In this paper, we investigate a nonlinear generalized fractional differential equation with two-point and integral boundary conditions in the frame of $ \kappa $-Hilfer fractional derivative. The existence and uniqueness results are obtained using Krasnoselskii and Banach's fixed point theorems. We analyze different types of stability results of the proposed problem by using some mathematical methodologies. At the end of the paper, we present a numerical example to demonstrate and validate our findings.

The purpose of this research was to discover a novel method to recover k-fractional integral inequalities of the Pólya–Szegö-type. We employ these generalized inequalities to investigate some new fractional integral inequalities of the Grüss-type. More precisely, we generalize the proportional fractional operators with respect to another strictly increasing continuous function $\psi$. Then, we state and prove some of its properties and special cases. With the help of this generalized operator, we investigate some Pólya–Szegö- and Grüss-type fractional integral inequalities. The functions used in this work are bounded by two positive functions to obtain Pólya–Szegö- and Grüss-type k-fractional integral inequalities in a new sense. Moreover, we discuss some new special cases of the Pólya–Szegö- and Grüss-type inequalities through this work.

This research paper is dedicated to an investigation of an evolution problem under a new operator ($\mathfrak{g}$-Atangana–Baleanu–Caputo type fractional derivative)(for short, $\mathfrak{g}$-ABC). For the proposed problem, we construct sufficient conditions for some properties of the solution like existence, uniqueness and stability analysis. Existence and uniqueness results are proved based on some fixed point theorems such that Banach and Krasnoselskii. Furthermore, through mathematical analysis techniques, we analyze different types of stability results. The symmetric properties aid in identifying the best strategy for getting the correct solution of fractional differential equations. An illustrative example is discussed for the control problem.

In this research paper, we improve some fractional integral inequalities of Minkowski-type. Precisely, we use a proportional fractional integral operator with respect to another strictly increasing continuous function ψ. The functions used in this work are bounded by two positive functions to get reverse Minkowski inequalities in a new sense. Moreover, we introduce new fractional integral inequalities which have a close relationship to the reverse Minkowski-type inequalities viaψ-proportional fractional integral, then with the help of this fractional integral operator, we discuss some new special cases of reverse Minkowski-type inequalities through this work. An open issue is covered in the conclusion section to extend the current findings to be more general.

The main objective of this paper is to obtain a generalization of some Grüss-type inequalities in case of functional bounds by using a generalized Katugampola fractional integral. We obtained new Grüss type inequalitys with functional bounds via the generalized fractional integral operators having same and different parameters. Results obtained are more generalized in nature.