A Note on Lacunary Sequence Spaces of Fractional Difference Operator of Orderα,β

Abstract

In the present paper, we defined lacunary sequence spaces of fractional difference operator of order$\left(\alpha ,\beta \right)$over$n$-normed spaces via Musielak-Orlicz function$\mathcal{M}=\left({\mathfrak{I}}_k\right)$. Our aim in this paper is to study some topological properties and inclusion relation between the spaces$\mathcal{I}-N_{\alpha }^{\beta }{\left(A,\mathcal{M},u,\theta ,{\Delta }^{\left(\gamma \right)},‖·,\cdots ,·‖\right)}_0$,$\mathcal{I}-N_{\alpha }^{\beta }\left(A,\mathcal{M},u,\theta ,{\Delta }^{\left(\gamma \right)},‖·,\cdots ,·‖\right)$, and$\mathcal{I}-N_{\alpha }^{\beta }{\left(A,\mathcal{M},u,\theta ,{\Delta }^{\left(\gamma \right)},‖·,\cdots ,·‖\right)}_{\infty }$.