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

In local terms on finite and infinite intervals we obtain necessary and sufficient conditions for the conjugacy and disconjugacy of the following second order half-linear difference equation ∆(ρi|∆yi|∆yi) + vi|yi+1|yi+1 = 0, i = 0, 1, 2, . . . , where 1 < p < ∞, ∆yi = yi+1 − yi, {ρi} and {vi} are sequences of positive and nonnegative real numbers, respectively. Moreover, we study oscillation and non-oscillation properties of this equation.


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 where 1 < p < ∞, ∆y i = y i+1 − y i , {ρ i } and {v i } are sequences of positive and non-negative real numbers, respectively.
Let N and Z be 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.
-If there exists a non-trivial solution y = {y i } of the equation (1.1) such that y m = 0 and y m y m+1 < 0, then we say that the solution y has a generalized zero on the interval (m, m + 1]. -A non-trivial solution y of 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, otherwise 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 ỹ with the initial condition ỹm = 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 "roundabout 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 "variational 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].
Let y = {y i } ∞ i=0 be a sequence of real numbers.Denote that supp y := {i ≥ 0 : When n = ∞ we suppose that for any y there exists an integer (1.5) The inequality (1.3) is the discrete Hardy inequality where 0 < C ≤ 1 and C is 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 ∑ m i=k for m < k and ∑ i∈Ω for empty Ω are equal to zero.Moreover, 1 < p < ∞ and 1 p + 1 p = 1.The numbers m, n, t, s, x and z with and without indexes are integers.

Main results
Let 0 ≤ m < n ≤ ∞.Let us introduce the notations   (1.5) holds.This means that the inequality (1.6) is not valid for all y ∈ Y(m, n) when C ≤ 1, i.e., the least constant C in the inequality (1.6) must be larger than one.Then from (2.1) it follows that 2α p B p (m, n) > 1.
Inversely, let B p (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 ỹ ∈ Y(m, n) such that the inequality (1.5) holds.Consequently, by Lemma 1.2 the equation (1.1) is conjugate on the interval [m, n].The proof of the statement (ii) is complete.Thus, the proof of Theorem 2.4 is complete.(ii) if the equation (1.1) is conjugate or disconjugate on the interval [m, n], then there exist integers t, s : In particular, from (2.3) we have the following simple condition of the conjugacy of the equation ( 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).
(i) For the equation (1.1) to be non-oscillatory the condition B p (m, ∞) ≤ 1 for some m ≥ 0 is necessary and the condition 2α p B p (n, ∞) ≤ 1 for some n ≥ 0 is sufficient; (ii) For the equation (1.1) to be oscillatory the condition 2α p lim sup m→∞ B p (m, ∞) ≥ 1 is necessary and the condition lim sup m→∞ B p (m, ∞) > 1 is sufficient.
Proof.The statement (i) directly follows from the statement (i) of Theorem 2.4.Let us prove the statement (ii).
Let the equation (1.1) be oscillatory.Then there exists an integer Inversely, let lim sup m→∞ B p (m, ∞) > 1.Then there exists an increasing sequence of numbers 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 { mk } ⊂ {m k } such that on all intervals [ mk , mk+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 Theorem 2.6 is complete.
From Theorem 2.6 we have the following corollary.
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, 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 The condition (2.6) coincides with the condition of Corollary 2 from [16].For example, from (2.6) for s k − 1 = t k we have Whence it follows that if v i = 0, i = t k , v t k = 0 and (2.7) holds, then under the conditions of Corollary 2.7 the equation (1.1) is oscillatory.
In the case oscillation properties of the equation (1.1) are studied in the work [3] on the basis of the following lemma.
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 Lemma 2.8 in 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 C in (2.9) are presented.
Let us use the following notations: From [13,Theorem 7] we have the following theorem.
Theorem À.The inequality (2.9) holds if and only if 3) the following estimates (2.10) hold.
In the case (2.8) by Lemma 2.8 for the least constant C in 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 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 oscillatory, where As an application of Theorem 2.4 let us consider the following example.
Proof.(i) Let β < α.In the case β ≥ 0 we have In the case β < 0 we have For s > t the function s−t k αs has a maximum at the point s = t + 1 α ln k .Therefore, In view of β − α < 0, the last inequality gives that (2.14) holds for some m ∈ N, i.e., by Theorem 2.4 the equation (2.13) is non-oscillatory.
(ii) Let β > α.Then we have the following estimates Consequently, in view of β − α > 0, we have that B p (m, ∞) > 1 for all m ∈ N. Hence, on the basis of Theorem 2.4 the equation (2.13) is oscillatory.The proof of Proposition 2.11 is complete.

Proof of Theorem 2.1
Proof.Necessity.Let the inequality (1.6) hold with the least constant C > 0. Let α, t, s and β be integers satisfying the condition m < α ≤ t ≤ s ≤ β < n.
We construct a test sequence y = {y k } in the following way It is obvious that y ∈ Y(m, n).Let us calculate ∆y k .
Due to independence of the left-hand side of the last estimate from α : m < α ≤ t and β : s ≤ β < n and independence of the constant C from t, s : m < t ≤ s < n, we have Without loss of generality, we denote that y i ≥ 0, i = 0, 1, 2, . . .Let λ > 1.For any k ∈ Z we define the set Due to boundedness of the set {y i } there exists a number τ = τ(y, λ) ∈ Z such that T τ = and The definition of T k and the relation T τ = give that T k = for all k ≤ τ.Let k < τ.We present the set T k in the form .
Whence it follows that Let Similarly, if Combining the inequalities (3.6), (3.7), (3.8) and (3.9), we write them in the following way where We will use the following notations: It is obvious that Now we ready to estimate the left-hand side of the inequality (1.6).If Due to (3.13) we have ∆ + k,2 , there exist integers The left-hand side of this inequality does not depend on λ > Then there exists a point λ p such that and for p = 2 the estimate holds.

Corollary 2 . 5 .
Let the conditions of Theorem 2.4 hold.Then

Corollary 2.3. Let
The inequality(1.6)holdsifand only if B p (m, n) < ∞.Moreover, B p (m, n) ≤ C ≤ 2α p B p (m, n), 0 ≤ m < n ≤ ∞.If (1.3) holds, then B p (m, n) ≤ 1; and if 2α p B p (m, n) ≤ 1, then (1.3) holds.Applying Corollary 2.3, Lemmas 1.1 and 1.2 to the problem of the conjugacy and disconjugacy of the equation (1.2) on the interval [m, n], we get the following theorem (let us remind that Hence, by Lemma 1.1 the equation (1.1) is disconjugate on the interval [m, n].The proof of the statement (i) is complete.
for the disconjugacy of the equation (1.1) on the interval [m, n] the condition B p (m, n) ≤ 1 is necessary and the condition 2α p B p (m, n) ≤ 1 is sufficient; (ii) for the conjugacy of the equation (1.1) on the interval [m, n] the condition 2α p B p (m, n) > 1 is necessary and the condition B p (m, n) > 1 is sufficient.Proof.If the equation (1.1) is disconjugate on the interval [m, n], then by Lemma 1.1 the inequality (1.3) holds.Hence, by Corollary 2.3 we have B p (m, n) ≤ 1. Inversely, if 2α p B p (m, n) ≥ 1, then by Corollary 2.3 the inequality (1.3) holds.Let the equation (1.1) be conjugate on the interval [m, n].Then by Lemma 1.2 there exists ỹ 1. Hence, taking infimum with respect to λ > 1 in the right-hand side, we have C ≤ 2α p B p (m, n) for the least constant C. The last estimate together with (3.3) gives (2.1).The estimate (2.2) follows from Lemma 3.1 proved below.The proof of Theorem 2.1 is complete.Lemma 3.1.Let f