International Journal of Differential Equations
Volume 2017 (2017), Article ID 5042421, 8 pages
https://doi.org/10.1155/2017/5042421

Research Article

Critical Oscillation Constant for Euler Type Half-Linear Differential Equation Having Multi-Different Periodic Coefficients

Adil Misir and Banu Mermerkaya

Department of Mathematics, Faculty of Science, Gazi University, Teknikokullar, 06500 Ankara, Turkey

Correspondence should be addressed to Adil Misir; adilm@gazi.edu.tr

Received 15 July 2016; Accepted 12 October 2016; Published 27 February 2017

Academic Editor: Said R. Grace

Copyright © 2017 Adil Misir and Banu Mermerkaya. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

We compute explicitly the oscillation constant for Euler type half-linear second-order differential equation having multi-different periodic coefficients.

1. Introduction

In literature, half-linear second-order differential equations are given by

where are continuous functions and It is well known that oscillation theory of (1) is very similar to that of the linear Sturm-Liouville differential equation, which is the special case of in (1); see [1].

In particular, (1) with instead of is said to be conditionally oscillatory if there exists a constant such that this equation is oscillatory for and nonoscillatory for . is called the critical oscillation constant of this equation; see [2].

The half-linear Euler differential equation

with the so-called critical oscillation constant , plays an important role in the conditionally oscillatory half-linear differential equation.

Equation (2) can be regarded as a good comparative equation in the sense that (2) with instead of is oscillatory if and only if (see [3]) and if in (1), then this equation is oscillatory provided

and nonoscillatory if

see [4].

In [5], perturbations of (2) being of the form

are investigated when for constant Here the notation

is used. It is shown that the constant plays a crucial role in (5). In particular, if in (5) this equation reduces to the so-called Riemann-Weber half-linear differential equation, and this equation is oscillatory if and nonoscillatory otherwise. In general, if for , then (5) is oscillatory if and only if

One of the typical problems in the qualitative theory of various differential equations is to study what happens when constants in an equation are replaced by periodic functions which have same periods and different periods. Our investigation follows this line and it is mainly motivated by the paper [6].

In [7], the half-linear differential equation being of the form

is investigated for and are constants and the following result is obtained.

Theorem 1. Suppose that there exists such that

and Then (7) is oscillatory if and nonoscillatory if

In [8], the half-linear differential equation being of the form

is considered for periodic positive functions and and it is shown that (9) is oscillatory if and nonoscillatory if , where is given by

for and are conjugate numbers; that is, .

In [9], (9) and the half-linear differential equation being of the form

are considered for , and are periodic, positive functions defined on and it is shown that (9) is nonoscillatory if and only if , where is given by

In the limiting case (11) is nonoscillatory if and it is oscillatory if , where is given by

In [10], the half-linear differential equation being of the form

is considered for , (), is a continuous function for which mean value exists and for which

holds and , (), is a continuous function having mean value and it was shown that (14) is oscillatory if and nonoscillatory if , where is given by

In [6], the half-linear differential equation being of the form

is considered for -periodic functions , and , and and the following result was obtained.

Theorem 2. Let , and be -periodic continuous functions, , and their mean values over the period are denoted by , and .

If , then (17) is oscillatory and if , then it is nonoscillatory.Let . If there exists such that

 (if ), and , then (17) is oscillatory if

 and nonoscillatory if

Our research is motivated by the paper [6], where the oscillation constant is computed for (17) with the periodic coefficients having same period. However, if these periodic functions have different periods what would be the oscillation constant is not investigated. Thus, in this paper we investigate the oscillation constant for (17) with periodic coefficients having different periods. In this paper we consider two types of periodic coefficients which have different periods for (17). In the first type we consider these periodic coefficient functions having the least common multiple and in the second type, we consider these periodic coefficient functions which do not have least common multiple. We give some corollaries which illustrate the first type’s cases that our results compile the known results in [6] but in the second type only our results can be applied.

In Section 2, we recall the concept of half-linear-trigonometric functions and their properties. In Section 3 we compute the oscillation constant for (17) with periodic coefficients which have different periods. Additionally we show that if the different periods coincide, then our results compile with the known results in [6]. Thus, our results extend and improve the results of [6].

2. Preliminaries

We start this section with recalling the concept of half-linear-trigonometric functions; see [1] or [4]. Consider the following special half-linear equation being of the form

and denote its solution by given by the initial conditions , . We see that the behavior of this solution is very similar to that of the classical sine function. We denote this solution by and its derivative by . These functions are periodic, where , and satisfy the half-linear Pythagorean identity

Every solution of (21) is of the form , where and are real constants; that is, it is bounded together with its derivative and periodic with the period . The function is a solution to the reciprocal equation of (21);

which is an equation of the form as in (21), so the functions and are also bounded.

Let be a nontrivial solution of (1) and we consider the half-linear Prüfer transformation which is introduced using the half-linear-trigonometric functions

where and Prüfer angle is a continuous function defined at all points where

Then satisfies the following differential equation:

see [9].

3. Main Results

We need the following lemma in order to prove our main Theorem 4.

Lemma 3. Let ( is a suitable constant) be a solution of the equation

where

with , and are periodic functions having different , and periods, respectively, and and

where is one of the following , and periods. Then is a solution of

where

and as .

Proof. Taking derivative of , we have

Using integration by parts, we get

Let be a continuous periodic function and ; then integration by parts yields

By using (33) and for any -periodic function and Pythagorean identity, the expressions

are bounded. Thus we get

If we add and subtract the below terms in the right side of this equation

we can rewrite this equation as

And using the half-linear-trigonometric functions, we have

By the Mean Value Theorem we can write

for , , , , and ; thus

This implies that

And using , , , , and (33), we get

The term can be written as ; hence we get

Now since all the terms of are as for , then all these terms are asymptotically less than . Hence we get

The main result of this paper is as follows.

Theorem 4. Let , and , are periodic functions which have different , and , periods, respectively, and in (17).

(17) is oscillatory if and nonoscillatory if , where and are defined in Lemma 3.Let . If there exists such that

 and , then (17) is oscillatory if

 and nonoscillatory if

 where and , are defined in Lemma 3.

Proof. The statement (i) is proved in [10]. It remains to prove the statement (ii) in full generality.
We consider (17); let be the nontrivial solution of (17) and is the Prüfer angle of (17) given in (24). Then

is a solution of

where

By the help of Lemma 3, is a solution of

where , and , are given in Lemma 3.
This equation is a “Prüfer angle” equation for the following second-order half-linear differential equation

which is the same as the following equation:

Suppose that assumption (ii) of Theorem 4 is satisfied and that (46) holds for . Then (53) is oscillatory as a direct consequence of Theorem 1. If (46) holds for , let be so small that still

and consider the following equation:

where . This equation is a Sturmian minorant for sufficiently large in (53) and (54) and Theorem 1 implies that this minorant equation is oscillatory and hence (53) is oscillatory as well. This means that the Prüfer angle of the solution of (52) is unbounded and by Lemma 3 the Prüfer angle of the solution of (17) is unbounded as well. Thus, (17) is oscillatory. A slightly modified argument implies that (17) is nonoscillatory provided that (47) holds.

Corollary 5. If the periods of the functions , and , in (17) coincide with -period, which is given in [6], then our oscillation constants overlap to their oscillation constants and our main result compiles with the result given in [6].

Corollary 6. If there exists a , and the period which is given in [6] is chosen as , then our oscillation constants overlap to their oscillation constants and our main result compiles with the result given in [6].

Remark 7. If for is not defined, then only our result can be applied whereas the result given in [6] can not.

Example 8. Consider the nonlinear equation (17) for , , , , , , and . In this case , , , , , and are periods of these functions, respectively. Because of these functions being periodic functions and positive defined we can use Theorem 4 for all and we obtain

Thus we get for all and considered equation is oscillatory. Here the important point to note is that while we cannot apply Theorem 2 which is given in [6] for this example if we choose , then lcm is not defined, we can apply our Theorem 4.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

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