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In the present paper we introduce some vector-valued statistical convergent sequence spaces defined by a sequence of modulus functions associated with multiplier sequences and we also make an effort to study some topological properties and inclusion relation between these spaces.

In this paper we introduce a new generalized vector-valued paranormed sequence spaces \(N_p\left(E_k, \triangle_u^m, f, s\right)\) using modulus function \(f\), where \(p=\left(p_k\right)\) is a bounded sequence of positive real numbers such that \(\inf _k p_k>\) \(0,\left(E_k, q_k\right)\) is a sequence of seminormed spaces with \(E_{k+1} \subseteq E_k\) for each \(k \in N\) and \(s \geq 0\). We prove results regarding completeness, \(K\)-space, normality, inclusion relation are derived. These are more general than those of Ruckle [7], Maddox [5], Ozturk and Bilgin [6], Sahiner [8], Atlin et al. [1] and Srivastava and Kumar [9].

In this paper we introduce a new generalized vector-valued paranormed sequence spaces \(N_p\left(E_k, \triangle_u^m, f, s\right)\) using modulus function \(f\), where \(p=\left(p_k\right)\) is a bounded sequence of positive real numbers such that \(\inf _k p_k>\) \(0,\left(E_k, q_k\right)\) is a sequence of seminormed spaces with \(E_{k+1} \subseteq E_k\) for each \(k \in N\) and \(s \geq 0\). We prove results regarding completeness, \(K\)-space, normality, inclusion relation are derived. These are more general than those of Ruckle [7], Maddox [5], Ozturk and Bilgin [6], Sahiner [8], Atlin et al. [1] and Srivastava and Kumar [9].

In the present paper we introduce some vector-valued statistical convergent sequence spaces defined by a sequence of modulus functions associated with multiplier sequences and we also make an effort to study some topological properties and inclusion relation between these spaces.

In this paper we study sequence spaces equations (SSE) with operators, which are determined by an identity whose each term is a sum or a sum of products of sets of the form $\chi _{a}\left( T\right)$ and $\chi _{f\left( x\right) }\left( T\right)$ where $f$ maps $U^{+}$ to itself, $\chi $ is either of the symbols $s$, $s^{0}$, or $s^{\left( c\right) }$. Then we solve five (SSE) of the form $\chi _{a}+\chi _{x}^{\prime }=\chi _{b}^{\prime }$, where $\chi $, $\chi ^{\prime }$ are either $s^{{{}0}}$, $s^{\left( c\right) }$, or $s$. We apply the previous results to the solvability of the systems $s_{a}^{0}+s_{x}\left( \Delta \right) =s_{b}$, $s_{x}\supset s_{b}$ and $s_{a}+s_{x}^{\left( c\right) }\left( \Delta \right) =s_{b}^{\left( c\right) }$, $s_{x}^{\left( c\right) }\supset s_{b}^{\left( c\right) }$. Finally we solve the (SSE) with operators defined by $\chi _{a}\left( C\left( \lambda \right) D_{\tau }\right) +s_{x}^{\left( c\right) }\left( C\left( \mu \right) D_{\tau }\right) =s_{b}^{\left( c\right) }$ where $\chi $ is either $s^{0}$, or $s$.