Singular regularization of operator equations in L1 spaces via fractional differential equations
Karakostas, George L.
Purnaras, Ioannis K.
Texas State University, Department of Mathematics
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.  in which regularization via ordinary differential equations is used.
Causal operator equations, Fractional differential equations, Regularization, Banach space
Karakostas, G. L., & Purnaras, I. K. (2016). Singular regularization of operator equations in L1 spaces via fractional differential equations. <i>Electronic Journal of Differential Equations, 2016</i>(01), pp. 1-15.