We prove that the existence of the mean values of coefficients is sufficient for second-order half-linear Euler-type differential equations to be conditionally oscillatory. We explicitly find an oscillation constant even for the considered equations whose coefficients can change sign. Our results cover known results concerning periodic and almost periodic positive coefficients and extend them to larger classes of equations. We give examples and corollaries which illustrate cases that our results solve. We also mention an application of the presented results in the theory of partial differential equations.

1. Introduction

In this paper, we analyse oscillatory properties of the half-linear differential equation where and are continuous functions, is positive, and . To describe our main interest, let us consider (1) with for a continuous function and . We say that such an equation is conditionally oscillatory if there exists the so-called oscillation constant such that the equation under consideration is oscillatory for and nonoscillatory for . 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 . 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 [48].

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 in [11] (see also [12]). Later, this result has been extended in a number of papers (e.g., [1317]), 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 is treated for positive asymptotically almost periodic functions and satisfying 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 [2325] 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.

2. Preliminaries

Let be arbitrarily given and let be the real number conjugated with satisfying As usual, for given , the symbol 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 . 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.

Theorem 1. Equation (1) is nonoscillatory if and only if there exists a function which solves (11) on some interval .

Proof. The theorem is a consequence of the well-known roundabout theorem (see, e.g., [10, Theorem ]).

We will also use the Sturmian comparison theorem in the form given below.

Theorem 2. Let , be continuous functions satisfying for all sufficiently large . Let one consider (1) and the equation (i)If (1) is oscillatory, then (12) is oscillatory as well.(ii)If (12) is nonoscillatory, then (1) is nonoscillatory 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 half-linear equations.

Definition 3. Let continuous function be such that the limit is finite and exists uniformly with respect to . The number is called the mean value of .

In fact, we will study (1) in the form where is a continuous function having mean value and satisfying and is a continuous function having mean value . 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).

The Riccati equation associated to (14) has the form (see (11)) Finally, using the substitution , we obtain the adapted Riccati equation which will play a crucial role in the proof of the announced result (see the below given Theorem 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 , then (14) is nonoscillatory.

Proof. A solution of (17) on an interval gives the solution of (16) on the same interval. Thus, the lemma follows from Theorem 1.

Lemma 5. Let (14) be nonoscillatory and let there exist such that For any solution of (16) on , it holds

Proof. The lemma follows, for example, from [10, Theorem ], where it suffices to use (15).

Lemma 6. If (14) is nonoscillatory, then there exists a solution of (17) on some interval with the property that for all and for some .

Proof. Considering Theorem 1, the nonoscillation of (14) implies that there exists a solution of (16) on some interval which gives the solution of (17) on the interval. We show that this solution is bounded above.
At first, we prove the convergence of the integral and the inequality Evidently, it suffices to prove (20) and
Let be such that where we use directly Definition 3 (the existence of ). The symbols and will denote the positive and negative parts of function , respectively. We choose . We can express For an arbitrarily given positive integer , we have if , and if . Using and using (23), (25), and (26), we obtain the existence of such that it holds Since is arbitrary, it also holds for all sufficiently large .
Hence, the integral is convergent because where is sufficiently large. Particularly, Moreover, we have (see (29)) Thus, (22) is valid; that is, there exists for which (21) is valid.
Integrating (16), we obtain We know that Indeed, considering (18) together with (31), one can get (19) from Lemma 5. From (20) and (33) it follows that there exists the limit . In addition, the convergence of the integral in (34) gives Again, we consider arbitrarily given . We can rewrite (33) into (or see directly (16)) Putting , from (20), (34), (35), and (36), we obtain
Finally, let us denote , where We know that (see (21) and (37)) We denote . If is positive, then the statement of the lemma is true for all . Therefore, we can assume that . Since is nonincreasing and , function is nonnegative. From (39) it follows Hence, we have and, consequently, we obtain that , . It means that the statement of the lemma is valid for .

Remark 7. Let (14) be nonoscillatory. If the considered function is positive for all , then the statement of Lemma 6 is true for a negative solution of (17). See, for example, [10, Lemma ].

Theorem 8. Equation (14) is oscillatory if and nonoscillatory if .

Proof. The proof is organized as follows. In the first part, we derive upper bounds for two integrals involving function . 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 with the property that and, consequently, there exists with the property that We can rewrite (43) into the form Using (42), we obtain that is, Combining (44) and (47), we have where . Since the function is decreasing and positive on , it holds for all and for some . Analogously, for any , there exists such that Hence, from (48) it follows
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 . Evidently, we can assume that .
We show that there exists satisfying On the contrary, let us assume that . Let for all from some interval , where for some , and let be such that (see (15)) Indeed, . We can assume that is increasing for . Using (17), (51), and (53), it holds Thus, for all which proves (52). Indeed, it suffices to consider . In addition, we can assume that ; that is, Thus (see directly (17) and (51)), we have for all , where . The previous inequality implies
Considering Definition 3 and , there exist and such that and, at the same time, such that For such an integer , we define Since we have Hence, to prove the first implication in the statement of the theorem, it suffices to show that (62) is not true.
From (57) it follows where Particularly (see (60)), (63) gives
Next, we consider the function If for some , then . Henceforth (in this paragraph), we consider the case when , . Let us define It can be directly verified that function has the global minimum It means that , . Particularly, it gives the inequality Considering (59) and (69), we have Applying (63), the inequalities , , and the fact that the function has the Lipschitz property on any bounded set, there exists such that Hence (see also (59)), we get From (70) and (72), we know that Of course, (73) remains true for as well.
Let us consider for which Note that the existence of such a number follows from (65). It is seen that (73) and (74) imply
Evidently, we can consider the solution in an arbitrarily given neighbourhood of . Hence, we can assume that From (48) and (76), we see that from (55) and (77), we have and, analogously, from (15), (55), and (78) it follows For all , using (58), (75), (79), (80), and (81), we obtain Thus, it holds Since we obtain that . The contradiction with (62) proves the first implication.
In the nonoscillatory part of the proof, we consider . Let and satisfy Let us consider solution of (17) given by for some sufficiently large . Since the right-hand side of (17) is continuous, the considered solution can be defined on an interval , where . In addition, if , we can assume that If , then the considered solution of (17) satisfies the condition of Lemma 4. It means that it suffices to find for which
As in the oscillatory part of the proof (see (52)), we can prove that for some and for all . Indeed, we can analogously show that the inequality cannot be valid for any , where is taken from (48) and from (53). We want to prove that . On the contrary, let (86) be valid for some . Particularly, solution has to be positive on some interval in this case.
We denote and we compute
We know that is negative on an interval . Let have the property that . For all , , we have (see (51)) Thus, for general satisfying , we have We can assume that is so large that if (see (88)). Particularly (), we can define the function for all and for all when . Particularly, let be so large that .
We repeat that we assume the positivity of which implies the inequality for from some interval. The continuity of gives the existence of such that From (91) it follows that, for any , one can choose so large that Thus (see (93)), we can assume that Consequently, let At the same time, we can assume that was chosen in such a way that it is valid
Using (97) and (98), we have Since (see (89), (94)) we have
Let be so large that (see (48) and also (79)) and (see (15) together with (96) and also (81))
Considering (85), (101), (102), and (103), we obtain This contradiction (see (94)) means that (87) is true for and . Since (86) cannot be valid for any , the considered solution exists on interval . 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 be a continuous function, for which mean value exists and for which it holds and let be a continuous function having mean value . Let Consider the equation Equation (107) is oscillatory if and nonoscillatory if .

Proof. Let . Equation(107) can be rewritten into the form that is, Equation (109) has the form of (14) for Note that and and that (15) follows from (105). Thus (see Theorem 8), (107) is oscillatory for and nonoscillatory if the opposite inequality holds.
It remains to consider the case when . Of course, there exists such that . We know that the equation is nonoscillatory. Now it suffices to use Theorem 2 (ii).

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 , 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 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 and (107) is oscillatory. At the same time, there exist different almost periodic functions , satisfying 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.

Example 12. For ,