Developments of some new results that weaken certain conditions of fractional type differential equations

We introduce double and triple F-expanding mappings. We prove related fixed point theorems. Based on our obtained results, we also prove the existence of a solution for fractional type differential equations by using a weaker condition than the sufficient small Lipschitz constant studied by Mehmood and Ahmad (AIMS Math. 5:385–398, 2019) and Hanadi et al. (Mathematics 8:1168, 2020). As applications, we ensure the existence of a unique solution of a boundary value problem for a second-order differential equation.