Study of the $\phi$-generalized type $k$-fractional integrals or derivatives and some of their properties

Authors: MUSTAFA AYDIN, NAZIM I. MAHMUDOV

Abstract: A novel fractional integral in the sense of Riemann-Liouville integral and two new fractional derivatives in the sense of Riemann-Liouville derivative and Caputo derivative with respect to another function and two parameters are introduced. Some significant properties of them are presented like semigroup property, inverse property, etc. The solution of the Cauchy-type problem for the nonhomogenous linear differential equation with the $\phi$-generalized Caputo $k$-fractional derivative is given by using the method of successive approximation.

Keywords: $\phi$-generalized Riemann-Liouville $k$-fractional integral, $\phi$-generalized Riemann-Liouville $k$-fractional derivative, $\phi$-generalized Caputo $k$-fractional derivative, Cauchy type problem

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