Existence and uniqueness results for nonlinear fractional differential equations via new Q-function

In this paper, we first introduce a new concept of Q-function on partial metric spaces. Also, we give a new definition of $$(\alpha ,\phi ,q)$$ -contractive mapping by considering the new kind of Q-function. Then we obtain some best proximity point results for such mappings. Thus, we improve and unify many well-known results in the literature. Moreover, we provide some illustrative and nontrivial examples. Therefore, we show that the approach of Haghi et al. (Topol Appl 160:450–454, 2013) cannot be applied to our results. Finally, we obtain a solution of nonlinear fractional differential equations using the new Q-function.