Conformable fractional semigroups of operators

Abstract

Let X be a Banach space, and T : [0,∞) → L(X,X), the bounded linear operators on X. A family {T(t)}t≥0⊆ L(X,X) is called a one-parameter semigroup if T(s+t) = T(s)T(t), and T(0) = I, the identity operator on X. The infinitesimal generator of the semigroup is the derivative of the semigroup at t = 0. The object of this paper is to introduce a (conformable) fractional semigroup of operators whose generator will be the fractional derivative of the semigroup at t = 0. The basic properties of such semigroups will be studied.