In this paper, we discuss the existence, uniqueness and stability of boundary value problems for $ \psi $-Hilfer fractional integro-differential equations with mixed nonlocal (multi-point, fractional derivative multi-order and fractional integral multi-order) boundary conditions. The uniqueness result is proved via Banach's contraction mapping principle and the existence results are established by using the Krasnosel'ski${\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over i} }}$'s fixed point theorem and the Larey-Schauder nonlinear alternative. Further, by using the techniques of nonlinear functional analysis, we study four different types of Ulam's stability, i.e., Ulam-Hyers stability, generalized Ulam-Hyers stability, Ulam-Hyers-Rassias stability and generalized Ulam-Hyers-Rassias stability. Some examples are also constructed to demonstrate the application of main results.

In this paper, we developed a nonlinear mathematical model for the transmission of the monkeypox virus among populations of humans and rodents under the fractal-fractional operators in the context of Atangana-Baleanu. For the theoretical analysis, the renowned theorems of fixed points, like Banach's and Krasnoselskii's types, were used to prove the existence and uniqueness of the solutions. Additionally, some results regarding the stability of the equilibrium points and the basic reproduction number were provided. In addition, the numerical schemes of the considered model were established using the Adams-Bashforth method. Our analytical findings were supported by the numerical simulations to explain the effects of changing a few sets of fractional orders and fractal dimensions. Some graphic simulations were displayed with some parameters calculated from real data to understand the behavior of the model.

In this work, we introduce a novel idea of generalized $({{k}},\psi )$( k , ψ ) -Hilfer proportional fractional operators. The proposed operator combines the $({{k}},\psi )$( k , ψ ) -Riemann–Liouville and $({{k}},\psi )$( k , ψ ) -Caputo proportional fractional operators. Some properties and auxiliary results of the proposed operators are investigated. The ψ-Laplace transform and its properties of the proposed operators are established and utilized to solve Cauchy-type problems. Furthermore, the uniqueness result for a higher-order initial value problem under $({{k}},\psi )$( k , ψ ) -Hilfer proportional fractional operators is proved by using Picard’s iterative technique. At the end, examples are provided to present the theoretical results. This new type of proposed operator can help other researchers who are still working on real-world problems.

In this paper, we investigate the existence and Ulam–Hyers–Rassias stability results for a class of boundary value problems for implicit $\psi$-Caputo fractional differential equations with non-instantaneous impulses involving both retarded and advanced arguments. The results are based on the Banach contraction principle and Krasnoselskii’s fixed point theorem. In addition, the Ulam–Hyers–Rassias stability result is proved using the nonlinear functional analysis technique. Finally, illustrative examples are given to validate our main results.

A mathematical model of the nutrient-phytoplankton-zooplankton associated with viral infection in phytoplankton under the Atangana-Baleanu derivative in Caputo sense is investigated in this study. We prove the theoretical results for the existence and uniqueness of the solutions by using Banach’s and Sadovskii’s fixed point theorems. The notion of various Ulam’s stability is used to guarantee the context of the stability analysis. Furthermore, the equilibrium points and the basic reproduction numbers for the proposed model are provided. The Adams type predictor-corrector algorithm has been applied for the theoretical confirmation to establish the approximate solutions. A variety of numerical plots corresponding to various fractional orders between zero and one are presented to describe the dynamical behavior of the fractional model under consideration.

In this paper, we establish the existence and stability results for the $({\rho }_k,{\varphi }_k)$-Hilfer fractional integro-differential equations under instantaneous impulse with non-local multi-point fractional integral boundary conditions. We achieve the formulation of the solution to the $({\rho }_k,{\varphi }_k)$-Hilfer fractional differential equation with constant coefficients in term of the Mittag–Leffler kernel. The uniqueness result is proved by applying Banach’s fixed point theory with the Mittag–Leffler properties, and the existence result is derived by using a fixed point theorem due to O’Regan. Furthermore, Ulam–Hyers stability and Ulam–Hyers–Rassias stability results are demonstrated via the non-linear functional analysis method. In addition, numerical examples are designed to demonstrate the application of the main results.