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

#### Lemma 3.1

*Let φ = φ*_{1} + *φ*_{2} + *φ*_{3}, + *φ*_{4} + *Μ*, (*Μ is a suitable constant*)* be a solution of the equation*
$$\phi {}^{\prime}(t)={\phi}_{1}^{\prime}(t)+{\phi}_{2}^{\prime}(t)+{\phi}_{3}^{\prime}(t)+{\phi}_{4}^{\prime}(t),$$

*where*
$$\begin{array}{ll}{\phi}_{1}^{{}^{\prime}}& =\frac{1}{t}{r}^{1-q}(t)|{\mathrm{cos}}_{p}\phi (t){|}^{p},\\ {\phi}_{2}^{{}^{\prime}}& =-\frac{1}{t}\mathrm{\Phi}({\mathrm{cos}}_{p}\phi (t)){\mathrm{sin}}_{p}\phi (t),\\ {\phi}_{3}^{{}^{\prime}}& =\frac{c(t)}{(p-1)t}|{\mathrm{sin}}_{p}\phi (t){|}^{p},\\ {\phi}_{4}^{{}^{\prime}}& =\frac{d(t)}{(p-1)t{\mathrm{log}}^{2}t}|{\mathrm{sin}}_{p}\phi (t){|}^{p},\end{array}$$

*with r, c and d are positive defined functions having different periods β*_{1}*, β*_{2}*, and β*_{3}*, respectively and let*
$$\theta (t)=\frac{1}{{\beta}_{1}}\underset{t}{\overset{t+{\beta}_{1}}{\int}}{\phi}_{1}(s)ds+\frac{1}{\xi}\underset{t}{\overset{t+\xi}{\int}}{\phi}_{2}(s)ds+\frac{1}{{\beta}_{2}}\underset{t}{\overset{t+{\beta}_{2}}{\int}}{\phi}_{3}(s)ds+\frac{1}{{\beta}_{3}}\underset{t}{\overset{t+{\beta}_{3}}{\int}}{\phi}_{4}(s)ds,$$

*where ξ is one of the periods β*_{1}*, β*_{2}, or *β*_{3} *Then θ is a solution of*
$${\theta}^{\prime}=\frac{1}{t}\left[R|{\mathrm{cos}}_{p}\theta {|}^{p}-\mathrm{\Phi}({\mathrm{cos}}_{p}\theta ){\mathrm{sin}}_{p}\theta +\frac{1}{p-1}(C+\frac{D}{{\mathrm{log}}^{2}t})|{\mathrm{sin}}_{p}\theta {|}^{p}\right]+O\left(\frac{\mathrm{l}}{t{\mathrm{log}}^{2}t}\right),$$

*where*
$$R=\frac{1}{{\beta}_{1}}\underset{0}{\overset{{\beta}_{1}}{\int}}{r}^{1-q}(\tau )d\tau ,C=\frac{1}{(p-1){\beta}_{2}}\underset{0}{\overset{{\beta}_{2}}{\int}}c(\tau )d\tau \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathit{a}\mathit{n}\mathit{d}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}D=\frac{1}{(p-1){\beta}_{3}}\underset{0}{\overset{{\beta}_{3}}{\int}}d(\tau )d\tau $$

*and φ* (τ) — *θ* (*t*) = *o* (1) *as t* → ∞.

#### Proof

Taking derivative of *θ* (*t* ), we have
$$\begin{array}{ll}{\theta}^{\prime}(t)& =[\frac{1}{{\beta}_{1}}\underset{t}{\overset{t+{\beta}_{1}}{\int}}{\phi}_{1}^{{}^{\prime}}(s)ds+\frac{1}{\xi}\underset{t}{\overset{t+\xi}{\int}}{\phi}_{2}^{{}^{\prime}}(s)ds\\ & +\frac{1}{{\beta}_{2}}\underset{t}{\overset{t+{\beta}_{2}}{\int}}{\phi}_{3}^{{}^{\prime}}(s)ds+\frac{1}{{\beta}_{3}}\underset{t}{\overset{t+{\beta}_{3}}{\int}}{\phi}_{4}(s)ds]\\ & =\frac{1}{{\beta}_{1}}\underset{t}{\overset{t+{\beta}_{1}}{\int}}\frac{1}{s}{r}^{1-q}(s)|{\mathrm{cos}}_{p}\phi (s){|}^{p}ds\end{array}$$
$$\begin{array}{ll}& -\frac{1}{\xi}\underset{t}{\overset{t+\xi}{\int}}\frac{1}{s}\mathrm{\Phi}({\mathrm{cos}}_{p}\phi (s)){\mathrm{sin}}_{p}\phi (s)ds\\ & +\frac{1}{{\beta}_{2}}\underset{t}{\overset{t+{\beta}_{2}}{\int}}\frac{c(s)}{(p-1)s}|{\mathrm{sin}}_{p}\phi (s){|}^{p}ds\\ & +\frac{1}{{\beta}_{3}}\underset{t}{\overset{t+{\beta}_{3}}{\int}}\frac{d(s)}{(p-1)s{\mathrm{log}}^{2}s}|{\mathrm{sin}}_{p}\phi (s){|}^{p}ds.\end{array}$$

Using integration by parts, we get
$$\begin{array}{ll}& {\theta}^{\prime}(t)=\frac{1}{{\beta}_{1}t}\underset{t}{\overset{t+{\beta}_{1}}{\int}}{r}^{1-q}(\tau )|{\mathrm{cos}}_{p}\phi (\tau ){|}^{p}d\tau \\ & -\frac{1}{\xi t}\underset{t}{\overset{t+\xi}{\int}}\mathrm{\Phi}({\mathrm{cos}}_{p}\phi (\tau )){\mathrm{sin}}_{p}\phi (\tau )d\tau \\ & +\frac{1}{{\beta}_{2}t}\underset{t}{\overset{t+{\beta}_{2}}{\int}}\frac{c(\tau )}{(p-1)}|{\mathrm{sin}}_{p}\phi (\tau ){|}^{p}d\tau \\ & +\frac{1}{{\beta}_{3}t}\underset{t}{\overset{t+{\beta}_{3}}{\int}}\frac{d(\tau )}{(p-1){\mathrm{log}}^{2}\tau}|{\mathrm{sin}}_{p}\phi (\tau ){|}^{p}d\tau \\ & -\frac{1}{{\beta}_{1}}\underset{t}{\overset{t+{\beta}_{1}}{\int}}\frac{1}{{s}^{2}}\underset{s}{\overset{t+{\beta}_{1}}{\int}}{r}^{1-q}(\tau )|{\mathrm{cos}}_{p}\phi (\tau ){|}^{p}d\tau ds\\ & +\frac{1}{\xi}\underset{t}{\overset{t+\xi}{\int}}\frac{1}{{s}^{2}}\underset{s}{\overset{t+\xi}{\int}}\mathrm{\Phi}({\mathrm{cos}}_{p}\phi (\tau )){\mathrm{sin}}_{p}\phi (\tau )d\tau ds\\ & -\frac{1}{{\beta}_{2}}\underset{t}{\overset{t+{\beta}_{2}}{\int}}\frac{1}{{s}^{2}}\underset{s}{\overset{t+{\beta}_{2}}{\int}}\frac{c(\tau )}{(p-1)}|{\mathrm{sin}}_{p}\phi (\tau ){|}^{p}d\tau ds\\ & -\frac{1}{{\beta}_{3}}\underset{t}{\overset{t+{\beta}_{3}}{\int}}\frac{1}{{s}^{2}}\underset{s}{\overset{t+{\beta}_{3}}{\int}}\frac{d(\tau )}{(p-1){\mathrm{log}}^{2}\tau}|{\mathrm{sin}}_{p}\phi (\tau ){|}^{p}d\tau ds.\end{array}$$

By the fact that, ${\int}_{t}^{t+T}f(s)ds={\int}_{0}^{T}f(s)ds$
* ds* for any *T–*periodic function and half-linear Pythagorean identity, the expressions
$${r}^{1-q}(t)|{\mathrm{cos}}_{p}\phantom{\rule{thinmathspace}{0ex}}\phi {|}^{p},-\mathrm{\Phi}({\mathrm{cos}}_{p},\phantom{\rule{thinmathspace}{0ex}}\phi ){\mathrm{sin}}_{p,}\phi \frac{c(t)}{p-1}|{\mathrm{sin}}_{p}\phi {|}^{p}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\frac{d(t)}{p-1}|{\mathrm{sin}}_{p}\phi {|}^{p}$$

are bounded. Thus we get
$$\begin{array}{ll}{\theta}^{\prime}(t)=& \frac{1}{{\beta}_{1}t}\underset{t}{\overset{t+{\beta}_{1}}{\int}}{r}^{1-q}(\tau )|{\mathrm{cos}}_{p}\phi (\tau ){|}^{p}d\tau \\ & -\frac{1}{\xi t}\underset{t}{\overset{t+\xi}{\int}}\mathrm{\Phi}({\mathrm{cos}}_{p}\phi (\tau )){\mathrm{sin}}_{p}\phi (\tau )d\tau \\ & +\frac{1}{{\beta}_{2}t}\underset{t}{\overset{t+{\beta}_{2}}{\int}}\frac{c(\tau )}{(p-1)}|{\mathrm{sin}}_{p}\phi (\tau ){|}^{p}d\tau \\ & +\frac{1}{{\beta}_{3}t}\underset{t}{\overset{t+{\beta}_{3}}{\int}}\frac{d(\tau )}{(p-1){\mathrm{log}}^{2}\tau}|{\mathrm{sin}}_{p}\phi (\tau ){|}^{p}d\tau +O(\frac{\mathrm{l}}{t{\mathrm{log}}^{2}t}).\end{array}$$

We can rewrite this equation as
$$\begin{array}{ll}{\theta}^{\prime}(t)& =\frac{1}{{\beta}_{1}t}\underset{t}{\overset{t+{\beta}_{1}}{\int}}{r}^{1-q}(\tau )|{\mathrm{cos}}_{p}\theta (t){|}^{p}d\tau \\ & -\frac{1}{\xi t}\underset{t}{\overset{t+\xi}{\int}}\mathrm{\Phi}({\mathrm{cos}}_{p}\theta (t)){\mathrm{sin}}_{p}\theta (t)d\tau \\ & +\frac{1}{{\beta}_{2}t}\underset{t}{\overset{t+{\beta}_{2}}{\int}}\frac{c(\tau )}{(p-1)}|{\mathrm{sin}}_{p}\theta (t){|}^{p}d\tau \\ & +\frac{1}{{\beta}_{3}t}\underset{t}{\overset{t+{\beta}_{3}}{\int}}\frac{d(\tau )}{(p-1){\mathrm{log}}^{2}\tau}|{\mathrm{sin}}_{p}\theta (t){|}^{p}d\tau \\ & +\frac{1}{{\beta}_{1}t}\underset{t}{\overset{t+{\beta}_{1}}{\int}}{r}^{1-q}(\tau )\{|{\mathrm{cos}}_{p}\phi (\tau ){|}^{p}-|{\mathrm{cos}}_{p}\theta (t){|}^{p}\}d\tau \\ & -\frac{1}{\xi t}\underset{t}{\overset{t+\xi}{\int}}\{\mathrm{\Phi}({\mathrm{cos}}_{p}\phi (\tau )){\mathrm{sin}}_{p}\phi (\tau )-\mathrm{\Phi}({\mathrm{cos}}_{p}\theta (t))\mathrm{sin}\theta (t)\}d\tau \\ & +\frac{1}{{\beta}_{2}t}\underset{t}{\overset{t+{\beta}_{2}}{\int}}\frac{c(\tau )}{(p-1)}\{|{\mathrm{sin}}_{p}\phi (t){|}^{p}-|{\mathrm{sin}}_{p}\theta (t){|}^{p}\}d\tau \\ & +\frac{1}{{\beta}_{3}t}\underset{t}{\overset{t+{\beta}_{3}}{\int}}\frac{d(\tau )}{(p-1){\mathrm{log}}^{2}\tau}\{|{\mathrm{sin}}_{p}\phi (t){|}^{p}-|{\mathrm{sin}}_{p}\theta (t){|}^{p}\}d\tau \\ & +O\left(\frac{\mathrm{l}}{t{\mathrm{log}}^{2}t}\right).\end{array}$$

Similarly as in [6] if we use integration by parts, we get
$$\underset{t}{\overset{t+{\beta}_{3}}{\int}}\frac{d(s)}{{\mathrm{log}}^{2}s}ds=\frac{(p-1){\beta}_{3}D}{{\mathrm{log}}^{2}t}+O\left(\frac{\mathrm{l}}{t{\mathrm{log}}^{3}t}\right)$$

and by using the definition of *R, C* and *D* we get
$$\begin{array}{ll}{\theta}^{\prime}(t)& =\frac{1}{t}[R|{\mathrm{cos}}_{p}\theta (t){|}^{p}-\mathrm{\Phi}({\mathrm{cos}}_{p}\theta (t)){\mathrm{sin}}_{p}\theta (t)\\ & +\frac{1}{p-1}(C+\frac{D}{{\mathrm{log}}^{2}t})|{\mathrm{sin}}_{p}\theta (t){|}^{p}]\\ & +\frac{1}{{\beta}_{1}t}\underset{t}{\overset{t+{\beta}_{1}}{\int}}{r}^{1-q}(\tau )\{|{\mathrm{cos}}_{p}\phi (\tau ){|}^{p}-|{\mathrm{cos}}_{p}\theta (t){|}^{p}\}d\tau \\ & -\frac{1}{\xi t}\underset{t}{\overset{t+\xi}{\int}}\{\mathrm{\Phi}({\mathrm{cos}}_{p}\phi (\tau )){\mathrm{sin}}_{p}\phi (\tau )-\mathrm{\Phi}({\mathrm{cos}}_{p}\theta (t))\mathrm{sin}\theta (t)\}d\tau \\ & +\frac{1}{{\beta}_{2}t}\underset{t}{\overset{t+{\beta}_{2}}{\int}}\frac{c(\tau )}{(p-1)}\{|{\mathrm{sin}}_{p}\phi (\tau ){|}^{p}-|{\mathrm{sin}}_{p}\theta (t){|}^{p}\}d\tau \end{array}$$
$$\begin{array}{ll}& +\frac{1}{{\beta}_{3}t}\underset{t}{\overset{t+{\beta}_{3}}{\int}}\frac{d(\tau )}{(p-1){\mathrm{log}}^{2}\tau}\{|{\mathrm{sin}}_{p}\phi (t){|}^{p}-|{\mathrm{sin}}_{p}\theta (t){|}^{p}\}d\tau \\ & +O\left(\frac{\mathrm{l}}{t{\mathrm{log}}^{2}t}\right).\end{array}$$

And using the half-linear trigonometric functions, we have
$$\begin{array}{ll}\left||{\mathrm{cos}}_{p}\phi (\tau ){|}^{p}-|{\mathrm{cos}}_{p}\theta (t){|}^{p}\right|& \le \phantom{\rule{thinmathspace}{0ex}}p\phantom{\rule{thinmathspace}{0ex}}\left|\underset{\theta (t)}{\overset{\phi (\tau )}{\int}}|\mathrm{\Phi}({\mathrm{cos}}_{p}s)({\mathrm{cos}}_{p}s{)}^{\prime}|ds\right|\\ & \le \mathit{c}\mathit{o}\mathit{n}\mathit{s}\mathit{t}|\phi (\tau )-\theta (t)|,\end{array}$$
$$\begin{array}{ll}|\mathrm{\Phi}({\mathrm{cos}}_{p}\phi (\tau )){\mathrm{sin}}_{p}\phi (\tau )-\mathrm{\Phi}({\mathrm{cos}}_{p}\theta (t)){\mathrm{sin}}_{p}\theta (t)|& \le \left|\underset{\theta (t)}{\overset{\phi (\tau )}{\int}}|(\mathrm{\Phi}({\mathrm{cos}}_{p}s){\mathrm{sin}}_{p}s{)}^{\prime}|ds\right|\\ & \le \mathit{c}\mathit{o}\mathit{n}\mathit{s}\mathit{t}|\phi (\tau )-\theta (t)|\end{array}$$

and
$$||{\mathrm{sin}}_{p}\phi (t){|}^{p}-|{\mathrm{sin}}_{p}\theta (t){|}^{p}\le \mathit{c}\mathit{o}\mathit{n}\mathit{s}\mathit{t}|\phi (\tau )-\theta (t)|.$$

By the mean value theorem we can write
$$\theta (t)={\phi}_{1}({t}_{1})+{\phi}_{2}({t}_{2})+{\phi}_{3}({t}_{3})+{\phi}_{4}({t}_{4})$$

for *t*_{1}∈ [*t*, *t* + *β*_{1}]*, t*_{2} ∈ [*t, t + ξ*]*, t*_{3} ∈ [*t*, *t + β*_{2}], *t*_{4} ∈ [*t, t + β*_{3}]. Thus
$$|\phi (\tau )-\theta (t)|\le |{\phi}_{1}(\tau )-{\phi}_{1}({t}_{1})|+|{\phi}_{2}(\tau )-{\phi}_{2}({t}_{2})|+|{\phi}_{3}(\tau )-{\phi}_{3}({t}_{3})|+|{\phi}_{4}(\tau )-{\phi}_{4}({t}_{4})|.$$

This implies that
$$|{\displaystyle \phi (\tau )-\theta (t)|\le o\left(\frac{1}{t}\right)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{as}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}t\to \mathrm{\infty},\phi (\tau )-\theta (t)=o(1).}$$

Hence we get
$${\theta}^{\prime}=\frac{1}{t}\left[R|{\mathrm{cos}}_{p}\theta {|}^{p}-\mathrm{\Phi}({\mathrm{cos}}_{p}\theta ){\mathrm{sin}}_{p}\theta +\frac{1}{p-1}(C+\frac{D}{{\mathrm{log}}^{2}t})|{\mathrm{sin}}_{p}\theta {|}^{p}\right]+O(\frac{\mathrm{l}}{t{\mathrm{log}}^{2}t}).$$ □

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

#### Lemma 3.2

*Let ${\left(\underset{0}{\overset{{\beta}_{1}}{\int}}{r}^{1-q}(t)dt\right)}^{p-1}\underset{0}{\overset{{\beta}_{2}}{\int}}c(t)dt={\gamma}_{p}{\beta}_{1}^{p-1}{\beta}_{2}$
* *and θ is a solution of the differential equation*
$${\theta}^{\prime}=\frac{1}{t}\left[R|{\mathrm{cos}}_{p}\theta {|}^{p}-\mathrm{\Phi}({\mathrm{cos}}_{p}\theta ){\mathrm{sin}}_{p}\theta +\frac{1}{p-1}(C+\frac{D}{{\mathrm{log}}^{2}t})|{\mathrm{sin}}_{p}\theta {|}^{p}\right]+o\left(\frac{\mathrm{l}}{t{\mathrm{log}}^{2}t}\right),$$(13)

*where R,C, D are as in Lemma 3.1*.
$$\mathit{I}\mathit{f}{\left(\underset{0}{\overset{{\beta}_{1}}{\int}}{r}^{1-q}(t)dt\right)}^{p-1}\underset{0}{\overset{{\beta}_{3}}{\int}}d(t)dt>{\mu}_{p}{\beta}_{1}^{p-1}{\beta}_{3},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathit{t}\mathit{h}\mathit{e}\mathit{n}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\theta (t)\to \mathrm{\infty}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathit{a}\mathit{s}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}t\to \mathrm{\infty}$$

and
$$\mathit{I}\mathit{f}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\left(\underset{0}{\overset{{\beta}_{1}}{\int}}{r}^{1-q}(t)dt\right)}^{p-1}\underset{0}{\overset{{\beta}_{3}}{\int}}d(t)dt<{\mu}_{p}{\beta}_{1}^{p-1}{\beta}_{3},\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathit{t}\mathit{h}\mathit{e}\mathit{n}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\theta (t)\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathit{i}\mathit{s}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\mathit{b}\mathit{o}\mathit{u}\mathit{n}\mathit{d}\mathit{e}\mathit{d}.$$

#### Proof

We rewrite Eq.(13) in the form
$${\theta}^{\prime}=\frac{1}{t}\left[R|{\mathrm{cos}}_{p}\theta {|}^{p}-\mathrm{\Phi}({\mathrm{cos}}_{p}\theta ){\mathrm{sin}}_{p}\theta +\frac{1}{p-1}(C+\frac{D}{{\mathrm{log}}^{2}t})|{\mathrm{sin}}_{p}\theta {|}^{p}\right]$$
$$+\frac{o(1)}{t{\mathrm{log}}^{2}t}(|{\mathrm{cos}}_{p}\theta {|}^{p}+|{\mathrm{sin}}_{p}\theta {|}^{p}).$$

This is the equation for Prüfer angle *θ*, which corresponds to the differential equation
$${\left({\left(R+\frac{o(1)}{{\mathrm{log}}^{2}t}\right)}^{1-p}\mathrm{\Phi}\left({x}^{\prime}\right)\right)}^{\prime}+\frac{p-1}{{t}^{p}}\left(C+\frac{D+o(1)}{{\mathrm{log}}^{2}t}\right)\mathrm{\Phi}(x)=0,$$

which is the same (using the formula (1 + *x*)^{α} = 1 + *αx + ο* (*x*) as *x* →0) as the equation
$${\left(\left(1+{\displaystyle \frac{o(1)}{{\mathrm{log}}^{2}t}}\right)\mathrm{\Phi}\left({x}^{\prime}\right)\right)}^{\prime}+{\displaystyle \frac{p-1}{{t}^{p}}\left({R}^{p-1}C+\frac{{R}^{p-1}D+o(1)}{{\mathrm{log}}^{2}t}\right)\mathrm{\Phi}(x)=0,}$$(14)

i.e., the same as
$${\left(\left(1+\frac{o(1)}{{\mathrm{log}}^{2}t}\right)\mathrm{\Phi}\left({x}^{\prime}\right)\right)}^{\prime}+\frac{1}{{t}^{p}}\left({\gamma}_{p}+\frac{(p-1){R}^{p-1}D+o(1)}{{\mathrm{log}}^{2}t}\right)\mathrm{\Phi}(x)=0.$$

First, suppose that
$$\frac{1}{{\beta}_{1}^{p-1}}{\left(\underset{0}{\overset{{\beta}_{1}}{\int}}{r}^{1-q}(t)dt\right)}^{p-1}\frac{1}{{\beta}_{3}}\underset{0}{\overset{{\beta}_{3}}{\int}}d(t)dt<{\mu}_{p}$$

and denote
$$\delta :=\frac{1}{2}\left[{\mu}_{p}-\frac{1}{{\beta}_{1}^{p-1}}{\left(\underset{0}{\overset{{\beta}_{1}}{\int}}{r}^{1-q}(t)dt\right)}^{p-1}\frac{1}{{\beta}_{3}}\underset{0}{\overset{{\beta}_{3}}{\int}}d(t)dt\right].$$

Let *ε* > 0 be sufficiently small and let *Τ* be so large that
$$|o(1)|<\u03f5\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}{\mu}_{p}-\frac{1}{{\beta}_{1}^{p-1}}{\left(\underset{0}{\overset{{\beta}_{1}}{\int}}{r}^{1-q}(t)dt\right)}^{p-1}\frac{1}{{\beta}_{3}}\underset{0}{\overset{{\beta}_{3}}{\int}}d(t)dt+o(1)>\delta $$

for *t* ≥ *T*. Then, the equation
$${\left(\left(1-{\displaystyle \frac{\u03f5}{{\mathrm{log}}^{2}t}}\right)\mathrm{\Phi}\left({x}^{\mathrm{\prime}}\right)\right)}^{\prime}+{\displaystyle \frac{1}{{t}^{p}}\left({\gamma}_{p}+\frac{{\mu}_{p}-\delta}{{\mathrm{log}}^{2}t}\right)\mathrm{\Phi}(x)=0}$$(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 $h(t)={t}^{\frac{p-1}{p}}{\mathrm{log}}^{\frac{1}{p}}t$
satisfies
$${\left(\left(1-{\displaystyle \frac{\u03f5}{{\mathrm{log}}^{2}t}}\right)\mathrm{\Phi}\left({h}^{\prime}\right)\right)}^{\prime}+\frac{1}{{t}^{p}}\left({\gamma}_{p}+\frac{{\mu}_{p}-\delta}{{\mathrm{log}}^{2}t}\right)\mathrm{\Phi}(h)\le 0$$(16)

for large *t*, then nonoscillation of Eq.(15) follows from Lemma 2.1. According to [5] ,we have for $h(t)={t}^{\frac{p-1}{p}}{\mathrm{log}}^{\frac{1}{p}}t$
$$h(t)\left\{{\left(\mathrm{\Phi}\left({h}^{\prime}\right)\right)}^{\prime}+\frac{1}{{t}^{p}}\left({\gamma}_{p}+\frac{{\mu}_{p}}{{\mathrm{log}}^{2}t}\right)\mathrm{\Phi}(h)\right\}\sim \frac{H}{t{\mathrm{log}}^{2}t}$$

for *t* →∞, where *Η* is a real constant. At the same time, by a direct computation,
$$-h{\left(\frac{\u03f5\mathrm{\Phi}({h}^{\prime})}{{\mathrm{log}}^{2}t}\right)}^{\prime}-\frac{\delta {h}^{p}}{{t}^{p}{\mathrm{log}}^{2}t}\sim \frac{\u03f5{\gamma}_{p}-\delta}{t\mathrm{log}t}<0$$

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
$$\frac{1}{{\beta}_{1}^{p-1}}{\left(\underset{0}{\overset{{\beta}_{1}}{\int}}{r}^{1-q}(t)dt\right)}^{p-1}\frac{1}{{\beta}_{3}}\underset{0}{\overset{{\beta}_{3}}{\int}}d(t)>{\mu}_{p}$$

and denote
$$\delta :=\frac{1}{2}\left[\frac{1}{{\beta}_{1}^{p-1}}{\left(\underset{0}{\overset{{\beta}_{1}}{\int}}{r}^{1-q}(t)dt\right)}^{p-1}\frac{1}{{\beta}_{3}}\underset{0}{\overset{{\beta}_{3}}{\int}}d(t)dt-{\mu}_{p}\right].$$

Let *ε >* 0 be again sufficiently small and let *Τ* be so large that
$$o(1)<\epsilon \phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{and}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \frac{1}{{\beta}_{1}^{p-1}}{\left(\stackrel{{\beta}_{1}}{\int}{r}^{1-q}(t)dt\right)}^{p-1}\frac{1}{{\beta}_{3}}\stackrel{{\beta}_{3}}{\int}d(t)dt-{\mu}_{p}+o(1)>\delta}$$

for *t* ≥ *T*. Then the equation
$${\left(\left(1+{\displaystyle \frac{\epsilon}{{\mathrm{log}}^{2}t}}\right)\mathrm{\Phi}\left({x}^{\mathrm{\prime}}\right)\right)}^{\prime}+{\displaystyle \frac{1}{{t}^{p}}\left({\gamma}_{p}+\frac{{\mu}_{p}+\delta}{{\mathrm{log}}^{2}t}\right)\mathrm{\Phi}(x)=0}$$(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
$${w}^{\prime}+\frac{1}{{t}^{p}}\left[{\gamma}_{p}+\frac{{\mu}_{p}+\delta}{{\mathrm{log}}^{2}t}\right]+(p-1)(1+R(t){)}^{1-q}|w{|}^{q}=0,$$

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

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

Then by a direct computation *v* is a solution of the equation
$$\begin{array}{ll}& {v}^{\prime}+\frac{{\mu}_{p}+\delta}{t{\mathrm{log}}^{2}t}+\frac{(p-1)}{t}(1+R(t){)}^{1-q}\left\{|v+{\lambda}_{p}{|}^{q}-q(1+R(t){)}^{q-1}{\mathrm{\Phi}}^{-1}({\lambda}_{p})v-{\lambda}^{q}\right.\\ & \left.-q(1+R(t){)}^{q-1}{\mathrm{\Phi}}^{-1}({\lambda}_{p})v-{\lambda}^{q}\right\}+\frac{(p-1)}{t}{\lambda}_{p}^{q}\left[1-(1+R(t){)}^{1-q}\right]=0.\end{array}$$(18)

Denote now ƒ (*t*) := (1 *+ R*(*t*)^{q-1}, and $F(v):=|v+{\lambda}_{p}{|}^{q}-q{\mathrm{\Phi}}^{-1}({\lambda}_{p})fv-{\lambda}_{p}^{q}.$
Then we have
$${F}^{\prime}(v)=q[{\mathrm{\Phi}}^{-1}(v+{\lambda}_{p})-{\mathrm{\Phi}}^{-1}({\lambda}_{p})f]=0\iff v={v}^{\ast}={\lambda}_{p}(\mathrm{\Phi}(f)-1).$$

At this extremal point
$$F({v}^{\ast})={\lambda}_{p}^{q}[{f}^{p}(1-q)+fq-1].$$

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

we obtain
$$\begin{array}{ll}F({v}^{\ast})& ={\lambda}_{p}^{q}\left[(1+R(t){)}^{p(q-1)}(1-q)+q(1+R(t){)}^{q-1}-1\right]\\ & ={\lambda}_{p}^{q}o(R)=o(R),\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\text{as}\phantom{\rule{thinmathspace}{0ex}}\phantom{\rule{thinmathspace}{0ex}}t\to \mathrm{\infty}.\end{array}$$

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

we get
$$\begin{array}{ll}& {v}^{\prime}+\frac{{\mu}_{p}+\delta}{t{\mathrm{log}}^{2}t}+\frac{(p-1)}{t}{\left(1+\frac{\epsilon}{{\mathrm{log}}^{2}t}\right)}^{1-q}\left\{|v+{\lambda}_{p}{|}^{q}-{\left(1+\frac{\epsilon}{{\mathrm{log}}^{2}t}\right)}^{q-1}{\mathrm{\Phi}}^{-1}({\lambda}_{p})v-{\lambda}_{p}^{q}\}\right.\\ & \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}+\frac{(p-1)}{t}{\lambda}_{p}^{q}[1-1+\frac{(q-1)\epsilon}{{\mathrm{log}}^{2}t}+\frac{o(1)}{{\mathrm{log}}^{2}t}]=0\end{array}$$

and we see that
$${v}^{\prime}+\frac{{\mu}_{p}+\delta}{t{\mathrm{log}}^{2}t}-\frac{\zeta}{{\mathrm{log}}^{2}t}\le 0,$$

for large *t*, where the constant *ζ* > 0 depends on *ε* and *η* and *μ*_{ρ} + *δ* – *ζ* > 0 if *ε, η* are sufficiently small. Hence *v'* (*t*) *<* 0 for large *t*, which means that there exists the limit *v*_{0}= lim_{t→∞} *v* (*t*). This limit is finite, because of Lemma 2.1, *w* (*t*) > 0, i.e., *v* (*t*) = *h*^{p} (*t*) (*w* (*t*) – *w*_{h} (*t*)) ≥ *-h*^{p} (*t*)* w*_{h}(*t*) = –*λ*_{ρ}.

Next, we show that *v*_{0} = 0. If *v*_{0} ≠0, then $|{v}_{0}+{\lambda}_{p}{|}^{q}-q{\mathrm{\Phi}}^{-1}({\lambda}_{p}){v}_{0}-{\lambda}_{p}^{q}=:\omega >0,$
i.e.,
$$|v(t)+{\lambda}_{p}{|}^{q}-q{\mathrm{\Phi}}^{-1}({\lambda}_{p})v(t)-{\lambda}_{p}^{q}>\frac{\omega}{2}$$

for large *t*. From Eq.(18), using the fact that
$$F(v)=|v+{\lambda}_{p}{|}^{q}-q{\mathrm{\Phi}}^{-1}({\lambda}_{p})v-{\lambda}_{p}^{q}+\left[\frac{q(q-1){\mathrm{\Phi}}^{-1}({\lambda}_{p})\u03f5}{{\mathrm{log}}^{2}t}+\frac{o(1)}{{\mathrm{log}}^{2}t}\right]v$$

we have
$$v(t)-v(T)+\underset{T}{\overset{t}{\int}}\frac{{\mu}_{p}+\delta}{s{\mathrm{log}}^{2}s}ds+\underset{T}{\overset{t}{\int}}\frac{(p-1)\omega}{2s}ds+\underset{T}{\overset{t}{\int}}\frac{O(1)}{s{\mathrm{log}}^{2}s}\le 0$$

i.e., since *v* (*t*) > –* λ*_{ρ}
$$\underset{T}{\overset{t}{\int}}\frac{{\mu}_{p}+\delta}{s{\mathrm{log}}^{2}s}ds+\underset{T}{\overset{t}{\int}}\frac{(p-1)\omega}{2s}ds+\underset{T}{\overset{t}{\int}}\frac{O(1)}{s{\mathrm{log}}^{2}s}\le {\lambda}_{p}+v(T)$$

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 *v*_{0} = 0. Using second order Taylor’s expansion , we have
$$(p-1)\left\{|v+{\lambda}_{p}{|}^{q}-q{\mathrm{\Phi}}^{-1}({\lambda}_{p})v-{\lambda}_{p}^{q}\right\}\ge \frac{(1-\epsilon )q}{2}{\lambda}_{p}^{q-2}{v}^{2}$$

for |*v*| sufficiently small. Hence, for *t* sufficiently large
$${v}^{\prime}+\frac{{\mu}_{p}+\delta}{t{\mathrm{log}}^{2}t}+\frac{(1-\epsilon )q}{2t}{\lambda}_{p}^{q-2}{v}^{2}+\frac{o(1)}{t{\mathrm{log}}^{2}t}v\le 0.$$

The last inequality is the Riccati type inequality associated with the linear second order differential equation
$$(t{x}^{\prime}{)}^{\prime}+{\displaystyle \frac{o(1)}{{\mathrm{log}}^{2}t}{x}^{\prime}+\frac{(1-\epsilon )q{\lambda}_{p}^{q-2}({\mu}_{p}+\delta )}{2t{\mathrm{log}}^{2}t}x=0,}$$(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
$$\frac{q}{2}{\left(\frac{p-1}{p}\right)}^{(p-1)(q-2)}\frac{1}{2}{\left(\frac{p-1}{p}\right)}^{(p-1)}\left(1+\frac{\delta}{{\mu}_{p}}\right)(1-\epsilon )=\frac{1}{4}\left(1+\frac{\delta}{{\mu}_{p}}\right)(1-\epsilon )$$

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

#### Example 3.7

*Consider the equation*
$${\left(\left({\displaystyle \frac{1}{2+\mathrm{cos}6t}}\right)\mathrm{\Phi}\left({x}^{\mathrm{\prime}}\right)\right)}^{\prime}+{\displaystyle \frac{1}{{t}^{p}}\left[\gamma (2+\mathrm{cos}8t)+\frac{\mu (2+\mathrm{cos}(ax+b))}{{\mathrm{log}}^{2}t}\right]\mathrm{\Phi}(x)=0}$$(20)

*which is Eq.(7) for $q=3,r(t)={\displaystyle \frac{1}{2+\mathrm{cos}6t},c(t)=2+\mathrm{cos}8t}$
* *and d* (*t*) = 2 + cos (*ax + b*),(*a,b ∈* ℝ). *In this* * case $r(t)={\displaystyle \frac{1}{2+\mathrm{cos}6t}}$
defined for all t ∈* ∞ *with period $\frac{\pi}{3},c(t)=2+\mathrm{cos}8t$
* *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)*
$$\begin{array}{ll}{\mu}_{\ast}& =\frac{{\beta}_{1}^{p-1}{\beta}_{3}{\mu}_{p}}{(\underset{0}{\overset{{\beta}_{1}}{\int}}{r}^{1-q}(t)dt{)}^{p-1}\underset{0}{\overset{{\beta}_{3}}{\int}}d(t)dt}\\ & =\frac{(\frac{\pi}{3}{)}^{\frac{1}{2}}\frac{2\pi}{|a|}\frac{1}{2}(\frac{1}{3}{)}^{\frac{1}{2}}}{{\left(\underset{0}{\overset{\frac{\pi}{3}}{\int}}(2+\mathrm{cos}6t{)}^{2}dt\right)}^{\frac{1}{2}}\underset{0}{\overset{\frac{2\pi}{|a|}}{\int}}(2+\mathrm{cos}(ax+b))dt}\\ & =\frac{1}{12\sqrt{2}}.\end{array}$$

*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 $\alpha =\text{1cm}\left({\displaystyle \frac{\pi}{3},\frac{\pi}{4},\frac{2\pi}{|a|}}\right)$
* *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*
$$\begin{array}{ll}{\mu}_{rd}& =\frac{{\alpha}^{p}{\mu}_{p}}{{\left(\underset{0}{\overset{\alpha}{\int}}{r}^{1-q}(t)dt\right)}^{p-1}\underset{0}{\overset{\alpha}{\int}}d(t)dt}\\ & \frac{\frac{1}{2\sqrt{3}}{\left(s\frac{2\pi}{|a|}\right)}^{\frac{3}{2}}}{{\left(l\underset{0}{\overset{\frac{\pi}{3}}{\int}}(2+\mathrm{cos}6t{)}^{2}dt\right)}^{\frac{1}{2}}\left(s\underset{0}{\overset{\frac{2\pi}{|a|}}{\int}}(2+\mathrm{cos}(ax+b))dt\right)}\\ & =\frac{1}{12\sqrt{2}}.\end{array}$$

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

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