Oscillation of a second order half-linear difference equation and the discrete Hardy inequality

Oscillation of a second order half-linear difference equation and the discrete Hardy inequality

Aigerim Kalybay

¹

, Danagul Karatayeva

²

, Ryskul Oinarov

^B2

and Ainur Temirkhanova

²

1KIMEP University, 4 Abai Avenue, Almaty 050010, Kazakhstan

2L. N. Gumilyov Eurasian National University, 5 Munaytpasov Street, Astana 010008, Kazakhstan

Received 3 October 2016, appeared 22 May 2017 Communicated by Zuzana Došlá

Abstract. In local terms on finite and infinite intervals we obtain necessary and suf- ficient conditions for the conjugacy and disconjugacy of the following second order half-linear difference equation

∆(ρ_i|∆yi|^p−2∆yi) +vi|yi+1|^p−2yi+1=0, i=0, 1, 2, . . . ,

where 1 < p < _{∞, ∆yi} = y_i+1−y_i,{ρ_i}and {v_i}are sequences of positive and non- negative real numbers, respectively. Moreover, we study oscillation and non-oscillation properties of this equation.

Keywords: half-linear equation, conjugate, disconjugate, oscillation, non-oscillation, discrete Hardy inequality.

2010 Mathematics Subject Classification: 34C10, 39A10, 26D15.

1 Introduction

During the last several decades the oscillation properties of the half-linear difference equation have been intensively investigated. There is a lot of works devoted to this problem (see [2–4,7–12,16–22] and references given there).

We consider the following second order half-linear difference equation

∆(ρ_i|_∆yi|^p−2_∆yi) +v_i|y_i+1|^p−2y_i+1 =0, i=0, 1, 2, . . . , (1.1) where 1< p < _{∞, ∆yi} = y_i+1−y_i, {ρ_i}and{v_i}are sequences of positive and non-negative real numbers, respectively.

LetNandZbe the sets of natural and integer numbers, respectively.

Let us remind some notions and statements related to (1.1). Let m ≥ 0 be an integer number.

BCorresponding author. Email: o_ryskul@mail.ru

– If there exists a non-trivial solutiony = {y_i}of the equation (1.1) such thaty_m 6= 0 and ymym+₁<0, then we say that the solutionyhas a generalized zero on the interval(m,m+1]. – A non-trivial solutionyof the equation (1.1) is called oscillatory if it has infinite number of generalized zeros, otherwise it is called non-oscillatory.

– The equation (1.1) is called oscillatory if all its non-trivial solutions are oscillatory, oth- erwise it is called non-oscillatory.

– Due to Sturm’s separation theorem [18, Theorem 3], the equation (1.1) is oscillatory if one of its non-trivial solutions is oscillatory.

– The equation (1.1) is called disconjugate on the discrete interval [m,n], 0 ≤ m < n, (further just “interval”) if its any non-trivial solution has no more than one generalized zero on the interval(m,n+1]and its non-trivial solution ˜ywith the initial condition ˜ym =0 has not a generalized zero on the interval (m,n+1], otherwise it is called conjugate on the interval [m,n].

– The equation (1.1) is called disconjugate on the interval [m,∞) if for any n > m it is disconjugate on the interval[m,n].

The main properties of solutions of the equation (1.1) are described by so-called “round- about theorem” [18, Theorem 1] that gives two important methods [8] of the investigation of oscillation properties of the equation (1.1). Here we use one of these methods called “vari- ational method”. This method is based on the lemma given below that follows from the equivalence of the statements(i)and(ii)of Theorem 1 from [18].

Lemma A. Let 0 ≤ m < n < _∞, m and n be integers. The equation (1.1) is disconjugate on the interval[m,n]if and only if

∑

n i=m

(ρ_i|_∆yi|^p−v_i|y_i+1|^p)≥0 (1.2) for all non-trivial y={y_k}ⁿ_k=⁺_m¹, y_m =0and y_n+1=0.

Let y = {y_i}^∞_i=0 be a sequence of real numbers. Denote that suppy := {i ≥ 0 : y_i 6= 0}. Let 0 ≤ m < n ≤ _∞. Denote by ˚Y(m,n)the set of all non-trivial sequences of real numbers y = {y_i}^∞_i=0 such that suppy ⊂ [m+1,n], n < _{∞. When} n = _∞ we suppose that for any y there exists an integerk=k(y): m<k<_∞such that suppy⊂[m+1,k].

Lemma 1.1. Let0≤ m< n≤∞. The equation(1.1)is disconjugate on the interval[m,n]([m,n] = [m,∞)when n=∞) if and only if

∑

n i=m

v_i−1|y_i|^p ≤

∑

n i=m

ρ_i|_∆yi|^p (1.3)

for all y∈Y^˚(m,n), where v−1 =0.

Proof. Let the equation (1.1) be disconjugate on the interval[m,n],n< ∞. Then by LemmaA the condition (1.2) is valid for ally∈Y^˚(m,n). Since fromy_m =y_n+1=0 it follows that

∑

n i=m

v_i|y_i+1|^p=

n+₁ i=

∑

m+1

v_i−1|y_i|^p =

∑

n i=m

v_i−1|y_i|^p (1.4) and the sum in (1.2) is finite, then (1.2) is equivalent to the inequality (1.3). Letn=_{∞. Let an} arbitrary integer numbern₁be such thatm<n₁. Then the equation (1.1) is disconjugate on the interval[m,n₁] and by LemmaAthe inequality (1.3) is valid for ally ∈ Y^˚(m,n₁)⊂ Y^˚(m,∞)_.

Whence it appears that due to arbitrariness of the numbern₁the inequality (1.3) is correct for ally∈Y^˚(m,∞).

Inversely, let (1.3) be valid for ally ∈ Y^˚(m,n). For n < _∞ due to (1.4) we have that (1.2) holds. Then by LemmaAthe equation (1.1) is disconjugate on the interval [m,n]. For n =_∞ we have that (1.3) is correct for ally ∈ Y^˚(m,n₁) ⊂ Y^˚(m,∞)and arbitrary n₁ > m. Then by LemmaAthe equation (1.1) is disconjugate on the interval [m,n₁]. Due to arbitrariness of n₁ we have that the equation (1.1) is disconjugate on the interval[m,∞). The proof of Lemma1.1 is complete.

The dual statement to Lemma1.1is the following lemma.

Lemma 1.2. Let0≤ m< n≤∞. The equation(1.1)is conjugate on the interval[m,n]if and only if there existsy˜∈Y^˚(m,n)such that

∑

n i=m

v_i−1|y˜_i|^p>

∑

n i=m

ρ_i|_∆y˜_i|^p. (1.5)

The inequality (1.3) is the discrete Hardy inequality

∑

n i=m

v_i−1|y_i|^p ≤C

∑

n i=m

ρ_i|_∆y_i|^p, y∈Y^˚(m,n)_{, (1.6)} where 0<C≤1 andCis the least constant in (1.6).

A continuous analogue of the inequality (1.6) is investigated in many works (see e.g. [1], [14] and [15]). The resume of these works is given in [13]. Here we study the inequality (1.6) by methods different from the methods used for the continuous case in the mentioned works.

The paper is organized as follows. In Section 2 on the basis of the Hardy inequality (1.6) we find necessary and sufficient conditions for the conjugacy and disconjugacy of the equation (1.1) on the interval[m,n]. Moreover, in the same Section 2 on the basis of the first results we give necessary and sufficient conditions for the oscillation and non-oscillation of the equation (1.1). In Section 3 we present proofs of the results on the validity of the Hardy inequality (1.6).

Hereinafter “sequence” means a sequence of real numbers. The sums∑^mi=k form<k and

∑i∈_Ωfor emptyΩare equal to zero. Moreover, 1< p<_∞and ¹_p+ _p¹0 =1. The numbersm,n, t,s,xandzwith and without indexes are integers.

2 Main results

Let 0≤m<n≤ ∞. Let us introduce the notations Bp(m,n) = sup

m≤t≤s≤n

∑^s_i=⁻t¹v_i ∑^ti=mρ^1−p

0 i

1−p

+_∑ⁿ_i=sρ^1−p

0 i

1−p ,

α_p= inf

λ>1

λ^p(λ^p−1)

(λ−₁)^{p , γ0} =γ₀(p) = ^2p p+₁

2p p+₁

p 2p p−₁

p−1

,

γ₁ =γ₁(p) = p 2p

p+1 p

2p p−1

p−1

, γ₂ =γ₂(p) = pp^p p

p−1 p−1

.

Theorem 2.1. Let0≤ m< n≤_∞and∑^∞i=1ρ^1−p

0

i <_∞for n= ∞. The inequality(1.6)holds if and only if Bp(m,n)<∞. Moreover,

B_p(m,n)≤C≤2α_pB_p(m,n), (2.1) i.e., if the inequality(1.6)holds with C, then Bp(m,n)≤C; if Bp(m,n)<∞, then the inequality(1.6) holds with the estimate C≤2α_pB_p(m,n), where

α₂= ³

√5+7

√5−1 , γ₀< α_p <min{γ₁,γ₂, 4^p} for p6=2, (2.2) and C is the least constant in(1.6).

Remark 2.2. We do not use the condition∑^∞i=1ρ^1−p

0

i <_∞for the proof of Theorem2.1. How- ever, whenn = _∞ and∑^∞i=1ρ^1−p

0

i =∞, there exists an other method that estimates the least constantCin (1.6) better than (2.1). We turn back to this problem at the end of this section.

The proof of Theorem 2.1 is in the last Section 3. Now we study oscillation properties of the equation (1.1) that follow from Theorem 2.1, Lemmas 1.1 and 1.2. The relation (2.1) obviously gives the following corollary.

Corollary 2.3. Let 0 ≤ m < n ≤ _∞. If (1.3) holds, then B_p(m,n) ≤ 1; and if2α_pB_p(m,n) ≤ 1, then(1.3)holds.

Applying Corollary2.3, Lemmas1.1 and1.2 to the problem of the conjugacy and discon- jugacy of the equation (1.2) on the interval[m,n], we get the following theorem (let us remind that[m,n] = [m,∞)whenn=_∞).

Theorem 2.4. Let0≤m<n≤_∞and∑^∞i=1ρ^1−p

0

i <_∞for n=_{∞. Then}

(i) for the disconjugacy of the equation(1.1) on the interval [m,n]the condition Bp(m,n) ≤ 1 is necessary and the condition2α_pB_p(m,n)≤1is sufficient;

(ii) for the conjugacy of the equation(1.1)on the interval [m,n]the condition 2α_pB_p(m,n)> 1is necessary and the condition Bp(m,n)>1is sufficient.

Proof. If the equation (1.1) is disconjugate on the interval [m,n], then by Lemma 1.1 the in- equality (1.3) holds. Hence, by Corollary2.3we have Bp(m,n)≤1.

Inversely, if 2α_pB_p(m,n) ≥ 1, then by Corollary2.3 the inequality (1.3) holds. Hence, by Lemma1.1the equation (1.1) is disconjugate on the interval[m,n]. The proof of the statement (i)is complete.

Let the equation (1.1) be conjugate on the interval[m,n]. Then by Lemma1.2 there exists

˜

y ∈ Y^˜(m,n) such that (1.5) holds. This means that the inequality (1.6) is not valid for all y ∈ Y^˚(m,n)whenC ≤ 1, i.e., the least constantC in the inequality (1.6) must be larger than one. Then from (2.1) it follows that 2α_pB_p(m,n)>1.

Inversely, let Bp(_m,n) > 1. Then from (2.1) we have that the least constant C in the inequality (1.6) is larger than one, i.e., the inequality (1.3) is not valid for all y ∈ Y^˚(m,n). Therefore, there exists ˜y ∈ Y^˚(m,n) such that the inequality (1.5) holds. Consequently, by Lemma1.2 the equation (1.1) is conjugate on the interval [m,n]. The proof of the statement (ii)is complete. Thus, the proof of Theorem2.4is complete.

Corollary 2.5. Let the conditions of Theorem2.4hold. Then

(i) if there exist integers t, s: m≤t<s ≤n such that

s−1

∑

i=t

v_i >

∑

t i=m

ρ^1−p

0 i

!1−p

+

∑

n i=s

ρ^1−p

0 i

!1−p

, (2.3)

then the equation(1.1)is conjugate on the interval[m,n];

(ii) if the equation(1.1)is conjugate or disconjugate on the interval[m,n], then there exist integers t, s: m≤t <s≤ n such that

s−1

∑

i=t

v_i > ¹ 2αp





∑

t i=m

ρ^1−p

0 i

!1−p

+

∑

n i=s

ρ^1−p

0 i

!1−p

 or

s−1

∑

i=t

v_i ≤

∑

t i=m

ρ^1−p

0 i

!1−p

+

∑

n i=s

ρ^1−p

0 i

!1−p

, respectively.

In particular, from (2.3) we have the following simple condition of the conjugacy of the equation (1.1) on the interval[m,n]

n−1 i

∑

=m

v_i >ρ_m+ρ_n. (2.4)

The condition (2.4) coincides with the condition of Theorem 5 from [16].

Now we consider oscillation and non-oscillation properties of the equation (1.1).

Theorem 2.6. Let∑^∞i=1ρ^1−p

0

i < _∞.

(i) For the equation (1.1) to be non-oscillatory the condition B_p(m,∞) ≤ 1 for some m ≥ 0 is necessary and the condition2αpBp(n,∞)≤1for some n≥0is sufficient;

(ii) For the equation(1.1)to be oscillatory the condition2α_plim sup_m→∞B_p(m,∞)≥1is necessary and the conditionlim sup_m→∞B_p(m,∞)>1is sufficient.

Proof. The statement(i)directly follows from the statement (i)of Theorem2.4. Let us prove the statement(ii).

Let the equation (1.1) be oscillatory. Then there exists an integerk : 0≤k< _∞such that for allm>k the equation (1.1) is conjugate on the interval [m,∞). Therefore, by Theorem2.4we have that 2α_pB_p(m,∞)>1 for allm>k. Whence it follows that 2α_plim sup_m→∞B_p(m,∞)≥1.

Inversely, let lim sup_m→∞Bp(m,∞)> 1. Then there exists an increasing sequence of num- bers {m_k}^∞_k=1 ⊂ _N such thatm_k → _∞ for k → _∞ and B_p(m_k,∞) > 1 for all k ≥ 1. Then by Theorem 2.4 the equation (1.1) is conjugate on the interval [m_k,∞) _{for all} k ≥ 1, i.e., for all k≥ 1 there exists a non-trivial solution of the equation (1.1) that has at least two generalized zeros on the interval [m_k,∞). Hence, there exists a sequence {m˜_k} ⊂ {m_k} such that on all intervals[m˜_k, ˜m_k+1−₁]some non-trivial solution of the equation (1.1) has two zeros. Then by Sturm’s separation theorem [18, Theorem 2] there exists a non-trivial solution of the equation (1.1) that has at least one generalized zero on each interval [m_k,m_k+1−1], k ≥ 1. Thus, this solution of the equation (1.1) is oscillatory. The proof of Theorem2.6is complete.

From Theorem2.6 we have the following corollary.

Corollary 2.7. Let∑^∞i=1ρ^1−p

0 i <_∞.

(i) If there exist sequences of integers m_k, t_k and s_k, k≥ 1, such that0< m_k ≤ t_k <s_k, m_k → _∞ for k→_∞and

sk−1 i

∑

=t_k

v_i >

tk

i=

∑

m_k

ρ^1−p

0 i

!1−p

+

∑

∞ i=s_k

ρ^1−p

0 i

!1−p

(2.5) for sufficiently large k, then the equation(1.1)is oscillatory.

(ii) If the equation(1.1) is oscillatory, then there exist sequences of integers m_k, t_k and s_k, k ≥ 1, such that0<m_k ≤t_k <s_k, m_k →_∞for k →_∞and

s_k−1 i

∑

=tk

v_i > ¹ 2α_p





t_k i=

∑

mk

ρ^1−p

0 i

!1−p

+

∑

∞ i=sk

ρ^1−p

0 i

!1−p

.

In particular, from (2.6) under the conditions of Corollary 2.7 for the equation (1.1) to be oscillatory we have the following condition

sk−1 i

∑

=tk

v_i >ρ_tk+ρ_sk, t_k <s_k, t_k →_∞. (2.6) The condition (2.6) coincides with the condition of Corollary 2 from [16]. For example, from (2.6) fors_k−1= t_k we have

vt_k > ρt_k+ρ_tk+1, t_k →_∞. (2.7) Whence it follows that ifv_i = 0, i6= t_k,vt_k 6= 0 and (2.7) holds, then under the conditions of Corollary2.7the equation (1.1) is oscillatory.

In the case

∑

∞ i=1

ρ^1−p

0

i = _{∞ (2.8)}

oscillation properties of the equation (1.1) are studied in the work [3] on the basis of the following lemma.

Lemma 2.8. Let n = _∞ and∑^∞i=1ρ^1−p

0

i = ∞. Then the inequality (1.6)is equivalent to the discrete Hardy inequality

∑

∞ k=m

v_k

∑

k i=m

a_i

p

≤ C

∑

∞ k=m

ρ_k|a_k|^p (2.9)

for all sequences{a_k}^∞_k=m of real numbers. Moreover, the least constants in(1.6)and(2.9)coincide.

For complete presentation we prove Lemma2.8in the next Section 3 by a method different from those in [3].

The inequality (2.9) is well-studied. The main results on the inequality (2.9) are obtained in the works [5] and [6]. In [13] the summary of these results and estimates of the least constant Cin (2.9) are presented.

Let us use the following notations:

A₁(n) =

∑

∞ k=n

v_k

∑

n i=m

ρ^1−p

0 i

!1−p

, A₂(n) =

∑

n i=m

ρ^1−p

0 i

!−1

∑

n k=m

v_k

∑

k i=m

ρ^1−p

0 i

!p

,

A3(n) =

∑

∞ k=n

v_k

!1−p



∑

n i=m

ρ^1−p

0 i

∑

∞ k=i

v_k

!p⁰



p−1

. From [13, Theorem 7] we have the following theorem.

Theorem À. The inequality(2.9) holds if and only if A_i ≡ A_i(m) =sup_n≥mA_i(n)< _∞for i =1, i=2or i=3. Moreover, for the least constant C in(2.3)the following estimates

A₁≤C≤ p p

p−1 p−1

A₁, (2.10)

1

pA₂≤ C≤(p⁰)^pA₂ (2.11)

and

1 p⁰

p−1

A₃ ≤C≤ p^pA₃ (2.12)

hold.

In the case (2.8) by Lemma2.8 for the least constantCin the inequality (1.6) the estimates (2.10), (2.11) and (2.12) hold. Therefore, the following theorem is correct (see [3, Theorems 2 and 3]).

Theorem 2.9. Let(2.8)hold. Then

(i) the condition lim_m→_∞A_i(m) ≤ k_i for all i = 1, 2, 3 is necessary and the condition lim_m→_∞A_i(m) ≤ k_i for i = 1, i = 2 or i = 3 is sufficient for the equation (1.1) to be non-oscillatory;

(ii) the condition limm→_∞A_i(m) > K_i for all i = 1, 2, 3 is necessary and the condition lim_m→_∞A_i(m) > K_i for i = 1, i = 2 or i = 3 is sufficient for the equation (1.1) to be os- cillatory, where k₁ =1, k₂= p, k₃ = (p⁰)^p−1,K₁ = ¹_p ^p−_p^1p−1, K₂= _p¹0

p

and K₃= ¹_p^p. As an application of Theorem2.4let us consider the following example.

Example 2.10.

∆(k^αi|_∆yi|^p−2_∆yi) +k^βi|y_i+1|^p−2y_i+1 =0, (2.13) wherek,α,β∈_R,α> p−1 andk >1. From the conditionα> p−1 it follows that

∑

∞ i=m

k^α(1−p0) <_∞ for any m∈_N.

Denote that

a(t,s) = ^∑

s−1 i=t k^βi

∑^t_i=_mk^α(1−p0)1−p

+ _∑^∞_i=sk^α(1−p0)1−p. Then

B_p(m,∞) = _sup

m<t<s

a(t,s)_.

Proposition 2.11. Letα> p−1and1< p <∞. Then the equation(2.13)is (i) non-oscillatory forβ<α;

(ii) oscillatory forβ>α.

Proof. (i)Letβ<α. In the case β≥0 we have

s−1

∑

i=t

k^βi =k^βtk^(s−t)β−1 k^β−1 ≤ ^k

sβ

k^p−1,

∑

∞ i=s

k^α(1−p0)

!1−p

=k^αs(1−k^α(1−p0))^p−1 and

a(t,s)< ^k

(β−α)s

(k^β−1)(1−k^α(1−p0))^p−1 = b(s).

Since β−α < 0 and k > 1, then b(s) ↓ 0 fors → ∞. Therefore, there exists m ∈ _Nsuch that

2α_pB_p(m,∞)<2α_pb(m)<1. (2.14) In the case β<0 we have

s−1

∑

i=t

k^βi <k^βt(s−t) and

a(t,s)<k^βt s−t

k^αs(₁−k^α(1−p0))^p−1. Fors >tthe function ^s_k⁻αs^t has a maximum at the points=t+ ¹

αlnk. Therefore, sup

t<s

a(t,s)< ^k

(β−α)tk^−lnk1 αlnk(1−k^α(1−p0))^p−1.

In view of β−α < 0, the last inequality gives that (2.14) holds for some m ∈ _N, i.e., by Theorem2.4 the equation (2.13) is non-oscillatory.

(ii)Letβ>α. Then we have the following estimates

s−1

∑

i=t

k^βi >k^βt(s−t),

∑

t i=m

k^α(1−p0)i

!₁−p

+

∑

∞ i=s

k^α(1−p0)i

!₁−p

< k^αt+k^αs <2k^αs fors >t. Therefore,

a(t,s)> ^k

βt(s−t) 2k^αs . As above, the last inequality gives that

sup

t<s

a(t,s)> ¹ 2sup

t<s

k^βt(s−t) k^αs = ¹

2

k^(β−α)tk^−lnk1 αlnk .

Consequently, in view of β−α > 0, we have that Bp(m,∞) > 1 for all m ∈ _{N. Hence,} on the basis of Theorem2.4 the equation (2.13) is oscillatory. The proof of Proposition 2.11is complete.

3

Proof of Theorem2.1

Proof. Necessity. Let the inequality (1.6) hold with the least constantC > 0. Letα, t, s and β be integers satisfying the conditionm<α≤ t≤s≤ β<_n.

We construct a test sequencey= {y_k}in the following way

y_k =



















∑^k_i=⁻α¹−1ρ^1−p

0 i

∑^t_i⁻=¹α−1ρ^1−p

0 i

−1

, α≤ k≤t,

1, t≤k ≤s,

∑^β_i=_kρ^1−p

0 i

∑_i^β=sρ^1−p

0 i

−1

, s≤k ≤β,

0, m≤ k<α or β<k≤ n.

It is obvious that y∈Y^˚(m,n). Let us calculate∆yk.

∆yk =

















 ρ^1−p

0 k

∑_i^t=⁻α¹−1ρ^1−p

0 i

−1

, α−1≤k≤ t−1,

0, t≤ k≤s,

−ρ^1−p

0 k

∑_i^β=sρ^1−p

0 i

−1

, s≤ k≤ β,

0, m≤k<α−1 or β< k≤n.

Then

∑

n k=m

ρ_k|_∆yk|^p =

t−1 i=

∑

α−1

ρ^1−p

0 i

!1−p

+

∑

β i=s

ρ^1−p

0 i

!1−p

(3.1)

and n

i

∑

=m

v_i−1|y_i|^p ≥

∑

s i=t

v_i−1=

s−1 i=

∑

t−1

v_i. (3.2)

From (3.1), (3.2) and (1.6) we have

s−1 i=

∑

t−1

v_i ≤C





t−1 i=

∑

α−1

ρ^1−p

0 i

!1−p

+

∑

β i=s

ρ^1−p

0 i

!1−p

.

Due to independence of the left-hand side of the last estimate from α : m < α ≤ t and β: s≤β< nand independence of the constantCfromt,s: m<t≤ s<n, we have

s−1 i=

∑

t−1

v_i ≤ C





t−1 i

∑

=m

ρ^1−p

0 i

!1−p

+

∑

n i=s

ρ^1−p

0 i

!1−p



or

B_p(m,n)≤C. (3.3)

Sufficiency. Let B_p(m,n)< _{∞. Let} y ={y_i} ∈ Y^˚(m,n). Without loss of generality, we denote that y_i ≥ 0, i = 0, 1, 2, . . . Let λ > 1. For any k ∈ _Zwe define the set T_k ≡ T_k(λ) = {i ≥ m: y_i >λ^k}. Due to boundedness of the set{y_i}there exists a numberτ=τ(y,λ)∈_Zsuch that T_τ 6=_andT_τ+1=_{. Let∆}⁻T_k :=T_k−T_k+1. Then

[m,n] =

τ

[

k=−_∞

T_k =

τ

[

k=−_∞

∆⁻T_k. (3.4)

The definition ofT_k and the relation T_τ 6= give that T_k 6= for all k ≤ τ. Let k < τ. We present the set T_k in the form T_k = ^S_j[t^j_k,s^j_k], [t^j_k,s^j_k]^T[tⁱ_k,sⁱ_k] = for i 6= j. We denote that M_k^j = T_k+1T

[t^j_k,s^j_k], Ωk = {j: M_k^j 6= }. Moreover, for j∈ _Ωk we define x_k^j = minM_k^j and z_k^j =maxM^j_k. Thent^j_k ≤x^j_k,z^j_k ≤s^j_k and

T_k+1⊂ ^[

j∈_Ωk

[x_k^j,z_k^j], ∆⁻T_k ⊃ ^[

j∈_Ωk

[t^j_k,x_k^j −1]^[[z_k^j +1,s_k^j]. (3.5)

Lett^j_k <x^j_k. Theny

t^j_k−1 ≤λ^k,y

x^j_k >λ^k+1and

λ^k(λ−1) =λ^k+1−λ^k ≤y_xj k

−y_tj

k−1 =

x_k^j−1

∑

i=t^j_k

∆yi ≤





x_k^j−1

∑

i=t^j_k

ρ^1−p

0 i





1

p 



x_k^j−1

∑

i=t^j_k

ρ_i|_∆yi|^p





1 p

.

Whence it follows that

λ^pk





x_k^j−1

∑

i=t^j_k

ρ^1−p

0 i





1−p

≤ ¹

(λ−1)^p

x^j_k−1

∑

i=t^j_k

ρ_i|_∆yi|^p. (3.6)

Similarly, ifz_k^j <_sk^j_{, then}

λ^pk





s^j_k

∑

i=z_k^j

ρ^1−p

0 i





1−p

≤ ¹

(λ−1)^p

s^j_k

∑

i=z^j_k

ρ_i|_∆yi|^p. (3.7)

Letz_k^j = s_k^j. Theny

s^j_k =y

z^j_k >λ^k+1,y

z^j_k+1≤λ^k and

λ^k(λ−1)≤ y_zj k

−y_zj

k+1=−_∆y

z_k^j =

s^j_k

∑

i=z^j_k

(−_∆yi)

or

λ^pk





s^j_k

∑

i=z_k^j

ρ^1−p

0 i





1−p

≤ ¹

(λ−₁)^p

s^j_k

∑

i=z^j_k

ρ_i|_∆yi|^p. (3.8) Similarly, ift_k^j = x_k^j, then

λ^pk





x^j_k−1

∑

i=t_k^j−1

ρ^1−p

0 i





1−p

≤ ¹

(λ−1)^p

x^j_k−1

∑

i=t_k^j−1

ρ_i|_∆yi|^p. (3.9) Combining the inequalities (3.6), (3.7), (3.8) and (3.9), we write them in the following way

λ^pk





x_k^j−1

∑

i=˜t^j_k

ρ^1−p

0 i





1−p

≤ ¹

(λ−1)^p

x^j_k−1

∑

i=t˜^j_k

ρ_i|_∆yi|^p, (3.10)

λ^pk





s^j_k

∑

i=t˜_k^j

ρ^1−p

0 i





1−p

≤ ¹

(λ−1)^p

s^j_k

∑

i=z^j_k

ρ_i|_∆yi|^p, (3.11)

where ˜t_k^j = t_k^j fort^j_k <x_k^j and ˜t^j_k =t^j_k−_{1 for}t_k^j = x_k^j. FromB_p(m,n)<_{∞we have}

z^j_k−1

∑

i=x^j_k−1

v_i ≤ B_p(m,n)











x_k^j−1

∑

i=˜t^j_k

ρ^1−p

0 i





1−p

+





s^j_k

∑

i=z^j_k

ρ^1−p

0 i





1−p



. (3.12)

We will use the following notations:

ω⁺_k =ⁿj∈ _Ωk :t^j_k <x^j_k, z_k^j <s_k^jo

, ω_k,1=ⁿj∈_Ωk :t^j_k =x^j_k, z^j_k <s_k^jo , ω_k,2 =ⁿj∈ _Ωk :t^j_k <x^j_k, z_k^j =s_k^jo

, ω_k⁻=ⁿj∈_Ωk :t^j_k =x^j_k, z^j_k =s_k^jo ,

∆⁺_k,1 =ω_k⁺ [

ω_k,2, ∆⁺_k,2= ω⁺_k [

ω_k,1, ∆⁻_k,1=ω⁻_k [

ω_k,1, ∆⁻_k,2 =ω_k⁻ [

ω_k,2. It is obvious that Ωk =ω_k^+Sω_k,1^Sω_k,2^Sω_k⁻. The relation for ∆⁻T_k from (3.5) gives that

∆⁻T_k ⊃



 [

j∈_∆⁺_k,1

[t^j_k,x_k^j −₁]



 [



 [

j∈_∆⁺_k,2

[z_k^j +_1,s_k^j]



. (3.13)

Now we ready to estimate the left-hand side of the inequality (1.6). If ∆⁻T_k 6= , then λ^k <y_i ≤ λ^k+1 fori∈ _∆⁻T_k and

i∈

∑

_∆⁻Tk

v_i−1|y_i|^p≤ λ^p(k+1)

∑

i∈_∆⁻Tk

v_i−1. (3.14)

If∆⁻T_k =, then by assumption the inequality (3.14) is also correct.

Using (3.4), (3.5), (3.14) and the equalityλ^pk= (₁−λ^−p)_∑^k_t=−∞λ^pt, we get F≡

∑

n k=m

v_k−1|y_k|^p =

τ−1 k=−

∑

_∞

∑

i∈_∆⁻T_k+1

v_i−1|y_i|^p ≤

τ−1 k=−

∑

_∞

λ^q(k+2)

∑

i∈_∆⁻T_k+1

v_i−1

=λ^2p

τ−1 k=−

∑

_∞

λ^pk

∑

i∈_∆⁻Tk+1

v_i−1=λ^2p(1−λ^−p)

τ−1 k=−

∑

_∞

∑

i∈_∆⁻Tk+1

v_i−1

∑

k t=−_∞

λ^pt

≤λ^p(λ^p−1)

τ−1 k=−

∑

_∞

λ^pk

∑

j∈_Ωk z^j_k

∑

i=x_k^j

v_i−1= λ^p(λ^p−1)

τ−1 k=−

∑

_∞

λ^pk

∑

j∈_Ωk z^j_k−1

∑

i=x_k^j−1

v_i. (3.15)

Substituting (3.12) in (3.15) and using (3.10) and (3.11), we obtain

F≤λ^p(λ^p−1)B_p(m,n)

τ−1 k=−

∑

_∞

∑

j∈_Ωk





λ^pk





x^j_k−1

∑

i=t˜_k^j

ρ^1−p

0 i





1−p

+λ^pk





s^j_k

∑

i=z^j_k

ρ^1−p

0 i





1−p





≤ ^λ

p(λ^p−₁)

(λ−1)^{p Bp}(m,n)

τ−1 k=−

∑

_∞

∑

j∈_Ωk





x_k^j−1

∑

i=˜t^j_k

ρ_i|_∆y_i|^p+

s_k^j

∑

i=z_k^j

ρ_i|_∆y_i|^p



. (3.16)