On Hilfer cotangent fractional derivative and a particular class of fractional problems

In this work, a novel Hilfer cotangent fractional derivative is presented. This derivative combines the characteristics of the Riemann-Liouville cotangent fractional derivative and the Caputo cotangent fractional derivative. The essential properties of the newly introduced derivative are discussed. By utilizing this derivative, a nonlinear fractional differential problem with a nonlocal initial condition is investigated, and its equivalence to a cotangent Volterra integral equation is demonstrated. The uniqueness and existence of solutions are established by employing fixed-point theorems. Additionally, two illustrative examples are provided to illustrate the obtained results.