Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2007, Article ID 94325, 15 pages
doi:10.1155/2007/94325
Research Article
Relations between Limit-Point and Dirichlet Properties of
Second-Order Difference Operators
A. Delil
Received 24 July 2006; Revised 6 March 2007; Accepted 11 April 2007
Dedicated to Professor W. D. Evans on the occasion of his 65th birthday
Recommended by Martin J. Bohner
We consider second-order difference expressions, with complex coefficients, of the form
w
−1
n
[−Δ(p
n−1
Δx
n−1
)+q
n
x
n
] acting on infinite sequences. The discrete analog of some
known relationships in the theory of differential operators such as Dirichlet, conditional
Dirichlet, weak Dirichlet,andstrong limit-point is considered. Also, connections and some
relationships between these properties have been established.
Copyright © 2007 A. Delil. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and repro-
duction in any medium, provided the original work is properly cited.
1. Introduction

Δx
n−1

+ q
n
x
n

, n ≥ 0,
−
p
−1
w
−1
Δx
n
, n =−1,
(1.1)
with complex coefficients p
={p
n
}
∞
−
1
, q ={q
n
}
∞
−

w
aretheHilbertspaces

2
=

x =

x
n

∞
−
1
:
∞

n=−1


x
n


2
< ∞

,

2

(x, y)
=
∞

n=−1
x
n
y
n
,(x, y) =
∞

n=−1
x
n
y
n
w
n
, (2.2)
respectively . If
{x
n
}
∞
−
1
∈ 
1
but

w
n

−
Δ

p
n−1
Δx
n−1

+ q
n
x
n

, n ≥ 0,
−
p
−1
w
−1
Δx
n
, n =−1,
(2.3)
where Δx
n
= x
n+1

T(M)
into 
2
w
as

T(M)x

n
= T(M)x
n
:= Mx
n
, n =−1,0,1, , (2.6)
D
T(M)
:=

x =

x
n

∞
−
1
∈ 
2
w
:

k
−
m

n=k
y
n+1
Δx
n
, k ≤ m, k,m ∈ N, (2.8)
A. Delil 3
gives rise to the equalities
m

n=0
x
n
My
n
w
n
=
m

n=0
q
n
y
n
x

∞

n=0

p
n
Δy
n
Δx
n
+ q
n
y
n
x
n

=
∞

n=0

x
n
T(M)y
n

w
n
+lim

T(M)x
n

w
n
= lim
m→∞
p
m

Δx
m
y
m+1
−Δ y
m
x
m+1

−
p
−1

Δx
−1
y
0
−Δ y
−1
x


=
∞

n=0

x
n
T(M)x
n

w
n
+lim
m→∞
p
m
Δx
m
x
m+1
− p
−1
Δx
−1
x
0
.
(2.12)
An immediate consequence of (2.11) together with (2.7)isthat

∞
−
1
be linearly independent solutions of (2.5) and suppose
that [φ,ψ]
n
:= p
n
[(Δφ
n
)ψ
n+1
− (Δψ
n
)φ
n+1
] = 1foralln.Then,Φ ={Φ
n
}
∞
−
1
defined by
Φ
n
=
n

m=0


0
= 0. (2.15b)
Any solution of (2.15a)isoftheform
Ψ
= Φ + Aφ + Bψ (2.16)
for some constants A, B
∈ C.
4AdvancesinDifference Equations
Definit ion 2.1. If there is precisely one 
2
w
solution (up to constant multiples) of (2.5)
for
(λ) = 0, then the expression M is said to be in the limit-point (LP) case; otherwise
all solutions of (2.5)arein
2
w
for all λ ∈ C and M is said to be in the limit-circle (LC)
case, see Atkinson [11] a nd Hinton and Lewis [6]. Note that in the limit-circle (LC)case,
the defect numbers are equal and the limit-point case does not hold. An alternative but
equivalent characterization of M being LP is that
lim
m→∞
p
m

Δx
m
y
m+1

condition is equivalent to saying that
lim
m→∞
p
m

Δx
m
x
m+1
− Δx
m
x
m+1

=
0 (2.18)
or
lim
m→∞
p
m

x
m
x
m+1
− x
m+1
x

x
m+1
= 0 ∀x, y ∈ D
T(M)
. (2.19)
Definit ion 2.3. M is said to be
(i) Dirichlet (D)onD
T(M)
if



p
n


1/2
Δx
n

∞
−
1
,



q
n


∈ 
2
,
∞

n=0
q
n


x
n


2
is convergent ∀x ∈ D
T(M)
, (2.21)
(iii) weak Dirichlet ( WD)onD
T(M)
if
∞

n=0

p
n
Δx
n
Δy

T(M)
. (2.23)
Also, by Dirichlet formula (2.10), it is seen that the WD property, (2.22), is equivalent to
lim
m→∞
p
m
Δy
m
x
m+1
exists and is finite ∀x, y ∈ D
T(M)
, (2.24)
and this is equivalent to
lim
m→∞
p
m
Δx
m
x
m+1
exists and is finite ∀x ∈ D
T(M)
. (2.25)
Note also that in (iii), for all x, y
∈ D
T(M)
,

1
⇐⇒

p
n
Δx
n
Δy
n

∞
−
1
∈ 
1
. (2.26)
Following the above definitions and subsequent comments, we have the following .
Corollary 2.4. The following implications hold for all x, y
∈ D
T(M)
:
(a) D
⇒ CD ⇒ WD;
(b) SLP
⇒ WD;
(c) SLP
⇒ LP.
3. Statement of results
In this section, we would like to obtain some implications additional to Corollary 2.4 by
imposing conditions on p, q,andw which are as weak as possible. The motivation of the

T(M)
.Letx, y ∈ D
T(M)
then, by
(2.10),
α :
= lim
m→∞
p
m
Δy
m
x
m+1
< ∞. (3.1)
We need to prove that α
= 0 under the conditions in the hypothesis. Suppose the contrary
that α
= 0, then for some m
0
∈ N,


p
m
Δy
m
x
m+1


m+1




∀
m ≥ m
0
, ∀x, y ∈ D
T(M)
. (3.3)
6AdvancesinDifference Equations
However , M is CD and this implies that, summing over m, the left-hand side of (3.3)
belongs to 
1
.Thus,
∞

n=−1




Δx
n
x
n+1




Δx
n
x
n+1





∼




Δx
n
x
n+1




(3.5)
since
lim
t→0
log(1 − t)
t
=−1. (3.6)
Hence,

n=m
0
log
x
n+1
x
n
exists for m
0
∈ N.
(3.7)
This implies that
lim
N→∞
N

n=m
0
Δ

logx
n

=
lim
N→∞

logx
N+1
− logx

and, for some m
0
∈ N,


p
m

Δy
m

2


≥
1
4


αβ
−1


2


p
−1
m


m
x
m+1
Δx
m
= 0. (3.12)
A. Delil 7
Then, lim
m→∞
x
m
= β = 0 exists and it follows that
lim
m→∞
p
m
Δx
m
= αβ
−1
= 0 =⇒ lim
m→∞
Δx
m
= lim
m→∞
αβ
−1
p
−1

. (3.14)
Now, since x
∈ D
T(M)
, using Cauchy-Schwarz inequality in 
2
,wehave
∞

n=−1


x
n
w
1/2
n

−
Δ

p
n−1
Δx
n−1

+ q
n
x
n

−
Δ

p
n−1
Δx
n−1

+ q
n
x
n

w
−1/2
n


2

1/2
(3.15)
which gives
∞

n=−1


x
n


p
n−1
Δx
n−1

+ q
n
x
n



< ∞. (3.17)
Now,
∞

n=0

−
Δ

p
n−1
Δx
n−1

+ q
n
x

= lim
m→∞
p
m
Δx
m
− p
−1
Δx
−1
+
∞

n=0

−
Δ

p
n−1
Δx
n−1

+ q
n
x
n

, (3.19)
which proves the convergence of the sum

n

q
n
x
n

=
1
x
m+1
m

s=k−1
q
s
x
s
−
1
x
k
k
−1

s=k−1
q
s
x
s

−
q
k−1
x
k−1
x
k
+
m

n=k

n

s=k−1
q
s
x
s


Δx
n
x
n+1
x
n

.
(3.20)

n

x
n+1
− x
n

=

−
λw
n
+ q
n

x
n
+ p
n−1

x
n
− x
n−1

, (3.21)
which is equivalent to (2.5). So, taking
X
n
=

n
p
n−1
⎞
⎟
⎟
⎟
⎟
⎠
, (3.22)
we get
X
n
=

I + A
n

X
n−1
, n = 0,1, 2, , (3.23)
where I is the identity matrix and
x
n
= x
n−1
+
y
n−1
p

|x
n
|
2
w
n
< ∞ holds. Moreover, since

∞
n=−1
w
n
< ∞,itissufficient to prove that
all solutions of (3.21), with λ
= 0, are bounded. For this purpose, we make use of the
following theorem due to Atkinson [11, page 447].
Theorem 3.2 (Atkinson). Let the sequence of k-by-k matrices,
A
n
, n = 0,1,2,3, ; A
n
=

a
nrs

, r, s = 1,2,3, ,k, (3.25)
satisfy
∞


n
− X
n−1
= A
n−1
X
n−1
, n = 0,1,2, , (3.27)
where X
n
is a k-vector, converge as n →∞. If in addition the matrices I + A
n
are all nonsin-
gular, then lim
n→∞
X
n
= 0,unlessalltheX
n
are zero vectors.
A. Delil 9
So, applying this theorem to our case,
{X
n
}
∞
0
is convergent, that is, the entries of X
n
,

n
Δx
n

∞
0
, (3.28)
are convergent, so they are bounded and hence (i) of condition (c) is proved.
(ii) The D case. We will state the proof for λ
= 0only,buttheproofalsoappliestoall
λ
∈ C.Letx ∈ D
T(M)
and define f ={f
n
}
∞
−
1
by
f
n
= Mx
n
. (3.29)
Then

∞
n=−1
| f

n
Δϕ
n−1

=
1 ∀n ∈ N, (3.30)
then any solution of
Mx
n
= λx
n
+ f
n
(3.31)
is of the form
x
n
= Φ
n
+ Aϕ
n
+ Bψ
n
(3.32)
in which A and B are constants, and
Φ
n
=
n


, it follows that


Φ
n


≤
C
n

m=0
w
m


f
m


, (3.34)
where C is a positive constant. Hence, Φ is bounded. This implies that
{x
n
}
∞
−
1
is bounded
from the fact that

n


2
< ∞. (3.35)
We also need to prove that

∞
n=0
|p
n
||Δx
n
|
2
< ∞.For,from(3.32),
p
n
Δx
n
= p
n
ΔΦ
n
+ p
n
Δ

Aϕ
n

w
m
f
m
;
(3.36)
10 Advances in Difference Equations
and since
{p
n
Δϕ
n
}
∞
−
1
, {p
n
Δψ
n
}
∞
−
1
, {ϕ
n
}
∞
−
1

∞

n=0


p
n




Δx
n


2
=
∞

n=0



p
n




Δx

implies that the sum

∞
n=0
(p
n
|Δx
n
|
2
+ q
n
|x
n
|
2
) is convergent for all x ∈ D
T(M)
.(2)Under
the condit ions of Theorem 3.1(a)-(b), D
⇒ CD ⇒ SLP ⇒ LP on D
T(M)
.
Remarks 3.4. (1) When w, p
−1
,q ∈ 
1
, it is proved by Atkinson [11, page 134] that M is
LC. We have additionally proved that M is also D. (2) The condition imposed on q in
Theorem 3.1(a) is in general weaker than q

1
so that the sequence

R
n

∞
0
=

n

k=0
r
k

∞
0
+ C
1
(3.38)
be positive, that is, R
n
> 0forall,n = 0,1,2, Then{R
n
}
∞
0
is bounded, for p
n

≥ x
0
(3.39)
is also positive. Note that
{x
n
}
∞
−
1
is monotonic increasing, that is, x
n+1
≥ x
n
for all n,from
the fact that x
n
are the sum of positive numbers. Now,
X
= lim
n→∞
x
n
exists (3.40)
since
{R
n
}
∞
−

is given by
q
n
=
r
n
x
n
, n ≥ 0, q
−1
= 0, (3.41)
A. Delil 11
then
{x
n
}
∞
−
1
is a s olution of (2.5)withλ = 0. Note that, in


q
n


=


r

gent. Now, summation-by-parts formula gives, for all N
∈ N,
N

n=0
q
n
=
N

n=0
r
n
x
n
=
R
N
x
N
−
N−1

n=−1
R
n
x
n+1
+
N−1

}
∞
−
1
is positive and decreasing, both

N
n
=−1
(R
n
/x
n+1
)and

N
n
=−1
(R
n
/x
n
)areconver-
gent, and therefore

∞
n=0
q
n
is convergent. Now, let {y


y
n−1
x
n−1

=
1
p
n−1
x
n
x
n−1
=⇒ y
n
= x
n
n

k=0
1
p
k−1
x
k
x
k−1
. (3.44)
So, since

−1
∈ 
1
.So,y ∈ 
2
w
since w ∈ 
1
.WealsoseethatMy
n
= 0. Hence, we have shown that
M is LC, and hence not SLP since x, y
∈ 
2
w
and x, y are linearly independent solutions of
Mx
n
= λx
n
, λ ∈ C. We now show that M is CD. Since, from the identity (2.12), the CD
property is equivalent to
(a)
{p
n
|Δz
n
|
2
}

Mz
n

∞
−
1
=

f
n

∞
−
1
∈ 
2
w
, w ∈ 
1
. (3.46)
The method of variation of parameters gives
z
n
= Ax
n
+ By
n
+
n


m
− y
n
x
m
) f
m
w
m
< ∞,(3.40)and
(3.45)togetherimplythat
lim
n→∞
z
n
exists. (3.48)
12 Advances in Difference Equations
We see that
{p
1/2
n
Δx
n
}
∞
−
1
, {p
1/2
n

y
m

p
1/2
n
Δx
n

−
x
m

p
1/2
n
Δy
n

f
m
w
m
≤
C
p
1/2
n

n


∞
−
1
∈ 
2
. (3.50)
Finally,
(i) lim
n→∞
p
n
Δx
n
= lim
n→∞
R
n
< ∞,
(ii) lim
n→∞
p
n
Δy
n
= lim
n→∞
[1/x
n
+(p

k−1
x
k
x
k−1
)isabsolutelycon-
vergent,
(iii) For K<
∞,
lim
n→∞





p
n
Δx
n
n

m=0
y
m

w
m
f
m


1/2
< ∞, (3.51)
(iv) lim
n→∞
|p
n
Δy
n

n
m
=0
x
m
(w
m
f
m
)|≤C lim
n→∞
|p
n
Δy
n

n
m
=0
w

}
∞
−
1
may still be complex. If eithe r
{w
m

m
n
=−1
p
−1
n
}
∞
m=−1
/∈ 
1
or {q
n
}
∞
−
1
/∈ 
1
, then
M is D on D
T(M)

−
1
∈ 
2
for all x ∈ D
T(M)
, we only need to
prove the other implication. So, suppose that
{|q
n
|
1/2
x
n
}
∞
−
1
∈ 
2
for all x ∈ D
T(M)
.Inthe
formula
m

n=0
p
n


q
n


x
n


2
, (3.54)
thesumsontherightconvergeasm
→∞. Thus, we see that {p
1/2
n
|Δx
n
|}
∞
−
1
/∈ 
2
only if
lim
m→∞
p
m
Δx
m
x


x
m



≤
p
m
Δ



x
m


2

, (3.55)
A. Delil 13
and hence
lim
m→∞
p
m
Δ




/∈ 
1
or {q
n
}
∞
−
1
∈ 
1
.If
{q
n
}
∞
−
1
/∈ 
1
, then we get a contradiction to the assumption since this would imply that
{|q
n
|
1/2
x
n
}
∞
−
1

m


2
≥


x
m


2
−


x
m
0
−1


2
>
m

n=m
0
p
−1
n

0
p
−1
k
−1

, (3.58)
which is a contradiction to the assumption that
{w
m

m
n
=−1
p
−1
n
}
∞
m=−1
/∈ 
1
, and hence
{p
1/2
n
|Δx
n
|}
∞

/∈ 
1
.
(2) If w
∈ 
1
,then,foranym ∈ N ∪{−1},
m

n=−1
w
n

n

k=−1
p
−1
k

=
m

n=−1
p
−1
n

m


p
−1
n
(m − n) =∞. (3.62)
Theorem 3.8. Suppose that p
n
> 0 for all n, w/p ∈ 
1
,and

w
n
/w
n+1

∞
−
1
is bounded above.
Then, M is SLP on D
T(M)
if and only if M is WD on D
T(M)
.
14 Advances in Difference Equations
Proof. Since SLP always implies WD by Corollary 2.4, we only need to prove that WD
⇒
SLP under the conditions in the hypothesis. So, suppose that M satisfies the WDproperty,
that is, β
= lim

m+1
Δx
m
= x
2
m+1
− x
m
x
m+1
, (3.64)
we have
∞

m=0

βp
m
Δx
m
x
m+1

w
m
p
−1
m
= β


w
m
w
m+1

1/2

.
(3.65)
Under the conditions of the hypothesis, the left-hand side of this equality is
∞ while the
right-hand side is finite. This contradiction leads us to say that β
= 0andM is SLP on
D
T(M)
. Hence the theorem is proved. 
Remark 3.9. As a final remark, Theorem 3.1(c) demonstrates that when w, p
−1
,q ∈ 
1
WD does not imply SLP or even LP. Thus, for the equivalency of WD and SLP,the
hypothesis of Theorem 3.8 is needed. For example, when w
= 1, the requirements for the
result SLP
⇐⇒ WD become

∞
n=−1
p
−1

[11] F. V. Atkinson, Discrete and Continuous Boundary Problems, vol. 8 of Mathematics in Science and
Engineering, Academic Press, New York, NY, USA, 1964.
[12] A. Delil, “
˙
Ikinci mertebe fark ifadesinin Dir ichlet ve limit-nokta
¨
ozellikleri,” in 17th National
Symposium of Mathematics, pp. 26–31, Bolu, Turkey, August 2004.
A. Delil: E
ˇ
gitim Fak
¨
ultesi, Celal Bayar
¨
Universitesi, 45900 Demirci, Manisa, Turkey
Email address: [email protected]