Some new classes of ideal convergent difference multiple sequences of fuzzy real numbers

Abstract

The idea of difference sequence spaces (for single sequences) was first introduced by Kizmaz in 1981 and the idea of triple sequences was first introduced by Sahiner et al. 2007. In this article, we introduce some new classes of ideal convergent difference multiple sequence spaces ${}_3\left(c^{I\left(F\right)}\right)(\Delta ,p),{}_3\left(c_0^{I\left(F\right)}\right)(\Delta ,p)$ and ${}_3\left(c_0^{I\left(F\right)}\right)(\Delta ,p),$of fuzzy real numbers using a difference operator Δ, where p =〈 p _nlk  〉 is a triple sequence of bounded strictly positive numbers. We study some basic algebraic and topological properties of these spaces. We also investigate the relations related to these spaces. It is shown that the sequence spaces ${}_3\left(c^{I\left(F\right)}\right)(\Delta ,p),{}_3\left(c_0^{I\left(F\right)}\right)(\Delta ,p),{}_3\left(m^{I\left(F\right)}\right)(\Delta ,p)$ and ${}_3\left(m_0^{I\left(F\right)}\right)(\Delta ,p)$ are closed under addition and scalar multiplication also these spaces are sequence algebras. We have proved that the sequence space ${}_3\left(m_0^{I\left(F\right)}\right)(\Delta ,p)$ is solid as well as monotone. We have obtained the inclusion relation ${}_3\left(m_0^{I\left(F\right)}\right)(\Delta ,p)\subset {}_3\left(m^{I\left(F\right)}\right)(\Delta ,p)\subset {}_3\left({\ell }_{\infty }^F\right)(\Delta ,p)$ where the inclusions are strict. We have also proved that the sequence spaces ${}_3\left(m_0^{I\left(F\right)}\right)(\Delta ,p)$ and ₃ (m^I(F)) (Δ, p) are nowhere dense subsets of ${}_3\left({\ell }_{\infty }^F\right)(\Delta ,p).$