On generalized fractional integration by parts formulas and their applications to boundary value problems (Q1983495)

In this paper, the author derives an integration by parts formula for the Hilfer fractional derivative which generalizes the existing integration by parts formula for Riemann-Liouville and Caputo fractional derivatives mentioned in the book by \textit{S. G. Samko} et al. [Fractional integrals and derivatives: theory and applications. Transl. from the Russian. New York, NY: Gordon and Breach (1993; Zbl 0818.26003)]. The results are proved with the help of the well-known Hölder inequality and the concepts of fractional calculus. As an application, boundary value problems involving mixed fractional derivatives (in the Hilfer and Riemann-Liouville sense) are considered. By defining a weak solution to such problems, the existence and uniqueness of a weak solution are shown with the help of Lax-Milgram theorem.