A new class of nonlocal boundary value problems consisting of multi-term delay fractional differential equations and multipoint-integral boundary conditions is studied in this paper. We derive a more general form of the solution for the given problem by applying a fractional integral operator of an arbitrary order ${\beta }_{\xi }$ instead of ${\beta }_1$; for details, see Lemma 2. The given problem is converted into an equivalent fixed-point problem to apply the tools of fixed-point theory. The existence of solutions for the given problem is established through the use of a nonlinear alternative of the Leray-Schauder theorem, while the uniqueness of its solutions is shown with the aid of Banach's fixed-point theorem. We also discuss the stability criteria, icluding Ulam-Hyers, generalized Ulam-Hyers, Ulam-Hyers-Rassias, and generalized Ulam-Hyers-Rassias stability, for solutions of the problem at hand. For illustration of the abstract results, we present examples. Our results are new and useful for the discipline of multi-term fractional differential equations related to hydrodynamics. The paper concludes with some interesting observations.

In this work, we studied the multiplicity of solutions for a Kirchhoff problem involving the $\kappa (\xi )$-fractional derivative and critical exponent. More precisely, we transformed the studied problem into an integral equation that lead to the study of the critical point for the energy functional; after that, we presented and proved some properties related to this functional and demonstrated that the energy functional satisfied the geometry of the mountain pass geometry. Finally, by applying the mountain pass theorem for the even functional, we proved that this functional admitted infinitely many critical points, which means that the studied problem has infinitely many solutions.