Oscillation and non-oscillation of half-linear differential equations with coeffcients determined by functions having mean values

Definition 2.1

Let a continuous function f : [T, ∞) → ℝ be such that the limit

is finite and exists uniformly with respect to t ∈ [T, ∞). The number f is called the mean value of f.

Obviously, the mean value of any β-periodic continuous function f is

where τ is arbitrary.

To prove our results, we use the generalized Riccati technique which is described below. We consider the second-order half-linear differential equation

where α ≤ 0 and r, s are continuous functions such that the mean values of functions r^1/(1−p) and s exist, the mean value s is positive, and

We denote by q the number conjugated with the given number p > 1, i.e.,

Immediately, we obtain the inverse function to Φ in the form Φ⁻¹(x) = |x|^{q − 1}sgn x.

The basis of our method is the transformation to the Riccati half-linear equation which can be introduced as follows. We consider a non-zero solution x of Eq. (5) and we define

Considering Eq. (5), the differentiation of (8) leads to the Riccati half-linear equation

The form of Eq. (9) is not sufficient enough for our method. Hence, we apply the transformation ζ(t) = −t^{p − α − 1}w(t) which leads to the equation

Eq. (10) is called the adapted generalized Riccati equation for the consistency with similar cases in the literature.

Further, in this section, we formulate auxiliary results that will be needed in the following section within the proof of Theorem 3.1 below. We begin with properties of functions having mean values.

Lemma 2.2

Let a continuous function f : [T, ∞) ⊂ (0, ∞) → ℝ have mean value f. For an arbitrarily given a > 0, there exists a constant M(f) > 0 such that

and

for all t ≥ T, b ∈ [0, a].

Proof

The statement of the lemma follows from the beginning of the proof of [1, Theorem 8] (see directly inequalities (48) and (51) in [1]). □

Next, we recall the well-known Sturm half-linear (also called the Sturm–Picone) comparison theorem.

Theorem 2.3

Let r̂, r̃, ŝ, s̃ be continuous functions satisfying r̂ (t) ≥ r̃ (t) > 0 and s̃(t) ≥ ŝ(t) for all sufficiently large t. Consider the pair of equations

1. If Eq. (14)is non-oscillatory, then Eq. (13)is non-oscillatory.

2. If Eq. (13)is oscillatory, then Eq. (14)is oscillatory.

Proof

See, e.g., [3, Theorem 1.2.4]. □

Now, we formulate the condition which is equivalent to the non-oscillation of Eq. (5).

Theorem 2.4

Eq. (5)is non-oscillatory if and only if there exists a ∈ ℝ and a solution w: [a, ∞) → ℝ of Eq. (9)satisfying either

or

for all t ≥ a.

Proof

See, e.g., [3, Theorems 2.2.4 and 2.2.5], where it suffices to consider that the used divergence of ∫^∞ (τ^α r(τ))^{1 − q} dτ follows from α ≤ 0 and from (6) and the used convergence of ∫^∞τ^{α − p}s(τ) dτ is proved as (20) in [1]. □

The upcoming lemma describes the connection between the behaviour of solutions of Eq. (5) and the adapted Riccati equation (10).

Lemma 2.5

If Eq. (5)is non-oscillatory, then there exists a solution ζ of the associated adapted generalized Riccati equation(10)such that ζ(t) ≤ 0 for all large t ∈ ℝ.

Proof

We apply Theorem 2.4. From the positivity of s (see also [1]), it is seen that (16) cannot be valid for all large t, i.e., we obtain (15). Hence, there exists a non-negative solution w of Eq. (9) on some interval [T, ∞), i.e., there exists a non-positive solution ζ of Eq. (10) on [T, ∞). □

The last lemma contains the opposite implication to the one in Lemma 2.5.

Lemma 2.6

If there exists a solution ζ of Eq. (10)for all large t ∈ ℝ, then Eq. (5)is non-oscillatory.

Proof

The statement of the lemma follows directly from the half-linear version of the Reid roundabout theorem (see [3, Theorem 1.2.2] and also [1, Lemma 4]). □

Theorem 3.1

Let us consider Eq. (17).

1. If

   pp−α−1ps¯r¯p−1>1,(18)

   then Eq. (17)is oscillatory.

2. If

   pp−α−1ps¯r¯p−1<1,(19)

   then Eq. (17)is non-oscillatory.

Proof

In the both parts of the proof, we will consider such a number a > 2 for which (see Definition 2.1 together with (18) and (19))

or

for all considered t and for some ε ∈ (0, 1). We can rewrite (20) and (21) into the following forms

and

Using (6), from (22) and (23), we obtain the existence of L > 0, for which

and

for all considered t. Indeed, one can put

in the both cases.

We will consider the associated adapted generalized Riccati equation in the form of Eq. (10) which corresponds Eq. (17), i.e., the equation

At first, we show that any solution ζ of Eq. (26) defined for t ≥ T is bounded from below, i.e., we show that there exists K > 1 satisfying

On the contrary, let us assume that

or

for some T₀ ∈ (T, ∞). For given a and function s, let us consider M(s) from Lemma 2.2. Let P > 0 be an arbitrary number such that

In particular, considering inf_{t ≥ T}ζ (t) = −∞, the continuity of ζ implies the existence of an interval [t₁, t₂] such that ζ (t₁) ≤ − P, ζ(t) < −P for all t ∈ (t₁, t₂], and t₂ − t₁ ∈ (0, a]. Without loss of generality, we consider T > 2. Using (12) in Lemma 2.2, the form of Eq. (26), and (30), we have

This inequality proves that (29) cannot be valid for any T₀ ∈ ℝ (consider that a > 2 is given). Therefore, there exist arbitrarily long intervals, where ζ(t) ≤ −P. Let I = [t₃, t₄] be such an interval whose length is at least 2 (i.e., t₄ − t₃ ≥ 2) and t₄ − t₃ ≤ a. As in (31) (consider that t₃ > 2), we obtain

Of course, (32) means that ζ(t₄) > ζ(t₃). In fact, (31) and (32) guarantee that

which contradicts (28). Hence, (27) is valid.

In the both parts of the proof, we will also apply the estimation

for all large t and M(s) from Lemma 2.2. We use the mean value theorem of the integral calculus to get this estimation. More precisely, considering t ∈ [t₁, t₂], where t₁ is sufficiently large, since s is integrable and x(t) = t^{− 1} is monotone for t ∈ [t₁, t₂], there exists t₃ ∈ [t₁, t₂] such that

Immediately, from (34), we obtain (see (11) in Lemma 2.2)

where b ∈ [0, a]. It is seen that (35) gives (33).

Part (I). Evidently (see (18)), s > 0. By contradiction, let us suppose that Eq. (17) is non-oscillatory. From Lemma 2.5, we know that there exists a non-positive solution ζ of Eq. (26) on some interval [T, ∞). For this solution ζ, we introduce the averaging function ζ_ave by

We know that (see (27))

For t > T (see Eq. (26)), we have

For t > T, we also have (see (27) and (33))

where

For t > T, we obtain (see (38), (39), and (40))

If we put

for t > T, then we have (see (41))

Taking into account (43), for large t, we will show the inequalities

The first inequality (44) is valid for all t ≥ 3N/L. Hence, we can approach to (45). It holds (see (36) and (42))

We recall the well-known Young inequality which says that

holds for all non-negative numbers A, B. We take A = (p t X(t))^1/p and B = (qtY(t))^1/q. Hence,

and (see (37))

Finally, considering (48) and (49), we have

which proves (45).

Next, (46) is valid. Indeed, we have (see (24) and (42))

for all considered t.

To prove (47), we use the form of Eq. (26) together with (6), (12) from Lemma 2.2, and with (27) which immediately give

for t > T and i, j ∈ [0, a], i ≤ j, where

Hence, we have

which implies (see (36))

for all t > T and τ ∈ [t, t+a].

Further, since the function x(t) = |t|^q is continuously differentiable on [−K, 0], there exists C > 0 for which

Thus, we have (see (6), (7), (27), (37), (42), (51), and (52))

for t > T which gives (47) for all sufficiently large t.

Altogether, (43) together with (44), (45), (46), and (47) guarantee

for all large t. From (53) it follows that lim_{t → ∞}ζ_ave (t) = ∞. In particular (see (36)), ζ is positive at least in one point which is a contradiction. The proof of part (I) is complete.

Part (II). Without loss of generality, we can assume that s > 0. Indeed, for s ≤ 0, it suffices to replace function s by function s + k for a constant k > 0 such that s + k > 0 and

and to use Theorem 2.3.

Let t₀ be a sufficiently large number. We denote (see (6))

Let ζ be the solution of the adapted generalized Riccati equation (26) satisfying

Based on Lemma 2.6, it suffices to prove that this solution exists for all t ∈ [t₀, ∞). Let [t₀, T) be the maximal interval, where the solution ζ exists. Note that T ∈ (t₀, ∞) ∪ {∞}. In fact, considering the continuity and the boundedness from below of ζ (see (27)), it suffices to show that ζ(t) < 0 for all t ∈ [t₀, T).

Let us consider an interval J := [t₀, t₁) such that ζ(t) ∈ (−2 Z, 0) for t ∈ J. For any t₂, t₃ ∈ J ∩ [t₀, t₀ + a], t₂ ≤ t₃, we have (see (6) and (12) in Lemma 2.2, cf. (50))

where

Since t₀ is given as a sufficiently large number, from (55), we see that

In addition, (55) means that

Similarly as in the first part of the proof, we introduce

for t from a neighbourhood of t₀. It holds (see (56), (57), and (58))

We have (see Eq. (26), cf. (38))

In addition, as in the first part of the proof (see (39)), we have

where N is defined in (40) (we can also put K = 2Z).

Hence, from (60) and (61), we obtain (cf. (41))

If we put (cf. (42))

then we have (see (62))

The aim of our process is to prove that ζave′(t0)<0. To obtain this inequality, it suffices to show (see (65))

Evidently, (69) is valid, because t₀ is sufficiently large. We show that (66) is valid. We have (see (25), (63))

i.e., we obtain (66).

Now we prove (67). Let D > 0 be such that (cf. (52))

Using (6), (7), (54), (56), (57), (64), and (70), we have

For a sufficiently large number t₀, inequality (67) follows from (71).

It remains to prove (68). We have (see (58), (63), and (64))

Let us assume that (see (54))

Then, (72) gives