# Existence of Solutions for Boundary Value Problem of a Caputo Fractional Difference Equation

Authors: Zhiping Liu; Shugui Kang; Huiqin Chen; Jianmin Guo; Yaqiong Cui; Caixia Guo
Journal: Discrete Dynamics in Nature and Society
 (2015)

---

## Abstract

We investigate the existence of solutions for a Caputo fractional difference equation boundary value problem. We use Schauder fixed point theorem to deduce the existence of solutions. The proofs are based upon the theory of discrete fractional calculus. We also provide some examples to illustrate our main results.


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## Body

## 1. Introduction

The theory of fractional difference equations and their applications have been receiving intensive attention. In the last ten years, new research achievements kept emerging (see [1–16] and the references therein). In [1–3], the authors introduced fractional sum and difference operators, studied their behavior, and developed a complete theory governing their compositions. Abdeljawad [3] considered the initial value problem to Caputo fractional difference equation. The authors explored BVP of fractional difference equation, and they deduced the existence of one or more positive solutions in [4–7]. Abdeljawad and Baleanu [8] introduced the fractional differences and integration by parts. In [9] the authors studied the stability of discrete nonautonomous systems within the frame of the Caputo fractional difference by using Lyapunov’s direct method. They discussed the conditions for uniform stability, uniform asymptotic stability, and uniform global stability.In [10, 11], the authors studied multiple solutions to fractional difference boundary value problems by means of Krasnosel’skii theorem and Schauder fixed point theorem. They obtained sufficient conditions of the existence of two positive solutions for the boundary value problem of fractional difference equations depending on parameters in [12]. Chen et al. [13] presented the existence of at least one positive solution for Caputo fractional boundary value problems. In [14], Kang et al. discussed existence of positive solutions for a system of Caputo fractional difference equations depending on parameters on the basis of [13].Recently, Wu and Baleanu introduced some applications of the Caputo fractional difference to discrete chaotic maps in [15, 16]. Thus, the fractional difference equation has recently attracted increasing attention from a growing number of researchers.Following this trend, we investigate the following boundary value problem of fractional difference equation (FBVP):(1)
Δ
C
ν
y
t
=
-
f
t
+
ν
-
1
,
y
t
+
ν
-
1
,
y
ν
-
3
=
Δ
y
b
+
ν
=
Δ
2
y
ν
-
3
=
0
,where t
∈
{
0,1
,
…
,
b
+
1
}
≔
[
0
,
b
+
1
]
N
0, b
⩾
5, is an integer. f
:
[
ν
-
2
,
b
+
ν
]
N
ν
-
2
×
R
→
R is continuous and f is not identically zero, 2
<
ν
⩽
3, and Δ
C
ν
y
(
t
) is the standard Caputo difference.The reason for studying (1) is that, in [13], Chen et al. studied similar FBVP (1) by using the cone fixed point theorem. In this note, we consider existence of solutions to FBVP (1) by using Schauder theorem. Here, the nonnegative function f is not necessary, but f is nonnegative in [13]. We extend the domain of f.This paper is organized as follows. Section1 introduces the developing of fractional difference equations in a simple way. We present some necessary definitions and lemmas in Section 2. In Section 3, we prove existence of solution to FBVP (1). Finally, we provide some examples to illustrate our main results.


## 2. Preliminaries

We first introduce some theory about fractional sums and differences. The definitions and some basic results about fractional sums and differences can be seen in [1–7], so we omit their proof.Definition 1 (see [3]).
For anyt and ν, one defines (2)
t
ν
_
=
Γ
t
+
1
Γ
t
+
1
-
νfor which the right-hand side is defined. One appeals to the convention that if t
+
1
-
ν is a pole of the Gamma function and t
+
1 is not a pole, then t
ν
_
=
0.Definition 2 (see [3]).
Theνth fractional sum of a function f is (3)
Δ
-
ν
f
t
=
1
Γ
ν
∑
s
=
a
t
-
ν
t
-
s
-
1
ν
-
1
_
f
s
,for ν
>
0 and t
∈
a
+
ν
,
a
+
ν
+
1
,
…
≔
N
a
+
ν
. One also defines the νth Caputo fractional difference for ν
>
0 by (4)
Δ
C
ν
f
t
=
Δ
-
n
-
ν
Δ
n
f
t
=
1
Γ
n
-
ν
∑
s
=
a
t
-
n
-
ν
t
-
s
-
1
n
-
ν
-
1
_
Δ
a
n
f
s
,where n
-
1
<
ν
⩽
n
.Lemma 3 (see [3]).
Assume thatν
>
0 and f is defined on domains N
a, then (5)
Δ
a
+
n
-
ν
-
ν
Δ
C
ν
f
t
=
f
t
-
∑
k
=
0
n
-
1
c
k
t
-
a
k
_
,where c
k
∈
R, k
=
0,1
,
…
,
n
-
1, and n
-
1
<
ν
⩽
n
.Lemma 4 (see [13]).
Let2
<
ν
⩽
3 and g
:
[
ν
-
2
,
b
+
ν
]
N
ν
-
2
→
R be given. Then the solution of FBVP,(6)
Δ
C
ν
y
t
=
-
g
t
+
ν
-
1
,
y
ν
-
3
=
Δ
y
b
+
ν
=
Δ
2
y
ν
-
3
=
0
,is given by (7)
y
t
=
∑
s
=
0
b
+
1
G
t
,
s
g
s
+
ν
-
1
,where Green’s function G
:
[
ν
-
2
,
b
+
ν
]
N
ν
-
2
×
[
0
,
b
+
1
]
N
0
→
R is defined by (8)
G
t
,
s
=
1
Γ
ν
ν
-
1
t
-
ν
+
3
b
+
ν
-
s
-
1
ν
-
2
_
-
t
-
s
-
1
ν
-
1
_
,
0
⩽
s
<
t
-
ν
+
1
⩽
b
+
1
,
ν
-
1
t
-
ν
+
3
b
+
ν
-
s
-
1
ν
-
2
_
,
0
⩽
t
-
ν
+
1
⩽
s
⩽
b
+
1
.Remark 5.
Notice thatG
(
ν
-
3
,
s
)
=
0, G
(
t
,
b
+
2
)
=
0. G could be extended to [
ν
-
3
,
b
+
ν
]
N
ν
-
3
×
[
0
,
b
+
2
]
N
0, so we only discuss (
t
,
s
)
∈
[
ν
-
2
,
b
+
ν
]
N
ν
-
2
×
[
0
,
b
+
1
]
N
0
.Lemma 6 (see [13]).
The Green functionG has the following properties:(i)
G
(
t
,
s
)
>
0, (
t
,
s
)
∈
[
ν
-
2
,
b
+
ν
]
N
ν
-
2
×
[
0
,
b
+
1
]
N
0
.
(ii)
m
a
x
t
∈
[
ν
-
2
,
b
+
ν
]
N
ν
-
2
G
(
t
,
s
)
=
G
(
b
+
ν
,
s
), s
∈
[
0
,
b
+
1
]
N
0
.Let (9)
K
=
y
∣
y
:
ν
-
3
,
ν
+
b
N
ν
-
3
⟶
R
,
y
ν
-
3
=
Δ
y
ν
+
b
=
Δ
2
y
ν
-
3
=
0
.
It is clear thatK is a Banach space with the norm y
=
m
a
x
t
∈
[
ν
-
2
,
ν
+
b
]
N
ν
-
2
y
t
. Now consider the operator T defined by(10)
T
y
t
=
∑
s
=
0
b
+
1
G
t
,
s
f
s
+
ν
-
1
,
y
s
+
ν
-
1
,where t
∈
[
ν
-
2
,
b
+
ν
]
N
ν
-
2, f
:
[
ν
-
2
,
b
+
ν
]
N
ν
-
2
×
R
→
R is continuous, and f is not identically zero. It is easy to see that y
=
y
(
t
) is a solution of FBVP (1) if and only if y
=
y
(
t
) is a fixed point of T.Lemma 7 (see [17] (Schauder fixed point theorem)).
Suppose thatK is a Banach space. Let B be a bounded closed convex set of K, and let T
:
B
→
B be a complete continuous operator. Then T has at least one fixed point in B.


## 3. Main Results

In this section, we give the main result of this paper. We will prove this result by using Schauder fixed point theorem and provide some examples to illustrate our main results.For the sake of convenience, we write out the conditions as follows:(
H
1
)
There exists a nonnegative functiona
∈
C
(
[
ν
-
2
,
b
+
ν
]
N
ν
-
2
) and a constant c such that |
f
(
t
,
y
)
|
⩽
a
(
t
)
+
c
|
y
|
ρ, where c
⩾
0, 0
<
ρ
<
1
.
(
H
2
)
|
f
(
t
,
y
)
|
⩽
c
|
y
|
ρ, where c
>
0, ρ
>
1
.Theorem 8.
Letf
:
[
ν
-
2
,
b
+
ν
]
N
ν
-
2
×
R
→
R be a continuous function. Suppose that one of conditions (
H
1
) and (
H
2
) is satisfied. Then FBVP (1) has at least one solution.Proof.
First, suppose that condition(
H
1
) is satisfied. Let (11)
B
=
y
t
∣
y
t
∈
K
,
y
⩽
R
,
t
∈
ν
-
2
,
b
+
ν
N
ν
-
2
,where (12)
R
⩾
max
⁡
2
∑
s
=
0
b
+
1
G
b
+
ν
,
s
a
s
+
ν
-
1
,
2
c
Γ
b
+
ν
+
1
Γ
b
+
1
Γ
ν
1
ν
+
b
+
3
b
+
1
1
/
1
-
ρ
.Obviously B is the ball in the Banach space.
Now we prove thatT
:
B
→
B. For any y
∈
B, then (13)
T
y
t
⩽
∑
s
=
0
b
+
1
G
t
,
s
a
s
+
ν
-
1
+
c
y
s
+
ν
-
1
ρ
⩽
∑
s
=
0
b
+
1
G
t
,
s
a
s
+
ν
-
1
+
∑
s
=
0
b
+
1
c
y
ρ
·
G
t
,
s
⩽
∑
s
=
0
b
+
1
G
b
+
ν
,
s
a
s
+
ν
-
1
+
c
R
ρ
∑
s
=
0
t
-
ν
G
t
,
s
+
∑
s
=
t
-
ν
+
1
b
+
1
G
t
,
s
⩽
∑
s
=
0
b
+
1
G
b
+
ν
,
s
a
s
+
ν
-
1
+
c
R
ρ
Γ
ν
∑
s
=
0
t
-
ν
t
-
s
-
1
ν
-
1
_
+
c
R
ρ
Γ
ν
∑
s
=
0
b
+
1
ν
-
1
·
t
-
ν
+
3
b
+
ν
-
s
-
1
ν
-
2
_
⩽
∑
s
=
0
b
+
1
G
b
+
ν
,
s
·
a
s
+
ν
-
1
+
c
R
ρ
m
a
x
t
∈
ν
-
2
,
ν
+
b
N
ν
-
2
1
Γ
ν
·
∑
s
=
0
t
-
ν
t
-
s
-
1
ν
-
1
_
+
c
R
ρ
m
a
x
t
∈
ν
-
2
,
ν
+
b
N
ν
-
2
t
-
ν
+
3
Γ
ν
-
1
·
∑
s
=
0
b
+
1
b
+
ν
-
s
-
1
ν
-
2
_
.As (14)
1
Γ
ν
∑
s
=
0
t
-
ν
t
-
s
-
1
ν
-
1
_
=
1
Γ
ν
-
1
ν
t
-
s
ν
_
s
=
0
t
-
ν
+
1
=
t
ν
_
Γ
ν
+
1
≤
Γ
b
+
ν
+
1
Γ
b
+
1
Γ
ν
+
1
,
t
-
ν
+
3
Γ
ν
-
1
∑
s
=
0
b
+
1
b
+
ν
-
s
-
1
ν
-
2
_
⩽
b
+
3
Γ
ν
-
1
-
1
ν
-
1
b
+
ν
-
s
ν
-
1
_
s
=
0
b
+
2
=
b
+
3
Γ
b
+
ν
+
1
Γ
b
+
2
Γ
ν
,it follows that (15)
T
y
t
⩽
∑
s
=
0
b
+
1
G
b
+
ν
,
s
a
s
+
ν
-
1
+
c
R
ρ
Γ
b
+
ν
+
1
Γ
b
+
1
Γ
ν
+
1
+
c
R
ρ
b
+
3
Γ
b
+
ν
+
1
Γ
b
+
2
Γ
ν
=
∑
s
=
0
b
+
1
G
b
+
ν
,
s
a
s
+
ν
-
1
+
c
R
ρ
Γ
b
+
ν
+
1
Γ
b
+
1
Γ
ν
1
ν
+
b
+
3
b
+
1
⩽
R
2
+
R
2
=
R
.Hence, T
y
⩽
R
. Namely, T
:
B
→
B
.
Second, let condition(
H
2
) be valid. Choose (16)
0
<
R
⩽
Γ
ν
Γ
b
+
1
c
Γ
b
+
ν
+
1
1
ν
+
b
+
3
b
+
1
-
1
1
/
ρ
-
1
.
Repeating the course of the above, we obtainT
y
⩽
R
. Consequently, we get T
:
B
→
B. By means of the continuity of G and f, it is easy to see that operator T is continuous. Next, we show that T is a completely continuous operator. For this, we take (17)
M
=
m
a
x
t
∈
ν
-
2
,
b
+
ν
N
ν
-
2
f
t
,
y
t
.
For anyy
∈
B, let t
,
τ
∈
[
ν
-
2
,
b
+
ν
]
N
ν
-
2 such that t
<
τ; then (18)
T
y
t
-
T
y
τ
=
∑
s
=
0
b
+
1
G
t
,
s
-
G
τ
,
s
·
f
s
+
ν
-
1
,
y
s
+
ν
-
1
⩽
M
∑
s
=
0
t
-
ν
G
t
,
s
-
G
τ
,
s
+
∑
s
=
t
-
ν
+
1
τ
-
ν
G
t
,
s
-
G
τ
,
s
+
∑
s
=
τ
-
ν
+
1
b
+
1
G
t
,
s
-
G
τ
,
s
=
M
Γ
ν
∑
s
=
0
t
-
ν
ν
-
1
·
t
-
ν
+
3
b
+
ν
-
s
-
1
ν
-
2
_
-
t
-
s
-
1
ν
-
1
_
-
ν
-
1
τ
-
ν
+
3
b
+
ν
-
s
-
1
ν
-
2
_
+
τ
-
s
-
1
ν
-
1
_
+
M
Γ
ν
∑
s
=
t
-
ν
+
1
τ
-
ν
ν
-
1
t
-
ν
+
3
·
b
+
ν
-
s
-
1
ν
-
2
_
-
ν
-
1
τ
-
ν
+
3
·
b
+
ν
-
s
-
1
ν
-
2
_
+
M
Γ
ν
∑
s
=
τ
-
ν
+
1
b
+
1
ν
-
1
·
t
-
ν
+
3
b
+
ν
-
s
-
1
ν
-
2
_
-
ν
-
1
τ
-
ν
+
3
·
b
+
ν
-
s
-
1
ν
-
2
_
⩽
M
Γ
ν
∑
s
=
0
t
-
ν
ν
-
1
·
b
+
ν
-
s
-
1
ν
-
2
_
t
-
τ
+
τ
-
s
-
1
ν
-
1
_
-
t
-
s
-
1
ν
-
1
_
+
M
Γ
ν
∑
s
=
t
-
ν
+
1
τ
-
ν
ν
-
1
·
b
+
ν
-
s
-
1
ν
-
2
_
t
-
τ
+
τ
-
s
-
1
ν
-
1
_
+
M
Γ
ν
·
∑
s
=
τ
-
ν
+
1
b
+
1
ν
-
1
b
+
ν
-
s
-
1
ν
-
2
_
t
-
τ
=
M
Γ
ν
∑
s
=
0
b
+
1
ν
-
1
b
+
ν
-
s
-
1
ν
-
2
_
t
-
τ
+
∑
s
=
0
τ
-
ν
τ
-
s
-
1
ν
-
1
_
-
∑
s
=
0
t
-
ν
t
-
s
-
1
ν
-
1
_
=
M
Γ
ν
-
b
+
ν
-
s
ν
-
1
_
s
=
0
b
+
2
t
-
τ
+
-
1
ν
τ
-
s
ν
_
s
=
0
τ
-
ν
+
1
-
-
1
ν
t
-
s
ν
_
s
=
0
t
-
ν
+
1
=
M
Γ
ν
-
ν
-
2
ν
-
1
_
+
b
+
ν
ν
-
1
_
t
-
τ
+
1
ν
τ
ν
_
-
t
ν
_
=
M
Γ
ν
b
+
ν
ν
-
1
_
t
-
τ
+
1
ν
τ
ν
_
-
t
ν
_
.
Since functionst
ν
_, t are uniformly continuous on interval [
ν
-
2
,
b
+
ν
]
N
ν
-
2, we conclude that T
(
B
) is an equicontinuous set. Obviously, it is uniformly bounded since T
(
B
)
⊂
B
. Thus, we know T is completely continuous.
Consequently, it follows at once by Schauder fixed point theorem thatT has a fixed point y; namely, y is a solution of (1). The theorem is proved.Remark 9.
In this paper,f is only continuous function, without nonnegative assumptions on function f
.Remark 10.
Ifρ
=
1 in (
H
1
), we need condition c
(
Γ
ν
+
b
+
1
/
Γ
b
+
1
Γ
ν
)
1
/
ν
+
(
b
+
3
)
/
(
b
+
1
)
⩽
1
/
2
. At this moment, choose (19)
R
⩾
2
∑
s
=
0
b
+
1
G
s
+
ν
-
1
,
s
a
s
+
ν
-
1
.Ifρ
=
1 in (
H
2
), we only need condition c
Γ
(
ν
+
b
+
1
)
/
Γ
(
b
+
1
)
Γ
(
ν
)
1
/
ν
+
(
b
+
3
)
/
(
b
+
1
)
⩽
1
. Then the conclusion of Theorem 8 remains true.Example 11.
Consider the following Caputo fractional difference boundary value problem:(20)
Δ
C
9
/
4
y
t
=
-
s
i
n
π
t
+
5
4
1
+
y
1
/
2
t
+
5
/
4
y
2
t
+
5
/
4
+
50
,
y
-
3
4
=
Δ
y
45
4
=
Δ
2
y
-
3
4
=
0
,where ν
=
9
/
4, b
=
9, f
(
t
,
y
)
=
s
i
n
π
t
·
1
+
y
1
/
2
/
y
2
+
50
. Since (21)
f
t
,
y
⩽
1
+
1
50
y
1
/
2
,then a
(
t
)
=
1, c
=
1
/
50, and ρ
=
1
/
2. At this moment, we take R
=
270.918. The condition of (
H
1
) in Theorem 8 is satisfied. Applying Theorem 8, FBVP (20) has at least one solution y, and y
⩽
R.Example 12.
Consider the following Caputo fractional difference boundary value problem:(22)
Δ
C
5
/
2
y
t
=
-
2
sin
⁡
π
t
+
3
/
2
2
t
+
323
y
2
t
+
3
2
,
y
-
1
2
=
Δ
y
23
2
=
Δ
2
y
-
1
2
=
0
,where t
∈
[
0,10
]
N
0 and ν
=
5
/
2, b
=
9, and f
(
t
,
y
)
=
sin
⁡
π
t
/
t
+
160
y
2
. As (23)
f
t
,
y
⩽
1
160
y
2
,then a
(
t
)
=
0, c
=
1
/
160, and ρ
=
2
>
1
. At this moment, we take R
=
0.3525
. The condition of (
H
2
) in Theorem 8 is satisfied. Applying Theorem 8, FBVP (22) has at least one solution y, and |
y
|
⩽
R.



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*Source: 206261-2015-08-04.xml*