Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 879657, 14 pages
doi:10.1155/2012/879657
Research Article
Asymptotic Stability Results for Nonlinear
Fractional Difference Equations
Fulai Chen and Zhigang Liu
Department of Mathematics, Xiangnan University, Chenzhou 423000, China
Correspondence should be addressed to Fulai Chen, cflmath@163.com
Received 1 August 2011; Revised 27 December 2011; Accepted 2 January 2012
Academic Editor: Michela Redivo-Zaglia
Copyright q 2012 F. Chen and Z. Liu. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We present some results for the asymptotic stability of solutions for nonlinear fractional difference
equations involvingRiemann-Liouville-like difference operator. The results are obtained by using
Krasnoselskii’s fixed point theorem and discrete Arzela-Ascoli’s theorem. Three examples are also
provided to illustrate our main results.
1. Introduction
In this paper we consider the asymptotic stability of solutions for nonlinear fractional
difference equations:
Δα xt ft α, xt α,
α−1
Δ
t ∈ N0 , 0 < α ≤ 1,
xt|t0 x0 ,
1.1
where Δα is aRiemann-Liouville-like discrete fractional difference, f : 0, ∞ × R → R is
continuous with respect to t and x, Na {a, a 1, a 2, . . .}.
Fractional differential equations have received increasing attention during recent years
since these equations have been proved to be valuable tools in the modeling of many
phenomena in various fields of science and engineering. Most of the present works were
focused on fractional differential equations, see 1–12 and the references therein. However,
very little progress has been made to develop the theory of the analogous fractional finite
difference equation 13–19.
Due to the lack of geometry interpretation of the fractional derivatives, it is difficult to
find a valid tool to analyze the stability of fractional difference equations. In the case that it
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is difficult to employ Liapunov’s direct method, fixed point theorems are usually considered
in stability 20–25. Motivated by this idea, in this paper, we discuss asymptotic stability of
nonlinear fractional difference equations by using Krasnoselskii’s fixed point theorem and
discrete Arzela-Ascoli’s theorem. Different from our previous work 18, in this paper, the
sufficient conditions of attractivity are irrelevant to the initial value x0 .
2. Preliminaries
In this section, we introduce preliminary facts of discrete fractional calculus. For more details,
see 14.
Definition 2.1 see 14. Let ν > 0. The ν-th fractional sum x is defined by
Δ−ν ft t−ν
1 t − s − 1ν−1 fs,
Γν sa
2.1
where f is defined for s a mod 1 and Δ−ν f is defined for t a ν mod 1, and tν Γt1/Γt−ν 1. The fractional sum Δ−ν maps functions defined on Na to functions defined
on Naν .
Definition 2.2 see 14. Let μ > 0 and m − 1 < μ < m, where m denotes a positive integer,
m μ, · ceiling of number. Set ν m − μ. The μ-th fractional difference is defined as
Δμ ft Δm−ν ft Δm Δ−ν ft .
2.2
Theorem 2.3 see 15. Let f be a real-value function defined on Na and μ, ν > 0, then the following
equalities hold:
i Δ−ν Δ−μ ft Δ−μν ft Δ−μ Δ−ν ft;
ii Δ−ν Δft ΔΔ−ν ft −
t − aν−1
fa.
Γν
Lemma 2.4 see 15. Let μ / 1 and assume μ ν 1 is not a nonpositive integer, then
−ν μ
Δ t
Γ μ1
tμν .
Γ μν1
2.3
Lemma 2.5 see 15. Assume that the following factorial functions are well defined:
i If 0 < α < 1, then tαγ ≥ tγ α ;
ii tβγ t − γβ tγ .
Lemma 2.6 see 13. Let μ > 0 be noninteger, m μ, ·, ν m − μ, thus one has
t−μ
saν
t − s − 1μ−1 t − a − νμ
.
μ
2.4
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Lemma 2.7. The equivalent fractional Taylor’s difference formula of 1.1 is
xt t−α
x0 α−1
1 t
t − s − 1α−1 fs α, xs α,
Γα
Γα s0
t ∈ Nα .
2.5
Proof. Apply the Δ−α operator to each side of the first formula of 1.1 to obtain
Δ−α Δα xt Δ−α ft α, xt α,
t ∈ Nα .
2.6
Apply Theorem 2.3 to the left-hand side of 2.6 to obtain
Δ−α Δα xt Δ−α ΔΔ−1−α xt ΔΔ−α Δ−1−α xt −
x0 α−1
xt −
t
.
Γα
tα−1
xα − 1
Γα
2.7
So, applying Definition 2.1 to the right-hand side of 2.6, for t ∈ Nα we obtain 2.5.
The recursive iteration to this Taylor’s difference formula implies that 2.5 represents the
unique solution of the IVP 1.1. This completes the proof.
Lemma 2.8 see 4, 1.5.15. The quotient expansion of two gamma functions at infinityis
Γz a
1
za−b 1 O
,
Γz b
z
argz a < π, |z| −→ ∞ .
2.8
Corollary 2.9. One has
t−β > t α−β
for α, β, t > 0.
2.9
Proof. According to Lemma 2.8,
t−β
t α−β
Γ tαβ1
Γt 1
·
Γt α 1
Γ tβ1
Γ tαβ1
Γt 1
·
Γt α 1
Γ tβ1
α
1
1
−α
· tβ 1O
t 1O
t
tβ
β α
1
1
1O
1
1O
t
t
tβ
> 1.
Then, t−β > t α−β for α, β, t > 0. This completes the proof.
2.10
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Definition 2.10. The solution x ϕt of the IVP 1.1 is said to be
i stable if for any ε > 0 and t0 ∈ R , there exists a δ δt0 , ε > 0 such that
xt, x0 , t0 − ϕt < ε
2.11
for |x0 − ϕt0 | ≤ δt0 , ε and all t ≥ t0 ;
ii attractive if there exists ηt0 > 0 such that x0 ≤ η implies
lim xt, x0 , t0 0;
t→∞
2.12
iii asymptotically stable if it is stable and attractive.
The space ln∞0 is the set of real sequences defined on the set of positive integers where
any individual sequence is bounded with respect to the usual supremum norm. It is well
known that under the supremum norm ln∞0 is a Banach space 26.
Definition 2.11 see 27. A set Ω of sequences in ln∞0 is uniformly Cauchy or equi-Cauchy,
if for every ε > 0, there exists an integer N such that |xi − xj| < ε, whenever i, j > N for
any x {xn} in Ω.
Theorem 2.12 see 27, discrete Arzela-Ascoli’s theorem. A bounded, uniformly Cauchy
subset Ω of ln∞0 is relatively compact.
Theorem 2.13 see 20, Krasnoselskii’s fixed point theorem. Let S be a nonempty, closed,
convex, and bounded subset of the Banach space X and let A : X → X and B : S → X be two
operators such that
a A is a contraction with constant L < 1,
b B is continuous, BS resides in a compact subset of X,
c x Ax By, y ∈ S ⇒ x ∈ S.
Then the operator equation Ax Bx x has a solution in S.
3. Main Results
∞
Let lα∞ be the set ofall real sequences x {xt}∞
tα with norm x supt∈Nα |xt|, then lα is a
Banach space.
Define the operator
t−α
1 x0 α−1
t
t − s − 1α−1 fs α, xs α,
Γα
Γα s0
x0 α−1
t
,
Axt Γα
t−α
1 Bxt t − s − 1α−1 fs α, xs α, t ∈ Nα .
Γα s0
P xt 3.1
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5
Obviously, P x Ax Bx, the operator A is a contraction with the constant 0, which implies
that condition a of Theorem 2.13 holds, and xt is a solution of 1.1 if it is a fixed point of
the operator P .
Lemma 3.1. Assume that the following condition is satisfied:
H1 there exist constants β1 ∈ α, 1 and L1 ≥ 0 such that
ft, xt ≤ L1 t−β1 for t ∈ Nα .
3.2
Then the operator B is continuous and BS1 is a compact subset of R for t ∈ Nαn1 , where
S1 xt : |xt| ≤ t−γ1 for t ∈ Nαn1 ,
3.3
γ1 −1/2α − β1 , and n1 ∈ N satisfies that
L1 Γ 1 − β1 1/2αβ1 −1
−γ1 |x0 | ≤ 1.
α n1 γ1
α n1 γ1
Γα
Γ 1 α − β1
3.4
Proof. For t ∈ Nα , apply Lemma 2.8 and γ1 > 0,
Γt 1
1
,
t−γ1 t−γ1 1 O
t
Γ t γ1 1
3.5
and we have that t−γ1 → 0 as t → ∞, then there exists a n1 ∈ N such that inequality 3.4
holds, which implies that the set S1 exists.
We firstly show that B maps S1 in S1 .
It is easy to know that S1 is a closed, bounded, and convex subset of R.
Apply condition H1 , Lemma 2.5, Corollary 2.9 and 3.4, for t ∈ Nαn1 , we have
|Bxt| ≤
≤
t−α
1 t − s − 1α−1 fs α, xs α
Γα s0
t−α
1 t − s − 1α−1 L1 s α−β1 Γα s0
L1 Δ−α t α−β1 L1 Γ 1 − β1
t αα−β1 Γ 1 α − β1
L1 Γ 1 − β1 α−β1 < t
Γ 1 α − β1
L1 Γ 1 − β1 −γ1 −γ1 t
t γ1
Γ 1 α − β1
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L1 Γ 1 − β1 −γ1 −γ1 t
≤ α n1 γ1
Γ 1 α − β1
≤ t−γ1 ,
3.6
which implies that BS1 ⊂ S1 for t ∈ Nαn1 .
Nextly, we show that B is continuous on S1 .
Let ε > 0 be given then there exist T1 ∈ N and T1 ≥ n1 such that t ∈ NαT1 implies that
L1 Γ 1 − β1 α−β1 ε
< .
t
2
Γ 1 α − β1
3.7
Let {xn } be a sequence such that xn → x. For t ∈ {α n1 , α n1 1, . . . , α T1 − 1},
applying the continuity of f and Lemma 2.6, we have
t−α
1 t − s − 1α−1 fs α, xn s α − fs α, xs α
Γα s0
≤
max fs α, xn s α − fs α, xs α
|Bxn t − Bxt| ≤
s∈{0,1,...,T1 −1}
×
t−α
1 t − s − 1α−1
Γα s0
tα
max fs α, xn s α − fs α, xs α
Γα 1 s∈{0,1,...,T1 −1}
α T1 − 1α
≤
max fs α, xn s α − fs α, xs α
Γα 1 s∈{0,1,...,T1 −1}
−→ 0
3.8
as n −→ ∞.
For t ∈ NαT1 ,
|Bxn t − Bxt| ≤
≤
t−α
1 t − s − 1α−1 fs α, xn s α fs α, xs α
Γα s0
t−α
2L1 t − s − 1α−1 s α−β1 Γα s0
2L1 Δ−α t α−β1 2L1 Γ 1 − β1
t αα−β1 Γ 1 α − β1
2L1 Γ 1 − β1 α−β1 < t
Γ 1 α − β1
< ε.
3.9
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Thus, for all t ∈ Nαn1 , we have
|Bxn t − Bxt| −→ 0
as n → ∞.
3.10
which implies that B is continuous.
Lastly, we show that BS1 is relatively compact.
Let t1 , t2 ∈ NαT1 and t2 > t1 , thus we have
1 t
2 −α
t − s − 1α−1 fs α, xs α
|Bxt2 − Bxt1 | Γα s0 2
1 −α
1 t
α−1
−
fs α, xs α
t1 − s − 1
Γα s0
≤
2 −α
1 t
t2 − s − 1α−1 fs α, xs α
Γα s0
1 −α
1 t
t1 − s − 1α−1 fs α, xs α
Γα s0
L1 Γ 1 − β1
L1 Γ 1 − β1
α−β1 ≤ t2 α
t1 αα−β1 Γ 1 α − β1
Γ 1 α − β1
L1 Γ 1 − β1
L1 Γ 1 − β1
α−β1 < t2
t1 α−β1 Γ 1 α − β1
Γ 1 α − β1
3.11
< ε.
Thus, {Bx : x ∈ S1 } is a bounded and uniformly Cauchy subset by Definition 2.11, and BS1
is relatively compact by means of Theorem 2.12. This completes the proof.
Lemma 3.2. Assume that condition H1 holds, then a solution of 1.1 is in S1 for t ∈ Nαn1 .
Proof. Notice if that xt is a fixed point of P , then it is a solution of 1.1. To prove this, it
remains to show that, for fixed y ∈ S1 , x Ax By ⇒ x ∈ S1 holds.
If x Ax By, applying condition H1 and 3.4, for t ∈ Nαn1 , we have
|xt| ≤ |Axt| Byt
t−α
1 |x0 | α−1
t
t − s − 1α−1 f s α, ys α Γα
Γα s0
L1 Γ 1 − β1
|x0 | α−1
≤
t
t αα−β1 Γα
Γ 1 α − β1
L1 Γ 1 − β1 α−β1 |x0 | α−1
t
<
t
Γα
Γ 1 α − β1
≤
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L1 Γ 1 − β1 1/2αβ1 −1
−γ1 −γ1 |x0 | t
t γ1
t γ1
Γα
Γ 1 α − β1
L1 Γ 1 − β1 1/2αβ1 −1
−γ1 −γ1 |x0 | t
≤
α n1 γ1
α n1 γ1
Γα
Γ 1 α − β1
≤ t−γ1 .
3.12
Thus, xt ∈ S1 for t ∈ Nαn1 . According to Theorem 2.13 and Lemma 3.1, there exists a x ∈ S1
such that x AxBx, that is, P has a fixed point in S1 which is a solution of 1.1 for t ∈ Nαn1 .
This completes the proof.
Theorem 3.3. Assume that condition H1 holds, then the solutions of 1.1 is attractive.
Proof. By Lemma 3.2, the solutions of 1.1 exist and are in S1 . All functions xt in S1 tend to
0 as t → ∞. Then the solutions of 1.1 tend to zero as t → ∞. This completes the proof.
Theorem 3.4. Assume that the following condition is satisfied:
H2 there exist constants β2 ∈ α, 1 and L2 ≥ 0 such that
ft, xt − f t, yt ≤ L2 t−β2 x − y
for t ∈ Nα .
3.13
Then the solutions of 1.1 are stable provided that
L2 Γ1 αΓ 1 − β2
c : < 1.
Γ 1 α − β2 Γ 1 β2
3.14
Proof. Let xt be a solution of 1.1, and let xt
be a solution of 1.1 satisfying the initial
value condition x0
x0 . For t ∈ Nα , applying condition H2 , we have
≤
|xt − xt|
≤
t−α
tα−1
1 t − s − 1α−1
|x0 − x0 | Γα
Γα s0
α
× fs α, xs α − fs α, xs
t−α
tα−1
L2 t − s − 1α−1 s α−β2 x − x
|x0 − x0 | Γα
Γα s0
tα−1
|x0 − x0 | L2 Δ−α t α−β2 x − x
Γα
L2 Γ 1 − β2
tα−1
t αα−β2 x − x
|x0 − x0 | Γα
Γ 1 α − β2
L2 Γ 1 − β2
αα−1
≤
αα−β2 x − x
|x0 − x0 | Γα
Γ 1 α − β2
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L2 Γ1 αΓ 1 − β2
α|x0 − x0 | x − x
Γ 1 α − β2 Γ 1 β2
α|x0 − x0 | cx − x,
3.15
which implies that
≤
x − x
α
|x0 − x0 |.
1−c
3.16
For any given ε > 0, let δ 1 − c/αε, |x0 − x0 | < δ follows that x − x
< ε, which
yields that the solutions of 1.1 are stable. This completes the proof.
Theorem 3.5. Assume that conditions H1 and H2 hold, then the solutions of 1.1 are
asymptotically stable provided that 3.14 holds.
Theorem 3.5 is the simple consequence of Theorems 3.3 and 3.4.
Theorem 3.6. Assume that the following condition is satisfied:
H3 there exist constants β3 ∈ α, 1/21 α, γ2 1/21 − α, and L3 ≥ 0 such that
ft, xt ≤ L3 t γ2 −β3 |xt|
for t ∈ Nα .
3.17
Then the solutions of 1.1 is attractive.
Proof. Set
S2 xt : |xt| ≤ t−γ2 for t ∈ Nαn2 ,
3.18
where n2 ∈ N satisfies that
L3 Γ 1 − β3 − γ2 −γ2 α−β3 |x0 | ≤ 1.
α n2 γ2
α n2 γ2
Γα
Γ 1 α − β3 − γ2
3.19
We first prove condition c of Theorem 2.13, that is, for fixed y ∈ S2 and for all x ∈ R,
x Ax By ⇒ x ∈ S2 holds.
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If x Ax By, applying condition H3 and 3.19, for t ∈ Nαn2 , we have
|xt| ≤ |Axt| Byt
≤
t−α
1 |x0 | α−1
t
t − s − 1α−1 f s α, ys α Γα
Γα s0
≤
t−α
−β3 1 |x0 | α−1
ys α
t
t − s − 1α−1 L3 s α γ2
Γα
Γα s0
≤
t−α
−β3 L3 |x0 | α−1
t
t − s − 1α−1 s α γ2
s α−γ2 Γα
Γα s0
t−α
L3 |x0 | α−1
t
t − s − 1α−1 s α−β3 −γ2 Γα
Γα s0
L3 Γ 1 − β3 − γ2
|x0 | α−1
t
≤
t αα−β3 −γ2 Γα
Γ 1 α − β3 − γ2
L3 Γ 1 − β3 − γ2 α−β3 −γ2 |x0 | α−1
t
<
t
Γα
Γ 1 α − β3 − γ2
L3 Γ 1 − β3 − γ2 −γ2 α−β3 −γ2 |x0 | t
t γ2
t γ2
Γα
Γ 1 α − β3 − γ2
L3 Γ 1 − β3 − γ2 −γ2 α−β3 −γ2 |x0 | t
≤
α n2 γ2
α n2 γ2
Γα
Γ 1 α − β3 − γ2
≤
3.20
≤ t−γ2 .
Thus, condition c of Theorem 2.13 holds.
The proof of condition b of Theorem 2.13 is similar to that of Lemma 3.1, and we
omit it. Therefore, P has a fixed point in S2 by using Theorem 2.13, that is, the IVP 1.1
has a solution in S2 . Moreover, all functions in S2 tend to 0 as t → ∞, then the solution of
1.1 tends to zero as t → ∞, which shows that the zero solution of 1.1 is attractive. This
completes the proof.
Theorem 3.7. Assume that conditions H2 and H3 hold, then the solutions of 1.1 are
asymptotically stable provided that 3.14 holds.
Theorem 3.8. Assume that the following condition is satisfied:
H4 there exist constants η ∈ 0, 1, β4 ∈ α, 2 αη/2 η, and L4 ≥ 0 such that
ft, xt ≤ L4 t 1−β4 |xt|η
for t ∈ Nα .
3.21
Then the solutions of 1.1 is attractive.
Proof. Set
S3 xt : |xt| ≤ t−γ3 for t ∈ Nαn3 ,
3.22
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11
where γ3 1/2β4 − α, and n3 ∈ N satisfies that
L4 Γ 1 − β4 − γ3 η α−1γ3 −γ
|x0 | α n3 γ3
α n3 γ3 3 ≤ 1.
Γα
Γ 1 α − β4 − γ3 η
3.23
Here we only prove that condition c of Theorem 2.13 holds, and the remaining part
of the proof is similar to that of Theorem 3.6.
Since η ∈ 0, 1, β4 ∈ α, 2 αη/2 η, and γ3 1/2β4 − α, then γ3 , γ3 η, α γ3 ∈
0, 1, β4 γ3 η ∈ α, 1.
If x Ax By, applying condition H4 , Lemma 2.5 and 3.23, for t ∈ Nαn3 , we have
t−α
1 |x0 | α−1
t
t − s − 1α−1 f s α, ys α |xt| ≤ |Axt| Byt ≤
Γα
Γα s0
≤
t−α
η
1 |x0 | α−1
t
t − s − 1α−1 L4 s α 1−β4 ys α
Γα
Γα s0
≤
t−α
η
−β4 L4 |x0 | α−1
t
t − s − 1α−1 s α γ3 η
s α−γ3 Γα
Γα s0
≤
t−α
−β4 L4 |x0 | α−1
t
t − s − 1α−1 s α γ3 η
s α−γ3 η
Γα
Γα s0
t−α
L4 |x0 | α−1
t
t − s − 1α−1 s α−β4 −γ3 η
Γα
Γα s0
L4 Γ 1 − β4 − γ3 η
|x0 | α−1
t
≤
t αα−β4 −γ3 η
Γα
Γ 1 α − β4 − γ3 η
L4 Γ 1 − β4 − γ3 η α−β4 −γ3 η
|x0 | α−1
t
<
t
Γα
Γ 1 α − β4 − γ3 η
L4 Γ 1 − β4 − γ3 η α−β4 |x0 | α−1
t
≤
t
Γα
Γ 1 α − β4 − γ3 η
L4 Γ 1 − β4 − γ3 η α−1γ3 −γ3 −γ3 |x0 | t
t γ3
t γ3
Γα
Γ 1 α − β4 − γ3 η
α−1γ3 |x0 | ≤
α n3 γ3
Γα
L4 Γ 1 − β4 − γ3 η −γ3 −γ3 t
α n3 γ3
Γ 1 α − β4 − γ3 η
≤ t−γ3 .
3.24
Thus, condition c of Theorem 2.13 holds. This completes the proof.
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4. Examples
Example 4.1. Consider
Δ0.5 xt 0.2t 0.5−0.75 sinxt 0.5, t ∈ N0 ,
Δ−0.5 xt|t0 x0 ,
4.1
where ft, xt 0.2t−0.75 sinxt, t ∈ N0.5 .
Since
ft, xt 0.2t−0.75 sinxt ≤ 0.2t−0.75 ,
4.2
this implies that condition H1 holds.
In addition,
ft, xt − f t, yt ≤ 0.2t−0.75 x − y.
4.3
Thus, condition H2 is satisfied.
Moreover, from L2 0.2, α 0.5, and β2 0.75, we have
L2 Γ1 αΓ 1 − β2
0.2Γ1.5Γ0.25
≈ 0.7716 < 1,
c Γ1.25Γ1.75
Γ 1 α − β2 Γ 1 β2
4.4
which implies that inequality 3.14 holds.
Thus the solutions of 4.1 are asymptotically stable by Theorem 3.5.
Example 4.2. Consider
Δ0.5 xt 0.2t 1.5−0.6 xt 0.5, t ∈ N0 ,
Δ−0.5 xt|t0 x0 ,
4.5
where ft, xt 0.2t 1−0.6 xt, t ∈ N0.5 .
Since β3 0.6, α 0.5, we have that β3 ∈ α, 1/21 α, γ2 0.25 and
ft, xt 0.2t 1−0.6 xt ≤ 0.2t 0.25−0.6 |xt|,
4.6
which implies that condition H3 is satisfied.
Meanwhile,
ft, xt − f t, yt ≤ 0.2t 1−0.6 x − y ≤ 0.2t−0.6 x − y,
which implies that condition H2 is satisfied.
4.7
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13
From L2 0.2, α 0.5, and β2 0.6, we have
L2 Γ1 αΓ 1 − β2
0.2Γ1.5Γ0.4
c ≈ 0.4120 < 1,
Γ0.9Γ1.6
Γ 1 α − β2 Γ 1 β2
4.8
which implies that inequality 3.14 holds.
Thus the solutions of 4.5 are asymptotically stable by Theorem 3.7.
Example 4.3. Consider
Δ0.5 xt t 1.5−0.6 x1/3 t 0.5, t ∈ N0 ,
Δ−0.5 xt|t0 x0 ,
4.9
where ft, xt t 1−0.6 x1/3 t, t ∈ N0.5 .
Since α 0.5, β4 0.6, η 1/3, we have that η ∈ 0, 1, β4 ∈ α, 2 αη/2 η and
ft, xt ≤ t 1−0.6 |xt|1/3 ,
4.10
then condition H4 is satisfied.
The solutions of 4.9 are attractive by Theorem 3.8.
Acknowledgments
This research was supported by the NSF of Hunan Province 10JJ6007, 2011FJ3013,
the Scientific Research Foundation of Hunan Provincial Education Department, and the
Construct Program of the Key Discipline in Hunan Province.
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