HALF–LINEAR DISCRETE OSCILLATION THEORY

PAVEL ŘEHÁK

Abstract. Oscillatory properties of the second order half-linear difference equation

Δ(r_{k}|Δy_{k}|^{α−2}Δy_{k}) + p_{k}|y_{k+1}|^{α−2}y_{k+1} = 0,

where α > 1, are investigated. A particular attention is devoted to the connection with oscillation theory of its continuous counterpart, half–linear differential equation and also with the theory of linear differential and linear difference equations. We present not only the overview of the existing results but we also establish new oscillation and nonoscillation criteria.

1. Introduction

The aim of this contribution is to present a survey of the recent results of the oscillation theory of the second order half–linear difference equation

Δ(r_{k}Φ(Δy_{k})) + p_{k}Φ(y_{k+1}) = 0,  (1)

where p_{k} and r_{k} are real-valued sequences with r_{k} ≠ 0 and Φ(y) := |y|^{α−1} sgny = |y|^{α−2}y, Φ(0) = 0, with α > 1. We will discuss the application of various methods in the oscillation theory of (1) which come from theory of linear differential equations. It is known, see [15], that the basic oscillatory properties of half–linear difference equation (1) (≡ HLΔE) are essentially the same as those of the linear difference equation (≡ LΔE)

Δ(r_{k}Δy_{k}) + p_{k}y_{k+1} = 0,  (2)

which is the special case of (1) for α = 2. Moreover, there exists a considerable similarity between the theories of difference and differential equations. This means that in this connection we are also interested in the second order half–linear differential equation (≡ HLDE)

(r(t)Φ(y′))′ + p(t)Φ(y) = 0,  (3)

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 24, p. 1 which was very frequently investigated in the last years, see e.g. [9, 2, 3, 4, 5, 7, 8, 10, 11, 12], and its special case, namely the well–known Sturm–Liouville linear differential equation (≡ LDE)

1991 Mathematics Subject Classification. 39A10.

Key words and phrases. Half-linear difference equation, Roundabout Theorem, Sturmian theory, Riccati technique, variational principle, reciprocity principle, discrete oscillation and nonoscillation criteria.

Supported by the Grants No. 201/98/0677 and No. 201/99/0295 of the Czech Grant Agency (Prague)

and No. 384/1999 (Development of Czech Universities).

This paper is in final form and no version of it will be submitted for publication elsewhere.

(r(t)y′)′ + p(t)y = 0,  (4)

where the functions r and p are mostly considered to be continuous with r(t) > 0. Note that a natural requirement for the sequence r_{k} in (1), (2) is only to be nonzero. There exist several viewpoints which provide an explanation of this discrepancy between the continuous and the discrete cases, see e.g. [1]. It seems to be natural to require the assumption r_{k} ≠ 0 also for equation (1).

Thus, our approach to the investigating of the qualitative properties of (1) can look as follows:

LDE ⟶^{HL} HLDE
Δ↓  ↓^{Δ}
LΔE ⟶^{HL} HLΔE,
HL ≡ generalization in a half–linear sense,
Δ ≡ discretization,

which means that we motivate ourselves by the linear continuous theory that offers many interesting topics for the discretization and the extension to the half–linear discrete case. This approach brings two types of interesting problems:

1) The first type relates to the discretization. The techniques of proofs that are needed in the discrete case are often different from the continuous case and also more complicated. This is due to the absence of the chain rule for the difference of the composite sequences and also due to some other specific properties of difference calculus.

2) The second type of problems is related to the extension to the half–linear case (both, continuous and discrete). There exist certain limitations in the use of the linear approach to the investigating of half–linear equations. These limitations are first of all the absence of transformation theory similar to that for linear equations or the impossibility of the extension of the so–called Casoratian to the half–linear discrete case. Casoratian is the discrete counterpart of Wronskian from the theory of linear differential equations.

Being motivated by the linear continuous case, we will show that one can extend some basic methods and results of oscillation theory of (4) to equation (1). In particular, this is the discrete half–linear version of the so called Roundabout theorem, see Section 2, which provides not only the Sturm type separation and comparison theorems but also two important tools for the investigating of oscillation and nonoscillation of (1). These methods are the Riccati technique and the variational principle, see Section 3. The reciprocity principle is also available in the oscillation theory of (1) and

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 24, p. 2 it is discussed in Section 3. Sections 4 and 5 contain an application of these methods, namely oscillation and nonoscillation criteria for (1).

At the end of this introductory section we recall some important concepts with some comments.

Definition 1. An interval (m, m + 1] is said to contain the generalized zero of a solution y of (1), if y_{m} ≠ 0 and r_{m}y_{m}y_{m+1} ≤ 0.

Definition 2. A nontrivial solution of (1) is called oscillatory if it has infinitely many generalized zeros. In view of the fact that Sturm type Separation Theorem extends to (1) (this follows from Theorem 1 – hereafter mentioned), we have the following equivalence: Any solution of (1) is oscillatory if and only if every solution of (1) is oscillatory. Hence we can speak about oscillation or nonoscillation of equation (1).

Note that authors who studied equation (2) with similar definition of generalized zero as above (but with the assumption y_{m}y_{m+1} ≤ 0 instead of r_{m}y_{m}y_{m+1} ≤ 0 – this lead to the Hartman’s definition of the so called node, had problems with Sturmian theory since there exist simple examples such as the Fibonacci recurrence relation, where one solution seems to be oscillatory while another is nonoscillatory. This fact does not occur in the continuous case and our definition of generalized zero brings these exceptional cases into the general theory. On the other hand, it is obvious that the presence of the sequence r_{k} in Definition 1 is the result of the assumption r_{k} ≠ 0 for equation (2) (or for (1)). Note that we cannot rewrite the Fibonacci equation into the self–adjoint form (2) with a positive r_{k}.

2. Roundabout Theorem

In this section we present the half–linear discrete version of the so called Roundabout Theorem (for its linear continuous and discrete version see e.g. [14] and [1], respectively, the proofs of some (nontrivial) parts of its half–linear continuous version can be found in [8, 12]) that is of the basic importance in the oscillation theory of (1). This theorem shows the connections between such concepts as disconjugacy of (1), existence of a solution of the generalized Riccati difference equation and positive definiteness of certain functional (see Definition 3 below), and therefore it provides powerful tools (see Section 3) for the investigation of oscillatory properties of equation (1). Note that by the term generalized Riccati difference equation we mean the following equation

Δw_{k} + p_{k} + S(w_{k}, r_{k}) = 0,  (5)

or, equivalently,

w_{k+1} = −p_{k} + S̄(w_{k}, r_{k}),

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 24, p. 3 where

S(w_{k}, r_{k}) = w_{k} (1 − \frac{r_{k}}{Φ(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k}))}) ,  (6)

S̄(w_{k}, r_{k}) = w_{k}−S(w_{k}, r_{k}) and the function Φ^{−1} being the inverse of Φ, i.e., Φ^{−1}(x) = |x|^{β−1} sgnx, where β is the conjugate number of α, i.e., 1/α + 1/β = 1. Equation (5) is related to (1) by the Riccati type substitution

w_{k} = \frac{r_{k}Φ(Δy_{k})}{y_{k}}.

Now we define and recall some important concepts

Definition 3. Equation (1) is said to be disconjugate on [m, n] provided any solution of this equation has at most one generalized zero on (m, n + 1] and the solution ỹ satisfying ỹ_{m} = 0 has no generalized zeros on (m, n + 1]. Define a class U of the so called admissible sequences by

U = {ξ | ξ : [m, n + 2] ⟶ ℝ such that ξ_{m} = ξ_{n+1} = 0}.

Then define an ‘α-degree” functional F on U by

F(ξ; m, n) = ∑_{k=m}^{n}[r_{k}|Δξ_{k}|^{α} − p_{k}|ξ_{k+1}|^{α}].

We say F is positive definite on U provided F(ξ) ≥ 0 for all ξ ∈ U and F(ξ) = 0 if and only if ξ = 0.

Theorem 1 (Roundabout Theorem, [15]). The following statements are equivalent: (i) Equation (1) is disconjugate on [m, n]. (ii) Equation (1) has a solution y without generalized zeros on [m, n + 1]. (iii) The generalized Riccati difference equation (5) has a solution w_{k} on [m, n] with

r_{k} + w_{k} > 0. (iv) F is positive definite on U.

Proof. The proof of this theorem can be found in [15]. Note only that we use here the usual “roundabout method” that (i) ⇒ (ii) ⇒ (iii) ⇒ (iv) ⇒ (i) and for the proof of the implication (iii) ⇒ (iv) we use the generalized Picone identity.

3. Methods of half–linear discrete oscillation theory

In this section we describe three methods which are available in the oscillation theory of (1).

The first method is the so called Riccati technique. This method uses the following idea: Suppose (by contradiction) that equation (1) is nonoscillatory, i.e., there exists a solution y_{k} without generalized zeros (for k sufficiently large) and therefore there

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 24, p. 4 exists a solution of equation (5) with r_{k} + w_{k} > 0 according to Theorem 1 and vice versa, the existence of a solution of (5) with r_{k} + w_{k} > 0 in a neighborhood of infinity guarantees nonoscillation of (1). Actually, according to the following lemma, for nonoscillation of (1), it is sufficient to find a solution of the generalized Riccati difference inequality

Δw_{k} + p_{k} + S(w_{k}, r_{k}) ≤ 0  (7)

satisfying r_{k} + w_{k} > 0. However, this is not included in the Roundabout Theorem.

Lemma 1 ([6]). Equation (1) is nonoscillatory if and only if there exists a sequence w_{k} satisfying the inequality (7) with r_{k} + w_{k} > 0 for k ≥ m with suitable m ∈ ℕ.

Another method is the variational principle which is based on the following fact. Equation (1) is nonoscillatory if and only if the functional F is positive definite on the class U. But (1) is nonoscillatory if and only if it is disconjugate on a certain half–bounded discrete interval and therefore we have the equivalence (i) ⇔ (iv) from Theorem 1.

The third method which is available in the half–linear discrete oscillation theory is the reciprocity principle. Here we suppose that r_{k} > 0, p_{k} > 0. If we denote u_{k} = r_{k}Φ(Δy_{k}), where y is a solution of (1) then (as one can easily verify) u_{k} satisfies the reciprocal equation

Δ(p_{k}^{1−β}Φ_{β}(Δu_{k})) + r_{k+1}^{1−β}Φ_{β}(u_{k+1}) = 0,  (8)

where Φ_{β}(x) = |x|^{β−2}x and β is the conjugate number of α, i.e., 1/α + 1/β = 1. Conversely, if y_{k} = p_{k−1}^{1−β}Φ_{β}(Δu_{k−1}), where u_{k} is a solution of (8), then y_{k} solves the original equation (1). Since the discrete version of the Rolle mean value theorem holds, we have the following equivalence: (1) is oscillatory [nonoscillatory] if and only if (8) is oscillatory [nonoscillatory]. Indeed, if y_{k} is an oscillatory solution of (1) then its difference and hence also u_{k} = r_{k}Φ(Δy_{k}) oscillates. Conversely, if u_{k} oscillates then y_{k} = p_{k−1}^{1−β}Φ_{β}(Δu_{k−1}) oscillates as well.

Remark 1. Note that important concept of the oscillation theory of linear difference equations is the concept of recessive solution (the so called principal solution in the “continuous terminology”). Since the construction of principal solution and its application in oscillation theory of (3) has been already successfully made, see e.g. [2, 3, 7, 13], we would like to construct the recessive solution of half-linear difference equation and possibly apply it in oscillation theory of (1).

4. Oscillation criteria

In this section we present some already existing oscillation criteria for (1), where r_{k} > 0 for large k. We will also give one new criterion and mention some comments

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 24, p. 5 on the proofs and related open problems. The following notation will be used

P_{k} = ∑^{k} p_{j}, P̃_{k} = ∑_{j=k}^{∞} p_{j}, R_{k} = ∑^{k} r_{j}^{1−β}, R̃_{k} = ∑_{j=k}^{∞} r_{j}^{1−β},  (9)

β is the conjugate number of α, i.e., 1/α + 1/β = 1.

We start with the half–linear version of well–known criterion.

Theorem 2 (Leighton-Wintner type oscillation criterion, [15]). Suppose that

R_{∞} = ∞  (10)
P_{∞} = ∞.  (11)

Then (1) is oscillatory.

Proof. This statement is proved via the variational principle. Note only that according to Theorem 1, it is sufficient to find for any K ≥ m a sequence y satisfying y_{k} = 0 for k ≤ K and k ≥ N + 1, where N > K, such that

F(y, K, N) = ∑_{k=K}^{N}[r_{k}|Δy_{k}|^{α} − p_{k}|y_{k+1}|^{α}] ≤ 0.

Remark 2. In order to compare the similarity between some qualitative properties of our equation (1) and of the above mentioned equations we present here also the sufficient conditions of the same type for equations (2), (4) and (3), respectively. They are ∑^{∞} r_{j}^{−1} = ∞ = ∑^{∞} p_{k}, ∫^{∞} r^{−1}(t)dt = ∞ = ∫^{∞} p(t)dt and ∫^{∞} r^{1−β}(t)dt = ∞ = ∫^{∞} p(t)dt, respectively.

In the case when

lim_{k→∞} ∑^{k} p_{j} is convergent,  (12)

we can use the following criterion which is also proved via the variational principle.

Theorem 3 (Hinton-Lewis type oscillation criterion, [15]). Suppose that the conditions (10) and (12) hold and

lim_{k→∞} R_{k}^{α−1}P̃_{k} > 1.  (13)

Then (1) is oscillatory.

Remark 3. In the above criteria, equation (1) is essentially viewed as a perturbation of the nonoscillatory equation

Δ(r_{k}Φ(Δy_{k})) = 0.

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 24, p. 6 In the continuous case it is known (see e.g. [4, 5, 11]) that one can investigate equation (3) not as a perturbation of the nonoscillatory equation

(r(t)Φ(y′))′ = 0

but as a perturbation of a more general equation

(r(t)Φ(y′))′ + p̃(t)Φ(y) = 0,

for example, of the generalized Euler equation

(r(t)Φ(y′))′ + λt^{−α}Φ(y) = 0,

where λ = ((α − 1)/α)^{α} is the so–called critical constant. We conjecture that it is possible to prove some stronger criteria for (1), e.g., the assumption (13) can be replaced by the weaker one

lim inf_{k→∞} R_{k}^{α−1}P̃_{k} > \frac{1}{α} (\frac{α − 1}{α})^{α−1}  (14)

(note we know that lim in Theorem 3 can be replaced by lim inf). From many other open problems we mention here e.g. the following: What are the additional conditions guaranteeing oscillation of (1) if (14) does not hold? Such types of criteria for equation (3) (with r(t) ≡ 1) are presented in [9]. See also Remark 6 for some other relates problems.

The next criterion is the “complementary” case of Hinton–Lewis type criterion – in the sense of the following convergence

∑^{∞} r_{j}^{1−β} < ∞.  (15)

Here we suppose not only r_{k} > 0 but also p_{k} > 0 for large k.

Theorem 4. Suppose that (15) holds and

lim_{k→∞} R̃_{k+1}^{α−1}P_{k} > 1.  (16)

Then (1) is oscillatory.

Proof. We will use the reciprocity principle. From (16) one can observe that

∑^{∞} p_{j}^{(1−β)(1−α)} = ∑^{∞} p_{j} = ∞.

Therefore according to Theorem 3, equation (8) is oscillatory if

lim_{k→∞} (∑^{k} p_{j}^{(1−β)(1−α)})^{β−1}(∑_{j=k}^{∞} r_{j+1}^{1−β}) > 1,

which is equivalent to (16). From here (1) is also oscillatory.

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 24, p. 7

5. Nonoscillation criteria

This section contains nonoscillation criteria for (1), where, again, it is supposed r_{k} > 0 for large k. The first two are already proved “nonoscillatory supplements” of the meanwhile unproved Theorem 3 (but with (14) instead of (13)). Criteria that are included in Theorems 8 – 11 are new, some of them even in the linear case. Note we still use notation (9).

We start with the half–linear discrete version of quite well–known criterion for equation (4). The proof of this theorem is based on the variational principle. Note that the discrete “half–linear” version of a Wirtinger type inequality is used there.

Theorem 5 ([6]). Suppose that (10) holds, ∑^{∞} p_{j}^{+} < ∞, p^{+} := max{0, p} and

φ_{N} := (sup_{k≥N} \frac{R_{k}}{R_{k−1}} )^{α(α−1)}< ∞, ψ_{N} := sup_{k≥N} (\frac{r_{k}}{r_{k−1}} )^{1−β} < ∞.

Further suppose that

0 < lim sup_{N→∞}(1 + ψ_{N})^{α−1}φ_{N} =: Ψ < ∞.  (17)

If

lim sup_{k→∞} R_{k−1}^{α−1} ∑_{j=k}^{∞} p_{j}^{+} < \frac{1}{αμ^{α−1}} (\frac{α − 1}{α})^{α−1} \frac{1}{Ψ},  (18)

where

μ := \{ \begin{matrix}sup_{t>s>0} \frac{1}{t−s} [Φ^{−1} (\frac{t^{α}−s^{p}}{α(t−s)}) − s] , & α ≥ 2, \\ sup_{t>s>0} \frac{1}{t−s} [t − Φ^{−1} (\frac{t^{α}−s^{α}}{α(t−s)})] , & α ≤ 2, \end{matrix}

then (1) is nonoscillatory.

The Riccati technique is used in the proof of the following theorem. More precisely, we use Lemma 1.

Theorem 6 ([6]). Suppose that (10) and (12) hold and

lim_{k→∞} \frac{r_{k}^{1−β}}{R_{k−1}} = 0.  (19)

If

lim sup_{k→∞} R_{k−1}^{α−1}P̃_{k} < \frac{1}{α} (\frac{α − 1}{α})^{α−1}  (20)

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 24, p. 8 and

lim inf_{k→∞} R_{k−1}^{α−1}P̃_{k} > −\frac{2α − 1}{α} (\frac{α − 1}{α})^{α−1}  (21)

then (1) is nonoscillatory.

Remark 4. 1) The above two criteria are not in fact completely the supplements of Theorem 3 (with (14)) since in contrast to the linear cases and half–linear continuous case we require additional restrictions on the sequence r_{k}, namely (17) and (19). However, we have an open problem in this connection: Is there really a need of these additional conditions?

2) We can see that the use of the different methods gives the different results in this case. The condition (17) is weaker than (19) but the constant at the right–hand side of (18) is less than constant in (20).

The following theorem complements the previous statement in the sense of the “complementary” case (15).

Theorem 7 ([6]). Suppose that (15) holds and

lim_{k→∞} \frac{r_{k}^{1−β}}{R̃_{k}} = 0.  (22)

If

lim sup_{k→∞} R̃_{k}^{α−1}P_{k−1} < \frac{1}{α} (\frac{α − 1}{α})^{α−1}

and

lim inf_{k→∞} R̃_{k}^{α−1}P_{k−1} > −\frac{2α − 1}{α} (\frac{α − 1}{α})^{α−1}  (23)

then (1) is nonoscillatory.

Next, denote by θ(C^{[0]}) the greatest root of the equation

|x|^{\frac{1}{β}} + x + C^{[0]} = 0,

for certain C^{[0]} which will be determined by the next statement and set A_{k} = R_{k−1}^{α−1}P̃_{k}. In the case when (21) does not hold, we can use the following criterion which completes Theorem 6.

Theorem 8. Suppose that (10), (12) and (19) hold. If

lim sup_{k→∞} A_{k} < [θ (lim inf_{k→∞} A_{k})]^{\frac{1}{β}} − θ (lim inf_{k→∞} A_{k})

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 24, p. 9 and

−∞ < lim inf_{k→∞} A_{k} ≤ −\frac{2α − 1}{α} (\frac{α − 1}{α})^{α−1}

then (1) is nonoscillatory.

Proof. The proof is similar to that of the continuous case, see [9].

Remark 5. It is clear that by the same way as above one can show there exists similar complementary case (i.e., when (23) fails to hold) also for Theorem 7.

The following statement is essentially a generalization of the criterion presented in [16] and at the same time the discretization of the criterion presented in [9]. First we introduce some notation. Set

B_{k} = R_{k−1}^{−1} ∑^{k−1} (R_{j−1}^{α}p_{j}) .

Let

λ(α) = x^{[0]} + \frac{2α − 1}{α} (\frac{α − 1}{α})^{α−1} ,

where x^{[0]} is the least root of equation

(α − 1)|x|^{β} + αx + \frac{2α − 1}{α} (\frac{α − 1}{α})^{α−1} = 0.  (24)

Theorem 9. Suppose that (10), (12) and (19) hold. If

lim sup_{k→∞} B_{k} < (\frac{α − 1}{α})^{α}  (25)

and

lim inf_{k→∞} B_{k} > λ(α),  (26)

then (1) is nonoscillatory.

Proof. We show that the generalized Riccati difference inequality (7) has a solution w with r_{k} + w_{k} > 0 in a neighbourhood of infinity. Set

w_{k} = R_{k−1}^{1−α}(C − B_{k}).

By the Lagrange mean value theorem we have

ΔR_{k−1}^{1−α} = (1 − α)r_{k}^{1−β}η_{k}^{−α}

and

ΔR_{k−1}^{α} = αr_{k}^{1−β}μ_{k}^{α−1},

where η_{k} and μ_{k}, respectively, are between R_{k−1} and R_{k}. Similarly,

w_{k} [Φ(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k})) − r_{k}] = (α − 1)|ξ_{k}|^{α−2}|w_{k}|^{β},

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 24, p. 10 where ξ_{k} is between Φ^{−1}(r_{k}) and Φ^{−1}(r_{k}) + Φ^{−1}(w_{k}), hence

r_{k}^{β−1} − |w_{k}|^{β−1} ≤ ξ_{k} ≤ r_{k}^{β−1} + |w_{k}|^{β−1}.

Let

C = \frac{2α − 1}{α} (\frac{α − 1}{α})^{α−1} .

Then

\frac{|w_{k}|}{r_{k}} = \frac{|R_{k−1}^{1−α}(C − B_{k})|}{r_{k}} = (\frac{r_{k}^{1−β}}{R_{k−1}} )^{α−1} |C − B_{k}| → 0

as k → ∞ according to (19) and hence r_{k} + w_{k} = r_{k}(1 + w_{k}/r_{k}) > 0 for large k. Further, assumptions (25) and (26) imply the existence of ε_{1} > 0 such that

λ(α) + ε_{1} < B_{k} < (\frac{α − 1}{α})^{α} − ε_{1}

for k sufficiently large, say k ≥ K_{1}. From here,

λ(α) − C + ε_{1} = x^{[0]} < B_{k} − C < x̄^{[0]} − ε̄_{1},  (27)

where x̄^{[0]} = − ((α − 1)/α)^{α−1} is the greatest root of equation (24). Therefore, from (27) it follows that there exists ε_{2} > 0 such that

(α − 1)|B_{k} − C|^{β} + α(B_{k} − C) + C + ε_{2} < 0

for k ≥ K_{1}, and, finally, this implies the existence of ε > 0 such that

(α − 1)|C − B_{k}|^{β}(1 + ε) − (α − 1)C(1 − ε) + αB_{k}(1 + ε sgnB_{k}) < 0.

Multiplying this inequality by r_{k}^{1−β}R_{k−1}^{−α} we obtain

(α − 1)r_{k}^{1−β}R_{k−1}^{−α}|C − B_{k}|^{β}(1 + ε) − (α − 1)r_{k}^{1−β}CR_{k−1}^{−α}(1 − ε) +

+ αr_{k}^{1−β}B_{k}R_{k−1}^{−α}(1 + ε sgnB_{k}) < 0 (28)

for k ≥ K_{1}. We can suppose that ε is at the same time such that we may add the term p_{k} − p_{k}(1 − ε sgnp_{k}) to the left–hand side of (28).

Now, for a given ε > 0 there exists K_{2} ∈ ℕ such that we have r_{k} > |w_{k}|,

(\frac{R_{k−1}}{η_{k}} )^{α} > 1 − ε, (sgnp_{k})(\frac{R_{k−1}}{R_{k}} )^{α}> (sgnp_{k})(1 − ε sgnp_{k}),

(sgnB_{k})(1 + ε sgnB_{k}) > (sgnB_{k}) (\frac{μ_{k}}{R_{k}} )^{α}= \frac{(sgnB_{k})R_{k−1}^{α}μ_{k}^{α−1}}{R_{k}^{α}R_{k−1}^{α−1}}

and

\frac{|ξ_{k}|^{α−2}r_{k}^{β−1}}{(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k}))^{α−1}} < \begin{matrix}(1 + Φ^{−1}(|w_{k}|/r_{k})) & α−2 \\ (1 + Φ^{−1}(w_{k}/r_{k})) & α−1 \end{matrix} < 1 + ε

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 24, p. 11 for k ≥ K_{2}. Using the above estimates, we obtain from (28)

0 > (α − 1)r_{k}^{1−β}R_{k−1}^{−α}|C − B_{k}|^{β}(1 + ε) −

−(α − 1)r_{k}^{1−β}CR_{k−1}^{−α}(1 − ε) +

+αr_{k}^{1−β}B_{k}R_{k−1}^{−α}(1 + ε sgnB_{k}) + p_{k} − p_{k}(1 − ε sgnp_{k})

> −(α − 1)r_{k}^{1−β}Cη_{k}^{−α} − p_{k} \frac{R_{k−1}^{2α}}{R_{k−1}^{α}R_{k}^{α}} +

+\frac{αr_{k}^{1−β}μ_{k}^{α−1}B_{k}}{R_{k−1}^{α−1}R_{k}^{α}} + p_{k} +

+\frac{(α − 1)R_{k−1}^{−α}|C − B_{k}|^{β}|ξ_{k}|^{α−2}}{(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k}))^{α−1}}

= Δw_{k} + p_{k} + \frac{(α − 1)|ξ_{k}|^{α−2}|w_{k}|^{β}}{(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k}))^{α−1}}
= Δw_{k} + p_{k} + w_{k} (1 − \frac{r_{k}}{(Φ^{−1}(r_{k}) + Φ^{−1}(w_{k}))^{α−1}} ) ,

which means that inequality (7) has a solution w with r_{k} + w_{k} > 0 in a neighbourhood of infinity and hence equation (1) is nonoscillatory by Lemma 1.

Denote

B̃_{k} = R̃_{k}^{−1} ∑_{k}^{∞} (R̃_{j}^{α}p_{j}) .

The following statement complements the previous one in the sense of the “complementary” case (15).

Theorem 10. Suppose that (15) and (22) hold. If

lim sup_{k→∞} B̃_{k} < (\frac{α − 1}{α})^{α}  (29)

and

lim inf_{k→∞} B̃_{k} > λ(α),  (30)

then (1) is nonoscillatory.

Proof. One can show by the similar way as in the proof of the previous theorem that under the assumptions (15), (22), (29) and (30) the sequence

w_{k} = − \binom{}{R̃_{k}^{1−α}} (C − B̃_{k}) ,

where C = \frac{2α−1}{α} (\frac{α−1}{α})^{α−1}, solves the inequality (7) with r_{k} + w_{k} > 0.

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 24, p. 12 Remark 6. Being motivated by the half–linear continuous case, see [9, 2, 10], we conjecture that one can prove oscillatory supplements of the above criteria, namely lim inf_{k→∞} B_{k} > (\frac{α−1}{α})^{α} or lim inf_{k→∞} B̃_{k} > (\frac{α−1}{α})^{α} with appropriate additional conditions. Similarly as in the case of sequence R_{k}^{α−1}P̃_{k}, see Remark 3, we can look for further oscillation criteria containing also sequence B_{k}. For such types of criteria for equation (3) see [9].

Denote Λ(C^{[1]}) the greatest root of equation

(α − 1)|x|^{β} + αx + C^{[1]} = 0

and Γ(C^{[2]}) the greatest root of equation

(α − 1)|x|^{β} − (α − 1)x + C^{[2]} = 0

for certain C^{[1]} and C^{[2]} which are determined by the following statement that completes Theorem 9 in the case when (26) fails to hold.

Theorem 11. Suppose that (10), (12) and (19) hold. If

lim sup_{k→∞} B_{k} < lim inf_{k→∞} B_{k} + Γ (lim inf_{k→∞} B_{k}) + Λ [lim inf_{k→∞} B_{k} + Γ (lim inf_{k→∞} B_{k})]

and

−∞ < lim inf_{k→∞} B_{k} ≤ λ(α),

then (1) is nonoscillatory.

Proof. The proof is similar to that of the continuous case, see [9].

Remark 7. By the same way as above one can show that there exists similar complementary case (i.e., when (30) fails to hold) also for Theorem 10.

Remark 8. Under the assumption r_{k}^{1−β} → ∞ as k → ∞, the condition (19) can be replaced by more simple (but stronger) one, namely lim_{k→∞} r_{k+1}/r_{k} = 1 in all above criteria where is presented. Similarly in the case when (15) holds we can suppose the same condition instead of (22).

References

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[3] O. Došlý, A remark on conjugacy of half-linear second order differential equations, Math. Slovaca 50 (2000), 67–79.

[4] O. Došlý, Oscillation criteria for half-linear second order differential equations, Hiroshima J.

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[12] H. J. Li, C. C. Yeh, Sturmian comparison theorem for half-linear second order differential equations, Proc. Roy. Soc. Edinburgh 125A (1996), 1193–1204.

[13] J. D. Mirzov, Principal and nonprincipal solutions of a nonoscillatory system, Tbiliss. Gos. Univ. Inst. Prikl. Mat. Trudy 31 (1988), 100–117.

[14] W. T. Reid, Sturmian Theory for Ordinary Differential Equations, Springer-Verlag, New York - Berlin - Heidelberg, 1980.

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[16] P. Řehák, Oscillation and nonoscillation criteria for second order linear difference equations, to appear in Fasc. Mathematici.

Department of Mathematical Analysis, Faculty of Science, Masaryk University Brno, Janáčkovo nám. 2a, CZ-66295 Brno, Czech Republic

E-mail address: rehak@math.muni.cz

EJQTDE, Proc. 6th Coll. QTDE, 2000 No. 24, p. 14