Geodesic
Singular regularization of operator equations in \(L_1\) spaces via fractional differential equations
Electronic journal of differential equations, Tome 2016 (2016)
An abstract causal operator equation y=Ay defined on a space of the form

$ \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.
Classification : 34K35, 34A08, 65J20
Keywords: causal operator equations, fractional differential equations, regularization, Banach space 
@article{EJDE_2016__2016__a19,
     author = {Karakostas,  George L. and Purnaras,  Ioannis K.},
     title = {Singular regularization of operator equations in {\(L_1\)} spaces via fractional differential equations},
     journal = {Electronic journal of differential equations},
     year = {2016},
     volume = {2016},
     zbl = {1329.34121},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2016__2016__a19/}
}
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Karakostas,  George L.; Purnaras,  Ioannis K. Singular regularization of operator equations in \(L_1\) spaces via fractional differential equations. Electronic journal of differential equations, Tome 2016 (2016). http://geodesic.mathdoc.fr/item/EJDE_2016__2016__a19/