Singular regularization of operator equations in L1 spaces via fractional differential equations

Abstract

An abstract causal operator equation y=Ay defined on a space of the form L1([0,τ],X), with X a Banach space, is regularized by the fractional differential equation ε(Dα0yε)(t) = -yε(t) + (Ayε)(t), t ∈ [0,τ], where Dα0 denotes the (left) Riemann-Liouville derivative of order α ∈ (0,1). The main procedure lies on properties of the Mittag-Leffler function combined with some facts from convolution theory. Our results complete relative ones that have appeared in the literature; see, e.g. [5] in which regularization via ordinary differential equations is used.