Critical Oscillation Constant for Euler Type Half-Linear Differential Equation Having Multi-Different Periodic Coefficients
We compute explicitly the oscillation constant for Euler type half-linear second-order differential equation having multi-different periodic coefficients.
In literature, half-linear second-order differential equations are given bywhere 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 .
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 .
The half-linear Euler differential equationwith 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 ) and if in (1), then this equation is oscillatory providedand nonoscillatory ifsee .
In , perturbations of (2) being of the formare investigated when for constant Here the notationis 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 .
In , the half-linear differential equation being of the formis 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 , the half-linear differential equation being of the formis 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) and the half-linear differential equation being of the formare 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 , the half-linear differential equation being of the form is considered for , (), is a continuous function for which mean value exists and for whichholds 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 , the half-linear differential equation being of the formis 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 , 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  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 . Thus, our results extend and improve the results of .
We start this section with recalling the concept of half-linear-trigonometric functions; see  or . 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 identityEvery 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 functionswhere and Prüfer angle is a continuous function defined at all points where
Then satisfies the following differential equation:see .
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 equationwherewith , and are periodic functions having different , and periods, respectively, and andwhere is one of the following , and periods. Then is a solution ofwhereand as .
Proof. Taking derivative of , we haveUsing integration by parts, we getLet be a continuous periodic function and ; then integration by parts yields By using (33) and for any -periodic function and Pythagorean identity, the expressionsare bounded. Thus we getIf we add and subtract the below terms in the right side of this equationwe can rewrite this equation asAnd using the half-linear-trigonometric functions, we haveBy the Mean Value Theorem we can writefor , , , , and ; thusThis implies that And using , , , , and (33), we getThe 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 . 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 ofwhereBy the help of Lemma 3, is a solution ofwhere , 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 , then our oscillation constants overlap to their oscillation constants and our main result compiles with the result given in .
Corollary 6. If there exists a , and the period which is given in  is chosen as , then our oscillation constants overlap to their oscillation constants and our main result compiles with the result given in .
Remark 7. If for is not defined, then only our result can be applied whereas the result given in  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 obtainThus 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  for this example if we choose , then lcm is not defined, we can apply our Theorem 4.
The authors declare that there is no conflict of interests regarding the publication of this paper.
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