In this article, we established a wide range of fractional mean-type integral inequalities for notable Hilfer fractional derivative using twice differentiable convex and $s$-convex functions for $s\in(0,1]$ with related identities. Also the results for Caputo fractional derivatives are derived as a special case of our general results.

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.

In this paper we apply the fractional integrals with arbitrary order depending on the fractional operators of Riemann type (ABR) and Caputo type (ABC) with kernels of Mittag Lefller in three parameters $E_{\alpha,\mu}^\gamma(\lambda,t)$ for solving the time fractional parabolic nonlinear equation. We utilize these operators with homotopy analysis method (HAM) for constructing the new scheme for generating the successive approximations. This procedure are used successfully on two examples for finding the solutions. The effectiveness and accuracy are verified by clarifying the convergence region in the $h$-curves as well as by calculating the residual error and the results were accurate. Depending on this results, this treatment can be used to find the approximate solutions to many fractional differential equations.