FRACTIONAL DIFFERENTIAL EQUATIONS AND FRACTIONAL SEMIGROUPS OF OPERATORS

In this paper, the authors investigate the solution of a second-order fractional differential equation. They consider cases where either the variable u or v is missing. The paper also introduces the concept of a fractional semigroup of operators. This involves a mapping that represents bounded linear operators on a Banach space U. The one-parameter semigroup satisfies certain properties, such as the semigroup property and having an identity operator at y = 0. The derivative of this semigroup at y = 0 is called the infinitesimal generator. The main objective of the paper is to explore the fundamental properties of these fractional semigroups and their connection to the fractional derivative of the semigroup at y = 0. We confirm that we are using the definition of Al-Sharif and Malkawi [22] for the conformable fractional derivative.