# American Institute of Mathematical Sciences

March  2020, 13(3): 351-375. doi: 10.3934/dcdss.2020020

## Fractional operators with boundary points dependent kernels and integration by parts

 1 Department of Mathematics and General Sciences, Prince Sultan University, P. O. Box 66833, 11586 Riyadh, Saudi Arabia 2 Department of Medical Research, China Medical University, Taichung 40402, Taiwan 3 Department of Computer Science and Information Engineering, Asia University, Taichung, Taiwan

Received  April 2018 Revised  May 2018 Published  March 2019

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.

Citation: Thabet Abdeljawad. Fractional operators with boundary points dependent kernels and integration by parts. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 351-375. doi: 10.3934/dcdss.2020020
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