doi:10.1155/2009/521058
Research Article
Summation Characterization of the Recessive
Solution for Half-Linear Difference Equations
Ondˇrej Do ˇsl ´y
1and Simona Fi ˇsnarov ´a
21Department of Mathematics and Statistics, Masaryk University, Kotl´aˇrsk´a 2, 611 37 Brno, Czech Republic 2Department 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,[email protected]
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
ΔrkΦΔxk ckΦxk1 0, Φx : |x|p−2x,p > 1, wherer, c are real-valued sequences, is
closely related to the divergence of the infinite series∞rkxkxk1|Δxk|p−2 −1
.
Copyrightq2009 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
ΔrkΦΔxk ckΦxk1 0, Φx:|x|p−2x, p >1, 1.1
wherer, care real-valued sequences andrk >0,and we investigate properties of itsrecessive solution.
Qualitative theory of1.1was established in the series of the papers of ˘Reh´ak 1–5 and it is summarized in 6, Chapter 3. It was shown there that the oscillation theory of1.1 is very similar to that of the linear equation
ΔrkΔxk ckxk10, 1.2
which is the special casep 2 in1.1. We will recall basic facts of the oscillation theory of
The concept of the recessive solution of 1.1 has been introduced in 7. There are several attempts in literature to find a summation characterization of this solution, see 8 and also related references 9,10, which are based on the asymptotic analysis of solutions of
1.1. However, this approach requires the sign restriction of the sequenceckand additional
assumptions on the convergencedivergenceof certain infinite series involving sequencesr andc,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 of1.1is a discrete counterpart of the concept of the principal solution of the half-linear differential equation
rtΦxctΦx 0, 1.3
which attracted considerable attention in recent years, we refer to the work in 11–15and the references given therein.
Let us recall the main result of 11whose discrete version we are going to prove in this paper.
Proposition 1.1. Letxbe a solution of 1.3such thatxt/0for larget.
iLetp∈1,2. If
Ix: ∞
dt
rtx2t|xt|p−2 ∞, 1.4
thenxis the principal solution of 1.3.
iiIfp≥2andIx<∞, thenxis 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 of1.1are defined using the concept of the generalized zero which is defined in the same way as for1.2, see, for example, 6, Chapter 3,or 16, Chapter 7. A solutionxof1.1has a generalized zeroin an intervalm, m1ifxm/0 andxmxm1rm ≤0.
Since we suppose thatrk > 0 oscillation theory of 1.1 generally requires onlyrk/0, a
Ifxis a solution of1.1such thatxk/0 in some discrete interval m,∞,thenwk
rkΦΔxk/xkis a solution of the associated Riccati type equation
Δwkckwk If we suppose that 1.1is nonoscillatory, among all solutions of 2.1there exists the so-called distinguished solutionw which has the property that there exists an interval m,∞ such that any other solutionwof2.1for whichΦ−1rk Φ−1wk>0,k∈ m,∞, satisfies wk > wk, k ∈ m,∞. Therefore, the distinguished solution of 2.1is, in a certain sense,
minimal solution of this equation near∞,and sometimes it is called theminimalsolution of
2.1. Ifwis the distinguished solution of2.1, then the associated solution of1.1given by
At the end of this section, for the sake of comparison, we recall the main results of 8, 17, where summation characterizations of recessive solutions of 1.1 are investigated using the asymptotic analysis of the solution space of1.1.
Ifxis the recessive solution of 1.1, then
∞
1
rkxkxk1|Δxk|p−2
∞. 2.6
iiiSuppose thatck > 0,∞ck <∞, and∞rk1−q < ∞. Thenxis the recessive solution if and only if 2.4holds.
In casesiand 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 condition2.4. We refer to
8,17and also to 9,10for more details.
3. Technical Results
Throughout the rest of the paper we suppose that1.1is nonoscillatory andhis its solution. Denote
v∗k:rkhkΦhk ΦΔhk, Rk:
2
qrkhkhk1|Δhk|
p−2,
Gk:rkhkΦΔhk,
3.1
and define the function
Hk, v:vrkhk1ΦΔhk− rkvGk|hk1|
p
Φ|hk|qΦ−1rk Φ−1vGk. 3.2
Lemma 3.1. Put
vk:|hk|pwk−wk, 3.3
wherewkrkΦΔhk/hkis a solution of 2.1andwkis any sequence satisfyingrkwk/0. Then the following statements hold:
iwkis a solution of 2.1if and only ifvkis a solution of
ΔvkHk, vk 0; 3.4
iiHk, v≥0forv >−vk∗with the equality if and only ifv0;
iiirkwk>0if and only ifvkv∗k>0;
Proof. The statementsi,iiare consequences of 18, Lemma 2.5.
Denote byAthe expression in brackets, then
sgnAsgn
Lemma 3.2. Letv∗, R, G, Hbe defined by3.1,3.2and suppose thathkΔhk<0for largek. Then one has the following inequalities for largek.
Ifp∈1,2, thenv∗k≤Rkand
v−Hk, v≤ Rkv Rkv
forv∈−vk∗,0. 3.9
Ifp≥2, thenvk∗≥Rkand
v−Hk, v≥ Rkv Rkv
forv∈−Rk,0. 3.10
Proof. We havewith using the Lagrange mean value theorem
vk∗rkhkΦhk ΦΔhk
rkhkΦhk1
Φ
hk
hk1
−Φ
−Δhk
hk1
rkhkΦhk1Φξ,
3.11
where−Δhk/hk1≤ξ≤hk/hk1and henceξ≥ |Δhk/hk1|. Thus, ifp∈1,2,
v∗kp−1rkhkΦhk1|ξ|p−2≤
p−1rkhkΦhk1
Δhk
hk1
p−2
1
q−1rkhkhk1|Δhk|
p−2≤R
k,
3.12
and in the casep≥2,we obtain
v∗k≥Rk. 3.13
Next we proceed similarly as in 18, Lemma 2.6. Inequalities3.9,3.10can be written in the equivalent forms:
RkvHk, v≥v2, v∈
−vk∗,0forp∈1,2, 3.14
DenoteFk, v: RkvHk, v−v2and letv >−v∗k. Then
where
Ak, v:−2|hk|qΦ−1rk−2Φ−1vGk q|vGk|q−2Rkv
q−2Φ−1vGk qRk−Gk|vGk|q−2−2|hk|qΦ−1rk.
3.20
Hence
sgnAk, v sgnFvvk, v forv >−v∗k, 3.21
and from3.18
sgn Ak, v sgnq−2 3.22
in some left neighborhood ofv0. Moreover, forv <0
Avk, v q−2sgnvGk|vGk|q−3q−1vGk qRk−Gk
−q−2|vGk|q−3q−1v−GkqRk
,
3.23
andAvk, v 0forv <0if and only if
vvk: 1
q−1
Gk−qRk
− 1
q−1rkhk|Δhk|
p−2h
khk1. 3.24
Next we distinguish between the casesp∈1,2andp≥2. Ifp∈1,2, then using3.12,
vk≤ −
1
q−1rkhkhk1|Δhk|
p−2 ≤ −v∗
k, 3.25
hence Ak, v is decreasing on −v∗k,0 and in view of 3.22 it means that Ak, v and consequently from3.21alsoFvvk, vis positive forv∈−vk∗,0. Hence,3.14holds.
Similarly, ifp≥2, then
vk≤ − 1
q−1rkhkhk1|Δhk|
p−2≤ −R
k, 3.26
henceAk, vis increasing forv∈−Rk,0and from3.22we have thatAk, vand hence
4. Main Results
Theorem 4.1. Supposep∈1,2and lethbe a solution of 1.1such thathkΔhk<0for largek. If
∞
1
rkhkhk1|Δhk|p−2
∞, 4.1
thenhis the recessive solution.
Proof. Denote bywkrkΦΔhk/hkthe associated solution of2.1and letwkbe a solution
of2.1generated by another solutionlinearly independent ofhof 1.1. Then, it follows fromLemma 3.1thatvk|hk|pwk−wkis a solution of3.4, that is,
vk1vk−Hk, vk, 4.2
and suppose that this solution satisfies the conditionvN <0.This means thatwN <wNand
to prove thathis the recessive solution of1.1, we need to show that there existsm≥Nsuch thatrmwm≤0, that is, according toLemma 3.1,vmv∗m≤0. Suppose by contradiction that
vkvk∗ > 0 fork ≥N. According toLemma 3.1iv, it means thatvk <0 fork ≥N, that is,
vk∈−v∗k,0. Then we have fromLemma 3.2thatvkRk>0 and
vk1≤ Rkvk Rkvk
fork≥N. 4.3
Next, consider the equation
uk1 Rkuk
Rkuk
, 4.4
and letukbe its solution satisfyinguNvN. However,4.4is equivalent to
−Δuk
u2
k
Rkuk,
4.5
that is,
− Δuk
ukuk1
uk
uk1Rkuk
1 Rk
, 4.6
where we have substituted foruk1from4.4in the denominator. Hence
1 uk1
1 uk
1 Rk
and we obtain
uk
1 1/uN
k−1 jN
1/Rj
. 4.8
Condition4.1implies that there existsm≥Nsuch thatum <0 and eitherum1>0 orum1 is not defined. This means thatRmum ≤ 0from4.4. On the other hand,4.3together
with 4.4and the fact that Rkx/Rkx is increasing with respect toxon−v∗k,0 imply
that vk ≤ uk for k ≥ N.Since vk Rk > 0 for k ≥ N,we have uk Rk > 0 for k ≥ N,
a contradiction.
Theorem 4.2. Supposep≥2and lethbe a solution of 1.1such thathkΔhk<0for largek. If
∞
1
rkhkhk1|Δhk|p−2
<∞, 4.9
thenhis not the recessive solution.
Proof. Similarly, as in the proof ofTheorem 4.1, denotewk rkΦΔhk/hkand letwk be a
solution of 2.1 generated by another solutionlinearly independent of h of 1.1. Then vk|hk|pwk−wkis a solution of3.4, that is,
vk1vk−Hk, vk, 4.10
and suppose that this solution satisfies the conditionvN < 0, |vN|being sufficiently small will be specified later. HencewN < wN and we have to show thatrkwk >0 fork ≥N,
that is,vkvk∗>0 fork≥N.
Letukbe a solution of4.4and suppose thatuNvN. Hence, similarly as in the proof
ofTheorem 4.1, we obtain
uk
1 1/uN
k−1 jN
1/Rj
. 4.11
If|uN|is sufficiently small, then condition4.9implies thatuk<0 fork≥Nand from4.4,
we haveRkuk>0 fork≥N. Consequently, fromLemma 3.2we obtain thatv∗k≥Rkand
uk−Hk, uk≥
Rkuk
Rkuk
uk1 fork≥N. 4.12
Moreover, sincex−Hk, xis increasing with respect toxon−Rk,0,we obtain from4.12
thatvk≥ukfork≥N.HenceRkvk>0 fork≥Nand hence alsov∗kvk>0 fork≥N.
5. Applications and Open Problems
prove inequalities3.9,3.10only whenGrhΦΔh<0. We conjecture that Theorems4.1 and4.2remain to hold foreverysolution of1.1for whichΔhk/0 for largek. To justify this
conjecture, consider the function
Fkv Hk, v/vv−Hk, v. 5.1
By an easy computation one can find that inequalities 3.9, 3.10 are equivalent to the inequalities
Fkv≥ 1 Rk
, p∈1,2, Fkv≤ 1 Rk
, p∈ 2,∞. 5.2
However, ifGk>0, that is,−Gk<0, we have
Fk−Gk 1
rkhkhk1|Δhk|p−2
2
qRk
, 5.3
so inequalities3.9,3.10are no longer valid in this case. Numerical computations together with a closer examination of the graph of the functionFlead to the following conjecture.
Conjecture 5.1. Let hk, hk1 > 0, Δhk/0, and R∗k : q−1rkhkhk1|Δhk|p−2.Then for v ∈
−v∗k,∞one has
Fkv≥ 1
R∗k forp∈1,2, Fkv≤ 1
R∗k forp∈ 2,∞. 5.4
To explain this conjecture in more details, consider the casep ∈ 1,2, the casep ≥ 2 can be treated analogically. We havewe skip the indexk, only indices different fromkare written explicitly
F∞: lim
v→ ∞Fv
1
rhk1 Φhk1−ΦΔh
1
rΦhhk1 Φhk1/h−ΦΔh/h
q−1|ξ|2−p rΦhhk1 ,
5.5
whereΔh/h≤ξ≤hk1/h.IfΔh >0, the direct substitution yields
F∞≥
q−1 rhhk1|Δh|p−2
≥ 1
q−1rhhk1|Δh|p−2
1
IfΔh <0, then|Δh|< hand we proceed as follows. Forp ∈1,2, the functionΦis concave for nonnegative arguments, so forx, y≥0,we have the inequality
Φ
the graph of the functiont → 2·2−1/tshows that the required inequality really holds, so we
have
F∞≥ 1
q−1rhk1h|Δh|p−2
1
R∗. 5.10
By a similar computation we find that
F0 lim
These computations lead to the conjecture that Fattains its global minimum at a point in
−v∗,−GifG >0 and at a point in−G,∞ifG < 0. Numerical computations suggest that this minimum is 1/crhhk1|Δh|p−2, where 1≤c≤q−1.
iiA typical example of1.1to which Theorems4.1and4.2apply is1.1with
since under these assumption all positive solutions of1.1are decreasing, see 19. However, one can applyindirectlyTheorems4.1and4.2also to1.1with
∞
rk1−q ∞, ck>0 5.13
and∞ck <∞, otherwise1.1would be oscillatory, see 16, Theorem 8.2.14 , even if all
positive solutions of1.1areincreasingin this case. The method which enables to overcome this difficulty is the so-calledreciprocity principle, which can be explained as follows.
Suppose thatck/0 in1.1and letuk:rkΦΔxk. Then by a direct computation one
can verify thatusolves the so-calledreciprocal equation:
Δ
nonoscillatory, see 9. The following statement relates recessive solutions of1.1and5.14. A similar statement can be found in 9, but our proof differs from that given in 9.
Theorem 5.2. Suppose that1.1is nonoscillatory and5.12or5.13holds. If a solutionhof1.1 is recessive, thenu:rΦΔhis the recessive solution of 5.14.
Proof. First suppose that5.13holds and letw rΦΔh/hbe the distinguished solution of
2.1. Assumption5.13implies thatwk >0 for largek, see 7. The solutionvof the Riccati relationship between solutions of5.15and2.1 no index means again the indexk:
Since the function
andvis any other solution of5.15. Consequently,vis the distinguished solution of5.15 and henceuis the recessive solution of5.14.
Now suppose that5.12holds. Then all solutionswof2.1satisfyingrkwk>0 for
largekare negativesee 19, that is, 0> wk >wk. Then using the same argument as in the
first part of the proof we have 0<vk < vkfor largekfor any solutionvof5.15, that is,uis
the recessive solution of5.14.
iiiIn 18, we posed the question whether the sequencehk :kp−1/p is the recessive
solution of the difference equation
assumption5.12is satisfiedwithq,c1−q, and 1 instead ofp,r, andc, resp., hence positive
solutions of5.21are decreasing, that is, Theorems4.1and4.2apply to this case. By a direct computation, we have
c1k−qukuk1|Δuk|q−2∼k−p1−qk−2p−1/pk−2p1q−2/pk. 5.24
This means, byTheorem 4.1, that ifq∈1,2, thenuis the recessive solution of5.21 and hence yk hk1 is the recessive solution of 5.22. Consequently, hk kp−1/p is the
recessive solution of5.19ifp≥2.
Acknowledgments
This research is supported by the Grant 201/07/0145 of the Czech Grant Agency of the Czech Republic, and the Research Project MSM0022162409 of the Czech Ministry of Education.
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