Affiliation:
1. Department of Mathematics, Northwest Normal University , Lanzhou , P.R. China
2. Department of Mathematical Analysis, University of Rousse , 7017 Rousse , Bulgaria
Abstract
Abstract
In this article, we consider a discrete nonlinear third-order boundary value problem
Δ
3
u
(
k
−
1
)
=
λ
a
(
k
)
f
(
k
,
u
(
k
)
)
,
k
∈
[
1
,
N
−
2
]
Z
,
Δ
2
u
(
η
)
=
α
Δ
u
(
N
−
1
)
,
Δ
u
(
0
)
=
−
β
u
(
0
)
,
u
(
N
)
=
0
,
\left\{\phantom{\rule[-1.25em]{}{0ex}}\begin{array}{l}{\Delta }^{3}u\left(k-1)=\lambda a\left(k)f\left(k,u\left(k)),\hspace{1em}k\in {\left[1,N-2]}_{{\mathbb{Z}}},\hspace{1.0em}\\ {\Delta }^{2}u\left(\eta )=\alpha \Delta u\left(N-1),\Delta u\left(0)=-\beta u\left(0),\hspace{1em}u\left(N)=0,\hspace{1.0em}\end{array}\right.
where
N
>
4
N\gt 4
is an integer,
λ
>
0
\lambda \gt 0
is a parameter.
f
:
[
1
,
N
−
2
]
Z
×
[
0
,
+
∞
)
→
[
0
,
+
∞
)
f:{\left[1,N-2]}_{{\mathbb{Z}}}\times \left[0,+\infty )\to \left[0,+\infty )
is continuous,
a
:
[
1
,
N
−
2
]
Z
→
(
0
,
+
∞
)
a:{\left[1,N-2]}_{{\mathbb{Z}}}\to \left(0,+\infty )
,
α
∈
0
,
1
N
−
1
\alpha \in \left[0,\frac{1}{N-1}\right)
,
β
∈
0
,
2
(
1
−
α
(
N
−
1
)
)
N
(
2
−
α
(
N
−
1
)
)
\beta \in \left[0,\frac{2\left(1-\alpha \left(N-1))}{N\left(2-\alpha \left(N-1))}\right)
, and
η
∈
N
−
2
2
+
1
,
N
−
2
Z
\eta \in {\left[\left[\frac{N-2}{2}\right]+1,N-2\right]}_{{\mathbb{Z}}}
. With the sign-changing Green’s function, we obtain not only the existence of positive solutions but also the multiplicity of positive solutions to this problem.
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