Study of the Relationship Between Two Types of Mixed-Order Evolution Equations and the Existence of Their Solutions

The purpose of this article is to study the relationship between two types of mixed-order evolution equations and the existence of their solutions. Initially, the mild solutions of the two system are obtained by Laplace transform. Then, we use semigroup theorem and sector operator theorem to get the norm estimation results of under- standing operator through special paths in the complex plane. Further, the sufficient conditions for existence and uniqueness of mild solution of the proposed system are verified by applying fixed point theorems. Finally, examples are provided to illustrate the main results.