Electron. J. Diff. Equ., Vol. 2016 (2016), No. 01, pp. 1-15.

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

George L. Karakostas, Ioannis K. Purnaras

Abstract:
An abstract causal operator equation y=Ay defined on a space of the form $L_1([0,\tau],X)$, with X a Banach space, is regularized by the fractional differential equation
$$ \varepsilon(D_0^{\alpha}y_{\varepsilon})(t) =-y_{\varepsilon}(t)+(Ay_{\varepsilon})(t), \quad t\in[0,\tau], $$
where $D_0^{\alpha}$ denotes the (left) Riemann-Liouville derivative of order $\alpha\in(0,1)$. 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.

Submitted June 8, 2015. Published January 4, 2016.
Math Subject Classifications: 34K35, 34A08, 47045, 65J20.
Key Words: Causal operator equations; fractional differential equations; regularization; Banach space.

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George L. Karakostas
Department of Mathematics
University of Ioannina
451 10 Ioannina, Greece
email: gkarako@uoi.gr
Ioannis K. Purnaras
Department of Mathematics
University of Ioannina
451 10 Ioannina, Greece
email: ipurnara@uoi.gr

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