Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 707631, 15 pages
doi:10.1155/2012/707631
Review Article
The Existence of Solutions to a System of Discrete
Fractional Boundary Value Problems
Yuanyuan Pan, Zhenlai Han, Shurong Sun, and Yige Zhao
School of Mathematical Sciences, University of Jinan, Jinan, Shandong 250022, China
Correspondence should be addressed to Zhenlai Han, hanzhenlai@163.com
Received 7 December 2011; Revised 26 December 2011; Accepted 1 January 2012
Academic Editor: Ferhan M. Atici
Copyright q 2012 Yuanyuan Pan et al. 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 study the existence of solutions for the boundary value problem −Δν y1 t fy1 tν−1, y2 t
μ − 1, −Δμ y2 t gy1 t ν − 1, y2 t μ − 1, y1 ν − 2 Δy1 ν b 0, y2 μ − 2 Δy2 μ b 0, where 1 < μ, ν ≤ 2, f, g : R × R → R are continuous functions, b ∈ N0 . The existence of
solutions to this problem is established by the Guo-Krasnosel’kii theorem and the Schauder fixedpoint theorem, and some examples are given to illustrate the main results.
1. Introduction
In recent years, fractional differential equations have been of great interest. It is caused
both by the intensive development of the theory of fractional calculus itself and by the
applications of such constructions in various sciences such as physics, mechanics, chemistry,
and engineering. The continuous fractional calculus has seen tremendous growth within the
last ten years or so. Some of the recent progress in the continuous fractional calculus has
included a paper 1 in which the authors explored a continuous fractional boundary value
problem of conjugate type. Using cone theory, they then deduced the existence of one or
more positive solutions. Other recent work in the direction of those articles may be found,
see 2–14.
Recently, there appeared a number of papers on the discrete fractional calculus, such
as 15–32, which has helped to build up some of the basic theories of this area. For example,
Atici and Eloe discussed properties of the generalized falling function, a corresponding
power rule for fractional delta-operators, and the commutivity of fractional sums in 15.
And Goodrich studied a two-point fractional boundary value problem in 24, which gave
the existence results for a certain two-point boundary value problem of right-focal type for a
fractional difference equation.
2
Abstract and Applied Analysis
From the above works, we can see a fact, although the discrete fractional boundary
value problem has been studied by some authors, to the best of our knowledge, systems of
discrete fractional boundary value problem are seldom considered.
Goodrich in 24 considered a discrete fractional boundary value problem
−Δν yt f t ν − 1, yt ν − 1 ,
yν − 2 0 Δyν b,
1.1
where t ∈ 0, b 1N0 , f : ν − 1, ν b − 1Nν−1 × R → R, and 1 < ν ≤ 2, which gave a
representation for the solution to this problem and established the existence and uniqueness
of solution to this problem by the Guo-Krasnosel’kii theorem.
Goodrich in 25 examined a system of discrete fractional difference equations subject
to nonlocal boundary conditions
−Δν1 y1 t λ1 a1 t ν1 − 1f1 y1 t ν1 − 1, y2 t ν2 − 1 ,
−Δν2 y2 t λ2 a2 t ν2 − 1f2 y1 t ν1 − 1, y2 t ν2 − 1 ,
1.2
for t ∈ 0, bN0 , subject to the conditions
y1 ν1 − 2 ψ1 y1 ,
y1 ν1 b φ1 y1 ,
y2 ν2 − 2 ψ2 y2 ,
y2 ν2 b φ2 y2 ,
1.3
where λi > 0, ai : R → 0, ∞, νi ∈ 1, 2, and for each i 1, 2, ψi , φi : Rb3 → R are
given functionals, and fi : 0, ∞ × 0, ∞ → 0, ∞ are continuous for each admissible
i. This paper gives the existence of a positive solution on discrete fractional boundary value
problems.
Motivated by all the works above, in this paper, we discuss the existence of solutions
to a system of discrete fractional boundary value problem
−Δν y1 t f y1 t ν − 1,
−Δμ y2 t g y1 t ν − 1,
y1 ν − 2 Δy1 ν b 0,
y2 μ − 2 Δy2 μ b 0,
y2 t μ − 1 ,
y2 t μ − 1 ,
for t ∈ 0, b 1N0 ,
for t ∈ 0, b 1N0 ,
1.4
where 1 < μ, ν ≤ 2, f, g : R × R → R are continuous functions, b ∈ N0 .
The plan of the paper is as follows. In Section 2, we will present some lemmas in order
to prove our main results. Section 3, establishes and proves our main results. In Section 4 we
give some examples to illustrate the main results.
2. Preliminaries
For the convenience of the reader, we give some background materials from fractional
difference theory to facilitate the analysis of problem 1.4. These and other related results
and their proof can be found in 15–18.
Abstract and Applied Analysis
3
Definition 2.1. One defines tν : Γt 1/Γt 1 − ν, for any t and ν for which the right-hand
side is defined. One also appeals to the convention that if t 1 − ν is a pole of the Gamma
function and t 1 is not a pole, then tν 0.
Definition 2.2. The νth fractional sum of a function f, for ν > 0, is defined by
Δ−ν ft; a :
t−ν
1 t − s − 1ν−1 fs,
Γν sa
2.1
for t ∈ {a ν, a ν 1, . . .} : Naν . One also defines the νth fractional difference for ν > 0 by
Δν ft : ΔN Δν−N ft, where t ∈ Naν and ν ∈ N is chosen so that 0 ≤ N − 1 < ν ≤ N.
Lemma 2.3. Let t and ν be any numbers for which tν and tν−1 are defined, then
Δtν νtν−1 .
2.2
Lemma 2.4. Let 0 ≤ N − 1 < ν ≤ N, where N ∈ N and N − 1 ≥ 0, then
Δ−ν Δν yt yt C1 tν−1 C2 tν−2 · · · CN tν−N ,
2.3
for some Ci ∈ R, with 1 ≤ i ≤ N.
Before stating the next useful lemma, let us introduce the following notation, which
will be important in the sequel:
T1 : t, s ∈ ν − 1, ν b 1Nν−1 × 0, b 1N0 : 0 ≤ s < t − ν 1 ≤ b 2 ,
T2 : t, s ∈ ν − 1, ν b 1Nν−1 × 0, b 1N0 : 0 ≤ t − ν 1 ≤ s ≤ b 2 ,
L1 : t, s ∈ μ − 1, μ b 1 Nμ−1 × 0, b 1N0 : 0 ≤ s < t − μ 1 ≤ b 2 ,
L2 : t, s ∈ μ − 1, μ b 1 Nμ−1 × 0, b 1N0 : 0 ≤ t − μ 1 ≤ s ≤ b 2 .
2.4
Lemma 2.5. (24) The unique solution of the discrete fractional boundary value problem
−Δν yt ht ν − 1,
yν − 2 0 Δyν b
2.5
is given by
yt :
b1
G1 t, shs ν − 1,
s0
2.6
4
Abstract and Applied Analysis
where 1 < ν ≤ 2, G1 t, s is given by
⎧
Γb 3tν−1
⎪
⎪
⎪
ν b − s − 1ν−2 − t − s − 1ν−1 ,
⎪
⎪
⎨ Γν b 1
1
G1 t, s :
Γν ⎪
⎪
⎪
Γb 3tν−1
⎪
⎪
ν b − s − 1ν−2 ,
⎩
Γν b 1
t, s ∈ T1 ,
2.7
t, s ∈ T2 .
Remark 2.6. Let us note that in case we put μ replacing ν in Lemma 2.5, it follows that
yt :
b1
G2 t, sh s μ − 1 ,
2.8
s0
where 1 < μ ≤ 2, G2 t, s is given by
⎧
μ−1
μ−2
⎪
Γb 3t μ−1
⎪
⎪ − t − s − 1 ,
μb−s−1
⎪
⎪
⎪
Γ
μ
b
1
1 ⎨
G2 t, s : Γ μ ⎪
⎪
μ−1
⎪
μ−2
⎪
Γb 3t ⎪
⎪
,
μb−s−1
⎩ Γ μb1
t, s ∈ L1 ,
2.9
t, s ∈ L2 .
Lemma 2.7. (24) The Green function G1 t, s given in Lemma 2.5 satisfies
i G1 t, s ≥ 0 for t, s ∈ ν − 1, ν b 1Nν−1 × 0, b 1N0
ii maxt∈ν−1,νb1Nν−1 G1 t, s G1 s ν − 1, s, for s ∈ 0, bN0
iii there exists a number γ1 ∈ 0, 1 such that
min
t∈bν/4, 3bν/4
G1 t, s ≥ γ1
max
t∈ν−1, νb1Nν−1
G1 t, s γ1 G1 s ν − 1, s, for s ∈ 0, b 1N0 .
2.10
Similarly, G2 t, s satisfies
I G2 t, s ≥ 0 for t, s ∈ μ − 1, μ b 1Nμ−1 × 0, b 1N0
II maxt∈μ−1,μb1Nμ−1 G2 t, s G2 s μ − 1, s, for s ∈ 0, bN0
III there exists a number γ2 ∈ 0, 1 such that
G2 t, s ≥ γ2
max
G2 t, s γ2 G2 s μ − 1, s , for s ∈ 0, b 1N0 .
min
t∈bμ/4, 3bμ/4
t∈μ−1, μb1N
μ−1
2.11
Let B1 and B2 represent the Banach space of all maps from ν − 1, ν bNν−1 into R and
μ − 1, μ bNμ−1 into R, respectively. And · is the usual maximum norm. Define
X : B1 × B2 .
2.12
Abstract and Applied Analysis
5
By equipping X with the norm
y1 , y2 : y1 y2 ,
2.13
it follows that X, · is a Banach space, see 26, 27.
Next, we wish to develop a representation for a solution of 1.4 as the fixed point of an
appropriate operator on X. To accomplish this, we present some straightforward adaptations
of results from 24 that will be of use here. Since the proofs of these adaptations are evident,
we do not include them.
Now, consider the operator S : X → X defined by
S y1 , y2 t1 , t2 : S1 y1 , y2 t1 , S2 y1 , y2 t2 ,
2.14
where we define S1 : X → B1 by
b1
S1 y1 , y2 t1 :
G1 t1 , sf y1 s ν − 1, y2 s μ − 1 ,
2.15
s0
and S2 : X → B2 by
b1
S2 y1 , y2 t2 :
G2 t2 , sf y1 s ν − 1, y2 s μ − 1 .
2.16
s0
We claim that whenever y1 , y2 ∈ X is a fixed point of the operator S, it follows that the pair
of functions y1 t and y2 t are a solution to problem 1.4.
Lemma 2.8. Let f, g : R × R → R. If y1 , y2 ∈ X is a fixed point of S, then y1 , y2 is a solution to
problem 1.4.
Proof. Suppose that the operator S has a fixed point, say y1 , y2 ∈ X. Let t1 , t2 ∈ Nν−1 × Nμ−1 ,
then we have
y1 S1 y1 , y2 t1 ,
2.17
6
Abstract and Applied Analysis
where S1 is defined as in 2.15. It is easy to check that S1 y1 , y2 ν − 2 0 and
ΔS1 y1 , y2 ν b S1 y1 , y2 ν b 1 − S1 y1 , y2 ν b
b1
Γb 3ν b 1ν−1
ν−2
ν−1
ν b − s − 1 − ν b − s
Γν b 1
s0
× f y1 s ν − 1, y2 s μ − 1
b1
Γb 3ν bν−1
ν−2
ν−1
−
ν b − s − 1 − ν b − s − 1
Γν b 1
s0
× f y1 s ν − 1, y2 s μ − 1
b1 ν b 1ν b − s − 1ν−2 − b 2ν b − s − 1ν−2
s0
× f y1 s ν − 1, y2 s μ − 1
b1 ν b − sΓν b − s Γν b − s
−
−
b − s 1Γb − s 1 Γb − s 1
s0
× f y1 s ν − 1, y2 s μ − 1
b1 ν − 1Γν b − s
ν − 1ν b − s − 1ν−2 −
b − s 1Γb − s 1
s0
× f y1 s ν − 1, y2 s μ − 1
b1 Γν b − s ν − 1Γν b − s
−
ν − 1
Γb − s 2
Γb − s 2
s0
× f y1 s ν − 1, y2 s μ − 1 0,
2.18
so the boundary conditions are satisfied. Furthermore, we can check that S1 y1 , y2 t1 satisfies the difference equation in 1.4. Since similar verifications may be made for
S2 y1 , y2 t2 , the claim follows, which completes the proof.
3. Main Results
In this section, we will show that under certain conditions, problem 1.4 has at least one
solution.
Let us now present the conditions that we will assume henceforth. We note that
conditions F1-F2 are similar to the conditions given by Henderson et al. 26
F1 :f, g ∈ C0, ∞, 0, ∞
f y1 , y2
g y1 , y2
∗
f ,
lim
g∗,
lim
F2 :
y1 ,y2 → 0 ,0 y1 y2
y1 ,y2 → 0 ,0 y1 y2
f y1 , y2
g y1 , y2
∗∗
f ,
lim
g ∗∗ ,
lim
y1 ,y2 → ∞,∞ y1 y2
y1 ,y2 → ∞,∞ y1 y2
3.1
Abstract and Applied Analysis
7
satisfying
b1
1
G1 s ν − 1, s f ∗ ε ≤ ,
2
s0
b1
1
γG1 s ν − 1, s f ∗∗ − ε ≥ ,
2
s0
b1
1
G2 s μ − 1, s g ∗ ε ≤ ,
2
s0
b1
1
γG2 s μ − 1, s g ∗∗ − ε ≥ ,
2
s0
3.2
for some 0 < ε < min{f ∗∗ , g ∗∗ }.
Define the cone by
K :
y1 , y2 ∈ X : y1 , y2 ≥ 0,
min
t1 ,t2 ∈ν−1/4, 3νb/4×μ−1/4, 3μb/4
,
× y1 t1 y2 t2 ≥ γ y1 , y2
3.3
where γ min{γ1 , γ2 }, for i 1, 2, γi is defined as in Lemma 2.7. Clearly, K ⊆ X. In order to
show that S has a fixed point in K, we must first demonstrate that K is invariant under S,
that is, S : K → K. We show this now.
Lemma 3.1. Assume that f, g are nonnegative functions. Let S : X → X be the operator defined as
2.14, then S : K → K.
Proof. Suppose that y1 , y2 ∈ K. We claim that
S1 y1 , y2 t1 S2 y1 , y2 t2 ≥ γ S y1 , y2 . 3.4
3μb/4
min
t1 ,t2 ∈ν−1/4, 3νb/4×μ−1/4,
In fact,
min
S1 y1 , y2 t1 t1 ∈ν−1/4, 3νb/4
≥
b1
G1 t1 , sf y1 s ν − 1, y2 s μ − 1
min
t1 ∈ν−1/4, 3νb/4
s0
b1
≥ γ1 G1 s ν − 1, sf y1 s ν − 1, y2 s μ − 1
3.5
s0
γ1
max
t1 ∈ν−1,νbNν−1
b1
G1 t1 , sf y1 s ν − 1, y2 s μ − 1
s0
γ1 S1 y1 , y2 .
Similarly, we get
min
S2 y1 , y2 t2 ≥ γ2 S2 y1 , y2 .
t2 ∈μ−1/4, 3μb/4
3.6
8
Abstract and Applied Analysis
Put γ min{γ1 , γ2 }. Consequently, from 3.5 and 3.6, we obtain
min
t1 ,t2 ∈ν−1/4, 3νb/4×μ−1/4, 3μb/4
S1 y1 , y2 t1 S2 y1 , y2 t2 S1 y1 , y2 t1 t1 ∈ν−1/4, 3νb/4
min
S2 y1 , y2 t2 t2 ∈μ−1/4, 3μb/4
≥ γ1 S1 y1 , y2 γ2 S2 y1 , y2 ≥ γ S1 y1 , y2 S2 y1 , y2 γ S y1 , y2 .
≥
min
3.7
So, for y1 , y2 ∈ K, we find
S1 y1 , y2 t1 S2 y1 , y2 t2 ≥ γ S y1 , y2 . 3.8
t1 ,t2 ∈ν−1/4, 3νb/4×μ−1/4, 3μb/4
min
Therefore, from 3.8, whenever we get y1 , y2 ∈ K, it follows that Sy1 , y2 ∈ K. This
completes the proof.
We next recall the Guo-Krasnosel’kii fixed-point theorem and Schauder fixed-point
theorem, which we will use to prove the main results.
Lemma 3.2. ([21]) Let B be a Banach space, and let K ⊆ B be a cone. Assume that Ω1 and Ω2 are open
subsets contained in B such that 0 ∈ Ω1 and Ω1 ⊆ Ω2 . Assume, further, that T : K ∩ Ω2 \ Ω1 → K
is a completely continuous operator. If either
1 T y ≤ y for y ∈ K ∩ ∂Ω1 and T y ≥ y for y ∈ K ∩ ∂Ω2 ,
2 or T y ≥ y for y ∈ K ∩ ∂Ω1 and T y ≤ y for y ∈ K ∩ ∂Ω2 ,
then T has at least one fixed point in K ∩ Ω2 \ Ω1 .
Lemma 3.3. ([22]) (Schauder fixed-point theorem) Suppose that X is a Banach space. Let K be a
bounded closed-convex set of X, and let T : K → K be a completely continuous operator, then T has
at least one fixed point in K.
Theorem 3.4. Suppose that conditions (F1)-(F2) hold, then problem 1.4 has at least one solution.
Proof. From Lemma 3.1, we know that S : K → K. Note that S is a summation operator on
a discrete finite set. Hence, S is a completely continuous operator.
By conditions F1 and F2, for ε given in F2, there exists some constant r1 > 0, such
that
f y1 , y2 ≤ f ∗ ε y1 y2 ,
g y1 , y2 ≤ g ∗ ε y1 y2 ,
whenever y1 , y2 < r1 .
3.9
3.10
Abstract and Applied Analysis
9
Setting
Ω1 :
y1 , y2 ∈ X : y1 , y2 < r1 ,
3.11
Then, for y1 , y2 ∈ K ∩ ∂Ω1 , by Lemma 2.7, F2 and 3.9, we find that
S1 y1 , y2 ≤
b1
max
t1 ∈ν−1,νbNν−1
b1
G1 t1 , sf y1 s ν − 1, y2 s μ − 1
s0
G1 s ν − 1, s f ∗ ε y1 y2
s0
b1
≤ y1 , y2 G1 s ν − 1, s f ∗ ε
3.12
s0
1
1 ≤ r1 y1 , y2 .
2
2
Similar, by 3.10, it can be shown that
S2 y1 , y2 ≤ 1 y1 , y2 .
2
3.13
Thus, putting 3.12 and 3.13 together, for y1 , y2 ∈ K ∩ ∂Ω1 , we have
S y1 , y2 ≤ 1 y1 , y2 2
1
y1 , y2 y1 , y2 .
2
3.14
Now, for the above ε, by F2, we can find a constant r ∗∗ > 0 such that
f y1 , y2 ≥ f ∗∗ − ε y1 y2 ,
g y1 , y2 ≥ g ∗∗ − ε y1 y2 ,
3.15
3.16
whenever y1 y2 ≥ r ∗∗ . Let
b1
b1
∗∗
∗∗
∗∗ ∗∗
1 ∗∗
.
r ≤ r2 ≤ min
G1 s ν − 1, s f − ε γr , G2 s μ − 1, s g − ε γr
2
s0
s0
3.17
Moreover, put
Ω2 :
y1 , y2 ∈ X : y1 , y2 < 2r2 .
3.18
If y1 , y2 ∈ K ∩ ∂Ω2 , then
y1 t1 y2 t2 ≥
min
t1 ,t2 ∈ν−1/4, 3νb/4×μ−1/4,3μb/4
y1 t1 y2 t2 ≥ γ y1 , y2 .
3.19
10
Abstract and Applied Analysis
Thus, from 3.15 and 3.19, we obtain
S1 y1 , y2 ≥
max
t1 ∈ν−1,νbNν−1
b1
b1
G1 t1 , sf y1 s ν − 1, y2 s μ − 1
s0
G1 s ν − 1, s f ∗∗ − ε y1 y2
s0
≥
b1
G1 s ν − 1, s f ∗∗ − ε γ y1 , y2 s0
≥ r2 3.20
1
y1 , y2 .
2
Similarly, by 3.16 and 3.19, we have
S2 y1 , y2 ≥ 1 y1 , y2 .
2
3.21
So, from 3.20 and 3.21, we get
S y1 , y2 ≥ y1 , y2 ,
3.22
whenever y1 , y2 ∈ K ∩ ∂Ω2 .
Thus, by 3.14 and 3.22, we get that all of the hypotheses of Lemma 3.2 are satisfied.
Consequently, we conclude that S has a fixed point, say y1∗ , y2∗ ∈ K. Then the theorem is
proved.
Theorem 3.5. Let f, g : R × R → R be continuous functions. Suppose that the following conditions
are satisfied:
i there exist nonnegative functions a, h, p, q ∈ CR, R such that
f y1 , y2 ≤ a y1 p y2 , g y1 , y2 ≤ h y1 q y2 ,
3.23
ii Consider the following:
p y
< A,
|y| → ∞ y lim
q y
< B,
|y| → ∞ y lim
3.24
b1
−1
−1
where A 2maxt∈ν−1,νb1 b1
s0 G1 t, s , B 2maxt∈μ−1,μb1
s0 G2 t, s , then
∗∗
∗∗
boundary value problem 1.4 has at least one solution y1 , y2 .
Proof. Let ε1 1/2A − lim|y| → ∞ py/|y|. By ii, there exists a c1 > 0, such that py2 <
A − ε1 |y2 |, for |y2 | ≥ c1 . Set M max{py2 : |y2 | ≤ c1 }, then there exists a c2 > c1 , such that
M/c2 ≤ A − ε1 , so we have py2 < A − ε1 c2 , for |y2 | ≤ c2 .
Abstract and Applied Analysis
11
Let ε2 1/2B − lim|y| → ∞ qy/|y|. In the same way, there exists a constant c3 > 0,
such that qy2 < B − ε2 c3 , |y2 | ≤ c3 . Define
e1 max
b1
G1 t1 , sa y1 ,
e2 max
b1
G2 t2 , sh y1 ,
t2 ∈μ−1,μb1 s0
2A
2B
e1 ,
e2 ,
c max c2 , c3 ,
ε1
ε2
U y1 , y2 | y1 , y2 ∈ X, y1 , y2 ≤ c, t1 , t2 ∈ ν − 1, ν b 1Nν−1
× μ − 1, μ b 1 Nμ−1 .
t1 ∈ν−1,νb1
s0
3.25
Then U is a bounded closed-convex set of X × X, and for any y1 , y2 ∈ U, we have py2 <
A − ε1 c, qy2 < B − ε2 c. Next, we prove S : U → U, where S is defined as 2.14.
For y1 , y2 ∈ U, from i and sii, we obtain
b1
S1 y1 , y2 t1 G1 t1 , sf y1 , y2
s0
≤
b1
b1
G1 t1 , sa y1 G1 t1 , sp y2
s0
≤ e1 A − ε1 c
s0
b1
3.26
G1 t1 , s
s0
≤
1
ε1 1
ε1
c
1−
c c,
2A
2
A
2
similarly,
1
ε2 1
ε2
c
1−
c c.
S2 y1 , y2 t2 ≤
2B
2
B
2
3.27
Thus, from 3.26 and 3.27, we have Sy1 , y2 ≤ c. Then, S : U → U.
Note that S is a summation operator on a discrete finite set. Hence, S is trivially
completely continuous. By Schauder fixed-point theorem, the boundary problem value
problem 1.4 has at least one solution, say y1∗∗ , y2∗∗ . This completes the proof.
4. Examples
In this section, we will present some examples to illustrate the main results.
12
Abstract and Applied Analysis
Example 4.1. Suppose that ν 3/2, μ 5/4, η1 1/1000, η2 10/γ, γ min{γ1 , γ2 }, and γ1 ,
γ2 are defined as in Lemma 2.7, b 2. Let fy1 , y2 : 1/1000e−y2 1000/γe1/y1 y1 y2 and gy : 1/2000e−y1 1000/γe1/y2 y1 y2 , then 1.4 becomes
1
−Δ3/2 y1 t f y1 t , y2 t 2
1
−Δ5/4 y2 t g y1 t , y2 t 2
1
7
y1 −
Δy1
0,
2
2 3
13
Δy2
0.
y2 −
4
4
1
, for t ∈ 0, 3N0 ,
4
1
, for t ∈ 0, 3N0 ,
4
4.1
In this case,
f y1 , y2
1
lim f ∗,
y1 ,y2 → 0 ,0 y1 y2
1000
f y1 , y2
1000
f ∗∗ ,
lim
y1 ,y2 → ∞,∞ y1 y2
γ
g y1 , y2
1
g∗,
lim y1 ,y2 → 0 ,0 y1 y2
2000
g y1 , y2
1000
g ∗∗ ,
lim
y1 ,y2 → ∞,∞ y1 y2
γ
4.2
b1
b1
Γb 3s ν − 1ν−1
G1 s ν − 1, s ν b − s − 1ν−2
Γν
b
1
s0
s0
b1
Γb 3Γs νΓν b − s
Γν
b 1Γs 1Γb − s 2
s0
b1
b 2b 1 · · · b − s 2s ν − 1 · · · s 1
s0
≤
ν bν b − 1 · · · ν b − s
b1
s ν − 1 · · · s 1
s0
≤ b 2b νb 2 4 ×
7
× 4 56.
2
4.3
Similarly, we get
b1
G2 s μ − 1, s ≤ b 2 b μ b 2 52.
s0
4.4
Abstract and Applied Analysis
13
Thus,
b1
∗
1
1
1
≤ ,
G1 s ν − 1, s f η1 ≤ 56 ×
1000
1000
2
s0
b1
∗
1
1
1
≤ .
G2 s ν − 1, s g η1 ≤ 52 ×
2000
1000
2
s0
4.5
On the other hand,
b1
b1
b 2b 1 · · · b − s 2s ν − 1 · · · s 1
G1 s ν − 1, s ν bν b − 1 · · · ν b − s
s0
s0
≥
b1
b 2b 1 · · · b − s 2s ν − 1 · · · s 1
4.6
s0
≥ b 2b 2ν − 1 2 × 2 ×
1
2,
2
similarly,
b1
1
G2 s μ − 1, s ≥ b 2b 2 μ − 1 ≥ 4 × 4 × 1.
4
s0
4.7
b1
1000 10
1
−
1980 ≥ ,
γG1 s ν − 1, s f ∗∗ − η2 ≥ 2γ ×
γ
γ
2
s0
b1
1
1000 10
−
990 ≥ .
γG2 s μ − 1, s g ∗∗ − η2 ≥ γ ×
γ
γ
2
s0
4.8
Then,
Thus, the conditions of Theorem 3.4 are satisfied. So, we obtain that the problem 4.1
has at least one solution.
Example 4.2. Suppose that ν 5/3, μ 7/4, and b 3. Let fy1 , y2 : y1 A/4y2 e−|y2 | ,
gy1 , y2 : y1 B/5y2 e−|y2 | , ay1 |y1 |, hy1 2|y1 |, py2 A/3|y2 |e−|y2 | 1, and
qy2 B/4|y2 |e−|y2 | 1,y1 , y2 ∈ R , where
A
b1
2maxt∈ν−1,νb1 G1 t, s
s0
−1
,
B
2maxt∈μ−1,μb1
b1
s0
−1
G2 t, s
.
4.9
14
Abstract and Applied Analysis
Then problem 1.4 is
A
2
3 −|y2 t3/4|
y2 t e
,
−Δ5/3 y1 t y1 t 3
4
4
2
3 −|y2 t3/4|
B
−Δ7/4 y1 t y1 t ,
y2 t e
3
5
4
1
14
y1 −
Δy1
0,
3
3
1
19
y2 −
Δy2
0.
4
4
for t ∈ 0, 4N0 ,
for t ∈ 0, 4N0 ,
4.10
It is clear that f, g, a, h, p, and q satisfy the conditions of Theorem 3.5. Thus, by
Theorem 3.5, we deduce that problem 4.10 has at least one solution.
Acknowledgments
The authors sincerely thank the reviewers for their valuable suggestions and useful
comments that have lead to the present improved version of the original paper. This research
is supported by the Natural Science Foundation of China 11071143, 60904024, 61174217,
Natural Science Outstanding Youth Foundation of Shandong Province JQ201119, Shandong
Provincial Natural Science Foundation ZR2010AL002, ZR2009AL003, and the Natural
Science Foundation of Educational Department of Shandong Province J11LA01.
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