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In this paper, a new Monod type chemostat model with delay and impulsive input on two substrates is considered. By using the global attractivity of a k times periodically pulsed input chemostat model, we obtain the bound of the system. By the means of a fixed point in a Poincaré map for the discrete dynamical system, we obtain a semi-trivial periodic solution; further, we establish the sufficient conditions for the global attractivity of the semi-trivial periodic solution. Using the theory on delay functional and impulsive differential equations, we obtain a sufficient condition with time delay for the permanence of the system.

This paper studies a class of nonlinear neutral set-valued functional differential equations. The globally asymptotic stability theorem with necessary and sufficient conditions is obtained via the fixed point method. Meanwhile, we give an example to illustrate the obtained result.

This paper introduces the notions of partially equi-integral stability and partially equi-integral φ 0 -stability for two differential systems, and establishes some criteria on stability relative to the xcomponent by using the cone-valued Lyapunov functions and the comparison technique. An example is also given to illustrate our main results.

In this paper, we investigate the stability of set differential equations in Fréchet space F . Some comparison principles and stability criteria are established for set differential equations with the fact that every Fréchet space F is a projective limit of Banach spaces.