Conditional Oscillation of Half-Linear Differential Equations with Coefficients Having Mean Values

and Applied Analysis 3 Finally, using the substitutionw(t) = −ζ(t)t, we obtain the adapted Riccati equation ζ󸀠 (t) = 1 t [(p − 1) ζ (t) + s (t) + (p − 1) r (t) 󵄨󵄨󵄨ζ (t) 󵄨󵄨󵄨 q] , (17) which will play a crucial role in the proof of the announced result (see the below givenTheorem 8). 3. Results To prove the announced result, we need the following lemmas. Lemma 4. If there exists a solution of (17) on some interval [T,∞), then (14) is nonoscillatory. Proof. A solution ζ of (17) on an interval [T,∞) gives the solution w(t) = −ζ(t)t of (16) on the same interval. Thus, the lemma follows fromTheorem 1. Lemma 5. Let (14) be nonoscillatory and let there existM > 0 such that 󵄨󵄨󵄨󵄨󵄨󵄨 ∫ c b s (τ) τp dτ 󵄨󵄨󵄨󵄨󵄨󵄨 < M, a ≤ b < c ≤ ∞. (18) For any solution w of (16) on [T,∞), it holds ∫ ∞ T r (τ) |w (τ)|qdτ < ∞. (19) Proof. The lemma follows, for example, from [10, Theorem 2.2.3], where it suffices to use (15). Lemma 6. If (14) is nonoscillatory, then there exists a solution ζ of (17) on some interval [T,∞) with the property that ζ(t) < A for all t ≥ T and for some A > 0. Proof. Considering Theorem 1, the nonoscillation of (14) implies that there exists a solutionw of (16) on some interval [T,∞) which gives the solution ζ(t) = −w(t)t of (17) on the interval. We show that this solution ζ is bounded above. At first, we prove the convergence of the integral ∫ ∞ T s (τ) τp dτ ∈ R (20) and the inequality sup t≥T 󵄨󵄨󵄨󵄨󵄨󵄨 tp−1 ∫ ∞ t s (τ) τp dτ 󵄨󵄨󵄨󵄨󵄨󵄨 < L for some L > 0. (21) Evidently, it suffices to prove (20) and lim sup t→∞ 󵄨󵄨󵄨󵄨󵄨󵄨 tp−1 ∫ ∞ t s (τ) τp dτ 󵄨󵄨󵄨󵄨󵄨󵄨 < ∞. (22) Let b > 0 be such that ∫ t+b t s (τ) dτ > 0, 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 ∫ t+b t s (τ) dτ − bM (s) 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨 < b, t ≥ T, (23) where we use directly Definition 3 (the existence of M(s) > 0). The symbols [f(⋅)]+ and [f(⋅)]− will denote the positive and negative parts of functionf, respectively.We choose t 0 ≥ T. We can express ∫ t 0 +kb t 0 +(k−1)b s (τ) τp dτ = ∫ t 0 +kb t 0 +(k−1)b [s (τ)]+ τp dτ


Introduction
In this paper, we analyse oscillatory properties of the halflinear differential equation

[𝑟 (𝑡) Φ (𝑥 󸀠 )]
+  () Φ () = 0, Φ () = || −1 sgn , where  and  are continuous functions,  is positive, and  > 1.To describe our main interest, let us consider (1) with () = () − for a continuous function  and  ∈ R. We say that such an equation is conditionally oscillatory if there exists the so-called oscillation constant Γ ∈ R such that the equation under consideration is oscillatory for  > Γ and nonoscillatory for  < Γ.In fact, the oscillation constant depends on coefficients  and .
Looking back to the history (according to our best knowledge), the first attempt to this problem was made by Kneser in [1], where the oscillation constant for the linear equation has been identified as Γ = 1/4.Later, in [2,3], it has been shown that the conditional oscillation remains preserved also for periodic coefficients.More precisely, the equation where ,  are positive -periodic continuous functions, is conditionally oscillatory for We also refer to more general results in [4][5][6][7][8].
Since a lot of results from the linear oscillation theory are extendable to the half-linear case (see, e.g., [9,10]), it is reasonable to suppose that the oscillation constant can be found for the corresponding Euler-type half-linear equations as well.This hypothesis has been shown to be true for

[Φ (𝑥 󸀠 )]
+    Φ () = 0, Γ = (  − 1  )  (5) in [11] (see also [12]).Later, this result has been extended in a number of papers (e.g., [13][14][15][16][17]), where equations of the below given form (6) have been treated with coefficients ,  replaced by perturbations consisting of constant or periodic functions and iterated logarithms.Nevertheless, the most general result (concerning the topic of this paper) can be found in [18], where the equation It is shown that ( 6) is conditionally oscillatory with the oscillation constant where () stands for the mean value of function .This result is the main motivation of our current research.Our goal is to remove the condition of positivity of function  and, at the same time, to extend the class of functions ,  as much as possible applying the used methods.We present an oscillation criterion which is new in the half-linear case as well as in the linear one.We should mention some relevant references from the discrete and time scale theory.In this paper, we give only the most relevant references concerning the topic.The reader can find more comprehensive literature overview together with historical references in our previous article [18].Here, we refer at least to [19,20] for the corresponding results about difference equations (see also, e.g., [21,22]) and to [23][24][25] for results about dynamical equations on time scales.
The paper is organized as follows.In the next section, we mention the necessary background and we recall the basics of the Riccati technique.In Section 3, we prove preparatory lemmas and our results.We also state several corollaries, concluding remarks, and examples.In the last section, we give an application in the theory of partial differential equations.

Preliminaries
Let  > 1 be arbitrarily given and let  > 1 be the real number conjugated with  satisfying As usual, for given  > 0, the symbol R  stands for [, ∞).
To prove the main results, we will apply the Riccati technique for (1), where the transformation leads to the half-linear Riccati differential equation whenever () ̸ = 0.For details, we refer to [10].The fundamental connection between the nonoscillation of (1) and the solvability of ( 11) is described by the following theorem.
We will also use the Sturmian comparison theorem in the form given below.
Theorem 2. Let ,  : R  → R be continuous functions satisfying () ≥ () for all sufficiently large .Let one consider (1) and the equation is oscillatory, then (12) is oscillatory as well.
Proof.The theorem follows, for example, from [10, Theorem Now we recall the concept of mean values which is necessary to find an explicit oscillation constant for general halflinear equations.Definition 3. Let continuous function  : R  → R be such that the limit is finite and exists uniformly with respect to  ∈ R  .The number () is called the mean value of .
In fact, we will study (1) in the form where  : R  → R is a continuous function having mean value () = 1 and satisfying and  : R  → R is a continuous function having mean value () > 0. We repeat that the basic motivation comes from [18], where asymptotically almost periodic half-linear equations are analysed.Since positive nonvanishing asymptotically almost periodic functions have positive mean values and they are bounded, we will consider more general equations (cf.( 15) with ( 7) as well).

Results
To prove the announced result, we need the following lemmas.
Proof.Considering Theorem 1, the nonoscillation of (14) implies that there exists a solution  of ( 16) on some interval [, ∞) which gives the solution () = −() −1 of (17) on the interval.We show that this solution  is bounded above.At first, we prove the convergence of the integral where we use directly Definition 3 (the existence of () > 0).The symbols [(⋅)] + and [(⋅)] − will denote the positive and negative parts of function , respectively.We choose  0 ≥ .We can express For an arbitrarily given positive integer , we have if   > 0, and and using (23), (25) Moreover, we have (see (29)) Thus, ( 22) is valid; that is, there exists  > 0 for which ( 21) is valid.
Then we prove the oscillatory part and, finally, the nonoscillatory part.
At first, we use the existence of () and the continuity of function .Considering Definition 3, there exists  ≥ 1 with the property that Combining ( 44) and (47), we have where  := max{, 2[() + 2]}.Since the function () = 1/ is decreasing and positive on R  , it holds for all  2 ≥  1 ≥  and for some  3 ∈ [ Now we prove the oscillatory part.Let () >  − .By contradiction, in this part of the proof, we will suppose that ( 14) is nonoscillatory.Lemma 6 says that there exists a solution  of (17) on some interval [, ∞) and that () <  for all  ≥  and for a certain number  > 0. Evidently, we can assume that  > 1.
We show that there exists  < −1 satisfying On the contrary, let us assume that lim inf  → ∞ () = −∞.
From (57) it follows where If () ≥ 0 for some  ≥ , then  1 () > 0. Henceforth (in this paragraph), we consider the case when () < 0,  ≥ .Let us define It can be directly verified that function  has the global minimum It means that () ≥ 0,  ≤ 0. Particularly, it gives the inequality Thus, it holds we obtain that lim  → ∞ () = ∞.The contradiction with (62) proves the first implication.
In the nonoscillatory part of the proof, we consider () <  − .Let  ∈ N and  > 0 satisfy Let us consider solution  of (17) given by ( 0 ) = −1 for some sufficiently large  0 ≥ .Since the right-hand side of ( 17) is continuous, the considered solution  can be defined on an interval [ 0 ,  1 ), where  0 <  1 ≤ ∞.In addition, if  1 < ∞, we can assume that lim sup If  1 = ∞, then the considered solution of ( 17) satisfies the condition of Lemma 4. It means that it suffices to find ,  ∈ R for which As in the oscillatory part of the proof (see ( 52)), we can prove that () >  for some  < −1 and for all  ∈ [ 0 ,  1 ).Indeed, we can analogously show that the inequality () < − −  −1 0 cannot be valid for any  ∈ [ 0 ,  1 ), where  is taken from (48) and  from (53).We want to prove that  1 = ∞.On the contrary, let (86) be valid for some  1 ∈ R. Particularly, solution  has to be positive on some interval [ 2 ,  3 ] ⊂ [ 0 ,  1 ) in this case.
We repeat that we assume the positivity of  which implies the inequality () > t for  from some interval.The continuity of  gives the existence of  >  0 such that  () = t,   () ≥ 0. (94) From (91) it follows that, for any  > 0, one can choose  0 so large that      ( This contradiction (see (94)) means that (87) is true for  =  and  = 0. Since (86) cannot be valid for any  1 < ∞, the considered solution  exists on interval [ 0 , ∞).We repeat that the nonoscillation of ( 14) actually follows from Lemma 4.
The following theorem is a version of Theorem 8 which is ready for applications to the half-linear equations written in the form common in the literature.Theorem 9. Let  : R  → R be a continuous function, for which mean value ( 1− ) exists and for which it holds and let  : R  → R be a continuous function having mean value ().Let Consider the equation Equation ( 107) is oscillatory if () > Γ and nonoscillatory if () < Γ.
Remark 10.For reader's convenience, we consider (107) (instead of ( 14)) in Theorem 9.The form of (107) shows how the presented result improves the known ones (see Section 1).Particularly, we get new results in two important cases, when function  changes sign and when it is unbounded.For details, we refer to our previous paper [18].
Remark 11.For () = Γ, it is not possible to decide whether (107) is oscillatory or nonoscillatory for general functions ,  satisfying the conditions from the statement of Theorem 9.It follows, for example, from the main results of [13,16].One of the most studied classes of functions which have mean values is formed by almost periodic functions.Based on the constructions from [26], it is conjectured in [18] that the case () = Γ is not generally solvable (in the sense whether it is oscillatory or nonoscillatory) even for almost periodic coefficients of (107).It means that there exist almost periodic functions ,  such that () = Γ and (107) is oscillatory.At the same time, there exist different almost periodic functions ,  satisfying () = Γ with the property that (107) is nonoscillatory.We add that the case of periodic functions ,  was proved to be nonoscillatory (see again [13,16]).
To illustrate Theorem 9, we mention at least two examples.