Here, we consider the space \(\Upsilon :=\ell _{\nu +1}^{\infty }\) of functions x that consist of the set of all real sequences \(\{x(\mathrm{z}) \}_{\mathrm{z}=\nu +1}^{\infty }\). Note that ϒ is a Banach space under the norm \(\Vert x \Vert :=\sup_{\mathrm{z}\in \mathrm{N}_{ \nu +1}} x(\mathrm{z}) \). Also, we define
$$\begin{aligned} \mathtt{S}:=\bigl\{ x\in \Upsilon ; \bigl\vert x( \mathrm{z}) \bigr\vert \leqq ( \mathrm{z}1)^{(\gamma )}\ \forall \mathrm{z} \in \mathrm{N}_{ \nu +1}, \gamma >0 \bigr\} . \end{aligned}$$
(3.1)
It is clear that the set S is a nonempty bounded and closed subset of ϒ.
Theorem 3.1
Let the following condition on the function ψ hold true:

(C1)
Suppose that there exist positive constants C and β, with \(\nu +\beta =1\) and \(\beta >\nu \), such that
$$ \bigl\vert \psi (\mathrm{z},y) \bigr\vert \leqq \mathtt{C} \mathrm{z}^{(\beta )}\quad (\forall \mathrm{z}\in \mathrm{N}_{\nu +1} ). $$
(3.2)
Then, the operator B is continuous and \(\mathtt{B}\mathtt{S}_{1}\) is a relatively compact subset of \(\mathtt{S}_{1}\) for \(\mathrm{z}\in \mathrm{N}_{\nu +n}\), where
$$ \mathtt{S}_{1}:=\bigl\{ x\in \Upsilon ; \bigl\vert x( \mathrm{z}) \bigr\vert \leqq (\mathrm{z}1)^{(\gamma )}\ \forall \mathrm{z} \in \mathrm{N}_{\nu +n}, \gamma >0 \bigr\} , $$
\(\gamma =\frac{\beta \nu }{2}\) and n satisfies the following condition:
$$ \frac{(\nu +n+\gamma 1)^{(0.5)}}{\Gamma (\nu )} \vert y_{0} \vert + \mathtt{C} \frac{\Gamma (1\beta )}{\Gamma (1+\nu \beta )} (\nu +n+\gamma 1)^{( \gamma )}\leqq 1. $$
(3.3)
Proof
From the definition (2.8) of the operator B, Lemma 2.2 and assumption (C1), we have, for \(\mathrm{z}\in \mathrm{N}_{\nu +n}\),
$$\begin{aligned} \bigl\vert (\mathtt{B}y) (\mathrm{z}) \bigr\vert &{{\leqq }} \frac{1}{\Gamma (\nu )}\sum_{\kappa =0}^{\mathrm{z}\nu } \bigl( \mathrm{z}\sigma (\kappa ) \bigr)^{(\nu 1)} \bigl\vert \psi \bigl( \kappa + \nu 1,y(\kappa +\nu 1) \bigr) \bigr\vert \\ &\leqq \frac{\mathtt{C}}{\Gamma (\nu )}\sum_{\kappa =0}^{\mathrm{z} \nu } \bigl(\mathrm{z}\sigma (\kappa ) \bigr)^{(\nu 1)}(\kappa +\nu 1)^{( \beta )} \\ &=\mathtt{C} \bigl({}_{0}{\Delta }^{\nu }(\kappa +\nu 1)^{( \beta )} \bigr) (\mathrm{z}) \\ &=\mathtt{C}\frac{\Gamma (1\beta )}{\Gamma (1+\nu \beta )}( \mathrm{z}+\nu 1)^{(\nu \beta )} \quad \text{provided that } \nu + \beta =1. \end{aligned}$$
Considering ν, \(\beta \nu \) and \(\mathrm{z}1\) are all positive for \(\mathrm{z}\in \mathrm{N}_{\nu +n}\), \(n=1,2,\ldots \) , by Lemmas 2.1 and 2.2, and assumption (3.3), we have
$$\begin{aligned} \bigl\vert (\mathtt{B}y) (\mathrm{z}) \bigr\vert &< \mathtt{C} \frac{\Gamma (1\beta )}{\Gamma (1+\nu \beta )}(\mathrm{z}1)^{(\nu  \beta )} \\ &=\mathtt{C}\frac{\Gamma (1\beta )}{\Gamma (1+\nu \beta )}( \mathrm{z}+\gamma 1)^{(\gamma )}( \mathrm{z}1)^{(\gamma )} \\ &\leqq \mathtt{C}\frac{\Gamma (1\beta )}{\Gamma (1+\nu \beta )}( \nu +\gamma +n1)^{(\gamma )}( \mathrm{z}1)^{(\gamma )} \\ &\leqq (\mathrm{z}1)^{(\gamma )}. \end{aligned}$$
(3.4)
This means that \(y\in \mathtt{S}_{1}\) and thus \(\mathtt{B}\mathtt{S}_{1}\subseteq \mathtt{S}_{1}\).
For the continuity of B on \(\mathtt{S}_{1}\), we let \(\epsilon >0\) be given. Then, by using Lemmas 2.1 and 2.2, there exists \(m\geq n\) in \(\mathbb{N}_{1}\) such that
$$ \mathtt{C} \frac{\Gamma (1\beta )}{\Gamma (1+\nu \beta )} ( \mathrm{z}1)^{(\nu \beta )}< \frac{\epsilon }{2} \quad \text{for } \mathrm{z}\in \mathrm{N}_{\nu +m}. $$
(3.5)
Let \(\{y_{j} \}_{j=\nu +n}^{\infty }\) be a sequence defined on \(\mathtt{S}_{1}\) that converges to y. For \(\mathrm{z}\in \mathrm{N}_{\nu +m}\), it follows from assumption (C1) and (3.5) that
$$\begin{aligned} & \bigl\vert (\mathtt{B}y_{j}) (\mathrm{z})(\mathtt{B}y) ( \mathrm{z}) \bigr\vert \\ &\quad \leqq \frac{1}{\Gamma (\nu )}\sum_{\kappa =0}^{\mathrm{z}\nu } \bigl(\mathrm{z}\sigma (\kappa ) \bigr)^{(\nu 1)} \bigl[ \bigl\vert \psi \bigl(\kappa +\nu 1,y_{j}(\kappa +\nu 1) \bigr) \bigr\vert \\ &\qquad {}+ \bigl\vert \psi \bigl(\kappa +\nu 1,y(\kappa +\nu 1) \bigr) \bigr\vert \bigr] \\ &\quad \leqq \frac{2\mathtt{C}}{\Gamma (\nu )}\sum_{\kappa =0}^{\mathrm{z} \nu } \bigl(\mathrm{z}\sigma (\kappa ) \bigr)^{(\nu 1)}(\kappa +\nu 1)^{( \beta )} \\ &\quad =2\mathtt{C} \bigl({}_{0}{\Delta }^{\nu }(\kappa +\nu 1)^{( \beta )} \bigr) (\mathrm{z}) \\ &\quad =2\mathtt{C}\frac{\Gamma (1\beta )}{\Gamma (1+\nu \beta )}( \mathrm{z}+\nu 1)^{(\nu \beta )} \\ &\quad < 2\mathtt{C}\frac{\Gamma (1\beta )}{\Gamma (1+\nu \beta )}( \mathrm{z}1)^{(\nu \beta )} \\ &\quad < \epsilon . \end{aligned}$$
For the rest of the interval \(\mathrm{z}\in \{\nu +n,\nu +n+1,\ldots ,\nu +m1 \}\), we use the continuity of ψ and Lemma 2.1 to obtain
$$\begin{aligned} & \bigl\vert (\mathtt{B}y_{j}) (\mathrm{z})(\mathtt{B}y) ( \mathrm{z}) \bigr\vert \\ &\quad \leqq \frac{1}{\Gamma (\nu )}\sum_{\kappa =0}^{\mathrm{z}\nu } \bigl(\mathrm{z}\sigma (\kappa ) \bigr)^{(\nu 1)} \bigl\vert \psi \bigl( \kappa +\nu 1,y_{j}(\kappa +\nu 1) \bigr)\psi \bigl(\kappa +\nu 1,y( \kappa +\nu 1) \bigr) \bigr\vert \\ &\quad \leqq \frac{1}{\Gamma (\nu )}\sum_{\kappa =0}^{\mathrm{z}\nu } \bigl(\mathrm{z}\sigma (\kappa ) \bigr)^{(\nu 1)} \\ &\qquad {}\times \max_{{{\kappa \in }} \{\nu +n,\nu +n+1, \ldots ,\nu +m1 \}} \bigl\vert \psi \bigl(\kappa +\nu 1,y_{j}(\kappa + \nu 1) \bigr)\psi \bigl(\kappa +\nu 1,y(\kappa +\nu 1) \bigr) \bigr\vert \\ &\quad = \bigl({}_{0}{\Delta }^{\nu }r^{(0)} \bigr) ( \mathrm{z}) \\ &\qquad {}\times \max_{{{\kappa \in }} \{\nu +n,\nu +n+1, \ldots ,\nu +m1 \}} \bigl\vert \psi \bigl(\kappa +\nu 1,y_{j}(\kappa + \nu 1) \bigr)\psi \bigl(\kappa +\nu 1,y(\kappa +\nu 1) \bigr) \bigr\vert \\ &\quad =\frac{\mathrm{z}^{(\nu )}}{\Gamma (\nu +1)} \\ &\qquad {}\times \max_{{{\kappa \in }} \{\nu +n,\nu +n+1, \ldots ,\nu +m1 \}} \bigl\vert \psi \bigl(\kappa +\nu 1,y_{j}(\kappa + \nu 1) \bigr)\psi \bigl(\kappa +\nu 1,y(\kappa +\nu 1) \bigr) \bigr\vert \\ &\quad \leqq \frac{(\nu +m1)^{(\nu )}}{\Gamma (\nu +1)} \\ &\qquad {}\times \max_{{{\kappa \in }} \{\nu +n,\nu +n+1, \ldots ,\nu +m1 \}} \bigl\vert \psi \bigl(\kappa +\nu 1,y_{j}(\kappa + \nu 1) \bigr)\psi \bigl(\kappa +\nu 1,y(\kappa +\nu 1) \bigr) \bigr\vert \\ &\quad =\frac{\Gamma (\nu +m)}{\Gamma (\nu +1){{\Gamma (m)}}} \\ &\qquad {}\times \max_{{{\kappa \in }} \{\nu +n,\nu +n+1, \ldots ,\nu +m1 \}} \bigl\vert \psi \bigl(\kappa +\nu 1,y_{j}(\kappa + \nu 1) \bigr)\psi \bigl(\kappa +\nu 1,y(\kappa +\nu 1) \bigr) \bigr\vert , \end{aligned}$$
which approaches zero when \(j\to \infty \). Therefore, we have proved for each \(\mathrm{z}\in \mathrm{N}_{\nu +n}\),
$$ \bigl\vert (\mathtt{B}y_{n}) (\mathrm{z})( \mathtt{B}y) (\mathrm{z}) \bigr\vert \to 0\quad \text{as } n\to \infty , $$
(3.6)
and thus the operator B is continuous. In the following, we prove that the operator B is also relatively compact in \(\mathtt{S}_{1}\). Let \(\mathrm{z}_{1}, \mathrm{z}_{2}\in \mathrm{N}_{\nu +n}\) with \(\mathrm{z}_{2}>\mathrm{z}_{1}\), yielding
$$\begin{aligned} & \bigl\vert (\mathtt{B}y) (\mathrm{z}_{1})(\mathtt{B}y) ( \mathrm{z}_{2}) \bigr\vert \\ &\quad \leqq \frac{1}{\Gamma (\nu )}\sum_{\kappa =0}^{\mathrm{z}_{1}\nu } \bigl(\mathrm{z}_{1}\sigma (\kappa ) \bigr)^{(\nu 1)} \bigl\vert \psi \bigl(\kappa +\nu 1,y(\kappa +\nu 1) \bigr) \bigr\vert \\ &\qquad {}+\frac{1}{\Gamma (\nu )}\sum_{\kappa =0}^{\mathrm{z}_{2}\nu } \bigl( \mathrm{z}_{2}\sigma (\kappa ) \bigr)^{(\nu 1)} \bigl\vert \psi \bigl( \kappa +\nu 1,y(\kappa +\nu 1) \bigr) \bigr\vert \\ &\quad =\mathtt{C}_{1} \bigl({}_{0}{\Delta }^{\nu }(\kappa +\nu 1)^{( \beta )} \bigr) (\mathrm{z}_{1}) +\mathtt{C}_{2} \bigl( {}_{0}{\Delta }^{\nu }( \kappa +\nu 1)^{(\beta )} \bigr) ( \mathrm{z}_{2}) \\ &\quad =\mathtt{C}_{1}\frac{\Gamma (1\beta )}{\Gamma (1+\nu \beta )}( \mathrm{z}_{1}+ \nu 1)^{(\nu \beta )} +\mathtt{C}_{2} \frac{\Gamma (1\beta )}{\Gamma (1+\nu \beta )}( \mathrm{z}_{2}+\nu 1)^{( \nu \beta )} \\ &\quad < \underbrace{\frac{\epsilon }{2}+\frac{\epsilon }{2}}_{ \text{according to (3.5)}}= \epsilon . \end{aligned}$$
Therefore, \(\{\mathtt{B}y: y\in \mathtt{S}_{1} \}\) is a bounded and uniformly Cauchy subset by Definition 2.1. Moreover, \(\mathtt{B}\mathtt{S}_{1}\) is relatively compact in view of Theorem 2.1. Thus, the conclusion follows. □
Theorem 3.2
Assume that a function ψ of two variables satisfies the assumption (C1) stated in Theorem 3.1. Then, there exists at least one solution \(y(\mathrm{z})\) of the difference equation (1.2) for \(\mathrm{z}\in \mathrm{N}_{\nu +1}\) in \(\mathtt{S}_{1}\).
Proof
It is enough to show that \(y(\mathrm{z})\) is a fixed point of P in \(\mathtt{S}_{1}\). Let \(z\in \mathtt{S}_{1}\) be fixed. If \(y:=\mathtt{A}y+\mathtt{B}z\), then we shall show that y is in \(\mathtt{S}_{1}\). By means of (C1), Lemmas 2.1, 2.2 and 2.3(iii) one has for \(\mathrm{z}\in \mathrm{N}_{\nu +n}\):
$$\begin{aligned} \bigl\vert y(\mathrm{z}) \bigr\vert &\leqq \bigl\vert (\mathtt{A}y) ( \mathrm{z}) \bigr\vert + \bigl\vert (\mathtt{B}z) (\mathrm{z}) \bigr\vert \\ &\leqq \frac{\mathrm{z}^{(\nu 1)}}{\Gamma (\nu )} \vert y_{0} \vert + \frac{1}{\Gamma (\nu )}\sum_{\kappa =0}^{\mathrm{z}\nu } \bigl( \mathrm{z}\sigma (\kappa ) \bigr)^{(\nu 1)} \bigl\vert \psi \bigl( \kappa + \nu 1,z(\kappa +\nu 1) \bigr) \bigr\vert \\ &\leqq \frac{\mathrm{z}^{(\nu 1)}}{\Gamma (\nu )} \vert y_{0} \vert +\mathtt{C} \frac{\Gamma (1\beta )}{\Gamma (1+\nu \beta )}(\mathrm{z}+\nu 1)^{( \nu \beta )} \\ &< \frac{(\mathrm{z}1)^{(\nu 1)}}{\Gamma (\nu )} \vert y_{0} \vert +\mathtt{C} \frac{\Gamma (1\beta )}{\Gamma (1+\nu \beta )}(\mathrm{z}1)^{(\nu  \beta )}. \end{aligned}$$
By considering the condition (3.3), and Lemmas 2.1 and 2.3(ii), it follows that
$$\begin{aligned} \bigl\vert y(\mathrm{z}) \bigr\vert &\leqq \biggl[ \frac{(\mathrm{z}+\gamma 1)^{(0.5)}}{\Gamma (\nu )} \vert y_{0} \vert + \mathtt{C}\frac{\Gamma (1\beta )}{\Gamma (1+\nu \beta )}(\mathrm{z}+ \gamma 1)^{(\gamma )} \biggr](\mathrm{z}1)^{(\gamma )} \\ &\leqq \biggl[\frac{(\nu +n+\gamma 1)^{(0.5)}}{\Gamma (\nu )} \vert y_{0} \vert + \mathtt{C} \frac{\Gamma (1\beta )}{\Gamma (1+\nu \beta )}(\nu +n+ \gamma 1)^{(\gamma )} \biggr]( \mathrm{z}1)^{(\gamma )} \\ &\leqq (\mathrm{z}1)^{(\gamma )}, \end{aligned}$$
which means that \(y(\mathrm{z})\in \mathtt{S}_{1}\) for \(\mathrm{z}\in \mathrm{N}_{\nu +n}\). By Theorem 3.1 and 2.2, therefore, P has a fixed point in \(\mathtt{S}_{1}\), which means that there exists at least one solution of the difference equation (1.2) on \(\mathrm{z}\in \mathrm{N}_{\nu +n}\). The proof is now completed. □
Theorem 3.3
Assume that a function ψ of two variables ψ satisfies the assumption (C1) stated in Theorem 3.1. Then, the solutions \(y(\mathrm{z})\) of the difference equation (1.2) are attractive in \(\mathtt{S}_{1}\).
Proof
By means of Theorem 3.2, the solutions of the difference equation (1.2) exist in \(\mathtt{S}_{1}\). Moreover, each of the functions \(y(\mathrm{z})\) tend to 0 as \({{\mathrm{z}\to \infty }}\). Therefore, the solutions of the difference equation (1.2) tend to 0 as \({{\mathrm{z}\to \infty }}\). The proof is complete. □
Theorem 3.4
Let the following condition on the function ψ hold true:

(C2)
There exist positive constants K and β, with \(\nu +\beta =1\) and \(\beta >\nu \), such that
$$ {{ \bigl\vert \psi \bigl(\mathrm{z},y_{1}( \mathrm{z}) \bigr) \psi \bigl(\mathrm{z},y_{2}(\mathrm{z}) \bigr) \bigr\vert \leqq \mathtt{K} \mathrm{z}^{(\beta )} \Vert yz \Vert \quad ( \forall \mathrm{z}\in \mathrm{N}_{\nu +1} ).}} $$
(3.7)
Then, the solutions of the difference equation (1.2) are stable if
$$ \ell :=\mathtt{K} \frac{\Gamma (1\beta )}{\Gamma (1+\nu \beta )} \frac{\Gamma (1+\nu )}{\Gamma (1+\beta )}< 1. $$
(3.8)
Proof
Let ω, ϖ be two solutions of the difference equation (1.2) and let \(\epsilon >0\). From the assumption (C2) and Lemmas 2.1, 2.2 and 2.3, one has the following for \(\mathrm{z}\in \mathrm{N}_{\nu +1}\):
$$\begin{aligned} \bigl\vert \omega (\mathrm{z})\varpi (\mathrm{z}) \bigr\vert &\leqq \frac{\mathrm{z}^{(\nu 1)}}{\Gamma (\nu )} \vert \omega _{0}\varpi _{0} \vert \\ &\quad {}+\frac{1}{\Gamma (\nu )}\sum_{\kappa =0}^{\mathrm{z}\nu } \bigl( \mathrm{z}\sigma (\kappa ) \bigr)^{(\nu 1)} \bigl\vert \psi \bigl( \kappa + \nu 1,\omega (\kappa +\nu 1) \bigr) \\ &\quad {}\psi \bigl(\kappa +\nu 1, \varpi ( \kappa +\nu 1) \bigr) \bigr\vert \\ &\leqq \frac{\mathrm{z}^{(\nu 1)}}{\Gamma (\nu )} \vert \omega _{0} \varpi _{0} \vert + \frac{ \Vert \omega \varpi \Vert }{\Gamma (\nu )}\mathtt{K}\sum _{ \kappa =0}^{\mathrm{z}\nu } \bigl(\mathrm{z}\sigma (\kappa ) \bigr)^{( \nu 1)}(\kappa +\nu 1)^{(\beta )} \\ &=\frac{\mathrm{z}^{(\nu 1)}}{\Gamma (\nu )} \vert \omega _{0} \varpi _{0} \vert +\mathtt{K} \frac{\Gamma (1\beta )}{\Gamma (1+\nu \beta )}(\mathrm{z}+\nu 1)^{( \nu \beta )} \Vert \omega \varpi \Vert \\ &\leqq \frac{(\nu +1)^{(\nu 1)}}{\Gamma (\nu )} \vert \omega _{0} \varpi _{0} \vert +\mathtt{K} \frac{\Gamma (1\beta )}{\Gamma (1+\nu \beta )} \nu ^{(\nu \beta )} \Vert \omega \varpi \Vert \\ &=\frac{\nu (\nu +1)}{2} \vert \omega _{0}\varpi _{0} \vert + \mathtt{K}\frac{\Gamma (1\beta )}{\Gamma (1+\nu \beta )} \frac{\Gamma (1+\nu )}{\Gamma (1+\beta )} \Vert \omega \varpi \Vert . \end{aligned}$$
By using (3.8), it follows that
$$ \Vert \omega \varpi \Vert \leqq \frac{\nu (\nu +1)}{2(1\ell )} \vert \omega _{0}\varpi _{0} \vert . $$
Now, chose \(\delta =\frac{2(1\ell )\epsilon }{\nu (\nu +1)}\). Therefore,
$$\begin{aligned} \Vert \omega \varpi \Vert &< \frac{\nu (\nu +1)}{2(1\ell )}\cdot \delta \quad \text{whenever } \vert \omega _{0}\varpi _{0} \vert < \delta \\ &=\epsilon . \end{aligned}$$
Thus, it is proven that the solutions of the difference equation (1.2) are stable. □
Corollary 3.1
Assume that a function ψ of two variables satisfies the assumptions (C1) and (C2) stated in Theorems 3.1and 3.4, respectively. Then, the solutions of the difference equation (1.2) are asymptotically stable.
Proof
Corollary 3.1 follows from Theorems 3.3 and 3.4. □
Remark 3.1
It is important to state explicitly that the power rule (2.4) is used mistakenly in [27, 28, 32–34] as follows:
$$ \bigl({}_{0}{\Delta }^{\nu }(\kappa +\nu )^{(\beta )} \bigr) ( \mathrm{z}) =\frac{\Gamma (1\beta )}{\Gamma (1+\nu \beta )}( \mathrm{z}+\nu )^{(\nu \beta )}. $$
In fact, according to Lemma 2.2, it is valid only when \(\nu =\beta \), which contradicts the positivity of ν and β. That is why we have chosen to study such a difference equation of the type (1.2). In this case, we have obtained
$$ \bigl({}_{0}{\Delta }^{\nu }(\kappa +\nu 1)^{(\beta )} \bigr) (\mathrm{z}) = \frac{\Gamma (1\beta )}{\Gamma (1+\nu \beta )}(\mathrm{z}+\nu 1)^{( \nu \beta )}, $$
for which we need \(\nu +\beta =1\) according to Lemma 2.2, as we have established in Theorems 3.1 to 3.4.
We now prove a new attractiveness of the solutions of the difference equation (1.2) with a new condition in the following theorem.
Theorem 3.5
Let the following condition on the function ψ hold true:

(C3)
There exist positive constants \(\mathtt{C}_{2}\), β and γ, with \(\nu +\beta +\gamma =1\) and \(\beta >\nu \), such that
$$ {{ \bigl\vert \psi \bigl(\mathrm{z},y(\mathrm{z})\bigr) \bigr\vert \leqq \mathtt{C}_{2} (\mathrm{z}+\gamma )^{(\beta )} \bigl\vert y(\mathrm{z}+1) \bigr\vert \quad (\forall \mathrm{z}\in \mathrm{N}_{\nu +1} ).}} $$
(3.9)
Then, the solutions of the difference equation (1.2) are attractive.
Proof
To prove this theorem, we will verify the conditions of Theorem 2.2. The first condition is clear because A is a contraction as we discussed before. Also, the second condition is very similar to the one we proved in Theorem 3.1, so we omit it. Here, we prove the last condition so that \(y(\mathrm{z})\) will be a fixed point of P in \(\mathtt{S}_{2}\), where
$$ \mathtt{S}_{2}:=\bigl\{ x\in \Upsilon ; \bigl\vert x( \mathrm{z}) \bigr\vert \leqq (\mathrm{z}1)^{(\gamma )}\ \forall \mathrm{z} \in \mathrm{N}_{\nu +n}, \gamma >0 \bigr\} , $$
where \(n\in \mathbb{N}_{1}\) satisfies the condition that
$$ \frac{(\nu +n+\gamma 1)^{(\beta )}}{\Gamma (\nu )} \vert y_{0} \vert + \mathtt{C}_{2} \frac{\Gamma (1\beta \gamma )}{\Gamma (1+\nu \beta \gamma )}(\nu +n+ \gamma 1)^{(\nu \beta )}\leqq 1. $$
(3.10)
Let \(w\in \mathtt{S}_{2}\) be fixed. Now, if \(y:=\mathtt{A}y+\mathtt{B}w\), then we shall show that y is in \(\mathtt{S}_{2}\). By using assumption (C3), Lemmas 2.1, 2.2 and 2.3, we have for \(\mathrm{z}\in \mathrm{N}_{\nu +n}\):
$$\begin{aligned} \bigl\vert y(\mathrm{z}) \bigr\vert &\leqq \bigl\vert (\mathtt{A}y) ( \mathrm{z}) \bigr\vert + \bigl\vert (\mathtt{B}w) (\mathrm{z}) \bigr\vert \\ &\leqq \frac{\mathrm{z}^{(\nu 1)}}{\Gamma (\nu )} \vert y_{0} \vert + \frac{1}{\Gamma (\nu )}\sum_{\kappa =0}^{\mathrm{z}\nu } \bigl( \mathrm{z}\sigma (\kappa ) \bigr)^{(\nu 1)} \bigl\vert \psi \bigl( \kappa + \nu 1,w(\kappa +\nu 1) \bigr) \bigr\vert \\ &\leqq \frac{\mathrm{z}^{(\nu 1)}}{\Gamma (\nu )} \vert y_{0} \vert + \frac{\mathtt{C}_{2}}{\Gamma (\nu )}\sum_{\kappa =0}^{\mathrm{z}\nu } \bigl(\mathrm{z}\sigma (\kappa ) \bigr)^{(\nu 1)} (\kappa +\nu + \gamma 1)^{(\beta )}{{ \bigl\vert w(\kappa +\nu ) \bigr\vert }} \\ &\leqq \frac{\mathrm{z}^{(\nu 1)}}{\Gamma (\nu )} \vert y_{0} \vert + \frac{\mathtt{C}_{2}}{\Gamma (\nu )}\sum_{\kappa =0}^{\mathrm{z}\nu } \bigl(\mathrm{z}\sigma (\kappa ) \bigr)^{(\nu 1)} (\kappa +\nu + \gamma 1)^{(\beta )}(\kappa +\nu 1)^{(\gamma )} \\ &\leqq \frac{\mathrm{z}^{(\nu 1)}}{\Gamma (\nu )} \vert y_{0} \vert + \frac{\mathtt{C}_{2}}{\Gamma (\nu )}\sum_{\kappa =0}^{\mathrm{z}\nu } \bigl(\mathrm{z}\sigma (\kappa ) \bigr)^{(\nu 1)} (\kappa +\nu 1)^{( \beta \gamma )} \\ &\leqq \frac{\mathrm{z}^{(\nu 1)}}{\Gamma (\nu )} \vert y_{0} \vert + \mathtt{C}_{2} \frac{\Gamma (1\beta \gamma )}{\Gamma (1+\nu \beta \gamma )}( \mathrm{z}+\nu 1)^{(\nu \beta \gamma )}\quad \text{such that } \nu +\beta +\gamma =1 \\ &< \frac{(\mathrm{z}1)^{(\nu 1)}}{\Gamma (\nu )} \vert y_{0} \vert + \mathtt{C}_{2} \frac{\Gamma (1\beta \gamma )}{\Gamma (1+\nu \beta \gamma )}( \mathrm{z}1)^{(\nu \beta \gamma )}. \end{aligned}$$
By considering condition (3.10), and Lemmas 2.1 and 2.3(ii), it follows that
$$\begin{aligned} \bigl\vert y(\mathrm{z}) \bigr\vert &\leqq \biggl[ \frac{(\mathrm{z}+\gamma 1)^{(\beta )}}{\Gamma (\nu )} \vert y_{0} \vert + \mathtt{C}_{2} \frac{\Gamma (1\beta \gamma )}{\Gamma (1+\nu \beta \gamma )}( \mathrm{z}+\gamma 1)^{(\nu \beta )} \biggr]( \mathrm{z}1)^{( \gamma )} \\ &\leqq \biggl[\frac{(\nu +n+\gamma 1)^{(\beta )}}{\Gamma (\nu )} \vert y_{0} \vert + \mathtt{C}_{2} \frac{\Gamma (1\beta \gamma )}{\Gamma (1+\nu \beta \gamma )}(\nu +n+ \gamma 1)^{(\nu \beta )} \biggr](\mathrm{z}1)^{(\gamma )} \\ &\leqq (\mathrm{z}1)^{(\gamma )}. \end{aligned}$$
This completes the required result. Therefore, by Theorem 3.1 and 2.2, P has a fixed point in \(\mathtt{S}_{2}\), which means that there exists at least one solution of the difference equation (1.2) on \(\mathrm{z}\in \mathrm{N}_{\nu +n}\). Moreover, by means of Theorem 3.2, each of the functions \(y(\mathrm{z})\) in \(\mathtt{S}_{2}\) tend to zero as \(\mathrm{z}\to \infty \). Therefore, the solutions of the difference equation (1.2) tend to zero as \(\mathrm{z}\to \infty \). This completes the proof. □
Corollary 3.2
Assume that a function ψ of two variables satisfies the assumptions (C2) and (C3) stated in Theorems 3.4and 3.5, respectively. Then, the solutions of the difference equation (1.2) are asymptotically stable such that (3.8) holds true.
Proof
This follows from Theorems 3.4 and 3.5. □
Theorem 3.6
Let the following condition on ψ hold true:

(C4)
There exist \(\eta \in (0,1)\) and the positive constants \(\mathtt{C}_{3}\) and β such that
$$ {{ \bigl\vert \psi \bigl(\mathrm{z},y(\mathrm{z})\bigr) \bigr\vert \leqq \mathtt{C}_{3} (\mathrm{z}+1)^{(\beta )} \bigl\vert y(\mathrm{z}+1) \bigr\vert ^{\eta } \quad (\forall \mathrm{z}\in \mathrm{N}_{\nu +1} ).}} $$
(3.11)
Then, the solutions of the difference equation (1.2) are attractive.
Proof
We proceed with the same method as that used in Theorem 3.5. We only prove the last condition in 2.2 so that \(y(\mathrm{z})\) will be a fixed point of P in \(\mathtt{S}_{3}\), where
$$ \mathtt{S}_{3}:=\bigl\{ x\in \Upsilon ; \bigl\vert x( \mathrm{z}) \bigr\vert \leqq (\mathrm{z}1)^{(\gamma )}\ \forall \mathrm{z} \in \mathrm{N}_{\nu +n}, \gamma >0 \bigr\} , $$
where \(\nu +\beta +\gamma \eta =1\), \(\beta >\nu \), \(\nu +\gamma \in (0,1)\), \(\gamma =\frac{\beta \nu }{2}\) and \(n\in \mathbb{N}_{1}\) satisfies the condition that
$$ \frac{(\nu +n+\gamma 1)^{(\nu +\gamma 1)}}{\Gamma (\nu )} \vert y_{0} \vert + \mathtt{C}_{3} \frac{\Gamma (1\beta \gamma \eta )}{\Gamma (1+\nu \beta \gamma \eta )}( \nu +n+\gamma 1)^{(\gamma )}\leqq 1. $$
(3.12)
Let \(w\in \mathtt{S}_{3}\) be fixed. Now, if \(y:=\mathtt{A}y+\mathtt{B}w\), then we shall show that y is in \(\mathtt{S}_{3}\). By using assumption (C4), \(\nu <\beta +\gamma \eta <1\), Lemmas 2.1, 2.2 and 2.3(ii)–(iv), we have for \(\mathrm{z}\in \mathrm{N}_{\nu +n}\):
$$\begin{aligned} \bigl\vert y(\mathrm{z}) \bigr\vert &\leqq \bigl\vert (\mathtt{A}y) ( \mathrm{z}) \bigr\vert + \bigl\vert (\mathtt{B}w) (\mathrm{z}) \bigr\vert \\ &\leqq \frac{\mathrm{z}^{(\nu 1)}}{\Gamma (\nu )} \vert y_{0} \vert + \frac{1}{\Gamma (\nu )}\sum_{\kappa =0}^{\mathrm{z}\nu } \bigl( \mathrm{z}\sigma (\kappa ) \bigr)^{(\nu 1)} \bigl\vert \psi \bigl( \kappa + \nu 1,w(\kappa +\nu 1) \bigr) \bigr\vert \\ &\leqq \frac{\mathrm{z}^{(\nu 1)}}{\Gamma (\nu )} \vert y_{0} \vert + \frac{\mathtt{C}_{3}}{\Gamma (\nu )}\sum_{\kappa =0}^{\mathrm{z}\nu } \bigl(\mathrm{z}\sigma (\kappa ) \bigr)^{(\nu 1)} (\kappa +\nu )^{( \beta )}{{ \bigl\vert w(\kappa +\nu ) \bigr\vert ^{\eta }}} \\ &\leqq \frac{\mathrm{z}^{(\nu 1)}}{\Gamma (\nu )} \vert y_{0} \vert + \frac{\mathtt{C}_{3}}{\Gamma (\nu )}\sum_{\kappa =0}^{\mathrm{z}\nu } \bigl(\mathrm{z}\sigma (\kappa ) \bigr)^{(\nu 1)} (\kappa +\nu + \gamma \eta 1)^{(\beta )} \bigl[(\kappa +\nu 1)^{(\gamma )} \bigr]^{\eta } \\ &\leqq \frac{\mathrm{z}^{(\nu 1)}}{\Gamma (\nu )} \vert y_{0} \vert + \frac{\mathtt{C}_{3}}{\Gamma (\nu )}\sum_{\kappa =0}^{\mathrm{z}\nu } \bigl(\mathrm{z}\sigma (\kappa ) \bigr)^{(\nu 1)} (\kappa +\nu + \gamma \eta 1)^{(\beta )}(\kappa +\nu 1)^{(\gamma \eta )} \\ &\leqq \frac{\mathrm{z}^{(\nu 1)}}{\Gamma (\nu )} \vert y_{0} \vert + \frac{\mathtt{C}_{3}}{\Gamma (\nu )}\sum_{\kappa =0}^{\mathrm{z}\nu } \bigl(\mathrm{z}\sigma (\kappa ) \bigr)^{(\nu 1)} (\kappa +\nu 1)^{( \beta \gamma \eta )} \\ &\leqq \frac{\mathrm{z}^{(\nu 1)}}{\Gamma (\nu )} \vert y_{0} \vert + \mathtt{C}_{3} \frac{\Gamma (1\beta \gamma \eta )}{\Gamma (1+\nu \beta \gamma \eta )} ( \mathrm{z}+\nu 1)^{(\nu \beta \gamma \eta )} \quad \text{such that } \nu +\beta +\gamma \eta =1 \\ &\leqq \frac{(\mathrm{z}1)^{(\nu 1)}}{\Gamma (\nu )} \vert y_{0} \vert + \mathtt{C}_{3} \frac{\Gamma (1\beta \gamma \eta )}{\Gamma (1+\nu \beta \gamma \eta )} ( \mathrm{z}1)^{(\nu \beta )}. \end{aligned}$$
Considering condition (3.12), \(\nu +\gamma \in (0,1)\), \(\beta \nu =2\gamma \), and Lemmas 2.1 and 2.3(ii), it follows that
$$\begin{aligned} \bigl\vert y(\mathrm{z}) \bigr\vert &\leqq \biggl[ \frac{(\mathrm{z}+\gamma 1)^{(\nu +\gamma 1)}}{\Gamma (\nu )} \vert y_{0} \vert +\mathtt{C}_{3} \frac{\Gamma (1\beta \gamma \eta )}{\Gamma (1+\nu \beta \gamma \eta )}( \mathrm{z}+\gamma 1)^{(\gamma )} \biggr]( \mathrm{z}1)^{(\gamma )} \\ &\leqq \biggl[ \frac{(\nu +n+\gamma 1)^{(\nu +\gamma 1)}}{\Gamma (\nu )} \vert y_{0} \vert + \mathtt{C}_{3} \frac{\Gamma (1\beta \gamma \eta )}{\Gamma (1+\nu \beta \gamma \eta )}( \nu +n+\gamma 1)^{(\gamma )} \biggr] \\ &\quad {}\times (\mathrm{z}1)^{(\gamma )} \\ &\leqq (\mathrm{z}1)^{(\gamma )}. \end{aligned}$$
This proves the required condition (iii) in Theorem 2.2 and thus the proof is completed. □
Remark 3.2
The same mistakes of the power rule, as we discussed in Remark 3.1, are made in Theorems 3.6 and 3.8 in [32]. In those theorems, the used power rule would have been true when \(\nu +\beta _{3}+\gamma _{2}=0\) and \(\nu +\beta _{3}+\gamma _{2}\eta =0\), respectively. However, these contradict the positivity of ν, β, γ and η. The chosen parameters here are such that \(\nu +\beta +\gamma =1\) in Theorem 3.5 and \(\nu +\beta +\gamma \eta =1\) in Theorem 3.6 have appropriately corrected the above mistakes.