Oscillation and nonoscillation of half-linear Euler type differential equations with different periodic coefficients

Abstract In this paper, we compute explicitly the oscillation constant for certain half-linear second-order differential equations having different periodic coefficients. Our result covers known result concerning half-linear Euler type differential equations with α—periodic positive coefficients. Additionally, our result is new and original in case that the least common multiple of these periods is not defined. We give an example and corollaries which illustrate cases that are solved with our result.


Introduction
An equation of the form r .t/ˆ x 0 0 C c .t/ˆ.x/ D 0;ˆ.s/ D jsj p 2 s; p > 1; is called half-linear differential equation, where r; c are continuous functions and r .t/ > 0 was introduced for the first time in [1]. During the last decades, these equations have been widely studied in the literature. The name halflinear equation was introduced in [2]. This term is motivated by the fact that the solution space of these equations is homogeneous (likewise in the linear case) but not additive. Since the linear Sturmian theory extends verbatim to half-linear case (for details, we refer to Section 1.2 in [3]), we can classify Eq.(1) as oscillatory or nonoscillatory. It is well known that oscillation theory of Eq.(1) is very similar to that of the linear Sturm-Liouville differential equation, which is the special case of Eq.(1) for p D 2 ( see [4]). Actually, we are interested in the conditional oscillation of half-linear differential equations with different periodic coefficients. We say that the equation  2) is oscillatory for all > 0 and nonoscillatory for all < 0 : The constant 0 is called an oscillation constant (more precisely, oscillation constant of c with respect to r) of this equation. Considerable effort has been made over the years to extend oscillation constant theory of half linear differential equation (1), see [5][6][7][8][9][10] and reference therein. According to our knowledge, the first attempt to this problem was made by Kneser in [11], where the oscillation constant for Cauchy-Euler differential equation (which is special case of Eq.(1) for p D 2; r .t / D 1 and c .t/ D 1 t 2 ) has been identified 0 D 1 4 and Eq.
(3) is oscillatory if > 1 4 , nonoscillatory if < 1 4 . Additionally, x .t/ D c 1 p t C c 2 p t log t is the general solution of Eq.(3) and nonoscillatory for D 1 4 : The conditional oscillation of linear equations is studied, e.g., in [9,10]. In [9,12], the oscillation constant was obtained for linear equation with periodic coefficients. Using the notion of the principle solution, the main result of [9] was generalized in [8], where periodic half-linear equations were considered.
In [7] the half-linear Euler differential equation of the form and the half-linear Riemann-Weber differential equation of the form was considered for r; c being˛ periodic, positive functions and it was shown that Eq.(6) is oscillatory if > K and nonoscillatory if < K; where K is given by for p and q are conjugate numbers, i.e., 1 p C 1 q D 1: If the functions r; c are positive constants, then Eq.(6) is reduced to the half-linear Euler equation (4), whose oscillatory properties were studied in detail [4,7] and references given therein.
In [6], Eq.(6) and the half-linear differential equation of the form  (7) is reduced to the half-linear Rimann-Weber equation (5), whose oscillatory properties are studied in detail [4,7] and references given therein. In [13], the half-linear differential equation of the form Our goal is to find the explicit oscillation constant for Eq.(7) with periodic coefficients having different periods. We point out that the main motivation of our research comes from the paper [6], where the oscillation constant was computed for Eq.(7) with the periodic coefficients having the same˛ period. But in that paper the oscillation constant wasn't obtained for the periodic functions having different periods and consequently for the case when the least common multiple of these periodic coefficients is not defined. Thus in this paper we investigate the oscillation constant for Eq.(7) with periodic coefficients having different periods. For the sake of simplicity, we usually use the same notations as in the paper [6].
This paper is organized as follows. In section 2, we recall the concept of half-linear trigonometric functions and their properties. In section 3 we compute the oscillation constant for Eq.(7) with periodic coefficients having different periods. Additionally, we show that if the same periods are taken, then our result compiles with the known result in [4] or if the same period˛, given in [6] can be chosen as the least common multiple of periods of the coefficients (which have different periods), then our result coincides with the result of [6]. Thus, our results extend and improve the results of [6]. Finally in the last section, we give an example to illustrate the importance of our result.

Preliminaries
We start this section with the recalling the concept of half-linear-trigonometric functions, see [3] or [4]. Consider the following special half-linear equation of the form and denote by x D x .t/ its solution given by the initial conditions x .0/ D 0; x 0 .0/ D 1: We see that the behavior of this solution is very similar to the classical sine function. We denote this solution by sin p t and its derivative as .sin p t / 0 D cos p t: These functions are 2 p periodic; where p WD which is an equation of the form as in Eq.(9), so the functions u and u 0 are also bounded. Let x be a nontrivial solution of Eq.(1) and we consider the half-linear Prüfer transformation which is introduced using the half-linear trigonometric functions ; then by using Eq.(11) we obtain v Dˆ.cot p '/ ; where sin p ' : If we use the fact that sin p t is a solution of Eq.(9), the function v satisfies the associated Riccati type At the same time, using the fact that w solves Eq. (12), we obtain Combinig the last equation with Eq.(12) we get Multiplying both sides of this equation by jsinp 'j p p 1 and using the half-linear Pythagorean identity Eq.(10), we obtain the equation which will play the fundamental role in our investigation. It is well-known that the nonoscillation of Eq. (7) is equivalent to the boundedness from above of the Prüfer angle ' given by Eq.(11) (see [6,10]). Next, we briefly mention the principal solution of nonoscillatory equation Eq.(1) in [5], which is defined via the minimal solution of the associated Riccati equation Eq. (12). Nonoscillation of Eq.(1) implies that there exist T 2 R and a solution We finish this section with a lemma without proof, to be used in the next section.
(ii) Let > 1 4 ; then for every positive 2; t he linear differential equation (1) and this equation is non-oscillatory. Then the minimal solution of associate Riccati equation (12) is positive for large t [6].

Main results
To prove the main result, we need the following lemmas. where with r, c and d are positive defined functions having different periodsˇ1;ˇ2;andˇ3 respectively and let where is one of the periodsˇ1;ˇ2;orˇ3: Then Â is a solution of Using integration by parts, we get .p 1/ log 2 jsin p ' . /j p d ds: By the fact that, We can rewrite this equation as Similarly as in [6] if we use integration by parts, we get and by using the definition of R; C and D we get This implies that Hence we get The computation of oscillation constant in Eq. (7) is based on the following lemma 1ˇ2 and Â is a solution of the differential equation where R; C; D are as in Lemma 3.1.
Proof. We rewrite Eq. (13) in the form This is the equation for Prüfer angle Â , which corresponds to the differential equation which is the same (using the formula .1 C x/˛D 1 C˛x C o .x/ as x ! 0) as the equation i.e., the same as x/ D 0: First, suppose that Let " > 0 be sufficiently small and let T be so large that jo .1/j < " and p is a Sturmian majorant of Eq. (14), i.e., nonoscillation of Eq.(15) implies nonoscillation of Eq. (14).
We will show that the function h .t/ D t for large t , then nonoscillation of Eq.(15) follows from Lemma 2.1. According to [5] ,we have for h .t / D t where H is a real constant. At the same time, by a direct computation, for large t; if " < ı p ; so we see that Eq.(16) really holds, hence Eq.(13) is non-oscillatory, i.e., the "Prüfer angle " of its solution is bounded. Let " > 0 be again sufficiently small and let T be so large that o .1/ < " and 1 for t T: Then the equation is a Sturmian minorant of Eq. (14), i.e., oscillation of Eq.(17) implies oscillation of Eq. (14). Suppose, by contradiction, that Eq.(17) is nonoscillatory and let w be the minimal solution of its associated Riccati equation and v D h p .w w h / : Then by a direct computation v is a solution of the equation Denote now f .t/ WD .1 C R .t // q 1 ; and F .v/ WD jv C p j q qˆ 1 . p / f v q p : Then we have At this extremal point F v D q p f p .1 q/ C f q 1 : Consequently, F .v/ Á log 2 t for sufficiently large t, where Á is any positive constant. Using this estimate in Eq.(18) we get v 0 C p Cı t log 2 t C .p 1/ t 1 C " and we see that for large t, where the constant > 0 depends on " and Á and p C ı > 0 if "; Á are sufficiently small. Hence v 0 .t / < 0 for large t , which means that there exists the limit v 0 D lim t !1 v .t/ : This limit is finite, because of for large t: From Eq.(18), using the fact that q .q 1/ˆ 1 . p / " for sufficiently large T . When t ! 1 , we obtain the convergence of the integral R 1 t 1 dt which is a contradiction, so necessarily ! must be equal to zero, which means that v 0 D 0. Using second order Taylor's expansion , we have for jvj sufficiently small. Hence, for t sufficiently large The last inequality is the Riccati type inequality associated with the linear second order differential equation Proof. The statement (i) is proved in [13] when ¤ .
(ii) We consider Eq. (7), let x be the nontrivial solution of Eq. (7) and ' is the Prüfer angle of Eq. (7) given by Eq. (11). Then ' is a solution of By the help of Lemma 3.1, Â is a solution of where R; C and D are as in Lemma 3.  (7) is oscillatory. This result also shows that Eq.(6) is nonoscillatory in the limiting case D since this case corresponds to Eq.(7) with D 0 < :The proof is now complete.
Corollary 3.4. If the periods of the functions r, c and d in Eq.(7) coincide with˛-period, which is given in [6] we get forˇ1 Dˇ2 Dˇ3 D˛ W Dˇp Thus in this case our oscillation constant reduces to rd given in [6] and the main result compiles with the result given by [6].
Corollary 3.5. In a similar way it is easy to see that if there exists a lcm .ˇ1;ˇ2;ˇ3/ and the period˛, given in [6] is chosen as the number lcm .ˇ1;ˇ2;ˇ3/, then there exist some natural numbers m; l; and s such as˛D mˇ1 D Thus in this case our, oscillation constant reduces to rd given in [6] and the main result compiles with the result given by [6].
Remark 3.6. If lcm .ˇ1;ˇ2;ˇ3/ is not defined, then we can not use the results of Remark 3.4 and Remark 3.5. However, only our result can be applied while the result given in [6,8] cannot be applied. Here the important point to note is that while we cannot apply the Theorem 3.1 in [6] for this example if we choose a D p 3 then lcm 3 ; 4 ; 2 jaj Á is not defined, we can apply our theorem Theorem 3.3.
Finally, as a future work this paper can be improved if we replace the periodic coefficient functions, having different periods, with asymptotically almost periodic coefficients or having different mean values coefficients functions. The conditional oscillation of half-linear equations with asymptotically almost periodic coefficient or coefficients having mean values are studied in [13][14][15].