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This article considered two generalized Nicholson's blowflies equations with iteration term and time delay, as well as with immigration, and Nicholson's blowflies equation with iteration term and time delay, as well as harvesting term, respectively. Under appropriate conditions, the existence and uniqueness of almost periodic positive solutions for these two equations were established, respectively, by employing Banach's fixed point theorem. These results were brand new. [ABSTRACT FROM AUTHOR]

In the present paper sufficient conditions about the existence of almost periodic solutions of systems of impulsive differential-difference equations are obtained. The investigations are carried out by means piecewise continuous Lyapunov functions and Razumikhin techniques. [ABSTRACT FROM AUTHOR]

In the present paper sufficient conditions for tile existence of almost periodic solutions of systems of impulsive differential equations with time varying delay obtained. The investigations are carried out by means piecewise continuous Lyapunov functions and Razumikhin techniques. [ABSTRACT FROM AUTHOR]

In this paper we study the dynamical behavior of the differential equation x ″ + a x + − b x − = f ( t ) , where x + = max ⁡ { x , 0 } , x − = max ⁡ { − x , 0 } , a and b are two different positive constants, f ( t ) is a real analytic almost periodic function. For this purpose, firstly, we have to establish some variants of the invariant curve theorem of planar almost periodic mappings, which was proved recently by the authors (see [11] ). Then we will discuss the existence of almost periodic solutions and the boundedness of all solutions for the above asymmetric oscillation. [ABSTRACT FROM AUTHOR]

In this paper, we discuss the bounded solutions, stationary solutions, periodic solutions, almost periodic solutions, almost automorphic solutions for stochastic reaction–diffusion equation with cubic nonlinearities. The main difficulty is the cubic nonlinearities, we overcome this difficulty by the semigroup approach, the energy estimate method and refined inequality technique. [ABSTRACT FROM AUTHOR]

In this paper, neutral-type competitive neural networks with mixed time-varying delays and leakage delays on time scales are proposed. Based on the contraction fixed-point theorem, some sufficient conditions that are independent of the backwards graininess function of the time scale are obtained for the existence and global exponential stability of almost periodic solutions of neural networks under consideration. The results obtained are brand new, indicating that the continuous time and discrete-time conditions of the network share the same dynamic behavior. Finally, two examples are given to illustrate the validity of the results obtained. [ABSTRACT FROM AUTHOR]

In this paper, limit periodic and almost periodic homogeneous linear difference systems are studied. The coefficient matrices of the considered systems belong to a given commutative group. We find a condition on the group under which the systems, whose fundamental matrices are not almost periodic, form an everywhere dense subset in the space of all considered systems. The treated problem is discussed for the elements of the coefficient matrices from an arbitrary infinite field with an absolute value. Nevertheless, the presented results are new even for the field of complex numbers. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we first present a notion of almost periodic functions on time scales and study their basic properties. Then by means of Liapunov functionals, we study the existence of almost periodic solutions for an almost periodic dynamic equation on time scales. [Copyright &y& Elsevier]

In this paper, by using Mawhin's continuation theorem of coincidence degree theory, sufficient conditions for the existence of positive almost periodic solutions are obtained for the predator-prey Lotka-Volterra competition system with delays {dui(t)/dt = ui(t)[ai(t) - ... ail(t)ul(t - σil(t)) - ... bij(t)vj(t - τij(t))], i = 1,..., n, {dvj(t)/dt = vj(t) [ - rj(t) + ... djl(t)ul(t)ul(t - δjl(t)) - ... ejh(t)vh(t - ϑjh(t))], j = 1,..., m, where ai, rj, ail, bij, djl, ejh ∈ C (ℝ, (0, ∞)), σil, τij, δjl, ϑjh, ∈ c (ℝ, ℝ) (i, l = 1,..., n, j, h = l,..., m) are almost periodic functions. [ABSTRACT FROM AUTHOR]

Abstract: This paper considers the fourth-order nonlinear differential equations with a deviating argument. Sufficient conditions for the existence and exponential stability of the almost periodic solutions are established, which are new and complement previously known results. [ABSTRACT FROM AUTHOR]

Abstract: This paper considers the Liénard-type systems with multiple varying time delays. Some sufficient conditions for the existence and exponential stability of the almost periodic solutions are established, which are new and complement previously known results. [Copyright &y& Elsevier]

Abstract: This paper is concerned with some extensions of the classical Liouville theorem for bounded harmonic functions to solutions of more general equations. We deal with entire solutions of periodic and almost periodic parabolic equations including the elliptic framework as a particular case. We derive a Liouville type result for periodic operators as a consequence of a result for operators periodic in just one variable, which is new even in the elliptic case. More precisely, we show that if and , , c, f are periodic in the same space direction or in time, with the same period, then any bounded solution u of is periodic in that direction or in time. We then derive the following Liouville type result: if , and , , c are periodic in all the space/time variables, with the same periods, then the space of bounded solutions of the above equation has at most dimension one. In the case of the equation , with L periodic elliptic operator independent of t, the hypothesis can be weakened by requiring that the periodic principal eigenvalue of −L is nonnegative. Instead, the periodicity assumption cannot be relaxed, because we explicitly exhibit an almost periodic function b such that the space of bounded solutions of in has dimension 2, and it is generated by the constant solution and a non-almost periodic solution. The above counterexample leads us to consider the following problem: under which conditions are bounded solutions necessarily almost periodic? We show that a sufficient condition in the case of the equation is: f is almost periodic and L is periodic with . Finally, we consider problems in general periodic domains under either Dirichlet or Robin boundary conditions. We prove analogous properties as in the whole space, together with some existence and uniqueness results for entire solutions. [Copyright &y& Elsevier]

Abstract: In this paper we study a non-autonomous neutral functional differential equation in a Banach space. Applying the theory of semigroups of operators to evolution equations and Krasnoselskii’s fixed point theorem we establish the existence and uniqueness of a mild almost periodic solution of the problem under consideration. [Copyright &y& Elsevier]

Abstract: The paper is devoted to the study of non-autonomous evolution equations: invariant manifolds, compact global attractors, almost periodic and almost automorphic solutions. We study this problem in the framework of general non-autonomous (cocycle) dynamical systems. First, we prove that under some conditions such systems admit an invariant continuous section (an invariant manifold). Then, we obtain the conditions for the existence of a compact global attractor and characterize its structure. Third, we derive a criterion for the existence of almost periodic and almost automorphic solutions of different classes of non-autonomous differential equations (both ODEs (in finite and infinite spaces) and PDEs). [Copyright &y& Elsevier]

Abstract: This paper considers the Liénard-type systems with continuously distributed delays. Some sufficient conditions for the existence and exponential stability of the almost periodic solutions are established, which are new and complement previously known results. [Copyright &y& Elsevier]

Abstract: In this paper, we study almost periodic logistic delay differential equations. The existence and module of almost periodic solutions are investigated. In particular, we extend some results of Seifert in [G. Seifert, Almost periodic solutions of certain differential equations with piecewise constant delays and almost periodic time dependence, J. Differential Equations 164 (2000) 451–458]. [Copyright &y& Elsevier]

In this paper, we consider an almost periodic commensal symbiosis model with nonlinear harvesting on time scales. We establish a criterion for the existence and uniformly asymptotic stability of unique positive almost periodic solution of the system. Our results show that the continuous system and discrete system can be unify well. Examples and their numerical simulations are carried out to illustrate the feasibility of our main results.

In this paper, limit periodic and almost periodic homogeneous linear difference systems are studied. The coefficient matrices of the considered systems belong to a given commutative group. We find a condition on the group under which the systems, whose fundamental matrices are not almost periodic, form an everywhere dense subset in the space of all considered systems. The treated problem is discussed for the elements of the coefficient matrices from an arbitrary infinite field with an absolute value. Nevertheless, the presented results are new even for the field of complex numbers.

The aim of this paper is to establish a decay estimate for viscosity solutions of nonlinear PDEs. As an application we prove existence and uniqueness for time almost periodic viscosity solutions.

In this paper, neutral-type competitive neural networks with mixed time-varying delays and leakage delays on time scales are proposed. Based on the contraction fixed-point theorem, some sufficient conditions that are independent of the backwards graininess function of the time scale are obtained for the existence and global exponential stability of almost periodic solutions of neural networks under consideration. The results obtained are brand new, indicating that the continuous time and discrete-time conditions of the network share the same dynamic behavior. Finally, two examples are given to illustrate the validity of the results obtained.

In this paper, we first present a notion of almost periodic functions on time scales and study their basic properties. Then by means of Liapunov functionals, we study the existence of almost periodic solutions for an almost periodic dynamic equation on time scales.

In order to obtain the conditions for the existence of periodic and almost periodic solutions of Volterra difference equations, x(n+1) = f(n; x(n))+ Σns=-∞ F(n,s, x(n + s), x(n)), we consider certain stability properties, which are referred to as (K, ρ)-weakly uniformly-asymptotic stability and (K, ρ)-uniformly asymptotic stability. Moreover, we discuss the relationship between the ρ-separation condition and the uniformly-asymptotic stability property in the ρ sense. [ABSTRACT FROM AUTHOR]