Existence and Uniqueness of Solutions to Abstract Discrete-Time Cauchy Problems in Vector-Valued Weighted Spaces

Abstract

This article studies the abstract discrete-time Cauchy problem involving the Riemann–Liouville type difference operator. Sufficient conditions for the existence of unique solution to the semilinear Cauchy problem in Lebesgue and weighted Lebesgue vector-valued spaces are shown. Finally, some examples are presented to illustrate the main results.

1. Introduction

In 2020, He et al. [1] investigated the solutions of the nonlinear abstract fractional difference equation

$$\left\{\begin{array}{c}{\Delta }^{\upsilon }x\left(t\right)=Ax(t+1)+\mathcal{F}(t,x\left(t\right)),t\in {\mathbb{N}}_0,\hfill \\ x\left(0\right)=x_0\in \mathcal{Y},\hfill \end{array}\right.$$

(1)

where $0<\upsilon \le 1$, ${\Delta }^{\upsilon }$ denotes the Riemann–Liouville type fractional difference operator of order $\upsilon$ as defined in [2], A denotes the infinitesimal generator of a bounded $C_0$ semigroup on a Banach space $\mathcal{Y}$ with domain $D\left(A\right)$, $\mathcal{F}:{\mathbb{N}}_0\times \mathcal{Y}\to \mathcal{Y}$.

It was proved in Theorem 5.1 [1] that, provided $\mathcal{F}$ satisfies a Lipschitz-type condition, Equation (1) admits a unique solution in the vector-valued sequence space ${\ell }^{\infty }({\mathbb{N}}_0;\mathcal{Y})$ for every initial value $x_0$ in $D\left(A\right)$. Furthermore, Theorem 5.2 [1] shows that if the nonlinearity f is not Lipschitz but is instead ${\ell }^{\infty }$-bounded w.r.t. the discrete variable and linearly bounded in the continuous variable, then, for the initial condition $x_0=0$, Equation (1) admits at least one solution in the weighted vector-valued space ${\ell }_w^{\infty }({\mathbb{N}}_0;\mathcal{Y})$, where $w\left(t\right)=tt!,$ provided the ${\ell }^{\infty }$-bound is sufficiently small. The existence of a class of periodic solutions to (1) was proved in Theorem 4.5 [3]. After that, the existence of stable solutions to (1) in the space ${\ell }^{\infty }({\mathbb{N}}_0;\mathcal{Y})$ was studied in Theorem 3.1 [4]. Generalizations of the model (1) to discrete-time abstract Volterra equations have been studied but always restricted to the space ${\ell }^{\infty }({\mathbb{N}}_0;\mathcal{Y}),$ in Theorems 3.4 and 3.5 [5]. However, despite this important variety of consequences and applications, the extension of the results in [1] to Lebesgue spaces ${\ell }^p({\mathbb{N}}_0;\mathcal{Y})$ in the range $1\le p<\infty$ or to other classes of weighted Lebesgue spaces of sequences remains an important open problem in need of prompt attention.

More precisely, the results in [1] establish well-posedness primarily in the ${\ell }^{\infty }({\mathbb{N}}_0;\mathcal{Y})$ setting (and in a very specific weighted ${\ell }_w^{\infty }$ framework) but do not provide a finite-p theory in ${\ell }^p({\mathbb{N}}_0;\mathcal{Y})$ nor a systematic treatment in weighted Lebesgue sequence spaces. The main difficulty is that the discrete variation-of-constants representation involves a fractional convolution kernel whose summability depends delicately on the parameter $\upsilon$, so the admissible Lipschitz profiles in the discrete variable must be tuned to the target norm.

The aim of this paper is to address this open problem. We establish both existence and uniqueness of solutions in the following sequence spaces:

(i)

${\ell }^p({\mathbb{N}}_0;\mathcal{Y}),0<\upsilon <1,{\displaystyle \frac{1}{1-\upsilon }}<p<\infty$;

(ii)

${\ell }_w^1({\mathbb{N}}_0;\mathcal{Y})$ where the weight has the form $w\left(t\right)=a^t$ with $a>1$;

(iii)

${\ell }_w^{\infty }({\mathbb{N}}_0;\mathcal{Y})$ where $w\left(t\right)=k^{\upsilon +1}\left(t\right)$;

(iv)

${\ell }_w^p({\mathbb{N}}_0;\mathcal{Y}),1\le p<\infty$ where $w\left(t\right)=a^t{k}^{\upsilon +1}\left(t\right)$ with $a>1$.

These four items are obtained, respectively, in Theorems 4 (${\ell }^p$), 5 (${\ell }_{a^t}^1$), 6 (${\ell }_{k^{\upsilon +1}\left(t\right)}^{\infty }$), and 7 (${\ell }_{a^t{k}^{\upsilon +1}\left(t\right)}^p$). Our approach is based on resolvent sequences and the discrete variation-of-constants Formula (6), combined with sharp convolution estimates in ${\ell }^p$ and ${\ell }_w^p$ spaces. A key advantage over earlier ${\ell }^{\infty }$-based approaches is that this framework makes explicit how the fractional order $\upsilon$ controls the admissible growth of the nonlinearity in the discrete variable through the Lipschitz-type profiles.

For the reader’s convenience we summarize below the ambient spaces, admissible initial data, and typical Lipschitz profiles.

Setting	Space	Weight $w\left(t\right)$	Initial data	Reference/Theorem
Classical	${\ell }^{\infty }$	1	$x_0\in D\left(A\right)$	Theorem  5.1 [1]
Weighted (specific)	${\ell }_w^{\infty }$	$tt!$	$x_0=0$ (existence)	Theorem 5.2 [1]
New	${\ell }^p$	1	$x_0=0$	Theorem 4
New	${\ell }_w^1$	$a^t$ ($a>1$)	$x_0=0$	Theorem 5
New	${\ell }_w^{\infty }$	$k^{\upsilon +1}\left(t\right)$	$x_0\in D\left(A\right)$	Theorem 6
New	${\ell }_w^p$	$a^t{k}^{\upsilon +1}\left(t\right)$	$x_0\in D\left(A\right)$	Theorem 7

We will need to assume different Lipschitz-type conditions on the nonlinearity $\mathcal{F}$ in each case, which, in contrast to those obtained in [1], reveal the importance of the fractional parameter $\upsilon$, which appears as a measure of the size of the growth of $\mathcal{F}$ in the discrete variable. This is a very attractive aspect of our results, because they provide new insights to the study of the abstract discrete-time Cauchy problem (1).

In the integer case, i.e., $\upsilon =1,$ the abstract discrete-time Cauchy problem (1) models mixed differential-difference equations which are present in many problems in physics and engineering. There are classical texts that consider this kind of models. We cite here, for instance, the books of Agarwal (Chapter 2 [6]) and Elaydi (Chapters 2 and 5 [7]). Some models are closely related to numerical methods for partial and integro-differential equations (Chapter 6 [7]). Mixed models also arise in time discretizations of partial differential equations, traffic flow dynamics, probability theory, and the theory of chain processes in chemistry and radioactivity [8].

When $0<\upsilon <1,$ we are talking about discrete-time fractional calculus [9,10,11,12]. Over the past twenty years, this area has attracted growing interest, largely driven by its wide range of applications [12,13,14]. Discrete-time fractional models offer additional degrees of freedom, enabling them to represent latent features of natural phenomena that exhibit memory effects. It has been successfully applied to computational biology, economics, physics, imaging processing, etc., see e.g., [12,15]. In recent years, abstract fractional difference equations involving unbounded linear operators have attracted growing attention, with many mathematicians employing techniques from operator theory and functional analysis to study them [16,17,18]. The goal is to develop a theory that in some sense parallels those of $C_0$ semigroups of operators [19] and resolvent families for integral equations of convolution type [20]. Such a discrete theory is able to elucidate and clarify mathematical aspects that are more complicated when considered in the continuous case. One of the attractive aspects is that distributional solutions are not longer present in the discrete case, and also that differentiability is always guaranteed, avoiding additional analytical efforts on regularity, for example. This is particularly important because it shows that many deep continuous-time problems can take a simpler form and be better analyzed when considered in discrete-time.

This article is organized as follows. In Section 2, we introduce the preliminaries required for this work. In Section 3, we present our main findings on the existence and uniqueness of solutions of the initial value problem (1). Our main results are Theorems 4–7. Lastly, we include illustrative examples in Section 4 to highlight the main results.

2. Preliminaries

Represent by $\mathbb{N}=\{1,2,3,\dots \}$; ${\mathbb{N}}_0=\{0,1,2,\dots \}$; $\mathcal{Y}$, a complex Banach space with norm $\parallel ·\parallel$; $\mathcal{B}\left(\mathcal{Y}\right)$, the space of all bounded linear operators from $\mathcal{Y}$ into $\mathcal{Y}$ with the standard norm

$${\parallel \mathcal{Q}\parallel }_{\mathcal{B}\left(\mathcal{Y}\right)}=sup\{\parallel \mathcal{Q}\left(z\right)\parallel :\parallel z\parallel =1\}.$$

Represent by $\left[D\right(A\left)\right]$, the domain $D\left(A\right)$ with the graph norm ${\parallel z\parallel }_A:=\parallel z\parallel +\parallel Az\parallel$, if A is a closed linear operator on $\mathcal{Y}$. Then, $\left[D\right(A\left)\right]$ is a Banach space.

We denote by $s\left({\mathbb{N}}_0;\mathcal{Y}\right)$ the vectorial space consisting of all vector-valued sequences $x:{\mathbb{N}}_0\to \mathcal{Y}$. The first-order forward difference operator $\Delta :s\left({\mathbb{N}}_0;\mathcal{Y}\right)\to s\left({\mathbb{N}}_0;\mathcal{Y}\right)$ is defined by

$$\Delta x\left(t\right)=x(t+1)-x\left(t\right),t\in {\mathbb{N}}_0.$$

The finite convolution * of two sequences x, y (one of them can be vector-valued or operator-valued) is defined

$$(x\ast y)\left(t\right)=\sum _{\iota =0}^{t}x(t-\iota )y\left(\iota \right),t\in {\mathbb{N}}_0.$$

We define the scalar sequence ${\left\{k^{\beta }\left(t\right)\right\}}_{t\in {\mathbb{N}}_0}$, $\beta >0$, by

$$k^{\beta }\left(t\right):=\left\{\begin{array}{cc}{\displaystyle \frac{\Gamma (\beta +t)}{\Gamma \left(\beta \right)\Gamma (t+1)}}& t\in {\mathbb{N}}_0,\beta \in \mathbb{R}\backslash {\mathbb{Z}}^{-};\hfill \\ & \\ {({\delta }_0-{\delta }_1)}^{\ast (-\beta )}\left(t\right)& t\in {\mathbb{N}}_0,\beta \in {\mathbb{Z}}^{-}.\hfill \end{array}\right.$$

Here $\Gamma$ denotes the gamma function, $p^{\ast t}=\underset{t\text{-}\mathrm{times}}{\underbrace{p\ast p\ast \dots \ast p}}$ where * denotes convolution of sequences, and ${\delta }_j\left(t\right)$ denotes the Kronecker delta. Lizama in [2,21] introduced this sequence within the framework of fractional differences and satisfies several important properties (Section 3 [22]).

Remark 1.

The following properties hold.

(1) 

${\displaystyle \left(k^{\upsilon }\ast k^{\beta }\right)\left(t\right)=k^{\upsilon +\beta }\left(t\right),t\in {\mathbb{N}}_0}$;

(2) 

For all $t\in {\mathbb{N}}_0$ and for any $\upsilon \in (0,1]$, $k^{\upsilon }\left(t\right)\in (0,1]$; $k^{\upsilon }\left(t\right)$ is a non-increasing sequence and $k^{\upsilon }\left(t\right)\to 0$ as $t\to \infty .$

(3) 

Note that item (2) applies only to $\upsilon \in (0,1]$. In particular, when $\upsilon \in (0,1]$ the sequence $k^{\upsilon +1}\left(t\right)$ is increasing (see Lemma 1); hence, it is a non-decreasing weight bounded below by 1.

For other properties, we refer to [22].

Definition 1

([2]). For $x\in s\left({\mathbb{N}}_0;\mathcal{Y}\right)$, the ${\upsilon }^{\mathit{th}}$-order, $\upsilon >0$, fractional sum of x is

$${\Delta }^{-\upsilon }x\left(t\right)=\left(k^{\upsilon }\ast x\right)\left(t\right)=\sum _{\iota =0}^t{k}^{\upsilon }(t-\iota )x\left(\iota \right),t\in {\mathbb{N}}_0.$$

Definition 2

([2]). For $x\in s\left({\mathbb{N}}_0;\mathcal{Y}\right)$, the ${\upsilon }^{\mathit{th}}$-order, $0<\upsilon \le 1$, fractional difference of x in Riemann–Liouville-like sense is

$${\Delta }^{\upsilon }x\left(t\right)=\Delta {\Delta }^{-(1-\upsilon )}x\left(t\right),t\in {\mathbb{N}}_0.$$

The following definition was introduced by Lizama in [1,2,21]. They are the counterpart of resolvent families for abstract fractional evolution equations developed in [23].

Definition 3.

Consider a closed linear operator A defined on $\mathcal{Y}$, with domain $D\left(A\right)$. An operator-valued sequence ${\left\{{\mathrm{S}}_{\upsilon }\left(t\right)\right\}}_{t\in {\mathbb{N}}_0}\subset \mathcal{B}\left(\mathcal{Y}\right)$, $\upsilon >0$, is called a υ-resolvent sequence generated by A if it satisfies the following conditions:

(1) 

${\mathrm{S}}_{\upsilon }\left(t\right)x\in D\left(A\right)$ for all $x\in \mathcal{Y}$ and $A{\mathrm{S}}_{\upsilon }\left(t\right)x={\mathrm{S}}_{\upsilon }\left(t\right)Ax$ for all $t\in {\mathbb{N}}_0$ and $x\in D\left(A\right)$;

(2) 

${\mathrm{S}}_{\upsilon }\left(t\right)x=k^{\upsilon }\left(t\right)x+A\left(k^{\upsilon }\ast {\mathrm{S}}_{\upsilon }\right)\left(t\right)x$ for all $t\in {\mathbb{N}}_0$ and $x\in \mathcal{Y}$.

An important consequence of Theorem 4.5 [24] (see also Theorem 3.1 [3]) and Theorem 3.2 [25] is the following characterization.

Theorem 1.

There exists a υ-resolvent sequence ${\left\{{\mathrm{S}}_{\upsilon }\left(t\right)\right\}}_{t\in {\mathbb{N}}_0}$ generated by A if and only if $1\in \rho \left(A\right).$ Moreover,

$${\mathrm{S}}_{\upsilon }\left(t\right)x=\sum _{\iota =0}^{\infty }{\ell }_{\upsilon }(t,\iota ){(I-A)}^{-(\iota +1)}x,t\in {\mathbb{N}}_0,x\in \mathcal{Y},0<\upsilon <1.$$

Here, ${\ell }_{\upsilon }(t,\iota )$ is the Lévy $\upsilon$-stable distribution (Definition 3.1 and Proposition 3.4 (iii) [24]) defined by

$${\ell }_{\upsilon }(t,\iota ):=\sum _{j=1}^{\iota }\left({\displaystyle \genfrac{}{}{0pt}{}{\iota }{j}}\right){(-1)}^j{k}^{-\upsilon j}\left(t\right),0<\upsilon <1,t,\iota \in {\mathbb{N}}_0.$$

In case that A is the generator of a $C_0$ semigroup, we can subordinate the continuous to the discrete case as follows.

Theorem 2

([1]). If A generates a bounded $C_0$ semigroup ${\left\{\mathcal{Q}\left(t\right)\right\}}_{t\ge 0}$ on $\mathcal{Y}$, then A generates a υ-resolvent sequence ${\left\{{\mathrm{S}}_{\upsilon }\left(t\right)\right\}}_{t\in {\mathbb{N}}_0}$, $0<\upsilon \le 1$, given by

$${\mathrm{S}}_{\upsilon }\left(t\right)x={\int }_0^{\infty }{\int }_0^{\infty }{p}_n\left(t\right){\mathrm{F}}_{s,\upsilon }\left(t\right)\mathcal{Q}\left(s\right)xdsdt,t\in {\mathbb{N}}_0,x\in \mathcal{Y}.$$

(2)

In the above, we considered the stable Lévy distribution defined by

$${\mathrm{F}}_{t,\upsilon }\left(\lambda \right)={\displaystyle \frac{1}{2\pi i}}{\int }_{\sigma -i\infty }^{\sigma +i\infty }{e}^{z\lambda -t{z}^{\upsilon }}dz,0<\upsilon <1,\lambda \ge 0,\sigma >0,t>0.$$

Here the branch of $z^{\upsilon }$ is chosen so that $Re\left(z^{\upsilon }\right)>0$ whenever $Re\left(z\right)>0$. With this choice, the branch is single-valued in the z-plane cut along the negative real axis.

In particular, every generator of a bounded $C_0$ semigroup generates a $\upsilon$-resolvent sequence for each $\upsilon \in (0,1]$.

As a consequence, we obtain the following important estimate.

Corollary 1

(Corollary 3.1 [1]). If A generates a bounded $C_0$ semigroup ${\left\{\mathcal{Q}\left(t\right)\right\}}_{t\ge 0}$ on $\mathcal{Y}$, we have

$$∥{\mathrm{S}}_{\upsilon }\left(t\right)x∥\le M\parallel x\parallel k^{\upsilon }\left(t\right),t\in {\mathbb{N}}_0,0<\upsilon \le 1,x\in \mathcal{Y},$$

(3)

where

$$M=\underset{t\in {\mathbb{R}}^{+}\cup \left\{0\right\}}{sup}{\parallel \mathcal{Q}\left(t\right)\parallel }_{\mathcal{B}\left(\mathcal{Y}\right)}.$$

We will need the following definition.

Definition 4

([1]). Take $h\in s\left({\mathbb{N}}_0;\mathcal{Y}\right)$. $x\in s\left({\mathbb{N}}_0;\mathcal{Y}\right)$ is a strong solution of

$$\left\{\begin{array}{c}{\Delta }^{\upsilon }x\left(t\right)=Ax(t+1)+h\left(t\right),t\in {\mathbb{N}}_0,\hfill \\ x\left(0\right)=x_0\in \mathcal{Y},\hfill \end{array}\right.$$

(4)

if $x\left(t\right)\in D\left(A\right)$ for all $t\in {\mathbb{N}}_0$ and x satisfies (4).

The next result completely solves the linear case.

Theorem 3

([1]). Let $0<\upsilon <1$, $h\in s\left({\mathbb{N}}_0;\mathcal{Y}\right)$, and A generates a bounded $C_0$ semigroup ${\left\{\mathcal{Q}\left(t\right)\right\}}_{t\ge 0}$ on $\mathcal{Y}$. Then, the nonhomogeneous abstract fractional difference Equation (4) has a strong solution x in $s\left({\mathbb{N}}_0;\left[D\left(A\right)\right]\right)$ given by

$$x\left(t\right)={\mathrm{S}}_{\upsilon }\left(t\right)(I-A)x_0+\left({\mathrm{S}}_{\upsilon }\ast h\right)(t-1),t\in \mathbb{N}.$$

(5)

3. The Semilinear Cauchy Problem

In this section, we present our main findings on the existence and uniqueness of solutions of the initial value problem (1). For this purpose, and being consistent with Theorem 3, we introduce the following definition. Note that the assumption $x_0\in D\left(A\right)$ is not required (see Definition 3(1)). It corresponds to the analogue of “mild” solution in the continuous case.

Definition 5.

Let A generates a bounded $C_0$ semigroup ${\left\{\mathcal{Q}\left(t\right)\right\}}_{t\ge 0}$ on $\mathcal{Y}$ and $0<\upsilon <1$. $x\in s\left({\mathbb{N}}_0;\mathcal{Y}\right)$ is a solution of (1) if x satisfies

$$x\left(t\right)=(I-A){\mathrm{S}}_{\upsilon }\left(t\right)x_0+\sum _{\iota =0}^{t-1}{\mathrm{S}}_{\upsilon }(t-1-\iota )\mathcal{F}(\iota ,x\left(\iota \right)),t\in \mathbb{N}.$$

(6)

Next, consider (1) on the vector-valued Lebesgue sequence space

$${\ell }^p\left({\mathbb{N}}_0;\mathcal{Y}\right)=\left\{x:{\mathbb{N}}_0\to \mathcal{Y},{\left(\sum _{t=0}^{\infty }{\parallel x\left(t\right)\parallel }^p\right)}^{{\displaystyle \frac{1}{p}}}<\infty \right\},1<p<\infty ,$$

endowed with the usual norm

$${\parallel x\parallel }_{{\ell }^p}={\left(\sum _{t=0}^{\infty }{\parallel x\left(t\right)\parallel }^p\right)}^{{\displaystyle \frac{1}{p}}}.$$

Remark 2.

Let $1<p<\infty$ and $0<\upsilon <1$. Denote by

$$a\left(t\right):={\left[k^{1+\upsilon -\beta }\left(t\right)\right]}^p,t\in \mathbb{N},$$

where $\upsilon +1/p<\beta <1.$ Clearly, $a\left(t\right)>0,t\in \mathbb{N}.$ Since

$$\begin{array}{cc}\hfill {\displaystyle \frac{a\left(t\right)}{a(t+1)}}& ={\displaystyle \frac{{\left[k^{1+\upsilon -\beta }\left(t\right)\right]}^p}{{\left[k^{1+\upsilon -\beta }(t+1)\right]}^p}}={\left[{\displaystyle \frac{\Gamma (t+1+\upsilon -\beta )}{\Gamma (1+\upsilon -\beta )\Gamma (t+1)}}\right]}^p{\left[{\displaystyle \frac{\Gamma (1+\upsilon -\beta )\Gamma (t+2)}{\Gamma (t+2+\upsilon -\beta )}}\right]}^p\hfill \\ & ={\left[{\displaystyle \frac{t+1}{t+1+\upsilon -\beta }}\right]}^p={\left[1+{\displaystyle \frac{\beta -\upsilon }{t+1+\upsilon -\beta }}\right]}^p.\hfill \end{array}$$

We have

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim}t\left({\displaystyle \frac{a\left(t\right)}{a(t+1)}}-1\right)& =\underset{t\to \infty }{lim}{\displaystyle \frac{\left({\left[1+\frac{\beta -\upsilon }{t+1+\upsilon -\beta }\right]}^p-1\right)}{\frac{1}{t}}}\hfill \\ & =\underset{t\to \infty }{lim}{\displaystyle \frac{p{\left[1+\frac{\beta -\upsilon }{t+1+\upsilon -\beta }\right]}^{p-1}\frac{-(\beta -\upsilon )}{{(t+1+\upsilon -\beta )}^2}}{-\frac{1}{t^2}}}\hfill \\ & =p(\beta -\upsilon )\underset{t\to \infty }{lim}{\left[1+{\displaystyle \frac{\beta -\upsilon }{t+1+\upsilon -\beta }}\right]}^{p-1}{\displaystyle \frac{t^2}{{(t+1+\upsilon -\beta )}^2}}\hfill \\ & =p(\beta -\upsilon )>1.\hfill \end{array}$$

Then, by Raabe’s Test, the infinite series

$$N:=\sum _{t=0}^{\infty }a\left(t\right)=\sum _{t=0}^{\infty }{\left[k^{1+\upsilon -\beta }\left(t\right)\right]}^p,$$

(7)

converges.

In order to state our first existence result, we will need the following assumptions.

(A1)

A is the generator of a bounded $C_0$ semigroup ${\left\{\mathcal{Q}\left(t\right)\right\}}_{t\ge 0}$ defined on $\mathcal{Y}$ with

$$M=\underset{t\in {\mathbb{R}}^{+}\cup \left\{0\right\}}{sup}{\parallel \mathcal{Q}\left(t\right)\parallel }_{\mathcal{B}\left(\mathcal{Y}\right)};$$

(A2)

$\mathcal{F}(t,0)=0$ for all $t\in {\mathbb{N}}_0,$ ${\displaystyle \frac{1}{1-\upsilon }}<p<\infty$ and there exist $\beta \in \left(\upsilon +{\displaystyle \frac{1}{p}},1\right)$ and $0<L<{\displaystyle \frac{1}{{MN}^{\frac{1}{p}}}}$ so that

$$\parallel \mathcal{F}(t,u)-\mathcal{F}(t,v)\parallel \le L{k}^{1-\beta }\left(t\right)\parallel u-v\parallel ,u,v\in \mathcal{Y},t\in {\mathbb{N}}_0.$$

Note that (A2) has been considered before for existence of solutions in the case $p=\infty$, see e.g., [1]. In particular, it is implied by Remark 1 that $\parallel \mathcal{F}(t,·)\parallel \to 0$ as $t\to \infty .$

We prove the following result related to (i) of the introduction.

Theorem 4.

Let $0<\upsilon <1$ and ${\displaystyle \frac{1}{1-\upsilon }}<p<\infty$ be given. Assume A and $\mathcal{F}$ satisfy (A1) and (A2), respectively. Then, Equation (1) has a unique solution x in ${\ell }^p\left({\mathbb{N}}_0;\mathcal{Y}\right)$ for $x_0=0$. In particular, $\parallel x\left(t\right)\parallel \to 0$ as $t\to \infty .$

Proof. 

Define the map $\mathcal{S}:{\ell }^p\left({\mathbb{N}}_0;\mathcal{Y}\right)\to {\ell }^p\left({\mathbb{N}}_0;\mathcal{Y}\right)$ by

$$\left(\mathcal{S}x\right)\left(t\right):=\sum _{\iota =0}^{t-1}{\mathrm{S}}_{\upsilon }(t-1-\iota )\mathcal{F}(\iota ,x\left(\iota \right)),t\in \mathbb{N},$$

and $\left(\mathcal{S}x\right)\left(0\right):=0$. First, we show that $\mathcal{S}$ is well defined. Indeed, let $x\in {\ell }^p\left({\mathbb{N}}_0;\mathcal{Y}\right)$. It follows from (A2) that

$$\parallel \mathcal{F}(t,x\left(t\right))\parallel \le L{k}^{1-\beta }\left(t\right)\parallel x\left(t\right)\parallel ,x\in {\ell }^p\left({\mathbb{N}}_0;\mathcal{Y}\right),t\in {\mathbb{N}}_0.$$

Using the above estimate, (3), and (1) of Remark 1, we obtain

$$\begin{array}{cc}\hfill {\left({\displaystyle \sum _{t=1}^{\infty }}{\parallel \left(\mathcal{S}x\right)\left(t\right)\parallel }^p\right)}^{{\displaystyle \frac{1}{p}}}& ={\left({\displaystyle \sum _{t=1}^{\infty }}{∥{\displaystyle \sum _{\iota =0}^{t-1}}{\mathrm{S}}_{\upsilon }(t-1-\iota )\mathcal{F}(\iota ,x\left(\iota \right))∥}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \le {\left({\displaystyle \sum _{t=1}^{\infty }}{\left[{\displaystyle \sum _{\iota =0}^{t-1}}∥{\mathrm{S}}_{\upsilon }(t-1-\iota )∥∥\mathcal{F}(\iota ,x(\iota \left)\right)∥\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \le LM{\left({\displaystyle \sum _{t=1}^{\infty }}{\left[{\displaystyle \sum _{\iota =0}^{t-1}}{k}^{\upsilon }(t-1-\iota )k^{1-\beta }\left(\iota \right)\parallel x\left(\iota \right)\parallel \right]}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \le LM{\left({\displaystyle \sum _{t=1}^{\infty }}{\left[{\displaystyle \sum _{\iota =0}^{t-1}}{k}^{\upsilon }(t-1-\iota )k^{1-\beta }\left(\iota \right){\parallel x\parallel }_{{\ell }^p}\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & =LM{\left({\displaystyle \sum _{t=1}^{\infty }}{\left[(k^{\upsilon }\ast k^{1-\beta })(t-1){\parallel x\parallel }_{{\ell }^p}\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & ={LM\parallel x\parallel }_{{\ell }^p}{\left({\displaystyle \sum _{t=1}^{\infty }}{\left[k^{1+\upsilon -\beta }(t-1)\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & ={LM\parallel x\parallel }_{{\ell }^p}{\left({\displaystyle \sum _{t=0}^{\infty }}{\left[k^{1+\upsilon -\beta }\left(t\right)\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}.\hfill \end{array}$$

(8)

It follows from (7) and (8) that

$${\parallel \mathcal{S}x\parallel }_{{\ell }^p}={\left(\sum _{t=0}^{\infty }{\parallel \left(\mathcal{S}x\right)\left(t\right)\parallel }^p\right)}^{{\displaystyle \frac{1}{p}}}\le {LMN}^{{\displaystyle \frac{1}{p}}}{\parallel x\parallel }_{{\ell }^p}.$$

Set $q:={LMN}^{1/p}$. By hypothesis $0<q<1$; hence, the fixed point map is a contraction.

It proves the claim.

For any x, $y\in {\ell }^p\left({\mathbb{N}}_0;\mathcal{Y}\right)$, by (A2), (3) and Remark 1 we get

$$\begin{array}{cc}& {\left({\displaystyle \sum _{t=1}^{\infty }}{\parallel \left(\mathcal{S}x\right)\left(t\right)-\left(\mathcal{S}y\right)\left(t\right)\parallel }^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & ={\left({\displaystyle \sum _{t=1}^{\infty }}{∥{\displaystyle \sum _{\iota =0}^{t-1}}{\mathrm{S}}_{\upsilon }(t-1-\iota )\left[\mathcal{F}(\iota ,x(\iota \left)\right)-\mathcal{F}(\iota ,y(\iota \left)\right)\right]∥}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \le LM{\left({\displaystyle \sum _{t=1}^{\infty }}{\left[{\displaystyle \sum _{\iota =0}^{t-1}}{k}^{\upsilon }(t-1-\iota )k^{1-\beta }\left(\iota \right)\parallel x\left(\iota \right)-y\left(\iota \right)\parallel \right]}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \le {LM\parallel x-y\parallel }_{{\ell }^p}{\left({\displaystyle \sum _{t=1}^{\infty }}{\left[{\displaystyle \sum _{\iota =0}^{t-1}}{k}^{\upsilon }(t-1-\iota )k^{1-\beta }\left(\iota \right)\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & ={LM\parallel x-y\parallel }_{{\ell }^p}{\left({\displaystyle \sum _{t=1}^{\infty }}{\left[k^{1+\upsilon -\beta }(t-1)\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & ={LM\parallel x-y\parallel }_{{\ell }^p}{\left({\displaystyle \sum _{t=0}^{\infty }}{\left[k^{1+\upsilon -\beta }\left(t\right)\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}={LMN}^{{\displaystyle \frac{1}{p}}}{\parallel x-y\parallel }_{{\ell }^p},\hfill \end{array}$$

for all $t\in \mathbb{N}$. Hence, it follows that

$${\parallel \mathcal{S}x-\mathcal{S}y\parallel }_{{\ell }^p}\le {LMN}^{{\displaystyle \frac{1}{p}}}{\parallel x-y\parallel }_{{\ell }^p}.$$

In view of ${LMN}^{{\displaystyle \frac{1}{p}}}<1$, the Banach theorem provides the conclusion. □

Remark 3.

Compared to the case $p=\infty$ treated in Theorem 5.1 [1], we now have to assume that $x_0=0.$ This is a natural assumption taking into account that by (3), (2) of Remark 1 and assumption (A1), we necessarily have $\parallel {\mathrm{S}}_{\upsilon }\left(t\right)\parallel \to 0$ as $t\to \infty .$

The restriction $x_0=0$ in Theorem 4 is imposed to obtain a fixed point argument entirely within ${\ell }^p({\mathbb{N}}_0;\mathcal{Y})$ without additional regularity assumptions on $x_0$. If one assumes, in addition, that the “free term” ${\left\{(I-A){\mathrm{S}}_{\upsilon }\left(t\right)x_0\right\}}_{t\ge 1}$ belongs to ${\ell }^p({\mathbb{N}}_0;\mathcal{Y})$ (for instance under suitable regularity/compatibility on $x_0$), then the same contraction argument applies and yields existence/uniqueness for $x_0\ne 0$ as well. An analogous comment applies to Theorem 5 in ${\ell }_{a^t}^1$.

Remark 4.

The summability condition in Remark 2 (obtained via Raabe’s test) requires $p(\beta -\upsilon )>1$. In particular, for $p=1$ this would force $\beta -\upsilon >1$, which is incompatible with the natural parameter range $\beta <1$ when $0<\upsilon <1$. Equivalently, the series in (7) fails to converge in the $p=1$ regime. This is the reason Theorem 4 does not cover $p=1$ and motivates the weighted settings introduced below.

However, if we consider the setting of weighted Banach spaces, we can overcome this drawback. We recall that given $w\in s({\mathbb{N}}_0;\mathbb{R})$ and $1\le p<\infty$ we denote by

$${\ell }_w^p\left({\mathbb{N}}_0;\mathcal{Y}\right)=\left\{x:{\mathbb{N}}_0\to \mathcal{Y},{\left(\sum _{t=0}^{\infty }{\left[{\displaystyle \frac{∥x\left(t\right)∥}{w\left(t\right)}}\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}<\infty \right\},$$

the vector-valued weighted Lebesgue sequence spaces endowed with the natural norm

$${\parallel x\parallel }_{{\ell }_w^p}={\left(\sum _{t=0}^{\infty }{\left[{\displaystyle \frac{∥x\left(t\right)∥}{w\left(t\right)}}\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}.$$

In case $p=\infty$

$$l_w^{\infty }\left({\mathbb{N}}_0;\mathcal{Y}\right)=\left\{x:{\mathbb{N}}_0\to \mathcal{Y},\underset{t\in {\mathbb{N}}_0}{sup}{\displaystyle \frac{\parallel x\left(t\right)\parallel }{w\left(t\right)}}<\infty \right\},$$

with the norm

$${\parallel x\parallel }_{\infty }=\underset{t\in {\mathbb{N}}_0}{sup}{\displaystyle \frac{\parallel x\left(t\right)\parallel }{w\left(t\right)}}.$$

For any $a>1$, we denote by

$$a_1\left(t\right):={\displaystyle \frac{k^{1+\upsilon -\beta }\left(t\right)}{a^t}},t\in \mathbb{N},$$

where $0<\beta <1+\upsilon$. Clearly, $a_1\left(t\right)>0,t\in \mathbb{N},$ and we have

$$\begin{array}{cc}\hfill \underset{t\to \infty }{lim}{\displaystyle \frac{a_1(t+1)}{a_1\left(t\right)}}& =\underset{t\to \infty }{lim}{\displaystyle \frac{a^t{k}^{1+\upsilon -\beta }(t+1)}{a^{t+1}{k}^{1+\upsilon -\beta }\left(t\right)}}\hfill \\ & ={\displaystyle \frac{1}{a}}\underset{t\to \infty }{lim}\left[{\displaystyle \frac{\Gamma (t+2+\upsilon -\beta )}{\Gamma (1+\upsilon -\beta )\Gamma (t+2)}}\right]\left[{\displaystyle \frac{\Gamma (1+\upsilon -\beta )\Gamma (t+1)}{\Gamma (t+1+\upsilon -\beta )}}\right]\hfill \\ & ={\displaystyle \frac{1}{a}}\underset{t\to \infty }{lim}\left[{\displaystyle \frac{t+1+\upsilon -\beta }{t+1}}\right]={\displaystyle \frac{1}{a}}<1.\hfill \end{array}$$

Then, by Ratio Test, the series

$$N_0:=\sum _{t=0}^{\infty }{a}_1\left(t\right)=\sum _{t=0}^{\infty }{\displaystyle \frac{k^{1+\upsilon -\beta }\left(t\right)}{a^t}},$$

(9)

converges. We need the following assumption to state our next existence result.

(A3)

$\mathcal{F}(t,0)\equiv 0$. There exist $0<\beta <1$, $L\in \left(0,{\displaystyle \frac{1}{M{N}_0}}\right)$ with

$$\parallel \mathcal{F}(t,u)-\mathcal{F}(t,v)\parallel \le {\displaystyle \frac{L{k}^{1-\beta }\left(t\right)}{a^t}}\parallel u-v\parallel ,u,v\in \mathcal{Y},t\in {\mathbb{N}}_0.$$

Theorem 5.

Take $a>1$, $w\left(t\right)=a^t.$ Assume that A satisfies (A1) and $\mathcal{F}$ satisfies (A3). Then, for $x_0=0,$ the problem (1) has a unique solution x in ${\ell }_w^1\left({\mathbb{N}}_0;\mathcal{Y}\right).$

Proof. 

We define $\mathcal{S}:{\ell }_w^1\left({\mathbb{N}}_0;\mathcal{Y}\right)\to {\ell }_w^1\left({\mathbb{N}}_0;\mathcal{Y}\right)$ by

$$\left(\mathcal{S}x\right)\left(t\right):=\sum _{\iota =1}^{t-1}{\mathrm{S}}_{\upsilon }(t-1-\iota )\mathcal{F}(\iota ,x\left(\iota \right)),t\in \mathbb{N},$$

and $\left(\mathcal{S}x\right)\left(0\right):=0$. First, we prove that $\mathcal{S}$ is well-defined. Indeed, let $x\in {\ell }_w^1\left({\mathbb{N}}_0;\mathcal{Y}\right)$. From (A3), we have

$$\parallel \mathcal{F}(t,x\left(t\right))\parallel \le {\displaystyle \frac{L{k}^{1-\beta }\left(t\right)}{a^t}}\parallel x\left(t\right)\parallel ,x\in {\ell }_w^1\left({\mathbb{N}}_0;\mathcal{Y}\right),t\in {\mathbb{N}}_0.$$

Using the above estimate and (3), we have

$$\begin{array}{cc}\hfill {\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{\parallel \left(\mathcal{S}x\right)\left(t\right)\parallel }{a^t}}& ={\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{a^t}}∥{\displaystyle \sum _{\iota =1}^{t-1}}{\mathrm{S}}_{\upsilon }(t-1-\iota )\mathcal{F}(\iota ,x\left(\iota \right))∥\hfill \\ \hfill & \le {\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{a^t}}\left[{\displaystyle \sum _{\iota =1}^{t-1}}∥{\mathrm{S}}_{\upsilon }(t-1-\iota )∥∥\mathcal{F}(\iota ,x(\iota \left)\right)∥\right]\hfill \\ \hfill & \le LM{\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{a^t}}\left[{\displaystyle \sum _{\iota =1}^{t-1}}{k}^{\upsilon }(t-1-\iota )k^{1-\beta }\left(\iota \right){\displaystyle \frac{\parallel x\left(\iota \right)\parallel }{a^{\iota }}}\right]\hfill \\ \hfill & \le {LM\parallel x\parallel }_{{\ell }_w^1}{\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{a^t}}\left[{\displaystyle \sum _{\iota =1}^{t-1}}{k}^{\upsilon }(t-1-\iota )k^{1-\beta }\left(\iota \right)\right]\hfill \\ & ={LM\parallel x\parallel }_{{\ell }_w^1}{\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{k^{1+\upsilon -\beta }(t-1)}{a^t}}=LM{\parallel x\parallel }_{{\ell }_w^1}{\displaystyle \sum _{t=0}^{\infty }}{\displaystyle \frac{k^{1+\upsilon -\beta }\left(t\right)}{a^t}}.\hfill \end{array}$$

(10)

It follows from (9) and (10) that

$${\parallel \mathcal{S}u\parallel }_{{\ell }_w^1}=\sum _{t=0}^{\infty }\parallel \left(\mathcal{S}x\right)\left(t\right)\parallel \le LM{N}_0{\parallel x\parallel }_{{\ell }_w^1}.$$

For any x, $y\in {\ell }_w^1\left({\mathbb{N}}_0;\mathcal{Y}\right)$, by (A3), (3) and Remark 1 we get

$$\begin{array}{cc}& {\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{\parallel \left(\mathcal{S}x\right)\left(t\right)-\left(\mathcal{S}y\right)\left(t\right)\parallel }{a^t}}\hfill \\ & ={\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{a^t}}∥{\displaystyle \sum _{\iota =1}^{t-1}}{\mathrm{S}}_{\upsilon }(t-1-\iota )\left[\mathcal{F}(\iota ,x(\iota \left)\right)-\mathcal{F}(\iota ,y(\iota \left)\right)\right]∥\hfill \\ \hfill & \le LM{\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{a^t}}\left[{\displaystyle \sum _{\iota =1}^{t-1}}{k}^{\upsilon }(t-1-\iota )k^{1-\beta }\left(\iota \right){\displaystyle \frac{\parallel x\left(\iota \right)-y\left(\iota \right)\parallel }{a^{\iota }}}\right]\hfill \\ \hfill & \le {LM\parallel x-y\parallel }_{{\ell }_w^1}{\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{a^t}}\left[{\displaystyle \sum _{\iota =1}^{t-1}}{k}^{\upsilon }(t-1-\iota )k^{1-\beta }\left(\iota \right)\right]\hfill \\ \hfill & ={LM\parallel x-y\parallel }_{{\ell }_w^1}{\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{k^{1+\upsilon -\beta }(t-1)}{a^t}}\hfill \\ & ={LM\parallel x-y\parallel }_{{\ell }_w^1}{\displaystyle \sum _{t=0}^{\infty }}{\displaystyle \frac{k^{1+\upsilon -\beta }\left(t\right)}{a^t}}=LM{N}_0{\parallel x-y\parallel }_{{\ell }_w^1},\hfill \end{array}$$

for all $t\in \mathbb{N}$. Hence, it follows that

$${\parallel \mathcal{S}x-\mathcal{S}y\parallel }_{{\ell }_w^1}\le LM{N}_0{\parallel x-y\parallel }_{{\ell }_w^1}.$$

Set $q:=LM{N}_0$. By hypothesis $0<q<1$; hence, the fixed point map is a contraction.

By (A3) and the Banach fixed point theorem, the proof is finished. □

Denote by

$$w\left(t\right):=k^{\upsilon +1}\left(t\right),t\in {\mathbb{N}}_0.$$

The following result follows from Proposition 3.1 (iv) [22] and the fact that $w\left(0\right)=1.$

Lemma 1.

For all $t\in {\mathbb{N}}_0$ and for any $\upsilon \in (0,1]$, $w\left(t\right)$ is an increasing sequence and $w\left(t\right)\in [1,\infty )$.

Our next result concerns point (iii) of the introduction. To do so, instead of (A3) we will need the following assumption.

(A4)

$\mathcal{F}(t,0)\equiv 0$, and there exist $0<L<{\displaystyle \frac{1}{M}}$ such that

$$\parallel \mathcal{F}(t,x)-\mathcal{F}(t,y)\parallel \le {\displaystyle \frac{L}{k^{\upsilon +1}\left(t\right)}}\parallel x-y\parallel ,x,y\in \mathcal{Y},t\in {\mathbb{N}}_0.$$

Since $w\left(t\right)=k^{\upsilon +1}\left(t\right)$ is non-decreasing and $w\left(t\right)\ge 1$, the weighted space ${\ell }_w^{\infty }({\mathbb{N}}_0;\mathcal{Y})$ allows solutions with controlled (at most w-type) growth, and the ratio $w(t-1)/w\left(t\right)={\displaystyle \frac{t}{t+\upsilon }}<1$ is precisely what is needed in the proof of Theorem 6 to obtain a uniform contraction estimate.

Our main result is the following Theorem. Note that we do not assume $x_0=0.$

Theorem 6.

Assume A and $\mathcal{F}$ satisfy (A1) and (A4), respectively. Then, Equation (1) has a unique solution x in $l_w^{\infty }\left({\mathbb{N}}_0;\mathcal{Y}\right)$, for any $x_0\in D\left(A\right)$.

Proof. 

We define $\mathcal{S}:l_w^{\infty }\left({\mathbb{N}}_0;\mathcal{Y}\right)\to l_w^{\infty }\left({\mathbb{N}}_0;\mathcal{Y}\right)$ by

$$\left(\mathcal{S}x\right)\left(t\right)={\mathrm{S}}_{\upsilon }\left(t\right)(I-A)x_0+\sum _{\iota =1}^{t-1}{\mathrm{S}}_{\upsilon }(t-1-\iota )\mathcal{F}(\iota ,x\left(\iota \right)),t\in \mathbb{N},$$

(11)

and $\left(\mathcal{S}x\right)\left(0\right)=x_0$. First, we prove that $\mathcal{S}$ is well-defined. Take $x\in l_w^{\infty }\left({\mathbb{N}}_0;\mathcal{Y}\right)$. From (A4), we obtain

$$\parallel \mathcal{F}(t,x\left(t\right))\parallel \le {\displaystyle \frac{L}{w\left(t\right)}}\parallel x\left(t\right)\parallel ,t\in {\mathbb{N}}_0,x\in l_w^{\infty }\left({\mathbb{N}}_0;\mathcal{Y}\right).$$

Now, for any $t\in \mathbb{N}$, we have

$$\begin{array}{cc}\hfill {\displaystyle \frac{\parallel \left(\mathcal{S}x\right)\left(t\right)\parallel }{w\left(t\right)}}& \le {\displaystyle \frac{1}{w\left(t\right)}}\left[∥{\mathrm{S}}_{\upsilon }\left(t\right)(I-A)x_0∥+∥{\displaystyle \sum _{\iota =1}^{t-1}}{\mathrm{S}}_{\upsilon }(t-1-\iota )\mathcal{F}(\iota ,x\left(\iota \right))∥\right]\hfill \\ & \le {\displaystyle \frac{1}{w\left(t\right)}}\left[∥{\mathrm{S}}_{\upsilon }\left(t\right)∥∥(I-A)x_0∥+{\displaystyle \sum _{\iota =1}^{t-1}}∥{\mathrm{S}}_{\upsilon }(t-1-\iota )∥∥\mathcal{F}(\iota ,x(\iota \left)\right)∥\right]\hfill \\ & \le {\displaystyle \frac{1}{w\left(t\right)}}\left[M∥(I-A)x_0∥k^{\upsilon }\left(t\right)+ML{\displaystyle \sum _{\iota =1}^{t-1}}{k}^{\upsilon }(t-1-\iota ){\displaystyle \frac{\parallel x\left(\iota \right)\parallel }{w\left(\iota \right)}}\right]\hfill \\ & \le {\displaystyle \frac{1}{w\left(t\right)}}\left[M∥(I-A)x_0∥k^{\upsilon }\left(t\right)+ML{\parallel x\parallel }_{\infty }{\displaystyle \sum _{\iota =1}^{t-1}}{k}^{\upsilon }(t-1-\iota )\right]\hfill \\ & ={\displaystyle \frac{1}{w\left(t\right)}}\left[M∥(I-A)x_0∥k^{\upsilon }\left(t\right)+ML{\parallel x\parallel }_{\infty }(k^{\upsilon }\ast k^1)(t-1)\right]\hfill \\ & =M∥(I-A)x_0∥\left({\displaystyle \frac{\upsilon }{t+\upsilon }}\right)+ML{\parallel x\parallel }_{\infty }{\displaystyle \frac{w(t-1)}{w\left(t\right)}}\hfill \\ & \le M∥(I-A)x_0∥+ML{\parallel x\parallel }_{\infty },\hfill \end{array}$$

implying that $\mathcal{S}$ is well-defined. For x, $y\in l_w^{\infty }\left({\mathbb{N}}_0;\mathcal{Y}\right)$, by (A4), we get

$$\begin{array}{cc}\hfill {\displaystyle \frac{\parallel \left(\mathcal{S}x\right)\left(t\right)-\left(\mathcal{S}y\right)\left(t\right)\parallel }{w\left(t\right)}}& \le {\displaystyle \frac{1}{w\left(t\right)}}\left[M{\displaystyle \sum _{\iota =1}^{t-1}}{k}^{\upsilon }(t-1-\iota )\parallel \mathcal{F}(\iota ,x\left(\iota \right))-\mathcal{F}(\iota ,y\left(\iota \right))\parallel \right]\hfill \\ & \le {\displaystyle \frac{1}{w\left(t\right)}}\left[ML{\displaystyle \sum _{\iota =1}^{t-1}}{k}^{\upsilon }(t-1-\iota ){\displaystyle \frac{\parallel x\left(\iota \right)-y\left(\iota \right)\parallel }{w\left(\iota \right)}}\right]\hfill \\ & \le {\displaystyle \frac{1}{w\left(t\right)}}\left[{ML\parallel x-y\parallel }_{\infty }{\displaystyle \sum _{\iota =1}^{t-1}}{k}^{\upsilon }(t-1-\iota )\right]\hfill \\ & ={ML\parallel x-y\parallel }_{\infty }{\displaystyle \frac{w(t-1)}{w\left(t\right)}}\le ML{\parallel x-y\parallel }_{\infty },\hfill \end{array}$$

for every $t\in \mathbb{N}$. Thus, we have

$${\parallel \mathcal{S}x-\mathcal{S}y\parallel }_{\infty }\le ML{\parallel x-y\parallel }_{\infty }.$$

Since $ML<1$, we obtain the result by the Banach theorem. □

Remark 5.

Compared to [1] where the weighted Lebesgue space of sequences with $w\left(t\right)=tt!$ is treated, we have imposed in (A4) a condition revealing a dependence between the size of the nonlinearity $\mathcal{F}$ in the discrete variable t and the fractional order υ. Noting that $k^1\left(t\right)\equiv 1$ and $k^2\left(t\right)=t+1$ for all $t\in {\mathbb{N}}_0,$ we deduce that $\parallel \mathcal{F}(t,·)\parallel$ must vary between ${\displaystyle \frac{1}{t+1}}$ and 1 as $0<\upsilon <1.$

Finally, we consider the problem (1) on ${\ell }_w^p\left({\mathbb{N}}_0;\mathcal{Y}\right)$ for $1\le p<\infty ,$ with weight

$$w\left(t\right):=a^t{k}^{\upsilon +1}\left(t\right),$$

where $a>1.$ For any $p\ge 1,$ we denote

$$d\left(t\right):={\left[{\displaystyle \frac{k^{\upsilon +1}(t-1)}{a^t{k}^{\upsilon +1}\left(t\right)}}\right]}^p={\left[{\displaystyle \frac{t}{a^t(t+\upsilon )}}\right]}^p,t\in \mathbb{N}.$$

Clearly, $d\left(t\right)>0,t\in \mathbb{N}.$ Also,

$$\underset{t\to \infty }{lim}{\displaystyle \frac{d(t+1)}{d\left(t\right)}}=\underset{t\to \infty }{lim}{\left[{\displaystyle \frac{t+1}{a^{t+1}(t+1+\upsilon )}}\right]}^p{\left[{\displaystyle \frac{a^t(t+\upsilon )}{t}}\right]}^p={\left[{\displaystyle \frac{1}{a}}\right]}^p<1.$$

Then, by ratio test, the series ${\sum }_{t=1}^{\infty }d\left(t\right)$ converges. We denote

$$N_2:=\sum _{t=1}^{\infty }d\left(t\right).$$

(12)

Our last theorem is able to treat the case $1\le p<\infty$ but at the cost of a more restricted admissible range of values for the nonlinearity $\mathcal{F}$ in the discrete variable. We will need the following assumption.

(A5)

$\mathcal{F}(t,0)\equiv 0$, and there exist $0<L<{\displaystyle \frac{1}{M{\left(N_2\right)}^{\frac{1}{p}}}}$ such that

$$\parallel \mathcal{F}(t,x)-\mathcal{F}(t,y)\parallel \le {\displaystyle \frac{L}{a^t{k}^{\upsilon +1}\left(t\right)}}\parallel x-y\parallel ,x,y\in \mathcal{Y},t\in {\mathbb{N}}_0.$$

We prove the following theorem.

Theorem 7.

Let $w\left(t\right):=a^t{k}^{\upsilon +1}\left(t\right),$ where $a>1.$ Assume A and $\mathcal{F}$ satisfy (A1) and (A5), respectively. Then, Equation (1) has a unique solution x in ${\ell }_w^p\left({\mathbb{N}}_0;\mathcal{Y}\right)$, for $x_0\in D\left(A\right)$.

Proof. 

We define $\mathcal{S}:{\ell }_w^p\left({\mathbb{N}}_0;\mathcal{Y}\right)\to {\ell }_w^p\left({\mathbb{N}}_0;\mathcal{Y}\right)$ by (11) and $\left(\mathcal{S}x\right)\left(0\right)=x_0$. We prove that $\mathcal{S}$ is well-defined. Take $x\in {\ell }_w^p\left({\mathbb{N}}_0;\mathcal{Y}\right)$. From (A5), we get

$$\parallel \mathcal{F}(t,x\left(t\right))\parallel \le {\displaystyle \frac{L}{a^{t}b\left(t\right)}}\parallel x\left(t\right)\parallel ,x\in {\ell }_w^p\left({\mathbb{N}}_0;\mathcal{Y}\right),t\in {\mathbb{N}}_0,$$

where $b\left(t\right):=k^{\upsilon +1}\left(t\right).$ We have

$$\begin{array}{cc}\hfill & {\left({\displaystyle \sum _{t=1}^{\infty }}{\left[{\displaystyle \frac{∥\left(\mathcal{S}x\right)\left(t\right)∥}{a^{t}b\left(t\right)}}\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & ={\left({\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{{\left[a^{t}b\left(t\right)\right]}^p}}{∥{\mathrm{S}}_{\upsilon }\left(t\right)(I-A)x_0+{\displaystyle \sum _{\iota =1}^{t-1}}{\mathrm{S}}_{\upsilon }(t-1-\iota )\mathcal{F}(\iota ,x\left(\iota \right))∥}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \le {\left({\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{{\left[a^{t}b\left(t\right)\right]}^p}}{∥{\mathrm{S}}_{\upsilon }\left(t\right)(I-A)x_0∥}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & +{\left({\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{{\left[a^{t}b\left(t\right)\right]}^p}}{∥{\displaystyle \sum _{\iota =1}^{t-1}}{\mathrm{S}}_{\upsilon }(t-1-\iota )\mathcal{F}(\iota ,x\left(\iota \right))∥}^p\right)}^{{\displaystyle \frac{1}{p}}},\hfill \end{array}$$

(13)

where

$$\begin{array}{cc}\hfill & {\left({\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{{\left[a^{t}b\left(t\right)\right]}^p}}{∥{\mathrm{S}}_{\upsilon }\left(t\right)(I-A)x_0∥}^p\right)}^{{\displaystyle \frac{1}{p}}}\le ∥(I-A)x_0∥{\left({\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{{\left[a^{t}b\left(t\right)\right]}^p}}{∥{\mathrm{S}}_{\upsilon }\left(t\right)∥}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le ∥(I-A)x_0∥{\left({\displaystyle \sum _{t=1}^{\infty }}{\left[{\displaystyle \frac{M{k}^{\upsilon }\left(t\right)}{a^{t}b\left(t\right)}}\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}=M∥(I-A)x_0∥{\left({\displaystyle \sum _{t=1}^{\infty }}{\left[{\displaystyle \frac{k^{\upsilon }\left(t\right)}{a^{t}b\left(t\right)}}\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}.\hfill \end{array}$$

(14)

Denote by

$$c\left(t\right)={\left[{\displaystyle \frac{k^{\upsilon }\left(t\right)}{a^{t}b\left(t\right)}}\right]}^p,t\in \mathbb{N}.$$

Clearly,

$$c\left(t\right)>0,t\in \mathbb{N}.$$

Also,

$$\underset{t\to \infty }{lim}{\displaystyle \frac{c(t+1)}{c\left(t\right)}}=\underset{t\to \infty }{lim}{\left[{\displaystyle \frac{1}{a}}\right]}^p{\left[{\displaystyle \frac{t+\upsilon }{t+1+\upsilon }}\right]}^p<1.$$

Then, by ratio test, the series

$${\displaystyle \sum _{t=1}^{\infty }}c\left(t\right)={\displaystyle \sum _{t=1}^{\infty }}{\left[{\displaystyle \frac{k^{\upsilon }\left(t\right)}{a^{t}b\left(t\right)}}\right]}^p,$$

converges. Let

$$N_1:={\displaystyle \sum _{t=1}^{\infty }}{\left[{\displaystyle \frac{k^{\upsilon }\left(t\right)}{a^{t}b\left(t\right)}}\right]}^p.$$

(15)

Using (15) in (14), we obtain

$${\left({\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{{\left[a^{t}b\left(t\right)\right]}^p}}{∥{\mathrm{S}}_{\upsilon }\left(t\right)(I-A)x_0∥}^p\right)}^{{\displaystyle \frac{1}{p}}}\le M∥(I-A)x_0∥{\left(N_1\right)}^{{\displaystyle \frac{1}{p}}}.$$

(16)

Now, we also have the following inequalities

$$\begin{array}{cc}\hfill & {\left({\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{{\left[a^{t}b\left(t\right)\right]}^p}}{∥{\displaystyle \sum _{\iota =1}^{t-1}}{\mathrm{S}}_{\upsilon }(t-1-\iota )\mathcal{F}(\iota ,x\left(\iota \right))∥}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \le {\left({\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{{\left[a^{t}b\left(t\right)\right]}^p}}{\left[{\displaystyle \sum _{\iota =1}^{t-1}}∥{\mathrm{S}}_{\upsilon }(t-1-\iota )∥∥\mathcal{F}(\iota ,x(\iota \left)\right)∥\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \le LM{\left({\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{{\left[a^{t}b\left(t\right)\right]}^p}}{\left[{\displaystyle \sum _{\iota =1}^{t-1}}{k}^{\upsilon }(t-1-\iota ){\displaystyle \frac{\parallel x\left(\iota \right)\parallel }{a^{\iota }b\left(\iota \right)}}\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \le {LM\parallel x\parallel }_{{\ell }_w^p}{\left({\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{{\left[a^{t}b\left(t\right)\right]}^p}}{\left[{\displaystyle \sum _{\iota =1}^{t-1}}{k}^{\upsilon }(t-1-\iota )\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & ={LM\parallel x\parallel }_{{\ell }_w^p}{\left({\displaystyle \sum _{t=1}^{\infty }}{\left[{\displaystyle \frac{b(t-1)}{a^{t}b\left(t\right)}}\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}=LM{\parallel x\parallel }_{{\ell }_w^p}{\left(N_2\right)}^{{\displaystyle \frac{1}{p}}}.\hfill \end{array}$$

(17)

It follows from (13), (16) and (17) that

$$\begin{array}{cc}\hfill {\parallel \mathcal{S}x\parallel }_{{\ell }_w^p}& ={\left({\displaystyle \sum _{t=0}^{\infty }}{\left[{\displaystyle \frac{∥\left(\mathcal{S}x\right)\left(t\right)∥}{a^{t}b\left(t\right)}}\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}={\displaystyle \frac{∥\left(\mathcal{S}x\right)\left(0\right)∥}{b\left(0\right)}}+{\left({\displaystyle \sum _{t=1}^{\infty }}{\left[{\displaystyle \frac{∥\left(\mathcal{S}x\right)\left(t\right)∥}{a^{t}b\left(t\right)}}\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & \le ∥x_0∥+M∥(I-A)x_0∥{\left(N_1\right)}^{{\displaystyle \frac{1}{p}}}+LM{\parallel x\parallel }_{{\ell }_w^p}{\left(N_2\right)}^{{\displaystyle \frac{1}{p}}}<\infty ,\hfill \end{array}$$

implying that $\mathcal{S}$ is well defined. For x, $y\in {\ell }_w^p\left({\mathbb{N}}_0;\mathcal{Y}\right)$, by (A5), we get,

$$\begin{array}{cc}& {\left({\displaystyle {\displaystyle \sum _{t=1}^{\infty }}}{\left[{\displaystyle \frac{∥\left(\mathcal{S}x\right)\left(t\right)-\left(\mathcal{S}y\right)\left(t\right)∥}{a^{t}b\left(t\right)}}\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & ={\left({\displaystyle {\displaystyle \sum _{t=1}^{\infty }}}{\displaystyle \frac{1}{{\left[a^{t}b\left(t\right)\right]}^p}}{∥{\displaystyle {\displaystyle \sum _{\iota =1}^{t-1}}}{\mathrm{S}}_{\upsilon }(t-1-\iota )\left[\mathcal{F}(\iota ,x(\iota \left)\right)-\mathcal{F}(\iota ,y(\iota \left)\right)\right]∥}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \le {\left({\displaystyle {\displaystyle \sum _{t=1}^{\infty }}}{\displaystyle \frac{1}{{\left[a^{t}b\left(t\right)\right]}^p}}{\left[{\displaystyle {\displaystyle \sum _{\iota =1}^{t-1}}}∥{\mathrm{S}}_{\upsilon }(t-1-\iota )∥∥\mathcal{F}(\iota ,x(\iota \left)\right)-\mathcal{F}(\iota ,y(\iota \left)\right)∥\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \le LM{\left({\displaystyle {\displaystyle \sum _{t=1}^{\infty }}}{\displaystyle \frac{1}{{\left[a^{t}b\left(t\right)\right]}^p}}{\left[{\displaystyle {\displaystyle \sum _{\iota =1}^{t-1}}}{k}^{\upsilon }(t-1-\iota ){\displaystyle \frac{\parallel x\left(\iota \right)-y\left(\iota \right)\parallel }{a^{\iota }b\left(\iota \right)}}\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ \hfill & \le {LM\parallel x-y\parallel }_{{\ell }_w^p}{\left({\displaystyle {\displaystyle \sum _{t=1}^{\infty }}}{\displaystyle \frac{1}{{\left[a^{t}b\left(t\right)\right]}^p}}{\left[{\displaystyle {\displaystyle \sum _{\iota =1}^{t-1}}}{k}^{\upsilon }(t-1-\iota )\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & ={LM\parallel x-y\parallel }_{{\ell }_w^p}{\left({\displaystyle {\displaystyle \sum _{t=1}^{\infty }}}{\left[{\displaystyle \frac{b(t-1)}{a^{t}b\left(t\right)}}\right]}^p\right)}^{{\displaystyle \frac{1}{p}}}\hfill \\ & =LM{\left(N_2\right)}^{{\displaystyle \frac{1}{p}}}{\parallel x-y\parallel }_{{\ell }_w^p},\hfill \end{array}$$

for every $t\in \mathbb{N}$. Then,

$${\parallel \mathcal{S}x-\mathcal{S}y\parallel }_{{\ell }_w^p}\le LM{\left(N_2\right)}^{{\displaystyle \frac{1}{p}}}{\parallel x-y\parallel }_{{\ell }_w^p}.$$

Since $LM{\left(N_2\right)}^{{\displaystyle \frac{1}{p}}}<1$, the Banach theorem provides the conclusion. □

4. Examples

In this section, we provide a few examples to illustrate our main results.

Example 1.

Consider (1) with $\upsilon =0.3$ and

$$\mathcal{F}(t,\xi )=\left(0.3\right)k^{0.2}\left(t\right)sin\xi ,$$

for all $t\in {\mathbb{N}}_0$. If A generates ${\left\{\mathcal{Q}\left(t\right)\right\}}_{t\ge 0}$, Theorem 2 concludes that A generates ${\left\{{\mathrm{S}}_{0.3}\left(t\right)\right\}}_{t\in {\mathbb{N}}_0}$ with $M=1$. Hence, (A1) holds. We see that $\mathcal{F}(t,0)=0$ for all $t\in {\mathbb{N}}_0$. Take $p=4$. Clearly, $1.25={\displaystyle \frac{1}{1-0.2}}<4,$ $L=0.3$ and $\beta =0.8\in \left(0.3+{\displaystyle \frac{1}{4}},1\right)$. Also,

$$\begin{array}{cc}\hfill N& ={\displaystyle \sum _{t=0}^{\infty }}{\left[k^{1+0.3-0.8}\left(t\right)\right]}^4={\displaystyle \sum _{t=0}^{\infty }}{\left[k^{0.5}\left(t\right)\right]}^4={\left[k^{0.5}\left(0\right)\right]}^4+{\displaystyle \sum _{t=1}^{\infty }}{\left[k^{0.5}\left(t\right)\right]}^4\hfill \\ & =1+{\displaystyle \frac{1}{{\Gamma }^4\left(0.5\right)}}{\displaystyle \sum _{t=1}^{\infty }}{\left[{\displaystyle \frac{\Gamma (t+0.5)}{\Gamma (t+1)}}\right]}^4\le 1+{\displaystyle \frac{1}{{\Gamma }^4\left(0.5\right)}}{\displaystyle \sum _{t=1}^{\infty }}{\left[{\displaystyle \frac{1}{t^{0.5}}}\right]}^4\hfill \\ & =1+{\displaystyle \frac{1}{{\Gamma }^4\left(0.5\right)}}{\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{t^2}}=1+{\displaystyle \frac{1}{{\pi }^2}}\left[{\displaystyle \frac{{\pi }^2}{6}}\right]={\displaystyle \frac{7}{6}}.\hfill \end{array}$$

Since ${LMN}^{{\displaystyle \frac{1}{4}}}\le \left(0.3\right)\left(1\right){\left({\displaystyle \frac{7}{6}}\right)}^{{\displaystyle \frac{1}{4}}}\approx 0.3118<1$, (A2) holds. Thus, by Theorem 4, the problem (1) with $x_0=0$ and $\mathcal{F}$ as above, has a unique solution in ${\ell }^4\left({\mathbb{N}}_0;\mathcal{Y}\right)$.

Example 2.

Consider (1) with $\upsilon =0.3$ and

$$\mathcal{F}(t,\xi )={\displaystyle \frac{\left(0.3\right)k^{0.2}\left(t\right)sin\xi }{2^t}},$$

for all $t\in {\mathbb{N}}_0$. If A generates ${\left\{\mathcal{Q}\left(t\right)\right\}}_{t\ge 0}$, then (A1) holds. We see that $\mathcal{F}(t,0)=0$ for all $t\in {\mathbb{N}}_0$. Clearly, $L=0.3$, $a=2$ and $\beta =0.8\in \left(0,1\right)$. Also,

$$\begin{array}{cc}\hfill N_0& ={\displaystyle \sum _{t=0}^{\infty }}{\displaystyle \frac{k^{1+0.3-0.8}\left(t\right)}{2^t}}={\displaystyle \sum _{t=0}^{\infty }}{\displaystyle \frac{k^{0.5}\left(t\right)}{2^t}}={\displaystyle \frac{k^{0.5}\left(0\right)}{2^0}}+{\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{k^{0.5}\left(t\right)}{2^t}}\hfill \\ & =1+{\displaystyle \frac{1}{\Gamma \left(0.5\right)}}{\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{2^t}}{\displaystyle \frac{\Gamma (t+0.5)}{\Gamma (t+1)}}\le 1+{\displaystyle \frac{1}{\Gamma \left(0.5\right)}}{\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{2^t}}{\displaystyle \frac{1}{\sqrt{t}}}\hfill \\ & \le 1+{\displaystyle \frac{1}{\Gamma \left(0.5\right)}}{\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{2^t}}=1+{\displaystyle \frac{1}{\sqrt{\pi }}}\approx 1.5642.\hfill \end{array}$$

Since ${LMN}_0\le \left(0.3\right)\left(1\right)\left(1.5642\right)=0.4693<1$, (A3) holds. Thus, by Theorem 5, the problem (1) with $x_0=0$ has a unique solution in ${\ell }_w^1\left({\mathbb{N}}_0;\mathcal{Y}\right)$.

Example 3.

Consider (1) with $\upsilon =0.3$ and

$$\mathcal{F}(t,\xi )={\displaystyle \frac{\left(0.3\right)sin\xi }{k^{1.3}\left(t\right)}},$$

for all $t\in {\mathbb{N}}_0$. If A generates a contraction $C_0$ semigroup ${\left\{\mathcal{Q}\left(t\right)\right\}}_{t\ge 0}$ defined on $\mathcal{Y}$, then (A1) holds. We see that $\mathcal{F}(t,0)=0$ for all $t\in {\mathbb{N}}_0$. Clearly, $L=0.3$ and $b\left(t\right)=k^{1.3}\left(t\right)$ for $t\in {\mathbb{N}}_0$. Since $LM=0.3<1$, (A4) holds. Thus, by Theorem 6, the problem (1) has a unique solution in ${\ell }_w^{\infty }\left({\mathbb{N}}_0;\mathcal{Y}\right)$ for any $x_0\in D\left(A\right).$

Example 4.

Consider (1) with $\upsilon =0.3$ and

$$\mathcal{F}(t,\xi )={\displaystyle \frac{\left(0.3\right)sin\xi }{2^t{k}^{1.3}\left(t\right)}},$$

for all $t\in {\mathbb{N}}_0$. Again, we assume that A generates ${\left\{\mathcal{Q}\left(t\right)\right\}}_{t\ge 0}$, and hence (A1) holds. We see that $\mathcal{F}(t,0)=0$ for all $t\in {\mathbb{N}}_0$. Take $p=1$. Clearly, $L=0.3$, $a=2$ and $b\left(t\right)=k^{1.3}\left(t\right)$ for $t\in {\mathbb{N}}_0$. Also,

$$\begin{array}{cc}\hfill N_2& ={\displaystyle \sum _{t=1}^{\infty }}\left[{\displaystyle \frac{b(t-1)}{2^{t}b\left(t\right)}}\right]={\displaystyle \sum _{t=1}^{\infty }}\left[{\displaystyle \frac{t}{2^t(t+0.3)}}\right]\le {\displaystyle \sum _{t=1}^{\infty }}{\displaystyle \frac{1}{2^t}}=1.\hfill \end{array}$$

Since $LM{N}_2\le \left(0.3\right)\left(1\right)\left(1\right)=0.3<1$, (A5) holds. Thus, by Theorem 7, the problem (1) has a unique solution in ${\ell }_w^1\left({\mathbb{N}}_0;\mathcal{Y}\right)$ with $w\left(t\right):=2^t{k}^{1.3}\left(t\right)$.

5. Conclusions

We established existence and uniqueness of solutions to the semilinear abstract fractional difference Equation (1) in several new vector-valued sequence settings beyond the classical ${\ell }^{\infty }$ framework. In particular, we obtained well-posedness in ${\ell }^p({\mathbb{N}}_0;\mathcal{Y})$ for a finite range of p, and in weighted Lebesgue spaces with exponential and fractional weights, where the admissible Lipschitz profiles explicitly reflect the influence of the fractional order $\upsilon$.

Several directions seem natural for future work:

(i)

extensions to other discrete fractional operators (e.g., Caputo-type differences);

(ii)

weaker nonlinear hypotheses (local Lipschitz, growth conditions, or monotonicity methods);

(iii)

stability/periodicity theory in ${\ell }^p$ and weighted ${\ell }_w^p$ spaces;

(iv)

analogous results for discrete-time abstract Volterra equations and related memory models.