Author:
Sun Jian-Ping,Fang Li,Zhao Ya-Hong,Ding Qian
Abstract
AbstractIn this paper, we consider the existence and uniqueness of solutions for the following nonlinear multi-order fractional differential equation with integral boundary conditions $$ \textstyle\begin{cases} ({}^{C}D_{0+}^{\alpha}u)(t)+\sum_{i=1}^{m}\lambda _{i}(t)({}^{C}D_{0+}^{\alpha _{i}}u)(t)+ \sum_{j=1}^{n}\mu _{j}(t)({}^{C}D_{0+}^{\beta _{j}}u)(t)\\ \quad{}+\sum_{k=1}^{p}\xi _{k}(t)({}^{C}D_{0+}^{\gamma _{k}}u)(t)+\sum_{l=1}^{q}\omega _{l}(t)({}^{C}D_{0+}^{\delta _{l}}u)(t)\\ \quad{}+\sigma (t)u(t)+f(t,u(t))=0,\quad t\in [0,1],\\ u^{\prime \prime}(0)=u^{\prime \prime \prime}(0)=0,\qquad u^{\prime}(0)=\eta _{1}\int _{0}^{1}u(s)\,ds,\qquad u(1)=\eta _{2}\int _{0}^{1}u(s)\,ds, \end{cases} $$
{
(
C
D
0
+
α
u
)
(
t
)
+
∑
i
=
1
m
λ
i
(
t
)
(
C
D
0
+
α
i
u
)
(
t
)
+
∑
j
=
1
n
μ
j
(
t
)
(
C
D
0
+
β
j
u
)
(
t
)
+
∑
k
=
1
p
ξ
k
(
t
)
(
C
D
0
+
γ
k
u
)
(
t
)
+
∑
l
=
1
q
ω
l
(
t
)
(
C
D
0
+
δ
l
u
)
(
t
)
+
σ
(
t
)
u
(
t
)
+
f
(
t
,
u
(
t
)
)
=
0
,
t
∈
[
0
,
1
]
,
u
″
(
0
)
=
u
‴
(
0
)
=
0
,
u
′
(
0
)
=
η
1
∫
0
1
u
(
s
)
d
s
,
u
(
1
)
=
η
2
∫
0
1
u
(
s
)
d
s
,
where $0<\delta _{1}<\delta _{2}<\cdots <\delta _{q}<1<\gamma _{1}<\gamma _{2}<\cdots <\gamma _{p}<2<\beta _{1}<\beta _{2}<\cdots <\beta _{n}<3<\alpha _{1}<\alpha _{2}<\cdots <\alpha _{m}<\alpha <4$
0
<
δ
1
<
δ
2
<
⋯
<
δ
q
<
1
<
γ
1
<
γ
2
<
⋯
<
γ
p
<
2
<
β
1
<
β
2
<
⋯
<
β
n
<
3
<
α
1
<
α
2
<
⋯
<
α
m
<
α
<
4
and $\eta _{1}+2(1-\eta _{2})\neq 0$
η
1
+
2
(
1
−
η
2
)
≠
0
. Using a fixed point theorem and Banach contractive mapping principle, we obtain some existence and uniqueness results.
Funder
National Natural Science Foundation of China
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory,Analysis
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