Continuity and Fixed Point of a new extension of $F$-Suzuki-Contraction Mappings in b-metric Spaces with Application

In this paper, firstly, we introduce a new extension of F-Suzuki-contraction mappings namely generalized $F_p$-Suzuki contraction. Moreover, we prove a fixed point theorem for such contraction mappings even without considering the completeness condition of space. In the following, we respond the open question of Rhoades(see Rhoades \cite{23}, p.242) regarding existence of a contractive definition which is strong enough to generate a fixed point but dose not force the mapping to be continuous at the fixed point.  Also, we provide some examples show that our main theorem is a generalization of previous results. Finally, we give an application to the boundary value problem of a nonlinear fractional differential equation for our results.