Representations and properties of a new family of ω-Caputo fractional derivatives

In the most general case of ω-weights, some normed functional spaces X p ω a, b 1 ≤ p ≤ ∞ , ACn γ,ω[a, b] and a generalization of the fractional integro-differentiation operator are introduced and analyzed. The boundedness of the ω-weighted fractional operator over X p ω a, b is proved. Some theorems and lemmas on the properties of the invertions of the mentioned operator and several representations of functions from ACn γ,ω[a, b] are established. A general ω-weighted Caputo fractional derivative of order α is studied over ACn γ,ω[a, b]. Some representations and other properties of this fractional derivative are proved. Some conclusions are presented.