Abstract
In this paper, we study a general class of half-linear difference equations. Applying a version of the discrete Riccati transformation, we prove a non-oscillation criterion for the analyzed equations. In the formulation of the criterion, we do not use auxiliary sequences, but we consider directly the coefficients of the treated equations. Since the obtained criterion is new in many cases, we also formulate new simple corollaries and mention illustrative examples.
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1 Introduction
We study the so-called half-linear difference equations, i.e., the equations in the form
where \(\Phi (x) = |x|^{p -1} \textrm{sgn}\, x\) for some \(p > 1\) and \(r_k > 0\), \(k \in \mathbb {N}\). Let q be the number conjugated with p (i.e., \(p+q=pq\)). We add that Eq. (1.1) gives the linear equation for \(p=2\) and that the inverse function \(\Phi ^{ - 1}\) of \(\Phi \) takes the form \(\Phi ^{ -1}(x) = |x|^{q -1} {\textrm{sgn}\, }x\).
We contribute to the oscillation theory of Eq. (1.1). Hence, we recall the needed basic definitions. For any \(l \in \mathbb {N}\), we say that an interval \((l, l +1]\) contains the generalized zero of a solution \(\{x_k\}\) of Eq. (1.1) if \(x_l \ne 0\) and \( x_{l}x_{l + 1} \le 0\). For \(n \in \mathbb {N}\), Eq. (1.1) is called disconjugate on \(\{l,l+1,\dots ,l + n \}\) if every solution of Eq. (1.1) has at most one generalized zero on \((l, l + n + 1 ]\) and every solution \(\{x_k\}\) satisfying \(x_l = 0\) has no generalized zero on \((l,l + n+1]\). Equation (1.1) is called conjugate on \(\{l, l +1, \dots ,l + n \}\) in the opposite case. Finally, Eq. (1.1) is called non-oscillatory if and only if there exists \( l \in \mathbb {N}\) such that Eq. (1.1) is disconjugate on \(\{l, l+ 1, \dots , l + n\}\) for all \(n \in \mathbb {N}\) (note that Eq. (1.1) is called oscillatory in the opposite case).
The goal of our research is to obtain a new non-oscillation criterion for half-linear equations. Our basic motivation is given by the following result.
Theorem 1
Let us consider the equation
where \(\{r_k\}_{k = 1}^\infty \) is a positive and \(\alpha \)-periodic sequence (\(\alpha \in \mathbb {N}\)) and \(\{c_k\}_{k = 1}^\infty \) is an arbitrary sequence with the property that
If there exist \(n \in \mathbb {N}\) and \(\vartheta > 0\) such that the inequality
holds for all large \(k \in \mathbb {N}\), then Eq. (1.2) is non-oscillatory.
Proof
See [4]. \(\square \)
The concrete aim of this paper is to extend Theorem 1 into the half-linear case (see Theorem 5 below). Concerning the oscillation and non-oscillation of (linear and) half-linear difference equations, the fundamental background can be found, e.g., in [1, Chapter 3] and in [2, Chapter 8]. The presented research is motivated by a lot of recent and classic papers. In the linear case, we point out at least articles [10,11,12, 16, 17, 24,25,26]. In the half-linear case, we highlight relevant oscillation and non-oscillation results proved in [3, 5,6,7,8,9, 13,14,15, 18,19,23, 27,28,29].
2 Adapted Riccati Equation
In this section, we mention the used version of the adapted Riccati equation. Using the Riccati substitution
from Eq. (1.1), we obtain the classic Riccati equation
Considering [1, Lemma 3.2.6, (I\(_8\))], we have the following implication. If \(w_k+ r_k > 0\) for the considered \(k\in \mathbb {N}\), then
where \(\beta _k\) is between \(\Phi ^{-1}(r_k)\) and \(\Phi ^{-1}(r_k) + \Phi ^{-1}(w_k)\). Therefore, if \(w_k+ r_k > 0\) for the considered \(k\in \mathbb {N}\), then we obtain the Riccati equation in the form
where \(\beta _k\) is between \(\Phi ^{-1}(r_k)\) and \(\Phi ^{-1}(r_k) + \Phi ^{-1}(w_k)\).
We recall the following two famous results.
Theorem 2
Let us consider Eq. (1.1) together with the equation
Let \(\{r_k\}_{k = 1}^\infty , \{\tilde{r}_k\}_{k = 1}^\infty \subset (0, \infty )\) and \( \{c_k\}_{k = 1}^\infty , \{\tilde{c}_k\}_{k = 1}^\infty \) satisfy the inequalities
If Eq. (1.1) is non-oscillatory, then Eq. (2.2) is non-oscillatory as well.
Proof
See [1, Theorem 3.3.5] or [2, Theorem 8.2.3]. \(\square \)
Theorem 3
Equation (1.1) is non-oscillatory if and only if there exists a solution \(\{w_k\}\) of Eq. (2.1) satisfying \(w_k+ r_k>0\) for all large \(k \in \mathbb {N}\).
Proof
See [1, Theorem 3.3.4] or [2, Theorem 8.2.5]. \(\square \)
Now, we present the announced modification of the adapted Riccati equation. Applying the transformation
to Eq. (2.1), we have
where \(\beta _k\) is between \(\Phi ^{-1}(r_k)\) and \(\Phi ^{-1}(r_k) + \Phi ^{-1}\left( -\frac{\zeta _k}{k^{p-1}}\right) \). Therefore, we obtain the adapted Riccati equation as
where \(\beta _k\) is between \(\Phi ^{-1}(r_k)\) and \(\Phi ^{-1}(r_k) - \Phi ^{-1}\left( \frac{\zeta _k}{k^{p-1}}\right) \).
For Eq. (2.4), we formulate the corresponding part of Theorem 3 which we will use later.
Theorem 4
If Eq. (2.4) possesses a negative solution \(\{\zeta _k\} \) for all large \(k \in \mathbb {N}\), then Eq. (1.1) is non-oscillatory.
Proof
Let \(\{\zeta _k\}_{k = k_0}^\infty \) be a negative solution of Eq. (2.4). Then (see (2.3)), the sequence
is a positive solution of Eq. (2.1). Thus, the statement of this theorem follows from Theorem 3. \(\square \)
3 Results and Examples
Now, we formulate our main result.
Theorem 5
Let us consider Eq. (1.1), where \(\{r_k\}_{k = 1}^\infty \) is a positive and \(\alpha \)-periodic sequence (\(\alpha \in \mathbb {N}\)) and \(\{c_k\}_{k = 1}^\infty \) is an arbitrary sequence with the property that
If there exist \(n \in \mathbb {N}\) and \(\varepsilon > 0\) such that the inequality
holds for all large \(k \in \mathbb {N}\), then Eq. (1.1) is non-oscillatory.
Remark 1
We point out that the case given by the limit
is not covered by any known result, i.e., the oscillation behavior of Eq. (1.1) is not generally known if (3.3) is valid. Nevertheless, based on criteria proved in [8, 13, 15], we conjecture that there exist sequences \(\{r_k^1\}, \{r_k^2\} \subset (0, \infty )\), \(\{c_k^1\}, \{c_k^2\}\) satisfying (3.3) such that Eq. (1.1) is oscillatory for \(\{r_k\} \equiv \{r_k^1\}\) and \(\{c_k\} \equiv \{c_k^1\}\) and non-oscillatory for \(\{r_k\} \equiv \{r_k^2\}\) and \(\{c_k\} \equiv \{c_k^2\}\).
Remark 2
In view of the proof of Theorem 5 (see (4.6) below), one can replace (3.2) by
in the statement of Theorem 5.
Remark 3
One can easily verify that Theorem 5 is the generalization of Theorem 1 into the half-linear case (it suffices to put \(p = 2\) in the statement of Theorem 5). Concerning linear equations, the most relevant results are proved in [4, 10], for half-linear equations, we refer to [7,8,9, 13, 15, 23]. From the main results of the articles mentioned above (with exception [4, 23]), it follows that the value 1/4 in Theorem 1 and, at the same time, the value \([(p-1)/p]^p\) in Theorem 5 are the best constants in the following sense. If \(\alpha \)-periodic positive sequences \(\{r_k\}\), \(\{c_k\}\) satisty
for all large \(k \in \mathbb {N}\) and some \(\varrho > 0\), then Eq. (1.2) is oscillatory. Analogously, if \(\alpha \)-periodic positive sequences \(\{r_k\}\), \(\{c_k\}\) satisty
for all large \(k \in \mathbb {N}\) and some \(\varrho > 0\), then Eq. (1.1) is oscillatory. See Remark 2.
Next, we mention the following illustrative examples and new results which are not covered by any known non-oscillation criteria.
Example 1
Let us consider
where
and \(\alpha , \beta \in \mathbb {N}\). Then, Eq. (1.1) becomes
It is clear that \(\{r_k\}_{k = 1}^\infty \) is positive and \(\alpha \)-periodic and that \(\{c_k\}_{k = 1}^\infty \) satisfies (3.1). We have
Moreover, we obtain
because
holds for any \(l\in \mathbb {N}\). Hence, we have (3.4) for \(n = 2 \beta \). From Theorem 5 and Remark 2, it follows that Eq. (3.5) is non-oscillatory.
Corollary 1
Let us consider the equation
where \(\{c_k\}_{k = 1}^\infty \) satisfies (3.1). If there exists \(n \in \mathbb {N}\) such that
then Eq. (3.6) is non-oscillatory.
Proof
It suffices to put \(\{r_k\}_{k = 1}^\infty \equiv \{1\}_{k = 1}^\infty \) in Theorem 5. \(\square \)
Corollary 2
Let us consider Eq. (3.6), where \(\{ c_k \}_{k=1}^\infty \) satisfies (3.1). If there exists \(n\in \mathbb {N}\) such that the inequality
holds for all large \(k \in \mathbb {N}\), then Eq. (3.6) is non-oscillatory.
Proof
The assertion follows immediately from Corollary 1. Indeed, (3.8) guarantees (3.7). \(\square \)
Corollary 3
Let \(\{d_k\}_{k=1}^\infty \) be a \(\beta \)-periodic sequence (\(\beta \in \mathbb {N}\)) and \(\{h_k\}_{k=1}^\infty \) a non-decreasing positive sequence such that
Let us consider Eq. (3.6), where \( c_k={d_k}/{h_k}\), \(k \in \mathbb {N}\). If
then Eq. (3.6) is non-oscillatory.
Proof
Evidently, (3.1) follows from \(\lim _{k\rightarrow \infty }h_k/k^{p-1}=\infty \). We put
Then, we have
Therefore,
for all large \(k \in \mathbb {N}\). Hence,
holds for all large \(k \in \mathbb {N}\). Thus, it suffices to use Corollary 2. \(\square \)
Example 2
Let \(c_k=(-1)^{\lfloor (2k+1)/3 \rfloor }k^{-p} \log \left( k+1\right) \), \(k \in \mathbb {N}\), where \(\lfloor z \rfloor =\max \{n\in \mathbb {Z}\mid z\ge n\}\) is the integer part of \(z\in \mathbb {R}\). Then, Eq. (3.6) becomes the equation
Since
where \(k, m\in \mathbb {N}\), we have
where \(k, m\in \mathbb {N}\). Applying Corollary 3 for \(\beta =3\) and
we obtain the non-oscillation of Eq. (3.9).
4 Proof of Main Result
This section contains only the proof of the main result of this paper.
Proof of Theorem 5. We prove that the solution \(\{\zeta _{k}\}_{k = k_0}^\infty \) of Eq. (2.4) given by the initial condition
is negative if \(k_0 \in \mathbb {N}\) is sufficiently large. Then, Theorem 4 guarantees the non-oscillation of Eq. (1.1).
In the first part of the proof, we identify sufficiently large \(k_0\). For simplicity, without loss of generality, we assume that (3.2) is valid for all \(k \in \mathbb {N}\). In addition, we can assume that
Indeed, it suffices to consider the limit (see (3.1))
and Theorem 2, because we can redefine \(c_k, \dots , c_{k + n\alpha -1}\) in such a way that (4.2) is true and (3.2) remains valid (on the right side of (3.2), there is a positive constant; on the left side, we have an arbitrarily small value \(\varepsilon > 0\)).
Now, we consider (3.1). Evidently, there exists a continuous non-increasing function \(f: [1, \infty ) \rightarrow (0, \infty )\) satisfying
such that
In addition, considering the well known fact that \( \lim _{k \rightarrow \infty } ({k+l})^{p-1}/{k^{p-1}} = 1\) for any \(l \in \mathbb {N}\) together with (4.3) and (4.4), we obtain
We have (see (4.5))
where \(A > 0\) is a constant. Hence, without loss of generality, we will also assume that
Next (in the first part), we denote
and
We also put
The power function \(y = |x|^q\) has a continuous derivative on \([-N, 0]\). Thus, there exists \(L >0\) such that
Next, without loss of generality, we assume that (see (4.9) and (4.10))
Finally, we put
and
From the well known limits
we obtain \(\tilde{ k} \in \mathbb {N}\) for which (see also (4.9) and (4.10))
Further, from the limits (consider (4.8))
we obtain \(\hat{k} \in \mathbb {N}\) satisfying (see (4.9), (4.10), and (4.12))
where
With regard to (see (4.3) and (4.14))
let \(\overline{k} \in \mathbb {N}\) be so large that (see (4.1), (4.3), (4.9), (4.11), (4.13), and (4.14))
For
we will consider the solution \(\{\zeta _k\}_{k = k_0}^\infty \) of Eq. (2.4) given by the initial condition (4.1) and we will prove that this solution is negative for all \(k \ge k_0\), \(k \in \mathbb {N}\).
In the second part of the proof, we will estimate \(\left| \Delta \zeta _k\right| \) if \(\zeta _k \in (- N, 0)\). Let an integer \(k \ge k_0\) be such that
Note that (see (4.1) and (4.10))
Considering (4.15) and (4.26), we have
Applying the inequalities
and the identity
together with (4.16), (4.17), (4.18), (4.19), (4.20), and (4.26), we obtain
i.e., we obtain
if \(\zeta _k \in (- N, 0)\).
From (4.28) and (4.29), we obtain (see also Eq. (2.4))
whenever \(\zeta _k \in (- N, 0)\). If \(\zeta _k \in (- N, 0)\), then (4.4) and (4.30) give (see also (4.9) and (4.14))
Especially, using (4.1), (4.10), (4.23), (4.27), and (4.31), we have
and, consequently,
In the third part of the proof, for the considered sequence \(\{\zeta _k\}_{k = k_0}^\infty \), we will prove its boundedness from below by \(- N\). If
then (see (4.4), (4.8), (4.9), (4.12), and (4.30))
In other words, if (4.34) is true, then \(\Delta \zeta _k > -f(1)\), i.e., \(\zeta _{k+1} > \zeta _k - f (1)\). Analogously as in (4.35), if (4.34) is valid, then we have (see (4.2))
and, at the same time,
i.e., \(\zeta _{k+n \alpha } > \zeta _k \) if (4.34) is true. Altogether, considering (4.10), one can see that \(\zeta _k\) is bounded from below by \(-N\) if \(\{\zeta _i\}_{i = k_0}^k\) is negative.
In the last part of the proof, we will show that \(\zeta _k < 0\) for all \(k \ge k_0\), \(k \in \mathbb {N}\). On contrary, we assume that there exists \(l \in \mathbb {N}\) such that (see also (4.33))
We introduce the auxiliary sequence given by
Obviously,
Using (4.30) and (4.37), we have
From (4.39) and from (see (4.7), (4.9), (4.14), (4.21), and (4.33))
it follows
where
and
Analogously as in (4.32), considering (4.31) and (4.37), we obtain
Especially,
Now, we consider the three parts of the right hand side of (4.40) given by the three sums. Applying (4.9), (4.11), (4.25), (4.33), (4.38), (4.41), and (4.43), we have
In fact, in the estimations above, we do not use the concrete initial value \(- \zeta _{0}\) (see (4.1)). Hereafter, we will consider the more general initial condition
In particular (see (4.13)), (4.46) means that
We have (see (4.1), (4.9), (4.11), (4.24), (4.25), (4.33), (4.37), (4.38), (4.41), (4.42), (4.44), and (4.47))
i.e.,
We recall that (see (3.2) and (4.42))
Finally, considering (4.40), (4.45), (4.48), and (4.49), we obtain
In particular,
i.e, \(\zeta _{k_0+n\alpha } < \zeta _{k_0}\).
In the estimations above, one can replace \(k_0\) by an arbitrary integer \(k \ge k_0\). Therefore (see (4.33)), we have proved that
if
for an arbitrary integer \(k \ge k_0\). Of course, (4.22), (4.31), and the implication (4.51) \(\Rightarrow \) (4.50) give the contradiction \(\zeta _{l} < 0 \) (see (4.36)). Thus, the considered solution \(\{\zeta _{k}\}_{k = k_0}^\infty \) of Eq. (2.4) satisfying the initial condition (4.1) is negative for all \(k \ge k_0\), \( k \in \mathbb {N}\). \(\square \)
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Funding
Kōdai Fujimoto is supported by JSPS KAKENHI Grant number JP22K13942. Petr Hasil and Michal Veselý are supported by Grant GA20-11846 S of the Czech Science Foundation.
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Fujimoto, K., Hasil, P. & Veselý, M. Riccati Transformation and Non-Oscillation Criterion for Half-Linear Difference Equations. Bull. Malays. Math. Sci. Soc. 47, 146 (2024). https://doi.org/10.1007/s40840-024-01745-w
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DOI: https://doi.org/10.1007/s40840-024-01745-w