Electronic Journal of Differential Equations, Vol. 2020 (2020), No. 72, pp. 1–20.

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

REPRESENTATION OF SOLUTIONS OF A SECOND ORDER

DELAY DIFFERENTIAL EQUATION

KEE QIU, JINRONG WANG

Abstract. In this article, we study an inhomogeneous second order delay

differential equation on the fractal set Rαn (0 < α ≤ 1), based on the theory

of local calculus. We introduce delay cosine and sine type matrix functions and
give their properties on the fractal set. We give the representation of solutions

to second order differential equations with pure delay and two delays.

1. Introduction

In 2003, Khusainov and Shuklin [5] introduced the useful notation of delayed
exponential matrix functions, which is used to represent solutions of linear au-
tonomous time-delay systems with permutation matrices. Khusainov and Dibĺık
[4] transferred this idea for solving the Cauchy problem for an oscillating system
with second order and pure delay, by constructing special delayed matrix of co-
sine and sine type. These pioneer works led to many new results in integer and
fractional order differential equations with delays and discrete delayed system; see
[1, 2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 24, 25].

In 2012, Yang [20] transferred the standard calculus to local calculus on a fractal
set, which is utilized in various non-differentiable problems that appear in complex
systems of real-world phenomena. Furthermore, the non-differentiability occurring
in science and engineering was modeled by the local fractional ordinary or partial
differential equations [19, 21, 23]. As an effective research tool for continuous non-
differentiable function, local fractional calculus has attracted a lot of attention, see
[22].

In light of the above mentioned theory of local fractional calculus and delayed
matrix of cosine and sine type on real set, we shall introduce the notation of delayed
matrix of cosine and sine type on the fractal set Rαn (0 < α ≤ 1). The potential
applications of the delayed cosine and sine type matrix function on a fractal set
will be effective for homogeneous or inhomogeneous delay differential equation ona
fractal set with constant matrix coefficients. In this article, we use two new special
matrix functions to derive the representation of the solution to the following second

2010 Mathematics Subject Classification. 26A33, 28A80, 34A34.
Key words and phrases. Second order delay differential equations; delayed matrix functions;

fractal set.
c©2020 Texas State University.

Submitted March 24, 2020. Published July 7, 2020.

1



2 K. QIU, J. WANG EJDE-2020/72

order inhomogeneous delay differential equations on a fractal set:

y(2α)(x) +A2y(x− τ) = f(x), y(x) ∈ Rαn, x ≥ 0, τ > 0,

y(x) ≡ φ(x), y(α)(x) ≡ φ(α)(x), −τ ≤ x ≤ 0,
(1.1)

and

y(2α)(x) +A2y(x− τ1) +B2y(x− τ2) = f(x),

y(x) ∈ Rαn, x ≥ 0, τ1, τ2 > 0,

y(x) = φ(x), y(α)(x) = φ(α)(x), −τ ≤ x ≤ 0,

(1.2)

where y(nα)(x) is the nα-local fractional derivative on the fractal set Rαn (0 < α ≤
1), and f : R+

0 → Rαn is a given function, the matrices A = (aαij)n and B = (bαij)n
are permutable constant matrices on a fractal set with detA 6= 0 and detB 6= 0,
and φ(x) is an arbitrary twice local continuously differentiable vector function on
the fractal set, i.e., φ ∈ C2α([−τ, 0],Rαn).

Following the approach in [4, 5, 13], the main contribution of this article is
deriving the representation of (1.1) and (1.2) involving special matrix functions on
the fractal set. Section 2 introduces the concepts of matrix functions called delay
cosine and sine type on a fractal set, and gives their properties. Section 3 gives
the representation of solution to (1.1). The final section gives the representation of
solution to (1.2).

2. Preliminaries

We recall some basic definitions of local fractional calculus from [20, 22]. Let Rα
(0 < α ≤ 1) be α-type set of the real line. If aα, bα, cα ∈ Rα, then

(i) aα + bα ∈ Rα, aαbα ∈ Rα.
(ii) aα + bα = bα + aα = (a+ b)α = (b+ a)α and (a− b)α = aα − bα.

(iii) aα + (bα + cα) = (a+ b)α + cα.
(iv) aαbα = bαaα = (ab)α = (ba)α.
(v) aα(bαcα) = (aαbα)cα.

(vi) aα(bα + cα) = aαbα + aαcα.
(vii) aα + 0α = 0α + aα = aα and aα1α = 1αaα = aα.

Definition 2.1. A function f : R → Rα is called local fractional continuous at
x = x0, if for each ε > 0, there exists δ > 0 such that

|f(x)− f(x0)| < εα

holds whenever |x − x0| < δ, where ε, δ ∈ R. If f(x) is local fractional continuous
in the domain (a, b), then, we denote f(x) ∈ Cα(a, b).

Definition 2.2. Suppose that f ∈ Cα(a, b), 0 < α ≤ 1, and that for δ > 0 and
0 < |x− x0| < δ, the limit

D(α)f(x0) =
dαf(x)

dxα

∣∣∣
x=x0

= lim
x→x0

Γ(1 + α)(f(x)− f(x0))

(x− x0)α
,

exists and is finite. Then D(α)f(x0) is said to be the local fractional derivative of
f of order α at x = x0. It is convenient to denote the local fractional derivative as
f (α)(x0).



EJDE-2020/72 SECOND ORDER DELAY DIFFERENTIAL EQUATION 3

Let f(u, x) be defined in a domain ℘ of the ux-plane. The local fractional partial
derivative operator of f(u, x) of order α with respect to u in a domain ℘ is defined
by

f (α)(u0, x) =
∂αf(u, x)

∂uα

∣∣∣
u=u0

= lim
u→u0

Γ(1 + α)(f(u, x)− f(u0, x))

(u− u0)α
.

Similarly, the local fractional partial derivative operator of f(u, x) of higher order
nα with respect to u in a domain ℘ is defined by

f (nα)(u0, x) =
∂nαf(u, x)

∂uα

∣∣∣
u=u0

=

n times︷ ︸︸ ︷
∂α

∂uα
. . .

∂α

∂uα
f(u, x)

∣∣∣
u=u0

,

where n is a positive integer.

Definition 2.3. Let f ∈ Cα[a, b]. Then the local fractional integral of function f
of order α is defined by

aI
(α)
b f(x) =

1

Γ(1 + α)

∫ b

a

f(t)(dt)α =
1

Γ(1 + α)
lim

∆tj→0

N−1∑
j=0

f(tj)(∆tj)
α,

where ∆tj = tj+1 − tj with a = t0 < t1 < · · · < tN−1 < tN = b, [tj , tj+1] is a

partition of the interval [a, b]. Note that aI
(α)
a f(x) = 0 and aI

(α)
b f(x) = −bI(α)

a f(x)
if a < b.

Now we introduce the concepts matrix functions called delay cosine and sine
type on the fractal set Rαn (0 < α ≤ 1).

Definition 2.4. The delayed cosine type matrix function is deifined as

cosτ (Axα) :=



Θ, −∞ < x < −τ,
I −τ ≤ x < 0,

I −A2 x2α

Γ(1+2α) +A4 (x−τ)4α

Γ(1+4α) + . . .

+(−1)kA2k (x−(k−1)τ)2kα

Γ(1+2kα) , (k − 1)τ ≤ x < kτ, k ∈ N,

and the delayed sine type matrix function as

sinτ (Axα) :=


Θ, −∞ < x < −τ,
A (x+τ)α

Γ(1+α) , −τ ≤ x < 0,

A (x+τ)α

Γ(1+α) −A
3 x3α

Γ(1+3α) + . . .

+(−1)kA2k+1 (x−(k−1)τ)(2k+1)α

Γ(1+(2k+1)α) , (k − 1)τ ≤ x < kτ, k ∈ N,

where A = (aαij)n is a constant matrix on the fractal set, Θ is the null matrix and
I is the identity matrix. Moreover, N denotes the set of all nonnegative integers.

Next, we introduce two functions via an analogous delayed sine and cosine type
matrix functions on the fractal set, which are tools for solving differential equation
with two delays.

Definition 2.5. We define UA,Bτ1,τ2(x), V A,Bτ1,τ2(x) : R→ L(Rαn) as follows:

UA,Bτ1,τ2(x) =
∑
i,j≥0

iτ1+jτ2≤x

(−1)i+jCii+jA
2iB2j (x− iτ1 − jτ2)2(i+j)α

Γ(1 + 2(i+ j)α)
,



4 K. QIU, J. WANG EJDE-2020/72

V A,Bτ1,τ2(x) =
∑
i,j≥0

iτ1+jτ2≤x

(−1)i+jCii+jA
2iB2j (x− iτ1 − jτ2)(2(i+j)+1)α

Γ(1 + (2(i+ j) + 1)α)
,

where τ1, τ2 > 0, A,B are n × n constant matrixes on the fractal set Rαn, by
definition UA,Bτ1,τ2(x) = 0, V A,Bτ1,τ2(x) = 0 if x < 0.

Some properties of UA,Bτ1,τ2(x), V A,Bτ1,τ2(x) are established in Lemma 2.12 below.
Now, we give some properties associated with the local fractional derivatives and
the local fractional integrals on the fractal set, see [20, 22].

Lemma 2.6. (i) Suppose that g(α)(x) = f(x) ∈ Cα[a, b], then

aI
(α)
b f(x) = g(b)− g(a).

(ii) Suppose that f, g ∈ Cα[a, b], and , g ∈ Dα(a, b), then

aI
(α)
b (f(x)g(α)(x)) = f(x)g(x)

∣∣∣b
a
− aI

(α)
b (f (α)(x)g(x)).

(iii) Suppose that f ∈ Cα[a, b], then

dα

dxα

∫ x

a

f(ξ)(dξ)α = Γ(1 + α)f(x), x ∈ (a, b),

dα

dxα

∫ u(x)

a

f(ξ)(dξ)α = Γ(1 + α)f(u(x))(u′(x))α,

for x ∈ [a, b] and u ∈ C1[a, b].

(iv) Suppose that f(u, x) ∈ Cα([a, b], [c, d]), ∂α

∂uα f ∈ Cα([a, b], [c, d]), then

φ(u) =
1

Γ(1 + α)

∫ b

a

f(u, x)(dx)α

is a local fractional derivative on [a, b], and

dκ

duκ
φ(u) =

1

Γ(1 + α)

∫ b

a

dκf(u, x)

duκ
(dx)α, 0 < κ ≤ 1.

(v) Suppose that f(u, x) ∈ Cα([a, b], [c, d]), ∂α

∂uα f ∈ Cα([a, b], [c, d]), c(u), d(u) ∈
C1[a, b], c ≤ c(u) ≤ d, c ≤ d(u) ≤ d for any u ∈ [a, b], then

φ(u) =
1

Γ(1 + α)

∫ d(u)

c(u)

f(u, x)(dx)α

is a local fractional derivative on [a, b], and

dκ

duκ
φ(u) =

1

Γ(1 + α)

∫ d(u)

c(u)

dκf(u, x)

duκ
(dx)α

+ f(u, d(u))(d′(u))α − f(u, c(u))(c′(u))α, 0 < κ ≤ 1.

Lemma 2.7. We have

dαxkα

dxα
=

Γ(1 + kα)

Γ(1 + (k − 1)α)
x(k−1)α,

1

Γ(1 + α)

∫ b

a

xkα(dx)α =
Γ(1 + kα)

Γ(1 + (k + 1)α)
(b(k+1)α − a(k+1)α), k > 0.



EJDE-2020/72 SECOND ORDER DELAY DIFFERENTIAL EQUATION 5

Lemma 2.8. Suppose that f(x), g(x) ∈ Dα(a, b), λ, γ ∈ R. The local fractional
differentiation rules of non-differentiable functions defined on fractal set are listed
as follows:

(i) (λf(x)± γg(x))(α) = λf (α)(x) + γg(α)(x).
(ii) (f(x)g(x))(α) = f (α)(x)g(x) + f(x)g(α)(x).
(iii) (f(x)/g(x))(α) = (f (α)(x)g(x)− f(x)g(α)(x))/g2(x), provided g(x) 6= 0.

Suppose that g(x) = f(u(x)), and f (α)(u) and u′(x) exist. Then

g(α)(x) = f(u(x))(α) = f (α)(u)(u′(x))α.

Lemma 2.9. Suppose that f(x), g(x) ∈ Cα[a, b], λ, γ ∈ R. The local fractional
integral rules of non-differentiable functions defined on a fractal set are listed as
follows:

(i) aI
(α)
b (λf(x)± γg(x)) = λaI

(α)
b f(x) + γaI

(α)
b g(x).

(ii) aI
(α)
b f(x) = aI

(α)
c f(x) + cI

(α)
b f(x), provided a < c < b.

It should be noted that the fractional derivative in the following represent the
one-side derivative in nodes x = kτ, k = 0, 1, 2, . . . and x = τ1, τ2.

Lemma 2.10. For delayed cosine type matrix function cosτ (Axα), one has(
cosτ (Axα)

)(α)

= −A sinτ (A(x− τ)α),(
cosτ (Axα)

)(2α)

= −A2 cosτ (A(x− τ)α).

(2.1)

In other words, the delayed cosine type matrix function is a solution of differential
equation of the second order with pure delay on fractal set

y(2α)(x) +A2y(x− τ) = 0,

subject to initial value condition y(x) = I, −τ ≤ x ≤ 0.

Proof. Let A and τ are fixed. Firstly, for arbitrary x ∈ (−∞,−τ), cosτ (Axα) =
sinτ (A(x− τ)α) = cosτ (A(x− τ)α) = Θ. Obviously, (2.1) holds.

Secondly, cosτ (Axα) = I, sinτ (A(x − τ)α) = Θ, cosτ (A(x − τ)α) = Θ, which

reduces to
(

cosτ (Axα)
)(α)

= I(α) = Θ = sinτ (A(x − τ)α) and
(

cosτ (Axα)
)(2α)

=

I(2α) = Θ = cosτ (A(x− τ)α) for arbitrary x ∈ [−τ, 0), then (2.1) holds.
Finally, for an arbitrary x : (k − 1)τ ≤ x < kτ), we have(

cosτ (Axα)
)(α)

=
(
I −A2 x2α

Γ(1 + 2α)
+A4 (x− τ)4α

Γ(1 + 4α)

+ · · ·+ (−1)kA2k (x− (k − 1)τ)2kα

Γ(1 + 2kα)

)(α)

,

(2.2)

applying Lemmas 2.7 and 2.8, we have(
cosτ (Axα)

)(α)

= −A2 xα

Γ(1 + α)
+A4 (x− τ)3α

Γ(1 + 3α)
+ · · ·+ (−1)kA2k (x− (k − 1)τ)(2k−1)α

Γ(1 + (2k − 1)α)

= −A
(
A

xα

Γ(1 + α)
−A3 (x− τ)3α

Γ(1 + 3α)
+ · · ·+ (−1)k−1A2k−1 (x− (k − 1)τ)(2k−1)α

Γ(1 + (2k − 1)α)

)



6 K. QIU, J. WANG EJDE-2020/72

= −A sinτ (A(x− τ)α).

Then(
cosτ (Axα)

)(2α)

=
((

cosτ (Axα)
)(α))(α)

= −A
(

sinτ (A(x− τ)α)
)(α)

= −A
(
A

xα

Γ(1 + α)
−A3 (x− τ)3α

Γ(1 + 3α)
+ · · ·+ (−1)k−1A2k−1 (x− (k − 1)τ)(2k−1)α

Γ(1 + (2k − 1)α)

)(α)

= −A
(
A−A3 (x− τ)2α

Γ(1 + 2α)
+ · · ·+ (−1)k−1A2k−1 (x− (k − 1)τ)2(k−1)α

Γ(1 + 2(k − 1)α)

)
= −A2

(
I −A2 (x− τ)2α

Γ(1 + 2α)
+ · · ·+ (−1)k−1A2(k−1) (x− (k − 1)τ)2(k−1)α

Γ(1 + 2(k − 1)α)

)
= −A2 cosτ (A(x− τ)α).

This completes the proof. �

Remark 2.11. Using a method similar to the one in the proof of Lemma 2.10, the
following rule of fractional differentiation is true for the sine type matrix function.(

sinτ (Axα)
)(α)

= A cosτ (Axα),
(

sinτ (Axα)
)(2α)

= −A2 sinτ (A(x− τ)α).

In this case, the delayed sine type matrix function is a solution of differential system
of the second order with pure delay on fractal set

y(2α)(x) +A2y(x− τ) = 0,

that satisfies the initial conditions y(x) = A (x+τ)α

Γ(1+α) for −τ ≤ x ≤ 0.

Lemma 2.12. Let τ1, τ2 > 0, A = (aαij)n, B = (bαij)n be permutable constant

matrices on fractal set with detA 6= 0,detB 6= 0. Then both UA,Bτ1,τ2(x) and V A,Bτ1,τ2(x)
satisfy

y(2α)(x) +A2y(x− τ1) +B2y(x− τ2) = 0. (2.3)

for any x ∈ R.

Proof. (i) If τ := τ1 = τ2, then

UA,Bτ1,τ2(x) =
∑
i,j≥0

iτ1+jτ2≤x

(−1)i+jCii+jA
2iB2j (x− iτ1 − jτ2)2(i+j)α

Γ(1 + 2(i+ j)α)

=
∑
k≥0
kτ≤x

∑
i,j≥0
i+j=k

(−1)i+jCii+jA
2iB2j (x− (i+ j)τ)2(i+j)α

Γ(1 + 2(i+ j)α)

=
∑
k≥0
kτ≤x

(−1)k(A2 +B2)k
(x− kτ)2kα

Γ(1 + 2kα)

= cosτ

(√
A2 +B2(x− τ)α

)
,



EJDE-2020/72 SECOND ORDER DELAY DIFFERENTIAL EQUATION 7

where k = i+ j. Using Lemma 2.10, we have cosτ
(√
A2 +B2(x−τ)α

)
is a solution

of y(2α)(x) + (A2 +B2)y(x− τ) = 0, i.e.(
UA,Bτ1,τ2(x)

)(2α)

+A2UA,Bτ1,τ2(x− τ1) +B2UA,Bτ1,τ2(x− τ2) = 0.

(ii) If τ1 6= τ2, suppose that τ1 < τ2. Firstly, we suppose that x < τ1, so that

UA,Bτ1,τ2(x) = I, UA,Bτ1,τ2(x− τ1) = 0, UA,Bτ1,τ2(x− τ2) = 0,

since iτ1 + jτ2 ≤ x < τ1, Definition 2.5 indicates that i = 0 and j = 0. Thus, (2.3)
holds.

Secondly, we suppose that τ1 ≤ x < τ2, i.e., x− τ2 < 0, then

UA,Bτ1,τ2(x− τ2) = 0,

and

UA,Bτ1,τ2(x) =
∑
i≥0
iτ1≤x

(−1)iA2i (x− iτ1)2iα

Γ(1 + 2iα)
= cosτ (A(x− τ1)α),

since iτ1 + jτ2 ≤ x < τ2 in the Definition 2.5 indicates that j = 0, using Lemma
2.10, we obtain (2.3).

Finally, we suppose that x ≥ τ2. It suffices to note that

UA,Bτ1,τ2(x) := I + ω1(x) + ω2(x) + ω3(x), (2.4)

where

ω1(x) =
∑
i≥1
iτ1≤x

(−1)iA2i (x− iτ1)2iα

Γ(1 + 2iα)
= −A2 (x− τ1)2α

Γ(1 + 2α)
+A4 (x− 2τ1)4α

Γ(1 + 4α)
− . . . ,

ω2(x) =
∑
j≥1
jτ2≤x

(−1)jB2j (x− jτ2)2jα

Γ(1 + 2jα)
= −B2 (x− τ2)2α

Γ(1 + 2α)
+B4 (x− 2τ2)4α

Γ(1 + 4α)
− . . . ,

ω3(x) =
∑
i,j≥1

iτ1+jτ2≤x

(−1)i+jCii+jA
2iB2j (x− iτ1 − jτ2)2(i+j)α

Γ(1 + 2(i+ j)α)
.

Calculating the second local fractal derivative of ω1(x), we have

ω
(2α)
1 (x) =

∑
i≥1
iτ1≤x

(−1)iA2i (x− iτ1)2(i−1)α

Γ(1 + 2(i− 1)α)

= −A2
∑
i≥1
iτ1≤x

(−1)i−1A2(i−1) (x− τ1 − (i− 1)τ1)2(i−1)α

Γ(1 + 2(i− 1)α)

= −A2
∑
i≥0

iτ1≤x−τ1

(−1)iA2i (x− τ1 − iτ1)2iα

Γ(1 + 2iα)

= −A2 −A2
∑
i≥1

iτ1≤x−τ1

(−1)iA2i (x− τ1 − iτ1)2iα

Γ(1 + 2iα)

= −A2 −A2ω1(x− τ1).



8 K. QIU, J. WANG EJDE-2020/72

Analogously, we have

ω
(2α)
2 (x) = −B2 −B2ω2(x− τ2).

By using the properties of binomial numbers Cmn+1 = Cmn +Cm−1
n and Ckn = Cn−kn ,

for n,m ≥ 1, we find that

ω
(2α)
3 (x)

=
∑
i,j≥1

iτ1+jτ2≤x

(−1)i+jCii+jA
2iB2j (x− iτ1 − jτ2)2(i+j−1)α

Γ(1 + 2(i+ j − 1)α)

=
∑
i,j≥1

iτ1+jτ2≤x

(−1)i+jCi−1
i+j−1A

2iB2j (x− iτ1 − jτ2)2(i+j−1)α

Γ(1 + 2(i+ j − 1)α)

+
∑
i,j≥1

iτ1+jτ2≤x

(−1)i+jCii+j−1A
2iB2j (x− iτ1 − jτ2)2(i+j−1)α

Γ(1 + 2(i+ j − 1)α)

=
∑
i,j≥1

iτ1+jτ2≤x

(−1)i+jCi−1
i−1+jA

2iB2j (x− τ1 − (i− 1)τ1 − jτ2)2(i−1+j)α

Γ(1 + 2(i− 1 + j)α)

+
∑
i,j≥1

iτ1+jτ2≤x

(−1)i+jCj−1
i+j−1A

2iB2j (x− iτ1 − jτ2)2(i+j−1)α

Γ(1 + 2(i+ j − 1)α)

=
∑
i,j≥1

iτ1+jτ2≤x

(−1)i+jCi−1
i−1+jA

2iB2j (x− τ1 − (i− 1)τ1 − jτ2)2(i−1+j)α

Γ(1 + 2(i− 1 + j)α)

+
∑
i,j≥1

iτ1+jτ2≤x

(−1)i+jCj−1
i+j−1A

2iB2j (x− τ2 − iτ1 − (j − 1)τ2)2(i+j−1)α

Γ(1 + 2(i+ j − 1)α)

= −A2
∑
i,j≥1

iτ1+jτ2≤x

(−1)i−1+jCi−1
i−1+jA

2(i−1)B2j (x− τ1 − (i− 1)τ1 − jτ2)2(i−1+j)α

Γ(1 + 2(i− 1 + j)α)

−B2
∑
i,j≥1

iτ1+jτ2≤x

(−1)i+j−1Cj−1
i+j−1A

2iB2(j−1) (x− τ2 − iτ1 − (j − 1)τ2)2(i+j−1)α

Γ(1 + 2(i+ j − 1)α)
.

We now replace i − 1 by i in the first sum and j − 1 → j in the second sum
above, then we have

ω
(2α)
3 (x) = −A2

∑
i≥0,j≥1

iτ1+jτ2≤x−τ1

(−1)i+jCii+jA
2iB2j (x− τ1 − iτ1 − jτ2)2(i+j)α

Γ(1 + 2(i+ j)α)

−B2
∑

i≥1,j≥0
iτ1+jτ2≤x−τ2

(−1)i+jCji+jA
2iB2j (x− τ2 − iτ1 − jτ2)2(i+j)α

Γ(1 + 2(i+ j)α)
.



EJDE-2020/72 SECOND ORDER DELAY DIFFERENTIAL EQUATION 9

Further, we split the first sum into i = 0 and i ≥ 1 and the second sum into j = 0
and j ≥ 1, then

ω
(2α)
3 (x) = −A2

∑
j≥1

jτ2≤x−τ1

(−1)jB2j (x− τ1 − jτ2)2jα

Γ(1 + 2jα)

−A2
∑
i,j≥1

iτ1+jτ2≤x−τ1

(−1)i+jCii+jA
2iB2j (x− τ1 − iτ1 − jτ2)2(i+j)α

Γ(1 + 2(i+ j)α)

−B2
∑
i≥1

iτ1≤x−τ2

(−1)iA2i (x− τ2 − iτ1)2iα

Γ(1 + 2iα)

−B2
∑
i,j≥1

iτ1+jτ2≤x−τ2

(−1)i+jCji+jA
2iB2j (x− τ2 − iτ1 − jτ2)2(i+j)α

Γ(1 + 2(i+ j)α)

= −A2ω2(x− τ1)−A2ω3(x− τ1)−B2ω1(x− τ2)−B2ω3(x− τ2).

Substituting the formulas for ω
(2α)
1 (x), ω

(2α)
2 (x), ω

(2α)
3 (x) and calculating the second

fractal derivative both sides of (2.4), we obtain(
UA,Bτ1,τ2(x)

)(2α)

= −A2
(
I + ω1(x− τ1) + ω2(x− τ1) + ω3(x− τ1)

)
−B2

(
I + ω1(x− τ2) + ω2(x− τ2) + ω3(x− τ2)

)
= −A2UA,Bτ1,τ2(x− τ1)−B2UA,Bτ1,τ2(x− τ2).

Thus, we arrive at the relation. Further, we proceed by analogy with V A,Bτ1,τ2(x).

Statement holds with V A,Bτ1,τ2(x) instead of UA,Bτ1,τ2(x). Therefore, we have the results.
�

3. Solutions of differential equation with pure delay on fractal set

We study the linear homogeneous differential delay equations on fractal sets,

y(2α)(x) +A2y(x− τ) = 0, y(x) ∈ Rαn, x ≥ 0, τ > 0,

y(x) = φ(x), y(α)(x) = φ(α)(x), −τ ≤ x ≤ 0.
(3.1)

Theorem 3.1. Suppose that the matrix A = (aαij)n is a constant matrix on a fractal
set with detA 6= 0, and φ(x) ∈ C2α([−τ, 0],Rαn). Then the solution y(x) of (3.1)
can be expressed as

y(x) = (cosτ (Axα))φ(−τ) +A−1(sinτ (Axα))φ(α)(−τ)

+
A−1

Γ(1 + α)

∫ 0

−τ
sinτ (A(x− τ − s)α)φ(2α)(s)(ds)α.

(3.2)

Proof. We seek for a solution of (3.1) in the form

y(x) = (cosτ (Axα))c1 + (sinτ (Axα))c2

+
1

Γ(1 + α)

∫ 0

−τ
sinτ (A(x− τ − s)α)z(2α)(s)(ds)α,

(3.3)



10 K. QIU, J. WANG EJDE-2020/72

where c1, c2 are unknown constant vectors on Rαn and z(x) : [−τ,+∞) → Rαn is
an unknown twice continuously differentiable vector function. From Lemma 2.10
and Remark 2.11 , i.e., due to linearity, we know that (3.3) is a solution of (3.1)
for arbitrary c1, c2 and vector function z(x) ∈ C2α([−τ, 0],Rαn). Now we try to fix
the constants c1, c2 and the vector function z(x) in such manner that the initial
conditions y(x) = φ(x), y(α)(x) = φ(α)(x),−τ ≤ x ≤ 0, are satisfied.

We use (3.3) to represent the first initial condition y(x) = φ(x), −τ ≤ x ≤ 0,
i.e.,

(cosτ (Axα))c1 + (sinτ (Axα))c2

+
1

Γ(1 + α)

∫ 0

−τ
sinτ (A(x− τ − s)α)z(2α)(s)(ds)α = φ(x).

This leads to

c1 + c2A
(x+ τ)α

Γ(1 + α)
+

1

Γ(1 + α)

∫ 0

−τ
sinτ (A(x− τ − s)α)z(2α)(s)(ds)α = φ(x), (3.4)

where

(cosτ (Axα)) = I, (sinτ (Axα)) = A
(x+ τ)α

Γ(1 + α)
.

Since

sinτ (A(x− τ − s)α) =

{
0, x ≤ s ≤ 0,

A (x−s)α
Γ(1+α) , −τ ≤ s ≤ x,

it follows that

1

Γ(1 + α)

∫ 0

−τ
sinτ (A(x− τ − s)α)z(2α)(s)(ds)α

=
A(

Γ(1 + α)
)2

∫ x

−τ
(x− s)αz(2α)(s)(ds)α.

(3.5)

Using (i) and (ii) of Lemma 2.6 and Lemma 2.7 to the right-hand side of (3.5),
i.e., using local fractional integration by parts and local fractional derivative for
the right of (3.5), it is necessary to verify that

A(
Γ(1 + α)

)2

∫ x

−τ
(x− s)αz(2α)(s)(ds)α

=
A

Γ(1 + α)

(
(x− s)αz(α)(s)

∣∣∣x
−τ
− 1

Γ(1 + α)

∫ x

−τ
z(α)(s)((x− s)α)(α)(ds)α

)
=

A

Γ(1 + α)

(
(x− s)αz(α)(s)

∣∣∣x
−τ

+
1

Γ(1 + α)

∫ x

−τ
z(α)(s)(ds)αΓ(1 + α)

)
=

A

Γ(1 + α)

(
− (x+ τ)αz(α)(−τ) +

∫ x

−τ
z(α)(s)(ds)α

)
= − A

Γ(1 + α)
(x+ τ)αz(α)(−τ) +

A

Γ(1 + α)

∫ x

−τ
z(α)(s)(ds)α

= − A

Γ(1 + α)
(x+ τ)αz(α)(−τ) +Az(s)

∣∣∣x
−τ

= − A

Γ(1 + α)
(x+ τ)αz(α)(−τ) +Az(x)−Az(−τ).



EJDE-2020/72 SECOND ORDER DELAY DIFFERENTIAL EQUATION 11

submitting this and (3.5) into (3.4), it follows that

c1 + c2A
(x+ τ)α

Γ(1 + α)
− A

Γ(1 + α)
(x+ τ)αz(α)(−τ) +Az(x)−Az(−τ) = φ(x). (3.6)

Let us rewrite the above equalityin the form(
c1 −Az(−τ)

)
+
(
c2 − z(α)(−τ)

)A(x+ τ)α

Γ(1 + α)
+Az(x) = φ(x). (3.7)

Applying Lemmas 2.7 and 2.8 to both sides of (3.7) and paying attention to the
second initial condition y(α)(x) = φ(α)(x), −τ ≤ x ≤ 0, we have

A
(
c2 − z(α)(−τ)

)
+Az(α)(x) = φ(α)(x). (3.8)

In this case, a combination of (3.7) and (3.8), one has

c1 = φ(−τ),c2 = A−1φ(α)(−τ), z(x) = A−1φ(x),

since det(A) 6= 0. Putting c1, c2 and z(x) into (3.3), we obtain (3.2). �

Remark 3.2. To obtain some alternative conclusions, with the assumptions in
Theorem 3.1, one can apply integration by parts via Lemma 2.6. We have

A−1

Γ(1 + α)

∫ 0

−τ
sinτ (A(x− τ − s)α)φ(2α)(s)(ds)α

= A−1 1

Γ(1 + α)

∫ 0

−τ
sinτ (A(x− τ − s)α)(dφ(α)(s))α

= A−1
(

sinτ (A(x− τ − s)α)φ(α)(s)
∣∣∣0
−τ

+
1

Γ(1 + α)

∫ 0

−τ
φ(α)(s)A cosτ (A(x− τ − s)α)(ds)α

)
= A−1 sinτ (A(x− τ)α)φ(α)(0)−A−1 sinτ (Axα)φ(α)(−τ)

+
1

Γ(1 + α)

∫ 0

−τ
A cosτ ((x− τ − s)α)(dφ(s))α

= A−1 sinτ (A(x− τ)α)φ(α)(0)−A−1 sinτ (Axα)φ(α)(−τ)

+ cosτ (A(x− τ − s)α)φ(s)
∣∣∣0
−τ

− A

Γ(1 + α)

∫ 0

−τ
φ(s) sinτ ((x− 2τ − s)α)(ds)α

= A−1 sinτ (A(x− τ)α)φ(α)(0)−A−1 sinτ (Axα)φ(α)(−τ)

+ cosτ (A(x− τ)α)φ(0)

− cosτ (Axα)φ(−τ)− A

Γ(1 + α)

∫ 0

−τ
sinτ ((x− 2τ − s)α)φ(s)(ds)α.

This implies that the conclusion of Theorem 3.1 can be expressed as

y(x) = (cosτ (A(x− τ)α))φ(0) +A−1(sinτ (A(x− τ)α))φ(α)(0)

− A

Γ(1 + α)

∫ 0

−τ
sinτ (A(x− 2τ − s)α)φ(s)(ds)α.



12 K. QIU, J. WANG EJDE-2020/72

To end this section, we consider the inhomogeneous differential delay system on
a fractal set

y(2α)(x) +A2y(x− τ) = f(x), y(x) ∈ Rαn, x ≥ 0, τ > 0,

y(x) = 0, −τ ≤ x ≤ 0.
(3.9)

Theorem 3.3. Suppose that the matrix A = (aαij)n is a constant matrix on a fractal

set with detA 6= 0, and f : R+
0 → Rαn is a given function. Then the solution y0(x)

of the inhomogeneous equation (3.9) can be expressed as

y0(x) =
A−1

Γ(1 + α)

∫ x

0

sinτ (A(x− τ − s)α)f(s)(ds)α.

Proof. We will try to seek a particular solution y0(x) of the inhomogeneous equation
(3.9), employing the method of variation of an arbitrary constant in the form

y0(x) =
1

Γ(1 + α)

∫ x

0

sinτ (A(x− τ − s)α)C(s)(ds)α,

where C(s), 0 ≤ s ≤ x, is an unknown function. Local fractional differentiating the
function y0(x), we obtain

y
(α)
0 (x)

=
1

Γ(1 + α)

∫ x

0

A cosτ (A(x− τ − s)α)C(s)(ds)α + sinτ (A(x− τ − s)α)C(s)
∣∣∣
s=x

=
A

Γ(1 + α)

∫ x

0

cosτ (A(x− τ − s)α)C(s)(ds)α,

and

y
(2α)
0 (x)

=
A

Γ(1 + α)

∫ x

0

(
(A sinτ (A(x− 2τ − s)α)C(s)

)
(ds)α

+A cosτ (A(x− τ − s)α)C(s)
∣∣∣
s=x

= − A2

Γ(1 + α)

∫ x

0

sinτ (A(x− 2τ − s)α)C(s)(ds)α +A cosτ (A(−τ)α)C(x)

= − A2

Γ(1 + α)

∫ x

0

sinτ (A(x− 2τ − s)α)C(s)(ds)α +AC(x),

since sinτ (A(x− 2τ − s)α) = 0, when x− τ ≤ s ≤ x. Hence,

y
(2α)
0 (x) = − A2

Γ(1 + α)

∫ x−τ

0

sinτ (A(x− 2τ − s)α)C(s)(ds)α +AC(x).

Substituting y
(2α)
0 (x) and y0(x− τ) into system (3.9), we obtain

− A2

Γ(1 + α)

∫ x−τ

0

sinτ (A(x− 2τ − s)α)C(s)(ds)α +AC(x)

+
A2

Γ(1 + α)

∫ x−τ

0

sinτ (A(x− 2τ − s)α)C(s)(ds)α = f(x).

Since detA 6= 0, we obtain C(x) = A−1f(x). Thus, we arrive at the results in
Theorem 3.3. �



EJDE-2020/72 SECOND ORDER DELAY DIFFERENTIAL EQUATION 13

As we know, the solution of system (1.1) is the sum of solution of homogeneous
problem (3.1) and a particular solution of (3.9). Therefore, collecting the results
of Theorem 3.1, Remark 3.2 and Theorem 3.3, we obtain the following results.

Corollary 3.4. Solution of (1.1) can be represented in the form

y(x) =



φ(x), −τ ≤ x ≤ 0,

(cosτ (Axα))φ(−τ)

+ A−1

Γ(1+α)

∫ 0

−τ sinτ (A(x− τ − s)α)φ(2α)(s)(ds)α

+A−1(sinτ (Axα))φ(α)(−τ)

+ A−1

Γ(1+α)

∫ x
0

sinτ (A(x− τ − s)α)f(s)(ds)α, x ≥ 0,

or

y(x) =



φ(x), −τ ≤ x ≤ 0,

(cosτ (A(x− τ)α))φ(0)

− A
Γ(1+α)

∫ 0

−τ sinτ (A(x− 2τ − s)α)φ(s)(ds)α

+A−1(sinτ (A(x− τ)α))φ(α)(0)

+ A−1

Γ(1+α)

∫ x
0

sinτ (A(x− τ − s)α)f(s)(ds)α, x ≥ 0.

4. Solutions of differential equation with two delays on a fractal
set

In this section, we deduce the representation of a solution of system (1.2) by using
matrix functions UA,Bτ1,τ2(x), V A,Bτ1,τ2(x) which is counterpart of formulas in Corollary
3.4.

Theorem 4.1. Suppose that the matrix A = (aαij)n, B = (bαij)n are permutable
constant matrix on fractal set with detA 6= 0, detB 6= 0. Let τ1, τ2 > 0, τ :=
max{τ1, τ2}, φ ∈ Cα([−τ, 0],Rαn), and f : [0,∞)→ Rαn be a given function. Then
the solution y(x) of (1.2) has the form

y(x) =



φ(x), −τ ≤ x ≤ 0,

U(x)φ(0) + V (x)φ(α)(0)

−A2 1
Γ(1+α)

∫ 0

−τ1 V (x− τ1 − s)φ(s)(ds)α

−B2 1
Γ(1+α)

∫ 0

−τ2 V (x− τ2 − s)φ(s)(ds)α

+ 1
Γ(1+α)

∫ x
0
V (x− s)f(s)(ds)α, x ≥ 0,

(4.1)

where U(x) = UA,Bτ1,τ2(x), V (x) = V A,Bτ1,τ2(x).

Proof. The main steps on the proof are as follows:
Step1: we show that Theorem 4.1 hold by using Lemma 2.12 and Corollary 3.4

if τ1 = τ2.
Step2: let τ1 < τ2, we show that y(x) satisfies the initial value condition on

[−τ, 0] and y(0) = φ(0), y(α)(0) = φ(α)(0) from the form of y(x) and the calculating
of local fractal derivative of y(x) .

Step3: we show that y(x) is a solution of system (1.2) from the following three
cases because x ≥ 0: 0 ≤ x < τ1, τ1 ≤ x < τ2 and x ≥ τ2.

The detailed proof process is as below:



14 K. QIU, J. WANG EJDE-2020/72

(i) We consider only the case τ1 6= τ2 because if τ1 = τ2, then one can use Lemma
2.12 and Corollary 3.4 to show that Theorem 4.1 holds.

(ii) We show that y(x) satisfies the initial value condition on [−τ, 0] and y(0) =
φ(0), y(α)(0) = φ(α)(0). Due to the form of y(x), if x ≥ 0, suppose that x <
min{τ1, τ2}, then we have

U(x) = I, V (x) =
xα

Γ(1 + α)
,

V (x− τi − s) =

{
0, s ∈ [x− τi, 0],
(x−τi−s)α

Γ(1+α) , s ∈ [−τi, x− τi],
V (x− s) =

(x− s)α

Γ(1 + α)
,

for s ∈ [0, x], imply that x − s ∈ [0, x] ⊂ [0,min{τ1, τ2}]. After some calculation,
we obtain

y(x)

= φ(0) +
xα

Γ(1 + α)
φ(α)(0)− A2

Γ(1 + α)

∫ x−τ1

−τ1

(x− τ1 − s)α

Γ(1 + α)
φ(s)(ds)α

− B2

Γ(1 + α)

∫ x−τ2

−τ2

(x− τ2 − s)α

Γ(1 + α)
φ(s)(ds)α +

1

Γ(1 + α)

∫ x

0

(x− s)α

Γ(1 + α)
f(s)(ds)α.

Calculating local fractal derivative of y(x), from Lemmas 2.6, 2.7, 2.8, 2.9, we have

y(α)(x) = φ(α)(0)− A2

Γ(1 + α)

∫ x−τ1

−τ1
φ(s)(ds)α

− B2

Γ(1 + α)

∫ x−τ2

−τ2
φ(s)(ds)α +

1

Γ(1 + α)

∫ x

0

f(s)(ds)α.

Let x→ 0+, then

lim
x→0+

y(x) = φ(0), lim
x→0+

y(α)(x) = φ(α)(0).

Obviously, y(x) satisfies the initial condition [−τ, 0], which completes the proof for
this case.

(iii) Now we show that y(x) is a solution of system (1.2). Since τ1 6= τ2, let
τ1 < τ2. Firstly, if 0 ≤ x < τ1, then

y(2α)(x) = −A2φ(x− τ1)−B2φ(x− τ2) + f(x),

at the sometime, φ(x− τ1) = y(x− τ1), φ(x− τ2) = y(x− τ2) while 0 ≤ x < τ1 < τ2,
we have x− τ1 < 0, x− τ2 < 0. We find that y(x) is a solution of system (1.2).

Secondly, if τ1 ≤ x < τ2, then for any s ∈ [x− τ2, 0], V (x− τ2 − s) = 0, and we
have

y(x) = U(x)φ(0) + V (x)φ(α)(0)− A2

Γ(1 + α)

∫ 0

−τ1
V (x− τ1 − s)φ(s)(ds)α

− B2

Γ(1 + α)

∫ x−τ2

−τ2
V (x− τ2 − s)φ(s)(ds)α

+
1

Γ(1 + α)

∫ x

0

V (x− s)f(s)(ds)α.



EJDE-2020/72 SECOND ORDER DELAY DIFFERENTIAL EQUATION 15

Calculating the local fractal derivative of y(x), from Lemma 2.6 and the prop-
erties of U(x) = UA,Bτ1,τ2(x), V (x) = V A,Bτ1,τ2(x), we have

y(α)(x)

= U (α)(x)φ(0) + V (α)(x)φ(α)(0)− A2

Γ(1 + α)

∫ 0

−τ1
V (α)(x− τ1 − s)φ(s)(ds)α

− B2

Γ(1 + α)

∫ x−τ2

−τ2
V (α)(x− τ2 − s)φ(s)(ds)α

−B2V (x− τ2 − (x− τ2))φ(x− τ2)((x− τ2)′)α

+B2V (x− τ2 − (−τ2))φ(−τ2)((−τ2)′)α

+
1

Γ(1 + α)

∫ x

0

V (α)(x− s)f(s)(ds)α + V (x− x)f(x)(x′)α

− V (x− 0)f(0)(0′)α

= U (α)(x)φ(0) + V (α)(x)φ(α)(0)− A2

Γ(1 + α)

∫ 0

−τ1
V (α)(x− τ1 − s)φ(s)(ds)α

− B2

Γ(1 + α)

∫ x−τ2

−τ2
V (α)(x− τ2 − s)φ(s)(ds)α −B2V (0)φ(x− τ2)

+
1

Γ(1 + α)

∫ x

0

V (α)(x− s)f(s)(ds)α + V (0)f(x)

Clearly, V (0) = 0 from Definition 2.5, thus

y(α)(x)

= U (α)(x)φ(0) + V (α)(x)φ(α)(0)− A2

Γ(1 + α)

∫ 0

−τ1
V (α)(x− τ1 − s)φ(s)(ds)α

− B2

Γ(1 + α)

∫ x−τ2

−τ2
V (α)(x− τ2 − s)φ(s)(ds)α

+
1

Γ(1 + α)

∫ x

0

V (α)(x− s)f(s)(ds)α

= U (α)(x)φ(0) + V (α)(x)φ(α)(0)− A2

Γ(1 + α)

∫ 0

−τ1
U(x− τ1 − s)φ(s)(ds)α

− B2

Γ(1 + α)

∫ x−τ2

−τ2
U(x− τ2 − s)φ(s)(ds)α +

1

Γ(1 + α)

∫ x

0

U(x− s)f(s)(ds)α,

because V (α)(x) = U(x) follows from Definition 2.5. Using a method similar to the
calculation of y(α)(x), we have

y(2α)(x) = U (2α)(x)φ(0) + V (2α)(x)φ(α)(0)

− A2

Γ(1 + α)

∫ 0

−τ1
U (α)(x− τ1 − s)φ(s)(ds)α

− B2

Γ(1 + α)

∫ x−τ2

−τ2
U (α)(x− τ2 − s)φ(s)(ds)α −B2U(0)φ(x− τ2)

+
1

Γ(1 + α)

∫ x

0

U (α)(x− s)f(s)(ds)α + U(0)f(x).



16 K. QIU, J. WANG EJDE-2020/72

Applying Lemma 2.12 to y(2α)(x) and noticing that U(x− τ2) = 0, V (x− τ2) = 0
because x < τ2 and U(0) = 1, V (α)(x) = U(x) from Definition 2.5, we have

y(2α)(x)

=
(
−A2U(x− τ1)−B2U(x− τ2)

)
φ(0)

+
(
−A2V (x− τ1)−B2V (x− τ2)

)
φ(α)(0)

− A2

Γ(1 + α)

∫ 0

−τ1
V (2α)(x− τ1 − s)φ(s)(ds)α

− B2

Γ(1 + α)

∫ x−τ2

−τ2
V (2α)(x− τ2 − s)φ(s)(ds)α

−B2φ(x− τ2) +
1

Γ(1 + α)

∫ x

0

V (2α)(x− s)f(s)(ds)α + f(x)

= −A2U(x− τ1)φ(0)−A2V (x− τ1)φ(α)(0)

− A2

Γ(1 + α)

∫ 0

−τ1

(
−A2V (x− 2τ1 − s)−B2V (x− τ1 − τ2 − s)

)
φ(s)(ds)α

− B2

Γ(1 + α)

∫ x−τ2

−τ2

(
−A2V (x− τ1 − τ2 − s)

−B2V (x− 2τ2 − s)
)
φ(s)(ds)α −B2φ(x− τ2)

+
1

Γ(1 + α)

∫ x

0

(
−A2V (x− τ1 − s)−B2V (x− τ2 − s)

)
f(s)(ds)α + f(x).

Since V (x) = 0 for x < 0, we have V (x − τ1 − τ2 − s) = 0, V (x − 2τ2 − s) = 0,
V (x− τ2 − s) = 0 if τ1 ≤ x < τ2, and we have

y(2α)(x)

= −A2U(x− τ1)φ(0)−A2V (x− τ1)φ(α)(0)

+
A4

Γ(1 + α)

∫ 0

−τ1
V (x− 2τ1 − s)φ(s)(ds)α

+
A2B2

Γ(1 + α)

∫ x−τ2

−τ2
V (x− τ1 − τ2 − s)φ(s)(ds)α

−B2φ(x− τ2)− A2

Γ(1 + α)

∫ x

0

V (x− τ1 − s)f(s)(ds)α + f(x)

= −A2
(
U(x− τ1)φ(0) + V (x− τ1)φ(α)(0)

− A2

Γ(1 + α)

∫ 0

−τ1
V (x− 2τ1 − s)φ(s)(ds)α

− B2

Γ(1 + α)

∫ x−τ1−τ2

−τ2
V (x− τ1 − τ2 − s)φ(s)(ds)α

+
1

Γ(1 + α)

∫ x−τ1

0

V (x− τ1 − s)f(s)(ds)α
)

−B2φ(x− τ2) + f(x)



EJDE-2020/72 SECOND ORDER DELAY DIFFERENTIAL EQUATION 17

= −A2
(
U(x− τ1)φ(0) + V (x− τ1)φ(α)(0)

− A2

Γ(1 + α)

∫ 0

−τ1
V (x− 2τ1 − s)φ(s)(ds)α

− B2

Γ(1 + α)

∫ 0

−τ2
V (x− τ1 − τ2 − s)φ(s)(ds)α

+
1

Γ(1 + α)

∫ x−τ1

0

V (x− τ1 − s)f(s)(ds)α
)

−B2φ(x− τ2) + f(x)

= −A2y(x− τ1)−B2φ(x− τ2) + f(x).

Note φ(x− τ2) = y(x− τ2) if x < τ2. We have x− τ2 < 0 and y(x) is a solution of
system (1.2).

Finally, if x ≥ τ2, then we obtain

y(x) = U(x)φ(0) + V (x)φ(α)(0)− A2

Γ(1 + α)

∫ 0

−τ1
V (x− τ1 − s)φ(s)(ds)α

− B2

Γ(1 + α)

∫ 0

−τ2
V (x− τ2 − s)φ(s)(ds)α

+
1

Γ(1 + α)

∫ x

0

V (x− s)f(s)(ds)α.

Calculating the local fractal derivative of y(x), and using V (α)(x) = U(x) and
Lemma 2.12, we obtain

y(α)(x)

= U (α)(x)φ(0) + V (α)(x)φ(α)(0)− A2

Γ(1 + α)

∫ 0

−τ1
V (α)(x− τ1 − s)φ(s)(ds)α

− B2

Γ(1 + α)

∫ 0

−τ2
V (α)(x− τ2 − s)φ(s)(ds)α +

1

Γ(1 + α)

∫ x

0

V (α)(x− s)f(s)(ds)α

= U (α)(x)φ(0) + V (α)(x)φ(α)(0)− A2

Γ(1 + α)

∫ 0

−τ1
V (α)(x− τ1 − s)φ(s)(ds)α

− B2

Γ(1 + α)

∫ 0

−τ2
V (α)(x− τ2 − s)φ(s)(ds)α +

1

Γ(1 + α)

∫ x

0

U(x− s)f(s)(ds)α.

Then, we have

y(2α)(x)

= U (2α)(x)φ(0) + V (2α)(x)φ(α)(0)− A2

Γ(1 + α)

∫ 0

−τ1
V (2α)(x− τ1 − s)φ(s)(ds)α

− B2

Γ(1 + α)

∫ 0

−τ2
V (2α)(x− τ2 − s)φ(s)(ds)α

+
1

Γ(1 + α)

∫ x

0

U (α)(x− s)f(s)(ds)α + U(0)f(x)

= U (2α)(x)φ(0) + V (2α)(x)φ(α)(0)



18 K. QIU, J. WANG EJDE-2020/72

− A2

Γ(1 + α)

∫ 0

−τ1

(
−A2V (x− 2τ1 − s)−B2V (x− τ1 − τ2 − s)

)
φ(s)(ds)α

− B2

Γ(1 + α)

∫ 0

−τ2

(
−A2V (x− τ1 − τ2 − s)−B2V (x− 2τ2 − s)

)
φ(s)(ds)α

+
1

Γ(1 + α)

∫ x

0

U (2α)(x− s)f(s)(ds)α + f(x)

=
(
−A2U(x− τ1)−B2U(x− τ2)

)
φ(0)

+
(
−A2V (x− τ1)−B2V (x− τ2)

)
φ(α)(0)

+
A4

Γ(1 + α)

∫ 0

−τ1
V (x− 2τ1 − s)φ(s)(ds)α

+
A2B2

Γ(1 + α)

∫ 0

−τ1
V (x− τ1 − τ2 − s)φ(s)(ds)α

+
A2B2

Γ(1 + α)

∫ 0

−τ2
V (x− τ1 − τ2 − s)φ(s)(ds)α

+
B4

Γ(1 + α)

∫ 0

−τ2
V (x− 2τ2 − s)φ(s)(ds)α

− A2

Γ(1 + α)

∫ x

0

V (x− τ1 − s)f(s)(ds)α

− B2

Γ(1 + α)

∫ x

0

V (x− τ2 − s)f(s)(ds)α + f(x)

= −A2
(
U(x− τ1)φ(0) + V (x− τ1)φ(α)(0)

− A2

Γ(1 + α)

∫ 0

−τ1
V (x− 2τ1 − s)φ(s)(ds)α

− B2

Γ(1 + α)

∫ 0

−τ2
V (x− τ1 − τ2 − s)φ(s)(ds)α

+
1

Γ(1 + α)

∫ x−τ1

0

V (x− τ1 − τ2)f(s)(ds)α
)

−B2
(
U(x− τ2)φ(0) + V (x− τ2)φ(α)(0)

− A2

Γ(1 + α)

∫ 0

−τ1
V (x− τ1 − τ2 − s)φ(s)(ds)α

− B2

Γ(1 + α)

∫ 0

−τ2
V (x− 2τ2 − s)φ(s)(ds)α

+
1

Γ(1 + α)

∫ x−τ2

0

V (x− τ2 − s)f(s)(ds)α
)

+ f(x)

= −A2y(x− τ1)−B2y(x− τ2) + f(x).

From above, we can see that (4.1) is a solution of system (1.2). The proof is
complete. �



EJDE-2020/72 SECOND ORDER DELAY DIFFERENTIAL EQUATION 19

Concluding remarks. From the delayed cosine and sine type matrix function on
the fractal set Rαn(0 < α ≤ 1) corresponding to second order inhomogeneous delay
differential equations with permutable constant matrix coefficients, we provide a
representation of a solution to the second order inhomogeneous delay differential
equations with pure delay and two delays. It is worth mentioning that although
there are many continued contributions in a linear discrete/differential systems with
pure delay with permutable matrices, no results were obtained for such systems
with non permutable matrices on fractal set Rαn(0 < α ≤ 1). A representation of
a solution to delay discrete/differential systems with non permutable matrices on
fractal set Rαn(0 < α ≤ 1) is open at present, worthy our further study.

Acknowledgements. This work was supported by the Youth Science and Tech-
nology Talents Growth project of Guizhou Provincial Education Department ([2017]
205), by the Science and Technology Planning Project of Science and Technology
Department of Guizhou Province ([2018] 1121), and by the Doctoral Foundation
Project of Guizhou Education University (2018BS001).

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Kee Qiu

School of Mathematics and Big Data, Guizhou Education University, Guiyang, Guizhou
550018, China.

School of Mathematics and Information Sciences, Guangzhou University, Guangzhou

510006, China
Email address: qke456@sina.com

Jinrong Wang (corresponding author)
Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, China.

School of Mathematical Sciences, Qufu Normal University, Qufu 273165, Shandong,

China
Email address: jrwang@gzu.edu.cn


	1. Introduction
	2. Preliminaries
	3. Solutions of differential equation with pure delay on fractal set
	4. Solutions of differential equation with two delays on a fractal set
	Concluding remarks
	Acknowledgements

	References