (3.4), for each t,s ∈ I with |t − s| < δ we
have

‖x(t) − x(s)‖ ≤ ‖x(t) − y(t)‖ + ‖y(t) − y(s)‖ + ‖y(s) − x(s)‖

≤ 2(α + ε) + ‖y(t) − y(s)‖,
and, as δ > 0 can be arbitrarily small, we infer

ω(C) ≤ 2(α + ε) + ω
(

γ(I)
)

,

but, as ω
(

γ(I)
)

= 0 (by the Arzelà-Ascoli Theorem, γ(I) is equicontinuous),
letting ε → 0,
(3.5) ω(C) ≤ 2φd(C).

Then, from (3.3) and (3.5), we conclude that

(3.6) max
{

Φd(C),
1

2
ω(C)

}

≤ φd(C).

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G. Garćıa

On the other hand, fixed ε > 0 there is some δ > 0 such that for each t,s ∈ I
with |t − s| < δ we have ‖x(t) − x(s)‖ < ω(C) + ε for each x ∈ C. By the
compactness of I there is {t1, . . . , tn} ⊂ I such that I ⊂ ∪ni=1B(ti,δ) (the open
balls centered at ti and radius δ). So,

(3.7) C(t) ⊂
n
⋃

i=1

C(ti) + (ω(C) + ε)UX,

UX being the closed unit ball of X. Given α > Φd(C), let γi be an α-
dense curve in C(ti). Then, as each γi(I) is compact there is a finite set

{y(i)1 (ti), . . . ,y
(i)
ri (ti)} ⊂ γi(I), with y

(i)
ri ∈ C, for each i = 1, . . . ,n, such that

(3.8) C(ti) ⊂
n
⋃

i=1

γi(I) + αUX ⊂
n
⋃

i=1

{y(i)1 (ti), . . . ,y(i)ri (ti)} + (α + ε)UX,

and joining (3.7) and (3.8), we find

(3.9) C(t) ⊂
n
⋃

i=1

{y(i)1 (ti), . . . ,y(i)ri (ti)} + (ω(C) + α + 2ε)UX.

for each t ∈ I. Next, we claim:

(3.10) C ⊂
n
⋃

i=1

{y(i)1 , . . . ,y(i)ri } + (2ω(C) + α + 3ε)UC(I,X),

where UC(I,X) is the closed unit ball of C(I,X). In other case, there is x0 ∈ C
such that

(3.11) ‖x0 − y(i)ri ‖∞ > 2ω(C) + α + 3ε,
for each i = 1, . . . ,n. Take, for each i = 1, . . . ,n, t̃i ∈ I such that ‖x0(t̃i) −
y
(i)
ri (t̃i)‖ = ‖x0 − y

(i)
ri ‖∞, and let ti ∈ I be such that t̃i ∈ B(ti,δ). Then, in

view of (3.9)

‖x0 − y(i)ri ‖∞ = ‖x0(t̃i) − y
(i)
ri (t̃i)‖ ≤ ‖x0(t̃i) − y

(i)
ri (ti)‖

+‖y(i)ri (ti) − y
(i)
ri (t̃i)‖ ≤ ω(C) + α + 2ε + ω(C) + ε = 2ω(C) + α + 3ε,

which, is contradictory with (3.11). Therefore, (3.10) holds as claimed. Now,

if γ is a continuous map joining the functions y
(i)
1 , . . . ,y

(i)
ri for i = 1, . . . ,n

(this is possible because C is arc-connected), from (3.10), we find that γ is an
(2ω(C)+α+3ε)-dense curve in C. Letting ε → 0 and α → Φd(C), we conclude
that

(3.12) φd(C) ≤ 2ω(C) + Φd(C),
and consequently, the inequality (3.1) follows from (3.6) and (3.12).

To prove the inequality (3.2) of the statement, if C(t) is precompact for
every t ∈ I, then φd(C(t)) = 0 by (i) of Proposition 2.4. So, from (3.1),
φd(C) ≥ ω(C)/2. By [23, Theorem 1], κ(C) ≤ ω(C) and therefore, taking

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A quantitative version of the Arzelà-Ascoli theorem

into account Proposition 2.6, the inequality (3.2) holds. The proof is now
complete. �

Some comments are necessary before continuing.

(a) Replacing the Banach space X by an arbitrary complete metric space, the
above result remains true.

(b) The above result has been proved in [23, Theorem 1] for the Kuratowski
MNC κ. However, as we have pointed out in Section 2 (and, in particular,
in Example 2.12), the degree of nondensifiability is essentially different
from κ.

(c) As Theorem 1.2, our result “quantifies” the compactness of the bounded
and arc-connected subsets of BC(I,X). Indeed, if an arc-connected C ∈
BC(I,X) is precompact if, and only if, is equicontinuous (that is, ω(C) = 0)
and for each t ∈ I, C(t) is precompact (that is, φd(C(t)) = 0 by virtue of
Proposition 2.4).

(d) If C is equicontinuous then Theorem 1.2 remains true for the degree of
nondensifiability φd, as in such case ω(C) = 0.

In the following examples we show that the estimates given in the inequality
(3.2) of Theorem 3.2 are the best possible.

Example 3.3 (see [23, Example 2]). Define the sequence of functions xn :
I −→ R as

xn(t) :=























0, for 0 ≤ t ≤ 1/2 − 1/n

(t − 1/2 + 1/n)n/2, for 1/2 − 1/n ≤ t ≤ 1/2 + 1/n

1, for 1/2 + 1/n ≤ t ≤ 1
for every integer n ≥ 2. Then for B := {xn : n ≥ 2} we find κ(B) = 1/2 and
ω(B) = 1.

Putting C := Conv(B), by Proposition 2.6, we have φd(C) ≤ κ(C) = κ(B) =
1/2. Then, noticing (3.2) of Theorem 3.2, we have

φd(C) =
1

2
=

1

2
ω(C).

Example 3.4. Consider the following (closed and convex) set

C :=
{

x ∈ C(I,R) : 0 = x(0) ≤ x(t) ≤ x(1) = 1, for each t ∈ I
}

.

Then, ω(C) = 1 (see also the above example). We will prove in the following
lines that φd(C) = 1.

As Diam(C) = 1 we have φd(C) ≤ 1. If φd(C) < 1, we can take an α-dense
curve in C, put γ, with 0 < α < 1. Noticing that γ(I) is a compact subset of
C, by the Arzelà-Ascoli theorem, it is equicontinuous. So, for 0 < 3ε < 1 − α
there is δ1 > 0 such that

(3.13) |y(t) − y(t′)| ≤ ε for all y ∈ γ(I),

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G. Garćıa

whenever |t − t′| ≤ δ1. Taking into account that y(1) = 1 for each y ∈ γ(I),
again by the equicontinuity of the set γ(I), there is δ2 > 0 such that

(3.14) y(t′) ≥ 1 − ε for all y ∈ γ(I),
whenever 1 − t′ ≤ δ2. Therefore, taking 0 < t′ < t < 1 satisfying 1 − t′ ≤
min{δ1,δ2} and an integer n ≥ 1 such that f(t) := tn ≤ ε, as γ is an α-dense
curve in C we can pick y ∈ γ(I) with
(3.15) ‖f − y‖∞ ≤ α.

Finally, from (3.13), (3.14) and (3.15) we infer

α ≥ ‖f − y‖∞ ≥ |f(t) − y(t′)| − |y(t′) − y(t)| ≥ 1 − 2ε − |y(t′) − y(t)| ≥ 1 − 3ε,
which is contradictory with the choice of α. So, we conclude that φd(C) = 1.

On the other hand, we can derive from Theorem 3.2 some bounds for the
MNCs of arc-connected subsets of BC(I,X). In view of Proposition 2.7, our first
corollary is the following:

Corollary 3.5. Let µ : BC(I,X) −→ R+ be a MNC, and C ∈ BC(I,X) arc-
connected. Then, we have

µ(C) ≤ µ(UC(I,X))
[

Φd(C) + 2ω(C)
]

,

UC(I,X) being the closed unit ball of C(I,X), and Φd(C) defined in Theorem
3.2. In particular, if C(t) is precompact for every t ∈ I, then

µ(C) ≤ µ(UC(I,X))ω(C).

Example 3.6. Let X be an infinite dimensional Banach space, B ∈ BC(I,X)
convex and k : I2 −→ R, f : I × X −→ X continuous. Assume that there is
M > 0 with ‖f(s,x(s))‖ ≤ M for every s ∈ I and x ∈ B. Consider the set

C :=
{

∫ t

0

k(t,s)f(s,x(s))ds : x ∈ C
}

,

meaning the above integral in the Bochner sense (see, for instance, [27]). From
the convexity of B and the continuity of f, it follows that C is arc-connected.
Also, given 0 ≤ t′ < t ≤ 1 we have

∥

∥

∥

∫ t

0

k(t,s)f(s,x(s))ds −
∫ t′

0

k(t′,s)f(s,x(s))ds
∥

∥

∥
≤

∥

∥

∥

∫ t

0

(k(t,s) − k(t′,s))f(s,x(s))ds
∥

∥

∥
+
∥

∥

∥

∫ t

t′
k(t′,s)f(s,x(s))ds

∥

∥

∥
≤

M

∫ t

0

|k(t,s) − k(t′,s)|ds + MMk(t − t′),

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A quantitative version of the Arzelà-Ascoli theorem

and therefore, the set C is equicontinuous, Mk denoting the maximum of |k|
over I2. Then, from Corollary 3.5, we find the inequalities

χ(C) ≤ Φd(C) and κ(C) ≤ 2Φd(C),
with Φd(C) as in Theorem 3.2.

Recalling that two MNCs µ and ν are said to be comparable (see, for in-
stance, [3, Definition 1.1, p. 168]) if there are a,b > 0 such that

aν(B) ≤ µ(C) ≤ bν(B),
for every B ∈ BX, the next result follows directly from Proposition 2.5 and
Theorem 3.2:

Corollary 3.7. Assume µ : BC(I,X) −→ R+ is a comparable MNC with the
Hausdorff MNC χ, that is to say, there are a,b > 0 such that

aχ(B) ≤ µ(B) ≤ bχ(B),
for every B ∈ BC(I,X). Let C ∈ BC(I,X) be arc-connected. Then, we have

a

2
max

{

Φd(C),
1

2
ω(C)

}

≤ µ(C) ≤ b
[

Φd(C) + 2ω(C)
]

,

with Φd(C) as in Theorem 3.2. In particular, if C(t) is precompact for every
t ∈ I, then

a

4
ω(C) ≤ µ(C) ≤ bω(C).

Roughly speaking, the above corollaries provide us bounds for µ(C) (of an
arc-connected C ∈ BC(I,X)) from the degree of nondensifiability of C(t) or
from the uniform modulus of equicontinuity of C, depending if C(t) is, or not,
precompact for each t ∈ I.

4. Application to Volterra integral equations

For X := Rn one of the most important example of compact operators in
C(I,X) (i.e. continuous maps that transform bounded sets into precompact
sets) are those defined by integral operators with sufficient regular kernels (see,
for intance, [16]). However, as it is shown in [15] by several examples, if X
is an infinite dimensional Banach space, the situation is very different. It is
due, essentially, to the fact that in infinite dimensional Banach spaces bounded
subsets are not necessarily precompact. So, the MNCs are a useful tool to
analyze the existence of solutions of differential and integral equations posed
in infinite dimensional Banach spaces, see, for instance, [4, 21, 22, 24] and
references therein. In this section, we will use the results exposed in previous
sections to prove the existence of solutions of certain Volterra integral equation,
which can not be solved by the MNCs techniques.

Specifically, we consider the following Volterra integral equation

(4.1) x(t) = x0(t) +

∫ t

0

k(t,s)f(s,x(s))ds,

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G. Garćıa

where x0 ∈ C(I,X), k : I2 −→ R and f : I × X −→ X are known, and the
integral stands in the Bochner sense (see, for instance, [27]).

When X is a sequence Banach space (such as c0, ℓp, etc.), the above integral
equation is equivalent to certain infinite systems of second-order differential
equations (see [21] and references therein).

Let the following conditions:

(C1) The functions k : I2 −→ R and f : I × X −→ X are continuous.
(C2) There is R > 0 such that

‖x0(t)‖ + ‖
∫ t

0
k(t,s)f(s,x(s))ds‖
R

≤ 1 for all t ∈ I,

whenever ‖x‖∞ ≤ R.
(C3) For each B ∈ BC(I,X), there is c : I −→ R+, bounded and Lebesgue

integrable, and ψ : R+ −→ R+ continuous and increasing with ψ(r) < r
for every r > 0 and ψ(0) = 0, such that

φd
({

f(s,x(s)) : x ∈ B
})

≤ c(s)ψ
(

φd
({

x(s) : x ∈ B
}))

for all s ∈ I.
(C4) The functions c and k obey

∫ t

0

|k(t,s)|c(s)ds ≤ 1, for all t ∈ I.

Regarding the above conditions, we can do the following observations.

• Unlike many of the results based in MNCs for integral equations of
type (4.1), we do not require the uniform continuity of the family of
functions (f(s,x(s))s∈I (see the above cited references).

• Similar conditions to (C1)-(C4) are assumed in [12] to prove the exis-
tence of solutions of certain integral equations. In fact, condition (C3)
is required in [12] by taking c(s) equal to a positive constant.

• From Example 2.13 and [12, Examples 3.2 and 3.4], not detailed here
for lack of space, we can define some maps f(t,x) such that condition
(C3) is satisfied by the φd but not by the Hausdorff MNC χ. This fact
will be also evidenced in Example 4.2.

• In particular, conditions (C3) and (C4) are assumed to prove that the
operator which defined the integral equation (4.1) is φd-condensing and
then, we can apply Theorem 2.11. Let us note that, as the concepts of
φd-condensing and µ-condensing are essentially different (see Example
2.13), Sadovskĭı fixed point theorem and Theorem 2.11 too.

Now, we can state and prove the main result of this section.

Theorem 4.1. Assume conditions (C1)-(C4). Then, equation (4.1) has some
solution x∗ ∈ C(I,X).
Proof. Clearly, the existence of solutions of (4.1) is equivalent to the existence
of fixed points of the map

T(x)(t) := x0(t) +

∫ t

0

k(t,s)f(s,x(s))ds for all x ∈ C(I,X),

c© AGT, UPV, 2019 Appl. Gen. Topol. 20, no. 1 276



A quantitative version of the Arzelà-Ascoli theorem

which, from condition (C1), satisfies T(x) ∈ C(I,X) for every x ∈ C(I,X).
Also, a routine check shows us that T is continuous. We will apply Theorem
2.11 to prove the existence of fixed points of T .

From condition (C2), there is R > 0 such that T(C) ⊂ C, C := RUC(I,X).
Also, from condition (C3) and Theorem 3.2, for every s ∈ I the degree of
nondensifiablity of the set {f(s,x(s)) : x ∈ C} remains bounded by sup{c(t) :
t ∈ I}ψ(φd(C)) + 2ω(C) and, therefore, the set {f(s,x(s)) : x ∈ C} must be
bounded. So, in view of Example 3.6, we find that T(C) is equicontinuous.

Next, let B ⊂ C be non-empty, non-precompact and convex. By Theorem
3.2 and the above considerations, we have

(4.2) φd
(

T(B)
)

≤ Φd
(

T(B)
)

+ 2ω
(

T(B)
)

= sup{φd
(

T(B)(t) : t ∈ I
}

,

meaning T(B)(t) := {T(x)(t) : x ∈ B}.
On the other hand, by condition (C3), for each s ∈ I, if αs := φd(B(s)) > 0

then for an arbitrarily small ε > 0, there is an (ψ(αs) + ε)-dense curve in
{f(s,x(s)) : x ∈ B}, put τ ∈ I 7−→ γτ(s). So, given x ∈ B there is τ ∈ I such
that

(4.3) ‖f(s,x(s)) − γτ (s)‖ ≤ c(s)ψ(αs),

for each s ∈ I. Now, consider the map

τ ∈ I 7−→ γ̃τ (t) := x0(t) +
∫ t

0

k(t,s)γτ(s)ds, for all t ∈ I,

which is continuous and γ(I) ⊂ T(B).
Then, noticing (4.3) and condition (C4), for every t ∈ I, we have

‖T(x)(t) − γ̃τ (t)‖ ≤
∫ t

0

|k(t,s)|‖f(s,x(s)) − γτ(s)‖ds ≤

∫ t

0

|k(t,s)|c(s)(ψ(αs) + ε)ds ≤ sup{ψ(αs) : s ∈ I} + ε,

and letting ε → 0

‖T(x)(t) − γ̃τ (t)‖ ≤ sup{ψ(αs) : t ∈ I} = sup{ψ(αt) : t ∈ I},

or, in other words, γ̃ is a sup{ψ
(

φd(B(t))
)

: t ∈ I}-sense curve in T(B)(t).
Then, noticing (4.2), the properties of ψ and Theorem 3.2

φd
(

T(B)
)

≤ sup
{

ψ
(

φd(B(t))
)

: t ∈ I
}

≤ ψ
(

sup{φd(B(t)) : t ∈ I}
)

<

sup
{

φd(B(t)) : t ∈ I
}

≤ max
{

sup{φd(B(t)) : t ∈ I}, 12ω(B)
}

≤ φd(B).

So, T is a φd-condensing map and the proof is complete. �

We conclude our exposition with an example.

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G. Garćıa

Example 4.2. Consider the following integral equation in the Banach space
c0 of the null sequences:

(4.4) xn(t) =
4n

3n + 1

∫ t

0

3ts3 arctan
(

|xn(s)|
)

ds. for all n ≥ 1.

In this case, x0 ≡ 0, f(s,x) := ( 4n3n+1 arctan(|xn|))n≥1 and k(t,s) := 3ts
3.

So, condition (C1) is trivially satisfied and (C2) taking, for instance, R := π.
From the elementary properties of the arctan function, it is immediate to

check that, if γ is an α-dense curve in B, then f(s,γ(s)) is an 4
3
arctan(α)-dense

curve in the set {f(s,x(s)) : s ∈ B}, for every B ∈ BC(I,c0). So, condition (C3)
is fulfilled for c(s) := 4

3
and ψ(r) := arctan(r). Also,

4

3

∫ t

0

3ts3ds = t5 ≤ 1 for all t ∈ I,

and therefore condition (C4) holds. Then, by Theorem 4.1, the above integral
equation has some continuous solution.

On the other hand, from the following formula (see, for instance, [1, p. 5])

χ(B) = lim
n

sup
x∈B

{

sup{|xi| : i ≥ n}
}

,

for every B ∈ BC(I,c0), we deduce that

χ({en : n ≥ 1}) = 1 <
π

3
= χ({f(s,en) : n ≥ 1}),

en ∈ c0 being the vector with 1 in the n-th position and zeros otherwise. In
other words, condition (C3) is not satisfied for the Hausdorff MNC χ and the
above functions c(s) and ψ(r). Moreover, Sadovskĭı fixed point theorem (see
Theorem 2.11) can not be used in this example for χ as, by the above inequality,
T is not χ-condensing.

Acknowledgements. To the anonymous referee, for his/her useful comments
and suggestions to improve the quality of the paper. Also, to my beloved Loli,
for her carefully revision of the English grammar to improve the presentation.

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