Fractional integration and differentiation of variable order
The paper deals with the generalized fractional integration and differentiation operators of variable order \(\alpha (x)\): \[ I_+^{\alpha (x)} \varphi= {1\over {\Gamma (\alpha (x))}} \int^x_{-\infty} {{\varphi (t)dt} \over {(x-t)^{1- \alpha(x)}}} \;(\alpha (x)>0) \;\text{ and } \;D_+^{\alpha (x)}= \lim_{\varepsilon \to 0} D^{\alpha (x)}_{+,\varepsilon}f\;\;(0< \alpha (x)< 1) \] where \[ D^{\alpha (x)}_{+, \varepsilon} f= {{\alpha(x)} \over {\Gamma (1- \alpha (x))}} \int^\infty_\varepsilon {{f(x)- f(x-t)} \over {t^{1+ \alpha (x)}}} dt. \] When \(\alpha (x)= \alpha\) is a constant, \(I_+^\alpha \varphi\) is the classical Liouville fractional integral and \(D_+^\alpha f\) the Marchaud fractional derivative [see Section 5 in the book by the author, the reviewer and \textit{O. I. Marichev}: ``Integrals and derivatives of fractional order and some of their applications'' (1987; Zbl 0617.26004; English translation 1993; Zbl 0818.26003)]. It is known (see Section 13 in the books above) that \[ \lim_{\varepsilon \to 0} D^\alpha_{+, \varepsilon} I^\alpha_+ \varphi= \varphi \] in \(L_p\)-norm and hence \(D^\alpha_+= (I^\alpha_+ )^{-1}\). The present paper is devoted to study the comparison between the operators \(D_+^{\alpha (x)}\) and \((I_+^{\alpha (x)} )^{-1}\). Such a comparison is based on the following results: the integral representation for the composition \(D^{\alpha (x)}_{+, \varepsilon} I_+^{\alpha (x)} \varphi\), the relation \[ \lim_{\varepsilon\to 0} D^{\alpha (x)}_{+, \varepsilon} I_+^{\alpha (x)} \varphi= \varphi(x)+ \int^x_{-\infty} K(x, x-y) \varphi (y) dy, \] and the compactness of the integral operator above in the space \(L^p (-\infty, N)\) for \(N< \infty\) and \(1<p< (\alpha (-\infty) )^{-1}\).
generalized fractional integration and differentiation operators of variable order
scientific article; zbMATH DE number 817241
scientific article