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formerly Central European Journal of Mathematics

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

• Corresponding author
• Department of Mathematics, Faculty of Science, Gazi University, Teknikokullar, 06500 Ankara, Turkey
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• Other articles by this author:
/ Banu Mermerkaya
• Department of Mathematics, Graduate School of Natural and Applied Sciences, Gazi University, Teknikokullar, 06500 Ankara, Turkey
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Published Online: 2017-05-04 | DOI: https://doi.org/10.1515/math-2017-0046

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.

MSC 2010: 34C10

1 Introduction

An equation of the form $(r(t)Φ(x′))′+c(t)Φ(x)=0,Φ(s)=|s|p−2s,p>1,$(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 half-linear 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 = 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 $(r(t)Φ(x′))′+γc(t)Φ(x)=0$(2)

with positive coefficients is conditionally oscillatory if there exists a constant γ0 such that Eq.(2) is oscillatory for all γ > γ0 and nonoscillatory for all γ < γ0. The constant γ0 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 [5-10] 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 $x″+γt2x=0,$(3)

(which is special case of Eq.(1) for $p=2,r\left(t\right)=1\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}c\left(t\right)=\frac{1}{{t}^{2}}$ has been identified ${\gamma }_{0}=\frac{1}{4}$ and Eq.(3) is oscillatory if $\gamma >\frac{1}{4},$ nonoscillatory if $\gamma >\frac{1}{4}.$ Additionally, $x\left(t\right)={c}_{1}\sqrt{t}+{c}_{2}\sqrt{t}\mathrm{log}t$ is the general solution of Eq.(3) and nonoscillatory for $\gamma >\frac{1}{4}.$

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 $(Φ(x′))′+γtpΦ(x)=0,$(4)

and the half-linear Riemann-Weber differential equation of the form $(Φ(x′))′+1tpγ+μlog2⁡tΦ(x)=0,$(5)

was considered and it was shown that Eq.(4) is nonoscillatory if and only if $\gamma \le {\gamma }_{p}:={\left(\frac{p-1}{p}\right)}^{p}$ and Eq.(5), with is nonoscillatory if $\mu <{\mu }_{p}:=\frac{1}{2}{\left(\frac{p-1}{p}\right)}^{p-1}$ and oscillatory if μ > μp:

In [8], the half-linear differential equation of the form $(r(t)Φ(x′))′+γc(t)tpΦ(x)=0,$(6)

was considered for r, c being α—periodic, positive functions and it was shown that Eq.(6) is oscillatory if γ > K and nonoscillatory if γ < K, where K is given by $K=q−p1α∫0αdτrq−11−p1α∫0αc(τ)dτ−1$

for p and q are conjugate numbers, i.e., $\frac{1}{p}+\frac{1}{q}=1.$ 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 $(r(t)Φ(x′))′+1tpγc(t)+μd(t)log2⁡tΦ(x)=0,$(7)

was considered positive, α—periodic functions r; c and d; which are defined on [0, ∞) and it was shown that Eq.(6) is oscillatory if and only if γγrc , where γrc is given by $γrc:=αpγp∫0αr1−q(t)dtp−1∫0αc(t)dt,$

and in the limiting case γ = γrc Eq.(7) is nonoscillatory if μ < μrd and it is oscillatory if μ > μrd where μrd is given by $μrd=αpμp∫0αr1−q(t)dtp−1∫0αd(t)dt.$

If the functions r, c, and d are positive constants, then Eq.(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 $(r(t)Φ(x′))′+c(t)tpΦ(x)=0$(8)

was considered for r: [a, ∞) → ℝ ,(a > 0) where r is a continuous function for which mean value $M\left({r}^{1-q}\right):=\underset{t\to 1}{lim}\frac{1}{t}\underset{a}{\overset{a+t}{\int }}{r}^{1-q}\left(\tau \right)d\tau$ exists and for which $0

and c: [a, ∞) → ℝ ,(a > 0), be a continuous function having mean value $M\left(c\right):=\underset{t\to 1}{lim}\frac{1}{t}\underset{a}{\overset{a+t}{\int }}c\left(\tau \right)d\tau$ and it was shown that Eq.(8) is oscillatory if M (c) > Γ and nonoscillatory if M (c) > Γ, where Γ is given by $Γ=q−p[M(r1−q)]1−p.$

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 α—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 α, 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.

2 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 $(Φ(x′))′+(p−1)Φ(x)=0,$(9)

and denote by x = x (t) its solution given by the initial conditions x (0) = 0, x′(0) = 1: We see that the behavior of this solution is very similar to the classical sine function. We denote this solution by sinp t and its derivative as (sinp t)′ =cosp t. These functions are pperiodic; where ${\pi }_{p}:=\frac{2\pi }{p\mathrm{sin}\left(\frac{\pi }{p}\right)}$ and satisfy the half-linear Pythagorean identity $|sinp⁡t|p+|cosp⁡t|p=1,t∈R.$(10)

All solutions of Eq.(9) are of the form x (t) = C sinp (t + φ), where C, φ are real constants. All these solutions and their derivatives are bounded and periodic with the period p. The function u = Φ (cosp t) is a solution of the reciprocal equation to Eq.(9); $Φ−1(u′)′+(p−1)q−1Φ−1(u)=0,Φ−1(u)=|u|q−2u,q=pp−1,$

which is an equation of the form as in Eq.(9), so the functions u and u' 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 $x(t)=ρ(t)sinp⁡φ(t),x′(t)=r1−q(t)ρ(t)tcosp⁡φ(t),$(11)

where $\rho \left(t\right)=\sqrt{|x\left(t\right){|}^{p}+{r}^{q}\left(t\right)|{x}^{\prime }\left(t\right){|}^{p}}$ and Prüfer angle φ is a continuous function defined at all points where x (t)≠0.

Let $w\left(t\right)=r\left(t\right)\mathrm{\Phi }\left(\frac{{x}^{\prime }\left(t\right)}{x\left(t\right)}\right)$ then w(t) is the solution of Riccati equation $w′+c(t)+(p−1)r1−q(t)|w|q=0,$(12)

associated with Eq.(1).

Let $v\left(t\right)={t}^{p-1}w\left(t\right)={t}^{p-1}r\left(t\right)\mathrm{\Phi }\left(\frac{{x}^{\prime }\left(t\right)}{x\left(t\right)}\right),$ then by using Eq.(11) we obtain v = Φ (cotp φ), where ${\mathrm{cot}}_{p}\phi =\frac{{\mathrm{cos}}_{p}\phi }{{\mathrm{sin}}_{p}\phi }.$ If we use the fact that sinp t is a solution of Eq.(9), the function v satisfies the associated Riccati type equation $v′=[1−p+(1−p)|Φ(cotp)|q]φ′.$

At the same time, using the fact that w solves Eq.(12), we obtain $v′=(p−1)tp−2w−tp−1c(t)+(p−1)r1−q(t)|w|q=vt(p−1)−tp−1c(t)−(p−1)tp−1r1−q(t)t1−pwq=p−1tv−tpc(t)p−1−r1−q(t)|cotp⁡φ|p.$

Combinig the last equation with Eq.(12) we get $(p−1)1+|cosp⁡φ|p|sinp⁡φ|pφ′=p−1ttpc(t)p−1+r1−q(t)|cosp⁡φ|p|sinp⁡φ|p−Φ(cospφ)Φ(sinpφ).$

Multiplying both sides of this equation by $\frac{|{\mathrm{sin}}_{p}\phi {|}^{p}}{p-1}$ and using the half-linear Pythagorean identity Eq.(10), we obtain the equation $φ′=1t[r1−q(t)|cospφ|p−Φ(cospφ)sinp⁡φ+tpc(t)p−1|sinp⁡φ|p],$

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 φ 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 ∈ ℝ and a solution $\stackrel{~}{w}$ of Eq.(12) defined on some interval $\left[{T}_{\stackrel{~}{w}},\mathrm{\infty }\right).\stackrel{~}{w}$ is called the minimal solution of all solutions of Eq.(12) and it satisfies the inequality $w\left(t\right)>\stackrel{~}{w}\left(t\right)$ where w is any other solution of Eq.(12) defined on some interval [Tw, ∞) and then $\stackrel{~}{x}$ is the principal solution of Eq.(1) via the formula $\stackrel{~}{w}\left(t\right)=r\left(t\right)\mathrm{\Phi }\left(\frac{\stackrel{~}{x}\left(t\right)}{\stackrel{~}{x}\left(t\right)}\right).$

We finish this section with a lemma without proof, to be used in the next section.

Lemma 2.1

1. Eq.(1) is non-oscillatory if and only if there exists a positive differentiable function h such that $(r(t)Φ(h′))′+c(t)Φ(h)≤0$ for large t.

2. Let $\mu >\frac{1}{4},$ then for every positive,the linear differential equation $(tx′)′+∈log2⁡tx′+μtlog2⁡tx=0$ is oscillatory.

3. Suppose that $\int {r}^{1-q}\left(t\right)dt=\mathrm{\infty },c\left(t\right)>0$ for large t in Eq.(1) and this equation is non-oscillatory. Then the minimal solution of associate Riccati equation (12) is positive for large t [6].

3 Main results

To prove the main result, we need the following lemmas.

Lemma 3.1

Let φ = φ1 + φ2 + φ3, + φ4 + Μ, (Μ is a suitable constant) be a solution of the equation $φ′(t)=φ1′(t)+φ2′(t)+φ3′(t)+φ4′(t),$

where $φ1′=1tr1−q(t)|cosp⁡φ(t)|p,φ2′=−1tΦ(cosp⁡φ(t))sinp⁡φ(t),φ3′=c(t)(p−1)t|sinp⁡φ(t)|p,φ4′=d(t)(p−1)tlog2⁡t|sinp⁡φ(t)|p,$

with r, c and d are positive defined functions having different periods β1, β2, and β3, respectively and let $θ(t)=1β1∫tt+β1φ1(s)ds+1ξ∫tt+ξφ2(s)ds+1β2∫tt+β2φ3(s)ds+1β3∫tt+β3φ4(s)ds,$

where ξ is one of the periods β1, β2, or β3 Then θ is a solution of $θ′=1tR|cosp⁡θ|p−Φ(cosp⁡θ)sinp⁡θ+1p−1(C+Dlog2⁡t)|sinp⁡θ|p+Oltlog2⁡t,$

where $R=1β1∫0β1r1−q(τ)dτ,C=1(p−1)β2∫0β2c(τ)dτandD=1(p−1)β3∫0β3d(τ)dτ$

and φ (τ) — θ (t) = o (1) as t → ∞.

Proof

Taking derivative of θ (t ), we have $θ′(t)=[1β1∫tt+β1φ1′(s)ds+1ξ∫tt+ξφ2′(s)ds+1β2∫tt+β2φ3′(s)ds+1β3∫tt+β3φ4(s)ds]=1β1∫tt+β11sr1−q(s)|cosp⁡φ(s)|pds$ $−1ξ∫tt+ξ1sΦ(cosp⁡φ(s))sinp⁡φ(s)ds+1β2∫tt+β2c(s)(p−1)s|sinp⁡φ(s)|pds+1β3∫tt+β3d(s)(p−1)slog2⁡s|sinp⁡φ(s)|pds.$

Using integration by parts, we get $θ′(t)=1β1t∫tt+β1r1−q(τ)|cosp⁡φ(τ)|pdτ−1ξt∫tt+ξΦ(cosp⁡φ(τ))sinp⁡φ(τ)dτ+1β2t∫tt+β2c(τ)(p−1)|sinp⁡φ(τ)|pdτ+1β3t∫tt+β3d(τ)(p−1)log2⁡τ|sinp⁡φ(τ)|pdτ−1β1∫tt+β11s2∫st+β1r1−q(τ)|cosp⁡φ(τ)|pdτds+1ξ∫tt+ξ1s2∫st+ξΦ(cosp⁡φ(τ))sinp⁡φ(τ)dτds−1β2∫tt+β21s2∫st+β2c(τ)(p−1)|sinp⁡φ(τ)|pdτds−1β3∫tt+β31s2∫st+β3d(τ)(p−1)log2⁡τ|sinp⁡φ(τ)|pdτds.$

By the fact that, ${\int }_{t}^{t+T}f\left(s\right)ds={\int }_{0}^{T}f\left(s\right)ds$ ds for any T–periodic function and half-linear Pythagorean identity, the expressions $r1−q(t)|cospφ|p,−Φ(cosp,φ)sinp,⁡φc(t)p−1|sinp⁡φ|pandd(t)p−1|sinp⁡φ|p$

are bounded. Thus we get $θ′(t)=1β1t∫tt+β1r1−q(τ)|cosp⁡φ(τ)|pdτ−1ξt∫tt+ξΦ(cosp⁡φ(τ))sinp⁡φ(τ)dτ+1β2t∫tt+β2c(τ)(p−1)|sinp⁡φ(τ)|pdτ+1β3t∫tt+β3d(τ)(p−1)log2⁡τ|sinp⁡φ(τ)|pdτ+O(ltlog2⁡t).$

We can rewrite this equation as $θ′(t)=1β1t∫tt+β1r1−q(τ)|cosp⁡θ(t)|pdτ−1ξt∫tt+ξΦ(cosp⁡θ(t))sinp⁡θ(t)dτ+1β2t∫tt+β2c(τ)(p−1)|sinp⁡θ(t)|pdτ+1β3t∫tt+β3d(τ)(p−1)log2⁡τ|sinp⁡θ(t)|pdτ+1β1t∫tt+β1r1−q(τ){|cosp⁡φ(τ)|p−|cosp⁡θ(t)|p}dτ−1ξt∫tt+ξ{Φ(cosp⁡φ(τ))sinp⁡φ(τ)−Φ(cosp⁡θ(t))sin⁡θ(t)}dτ+1β2t∫tt+β2c(τ)(p−1){|sinp⁡φ(t)|p−|sinp⁡θ(t)|p}dτ+1β3t∫tt+β3d(τ)(p−1)log2⁡τ{|sinp⁡φ(t)|p−|sinp⁡θ(t)|p}dτ+Oltlog2⁡t.$

Similarly as in [6] if we use integration by parts, we get $∫tt+β3d(s)log2⁡sds=(p−1)β3Dlog2⁡t+Oltlog3⁡t$

and by using the definition of R, C and D we get $θ′(t)=1t[R|cosp⁡θ(t)|p−Φ(cosp⁡θ(t))sinp⁡θ(t)+1p−1(C+Dlog2⁡t)|sinp⁡θ(t)|p]+1β1t∫tt+β1r1−q(τ){|cosp⁡φ(τ)|p−|cosp⁡θ(t)|p}dτ−1ξt∫tt+ξ{Φ(cosp⁡φ(τ))sinp⁡φ(τ)−Φ(cosp⁡θ(t))sin⁡θ(t)}dτ+1β2t∫tt+β2c(τ)(p−1){|sinp⁡φ(τ)|p−|sinp⁡θ(t)|p}dτ$ $+1β3t∫tt+β3d(τ)(p−1)log2⁡τ{|sinp⁡φ(t)|p−|sinp⁡θ(t)|p}dτ+Oltlog2⁡t.$

And using the half-linear trigonometric functions, we have $|cosp⁡φ(τ)|p−|cosp⁡θ(t)|p≤p∫θ(t)φ(τ)|Φ(cosp⁡s)(cosp⁡s)′|ds≤const|φ(τ)−θ(t)|,$ $|Φ(cosp⁡φ(τ))sinp⁡φ(τ)−Φ(cosp⁡θ(t))sinp⁡θ(t)|≤∫θ(t)φ(τ)|(Φ(cosp⁡s)sinp⁡s)′|ds≤const|φ(τ)−θ(t)|$

and $||sinp⁡φ(t)|p−|sinp⁡θ(t)|p≤const|φ(τ)−θ(t)|.$

By the mean value theorem we can write $θ(t)=φ1(t1)+φ2(t2)+φ3(t3)+φ4(t4)$

for t1∈ [t, t + β1], t2 ∈ [t, t + ξ], t3 ∈ [t, t + β2], t4 ∈ [t, t + β3]. Thus $|φ(τ)−θ(t)|≤|φ1(τ)−φ1(t1)|+|φ2(τ)−φ2(t2)|+|φ3(τ)−φ3(t3)|+|φ4(τ)−φ4(t4)|.$

This implies that $|φ(τ)−θ(t)|≤o1tast→∞,φ(τ)−θ(t)=o(1).$

Hence we get $θ′=1tR|cosp⁡θ|p−Φ(cosp⁡θ)sinp⁡θ+1p−1(C+Dlog2⁡t)|sinp⁡θ|p+O(ltlog2⁡t).$ □

The computation of oscillation constant μ in Eq.(7) is based on the following lemma

Lemma 3.2

Let ${\left(\underset{0}{\overset{{\beta }_{1}}{\int }}{r}^{1-q}\left(t\right)dt\right)}^{p-1}\underset{0}{\overset{{\beta }_{2}}{\int }}c\left(t\right)dt={\gamma }_{p}{\beta }_{1}^{p-1}{\beta }_{2}$ and θ is a solution of the differential equation $θ′=1tR|cosp⁡θ|p−Φ(cosp⁡θ)sinp⁡θ+1p−1(C+Dlog2⁡t)|sinp⁡θ|p+oltlog2⁡t,$(13)

where R,C, D are as in Lemma 3.1. $If∫0β1r1−q(t)dtp−1∫0β3d(t)dt>μpβ1p−1β3,thenθ(t)→∞ast→∞$

and $If∫0β1r1−q(t)dtp−1∫0β3d(t)dt<μpβ1p−1β3,thenθ(t)isbounded.$

Proof

We rewrite Eq.(13) in the form $θ′=1tR|cosp⁡θ|p−Φ(cosp⁡θ)sinp⁡θ+1p−1(C+Dlog2⁡t)|sinp⁡θ|p$ $+o(1)tlog2⁡t(|cosp⁡θ|p+|sinp⁡θ|p).$

This is the equation for Prüfer angle θ, which corresponds to the differential equation $R+o(1)log2⁡t1−pΦx′′+p−1tpC+D+o(1)log2⁡tΦ(x)=0,$

which is the same (using the formula (1 + x)α = 1 + αx + ο (x) as x →0) as the equation $1+o(1)log2⁡tΦx′′+p−1tpRp−1C+Rp−1D+o(1)log2⁡tΦ(x)=0,$(14)

i.e., the same as $1+o(1)log2⁡tΦx′′+1tpγp+(p−1)Rp−1D+o(1)log2⁡tΦ(x)=0.$

First, suppose that $1β1p−1∫0β1r1−q(t)dtp−11β3∫0β3d(t)dt<μp$

and denote $δ:=12μp−1β1p−1∫0β1r1−q(t)dtp−11β3∫0β3d(t)dt.$

Let ε > 0 be sufficiently small and let Τ be so large that $|o(1)|<ϵandμp−1β1p−1∫0β1r1−q(t)dtp−11β3∫0β3d(t)dt+o(1)>δ$

for tT. Then, the equation $1−ϵlog2⁡tΦx′′+1tpγp+μp−δlog2⁡tΦ(x)=0$(15)

is a Sturmian majorant of Eq.(14), i.e., nonoscillation of Eq.(15) implies nonoscillation of Eq.(14).

We will show that the function $h\left(t\right)={t}^{\frac{p-1}{p}}{\mathrm{log}}^{\frac{1}{p}}t$ satisfies $1−ϵlog2⁡tΦh′′+1tpγp+μp−δlog2⁡tΦ(h)≤0$(16)

for large t, then nonoscillation of Eq.(15) follows from Lemma 2.1. According to [5] ,we have for $h\left(t\right)={t}^{\frac{p-1}{p}}{\mathrm{log}}^{\frac{1}{p}}t$ $h(t)Φh′′+1tpγp+μplog2⁡tΦ(h)∼Htlog2⁡t$

for t →∞, where Η is a real constant. At the same time, by a direct computation, $−hϵΦ(h′)log2⁡t′−δhptplog2⁡t∼ϵγp−δtlog⁡t<0$

for large t, if $\epsilon <\frac{\delta }{{\gamma }_{p}},$ so we see that Eq.(16) really holds, hence Eq.(13) is non-oscillatory, i.e., the “Prüfer angle” of its solution is bounded.

Now suppose that $1β1p−1∫0β1r1−q(t)dtp−11β3∫0β3d(t)>μp$

and denote $δ:=121β1p−1∫0β1r1−q(t)dtp−11β3∫0β3d(t)dt−μp.$

Let ε > 0 be again sufficiently small and let Τ be so large that $o(1)<εand1β1p−1∫β1r1−q(t)dtp−11β3∫β3d(t)dt−μp+o(1)>δ$

for tT. Then the equation $1+εlog2⁡tΦx′′+1tpγp+μp+δlog2⁡tΦ(x)=0$(17)

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 $w′+1tpγp+μp+δlog2⁡t+(p−1)(1+R(t))1−q|w|q=0,$

where we denote $R\left(t\right):=\frac{\epsilon }{{\mathrm{log}}^{2}t}.$

Let $h\left(t\right)=t{,}^{\frac{p-1}{p}}{w}_{h}=\mathrm{\Phi }\left(\frac{{h}^{\prime }}{h}\right)={\left(\frac{p-1}{p}\right)}^{p-1}{t}^{1-p}$ and denote ${\lambda }_{p}={\left(\frac{p-1}{p}\right)}^{p-1}$ and v = hp (w – wh).

Then by a direct computation v is a solution of the equation $v′+μp+δtlog2⁡t+(p−1)t(1+R(t))1−q|v+λp|q−q(1+R(t))q−1Φ−1(λp)v−λq−q(1+R(t))q−1Φ−1(λp)v−λq+(p−1)tλpq1−(1+R(t))1−q=0.$(18)

Denote now ƒ (t) := (1 + R(t)q-1, and $F\left(v\right):=|v+{\lambda }_{p}{|}^{q}-q{\mathrm{\Phi }}^{-1}\left({\lambda }_{p}\right)fv-{\lambda }_{p}^{q}.$ Then we have $F′(v)=q[Φ−1(v+λp)−Φ−1(λp)f]=0⇔v=v∗=λp(Φ(f)−1).$

At this extremal point $F(v∗)=λpq[fp(1−q)+fq−1].$

Substituting f (t) = (1 + R(t))q-1into F (v*) and using again the formula (1 + x)α = 1 +αx + o(x)as x → 0,

we obtain $F(v∗)=λpq(1+R(t))p(q−1)(1−q)+q(1+R(t))q−1−1=λpqo(R)=o(R),ast→∞.$

Consequently, $F\left(v\right)\ge -\frac{\eta }{{\mathrm{log}}^{2}t}$ for sufficiently large t, where η is any positive constant. Using this estimate in Eq.(18)

we get $v′+μp+δtlog2⁡t+(p−1)t1+εlog2⁡t1−q|v+λp|q−1+εlog2⁡tq−1Φ−1(λp)v−λpq}+(p−1)tλpq[1−1+(q−1)εlog2⁡t+o(1)log2⁡t]=0$

and we see that $v′+μp+δtlog2⁡t−ζlog2⁡t≤0,$

for large t, where the constant ζ > 0 depends on ε and η and μρ + δζ > 0 if ε, η are sufficiently small. Hence v' (t) < 0 for large t, which means that there exists the limit v0= limt→∞ v (t). This limit is finite, because of Lemma 2.1, w (t) > 0, i.e., v (t) = hp (t) (w (t) – wh (t)) ≥ -hp (t) wh(t) = –λρ.

Next, we show that v0 = 0. If v0 ≠0, then $|{v}_{0}+{\lambda }_{p}{|}^{q}-q{\mathrm{\Phi }}^{-1}\left({\lambda }_{p}\right){v}_{0}-{\lambda }_{p}^{q}=:\omega >0,$ i.e., $|v(t)+λp|q−qΦ−1(λp)v(t)−λpq>ω2$

for large t. From Eq.(18), using the fact that $F(v)=|v+λp|q−qΦ−1(λp)v−λpq+q(q−1)Φ−1(λp)ϵlog2⁡t+o(1)log2⁡tv$

we have $v(t)−v(T)+∫Ttμp+δslog2⁡sds+∫Tt(p−1)ω2sds+∫TtO(1)slog2⁡s≤0$

i.e., since v (t) > – λρ $∫Ttμp+δslog2⁡sds+∫Tt(p−1)ω2sds+∫TtO(1)slog2⁡s≤λp+v(T)$

for sufficiently large Τ. When t → ∞, we obtain the convergence of the integral ${\int }^{\mathrm{\infty }}{t}^{-1}dt$ which is a contradiction, so necessarily ω must be equal to zero, which means that v0 = 0. Using second order Taylor’s expansion , we have $(p−1)|v+λp|q−qΦ−1(λp)v−λpq≥(1−ε)q2λpq−2v2$

for |v| sufficiently small. Hence, for t sufficiently large $v′+μp+δtlog2⁡t+(1−ε)q2tλpq−2v2+o(1)tlog2⁡tv≤0.$

The last inequality is the Riccati type inequality associated with the linear second order differential equation $(tx′)′+o(1)log2⁡tx′+(1−ε)qλpq−2(μp+δ)2tlog2⁡tx=0,$(19)

which means that Eq.(19) is nonoscillatory. However, if we compute the value of the coefficient by x in Eq.(19), we get the value $q2p−1p(p−1)(q−2)12p−1p(p−1)1+δμp(1−ε)=141+δμp(1−ε)$

and the value of this constant is greater than $\frac{1}{4}$ if ε is sufficiently small, i.e., Eq.(19) is oscillatory by Lemma 2.1 which is the required contradiction implying that Eq.(13) is oscillatory.

The main results of this paper are as follows

Theorem 3.3

1. Let r, c and d be positive defined functions in Eq.(6) and Eq.(7) having different periods β1, β2 and β3, respectively. Then Eq.(6) is nonoscillatory if and only if $γ≤γ∗:=β1p−1β2γp∫0β1r1−q(t)dtp−1∫0β2c(t)dt,$

where ${\gamma }_{p}:={\left(\frac{p-1}{p}\right)}^{p}.$

2. In the limiting case γ = γ*, Eq.(7) is nonoscillatory if $μ<μ∗:=β1p−1β3μp∫0β1r1−q(t)dtp−1∫0β3d(t)dt$

and it is oscillatory if μ > μ*, where ${\mu }_{p}:=\frac{1}{2}{\left(\frac{p-1}{p}\right)}^{p-1}.$

Proof

The statement (i) is proved in [13] when γ γ*.

(ii) We consider Eq.(7), let x be the nontrivial solution of Eq.(7) and φ is the Prüfer angle of Eq.(7) given by Eq.(11). Then φ is a solution of $φ′=1tr1−q(t)|cosp⁡φ|p−Φ(cosp,φ)sinp⁡φ+1(p−1)γc(t)+μd(t)log2⁡t|sinp⁡φ|p.$

By the help of Lemma 3.1, θ is a solution of $θ′=1tR|cosp⁡θ|p−Φ(cosp⁡θ)sinp⁡θ+γC+μDlog2⁡t|sinp⁡φ|p+oltlog2⁡t,$

where R, C and D are as in Lemma 3.1. By the help of Lemma 3.2 if $μ<μ∗:=μpβ1p−1β3∫0β1r1−q(t)dtp−1∫0β3d(t)dt$

holds, then θ is bounded and by the help of Lemma 3.1, φ is bounded. Then from Eq.(11), Eq.(7) is nonoscillatory. Again by the help of Lemma 3.2, if $μ>μ∗:=μpβ1p−1β3∫0β1r1−q(t)dtp−1∫0β3d(t)dt,$

holds, then θ (t)⊒∞as t ⊒∞and by the help of Lemma 3.1, φ is unbounded. Then from Eq.(11), Eq.(7) is oscillatory. This result also shows that Eq.(6) is nonoscillatory in the limiting case γ = γ* since this case corresponds to Eq.(7) with μ = 0 < μ*. The proof is now complete.  □

Corollary 3.4

If the periods of the functions r, c and d in Eq.(7) coincide with α-period, which is given in [6] we get for β1 = β2 = β3 = α $μ>μ∗:=μpβ1p−1β3∫0β1r1−q(t)dtp−1∫0β3d(t)dt=αpμp∫0αr1−q(t)dtp−1∫0αd(t)dt=μrd.$

Thus in this case our oscillation constant μ* reduces to μrd 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 1cm (β1, β2, β3) and the period a, given in [6] is chosen as the number 1cm (β1, β2, β3), then there exist some natural numbers m,l, and s such as α = mβ1 = 2 = 3 and by the help of periodic functions properties we get $μ∗:=β1p−1β3μp∫0β1r1−q(t)dtp−1∫0β3d(t)dt=(αm)p−1(αs)μp1m∫0αr1−q(t)dtp−11s∫0αd(t)dt=μrd.$

Thus in this case our, oscillation constant μ* reduces to μrd given in [6] and the main result compiles with the result given by [6].

Remark 3.6

If 1cm(β1, β23) 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.

Example 3.7

Consider the equation $12+cos⁡6tΦx′′+1tpγ(2+cos⁡8t)+μ(2+cos⁡(ax+b))log2⁡tΦ(x)=0$(20)

which is Eq.(7) for $q=3,r\left(t\right)=\frac{1}{2+\mathrm{cos}6t},c\left(t\right)=2+\mathrm{cos}8t$ and d (t) = 2 + cos (ax + b),(a,b ∈ ℝ). In this case $r\left(t\right)=\frac{1}{2+\mathrm{cos}6t}$ defined for all t ∈with period $\frac{\pi }{3},c\left(t\right)=2+\mathrm{cos}8t$ is positive defined for all t ∈ R with period $\frac{\pi }{4},$ and d (t) = 2 + cos (ax + b) > 0 can be considered as positive defined function with period $\frac{2\pi }{|a|}.$

Thus we can apply Theorem 3.3 for all a ≠0 and we obtain an oscillation constant for the differential equation Eq.(20) $μ∗=β1p−1β3μp(∫0β1r1−q(t)dt)p−1∫0β3d(t)dt=(π3)122π|a|12(13)12∫0π3(2+cos⁡6t)2dt12∫02π|a|(2+cos⁡(ax+b))dt=1122.$

and equation Eq.(20) is nonoscillatory if and only if $\mu <\frac{1}{12\sqrt{2}}.$ This computation shows that we can compute the oscillation constant μ* for every period $\frac{2\pi }{|a|}>0.$ If the functions r (t) , c (t) , and d(t) are having an $\alpha =\text{1cm}\left(\frac{\pi }{3},\frac{\pi }{4},\frac{2\pi }{|a|}\right)$ period it is well known that there exists some m, l and s natural numbers such as $\alpha =m\frac{\pi }{4}=l\frac{\pi }{3}=s\frac{2\pi }{|a|}.$ In this case we use the fact of the remark 3.4 and we can apply Theorem 3.1 in [6] to the above example and we get oscillation constant as $μrd=αpμp∫0αr1−q(t)dtp−1∫0αd(t)dt123s2π|a|32l∫0π3(2+cos⁡6t)2dt12s∫02π|a|(2+cos⁡(ax+b))dt=1122.$

Here the important point to note is that while we cannot apply the Theorem 3.1 in [6] for this example if we choose $a=\sqrt{3}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathit{t}\mathit{h}\mathit{e}\mathit{n}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{lcm}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\left(\frac{\pi }{3},\frac{\pi }{4},\frac{2\pi }{|a|}\right)$ 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 [13-15].

Acknowledgement

The authors would like to express their sincere gratitude to the referees for a number of valuable comments and suggestions which led to significant improvement of the final version of the paper.

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Accepted: 2017-02-28

Published Online: 2017-05-04

Conflict of interestsThe authors declare that there is no conflict of interest regarding the publication of this paper.

Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 548–561, ISSN (Online) 2391-5455,

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