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

In literature, half-linear second-order differential equations are given by (r (t) Φ (x󸀠))󸀠 + c (t) Φ (x) = 0, Φ (s) = |s|p−2 s, p > 1, (1) where r, c are continuous functions and r(t) > 0. 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 p = 2 in (1); see [1]. In particular, (1) with λc(t) instead of c(t) is said to be conditionally oscillatory if there exists a constant λ0 such that this equation is oscillatory for λ > λ0 and nonoscillatory for λ < λ0. λ0 is called the critical oscillation constant of this equation; see [2]. The half-linear Euler differential equation (Φ (x󸀠))󸀠 + γp tpΦ (x) = 0, (2)


Introduction
In literature, half-linear second-order differential equations are given by ( () Φ (  )) +  () Φ () = 0, where ,  are continuous functions and () > 0. 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  = 2 in (1); see [1].
In particular, (1) with () instead of () is said to be conditionally oscillatory if there exists a constant  0 such that this equation is oscillatory for  >  0 and nonoscillatory for  <  0 . 0 is called the critical oscillation constant of this equation; see [2].
The half-linear Euler differential equation with the so-called critical oscillation constant   = (( − 1)/)  , 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 () = 1 in (1), then this equation is oscillatory provided lim →∞ inf    () >   (3) and nonoscillatory if lim →∞ sup    () <   ; see [4].
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  ∈ {2, . . ., } such that and 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, 1/ + 1/ = 1.
In [9], (9) and the half-linear differential equation being of the form are considered for , , and  are -periodic, positive functions defined on [0, ∞) 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 a continuous function for which mean value ( 1− ) exists and for which In [6], the half-linear differential equation being of the form is considered for -periodic functions , ,   , and   ,  = 1, 2, . . ., , and () > 0 and the following result was obtained.
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].

Preliminaries
We start this section with recalling the concept of half-lineartrigonometric 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 (0) = 0,   (0) = 1.We see that the behavior of this solution is very similar to that of the classical sine function.We denote this solution by sin   and its derivative by (sin  )  = cos  .These functions are 2  -periodic, where   fl 2/ sin(/), and satisfy the half-linear Pythagorean identity Every solution of ( 21) is of the form () =  sin  (+), where  and  are real constants; that is, it is bounded together with its derivative and periodic with the period 2  .The function  = Φ(cos  ) 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.

Main Results
We need the following lemma in order to prove our main Theorem 4. where where  is one of the following  1 , where and () − () = ∘(1) as  → ∞.
Proof.Taking derivative of (), we have By the Mean Value Theorem we can write for The term (1/) can be written as (|cos  |  +|sin  |  )(1/); hence we get Now since all the terms of (1/ log )/log 2   are (1/) as  → ∞ for  = 1, 2, . . ., , then all these terms are asymptotically less than (1)/log 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 where which is the same as the following equation: Suppose that assumption (ii) of Theorem 4 is satisfied and that (46) holds for  ∈ {1, 2, . . .,  − 1}.Then (53) is oscillatory as a direct consequence of Theorem 1.If (46) holds for  = , let  > 0 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.