In the present paper, we introduce the notion of α-type F-τ-contraction and establish related fixed point results in metric spaces. An example is also given to illustrate our main results and to show that our results are proper generalization of Altun et al. (2015), Miank et al. (2015), Altun et al. (2016) and Olgun et al. (2016). We also obtain fixed point results in the setting of partially ordered metric spaces. Finally, an application is given to set up the existence of positive definite solution of non-linear matrix equation.