On Ulam Stability of a Generalized Delayed Differential Equation of Fractional Order

We investigate Ulam stability of a general delayed differential equation of a fractional order. We provide formulas showing how to generate the exact solutions of the equation using functions that satisfy it only approximately. Namely, the approximate solution $$\phi $$ ϕ generates the exact solution as a pointwise limit of the sequence $$\varLambda ^n\phi $$ Λ n ϕ with some integral (possibly, nonlinear) operator $$\varLambda $$ Λ . We estimate the speed of convergence and the distance between those approximate and exact solutions. Moreover, we provide some exemplary calculations, involving the Chebyshev and Bielecki norms and some semigauges, that could help to obtain reasonable outcomes for such estimations in some particular cases. The main tool is the Diaz–Margolis fixed point alternative.

Persistence and extinction of Markov switched stochastic Nicholson’s blowflies delayed differential equation

In this paper, we study the persistence and extinction of Markov switched stochastic Nicholson’s blowflies delayed differential equation. We derive sufficient conditions of persistence and extinction for blowflies population, respectively, which solve one of open problems proposed by Zhu et al. [Stochastic Nicholson’s blowflies delay differential equation with regime switching, Appl. Math. Lett. 94 (2019) 187–195].

Analysis of a Crowley-Martin Type Chemostat with Delayed Growth Response and Pulsed Input

In this paper, we introduce and study a Crowley-Martin type Chemostat model with delayed growth response and pulsed input. We get that the existence and the global attractivity of a ‘microorganism-extinction’periodic solution. We prove that the system is permanent under appropriate conditions, by use of new computational techniques for impulsive and delayed differential equation.