Cauchy problem for non-autonomous fractional evolution equations with nonlocal conditions of order $(1,2)$

This article contracts through Cauchy problems in infinite-dimensional Banach spaces towards a system of nonlinear non-autonomous mixed type integro-differential fractional evolution equation by nonlocal conditions through noncompactness measure (MNC). We demonstrate the existence of novel mild solutions in the condition that the nonlinear function mollifies generally adequate, an MNC form and local growth form, using evolution families and fractional calculus theory, as well as the fixed-point theorem w.r.t. K-set-contractive operator and another MNC assessment procedure. Our findings simplify and improve upon past findings in this area. Finally, towards the end of this article, as an example of submissions, we use a fractional non-autonomous partial differential equation (PDE) with nonlocal conditions and a homogeneous Dirichlet boundary condition.