Singular Regularization of Operator Equations in L1 Spaces via Fractional Differential Equations

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(D 0 yε)(t) = −yε(t) + (Ayε)(t), t ∈ [0, τ ], where Dα 0 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.