3. Paranorm bounded variation sequence spaces

� � In this section we study double sequence spaces by using the double sequences of strictly positive real numbers p ¼ pij with the help of BV<sup>σ</sup> space and an Orlicz function M. We study some of its properties and prove some inclusion relations related to these new spaces. For m, n ≥ 0, we have

$$\begin{split} \, \_2BV\_{\sigma}^{I}(M, p) = \{ (\mathbf{x}\_{\dot{\mathbf{y}}}) \in \_2\boldsymbol{\omega} : \left\{ (i, j) : M \Big( \frac{|\phi\_{\text{min}}(\mathbf{x}) - L|}{\rho} \Big)^{p\_{\dot{\mathbf{y}}}} \ge \mathbf{c} \right\} \in I; \\ \text{for some } \mathbf{L} \in \mathbb{C}, \rho > \mathbf{0} \end{split} \tag{19}$$

$$\mathbb{P}\_2\left({}\_0\mathcal{B}V^I\_\sigma(M,p)\right) = \left\{ \left( \mathbf{x}\_{\vec{\eta}} \right) \in {}\_2\omega \, : \, \left\{ (i,j) : M\left(\frac{|\phi\_{m\vec{\eta}j}(\mathbf{x})|}{\rho}\right)^{p\_{\vec{\eta}}} \ge \epsilon \right\} \in I, \rho > 0 \right\},\tag{20}$$

$$\mathcal{C}\_2\left(\_{\infty}BV\_{\sigma}^{I}(M,p)\right) = \left\{ \left( \mathbf{x}\_{\overline{\eta}} \right) \in \_2\boldsymbol{\omega} : \left\{ (i,j) : \exists K \times \mathbf{0} : M \left( \frac{|\phi\_{m \text{inj}}(\mathbf{x})|}{\rho} \right)^{p\_{\overline{\eta}}} \succeq K \right\} \in I, \rho > \mathbf{0} \right\} \tag{21}$$

$$\mathcal{I}\_2 l\_{\infty}(M, p) = \left\{ (\mathbf{x}\_{\vec{\eta}}) \in \_2\boldsymbol{\omega} : \sup M \left( \frac{|\phi\_{m \vec{\eta} j}(\boldsymbol{\infty})|}{\rho} \right)^{p\_{\vec{\eta}}} < \infty, \rho > 0 \right\}. \tag{22}$$

We also denote

$$\iota\_2 \mathcal{M}^{I}\_{BV\_\sigma}(\mathcal{M}, p) = \,\_2BV^I\_\sigma(\mathcal{M}, p) \cap \,\_2l\_\infty(\mathcal{M}, p)$$

and

$$\_2\left(\_0\mathcal{M}\_{BV\_\sigma}^l(\mathcal{M},p)\right) = \_2\left(\_0\mathcal{B}V\_\sigma^I(\mathcal{M},p)\right) \cap \_2l\_\infty(\mathcal{M},p).$$

We can now state and proof the theorems based on these double sequence spaces which are as follows:

� � � � Theorem 3.1 Let p ¼ pij ∈ <sup>2</sup>l<sup>∞</sup> then the classes of double sequence 2M<sup>I</sup> BV<sup>σ</sup> ðM; pÞ and 2 0MBV I σ ðM; pÞ are paranormed spaces, paranormed by

$$\log \left( \mathbf{x}\_{\vec{\eta}} \right) = \inf\_{i,j \ge 1} \left\{ \rho^{\frac{p\_{\vec{\eta}}}{H}} : \sup\_{\vec{\eta}} M \left( \frac{|\phi\_{m m \vec{\eta}}(\mathbf{x})|}{\rho} \right)^{p\_{\vec{\eta}}} \le \mathbf{1}, \text{for some } \rho > \mathbf{0} \right\}$$

n o where <sup>H</sup> <sup>¼</sup> max <sup>1</sup>;supij pij . Proof. ð Þ P1 It is clear that g xð Þ¼ 0 if and only if x ¼ 0: P2 ð xÞ ¼ g x � � ð Þ g � ð Þ is obvious. <sup>P</sup><sup>3</sup> � � <sup>∈</sup> <sup>2</sup>MI <sup>ð</sup> . Now for <sup>ρ</sup><sup>1</sup> ð Þ Let <sup>x</sup> <sup>¼</sup> xij , y <sup>¼</sup> <sup>y</sup> , <sup>ρ</sup><sup>2</sup> <sup>&</sup>gt; 0, we denote ij BV<sup>σ</sup> <sup>M</sup>; <sup>p</sup><sup>Þ</sup>

$$A\_1 = \left\{ \rho\_1 : \sup\_{\vec{\eta}} M\left(\frac{|\phi\_{mii\vec{\jmath}}(\mathbf{x})|}{\rho}\right)^{p\_{\vec{\eta}}} \le 1 \right\} \tag{23}$$

$$A\_2 = \left\{ \rho\_2 : \sup\_{\vec{\eta}} M\left(\frac{|\phi\_{m\vec{\eta}j}(\mathbf{x})|}{\rho}\right)^{p\_{\vec{\eta}}} \le 1 \right\} \tag{24}$$

Let us take ρ<sup>3</sup> ¼ ρ<sup>1</sup> þ ρ2. Then by using the convexity of M, we have

Applied Mathematics

$$M\left(\frac{|\phi\_{m\text{inj}}(\mathbf{x}+\mathbf{y})|}{\rho}\right) \le \frac{\rho\_1}{\rho\_1 + \rho\_2} M\left(\frac{|\phi\_{m\text{inj}}(\mathbf{x})|}{\rho\_1}\right) + \frac{\rho\_2}{\rho\_1 + \rho\_2} M\left(\frac{|\phi\_{m\text{inj}}(\mathbf{y})|}{\rho\_2}\right)$$

which in terms give us

$$\sup\_{\vec{\eta}} M\left(\frac{|\phi\_{m\vec{\eta}}(\mathfrak{x}+\mathfrak{y})|}{\rho}\right)^{p\_{\vec{\eta}}} \le 1$$

and

$$\begin{split} \mathrm{g}\left(\mathbf{x}\_{\circ j} + \mathbf{y}\_{\circ j}\right) &= \inf \left\{ (\rho\_1 + \rho\_2)^{\frac{p\_{\vec{\imath}}}{\mathsf{H}}} : \rho\_1 \in A\_1, \rho\_2 \in A\_2 \right\} \\ &\leq \inf \left\{ (\rho\_1)^{\frac{p\_{\vec{\imath}}}{\mathsf{H}}} : \rho\_1 \in A\_1 \right\} + \inf \left\{ (\rho\_2)^{\frac{p\_{\vec{\imath}}}{\mathsf{H}}} : \rho\_2 \in A\_2 \right\} \\ &= \mathrm{g}\left(\mathbf{x}\_{\circ j}\right) + \mathrm{g}\left(\mathbf{y}\_{\circ j}\right) . \end{split}$$

Therefore g xð þ yÞ≤ g xð Þþ g yð Þ:

� � � � ð Þ P4 Let λij be a double sequence of scalars with λij ! λ ði; j ! ∞Þ and � �, L <sup>∈</sup> <sup>2</sup>MI xij <sup>ð</sup>M; <sup>p</sup><sup>Þ</sup> such that BV<sup>σ</sup>

$$
\mathfrak{x}\_{\vec{\eta}} \to L \ (i, j \to \infty),
$$

in the sense that

$$(\mathbf{g}(\kappa\_{i\dot{\jmath}} - L) \to \mathbf{0} \ (i, j \to \infty) .$$

Then, since the inequality

$$\mathbf{g}\left(\mathbf{x}\_{\vec{\eta}}\right) \le \mathbf{g}\left(\mathbf{x}\_{\vec{\eta}} - L\right) + \mathbf{g}(L)$$

� � holds by subadditivity of g, the sequence g xij is bounded. Therefore,

$$\begin{split} \mathbb{g}\left[\left(\lambda\_{\vec{\eta}}\mathbf{x}\_{\vec{\eta}}-\lambda L\right)\right] &= \mathbb{g}\left[\left(\lambda\_{\vec{\eta}}\mathbf{x}\_{\vec{\eta}}-\lambda\mathbf{x}\_{\vec{\eta}}+\lambda\mathbf{x}\_{\vec{\eta}}-\lambda L\right)\right] \\ &= \mathbb{g}\left[\left(\lambda\_{\vec{\eta}}-\lambda\right)\mathbf{x}\_{\vec{\eta}}+\lambda\left(\mathbf{x}\_{\vec{\eta}}-L\right)\right] \\ &\leq \mathbb{g}\left[\left(\lambda\_{\vec{\eta}}-\lambda\right)\mathbf{x}\_{\vec{\eta}}\right] + \mathbb{g}\left[\lambda\left(\mathbf{x}\_{\vec{\eta}}-L\right)\right] \\ &\leq \left|\left(\lambda\_{\vec{\eta}}-\lambda\right)\right|^{\frac{r\_{\vec{\eta}}}{M}}\mathbf{g}\left(\mathbf{x}\_{\vec{\eta}}\right) + \left|\lambda\right|^{\frac{r\_{\vec{\eta}}}{M}}\mathbf{g}\left(\mathbf{x}\_{\vec{\eta}}-L\right) \longrightarrow \mathbf{0} \end{split}$$

� � as ði; j ! ∞Þ. That implies that the scalar multiplication is continuous. Hence <sup>2</sup>M<sup>I</sup> <sup>ð</sup>M; <sup>p</sup><sup>Þ</sup> is <sup>a</sup> paranormed space. For another space 2 0M<sup>I</sup> <sup>ð</sup>M; <sup>p</sup><sup>Þ</sup> , the result is BV<sup>σ</sup> BV<sup>σ</sup> similar.

We shall see about the separability of these new defined double sequence spaces in the next theorem.

� � Theorem 3.2 The spaces <sup>2</sup>M<sup>I</sup> <sup>ð</sup>M; <sup>p</sup><sup>Þ</sup> and 2 0M<sup>I</sup> <sup>ð</sup>M; <sup>p</sup><sup>Þ</sup> are not separable. BV<sup>σ</sup> BV<sup>σ</sup>

Example 3.1 By counter example, we prove the above result for the space <sup>2</sup>M<sup>I</sup> <sup>ð</sup>M; <sup>p</sup>Þ. BV<sup>σ</sup>

Let A be an infinite subset of increasing natural numbers, i.e., A ⊆ N � N such that A ∈ I.

Let

$$p\_{\vec{\eta}} = \begin{cases} \mathbf{1}, & \text{if } (i, j) \in A \\ \mathbf{2}, & \text{otherwise.} \end{cases}$$

A Study of Bounded Variation Sequence Spaces DOI: http://dx.doi.org/10.5772/intechopen.81907

�� � � Let P<sup>0</sup> ¼ xij : xij ¼ 0 or 1;for i; j∈ M and xij ¼ 0; otherwise : Since A is infinite, so P<sup>0</sup> is uncountable. Consider the class of open balls

$$B\_1 = \left\{ B\left(z, \frac{1}{2}\right) : z \in P\_0 \right\}.$$

Let <sup>C</sup><sup>1</sup> be an open cover of <sup>2</sup>M<sup>I</sup> <sup>ð</sup>M; <sup>p</sup><sup>Þ</sup> containing <sup>B</sup>1. BV<sup>σ</sup>

Since B<sup>1</sup> is uncountable, so C<sup>1</sup> cannot be reduced to a countable subcover for 2M<sup>I</sup> BV<sup>σ</sup> ðM; pÞ. Thus <sup>2</sup>MBV I σ ðM; pÞ is not separable.

We shall now introduce a theorem which improves our work.

� � � � � � � � Theorem 3.3 Let p and be two double sequences of positive real ij qij numbers. Then 2 0MI <sup>ð</sup>M; <sup>p</sup><sup>Þ</sup> <sup>⊇</sup>2 0M<sup>I</sup> <sup>ð</sup>M; <sup>q</sup><sup>Þ</sup> if and only if limi,j <sup>∈</sup><sup>K</sup> inf <sup>p</sup> qij ij >0, BV<sup>σ</sup> BV<sup>σ</sup> where <sup>K</sup><sup>c</sup> <sup>⊆</sup> <sup>N</sup> � <sup>N</sup> such that <sup>K</sup> <sup>∈</sup>I.

� � � � Proof. Let limi,j <sup>∈</sup><sup>K</sup> inf <sup>p</sup> qij ij >0 and xij ∈2 0M<sup>I</sup> BV ðM; qÞ : Then, there exists β > 0 <sup>σ</sup> such that pij > β qij for sufficiently large ð Þ i; j ∈K:

� � � � Since xij <sup>∈</sup> 2 0M<sup>I</sup> <sup>ð</sup>M; <sup>q</sup><sup>Þ</sup> : For <sup>a</sup> given <sup>ϵ</sup> > 0, there exist <sup>ρ</sup> <sup>&</sup>gt; <sup>0</sup> such that BV<sup>σ</sup>

$$B\_0 = \left\{ (i, j) \in \mathbb{N} \times \mathbb{N} : \mathcal{M} \left( \frac{|\phi\_{miij}(\mathbf{x})|}{\rho} \right)^{q\_{ij}} \ge \mathbf{c} \right\} \in I.$$

Let <sup>G</sup><sup>0</sup> <sup>¼</sup> <sup>K</sup><sup>c</sup> <sup>∪</sup> <sup>B</sup>0: Then for all sufficiently large ð Þ <sup>i</sup>; <sup>j</sup> <sup>∈</sup> <sup>G</sup>0:

$$\left\{ (i,j) : M\left(\frac{|\phi\_{m\text{nij}}(\mathbf{x})|}{\rho}\right)^{p\_{\bar{\mathbf{y}}}} \succeq \epsilon \right\} \subseteq \left\{ (i,j) : M\left(\frac{|\phi\_{m\text{nij}}(\mathbf{x})|}{\rho}\right)^{\beta q\_{\bar{\mathbf{y}}}} \succeq \epsilon \right\} \in I.$$

� � � � Therefore, xij <sup>∈</sup> 2 0M<sup>I</sup> <sup>ð</sup>M; <sup>p</sup><sup>Þ</sup> . The converse part of the result follows obvi- BV<sup>σ</sup> ously.

� � � � � � � � � � � � Remark 3.1 Let p and be two double sequences of positive real num- ij qij BV<sup>σ</sup> <sup>⊇</sup>2 0M<sup>I</sup> qij bers. Then 2 0M<sup>I</sup> <sup>ð</sup>M; <sup>q</sup><sup>Þ</sup> <sup>ð</sup>M; <sup>p</sup><sup>Þ</sup> if and only if limi,j∈<sup>K</sup> inf pij BV >0 and <sup>σ</sup> 2 0M<sup>I</sup> BV<sup>σ</sup> <sup>ð</sup>M; <sup>q</sup>Þ ¼2 0M<sup>I</sup> <sup>ð</sup>M; <sup>p</sup><sup>Þ</sup> if and only if limi,j <sup>∈</sup><sup>K</sup> inf pij BV >0 and <sup>σ</sup> qij limi,j∈<sup>K</sup> inf p qij ij >0, where <sup>K</sup><sup>c</sup> <sup>⊆</sup> <sup>N</sup> � <sup>N</sup> such that <sup>K</sup> <sup>∈</sup>I.

Theorem 3.4 The set <sup>2</sup>MI <sup>ð</sup>M; <sup>p</sup><sup>Þ</sup> is closed subspace of <sup>2</sup>l∞ðM; <sup>p</sup>Þ. BV<sup>σ</sup>

� � � � ð Þ pq Proof. Let <sup>x</sup> be <sup>a</sup> Cauchy double sequence in <sup>2</sup>M<sup>I</sup> <sup>ð</sup>M; <sup>p</sup><sup>Þ</sup> such that ij BV<sup>σ</sup> ð Þ <sup>x</sup>ð Þ pq pq ! <sup>x</sup>. We show that <sup>x</sup><sup>∈</sup> <sup>2</sup>M<sup>I</sup> <sup>ð</sup>M; <sup>p</sup>Þ: Since, <sup>x</sup> <sup>∈</sup> <sup>2</sup>MI <sup>ð</sup>M; <sup>p</sup>Þ, then there BV<sup>σ</sup> ij BV<sup>σ</sup> exists apq, and ρ>0 such that

$$\left\{ (i,j) : \mathcal{M} \left( \frac{|\phi\_{mnj}(\mathfrak{x}^{pq}) - a\_{pq}|}{\rho} \right)^{p\_{ij}} \ge \epsilon \right\} \in I. $$

We need to show that

� � (1) apq converges to a.

$$\text{(2) If } U = \left\{ (i, j) : M\left(\frac{|\phi\_{\text{unij}}(\mathbf{x}^{\text{pq}}) - a|}{\rho}\right)^{p\_{ij}} < \epsilon \right\}, \text{ then } U^{\epsilon} \in I.$$

� � ð Þ pq Since x be a Cauchy double sequence in <sup>2</sup>M<sup>I</sup> ij BV ðM; pÞ then for a given ϵ>0 <sup>σ</sup> there exists k<sup>0</sup> ∈ N such that

$$\sup\_{\vec{\eta}} M\left(\frac{|\phi\_{m\vec{n}\vec{j}}(\mathbf{x}^{pq}) - \phi\_{m\vec{n}\vec{j}}(\mathbf{x}^{r})|}{\rho}\right)^{p\_{\vec{j}}} < \frac{\epsilon}{3}, \text{ for all } p, q, r, s \ge k\_0.$$

For a given ϵ>0, we have

$$\begin{split} B\_{pqrs} &= \left\{ (i,j) : M\left(\frac{|\phi\_{mnj}(\mathbf{x}^{pq}) - \phi\_{mnj}(\mathbf{x}^{r})|}{\rho}\right)^{p\_{\bar{j}}} < \left(\frac{\epsilon}{3}\right)^{M} \right\}, \\ B\_{pq} &= \left\{ (i,j) : M\left(\frac{|\phi\_{mnj}(\mathbf{x}^{pq}) - a\_{pq}|}{\rho}\right)^{p\_{\bar{j}}} < \left(\frac{\epsilon}{3}\right)^{M} \right\}, \\ B\_{rs} &= \left\{ (i,j) : M\left(\frac{|\phi\_{mnj}(\mathbf{x}^{r}) - a\_{rs}|}{\rho}\right)^{p\_{\bar{j}}} < \left(\frac{\epsilon}{3}\right)^{M} \right\}. \end{split}$$

Then Bc , Bc , Bc <sup>∈</sup>I. Let Bc <sup>¼</sup> Bc <sup>∩</sup> Bc <sup>∩</sup> Bc , pqrs pq rs pqrs pq rs <sup>n</sup> � � o : <sup>M</sup> <sup>∣</sup>apq�ars<sup>∣</sup> pij where <sup>B</sup> <sup>¼</sup> ð Þ <sup>i</sup>; <sup>j</sup> <sup>&</sup>lt; <sup>ϵ</sup> , then Bc <sup>∈</sup>I. We choose <sup>k</sup><sup>0</sup> <sup>∈</sup>Bc , then for <sup>ρ</sup> each p, q,r, s≥ k0, we have

$$\begin{split} \left\{ (i,j) : \mathcal{M} \left( \frac{|a\_{\mathcal{V}} - a\_{n}|}{\rho} \right)^{p\_{\mathcal{V}}} \prec \mathsf{c} \right\} &\supseteq \left[ \left\{ i, j \in \mathbb{N} : \mathcal{M} \left( \frac{|\phi\_{\mathrm{mij}}(\mathbf{x}^{\mathsf{pr}}) - \phi\_{\mathrm{mij}}(\mathbf{x}^{\mathsf{r}})|}{\rho} \right)^{p\_{\mathcal{V}}} < \left( \frac{\mathsf{c}}{3} \right)^{M} \right\} \right] \\ &\cap \left\{ (i,j) : \mathcal{M} \left( \frac{|\phi\_{\mathrm{mij}}(\mathbf{x}^{\mathsf{pr}}) - \mathbf{a}\_{\mathsf{pq}}|}{\rho} \right)^{p\_{\mathcal{V}}} < \left( \frac{\mathsf{c}}{3} \right)^{M} \right\} \\ &\cap \left\{ (i,j) : \mathcal{M} \left( \frac{|\phi\_{\mathrm{mij}}(\mathbf{x}^{\mathsf{r}}) - \mathbf{a}\_{\mathsf{r}}|}{\rho} \right)^{p\_{\mathcal{V}}} < \left( \frac{\mathsf{c}}{3} \right)^{M} \right\} . \end{split}$$

� � � � Then apq is a Cauchy double sequence in C. So, there exists a scalar a∈ C such that apq ! a, as p, q ! ∞:

(2) For the next step, let 0 < δ < 1 be given. Then, we show that if

$$U = \left\{ (i, j) : M\left(\frac{|\phi\_{m\bar{n}j}(\mathbf{x}^{pq}) - a|}{\rho}\right)^{p\_{\bar{\eta}}} \le \delta \right\}$$

ð Þ then <sup>U</sup><sup>c</sup> <sup>∈</sup>I: Since <sup>x</sup> pq ! <sup>x</sup>, then there exists <sup>p</sup>0, q<sup>0</sup> <sup>∈</sup> <sup>N</sup> such that,

$$P = \left\{ (i, j) : M \left( \frac{|\phi\_{miij}(\mathbf{x}^{p\_0q\_0}) - \phi\_{miij}(\mathbf{x})|}{\rho} \right)^{p\_{\bar{\eta}}} < \left( \frac{\delta}{3D} \right)^H \right\} \tag{25}$$

� � � � n o <sup>1</sup> where <sup>D</sup> <sup>¼</sup> max <sup>1</sup>; <sup>2</sup><sup>G</sup>� , G <sup>¼</sup> supij pij <sup>≥</sup> <sup>0</sup> and <sup>H</sup> <sup>¼</sup> max <sup>1</sup>;supij pij implies P<sup>c</sup> ∈I. The number p0; q<sup>0</sup> can be so chosen that together with (25), we have

$$Q = \left\{ (i, j) : M \left( \frac{|a\_{p\_0 q\_0} - a|}{\rho} \right)^{p\_{\bar{\eta}}} < \left( \frac{\delta}{3D} \right)^H \right\}.$$

� � ð Þ pq such that <sup>Q</sup><sup>c</sup> <sup>∈</sup> <sup>I</sup>: Since <sup>x</sup> <sup>∈</sup> <sup>2</sup>M<sup>I</sup> <sup>ð</sup>M; <sup>p</sup>Þ. ij BV<sup>σ</sup> We have

$$\left\{ (i,j) : \mathcal{M} \Big( \frac{|\phi\_{m:ij}(\mathfrak{x}^{p\_0q\_0}) - a\_{p\_0q\_0}|}{\rho} \Big)^{p\_{\bar{\eta}}} \ge \delta \right\} \in I. \mathbb{1}$$

Then we have <sup>a</sup> subset <sup>S</sup> <sup>⊆</sup> <sup>N</sup> � <sup>N</sup> such that <sup>S</sup><sup>c</sup> <sup>∈</sup> <sup>I</sup>, where

$$\mathcal{S} = \left\{ (i, j) : \mathcal{M} \left( \frac{|\phi\_{mnij}(\mathcal{X}^{p\_0q\_0}) - a\_{p\_0q\_0}|}{\rho} \right)^{p\_{\bar{\eta}}} < \left( \frac{\delta}{3D} \right)^H \right\}.$$

Let <sup>U</sup><sup>c</sup> <sup>¼</sup> <sup>P</sup><sup>c</sup> <sup>∪</sup> <sup>Q</sup><sup>c</sup> <sup>∪</sup> <sup>S</sup><sup>c</sup> , where

$$U = \left\{ (i, j) : M \left( \frac{|\phi\_{m \bar{n} j}(\infty) - a|}{\rho} \right)^{p\_{\bar{y}}} < \delta \right\}.$$

Therefore, for ð Þ <sup>i</sup>; <sup>j</sup> <sup>∈</sup> <sup>U</sup> , we have <sup>c</sup>

$$\begin{split} & \left\{ (i, j) : \mathcal{M} \Big( \frac{|\phi\_{\min\bar{\jmath}}(\mathbf{x}) - a|}{\rho} \Big)^{p\_{\bar{\jmath}}} < \delta \right\} \\ \geq & \left[ \left\{ (i, j) : \mathcal{M} \Big( \frac{|\phi\_{\min\bar{\jmath}}(\mathbf{x}^{p\_{0}q\_{0}}) - \phi\_{\min\bar{\jmath}}(\mathbf{x})|}{\rho} \Big)^{p\_{\bar{\jmath}}} < \left( \frac{\delta}{3D} \right)^{H} \right\} \\ \subset & \left\{ (i, j) : \mathcal{M} \Big( \frac{|a\_{p\_{0}q\_{0}} - a|}{\rho} \Big)^{p\_{\bar{\jmath}}} < \left( \frac{\delta}{3D} \right)^{M} \right\} \\ \subset & \left\{ (i, j) : \mathcal{M} \Big( \frac{|\phi\_{\min\bar{\jmath}}(\mathbf{x}^{p\_{0}q\_{0}}) - a\_{p\_{0}q\_{0}}|}{\rho} \Big)^{p\_{\bar{\jmath}}} < \left( \frac{\delta}{3D} \right)^{H} \right\}. \end{split}$$

Hence the result <sup>2</sup>M<sup>I</sup> <sup>ð</sup>M; <sup>p</sup>Þ<sup>⊂</sup> <sup>2</sup>l∞ðM; <sup>p</sup><sup>Þ</sup> follows. BV<sup>σ</sup>

� � Since the inclusions <sup>2</sup>M<sup>I</sup> <sup>ð</sup>M; <sup>p</sup> ð Þ and 2 0M<sup>I</sup> <sup>ð</sup>M; <sup>p</sup><sup>Þ</sup> ð Þ BV<sup>σ</sup> <sup>Þ</sup><sup>⊂</sup> <sup>2</sup>l<sup>∞</sup> <sup>M</sup>; <sup>p</sup> BV<sup>σ</sup> <sup>⊂</sup> <sup>2</sup>l<sup>∞</sup> <sup>M</sup>; <sup>p</sup>

are strict so in view of Theorem (3.3), we have the following result.

The above theorem is interesting and itself will have various applications in our future work.
