Search

In the present survey paper, we prove some results concerning the existence of solutions and weak solutions for some classes of Hadamard and Hilfer fractional differential equations. Some attractivity and Ulam stability results are presented. Our results are provided by applying the fixed point theory, and the technique of strong and weak measures of noncompactness. [ABSTRACT FROM AUTHOR]

Abstract: By using a fixed point theorem of strict-set-contraction, some criteria are established for the existence of positive periodic solutions for a functional differential equation with delays and impulses on time scales. [Copyright &y& Elsevier]

This note presents a new method to design adaptive feedback controller for nonlinear time-delay systems. First, by using the LaSalle-Razumikhin theorem, a sufficient condition is derived that ensures the convergence of a part of the solution with stability for a class of functional differential equations. Then, using this condition, the adaptive stabilization problem is solved for nonlinear time-delay systems. Moreover, it is shown that for a class of nonlinear time-delay systems with triangular structure the adaptive stabilizing controller can be obtained by recursively constructing the Lyapunov-Razumikhin function. It will be shown that the provided recursive design approach, which is obviously motivated by the typical backstepping method, is not a trivial extension of the existing design method. [ABSTRACT FROM AUTHOR]

• We considered an opinion model of threshold-type delay differential equations. • We proved that the model with either a transcritical forward or a backward bifurcation, can exhibit Hopf bifurcation. • We showed that the obtained Hopf bifurcation appears only on upper bifurcated branches of the equilibria. • We performed numerical simulations to illustrate and support our theoretical results. In a recent paper [R. Qesmi, Dynamics of an opinion model with threshold-type delay, Chaos, Solitons & Fractals 98 (2020). (https://doi.org/10.1016/j.chaos.2020.110379)], we proposed a mathematical model of threshold-type delay differential equations describing the relationship between two subpopulations with opposite opinions and the opinion spread dynamics. The study there showed the possibility of a transcritical forward and backward bifurcations of positive equilibria. In the present paper, we show that the opinion model undergoes a Hopf bifurcation through which one of the bifurcation branches loses the stability and periodic solutions appear. One of the important consequences of the obtained dynamics is that the consensus of the both opinions could be lost by maintaining the balance between the time taken for an individual to become convinced of the outsider opinion, which need be short, and the number of individuals converted to the local opinion which need be low. Finally, we provide numerical simulations to illustrate and support our theoretical results. [ABSTRACT FROM AUTHOR]

• We developed a mathematical model of threshold-type delay differential equations • The model describes the relationship between two subpopulations with opposite opinions and the opinion spread dynamics. • We performed mathematical analysis and simulations of the model. • The model exhibits either a forward transcritical or a backward bifurcation. • The model can predict the threshold level of convinced individuals of one opinion that are needed to achieve consensus. • The model is able to study the impact of relevant parameters of the model on the achievement of the consensus. The process of opinion formation rarely boils down to accepting or rejecting the social consensus of others, despite the considerable research (since Sch, 1956 [5]) that has been focused on such situations. In this paper, we propose a mathematical model of threshold-type delay differential equations describing the relationship between two subpopulations with opposite opinions and the opinion spread dynamics. We study the quantitative as well as qualitative properties of our model, and we present results on positivity and boundedness, local stability of equilibria and global stability of the boundary state under certain conditions. Further analysis of the model show that the system exhibits either a forward transcritical or a backward bifurcation. Furthermore, numerical simulations and sensitivity analysis are performed to study the impact of the relevant parameters of the model on the achievement of the consensus. [ABSTRACT FROM AUTHOR]

Functional differential equations have been widely used for modeling real-world phenomena in distinct areas of science. However, classical calculus can not provide always the best description of some complex phenomena, namely those observed in biological systems and medicine. This paper proposes a new numerical method for solving variable order fractional functional differential equations (VO-FFDE). Firstly, the shifted fractional Jacobi collocation method (SF-JC) is applied to solve the VO-FFDE with initial conditions. Then, the SF-JC is applied to the VO-FFDE with boundary conditions. Several numerical examples with different types of VO-FFDE demonstrate the superiority of the proposed method. [ABSTRACT FROM AUTHOR]