Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 65, pp. 1–27.

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

CONVERGENCE OF DELAY EQUATIONS DRIVEN BY A

HÖLDER CONTINUOUS FUNCTION OF ORDER 1/3 < β < 1/2

MIREIA BESALÚ, GIULIA BINOTTO, CARLES ROVIRA

Abstract. In this article we show that, when the delay approaches zero, the
solution of multidimensional delay differential equations driven by a Hölder

continuous function of order 1/3 < β < 1/2 converges with the supremum

norm to the solution for the equation without delay. Finally we discuss the
applications to stochastic differential equations

1. Introduction

Hu and Nualart [9] used fractional calculus to establish the existence and unique-
ness of a solution for the dynamical system

dxt = f(xt) dyt,

where y is a Hölder continuous function of order 1/3 < β < 1/2. They give an

explicit expression for the integral
∫ t

0
f(xs)dys that depends on the functions x,

y and a quadratic multiplicative functional x ⊗ y. As an example of a path-wise
approach to classical stochastic calculus, they apply these results to solve stochastic
differential equations driven by a multidimensional Brownian motion. Using the
same approach, Besalú and Nualart [2] obtained estimates for the supremum norm
of the solution and Besalú et al. [1] studied delay equations with non-negativity
constraints.

The work by Hu and Nualart [9] is an extension of the previous paper of Nualart
and Răşcanu [16], where they study the dynamical systems dxt = f(xt)dyt and
the control function y is Hölder continuous of order β > 1/2. In this case the

Riemann-Stieltjes integral
∫ t

0
f(xs)dys can be expressed as a Lebesgue integral using

fractional derivatives following the ideas by Zähle [19].
All these papers have to be seen in the framework of the theory of rough path

analysis and the path-wise approach to classical stochastic calculus. This theory
has been developed from the initial paper by Lyons [12] and has generated many
publications (see, for instance, Lyons and Qian [13], Friz and Victoir [5], Lejay [11]
or Gubinelli [8]). We refer to Coutin and Lejay [3], Friz and Victoir [6], Friz [7]
and Ledoux et al. [14] for some applications of rough path analysis to stochastic
calculus.

2010 Mathematics Subject Classification. 60H05, 60H07.
Key words and phrases. Delay equation; stochastic differential equation; convergence;

fractional integral.
c©2020 Texas State University.

Submitted October 30, 2018. Published June 26, 2020.

1



2 M. BESALÚ, G. BINOTTO, C. ROVIRA EJDE-2020/65

Delay differential equations rise from the need to study models that behave more
like real processes. They find their applications in dynamical systems with afteref-
fects or when the dynamics are subjected to propagation delay. Some examples are
epidemiological models with incubation periods that postpone the transmission of
disease, or neuronal models where the spatial distribution of neurons can cause a
delay in the transmission of the impulse. Sometimes the delay avoids some usual
problems, but in general, it adds difficulties and cumbersome notations.

The purpose of our paper is to consider the differential equation with delay

xrt = η0 +

∫ t

0

b(u, xr) du+

∫ t

0

σ(xru−r) dyu, t ∈ (0, T ],

xrt = ηt, t ∈ [−r, 0],

where r denotes a strictly positive time delay, η : [−r, 0]→ Rd is a smooth function,
y is a Hölder continuous function of order β ∈ ( 1

3 ,
1
2 ) and the hereditary term b(u, x)

depends on the path {xs, 0 ≤ s ≤ u}. From Hu and Nualart [9] and Besalú et al.
[1] it is easy to check that there exists a unique solution to this equation. Our aim
is to prove that it converges almost surely in the supremum norm to the solution
of the differential equation without delay

xt = η0 +

∫ t

0

b(u, xu) du+

∫ t

0

σ(xu) dyu, t ∈ [0, T ],

when the delay tends to zero. Our approach is based on the techniques of the
classical fractional calculus and it is inspired by [9]. Finally, we apply these results
to stochastic differential equations driven by Brownian motion.

The case when β > 1/2 has been studied by Ferrante and Rovira in [4]. They
proved that the solution to the delay equation converges, almost surely and in Lp,
to the solution to the equation without delay and then apply the result pathwise
to fractional Brownian motion with Hurst parameter H > 1/2.

With a different approach based on a slight variation of the Young integration
theory, called algebraic integration, León and Tindel [10] prove the existence of a
unique solution for a general class of delay differential equations driven by a Hölder
continuous function with parameter greater than 1/2. They obtain some estimates
of the solution which allow to show that the solution to a delay differential equation
driven by a fractional Brownian motion with Hurst parameter H > 1/2 has a C∞-
density.

When β < 1/2 more difficulties appear. In the literature we find results only up
to the value β > 1/3, eventually extended to β > 1/4. In [15], Neuenkirch, Nourdin
and Tindel consider delay differential equation driven by a β-Hölder continuous
function with β > 1/3. The authors show the existence of a unique solution for
these equations under suitable hypothesis. Then, they apply these results to a delay
differential equation driven by a fractional Brownian motion with Hurst parameter
H > 1/3. These results are extended by Tindel and Torrecilla in [18] to the
deterministic case with β > 1/4 and the corresponding stochastic case with Hurst
parameter H > 1/4.

This article is organized as follows. The following section is devoted to introduce
some notation. In section 3 we define the equations and the solutions we work
with and we describe our main result. Section 4 contains technical estimates for
the study of the integrals. In section 5 we give some estimates for the solutions
of our equations. In section 6 we give the proof of the main theorem. In the last



EJDE-2020/65 CONVERGENCE OF DELAY EQUATIONS 3

section we give an example of applications of the main theorem, studying stochastic
differential equations driven by Brownian motion.

2. Preliminaries

First, we recall some definitions and results presented in Hu and Nualart [9]. Fix
a time interval [0, T ] and 0 < β ≤ 1. For any function x : [0, T ]→ Rd, the β-Hölder
norm of x on the interval [s, t] ⊂ [0, T ] will be denoted by

‖x‖β(s,t) = sup
s≤u<v≤t

|xv − xu|
(v − u)β

.

If ∆T := {(s, t) : 0 ≤ s < t ≤ T}, for any (s, t) ∈ ∆T and for any g : ∆T → Rd we
set

‖g‖β(s,t) = sup
s≤u<v≤t

|gu,v|
(v − u)β

.

We will also set ‖x‖β = ‖x‖β(0,T ) and ‖x‖β(r) = ‖x‖β(−r,T ). Moreover, ‖ · ‖∞(s,t)

will denote the supremum norm in the interval [s, t], and for simplicity ‖x‖∞ =
‖x‖∞(0,T ) and ‖x‖∞(r) = ‖x‖∞(−r,T ).

Hu and Nualart [9] proved an explicit formula for the integral
∫ b
a
f(xu) dyu in

terms of x, y and x ⊗ y, and transformed the dynamical system dxt = f(xt) dyt
into a closed system of equations involving only x, x ⊗ y and x ⊗ (y ⊗ y). From
Lyons [12] we introduce the definition of x⊗ y.

Definition 2.1. We say that (x, y, x⊗ y) is a (d,m)-dimensional β-Hölder contin-
uous multiplicative functional if:

(1) x : [0, T ]→ Rd and y : [0, T ]→ Rm are β-Hölder continuous functions,
(2) x ⊗ y : ∆T → Rd ⊗ Rm is a continuous function satisfying the following

properties:
(a) (Multiplicative property) For all s ≤ u ≤ t we have

(x⊗ y)s,u + (x⊗ y)u,t + (xu − xs)⊗ (yt − yu) = (x⊗ y)s,t.

(b) For all (s, t) ∈ ∆T , |(x⊗ y)s,t| ≤ c|t− s|2β .

We denote by Mβ
d,m(0, T ) the space of (d,m)-dimensional β-Hölder continuous

multiplicative functionals. Furthermore, we will denote by Mβ
d,m(a, b) the obvious

extension of the definition Mβ
d,m(0, T ) to a general interval [a, b]. We introduce the

following functional defined on Mβ
d,m(0, T ) for (a, b) ∈ ∆T :

Φβ(a,b)(x, y) = ‖x⊗ y‖2β(a,b) + ‖x‖β(a,b)‖y‖β(a,b). (2.1)

Moreover, if (x, y, x⊗ y) and (y, z, y ⊗ z) belongs to Mβ
d,m(0, T ) we define

Φβ(a,b)(x, y, z) = ‖x‖β(a,b)‖y‖β(a,b)‖z‖β(a,b) + ‖z‖β(a,b)‖x⊗ y‖2β(a,b)

+ ‖x‖β(a,b)‖y ⊗ z‖2β(a,b).
(2.2)

From these definitions it follows that

‖(x⊗ y)·,b‖β(a,b) ≤ Φβ(a,b)(x, y)(b− a)β , (2.3)∥∥x⊗ (y ⊗ z)·,b
∥∥

2β(a,b)
≤ KΦβ(a,b)(x, y, z)(b− a)β (2.4)

which are equations (3.29) and (3.30) in [9]. We refer to [9] and [12] for a more
detailed presentation on β-Hölder continuous multiplicative functionals.



4 M. BESALÚ, G. BINOTTO, C. ROVIRA EJDE-2020/65

To define the integral
∫ b
a
f(xu) dyu we use the construction of the integral given

by Hu and Nualart in [9], who were inspired by the work of Zälhe [19] and use
fractional derivatives. We refer to Hu and Nualart in [9] for the details.

In the sequel, K denotes a generic constant that may depend on the parameters
β, α, λ and T and can vary from line to line.

3. Main result

Consider the differential equation with delay on Rd,

xrt = η0 +

∫ t

0

b(u, xru) du+

∫ t

0

σ(xru−r) dyu, t ∈ [0, T ],

xrt = ηt, t ∈ [−r, 0),

(3.1)

where x and y are Hölder continuous functions of order β ∈ ( 1
3 ,

1
2 ), η is a contin-

uous function and r denotes a strictly positive time delay. We set the following
hypotheses:

(H1) σ : Rd → Rd × Rm is a bounded and continuously twice differentiable
function such that σ′ and σ′′ are bounded and λ-Hölder continuous for
λ > 1

β − 2.

(H2) b : [0, T ] × Rd → Rd is a measurable function such that there exists b0 ∈
Lρ(0, T ;Rd) with ρ ≥ 2 and ∀N ≥ 0 there exists LN > 0 such that:
(1) |b(t, xt)− b(t, yt)| ≤ LN |xt − yt|, ∀x, y such that |xt| ≤ N , |yt| ≤ N
∀t ∈ [0, T ],

(2) |b(t, xt)| ≤ L0|xt|+ b0(t), ∀t ∈ [0, T ].
(H3) σ and b are bounded functions.

Conditions (H1) and (H2) are a particular case of the hypotheses for the proof
of existence and uniqueness of solution to the delay equation (3.1), while condition
(H3) is necessary to prove that the solution is bounded.

We denote by (x, y, x⊗y) ∈Mβ
d,m(0, T ) the solution to the stochastic differential

equation on Rd without delay

xt = η0 +

∫ t

0

b(u, xu) du+

∫ t

0

σ(xu) dyu, t ∈ [0, T ]. (3.2)

Assuming that σ : Rd → Rd × Rm is a continuously differentiable function such
that σ′ is λ-Hölder continuous with λ > 1

β − 2, σ and σ′ are bounded and (y, y, y⊗
y) ∈Mβ

m,m(0, T ), Hu and Nualart [9] prove the existence of a bounded solution in

Mβ
d,m(0, T ) for the differential equation (3.2) with b ≡ 0. Moreover, if σ is twice

continuously differentiable with bounded derivatives and σ′′ is λ-Hölder continuous,
with λ > 1

β − 2, the solution is unique. Here the authors deal with the equation

without the hereditary term, but the results can be easily extended to the case
where the hereditary term does not vanish. If (y, y, y ⊗ y) ∈ Mβ

m,m(0, T ), then we
can consider

(x⊗ y)s,t =

∫ t

s

(yt − yu)b(u, xu) du+

∫ t

s

σ(xu) du(y ⊗ y)·,t . (3.3)

And (x, y, x ⊗ y) ∈ Mβ
d,m(0, T ) will be a solution to (3.2) for x and (x ⊗ y) such

that (3.2) and (3.3) hold, respectively.



EJDE-2020/65 CONVERGENCE OF DELAY EQUATIONS 5

On the other hand, following the ideas contained in [1], it is easy to show

that there exists a unique solution to the delay equation (3.1) in Mβ
d,m(−r, T ).

It can be easily proved assuming that σ and b satisfy the hypotheses (H1) and

(H2), respectively, with ρ ≥ 1
1−β and that (η·−r, y, η·−r ⊗ y) ∈ Mβ

d,m(0, r) and

(y·−r, y, y·−r ⊗ y) ∈ Mβ
m,m(r, T ). If we also assume that hypothesis (H3) is satis-

fied, we can obtain that the solution is bounded. So, (xr, y, xr ⊗ y) ∈Mβ
d,m(−r, T )

is the unique solution to (3.1) for xr such that (3.1) holds and (xr⊗ y)s,t is defined
as follows:

• for s < t ∈ [−r, 0),

(xr ⊗ y)s,t = (η ⊗ y)s,t =

∫ t

s

(yt − yu) dηu,

• for s ∈ [−r, 0) and t ∈ [0, T ],

(xr ⊗ y)s,t = (η ⊗ y)s,0 +

∫ t

0

(yt − yu)b(u, xru) du

+

∫ t

0

σ(xru−r) du(y ⊗ y)·,t + (η0 − ηs)⊗ (yt − y0),

• for 0 ≤ s < t ≤ T ,

(xr ⊗ y)s,t =

∫ t

s

(yt − yu)b(u, xru) du+

∫ t

s

σ(xru−r) du(y ⊗ y)·,t.

Let β ∈ ( 1
3 ,

1
2 ) and set β′ = β − ε, where ε > 0 is such that β − 2ε > 0 and

λ > 1
β−ε − 2. Set r0 ∈ (0, T ). The main result of this article is the following

theorem.

Theorem 3.1. Suppose that (y, y, y⊗y) belongs to Mβ
m,m(0, T ) and (y·−r, y, y·−r⊗

y) belongs to Mβ
m,m(r, T ) for all 0 < r ≤ r0. Assume that σ and b satisfy (H1)

and (H2), respectively, and both satisfy (H3). Assume also that (η·−r0 , y, η·−r0 ⊗
y) ∈ Mβ

d,m(0, r0), ‖η‖β(−r0,0) < ∞ and supr≤r0 Φβ(0,r)(η·−r, y) < ∞. Suppose that

‖(y−y·−r)⊗y‖2β′(r,T ) → 0 and ‖y·−r⊗(y−y·−r)‖2β′(r,T ) → 0 when r tends to zero.

Then, (x, y, x⊗ y) ∈Mβ
d,m(0, T ) the solution to the stochastic differential equation

without delay (3.2) and (xr, y, xr⊗y) ∈Mβ
d,m(−r, T ) the solutions to the stochastic

differential equations with delay (3.1) satisfy that

lim
r→0
‖x− xr‖∞ = 0 and lim

r→0
‖(x⊗ y)− (xr ⊗ y)‖∞ = 0.

4. Estimates of integrals

In this section we will give some estimates for the integrals appearing in our
equations. We begin recalling versions of [9, Propositions 3.4 and 3.9 ].

Proposition 4.1. Let (x, y, x⊗y) be in Mβ
d,m(0, T ). Assume that f : Rd → Rm is a

continuous differentiable function such that f ′ is bounded and λ-Hölder continuous,
where λ > 1

β − 2. Then, for any 0 ≤ a < b ≤ T , we have∣∣ ∫ b

a

f(xu) dyu
∣∣ ≤ K|f(xa)| ‖y‖β(a,b)(b− a)β +K Φβ(a,b)(x, y)

×
(
‖f ′‖∞ + ‖f ′‖λ‖x‖λβ(a,b)(b− a)λβ

)
(b− a)2β ,



6 M. BESALÚ, G. BINOTTO, C. ROVIRA EJDE-2020/65

where Φβ(a,b)(x, y) is defined by (2.1).

Proposition 4.2. Suppose that (x, y, x⊗ y) and (y, z, y⊗ z) belong to Mβ
d,m(0, T ).

Let f : Rd → Rm be a continuously differentiable function such that f ′ is λ-Hölder
continuous and bounded, where λ > 1

β − 2. Then, the following estimate holds∣∣ ∫ b

a

f(xu) du(y ⊗ z)·,b
∣∣

≤ K|f(xa)|Φβ(a,b)(y, z)(b− a)2β

+K
(
‖f ′‖∞ + ‖f ′‖λ‖x‖λβ(a,b)(b− a)λβ

)
Φβ(a,b)(x, y, z)(b− a)3β ,

where Φβ(a,b)(x, y, z) is defined in (2.2).

The following propositions give some useful estimates for proving Theorem 3.1.
First, we give an estimate for a function b that fulfills conditions (H2).

Proposition 4.3. Assume that b satisfies (H2). Let x, x̃ ∈ C(0, T ;Rd) such that
‖x‖∞ ≤ N and ‖x̃‖∞ ≤ N . Then, for 0 ≤ a < b ≤ T ,∣∣ ∫ b

a

[
b(u, xu)− b(u, x̃u)

]
du
∣∣ ≤ LN (b− a)‖x− x̃‖∞(a,b).

The proof of the above proposition follows easily using the Lipschitz property of
hypothesis (H2). To give some results for a function f under conditions (H1) we
need to introduce some notation. Let

G1
β(a,b)(f, x, x̃, y) = K

[
‖y‖β ‖f ′‖∞ +

(
‖f ′′‖∞ + ‖f ′′‖λ(‖x‖λβ(a,b)

+ ‖x̃‖λβ(a,b))(b− a)λβ
)(

Φβ(a,b)(x, y) + ‖y‖β ‖x̃‖β(a,b)

)]
,

G2
β(a,b)(f, x, x̃, y) = K

[
‖y‖β ‖f ′‖∞ + ‖f ′′‖∞

(
Φβ(a,b)(x, y)

+ ‖y‖β ‖x̃‖β(a,b)

)
(b− a)β

]
,

G3
β(a,b)(f, x̃) = K

[
‖f ′‖∞ + ‖f ′′‖∞‖x̃‖β(a,b)(b− a)β

]
.

The first result corresponds to Hu and Nualart [9, Proposition 6.4].

Proposition 4.4. Suppose that (x, y, x⊗ y) and (x̃, y, x̃⊗ y) belong to Mβ
d,m(0, T ).

Assume that f satisfies (H1). Then, for 0 ≤ a < b ≤ T ,∣∣ ∫ b

a

[f(xu)− f(x̃u)] dyu
∣∣ ≤ G1

β(a,b)(f, x, x̃, y)(b− a)2β‖x− x̃‖∞(a,b)

+G2
β(a,b)(f, x, x̃, y)(b− a)2β‖x− x̃‖β(a,b)

+G3
β(a,b)(f, x̃)(b− a)2β‖(x− x̃)⊗ y‖2β(a,b).

From this we can deduce the following estimate.

Proposition 4.5. Assume (x, y, x⊗y) and (x·−r, y, x·−r⊗y) belong to Mβ
d,m(0, T ),

and f satisfies (H1). Then, for 0 ≤ a < b ≤ T ,∣∣ ∫ b

a

[f(xu)− f(xu−r)] dyu
∣∣ ≤ G1

β(a,b)(f, x, x·−r, y)(b− a)2β‖x− x·−r‖∞(a,b)



EJDE-2020/65 CONVERGENCE OF DELAY EQUATIONS 7

+G2
β(a,b)(f, x, x·−r, y)(b− a)2β‖x− x·−r‖β(a,b)

+G3
β(a,b)(f, x·−r)(b− a)2β‖(x− x·−r)⊗ y‖2β(a,b).

The above proposition is a particular case of Proposition 4.4 with x̃ ≡ x·−r. Let
us introduce more useful notation:

G4
β(a,b)(f, x, x̃, y, z) = K

[
‖f ′‖∞Φβ(a,b)(y, z)

+
(
‖f ′′‖∞ + ‖f ′′‖λ(‖x‖λβ(a,b) + ‖x̃‖λβ(a,b))(b− a)λβ

)
×
(
Φβ(a,b)(x, y, z) + ‖x̃‖β(a,b)Φβ(a,b)(y, z)

)]
,

G5
β(a,b)(f, x, x̃, y, z) = K

[(
‖f ′‖∞ + ‖f ′′‖∞‖x̃‖β(a,b)(b− a)β

)
Φβ(a,b)(y, z)

+ ‖f ′′‖∞Φβ(a,b)(x, y, z)(b− a)β
]
,

G6
β(a,b)(f, x̃, z) = KG3

β(a,b)(f, x̃)‖z‖β(a,b).

From the previous results it is possible to prove the following two propositions.

Proposition 4.6. Suppose that (x, y, x ⊗ y), (x̃, y, x̃ ⊗ y) and (y, z, y ⊗ z) belong

to Mβ
d,m(0, T ). Assume that f satisfies (H1). Then, for 0 ≤ a < b ≤ T ,

∣∣ ∫ b

a

[f(xu)− f(x̃u)] du(y ⊗ z)·,b
∣∣ ≤ G4

β(a,b)(f, x, x̃, y, z)(b− a)3β‖x− x̃‖∞(a,b)

+G5
β(a,b)(f, x, x̃, y, z)(b− a)3β‖x− x̃‖β(a,b)

+G6
β(a,b)(f, x̃, z)(b− a)3β‖(x− x̃)⊗ y‖2β(a,b).

Proof. To simplify the proof we will assume d = m = 1. Observe that from in-
equalities (2.3) and (2.4) we obtain

Φβ(a,b)(x, y ⊗ z) ≤ KΦβ(a,b)(x, y, z)(b− a)β , (4.1)∥∥(x− x̃)⊗ (y ⊗ z)·,b
∥∥

2β(a,b)
≤ KΦβ(a,b)(y, z)(b− a)β‖x− x̃‖β(a,b)

+K‖z‖β(a,b)(b− a)β‖(x− x̃)⊗ y‖2β(a,b).
(4.2)

The proof of the proposition is obtained by applying Proposition 4.4 and using
inequalities (2.3), (2.4), (4.1) and (4.2). �

Proposition 4.7. Suppose that (x, y, x ⊗ y), (x·−r, y, x·−r ⊗ y) and (y, z, y ⊗ z)
belong to Mβ

d,m(0, T ). Assume that f satisfies (H1). Then, for 0 ≤ a < b ≤ T ,

∣∣ ∫ b

a

[f(xu)− f(xu−r)] du(y ⊗ z)·,b
∣∣

≤ G4
β(a,b)(f, x, x·−r, y, z)(b− a)3β‖x− x·−r‖∞(a,b)

+G5
β(a,b)(f, x, x·−r, y, z)(b− a)3β‖x− x·−r‖β(a,b)

+G6
β(a,b)(f, x·−r, z)(b− a)3β‖(x− x·−r)⊗ y‖2β(a,b).

(4.3)

The above proposition is a particular case of Proposition 4.6 with x̃ ≡ x·−r. We
conclude this section with a general result on β-Hölder functions.



8 M. BESALÚ, G. BINOTTO, C. ROVIRA EJDE-2020/65

Lemma 4.8. Let y : [0, T ] → Rm be a β-Hölder continuous function and β′ =
β − ε > 0 with ε > 0, then

‖y − y·−r‖∞(r,T ) ≤ ‖y‖βrβ , (4.4)

‖y − y·−r‖β′(r,T ) ≤ 2‖y‖βrε. (4.5)

Proof. On the one hand,

‖y − y·−r‖∞(r,T ) = sup
t∈[r,T ]

|yt − yt−r|
rβ

· rβ ≤ ‖y‖βrβ .

On the other hand,

sup
s<t∈[r,T ]
t−s≤r

|(y − y·−r)t − (y − y·−r)s|
(t− s)β′

≤ sup
s<t∈[r,T ]
t−s≤r

|yt − ys|
(t− s)β

· (t− s)β

(t− s)β′
+ sup
s<t∈[r,T ]
t−s≤r

|yt−r − ys−r|
(t− s)β

· (t− s)β

(t− s)β′

≤ 2‖y‖βrε

(4.6)

and

sup
s<t∈[r,T ]
t−s≥r

∣∣(y − y·−r)t − (y − y·−r)s
∣∣

(t− s)β′

≤ sup
s<t∈[r,T ]
t−s≥r

|yt − yt−r|
rβ

· rβ

(t− s)β′
+ sup
s<t∈[r,T ]
t−s≥r

|ys − ys−r|
rβ

· rβ

(t− s)β′

≤ 2‖y‖βrε.

(4.7)

The proof finishes by putting together (4.6) and (4.7). �

5. Estimates of the solutions

In this section we obtain some estimates on the solutions to our equations. Let
us recall that ε > 0 with β − 2ε > 0, λ > 1

β−ε − 2 and β′ = β − ε. First of all, let

us introduce x̂rt = xrt−r where xr is the solution to (3.1). Then (x̂r ⊗ y)s,t can be
expressed as follows:

• for s < t ∈ [0, r),

(x̂r ⊗ y)s,t = (η·−r ⊗ y)s,t =

∫ t

s

(yt − yu) dηu−r, (5.1)

• for s ∈ [0, r) and t ∈ [r, T ],

(x̂r ⊗ y)s,t = (η·−r ⊗ y)s,r +

∫ t

r

(yt − yu)b(u− r, x̂ru) du

+

∫ t

r

σ(x̂ru−r) du(y·−r ⊗ y)·,t + (η0 − ηs−r)⊗ (yt − yr),
(5.2)

• for s < t ∈ [r, T ],

(x̂r ⊗ y)s,t =

∫ t

s

(yt − yu)b(u− r, x̂ru) du+

∫ t

s

σ(x̂ru−r) du(y·−r ⊗ y)·,t.



EJDE-2020/65 CONVERGENCE OF DELAY EQUATIONS 9

We will prove that the norms ‖x̂r‖β′ and ‖x̂r⊗y‖2β′ are bounded and their upper
bound does not depend on r. To this aim, the following lemma will be useful.

Lemma 5.1. Let (η·−r, y, η·−r⊗y) ∈Mβ
d,m(0, r) and (y·−r, y, y·−r⊗y) ∈Mβ

d,d(r, T ).

Let (xr, y, xr ⊗ y) ∈Mβ
d,m(0, T ) be the solution to (3.1). Then

‖x̂r‖β′ ≤ ‖x̂r‖β′(0,r) + ‖x̂r‖β′(r,T ), (5.3)

‖x̂r ⊗ y‖2β′ ≤ ‖x̂r ⊗ y‖2β′(0,r) + ‖x̂r ⊗ y‖2β′(r,T ) + ‖η‖β′(−r,0)‖y‖β′ . (5.4)

Proof. On the one hand, observe that

‖x̂r‖β′ ≤ max
(

sup
0≤s<t<r

|x̂rt − x̂rs|
(t− s)β′

, sup
0≤s<r≤t≤T

|x̂rt − x̂rs|
(t− s)β′

, sup
r≤s<t≤T

|x̂rt − x̂rs|
(t− s)β′

)
,

and

sup
0≤s<r≤t≤T

|x̂rt − x̂rs|
(t− s)β′

≤ sup
r≤s<t≤T

|x̂rt − x̂rs|
(t− s)β′

+ sup
0≤s<t<r

|x̂rt − x̂rs|
(t− s)β′

.

So, we easily get (5.3).
On the other hand, observe that from the multiplicative property we obtain

sup
0≤s<r≤t≤T

|(x̂r ⊗ y)s,t|
(t− s)2β′

≤ sup
0≤s<r≤t≤T

[ |(x̂r ⊗ y)s,r|
(t− s)2β′

+
|(x̂r ⊗ y)r,t|
(t− s)2β′

+
|(x̂rr − x̂rs)⊗ (yt − yr)|

(t− s)2β′

]
,

and using the same argument as before (5.4) follows easily. �

Proposition 5.2. Let (η·−r, y, η·−r ⊗ y) ∈ Mβ
d,m(0, r) and (y·−r, y, y·−r ⊗ y) ∈

Mβ
m,m(r, T ) for all r ≤ r0. Assume that σ and b satisfy (H1) and (H2) respectively,

and both satisfy (H3). Let (xr, y, xr ⊗ y) ∈ Mβ
d,m(0, T ) be the solution to equation

(3.1). Assume also that ‖η‖β(−r0,0) <∞ and supr≤r0 ‖η·−r⊗ y‖2β(0,r) <∞. Then,
for r ≤ r0, we have the following estimates:

‖x̂r‖∞(0,T+r) ≤Mη,y, (5.5)

‖x̂r‖β′(0,T+r) ≤ Kρη,b,σΛy(1 + 2Mη,y), (5.6)

‖x̂r ⊗ y‖2β′(0,T+r) ≤ Kρη,b,σΛy
(
2 + (T + r0)(Kρη,b,σΛy)1/β

)
, (5.7)

where K ≥ 1 and

ρη,b,σ := 2‖η‖β(−r0,0) + ‖b‖∞T 1−β + ‖σ‖∞ + ‖σ′‖∞ + ‖σ′‖λ, (5.8)

Λy := ‖y‖β + max(1, ‖y‖2β + ‖y ⊗ y‖2β), (5.9)

Mη,y := |η0|+ (T + r0)(Kρη,b,σΛy)
1
β + 1. (5.10)

Proof. For simplicity, we assume that d = m = 1. Firstly, note that, if ‖η‖β(−r0,0) <
C and supr≤r0 ‖η·−r ⊗ y‖2β(0,r) < C ′, with C and C ′ two positive constants, then

‖η‖β′(−r0,0) < Crε0 and supr≤r0 ‖η·−r ⊗ y‖2β′(0,r) < C ′r2ε
0 .

Secondly, notice that by (5.1)

‖η·−r ⊗ y‖2β(0,r) = sup
s,t∈[0,r)

∣∣ ∫ t
s
(yt − yu) dηu−r

∣∣
(t− s)2β

≤ ‖η·−r‖β(0,r)‖y‖β
≤ ‖η‖β(−r0,0)‖y‖β .



10 M. BESALÚ, G. BINOTTO, C. ROVIRA EJDE-2020/65

To prove the result we will follow the ideas in [2, Theorem 4.1]. Consider the

mapping J : Mβ
1,1(0, T + r)→Mβ

1,1(0, T + r) given by J(x̂r, y, x̂r ⊗ y) = (J1, y, J2),

where J1 and J2 are the right-hand sides of the definition of x̂r and (x̂r ⊗ y),
respectively:

J1(x̂r, y, x̂r ⊗ y)(t) =

{
ηt−r for 0 ≤ t < r

η0 +
∫ t−r

0
b(u, x̂ru+r) du+

∫ t−r
0

σ(x̂ru) dyu for r ≤ t ≤ T,
J2(x̂r, y, x̂r ⊗ y)(s, t)

=



∫ t
s
(yt − yu) dηu−r for 0 ≤ s ≤ t < r,

(η0 − ηs−r)⊗ (yt − yr) +
∫ r
s

(yr − yu) dηu−r

+
∫ t
r
(yt − yu)b(u− r, x̂ru) du

+
∫ t
r
σ(x̂ru−r) du(y·−r ⊗ y)·,t for 0 ≤ s < r ≤ t ≤ T,∫ t

s
(yt − yu)b(u− r, x̂ru) du

+
∫ t
s
σ(x̂ru−r) du(y·−r ⊗ y)·,t for r ≤ s ≤ t ≤ T.

This mapping is well-defined because (J1, y, J2) is a real-valued β-Hölder continuous

multiplicative functional for each (x̂r, y, x̂r ⊗ y) ∈Mβ
1,1(0, T ).

Now we bound the Hölder norms of J1 and J2 using Proposition 4.1 and Propo-
sition 4.2. Let s < t ∈ [0, T ]. Then

• for s < t ∈ [0, r),

‖J1‖β(s,t) ≤ ‖η‖β(−r0,0), ‖J2‖2β(s,t) ≤ ‖η‖β(−r0,0)‖y‖β , (5.11)

• for s < t ∈ [r, T ],

‖J1‖β(s,t)

≤ ‖b‖∞(t− s)1−β +K‖σ‖∞ ‖y‖β

+K
(
‖σ′‖∞ + ‖σ′‖λ‖x̂r·−r‖λβ(s,t)(t− s)

λβ
)

Φβ(s,t)(x̂
r
·−r, y·−r)(t− s)β ,

‖J2‖2β(s,t)

≤ ‖b‖∞ ‖y‖β(t− s)1−β +K‖σ‖∞Φβ(s,t)(y·−r, y)

+K
(
‖σ′‖∞ + ‖σ′‖λ‖x̂r·−r‖λβ(s,t)(t− s)

λβ
)

Φβ(s,t)(x̂
r
·−r, y·−r, y)(t− s)β ,

(5.12)

• for s ∈ [0, r) and t ∈ [r, T ],

‖J1‖β(s,t)

≤ ‖J1‖β(s,r) + ‖J1‖β(r,t)

≤ ‖η‖β(−r0,0) + ‖b‖∞(t− r)1−β +K‖σ‖∞ ‖y‖β

+K
(
‖σ′‖∞ + ‖σ′‖λ‖x̂r·−r‖λβ(r,t)(t− r)

λβ
)

Φβ(r,t)(x̂
r
·−r, y·−r)(t− r)β ,

‖J2‖2β(s,t)

≤ ‖J2‖2β(s,r) + ‖J2‖2β(r,t)

≤ 2‖η‖β(−r0,0)‖y‖β + ‖b‖∞ ‖y‖β(t− r)1−β +K‖σ‖∞Φβ(r,t)(y·−r, y)

+K
(
‖σ′‖∞ + ‖σ′‖λ‖x̂r·−r‖λβ(r,t)(t− r)

λβ
)

Φβ(r,t)(x̂
r
·−r, y·−r, y)(t− r)β .



EJDE-2020/65 CONVERGENCE OF DELAY EQUATIONS 11

For s < t ∈ [r, T ], we set

(x̂r·−r ⊗ y·−r)s,t := (x̂⊗ y)s−r,t−r ,

which following [1, section 5] is a β-Hölder continuous multiplicative functional.
We divide the proof into two steps.

Step 1: We find a set Cy of elements (x̂r, y, x̂r⊗y) ∈Mβ
1,1(0, T ) such that J(Cy) ⊂

Cy. Let us recall the definitions of ρη,b,σ and Λy from (5.8) and (5.9), respectively,
and set

∆̃y :=
(
Kρη,b,σΛy

)−1/β
.

Let Cy be the set of elements (x̂r, y, x̂r ⊗ y) ∈ Mβ
1,1(0, T ) satisfying the following

conditions:

‖x̂r‖∞ ≤Mη,y, (5.13)

sup
0<t−s≤∆̃y

‖x̂r‖β(s,t) ≤ Kρη,b,σ(‖y‖β + 1), (5.14)

sup
0<t−s≤∆̃y

‖x̂r ⊗ y‖2β(s,t) ≤ Kρη,b,σ(‖y‖β + ‖y‖2β + ‖y·−r ⊗ y‖2β). (5.15)

If we take s, t ∈ [0, T ], s < t such that

t− s ≤ ∆̃y, (5.16)

then we have

(t− s)β ≤ ∆̃β
y ≤

1

Kρη,b,σ(‖y‖β + 1)
, (5.17)

(t− s)β ≤ ∆̃β
y ≤

1

Kρη,b,σ(‖y‖β + ‖y‖2β + ‖y·−r ⊗ y‖2β)
. (5.18)

Suppose that (x̂r, y, x̂r ⊗ y) ∈ Cy. Then using (5.14), (5.17) and (5.15), (5.18),
respectively, we have

(t− s)β‖x̂r‖β(s,t) ≤ 1, (5.19)

(t− s)β‖x̂r ⊗ y‖2β(s,t) ≤ 1. (5.20)

Now observe that, if s, t ∈ [r, T ] and s < t satisfy (5.16), then s − r, t − r ∈ [0, T ]
also satisfy this inequality. Thus,

(t− s)β‖x̂r·−r‖β(s,t) ≤ 1,

(t− s)β‖x̂r·−r ⊗ y·−r‖2β(s,t) ≤ 1.

From this inequality it easily follows that

Φβ(s,t)(x̂
r
·−r, y·−r, y)(t− s)β

=
[
‖x̂r·−r‖β(s,t)‖y·−r‖β(s,t)‖y‖β(s,t) + ‖y‖β(s,t)‖x̂r·−r ⊗ y·−r‖2β(s,t)

+ ‖x̂r·−r‖β(s,t)‖y·−r ⊗ y‖2β(s,t)

]
(t− s)β

≤ ‖y‖β + ‖y‖2β + ‖y·−r ⊗ y‖2β .

(5.21)

Observe also that if s ∈ [0, r) and t ∈ [r, T ] satisfy (5.16), then t − r ≤ ∆̃y and
all the previous inequalities are satisfied when we change the interval (s, t) by the
interval (r, t) for t ∈ [r, T ].



12 M. BESALÚ, G. BINOTTO, C. ROVIRA EJDE-2020/65

From expressions (5.11),(5.12) and (5.19)–(5.21), we easily get that

‖J1‖β(s,t) ≤ ‖η‖β(−r0,0) + ‖b‖∞T 1−β +K‖σ‖∞‖y‖β
+K (‖σ′‖∞ + ‖σ′‖λ) (‖y‖β + 1)

≤ Kρη,b,σ(‖y‖β + 1)

(5.22)

and

‖J2‖2β(s,t) ≤ 2‖η‖β(−r0,0)‖y‖β + ‖b‖∞ ‖y‖βT 1−β

+K‖σ‖∞(‖y‖2β + ‖y·−r ⊗ y‖2β)

+K (‖σ′‖∞ + ‖σ′‖λ) (‖y‖β + ‖y‖2β + ‖y·−r ⊗ y‖2β)

≤ Kρη,b,σ(‖y‖β + ‖y‖2β + ‖y·−r ⊗ y‖2β)

(5.23)

where K ≥ 1.
It only remains to prove that ‖J1‖∞ ≤Mη,y. SetN =

[
(T+r)∆̃−1

y

]
+1 and define

the partition 0 = t0 < t1 < · · · < tN = T + r, where ti = i∆̃y for i = 0, . . . , N − 1.
Estimates (5.17) and (5.22) imply

sup
u∈[ti−1,ti]

|(J1)u| ≤ |(J1)ti−1
|+ (ti − ti−1)β‖J1‖β(ti−1,ti) ≤ |(J1)ti−1

|+ 1.

Moreover,

sup
u∈[0,ti]

|(J1)u| ≤ sup
u∈[0,ti−1]

|(J1)u|+ 1,

and iterating this inequality we finally obtain

sup
u∈[0,T ]

|(J1)u| ≤ |η0|+N ≤ |η0|+ T ∆̃−1
y + 1 = Mη,y.

Hence, (J1, y, J2) ∈ Cy.

Step 2: We find a bound for the Hölder norms of x̂r and (x̂r⊗y). We can construct
a sequence of functions x̂r(n) and (x̂r ⊗ y)(n) such that, x̂r(0) = η0, (x̂r ⊗ y)(0) = 0
and

x̂r(n) = J1

(
x̂r(n−1), y, (x̂r ⊗ y)

(n−1)
)
,

(x̂r ⊗ y)
(n)

= J2

(
x̂r(n−1), y, (x̂r ⊗ y)

(n−1)
)
.

Notice that
(
x̂r(0), y, (x̂r ⊗ y)(0)

)
∈ Cy and, since we have proved in Step 1 that

J
(
Cy
)
⊂ Cy, we have that

(
x̂r(n), y, (x̂r ⊗ y)(n)

)
∈ Cy for each n. We estimate

‖x̂r(n)‖β as follows:

‖x̂r(n)‖β ≤ sup
0≤s<t≤T
t−s≤∆̃y

|x̂r(n)
t − x̂r(n)

s |
(t− s)β

+ sup
0≤s<t≤T
t−s≥∆̃y

|x̂r(n)
t − x̂r(n)

s |
(t− s)β

≤ Kρη,b,σ(‖y‖β + 1) + 2∆̃−βy ‖x̂r(n)‖∞
≤ Kρη,b,σΛy(1 + 2Mη,y).

(5.24)

It implies that the sequence of functions x̂r(n) is equicontinuous and bounded in
Cβ(0, T ) and the upper bound does not depend on r. So, there exists a subsequence
which converges in the β′-Hölder norm if β′ < β and such that the upper bound of
the β′-Hölder norm does not depend on r.



EJDE-2020/65 CONVERGENCE OF DELAY EQUATIONS 13

In a similar way we obtain the same result for (x̂r⊗y)(n). From inequality (5.20)
we obtain that

sup
ti−1≤s<t≤ti

|(x̂r ⊗ y)
(n)
s,t | ≤ ‖(x̂r ⊗ y)(n)‖2β(ti−1,ti)(ti − ti−1)2β ≤ ∆̃β

y ,

sup
0≤s<t≤T

|(x̂r ⊗ y)
(n)
s,t | ≤ N∆̃β

y ≤ T ∆̃β−1
y + ∆̃β

y .

As we did to obtain (5.24), we estimate ‖(x̂r ⊗ y)(n)‖2β as follows:

‖(x̂r ⊗ y)(n)‖2β ≤ Kρη,b,σ(‖y‖2β + ‖y‖β + ‖y·−r ⊗ y‖2β) + T ∆̃−β−1
y + ∆̃−βy

≤ Kρη,b,σΛy
(
2 + (T + r0)(Kρη,b,σΛy)

1
β
)
.

It implies that the sequence of functions (x̂r ⊗ y)
(n)

is bounded and equicontinuous
in the set of functions 2β-Hölder continuous on ∆T , and the upper bound does not
depend on r. So, there exists a subsequence which converges in the β′-Hölder norm
if β′ < β and such that the upper bound of the β′-Hölder norm does not depend
on r.

Now as n tends to infinity it is easy to see that the limit is a solution, and the
limit defines a β−Hölder continuous multiplicative functional (x̂r, y, x̂r ⊗ y) and
this functional satisfies estimates (5.5), (5.6) and (5.7). �

Remark 5.3. In Proposition 5.2 it is proved that ‖x̂r‖β′ ≤ Kρη,b,σΛy(1 + 2Mη,y).
Thus we have the same bound for ‖xr‖β′(r). Moreover, using the ideas in the proof
of Proposition 5.2 it is possible to prove that ‖xr ⊗ y‖2β′ is bounded and its bound
does not depend on r.

We are also interested in the behavior of (xr− x̂r) when r tends to zero. We can
write (x− xr)t as follows

(x− xr)t =

∫ t

0

[
b(u, xu)− b(u, xru)

]
du+

∫ t

0

[
σ(xu)− σ(xru)

]
dyu

+

∫ t

0

[
σ(xru)− σ(xru−r)

]
dyu.

(5.25)

Following the ideas in [1, Section 4], let us write
(
(x− xr)⊗ y

)
s,t
, for s, t ∈ [0, T ]:(

(x− xr)⊗ y
)
s,t

=

∫ t

s

(yt − yu)
[
b(u, xu)− b(u, xru)

]
du

+

∫ t

s

[
σ(xu)− σ(xru)

]
du(y ⊗ y)·,t +

∫ t

s

[
σ(xru)− σ(xru−r)

]
du(y ⊗ y)·,t.

(5.26)

It is also useful to write the following expressions:

• for s < t ∈ [0, r),

(xr − x̂r)t − (xr − x̂r)s = ηs−r − ηt−r +

∫ t

s

b(u, xru) du+

∫ t

s

σ(ηu−r) dyu, (5.27)



14 M. BESALÚ, G. BINOTTO, C. ROVIRA EJDE-2020/65

• for s ∈ [0, r) and t ∈ [r, T ],

(xr − x̂r)t − (xr − x̂r)s = ηs−r − η0 +

∫ t

t−r
b(u, xru) du−

∫ s

0

b(u, xru) du

+

∫ t

t−r
σ(xru−r) dyu −

∫ s

0

σ(ηu−r) dyu,

(5.28)

• for s < t ∈ [r, T ],

(xr − x̂r)t − (xr − x̂r)s =

∫ t

s

b(u, xru) du−
∫ t−r

s−r
b(u, xru) du

+

∫ t

s

σ(xru−r) dyu −
∫ t−r

s−r
σ(xru−r) dyu.

(5.29)

Finally, following the ideas in [1, Section 4], we define(
(xr − x̂r)⊗ y

)
s,t

:= (xr ⊗ y)s,t − (x̂r ⊗ y)s,t,

that is,

• for s < t ∈ [0, r),(
(xr − x̂r)⊗ y

)
s,t

=

∫ t

s

(yu − yt) dηu−r +

∫ t

s

(yt − yu)b(u, xru) du

+

∫ t

s

σ(ηu−r) du(y ⊗ y)·,t,

(5.30)

• for s ∈ [0, r) and t ∈ [r, T ],(
(xr − x̂r)⊗ y

)
s,t

=

∫ t

s

(yt − yu)b(u, xru) du+

∫ t

s

σ(xru−r) du(y ⊗ y)·,t

− (η·−r ⊗ y)s,r −
∫ t

r

(yt − yu)b(u− r, x̂ru) du

−
∫ t

r

σ(x̂ru−r) du(y·−r ⊗ y)·,t − (η0 − ηs−r)⊗ (yt − yr),

• for s < t ∈ [r, T ],(
(xr − x̂r)⊗ y

)
s,t

=

∫ t

s

(yu+r − yu)b(u, xru) du+

∫ t

s

σ(xru−r) du
(
(y − y·−r)⊗ y

)
·,t

+

∫ t

s

[
σ(xru−r)− σ(x̂ru−r)

]
du(y·−r ⊗ y)·,t.

(5.31)

The following proposition gives us a result about the behavior of (xr− x̂r) when
r tends to zero.

Proposition 5.4. Set β′ = β − ε, where ε > 0 is such that β − 2ε > 0 and
λ > 1

β−ε − 2. Suppose that (x, y, x ⊗ y), (xr, y, xr ⊗ y), (x̂r, y, x̂r ⊗ y) belong to

Mβ
d,m(0, T ) and (y, y, y ⊗ y) belongs to Mβ

m,m(0, T ). Assume that σ and b satisfy

(H1) and (H2) respectively, and both satisfy (H3). Assume also that ‖η‖β(−r0,0) <



EJDE-2020/65 CONVERGENCE OF DELAY EQUATIONS 15

∞ and supr≤r0 Φβ(0,r)(η·−r, y) < ∞ and suppose that ‖(y − y·−r) ⊗ y‖2β′(r,T ) → 0
and ‖y·−r ⊗ (y − y·−r)‖2β′(r,T ) → 0 when r tends to zero. Then

‖xr − x̂r‖∞ ≤ KρΛrβ
′

‖xr − x̂r‖β′ ≤ KρΛrε∥∥(xr − x̂r)⊗ y
∥∥

2β′
≤ KMρ3Λ3rε +KMρ3Λ2Λr

where K ≥ 1, M ≥ 1 are constants depending on β, β′, r0, T, σ, y and

ρ =
(

1 + 3‖b‖∞T 1−β′ + 3‖σ‖∞(1 + T β
′
) + 2‖σ′‖∞(1 + T β

′
) + 3‖σ′‖∞T β

′−ε

+ ‖σ′‖λ
(

2 sup
r≤r0
‖xr‖λβ′ + ‖η‖λβ′(−r0,0)

)
T (λ+1)β′−ε + ‖σ′′‖∞T β

′
(1 + T β

′
)

+ 2‖σ′′‖λ sup
r≤r0
‖x̂r‖λβ′T (λ+1)β′

)
(1 + T ε),

Λ = max
(

1, ‖η‖β(−r0,0), sup
r≤r0

Φβ′(0,r)(η·−r, y),Φβ(0,T )(y, y),

sup
r≤r0

Φβ′(r,T )(y·−r, y), sup
r≤r0

Φβ′(0,r)(η·−r, y, y), sup
r≤r0

Φβ′(0,T )(x
r, y),

sup
r≤r0

Φβ′(0,T )(x̂
r, y)

)(
1 + sup

r≤r0
‖xr‖β′(r)

)(
1 + ‖y‖β

)
,

Λr = max
(

1, sup
r≤r0
‖xr‖β′

)(
‖(y − y·−r)⊗ y‖2β′(r,T )

+ ‖y·−r ⊗ (y − y·−r)‖2β′(r,T )

)
.

Remark 5.5. Since ρ and Λ are finite (from Proposition 5.2) and Λr converges to
zero when r tends to zero (by hypothesis), Proposition 5.4 implies:

‖xr − x̂r‖∞
r↓0−−→ 0,

‖xr − x̂r‖β′
r↓0−−→ 0,∥∥(xr − x̂r)⊗ y
∥∥

2β′
r↓0−−→ 0.

Proof of Proposition 5.4. We start studying the supremum norm. On the one hand,
using Proposition 4.1, for r ≤ r0, we obtain

‖xr − x̂r‖∞(0,r) ≤ ‖η‖β′(−r,0)r
β′ + ‖b‖∞r +K‖σ‖∞‖y‖β′rβ

′

+KΦβ′(0,r)(η·−r, y)
(
‖σ′‖∞ + ‖σ′‖λ‖η·−r‖λβ′(0,r)r

λβ′
)
r2β′

≤
[
‖η‖β(−r0,0)T

ε + ‖b‖∞T 1−β′ +K‖σ‖∞‖y‖βT ε

+KΦβ′(0,r)(η·−r, y)
(
‖σ′‖∞ + ‖σ′‖λ‖η‖λβ′(−r0,0)T

λβ′
)
T β
′
]
rβ
′

where we have used that ‖η‖β′(−r,0) ≤ ‖η‖β(−r0,0)T
ε and ‖y‖β′ ≤ ‖y‖βT ε.

On the other hand, using Proposition 4.1 we obtain

‖xr − x̂r‖∞(r,T ) ≤ ‖b‖∞r +K‖σ‖∞‖y‖β′rβ
′

+KΦβ′(0,T )(x̂
r, y)

(
‖σ′‖∞ + ‖σ′‖λ‖x̂r‖λβ′(r,T )r

λβ′
)
r2β′

≤
[
‖b‖∞T 1−β′ +K‖σ‖∞‖y‖βT ε



16 M. BESALÚ, G. BINOTTO, C. ROVIRA EJDE-2020/65

+KΦβ′(0,T )(x̂
r, y)

(
‖σ′‖∞ + ‖σ′‖λ‖xr‖λβ′Tλβ

′)
T β
′
]
rβ
′
.

Hence, we have

‖xr − x̂r‖∞ ≤ KρΛrβ
′
. (5.32)

Now we study the Hölder norms. Following the proof of Lemma 5.1 we easily
obtain

‖xr − x̂r‖β′ ≤ ‖xr − x̂r‖β′(0,r) + ‖xr − x̂r‖β′(r,T ), (5.33)

‖(xr − x̂r)⊗ y‖2β′ ≤ ‖(xr − x̂r)⊗ y‖2β′(0,r) + ‖(xr − x̂r)⊗ y‖2β′(r,T )

+ ‖xr − x̂r‖β′(0,r)‖y‖β′ .
(5.34)

So we can study the Hölder norms independently in the intervals [0, r) and [r, T ].
We deal with the Hölder norm of (xr − x̂r). By (5.27) and Proposition 4.1 we have

‖xr − x̂r‖β′(0,r)
≤ ‖η‖β′(−r,0) + ‖b‖∞r1−β′ +K‖σ‖∞‖y‖β′(0,r)

+KΦβ′(0,r)(η·−r, y)
(
‖σ′‖∞ + ‖σ′‖λ‖η·−r‖λβ′(0,r)r

λβ′
)
rβ
′

≤
[
‖η‖β(−r0,0) + ‖b‖∞T 1−β +K‖σ‖∞‖y‖β +KΦβ′(0,r)(η·−r, y)

×
(
‖σ′‖∞ + ‖σ′‖λ‖η‖λβ′(−r0,0)T

λβ′
)
T β
′−ε
]
rε.

(5.35)

In the interval [r, T ], observe that

‖xr − x̂r‖β′(r,T ) ≤ max
(

sup
s<t∈[r,T ]
t−s≤r

|(xr − x̂r)t − (xr − x̂r)s|
(t− s)β′

,

sup
s<t∈[r,T ]
t−s≥r

|(xr − x̂r)t − (xr − x̂r)s|
(t− s)β′

)
.

(5.36)

On the one hand, by (5.29) and Proposition 4.1 we have

sup
s<t∈[r,T ] t−s≤r

|(xr − x̂r)t − (xr − x̂r)s|
(t− s)β′

≤ 2‖b‖∞r1−β′ + 2K‖σ‖∞‖y‖βrε

+ 2KΦβ′(r,T )(x̂
r, y)

(
‖σ′‖∞ + ‖σ′‖λ‖x̂r‖λβ′(r,T )r

λβ′
)
rβ
′

≤
[
2‖b‖∞T 1−β + 2K‖σ‖∞‖y‖β

+ 2KΦβ′(0,T )(x̂
r, y)

(
‖σ′‖∞ + ‖σ′‖λ‖xr‖λβ′Tλβ

′)
T β
′−ε
]
rε,

(5.37)

where we have used that sups<t∈[r,T ] t−s≤r ‖y‖β′(s,t) ≤ ‖y‖βrε.



EJDE-2020/65 CONVERGENCE OF DELAY EQUATIONS 17

On the other hand, with a similar computations, by Proposition 4.1 we have

sup
s<t∈[r,T ] t−s≥r

|(xr − x̂r)t − (xr − x̂r)s|
(t− s)β′

≤ 2‖b‖∞r1−β′ + 2K‖σ‖∞‖y‖βT εrε

+ 2KΦβ′(r,T )(x̂
r, y)

(
‖σ′‖∞ + ‖σ′‖λ‖x̂r‖λβ′(r,T )r

λβ′
)
rβ
′

≤
[
2‖b‖∞T 1−β + 2K‖σ‖∞‖y‖βT ε

+ 2KΦβ′(0,T )(x̂
r, y)

(
‖σ′‖∞ + ‖σ′‖λ‖xr‖λβ′Tλβ

′)
T β
′−ε
]
rε,

(5.38)

where we have used that supt∈[r,T ] ‖y‖β′(t−r,t) ≤ ‖y‖βrε. Then, by (5.33) and using

(5.35), (5.36), (5.37) and (5.38) it follows that

‖xr − x̂r‖β′ ≤ KρΛrε. (5.39)

Finally, we study the Hölder norm ‖(xr − x̂r)⊗ y‖2β′ . By definition (5.30) and
Proposition 4.2 we have

‖(xr − x̂r)⊗ y‖2β′(0,r)
≤ ‖η·−r‖β′(0,r)‖y‖β′(0,r) + ‖y‖β′(0,r)‖b‖∞r1−β′ +K‖σ‖∞Φβ′(0,r)(y, y)

+K(‖σ′‖∞ + ‖σ′‖λ‖η·−r‖λβ′(0,r)r
λβ′)Φβ′(0,r)(η·−r, y, y)rβ

′

≤
[
‖η‖β(−r0,0)‖y‖βT ε + ‖y‖β‖b‖∞T 1−β′ +K‖σ‖∞Φβ(0,T )(y, y)T ε

+K(‖σ′‖∞ + ‖σ′‖λ‖η‖λβ′(−r0,0)T
λβ′)Φβ′(0,r)(η·−r, y, y)T β

′−ε
]
rε

≤ KρΛrε,

(5.40)

where we have used that Φβ′(0,r)(y, y) ≤ Φβ(0,T )(y, y)r2ε.
Now we study the Hölder norm in the interval [r, T ]. Let a, b ∈ [r, T ], a < b.

Then by (5.31),

‖(xr − x̂r)⊗ y‖2β′(a,b)

≤ sup
s<t∈[a,b]

∣∣ ∫ t
s
(yu+r − yu)b(u, xru) du

∣∣
(t− s)2β′

+ sup
s<t∈[a,b]

∣∣ ∫ t
s
σ(xru−r) du

(
(y − y·−r)⊗ y

)
·,t

∣∣
(t− s)2β′

+ sup
s<t∈[a,b]

∣∣ ∫ t
s

[
σ(xru−r)− σ(x̂ru−r)

]
du(y·−r ⊗ y)·,t

∣∣
(t− s)2β′

=: A1 +A2 +A3.

(5.41)

Let us study A1, A2 and A3. It is easy to see that

A1 ≤ ‖y‖β ‖b‖∞T 1−β′ ≤ KρΛrε. (5.42)



18 M. BESALÚ, G. BINOTTO, C. ROVIRA EJDE-2020/65

By Proposition 4.2 we have

A2 ≤ K‖σ‖∞ Φβ′(a,b)(y − y·−r, y)

+K
(
‖σ′‖∞ + ‖σ′‖λ‖x̂r‖λβ′(a,b)T

λβ′
)

Φβ′(a,b)(x̂
r, y − y·−r, y)T β

′

= K‖y‖β′
(
‖σ‖∞ +

(
‖σ′‖∞ + ‖σ′‖λ‖x̂r‖λβ′Tλβ

′
)
‖x̂r‖β′T β

′
)

× ‖y − y·−r‖β′(r,T )

+K
(
‖σ‖∞ +

(
‖σ′‖∞ + ‖σ′‖λ‖x̂r‖λβ′Tλβ

′
)
‖x̂r‖β′T β

′
)

× ‖(y − y·−r)⊗ y‖2β′(r,T )

+K‖y‖β′T β
′
(
‖σ′‖∞ + ‖σ′‖λ‖x̂r‖λβ′Tλβ

′
)
‖x̂r ⊗ (y − y·−r)‖2β′(a,b).

(5.43)

Now we estimate the norm ‖x̂r ⊗ (y − y·−r)‖2β′(a,b). For s < t ∈ [a, b],(
x̂r ⊗ (y − y·−r)

)
s,t

=

∫ t

s

(yt − yt−r − yu + yu−r)b(u− r, x̂ru) du

+

∫ t

s

σ(x̂ru−r) du
(
y·−r ⊗ (y − y·−r)

)
·,t.

So, by Proposition 4.2 and Lemma 4.8 we have

‖x̂r ⊗ (y − y·−r)‖2β′(a,b)

≤ K
[(
‖σ′‖∞ + ‖σ′‖λ‖x̂r‖λβ′Tλβ

′)(
‖x̂r‖β′‖y‖β′ + ‖x̂r ⊗ y‖2β′

)
T β
′

+ ‖b‖∞T 1−β′ + ‖σ‖∞‖y‖β′
]
‖y‖βrε

+K
[
‖σ‖∞ +

(
‖σ′‖∞+ ‖σ′‖λ‖x̂r‖λβ′Tλβ

′)
‖x̂r‖β′T β

′]
× ‖y·−r ⊗ (y − y·−r)‖2β′(r,T ).

(5.44)

Putting together (5.43) and (5.44) and inequality (4.5) we obtain

A2 ≤ KρΛrε +Kρ2Λ2rε +Kρ2ΛΛr +KρΛr

≤ Kρ2Λ2rε +Kρ2ΛΛr,
(5.45)

where we have used that 1 ≤ ρ ≤ ρ2 and 1 ≤ Λ ≤ Λ2.
Finally, by Proposition 4.7 and inequalities (5.32) and (5.39) we obtain

A3 ≤ G4
β′(r,T )(σ, x̂

r, x̂r·−r, y·−r, y)T β
′
‖xr − x̂r‖∞

+G5
β′(r,T )(σ, x̂

r, x̂r·−r, y·−r, y)T β
′
‖xr − x̂r‖β′

+G6
β′(r,T )(σ, x̂

r
·−r, y)(b− a)β

′
‖(xr − x̂r)⊗ y‖2β′(a−r,b−r)

≤ Kρ2Λ2rε +G6
β′(r,T )(σ, x̂

r
·−r, y)(b− a)β

′

× ‖(xr − x̂r)⊗ y‖2β′(a−r,b−r)

(5.46)

where we have used that Giβ′(r,T )(σ, x̂
r, x̂r·−r, y·−r, y)T β

′ ≤ KρΛ for i = 4, 5. Ap-

plying the multiplicative property, it is easy to see that

‖(xr − x̂r)⊗ y‖2β′(a−r,b−r)
≤ ‖(xr − x̂r)⊗ y‖2β′(a−r,a) + ‖(xr − x̂r)⊗ y‖2β′(a,b) + ‖xr − x̂r‖β′‖y‖β′ .



EJDE-2020/65 CONVERGENCE OF DELAY EQUATIONS 19

On the one hand, by (5.39),

‖xr − x̂r‖β′‖y‖β′ ≤ ‖xr − x̂r‖β′‖y‖βT ε ≤ Kρ2Λ2rε.

On the other hand,

‖(xr − x̂r)⊗ y‖2β′(a−r,a) ≤ Kρ2Λ2rε +Kρ2ΛΛr,

where the result is obtained considering separately the two cases a ∈ [r, 2r) and
a ∈ [2r, T ] and applying multiplicative property, inequalities (5.40), (5.41), (5.42),

(5.45) and that G6
β′(r,T )(σ, x̂

r
·−r, y)T β

′ ≤ KρΛ. Therefore,

‖(xr − x̂r)⊗ y‖2β′(a−r,b−r) ≤ Kρ2Λ2rε +Kρ2ΛΛr + ‖(xr − x̂r)⊗ y‖2β′(a,b).
Then, it follows that

A3 ≤ G6
β′(r,T )(σ, x̂

r
·−r, y)(b− a)β

′
‖(xr − x̂r)⊗ y‖2β′(a,b) +Kρ3Λ3rε +Kρ3Λ2Λr,

(5.47)

where we have used again that G6
β′(r,T )(σ, x̂

r
·−r, y)T β

′ ≤ KρΛ. From inequalities

(5.41), (5.42), (5.45) and (5.47) we have

‖(xr − x̂r)⊗ y‖2β′(a,b) ≤ G6
β′(r,T )(σ, x̂

r
·−r, y)(b− a)β

′
‖(xr − x̂r)⊗ y‖2β′(a,b)

+Kρ3Λ3rε +Kρ3Λ2Λr,

where we have used that ρn ≤ ρn+1 and Λn ≤ Λn+1 for any n ∈ N. Set

∆̃ :=
(

2 sup
r≤r0

G6
β′(r,T )(σ, x̂

r
·−r, y)

)1/β′

. (5.48)

Observe that, if a, b, a < b are such that b− a ≤ ∆̃, then

‖(xr − x̂r)⊗ y‖2β′(a,b) ≤ Kρ3Λ3rε +Kρ3Λ2Λr. (5.49)

Now consider a partition r = t0 < · · · < tM = T such that (ti+1 − ti) ≤ ∆̃ for
i = 0 . . . ,M − 1. Then, using the multiplicative property iteratively, we have

‖(xr − x̂r)⊗ y‖2β′(r,T ) ≤
M−1∑
i=0

‖(xr − x̂r)⊗ y‖2β′(ti,ti+1) + (M − 1)‖xr − x̂r‖β′‖y‖β′ .

Using (5.39) and (5.49), we obtain

‖(xr − x̂r)⊗ y‖2β′(r,T ) ≤ KMρ3Λ3rε +KMρ3Λ2Λr +K(M − 1)ρ2Λ2rε

≤ KMρ3Λ3rε +KMρ3Λ2Λr.
(5.50)

Finally, putting together (5.34), (5.39), (5.40) and (5.50) we have

‖(xr − x̂r)⊗ y‖2β′ ≤ KMρ3Λ3rε +KMρ3Λ2Λr.

So the proof is complete. �

The following definitions will be useful for the next results:

G
i

β′ := sup
r≤r0

Giβ′(0,T )(σ, x, x
r, y) i = 1, 2; (5.51)

G
3

β′ := sup
r≤r0

G3
β′(0,T )(σ, x

r); (5.52)

G
j

β′ := sup
r≤r0

Gjβ′(0,T )(σ, x, x
r, y, y) j = 4, 5; (5.53)



20 M. BESALÚ, G. BINOTTO, C. ROVIRA EJDE-2020/65

G
6

β′ := sup
r≤r0

G6
β′(0,T )(σ, x

r, y). (5.54)

The following result gives us a bound for
∥∥(x − xr) ⊗ y

∥∥
2β′(a,b)

when the interval

(a, b) is small enough. Let

∆1
β′ =

(
2G

6

β′
)1/β′

.

Proposition 5.6. Suppose that (x, y, x⊗y), (xr, y, xr⊗y) and (x̂r, y, x̂r⊗y) belong

to Mβ
d,m(0, T ), (y, y, y⊗ y) belongs to Mβ

m,m(0, T ) and (y·−r, y, y·−r ⊗ y) belongs to

Mβ
m,m(r, T ). Assume that σ and b satisfy (H1) and (H2) respectively. Then, for all

0 ≤ a < b ≤ T such that (b− a) ≤ ∆1
β′ ,∥∥(x− xr)⊗ y

∥∥
2β′(a,b)

≤ 2
[
LN‖y‖β′(b− a)1−2β′ +G4

β′(a,b)(σ, x, x
r, y, y)

]
(b− a)β

′
‖x− xr‖∞(a,b)

+ 2G5
β′(a,b)(σ, x, x

r, y, y)(b− a)β
′
‖x− xr‖β′(a,b)

+ 2KρΛG4
β′(a,b)(σ, x

r, x̂r, y, y)(b− a)β
′
rβ
′

+ 2KρΛ
[
G5
β′(a,b)(σ, x

r, x̂r, y, y) +Mρ2Λ2G6
β′(a,b)(σ, x̂

r, y)
]
(b− a)β

′
rε

+ 2KMρ3Λ2G6
β′(a,b)(σ, x̂

r, y)(b− a)β
′
Λr.

Proof. The proposition is proved by applying first Propositions 4.3, 4.6 and 4.7 to
definition (5.26), and then Proposition 5.4, and observing that for a < b such that
(b− a) ≤ ∆1

β′

G6
β′(a,b)(σ, x

r, y)(b− a)β
′
≤ 1

2
.

�

6. Proof of main results

Proof of Theorem 3.1. We start by studying limr→0 ‖x− xr‖∞. As in Lemma 5.1,
we can study separately the intervals [0, r) and (r, T ).

First, we study the norm in the interval [0, r). We apply Proposition 4.3 and
Proposition 4.4 to (5.25) and obtain

‖x− xr‖β(0,r) ≤ LNr1−β‖x− xr‖∞(0,r) +G1
β(0,r)(σ, x, η·−r, y)rβ‖x− η·−r‖∞(0,r)

+G2
β(0,r)(σ, x, η·−r, y)rβ‖x− η·−r‖β(0,r)

+G3
β(0,r)(σ, η·−r)r

β‖(x− η·−r)⊗ y‖2β(0,r).

Using that the supremum norm of x is bounded and the bound does not depend on r,
we see that supr≤r0 G

i
β(0,r)(σ, x, η·−r, y) <∞ i = 1, 2 and supr≤r0 G

3
β(0,r)(σ, η·−r) <

∞. So the last expression clearly approaches zero when r tends to zero.
Now we work on the interval [r, T ]. Let r ≤ a < b ≤ T . Applying Propositions

4.3, 4.4, 4.5 and 5.4, we obtain

‖x− xr‖β′(a,b) ≤
[
LN (b− a)1−2β′ +G1

β′(a,b)(σ, x, x
r, y)

]
(b− a)β

′
‖x− xr‖∞(a,b)

+G2
β′(a,b)(σ, x, x

r, y)(b− a)β
′
‖x− xr‖β′(a,b)

+G3
β′(a,b)(σ, x

r)(b− a)β
′
‖(x− xr)⊗ y‖2β′(a,b)

+KρΛG1
β′(a,b)(σ, x

r, x̂r, y)(b− a)β
′
rβ
′



EJDE-2020/65 CONVERGENCE OF DELAY EQUATIONS 21

+
[
KρΛG2

β′(a,b)(σ, x
r, x̂r, y) +Kρ3Λ3G3

β′(a,b)(σ, x̂
r)
]
(b− a)β

′
rε

+Kρ3Λ2ΛrG
3
β′(a,b)(σ, x̂

r)(b− a)β
′
.

Set

Hr

= KρΛ
(
G1
β′(0,T )(σ, x

r, x̂r, y) + 2G3
β′(0,T )(σ, x

r)G4
β′(0,T )(σ, x

r, x̂r, y, y)T β
′)
T β
′
rβ
′

+
[
KρΛ

(
G2
β′(0,T )(σ, x

r, x̂r, y) + 2G3
β′(0,T )(σ, x

r)G5
β′(0,T )(σ, x

r, x̂r, y, y)T β
′)

+Kρ3Λ3
(
G3
β′(0,T )(σ, x̂

r) + 2G3
β′(0,T )(σ, x

r)G6
β′(0,T )(σ, x̂

r, y)T β
′)]
T β
′
rε

+Kρ3Λ2
(
G3
β′(0,T )(σ, x̂

r) + 2G3
β′(0,T )(σ, x

r)G6
β′(0,T )(σ, x̂

r, y)T β
′)]
T β
′
Λr.

Observe that Hr converges to zero when r tends to zero. We select a and b such
that

(b− a) ≤ ∆1
β′ (6.1)

and then we apply Proposition 5.6 to obtain

‖x− xr‖β′(a,b)

≤
[
LN
(
2‖y‖β′G3

β′(a,b)(σ, x
r)(b− a)β

′
+ 1
)
(b− a)1−2β′

+G1
β′(a,b)(σ, x, x

r, y) + 2G3
β′(a,b)(σ, x

r)G4
β′(a,b)(σ, x, x

r, y, y)(b− a)β
′
]

× (b− a)β
′
‖x− xr‖∞(a,b)

+
[
G2
β′(a,b)(σ, x, x

r, y) + 2G3
β′(a,b)(σ, x

r)G5
β′(a,b)(σ, x, x

r, y, y)

× (b− a)β
′
]
(b− a)β

′
‖x− xr‖β′(a,b)

+KρΛ
(
G1
β′(a,b)(σ, x

r, x̂r, y) + 2G3
β′(a,b)(σ, x

r)G4
β′(a,b)(σ, x

r, x̂r, y, y)

× (b− a)β
′)

(b− a)β
′
rβ
′

+
[
KρΛ

(
G2
β′(a,b)(σ, x

r, x̂r, y)

+ 2G3
β′(a,b)(σ, x

r)G5
β′(a,b)(σ, x

r, x̂r, y, y)(b− a)β
′)

+Kρ3Λ3
(
G3
β′(a,b)(σ, x̂

r) + 2G3
β′(a,b)(σ, x

r)G6
β′(a,b)(σ, x̂

r, y)(b− a)β
′)]

× (b− a)β
′
rε

+Kρ3Λ2
(
G3
β′(a,b)(σ, x̂

r) + 2G3
β′(a,b)(σ, x

r)G6
β′(a,b)(σ, x̂

r, y)(b− a)β
′)

× (b− a)β
′
Λr

≤
[
LN
(
2‖y‖β′G3

β′(a,b)(σ, x
r)T β

′
+ 1
)
T 1−2β′ +G1

β′(a,b)(σ, x, x
r, y)

+ 2G3
β′(a,b)(σ, x

r)G4
β′(a,b)(σ, x, x

r, y, y)T β
′
]
(b− a)β

′
‖x− xr‖∞(a,b)

+
[
G2
β′(a,b)(σ, x, x

r, y) + 2G3
β′(a,b)(σ, x

r)G5
β′(a,b)(σ, x, x

r, y, y)T β
′]

× (b− a)β
′
‖x− xr‖β′(a,b) +Hr.



22 M. BESALÚ, G. BINOTTO, C. ROVIRA EJDE-2020/65

For a and b such that[
G2
β′(a,b)(σ, x, x

r, y) + 2G3
β′(a,b)(σ, x

r)G5
β′(a,b)(σ, x, x

r, y, y)T β
′]

(b−a)β
′
≤ 1

2
, (6.2)

we obtain

‖x− xr‖β′(a,b)

≤ 2
[
LN
(
2‖y‖β′G3

β′(a,b)(σ, x
r)T β

′
+ 1
)
T 1−2β′

+G1
β′(a,b)(σ, x, x

r, y) + 2G3
β′(a,b)(σ, x

r)G4
β′(a,b)(σ, x, x

r, y, y)T β
′
]

× (b− a)β
′
‖x− xr‖∞(a,b) + 2Hr.

(6.3)

On the other hand, putting

‖x− xr‖∞(a,b) ≤ |xa − xra|+ (b− a)β
′
‖x− xr‖β′(a,b),

in (6.3) we obtain

‖x− xr‖∞(a,b)

≤ |xa − xra|+ 2
[
LN
(
2‖y‖β′G3

β′(a,b)(σ, x
r)T β

′
+ 1
)
T 1−2β′

+G1
β′(a,b)(σ, x, x

r, y) + 2G3
β′(a,b)(σ, x

r)G4
β′(a,b)(σ, x, x

r, y, y)T β
′
]

× (b− a)2β′‖x− xr‖∞(a,b) + 2T β
′
Hr .

For a and b, a < b such that

2
[
LN
(
2‖y‖β′G3

β′(a,b)(σ, x
r)T β

′
+ 1
)
T 1−2β′ +G1

β′(a,b)(σ, x, x
r, y)

+ 2G3
β′(a,b)(σ, x

r)G4
β′(a,b)(σ, x, x

r, y, y)T β
′
]
T β
′
(b− a)β

′
≤ 1

2

(6.4)

we obtain

‖x− xr‖∞(a,b) ≤ 2|xa − xra|+ 4T β
′
Hr ,

and hence

sup
0≤t≤b

|xt − xrt | ≤ 2 sup
0≤t≤a

|xt − xrt |+ 4T β
′
Hr . (6.5)

We define now ∆β′ such that all a, b with (b−a) ≤ ∆β′ fulfill (6.1), (6.2) and (6.4),
that is

∆β′ :=
(

16LNT
1−β′ + 16G

1

β′T
β′ + 4G

2

β′

+ 8G
3

β′
[
4LN‖y‖β′T + 4G

4

β′T
2β′ +G

5

β′T
β′
]

+ 2G
6

β′

)1/β′

.

Then, it is clear that (6.5) holds for all a and b such that b− a ≤ ∆β′ .
Now, we take a partition 0 = t0 < t1 < · · · < tM = T of the interval [0, T ] such

that (ti+1 − ti) ≤ ∆β′ . Then

sup
0≤t≤tM=T

|xt − xrt | ≤ 2 sup
0≤t≤tM−1

|xt − xrt |+ 4T β
′
Hr. (6.6)

Repeating the process M times we obtain

sup
0≤t≤T

|xt − xrt | ≤ 2M |x0 − xr0|+
(M−1∑
k=0

2k
)

4T β
′
Hr = 4(2M − 1)T β

′
Hr



EJDE-2020/65 CONVERGENCE OF DELAY EQUATIONS 23

that clearly converges to zero when r tends to zero. Following the same arguments,
obtain

lim
r→0
‖(x⊗ y)− (xr ⊗ y)‖∞ = lim

r→0
‖(x− xr)⊗ y‖∞.

�

7. Stochastic case

In this section we apply the results obtained in the deterministic case to Brownian
motion paths in order to get convergence of stochastic differential equations driven
by Brownian motion.

Suppose that B = {Bt = (B1
t , B

2
t , . . . , B

m
t ), t ≥ 0} is a m-dimensional Brownian

motion. Fix a time interval [0, T ]. Then, for s, t ∈ [0, T ] and i, j ∈ {1, . . . ,m}, we
consider the following tensor products:

(Bi ⊗Bj)s,t :=

∫ t

s

(Biu −Bis) d◦Bju −
1

2
(t− s)1{i=j} ,

(Bi ⊗Bj·−r)s,t :=

∫ t

s

(Biu −Bis) d◦B
j
u−r ,

where the stochastic integral is a Stratonovich integral (see Russo and Vallois [17]).
In [15] the authors show that we can choose a version (B⊗B·−r)s,t in such a way that
(B·−r, B,B⊗B·−r) constitutes a β-Hölder continuous multiplicative functional, for
a fixed β ∈ ( 1

3 ,
1
2 ). On the other hand, from Hu and Nualart [9] it follows that

(B,B,B ⊗B) is also a β-Hölder continuous multiplicative functional.
As an application of Theorem 3.1 we deduce the convergence when the delay

goes to zero of the solutions for the stochastic differential delay equations

Xr(t) = η(0) +

∫ t

0

b(u,Xr
u) du+

∫ t

0

σ(Xr
u−r) dBu, t ∈ (0, T ],

Xr(t) = η(t), t ∈ [−r, 0],

where the stochastic integral is a pathwise integral which depends on B and (B⊗B).
Set X ≡ X0 the solution without delay and fix β ∈ ( 1

3 ,
1
2 ). Then the theorem reads

as follows:

Theorem 7.1. Assume that σ and b satisfy (H1) and (H2) respectively, and both

satisfy (H3). Assume also that (η·−r0 , B, η·−r0⊗B) ∈Mβ
d,m(0, r0), ‖η‖β(−r0,0) <∞

and supr≤r0 Φβ(0,r)(η·−r, B) <∞ a.s. Then,

lim
r→0
‖X −Xr‖∞ = 0 a.s. and lim

r→0
‖(X ⊗B)− (Xr ⊗B)‖∞ = 0 a.s.

Applying Theorem 3.1 pathwise, the proof of Theorem 7.1 is an obvious conse-
quence of (7.2) and (7.3) of the following lemma.

Lemma 7.2. We have

‖B ⊗ (B −B·−r)‖2β′(r,T ) → 0 a.s. when r tends to 0, (7.1)

‖B·−r ⊗ (B −B·−r)‖2β′(r,T ) → 0 a.s. when r tends to 0, (7.2)

‖(B −B·−r)⊗B‖2β′(r,T ) → 0 a.s. when r tends to 0. (7.3)

Proof. Let us recall first that ‖B‖β <∞ a.s. We begin estimating (7.1) when i 6= j
(we will consider the case i = j at the end). By definition,

‖B ⊗ (B −B·−r)‖2β′(r,T )



24 M. BESALÚ, G. BINOTTO, C. ROVIRA EJDE-2020/65

= sup
s,t∈[r,T ]

1

(t− s)2β′

∣∣∣ ∫ t

s

(Biu −Bis) d◦B
j
u−r −

∫ t

s

(Biu −Bis) d◦Bju
∣∣∣

Assume first that t− s > r. Applying integration by parts, we have

‖B ⊗ (B −B·−r)‖2β′(r,T )

= sup
s,t∈[r,T ]
t−s>r

1

(t− s)2β′

∣∣∣(Bit −Bis)(Bjt−r −Bjs−r)− ∫ t

s

(Bju−r −B
j
s−r) d

◦Biu

+

∫ t

s

(Bju −Bjs) d◦Biu − (Bit −Bis)(B
j
t −Bjs)

∣∣∣
≤ sup
s,t∈[r,T ]
t−s>r

1

(t− s)2β′

∣∣(Bit −Bis)(Bjt−r −Bjt )∣∣
+ sup
s,t∈[r,T ]
t−s>r

1

(t− s)2β′

∣∣∣ ∫ t

s

(Bju −B
j
u−r) d

◦Biu

∣∣∣
= A1 +A2.

(7.4)

On the one hand, by (4.4),

A1 ≤ sup
s,t∈[r,T ]
t−s>r

‖B‖β′
|Bjt −B

j
t−r|

(t− s)β′

≤ sup
s,t∈[r,T ]
t−s>r

1

(t− s)β′
‖B‖β′‖B −B·−r‖∞

≤ sup
s,t∈[r,T ]
t−s>r

rβ

(t− s)β′
‖B‖2βT ε ≤ ‖B‖2βT εrε

that approaches zero when r tends to zero. On the other hand, we have that∫ t
s
(Bju−B

j
u−r) d

◦Biu is a continuous martingale, so it can be represented as a time-
changed Brownian motion: W∫ t

s
(Bju−Bju−r)2 du, where W is a Brownian motion. Now

we choose a ∈ (0, 1
2 ) such that 2β−2ε

2β+1 < a < 2β − 2ε. Applying Hölder property of

the Brownian motion, we have

A2 = sup
s,t∈[r,T ]
t−s>r

1

(t− s)2β′

∣∣W∫ t
s

(Bju−Bju−r)2 du

∣∣
≤ sup
s,t∈[r,T ]
t−s>r

Ca,T
(t− s)2β′

∣∣∣ ∫ t

s

(Bju −B
j
u−r)

2 du
∣∣∣a

≤ sup
s,t∈[r,T ]
t−s>r

Ca,T ‖B‖2aβ r2aβ(t− s)a−2β′

≤ Ca,T ‖B‖2aβ r2aβ+a−2β′ ,

that clearly appraoches zero as r tends to zero.



EJDE-2020/65 CONVERGENCE OF DELAY EQUATIONS 25

Now assume that t− s ≤ r. By integration by part formula, we have

‖B ⊗ (B −B·−r)‖2β′(r,T )

≤ sup
s,t∈[r,T ]
t−s≤r

1

(t− s)2β′

∣∣(Bit −Bis)(Bjt−r −Bjs−r)∣∣
+ sup
s,t∈[r,T ]
t−s≤r

1

(t− s)2β′

∣∣∣ ∫ t

s

(Bju−r −B
j
s−r) d

◦Biu

∣∣∣
+ sup
s,t∈[r,T ]
t−s≤r

1

(t− s)2β′

∣∣∣ ∫ t

s

(Biu −Bis) d◦Bju
∣∣∣

= V1 + V2 + V3.

(7.5)

The first term is easy to bound. Indeed,

V1 = sup
s,t∈[r,T ]
t−s≤r

|Bit −Bis|
(t− s)β

·
|Bjt−r −B

j
s−r|

(t− s)β
· (t− s)2β

(t− s)2β′
≤ ‖B‖2βr2ε.

For the other two terms we use inequality (5.8) of Hu and Nualart [9]. It states
that there exists a random variable Z such that, almost surely, for all s, t ∈ [0, T ]
we have ∣∣∣ ∫ t

s

(Biu −Bis) d◦Bju
∣∣∣ ≤ Z|t− s| log

1

|t− s|
.

Set M ′t =
∫ t
s
(Bju−r −B

j
s−r) d

◦Biu. Since the process {M ′t , t ∈ [s, T ]} is a continuous
martingale, we can follow the ideas in [9] to get that there exists a random variable
Z ′ such that, almost surely, for all s, t ∈ [0, T ] we have∣∣ ∫ t

s

(Bju−r −B
j
s−r) d

◦Biu
∣∣ ≤ Z ′|t− s| log

1

|t− s|
.

Hence

V2 ≤ Z ′(t− s)1−2β′ log
1

(t− s)
≤ Z ′r1−2β′ log

1

r

and V2 goes to zero when r tends to zero. V3 can be studied using the same
arguments.

It only remains to prove the case where i = j. To simplify the notation we will
not write the supra-index i.

For t − s ≤ r, we apply again the integration by parts formula and we obtain
that

‖B ⊗ (B −B·−r)‖2β′(r,T ) ≤ V ′1 + V ′2 + V ′3 + V ′4 ,

where V ′1 , V ′2 and V ′3 are the terms defined in (7.5) with i = j and

V ′4 := sup
s,t∈[r,T ]
t−s≤r

1

2
|t− s|1−β

′
≤ 1

2
r1−β′ .

So it only remains to study the terms V ′1 , V ′2 and V ′3 . Easily, for V ′1 we can repeat
the same arguments used for V1 and we also obtain that V ′1 ≤ ‖B‖2βr2ε. If we focus



26 M. BESALÚ, G. BINOTTO, C. ROVIRA EJDE-2020/65

in the second term, it can be written as

V ′2 = sup
s,t∈[r,T ]
t−s≤r

1

(t− s)2β′

∣∣∣ ∫ t

s

(Bu−r −Bs−r)dBu +
1

2

∫ t

s

Du(Bu−r −Bs−r)du
∣∣∣,

where Du denotes the de Malliavin derivative. It is easy to check that this Malliavin
derivative is zero. So, V ′2 is now a martingale and proceeding as in the case i 6= j
we obtain that

V ′2 ≤ Cr1−2β′ log
1

r
.

Finally, for the last term we have

V ′3 ≤ sup
s,t∈[r,T ]
t−s≤r

1

2(t− s)2β′
(Bt −Bs)2 ≤ 1

2
‖B‖2βr2ε.

Therefore, when t− s ≤ r, the three terms tend to zero when r tends to zero.
For the case t− s > r, by integration by parts formula we have

‖B ⊗ (B −B·−r)‖2β′(r,T )

≤ sup
s,t∈[r,T ]
t−s>r

1

(t− s)2β′

∣∣(Bt −Bs)(Bt−r −Bt)∣∣
+ sup
s,t∈[r,T ]
t−s>r

1

(t− s)2β′

∣∣∣ ∫ t

s

(Bu −Bu−r) d◦Bu −
1

2
(t− s)

∣∣∣.
The first term is analogous to the term A1 defined in (7.4), so it is bounded by
‖B‖2βT εrε.

For the second term, we can use the relation between Stratonovich and Itô
integrals, ∫ t

s

(Bu −Bu−r) d◦Bu =

∫ t

s

(Bu −Bu−r) dBu +
1

2
(t− s).

Set M ′′t =
∫ t
s
(Bu−Bu−r) dBu. Fixed s, the process {M ′′t , t ∈ [s, T ]} is a continuous

martingale. So following the ideas used for A2, we obtain

sup
s,t∈[r,T ]
t−s>r

1

(t− s)2β′

∣∣∣ ∫ t

s

(Bu −Bu−r) d◦Bu −
1

2
(t− s)

∣∣∣
≤ sup
s,t∈[r,T ]
t−s>r

1

(t− s)2β′

∣∣∣ ∫ t

s

(Bu −Bu−r) dBu
∣∣∣

≤ Ca,T ‖B‖2aβ r2aβ+a−2β′ ,

where a ∈ (0, 1
2 ) such that 2β−2ε

2β+1 < a < 2β − 2ε. Thus we obtain that ‖B ⊗ (B −
B·−r)‖2β′(r,T ) → 0 as we wish.

Inequality (7.2) can be proved with similar computations and the proof of (7.3)
follows immediately from the fact that

‖(B −B·−r)⊗B‖2β′(r,T ) ≤ V2 + V3.

�



EJDE-2020/65 CONVERGENCE OF DELAY EQUATIONS 27

Acknowledgements. This work was supported by the grant MTM2015-65092-P
from MINECO, Spain.

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Theory Relat. Fields, 111 (1998), 333–374.

Mireia Besalú
Departament de Genètica, Microbiologia i Estad́ıstica, Universitat de Barcelona,

Barcelona, Spain
Email address: mbesalu@ub.edu

Giulia Binotto

Departament de Matemàtiques, Universitat Autònoma de Barcelona, Barcelona, Spain
Email address: gbinotto@mat.uab.cat

Carles Rovira
Departament de Matemàtiques i Informàtica, Universitat de Barcelona, Barcelona,

Spain

Email address: carles.rovira@ub.edu


	1. Introduction
	2. Preliminaries
	3. Main result
	4. Estimates of integrals
	5. Estimates of the solutions
	6. Proof of main results
	7. Stochastic case
	Acknowledgements

	References