Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2009, Article ID 730484, 10 pages
doi:10.1155/2009/730484
Research Article
Solutions of 2nth-Order Boundary Value Problem
for Difference Equation via Variational Method
Qingrong Zou and Peixuan Weng
School of Mathematics, South China Normal University, Guangzhou 510631, China
Correspondence should be addressed to Peixuan Weng, wengpx@scnu.edu.cn
Received 7 July 2009; Accepted 15 October 2009
Recommended by Kanishka Perera
The variational method and critical point theory are employed to investigate the existence of
solutions for 2nth-order difference equation Δn pk−n Δn yk−n −1n1 fk, yk 0 for k ∈ 1, N
with boundary value condition y1−n y2−n · · · y0 0, yN1 · · · yNn 0 by constructing
a functional, which transforms the existence of solutions of the boundary value problem BVP
to the existence of critical points for the functional. Some criteria for the existence of at least one
solution and two solutions are established which is the generalization for BVP of the even-order
difference equations.
Copyright q 2009 Q. Zou and P. Weng. 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.
1. Introduction
Difference equations have been applied as models in vast areas such as finance insurance,
biological populations, disease control, genetic study, physical field, and computer application technology. Because of their importance, many literature deals with its existence and
uniqueness problems. For example, see 1–10.
We notice that the existing results are usually obtained by various analytical
techniques, for example, the conical shell fixed point theorem 1, 6, Banach contraction
map method 7, Leray-Schauder fixed point theorem 2, 10, and the upper and lower
solution method 3. It seems that the variational technique combining with the critical
point theory 11 developed in the recent decades is one of the effective ways to study the
boundary value problems of difference equations. However because the variational method
requires a “symmetrical” functional, it is hard for the odd-order difference equations to create
a functional satisfying the “symmetrical” property. Therefore, the even-order difference
equations have been investigated in most references.
Let a, b, N > 1, n ≥ 1, k be integers, and a < b, a, b : {a, a 1, . . . , b} be a discrete
interval in Z. Inspired by 5, 8, in this paper, we try to investigate the following 2nth-order
boundary value problem BVP of difference equation via variational method combining
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Advances in Difference Equations
with some traditional analytical skills:
Δn pk−n Δn yk−n −1n1 f k, yk 0 k ∈ 1, N,
1.1
y1−n y2−n · · · y0 0,
1.2
yN1 · · · yNn 0,
where Δn yk Δn−1 yk1 − Δn−1 yk n 1 is the forward difference operator; pk ∈ R for k ∈
1 − n, N and f ∈ C1, N × R, R. A variational functional for BVP 1.1-1.2 is constructed
which transforms the existence of solutions of the boundary value problem BVP to the
existence of critical points of this functional. In order to prove the existence criteria of critical
points of the functional, some lemmas are given in Section 2. Two criteria for the existence of
at least one solution and two solutions for BVP 1.1-1.2 are established in Section 3 which
is the generalization for BVP of the even-order difference equations. The existence results
obtained in this paper are not found in the references, to the best of our knowledge.
For convenience, we will use the following notations in the following sections:
Fk, u u
fk, sds,
0
p
max pk ,
p
k∈1−n,N
min pk .
k∈1−n,N
1.3
2. Variational Structure and Preliminaries
We need two lemmas from 12 or 11.
Lemma 2.1. Let H be a real reflexive Banach space with a norm · , and let φ be a weakly lower
(upper) semicontinuous functional, such that
lim φx ∞
x → ∞
or lim φx −∞ ,
x → ∞
2.1
then there exists x0 ∈ H such that
φx0 inf φx0 x∈H
or φx0 sup φx0 .
2.2
x∈H
Furthermore, if φ has bounded linear Gâteaux derivative, then φ x0 0.
Lemma 2.2 mountain-pass lemma. Let H be a real Banach space, and let φ : H →
R be continuously differential, satisfying the P-S condition. Assume that x0 , x1 ∈ H and Ω
∈ Ω. If max{φx0 , φx1 } < infx∈∂Ω φx, then c is an open neighborhood of x0 , but x1 /
infh∈Γ maxt∈0,1 φht is the critical value of φ, where
Γ {h | h : 0, 1 −→ H, h is continuous, h0 x0 , h1 x1 }.
This means that there exists x2 ∈ H, s.t. φ x2 0, φx2 c.
The following lemma will be used in the proof of Lemma 2.4.
2.3
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3
Lemma 2.3. If Am×m is a symmetric and positive-defined real matrix, Bm×n is a real matrix, BT is the
transposed matrix of B. Then BT AB is positive defined if and only if rank B n.
Proof. Since A is positive defined, then
BT AB is positive-defined ⇐⇒ ∀x /
0, xT BT ABx > 0
2.4
T
⇐⇒ ∀x / 0, Bx / 0 ⇐⇒ rank B n.
/ 0, Bx ABx > 0 ⇐⇒ ∀x Let H be a Hilbert space defined by
H y : 1 − n, N n −→ R | y1−n y2−n · · · y0 0, yN1 · · · yNn 0
2.5
with the norm
N
y y 2 ,
2.6
y ∈ H.
k
k1
2 1/q
, then one
Hence H is an N-dimensional Hilbert space. For any q > 1, let yq N
k1 yk can show that there exist constants q1 , q2 > 0, s.t. q1 y yq q2 y; that is, · q is an
equivalent norm of · see 9, page 68.
Lemma 2.4. There is
N
λx2 Δn xk 2 4n x2 ,
x ∈ H, where λ is a postive constant.
2.7
k1−n
Proof. Since x ∈ H, Δn−j xN1 Δn−j xj−n 0, j 1, 2, . . . , n. By using the inequality a − b2 2a2 b2 , a, b ∈ R, we have
N
Δn xk 2 k1−n
N
k1−n
2
N 2
Δn−1 xk1 − Δn−1 xk ≤ 2
Δn−1 xk1
k1−n
N1
Δ
n−1
xk
2
≤4×2
4×2
Δ
k2−n
N xk
2
Δ
n−2
xk1
2
k2−n
k3−n
Δ
xk
2
Δn−1 xk Δn−1 xk 2
2
k2−n
N
Δ
n−2
N Δ
k2−n
2.8
xk k2−n
n−2
N
4
k2−n
N N1
n−1
2
n−2
xk
2
2
42
N
2
Δn−2 xk .
k3−n
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Advances in Difference Equations
Repeating the above process, we obtain
N
Δn xk 2 4n
k1−n
N
xk 2 4n x2 .
2.9
k1
On the other hand, define bk Δn xk n
j
j j
j0 −1 Cn xkn−j , k ∈ 1 − n, N, where Cn is
{bk }N
1−n in a vector form, that is, b Bx, where
the combination number, then we can rewrite
b b1−n , b2−n , . . . , bN T , x x1 , x2 , . . . , xN T , and
⎛
⎞
1
⎜
⎜ c1 1
⎜
⎜
⎜.
..
⎜ ..
.
⎜
⎜
⎜c c
⎜ n n−1
⎜
⎜
cn
⎜
⎜
⎜
B⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎜
⎝
..
.
··· 1
· · · c1
1
.
. ..
..
.
..
..
.
cn cn−1 · · · 1
cn
· · · c1
1
.
. ..
..
.
..
cn cn−1
cn
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎟
⎠
,
2.10
Nn×N
where ci −1i Cni . Hence rank B N. Note that
N
Δn xk 2 bT b BxT Bx xT BT Bx,
2.11
k1−n
and by Lemma 2.3 with Am×m INn×Nn , we know that BT B is positive defined. Hence
all eigenvalues of BT B are real and positive. Let λ be the minimal eigenvalue of these N
2
n
2
eigenvalues, then λ > 0. Therefore xT BT Bx λxT x, that is, N
1−n Δ xk λx .
However, how to find the λ in Lemma 2.4 is a skillful and challenging task. The
following lemma from 13 offers some help for the estimation of λ.
Lemma 2.5 Brualdi 13. If A aij ∈ MN is weak irreducible, then each eigenvalue is contained
in the set
⎧
⎨
⎫
" ⎬
z∈C:
Ri .
|z − aii | ⎩
⎭
P ∈γ
P ∈γ
γ∈CA
"
i
i
2.12
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In Lemma 2.5, C is the complex number set, and the denotations CA, γ, Pi , Ri can
be found in 13. Since BT B is positive defined, all eigenvalues are positive real numbers.
Therefore, by Lemma 2.5, let
⎧
⎨
B
γ∈CBT B
⎫
" ⎬
z∈R:
Ri ,
|z − bii | ⎩
⎭
P ∈γ
P ∈γ
"
i
2.13
i
where BT B bij . B is a subset of R and can be calculated directly from BT B. Define λ max{0, min{B}}. If λ > 0, we can use this λ as λ in Lemma 2.4. If λ 0, then one needs to
calculate the eigenvalues directly.
Define the functional φ on H by
N 1
pk Δn yk 2 − F k, yk .
φ y 2
k1−n
2.14
Then φ is C1 with
N
& n n
'
φ y ,x pk Δ yk Δ xk − xk f k, yk ,
2.15
k1−n
where x {xk }Nn
k1−n ∈ H and ·, · is the inner product in H. In fact, we have
N
) & 2
'
1 ( n
pk Δ yk Δn xk − Δn yk 2 − F k, yk xk − F k, yk
φ yx −φ y 2
k1−n
N 1
pk Δn yk Δn xk pk Δn xk 2 − f k, yk θxk xk ,
2
k1−n
θ ∈ 0, 1.
2.16
The continuity of f and the right-hand side of the inequality in Lemma 2.4 lead to 2.15.
Furthermore, for any x ∈ H, we have Δn−j xN1 Δn−j x1−n 0, j 1, 2, . . . , n. By using
the following formula e.g., see 14, page 28:
n2
&
kn1
' n 1
gk Δfk fk1 Δgk fk gk n21 ,
2.17
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Advances in Difference Equations
we have
N
N
N1 pk Δn yk Δn xk pk−1 Δn yk−1 Δn−1 xk − Δ pk−1 Δn yk−1 Δn−1 xk
1−n
1−n
−
1−n
N
Δ pk−1 Δn yk−1 Δn−1 xk
1−n
N
N1 −Δ pk−2 Δn yk−2 Δn−2 xk Δ2 pk−2 Δn yk−2 Δn−2 xk
1−n
2.18
1−n
N
Δ2 pk−2 Δn yk−2 Δn−2 xk .
1−n
Repeating the above process, we obtain
N
N
pk Δn yk Δn xk −1n Δn pk−n Δn yk−n xk .
1−n
2.19
1−n
Let φ y, x 0, that is,
N
&
'
−1n Δn pk−n Δn yk−n − f k, yk xk
1−n
−1n
N (
)
Δn pk−n Δn yk−n −1n1 f k, yk xk
2.20
1−n
0.
Since x ∈ H is arbitrary, we know that the solution of BVP 1.1-1.2 corresponds to the
critical point of φ.
3. Main Results
Now we present our main results of this paper.
Theorem 3.1. If there exist M1 > 0, a1 > 0, a2 ∈ R, and σ > 2 s.t.
Fk, u a1 |u|σ a2 ,
∀ |u| > M1 ,
3.1
then BVP 1.1-1.2 has at least one solution.
Proof. In fact, we can choose a suitable a2 < 0 such that
Fk, u a1 |u|σ a2 ,
∀u ∈ R.
3.2
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Since there exists σ1 > 0 with σ1 y yσ , we have
N
N σ
σ
F k, yk a1 yk a2 N n a1 σ1σ y a2 N n.
1−n
3.3
1
Then by Lemma 2.4, we obtain
σ
p 2
φ y 4n y − a1 σ1σ y − a2 N n.
2
3.4
Noticing that σ > 2, we have limy → ∞ φy −∞. From Lemma 2.1, the conclusion of this
lemma follows.
Corollary 3.2. If there exists M2 > 0 s.t. ufk, u > 0 for all |u| > M2 , and
lim
inf
k∈1−n,N u → ∞
fk, u
|u|r
3.5
r1 ,
where r, r1 satisfy either r 1, r1 > 4n p or r > 1, r1 > 0, then BVP 1.1-1.2 has at least one
solution.
Proof. Assume that r 1, r1 > 4n p. Then for 1 r1 − 4n p/2 > 0, there exists M3 > M2 ,
such that |fk, y| r1 − 1 |y| as |y| > M3 . We have from the continuity of fk, u that there
is a K > 0 such that −K ≤ fk, u ≤ K for all k ∈ 1, N, |u| ≤ M3 . When y > 0, one has
fk, y r1 − 1 y > 0 for y ∈ M3 , ∞, then
y
fk, sds M3
0
fk, sds y
fk, sds −KM3 M3
0
r1 − 1 2 r1 − 1 2
y −
M3 ;
2
2
3.6
when y < 0, one has fk, y r1 − 1 y < 0 for y ∈ −∞, −M3 , then
y
0
fk, sds −M3
0
fk, sds y
−M3
fk, sds −KM3 Let c : −KM3 − r1 − 1 /2M32 , then we have
Therefore, we have
*y
0
r1 − 1 2 r1 − 1 2
y −
M3 .
2
2
3.7
fk, sds r1 − 1 /2y2 c for y ∈ R.
N n
p 2 r1 − 1 yk 2 − c 4 p − r1 1 y2 − c − 1 y2 − c,
φ y 4n y −
2
2
2
2
1
3.8
which implies limy → ∞ φy −∞, and by Lemma 2.1, the conclusion of this lemma follows.
Assume that r > 1, r1 > 0. Then for 2 r1 /2 > 0, there exists M4 > M2 , such
that |fk, y| r1 /2|y| as |y| > M4 . We have from the continuity of fk, u that there is
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Advances in Difference Equations
a K > 0 such that −K ≤ fk, u ≤ K for all k ∈ 1, N, |u| ≤ M4 . When y > 0, one has
fk, y r1 /2yr > 0, y ∈ M4 , ∞, then we have
y
fk, sds M4
0
fk, sds y
fk, sds
M4
0
r1
r1
yr1 −
Mr1 ,
−KM4 2r 1
2r 1 4
3.9
when y < 0, one has fk, y −r1 /2|y|r −r1 /2−yr < 0, y ∈ −∞, −M4 , then we have
y
0
fk, sds −M4
fk, sds 0
y
y
−M4
fk, sds
−KM4 r1
2
−KM4 r1 r1
r1
y
M4 r1 .
−
2r 1
2r 1
−M4
3.10
−sr d−s
*y
Let d : −KM4 − r1 /2r 1M4r1 , then we have 0 fk, sds r 1 /2r 1|y|r1 d for
|y| > M4 . Therefore, by Theorem 3.1, the conclusion of this lemma follows.
Theorem 3.3. Assume that pk > 0, k 1 − n, . . . , N, and
i supk∈1−n,N limu → 0 fk, u/u r2 < pλ, λ > 0 is defined in Lemma 2.4;
ii F satisfies 3.1 in Theorem 3.1 or f satisfies the assumptions in Corollary 3.2.
Then BVP 1.1-1.2 has at least two solutions.
m
Proof. We first show that φ satisfies the P-S condition. Let {ym }∞
}
m1 ⊂ H satisfy that {φy
m
m
is bounded and limm → ∞ φ y 0. If {y } is unbounded, it possesses a divergent
subseries, say ymk → ∞ as k → ∞. However from ii, we get 3.4 or 3.8, hence
φymk → −∞ as k → ∞, which is contradictory to the the fact that {φym } is bounded.
Next we use the mountain-pass lemma to finish the proof. By i, for 3 pλ − r2 /2 >
*y
0, there exists R1 > 0 such that fk, y/y r2 3 for |y| R1 . Then 0 fk, sds r2 3 /2y2 for |y| R1 . Now together with Lemma 2.3, we have
N
2
p 2 r2 3 |yk |2 y
φ y λy −
2
2 1−n
pλ − r2 − 3
2
3 y2 > 0 for y R1 ,
2
3.11
which implies that
3
φ y R21 > 0 φθ,
2
y ∈ ∂Ω,
3.12
where θ is the zero element in H, and Ω {y ∈ H | y < R1 }. Since we have from 3.4
∈ Ω, but
or 3.8 that limy → ∞ φy −∞, there exists y1 ∈ H with y1 > R1 , that is, y1 /
Advances in Difference Equations
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φy1 < φθ 0. Using Lemma 2.2, we have shown that ξ infh∈Γ maxt∈0,1 φht is the
critical value of φ, with Γ defined as
Γ h | h : 0, 1 −→ H, h is continuous, h0 θ, h1 y1 .
3.13
We denote y as its corresponding critical point.
On the other hand, by Theorem 3.1 or Corollary 3.2, we know that there exists y∗ ∈ H,
y, the theorem is proved. If on the contrary, y∗ y, that
s.t. φy∗ supy∈H φy. If y∗ /
is, supy∈H φy infh∈Γ maxt∈0,1 φht, that implies for any h ∈ Γ, maxt∈0,1 φht h2 in Γ with maxt∈0,1 φh1 t maxt∈0,1 φh2 t supy∈H φy,
supy∈H φy. Taking h1 /
by the continuity of φht, there exist t1 , t2 ∈ 0, 1 s.t. φh1 t1 maxt∈0,1 φh1 t,
φh2 t2 maxt∈0,1 φh2 t. Hence h1 t1 , h2 t2 are two different critical points of φ, that
is, BVP 1.1-1.2 has at least two different solutions.
4. An Example
Consider the 6th-order boundary value problem for difference equation
Δ6 yk−3 yk3 eyk −9 0,
2
y−2 y−1 y0 0,
k ∈ 1, 300,
y301 y302 y303 0.
4.1
Let fk, u u3 eu −9 , we have limu → 0 fk, u/u 0, limu → ∞ fk, u/u ∞. Hence
fk, u satisfies the conditions in Theorem 3.3, the boundary value problem 4.1 has at least
two solutions.
2
Acknowledgments
This research is partially supported by the NSF of China and NSF of Guangdong Province.
References
1 R. P. Agarwal and J. Henderson, “Positive solutions and nonlinear eigenvalue problems for thirdorder difference equations,” Computers & Mathematics with Applications, vol. 36, no. 10–12, pp. 347–355,
1998.
2 R. P. Agarwal and D. O’Regan, “Singular discrete n, p boundary value problems,” Applied
Mathematics Letters, vol. 12, no. 8, pp. 113–119, 1999.
3 R. P. Agarwal and F.-H. Wong, “Upper and lower solutions method for higher-order discrete
boundary value problems,” Mathematical Inequalities & Applications, vol. 1, no. 4, pp. 551–557, 1998.
4 R. P. Agarwal, K. Perera, and D. O’Regan, “Multiple positive solutions of singular and nonsingular
discrete problems via variational methods,” Nonlinear Analysis: Theory, Methods & Applications, vol.
58, no. 1-2, pp. 69–73, 2004.
5 Z. M. Guo and J. S. Yu, “Existence of periodic and subharmonic solutions for two-order superlinear
difference equations,” Science in China Series A, vol. 33, pp. 226–235, 2003.
6 D. Jiang, J. Chu, D. O’Regan, and R. P. Agarwal, “Positive solutions for continuous and discrete
boundary value problems to the one-dimension p-Laplacian,” Mathematical Inequalities & Applications,
vol. 7, no. 4, pp. 523–534, 2004.
7 L. T. Li and P. X. Weng, “Boundary value problems of second order functional difference equation,”
Journal of South China Normal University Natural Science Edition, no. 3, pp. 20–24, 2003.
10
Advances in Difference Equations
8 H. H. Liang and P. X. Weng, “Existence and multiple solutions for a second-order difference boundary
value problem via critical point theory,” Journal of Mathematical Analysis and Applications, vol. 326, no.
1, pp. 511–520, 2007.
9 H. H. Liang and P. X. Weng, “Existence of solutions for a fourth-order difference boundary value
problem and a critical point method,” Applied Mathematics A Journal of Chinese Universities, Series A,
vol. 23, no. 1, pp. 67–72, 2008.
10 P. J. Y. Wong and R. P. Agarwal, “Existence theorems for a system of difference equations with n, ptype conditions,” Applied Mathematics and Computation, vol. 123, no. 3, pp. 389–407, 2001.
11 P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations,
vol. 65, American Mathematical Society, Providence, RI, USA, 1986.
12 A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and
applications,” Journal of Functional Analysis, vol. 14, pp. 349–381, 1973.
13 R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.
14 M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Application,
Birkhäuser, Boston, Mass, USA, 2001.