Existence and uniqueness of fractal-fractional equations generated by a new fractal-fractional operator utilizing the advanced gamma function

Numerous special functions, including the beta function, hypergeometric functions, and other orthogonal polynomials, are closely connected to the gamma function. Recently, gamma function has been enhanced by adding a new parameter. As a consequence, this gamma is called the parametric gamma function or b -gamma function. By utilizing the enhanced gamma function (or the parametric gamma function), we have present a generalization for the special function Rabotnov function. Consequently, new fractal-fractional operators (derivative and integral) involving the generalized Rabotnov function are defined. Analysis is introduced to discover the main properties of the suggested operators. Since the main challenge in the calculus of fractal-fractional operators is to design examples, we illustrate a set of examples including power series. We explore the boundedness of the recommended operators. There are many gains for checking the boundedness of fractal operators in general and fractal-fractional operators in particular. Moreover, as an application, we establish the existence and uniqueness solution of abstract fractal-fractional equation. Examples are presented at the end of the effort. Graphics and computations are observed using MATHEMATICA 13.3 software. By using the parametric gamma function, the Rabotnov special function is generalized. New fractal-fractional operators are presented using the generalized Rabotnov function with examples; Boundedness of these operators is investigated, where bounded operators give a strong foundation for understanding linear transformations across normed spaces, with many applications in science and mathematics. Conditions of the existence and uniqueness of solutions of fractal-fractional differential abstract equation are established.