1 Introduction

We study the so-called half-linear difference equations, i.e., the equations in the form

$$\begin{aligned} \Delta \left[ r_k \Phi (\Delta x_k) \right] + c_k \Phi \left( x_{k+1}\right) = 0, \qquad k \in \mathbb {N}, \end{aligned}$$
(1.1)

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

$$\begin{aligned} \Delta \left[ r_k \Delta x_k \right] + c_k x_{k+1} = 0, \qquad k \in \mathbb {N}, \end{aligned}$$
(1.2)

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

$$\begin{aligned} \lim \limits _{k \rightarrow \infty } k c_k = 0. \end{aligned}$$

If there exist \(n \in \mathbb {N}\) and \(\vartheta > 0\) such that the inequality

$$\begin{aligned} \vartheta + k^2 \left( \frac{1}{n \alpha } \sum \limits _{i = k}^{k + n \alpha - 1} c_i\right) < \frac{1}{4}\left( \frac{1}{\alpha } \sum \limits _{i = 1}^{\alpha } \frac{1}{r_i} \right) ^{-1} \end{aligned}$$

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

$$\begin{aligned} w_k = r_k \Phi \left( \frac{\Delta x_k}{x_k}\right) , \end{aligned}$$

from Eq. (1.1), we obtain the classic Riccati equation

$$ \Delta w_k + c_k + w_k \left( 1 - \frac{r_k}{\Phi (\Phi ^{-1}(r_k) + \Phi ^{-1}(w_k))}\right) = 0. $$

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

$$w_k \left( 1 - \frac{r_k}{\Phi (\Phi ^{-1}(r_k) + \Phi ^{-1}(w_k))}\right) =\frac{(p-1) \left| w_k\right| ^q \left| \beta _k \right| ^{p-2}}{\Phi (\Phi ^{-1}(r_k) + \Phi ^{-1}(w_k))}, $$

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

$$\begin{aligned} \Delta w_k + c_k + \frac{(p-1) \left| w_k\right| ^q \left| \beta _k \right| ^{p-2}}{\Phi (\Phi ^{-1}(r_k) + \Phi ^{-1}(w_k))} = 0, \end{aligned}$$
(2.1)

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

$$\begin{aligned} \Delta [ \tilde{r}_k \Phi (\Delta x_k)] + \tilde{c}_k \Phi (x_{k+1}) = 0, \qquad k \in \mathbb {N}. \end{aligned}$$
(2.2)

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

$$\tilde{r}_k \ge r_k, \qquad c_k \ge \tilde{c}_k, \qquad k\in {\mathbb {N}}.$$

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

$$\begin{aligned} \zeta _k = - k^{p-1 } w_k \end{aligned}$$
(2.3)

to Eq. (2.1), we have

$$\begin{aligned} \Delta \zeta _k&= - w_k \Delta k^{p -1 } - (k+1)^{p - 1 } \Delta w_k \\&= \frac{\zeta _k}{k^{p-1 }} \Delta k^{p -1} + (k+1)^{p -1 } c_k + (k+1)^{p - 1 } \frac{(p-1) \left| \zeta _k\right| ^q \left( k^{p - 1} \right) ^{-q} \left| \beta _k \right| ^{p-2}}{\Phi \left( \Phi ^{-1}(r_k) + \Phi ^{-1}\left( -\frac{\zeta _k}{k^{p-1 }}\right) \right) } , \end{aligned}$$

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

$$\begin{aligned} \begin{aligned} \Delta \zeta _k = \frac{\Delta k^{p -1 }}{k^{p-1 }}\,\zeta _k + (k+1)^{p -1} c_k + \frac{(p-1) k^{- p} (k+1)^{p - 1 } \left| \beta _k \right| ^{p-2}}{\Phi \left( \Phi ^{-1}(r_k) - \Phi ^{-1}\left( \frac{\zeta _k}{k^{p-1}}\right) \right) } \, \left| \zeta _k\right| ^q, \end{aligned} \end{aligned}$$
(2.4)

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

$$\{w_k\}_{k = k_0}^\infty \equiv \left\{ - \frac{\zeta _k}{k^{p-1 } }\right\} _{k = k_0}^\infty $$

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

$$\begin{aligned} \lim \limits _{k \rightarrow \infty } k^{p-1} c_k = 0. \end{aligned}$$
(3.1)

If there exist \(n \in \mathbb {N}\) and \(\varepsilon > 0\) such that the inequality

$$\begin{aligned} \varepsilon + k^p \left( \frac{1}{n \alpha } \sum \limits _{i = k}^{k + n \alpha - 1} c_i \right) < \left( \frac{p-1}{p}\right) ^p \left( \frac{1}{\alpha } \sum \limits _{i = 1}^{\alpha } {r_i}^{\frac{1}{1-p}} \right) ^{1-p} \end{aligned}$$
(3.2)

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

$$\begin{aligned} \lim \limits _{k \rightarrow \infty } k^p \left( \frac{1}{ \alpha } \sum \limits _{i = k}^{k + \alpha - 1} c_i \right) = \left( \frac{p-1}{p} \right) ^p \left( \frac{1}{\alpha } \sum \limits _{i = 1}^{\alpha } r_i^{\frac{1}{1-p}} \right) ^{1-p} \end{aligned}$$
(3.3)

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

$$\begin{aligned} \limsup _{k\rightarrow \infty } \frac{1}{n\alpha } \sum _{i=k}^{k+n\alpha -1}i^p c_i < \left( \frac{p-1}{p} \right) ^p \left( \frac{1}{\alpha }\sum _{i=1}^\alpha r_i^\frac{1}{1-p} \right) ^{1-p} \end{aligned}$$
(3.4)

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

$$ \frac{1}{\alpha } \sum \limits _{i = 1}^{\alpha } i^2 c_i > \frac{1}{4}\left( \frac{1}{\alpha } \sum \limits _{i = 1}^{\alpha } \frac{1}{r_i} \right) ^{-1} + \varrho $$

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

$$ \frac{1}{\alpha } \sum \limits _{i = 1}^{ \alpha } i^p c_i > \left( \frac{p-1}{p}\right) ^p \left( \frac{1}{\alpha } \sum \limits _{i = 1}^{\alpha } {r_i}^{\frac{1}{1-p}} \right) ^{1-p} + \varrho $$

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

$$\begin{aligned} r_k=\left( a+\sin {\frac{2\pi k}{\alpha }} \right) ^{1-p},\qquad c_k=\frac{1}{k^p}\left( b+\sqrt{k}\cos {\frac{\pi k}{\beta }} \right) , \qquad k \in \mathbb {N}, \end{aligned}$$

where

$$\begin{aligned} a>1,\qquad b<\left( \frac{p-1}{p}\right) ^p a^{1-p}, \end{aligned}$$

and \(\alpha , \beta \in \mathbb {N}\). Then, Eq. (1.1) becomes

$$\begin{aligned} \Delta \left[ \left( a+\sin {\frac{2\pi k}{\alpha }} \right) ^{1-p} \Phi (\Delta x_k)\right] +\frac{1}{k^p}\left( b+\sqrt{k}\cos {\frac{\pi k}{\beta }} \right) \Phi (x_{k+1})=0, \qquad k \in \mathbb {N} . \end{aligned}$$
(3.5)

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

$$\begin{aligned} \left( \frac{1}{\alpha }\sum _{i=1}^\alpha r_i^\frac{1}{1-p} \right) ^{1-p}=a^{1-p}. \end{aligned}$$

Moreover, we obtain

$$\begin{aligned} \limsup _{k\rightarrow \infty }\frac{1}{2\alpha \beta }\sum _{i=k}^{k+2\alpha \beta -1}i^p c_i =b+\lim _{k\rightarrow \infty }\frac{1}{2\alpha \beta }\sum _{i=k}^{k+2\alpha \beta -1}\sqrt{i}\cos {\frac{\pi i}{\beta }}=b, \end{aligned}$$

because

$$\begin{aligned} \lim _{k\rightarrow \infty }\left( \sqrt{k+l}-\sqrt{k} \right) =0 \end{aligned}$$

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

$$\begin{aligned} \Delta \left[ \Phi (\Delta x_k) \right] + c_k \Phi \left( x_{k+1}\right) = 0, \qquad k \in \mathbb {N}, \end{aligned}$$
(3.6)

where \(\{c_k\}_{k = 1}^\infty \) satisfies (3.1). If there exists \(n \in \mathbb {N}\) such that

$$\begin{aligned} \limsup \limits _{k \rightarrow \infty } \frac{k^p}{n} \sum \limits _{i = k}^{k + n - 1} c_i < \left( \frac{p-1}{p} \right) ^p, \end{aligned}$$
(3.7)

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

$$\begin{aligned} \sum _{i=k}^{k+n-1}c_i\le 0 \end{aligned}$$
(3.8)

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

$$\begin{aligned} \lim _{k\rightarrow \infty }\frac{h_k}{k^{p-1}}=\infty ,\qquad \lim _{k\rightarrow \infty } \frac{h_k}{h_{k+ \beta -1}}=1. \end{aligned}$$

Let us consider Eq. (3.6), where \( c_k={d_k}/{h_k}\), \(k \in \mathbb {N}\). If

$$\sum _{i=1}^\beta d_i<0, $$

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

$$\begin{aligned} d^+:=\sum _{i=1}^\beta \max \{d_i,0\},\qquad d^-:=\sum _{i=1}^\beta \max \{-d_i,0\}. \end{aligned}$$

Then, we have

$$\begin{aligned} \frac{d^+}{d^-}<1=\lim _{k\rightarrow \infty } \frac{h_k}{h_{k+ \beta -1}}. \end{aligned}$$

Therefore,

$$\begin{aligned} \frac{d^+}{h_k}-\frac{d^-}{h_{k+ \beta -1}}\le 0 \end{aligned}$$

for all large \(k \in \mathbb {N}\). Hence,

$$\begin{aligned} \sum _{i=k}^{k+ \beta -1}c_i = \sum _{i=k}^{k+ \beta -1}\frac{d_i}{h_i}\le \frac{d^+}{h_k}-\frac{d^-}{h_{k+ \beta -1}}\le 0 \end{aligned}$$

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

$$\begin{aligned} \Delta \left[ \Phi (\Delta x_k) \right] +\frac{(-1)^{\left\lfloor \frac{2k+1}{3} \right\rfloor } \log \left( k+1\right) }{k^p}\,\Phi (x_{k+1})=0, \qquad k \in \mathbb {N}. \end{aligned}$$
(3.9)

Since

$$\begin{aligned} \left\lfloor \frac{2k+1}{3} \right\rfloor =\left\{ \begin{array}{ll} 2m, &{} k=3m,\\ 2m+1, &{} k\in \{3m+1,3m+2\}, \end{array}\right. \end{aligned}$$

where \(k, m\in \mathbb {N}\), we have

$$\begin{aligned} (-1)^{\left\lfloor \frac{2k+1}{3} \right\rfloor }=\left\{ \begin{array}{ll} 1, &{} k=3m,\\ -1, &{} k\in \{3m+1,3m+2\}, \end{array}\right. \end{aligned}$$

where \(k, m\in \mathbb {N}\). Applying Corollary 3 for \(\beta =3\) and

$$ d_k = (-1)^{\left\lfloor \frac{2k+1}{3} \right\rfloor }, \qquad h_k = \frac{k^p}{\log \left( k+1\right) }, \qquad k \in \mathbb {N}, $$

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

$$\begin{aligned} \zeta _{k_0} = - \zeta _0 := - \left( \frac{q}{n\alpha } \sum \limits _{i = 1}^{n\alpha } r_i^{\frac{1}{1-p}} \right) ^{1-p} = - \left( \frac{q}{\alpha } \sum \limits _{i = 1}^{\alpha } r_i^{{1-q}} \right) ^{1-p} \end{aligned}$$
(4.1)

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

$$\begin{aligned} \sum \limits _{i=k}^{k+n\alpha -1} (i+1)^{p-1} c_i \ge 0, \qquad k \in \mathbb {N}. \end{aligned}$$
(4.2)

Indeed, it suffices to consider the limit (see (3.1))

$$\begin{aligned} \lim \limits _{k \rightarrow \infty } k \left[ (k+1)^{p-1} c_k + (k+2)^{p-1} c_{k+1} + \cdots + (k+ n \alpha )^{p-1} c_{k+ n \alpha - 1} \right.&\\ \left. - \, k^{p-1} \left( c_k + c_{k+1} + \cdots + c_{k+n\alpha -1} \right) \right]&= 0 \end{aligned}$$

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

$$\begin{aligned} \lim _{t \rightarrow \infty } f(t) = 0 \end{aligned}$$
(4.3)

such that

$$\begin{aligned} (k+1)^{p-1} \left| c_k \right| < f (k), \qquad k \in \mathbb {N}. \end{aligned}$$
(4.4)

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

$$\begin{aligned} \lim \limits _{k \rightarrow \infty } k^{p-1} \left| c_{k + i} \right| = 0, \qquad i \in \{0, 1, \ldots , n \alpha -1 \}. \end{aligned}$$
(4.5)

We have (see (4.5))

$$\begin{aligned} \begin{aligned}&\limsup \limits _{k \rightarrow \infty } \left| k^p \left( \frac{1}{n \alpha } \sum \limits _{i = k}^{k + n \alpha - 1} c_i \right) - k \left( \frac{1}{n \alpha } \sum \limits _{i = k}^{k + n \alpha - 1} \left( i+1\right) ^{p-1} c_i \right) \right| \\ {}&\le \limsup \limits _{k \rightarrow \infty } k \left( \frac{1}{n \alpha } \sum \limits _{i = k}^{k + n \alpha - 1} \left| k^{p-1} - (i+1)^{p-1} \right| \left| c_i \right| \right) \\ {}&\le \limsup \limits _{k \rightarrow \infty } k \left( (k+n\alpha )^{p-1} - k^{p-1} \right) \left( \frac{1}{n \alpha } \sum \limits _{i = k}^{k + n \alpha - 1} \left| c_i \right| \right) \\ {}&\le \limsup \limits _{k \rightarrow \infty } A k^{p-1} \left( \frac{1}{n \alpha } \sum \limits _{i = k}^{k + n \alpha - 1} \left| c_i \right| \right) \\ {}&\le A \lim \limits _{k \rightarrow \infty } k^{p-1} \sum \limits _{i = k}^{k + n \alpha - 1} \left| c_i \right| = 0 , \end{aligned} \end{aligned}$$
(4.6)

where \(A > 0\) is a constant. Hence, without loss of generality, we will also assume that

$$\begin{aligned} \left| k^p \left( \frac{1}{n \alpha } \sum \limits _{i = k}^{k + n \alpha - 1} c_i \right) - k \left( \frac{1}{n \alpha } \sum \limits _{i = k}^{k + n \alpha - 1} \left( i+1\right) ^{p-1} c_i \right) \right| < \frac{\varepsilon }{16}, \qquad k \in \mathbb {N}. \end{aligned}$$
(4.7)

Next (in the first part), we denote

$$\begin{aligned} r_{\text {min}} := \min \{r_1, \dots , r_\alpha \} > 0, \qquad r_{\text {max}} := \max \{r_1, \dots , r_\alpha \} < \infty \end{aligned}$$
(4.8)

and

$$\begin{aligned} R_{\text {max}} := r_{\text {min}}^{1-q}, \qquad R_{\text {min}} := r_{\text {max}}^{1-q}. \end{aligned}$$
(4.9)

We also put

$$\begin{aligned} \begin{aligned} N := 1 + n \alpha f (1) + 2 \zeta _0 + \left( \frac{ p- 1 + \frac{9}{8} }{ (p-1) R_{\text {min}}} \right) ^{{p-1}} + \frac{1}{16 (p-1)} . \end{aligned} \end{aligned}$$
(4.10)

The power function \(y = |x|^q\) has a continuous derivative on \([-N, 0]\). Thus, there exists \(L >0\) such that

$$\begin{aligned} \left| \left| x_1\right| ^q - \left| x_2\right| ^q \right| \le L \left| x_1-x_2\right| , \qquad x_1, x_2 \in \left[ -N,0\right] . \end{aligned}$$
(4.11)

Next, without loss of generality, we assume that (see (4.9) and (4.10))

$$\begin{aligned} \varepsilon < \min \left\{ 16 (p-1)R_{\text {max}} N^q, 1 \right\} . \end{aligned}$$
(4.12)

Finally, we put

$$\begin{aligned} L_0 := \frac{\varepsilon }{16} \cdot \min \left\{ \frac{1}{(p - 1)R_{\text {max}} L }, \frac{1}{ p - 1 } \right\} \end{aligned}$$
(4.13)

and

$$\begin{aligned} B:= (p-1) R_{\text {max}} N^q + \left( p-1\right) N + \frac{\varepsilon }{8}. \end{aligned}$$
(4.14)

From the well known limits

$$ \lim \limits _{k \rightarrow \infty } \frac{{\Delta } k^{p-1 } }{k^{ p- 2 }} = p-1, \qquad \lim \limits _{k \rightarrow \infty } \frac{(k+1)^{p - 1 } }{k^{p - 1 }} = 1, $$

we obtain \(\tilde{ k} \in \mathbb {N}\) for which (see also (4.9) and (4.10))

$$\begin{aligned} \left| \frac{{\Delta } k^{p- 1 } }{k^{p -2}} - \left( p-1 \right) \right| <\frac{\varepsilon }{16N}, \qquad k \ge \tilde{ k}, \, k \in \mathbb {N} , \end{aligned}$$
(4.15)
$$\begin{aligned} 1< \frac{ (k+1)^{p-1 }}{k^{p-1 }} < \min \left\{ 1 + \frac{\varepsilon }{16\, f(1)}, \sqrt{ 1 + \frac{\varepsilon }{16 (p-1) R_{\text {max}} N^q} } \right\} , \qquad k \ge \tilde{k}, \, k \in \mathbb {N}. \end{aligned}$$
(4.16)

Further, from the limits (consider (4.8))

$$ \lim \limits _{k \rightarrow \infty } \frac{\Phi ^{-1}\left( r_k \right) }{\Phi ^{-1}\left( r_k\right) + \Phi ^{-1}\left( \frac{N}{k^{p- 1 }}\right) } = 1, $$
$$ \lim \limits _{k \rightarrow \infty } \frac{ r_k }{{\Phi \left( \Phi ^{-1}\left( r_k \right) + \Phi ^{-1}\left( \frac{N}{k^{p - 1 }}\right) \right) }} = 1, $$

we obtain \(\hat{k} \in \mathbb {N}\) satisfying (see (4.9), (4.10), and (4.12))

$$\begin{aligned} \sqrt{ 1- \frac{\varepsilon }{16 (p-1) R_{\text {max}} N^q} }< B_k < \sqrt{1 + \frac{\varepsilon }{16 (p-1) R_{\text {max}} N^q} }, \qquad k \ge \hat{k}, \, k \in \mathbb {N}, \end{aligned}$$
(4.17)
$$\begin{aligned} \sqrt{ 1- \frac{\varepsilon }{16 (p-1) R_{\text {max}} N^q} }< b_k < 1 , \qquad k \ge \hat{k}, \, k \in \mathbb {N}, \end{aligned}$$
(4.18)

where

$$\begin{aligned} B_k:= \left( \frac{\Phi ^{-1}\left( r_k\right) }{\Phi ^{-1}\left( r_k\right) + \Phi ^{-1}\left( \frac{N}{k^{p-1 }}\right) } \right) ^{2-p}, \qquad k \in \mathbb {N}, \end{aligned}$$
(4.19)
$$\begin{aligned} b_k:= \frac{ r_k}{\Phi \left( \Phi ^{-1}\left( r_k\right) + \Phi ^{-1}\left( \frac{N}{k^{p - 1 }}\right) \right) }, \qquad k \in \mathbb {N}. \end{aligned}$$
(4.20)

With regard to (see (4.3) and (4.14))

$$\begin{aligned} \lim \limits _{k \rightarrow \infty } f(k)+ \frac{B}{k} = 0, \end{aligned}$$

let \(\overline{k} \in \mathbb {N}\) be so large that (see (4.1), (4.3), (4.9), (4.11), (4.13), and (4.14))

$$\begin{aligned} \frac{n\alpha B}{k} < \frac{\varepsilon }{16}, \qquad k \ge \overline{k}, \, k \in \mathbb {N}, \end{aligned}$$
(4.21)
$$\begin{aligned} n\alpha f(k) + \frac{n\alpha B}{k} < L_0, \qquad k \ge \overline{k}, \, k \in \mathbb {N}, \end{aligned}$$
(4.22)
$$\begin{aligned} n\alpha f(k) + \frac{n\alpha B}{k} < {\zeta _0}, \qquad k \ge \overline{k}, \, k \in \mathbb {N}, \end{aligned}$$
(4.23)
$$\begin{aligned} (p-1) \, n\alpha \left( f(k)+ \frac{B}{k} \right) < \frac{\varepsilon }{16}, \qquad k \ge \overline{k}, \, k \in \mathbb {N}, \end{aligned}$$
(4.24)
$$\begin{aligned} {(p-1)R_{\text {max}} L} n \alpha \left( f(k)+ \frac{ B}{k}\right)< \frac{\varepsilon }{16} < \frac{\varepsilon }{4}, \qquad k \ge \overline{k}, \, k \in \mathbb {N}. \end{aligned}$$
(4.25)

For

$$ k_0:= \max \left\{ \tilde{ k}, \hat{k}, \overline{k}\right\} , $$

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

$$\begin{aligned} \zeta _k \in (- N, 0) . \end{aligned}$$
(4.26)

Note that (see (4.1) and (4.10))

$$\begin{aligned} \zeta _{k_0} = - \zeta _0 \in (- N, 0). \end{aligned}$$
(4.27)

Considering (4.15) and (4.26), we have

$$\begin{aligned} \left| \frac{ {\Delta } k^{p-1} }{k^{p-1 }} \, \zeta _k - \left( p-1 \right) \, \frac{ \zeta _k}{k} \right| = \frac{|\zeta _k|}{k} \cdot \left| \frac{\Delta k^{p -1} }{k^{p - 2}} - \left( p-1 \right) \right| < \frac{N}{k} \cdot \frac{\varepsilon }{16N} = \frac{\varepsilon }{16k } .\nonumber \\ \end{aligned}$$
(4.28)

Applying the inequalities

$$ 0 < \Phi ^{-1}\left( r_k \right) \le \beta _k \le \Phi ^{-1}\left( r_k \right) + \Phi ^{-1}\left( \frac{N}{k^{p - 1 }}\right) $$

and the identity

$$ \frac{r_k^{1-q}}{k} = \frac{ k^{- p} k^{p-1}\left( {\Phi ^{-1}\left( r_k \right) } \right) ^{p-2} }{r_k } $$

together with (4.16), (4.17), (4.18), (4.19), (4.20), and (4.26), we obtain

$$\begin{aligned}&\left| \frac{ k^{- p} (k + 1)^{p-1}|\beta _k|^{p-2} }{\Phi \left( \Phi ^{-1}\left( r_k \right) - \Phi ^{-1}\left( \frac{\zeta _k }{k^{p - 1 }}\right) \right) } - \frac{r_k^{1-q}}{k} \right| \\&= \frac{r_k^{1-q}}{k} \left| \frac{ \frac{(k + 1)^{p- 1}}{k^{p-1}} \cdot \frac{\beta _k^{p-2}}{\left( {\Phi ^{-1}\left( r_k \right) }\right) ^{p-2}} }{\frac{ \Phi \left( \Phi ^{-1}\left( r_k \right) - \Phi ^{-1}\left( \frac{\zeta _k }{k^{p - 1}}\right) \right) }{ r_k }} - 1 \right| \\ {}&< \frac{r_k^{1-q}}{k} \max \left\{ \left| \frac{ \sqrt{1+ \frac{\varepsilon }{16 (p-1) R_{\text {max}} N^q}} \cdot \sqrt{1+ \frac{\varepsilon }{16 (p-1) R_{\text {max}} N^q}} }{1} - 1 \right| ,\right. \\&\quad \left. \left| \frac{ 1 \cdot \sqrt{1- \frac{\varepsilon }{16 (p-1) R_{\text {max}} N^q}} }{\frac{1}{\sqrt{1- \frac{\varepsilon }{16 (p-1) R_{\text {max}} N^q}}}} - 1 \right| \right\} \\&= \frac{r_k^{1-q}}{k} \, \frac{\varepsilon }{16 (p-1) R_{\text {max}} N^q} \le \frac{\varepsilon }{16 k (p-1) N^q}, \end{aligned}$$

i.e., we obtain

$$\begin{aligned} \begin{aligned}&\left| \frac{(p-1)k^{- p} (k + 1)^{p-1 }|\beta _k |^{p-2}|\zeta _k |^q}{\Phi \left( \Phi ^{-1}\left( r_k \right) - \Phi ^{-1}\left( \frac{\zeta _k }{k^{p -1 }}\right) \right) } - \frac{(p-1) r_k^{1-q}|\zeta _k |^q}{k} \right| \\ {}&= (p-1) |\zeta _k |^q \cdot \left| \frac{ k^{ - p} (k + 1)^{p-1 }|\beta _k|^{p-2} }{\Phi \left( \Phi ^{-1}\left( r_k \right) - \Phi ^{-1}\left( \frac{\zeta _k }{k^{p - 1 }}\right) \right) } - \frac{r_k^{1-q}}{k} \right| \\ {}&< (p-1) |\zeta _k |^q \cdot \frac{\varepsilon }{16 k (p-1) N^q} < \frac{\varepsilon }{16 k} \end{aligned} \end{aligned}$$
(4.29)

if \(\zeta _k \in (- N, 0)\).

From (4.28) and (4.29), we obtain (see also Eq. (2.4))

$$\begin{aligned} \left| \Delta \zeta _k - (k+1)^{p-1} c_k - \frac{ (p-1) r_k^{1-q} |\zeta _k |^q + \left( p-1 \right) \zeta _k}{k} \right| < \frac{\varepsilon }{8 k} \end{aligned}$$
(4.30)

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))

$$\begin{aligned} \begin{aligned} \left| \Delta \zeta _k \right|&< (k+1)^{p-1} \left| c_k\right| + \frac{1}{k} \left[ (p-1) r_k^{1-q} |\zeta _k |^q + \left( p-1 \right) |\zeta _k | + \frac{\varepsilon }{8} \right] \\ {}&< f(k) + \frac{1}{k} \left[ (p-1) R_{\text {max}} N^q + \left( p-1 \right) N + \frac{\varepsilon }{8} \right] = f(k) + \frac{B}{k} . \end{aligned} \end{aligned}$$
(4.31)

Especially, using (4.1), (4.10), (4.23), (4.27), and (4.31), we have

$$\begin{aligned} \begin{aligned} \left| \zeta _k + \zeta _0 \right|&= \left| \sum \limits _{i = k_0}^{k-1} \Delta \zeta _i \right| \le \sum \limits _{i = k_0}^{k-1} \left| \Delta \zeta _i \right| \\&< n\alpha \left( f(k) + \frac{B}{k}\right) < \zeta _0, \qquad k \in \{k_0 + 1, \dots , k_0 + n \alpha \}, \end{aligned} \end{aligned}$$
(4.32)

and, consequently,

$$\begin{aligned} \zeta _k \in (- N, 0), \qquad k \in \{k_0, k_0 + 1, \dots , k_0 + n \alpha \} . \end{aligned}$$
(4.33)

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

$$\begin{aligned} -N + n \alpha f (1)< \zeta _k <- \max \left\{ 1 , \left( \frac{ p-1 + \frac{9}{8} }{ (p-1) R_{\text {min}}} \right) ^{p-1} \right\} , \end{aligned}$$
(4.34)

then (see (4.4), (4.8), (4.9), (4.12), and (4.30))

$$\begin{aligned} \begin{aligned} \Delta \zeta _k&> (k+1)^{p-1} c_k + \frac{1}{k} \left[ (p-1) r_k^{1-q} |\zeta _k|^q + \left( p-1 \right) \zeta _k - \frac{\varepsilon }{8} \right] \\ {}&> -f(k) + \frac{1}{k} \left[ (p-1) R_{\text {min}} |\zeta _k|^q - \left( p-1 \right) |\zeta _k| - \frac{1}{8} \right] \\&= -f(k)+ \frac{1}{k} \left[ |\zeta _k| \left( (p-1) R_{\text {min}} |\zeta _k|^{q-1} - \left( p-1 \right) \right) - \frac{1}{8} \right] \\&> -f(k)+ \frac{1}{k} \left[ 1 \cdot \left( (p-1) R_{\text {min}} \frac{ p- 1 + \frac{9}{8} }{ (p-1) R_{\text {min}}} - \left( p- 1 \right) \right) - \frac{1}{8} \right] \\ {}&= -f(k)+ \frac{1}{k} \left[ \frac{9}{8} - \frac{1}{8} \right] = -f(k)+ \frac{1}{k} > -f(1) . \end{aligned} \end{aligned}$$
(4.35)

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))

$$\begin{aligned} \sum \limits _{i = k}^{k + n\alpha -1 }\Delta \zeta _i&> \sum \limits _{i = k}^{k + n\alpha -1 } (i+1)^{p-1} c_i + \sum \limits _{i = k}^{k + n\alpha -1 } \frac{1}{i} \left[ (p-1) r_i^{1-q} |\zeta _i|^q + \left( p-1 \right) \zeta _i - \frac{\varepsilon }{8} \right] \\ {}&> \cdots > \sum \limits _{i = k}^{k + n \alpha -1 } (i+1)^{p-1} c_i \ge 0 \end{aligned}$$

and, at the same time,

$$ \sum \limits _{i = k}^{k + n\alpha -1 }\Delta \zeta _i = \zeta _{k+n \alpha } - \zeta _k, $$

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))

$$\begin{aligned} \zeta _{k_0}, \zeta _{k_0+1}, \ldots , \zeta _{k_0 + n \alpha }, \ldots , \zeta _{l -1} < 0, \qquad \zeta _{l} \ge 0 . \end{aligned}$$
(4.36)

We introduce the auxiliary sequence given by

$$\begin{aligned} \xi _k :=\frac{1}{n\alpha }\sum \limits _{i = k}^{k + n\alpha -1 }\zeta _i, \qquad k \in \{ k_0, k_0 + 1, \dots , l - n\alpha \}. \end{aligned}$$
(4.37)

Obviously,

$$\begin{aligned} \xi _k \in (-N, 0), \qquad k \in \{ k_0, k_0 + 1, \dots , l - n\alpha \}. \end{aligned}$$
(4.38)

Using (4.30) and (4.37), we have

$$\begin{aligned} \begin{aligned} \Delta \xi _{k_0}&=\frac{1}{n \alpha }\sum \limits _{i = k_0}^{k_0 + n\alpha -1 }\Delta \zeta _i\\&< \frac{1}{n \alpha }\sum \limits _{i = k_0}^{k_0 + n\alpha -1 } \left[ (i+1)^{p-1} c_i + \frac{1}{i} \left( {(p-1) r_i^{1-q} |\zeta _i|^q} + (p-1) \zeta _i + \frac{\varepsilon }{8} \right) \right] . \end{aligned} \end{aligned}$$
(4.39)

From (4.39) and from (see (4.7), (4.9), (4.14), (4.21), and (4.33))

$$\begin{aligned}&\left| \frac{1}{n\alpha }\sum \limits _{i = k_0}^{k_0 + n\alpha -1 } \left[ (i+1)^{p-1} c_i + \frac{1}{i} \left( {(p-1) r_i^{1-q} |\zeta _i|^q} + (p-1) \zeta _i + \frac{\varepsilon }{8} \right) \right] \right. \\&\qquad \left. - \frac{1}{k_0} \left[ \frac{1}{n\alpha }\sum \limits _{i = k_0}^{k_0 + n\alpha -1 } \left( k_0^p c_i + {(p-1) r_i^{1-q} |\zeta _i|^q} + (p-1) \zeta _i + \frac{\varepsilon }{8} \right) \right] \right| \\&< \frac{1}{n\alpha }\sum \limits _{i = k_0}^{k_0 + n\alpha -1 } \left( \frac{1}{k_0} - \frac{1}{i}\right) \left( (p-1)R_{\text {max}} N^q + (p- 1) N + \frac{\varepsilon }{8}\right) + \frac{\varepsilon }{16k_0} \\&< \frac{n\alpha }{k_0^2} \left( (p-1)R_{\text {max}} N^q + (p- 1) N + \frac{\varepsilon }{8} \right) + \frac{\varepsilon }{16k_0} = \frac{n\alpha B}{k_0^2} + \frac{\varepsilon }{16k_0} < \frac{\varepsilon }{8 k_0} , \end{aligned}$$

it follows

$$\begin{aligned} \Delta \xi _{k_0}&< \frac{1}{{k_0}} \left[ \frac{1}{n\alpha }\sum \limits _{i = k_0}^{ k_0 + n\alpha -1 } \left( k_0^p c_i + {(p-1) r_i^{1-q} |\zeta _i |^q} + (p-1)\zeta _i + \frac{\varepsilon }{4} \right) \right] \nonumber \\&= \frac{1}{{k_0}} \left[ \frac{1}{n\alpha }\sum \limits _{i = k_0}^{ k_0 + n\alpha -1 } {(p-1) r_i^{1-q} |\zeta _i |^q} - u \right. \nonumber \\&\qquad \qquad + \frac{1}{n\alpha }\sum \limits _{i = k_0}^{ k_0 + n\alpha -1 } (p-1) \zeta _i + u + v \nonumber \\&\qquad \qquad \qquad \qquad + \left. \frac{1}{n\alpha }\sum \limits _{i = k_0}^{ k_0 + n\alpha -1 } k_0^p {c_i } - v + \frac{\varepsilon }{4} \right] , \end{aligned}$$
(4.40)

where

$$\begin{aligned} u := (p-1)\left| \xi _{k_0} \right| ^q \left( \frac{1}{n\alpha }\sum \limits _{i = k_0}^{ k_0 + n\alpha -1 } r_i^{\frac{1}{1-p}} \right) = (p-1)\left| \xi _{k_0} \right| ^q \left( \frac{1}{\alpha }\sum \limits _{i = 1}^{\alpha } r_i^{{1-q}} \right) \end{aligned}$$
(4.41)

and

$$\begin{aligned} v := \left( \frac{p- 1}{p}\right) ^p \left( \frac{1}{n\alpha }\sum \limits _{i = k_0}^{ k_0 + n\alpha -1 } r_i^{\frac{1}{1-p}} \right) ^{1-p} = q^{-p} \left( \frac{1}{\alpha }\sum \limits _{i = 1}^{\alpha } r_i^{{1-q}} \right) ^{1-p}. \end{aligned}$$
(4.42)

Analogously as in (4.32), considering (4.31) and (4.37), we obtain

$$\begin{aligned} \begin{aligned} \left| \zeta _{i} - \xi _{k_0} \right|&\le \max \limits _{j \in \{ k_0, k_0 + 1, \dots , k_0 + n\alpha - 1\} } \left| \zeta _{i} - \zeta _{j} \right| \\ {}&\le \sum \limits _{i = k_0}^{k_0 + n \alpha -2} \left| \Delta \zeta _i \right| \\ {}&\le (n\alpha -1) \left( f(k_0)+ \frac{ B}{k_0}\right) \\ {}&< n\alpha \left( f(k_0)+ \frac{ B}{k_0}\right) , \qquad i \in \{ k_0, k_0 + 1, \dots , k_0 + n\alpha - 1\}. \end{aligned} \end{aligned}$$
(4.43)

Especially,

$$\begin{aligned} \left| \zeta _{k_0} - \xi _{k_0} \right| < n\alpha \left( f(k_0)+ \frac{ B}{k_0}\right) . \end{aligned}$$
(4.44)

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

$$\begin{aligned} \left| \frac{1}{n\alpha }\sum \limits _{i = k_0}^{ k_0 + n\alpha -1 } {(p-1)r_i^{1-q} |\zeta _i|^q} - u \right|&\le \frac{p-1}{n\alpha } \sum \limits _{i = k_0}^{ k_0 + n\alpha -1 } r_i^{1-q} \left| |\zeta _i|^q - \left| \xi _{k_0}\right| ^q \right| \nonumber \\&\le \frac{(p-1) R_{\text {max}}}{n\alpha } \sum \limits _{i = k_0}^{ k_0 + n\alpha -1 } \left| |\zeta _i|^q - \left| \xi _{k_0} \right| ^q \right| \nonumber \\&\le \frac{(p-1) R_{\text {max}}}{n\alpha } \sum \limits _{i = k_0}^{ k_0 + n\alpha -1 } L \left| \zeta _i - \xi _{k_0} \right| \nonumber \\&< {(p-1)R_{\text {max}} L} n \alpha \left( f(k_0)+ \frac{ B}{k_0}\right) < \frac{\varepsilon }{4} . \end{aligned}$$
(4.45)

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

$$\begin{aligned} \zeta _{k_0} \in \left( - \zeta _0 - L_0 , - \zeta _0 \right] . \end{aligned}$$
(4.46)

In particular (see (4.13)), (4.46) means that

$$\begin{aligned} \left| \zeta _{k_0} + \zeta _0 \right| < \frac{\varepsilon }{16} \cdot \min \left\{ \frac{1}{(p - 1)R_{\text {max}} L }, \frac{1}{ p - 1 } \right\} . \end{aligned}$$
(4.47)

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))

$$\begin{aligned}&\left| \frac{1}{n\alpha }\sum \limits _{i = k_0}^{ k_0 + n\alpha -1 } (p-1 ) \zeta _i + u + v \right| \\ {}&= \left| (p-1) \xi _{k_0} + (p-1) \left| \xi _{k_0}\right| ^q \left( \frac{1}{ \alpha } \sum \limits _{i = 1}^{\alpha } r_i^{1-q} \right) + q^{-p} \left( \frac{1}{\alpha }\sum \limits _{i = 1}^{\alpha } r_i^{1-q} \right) ^{1-p} \right| \\&\le \left| (p-1) \xi _{k_0} - (p-1) \zeta _{k_0} \right| + \left| (p-1) \zeta _{k_0} + (p-1) \zeta _0 \right| \\&\qquad + \left| (p-1) \left| \xi _{k_0} \right| ^q \left( \frac{1}{ \alpha } \sum \limits _{i = 1}^{\alpha } r_i^{1-q} \right) - (p-1) \left| \zeta _0 \right| ^q \left( \frac{1}{ \alpha } \sum \limits _{i = 1}^{\alpha } r_i^{1-q} \right) \right| \\&\qquad + \left| -(p-1) \zeta _{0} + (p-1) \left| {\zeta _0}\right| ^q \left( \frac{1}{ \alpha } \sum \limits _{i = 1}^{\alpha } r_i^{1-q} \right) + q^{-p} \left( \frac{1}{\alpha }\sum \limits _{i = 1}^{\alpha } r_i^{1-q} \right) ^{1-p} \right| \end{aligned}$$
$$\begin{aligned}&< (p-1) \, n\alpha \left( f(k_0)+ \frac{B}{k_0} \right) + (p-1) \, \frac{\varepsilon }{16 \left( p - 1\right) } \\ {}&\qquad + (p-1) R_{\text {max}} L \left( \left| \xi _{k_0} - \zeta _{k_0} \right| + \left| \zeta _{k_0} + \zeta _0 \right| \right) \\&\qquad + \left| -(p-1) q^{1-p} \left( \frac{1}{\alpha }\sum \limits _{i = 1}^{\alpha } r_i^{1-q} \right) ^{1-p} \right. \\ {}&\qquad \qquad \left. + \,(p-1) q^{-p} \left( \frac{1}{\alpha }\sum \limits _{i = 1}^{\alpha } r_i^{1-q} \right) ^{1-p} + q^{-p} \left( \frac{1}{\alpha }\sum \limits _{i = 1}^{\alpha } r_i^{1-q} \right) ^{1-p} \right| \qquad \quad \\&< (p-1) \, n\alpha \left( f(k_0)+ \frac{B}{k_0} \right) + \frac{\varepsilon }{16 } \\ {}&\qquad + (p-1) R_{\text {max}} L \left( n\alpha \left( f(k_0)+ \frac{B}{k_0} \right) + \frac{\varepsilon }{16 (p-1) R_{\text {max}} L } \right) \\&\qquad + q^{-p} \left( \frac{1}{\alpha }\sum \limits _{i = 1}^{\alpha } r_i^{1-q} \right) ^{1-p} \left| -(p -1) q + p -1 +1\right| \\&< \frac{\varepsilon }{16 } + \frac{\varepsilon }{16 } + \frac{\varepsilon }{16 } + \frac{\varepsilon }{16 } + 0 = \frac{\varepsilon }{4}, \end{aligned}$$

i.e.,

$$\begin{aligned} \left| \frac{1}{n\alpha }\sum \limits _{i = k_0}^{ k_0 + n\alpha -1 } (p-1 ) \zeta _i + u + v \right| < \frac{\varepsilon }{4}. \end{aligned}$$
(4.48)

We recall that (see (3.2) and (4.42))

$$\begin{aligned} \frac{1}{n\alpha }\sum \limits _{i = k_0}^{ k_0 + n\alpha -1 } k_0^p {c_i } - v = k_0^p \left( \frac{1}{n\alpha } \sum \limits _{i = k_0}^{k_0 + n \alpha - 1} c_i \right) - \left( \frac{p-1}{p}\right) ^p \left( \frac{1}{\alpha } \sum \limits _{i = 1}^{\alpha } r_i^{\frac{1}{1-p}} \right) ^{1-p} < - {\varepsilon }. \end{aligned}$$
(4.49)

Finally, considering (4.40), (4.45), (4.48), and (4.49), we obtain

$$ \Delta \xi _{k_0}< \frac{1}{k_0} \left[ \frac{\varepsilon }{4} + \frac{\varepsilon }{4} - {\varepsilon } + \frac{\varepsilon }{4} \right] = - \frac{\varepsilon }{4k_0} < 0. $$

In particular,

$$\begin{aligned} \Delta \xi _{k_0} = \frac{1}{n\alpha } \sum \limits _{i = k_0}^{k_0 + n\alpha - 1} \Delta \zeta _i = \frac{\zeta _{k_0+n\alpha } - \zeta _{k_0}}{n\alpha } < 0 , \end{aligned}$$

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

$$\begin{aligned} \zeta _{i} \in \left( -N, 0\right) , \qquad i \in \{k, k + 1, \dots , k + n\alpha \}, \qquad \zeta _{k + n\alpha } < \zeta _{k} \end{aligned}$$
(4.50)

if

$$\begin{aligned} \zeta _{k} \in \left( - \zeta _0 - L_0, -\zeta _0\right] \end{aligned}$$
(4.51)

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 \(k \ge k_0\), \( k \in \mathbb {N}\). \(\square \)