Partial differential equations is a many-faceted subject. Created to describe the mechanical behavior of objects such as vibrating strings and blowing winds, it has developed into a body of material that interacts with many branches of mathematics, such as differential geometry, complex analysis, and harmonic analysis, as well as a ubiquitous factor in the description and elucidation of problems in mathematical physics. The goal of this work is to make more precise the operator approach for some evolution partial differential equations and extend the theory to semilinear operator systems. More exactly,we shall precise basic properties, such as norm estimation and compactness, for the (linear)solution operator associated to the non-homogeneous linear evolution equations and we shall use them in order to apply the Banach, Schauder and Leray-Schauder theorems to the fixed point problems equivalent to Chaichy-Dirichlet problems for evolution equations. We extend these results to the corresponding semilinear operator system.