Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions

In this paper, we shall establish sufficient conditions for the existence of integral solutions and extremal integral solutions for some nondensely defined impulsive semilinear functional differential inclusions in separable Banach spaces. We shall rely on a fixed point theorem for the sum of completely continuous and contraction operators. The question of controllability of these equations and the topological structure of the solutions set are considered too.     

Existence and uniqueness of periodic solutions for a class of generalized Liénard systems with forcing term

By using topological degree theory and some analysis skills, we obtain some sufficient conditions for the existence and uniqueness of periodic solutions for a class of forced generalized Liénard systems.     

Sharp conditions for global stability of Lotka-Volterra systems with distributed delays

We give a criterion for the global attractivity of a positive equilibrium of n-dimensional non-autonomous Lotka-Volterra systems with distributed delays. For a class of autonomous Lotka-Volterra systems, we show that such a criterion is sharp, in the sense that it provides necessary and sufficient conditions for the global asymptotic stability independently of the choice of the delay functions. The global attractivity of positive equilibria is established by imposing a diagonal dominance of the instantaneous negative feedback terms, and relies on auxiliary results showing the boundedness of all positive solutions. The paper improves and generalizes known results in the literature, namely by considering systems with distributed delays rather than discrete delays.     

Periodic solutions for a Liénard type p-Laplacian differential equation

By using topological degree theory and some analysis skill, we obtain sufficient conditions for the existence and uniqueness of periodic solutions for Liénard type p$p$-Laplacian differential equation.     

New results of almost periodic solutions for recurrent neural networks

In this paper recurrent neural networks with time-varying delays and continuously distributed delays are considered. Without assuming the global Lipschitz conditions of activation functions, some sufficient conditions for the existence and local exponential stability of the almost periodic solutions are established, which are new and complement previously known results.     

Topological structure of solution sets for impulsive differential inclusions in Fréchet spaces

In this paper, we consider the existence of solutions as well as the topological and geometric structure of solution sets for first-order impulsive differential inclusions in some Fréchet spaces. Both the initial and terminal problems are considered. Using ingredients from topology and homology, the topological structures of solution sets (closedness and compactness) as well as some geometric properties (contractibility, acyclicity, AR and R_δ) are investigated. Some of our existence results are obtained via the method of taking the inverse system limit on noncompact intervals.     

Almost periodic solutions for shunting inhibitory cellular neural networks without global Lipschitz activation functions

In this paper the shunting inhibitory cellular neural networks (SICNNs) with continuously distributed delays are considered. Without assuming the global Lipschitz conditions of activation functions, sufficient conditions for the existence and local exponential stability of the almost periodic solutions are established by using a fixed point theorem, Lyapunov functional method and differential inequality techniques. The results of this paper are new and they complement previously known results.     

Periodic bi-directional Cohen–Grossberg neural networks with distributed delays

Some sufficient conditions are obtained for the global exponential stability of periodic solutions to periodic bi-directional Cohen–Grossberg neural networks involving distributed delays, by using the Lyapunov functional method and some analytical techniques. These results improve and generalize some previous works, and are helpful for designing a global exponential stable periodic bi-directional Cohen–Grossberg neural networks.     

Dynamics of a neutral delay equation for an insect population with long larval and short adult phases

We present a global study on the stability of the equilibria in a nonlinear autonomous neutral delay differential population model formulated by Bocharov and Hadeler. This model may be suitable for describing the intriguing dynamics of an insect population with long larval and short adult phases such as the periodical cicada. We circumvent the usual difficulties associated with the study of the stability of a nonlinear neutral delay differential model by transforming it to an appropriate non-neutral nonautonomous delay differential equation with unbounded delay. In the case that no juveniles give birth, we establish the positivity and boundedness of solutions by ad hoc methods and global stability of the extinction and positive equilibria by the method of iteration. We also show that if the time adjusted instantaneous birth rate at the time of maturation is greater than 1, then the population will grow without bound, regardless of the population death process.     

Global solutions of nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces

In this paper, by using the generalization of Darbo’s fixed point theorem, we establish the existence of global solutions of an initial value problem for a class of second-order impulsive integro-differential equations of mixed type in a real Banach space. Our results generalize and improve on the results of Guo et al. [F. Guo, L.S. Liu, Y.H. Wu, P. Siew, Global solutions of initial value problems for nonlinear second-order impulsive integro-differential equations of mixed type in Banach spaces, Nonlinear Anal. 61 (2005) 1363–1382] in the sense that the conditions for existence of global solution in our theorem is simpler and less strict. To demonstrate the application of the theorem, we give the global solutions of two mixed boundary value problems for two classes of fourth order impulsive integro-differential equations.     

Convergence of a class of discrete-time semiflows with application to neutral delay differential equations

In this paper, we introduce a class of pseudo-monotone maps on ordered topological spaces. By exploiting monotonicity methods and the invariance of the omega limit set, we establish a convergence principle for discrete-time semiflows generated by the maps introduced. The convergence principle is then applied to a class of periodic neutral delay differential equations, which leads to some novel and sharper results.     

On a topological description of solutions of complex differential equations

In this article we give a topological approach to the global behavior of arbitrary single-valued solutions in a simple connected domain of some general classes of complex differential equations with multi-valued coefficients. In particular, this permits us to describe certain globally multi-valued solutions as well as algebraic and algebroid solutions.     

Global existence for second order functional semilinear equations

In this paper we study the global existence of solutions for initial value problems for functional semilinear equations, where the linear operator in the differential equation is the infinitesimal generator of a strongly cosine family in a Banach spaceX. Using the Leray-Schauder Alternative, we derive conditions under which a solution exists globally.     

The Mackey–Glass model of respiratory dynamics: Review and new results

Since the celebrated Mackey–Glass model of respiratory dynamics was introduced in 1977, many results on its qualitative behavior have been obtained, including oscillation, stability and chaos. The paper reviews some known properties and presents new results for more general models: equations with time-dependent parameters, several delays, a positive periodic equilibrium and distributed delays. The problems considered in the paper involve existence, positivity and permanence of solutions, oscillation and global asymptotic stability. In addition, some general approaches to the study of nonlinear nonautonomous scalar delay equations are outlined. The paper generalizes and unifies existing results and provides an outlook on further studies.     

On the existence and asymptotic behaviour of solutions of an evolution equation and an application to the Feynman-Kac theorem

In this paper, we study the existence of mild solutions of a semilinear evolution equation on an unbounded interval. The rapidity of the growth of those solutions is characterized. Moreover, we investigate the local and global attractivity of solutions of an equation in question and we describe their asymptotic behaviour. Our investigations are conducted in a real separable Banach space.     

Global solutions for abstract neutral differential equations

We study the existence of global solutions for a class of abstract neutral differential equation defined on the whole real axis. Some concrete applications related to ordinary and partial differential equations are considered.     

Two periodic solutions of neutral difference systems depending on two parameters

In this paper, we establish existence, multiplicity and nonexistence of periodic solutions for a class of first-order neutral difference systems. Our approach is based on a fixed point theorem in cones as well as some analysis techniques.     

Inequalities for solutions of singular initial problems for Carathéodory systems via Ważewski’s principle

In this paper we give sufficient conditions for solvability of a singular initial problem formulated for Carathéodory systems of ordinary differential equations. The existence of solutions is proved by the supposition that corresponding auxiliary lower and upper singular problems have solutions. The proof technique uses a notion of a regular polyfacial subset which is developed for Carathéodory systems of ordinary differential equations and a modification of the topological method for such systems given by Palamides, Sficas and Staikos. An application concerning the existence of positive solutions for a special class of singular problems is given as well.     

The global attractor of a competitor–competitor–mutualist reaction–diffusion system with time delays

The aim of this paper is to investigate the asymptotic behavior of time-dependent solutions of a three-species reaction–diffusion system in a bounded domain under a Neumann boundary condition. The system governs the population densities of a competitor, a competitor–mutualist and a mutualist, and time delays may appear in the reaction mechanism. It is shown, under a very simple condition on the reaction rates, that the reaction–diffusion system has a unique constant positive steady-state solution, and for any nontrivial nonnegative initial function the corresponding time-dependent solution converges to the positive steady-state solution. An immediate consequence of this global attraction property is that the trivial solution and all forms of semitrivial solutions are unstable. Moreover, the state–state problem has no nonuniform positive solution despite possible spatial dependence of the reaction and diffusion. All the conclusions for the time-delayed system are directly applicable to the system without time delays and to the corresponding ordinary differential system with or without time delays.     

Multiplicity of positive periodic solutions to a generalized delayed predator–prey system with stocking

In this paper, by using the continuation theorem of coincidence degree theory, the existence of multiple positive periodic solutions for a generalized delayed predator–prey system with stocking is established. When our result is applied to a delayed predator–prey system with nonmonotonic functional response and stocking, we establish the sufficient condition for the existence of multiple positive periodic solutions for the system.