Volume 2009, Article ID 521058, 16 pagesdoi:10.1155/2009/521058 Research Article Summation Characterization of the Recessive Solution for Half-Linear Difference Equations Ondˇrej Do ˇsl
Trang 1Volume 2009, Article ID 521058, 16 pages
doi:10.1155/2009/521058
Research Article
Summation Characterization of the Recessive
Solution for Half-Linear Difference Equations
Ondˇrej Do ˇsl ´y1 and Simona Fi ˇsnarov ´a2
1 Department of Mathematics and Statistics, Masaryk University, Kotl´aˇrsk´a 2, 611 37 Brno, Czech Republic
2 Department of Mathematics, Mendel University of Agriculture and Forestry in Brno, Zemˇedˇelsk´a 1,
613 00 Brno, Czech Republic
Correspondence should be addressed to Ondˇrej Doˇsl ´y,dosly@math.muni.cz
Received 24 June 2009; Accepted 24 August 2009
Recommended by Martin J Bohner
We show that the recessive solution of the second-order half-linear difference equation
Δr k ΦΔx k c k Φx k1 0, Φx : |x| p−2x, p > 1, where r, c are real-valued sequences, is
closely related to the divergence of the infinite series∞
r kxkxk1|Δx k|p−2−1 Copyrightq 2009 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
We consider the second-order half-linear difference equation
Δr k ΦΔx 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, and we investigate properties of its recessive solution.
Qualitative theory of1.1 was established in the series of the papers of ˘Reh´ak 1 5
is very similar to that of the linear equation
which is the special case p 2 in 1.1 We will recall basic facts of the oscillation theory of
1.1 in the following section
Trang 2The concept of the recessive solution of 1.1 has been introduced in 7
several attempts in literature to find a summation characterization of this solution, see 8 and also related references 9,10
1.1 However, this approach requires the sign restriction of the sequence c kand additional assumptions on the convergencedivergence of certain infinite series involving sequences r and c, seeProposition 2.1in the following section Here we use a different approach which is based on estimates for a certain nonlinear function which appears in the Picone-type identity for1.1
The recessive solution of1.1 is a discrete counterpart of the concept of the principal solution of the half-linear differential equation
r tΦx
which attracted considerable attention in recent years, we refer to the work in 11–15 the references given therein
Let us recall the main result of 11
this paper
Proposition 1.1 Let x be a solution of 1.3 such that xt / 0 for large t.
I x :
∞
dt
r tx2t|xt| p−2 ∞, 1.4
then x is the principal solution of 1.3.
ii If p ≥ 2 and Ix < ∞, then x is not the principal solution of 1.3.
The paper is organized as follows InSection 2we recall elements of the oscillation theory of1.1.Section 3is devoted to technical statements which we use in the proofs of our main results which are presented inSection 4 Section 5contains formulation of open problems in our research
2 Preliminaries
Oscillatory properties of1.1 are defined using the concept of the generalized zero which is defined in the same way as for1.2, see, for example, 6, Chapter 3 16, Chapter 7
Since we suppose that r k > 0 oscillation theory of 1.1 generally requires only r k / 0, a
generalized zero of x in
between m and m 1 However, 1.1
the solution x of1.1 given by the initial condition x m 0, x m1 / 0 has no generalized zero
in 1.1 is said to be nonoscillatory if there exists m ∈ N such that it is
disconjugate on
Trang 3If x is a solution of1.1 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
Δw k c k w k
ΦΦ−1r k Φ−1w k
whereΦ−1x |x| q−2 x is the inverse function of Φ and q p/p − 1 is the conjugate number
to p Moreover, if x has no generalized zero in m, ∞, then Φ−1r kΦ−1w k > 0, k ∈ m, ∞.
If we suppose that 1.1 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 of2.1 for which Φ−1r k Φ−1w k > 0, k ∈ m, ∞, satisfies
w k > wk , k ∈ m, ∞ Therefore, the distinguished solution of 2.1 is, in a certain sense, minimal solution of this equation near∞, and sometimes it is called the minimal solution of
2.1 If w is the distinguished solution of2.1, then the associated solution of 1.1 given by the formula
x k k−1
jm
1 Φ−1
w j
r j
2.2
is said to be the recessive solution of1.1, see 7
x of 1.2 is recessive if and only if
∞
At the end of this section, for the sake of comparison, we recall the main results of
using the asymptotic analysis of the solution space of1.1
Proposition 2.1 Let x be a solution of 1.1 .
i Suppose that c k < 0, then x is the recessive solution of 1.1 if and only if
∞
ii Suppose that c k > 0,∞
r k1−q< ∞, and
∞
c kΦ
⎛
⎝ ∞
jk1
r j1−q
⎞
Trang 4If x is the recessive solution of 1.1, then
∞
r k x k x k1 |Δx k|p−2 ∞. 2.6
iii Suppose that c k > 0,∞
c k < ∞, and∞r k1−q < ∞ Then x is the recessive solution if
and only if 2.4 holds.
In casesi and iii, the previous proposition gives necessary and sufficient condition for a solution x to be recessive The reason why under assumptions in i or iii it is possible to formulate such a condition is that there is a substantial difference in asymptotic behavior of recessive and dominant solutionsi.e., solutions which are linearly independent
of the recessive solution This difference enables to “separate” the recessive solution from dominant ones and to formulate for it a necessary and sufficient condition 2.4 We refer to
8,17 9,10
3 Technical Results
Throughout the rest of the paper we suppose that1.1 is nonoscillatory and h is its solution.
Denote
v∗k: rk h k Φh k ΦΔh k , R k: 2
q r k h k h k1 |Δh k|p−2 ,
G k: rk h k ΦΔh k ,
3.1
and define the function
H k, v : v r k h k1 ΦΔh k − r k v G k |h k1|p
Φ|h k|qΦ−1r k Φ−1v G k. 3.2
Lemma 3.1 Put
v k: |hk|p w k− w k , 3.3
where wk r k ΦΔh k /h k is a solution of 2.1 and w k is any sequence satisfying r k w k / 0 Then
the following statements hold:
i w k is a solution of 2.1 if and only if v k is a solution of
ii Hk, v ≥ 0 for v > −v∗
k with the equality if and only if v 0;
iii r k w k > 0 if and only if v k v∗
k > 0;
iv let v be a solution of 3.4 and suppose that v m < 0 for some m ∈ N, that is, w m < wm , then v m1 > 0 if and only if v m v∗
m < 0.
Trang 5Proof The statementsi, ii are consequences of 18, Lemma 2.5
iii We have
r k w k r k |h k|−p v k w k
r k |h k|−p v k r kΦΔh k
h k
|h k|−p v k r k h k Φh k ΦΔh k
|h k|−pv k v∗
k
.
3.5
iv We have
v m1 v m − Hm, v m
r m h m1 Φh m1 v m G m
Φ|h m|qΦ−1r m Φ−1v m G m − ΦΔh m
r m h m1 Φh m1 w m
ΦΦ−1r m Φ−1w m − ΦΔh m
r m h m1 Φh m Φ
h m1
h m
w m
ΦΦ−1r m Φ−1w m − Φ
Δh
m
h m
r m h m1 Φh m
ΦΦ−1r m Φ−1w m
× Φ
h m1Φ−1w m
h m
− Φ
Δh
m
h m
ΦΦ−1r m Φ−1w m .
3.6
Denote by A the expression in brackets, then
sgn A sgn h m1Φ−1w m
h m −
h m1
h m − 1
Φ−1r m Φ−1w m
sgn Φ−1r m Φ−1w m −h m Δh mΦ−1r m
h m
sgnΦ−1w m − Φ−1 w m sgn v m −1.
3.7
Consequently,
v m1 > 0⇐⇒ Φ−1r m Φ−1w m < 0, 3.8 that is, the statement holds according to the statementiii of this lemma
Trang 6Lemma 3.2 Let v∗, R, G, H be defined by3.1, 3.2 and suppose that h k Δh k < 0 for large k Then one has the following inequalities for large k.
If p ∗k ≤ R k and
v − Hk, v ≤ R k v
R k v for v∈
−v∗
k , 0
If p ≥ 2, then v∗
k ≥ R k and
v − Hk, v ≥ R k v
R k v for v ∈ −R k , 0 3.10
Proof We havewith using the Lagrange mean value theorem
v k∗ r k h k Φh k ΦΔh k
r k h k Φh k1
Φ
h k
h k1
− Φ
−Δh k
h k1
r k h k Φh k1Φξ,
3.11
where−Δh k /h k1 ≤ ξ ≤ h k /h k1 and hence ξ ≥ |Δh k /h k1|
Thus, if p
v∗kp− 1r k h k Φh k1 |ξ| p−2≤p− 1r k h k Φh k1
Δh k
h k1
p−2
1
q− 1r k h k h k1 |Δh k|p−2 ≤ R k ,
3.12
and in the case p ≥ 2, we obtain
Next we proceed similarly as in 18, Lemma 2.6 3.9, 3.10 can be written in the equivalent forms:
R k vHk, v ≥ v2, v∈−v∗
k , 0
R k vHk, v ≤ v2, v ∈ −R k , 0 3.15
Trang 7Denote Fk, v : R k vHk, v − v2and let v > −v∗
k Then
H v k, v 1 − r
q
k |h k|q |h k1|p
|h k|qΦ−1r k Φ−1v G kp ,
H vv k, v qr
q
k |h k|q |h k1|p |v G k|q−2
|h k|qΦ−1r k Φ−1v G kp1 ,
H vvv k, 0 r2 q
k h2
k h2
k1 Δh k2p−3
q− 2h k1−2q− 1Δh k
.
3.16
Consequently, Fk, 0 F v k, 0 F vv k, 0 0 and
F vvv k, 0 R k H vvv k, 0 3H vv k, 0
r k h k h k1 ΦΔh k
q− 2h k1−2q− 1Δh k
r k h k h k1 |Δh k|p−2
r k h k h k1 ΦΔh k
2
q− 2h k Δh k 2− qΔh k
r k h k h k1 ΦΔh k h k h k Δh k
r k h k h k1 ΦΔh k h k h k1
3.17
Hence, in view of the assumption h k Δh k < 0, sgn F vvv k, 0 − sgnq − 2 It follows that
sgn Fk, v sgn F vv k, v sgnq− 2 3.18
in some left neighborhood of v 0, and the function F is positive, decreasing, and convex for
p
the inequalities3.14 and 3.15 are satisfied in some left neighborhood of v 0 The proof will be completed by showing that F vv k, v has constant sign on the given intervals By a
direct computation,
F vv k, v 2H v k, v R k vH vv k, v − 2
q
k |h k|q |h k1|p
|h k|qΦ−1r k Φ−1v G kp qr
q
k |h k|q |h k1|p |v G k|q−2 R k v
|h k|qΦ−1r k Φ−1v G kp1
q
k |h k|q |h k1|p
|h k|qΦ−1r k Φ−1v G kp1 A k, v,
3.19
Trang 8A k, v : −2|h k|qΦ−1r k − 2Φ−1v G k q|v G k|q−2 R k v
q− 2Φ−1v G k qR k − G k |v G k|q−2 − 2|h k|qΦ−1r k . 3.20
Hence
sgn Ak, v sgn F vv k, v for v > −v∗
and from3.18
in some left neighborhood of v 0
Moreover, for v < 0
A v k, v q− 2sgnv G k |v G k|q−3
q− 1v G k qR k − G k
−q− 2|v G k|q−3
q− 1v − G k qR k
,
3.23
and A v k, v 0 for v < 0 if and only if
v v k: 1
q− 1
G k − qR k
− 1
q− 1r k h k |Δh k|p−2 h k h k1 . 3.24
Next we distinguish between the cases p
v k≤ − 1
q− 1r k h k h k1 |Δh k|p−2 ≤ −v∗
hence Ak, v is decreasing on −v∗
k , 0 and in view of 3.22 it means that Ak, v and
consequently from3.21 also F vv k, v is positive for v ∈ −v∗
k , 0 Hence, 3.14 holds
Similarly, if p≥ 2, then
v k≤ − 1
q− 1r k h k h k1 |Δh k|p−2 ≤ −R k , 3.26
hence Ak, v is increasing for v ∈ −R k , 0 and from 3.22 we have that Ak, v and hence also F vv k, v is negative for v ∈ −R k , 0 This means that 3.15 is satisfied
Trang 94 Main Results
1.1 such that h k Δh k < 0 for large k If
∞
r k h k h k1 |Δh k|p−2 ∞, 4.1
then h is the recessive solution.
Proof Denote by wk r k ΦΔh k /h k the associated solution of 2.1 and let w kbe a solution
of2.1 generated by another solution linearly independent of h of 1.1 Then, it follows fromLemma 3.1that v k |h k|p w k− w k is a solution of 3.4, that is,
and suppose that this solution satisfies the condition v N < 0 This means that w N < wNand
to prove that h is the recessive solution of1.1, we need to show that there exists m ≥ N such that r m w m≤ 0, that is, according toLemma 3.1, v m v∗
m≤ 0 Suppose by contradiction that
v k v∗
k > 0 for k ≥ N According toLemma 3.1iv, it means that v k < 0 for k ≥ N, that is,
v k ∈ −v∗
k , 0 Then we have fromLemma 3.2that v k R k > 0 and
v k1≤ R k v k
R k v k
Next, consider the equation
u k1 R k u k
R k u k
and let u k be its solution satisfying u N v N However,4.4 is equivalent to
−Δu k u2k
that is,
− Δu k
u k u k1 u k
u k1 R k u k
1
where we have substituted for u k1from4.4 in the denominator Hence
1
u k1 1
u k 1
Trang 10and we obtain
u k 1
Condition4.1 implies that there exists m ≥ N such that u m < 0 and either u m1 > 0 or u m1
is not defined This means that R m u m ≤ 0 from 4.4 On the other hand, 4.3 together with 4.4 and the fact that R k x/ R k x is increasing with respect to x on −v∗
k , 0 imply
that v k ≤ u k for k ≥ N Since v k R k > 0 for k ≥ N, we have u k R k > 0 for k ≥ N,
a contradiction
Theorem 4.2 Suppose p ≥ 2 and let h be a solution of 1.1 such that h k Δh k < 0 for large k If
∞
r k h k h k1 |Δh k|p−2 < ∞, 4.9
then h is not the recessive solution.
Proof Similarly, as in the proof ofTheorem 4.1, denotewk r k ΦΔh k /h k and let w k be a solution of 2.1 generated by another solution linearly independent of h of 1.1 Then
v k |h k|p w k− w k is a solution of 3.4, that is,
and suppose that this solution satisfies the condition v N < 0, |v N| being sufficiently small
will be specified later Hence w N < wN and we have to show that r k w k > 0 for k ≥ N, that is, v k v∗
k > 0 for k ≥ N.
Let u kbe a solution of4.4 and suppose that u N v N Hence, similarly as in the proof
ofTheorem 4.1, we obtain
u k 1
1/u Nk−1 jN1/R j . 4.11
If|u N| is sufficiently small, then condition 4.9 implies that u k < 0 for k ≥ N and from 4.4,
we have R k u k > 0 for k ≥ N Consequently, fromLemma 3.2we obtain that v∗k ≥ R kand
u k − Hk, u k ≥ R k u k
R k u k u k1 for k ≥ N. 4.12
Moreover, since x − Hk, x is increasing with respect to x on −R k , 0 , we obtain from 4.12
that v k ≥ u k for k ≥ N Hence R k v k > 0 for k ≥ N and hence also v∗
k v k > 0 for k ≥ N.
5 Applications and Open Problems
iTheorems 4.1 and 4.2, as formulated in the previous section, apply only to positive decreasingor negative increasing solutions of 1.1 The reason is that we have been able to
Trang 11prove inequalities3.9, 3.10 only when G rhΦΔh < 0 We conjecture that Theorems4.1 and4.2remain to hold for every solution of1.1 for which Δh k / 0 for large k To justify this
conjecture, consider the function
By an easy computation one can find that inequalities 3.9, 3.10 are equivalent to the inequalities
Fk v ≥ 1
However, if G k > 0, that is, −G k < 0, we have
Fk −G k 1
r k h k h k1 |Δh k|p−2 2
so inequalities3.9, 3.10 are no longer valid in this case Numerical computations together with a closer examination of the graph of the functionF lead to the following conjecture
Conjecture 5.1 Let h k , h k1 > 0, Δh k / 0, and R∗
k : q − 1rk h k h k1 |Δh k|p−2 Then for v ∈
−v∗
k , ∞ one has
Fk v ≥ 1
R∗k for p Fk v ≤ 1
R∗k for p ∈ 2, ∞. 5.4
To explain this conjecture in more details, consider the case p
can be treated analogically We havewe skip the index k, only indices different from k are
written explicitly
F∞ : lim
rh k1 Φh k1
r Φhh k1 Φh k1 /h
q− 1|ξ|2−p
r Φhh k1 ,
5.5
whereΔh/h ≤ ξ ≤ h k1 /h If Δh > 0, the direct substitution yields
F∞ ≥
q− 1
rhh k1 |Δh| p−2 ≥
1
q− 1rhh k1 |Δh| p−2
1
...⎞
Trang 4If x is the recessive solution of< /i> 1.1, then
∞... dominant solutionsi.e., solutions which are linearly independent
of the recessive solution This difference enables to “separate” the recessive solution from dominant ones and to formulate for. .. 0.
Trang 5Proof The statementsi, ii are consequences of 18, Lemma 2.5
iii