On Bi-controlled fuzzy ϕ-normed linear spaces

Abstract

In this paper, from modifying the notion of the  Bag-Samanta's type fuzzy norm, we introduce the concepts of Bi-controlled $\varphi$-norm and Bi-controlled fuzzy $\varphi$-normed linear space, by utilizing a special function $\varphi$ which initiated by Gole$\c{t}$, and two  non-comparable functions $\omega$ and $\mu$ which initiated by N. Saleem et al., respectively. Furthermore, we establish some basic results of Bi-controlled fuzzy $\varphi$-normed linear spaces. Finally, we introduce the concept of $l^{\varphi}_{[\omega,\mu]}$-fuzzy convergent sequence, and investigate the completeness in finite dimensional Bi-controlled fuzzy $\varphi$-normed linear spaces.