A derivative concept with respect to an arbitrary kernel and applications to fractional calculus

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. KEYWORDS boundary value problem, conjugate kernels, fractional calculus, k-derivative x 0 (x − ) −1 ( ) d , > 0 and derivatives of order (D 0 )(x) = ( d dx ) n (I n− 0 )(x), Math Meth Appl Sci. 2019;42:137-160. wileyonlinelibrary.com/journal/mma

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