## Abstract

In this paper, we propose a new concept of derivative with respect to an arbitrary kernel function. Several properties related to this new operator, like inversion rules and integration by parts, are studied. In particular, we introduce the notion of conjugate kernels, which will be useful to guaranty that the proposed derivative operator admits a right inverse. The proposed concept includes as special cases Riemann-Liouville fractional derivatives, Hadamard fractional derivatives, and many other fractional operators. Moreover, using our concept, new fractional operators involving certain special functions are introduced, and some of their properties are studied. Finally, an existence result for a boundary value problem involving the introduced derivative operator is proved.

Original language | British English |
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Pages (from-to) | 137-160 |

Number of pages | 24 |

Journal | Mathematical Methods in the Applied Sciences |

Volume | 42 |

Issue number | 1 |

DOIs | |

State | Published - 15 Jan 2019 |

## Keywords

- boundary value problem
- conjugate kernels
- fractional calculus
- k-derivative