Picard $S$-type semi implicit mid-point method for fixed point approximation with applications

In this article, our goal is to propose and design a hybrid Picard $S$-type semi-implicit mid-point method to approximate the fixed point of a contractive-like mapping. The convergence result and stability of the proposed method are established under suitable assumptions. We implemented the newly constructed method to approximate the common element, which is the fixed point of a contractive-like mapping and simultaneously solves a general variational inequality. Finally, the significance of the proposed scheme is illustrated through the study of a fractional diffusion equation.