Fractional operators with boundary points dependent kernels and integration by parts

Abstract

Recently, U. N. Katugampola presented some generalized fractional integrals and derivatives by iterating a $ t^{\rho-1}- $weighted integral, $ \rho>0 $. The case $ \rho = 1 $ produces Riemann and Caputo fractional derivatives and the limiting case $ \rho\rightarrow 0^+ $ results in Hadamard type fractional operators. In this article, we discuss the differences between a new class of nonlocal generalized fractional derivatives generated by iterating left and right type conformable integrals weighted by $ (t-a)^{\rho-1} $ and $ (b-t)^{\rho-1} $ and the ones introduced by Katugampola. In fact, we will present very different integration by parts formulas by presenting new mixed left and right generalized fractional operators with boundary points dependent kernels. The properties of this new class of mixed fractional operators are analyzed in newly defined function spaces as well.