To illustrate the novelty of our results, we mention the following series of corollaries and examples that are not covered by any known non-oscillation (or oscillation) criteria. At first, we provide examples to illustrate the novelty of Theorem 1.1.
Example 1
For arbitrarily given real numbers \(a_{1} > 0\), \(a_{2}\), \(b_{1} > 0\), \(b_{2}\), let us consider the equation
$$ \biggl( \biggl(a_{1} + \frac{a_{2}}{\log ^{2} t} \biggr)^{1-p} \varPhi \bigl(x' \bigr) \biggr)' + \biggl(b_{1} + \sin t^{2} + \frac{b_{2}}{ \log ^{2} t} \biggr) \frac{\varPhi (x)}{t^{p}} = 0, $$
(5.1)
which is a special case of Eq. (1.2). The conditions in (1.3), (1.4), and (1.6) are satisfied for \(R_{1} = a_{1}\), \(S_{1} = b_{1}\), and any \(\alpha \in (0,1)\), because
$$ \sup_{t \ge 1} \frac{1}{{t^{\alpha }}} \int _{t}^{t +{t^{\alpha }}} \bigl\vert \sin \tau ^{2} \bigr\vert \,d \tau \le 1; $$
the integral \(\int ^{\infty } \sin \tau ^{2} \,d \tau \) is convergent which guarantees
$$ \sup_{t \ge \mathrm{e}} \int _{t}^{t + t^{\alpha }} \sin \tau ^{2} \,d \tau < \infty . $$
Applying Theorems 1.1 and 4.1, we find that Eq. (5.1) is oscillatory for \(a_{1}^{p-1} b_{1} > q^{-p}\) and non-oscillatory for \(a_{1}^{p-1} b_{1} < q^{-p}\) and, in the limiting case \(a_{1}^{p-1} b _{1} = q^{-p}\), we obtain a new result, which says that Eq. (5.1) is oscillatory for
$$ q \frac{b_{2}}{b_{1}} + p \frac{a_{2}}{a_{1}} > \frac{q^{2}}{2}, \quad \text{i.e.}, \quad 2q^{p+1} {b_{2}} + 2p{a_{2}} a_{1}^{-p} > q^{2} a_{1}^{1-p}, $$
and non-oscillatory for
$$ q \frac{b_{2}}{b_{1}} + p \frac{a_{2}}{a_{1}} < \frac{q^{2}}{2}. $$
Indeed, Eq. (5.1) has the form of Eq. (1.2) for \(r_{1} (t) = a_{1} \), \(r_{2} (t) = a_{2} \), \(s_{1} (t) = b_{1} + \sin t^{2} \), and \(s_{2} (t) = b_{2} \), where
$$ \frac{1}{t^{\alpha }} \int _{t}^{t + t^{\alpha }} r_{2} ( \tau ) \,d \tau = a_{2}, \qquad \frac{1}{t^{\alpha }} \int _{t}^{t + t^{\alpha }} s_{2}( \tau ) \,d \tau = b_{2} $$
for all \(t \in [\mathrm{e} , \infty )\) and \(\alpha \in (0, 1)\).
Example 2
Let \(a_{1} > 0\), \(b_{1} > 0\), \(a_{2}, a_{3}, b_{2}, b_{3}, b_{4} \in \mathbb{R}\), and \(a, b, c, d \ne 0\). The half-linear equation
$$ \begin{aligned}[b] & \biggl( \biggl( \frac{\log ^{2} t }{a_{1} \log ^{2} t + a_{2} + a_{3} \arctan (\sin (at) ) } \biggr)^{2} \bigl\vert x' \bigr\vert x' \biggr)' \\ &\quad {}+ \biggl(b_{1} + b_{2} \sin (b \sqrt{t} ) + b_{3} \sin (c t ) + \frac{b_{4} + \sin (d t )}{ \log ^{2} t} \biggr)\frac{ \vert x \vert x }{t^{3}} = 0 \end{aligned} $$
(5.2)
takes the form of Eq. (1.2) for \(p=3\) and
$$\begin{aligned}& r_{1} (t) = a_{1}, \qquad r_{2} (t) = a_{2} + a_{3} \arctan \bigl(\sin (at) \bigr), \\& s_{1} (t) = b_{1} + b_{2} \sin (b \sqrt{t} ) + b_{3} \sin (c t ), \qquad s_{2} (t) = b_{4} + \sin (d t ). \end{aligned}$$
For any \(\alpha \in (1/2, 1)\), based on the inequalities
$$ \limsup_{t \to \infty } \frac{1}{t^{\alpha }} \int _{t} ^{t + t^{\alpha }} \bigl\vert \sin (b \sqrt{\tau } ) \bigr\vert \,d \tau \le 1 $$
and
$$\begin{aligned} \biggl\vert \frac{1}{t^{\alpha }} \int _{t}^{t + t^{\alpha }} \sin (b \sqrt{\tau } ) \,d \tau \biggr\vert &\le \frac{1}{t ^{\alpha }} \biggl\vert \biggl[\frac{2 \sin (b \sqrt{\tau } ) - 2 b \sqrt{\tau } \cos (b \sqrt{\tau } )}{b^{2}} \biggr] _{t}^{t + t^{\alpha }} \biggr\vert \\ &\le \frac{1}{b^{2} t^{\alpha }} \bigl(4 + 2b \bigl(\sqrt{t + t ^{\alpha }} + \sqrt{t} \bigr) \bigr) < \frac{1}{\log ^{3} t} \end{aligned}$$
for all large t, it is easy to verify conditions (1.3), (1.4), and (1.6) for large t. Hence, we can apply Theorem 1.1. In conclusion, Theorems 1.1 and 4.1 give the oscillation of Eq. (5.2) if \(3^{3} a_{1}^{2} b_{1} > 2^{3}\) and the non-oscillation if \(3^{3} a_{1}^{2} b_{1} < 2^{3}\). In addition, in the threshold case \(3^{3} a_{1}^{2} b_{1} = 2^{3}\), which is studied in this paper, we have oscillation if
$$ 4 \frac{b_{4}}{b_{1}} + 8 \frac{a_{2}}{a_{1}} > 3, $$
and non-oscillation if the opposite sharp inequality holds.
Example 3
For arbitrary \(a > 1\), \(b > 1\), and \(c \in \mathbb{R}\), let us consider the equations
$$\begin{aligned} (p-1 ) \bigl\vert x' \bigr\vert ^{p - 2} x'' + \frac{1}{(t+1)^{p}} \biggl( {q^{-p} + \sin _{a} t + \sin _{b} t } + \frac{c}{\log ^{2} t} \biggr) \vert x \vert ^{p-2} x &= 0, \\ (p-1 ) \bigl\vert x' \bigr\vert ^{p - 2} x'' + \frac{1}{(t+1)^{p}} \biggl( {q^{-p} + \sin _{a} t + \cos _{b} t } + \frac{c}{\log ^{2} t} \biggr) \vert x \vert ^{p-2} x &= 0, \\ (p-1 ) \bigl\vert x' \bigr\vert ^{p - 2} x'' + \frac{1}{(t+1)^{p}} \biggl( {q^{-p} + \cos _{a} t + \cos _{b} t } + \frac{c}{\log ^{2} t} \biggr) \vert x \vert ^{p-2} x &= 0. \end{aligned}$$
Each one of these equations takes the form of Eq. (1.2). Applying Theorem 1.1, one can show that the equations are oscillatory for \(2c > q^{1-p}\) and non-oscillatory for \(2c < q^{1-p}\); this does not follow from known criteria.
Now, we focus on Theorem 4.3. In particular, Theorem 4.3 gives a new result in the linear case (for \(p=2\)) as well. Since linear equations form a very important class of equations, we mention the corresponding new corollary explicitly.
Corollary 5.1
Consider the linear equation
$$ \bigl( r (t) x' (t) \bigr)' + \frac{s(t)}{t^{2}} x (t) = 0, $$
(5.3)
where
\(r > 0\), s
are continuous functions and (4.30) is valid for some
\(\alpha \in [0, 1)\). If there exist
\(R, S > 0\)
satisfying
\(4 R S \le 1\)
and a continuous function
\(f: [\mathrm{e} , \infty ) \to \mathbb{R}\)
satisfying
\(\lim_{t \to \infty } f (t) = 0\)
for which
$$ \frac{1}{{t^{\alpha }}} { \int _{t}^{t + t^{\alpha }} \frac{1}{r (\tau )} \,d \tau } \le R + \frac{f(t)}{\log ^{2} t}, \quad t \in [\mathrm{e} , \infty ), $$
and
$$ \frac{1}{{t^{\alpha }}} { \int _{t}^{t + t^{\alpha }} s( \tau ) \,d \tau }\le S + \frac{f(t)}{\log ^{2} t}, \quad t \in [\mathrm{e} , \infty ), $$
then Eq. (5.3) is non-oscillatory.
Proof
The statement of the corollary follows from Theorem 4.1 and Theorem 4.3 for \(p=2\). □
Further, Corollary 5.1 is not covered by known results even in the case of constant coefficients in the main term. This case is embodied into the next corollary.
Corollary 5.2
Consider the linear equation
$$ x'' (t) + \frac{s(t)}{t^{2}} x (t) = 0, $$
(5.4)
where
s
is a continuous function and (4.30) is valid for some
\(\alpha \in [0, 1)\). If there exists a continuous function
\(f: [\mathrm{e} , \infty ) \to \mathbb{R}\)
satisfying
\(\lim_{t \to \infty } f (t) = 0\)
such that
$$ \frac{1}{{t^{\alpha }}} { \int _{t}^{t + t^{\alpha }} s( \tau ) \,d \tau }\le \frac{1}{4} + \frac{f(t)}{\log ^{2} t}, \quad t \in [\mathrm{e} , \infty ), $$
(5.5)
then Eq. (5.4) is non-oscillatory.
Proof
See Corollary 5.1 for \(r \equiv 1\) and \(R =1 \). □
To illustrate Corollary 5.2, we can consider simple types of equations whose non-oscillation does not follow from any previously known results. See the example below.
Example 4
For the function
$$ s (t) := \textstyle\begin{cases} \frac{1}{4} + \frac{t - 2^{n}}{n}, &t \in [2^{n}, 2^{n} + n ], n \in \mathbb{N}; \\ \frac{1}{4} + 1 - \frac{t - 2^{n} - n}{n}, &t \in (2^{n}+ n , 2^{n} + 2n ], n \in \mathbb{N}; \\ \frac{1}{4}, &t \notin [2^{n}, 2^{n} + 2n ], n \in \mathbb{N}, \end{cases} $$
(5.6)
we consider Eq. (5.4). For any \(\alpha \in (0, 1)\), we have
$$ \lim_{t \to \infty } \frac{1}{t^{\alpha }} \int _{t}^{t + t^{ \alpha }} \bigl\vert s (\tau ) \bigr\vert \,d \tau = \limsup_{n \to \infty } \frac{1}{2^{ \alpha n}} \int _{2^{n}}^{2^{n} + 2^{\alpha n}} s (\tau ) \,d \tau = \frac{1}{4}, $$
which gives (4.30). In addition, we obtain
$$ \frac{1}{t^{\alpha }} \int _{t}^{t + t^{\alpha }} s (\tau ) \,d \tau \le \frac{1}{4} + \frac{n + 1}{t^{\alpha }} $$
(5.7)
if \(\alpha \in (0, 1)\) and \(t \in [ 2^{n}, 2^{n+1} ]\), \(n \in \mathbb{N}\). Thus, if we put
$$ f(t) := \log ^{2} t \frac{1+ \log _{2} t}{t^{\alpha }}, \quad t \in [\mathrm{e} , \infty ) , $$
then \(\lim_{t \to \infty } f(t) = 0\) and (see (5.7))
$$ \frac{1}{t^{\alpha }} \int _{t}^{t + t^{\alpha }} s (\tau ) \,d \tau \le \frac{1}{4} + \frac{f (t)}{\log ^{2} t}, \quad t \in [\mathrm{e} , \infty ) , $$
for any \(\alpha \in (0, 1)\). Further, we have (5.5). Altogether, Corollary 5.2 states that Eq. (5.4) is non-oscillatory for s defined by (5.6). We repeat that the non-oscillation of Eq. (5.4) does not follow from any previously known results (or their combinations).
Although the aim of this paper is to prove Theorem 4.3, we obtain a more general result, which implies new criteria also for perturbed equations. Such a new corollary is as follows.
Corollary 5.3
Consider the half-linear equation
$$ \biggl( \biggl(1 + \frac{r(t)}{\log ^{2} t} \biggr)^{1-p} \varPhi \bigl(x' \bigr) \biggr)' + \biggl(q^{-p} + \frac{s(t)}{\log ^{2} t} \biggr)\frac{\varPhi (x)}{t^{p}} = 0, $$
(5.8)
where
r, s
are continuous functions and
$$ 1 + \frac{r (t)}{\log ^{2} t} > 0, \quad t \in [\mathrm{e} , \infty ) . $$
Assume that there exists
\(\alpha \in [0, 1)\)
with
$$ \limsup_{t \to \infty } \frac{1}{t^{\alpha }} \int _{t} ^{t + t^{\alpha }} \bigl\vert r (\tau ) \bigr\vert \,d \tau < \infty , \qquad \limsup_{t \to \infty } \frac{1}{t^{\alpha }} \int _{t} ^{t + t^{\alpha }} \bigl\vert s (\tau ) \bigr\vert \,d \tau < \infty . $$
-
(a)
If
$$ \frac{1}{t^{\alpha }} \int _{t}^{t + t^{\alpha }} r ( \tau ) \,d \tau \ge R, \qquad \frac{1}{t^{\alpha }} \int _{t}^{t + t^{\alpha }} s ( \tau ) \,d \tau \ge S $$
for all large
t
and some
\(R, S \in \mathbb{R}\)
and
$$ q^{p+1} S + p {R} > \frac{q^{2}}{2}, $$
then Eq. (5.8) is oscillatory.
-
(b)
If
$$ \frac{1}{t^{\alpha }} \int _{t}^{t + t^{\alpha }} r ( \tau ) \,d \tau \le R, \qquad \frac{1}{t^{\alpha }} \int _{t}^{t + t^{\alpha }} s ( \tau ) \,d \tau \le S $$
for all large
t
and some
\(R, S \in \mathbb{R}\)
and
$$ q^{p+1} {S} + p R < \frac{q^{2}}{2}, $$
then Eq. (5.8) is non-oscillatory.
Proof
It suffices to use Theorem 1.1 for \(R_{1} = 1\), \(S_{1} = q^{-p}\), \(R_{2} = R\), \(S_{2} = S\), \(r_{1} (t) = 1\), \(s_{1} (t) = q^{-p}\), \(r_{2} (t) = r(t)\), and \(s_{2} (t) = s(t)\). □
Corollary 5.3 is not covered by known results even for the Riemann–Weber type of (linear and half-linear) equations. Hence, we mention the next corollaries and examples.
Corollary 5.4
Consider the equation
$$ \bigl(\varPhi \bigl(x'\bigr) \bigr)' + \biggl(q^{-p} + \frac{s(t)}{\log ^{2} t} \biggr) \frac{\varPhi (x)}{t^{p}} = 0, $$
(5.9)
where
s
is a continuous function such that (4.30) holds for some
\(\alpha \in [0, 1)\).
-
(I)
If
$$ \liminf_{t \to \infty } \frac{1}{t^{\alpha }} { \int _{t}^{t + {t^{\alpha }}} s ( \tau ) \,d \tau } > \frac{q^{1-p}}{2}, $$
(5.10)
then Eq. (5.9) is oscillatory.
-
(II)
If
$$ \limsup_{t \to \infty } \frac{1}{t^{\alpha }} { \int _{t}^{t + {t^{\alpha }}} s ( \tau ) \,d \tau } < \frac{q^{1-p}}{2}, $$
(5.11)
then Eq. (5.9) is non-oscillatory.
Proof
The corollary follows from Theorem 1.1 for coefficients \(r_{1} (t) = 1\), \(r_{2} (t) = 0\), \(s_{1} (t) = q^{-p}\), \(s_{2} (t) = s (t)\). □
Example 5
Let \(a, b \in \mathbb{R}\) be arbitrary. Similarly as in Example 4 (see (5.6)), we define the function \(s : [8, \infty ) \to \mathbb{R}\) by the formula
$$ s (t) := \textstyle\begin{cases} a + b \sin \frac{\pi (t - 8^{n})}{4^{n}}, &t \in [8^{n}, 8^{n} + 4^{n} ], n \in \mathbb{N}; \\ a, &t \in (8^{n} + 4^{n}, 8^{n+1} ), n \in \mathbb{N}. \end{cases} $$
Let us consider Eq. (5.9) for this function s and \(t \ge 8\). Clearly,
$$ \limsup_{t \to \infty } \frac{1}{t^{\alpha }} \int _{t} ^{t + t^{\alpha }} \bigl\vert s (\tau ) \bigr\vert \,d \tau \le \vert a \vert + \vert b \vert < \infty $$
for any \(\alpha \in [0, 1)\), i.e., (4.30) is fulfilled. Now, we consider \(\alpha > 2/3\). For example, we choose \(\alpha = 3/4\). Then we obtain
$$\begin{aligned} \liminf_{t \to \infty } \frac{1}{\sqrt[4]{ t^{3}}} \int _{t}^{t + \sqrt[4]{ t^{3}} } s ( \tau ) \,d \tau &\ge \lim _{n \to \infty } \frac{1}{\sqrt[4]{ 8^{3n}} } \biggl( \int _{8^{n}}^{8^{n} + {4^{n}}} - \vert b \vert \,d \tau + \int _{8^{n}} ^{8^{n} + \sqrt[4]{ 8^{3n}} } a \,d \tau \biggr) \\ &= \lim_{n \to \infty } a - \vert b \vert \frac{4^{n} }{ \sqrt[4]{ 8^{3n}} } = \lim _{n \to \infty } a - \vert b \vert \biggl(\frac{4}{4 \sqrt[4]{2}} \biggr)^{n} = a \end{aligned}$$
and
$$\begin{aligned} \limsup_{t \to \infty } \frac{1}{\sqrt[4]{ t^{3}}} \int _{t}^{t + \sqrt[4]{ t^{3}} } s ( \tau ) \,d \tau &\le \lim _{n \to \infty } \frac{1}{\sqrt[4]{ 8^{3n}} } \biggl( \int _{8^{n}}^{8^{n} + {4^{n}}} \vert b \vert \,d \tau + \int _{8^{n}} ^{8^{n} + \sqrt[4]{ 8^{3n}} } a \,d \tau \biggr) \\ &= \lim_{n \to \infty } a + \vert b \vert \frac{4^{n} }{ \sqrt[4]{ 8^{3n}} } = a . \end{aligned}$$
Thus,
$$ \lim_{t \to \infty } \frac{1}{\sqrt[4]{ t^{3}}} \int _{t}^{t + \sqrt[4]{ t^{3}} } s ( \tau ) \,d \tau = a. $$
(5.12)
Applying Corollary 5.4 (see (5.10) and (5.11)) and considering (5.12), we obtain the oscillation of the considered equation for \(2a > q^{1-p}\) and its non-oscillation for \(2a < q^{1-p}\).
Corollary 5.5
Consider the linear equation
$$ x'' + \biggl(\frac{1}{4} + \frac{s(t)}{\log ^{2} t} \biggr)\frac{x}{t ^{2}} = 0, $$
(5.13)
where
s
is a continuous function satisfying (4.30) for some
\(\alpha \in [0, 1)\).
-
(a)
If the inequality
$$ \frac{1}{t^{\alpha }} \int _{t}^{t + t^{\alpha }} s ( \tau ) \,d \tau > \frac{1}{4} + \varepsilon $$
(5.14)
holds for all large
t
and some
\(\varepsilon > 0\), then Eq. (5.13) is oscillatory.
-
(b)
If the inequality
$$ \frac{1}{t^{\alpha }} \int _{t}^{t + t^{\alpha }} s ( \tau ) \,d \tau < \frac{1}{4} - \varepsilon $$
(5.15)
holds for all large
t
and some
\(\varepsilon > 0\), then Eq. (5.13) is non-oscillatory.
Proof
It suffices to consider Corollary 5.4 for \(p = 2\). □
Example 6
Let \(a \in \mathbb{R}\) and \(\alpha \in (1/3, 1)\) be arbitrary and let \(b: [\mathrm{e} , \infty ) \to \mathbb{R} \) be an arbitrary continuous function such that
$$ \limsup_{t \to \infty } \frac{1}{t^{\alpha }} \int _{t} ^{t + t^{\alpha }} \sqrt[3]{\tau } \bigl\vert b (\tau ) \bigr\vert \,d \tau < \infty . $$
(5.16)
We apply Corollary 5.5 to the equation
$$ x'' + \biggl(\frac{1}{4} + \frac{a+ \arctan ( t^{3} - 2 t^{2} + 1 ) + b (t) \sqrt[3]{t} \cos t + \frac{t^{5}}{t^{5} + 6}}{ \log (t+1) \log t} \biggr)\frac{x}{t^{2}} = 0, $$
(5.17)
i.e., for Eq. (5.13) with
$$ s(t) = \biggl(a + \arctan \bigl( t^{3} - 2 t^{2} + 1 \bigr) + b (t) \sqrt[3]{t} \cos t + \frac{t^{5}}{t^{5} + 6} \biggr) \frac{\log t}{ \log (t+1)}. $$
(5.18)
Since
$$ \lim_{t \to \infty } \arctan \bigl( t^{3} - 2 t^{2} + 1 \bigr) = \frac{\pi }{2}, \qquad \lim _{t \to \infty } \frac{t^{5}}{t^{5} + 6} = \lim_{t \to \infty } \frac{\log t}{ \log (t+1)} = 1, $$
(5.19)
and
$$ - \frac{2 \sqrt[3]{t + t^{\alpha }}}{t^{\alpha }} \le \frac{1}{t^{ \alpha }} \int _{t}^{t + t^{\alpha }} \sqrt[3]{\tau } \cos \tau \,d \tau \le \frac{2 \sqrt[3]{t + t^{\alpha }} }{t^{\alpha }} $$
for all large t, we have
$$ \lim_{t \to \infty } \frac{1}{t^{\alpha }} \int _{t}^{t + t^{\alpha }} s ( \tau ) \,d \tau = a + \frac{\pi }{2} + 1 . $$
(5.20)
To use Corollary 5.5, it remains to verify (4.30). Of course, (5.16), (5.18), and (5.19) guarantee that (4.30) is fulfilled. Considering (5.20) together with (5.14) and (5.15), Corollary 5.5 says that Eq. (5.17) is oscillatory if \(4a + 2\pi + 3 > 0\), and non-oscillatory if \(4a + 2\pi + 3 < 0\), i.e., if \(a < -2. 320 796 326 794 896 619\dots \)
In Example 6, one can see that it is possible to use the presented results asymptotically (also for other types of equations). It is formulated explicitly below in two new corollaries. Note that, for simplicity, we demonstrate such types of results only in a very special case. To prove Corollary 5.6 below, we need the following lemma.
Lemma 5.1
Let a function
\(F : [\mathrm{e} , \infty ) \to \mathbb{R}\)
be such that it can be expressed as a finite sum of continuous periodic functions. Then there exists a constant
\(E(F) > 0\)
such that the inequality
$$ \biggl\vert \frac{1}{\sqrt{t}} \int _{t}^{t+\sqrt{t}} F( \tau )\,d \tau - \overline{F} \biggr\vert \le \frac{E(F)}{\sqrt{t}} $$
holds for all
\(t \in [\mathrm{e} , \infty )\).
Proof
See [39, Lemma 3.5]. □
Corollary 5.6
For an arbitrary
\(p>1\), consider continuous functions
\(f_{1}, f_{2}, f _{3} : [\mathrm{e} , \infty ) \to \mathbb{R}\)
such that
$$ \lim_{t \to \infty } \frac{f_{1} (t)}{\log ^{2} t} = \lim_{t \to \infty } \frac{f_{2} (t)}{\log ^{2} t} = \lim_{t \to \infty } \frac{f_{3} (t)}{t^{p}} = 1 $$
and the half-linear differential equation
$$ \biggl( \biggl(1 + r_{1}(t)+ \frac{r_{2}(t)}{f_{1} (t) } \biggr)^{1-p} \varPhi \bigl(x' \bigr) \biggr)' + \biggl(q^{-p} + s_{1}(t) + \frac{s _{2}(t)}{f_{2} (t) } \biggr) \frac{\varPhi (x)}{f_{3} (t)} = 0, $$
(5.21)
where
\(r_{1}\), \(s_{1}\)
are finite sums of continuous periodic functions, \(r_{2}\), \(s_{2}\)
are continuous functions such that
$$ 1 + r_{1}(t)+ \frac{r_{2}(t)}{f_{1} (t)} > 0, \quad t \in [\mathrm{e} , \infty ), $$
the mean values
\(\overline{r_{1}}\), \(\overline{s_{1}}\)
are zero, and the inequalities
$$ \limsup_{t \to \infty } \frac{1}{\sqrt{t} } \int _{t} ^{t + \sqrt{t} } \bigl\vert r_{2} (\tau ) \bigr\vert \,d \tau < \infty , \qquad \limsup_{t \to \infty } \frac{1}{\sqrt{t} } \int _{t} ^{t + \sqrt{t} } \bigl\vert s_{2} (\tau ) \bigr\vert \,d \tau < \infty $$
hold. Let
\(R, S \in \mathbb{R}\).
-
(a)
If
$$ \frac{1}{\sqrt{t}} \int _{t}^{t + \sqrt{t}} r_{2} ( \tau ) \,d \tau \ge R, \qquad \frac{1}{\sqrt{t} } \int _{t}^{t + \sqrt{t}} s_{2}( \tau ) \,d \tau \ge S $$
for all large
t
and
\(2 q^{p+1} S + 2 p R > q^{2}\), then Eq. (5.21) is oscillatory.
-
(b)
If
$$ \frac{1}{\sqrt{t}} \int _{t}^{t + \sqrt{t}} r_{2} ( \tau ) \,d \tau \le R, \qquad \frac{1}{\sqrt{t}} \int _{t}^{t + \sqrt{t}} s_{2}( \tau ) \,d \tau \le S $$
for all large
t
and
\(2 q^{p+1} S + 2 p R < q^{2} \), then Eq. (5.21) is non-oscillatory.
Proof
The corollary is a consequence of Theorem 1.1, where it suffices to apply Lemma 5.1 and to put \(\alpha = 1/2\), \(R_{1} = 1\), and \(S_{1} = q^{-p}\). □
Corollary 5.6 gives a new result for \(p=2\) as well. This new result is formulated for constant coefficients in place of \(r_{2} \), \(s_{2}\).
Corollary 5.7
Let
\(R, S \in \mathbb{R}\)
and let
\(f_{1}, f_{2}, f_{3} : [\mathrm{e} , \infty ) \to \mathbb{R}\)
be arbitrary continuous functions satisfying
$$ \lim_{t \to \infty } \frac{f_{1} (t)}{\log ^{2} t} = \lim_{t \to \infty } \frac{f_{2} (t)}{\log ^{2} t} = \lim_{t \to \infty } \frac{f_{3} (t)}{t^{2}} = 1 . $$
Consider the linear differential equation
$$ \biggl(\frac{f_{1} (t)}{f_{1} (t) + f_{1} (t) r (t)+ R } x' \biggr)' + \biggl(\frac{1}{4} + s (t) + \frac{S}{f_{2} (t) } \biggr) \frac{x}{f _{3} (t)} = 0, $$
(5.22)
where
\(r > 0, s\)
are finite sums of continuous periodic functions with the property that
\(\overline{r} = \overline{s} = 0\).
-
(a)
If
\(4 S + R > 1\), then Eq. (5.22) is oscillatory.
-
(b)
If
\(4 S + R < 1\), then Eq. (5.22) is non-oscillatory.
Proof
See Corollary 5.6 for \(p=2\), \(r_{1} (t)= r(t) \), \(r_{2} (t)= R \), \(s_{1} (t)= s(t) \), \(s_{2} (t)= S\). □
Example 7
For \(a, b \in \mathbb{R}\), we can illustrate Corollary 5.7, e.g., by the equations
$$\begin{aligned}& \biggl(\frac{\log ^{2} t}{a+ (1+ \sin t) \log ^{2} t} x' \biggr)' + \biggl( \frac{1}{4} \biggl(1+ \sin t + \frac{b}{\log ^{2} (t+\sin t)} \biggr) \biggr) \frac{x}{t^{2}+ 2} = 0, \\& \biggl(\frac{\log ^{2} t}{a + \log ^{2} t} x' \biggr)' + \biggl( \frac{1}{4} \biggl(1+ \cos t + \frac{b}{\log ^{2} t} \biggr) \biggr) \frac{x}{t (t+ \sin t)} = 0, \\& \biggl(\frac{\log ^{2} (t+1)}{a+ \log ^{2} (t+1)} x' \biggr)' + \biggl( \frac{1}{4} \biggl(1+ \sin t \cos t + \frac{b}{\log t \log (t+\sqrt[3]{t} + 1 ) } \biggr) \biggr) \frac{t^{2} x}{t ^{4} + 1} = 0, \\& \biggl( \biggl(1 + \frac{a}{\log ^{2} (t+\sqrt{t}\sin t )} \biggr)^{-1} x' \biggr)' + \biggl(1 + \sin t + \frac{b}{\log ^{2} t} \biggr) \frac{x}{4(t+1)^{2}} = 0, \\& \biggl( \biggl(1 + \frac{a}{\log ^{2} t} \biggr)^{-1} x' \biggr)' + \biggl(\frac{1}{4} + \sin t + \sin (\sqrt{2}t ) + \frac{b}{4 \log ^{2} t} \biggr)\frac{x}{t(t+1)} = 0, \\& \biggl( \biggl(1 + \sin (\sqrt{2}t ) + \frac{a}{\log ^{2} (t+1)} \biggr)^{-1} x' \biggr)' + \biggl(1 + \sin (\sqrt{3}t ) + \frac{b}{\log ^{2} (t-1)} \biggr)\frac{x}{4 \sqrt[4]{t^{8} + 1}} = 0. \end{aligned}$$
One can easily verify that each one of the above equations is oscillatory for \(a+b > 1\) and non-oscillatory for \(a+b < 1\). To the best of our knowledge, the oscillatory behaviors of the above equations are not covered by any previously known criteria.
We end this paper by the following series of open problems which are connected to the research presented in this paper. Some of them are indicated in the Introduction; we describe them in detail here.
(I) Non-oscillation, principal solution, and boundary value problems. As claimed in the Introduction, non-oscillatory half-linear equations have important applications in boundary value problems on non-compact intervals. It is an open problem to study asymptotic properties of principal solutions of Euler type equations (1.2) and (4.1) under the non-oscillation condition mentioned in Theorem 4.2, (b). Along with this, it is of interest to apply these results to obtain effective criteria for the non-oscillation of principal solutions of some auxiliary half-linear equations that are used to find upper (lower) bounds for solutions of some boundary value problems on non-compact intervals by using a similar approach as in [7, 8].
(II) Discretization. Since the qualitative theory of difference equations follows the continuous theory frequently, the second open problem is devoted to the equations of the form
$$ \Delta \bigl(r_{k} \varPhi (\Delta x_{k} ) \bigr) + \frac{s _{k}}{k^{p}} \varPhi (x_{k+1})=0, $$
where \(\Delta x_{k} = x_{k+1}-x_{k}\) is the forward difference of the sequence \(\{x_{k}\}\). As we have already mentioned in the Introduction, some results concerning (non-perturbed) difference equations are already available (see, e.g., [36] or [34] and the references cited therein). The natural continuation is to extend known results to perturbed equations. Unfortunately, there is a problem with the method. Let us describe this problem thoroughly.
In this paper, we started with the Riccati technique. We recall that the core of this method is the equivalence of the non-oscillation of a studied equation and the existence of a solution of an associated Riccati type differential equation (or inequality) on an interval \([T, \infty )\) (see, e.g., [26,27,28, 30]). The standard Riccati technique and its variations (where the adapted Riccati equation (3.3) applied in this paper belongs) are used usually in the theory of non-perturbed equations. The technique itself is robust, clear, and straightforward but its “resolution” is not as precise as is needed for an analysis of the critical case and perturbed equations.
Hence, we combine the adapted Riccati equation with the Prüfer technique and involve the averaging method. Then we use the equivalence of the non-oscillation of the given equation and the boundedness of the Prüfer angle (for variations of such approach, see, e.g., [17, 33, 39, 45]).
Getting back to difference equations, the first approach (the Riccati technique) is available and frequently applied. The second method which involves the Prüfer transformation is available only in a “weak” version, which is a consequence of problems with the chain rule in the difference calculus (see [4]). Therefore, the acquisition of results concerning the conditional oscillation for difference equations with perturbations (or solve the critical case) remains a challenging open problem.
(III) Time scales. The third direction is towards dynamic equations on time scales, i.e., to equations of the form
$$ \bigl( r(t)\varPhi \bigl(x^{\Delta } \bigr) \bigr)^{\Delta } + \frac{s(t)}{t ^{p}} \varPhi \bigl(x^{\sigma }\bigr)=0, $$
where \(x^{\Delta }\) and \(x^{\sigma }\) denote the delta derivative and the forward jump operator applied to the function x, respectively. For more details, we refer, e.g., to [5]. Roughly speaking, the time scale calculus is a unification of the continuous and discrete calculus but a time scale itself is an arbitrary non-empty closed subset of real numbers. It means that much more cases are covered than only \(\mathbb{R}\) and \(\mathbb{Z}\). It has lead to the importance of dynamic equations in natural sciences, economy, and informatics and also to high interest of researchers. As well as in the discrete case (which is a special case of time scales), some results are already known (see, e.g., [38] with references therein). Of course, the problem with the fully functional Prüfer transformation is present as well. Therefore, the description from point (II) is valid also for dynamic equations on time scales. Moreover (see [50]), we note that the critical oscillation constant may be dependent on the graininess (a function that measures the distance between two consecutive points of the given time scale; it is identically zero for \(\mathbb{R}\) and one for \(\mathbb{Z}\)).
(IV) Modified equations. Since conditionally oscillatory equations are very useful, a natural direction of research is to find another types of equations that are conditionally oscillatory. As we have already mentioned in the Introduction, such equations are ideal for testing oscillatory properties of other equations via many comparison theorems. However, the form of the coefficients of known conditionally oscillatory equations may not be suitable for testing of certain equations. More precisely, let us consider Eq. (2.4), where the presence of \(t^{-p}\) in the potential term may be an obstacle. It is proved in [14] that the equation
$$ \bigl(t^{\beta -1} r(t) \varPhi \bigl(x' \bigr) \bigr)'+\frac{s(t)}{t ^{p-\beta +1}} \varPhi (x)=0, \quad \beta \neq p, $$
(5.23)
is conditionally oscillatory. The idea is to find other forms of differential equations and their perturbations that preserve the behavior of conditional oscillation. In this direction, we recall Theorem 2.3.
We remark that Eq. (5.23) with \(\beta =p\) is not conditionally oscillatory, which can be simply verified on the case of constant coefficients r, s. In this case (see [35]), the form of the corresponding conditionally oscillatory equation is
$$ \bigl( t^{p-1} r(t) \varPhi \bigl(x' \bigr) \bigr)'+\frac{s(t)}{t \log ^{p}t} \varPhi (x)=0 $$
and the conditionally oscillatory perturbed equation is
$$ \biggl(t^{p-1} \biggl(r_{1}(t) + \frac{r_{2}(t)}{\log ^{2} t} \biggr) ^{1-p } \varPhi \bigl(x' \bigr) \biggr)'+ \frac{1}{t \log ^{p}t} \biggl(s _{1}(t) + \frac{s_{2}(t)}{\log ^{2} t} \biggr) \varPhi (x)=0. $$
Continuation in this research direction has a potential to enlarge the set of conditionally oscillatory equations.
(V) Non-linear equations. Besides the applications within linear and half-linear equations (and “real world” applications in natural sciences or economics), half-linear equations can be considered a starting point to research in the field of non-linear equations. A natural way involves equations with the combination of two p-Laplacians such as
$$ \bigl(r(t)\varPhi _{a} \bigl(x^{\Delta } \bigr) \bigr)^{\Delta }+s(t) \varPhi _{b}\bigl(x^{\sigma }\bigr)=0, $$
where \(\varPhi _{a}\) and \(\varPhi _{b}\) stand for p-Laplacian with \(p=a\) and \(p=b\), respectively. Then one can proceed to equations of the form
$$ \bigl(r(t)\varPhi \bigl(x^{\Delta } \bigr) \bigr)^{\Delta }+s(t)g \bigl(x ^{\sigma }\bigr)=0, $$
where g satisfies the sign condition \(x g(x) >0\) whenever \(x\neq 0\). See, e.g., [55,56,57, 59].
(VI) Partial differential equations. Since half-linear equations are in fact PDE’s with one dimensional p-Laplacian, another natural way of research is towards elliptic partial differential equations in the form
$$ \mathrm{div} \bigl(A(x) \Vert \nabla u \Vert ^{p-2} \nabla u \bigr) + C(x) \varPhi (u)=0, $$
(5.24)
where A is an elliptic \(n\times n\) matrix function with differentiable components, C is a Hölder continuous function, and \(x \in \mathbb{R}^{n}\). See, e.g., [15, 18, 49] or [28], where an example of a result for Eq. (5.24) is presented using a theorem about conditionally oscillatory half-linear equations.
(VII) General coefficients. Whenever a result containing some restrictions on coefficients is proved, the main question is whether the limitations on coefficients can be removed or weakened at least. We conclude that the results of this paper cannot be significantly extended in this direction. We can illustrate this on Eq. (4.29), which is proved to be non-oscillatory in the critical case (see Theorem 4.3). If we allow functions r, s to be almost periodic, we can construct two sets of these functions, say \(r_{1}\), \(s_{1}\) and \(r_{2}\), \(s_{2}\), such that one of the obtained equations (e.g., with \(r_{1}\), \(s_{1}\)) is oscillatory and the second one (with \(r_{2}\), \(s_{2}\)) is non-oscillatory in the critical case. Constructions that can be used for obtaining these coefficients are described in [60] (and used, e.g., in [29, 31] in the discrete case).
