In this article, we study the controllability of ψ-Hilfer fractional differential equations with infinite delay. Sufficient conditions for controllability results are obtained by using the notion of the measure of noncompactness and the Mönch fixed point theorem. The novel feature of this study is to inquire into the controllability notion by using ψ-Hilfer fractional derivative, the generalized variant of the Hilfer derivative. Finally, we provide a numerical example to illustrate our main result.

We present a generalization of Darbo’s fixed point theorem in this article, and we use it to investigate the solvability of an infinite system of fractional order integral equations in space. The fundamental tool in the presentation of our proofs is the measure of noncompactness approach. The suggested fixed point theory has the advantage of relaxing the constraint of the domain of compactness, which is necessary for several fixed point theorems. To illustrate the obtained result in the sequence space, a numerical example is provided. Also, we have applied it to an integral equation involving fractional integral by another function that is the generalization of many fixed point theorems and fractional integral equations.

In this article, we investigated the controllability of fractional dynamical systems (FDS) involving the tempered Ψ‐Caputo fractional derivative (FD). First, we derived the solution representation for this generalized FD with the help of Laplace transform and Mittag–Leffler (M‐L) function. Furthermore, the controllability of the linear system is established using the controllability Gramian matrix. For nonlinear systems, we employ a Schauder’s fixed‐point theorem to determine sufficient conditions for controllability. The major innovation of this paper is the introduction and study of controllability based on the tempered Ψ‐Caputo FD that generalizes and combines some widely used FDs such as the tempered Caputo derivatives. These derivatives are particularly advantageous as they effectively model memory effects and enhance the accuracy of real‐world dynamical systems. To support the theoretical results, a numerical illustration is presented.