Sobolev-Type Nonlinear (k,ψ)−Hilfer Fractional Differential Equations With Control: Approximate Controllability Exploration

This paper is concerned with the approximate controllability of Sobolev-type (k,ψ)−Hilfer fractional differential equations (FDEs) with control and Sobolev-type (k,ψ)−Hilfer fractional initial conditions in Hilbert spaces. By means of two operators kSψα,β, kTψα and the k−probability density function, the definition of mild solutions for the studied problem was given. Then, via (k,ψ)−Hilfer fractional derivative and by combining the techniques of fractional calculus and the fixed point theorem, we analyzed the existence and uniqueness of mild solutions. With the help of a Cauchy sequence and approximate techniques, we established some sufficient conditions for the approximate controllability of the proposed control system. Finally, an example is presented for the demonstration of obtained results.