<abstract><p>In this manuscript, we study the existence and Ulam's stability results for impulsive multi-order Caputo proportional fractional pantograph differential equations equipped with boundary and integral conditions with respect to another function. The uniqueness result is proved via Banach's fixed point theorem, and the existence results are based on Schaefer's fixed point theorem. In addition, the Ulam-Hyers stability and Ulam-Hyers-Rassias stability of the proposed problem are obtained by applying the nonlinear functional analysis technique. Finally, numerical examples are provided to supplement the applicability of the acquired theoretical results.</p></abstract>

This manuscript investigates an extended boundary value problem for a fractional pantograph differential equation with instantaneous impulses under the Caputo proportional fractional derivative with respect to another function. The solution of the proposed problem is obtained using Mittag–Leffler functions. The existence and uniqueness results of the proposed problem are established by combining the well-known fixed point theorems of Banach and Krasnoselskii with nonlinear functional techniques. In addition, numerical examples are presented to demonstrate our theoretical analysis.

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.

This paper investigates existence, uniqueness, and Ulam’s stability results for a nonlinear implicit ψ-Hilfer FBVP describing Navier model with NIBCs. By Banach’s fixed point theorem, the unique property is established. Meanwhile, existence results are proved by using the fixed point theory of Leray-Schauder’s and Krasnoselskii’s types. In addition, Ulam’s stability results are analyzed. Furthermore, several instances are provided to demonstrate the efficacy of the main results.