Extended nonlinear variational like-inequalities driven by system of fractional evolutionary equations

In this article, we consider an extended evolutionary system involving fractional differential variational-like inequalities. The system consists of a nonlinear mixed variational-like inequality and an extended fractional differential equation in a complete normed linear space. We examine the non-emptiness, closedness, boundedness and convexity of solution set for the considered nonlinear mixed variational-like inequality in this setting. Furthermore, we establish upper semicontinuity and measurability of set valued solution map of the mixed quasi-variational inequality with respect to both state variable and time variable. Additionally, the existence of mild solutions for the system is shown by means of fractional operator theory, Bohnenblust-Karlin fixed point theorem for multivalued mappings and operator semigroup theory. We conclude by a numerical example, with the response times for different values of α.