Solvability of a Coupled System of Fractional Differential Equations with Periodic Boundary Conditions at Resonance
By using the coincidence degree theory, we study the existence of solutions for a coupled system of fractional differential equations with periodic boundary conditions. A new result on the existence of solutions of the indicated fractional boundary-value problem is obtained.
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| author | Hu, Zhigang Liu, Wenbin Ху, Чжиган Лю, Венбін |
| author_facet | Hu, Zhigang Liu, Wenbin Ху, Чжиган Лю, Венбін |
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| description | By using the coincidence degree theory, we study the existence of solutions for a coupled system of fractional differential equations with periodic boundary conditions. A new result on the existence of solutions of the indicated fractional boundary-value problem is obtained. |
| first_indexed | 2026-03-24T02:25:09Z |
| format | Article |
| fulltext |
UDC 517.9
Zhigang Hu, Wenbin Liu (China Univ. Mining and Technology, Xuzhou, China)
SOLVABILITY FOR A COUPLED SYSTEM
OF FRACTIONAL DIFFERENTIAL EQUATIONS
WITH PERIODIC BOUNDARY CONDITIONS AT RESONANCE*
РОЗВ’ЯЗНIСТЬ ЗВ’ЯЗАНОЇ СИСТЕМИ
ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ ДРОБОВОГО ПОРЯДКУ
З ПЕРIОДИЧНИМИ ГРАНИЧНИМИ УМОВАМИ ПРИ РЕЗОНАНСI
By using the coincidence degree theory, we study the existence of solutions for a coupled system of fractional differential
equations with periodic boundary conditions. A new result on the existence of solutions for above fractional boundary-value
problem is obtained.
Iз використанням теорiї збiгу степенiв дослiджено iснування розв’язкiв зв’язаних систем диференцiальних рiвнянь
дробового порядку з перiодичними граничними умовами. Встановлено новий результат щодо iснування розв’язкiв
граничної задачi дробового порядку.
1. Introduction. In recent years, the fractional differential equations have received more and more
attention. The fractional derivative has been occurring in many physical applications such as a non-
Markovian diffusion process with memory [1], charge transport in amorphous semiconductors [2],
propagations of mechanical waves in viscoelastic media [3], etc. Phenomena in electromagnetics,
acoustics, viscoelasticity, electrochemistry and material science are also described by differential
equations of fractional order (see [4 – 9]).
Recently, boundary-value problems for fractional differential equations have been studied in many
papers (see [10 – 19]). Moreover, the existence of solutions to a coupled systems of fractional differ-
ential equations have been studied by many authors (see [20 – 26]). But the existence of solutions for
a coupled system of fractional differential equations with periodic boundary conditions at resonance
has not been studied. We will fill this gap in the literature. In this paper, we consider the following
periodic boundary-value problem (PBVP for short) for a coupled system of fractional differential
equations given by:
Dα
0+u(t) = f
(
t, v(t), v′(t)
)
, t ∈ (0, 1),
Dβ
0+
v(t) = g
(
t, u(t), u′(t)
)
, t ∈ (0, 1), (1.1)
u(0) = u(1), u′(0) = u′(1), v(0) = v(1), v′(0) = v′(1),
where Dα
0+ , D
β
0+
are the standard Caputo fractional detivative, 1 < α ≤ 2, 1 < β ≤ 2 and
f, g : [0, 1]× R2 → R is continuous.
The rest of this paper is organized as follows. Section 2 contains some necessary notations,
definitions and lemmas. In Section 3, we establish a theorem on existence of solutions for PBVP (1.1)
under nonlinear growth restriction of f and g, basing on the coincidence degree theory due to Mawhin
(see [27]). Finally, in Section 4, an example is given to illustrate the main result.
* This research was supported by the Fundamental Research Funds for the Central Universities (Project 2013QNA33).
c© ZHIGANG HU, WENBIN LIU, 2013
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11 1463
1464 ZHIGANG HU, WENBIN LIU
2. Preliminaries. In this section, we will introduce some notations, definitions and preliminary
facts which are used throughout this paper.
Let X and Y be real Banach spaces and let L : domL ⊂ X → Y be a Fredholm operator with
index zero, and P : X → X, Q : Y → Y be projectors such that
ImP = KerL, KerQ = ImL,
X = KerL⊕KerP, Y = ImL⊕ ImQ.
It follows that
L|domL∩KerP : domL ∩KerP → ImL
is invertible. We denote the inverse by KP .
If Ω is an open bounded subset of X, and domL ∩ Ω 6= ∅, the map N : X → Y will be called
L-compact on Ω if QN(Ω) is bounded and KP (I − Q)N : Ω → X is compact. Where I is identity
operator.
Lemma 2.1 [27]. Let L : domL ⊂ X → Y be a Fredholm operator of index zero and N : X →
→ Y is L-compact on Ω. Assume that the following conditions are satisfied
(1) Lx 6= λNx for every (x, λ) ∈
[
(domL \KerL)
]
∩ ∂Ω× (0, 1);
(2) Nx 6∈ ImL for every x ∈ KerL ∩ ∂Ω;
(3) deg(QN |KerL,KerL∩Ω, 0) 6= 0, where Q : Y → Y is a projection such that ImL = KerQ.
Then the equation Lx = Nx has at least one solution in domL ∩ Ω.
Definition 2.1. The Riemann – Liouville fractional integral operator of order α > 0 of a func-
tion x is given by
Iα0+x(t) =
1
Γ(α)
t∫
0
(t− s)α−1x(s)ds,
provided that the right-hand side integral is pointwise defined on (0,+∞).
Definition 2.2. The Caputo fractional derivative of order α > 0 of a continuous function x is
given by
Dα
0+x(t) = In−α
0+
dnx(t)
dtn
=
1
Γ(n− α)
t∫
0
(t− s)n−α−1x(n)(s)ds,
where n is the smallest integer greater than or equal to α, provided that the right-hand side integral
is pointwise defined on (0,+∞).
Lemma 2.2 [28]. Assume that x ∈ C(0, 1)∩L(0, 1) with a Caputo fractional derivative of order
α > 0 that belongs to C(0, 1) ∩ L(0, 1). Then
Iα0+D
α
0+x(t) = x(t) + c0 + c1t+ c2t
2 + . . .+ cn−1t
n−1,
where ci ∈ R, i = 0, 1, 2, . . . , n− 1, here n is the smallest integer greater than or equal to α.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
SOLVABILITY FOR A COUPLED SYSTEM OF FRACTIONAL DIFFERENTIAL . . . 1465
In this paper, we denote X = C1[0, 1] with the norm ‖x‖X = max
{
‖x‖∞, ‖x′‖∞
}
and Y =
= C[0, 1] with the norm ‖y‖Y = ‖y‖∞, where ‖x‖∞ = maxt∈[0,1]
∣∣x(t)
∣∣. Then we denote X = X×
×X with the norm
∥∥(u, v)
∥∥
X
= max
{
‖u‖X , ‖v‖X
}
and Y = Y × Y with the norm
∥∥(x, y)
∥∥
Y
=
= max
{
‖x‖Y , ‖y‖Y
}
Obviously, both X and Y are Banach spaces.
Define the operator L1 : domL ⊂ X → Y by
L1u = Dα
0+u,
where
domL1 =
{
u ∈ X|Dα
0+u(t) ∈ Y, u(0) = u(1), u′(0) = u′(1)
}
.
Define the operator L2 : domL2 ⊂ X → Y by
L2v = Dβ
0+
v,
where
domL2 =
{
v ∈ X|Dβ
0+
v(t) ∈ Y, v(0) = v(1), v′(0) = v′(1)
}
.
Define the operator L : domL ⊂ X → Y by
L(u, v) = (L1u, L2v), (2.1)
where
domL =
{
(u, v) ∈ X|u ∈ domL1, v ∈ domL2
}
.
Let N : X → Y be the Nemytski operator
N(u, v) = (N1v,N2u),
where N1 : Y → X
N1v(t) = f
(
t, v(t), v′(t)
)
,
and N2 : Y → X
N2u(t) = g
(
t, u(t), u′(t)
)
.
Then PBVP (1.1) is equivalent to the operator equation
L(u, v) = N(u, v), (u, v) ∈ domL.
3. Main result. In this section, a theorem on existence of solutions for PBVP (1.1) will be given.
Theorem 3.1. Let f, g : [0, 1]× R2 → R be continuous. Assume that
(H1) there exist nonnegative functions pi, qi, ri ∈ C[0, 1], i = 1, 2, with
Γ(α+ 1)Γ(β + 1)− (α+ 1)(β + 1)(Q1 +R1)(Q2 +R2)
Γ(α+ 1)Γ(β + 1)
> 0
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
1466 ZHIGANG HU, WENBIN LIU
such that for all (u, v) ∈ R2, t ∈ [0, 1]∣∣f(t, u, v)
∣∣ ≤ p1(t) + q1(t)|u|+ r1(t)|v|,
and ∣∣g(t, u, v)
∣∣ ≤ p2(t) + q2(t)|u|+ r2(t)|v|,
where Pi = ‖pi‖∞, Qi = ‖qi‖∞, Ri = ‖ri‖∞, i = 1, 2;
(H2) there exists a constant B > 0 such that for all t ∈ [0, 1], |u| > B, v ∈ R either
uf(t, u, v) > 0, ug(t, u, v) > 0,
or
uf(t, u, v) < 0, ug(t, u, v) < 0;
(H3) there exists a constant D > 0 such that for every c1, c2 ∈ R satisfying min{c1, c2} > D
either
c1N1(c2) > 0, c2N2(c1) > 0
or
c1N1(c2) < 0, c2N2(c1) < 0.
Then PBVP (1.1) has at least one solution.
Now, we begin with some lemmas below.
Lemma 3.1. Let L be defined by (2.1), then
KerL = (KerL1,KerL2) =
{
(u, v) ∈ X|(u, v) = (a, b), a, b ∈ R
}
, (3.1)
ImL = (ImL1, ImL2) =
{
(x, y) ∈ Y |T1 = 0, T2 = 0
}
, (3.2)
where T1 =
∫ 1
0
(1− s)α−2x(s)ds, T2 =
∫ 1
0
(1− s)β−2y(s)ds.
Proof. By Lemma 2.2, L1u = Dα
0+u(t) = 0 has solution
u(t) = c0 + c1t, c0, c1 ∈ R.
Combining with the boundary-value conditions of PBVP (1.1), one has
KerL1 =
{
u ∈ X|u = a, a ∈ R
}
.
For x ∈ ImL1, there exists u ∈ domL1 such that x = L1u ∈ Y. By Lemma 2.2, we have
u(t) =
1
Γ(α)
t∫
0
(t− s)α−1x(s)ds+ c0 + c1t.
Then, we obtain
u′(t) =
1
Γ(α− 1)
t∫
0
(t− s)α−2x(s)ds+ c1.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
SOLVABILITY FOR A COUPLED SYSTEM OF FRACTIONAL DIFFERENTIAL . . . 1467
By conditions of PBVP (1.1), we can get that x satisfies
T1 =
1∫
0
(1− s)α−2x(s)ds = 0.
On the other hand, suppose x ∈ Y and satisfies
∫ 1
0
(1− s)α−2x(s)ds = 0. Let u(t) = Iα0+x(t)−
− µt, where µ = Iα0+x(t)|t=1, then u ∈ domL1 and Dα
0+u(t) = x(t). So that, x ∈ ImL1. Then we
have
ImL1 =
{
x ∈ Y |T1 = 0
}
.
Similarly, we can show that
KerL2 =
{
v ∈ X|v = b, b ∈ R
}
,
ImL2 =
{
y ∈ Y |T2 = 0
}
.
Lemma 3.1 is proved.
Lemma 3.2. Let L be defined by (2.1), then L is a Fredholm operator of index zero, and the
linear continuous projector operators P : X → X and Q : Y → Y can be defined as
P (u, v) = (P1u, P2v) =
(
u(0), v(0)
)
,
Q(x, y) = (Q1x,Q2y) =
(
(α− 1)T1, (β − 1)T2
)
.
Furthermore, the operator KP : ImL→ domL ∩KerP can be written by
KP (x, y) =
(
Iα0+x(t)− µt, Iβ
0+
y(t)− νt
)
,
where µ = Iα0+x(t)|t=1, ν = Iβ
0+
y(t)|t=1.
Proof. Obviously, ImP = KerL and P 2(u, v) = P (u, v). It follows from (u, v) =
(
(u, v) −
− P (u, v)
)
+ P (u, v) that X = KerP + KerL. By simple calculation, we can get that KerL ∩
∩KerP =
{
(0, 0)
}
. Then we get
X = KerL⊕KerP.
For (x, y) ∈ Y , we have
Q2(x, y) = Q
(
Q1x,Q2y)
)
= (Q2
1x,Q
2
2y).
By the definition of Q1, we can get
Q2
1x = Q1x · (α− 1)
1∫
0
(1− s)α−2ds = Q1x.
Similar proof can show that Q2
2y = Q2y. Thus, we obtain Q2(x, y) = Q(x, y).
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
1468 ZHIGANG HU, WENBIN LIU
Let (x, y) =
(
(x, y)−Q(x, y)
)
+Q(x, y), where (x, y)−Q(x, y) ∈ KerQ = ImL, Q(x, y) ∈
∈ ImQ. It follows from KerQ = ImL and Q2(x, y) = Q(x, y) that ImQ∩ ImL =
{
(0, 0)
}
. Then,
we have
Y = ImL⊕ ImQ.
Thus
dim KerL = dim ImQ = codim ImL.
This means that L is a Fredholm operator of index zero.
Now, we will prove that KP is the inverse of L|domL∩KerP . In fact, for (x, y) ∈ ImL, we have
LKP (x, y) =
(
Dα
0+(Iα0+x− µt), D
β
0+
(Iβ
0+
y − νt)
)
= (x, y). (3.3)
Moreover, for (u, v) ∈ domL ∩KerP, we get u(0) = 0, v(0) = 0 and
KPL(u, v) =
(
Iα0+D
α
0+u(t)−
{
Iα0+D
α
0+u(t)
}∣∣∣
t=1
t, Iβ
0+
Dβ
0+
v(t)−
{
Iβ
0+
Dβ
0+
v(t)
}∣∣∣
t=1
t
)
=
=
(
u(t) + c0 + c1t−
{
Iα0+D
α
0+u(t)
}∣∣∣
t=1
t, v(t) + c0 + c1t−
{
Iβ
0+
Dβ
0+
v(t)
}∣∣∣
t=1
t
)
,
which together with u(0) = u(1) and v(0) = v(1) yields that
KPL(u, v) = (u, v). (3.4)
Combining (3.3) with (3.4), we know that KP = (L|domL∩KerP )−1.
Lemma 3.2 is proved.
Lemma 3.3. Assume Ω ⊂ X is an open bounded subset such that domL ∩ Ω 6= ∅, then N is
L-compact on Ω.
Proof. By the continuity of f and g, we can get that QN(Ω) and KP (I−Q)N(Ω) are bounded.
So, in view of the Arzelà – Ascoli theorem, we need only prove that KP (I − Q)N(Ω) ⊂ X is
equicontinuous.
From the continuity of f and g, there exist constant Ai, Bi > 0, i = 1, 2, such that ∀(u, v) ∈ Ω∣∣(I −Q1)N1v
∣∣ ≤ A1,
∣∣Iα0+(I −Q1)N1v
∣∣ ≤ B1,∣∣(I −Q2)N2u
∣∣ ≤ A2,
∣∣Iα0+(I −Q2)N2u
∣∣ ≤ B2.
Furthermore for 0 ≤ t1 < t2 ≤ 1, (u, v) ∈ Ω, we have∣∣∣∣KP (I −Q)N
(
u(t2), v(t2)
)
−
(
KP (I −Q)N
(
u(t1), v(t1)
))∣∣∣∣ =
=
(
Iα0+(I −Q1)N1v(t2)− µt2, Iβ0+(I −Q2)N2u(t2)− νt2
)
−
−
(
Iα0+(I −Q1)N1v(t1)− µt1, Iβ0+(I −Q2)N2u(t1)− νt1
)
=
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
SOLVABILITY FOR A COUPLED SYSTEM OF FRACTIONAL DIFFERENTIAL . . . 1469
=
(
Iα0+(I −Q1)N1v(t2)− Iα0+(I −Q1)N1v(t1)− µ(t2 − t1),
Iβ
0+
(I −Q2)N2u(t2)− Iβ0+(I −Q2)N2u(t1)− ν(t2 − t1)
)
,
where µ =
{
Iα0+(I −Q1)N1v(t)
}∣∣
t=1
, ν =
{
Iα0+(I −Q2)N2u
}∣∣
t=1
.
By ∣∣∣Iα0+(I −Q1)N1v(t2)− Iα0+(I −Q1)N1v(t1)− µ(t2 − t1)
∣∣∣ ≤
≤ 1
Γ(α)
∣∣∣∣∣∣
t2∫
0
(t2 − s)α−1(I −Q1)N1v(s)ds−
−
t1∫
0
(t1 − s)α−1(I −Q1)N1v(s)ds
∣∣∣∣∣∣+B1|t2 − t1| ≤
≤ A1
Γ(α)
t1∫
0
(t2 − s)α−1 − (t1 − s)α−1ds+
t2∫
t1
(t2 − s)α−1ds
+B1|t2 − t1| =
=
A1
Γ(α+ 1)
(tα2 − tα1 ) +B1|t2 − t1|
and ∣∣∣(Iα0+(I −Q1)N1v
)′
(t2)−
(
Iα0+(I −Q1)N1v
)′
(t1)
∣∣∣ =
=
α− 1
Γ(α)
∣∣∣∣∣∣
t2∫
0
(t2 − s)α−2(I −Q1)N1v(s)ds−
−
t1∫
0
(t1 − s)α−2(I −Q1)N1v(s)ds
∣∣∣∣∣∣ ≤
≤ A1
Γ(α− 1)
t1∫
0
(t1 − s)α−2 − (t2 − s)α−2ds+
t2∫
t1
(t2 − s)α−2ds
≤
≤ A1
Γ(α)
[
tα−12 − tα−11 + 2(t2 − t1)α−1
]
.
Similar proof can show that∣∣∣Iβ0+(I −Q2)N2u(t2)− Iβ0+(I −Q2)N2u(t1)− ν(t2 − t1)
∣∣∣ ≤
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
1470 ZHIGANG HU, WENBIN LIU
≤ A2
Γ(β + 1)
(tβ2 − t
β
1 ) +B2|t2 − t1|,
∣∣∣(Iβ0+(I −Q2)N2u
)′
(t2)−
(
Iβ
0+
(I −Q2)N2u
)′
(t1)
∣∣∣ ≤
≤ A2
Γ(β)
[
tβ−12 − tβ−11 + 2(t2 − t1)β−1
]
.
Since tα, tα−1, tβ and tβ−1 are uniformly continuous on [0, 1], we can get that KP (I−Q)N(Ω) ⊂ X
is equicontinuous.
Thus, we get that KP (I −Q)N : Ω→ X is compact.
Lemma 3.3 is proved.
Lemma 3.4. Suppose (H1), (H2) hold, then the set
Ω1 =
{
(u, v) ∈ domL \KerL | L(u, v) = λN(u, v), λ ∈ (0, 1)
}
is bounded.
Proof. Take (u, v) ∈ Ω1, then N(u, v) ∈ ImL. By (3.2), we have
1∫
0
(1− s)α−2f
(
s, v(s), v′(s)
)
ds = 0,
1∫
0
(1− s)β−2g
(
s, u(s), u′(s)
)
ds = 0.
Then, by the integral mean value theorem, there exists constants ξ, η ∈ (0, 1) such that f
(
ξ, v(ξ), v′(ξ)
)
=
= 0 and g
(
η, u(η), u′(η)
)
= 0. So, from (H2), we get
∣∣v(ξ)
∣∣ ≤ B and
∣∣u(η)
∣∣ ≤ B. Hence
|u(t)| =
∣∣∣∣∣∣u(η) +
t∫
η
u′(s)ds
∣∣∣∣∣∣ ≤ B + ‖u′‖∞. (3.5)
That is
‖u‖∞ ≤ B + ‖u′‖∞. (3.6)
Similar proof can show that
‖v‖∞ ≤ B + ‖v′‖∞. (3.7)
By L(u, v) = λN(u, v), we have
u(t) =
λ
Γ(α)
t∫
0
(t− s)α−1f
(
s, v(s), v′(s)
)
ds+ u(0)− λµt
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
SOLVABILITY FOR A COUPLED SYSTEM OF FRACTIONAL DIFFERENTIAL . . . 1471
and
v(t) =
λ
Γ(β)
t∫
0
(t− s)β−1g
(
s, u(s), u′(s)
)
ds+ v(0)− λνt,
where µ = Iα0+f
(
t, v(t), v′(t)
)
|t=1, ν = Iβ
0+
g
(
t, u(t), u′(t)
)
|t=1.
Then we obtain
u′(t) =
λ
Γ(α− 1)
t∫
0
(t− s)α−2f
(
s, v(s), v′(s)
)
ds− λµ
and
v′(t) =
λ
Γ(β − 1)
t∫
0
(t− s)β−2g
(
s, u(s), u′(s)
)
ds− λν.
From (H1) and (3.7), we get that
|µ| =
∣∣∣Iα0+f(t, v(t), v′(t)
)∣∣
t=1
∣∣∣ =
=
1
Γ(α)
1∫
0
(1− s)α−1
∣∣∣f(s, v(s), v′(s)
)∣∣∣ds ≤
≤ 1
Γ(α)
1∫
0
(1− s)α−1
[
p1(s) + q1(s)
∣∣v(s)
∣∣+ r1(s)
∣∣v′(s)∣∣]ds ≤
≤ 1
Γ(α)
[
P1 +Q1B + (Q1 +R1)‖v′‖∞
] 1∫
0
(t− s)α−1ds ≤
≤ 1
Γ(α+ 1)
[
P1 +Q1B + (Q1 +R1)‖v′‖∞
]
.
So, we have
‖u′‖∞ ≤
1
Γ(α− 1)
t∫
0
(t− s)α−2
∣∣∣f(s, v(s), v′(s)
)∣∣∣ds+ |µ| ≤
≤ 1
Γ(α− 1)
t∫
0
(t− s)α−2
[
p1(s) + q1(s)
∣∣v(s)
∣∣+ r1(s)
∣∣v′(s)∣∣]ds+ |µ| ≤
≤ 1
Γ(α− 1)
[
P1 +Q1B + (Q1 +R1)‖v′‖∞
] t∫
0
(t− s)α−2ds+ |µ| ≤
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
1472 ZHIGANG HU, WENBIN LIU
≤
(
1
Γ(α)
+
1
Γ(α+ 1)
)[
P1 +Q1B + (Q1 +R1)‖v′‖∞
]
=
=
α+ 1
Γ(α+ 1)
[
P1 +Q1B + (Q1 +R1)‖v′‖∞
]
. (3.8)
Similarly, we can get
‖v′‖∞ ≤
β + 1
Γ(β + 1)
[
P2 +Q2B + (Q2 +R2)‖u′‖∞
]
. (3.9)
Together with (3.8), (3.9), we have
‖u′‖∞ ≤
≤ α+ 1
Γ(α+ 1)
{
P1 +Q1B + (Q1 +R1)
β + 1
Γ(β + 1)
[
P2 +Q2B + (Q2 +R2)‖u′‖∞
]}
.
Thus, from
Γ(α+ 1)Γ(β + 1)− (α+ 1)(β + 1)(Q1 +R1)(Q2 +R2)
Γ(α+ 1)Γ(β + 1)
> 0 and (3.9), we obtain
that
‖u′‖∞ ≤
(α+ 1)
[
Γ(β + 1)(P1 +Q1B) + (β + 1)(Q1 +R1)(P2 +Q2B)
]
Γ(α+ 1)Γ(β + 1)− (α+ 1)(β + 1)(Q1 +R1)(Q2 +R2)
:= M1
and
‖v′‖∞ ≤
β + 1
Γ(β + 1)
[
P2 +Q2B + (Q2 +R2)M1
]
:= M2.
Together with (3.6), (3.7), we get∥∥(u, v)
∥∥
X
≤ max{M1 +B,M2 +B} := M.
So Ω1 is bounded.
Lemma 3.4 is proved.
Lemma 3.5. Suppose (H3) holds, then the set
Ω2 =
{
(u, v)|(u, v) ∈ KerL,N(u, v) ∈ ImL
}
is bounded.
Proof. For (u, v) ∈ Ω2, we have (u, v) = (c1, c2), c1, c2 ∈ R. Then from N(u, v) ∈ ImL, we
obtain
1∫
0
(1− s)α−2f(s, c2, 0)ds = 0,
1∫
0
(1− s)β−2g(s, c1, 0)ds = 0,
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
SOLVABILITY FOR A COUPLED SYSTEM OF FRACTIONAL DIFFERENTIAL . . . 1473
which together with (H3) implies |c1|, |c2| ≤ D. Thus, we get
‖(u, v)‖X ≤ D.
Hence, Ω2 is bounded.
Lemma 3.5 is proved.
Lemma 3.6. Suppose the first part of (H3) holds, then the set
Ω3 =
{
(u, v) ∈ KerL|λ(u, v) + (1− λ)QN(u, v) = (0, 0), λ ∈ [0, 1]
}
is bounded.
Proof. For (u, v) ∈ Ω3, we have (u, v) = (c1, c2), c1, c2 ∈ R and
λc1 + (1− λ)(α− 1)
1∫
0
(1− s)α−2f(s, c2, 0)ds = 0, (3.10)
λc2 + (1− λ)(β − 1)
1∫
0
(1− s)β−2g(s, c1, 0)ds = 0. (3.11)
If λ = 0, then |c1|, |c2| ≤ D because of the first part of (H3). If λ = 1, then c1 = c2 = 0. For
λ ∈ (0, 1], we can obtain |c1|, |c2| ≤ D. Otherwise, if |c1| or |c2| > D, in view of the first part of
(H3), one has
λc21 + (1− λ)(α− 1)
1∫
0
(1− s)α−2c1f(s, c2, 0)ds > 0
or
λc22 + (1− λ)(β − 1)
1∫
0
(1− s)β−2c2g(s, c1, 0)ds > 0,
which contradict to (3.10) or (3.11). Therefore, Ω3 is bounded.
Lemma 3.6 is proved.
Remark 3.1. If the second part of (H3) holds, then the set
Ω′3 =
{
(u, v) ∈ KerL| − λ(u, v) + (1− λ)QN(u, v) = (0, 0), λ ∈ [0, 1]
}
is bounded.
Proof of Theorem 3.1. Set Ω =
{
(u, v) ∈ X|‖(u, v)‖X < max{M,D} + 1
}
. It follows from
Lemmas 3.2 and 3.3 that L is a Fredholm operator of index zero and N is L-compact on Ω. By
Lemmas 3.4 and 3.5, we get that the following two conditions are satisfied:
(1) L(u, v) 6= λN(u, v) for every
(
(u, v), λ
)
∈
[
(domL \KerL) ∩ ∂Ω
]
× (0, 1);
(2) Nx /∈ ImL for every (u, v) ∈ KerL ∩ ∂Ω.
ISSN 1027-3190. Укр. мат. журн., 2013, т. 65, № 11
1474 ZHIGANG HU, WENBIN LIU
Take
H
(
(u, v), λ
)
= ±λ(u, v) + (1− λ)QN(u, v).
According to Lemma 3.6 (or Remark 3.1), we know that H
(
(u, v), λ
)
6= 0 for (u, v) ∈ KerL ∩ ∂Ω.
Therefore
deg
(
QN |KerL,Ω ∩KerL, (0, 0)
)
= deg
(
H(·, 0),Ω ∩KerL, (0, 0)
)
=
= deg
(
H(·, 1),Ω ∩KerL, (0, 0)
)
= deg
(
± I,Ω ∩KerL, (0, 0)
)
6= 0.
So that, the condition (3) of Lemma 2.1 is satisfied. By Lemma 2.1, we can get that L(u, v) =
= N(u, v) has at least one solution in domL ∩ Ω. Therefore PBVP (1.1) has at least one solution.
Theorem 3.1 is proved.
4. Example.
Example 4.1. Consider the following PBVP:
D
3
2
0+
u(t) =
1
16
[
v(t)− 10
]
+
t2
16
e−|v
′(t)|, t ∈ [0, 1],
D
5
4
0+
v(t) =
1
12
[
u(t)− 8
]
+
t3
12
sin2 (u′(t)), t ∈ [0, 1], (4.1)
u(0) = u(1), u′(0) = u′(1), v(0) = v(1), v′(0) = v′(1).
Choose p1(t) =
11
16
, p2(t) =
3
4
, q1(t) =
1
16
, q2(t) =
1
12
, r1(t) = r2(t) = 0, B = D = 10.
By simple calculation, we can get that (H1), (H2) and the first part of (H3) hold.
By Theorem 3.1, we obtain that the problem PBVP (4.1) has at least one solution.
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|
| id | umjimathkievua-article-2526 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:25:09Z |
| publishDate | 2013 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/79/3135b51bd9e402821d3aa6beb8ebc279.pdf |
| spelling | umjimathkievua-article-25262020-03-18T19:25:49Z Solvability of a Coupled System of Fractional Differential Equations with Periodic Boundary Conditions at Resonance Розв'язність зв'язаної системи диференціальних рівнянь дробового порядку з періодичними граничними умовами при резонансі Hu, Zhigang Liu, Wenbin Ху, Чжиган Лю, Венбін By using the coincidence degree theory, we study the existence of solutions for a coupled system of fractional differential equations with periodic boundary conditions. A new result on the existence of solutions of the indicated fractional boundary-value problem is obtained. Із використанням теорії збігу степенів досліджено існування розв'язків зв'язаних систем диференціальних рівнянь дробового порядку з періодичними граничними умовами. Встановлено новий результат щодо існування розв'язків граничної задачі дробового порядку. Institute of Mathematics, NAS of Ukraine 2013-11-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2526 Ukrains’kyi Matematychnyi Zhurnal; Vol. 65 No. 11 (2013); 1463–1475 Український математичний журнал; Том 65 № 11 (2013); 1463–1475 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2526/1813 https://umj.imath.kiev.ua/index.php/umj/article/view/2526/1814 Copyright (c) 2013 Hu Zhigang; Liu Wenbin |
| spellingShingle | Hu, Zhigang Liu, Wenbin Ху, Чжиган Лю, Венбін Solvability of a Coupled System of Fractional Differential Equations with Periodic Boundary Conditions at Resonance |
| title | Solvability of a Coupled System of Fractional Differential Equations with Periodic Boundary Conditions at Resonance |
| title_alt | Розв'язність зв'язаної системи диференціальних рівнянь дробового порядку з періодичними граничними умовами при резонансі |
| title_full | Solvability of a Coupled System of Fractional Differential Equations with Periodic Boundary Conditions at Resonance |
| title_fullStr | Solvability of a Coupled System of Fractional Differential Equations with Periodic Boundary Conditions at Resonance |
| title_full_unstemmed | Solvability of a Coupled System of Fractional Differential Equations with Periodic Boundary Conditions at Resonance |
| title_short | Solvability of a Coupled System of Fractional Differential Equations with Periodic Boundary Conditions at Resonance |
| title_sort | solvability of a coupled system of fractional differential equations with periodic boundary conditions at resonance |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2526 |
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