We associate with the above-mentioned equa-tion a linear second-order difference equaequa-tion and we show that oscillatory properties of the above-mentioned one can be investigated usin
Trang 1fference Equations
Volume 2008, Article ID 438130, 18 pages
doi:10.1155/2008/438130
Research Article
Linearized Riccati Technique and (Non-)Oscillation Criteria for Half-Linear Difference Equations
Ondˇrej Do ˇsl ´y 1 and Simona Fiˇsnarov ´a 2
1 Department of Mathematics and Statistics, Masaryk University, Jan´aˇckovo n´am 2a,
66295 Brno, Czech Republic
2 Department of Mathematics, Mendel University of Agriculture and Forestry in Brno, Zemˇedˇelsk´a 1,
61300 Brno, Czech Republic
Correspondence should be addressed to Ondˇrej Doˇsl ´y, dosly@math.muni.cz
Received 23 August 2007; Accepted 26 November 2007
Recommended by John R Graef
We consider the half-linear second-order difference equation Δrk ΦΔx k c k Φx k1 0, Φx :
|x| p−2
x, p > 1, where r, c are real-valued sequences We associate with the above-mentioned
equa-tion a linear second-order difference equaequa-tion and we show that oscillatory properties of the above-mentioned one can be investigated using properties of this associated linear equation The main tool
we use is a linearization technique applied to a certain Riccati-type difference equation correspond-ing to the above-mentioned one.
Copyright q 2008 O Doˇsl´y and S Fiˇsnarov´a 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
In this paper, we deal with oscillatory properties of solutions of the half-linear second-order difference equation
Δr kΦΔx k c k Φx k1 0, Φx : |x| p−2 x, p > 1, 1.1
where r, c are real-valued sequences and r k > 0 This equation can be regarded as a discrete
counterpart of the half-linear di fferential equation
rtΦ
x
which attracted considerable attention in the recent years We refer to the books in 1,2
the references given therein The basic qualitative theory of1.1 has been established in the series of papers in 3 7 8, Chapter 3 2, Chapter 8
Trang 2It is known that oscillatory properties of 1.1 are very similar to those of the second-order Sturm-Liouville difference equation which is a special case of p 2 in 1.1:
Δr k Δx k
c k x k1 0. 1.3
In particular, the discrete linear Sturmian theory extends verbatim to1.1, and hence this equa-tion can be classified as oscillatory or nonoscillatory We will recall elements of the oscillaequa-tion theory of1.1 in more detail in the next section
The basic idea of the discrete linearization technique which we establish in this paper is motivated by the paper of Elbert and Schneider 9
ential equation
Φx
γ p
t p Φx 2
p − 1 p
p−1
δt
t p Φx 0, γ p :
p − 1 p
p
, 1.4
is viewed as a perturbation of the Euler-type half-linear differential equation
Φx
γ p
and oscillatory properties of1.4 are studied via the linear equation
ty
δt
under the assumption that∞
t δs/s ds ≥ 0 for large t In particular, the following statements
are presented in 9
i Let p ≥ 2 and let linear equation 1.6 be nonoscillatory Then 1.4 is also nonoscillatory.
1.4 be nonoscillatory Then linear equation 1.6
is also nonoscillatory.
The linearization technique for1.2 has been further developed in 10–12
given therein
In our paper, we introduce a similar linearization technique for the investigation of os-cillatory properties of1.1 This equation is regarded as a perturbation of the nonoscillatory equation of the same form:
Δr kΦΔx k
c kΦx k1
and oscillatory properties of solutions of1.1 are related to those of the linear second-order difference equation
ΔR k Δy k C k y k1 0, 1.8 where
R k 2
q r k h k h k1 Δh kp−2
, C k c k − c kh p k1 , 1.9
Trang 3with q p/p − 1 being the conjugate number of p, and with a certain distinguished solution
h of 1.7 This enables to apply the deeply developed linear oscillation theory when investi-gating oscillations of half-linear equation1.1 As we will see in the next sections, compared
to the continuous case, the linearization technique is technically more difficult in the discrete case since a nonlinear function which appears in the so-called modified Riccati equation is considerably more complicated in the discrete case
The paper is organized as follows In the next section, we recall basic oscillatory proper-ties of1.1, including a quadratization formula for a certain nonlinear function which plays an important role in subsequent sections of the paper InSection 3, we present a discrete version
of the above-mentioned result of Elbert and Schneider 9 Section 4, we show that under certain additional restriction on properties of solutions of1.7 we do not need to distinguish
of the results of the previous sections of the paper
2 Preliminaries
Oscillatory properties of1.1 are defined using the concept of the generalized zero which is defined in the same way as for1.3 see, e.g., 8, Chapter 3 2, Chapter 7
1.1 m / 0 and x m x m1 r m≤ 0 Since we suppose
that r k > 0 oscillation theory of 1.1 generally requires only r k / 0, a generalized zero of x in
initial condition x m 0, x m1 / 1.1 is said to
be nonoscillatory if there exists m ∈ N such that it is disconjugate on
it is said to be oscillatory in the opposite case.
If x is a solution of 1.1 such that x k / 0 in some discrete interval m, ∞, then w k
r k ΦΔx k /x k is a solution of the associated Riccati-type equation
R
w k : Δwk c k w k
1− r k
ΦΦ−1
r k
Φ−1
w k
0. 2.1
Moreover, if x has no generalized zero in m, ∞, then Φ−1r k Φ−1w k > 0, k ∈ m, ∞ If
we suppose that1.1 is nonoscillatory, among all solutions of 2.1 there exists the so-called
distinguished solution w which has the property that there exists an interval m, ∞ such that
any other solution w of 2.1 for which Φ−1r k Φ−1w k > 0, k ∈ m, ∞, satisfies w k > w k,
k ∈ m, ∞ Therefore, the distinguished solution of 2.1 is, in a certain sense, minimal solution
of this equation near∞ If w is the distinguished solution of 2.1, then the associated solution
of1.1 given by the formula
x k k−1
jm
1 Φ−1 w j
is said to be the recessive solution of1.1 see 13
x of 1.3 is recessive if and only if
∞
r k x k x k1 ∞. 2.3
Trang 4Our first statement presents a comparison theorem for distinguished solutions of2.1 and2.4 given below
Lemma 2.1 see 13 1.1 be nonoscillatory and let c k ≥ c k for large k Further, let w k , v k be distinguished solutions of the corresponding generalized Riccati equations2.1 and
R
v k : Δvk c k v k
1− r k
ΦΦ−1
r k
Φ−1
v k
0, 2.4
respectively Then there exists m ∈ Z such that wk ≥ v k for k ∈ m, ∞ In particular, if c k ≥ 0 and
∞
r k1−q ∞, then w k ≥ 0 for large k.
The next statement relates nonoscillation of1.1 to the existence of a certain solution of the Riccati inequality associated with2.1
Lemma 2.2 see 2, Theorem 8.2.7 1.1 is nonoscillatory if and only if there exists a
sequence w k satisfying r k w k > 0 and
R
for large k.
The next statement is the discrete version of the generalized Leighton-Wintner oscillation criterion In this criterion,1.1 is viewed as a perturbation of 1.7
∞
c k − c kh p k1 ∞, 2.6
then1.1 is oscillatory.
The last auxiliary oscillation results of this section are Hille-Neharinon-oscillation cri-teria for linear difference equation 1.3
Lemma 2.4 see 14 k ≥ 0, r k > 0,∞
r k−1 ∞, and∞c k < ∞ If
lim inf
k→∞
k−1
1
r j
∞
jk
c j
> 1
then1.3 is oscillatory If
lim sup
k→∞
k−1
1
r j
∞
jk
c j
< 1
then1.3 is nonoscillatory.
Trang 5For the remaining part of this section, we suppose that1.7 is nonoscillatory and we let
h be its solution such that h k > 0 for large k Further, put
G k : rk h kΦΔh k 2.9 and define the function
Hk, v : v r k h k1ΦΔh k− r k
v G k
h p k1
Φh q kΦ−1
r k
Φ−1
v G k
. 2.10
Lemma 2.5 Put
v k : hp kw k− w k
where wk r k ΦΔh k /h k is a solution of 2.4 and w k is any sequence satisfying r k w k / 0 Then
Δv kc k − c kh p k1 Hk, v k
h p k1 R
In particular, if w k is a solution of 2.1, then
Δv kc k − c kh p k1 Hk, v k
Moreover, Hk, v ≥ 0 for v > −r k h k ΦΔh k h p−1 k with the equality if and only if v 0.
Proof By a direct computation and using the fact that wkis a solution of2.4, we obtain
Δv k h p
k1
w k1− w k1
− v k
h p k1
w k1 c k− r k wk
ΦΦ−1
r k
Φ−1
w k
− v k
h p k1
w k1 c k − r kΦ
Δh
k
h k1
− v k
h p k1
w k1 c k− r k h k1ΦΔh k− v k
2.14
Next, since v k h p
k w k − G k , we have
r k
v k G k
Φh q kΦ−1
r k
Φ−1
v k G k
r k h p k w k
Φh q kΦ−1
r k
Φ−1
h p k w k
r k w k
ΦΦ−1
r k
Φ−1
w k
,
2.15
and hence
Δv kc k − c kh p k1 Hk, v k
h p k1
v k G k
Φh q kΦ−1
r k
Φ−1
v k G k
h p k1 R
w k
2.16
Trang 6If w k is a solution of2.1, then v k satisfies2.13 We prove the nonnegativity of the function
Hk, v for v > −r k h k ΦΔh k h p−1
k as follows By a direct computation, we have
H v
k, v
q
k h q k h p k1
h q kΦ−1
r k
Φ−1
v k G k
p ,
H vv k, v qr
q
k h q k h p k1v k G kq−2
h q kΦ−1
r k
Φ−1
v k G kp1 .
2.17
Hence H v k, v 0 if and only if v 0 and the function Hk, v is convex with respect to v for
v satisfying h −q k Φ−1r k Φ−1v G k > 0 which is equivalent to v > −r k h k ΦΔh k h p−1
k This proves the last statement ofLemma 2.5
Lemma 2.6 Let R, G be defined by 1.9 and 2.9, respectively, and suppose that G k > 0 for k ∈ N Then we have the following inequalities for v ≥ 0 and k ∈ N :
R k vHk, v ≥ v2,
Proof In this proof, we write explicitly an index by a sequence only if this index is different
from k; that is, no index means the index k In addition to 2.17, we have
H vvv k, 0 q
r2h2h2
k1 Δh 2p−3
q − 2h k1 − 2q − 1Δh 2.19
Denote Fk, v : R k vHk, v − v2 Then we have F v k, 0 0 F vv k, 0 and
F vvv k, 0 RH vvv k, 0 3H vv k, 0
rhh k1 Δh p−1
q − 2h k1 − 2q − 1Δh 3q
rhh k1 Δh p−2
rhh k1 Δh p−1
2q − 2h Δh 2 − qΔh
rhh k1 Δh p−1
q − 2 rhh k1 Δh p−1
h h k1
2.20
Consequently,
sgn Fk, v sgn q − 2 2.21
in some right neighborhood of v 0 Further, we have
F vv k, v 2H v k, v R vH vv k, v − 2
− 2r q h q h
p k1
h qΦ−1r Φ−1v Gp qr q h q h
p k1 v G q−2 R v
h qΦ−1r Φ−1v Gp1
r q h q h
p k1
h qΦ−1r Φ−1v Gp1
− 2r q−1 h q− 2Φ−1v G qv G q−2 R v
2.22
Trang 7Denote by Av the expression in brackets in the last expression By a direct computation, we
have
Av q − 2Φ−1v G qR − Gv G q−2 − 2r q−1 h q , 2.23
hence sgn Av sgn F vv k, v sgn q−2 for large v, and from the computation of F vvv k, 0,
we also haveq − 2Av > 0 in some right neighborhood of v 0 Since
Av q − 2v G q−3
q − 1v G qR − G 2.24 has no positive root observe that q − 1v G qR − G 0 if and only if v
−1/q − 1rhΔh p−2 h k1 h < 0, this means that q − 2Av and hence also
q − 2F vv k, v have a constant sign for v ∈ 0, ∞ Therefore, the function Fk, v is convex for
q ≥ 2 and concave for q ≤ 2, and this together with 2.21 implies the required inequalities
3 (Non-)oscillation criteria:
In this section, we suppose that1.7 is nonoscillatory and possesses a positive increasing
so-lution h We associate with 1.1 the linear Sturm-Liouville second-order difference equation
ΔR k Δy k
C k y k1 0, 3.1
where R and C are given by 1.9, that is,
R k 2
q r k h k h k1
Δh k
p−2
, C k c k − c k
h p k1 3.2 The results of this section can be regarded as a discrete version of the results given in 9
Theorem 3.1 Let p ≥ 2, c k ≥ c k for large k,
∞
1
and suppose that linear equation 3.1 with R, C given by 1.9 is nonoscillatory Then half-linear
equation1.1 is also nonoscillatory.
Proof The proof is based onLemma 2.2 Nonoscillation of3.1 implies the existence of a
solu-tion v of the associated Riccati equasolu-tion
Δv k C k v2k
such that R k v k > 0 for large k Moreover, since 3.3 holds and C k ≥ 0 for large k, by
Lemma 2.1v k ≥ 0 for large k ByLemma 2.6, we haveR k vHk, v ≤ v2; hence v is also a
solution of the inequality
Δv k C k Hk, v k
Now, substituting for v h p w − w, where w rΦΔh/h, we see fromLemma 2.5that w is
a solution of Riccati inequality2.5 Moreover, r k w k r k h −p k v k w k > 0 since v k ≥ 0 and h
is a nonoscillatory solution of1.7; that is, the corresponding solution of the associated Riccati equationw satisfies r k w k > 0 Therefore, 1.1 is nonoscillatory
Trang 8k ≥ c k for large k, and let h be the recessive solution of 1.7 If
half-linear equation1.1 is nonoscillatory, then linear equation 3.1 is also nonoscillatory.
Proof We proceed similarly as in the previous proof Nonoscillation of 1.1 implies the
ex-istence of the distinguished solution w of the associated Riccati equation 2.1 such that
w k r k > 0 for large k Put again v h p w − w, where w is the distinguished solution of
2.4 Then v solves the equation
Δv k C k Hk, v k
and byLemma 2.1, we have w k ≥ w k for large k, hence v k ≥ 0, and therefore R k v k > 0 for
large k ByLemma 2.6,
Δv k C k v
2
k
R k v k ≤ 0. 3.7
This means that3.1 is nonoscillatory byLemma 2.2
4 Criteria without restriction onp
Throughout this section, we suppose that R k , C k , and G k are given by1.9 and 2.9, respec-tively, and that1.7 is nonoscillatory
Theorem 4.1 Let c k ≥ c k for large k and let h k > 0 be the recessive solution of 1.7 such that
∞
Further, suppose that condition3.3 holds and
lim
k→∞ r k h kΦΔh k ∞. 4.2
If there exists ε > 0 such that the equation
ΔR k Δy k 1 − εC k y k1 0 4.3
is oscillatory, then1.1 is also oscillatory.
Proof Let ε > 0 be such that 4.3 is oscillatory i.e., ε < 1 Suppose, by contradiction,
that 1.1 is nonoscillatory, and let x k be its recessive solution Denote by w k r k ΦΔx k /
x k and w k r k ΦΔh k /h k the distinguished solutions of the Riccati equations 2.1 and 2.4,
respectively, and put v k : hp
k w k − w k Since c k ≥ c k for large k, it follows fromLemma 2.1
that w k ≥ w k , and hence also v k ≥ 0 for large k According toLemma 2.5, we have
Δv k −C k − Hk, v k
Hence v k is nonnegative and nonincreasing for large k, and this means that there exists a limit
of v k such that
0≤ lim
Trang 9Next, let N ∈ N be sufficiently large, k > N Summing 4.4 from N to k, we obtain
v N − v k1 k
jN
C jk
jN
H
j, v j
and hence
v N≥ k
jN
C jk
jN
H
j, v j
Letting k → ∞ and using condition 4.1, we have
∞
H
k, v k
Substituting z k v k /G k into Hk, v k, we obtain
H
k, G k z k
G k z k r k h k1ΦΔh k
− r k
z k 1h p k1
Φh k /Δh k Φ−1
z k 1 : H
k, z k
. 4.9
Now, it follows from conditions4.2 and 4.5 that z k → 0 as k → ∞ Hence we may
approx-imate the function Hk, z by the second-degree Taylor polynomial at the center z 0 k is
regarded as a parameter By a direct computation, we have
Hk, 0 0, Hz k, 0 0, Hzz k, 0 qr k h k
Δh kp
h k1 , 4.10 and hence
Hk, z qr k h k
Δh k
p
2h k1 z2 oz2
as z −→ 0. 4.11
The term oz2 is of the form H zzz k, ξz3for some ξ ∈ 0, z By a direct computation, we have
H zzz k, 0 qr k h k
Δh kp
h2
k1
q − 2h k1 − 2q − 1Δh k , 4.12 that is,
H zzz k, 0 ≤ qr k h k
Δh k
p
h k1
|q − 2| 2q − 1 4.13
Since H zzz k, z is continuous with respect to z near z 0, there exists a constant M > 0 such
that
H zzz k, ξ ≤ Mr k h k
Δh k
p
Trang 10and hence4.11 can be written in the form
Hk, z qr k h k
Δh k
p
2h k1 z2
1 o1 as z −→ 0 4.15
and the convergence o1 → 0 as z → 0 is uniform with respect to k This means that there exists N1such that
q − εr k h k
Δh kp
2h k1 z2
k < H
k, z k
< q εr k h k
Δh kp
2h k1 z2
k for k ≥ N1, 4.16 and consequently
∞ >∞H
k, v k
∞H
k, z k
> q − ε
2
∞
r k h k
Δh kp
h k1 z2k
q − ε 2
∞
r k h k
Δh kp
v2
k
h k1 G2k q − ε
2
∞
k
r k h k h k1
Δh kp−2
4.17
Taking into account condition 3.3, it follows that v k → 0 as k → ∞ Thus we can apply
Taylor’s formula to the function Fk, v : R k vHk, v at the center v 0 By a direct
computationsee also the proof ofLemma 2.6, we have k is regarded again as a parameter
Fk, 0 0, F v k, 0 0, F vv k, 0 2, 4.18 and hence
Fk, v v2 ov2
v2
1 o1 as v −→ 0. 4.19
Similarly as in the case of Hk, z, the convergence o1 → 0 as v → 0 is uniform with respect
to k because of 4.2 and 2.20 Hence
2
R k v
1 o1 as v −→ 0. 4.20
Consequently, there exists N2> N1such that
1− ε 2
v2
k
R k v k < H
k, v k
<
1 ε 2
v2
k
R k v k for k ≥ N2. 4.21 Since
R k 2
q G k
h k1
Δh k
2
q G k
1 h k
Δh k
> 2
q G k , 4.22 from conditions4.2 and 4.22 we have
v k
R k −→ 0 as k −→ ∞. 4.23
... sign for v ∈ 0, ∞ Therefore, the function Fk, v is convex for< /i>q ≥ and concave for q ≤ 2, and this together with 2.21 implies the required inequalities
3 (Non-)oscillation. .. be defined by 1.9 and 2.9, respectively, and suppose that G k > for k ∈ N Then we have the following inequalities for v ≥ and k ∈ N :
R...
and byLemma 2.1, we have w k ≥ w k for large k, hence v k ≥ 0, and therefore R k v k > for< /i>
large