General fractional operators with fixed memory length: definitions and properties

Abstract This paper proposes the definitions of general fractional operators with fixed memory length in several function spaces. Firstly, some properties of the general fractional integral with fixed memory length, such as boundedness, continuity, semigroup property, commutativity, and Leibniz rule, are discussed. Subsequently, the properties of the general fractional derivatives with fixed memory length, such as continuity and commutativity, are discussed. It is also proven that when the function μ satisfies certain conditions, the general fractional integral and derivatives with fixed memory length preserve the periodicity of periodic functions with same period T . Meanwhile, some illustrative examples are provided to enhance the comprehensibility of the results. Furthermore, fundamental theorem of calculus for general fractional operators with fixed memory length is explored.