In this manuscript, we study fractional Langevin equations(FLEs) involving Hadamard-Caputo’s derivative of distinct orders associated with nonlocal integral and nonperiodic boundary conditions. The Hyres-Ulam (HU) stability, existence and uniqueness(EU) of solutions are established for proposed equations. Our prospective is based on the Hadamard-Caputo’s derivatives and implementation of Krasnoselskii’s fixed point theorem and Banach contraction mapping principle. An application is offered to smooth the understanding of the theoretical results.

In this article, we discuss the controllability and stability of Hilfer fractional evolution equations in Banach space. We derive these results by first proving the existence and uniqueness of mild solutions for proposed system of equations. Existence and uniqueness results are obtained with the help of theory of propagation family, techniques of measure of non-compactness and fixed point theorem. An example is also given for the demonstration of obtained results.