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Abstract: We consider a scalar integral equation where , while has a significant singularity, but is convex when . We construct a Liapunov functional and show that and that pointwise as . Small perturbations are also added to the kernel. In addition, we consider both infinite and finite delay problems. This paper offers a first step toward treating discontinuous kernels with Liapunov functionals. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we study the existence of almost periodic solutions of a delay logistic model with fixed moments of impulsive perturbations. By using a comparison theorem and constructing a suitable Lyapunov functional, a set of sufficient conditions for the existence and global attractivity of a unique positive almost periodic solution is obtained. As applications, some special models are studied; our new results improve and generalize former results. [ABSTRACT FROM AUTHOR]

Abstract: Some theorems on complete instability of the zero solution relative to a set for nonautonomous nonlinear equations with infinite delay are provided. The right-hand side of the equation is assumed to be defined in a fading memory space and to satisfy conditions that allow the construction of limiting equations. We use conceptions of Lyapunov–Razumikhin pairs and limiting equations to obtain new instability results, which are applicable, in particular, to autonomous, periodic and almost periodic in delay differential equations. [Copyright &y& Elsevier]

Abstract: This paper investigates control for a class of nonlinear systems with time-varying delays and convex polytopic uncertainties. A new type of parameter-dependent Lyapunov–Krasovskii functional is introduced to derive delay-dependent sufficient conditions for the optimal control with exponential stability. All the conditions developed in this paper are formulated in terms of linear matrix inequalities. Finally, a numerical example shows the effectiveness of the proposed methodology. [Copyright &y& Elsevier]

Abstract: This paper deals with the behavior of solutions to the reaction–diffusion system under homogeneous Neumann boundary condition, which describes a prey–predator model with nonlocal delay. Sufficient conditions for the global stability of each equilibrium are derived by the Lyapunov functional and the results show that the introduction of stage-structure into predator positively affects the coexistence of prey and predator. Numerical simulations are performed to illustrate the results. [Copyright &y& Elsevier]

Abstract: For delay difference equations with infinite delay we consider the notion of nonuniform exponential dichotomy. This includes the notion of uniform exponential dichotomy as a very special case. Our main aim is to establish a stable manifold theorem under sufficiently small nonlinear perturbations. We also establish the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations. Finally, we characterize the nonuniform exponential dichotomies in terms of strict Lyapunov sequences. In particular, we construct explicitly a strict Lyapunov sequence for each exponential dichotomy. [Copyright &y& Elsevier]

Abstract: In this paper we mainly study the existence of periodic solutions for a system of delay differential equations representing a simple two-neuron network model of Hopfield type with time-delayed connections between the neurons. We first examine the local stability of the trivial solution, propose some sufficient conditions for the uniqueness of equilibria and then apply the Poincaré–Bendixson theorem for monotone cyclic feedback delayed systems to establish the existence of periodic solutions. In addition, a sufficient condition that ensures the trivial solution to be globally exponentially stable is also given. Numerical examples are provided to support the theoretical analysis. [Copyright &y& Elsevier]

Abstract: A sex-structured mathematical model for heterosexual transmission of HIV/AIDS with explicit incubation period is presented as a system of discrete delay differential equations. The epidemic threshold and equilibria for the model are determined and stabilities are examined. The disease-free equilibrium is shown to be locally and globally stable when the basic reproductive number is less than unity. We use the Lyapunov functional approach to show that the endemic equilibrium is locally asymptotically stable. Further comprehensive qualitative analysis of the model including persistence and permanence are investigated. [Copyright &y& Elsevier]

Abstract: The paper discusses the global exponential stability in the Lagrange sense for a non-autonomous Cohen–Grossberg neural network (CGNN) with time-varying and distributed delays. The boundedness and global exponential attractivity of non-autonomous CGNN with time-varying and distributed delays are investigated by constructing appropriate Lyapunov-like functions. Moreover, we provide verifiable criteria on the basis of considering three different types of activation function, which include both bounded and unbounded activation functions. These results can be applied to analyze monostable as well as multistable biology neural networks due to making no assumptions on the number of equilibria. Meanwhile, the results obtained in this paper are more general and challenging than that of the existing references. In the end, an illustrative example is given to verify our results. [Copyright &y& Elsevier]

Abstract: This paper deals with the stability of a class of impulsive infinite delay differential equations. By using the Razumikhin technique and Lyapunov functions, we obtain a new criterion on the uniform asymptotic stability and global stability for such differential equations. The result is less restrictive and conservative than that given in some earlier references. Also, our result shows that impulses can make unstable systems stable. An example is given to illustrate the feasibility and advantage of the result. [Copyright &y& Elsevier]

In this paper, criteria on uniform ultimate boundedness are derived for impulsive delay differential equations using Lyapunov functions and Razumikhin techniques. It should be noted that the boundedness criteria establish global existence of solutions as well as boundedness without assuming, a priori, that solutions can necessarily be continued to infinity. Those criteria are then used to obtain permanence results for population growth models. Some examples are discussed to illustrate the theorems. [Copyright &y& Elsevier]

Abstract: In this article we present some new nonlinear Bellman–Bihari type integral inequalities with delay for discontinuous functions (integro-sum inequalities; impulse integral inequalities). Some applications of the results are included: conditions of boundedness (uniformly), stability by Lyapunov (uniformly), practical stability by Chetaev (uniformly) for the solutions of impulsive differential and integro-differential systems of ordinary differential equations. [Copyright &y& Elsevier]

Abstract: This paper is devoted to linear parameter systems under linear fractional representations (LFR) of parameter-dependent nonlinear systems with real–rational nonlinearities and point-delayed dynamics. The robust global asymptotic stability of the system either independent of or dependent on the delay sizes is investigated. The associate matrix inequalities are related to the time-derivatives of appropriate Lyapunov functions at all the vertices of the polytope which contains the parameterized uncertainties. [Copyright &y& Elsevier]