Fractional operators with Kaniadakis logarithm kernels

Abstract

In this article, more general types of fractional operators with κ-deformed logarithm kernels are proposed. We analyse the new operators and prove various facts about them, including a semi group property. Results of existence are established in appropriate functional spaces. We prove that these results are valid at once for several standard fractional operators such as the Riemann-Liouville and Caputo operators, the Hadamard operators depending on the of the scaling function. We also show that our technique can beuseful to solve a wide range of Volterra integral equations. Finally, the solutions of theκ-fractional differential equations can be deduced from the solution representation of theCaputo or Riemann-Liouville versions via scaling.