$\mathcal{I}$-Convergence Sequence Paranormed Spaces of Order (α, β)

Abstract

In this paper, we introduce and rigorously define a novel class of difference sequence spaces, denoted by $w^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$, $w_0^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }, w_{\infty }^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$, and $w_{\infty }{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$. These spaces are constructed through the application of the concept of $\mathcal{I}$-convergence of sequences, combined with a Musielak–Orlicz function of order $(\alpha , \beta )$. The primary focus of our work is to thoroughly investigate the algebraic and topological properties of these defined sequence spaces. We explore their linearity, examine their structure within the framework of paranormed spaces, and analyze various other algebraic characteristics pertinent to these spaces. In addition, we examine the topological nature of these sequence spaces, identifying the conditions under which they exhibit specific topological properties. A significant part of our study is dedicated to examining the inclusion relationships between these sequence spaces, thereby providing a comprehensive understanding of how these spaces are interrelated. Our analysis contributes to the broader field of functional analysis and sequence space theory, offering new insights and potential applications of these advanced mathematical constructs.

1. Introduction and Preliminaries

Sequence spaces have been fundamental to the development of functional analysis from its early stages. Among the most significant spaces are ${\mathsf{\ell }}_{\infty }, c, c_0, {\mathsf{\ell }}_1$, and ${\mathsf{\ell }}_p$ which emerged as important specific cases of Banach spaces. These sequence spaces are widely utilized across various areas within functional analysis, including the theory of functions, locally convex spaces, summability invariance, and matrix transformations. Their significance in these areas is well-documented in the literature (see [1,2,3,4,5] and references therein), highlighting their crucial role in advancing both theoretical and applied aspects of the field.

In numerous investigations, including the work of Mursaleen et al. [6], sequence spaces defined by Orlicz functions have been a primary focus. The concept of $\mathcal{I}$-convergence, introduced by Kostyrko et al. [7], extends statistical convergence by utilizing an ideal on the set of natural numbers to establish this generalization. Subsequent studies, such as those by [8,9], have explored sequence spaces using the notion of $\mathcal{I}$-convergence. Additionally, Kuratowski [10] introduced the concept of an ideal on a nonempty set, which has further contributed to the development and understanding of these mathematical structures. Sequence spaces can be further generalized through the introduction of paranorms, as seen in paranormed sequence spaces, which extend traditional norm concepts by relaxing certain norm properties. Additionally, the incorporation of various convergence criteria, such as $\mathcal{I}$-convergence and Musielak–Orlicz functions, provides new insights into sequence behavior and broadens the scope of analysis. These extensions allow for the exploration of more complex relationships and properties within sequence spaces.

Definition 1.

A paranormed space is a vector space $(X,\parallel .\parallel )$ equipped with a function $\parallel .\parallel$ called a paranorm, which satisfies the following conditions:

1. 

$\parallel x\parallel \ge 0$ for all $x\in X$;

2. 

$\parallel x\parallel = 0$ if and only if $x=0$;

3. 

$\parallel x+y\parallel \le \parallel x\parallel +\parallel y\parallel$ for all $x,y\in X$;

4. 

$\parallel \alpha x\parallel =\left|\alpha \right|\parallel x\parallel$ for all $x\in X$.

An ideal $\mathcal{I}$ is defined as a nonempty set of subsets of X that has the following properties:

(i)

If $A\in \mathcal{I}$ and $B\subset A,$ then $B\in \mathcal{I}$;

(ii)

If $A\in \mathcal{I}$ and $B\in \mathcal{I},$ then $A\cup B\in \mathcal{I}$.

We define $\mathcal{I}$ as non-trivial if it does not equal the empty set and $X\notin \mathcal{I}$. Additionally, $\mathcal{I}$ is considered admissible if it satisfies the condition of being non-trivial and $\left\{x\right\}\in \mathcal{I}$ for each $x\in X.$ A non-trivial ideal $\mathcal{I}$ is maximal if there does not exist any nontrivial ideal $\mathcal{J}$ such that $\mathcal{I}\subseteq \mathcal{J}$.

Definition 2.

A sequence $({\xi }_k\in \omega )$ is said to be $\mathcal{I}$-convergent to a number L if for every $ϵ>0$, $\{k\in N:|{\xi }_k-L|\ge ϵ\}\in \mathcal{I}$. In this case we write $\mathcal{I}-lim{\xi }_k=L.$

The concept of difference sequence spaces was first introduced by Kızmaz [11], who studied the space $\mathcal{Z}(∆)=\{\xi =\left({\xi }_k\right)\in \omega :∆\xi \in Z\}$, where $\mathcal{Z}$ represents the spaces ${\mathsf{\ell }}_{\infty }, c$, and $c_0$, $∆\xi ={\xi }_k-{\xi }_{k+1}$ for all $k\in \mathbb{N}$ and $\omega$ denotes the space of all real or complex sequences. This concept was later expanded by Tripathy et al. [12], who introduced generalized difference operators. For non-negative integers n and m, they defined the sequence spaces $\mathcal{Z}\left({∆}_m^n\right)=\{\xi =\left({\xi }_k\right)\in \omega :{∆}_m^n\xi \in Z\}$, for $\mathcal{Z}={\mathsf{\ell }}_{\infty }, c, c_0$ where ${∆}_m^n\xi =\left({∆}_m^n{\xi }_k\right)={∆}_m^{n-1}{\xi }_k-{∆}_m^{n-1}{\xi }_{k+m}$ and ${∆}_m^0{\xi }_k={\xi }_k$ for each $k\in \mathbb{N}$. The difference operator ${∆}_m^n{\xi }_k$ can be expressed using the binomial formula as in [13] by

$${∆}_m^n{\xi }_k=\sum _n^{v=0}{(-1)}^v\left({\displaystyle \genfrac{}{}{0pt}{}{n}{v}}\right){\xi }_{k+mv}.$$

Further, in the context of paranormed spaces and Musielak–Orlicz function spaces, $\mathcal{I}$-convergence has been studied to explore new sequence spaces that offer more refined properties. Khan and Tuba [14] explored the application of ideal convergence within the context of paranormed sequence spaces, specifically those defined by the Jordan totient function. Their work contributes to the growing body of research that investigates the interplay between ideal convergence and functional spaces, expanding its applicability to new mathematical structures and providing deeper insights into the nature of convergence under different norms. The study of $\mathcal{I}$-convergence has also been enriched by examining its interaction with other sequence space constructions, such as those defined by Musielak–Orlicz functions and modulus functions. These developments have opened new avenues for research in both theoretical and applied mathematics. Mursaleen [15] introduced several sequence spaces by employing the concepts of ideal convergence and Musielak–Orlicz functions. The concept of $\mathcal{I}$-convergence has been significantly generalized and extended in various directions in recent years. For instance, Tripathy et al. [16] investigate generalized difference ideal convergence within the framework of generalized probabilistic n-normed spaces. Their study aims to bridge the gap between ideal convergence and probabilistic normed spaces, providing new insights into how sequences converge under different probabilistic norms.

Malik and Das [17] expanded on this concept by investigating the $\mathcal{I}$-convergence of sequences of subspaces in inner product spaces, a crucial area in functional analysis with implications for various applications, including quantum mechanics and signal processing. Their work offers a fresh perspective on the behavior of subspaces under ideal convergence, exploring how this generalized notion can capture convergence phenomena that are not observable through classical methods. For further details on ideal convergence, please refer to references ([18,19,20,21]) and references therein.

Tang and Xiong’s [22] study investigates how sequences in quasi-metric spaces can be analyzed through the lens of $\mathcal{I}$-convergence, focusing on the conditions under which a sequence is $\mathcal{I}$-convergent. They explore the implications of $\mathcal{I}$-convergence for the structure of quasi-metric spaces, providing new theoretical results and extending the application of $\mathcal{I}$-convergence to these more general spaces. These studies underscore the rich interplay between $\mathcal{I}$-convergence and paranormed spaces, expanding the theoretical framework and offering new insights into their practical applications. For more information about the ideal convergent sequence, we refer to ([23,24,25,26,27,28,29,30,31,32]) and references therein. In recent developments, various authors have extended the classical difference sequence spaces by incorporating more generalized difference operators and examining their interaction with various functional constructs such as Musielak–Orlicz functions and ideal convergence. These advancements have led to the creation of new sequence spaces that offer greater flexibility and applicability in both theoretical and practical settings. For more information about these, see ([33,34,35,36]) and references therein.

Suppose that X is a linear space. A function $f:X\to \mathbb{R}$ is known to be convex, if the following inequality holds:

$$f(\lambda u+(1-\lambda \left)v\right)\le \lambda f\left(u\right)+(1-\lambda )f\left(v\right)$$

for all $u, v\in X$, and $\lambda \in [0,1]$.

A sequence space X is solid (or normal) if $\left({\alpha }_n{\xi }_n\right)\in X$ whenever $\left({\xi }_n\right)\in X$ for all sequences $\left({\alpha }_n\right)$ of scalars with $|{\alpha }_n| \le 1$ for all $n\in \mathbb{N}$.

Theorem 1

([13,37]). Let f be a convex function with $f\left(0\right)=0$. Then $f\left(\lambda u\right)\le \lambda f\left(u\right),$ for all $\lambda \in [0,1]$.

An Orlicz function is a function $M:[0,\infty )\to [0,\infty ),$ which is continuous, non-decreasing, and convex, with $M\left(0\right)=0,M\left(u\right)>0$, and $M\left(u\right)\to \infty$, as $u\to \infty .$ An Orlicz function M is said to satisfy the ${∆}_2$-condition, if for each $u\in [0,\infty ),$ there exists a constant $Q\ge 0$ such that $M\left(2u\right)\le QM\left(u\right).$ Musielak–Orlicz functions are known to be sequence $\mathcal{M}=\left({\mathfrak{F}}_n\right)$ of Orlicz functions ([38,39]). A sequence $\mathcal{N}=\left({\mathfrak{N}}_n\right)$ is defined by

$${\mathfrak{N}}_n\left(v\right)=sup\left\{\right|v|u-{\mathfrak{F}}_n\left(u\right):u\ge 0\},n=1,2,\cdots$$

which is called the complementary function of a Musielak–Orlicz function $\mathcal{M}$; the Musielak–Orlicz sequence space $t_{\mathcal{M}}$ and its subspace $h_{\mathcal{M}}$ are defined as follows,

$$t_{\mathcal{M}}=\{u\in w:I_{\mathcal{M}}\left(cu\right)<\infty \mathrm{for}\mathrm{some}c>0\}$$

$$h_{\mathcal{M}}=\{u\in w:I_{\mathcal{M}}\left(cu\right)<\infty \mathrm{for}\mathrm{all}c>0\},$$

where w is the space of all real or complex sequences and $I_{\mathcal{M}}$ is a convex modular defined by

$$I_{\mathcal{M}}\left(u\right)=\sum _{n=1}^{\infty }{M}_n\left(u_n\right),u=\left(u_n\right)\in t_{\mathcal{M}}.$$

Theorem 2

([13,37]). An Orlicz function M satisfies the ${∆}_2$-condition if and only if for each $u\in [0,\infty )$ and each $l>1$ there exists a constant $S=S\left(l\right)\ge 0$ such that

$$M\left(lu\right)\le SM\left(u\right)\le SlM\left(u\right).$$

Notice that the function $M\left(u\right)=u^p$ where $u\in [0,\infty )$ and $1\le p<\infty$, is an Orlicz function that satisfies the ${∆}_2$-condition, as $M\left(2u\right)={\left(2u\right)}^p=2^p{u}^p=2^{p}M\left(u\right).$

Recent advancements have expanded this field by incorporating Musielak–Orlicz functions and ideal convergence. Musielak–Orlicz functions, which generalize the classical Orlicz functions, allow for a broader and more flexible approach to defining and studying sequence spaces. These functions enable the construction of sequence spaces that accommodate a wide range of growth conditions and convergence criteria. The primary objective of this paper is to present the concept of $\mathcal{I}$-convergence for sequences along with an investigation of Musielak–Orlicz functions of order $(\alpha ,\beta )$.

The sequence spaces we are defining play a crucial role in extending the theory of sequence spaces by incorporating $\mathcal{I}$-convergence and Musielak–Orlicz functions. These spaces are significant in several areas, including summability theory, where they refine the analysis of series convergence; functional analysis, by providing new frameworks for understanding operator behavior; and approximation theory, where they enhance methods for approximating functions. Their importance extends to practical applications such as mathematical modeling and numerical methods, where they offer sophisticated tools for handling various convergence criteria and sequence behaviors.

The spaces $w^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$, $w_0^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }, w_{\infty }^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$, and $w_{\infty }{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ are essential in various mathematical and applied fields. They are utilized in approximation theory to assess function convergence and approximation properties. In functional analysis, they help in understanding the behavior of linear operators and functionals, particularly in generalized settings where traditional norms might be too restrictive, and in the theory of differential equations to analyze solution properties and stability. Additionally, these spaces find applications in control theory for system design and signal processing for analyzing signal behavior and transformations. Their ability to handle different types of convergence and function spaces makes them valuable tools in both theoretical and practical contexts.

2. Outline

This paper is structured as follows: We begin with an introduction to the key concepts of $\mathcal{I}$-convergence, Musielak–Orlicz functions, and paranormed spaces, providing essential definitions and preliminaries. Next, we rigorously define the novel sequence spaces $w^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$, $w_0^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta },w_{\infty }^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$, and $w_{\infty }{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$, highlighting their construction and significance. We then explore the topological and algebraic properties of these spaces, including their paranormed structure and linearity, followed by an analysis of the inclusion relationships among these spaces and comparisons with classical sequence spaces. The paper also presents applications of these sequence spaces in various mathematical contexts, supported by non-trivial examples. Finally, we conclude with a summary of the findings and propose potential directions for future research.

3. Main Results

Let $\mathcal{M}=\left({\mathfrak{F}}_n\right)$ be a Musielak–Orlicz function and $r=\left(r_n\right)$ be a bounded sequence and $0<\alpha \le \beta \le 1.$ For each $\xi =\left({\xi }_n\right)\subset X.$ Then we define the following sequence spaces as the following:

$$\begin{array}{ccc}\hfill w^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }& =& \{\xi :\forall \epsilon >0,\left\{k\in \mathbb{N}:{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n-L}{\rho }}\right)\right]}^{r_n}\right]}^{\beta }\ge \epsilon \right\}\in \mathcal{I},\hfill \\ & & \mathrm{for}\mathrm{some}L\in X,\rho >0\},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill w_0^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }& =& \{\xi :\forall \epsilon >0,\left\{k\in \mathbb{N}:{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{\rho }}\right)\right]}^{r_n}\right]}^{\beta }\ge \epsilon \right\}\in \mathcal{I},\hfill \\ & & \mathrm{for}\mathrm{some}\rho >0{\displaystyle \}},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill w_{\infty }^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }& =& \{\xi :\exists Q>0:\left\{k\in \mathbb{N}:{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{\rho }}\right)\right]}^{r_n}\right]}^{\beta }\ge Q\right\}\in \mathcal{I},\hfill \\ & & \mathrm{for}\mathrm{some}\rho >0{\displaystyle \}},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill w_{\infty }{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }& =& \left\{\xi :\underset{k}{sup}{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{\rho }}\right)\right]}^{r_n}\right]}^{\beta }<\infty ,\mathrm{for}\mathrm{some}\rho >0\right\},\hfill \end{array}$$

$${\eta }^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }=w^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }\cap w_{\infty }{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$$

and

$${\eta }_0^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }=w_0^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }\cap w_{\infty }{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }.$$

In this section, we study some topological and algebraic properties of the above defined sequence spaces.

Firstly, we construct some examples related these spaces:

Example 1.

Consider the sequence $\xi =\left({\xi }_k\right)$ defined by ${\xi }_k={\displaystyle \frac{1}{k^{\alpha \beta }}}$ for $k\in \mathbb{N}$ and $0<\alpha \le \beta \le 1.$ Let $\mathcal{M}\left({\xi }_k\right)={\left|{\xi }_k\right|}^p$ be a Musielak–Orlicz function and $r={\displaystyle \frac{1}{\beta }}$. Then the sequence ξ belongs to the space $w^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ if

$$\sum _{k=1}^{\infty }\mathcal{M}\left({\displaystyle \frac{1}{k^{\alpha \beta }}}\right)<\infty .$$

This condition is satisfied if $p>{\displaystyle \frac{1}{\alpha \beta }}$, i.e., for sufficiently large p. Thus, $\xi ={\displaystyle \frac{1}{k^{\alpha \beta }}}\in w^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }.$

Example 2.

Consider the sequence $\chi =\left({\chi }_k\right)={\displaystyle \frac{{(-1)}^k}{k^{\alpha \beta }}}$ for $k\in \mathbb{N}$ and $0<\alpha \le \beta \le 1.$ Let $\mathcal{M}\left({\chi }_k\right)={\left|{\chi }_k\right|}^p$ be a Musielak–Orlicz function and $r={\displaystyle \frac{1}{\beta }}$. If $p>{\displaystyle \frac{1}{\alpha \beta }}$, then $\chi \in w_0^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ because

$$\sum _{k=1}^{\infty }\mathcal{M}\left({\displaystyle \frac{{(-1)}^k}{k^{\alpha \beta }}}\right)<\infty .$$

Example 3.

Let $\zeta =\left({\zeta }_k\right)={\displaystyle \frac{1}{log(k+1)}}$ for $k\in \mathbb{N}$ with Musielak–Orlicz function $\mathcal{M}\left({\zeta }_k\right)=log(1+|{\zeta }_k\left|\right)$ and $r={\displaystyle \frac{1}{\beta }}$. For $0<\alpha \le \beta \le 1$, the sequence $\left({\zeta }_k\right)$ is bounded and belongs to the space $w_{\infty }^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ because

$$\underset{k}{sup}\mathcal{M}\left({\displaystyle \frac{1}{log(k+1)}}\right)<\infty .$$

Example 4.

Take $\upsilon =\left({\upsilon }_k\right)=k^{\alpha }$ and Musielak–Orlicz function $\mathcal{M}\left({\upsilon }_k\right)={\displaystyle \frac{|{\upsilon }_k|}{1+|{\upsilon }_k|}}$ with $r={\displaystyle \frac{1}{\beta }}$. The sequence $\left({\upsilon }_k\right)\in w_{\infty }{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ if

$$\underset{k}{sup}\mathcal{M}\left(k^{\alpha }\right)<\infty .$$

Theorem 3.

If $\mathcal{M}=\left({\mathfrak{F}}_n\right)$ be a Musielak–Orlicz function, then $w^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ and $w_0^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ are linear spaces over the field of complex number $\mathbb{C}$.

Proof. 

Suppose that $\xi =\left({\xi }_n\right),\chi =\left({\chi }_n\right)\in w^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$, and $a,b\in \mathbb{C}$.

For $\epsilon >0$ to be given, we have to show that there exist $\rho >0$ and $L\in X$ such that

$$A=\left\{k\in \mathbb{N}:{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u(a{\xi }_n+b{\chi }_n)-L}{\rho }}\right)\right]}^{r_n}\right]}^{\beta }\ge \epsilon \right\}\in \mathcal{I}.$$

Since $\xi ,\chi \in w^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$, there exist two numbers ${\rho }_1>0$ and ${\rho }_2>0$ such that

$$A_1=\left\{k\in \mathbb{N}:{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n-L_1}{{\rho }_1}}\right)\right]}^{r_n}\right]}^{\beta }\ge {\displaystyle \frac{\epsilon }{2D}}\right\}\in \mathcal{I}\mathrm{for}\mathrm{some}{L}_1\in X,$$

and

$$A_2=\left\{k\in \mathbb{N}:{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\chi }_n-L_2}{{\rho }_2}}\right)\right]}^{r_n}\right]}^{\beta }\ge {\displaystyle \frac{\epsilon }{2D}}\right\}\in \mathcal{I}\mathrm{for}\mathrm{some}{L}_2\in X,$$

where $D=max\{1,2^{\mathcal{H}-1}\}$ and ${\displaystyle \mathcal{H}=\underset{n}{sup}{r}_n\ge r_n>0.}$

Assume that $\rho =max\left\{2\right|a|{\rho }_1,2|b\left|{\rho }_2\right\}$ and $L=a{L}_1+b{L}_2.$ Then, by using the triangular inequality and also the fact $\mathcal{M}$ is non-decreasing, we have

• ${\displaystyle \frac{1}{k^{\alpha }}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left(\frac{{∆}_v^u(a{\xi }_n+b{\chi }_n)-L}{\rho }\right)\right]}^{r_n}\right]}^{\beta }}$

$$\begin{array}{ccc}& =& {\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{a({∆}_v^u{\xi }_n-L_1)+b({∆}_v^u{\chi }_n-L_2)}{\rho }}\right)\right]}^{r_n}\right]}^{\beta }\hfill \\ & \le & {\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left(\left({\displaystyle \frac{{∆}_v^u{\xi }_n-L_1}{2{\rho }_1}}\right)+\left({\displaystyle \frac{{∆}_v^u{\chi }_n-L_2}{2{\rho }_2}}\right)\right)\right]}^{r_n}\right]}^{\beta }\hfill \end{array}$$

Since $\mathcal{M}$ is convex. Thus, we have

• ${\displaystyle \frac{1}{k^{\alpha }}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left(\frac{{∆}_v^u(a{\xi }_n+b{\chi }_n)-L}{\rho }\right)\right]}^{r_n}\right]}^{\beta }}$

$$\begin{array}{ccc}& \le & {\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\displaystyle \frac{1}{2}}{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n-L_1}{{\rho }_1}}\right)+{\displaystyle \frac{1}{2}}{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\chi }_n-L_2}{{\rho }_2}})\right)\right]}^{r_n}\right]}^{\beta }\hfill \\ & \le & {\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\displaystyle \frac{1}{2^{r_n}}}{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n-L_1}{{\rho }_1}}\right)+{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\chi }_n-L_2}{{\rho }_2}}\right)\right]}^{r_n}\right]}^{\beta }.\hfill \end{array}$$

Let $\rho ={\rho }_1+{\rho }_2.$ Then,

• ${\displaystyle \frac{1}{k^{\alpha }}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left(\frac{{∆}_v^u(a{\xi }_n+b{\chi }_n)-L}{\rho }\right)\right]}^{r_n}\right]}^{\beta }}$

$$\begin{array}{ccc}& \le & {\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\displaystyle \frac{D}{2^{r_n}}}\left\{{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n-L_1}{{\rho }_1}}\right)\right]}^{r_n}\right]}^{\beta }+{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\chi }_n-L_2}{{\rho }_2}}\right)\right]}^{r_n}\right]}^{\beta }\right\}\hfill \\ & \le & {\displaystyle \frac{D}{k^{\alpha }}}\left\{\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n-L_1}{{\rho }_1}}\right)\right]}^{r_n}\right]}^{\beta }+\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\chi }_n-L_2}{{\rho }_2}}\right)\right]}^{r_n}\right]}^{\beta }\right\}.\hfill \end{array}$$

If $k^{\alpha }\in A_1^c\cap A_2^c$, then from last inequality, we have

• ${\displaystyle \frac{1}{k^{\alpha }}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left(\frac{{∆}_v^u(a{\xi }_n+b{\chi }_n)-L}{\rho }\right)\right]}^{r_n}\right]}^{\beta }}$

$$\begin{array}{ccc}& \le & D\left\{{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n-L_1}{{\rho }_1}}\right)\right]}^{r_n}\right]}^{\beta }+{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\chi }_n-L_2}{{\rho }_2}}\right)\right]}^{r_n}\right]}^{\beta }\right\}\hfill \\ & <& D\left\{{\displaystyle \frac{\epsilon }{2D}}+{\displaystyle \frac{\epsilon }{2D}}\right\}=\epsilon ;\hfill \end{array}$$

this implies $k^{\alpha }\in A^c$. Thus, if ${(A_1\cup A_2)}^c=A_1^c\cap A_2^c\subset A^c$ and $A\subset A_1\cup A_2\in \mathcal{I}$, it follows that $A\in \mathcal{I}$. Therefore, $a\xi +b\chi \in w^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ and hence, $w^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ is a linear space. Similarly, we establish that $w_0^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ is a linear space. □

Theorem 4.

If $\mathcal{M}=\left({\mathfrak{F}}_n\right)$ is a Musielak–Orlicz function, then $w_{\infty }^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ is a linear space over the field of complex number $\mathbb{C}$.

Proof. 

Suppose that $\xi =\left({\xi }_n\right),\chi =\left({\chi }_n\right)\in w_{\infty }^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$, and $a,b\in \mathbb{C}.$ We have to show that there exist two positive numbers Q and $\rho$ such that

$$A=\left\{k\in \mathbb{N}:{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u(a{\xi }_n+b{\chi }_n)}{\rho }}\right)\right]}^{r_n}\right]}^{\beta }\ge Q\right\}\in \mathcal{I}.$$

By hypothesis, there exist four positive numbers $Q_1,Q_2,{\rho }_1$, and ${\rho }_2$ such that

$$A_1=\left\{k\in \mathbb{N}:{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{{\rho }_1}}\right)\right]}^{r_n}\right]}^{\beta }\ge {\displaystyle \frac{Q_1}{2D}}\right\}\in \mathcal{I}$$

and

$$A_2=\left\{k\in \mathbb{N}:{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{{\rho }_2}}\right)\right]}^{r_n}\right]}^{\beta }\ge {\displaystyle \frac{Q_2}{2D}}\right\}\in \mathcal{I},$$

where $D=max\{1,2^{\mathcal{H}-1}\}$ and ${\displaystyle \mathcal{H}=\underset{n}{sup}{r}_n\ge r_n>0}$.

Suppose that $Q={\displaystyle \frac{Q_1+Q_2}{2}}$ and $\rho =max\left\{2\right|a|{\rho }_1,2|b\left|{\rho }_2\right\}.$ Thus, we have

• ${\displaystyle \frac{1}{k^{\alpha }}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left(\frac{{∆}_v^u(a{\xi }_n+b{\chi }_n)}{\rho }\right)\right]}^{r_n}\right]}^{\beta }}$

$$\begin{array}{ccc}& \le & {\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{a\left({∆}_v^u{\xi }_n\right)}{\rho }}+{\displaystyle \frac{b\left({∆}_v^u{\chi }_n\right)}{\rho }}\right)\right]}^{r_n}\right]}^{\beta }\hfill \\ & \le & {\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{2{\rho }_1}}+{\displaystyle \frac{{∆}_v^u{\chi }_n}{2{\rho }_2}}\right)\right]}^{r_n}\right]}^{\beta }\hfill \\ & \le & {\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\displaystyle \frac{1}{2^{r_n}}}{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{{\rho }_1}}\right)+{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\chi }_n}{{\rho }_2}}\right)\right]}^{r_n}\right]}^{\beta }.\hfill \end{array}$$

Let $\rho ={\rho }_1+{\rho }_2.$ Then,

• ${\displaystyle \frac{1}{k^{\alpha }}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left(\frac{{∆}_v^u(a{\xi }_n+b{\chi }_n)}{\rho }\right)\right]}^{r_n}\right]}^{\beta }}$

$$\begin{array}{ccc}& \le & {\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\displaystyle \frac{D}{2^{r_n}}}\left\{{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{{\rho }_1}}\right)\right]}^{r_n}\right]}^{\beta }+{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\chi }_n}{{\rho }_2}}\right)\right]}^{r_n}\right]}^{\beta }\right\}\hfill \\ & \le & {\displaystyle \frac{D}{2^{r_n}}}\left\{\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{{\rho }_1}}\right)\right]}^{r_n}\right]}^{\beta }+\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\chi }_n}{{\rho }_2}}\right)\right]}^{r_n}\right]}^{\beta }\right\}.\hfill \end{array}$$

If $k^{\alpha }\in A_1^c\cap A_2^c,$ then from last inequality, we have

• ${\displaystyle \frac{1}{k^{\alpha }}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left(\frac{{∆}_v^u(a{\xi }_n+b{\chi }_n)}{\rho }\right)\right]}^{r_n}\right]}^{\beta }}$

$$\begin{array}{ccc}& \le & D\left\{{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{{\rho }_1}}\right)\right]}^{r_n}\right]}^{\beta }+{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\chi }_n}{{\rho }_2}}\right)\right]}^{r_n}\right]}^{\beta }\right\}\hfill \\ & <& D\left\{{\displaystyle \frac{Q_1}{2D}}+{\displaystyle \frac{Q_2}{2D}}\right\}=Q;\hfill \end{array}$$

this implies that $k^{\alpha }\in A^c$. Thus, if ${(A_1\cup A_2)}^c=A_1^c\cap A_2^c\subset A^c$ and hence, $A\subset A_1\cup A_2\in \mathcal{I}$, it follows that $A\in \mathcal{I}.$ Therefore $a\xi +b\chi \in w_{\infty }^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$. Hence, $w_{\infty }^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ is a linear space over the field of complex number $\mathbb{C}$. □

Corollary 1.

If $\mathcal{M}=\left({\mathfrak{F}}_n\right)$ be a Musielak–Orlicz function, then ${\eta }^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ and ${\eta }_0^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ are linear spaces over the field of complex number $\mathbb{C}$.

Theorem 5.

The spaces ${\eta }^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ and ${\eta }_0^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ are paranormed spaces with paranorm defined by

$${\displaystyle \phi \left(x\right)=inf\left\{{\rho }^{\frac{r_n}{\mathcal{H}}}:\underset{k}{sup}\frac{1}{k^{\alpha }}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left(\frac{{∆}_v^u{\xi }_n}{\rho }\right)\right]}^{r_n}\right]}^{\beta }\le 1,\rho >0\right\},}$$

where $\mathcal{H}=max\{1,{sup}_n{r}_n\}.$

Proof. 

It is clear that $\phi \left(x\right)=\phi (-x).$ Since ${\mathfrak{F}}_n\left(0\right)=0,$ we get $\phi \left(0\right)=0$. Let us take $\xi =\left({\xi }_n\right)$ and $\chi =\left({\chi }_n\right)$ in ${\eta }^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$.

Let ${\displaystyle A\left(\xi \right)=\left\{{\rho }^{\frac{r_n}{\mathcal{H}}}:\underset{k}{sup}\frac{1}{k^{\alpha }}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left(\frac{{∆}_v^u{\xi }_n}{\rho }\right)\right]}^{r_n}\right]}^{\beta }\le 1\right\}}$

• and

$${\displaystyle A\left(\chi \right)=\left\{{\rho }^{\frac{r_n}{\mathcal{H}}}:\underset{k}{sup}\frac{1}{k^{\alpha }}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left(\frac{{∆}_v^u{\chi }_n}{\rho }\right)\right]}^{r_n}\right]}^{\beta }\le 1\right\}.}$$

Let ${\rho }_1\in A\left(\xi \right)$ and ${\rho }_2\in A\left(\chi \right)$. If $\rho ={\rho }_1+{\rho }_2$, then we have

• ${\displaystyle \underset{k}{sup}\left\{\frac{1}{k^{\alpha }}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left(\frac{{∆}_v^u({\xi }_n+{\chi }_n}{\rho }\right)\right]}^{r_n}\right]}^{\beta }\right\}}$

$$\begin{array}{ccc}& \le & \left({\displaystyle \frac{{\rho }_1}{{\rho }_1+{\rho }_2}}\right)\underset{k}{sup}{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{\rho }}\right)\right]}^{r_n}\right]}^{\beta }\hfill \\ & +& \left({\displaystyle \frac{{\rho }_2}{{\rho }_1+{\rho }_2}}\right)\underset{k}{sup}{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\chi }_n}{\rho }}\right)\right]}^{r_n}\right]}^{\beta }.\hfill \end{array}$$

Thus ${\displaystyle \underset{k}{sup}\left\{\frac{1}{k^{\alpha }}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left(\frac{{∆}_v^u({\xi }_n+{\chi }_n}{\rho }\right)\right]}^{r_n}\right]}^{\beta }\right\}\le 1}$ and

$$\begin{array}{ccc}\hfill \phi (\xi +\chi )& \le & inf\{({\rho }_1+{\rho }_2)>0:{\rho }_1\in A\left(\xi \right),{\rho }_2\in A\left(\chi \right)\}\hfill \\ & \le & inf\{{\rho }_1>0:{\rho }_1\in A\left(\xi \right)\}+inf\{{\rho }_2>0:{\rho }_2\in A\left(\chi \right)\}\hfill \\ & =& \phi \left(\xi \right)+\phi \left(\chi \right).\hfill \end{array}$$

Suppose that $a^s\to a,$ where $a,a^s\in \mathbb{C}$, and $\phi ({\xi }_n-\xi )\to 0,$ as $s\to \infty$.

We have to show that $\phi (a^s{\xi }_n^s-a\xi )\to 0,$ as $s\to \infty$.

Let ${\displaystyle A\left({\xi }_n^s\right)=\left\{{\rho }_s^{\frac{r_n}{\mathcal{H}}}:\underset{k}{sup}\frac{1}{k^{\alpha }}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left(\frac{{∆}_v^u{\xi }_n^s}{\rho }\right)\right]}^{r_n}\right]}^{\beta }\le 1\right\},}$

• ${\displaystyle A({\xi }_n^s-\xi )=\left\{{\rho \prime }_s:\underset{k}{sup}\frac{1}{k^{\alpha }}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left(\frac{{∆}_v^u({\xi }_n^s-\xi )}{\rho }\right)\right]}^{r_n}\right]}^{\beta }\le 1\right\}}$.

If ${\rho }_s\in A\left({\xi }^s\right)$ and $\rho {\prime }_s\in A({\xi }_n^s-\xi )$, then we observe that

$$\begin{array}{ccc}\hfill {\mathfrak{F}}_n\left({\displaystyle \frac{|{∆}_v^u(a^s{\xi }_n^s-a\xi )|}{{\rho }_s|a^s-a|+\rho {\prime }_s\left|a\right|}}\right)& \le & {\mathfrak{F}}_n\left({\displaystyle \frac{|{∆}_v^u(a^s{\xi }_n^s-a{\xi }_n^s)|}{{\rho }_s|a^s-a|+\rho {\prime }_s\left|a\right|}}+{\displaystyle \frac{\left|\right(a{\xi }_n^s-a\xi \left)\right|}{{\rho }_s|a^s-a|+\rho {\prime }_s\left|a\right|}}\right)\hfill \\ & \le & {\displaystyle \frac{|a^s-a|{\rho }_s}{{\rho }_s|a^s-a|+\rho {\prime }_s\left|a\right|}}{\mathfrak{F}}_n\left({\displaystyle \frac{|{∆}_v^u{\xi }_n^s|}{{\rho }_s}}\right)\hfill \\ & +& {\displaystyle \frac{\left|a\right|\rho {\prime }_s}{{\rho }_s|a^s-a|+\rho {\prime }_s\left|a\right|}}{\mathfrak{F}}_n\left({\displaystyle \frac{|{∆}_v^u({\xi }_n^s-\xi )|}{\rho {\prime }_s}}\right).\hfill \end{array}$$

From the above inequality, it follows that

$${\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\mathfrak{F}}_n{\left({\displaystyle \frac{|{∆}_v^u(a^s{\xi }_n^s-a\xi )|}{{\rho }_s|a^s-a|+\rho {\prime }_s\left|a\right|}}\right)}^{r_n}\right]}^{\beta }\le 1$$

and consequently,

$$\begin{array}{ccc}\hfill \phi (a^s{\xi }_n^s-a\xi )& \le & inf\left\{\right({\rho }_s|a^s-a|+\rho {\prime }_s\left|a\right|)>0:{\rho }_s\in A\left({\xi }_n^s\right),\rho {\prime }_s\in A({\xi }_n^s-\xi )\}\hfill \\ & \le & \left(\right|a^s-a\left|\right)>0inf\{\rho >0:{\rho }_s\in A\left({\xi }_n^s\right)\}\hfill \\ & & +\left(\right|a\left|\right)>0inf\{{\left(\rho {\prime }_s\right)}^{{\displaystyle \frac{r_n}{\mathcal{H}}}}:\rho {\prime }_s\in A({\xi }_n^s-\xi )\}\hfill \\ & & ⟶0,\mathrm{as}s⟶\infty .\hfill \end{array}$$

This completes the proof. □

Theorem 6.

Let $\mathcal{M}=\left({\mathfrak{F}}_n\right)$ and ${\mathcal{M}}^{\prime }=\left({\mathfrak{F}}_n^{\prime }\right)$ be two Musielak–Orlicz functions which satisfies the ${∆}_2$-condition. The following statement holds:

1. 

If $r=\left(r_n\right)$ be a bounded sequence with $\mathcal{H}=sup{r}_n<\infty ,$ then $Z{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }\subseteq Z{({\mathcal{M}}^{\prime }\circ \mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ for $Z=w_0^{\mathcal{I}},w^{\mathcal{I}},w_{\infty }^{\mathcal{I}},{\eta }^{\mathcal{I}},{\eta }_0^{\mathcal{I}}.$

2. 

$Z{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }\cap Z{({\mathcal{M}}^{\prime },{∆}_v^u,r)}_{\alpha }^{\beta }\subseteq Z{(\mathcal{M}+{\mathcal{M}}^{\prime },{∆}_v^u,r)}_{\alpha }^{\beta }$ for $Z=w_0^{\mathcal{I}},w^{\mathcal{I}},w_{\infty }^{\mathcal{I}},{\eta }^{\mathcal{I}},{\eta }_0^{\mathcal{I}}.$

Proof. 

(1) Let $\xi =\left({\xi }_n\right)\in w_0^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$. There exists $\rho >0$ such that

$$\mathcal{I}-\underset{k\to \infty }{lim}{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{\rho }}\right)\right]}^{r_n}\right]}^{\beta }=0.$$

Given $\epsilon >0$. Since each ${\mathfrak{F}}_n$ is continuous at 0 from right, there exists $0<\delta <1$ such that $0\le t\le \delta$ implies that ${\mathfrak{F}}_n\left(t\right)<\epsilon .$ Suppose that ${\chi }_n={\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{\rho }}\right)$ for each $n\in \mathbb{N}$. Defining the sets $N_1=\{n\in \{1,2,\cdots ,k\}:{\chi }_n\le \delta \}$ and $N_2=\{n\in \{1,2,\cdots ,k\}:{\chi }_n>\delta \}.$ Observe that

$${\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n^{\prime }\left({\chi }_n\right)\right]}^{r_n}\right]}^{\beta }={\displaystyle \frac{1}{k^{\alpha }}}\sum _{n\in N_1}{\left[{\left[{\mathfrak{F}}_n^{\prime }\left({\chi }_n\right)\right]}^{r_n}\right]}^{\beta }+{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n\in N_2}{\left[{\left[{\mathfrak{F}}_n^{\prime }\left({\chi }_n\right)\right]}^{r_n}\right]}^{\beta }.$$

If $n\in N_1,$ then $0\le {\chi }_n\le 1.$ By Theorem $1.1$, we have ${\mathfrak{F}}_n^{\prime }\left({\chi }_n\right)\le {\chi }_n.{\mathfrak{F}}_n^{\prime }\left(1\right)$ and so,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{k^{\alpha }}}\sum _{n\in N_1}{\left[{\left[{\mathfrak{F}}_n^{\prime }\left({\chi }_n\right)\right]}^{r_n}\right]}^{\beta }& \le & {\displaystyle \frac{1}{k^{\alpha }}}\sum _{n\in N_1}{\left[{\left[{\mathfrak{F}}_n^{\prime }\left(1\right)·{\chi }_n\right]}^{r_n}\right]}^{\beta }\hfill \\ & \le & max\left\{{\left[{\mathfrak{F}}_n^{\prime }\left(1\right)\right]}^{\mathcal{H}}\right\}·{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n\in N_1}{\left[{\left[{\chi }_n\right]}^{r_n}\right]}^{\beta }.\hfill \end{array}$$

For $n\in N_2,$ we have ${\chi }_n<{\displaystyle \frac{{\chi }_n}{\delta }}<1+{\displaystyle \frac{{\chi }_n}{\delta }}.$ Since each ${\mathfrak{F}}_n^{\prime }$ satisfies the ${∆}_2$-condition and non-decreasing for each $n\in \mathbb{N}$, by Theorem $1.2$, there exists a constant $M>0$ such that

$$\begin{array}{ccc}\hfill {\mathfrak{F}}_n^{\prime }\left({\chi }_n\right)& <& {\mathfrak{F}}_n^{\prime }\left(1+{\displaystyle \frac{{\chi }_n}{\delta }}\right)={\mathfrak{F}}_n^{\prime }\left(1+{\displaystyle \frac{{\chi }_n}{\delta }}·1\right)\hfill \\ & \le & M·{\mathfrak{F}}_n^{\prime }\left(1\right)<M·{\mathfrak{F}}_n^{\prime }\left(1\right)·{\displaystyle \frac{{\chi }_n}{\delta }}.\hfill \end{array}$$

Thus,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{k^{\alpha }}}\sum _{n\in N_2}{\left[{\left[{\mathfrak{F}}_n^{\prime }\left({\chi }_n\right)\right]}^{r_n}\right]}^{\beta }& \le & {\displaystyle \frac{1}{k^{\alpha }}}\sum _{n\in N_2}{\left[{\left[M·{\mathfrak{F}}_n^{\prime }\left(1\right)·{\displaystyle \frac{{\chi }_n}{\delta }}\right]}^{r_n}\right]}^{\beta }\hfill \\ & \le & {\left({\displaystyle \frac{M}{\delta }}\right)}^{\mathcal{H}}·max\left\{{\left[{\mathfrak{F}}_n^{\prime }\left(1\right)\right]}^{\mathcal{H}}\right\}·{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n\in N_2}{\left[{\left[{\chi }_n\right]}^{r_n}\right]}^{\beta }.\hfill \end{array}$$

Therefore, from the above result, it follows that

$${\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n^{\prime }\left({\chi }_n\right)\right]}^{r_n}\right]}^{\beta }\le \left[1+{\left({\displaystyle \frac{M}{\delta }}\right)}^{\mathcal{H}}\right]·max\left\{{\left[{\mathfrak{F}}_n^{\prime }\left(1\right)\right]}^{\mathcal{H}}\right\}·{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\chi }_n\right]}^{r_n}\right]}^{\beta }.$$

Since ${\displaystyle \mathcal{I}-\underset{k\to \infty }{lim}\frac{1}{k^{\alpha }}\sum _{n=1}^k{\left[{\left[{\chi }_n\right]}^{r_n}\right]}^{\beta }=0,}$ we have

$${\displaystyle \mathcal{I}-\underset{k\to \infty }{lim}\frac{1}{k^{\alpha }}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n^{\prime }\left({\mathfrak{F}}_n\left(\frac{{∆}_v^u{\xi }_n}{\rho }\right)\right)\right]}^{r_n}\right]}^{\beta }=\mathcal{I}-\underset{k\to \infty }{lim}\frac{1}{k^{\alpha }}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n^{\prime }\left({\chi }_n\right)\right]}^{r_n}\right]}^{\beta }=0.}$$

Therefore, $\xi =\left({\xi }_n\right)\in w_0^{\mathcal{I}}{({\mathcal{M}}^{\prime }\circ \mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ and hence $w_0^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }\subseteq w_0^{\mathcal{I}}{({\mathcal{M}}^{\prime }\circ \mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }.$

(2) Let $\xi =\left({\xi }_n\right)\in w_0^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }\cap w_0^{\mathcal{I}}{({\mathcal{M}}^{\prime },{∆}_v^u,r)}_{\alpha }^{\beta }.$ There exist two numbers ${\rho }_1>0$ and ${\rho }_2>0$ such that

$$\mathcal{I}-\underset{k\to \infty }{lim}{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{{\rho }_1}}\right)\right]}^{r_n}\right]}^{\beta }=0$$

and

$$\mathcal{I}-\underset{k\to \infty }{lim}{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n^{\prime }\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{{\rho }_2}}\right)\right]}^{r_n}\right]}^{\beta }=0.$$

Let $\rho =max\{{\rho }_1,{\rho }_2\}$ and ${\mathfrak{F}}_n^{\prime \prime }={\mathfrak{F}}_n+{\mathfrak{F}}_n^{\prime }$. By applying Maddox’s inequality, we get

$${\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n^{\prime \prime }\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{\rho }}\right)\right]}^{r_n}\right]}^{\beta }\le D\left\{{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{{\rho }_1}}\right)\right]}^{r_n}\right]}^{\beta }+{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n^{\prime }\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{{\rho }_2}}\right)\right]}^{r_n}\right]}^{\beta }\right\}.$$

Therefore, from the above inequality, we get

$$\mathcal{I}-\underset{k\to \infty }{lim}{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[\left({\mathfrak{F}}_n+{\mathfrak{F}}_n^{\prime }\right)\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{\rho }}\right)\right]}^{r_n}\right]}^{\beta }=0.$$

Hence, $\xi =\left({\xi }_n\right)\in w_0^{\mathcal{I}}{(\mathcal{M}+{\mathcal{M}}^{\prime },{∆}_v^u,r)}_{\alpha }^{\beta }.$ Therefore,

$$w_0^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }\cap w_0^{\mathcal{I}}{({\mathcal{M}}^{\prime },{∆}_v^u,r)}_{\alpha }^{\beta }\subseteq w_0^{\mathcal{I}}{(\mathcal{M}+{\mathcal{M}}^{\prime },{∆}_v^u,r)}_{\alpha }^{\beta }.$$

Theorem 6 is proved. □

Theorem 7.

If $u,v\ge 1,$ then the following inclusion holds:

1. 

$w^{\mathcal{I}}{(\mathcal{M},{∆}_v^{u-1},r)}_{\alpha }^{\beta }\subseteq w^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$,

2. 

$w_0^{\mathcal{I}}{(\mathcal{M},{∆}_v^{u-1},r)}_{\alpha }^{\beta }\subseteq w_0^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$,

3. 

${\eta }^{\mathcal{I}}{(\mathcal{M},{∆}_v^{u-1},r)}_{\alpha }^{\beta }\subseteq {\eta }^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$,

4. 

${\eta }_0^{\mathcal{I}}{(\mathcal{M},{∆}_v^{u-1},r)}_{\alpha }^{\beta }\subseteq {\eta }_0^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$,

5. 

$w_{\infty }^{\mathcal{I}}{(\mathcal{M},{∆}_v^{u-1},r)}_{\alpha }^{\beta }\subseteq w_{\infty }^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$.

Proof. 

We have to prove $\left(1\right)$ only. The proof of remaining parts directly follows from $\left(1\right)$. Assume that $\xi =\left({\xi }_n\right)\in w^{\mathcal{I}}{(\mathcal{M},{∆}_v^{u-1},r)}_{\alpha }^{\beta }$. For $\epsilon >0$ to be given, we have to show that there exists $\rho >0$ and $L\in X$ such that

$$A=\left\{k\in \mathbb{N}:{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n-L}{\rho }}\right)\right]}^{r_n}\right]}^{\beta }\ge \epsilon \right\}\in \mathcal{I}.$$

Since $\xi =\left({\xi }_n\right)\in w^{\mathcal{I}}{(\mathcal{M},{∆}_v^{u-1},r)}_{\alpha }^{\beta }$, there exists ${\rho }_1>0$ and $L_1\in X$ such that

$$B=\left\{k\in \mathbb{N}:{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^{u-1}{\xi }_n-L_1}{\rho }}\right)\right]}^{r_n}\right]}^{\beta }\ge {\displaystyle \frac{\epsilon }{2D}}\right\}\in \mathcal{I},$$

where $D=max\{1,2^{\mathcal{H}-1}\}$ and ${\displaystyle \mathcal{H}=\underset{n}{sup}{r}_n\ge r_n>0.}$ Suppose $\rho =2{\rho }_1$ and $L=2L_1$. Since ${\mathfrak{F}}_n$ is non-decreasing and convex for each $n\in \mathbb{N}$. Put ${\chi }_n={\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n-L}{\rho }}\right)$, we have

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\chi }_n\right]}^{r_n}\right]}^{\beta }& \le & {\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left(\left({\displaystyle \frac{{∆}_v^{u-1}{\xi }_n-L_1}{2{\rho }_1}}\right)+\left({\displaystyle \frac{{∆}_v^{u-1}{\xi }_{n+v}-L_1}{2{\rho }_1}}\right)\right)\right]}^{r_n}\right]}^{\beta }\hfill \\ & \le & {\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\displaystyle \frac{1}{2}}{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^{u-1}{\xi }_n-L_1}{{\rho }_1}}\right)+{\displaystyle \frac{1}{2}}{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^{u-1}{\xi }_{n+v}-L_1}{{\rho }_1}}\right)\right]}^{r_n}\right]}^{\beta }\hfill \\ & \le & {\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^{u-1}{\xi }_n-L_1}{{\rho }_1}}\right)+{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^{u-1}{\xi }_{n+v}-L_1}{{\rho }_1}}\right)\right]}^{r_n}\right]}^{\beta }.\hfill \end{array}$$

Let $\rho ={\rho }_1+{\rho }_2.$ Then,

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\chi }_n\right]}^{r_n}\right]}^{\beta }& \le & D\{{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^{u-1}{\xi }_n-L_1}{{\rho }_1}}\right)\right]}^{r_n}\right]}^{\beta }\hfill \\ & & +{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^{u-1}{\xi }_{n+v}-L_1}{{\rho }_1}}\right)\right]}^{r_n}\right]}^{\beta }\}.\hfill \end{array}$$

If $k^{\alpha }\in B^c,$ then from above inequality, we have

$${\displaystyle \frac{1}{k^{\alpha }}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left(\frac{{∆}_v^u{\xi }_n-L}{\rho }\right)\right]}^{r_n}\right]}^{\beta }\le D\left\{\frac{\epsilon }{2D}+\frac{\epsilon }{2D}\right\}=\epsilon ,}$$

this implies that $k^{\alpha }\in A^c$. Therefore, $A\subset B\in \mathcal{I}$ and hence $A\in \mathcal{I}$. Therefore, $\xi =\left({\xi }_n\right)\in w^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ and so $w^{\mathcal{I}}{(\mathcal{M},{∆}_v^{u-1},r)}_{\alpha }^{\beta }\subseteq w^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$. □

Theorem 8.

The sequence spaces $w_{\infty }^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ and $w_0^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ are solid and hence monotone.

Proof. 

Let $\xi =\left({\xi }_n\right)\in w_{\infty }^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ and let $\left(a_n\right)$ be a sequence of scalars with $|a_n|\le 1$ for each $n\in \mathbb{N}$. There exist two positive numbers $\rho$ and Q such that

$$A=\left\{k\in \mathbb{N}:{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{\rho }}\right)\right]}^{r_n}\right]}^{\beta }\ge Q\right\}\in \mathcal{I}.$$

Let

$$B=\left\{k\in \mathbb{N}:{\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{a}_n{\xi }_n}{\rho }}\right)\right]}^{r_n}\right]}^{\beta }\ge Q\right\}.$$

If $k^{\alpha }\notin A$, then

$$\begin{array}{ccc}\hfill {\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{a}_n{\xi }_n}{\rho }}\right)\right]}^{r_n}\right]}^{\beta }& \le & {\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[|a_n{|}^{r_n}{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{\rho }}\right)\right]}^{r_n}\right]}^{\beta }\hfill \\ & \le & {\displaystyle \frac{1}{k^{\alpha }}}\sum _{n=1}^k{\left[{\left[{\mathfrak{F}}_n\left({\displaystyle \frac{{∆}_v^u{\xi }_n}{\rho }}\right)\right]}^{r_n}\right]}^{\beta }<Q,\hfill \end{array}$$

which implies that $k^{\alpha }\in B$. Thus, $B\subset A\in \mathcal{I}$ and hence, $B\in \mathcal{I}.$ This shows that $\left(a_n{\xi }_n\right)\in w_{\infty }^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ for all sequence of scalars $\left(a_n\right)$ with $|a_n|\le 1$ for each $n\in \mathbb{N}$, whenever $\left({\xi }_n\right)\in w_{\infty }^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$. Therefore, $w_{\infty }^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$ is solid. Similarly, we provided a proof for $w_0^{\mathcal{I}}{(\mathcal{M},{∆}_v^u,r)}_{\alpha }^{\beta }$. □

4. Conclusions

In this paper, we introduce novel difference sequence spaces utilizing the Musielak–Orlicz function and the concept of $\mathcal{I}$-convergence of sequences of order $(\alpha ,\beta )$. Furthermore, we analyze several topological properties of the resulting spaces, examining their linearity, paranormed structure, and solid and standard inclusion relations. This methodology presents a significant advancement in the study of sequence spaces, offering a robust framework that can be utilized to extend the analysis of existing operators and establish deeper connections with other metric and topological spaces. The introduction of Musielak–Orlicz functions within the context of $\mathcal{I}$-convergence opens up new avenues for research, particularly in the investigation of sequences that exhibit complex convergence behaviors. Additionally, the interplay between ideal convergence and modular functionals presents a rich area for further investigation, potentially leading to significant advancements in understanding the structure and properties of various functional spaces. Also, one can examine the dual spaces and functional operators associated with Musielak–Orlicz sequence spaces. Furthermore, explore the connections between ideal convergence with Musielak–Orlicz functions and other generalized convergence concepts, such as statistically convergent sequences or convergence in Banach spaces.