N_b $-fuzzy metric spaces with topological properties and applications

Our aim is to introduce the notion of $ N_{b} $-fuzzy metric space (FMS). We also define quasi $ N $-FMS, and pseudo $ N_{b} $-FMS with examples and counterexamples and prove a decomposition theorem for pseudo $ N_{b} $-FMS. We prove various theorems related to the convergence of sequences and analyze topology of symmetric $ N_{b} $-FMS. At last, we provide an application of $ q $-contraction mapping as a Banach contraction principle (BCP) in the structure of symmetric $ N_{b} $-FMS and applied it in the solution of integral equations and linear equations.