Convergence and stability for essentially strongly order-preserving semiflows

This paper is concerned with a class of essentially strongly order-preserving semiflows, which are defined on an ordered metric space and are generalizations of strongly order-preserving semiflows. For essentially strongly order-preserving semiflows, we prove several principles, which are analogues of the nonordering principle for limit sets, the limit set dichtomy and the sequential limit set trichotomy for strongly order-preserving semiflows. Then, under certain compactness hypotheses, we obtain some results on convergence, quasiconvergence and stability in essentially strongly order-preserving semiflows. Finally, some applications are made to quasimonotone systems of delay differential equations and reaction-diffusion equations with delay, and the main advantages of our results over the classical ones are that we do not require the delicate choice of state space and the technical ignition assumption.     

Convergence for pseudo monotone semiflows on product ordered topological spaces

In this paper, we consider a class of pseudo monotone semiflows, which only enjoy some weak monotonicity properties and are defined on product-ordered topological spaces. Under certain conditions, several convergence principles are established for each precompact orbit of such a class of semiflows to tend to an equilibrium, which improve and extend some corresponding results already known. Some applications to delay differential equations are presented.     

Positive almost periodic solution for a class of Nicholson’s blowflies model with multiple time-varying delays

In this paper, we study the existence and exponential convergence of positive almost periodic solutions for the generalized Nicholson’s blowflies model with multiple time-varying delays. Under proper conditions, we establish some criteria to ensure that the solutions of this model converge locally exponentially to a positive almost periodic solution. Moreover, we give some examples to illustrate our main results.     

Optimality conditions with state constraints for semilinear systems determined by operator valued measures

In this paper we develop the necessary conditions of optimality for a class of distributed parameter systems (partial differential equations) determined by operator valued measures and controlled by vector measures. Based on some recent results on existence of optimal controls from the space of vector measures, we develop necessary conditions of optimality for a class of control problems. The main results are the necessary conditions of optimality for problems without state constraints and those with state constraints. Also, a conceptual algorithm along with a brief discussion of its convergence is presented.     

Impulsive control for a class of neural networks with bounded and unbounded delays

In this paper, we study the problem of global exponential stability for a class of impulsive neural networks with bounded and unbounded delays and fixed moments of impulsive effect. We establish stability criteria by employing Lyapunov functions and Razumikhin technique. An illustrative example is given to demonstrate the effectiveness of the obtained results.     

Resolvent estimates in homogenisation of periodic problems of fractional elasticity

We provide operator-norm convergence estimates for solutions to a time-dependent equation of fractional elasticity in one spatial dimension, with rapidly oscillating coefficients that represent the material properties of a viscoelastic composite medium. Assuming periodicity in the coefficients, we prove operator-norm convergence estimates for an operator fibre decomposition obtained by applying to the original fractional elasticity problem the Fourier–Laplace transform in time and Gelfand transform in space. We obtain estimates on each fibre that are uniform in the quasimomentum of the decomposition and in the period of oscillations of the coefficients as well as quadratic with respect to the spectral variable. On the basis of these uniform estimates we derive operator-norm-type convergence estimates for the original fractional elasticity problem, for a class of sufficiently smooth densities of applied forces.     

Exponential convergence for high-order recurrent neural networks with a class of general activation functions

In this paper, we consider high-order recurrent neural networks with a class of general activation functions. By using some mathematical analysis techniques, we establish new results to ensure that all solutions of the networks converge exponentially to zero point.     

Almost periodic solutions for an impulsive delay Nicholson’s blowflies model

By means of the contraction mapping principle and Gronwall-Bellman’s inequality, we prove the existence and exponential stability of positive almost periodic solution for an impulsive delay Nicholson’s blowflies model. The main results are illustrated by an example.     

Time maps and exact multiplicity results for one-dimensional prescribed mean curvature equations

We investigate various properties of time maps for one-dimensional prescribed mean curvature equations. Using these properties, we obtain some exact multiplicity results of positive solutions and sign-changing solutions. As it turned out, these quasilinear problems show many different phenomena from semilinear problems. Our methods are based on a detailed analysis of time maps.     

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.