Relative Uniform Convergence of Quantum Difference Sequence of Functions Related to ℓp-Space Defined by Orlicz Function

Debbarma, Diksha; Tripathy, Binod Chandra

doi:10.2478/amsil-2024-0026

Full Article

1.

Introduction

A sequence (f_n(x)) of functions, defined on a compact domain D, converges relatively uniformly to a limit function f(x) if there exists a function μ(x), called a scale function, such that for every small positive number ɛ there is an integer n_ɛ such that for every n ≥ n_ɛ the inequality $| f (x) - f_{n} (x) | \leq ε | μ (x) |,$ \left| {f(x) - {f_n}(x)} \right| \le \varepsilon \left| {\mu (x)} \right|, holds uniformly in x on the interval D.

The above definition of a relatively uniform convergence of sequence of functions was formulated by Chittenden [3] using the notion which was first used by Moore [13]. Many additional scholars, including Demirci et al. ([4], [5]), Demirci and Orhan [6], Sahin and Dirik [21], Devi and Tripathy ([7], [8], [9]), as well as others, looked deeper into the idea. The term “calculus without limits” is used to refer to quantum calculus, also referred to as q-calculus. The fundamental q-calculus formulae were discovered by Euler in the seventeenth century. However, Jackson [11] may have been the first to introduce the notion of the definite q-difference operator which is defined by $\begin{matrix} D_{q} f (x) = \frac{f (q x) - f (x)}{x (q - 1)}, & x \neq 0, \end{matrix}$ \matrix{{{D_q}f(x) = {{f(qx) - f(x)} \over {x(q - 1)}},} & {x \ne 0,} \cr} and D_qf(0) = f^′(0), where q is a fixed number, q ∈ (0, 1). The function f is defined on a q-geometric set A ⊆ ℝ (or ℂ) that is qx ∈ A whenever x ∈ A. Due to its use in a variety of mathematical fields, including orthogonal polynomials, fundamental hypergeometric functions, combinatorics, the calculus of variations, and the theory of relativity, quantum difference operators play an intriguing role in mathematics.

The term “Orlicz function” refers to a continuous, non-decreasing, convex function that has the following characteristics: 𝒪(0) = 0, 𝒪(x) > 0, for x > 0 and 𝒪(x) → ∞, as x → ∞.

If there is a constant K > 0 such that 𝒪(2x) ≤ K𝒪(x), for all values of x ≥ 0, then an Orlicz function 𝒪 is considered to satisfy the δ₂-condition for all values x.

Lindenstrauss and Tzafriri [12] constructed the sequence space $ℓ_{O} = {x \in ω : \sum_{n = 1}^{\infty} O (\frac{| x_{n} |}{ν}) < \infty, for some ν > 0} .$ {\ell_{\cal O}} = \left\{{x \in \omega :\sum\limits_{n = 1}^\infty {{\cal O}\left({{{\left| {{x_n}} \right|} \over \nu}} \right) < \infty,\,\,\,{\rm{for}}\,{\rm{some}}\,\nu > 0}} \right\}. The norm using the idea of Orlicz function is as follows, $‖ x ‖ = inf {x \in ω : \sum_{n = 1}^{\infty} (O (\frac{| x_{n} |}{ν})) \leq 1} .$ \left\| x \right\| = \inf \left\{ {x \in \omega :\sum\limits_{n = 1}^\infty {\left({{\cal O}\left({{{\left| {{x_n}} \right|} \over \nu }} \right)} \right) \le 1} } \right\}. Numerous authors, such as Bhardwaj and Singh [1], Bilgin [2], Gung et al. [10], Tripathy and Mahanta [23], Parashar and Choudhary [19] are worked on Orlicz space. These motivated others to study different types of new sequence spaces defined by the Orlicz function.

In 1960, Sargent [22] proposed the m(ϕ) space, which is closely connected to the ℓ_p space. He examined a few m(ϕ) space characteristics. Afterward, it was examined from the sequence space point of view, by Rath and Tripathy [20], Tripathy [24], and Tripathy and Sen [25] and others.

We establish some geometric properties on the convexity of the space _rum_ϕ(𝒪, ∇_q, p). Banach spaces, which are complete, normed, linear, metric spaces with an additional characteristic of convexity of the norm, are the spaces which will be discussed in the present research. Japanese mathematician Nakano [18] introduced the concept of modular spaces. Also, many other researchers worked on modular space such as Musielak and Orlicz ([15], [16]), Musielak [14], Musielak and Wasak [17], Yildiz [26] etc.

Throughout the article ω_f, ξ_s, P (f_n) represents the space of sequences of functions, the subset of natural numbers of cardinality not greater than s, the permutation of the sequence of functions (f_n), respectively.

2.

Definitions and background

In this article, we shall use the following known sequence spaces defined by Orlicz functions, for 0 < p < 1, $\begin{array}{l} _{ru} ℓ_{p} (O, \nabla_{q}) = {(f_{n}) \in ω_{f} : \sum_{n = 1}^{\infty} {(O (\frac{| \nabla_{q} f_{n} (x) |}{ν}))}^{p} < ε | μ (x) |, for some ν > 0}, \\ _{ru} ℓ_{\infty} (O, \nabla_{q}) = {(f_{n}) \in ω_{f} : sup_{n \geq 1} O (\frac{| \nabla_{q} f_{n} (x) |}{ν}) < ε | μ (x) |, for some ν > 0}, \\ _{ru} c_{0} (O, \nabla_{q}) = {(f_{n}) \in ω_{f} : lim_{n \to \infty} O (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}) = 0, for some ν > 0}, \\ _{ru} c (O, \nabla_{q}) = {(f_{n}) \in ω_{f} : lim_{n \to \infty} O (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}) = L, for some ν > 0} . \end{array}$ \matrix{{_{ru}{\ell_p}({\cal O},{\nabla_q}) = \left\{{({f_n}) \in {\omega_f}:\sum\limits_{n = 1}^\infty {{{\left({{\cal O}\left({{{|{\nabla_q}{f_n}(x)|} \over \nu}} \right)} \right)}^p} < \varepsilon |\mu (x)|,\,\,\,{\rm{for}}\,{\rm{some}}\,\nu > 0}} \right\},} \hfill \cr {_{ru}{\ell_\infty}({\cal O},{\nabla_q}) = \left\{{({f_n}) \in {\omega_f}:\mathop {\sup}\limits_{n \ge 1} {\cal O}\left({{{|{\nabla_q}{f_n}(x)|} \over \nu}} \right) < \varepsilon |\mu (x)|,\,\,\,{\rm{for}}\,{\rm{some}}\,\nu > 0} \right\},} \hfill \cr {_{ru}{c_0}({\cal O},{\nabla_q}) = \left\{{({f_n}) \in {\omega_f}:\mathop {\lim}\limits_{n \to \infty} {\cal O}\left({{{|{\nabla_q}{f_n}(x)|} \over {\nu |\mu (x)|}}} \right) = 0,\,\,\,{\rm{for}}\,{\rm{some}}\,\nu > 0} \right\},} \hfill \cr {_{ru}c({\cal O},{\nabla_q}) = \left\{{({f_n}) \in {\omega_f}:\mathop {\lim}\limits_{n \to \infty} {\cal O}\left({{{|{\nabla_q}{f_n}(x)|} \over {\nu |\mu (x)|}}} \right) = L,\,\,\,{\rm{for}}\,{\rm{some}}\,\nu > 0} \right\}.} \hfill \cr}

In this article, we introduce the following spaces, for 0 < p < 1, $\begin{array}{l} _{ru} m_{ϕ} (O, \nabla_{q}, p) = {(f_{n}) \in ω_{f} : sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} O {(\frac{| \nabla_{q} f_{n} (x) |}{ν})}^{p} < ε | μ (x) |, for some ν > 0}, \\ _{ru} n_{ϕ} (O, \nabla_{q}, p) = {(f_{n}) \in ω_{f} : sup_{u_{n} \in P (f_{n})} \sum_{n = 1}^{\infty} {(O (\frac{| \nabla_{q} u_{n} (x) | Δ ϕ_{n}}{ν}))}^{p} < ε | μ (x) |, for some ν > 0}, \\ _{ru} m_{ϕ} (O, \nabla_{q}) = {(f_{n}) \in ω_{f} : sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} O (\frac{| \nabla_{q} f_{n} (x) |}{ν}) < ε | μ (x) |, for some ν > 0}, \\ _{ru} n_{ϕ} (O, \nabla_{q}) = {(f_{n}) \in ω_{f} : sup_{u_{n} \in P (f_{n})} \sum_{n = 1}^{\infty} O (\frac{| \nabla_{q} u_{n} (x) | Δ ϕ_{n}}{ν}) < ε | μ (x) |, for some ν > 0} . \end{array}$ \matrix{{_{ru}{m_\phi}({\cal O},{\nabla_q},p) = \left\{{({f_n}) \in {\omega_f}:\mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{\cal O}{{\left({{{|{\nabla_q}{f_n}(x)|} \over \nu}} \right)}^p} < \varepsilon |\mu (x)|,\,\,\,{\rm{for}}\,{\rm{some}}\,\nu > 0}} \right\},} \hfill \cr {_{ru}{n_\phi}({\cal O},{\nabla_q},p) = \left\{{({f_n}) \in {\omega_f}:\mathop {\sup}\limits_{{u_n} \in P({f_n})} \sum\limits_{n = 1}^\infty {{{\left({{\cal O}\left({{{|{\nabla_q}{u_n}(x)|\Delta {\phi_n}} \over \nu}} \right)} \right)}^p} < \varepsilon |\mu (x)|,\,\,\,{\rm{for}}\,{\rm{some}}\,\nu > 0}} \right\},} \hfill \cr {_{ru}{m_\phi}({\cal O},{\nabla_q}) = \left\{{({f_n}) \in {\omega_f}:\mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{\cal O}\left({{{|{\nabla_q}{f_n}(x)|} \over \nu}} \right) < \varepsilon |\mu (x)|,\,\,\,{\rm{for}}\,{\rm{some}}\,\nu > 0}} \right\},} \hfill \cr {_{ru}{n_\phi}({\cal O},{\nabla_q}) = \left\{{({f_n}) \in {\omega_f}:\mathop {\sup}\limits_{{u_n} \in P({f_n})} \sum\limits_{n = 1}^\infty {{\cal O}\left({{{|{\nabla_q}{u_n}(x)|\Delta {\phi_n}} \over \nu}} \right) < \varepsilon |\mu (x)|,\,\,\,{\rm{for}}\,{\rm{some}}\,\nu > 0}} \right\}.} \hfill \cr}

Definition 2.1

A subset Z_f ⊂ ω_f is said to be convergence free, if (f_n) ∈ Z_f implies (g_n) ∈ Z_f for any sequence (g_n) such that g_n(x) = 0 whenever f_n(x) = 0 for x ∈ D.

Definition 2.2

A subset Z_f ⊂ ω_f is said to be symmetric if (f_n) ∈ Z_f implies (f_π_(n)) ∈ Z_f, where π is a permutation of ℕ.

Definition 2.3

A subset Z_f ⊂ ω_f is said to be solid or normal, if (f_n) ∈ Z_f implies (g_n) ∈ Z_f for all sequence (g_n) such that |g_n(x)| ≤ |f_n(x)| for every n ∈ ℕ and for all x ∈ D.

Definition 2.4

Let Z_f ⊂ ω_f be a space of sequences of functions. Then Z_f is said to be sequence algebra if there is defined a product ∗ on Z_f such that $(f_{n}), (g_{n}) \in Z_{f} \Rightarrow (f_{n}) * (g_{n}) \in Z_{f} .$ ({f_n}),({g_n}) \in {Z_f} \Rightarrow ({f_n})*({g_n}) \in {Z_f}.

Lemma 2.5

(Tripathy and Sen [25, Theorem 7]). $ℓ_{p} \subseteq m (ϕ, p) \subseteq ℓ_{\infty} .$ {\ell_p} \subseteq m(\phi,p) \subseteq {\ell_\infty}.

In this section, we also give some known definitions of geometric terms related to normed spaces.

Definition 2.6

Let Z_f be a subset of a linear space. Then

(1)
Z_f is called convex if and only if (f_n), (g_n) ∈ Z_f, λ₁ + λ₂ = 1, λ₁ ≥ 0, λ₂ ≥ 0 implies (λ₁f_n + λ₂g_n) ∈ Z_f;
(2)
Z_f is called balanced if and only if (f_n) ∈ Z_f, |λ| ≤ 1 ⇒ (λf_n) ∈ Z_f ;
(3)
Z_f is a called absolutely convex if and only if (f_n), (g_n) ∈ Z_f, |λ₁|+|λ₂| ≤ 1 implies (λ₁f_n + λ₂g_n) ∈ Z_f.

Nakano [18] considered the definition of modular space as follows,

Definition 2.7

Let X be a linear space. A function ζ : X → [0, ∞) is called modular function if

(1)
ζ(x) = 0 if and only if x = θ,
(2)
ζ(αx) = ζ(x) for all scalars α with |α| = 1,
(3)
ζ(αx + βy) < ζ(x) + ζ(y), for all x, y ∈ X and α, β ≥ 0, |α| = |β| = 1. Further, the modular ζ is called convex if
(4)
ζ(αx + βy) ≤ αζ(x) + βζ(y) holds for all x, y ∈ X and all α, β ≥ 0 with α + β = 1.

For any modular ζ on X the space $X_{ζ} = {x \in X : lim_{λ \to 0^{+}} ζ (λ x) = 0}$ {X_\zeta} = \left\{{x \in X:\mathop {\lim}\limits_{\lambda \to {0^ +}} \zeta (\lambda x) = 0} \right\} is called the modular space. A sequence (x_n) of elements of X_ζ is called modular convergent to x ∈ X_ζ if there exists a λ > 0 such that ζλ(x_n − x) → 0, as n → ∞. If ζ is convex modular, then we have the following formula, ${‖ x ‖}_{L} = inf {λ > 0 : ζ (\frac{x}{λ}) \leq 1} .$ {\left\| x \right\|_L} = \inf \left\{{\lambda > 0:\zeta \left({{x \over \lambda}} \right) \le 1} \right\}.

Definition 2.8

A Banach space is said to be uniformly convex if, to each ɛ > 0, 0 < ɛ ≤ 2, there corresponds a ζ(ɛ) > 0 such that the following condition holds: $\begin{matrix} ‖ x ‖ = ‖ y ‖ = 1, & ‖ x - y ‖ \geq ε & \Rightarrow & \frac{1}{2} ‖ x + y ‖ \leq 1 - ζ (ε) . \end{matrix}$ \matrix{{\left\| x \right\| = \left\| y \right\| = 1,} & {\left\| {x - y} \right\| \ge \varepsilon} & \Rightarrow & {{1 \over 2}\left\| {x + y} \right\| \le 1 - \zeta (\varepsilon).} \cr}

3.

Main results

Theorem 3.1

The space _rum_ϕ(𝒪, ∇_q, p) is a linear space, for p > 0.

Proof

Let (f_n), (g_n) ∈ _rum_ϕ(𝒪, ∇_q, p) and α, β be two scalars. Then there exist positive numbers ν₁ and ν₂ such that $sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {\sum_{n \in σ} (O (\frac{| \nabla_{q} f_{n} (x) |}{ν_{1}}))}^{p} < ε_{1} | μ_{1} (x) |,$ \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{\sum\limits_{n \in \sigma} {\left({{\cal O}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {{\nu_1}}}} \right)} \right)}^p} < {\varepsilon_1}\left| {{\mu_1}(x)} \right|, and $sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {\sum_{n \in σ} (O (\frac{| \nabla_{q} g_{n} (x) |}{ν_{2}}))}^{p} < ε_{2} | μ_{2} (x) | .$ \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{\sum\limits_{n \in \sigma} {\left({{\cal O}\left({{{\left| {{\nabla_q}{g_n}(x)} \right|} \over {{\nu_2}}}} \right)} \right)}^p} < {\varepsilon_2}\left| {{\mu_2}(x)} \right|. Let ν₃ = max{|α|ν₁, |β|ν₂}. Since 𝒪 is a non-decreasing convex function, $\begin{array}{l} sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {\sum_{n \in σ} (O (\frac{| α \nabla_{q} f_{n} (x) + β \nabla_{q} g_{n} (x) |}{ν_{3}}))}^{p} \\ = sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {\sum_{n \in σ} (O (\frac{| α \nabla_{q} f_{n} (x) |}{ν_{3}} + \frac{| β \nabla_{q} g_{n} (x) |}{ν_{3}}))}^{p} \\ \leq sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {\sum_{n \in σ} (O (\frac{| α \nabla_{q} f_{n} (x) |}{ν_{3}}))}^{p} \\ + sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {\sum_{n \in σ} (O (\frac{| β \nabla_{q} g_{n} (x) |}{ν_{3}}))}^{p} \\ \leq sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {\sum_{n \in σ} (O (\frac{| \nabla_{q} f_{n} (x) |}{ν_{1}}))}^{p} \\ + sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {\sum_{n \in σ} (O (\frac{| \nabla_{q} g_{n} (x) |}{ν_{2}}))}^{p} \\ < max {ε_{1}, ε_{2}} max {| μ_{1} (x) |, | μ_{2} (x) |} < \infty . \end{array}$ \matrix{{\mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{{\sum\limits_{n \in \sigma} {\left({{\cal O}\left({{{\left| {\alpha {\nabla_q}{f_n}(x) + \beta {\nabla_q}{g_n}(x)} \right|} \over {{\nu_3}}}} \right)} \right)}}^p}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{{\sum\limits_{n \in \sigma} {\left({{\cal O}\left({{{\left| {\alpha {\nabla_q}{f_n}(x)} \right|} \over {{\nu_3}}} + {{\left| {\beta {\nabla_q}{g_n}(x)} \right|} \over {{\nu_3}}}} \right)} \right)}}^p}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \le \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{{\sum\limits_{n \in \sigma} {\left({{\cal O}\left({{{\left| {\alpha {\nabla_q}{f_n}(x)} \right|} \over {{\nu_3}}}} \right)} \right)}}^p}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{{\sum\limits_{n \in \sigma} {\left({{\cal O}\left({{{\left| {\beta {\nabla_q}{g_n}(x)} \right|} \over {{\nu_3}}}} \right)} \right)}}^p}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \le \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{{\sum\limits_{n \in \sigma} {\left({{\cal O}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {{\nu_1}}}} \right)} \right)}}^p}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{{\sum\limits_{n \in \sigma} {\left({{\cal O}\left({{{\left| {{\nabla_q}{g_n}(x)} \right|} \over {{\nu_2}}}} \right)} \right)}}^p}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, < \max \{{\varepsilon_1},{\varepsilon_2}\} \max \{|{\mu_1}(x)|,|{\mu_2}(x)|\} < \infty.} \hfill \cr} Thus, (α∇_qf_n + β∇_qg_n) ∈ _rum_ϕ(𝒪, ∇_q, p). Hence, _rum_ϕ(𝒪, ∇_q, p) is a linear space.

The following result is a consequence of the above Theorem 3.1

Corollary 3.2

The classes of sequences of functions _run_ϕ(𝒪, ∇_q, p), _rum_ϕ(𝒪, ∇_q), _run_ϕ(𝒪, ∇_q) are linear spaces.

Theorem 3.3

The space _rum_ϕ(𝒪, ∇_q, p) is a normed space, with the norm (3.1) ${‖ f ‖}_{_{_{ru}} m_{ϕ} (O, \nabla_{q}, p)} = inf {ν > 0 : \sum_{n = 1}^{\infty} O (\frac{| f_{n} (x) |}{ν | μ (x) |}) + sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {(\sum_{n \in σ} {(O (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}))}^{p})}^{\frac{1}{p}} \leq 1},$ {\left\| f \right\|_{_{_{ru}}{m_\phi}({\cal O},{\nabla_q},p)}} = \inf \left\{{\nu > 0:\sum\limits_{n = 1}^\infty {{\cal O}\left({{{\left| {{f_n}\left(x \right)} \right|} \over {\nu \left| {\mu \left(x \right)} \right|}}} \right)} + \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{{\left({\sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{|{\nabla_q}{f_n}(x)|} \over {\nu |\mu (x)|}}} \right)} \right)}^p}}} \right)}^{{1 \over p}}} \le 1} \right\}, for 1 ≤ p < ⧜.

The space _rum_ϕ(𝒪, ∇_q, p) is a p-normed space, with the norm (3.2) ${‖ f ‖}_{_{_{ru}} m_{ϕ} (O, \nabla_{q}, p)} = inf {ν > 0 : \sum_{n = 1}^{\infty} {(O (\frac{| f_{n} (x) |}{ν | μ (x) |}))}^{p} + sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} {(O (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}))}^{p} \leq 1},$ {\left\| f \right\|_{_{_{ru}}{m_\phi}({\cal O},{\nabla_q},p)}} = \inf \left\{{\nu > 0:\sum\limits_{n = 1}^\infty {{{\left({{\cal O}\left({{{\left| {{f_n}\left(x \right)} \right|} \over {\nu \left| {\mu \left(x \right)} \right|}}} \right)} \right)}^p}} + \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{|{\nabla_q}{f_n}(x)|} \over {\nu |\mu (x)|}}} \right)} \right)}^p}} \le 1} \right\}, for 0 < p < 1.

Proof

We check the conditions of norm:

(1)
Clearly, $\begin{matrix} {‖ f ‖}_{_{ru} m_{ϕ} (O, \nabla_{q}, p)} \geq 0, & for all (\nabla_{q} f_{n}) \in_{ru} m_{ϕ} (O, \nabla_{q}, p) . \end{matrix}$ \matrix{{{{\left\| f \right\|}_{_{ru}{m_\phi}({\cal O},{\nabla_q},p)}} \ge 0,} & {{\rm{for}}\,{\rm{all}}\,({\nabla_q}{f_n}) \in {\,_{ru}}{m_\phi}({\cal O},{\nabla_q},p).} \cr}
(2)
Clearly, ∥f∥_{_rum_ϕ_(𝒪,∇q,p)} = 0, if and only if $f_{n} (x) = \bar{θ}$ {f_n}(x) = \overline \theta , the null operator, for all n ∈ ℕ and x ∈ D.
(3)
We have, for f = (f_n) ∈ _rum_ϕ(𝒪, ∇_q, p) and any scalar λ, ${‖ λ \nabla_{q} f ‖}_{_{_{ru}} m_{ϕ} (O, \nabla_{q}, p)} = sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} {(O (\frac{| λ \nabla_{q} f_{n} (x) |}{ν | μ (x) |}))}^{p} = {| λ |}^{p} {‖ \nabla_{q} f ‖}_{_{_{ru}} m_{ϕ} (O, \nabla_{q}, p)} .$ {\left\| {\lambda {\nabla_q}f} \right\|_{_{_{ru}}{m_\phi}({\cal O},{\nabla_q},p)}} = \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{|\lambda {\nabla_q}{f_n}(x)|} \over {\nu |\mu (x)|}}} \right)} \right)}^p} = {{\left| \lambda \right|}^p}{{\left\| {{\nabla_q}f} \right\|}_{_{_{ru}}{m_\phi}({\cal O},{\nabla_q},p)}}}.
(4)
Let (f_n), (g_n) ∈ _rum_ϕ(𝒪, ∇_q, p). Then, $\begin{array}{l} sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} {(O (\frac{| \nabla_{q} (f_{n} (x) + g_{n} (x)) |}{ν | μ (x) |}))}^{p} \\ \leq sup_{s \geq 1, σ \in ξ_{s}, x \in D} \sum_{n \in σ} {(O (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}))}^{p} + sup_{s \geq 1, σ \in ξ_{s}, x \in D} \sum_{n \in σ} {(O (\frac{| \nabla_{q} g_{n} (x) |}{ν | μ (x) |}))}^{p} . \end{array}$ \matrix{{\mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{\left| {{\nabla_q}({f_n}(x) + {g_n}(x))} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \hfill \cr {\le \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} \sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}^p}} + \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} \sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{\left| {{\nabla_q}{g_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}^p}}.} \hfill \cr}

Therefore,

{‖ f + g ‖}_{_{ru} m_{^{ϕ}} (O, \nabla_{q}, p)} \leq {‖ f ‖}_{_{ru} m_{ϕ} (O, \nabla_{q}, p)} + {‖ g ‖}_{_{ru} m_{ϕ} (O, \nabla_{q}, p)} .

{\left\| {f + g} \right\|_{_{ru}{m_{^\phi}}({\cal O},{\nabla_q},p)}}\, \le {\left\| f \right\|_{_{ru}{m_\phi}({\cal O},{\nabla_q},p)}} + {\left\| g \right\|_{_{ru}{m_\phi}({\cal O},{\nabla_q},p)}}.

Thus, the space _rum_ϕ(𝒪, ∇_q, p) is a p-normed space with the norm (3.2). Similarly, for 1 ≤ p < ∞, the space _rum_ϕ(𝒪, ∇_q, p) is a norm space with the norm (3.1).

Theorem 3.4

If (Z, ru) is complete then the space _rum_ϕ(𝒪, ∇_q, p) is complete with the norm (3.1).

Proof

Let (f_n) ∈ _rum_ϕ(𝒪, ∇_q, p) be a Cauchy sequence of functions, where $f_{n} = (f_{n}^{i}) = ((f_{1}^{i}), (f_{2}^{i}), \dots) \in_{ru} m_{ϕ} (O, \nabla_{q}, p)$ {f_n} = (f_n^i) = ((f_1^i),(f_2^i), \ldots) \in {\,_{ru}}{m_\phi}({\cal O},{\nabla_q},p) , for each i ∈ ℕ. Let ρ > 0 and f₀ > 0 be fixed. Then, for each $\frac{ε}{ρ f_{0}} > 0$ {\varepsilon \over {\rho {f_0}}} > 0 there exists a positive integer n₀(ɛ) such that (3.3) $\begin{array}{l} \begin{array}{l} h (f^{i} - f^{j}) < \frac{ε}{ρ f_{0}}, & for all \end{array} i, j \geq n_{0}, \\ implies inf {ν > 0 : \sum_{n = 1}^{\infty} O (\frac{| f_{n}^{i} (x) - f_{n}^{j} (x) |}{ν | μ (x) |}) + sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {({\sum_{n = σ} (O (\frac{| \nabla_{q} (f_{n}^{i} (x) - f_{n}^{j} (x)) |}{ν | μ (x) |}))}^{p})}^{\frac{1}{p}} \leq 1} < \frac{ε}{ρ f_{0}}, \end{array}$ \matrix{{\matrix{{h({f^i} - {f^j}) < {\varepsilon \over {\rho {f_0}}},} \hfill & {{\rm{for}}\,{\rm{all}}} \hfill \cr} \,i,j \ge {n_0},} \hfill \cr {{\rm{implies}}\,\,\,\,\,\inf \left\{{\nu > 0:\sum\limits_{n = 1}^\infty {{\cal O}\left({{{\left| {f_n^i(x) - f_n^j(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} + \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{{\left({{{\sum\limits_{n = \sigma}^{} {\left({{\cal O}\left({{{\left| {{\nabla_q}(f_n^i(x) - f_n^j(x))} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}}^p}} \right)}^{{1 \over p}}} \le 1} \right\} < {\varepsilon \over {\rho {f_0}}},} \hfill \cr} for all i, j ≥ n₀. Then we have for all i, j ≥ n₀, $\sum_{n = 1}^{\infty} O (\frac{| f_{n}^{i} (x) - f_{n}^{j} (x) |}{h (f^{i} - f^{j}) | μ (x) |}) + sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {(\sum_{n = σ} {(O (\frac{| \nabla_{q} (f_{n}^{i} (x) - f_{n}^{j} (x)) |}{h (f^{i} - f^{j}) | μ (x) |}))}^{p})}^{\frac{1}{p}} \leq 1 .$ \sum\limits_{n = 1}^\infty {{\cal O}\left({{{\left| {f_n^i(x) - f_n^j(x)} \right|} \over {h({f^i} - {f^j})\left| {\mu (x)} \right|}}} \right)} + \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi _s},x \in D} {1 \over {{\phi _s}}}{\left({\sum\limits_{n = \sigma} {{{\left({{\cal O}\left({{{\left| {{\nabla _q}(f_n^i(x) - f_n^j(x))} \right|} \over {h({f^i} - {f^j})\left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \right)^{{1 \over p}}} \le 1 or, $\sum_{n = 1}^{\infty} O (\frac{| f_{n}^{i} (x) - f_{n}^{j} (x) |}{h (f^{i} - f^{j}) | μ (x) |}) \leq 1$ \sum\limits_{n = 1}^\infty {{\cal O}\left({{{\left| {f_n^i(x) - f_n^j(x)} \right|} \over {h({f^i} - {f^j})\left| {\mu (x)} \right|}}} \right) \le 1} and $sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {(\sum_{n = σ} {(O (\frac{| \nabla_{q} (f_{n}^{i} (x) - f_{n}^{j}) |}{h (f^{i} - f^{j}) | μ (x) |}))}^{p})}^{\frac{1}{p}} \leq 1 .$ \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{\left({\sum\limits_{n = \sigma} {{{\left({{\cal O}\left({{{\left| {{\nabla_q}(f_n^i(x) - f_n^j)} \right|} \over {h({f^i} - {f^j})\left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \right)^{{1 \over p}}} \le 1.

At first, we consider $\begin{matrix} O (\frac{| f_{n}^{i} (x) - f_{n}^{j} (x) |}{h (f^{i} - f^{j}) | μ (x) |}) \leq 1, & for all i, j \geq n_{0} \end{matrix} .$ \matrix{{{\cal O}\left({{{\left| {f_n^i(x) - f_n^j(x)} \right|} \over {h({f^i} - {f^j})\left| {\mu (x)} \right|}}} \right) \le 1,} & {{\rm{for}}\,{\rm{all}}\,i,j \ge {n_0}} \cr}. We can find ρ > 0 with $(\frac{ρ f_{0}}{2}) \geq max (1, ϕ_{1})$ \left({{{\rho {f_0}} \over 2}} \right) \ge \max \left({1,{\phi_1}} \right) , such that (3.4) $\begin{array}{l} \sum_{n = 1}^{\infty} O (\frac{| f_{n}^{i} (x) - f_{n}^{j} (x) |}{h (f^{i} - f^{j}) | μ (x) |}) \leq O (\frac{ρ f_{0}}{2}), \\ \frac{| f_{n}^{i} (x) - f_{n}^{j} (x) |}{| μ (x) |} < \frac{ρ f_{0}}{2} \cdot h (f^{i} - f^{j}), \\ \frac{| f_{n}^{i} (x) - f_{n}^{j} (x) |}{| μ (x) |} < \frac{ρ f_{0}}{2} \cdot \frac{ε}{ρ f_{0}} < \frac{ε}{2} . \end{array}$ \matrix{{\sum\limits_{n = 1}^\infty {{\cal O}\left({{{\left| {f_n^i(x) - f_n^j(x)} \right|} \over {h({f^i} - {f^j})\left| {\mu (x)} \right|}}} \right) \le {\cal O}\left({{{\rho {f_0}} \over 2}} \right),}} \hfill \cr {{{\left| {f_n^i(x) - f_n^j(x)} \right|} \over {\left| {\mu (x)} \right|}} < {{\rho {f_0}} \over 2} \cdot h({f^i} - {f^j}),} \hfill \cr {{{\left| {f_n^i(x) - f_n^j(x)} \right|} \over {\left| {\mu (x)} \right|}} < {{\rho {f_0}} \over 2} \cdot {\varepsilon \over {\rho {f_0}}} < {\varepsilon \over 2}.} \hfill \cr} Hence, ${(f_{n}^{i} (x))}_{i = 1}^{\infty}$ (f_n^i(x))_{i = 1}^\infty , for all n = 1, 2, 3,..., n is a Cauchy sequence in D, w.r.t. the scale function μ(x), for x ∈ D.

Now, from the second part we get $sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {(\sum_{n = σ} {(O (\frac{| \nabla_{q} (f_{n}^{i} (x) - f_{n}^{j} (x)) |}{h (f^{i} - f^{j}) | μ (x) |}))}^{p})}^{\frac{1}{p}} \leq 1,$ \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{\left({\sum\limits_{n = \sigma} {{{\left({{\cal O}\left({{{\left| {{\nabla_q}(f_n^i(x) - f_n^j(x))} \right|} \over {h({f^i} - {f^j})\left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \right)^{{1 \over p}}} \le 1, for all i, j ≥ n₀ and n ∈ ℕ, (3.5) $\begin{matrix} \begin{matrix} {(O (\frac{| \nabla_{q} (f_{n}^{i} (x) - f_{n}^{j} (x)) |}{h (f^{i} - f^{j}) | μ (x) |}))}^{p} \leq ϕ_{1}, & for all i, j \geq n_{0} and n \in ℕ, \end{matrix} \\ \begin{matrix} {(O (\frac{| \nabla_{q} (f_{n}^{i} (x) - f_{n}^{j} (x)) |}{h (f^{i} - f^{j}) | μ (x) |}))}^{p} \leq O (\frac{ρ f_{0}}{2}) & for all i, j \geq n_{0} and n \in ℕ, \end{matrix} \\ | \nabla_{q} f_{n}^{i} (x) - \nabla_{q} f_{n}^{j} (x) | < \frac{ρ f_{0}}{2} \cdot \frac{ε}{ρ f_{0}} = \frac{ε}{2} . \end{matrix}$ \matrix{{\matrix{{{{\left({{\cal O}\left({{{\left| {{\nabla_q}\left({f_n^i\left(x \right) - f_n^j\left(x \right)} \right)} \right|} \over {h\left({{f^i} - {f^j}} \right)\left| {\mu \left(x \right)} \right|}}} \right)} \right)}^p} \le {\phi_1},} & {{\rm{for}}\,{\rm{all}}\,i,j \ge {n_0}\,{\rm{and}}\,n \in {\mathbb N},} \cr}} \cr {\matrix{{{{\left({{\cal O}\left({{{\left| {{\nabla_q}\left({f_n^i\left(x \right) - f_n^j\left(x \right)} \right)} \right|} \over {h\left({{f^i} - {f^j}} \right)\left| {\mu \left(x \right)} \right|}}} \right)} \right)}^p} \le {\cal O}\left({{{\rho {f_0}} \over 2}} \right)} & {{\rm{for}}\,{\rm{all}}\,i,j \ge {n_0}\,{\rm{and}}\,n \in {\mathbb N},} \cr}} \cr {\left| {{\nabla_q}f_n^i\left(x \right) - {\nabla_q}f_n^j\left(x \right)} \right| < {{\rho {f_0}} \over 2} \cdot {\varepsilon \over {\rho {f_0}}} = {\varepsilon \over 2}.} \cr} Therefore, ${(\nabla_{q} f_{n}^{i})}_{n = 1}^{\infty}$ ({\nabla_q}f_n^i)_{n = 1}^\infty is a Cauchy sequence in D, w.r.t. the scale function μ(x), for x ∈ D. Hence, ${(\nabla_{q} f_{n}^{i})}_{n = 1}^{\infty}$ ({\nabla_q}f_n^i)_{n = 1}^\infty is convergent in D, w.r.t. the scale function μ(x), x ∈ D, for each n ∈ ℕ. By equations (3.4) and (3.5), the sequence of the functions $(f_{n}^{i})$ (f_n^i) is a Cauchy sequence in D, w.r.t. the scale function μ(x), for x ∈ D, for all n ∈ ℕ and it is convergent in D. Therefore for each n ∈ ℕ, there exists (f_n) ∈ (Z, ru) such that $(\frac{f_{n}^{i} - f_{n}}{| μ (x) |}) \to 0$ \left({{{f_n^i - {f_n}} \over {\left| {\mu (x)} \right|}}} \right) \to 0 , as i → ∞ for each n ∈ ℕ.

Using the continuity of 𝒪 we have $\sum_{n = 1}^{\infty} O (\frac{| f_{n}^{i} (x) - {lim}_{j \to \infty} f_{n}^{j} (x) |}{ν | μ (x) |}) + sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {(\sum_{n \in σ} {(O (\frac{| \nabla_{q} (f_{n}^{i} (x) - {lim}_{j \to \infty} f_{n}^{j} (x)) |}{ν | μ (x) |}))}^{p})}^{\frac{1}{p}} \leq 1,$ \sum\limits_{n = 1}^\infty {{\cal O}\left({{{\left| {f_n^i(x) - {{\lim}_{j \to \infty}}f_n^j(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} + \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{\left({\sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{\left| {{\nabla_q}(f_n^i(x) - {{\lim}_{j \to \infty}}f_n^j(x))} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \right)^{{1 \over p}}} \le 1, for some ν > 0, $\sum_{n = 1}^{\infty} O (\frac{| f_{n}^{i} (x) - f_{n} (x) |}{ν | μ (x) |}) + sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {(\sum_{n \in σ} {(O (\frac{| \nabla_{q} (f_{n}^{i} (x) - f_{n} (x)) |}{ν | μ (x) |}))}^{p})}^{\frac{1}{p}} \leq 1,$ \sum\limits_{n = 1}^\infty {{\cal O}\left({{{\left| {f_n^i(x) - {f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} + \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{\left({\sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{\left| {{\nabla_q}(f_n^i(x) - {f_n}(x))} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \right)^{{1 \over p}}} \le 1, for some ν > 0 and j → ∞. Taking the infimum value of ν^′s in the above and making use of (3.3) we get, $inf {ν > 0 : \sum_{n = 1}^{\infty} O (\frac{| f_{n}^{i} (x) - f_{n} (x) |}{ν | μ (x) |}) + sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {(\sum_{n \in σ} {(O (\frac{| \nabla_{q} (f_{n}^{i} (x) - f_{n} (x)) |}{ν | μ (x) |}))}^{p})}^{\frac{1}{p}} \leq 1} < \frac{ε}{ρ f_{0}},$ \inf \left\{{\nu > 0:\sum\limits_{n = 1}^\infty {{\cal O}\left({{{\left| {f_n^i(x) - {f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right) + \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{{\left({\sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{\left| {{\nabla_q}\left({f_n^i(x) - {f_n}(x)} \right)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \right)}^{{1 \over p}}} \le 1}} \right\} < {\varepsilon \over {\rho {f_0}}}, for all i ≥ n₀. Therefore, $(f_{n}^{i} (x) - f_{n} (x)) \in_{ru} m_{ϕ} (O, \nabla_{p}, p)$ (f_n^i(x) - {f_n}(x)) \in {\,_{ru}}{m_\phi }({\cal O},{\nabla _p},p) , for all n ≥ n₀. Let i ≥ n₀ and since _rum_ϕ(𝒪, ∇_q, p) is a linear space, therefore $(f_{n} (x)) = (f_{n}^{i} (x)) + (f_{n} (x) - f_{n}^{i} (x)) \in_{ru} m_{ϕ} (O, \nabla_{q}, p) .$ ({f_n}(x)) = (f_n^i(x)) + ({f_n}(x) - f_n^i(x)) \in {\,_{ru}}{m_\phi}({\cal O},{\nabla_q},p). Hence, the space _rum_ϕ(𝒪, ∇_q, p) is complete.

In view of Theorem 3.4, we formulate the following result without proof.

Theorem 3.5

The space _run_ϕ(𝒪, ∇_q, p) is complete.

Theorem 3.6

_ruℓ_p(𝒪, ∇_q) ⊆ _ruc₀(𝒪, ∇_q) ⊆ _ruc(𝒪, ∇_q) ⊆ _ruℓ_∞(𝒪, ∇_q).

The above theorem is easy to prove. That is why we avoid the proof of the above theorem.

Proposition 3.7

Let 0 < p < 1. Then the inclusion relation _ruℓ_p(𝒪, ∇_q) ⊂ _rum_ϕ(𝒪, ∇_q, p) holds and it is strict.

Proof

Taking $g_{n} (x) = O (\frac{| \nabla_{q} f_{n} (x) |}{ν})$ {g_n}(x) = {\cal O}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over \nu}} \right) , for all n ∈ ℕ, the inclusion follows from Lemma 2.5.

The inclusion is strict following the example below.

Example 3.8

Consider the sequence of functions f_n : [0, 1] → ℝ, defined by $\begin{matrix} f_{n} (x) = n x, & for all x \in [0, 1] . \end{matrix}$ \matrix{{{f_n}(x) = nx,} & {{\rm{for}}\,{\rm{all}}\,x \in [0,1].} \cr} Consider another non-decreasing sequence ϕ_n = n, for all n ∈ ℕ and also consider an Orlicz function 𝒪 : [0, ∞) → [0, ∞) such that, 𝒪(x) = x². Thus, (f_n) ∈ _rum_ϕ(𝒪, ∇_q, p) w.r.t. the scale function μ(x) = x², for all x ∈ [0, 1]. But (f_n) ∉ _ruℓ_p(𝒪, ∇_q). Hence, $_{ru} ℓ_{p} (O, \nabla_{q}) ⊉_{ru} m_{ϕ} (O, \nabla_{q}, p) .$ _{ru}{\ell_p}({\cal O},{\nabla_q})\, \nsupseteq {\,_{ru}}{m_\phi}({\cal O},{\nabla_q},p).

Theorem 3.9

Let 𝒪, 𝒪₁, 𝒪₂ be Orlicz functions satisfying ∆₂ condition. Then

(1)
_rum_ϕ(𝒪₁, ∇_q, p) ⊆ _rum_ϕ(𝒪 ◦ 𝒪₁, ∇_q, p),
(2)
_rum_ϕ(𝒪₁, ∇_q, p) ∩ _rum_ϕ(𝒪₂, ∇_q, p) ⊆ _rum_ϕ(𝒪₁ + 𝒪₂, ∇_q, p).

Proof

(1) Let (f_n) ∈ _rum_ϕ(𝒪₁, ∇_q, p). Then there exists ν > 0 such that, $sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {(\sum_{n \in σ} {(O (\frac{| \nabla_{q} (f_{n} (x)) |}{ν | μ (x) |}))}^{p})}^{\frac{1}{p}} < ε .$ \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{\left({\sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{\left| {{\nabla_q}({f_n}(x))} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \right)^{{1 \over p}}} < \varepsilon. Let η be such that 𝒪(t) < η, for all 0 ≤ t < η < 1. Note that $\sum_{n \in σ} O (O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |})) = \sum_{1} O (O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |})) + \sum_{2} O (O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |})),$ \sum\limits_{n \in \sigma} {{\cal O}\left({{{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)} = \sum\limits_1 {{\cal O}\left({{{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)} + \sum\limits_2 {{\cal O}\left({{{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}, where the first summation is over $O_{1} (\frac{| f_{n} (x) |}{ν | μ (x) |}) \leq η$ {{\cal O}_1}\left({{{\left| {{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right) \le \eta , and the second summation is over $O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}) > η$ {{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right) > \eta . Since 𝒪 is continuous, by the remark we have (3.6) $\sum_{1} O (O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |})) \leq O (1) \sum_{1} O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}) .$ \sum\limits_1 {{\cal O}\left({{{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)} \le {\cal O}\left(1 \right)\sum\limits_1 {{{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)}. For $O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}) > η$ {{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right) > \eta , we use the fact that $O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}) < O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}) η^{- 1} \leq 1 + O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}) η^{- 1} .$ {{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right) < {{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right){\eta^{- 1}} \le 1 + {{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right){\eta^{- 1}}. Because 𝒪 is convex and non-decreasing, $\begin{array}{l} O (O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |})) & < O (1 + O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}) \frac{1}{η}) \\ \leq \frac{1}{2} O (2) + \frac{1}{2} O (2 O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}) \frac{1}{η}) \\ = \frac{1}{2} O (2) + \frac{1}{2} O (2) (O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}) \frac{1}{η}) . \end{array}$ \matrix{{{\cal O}\left({{{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)} \hfill & {< {\cal O}\left({1 + {{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right){1 \over \eta}} \right)} \hfill \cr {} \hfill & {\le {1 \over 2}{\cal O}(2) + {1 \over 2}{\cal O}\left({2{{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right){1 \over \eta}} \right)} \hfill \cr {} \hfill & {= {1 \over 2}{\cal O}(2) + {1 \over 2}{\cal O}(2)\left({{{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right){1 \over \eta}} \right).} \hfill \cr} There exists K > 0 such that, given that 𝒪 meets the requirement of ∆₂, $\begin{array}{l} O (O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |})) & \leq \frac{1}{2} K O (2) O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}) \frac{1}{η} + \frac{1}{2} K O (2) O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}) \frac{1}{η} \\ = K O (2) O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}) \frac{1}{η} . \end{array}$ \matrix{{{\cal O}\left({{{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)} \hfill & {\le {1 \over 2}K{\cal O}\left(2 \right){{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right){1 \over \eta} + {1 \over 2}K{\cal O}\left(2 \right){{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right){1 \over \eta}} \hfill \cr {} \hfill & {= K{\cal O}\left(2 \right){{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right){1 \over \eta}.} \hfill \cr} Hence, (3.7) $\begin{array}{l} \sum_{1} O (O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |})) & \leq max {1, (K O (2) η^{- 1})} \frac{1}{ϕ_{s}} \sum_{2} O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}) \\ \leq max {1, (K O (2) η^{- 1})} \frac{1}{ϕ_{s}} \sum_{n \in σ} O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}) \end{array}$ \matrix{{\sum\limits_1 {{\cal O}\left({{{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}} \hfill & {\le \max \{1,(K{\cal O}(2){\eta^{- 1}})\} {1 \over {{\phi_s}}}\sum\limits_2 {{{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)}} \hfill \cr {} \hfill & {\le \max \{1,(K{\cal O}(2){\eta^{- 1}})\} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)}} \hfill \cr} From inequalities (3.6) and (3.7) it follows that $\begin{array}{l} sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}) \leq max {1, (K O (2) η^{- 1})} \frac{1}{ϕ_{s}} \sum_{2} O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}), \\ sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {(\sum_{n \in σ} {(O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}))}^{p})}^{\frac{1}{p}} \leq max {1, (K O (2) η^{- 1})} \frac{1}{ϕ_{s}} {(\sum_{2} {(O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}))}^{p})}^{\frac{1}{p}} < ε . \end{array}$ \matrix{{\mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \le \max \{1,(K{\cal O}(2){\eta^{- 1}})\} {1 \over {{\phi_s}}}\sum\limits_2 {{{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)},} \hfill \cr {\mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{{\left({\sum\limits_{n \in \sigma} {{{\left({{{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \right)}^{{1 \over p}}} \le \max \{1,(K{\cal O}(2){\eta^{- 1}})\} {1 \over {{\phi_s}}}{{\left({\sum\limits_2 {{{\left({{{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \right)}^{{1 \over p}}} < \varepsilon.} \hfill \cr} Hence, (f_n) ∈ _rum_ϕ(𝒪₁ ◦ 𝒪₂, ∇_q, p).

(2) Let (f_n) ∈ _rum_ϕ(𝒪₁, ∇_q, p) ∩ _rum_ϕ(𝒪₂, ∇_q, p). Therefore, (f_n) ∈ _rum_ϕ(𝒪₁, ∇_q, p) and (f_n) ∈ _rum_ϕ(𝒪₂, ∇_q, p). Consequently, there are ν₁ > 0 and ν₂ > 0 such that $sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {(\sum_{n \in σ} {(O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν_{1} | μ (x) |}))}^{p})}^{\frac{1}{p}} < \frac{ε}{2},$ \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{\left({\sum\limits_{n \in \sigma} {{{\left({{{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {{\nu_1}\left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \right)^{{1 \over p}}} < {\varepsilon \over 2}, and $sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {(\sum_{n \in σ} {(O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν_{2} | μ (x) |}))}^{p})}^{\frac{1}{p}} < \frac{ε}{2} .$ \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{\left({\sum\limits_{n \in \sigma} {{{\left({{{\cal O}_1}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {{\nu_2}\left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \right)^{{1 \over p}}} < {\varepsilon \over 2}. Let ν₃ = max{ν₁, ν₂}. Then $\begin{array}{l} sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {(\sum_{n \in σ} {((O_{1} + O_{2}) (\frac{| \nabla_{q} f_{n} (x) |}{ν_{1} | μ (x) |}))}^{p})}^{\frac{1}{p}} \\ \leq sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {(\sum_{n \in σ} {(O_{1} (\frac{| \nabla_{q} f_{n} (x) |}{ν_{1} | μ (x) |}))}^{p})}^{\frac{1}{p}} \\ + sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {(\sum_{n \in σ} {(O_{2} (\frac{| \nabla_{q} f_{n} (x) |}{ν_{1} | μ (x) |}))}^{p})}^{\frac{1}{p}} \\ < \frac{ε}{2} + \frac{ε}{2} = ε . \end{array}$ \matrix{{\mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi _s},x \in D} {1 \over {{\phi _s}}}{{\left({\sum\limits_{n \in \sigma} {{{\left({({{\cal O}_1} + {{\cal O}_2})\left({{{\left| {{\nabla _q}{f_n}(x)} \right|} \over {{\nu _1}\left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \right)}^{{1 \over p}}}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \le \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi _s},x \in D} {1 \over {{\phi _s}}}{{\left({\sum\limits_{n \in \sigma} {{{\left({{{\cal O}_1}\left({{{\left| {{\nabla _q}{f_n}(x)} \right|} \over {{\nu _1}\left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \right)}^{{1 \over p}}}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi _s},x \in D} {1 \over {{\phi _s}}}{{\left({\sum\limits_{n \in \sigma} {{{\left({{{\cal O}_2}\left({{{\left| {{\nabla _q}{f_n}(x)} \right|} \over {{\nu _1}\left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \right)}^{{1 \over p}}}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, < {\varepsilon \over 2} + {\varepsilon \over 2} = \varepsilon.} \hfill \cr} Therefore, (f_n) ∈ _rum_ϕ(𝒪₁ + 𝒪₂, ∇_q, p). Hence, $_{ru} m_{ϕ} (O_{1}, \nabla_{q}, p) \cap_{ru} m_{ϕ} (O_{2}, \nabla_{q}, p) \subseteq_{ru} m_{ϕ} (O_{1} + O_{2}, \nabla_{q}, p) .$ _{ru}{m_\phi}\left({{{\cal O}_1},{\nabla_q},p} \right) \cap {\,_{ru}}{m_\phi}\left({{{\cal O}_2},{\nabla_q},p} \right) \subseteq {\,_{ru}}{m_\phi}\left({{{\cal O}_1} + {{\cal O}_2},{\nabla_q},p} \right).

Lemma 3.10

_rum_ϕ(𝒪, ∇_q) ⊆ _rum_ϕ(𝒪, ∇_q, p).

Proof

Let (f_n) ∈ _rum_ϕ(𝒪, ∇_q). Then there exists ν > 0 such that $sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} O (\frac{| \nabla_{q} f_{n} (x) |}{ν}) < ε | μ (x) | .$ \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{\cal O}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over \nu}} \right) < \varepsilon \left| {\mu (x)} \right|}. Hence, for each fixed s, $\begin{matrix} \begin{matrix} \sum_{n = 1}^{\infty} O (\frac{| \nabla_{q} f_{n} (x) |}{ν}) < ε | μ (x) | ϕ_{s}, & σ \in ξ_{s}, \end{matrix} \\ \begin{matrix} sup_{s \geq 1, σ \in ξ_{s}, x \in D} {(\sum_{n \in σ} {(O (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}))}^{p})}^{\frac{1}{p}} < ε ϕ_{s}, & σ \in ξ_{s}, \end{matrix} \\ \begin{matrix} sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {(\sum_{n \in σ} {(O (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}))}^{p})}^{\frac{1}{p}} < ε, & σ \in ξ_{s}, \end{matrix} \\ (f_{n}) \in_{ru} m_{ϕ} (O, \nabla_{q}, p) . \end{matrix}$ \matrix{{\matrix{{\sum\limits_{n = 1}^\infty {{\cal O}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over \nu}} \right) < \varepsilon \left| {\mu (x)} \right|{\phi_s},}} & {\sigma \in {\xi_s},} \cr}} \cr {\matrix{{\mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {{\left({\sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \right)}^{{1 \over p}}} < \varepsilon {\phi_s},} & {\sigma \in {\xi_s},} \cr}} \cr {\matrix{{\mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{{\left({\sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \right)}^{{1 \over p}}} < \varepsilon ,} & {\sigma \in {\xi_s},} \cr}} \cr {({f_n}) \in {\,_{ru}}{m_\phi}({\cal O},{\nabla_q},p).} \cr} Hence, _rum_ϕ(𝒪, ∇_q) ⊆ _rum_ϕ(𝒪, ∇_q, p).

Remark 3.11

The class of sequences of functions _rum_ϕ(𝒪, ∇_q, p) is not convergence free.

This result follows from the following example.

Example 3.12

Consider an Orlicz function 𝒪 : [0, ∞) → [0, ∞) such that 𝒪(x) = x, for all x ∈ [0, ∞), ϕ_n = n, for all n ∈ ℕ, and we take a sequence of functions (f_n) defined as follows $f_{n} (x) = {\begin{array}{l} \frac{1}{n^{2} x}, & for x \neq 0, \\ 0, & for x = 0, \end{array}$ {f_n}(x) = \left\{{\matrix{{{1 \over {{n^2}x}},} \hfill & {{\rm{for}}\,x \ne 0,} \hfill \cr {0,} \hfill & {{\rm{for}}\,x = 0,} \hfill \cr}} \right. with respect to the scale function defined as $μ (x) = {\begin{array}{l} \frac{1}{x}, & for x \neq 0, \\ 0, & for x = 0 . \end{array}$ \mu (x) = \left\{{\matrix{{{1 \over x},} \hfill & {{\rm{for}}\,x \ne 0,} \hfill \cr {0,} \hfill & {{\rm{for}}\,x = 0.} \hfill \cr}} \right. Thus, (f_n) ∈ _rum_ϕ(𝒪, ∇_q, p). Now we consider another sequence of functions such as $g_{n} (x) = {\begin{array}{l} n / x, & for x \neq 0, \\ 0, & for x = 0 . \end{array}$ {g_n}(x) = \left\{{\matrix{{n/x,} \hfill & {{\rm{for}}\,x \ne 0,} \hfill \cr {0,} \hfill & {{\rm{for}}\,x = 0.} \hfill \cr}} \right. But (g_n) ∉ _rum_ϕ(𝒪, ∇_q, p).

Hence, the class of sequences of functions _rum_ϕ(𝒪, ∇_q, p) is not convergence free.

Remark 3.13

The class of sequences of functions _rum_ϕ(𝒪, ∇_q, p) is not symmetric.

This result follows from the following example.

Example 3.14

Let ϕ_n = n, for all n ∈ ℕ and also consider an Orlicz function 𝒪 : [0, ∞) → [0, ∞) such that 𝒪(x) = x, for all x ∈ [0, ∞). Now we consider the sequence of function (f_n) defined by $f_{n} (x) = {\begin{array}{l} nx, & for x \neq 0, \\ 0, & for x = 0, \end{array}$ {f_n}(x) = \left\{{\matrix{{nx,} \hfill & {{\rm{for}}\,x \ne 0,} \hfill \cr {0,} \hfill & {{\rm{for}}\,x = 0,} \hfill \cr}} \right. then (f_n) ∈ _rum_ϕ(𝒪, ∇_q, p) with respect to the scale function μ(x) such that, $μ (x) = {\begin{array}{l} x, & for x \neq 0, \\ 0, & for x = 0 . \end{array}$ \mu (x) = \left\{{\matrix{{x,} \hfill & {{\rm{for}}\,x \ne 0,} \hfill \cr {0,} \hfill & {{\rm{for}}x = 0.} \hfill \cr}} \right. Now, take the rearrangement (g_n) of (f_n) defined as follows $(g_{n}) = (f_{1}, f_{3}, f_{5}, f_{7}, f_{2}, \dots) .$ ({g_n}) = ({f_1},{f_3},{f_5},{f_7},{f_2}, \ldots). Then (g_n) ∉ _rum_ϕ(𝒪, ∇_q, p). Hence, _rum_ϕ(𝒪, ∇_q, p) is not symmetric.

Theorem 3.15

The class of sequences of functions _rum_ϕ(𝒪, ∇_q, p) is solid.

Proof

Let (f_n) ∈ _rum_ϕ(𝒪, ∇_q, p). Then $sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} {(O (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}))}^{p} < ε .$ \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}^p}} < \varepsilon. Let (λ_n) be any sequence of scalars with |λ_n| ≤ 1, for all n ∈ ℕ. Then $\begin{array}{l} sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} {(O (\frac{| λ_{n} \nabla_{q} f_{n} (x) |}{ν | μ (x) |}))}^{p} \\ \leq sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} {| λ_{n} |}^{p} {(O (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}))}^{p} \\ \leq sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} {(O (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}))}^{p} < ε . \end{array}$ \matrix{{\mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{\left| {{\lambda_n}{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \le \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{{\left| {{\lambda_n}} \right|}^p}{{\left({{\cal O}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \le \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}^p}} < \varepsilon.} \hfill \cr}

Theorem 3.16

_ruℓ₁(𝒪, ∇_q) ⊆ _rum_ϕ(𝒪, ∇_q) ⊆ _ruℓ_∞(𝒪, ∇_q).

Proof

Let (f_n) ∈ _ruℓ₁(𝒪, ∇_q). Then we have $\begin{matrix} \sum_{n = 1}^{\infty} O (\frac{| \nabla_{q} f_{n} (x) |}{ν}) < ε | μ (x) |, & for all ν > 0 . \end{matrix}$ \matrix{{\sum\limits_{n = 1}^\infty {{\cal O}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over \nu}} \right) < \varepsilon \left| {\mu (x)} \right|,}} & {{\rm{for}}\,{\rm{all}}\,\nu > 0.} \cr} Since (ϕ_n) is monotonically increasing, so we have $\frac{1}{ϕ_{s}} \sum_{n \in σ} O (\frac{| \nabla_{q} f_{n} (x) |}{ν}) \leq \frac{1}{ϕ_{1}} \sum_{n = 1}^{\infty} O (\frac{| \nabla_{q} f_{n} (x) |}{ν}) < ε | μ (x) | .$ {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{\cal O}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over \nu}} \right)} \le {1 \over {{\phi_1}}}\sum\limits_{n = 1}^\infty {{\cal O}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over \nu}} \right)} < \varepsilon \left| {\mu (x)} \right|. Hence, $sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} O (\frac{| \nabla_{q} f_{n} (x) |}{ν}) < ε | μ (x) | .$ \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{\cal O}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over \nu}} \right)} < \varepsilon \left| {\mu (x)} \right|. Thus, (f_n) ∈ _rum_ϕ(𝒪, ∇_q). Therefore, _ruℓ₁(𝒪, ∇_q) ⊆ _rum_ϕ(𝒪, ∇_q).

Let (f_n) ∈ _rum_ϕ(𝒪, ∇_q). Then we have $\begin{matrix} sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} O (\frac{| \nabla_{q} f_{n} (x) |}{ν}) < ε | μ (x) |, & for some ν > 0 \end{matrix}$ \matrix{{\mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{\cal O}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over \nu}} \right)} < \varepsilon \left| {\mu (x)} \right|,} & {{\rm{for}}\,{\rm{some}}\,\nu > 0} \cr} (on taking cardinally of σ to be 1). Therefore, (f_n) ∈ _ruℓ_∞(𝒪, ∇_q) and thus, _rum_ϕ(𝒪, ∇_q) ⊆ _ruℓ_∞(𝒪, ∇_q).

4.

Geometric properties

We introduced the space $_{ru} m_{ϕ} (O, \nabla_{q}) = {(f_{n}) \in ω_{f} : ζ (λ f_{n}) < ε for some λ > 0},$ _{ru}{m_\phi}({\cal O},{\nabla_q}) = \{({f_n}) \in {\omega_f}:\zeta (\lambda {f_n}) < \varepsilon \,{\rm{for}}\,{\rm{some}}\,\lambda > 0\}, where $ζ (f_{n} (x)) = sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {\sum_{n \in σ} (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |})}^{p},$ \zeta ({f_n}(x)) = \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{\sum\limits_{n \in \sigma} {\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)}^p}, ∇_qf_n(x) = f_n(x) − qf_n−₁(x) and p > 0. It is equipped with the norm defined by ${‖ f_{n} ‖}_{_{ru} m_{ϕ} (O, \nabla_{q})} = inf {ν > 0 : sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} O {(\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |})}^{p} \leq 1} .$ {\left\| {{f_n}} \right\|_{_{ru}{m_\phi}({\cal O},{\nabla_q})}} = \inf \left\{{\nu > 0:\mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{\cal O}{{\left({{{\left| {{\nabla_q}{f_n}\left(x \right)} \right|} \over {\nu \left| {\mu \left(x \right)} \right|}}} \right)}^p}} \le 1} \right\}.

Theorem 4.1

The space _rum_ϕ(𝒪, ∇_q, p) is a convex modular space with the modular $ζ (f_{n}) = sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} {(O (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}))}^{p} .$ \zeta ({f_n}) = \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}^p}.}

Proof

(1)
Clearly, ζ(f_n) = 0, if and only if $f_{n} = \bar{θ}$ {f_n} = \bar \theta , where $\bar{θ}$ \bar \theta represents the null sequence of functions.
(2)
Now for any scalar α with |α| = 1, $\begin{array}{l} ζ (α f_{n}) & = sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} {(O (\frac{| α \nabla_{q} f_{n} (x) |}{ν | μ (x) |}))}^{p} \\ = | α |^{p} sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} {(O (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}))}^{p} \\ = sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} {(O (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}))}^{p} \\ = ζ (f_{n}) . \end{array}$ \matrix{{\zeta (\alpha {f_n})} \hfill & {= \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{\left| {\alpha {\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \hfill \cr {} \hfill & {= |\alpha {|^p}\mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \hfill \cr {} \hfill & {=\mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \hfill \cr {} \hfill & {= \zeta ({f_n}).} \hfill \cr}
(3)
Let (f_n) and (g_n) be in _rum_ϕ(𝒪, ∇_q, p) and α, β ≥ 0 with α + β = 1. To prove the convexity of the function note that $\begin{array}{l} ζ (α f_{n} + β g_{n}) & = sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} {(O (\frac{| α \nabla_{q} f_{n} (x) + β \nabla_{q} g_{n} (x) |}{ν | μ (x) |}))}^{p} \\ \leq sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} {| α |}^{p} {(O (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}))}^{p} \\ + sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} {| β |}^{p} {(O (\frac{| \nabla_{q} g_{n} (x) |}{ν | μ (x) |}))}^{p} \\ \leq sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} {(O (\frac{| \nabla_{q} f_{n} (x) |}{ν | μ (x) |}))}^{p} \\ + sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} {(O (\frac{| \nabla_{q} g_{n} (x) |}{ν | μ (x) |}))}^{p} \\ = ζ (f_{n}) + ζ (g_{n}) . \end{array}$ \matrix{{\zeta (\alpha {f_n} + \beta {g_n})} \hfill & {= \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{\left| {\alpha {\nabla_q}{f_n}(x) + \beta {\nabla_q}{g_n}(x)} \right|} \over {\nu \left| {\mu (x)} \right|}}} \right)} \right)}^p}}} \hfill \cr {} \hfill & {\le \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{{\left| \alpha \right|}^p}{{\left({{\cal O}\left({{{|{\nabla_q}{f_n}(x)|} \over {\nu |\mu (x)|}}} \right)} \right)}^p}}} \hfill \cr {} \hfill & {+ \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{{\left| \beta \right|}^p}{{\left({{\cal O}\left({{{|{\nabla_q}{g_n}(x)|} \over {\nu |\mu (x)|}}} \right)} \right)}^p}}} \hfill \cr {} \hfill & {\le \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{|{\nabla_q}{f_n}(x)|} \over {\nu |\mu (x)|}}} \right)} \right)}^p}}} \hfill \cr {} \hfill & {+ \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{{\left({{\cal O}\left({{{|{\nabla_q}{g_n}(x)|} \over {\nu |\mu (x)|}}} \right)} \right)}^p}}} \hfill \cr {} \hfill & {= \zeta ({f_n}) + \zeta ({g_n}).} \hfill \cr} Hence, _rum_ϕ(𝒪, ∇_q, p) is a modular space. Also, it can be proven that ζ(αf_n + βg_n) ≤ αζ(f_n) + βζ(g_n), for α, β > 0 and α + β = 1. Thus, _rum_ϕ(𝒪, ∇_q, p) is a convex modular space.

Theorem 4.2

The space _rum_ϕ(𝒪, ∇_q, p) is uniform convex.

Proof

Fix 0 < ɛ ≤ 2. Let us consider two sequences (f_n), (g_n) of functions from the Banach space _rum_ϕ(𝒪, ∇_q) and assume that (4.1) $\begin{matrix} {‖ f_{n} ‖}_{_{ru} m_{ϕ} (O, \nabla_{q}, p)} = {‖ g_{n} ‖}_{_{ru} m_{ϕ} (O, \nabla_{q}, p)} = 1, & {‖ f_{n} - g_{n} ‖}_{_{ru} m_{ϕ} (O, \nabla_{q}, p)} \geq ε . \end{matrix}$ \matrix{{{{\left\| {{f_n}} \right\|}_{_{ru}{m_\phi}({\cal O},{\nabla_q},p)}}\, = {{\left\| {{g_n}} \right\|}_{_{ru}{m_\phi}({\cal O},{\nabla_q},p)}}\, = 1,} & {{{\left\| {{f_n} - {g_n}} \right\|}_{_{ru}{m_\phi}({\cal O},{\nabla_q},p)}}\, \ge \varepsilon.} \cr} Note that ${‖ f_{n} + g_{n} ‖}_{_{ru} m_{ϕ} (O, \nabla_{q}, p)}^{2} + {‖ f_{n} - g_{n} ‖}_{_{ru} m_{ϕ} (O, \nabla_{q}, p)}^{2} = 2 ({‖ f_{n} ‖}_{_{ru} m_{ϕ} (O, \nabla_{q}, p)}^{2} + {‖ g_{n} ‖}_{_{ru} m_{ϕ} (O, \nabla_{q}, p)}^{2}),$ \left\| {{f_n} + {g_n}} \right\|_{_{ru}{m_\phi}({\cal O},{\nabla_q},p)}^2 + \left\| {{f_n} - {g_n}} \right\|_{_{ru}{m_\phi}({\cal O},{\nabla_q},p)}^2 = 2(\left\| {{f_n}} \right\|_{_{ru}{m_\phi}({\cal O},{\nabla_q},p)}^2 + \left\| {{g_n}} \right\|_{_{ru}{m_\phi}({\cal O},{\nabla_q},p)}^2), and from (4.1) we get $\begin{matrix} {‖ f_{n} + g_{n} ‖}_{_{ru} m_{ϕ} (O, \nabla_{q}, p)}^{2} = 4 - {‖ f_{n} - g_{n} ‖}_{_{ru} m_{ϕ} (O, \nabla_{q}, p)}^{2} \\ \leq 4 - ε^{2} . \end{matrix}$ \matrix{{\left\| {{f_n} + {g_n}} \right\|_{_{ru}{m_\phi}({\cal O},{\nabla_q},p)}^2 = 4 - \left\| {{f_n} - {g_n}} \right\|_{_{ru}{m_\phi}({\cal O},{\nabla_q},p)}^2} \cr {\le 4 - {\varepsilon^2}.} \cr} Consequently, $\frac{1}{2} {‖ f_{n} + g_{n} ‖}_{_{r u} m_{ϕ} (O, \nabla_{q}, p)} \leq \sqrt{1 - \frac{ε^{2}}{4}} .$ {1 \over 2}{\left\| {{f_n} + {g_n}} \right\|_{_{ru}{m_\phi }({\cal O},{\nabla _q},p)}} \le \sqrt {1 - {{{\varepsilon ^2}} \over 4}}. So, there corresponds a ϑ(ɛ) > 0 such that $\frac{1}{2} {‖ f_{n} + g_{n} ‖}_{_{ru} m_{ϕ} (O, \nabla_{q}, p)} \leq 1 - ϑ (ε) .$ {1 \over 2}{\left\| {{f_n} + {g_n}} \right\|_{_{ru}{m_\phi}\left({{\cal O},{\nabla_q},p} \right)}} \le 1 - \vartheta (\varepsilon). Hence, the space _rum_ϕ(𝒪, ∇_q, p) is uniform convex.

Theorem 4.3

The space _rum_ϕ(𝒪, ∇_q, p) is convex.

Proof

Let (f_n), (g_n) ∈ _rum_ϕ(𝒪, ∇_q, p). By the definition we have $\begin{matrix} sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} O {(\frac{| \nabla_{q} f_{n} (x) |}{ν_{1}})}^{p} < ε_{1} | μ_{1} (x) |, & for some ν_{1} > 0 \end{matrix},$ \matrix{{\mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{\cal O}{{\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {{\nu_1}}}} \right)}^p}} < {\varepsilon_1}\left| {{\mu_1}(x)} \right|,} & {{\rm{for}}\,{\rm{some}}\,{\nu_1} > 0} \cr}, and $\begin{matrix} sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} {\sum_{n \in σ} O (\frac{| \nabla_{q} g_{n} (x) |}{ν_{2}})}^{p} < ε_{2} | μ_{2} (x) |, & for some ν_{2} > 0 \end{matrix},$ \matrix{{\mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}{{\sum\limits_{n \in \sigma} {{\cal O}\left({{{\left| {{\nabla_q}{g_n}(x)} \right|} \over {{\nu_2}}}} \right)}}^p} < {\varepsilon_2}\left| {{\mu_2}(x)} \right|,} & {{\rm{for}}\,{\rm{some}}\,{\nu_2} > 0} \cr}, Take ν = max{|λ₁|ν₁, |λ₂|ν₂}, where λ₁, λ₂ > 0, λ₁ + λ₂ = 1. Since λ₁, λ₂, ν₁, ν₂ > 0, then |λ₁|ν₁ > 0, |λ₂|ν₂ > 0. Now, $\begin{array}{l} sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} O {(\frac{| λ_{1} \nabla_{q} f_{n} (x) + λ_{2} \nabla_{q} g_{n} (x) |}{ν})}^{p} \\ = sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} O {(\frac{| λ_{1} \nabla_{q} f_{n} (x) + (1 - λ_{1}) \nabla_{q} g_{n} (x) |}{ν})}^{p} \\ \leq sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} O {(\frac{| λ_{1} \nabla_{q} f_{n} (x) |}{ν})}^{p} \\ + sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} O {(\frac{| (1 - λ_{1}) \nabla_{q} g_{n} (x) |}{ν})}^{p} \\ \leq | λ_{1} |^{p} sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} O {(\frac{| \nabla_{q} f_{n} (x) |}{ν_{1}})}^{p} \\ + | 1 - λ_{1} |^{p} sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} O {(\frac{| \nabla_{q} g_{n} (x) |}{ν_{2}})}^{p} \\ \leq | λ_{1} |^{p} sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} O {(\frac{| \nabla_{q} f_{n} (x) |}{ν_{1}})}^{p} \\ + | 1 - λ_{1} |^{p} sup_{s \geq 1, σ \in ξ_{s}, x \in D} \frac{1}{ϕ_{s}} \sum_{n \in σ} O {(\frac{| \nabla_{q} g_{n} (x) |}{ν_{2}})}^{p} \\ \leq | λ_{1} |^{p} ε_{1} | μ_{1} (x) | + (1 + | λ_{1} |^{p}) ε_{2} | μ_{2} (x) | \\ \leq | λ_{1} |^{p} (ε_{1} | μ_{1} (x) | + ε_{2} | μ_{2} |) + ε_{2} | μ_{2} (x) | \\ \leq ε | μ (x) |, \end{array}$ \matrix{{\mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{\cal O}{{\left({{{\left| {{\lambda_1}{\nabla_q}{f_n}(x) + {\lambda_2}{\nabla_q}{g_n}(x)} \right|} \over \nu}} \right)}^p}}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{\cal O}{{\left({{{\left| {{\lambda_1}{\nabla_q}{f_n}(x) + (1 - {\lambda_1}){\nabla_q}{g_n}(x)} \right|} \over \nu}} \right)}^p}}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \le \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{\cal O}{{\left({{{\left| {{\lambda_1}{\nabla_q}{f_n}(x)} \right|} \over \nu}} \right)}^p}}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{\cal O}{{\left({{{\left| {(1 - {\lambda_1}){\nabla_q}{g_n}(x)} \right|} \over \nu}} \right)}^p}}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \le |{\lambda_1}{|^p}\mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{\cal O}{{\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {{\nu_1}}}} \right)}^p}}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + |1 - {\lambda_1}{|^p}\mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{\cal O}{{\left({{{\left| {{\nabla_q}{g_n}(x)} \right|} \over {{\nu_2}}}} \right)}^p}}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \le |{\lambda_1}{|^p}\mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{\cal O}{{\left({{{\left| {{\nabla_q}{f_n}(x)} \right|} \over {{\nu_1}}}} \right)}^p}}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + |1 - {\lambda_1}{|^p}\mathop {\sup}\limits_{s \ge 1,\sigma \in {\xi_s},x \in D} {1 \over {{\phi_s}}}\sum\limits_{n \in \sigma} {{\cal O}{{\left({{{\left| {{\nabla_q}{g_n}(x)} \right|} \over {{\nu_2}}}} \right)}^p}}} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \le |{\lambda_1}{|^p}{\varepsilon_1}\left| {{\mu_1}(x)} \right| + (1 + |{\lambda_1}{|^p}){\varepsilon_2}\left| {{\mu_2}(x)} \right|} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \le |{\lambda_1}{|^p}({\varepsilon_1}\left| {{\mu_1}(x)} \right| + {\varepsilon_2}\left| {{\mu_2}} \right|) + {\varepsilon_2}\left| {{\mu_2}(x)} \right|} \hfill \cr {\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \le \varepsilon \left| {\mu (x)} \right|,} \hfill \cr} where $μ (x) = | λ_{1} |^{p} (ε_{1} | μ_{1} (x) | + ε_{2} | μ_{2} |) + ε_{2} | μ_{2} (x) | .$ \mu (x) = |{\lambda_1}{|^p}({\varepsilon_1}\left| {{\mu_1}(x)} \right| + {\varepsilon_2}\left| {{\mu_2}} \right|) + {\varepsilon_2}\left| {{\mu_2}(x)} \right|.

Theorem 4.4

The space _rum_ϕ(𝒪, ∇_q, p) is absolutely convex.

Proof

Let 0 < r < 1. V = {f = (f_n) ∈ ω_f : ∥f∥_{_rum_ϕ_{(𝒪,∇_q,p)}} ≤ r} is absolutely convex because if (f_n), (g_n) ∈ V and |λ₁| + |λ₂| ≤ 1, then ${‖ λ_{1} f_{n} + λ_{2} g_{n} ‖}_{_{ru} m_{ϕ} (O, \nabla_{q}, p)} \leq (| λ_{1} | + | λ_{2} |) r \leq r .$ {\left\| {{\lambda_1}{f_n} + {\lambda_2}{g_n}} \right\|_{_{ru}{m_\phi}({\cal O},{\nabla_q},p)}} \le (\left| {{\lambda_1}\left| + \right|{\lambda_2}} \right|)r \le r.

In view of Theorem 4.4, we formulate the following without proof.

Corollary 4.5

The space _rum_ϕ(𝒪, ∇_q, p) is balanced.

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