Paper-3-Acta24.2010. 137KB Jun 04 2011 12:03:14 AM

Acta Universitatis Apulensis
ISSN: 1582-5329

No. 24/2010
pp. 23-33

DIRICHLET BOUNDARY VALUE PROBLEMS OF NONLINEAR
FUNCTIONAL DIFFERENCE EQUATIONS WITH JACOBI
OPERATORS

WEI-PING LING, LIAN-WU YANG, HAI-PING SHI AND ZHI-GANG WANG
Abstract. In this paper, the solutions to Dirichlet boundary value problems
of nonlinear functional difference equations with Jacobi operators are investigated.
By using critical point theory, the existence and multiplicity results are obtained.
2000 Mathematics Subject Classification: 39A10, 65Q05.
1. INTRODUCTION
Let N, Z and R denote the sets of all natural numbers, integers and real numbers
respectively. For a, b ∈ Z, define Z(a) = {a, a + 1, · · · }, Z(a, b) = {a, a + 1, · · · , b}
when a ≤ b. ∆ is the forward difference operator defined by ∆un = un+1 − un . k is
a positive integer and * is the transpose sign for a vector.
Consider the second order functional difference equation

Lun = f (n, un+1 , un , un−1 ),

(1)

with boundary value conditions
u0 = A, uk+1 = B,

(2)

where the operator L is the Jacobi operator
Lun = an un+1 + an−1 un−1 + bn un ,
an and bn are real valued for each n ∈ Z, f ∈ C(R4 , R), A and B are constants.
We may think of Eq. (1) as being a discrete analogue of the second order
functional differential equation
Su(t) + f (t, u(t + 1), u(t), u(t − 1)) = 0, t ∈ R
23

(3)

W.P. Ling, L.W. Yang, H.P. Shi, Z.G. Wang - Dirichlet boundary value....

which includes the following equation
c2 u′′ (t) = V ′ (u(t + 1) − u(t)) − V ′ (u(t) − u(t − 1)), t ∈ R

(4)

where S is the Sturm-Liouville differential expression and f ∈ C(R4 , R). Eq. (4)
has been studied extensively by many scholars. For example, Smets and Willem
have obtained the existence of solitary waves of lattice differential equations, see
[21] and the references cited therein.
Jacobi operators appear in a variety of applications [22]. They can be viewed as
the discrete analogue of Sturm-Liouville operators and their investigation has many
similarities with Sturm-Liouville theory. Whereas numerous books about SturmLiouville operators have been written, only few on Jacobi operators exist. In particular, there are currently fewer researches available which cover some basic topics
(like positive solutions, periodic operators, boundary value problems, etc.) typically
found in textbooks on Sturm-Liouville operators [12].
It is well known that difference equations occur widely in numerous setting and
forms, both in mathematics itself and in its applications to statistics, computing,
electrical circuit analysis, dynamical systems, economics, biology and other fields, see
for examples [1,6,9,10,14,19]. Since the last decade, there has been much progress on
the qualitative properties of difference equations, which included results on stability

and attractivity and results on oscillation and other topics, see [1-4,6-11,14-16,1820,23-25]. However, to our best knowledge, no similar results are obtained in the
literature for the boundary value problem (BVP) (1) with (2). Since f in Eq. (1)
depends on un+1 and un−1 , the traditional ways of establishing the functional in
[2,23-25] are inapplicable to our case.
Our aim in this paper is to use the critical point theory to give some sufficient
conditions for the existence and multiplicity of the BVP (1) with (2). The main idea
in this paper is to transfer the existence of the BVP (1) with (2) into the existence
of the critical points [13] of some functional.
Our main results are as follows.
Let
pmax = max{an : n ∈ Z(0, k)}, pmin = min{an : n ∈ Z(0, k)},
qmax = max{bn + an−1 + an : n ∈ Z(1, k)}, qmin = min{bn + an−1 + an : n ∈ Z(1, k)}.
Theorem 1. Assume that the following hypotheses are satisfied:
(F1 ) there exists a constant M0 > 0 and a functional F (n, ·) ∈ C 1 (Z × R2 , R) with
F (0, ·) = 0 such that for any n ∈ Z(1, k),
∂F (n − 1, v2 , v3 ) ∂F (n, v1 , v2 )
+
= f (n, v1 , v2 , v3 ),
∂v2
∂v2

24

W.P. Ling, L.W. Yang, H.P. Shi, Z.G. Wang - Dirichlet boundary value....


 ∂F (n, v1 , v2 ) 

 ≤ M0 ,


∂v1



 ∂F (n, v1 , v2 ) 
 ≤ M0 ;



∂v2

(5)

(F2 ) for any n ∈ Z(0, k), an > 0; for any n ∈ Z(1, k), bn + an−1 + an > 0;

(F3 ) 4pmax < qmin .

Then the BVP (1) with (2) possesses at least one solution.
Remark 1. (5) implies that there exists a constant M1 > 0 such that
|F (n, v1 , v2 )| ≤ M1 + M0 (|v1 | + |v2 |), ∀n ∈ Z(1, k).

(6)

Corollary 1. Suppose that (F1 ) and (F3 ) are satisfied. And if
(F4 ) for any n ∈ Z(0, k), an < 0; for any n ∈ Z(1, k), bn + an−1 + an < 0.
Then the BVP (1) with (2) possesses at least one solution.
Corollary 2. Assume that (F1 )is satisfied. And if
(F5 ) for any n ∈ Z(0, k), an < 0; for any n ∈ Z(1, k), bn + an−1 + an > 0.
Then the BVP (1) with (2) possesses at least one solution.

Theorem 2. Suppose that the following hypotheses are satisfied:
(F6 ) there exists a functional F (n, ·) ∈ C 1 (Z × R2 , R) with F (0, ·) = 0 such that
q
F (n, v1 , v2 )
v12 + v22 , ∀n ∈ Z(1, k);
lim
=
0,
r
=
r→0
r2
(F7 ) there exists a constant β > 2 such that for any n ∈ Z(1, k),

∂F (n − 1, v2 , v3 ) ∂F (n, v1 , v2 )
+
= f (n, v1 , v2 , v3 ),
∂v2
∂v2
∂F (n, v1 , v2 )

∂F (n, v1 , v2 )
v1 +
v2 ≤ βF (n, v1 , v2 ) < 0, ∀(v1 , v2 ) 6= 0;
∂v1
∂v2
(F8 ) for any n ∈ Z(0, k), an > 0; for any n ∈ Z(1, k), bn + an−1 + an ≤ 0;

(7)

(F9 ) A = B = 0.

Then the BVP (1) with (2) possesses at least two nontrivial solutions.
Remark 2. (7) implies that there exist constants a1 > 0 and a2 > 0 such that
β
q
2
2
v1 + v2
+ a2 , ∀n ∈ Z(1, k).
(8)

F (n, v1 , v2 ) ≤ −a1
The rest of the paper is organized as follows. In Sect. 2 we shall establish the
variational framework for the BVP (1) with (2) in order to apply the critical point
method and give some useful lemmas. In Sect. 3 we shall complete the proof of the
main results and give an example to illustrate the result.
25

W.P. Ling, L.W. Yang, H.P. Shi, Z.G. Wang - Dirichlet boundary value....

2. VARIATIONAL STRUCTURE AND SOME LEMMAS
In order to apply the critical point theory, we shall establish the corresponding
variational framework for the BVP (1) with (2) and give some basic notations and
useful lemmas.
Let Rk be the real Euclidean space with dimension k. Define the inner product
on Rk as follows:
k
X
hu, vi =
uj vj , ∀u, v ∈ Rk ,
(9)

j=1

by which the norm k · k can be induced by


kuk = 

k
X
j=1

1

2

u2j 

, ∀u ∈ Rk .

(10)

On the other hand, we define the norm k · kr on Rk as follows:

1
r
k
X
r

kukr =
|uj |
,

(11)

j=1

for all u ∈ Rk and r > 1.
Since kukr and kuk2 are equivalent, there exist constants c1 , c2 such that c2 ≥
c1 > 0, and

c1 kuk2 ≤ kukr ≤ c2 kuk2 , ∀u ∈ Rk .
(12)

Clearly, kuk = kuk2 . For the BVP (1) with (2), consider the functional J on Rk
as follows:
J(u) =

k

k

k

n=0

n=1

n=1

X
1X
1X
F (n, un+1 , un ),
an (∆un )2 −
(bn + an−1 + an )u2n +
2
2

(13)

∀u = (u1 , u2 , · · · , uk )∗ ∈ Rk , u0 = A, uk+1 = B.
∗
Clearly, J ∈ C 1 (Rk , R) and for any u = {un }k+1
n=0 = (u0 , u1 , . . . , uk+1 ) , by using
u0 = A, uk+1 = B, we can compute the partial derivative as
∂J
= −an ∆un + an−1 ∆un−1 − (bn + an−1 + an )un + f (n, un+1 , un , un−1 )
∂un
= −Lun + f (n, un+1 , un , un−1 ), n ∈ Z(1, k).
26

W.P. Ling, L.W. Yang, H.P. Shi, Z.G. Wang - Dirichlet boundary value....
Thus, u is a critical point of J on Rk if and only if
Lun = f (n, un+1 , un , un−1 ), ∀n ∈ Z(1, k).
We reduce the existence of the BVP (1) with (2) to the existence of critical points
of J on Rk . That is, the functional J is just the variational framework of the BVP
(1) with (2).
Let E be a real Banach space, J ∈ C 1 (E, R), i.e., J is a continuously Fr´echetdifferentiable functional defined on E. J is said to be satisfying
the Palais-Smale




condition (P.S. condition for short) if any sequence u(k) ⊂ E for which J u(k)


is bounded and J ′ u(k) → 0(k → ∞) possesses a convergent subsequence in E.
Let Bρ denote the open ball in E about 0 of radius ρ and let ∂Bρ denote its
boundary.
Lemma 1 (Mountain Pass Lemma [17]). Let E be a real Banach space and J ∈
C 1 (E, R) satisfies the P.S. condition. If J(0) = 0 and
(J1 ) there exist constants ρ, a > 0 such that J|∂Bρ ≥ a, and
(J2 ) there exists e ∈ E \ Bρ such that J(e) ≤ 0.
Then J possesses a critical value c ≥ a given by
c = inf max J(g(s)),

(14)

Γ = {g ∈ C([0, 1], E)|g(0) = 0, g(1) = e}.

(15)

g∈Γ s∈[0,1]

where
Lemma 2. Suppose that (F6 ) − (F9 ) is satisfied. Then the functional J satisfies
the P.S. condition.



Proof. Let u(l) ∈ Rk , l ∈ Z(1) be such that J u(l)
is bounded. Then there
exists a positive constant M2 such that
 
−M2 ≤ J u(l) ≤ M2 , ∀l ∈ N.
By (1.8), we have


−M2 ≤ J u

(l)

k 
X



k
k
k


 2 X
1 X  (l) 2 1 X
(l)
(l)
(l)
=
F n, un+1 , un
an ∆un −
(bn +an−1 +an ) un +
2
2
n=0

n=1

n=1

"r
k

X


2  q
(l)
min
(l)
2
(l)
≤ pmax
−
un+1 + un
u
−a1
2
n=1
n=0

2

β

2 q


min
(l)
β
≤ 2pmax
u(l)
−
u
− a1 c1
u(l)
+ a2 k.
2
2



27

(l)

un+1

2



(l)

+ un

2

#β

+a2 k

W.P. Ling, L.W. Yang, H.P. Shi, Z.G. Wang - Dirichlet boundary value....

That is,


β 

qmin 



(l)
2
a1 cβ1
u(l)
− 2pmax −
u
≤ M2 + a2 k.
2
Since β > 2, there exists a constant M3 > 0 such that


(l)

u
≤ M3 , ∀l ∈ N.



Therefore, u(l) is bounded on Rk . As a consequence, u(l) possesses a convergence subsequence in Rk . And thus the P.S. condition is verified.
3. PROOF OF THE MAIN RESULTS
In this section, we shall complete the proof of Theorems 1 and 2.
3.1. Proof of Theorem 1
Proof. By (6), for any u = (u1 , u2 , · · · , uk )∗ ∈ Rk , we have
k

k

k

n=0

n=1

n=1

X
1X
1X
an (∆un )2 −
(bn + an−1 + an )u2n +
J(u) =
F (n, un+1 , un )
2
2
≤ pmax

k
X

k

u2n+1

+

n=0

≤ 2pmax

k
X

u2n




X
qmin
kuk2 + M0
(|un+1 | + |un |) + M1 k
−
2

u2n + pmax A2 + B

n=1

n=1



2

qmin
kuk2 + M0
−
2

2

k
X

n=1

!

|un | + |B|

+ M1 k

√


qmin 
kuk2 + 2M0 kkuk + M0 |B| + M1 k + pmax A2 + B 2
≤ 2pmax −
2
→ −∞, (kuk → +∞).
By continuity of J on Rk and above argument, there exists u0 ∈ Rk such that
n
o
J(u0 ) = max J(u)|u ∈ Rk .


Clearly, u0 is a critical point of the functional J. The proof of Theorem 1 is
complete.
3.2. Proof of Theorem 2
Proof. By (F6 ), for any ǫ = 18 pmin λ1 (λ1 can be referred to (16)), there exists ρ > 0,
such that


1
|F (n, v1 , v2 )| ≤ pmin λ1 v12 + v22 , ∀n ∈ Z(1, k),
8
28

W.P. Ling, L.W. Yang, H.P. Shi, Z.G. Wang - Dirichlet boundary value....

for

√
v12 + v22 ≤ 2ρ.
For any u = (u1 , u2 , · · · , uk )∗ ∈ Rk and kuk ≤ ρ, we have |un | ≤ ρ, n ∈ Z(1, k).
For any n ∈ Z(1, k),

p

J(u) =

k

k

k

n=0

n=1

n=1

X
1X
1X
an (∆un )2 −
(bn + an−1 + an )u2n +
F (n, un+1 , un )
2
2
k

k

n=0

n=1

X
X


1
1
(∆un )2 − pmin λ1
u2n+1 + u2n
≥ pmin
2
8
1
1
≥ pmin (u∗ Du) − pmin λ1 kuk2 ,
2
4

where u∗ = (u1 , u2 , · · · , uk ), u ∈ Rk ,

2 −1 0 · · ·
 −1 2 −1 · · ·

 0 −1 2 · · ·
D=
 ··· ··· ··· ···

 0
0
0 ···
0
0
0 ···

0
0
0
0
0
0
··· ···
2 −1
−1 2










k×k.

Clearly, D is positive definite. Let λ1 , λ2 , · · · , λk be the other eigenvalues of D.
Applying matrix theory, we know λj > 0, j = 1, 2, · · · , k. Without loss of generality,
we may assume that
0 < λ1 ≤ λ2 ≤ · · · ≤ λk ,
(16)
then

1
1
J(u) ≥ pmin λ1 kuk2 − pmin λ1 kuk2
2
4
1
= pmin λ1 kuk2 .
4

Take a 41 pmin λ1 kρk2 > 0. Therefore,
J(u) ≥ a > 0, ∀u ∈ ∂Bρ .
At the same time, we have also proved that there exist constants a > 0 and ρ > 0
such that J|∂Bρ ≥ a. That is to say, J satisfies the condition (J1 ) of the Mountain
Pass Lemma.
For our setting, clearly J(0) = 0. In order to exploit the Mountain Pass Lemma
in critical point theory, we need to verify other conditions of the Mountain Pass

29

W.P. Ling, L.W. Yang, H.P. Shi, Z.G. Wang - Dirichlet boundary value....

Lemma. By Lemma 2, J satisfies the P.S. condition. So it suffices to verify the
condition (J2 ).
From the proof of the P.S. condition, we know
J(u) ≤ 2pmax kuk2 −

qmin
kuk2 − a1 cβ1 kukβ + a2 k.
2

Since β > 2, we can choose u
¯ large enough to ensure that J(¯
u) < 0.
By the Mountain Pass Lemma, J possesses a critical value c ≥ a > 0, where
c = inf sup J(h(s)),
h∈Γ s∈[0,1]

and
Γ = {h ∈ C([0, 1], Rk ) | h(0) = 0, h(1) = u
¯}.

Let u
˜ ∈ Rk be a critical point associated to the critical value c of J, i.e., J(˜
u) = c.
Similar to the proof of the P.S. condition, we know that there exists u
ˆ ∈ Rk such
that
J(ˆ
u) = cmax = max J(h(s)).
s∈[0,1]

Clearly, u
ˆ 6= 0. If u
˜ 6= u
ˆ, then the conclusion of Theorem 2 holds. Otherwise,
u
˜=u
ˆ. Then c = J(˜
u) = cmax = max J(h(s)). That is,
s∈[0,1]

sup J(u) = inf sup J(h(s)).
u∈Rk

h∈Γ s∈[0,1]

Therefore,
cmax = max J(h(s)), ∀h ∈ Γ.
s∈[0,1]

By the continuity of J(h(s)) with respect to s, J(0) = 0 and J(¯
u) < 0 imply
that there exists s0 ∈ (0, 1) such that
J (h (s0 )) = cmax .
Choose h1 , h2 ∈ Γ such that {h1 (s) | s ∈ (0, 1)} ∩ {h1 (s) | s ∈ (0, 1)} is empty,
then there exists s1 , s2 ∈ (0, 1) such that
J (h1 (s1 )) = J (h2 (s2 )) = cmax .
Thus, we get two different critical points of J on Rk denoted by
u1 = h1 (s1 ) , u2 = h2 (s2 ) .

30

W.P. Ling, L.W. Yang, H.P. Shi, Z.G. Wang - Dirichlet boundary value....

The above argument implies that the BVP (1) with (2) possesses at least two nontrivial solutions. The proof of Theorem 2 is finished.
Remark 3. As an application of Theorem 2, finally, we give an example to illustrate
our result.
For n ∈ Z(1, k), assume that



β

β −1
2 2 −1
2
2
2
2
2un+1 + 2un−1 − 8un = −βun ϕ(n) un+1 + un
,
+ ϕ(n − 1) un + un−1

(17)

with boundary value conditions
u0 = 0, uk+1 = 0,

(18)

where β > 2, ϕ is continuously differentiable and ϕ(n) > 0, n ∈ Z(1, k) with
ϕ(0) = 0.
We have
an = an−1 ≡ 2, bn ≡ −8,



β

β
2
2 2 −1
2
2 2 −1
f (n, v1 , v2 , v3 ) = −βv2 ϕ(n) v1 + v2
+ ϕ(n − 1) v2 + v3

and

F (n, v1 , v2 ) = −ϕ(n) v12 + v22
Then


β
2

.




β

β
∂F (n − 1, v2 , v3 ) ∂F (n, v1 , v2 )
2
2 2 −1
2
2 2 −1
.
+ ϕ(n − 1) v2 + v3
+
= −βv2 ϕ(n) v1 + v2
∂v2
∂v2

It is easy to verify all the assumptions of Theorem 2 are satisfied and then the BVP
(17) with (18) possesses at least two nontrivial solutions.

References
[1] R.P. Agarwal, Difference Equations and Inequalities: Theory, Methods and
Applications, Marcel Dekker, New York, 1992.
[2] R.P. Agarwal, K. Perera and D. O’Regan, Multiple positive solutions of singular discrete p-Laplacian problems via variational methods, Adv. Difference Equ.
2005(2005), 93-99.
[3] D. R. Anderson, R.I. Avery and J. Henderson, Existence of solutions for a
one dimensional p-Laplacian on time-scales, J. Difference Equ. Appl. 10(2006),
889-896.
31

W.P. Ling, L.W. Yang, H.P. Shi, Z.G. Wang - Dirichlet boundary value....

[4] S. Chen, Disconjugacy, disfocality, and oscillation of second order difference
equations, J. Differential Equations 107(1994), 383-394.
[5] K. Deimling, Nonlinear Functional Analysis, Springer, New York, 1985.
[6] S. Elaydi, An Introduction to Difference Equations, Springer, New York, 1996.
[7] P. Eloe, A boundary value problem for a system of difference equations, Nonlinear Anal. 7(1983), 813-820.
[8] J. Henderson and H. B. Thompson, Existence of multiple solutions for secondorder discrete boundary value problems, Comput. Math. Appl. 43(2002), 1239-1248.
[9] W.G. Kelly and A.C. Peterson, Difference Equations: An Introduction with
Applications, Academic Press, New York, 1991.
[10] V.L. Kocic and G. Ladas, Global Behavior of Nonlinear Difference Equations
of Higher Order with Application, Kluwer Academic Publishers, Boston, 1993.
[11] H. Matsunaga, T. Hara and S. Sakata, Global attractivity for a nonlinear
difference equation with variable delay, Computers Math. Appl. 41(2001), 543-551.
[12] V. A. Marchenko, Sturm-Liouville Operators and Applications, Birkh¨auser,
Basel, 1986.
[13] J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems,
Springer, New York, 1989.
[14] R.E. Mickens, Difference Equations: Theory and Application, Van Nostrand
Reinhold, New York, 1990.
[15] A. Pankov and N. Zakharchenko, On some discrete variational problems,
Acta Appl. Math. 65(2001), 295-303.
[16] A. Peterson, Boundary value problems for an nth order linear difference
equations, SIAM J. Math. Anal. 15(1984), 124-132.
[17] P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, Amer. Math. Soc., Providence, RI, New York,
1986.
[18] Y. Rodrigues, On nonlinear discrete boundary value problems, J. Math.
Anal. Appl. 114(1986), 398-408.
[19] A.N. Sharkovsky, Y.L. Maistrenko and E.Y. Romanenko, Difference Equations and Their Applications, Kluwer Academic Publishers, Dordrecht, 1993.
[20] H.-P. Shi, W.-P. Ling, Y.-H. Long and H.-Q. Zhang, Periodic and subharmonic solutions for second order nonlinear functional difference equations, Commun.
Math. Anal. 5(2) (2008), 50-59.
[21] D. Smets and M. Willem, Solitary waves with prescribed speed on infinite
lattices, J. Funct. Anal. 149(1997), 266-275.
[22] G. Teschl, Jacobi Operators and Completely Integrable Nonlinear Lattices,
Amer. Math. Soc., Providence, RI, New York, 2000.

32

W.P. Ling, L.W. Yang, H.P. Shi, Z.G. Wang - Dirichlet boundary value....

[23] J.-S. Yu and Z.-M. Guo, Boundary value problems of discrete generalized
Emden-Fowler equation, Sci. China 49(A)(2006), 1303-1314.
[24] J.-S. Yu and Z.-M. Guo, On boundary value problems for a discrete generalized Emden-Fowler equation, J. Differential Equations, 231(2006), 18-31.
[25] J.-S. Yu, The minimal period problem for the classical forced pendulum equation, J. Differential Equations, 247(2009), 672-684.
Wei-Ping Ling
Department of Basic Courses,
Guangdong Baiyun Institute, Guangzhou 510450, Guangdong,
P. R. China
email:245635593@qq.com
Lian-Wu Yang
School of Mathematical and Computer Science,
Yichun University, Yichun 336000, Jiangxi,
P. R. China
email:ylwmath@126.com
Hai-Ping Shi
Department of Basic Courses,
Guangdong Baiyun Institute, Guangzhou 510450, Guangdong,
P. R. China
email:shp7971@163.com
Zhi-Gang Wang
School of Mathematics and Computing Science,
Hengyang Normal University, Hengyang 421008, Hunan,
P. R. China
email:zhigangwang@foxmail.com

33