Conformable Fractional Semigroups of Operators

TL;DR

This paper introduces conformable fractional semigroups of operators on Banach spaces, defining their generators as fractional derivatives at zero and exploring their fundamental properties.

Contribution

It presents a novel concept of fractional semigroups with generators defined via conformable fractional derivatives, expanding the theory of operator semigroups.

Findings

Defined conformable fractional semigroups of operators.

Established basic properties of these fractional semigroups.

Connected fractional derivatives to the generators of semigroups.

Abstract

Let $X$ be a Banach space, and $T:[0,\infty)\rightarrow {\mathcal{L}}(X,X),$ the bounded linear operators on $X.$ A family $\{T(t)\}_{t\ge 0}\subseteq {% \mathcal{L}}(X,X)$ 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.