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Traditional biological neural networks cannot simulate the real situation of the abrupt synaptic connections between neurons while modeling associative memory of human brains. In this paper, the memristive multidirectional associative memory neural networks (MAMNNs) with mixed time-varying delays are investigated in the sense of Filippov solution. First, three steps are given to prove the existence of the almost periodic solution. Two new lemmas are proposed to prove the boundness of the solution and the asymptotical almost periodicity of the solution by constructing Lyapunov function. Second, the uniqueness and global exponential stability of the almost periodic solution of memristive MAMNNs are investigated by a new Lyapunov function. The sufficient conditions guaranteeing the properties of almost periodic solution are derived based on the relevant definitions, Halanay inequality and Lyapunov function. The investigation is an extension of the research on the periodic solution and almost periodic solution of bidirectional associative memory neural networks. Finally, numerical examples with simulations are presented to show the validity of the main results. [ABSTRACT FROM AUTHOR]

In this paper, we establish the second Bogolyubov theorem and global averaging principle for stochastic partial differential equations (in short, SPDEs) with monotone coefficients. First, we prove that there exists a unique L²-bounded solution to SPDEs with monotone coefficients and this bounded solution is globally asymptotically stable in square-mean sense. Then we show that the L²-bounded solution possesses the same recurrent properties (e.g., periodic, quasi-periodic, almost periodic, almost automorphic, Birkhoff recurrent, Levitan almost periodic, etc.) in distribution sense as the coefficients. Third, we prove that the recurrent solution of the original equation converges to the stationary solution of averaged equation under the compact-open topology as the time scale goes to zero-in other words, there exists a unique recurrent solution to the original equation in a neighborhood of the stationary solution of averaged equation when the time scale is small. Finally, we establish the global averaging principle in weak sense, i.e., we show that the attractor of original system tends to that of the averaged equation in probability measure space as the time scale goes to zero. For illustration of our results, we give two applications, including stochastic reaction diffusion equations and stochastic generalized porous media equations. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence and uniqueness of pseudo-almost periodic C 0 -solutions to the evolution equations with nonlocal initial conditions in Banach space. Upon making some appropriate assumptions, the existence and the uniqueness of pseudo-almost periodic C 0 -solutions to the evolution equations with nonlocal initial conditions are obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of almost periodically forced harmonic oscillators x ¨ + τ 2 x = ϵ f (t , x) where τ ∈ A with A being a closed interval not containing zero, the forcing term f is real analytic almost periodic functions in t with the infinite frequency ω = (⋯ , ω λ , ⋯) λ ∈ Z . Using the modified Kolmogorov–Arnold–Moser (or KAM Arnold (Uspehi Mat. Nauk 18(5 (113)):13–40, 1963), Moser (Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II 1962:1–20 1962), Kolmogorov (Dokl. Akad. Nauk SSSR (N.S.) 98:527–530 1954)) theory about the lower dimensional tori, we show that there exists a positive Lebesgue measure set of τ contained in A such that the harmonic oscillators has almost periodic solutions with the same frequencies as f. The result extends the earlier research results with the forcing term f being quasi-periodic. [ABSTRACT FROM AUTHOR]

In this paper we are concerned with the existence of invariant curves of planar twist mappings which are almost periodic in a spatial variable. As an application of this result to differential equations we will prove the existence of almost periodic solutions and the boundedness of all solutions for superlinear Duffing's equation with an almost periodic external force. [ABSTRACT FROM AUTHOR]

In this paper, nearly integrable system under almost periodic perturbations is studied {x = ω0 + y + ƒ(t,x,y), y = g(t,x,y), where x ∈ Tn, y ∈ Rn, ω0 ∈ Rn is the frequency vector, and the perturbations ƒ,g are real analytic almost periodic functions in t with the infinite frequency ω = (⋯,ωλ,⋯)λ∈Z. We also assume that the above system is reversible with respect to the involution M0 : (x,y) → (−x,y). By KAM iterative method, we prove the existence of invariant tori for the above reversible system. As an application, we discuss the existence of almost periodic solutions and the boundedness of all solutions for a second-order nonlinear differential equation. [ABSTRACT FROM AUTHOR]

Stochastic periodic type solution is a powerful tool for studying qualitative analysis of stochastic dynamical systems. In this paper, we will establish the bounded solutions, stationary solutions, periodic solutions, almost periodic solutions, almost automorphic solutions for stochastic linearly coupled complex cubic-quintic Ginzburg–Landau equations under suitable conditions. The main novelty of this paper is dealing with cubic nonlinear terms and the quintic nonlinear terms which are not Lipschitz. We overcome this difficulty by the semigroup approach, stochastic analysis techniques, energy estimate method and refined inequality technique. [ABSTRACT FROM AUTHOR]

In this paper, we will establish the existence of the bounded solutions, periodic solutions, quasi-periodic solutions and almost periodic solutions for the derivative Ginzburg-Landau equation with time-dependent boundary external forces. A smoothing effect for the semigroup associated with linear Ginzburg-Landau operator is the crucial tool in establishing the main results. [ABSTRACT FROM AUTHOR]

In this paper, we are concerned with the existence of invariant tori in nearly integrable Hamiltonian systems H=h(y)+f(x,y,t),where y∈D⊆Rn with D being an open bounded domain, x∈Tn, f(x, y, t) is a real analytic almost periodic function in t with the frequency ω=(…,ωλ,…)λ∈Z∈RZ. As an application, we will prove the existence of almost periodic solutions and the boundedness of all solutions for the second-order differential equations with superquadratic potentials depending almost periodically on time. [ABSTRACT FROM AUTHOR]

In this paper, we use the variational approach to investigate recurrent properties of solutions for stochastic partial differential equations, which is in contrast to the previous semigroup framework. Consider stochastic differential equations with monotone coefficients. Firstly, we establish the continuous dependence on initial values and coefficients for solutions, which is interesting in its own right. Secondly, we prove the existence of recurrent solutions, which include periodic, almost periodic and almost automorphic solutions. Then we show that these recurrent solutions are globally asymptotically stable in square-mean sense. Finally, for illustration of our results we give two applications, i.e. stochastic reaction diffusion equations and stochastic porous media equations. [ABSTRACT FROM AUTHOR]

In this paper, we propose a class of quaternion-valued shunting inhibitory cellular neural networks of neutral type with time delays in the leakage term. In order to overcome the difficulty of the non-commutativity of quaternion multiplication, we first decompose the system under consideration into four real-valued systems. Then by using the exponential dichotomy theory of linear differential equations and Banach's fixed point theorem, we establish sufficient conditions to ensure the existence and global exponential stability of almost periodic solutions for this class of neural networks. Finally, a numerical example is given to demonstrate the effectiveness of the obtained result. [ABSTRACT FROM AUTHOR]

In this paper, we will establish the bounded solutions, periodic solutions, quasiperiodic solutions, almost periodic solutions, and almost automorphic solutions for linearly coupled complex cubic‐quintic Ginzburg‐Landau equations, under suitable conditions. The main difficulty is the nonlinear terms in the equations that are not Lipschitz‐continuity, traditional methods cannot deal with the difficulty in our problem. We overcome this difficulty by the Galerkin approach, energy estimate method, and refined inequality technique. [ABSTRACT FROM AUTHOR]

In this paper an impulsive model for the dynamics of competitive Lotka-Volterra systems using the Caputo fractional-order derivative is developed. The existence and uniqueness of almost periodic solutions are investigated. Applying the fractional Lyapunov method, we give sufficient conditions for global perfect uniform-asymptotic stability of the almost periodic solution and sufficient conditions to save these qualities at the uncertain case. Since the competitive relationship in ecosystem models is relevant in various contexts, including many population, neural nets or chemical kinetics problems, our results can be applied in the investigation of almost periodic processes in a wide range of competitive systems of diverse interest. [ABSTRACT FROM AUTHOR]

This paper presents a couple of existence results, related to the classes of functional equations of the form x+k" x = f , or d/dt [x+k " x] = f , with f , x ∊ APr(R, C) = the space of almost periodic functions defined by APr(R, C) = {f : f ≃ ∞Σj=1fjeiλjt, fj ∊ C, λ j ∊ R, ∞Σj=1 ∣fj∣r < ∞}, the norm being given by ∣f∣r = (Σ∞j=1 ∣fj∣r) 1/r, for each r ∊ [1,2]. The convolution porduct k " x, k ∊ L1 (R,C), x ∊ APr(R,C) is defined by (k" x)(t)= ∞Σj=1 xj(∫Rk(s)e-λjs eiλjt, where x(t) ≃ Σ∞j=1 xjeiλjt. [ABSTRACT FROM AUTHOR]

This paper is concerned with a nonautonomous time-delayed model for the regulation of the hematopoiesis with an oscillating circulation loss rate. We prove a fixed point theorem in cones to establish some sufficient conditions ensuring the existence of positive almost periodic solutions of the model with almost periodic coefficients. Moreover, some examples are given to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

This paper is concerned with almost periodic solutions for nonlinear non-instantaneous impulsive differential equations with variable structure. With the help of the notation of non-instantaneous impulsive Cauchy matrix, mild sufficient conditions are derived to guarantee the existence, uniqueness of asymptotically stable almost periodic solutions. Both example and numerical simulation are given to illustrate our effectiveness of the above results. As one expects, the results presented here have extended and improved some previous results for instantaneous impulsive differential equations. [ABSTRACT FROM AUTHOR]

This paper is concerned with a class of impulsive prey-predator model with Beddington-DeAngelis functional response. By fixed point theorem and constructing a Lyapunov function, some sufficient conditions are established to ensure the permanence and uniformly asymptotic stability of positive almost periodic solutions for the concerned system. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the existence and stability of almost periodic solutions of impulsive fractional-order differential systems with uncertain parameters. The impulses are realised at fixed moments of time. For the first time, we determine the impact of the uncertainties on the qualitative behaviour of such systems. The main criteria for the existence of almost periodic solutions are proved by employing the fractional Lyapunov method. The global perfect robust uniform-asymptotic stability of such solutions is also considered. We apply our results to uncertain impulsive neural network systems of fractional order. [ABSTRACT FROM AUTHOR]

In this paper, we consider the second KdV equation with the external parameters under zero mean-value periodic boundary conditions where $$M_\sigma $$ is a real Fourier multiplier. It is proved that the equations admit a Whitney smooth family of small amplitude, real analytic almost periodic solutions with all frequencies. The proof is based on a conserved quantity $$\int _0^{2\pi } u^2 dx$$ , Töplitz-Lipschitz property of the perturbation and an abstract infinite dimensional KAM theorem. By taking advantage of the conserved quantity $$\int _0^{2\pi } u^2 dx$$ and Töplitz-Lipschitz property of the perturbation, our normal form part is independent of angle variables in spite of the unbounded perturbation. This is the first attempt to prove the almost periodic solutions for the unbounded perturbation case. [ABSTRACT FROM AUTHOR]

In this paper, we study the problems of existence and global asymptotic stability of almost periodic solutions for a cellular neural network of fractional order with time-varying delays and impulses. The impulses are realized at fixed moments of time and can be considered as a control. The main results are obtained by employing the fractional Lyapunov method and fractional comparison principle. [ABSTRACT FROM AUTHOR]

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 this paper, a new impulsive Lasota-Wazewska model with patch structure and forced perturbed terms is proposed and analyzed on almost periodic time scales. For this, we introduce the concept of matrix measure on time scales and obtain some of its properties. Then, sufficient conditions are established which ensure the existence and exponential stability of positive almost periodic solutions of the proposed biological model. Our results are new even when the time scale is almost periodic, in particular, for periodic time scales on [ABSTRACT FROM AUTHOR]

In this paper, we consider a class of quaternion-valued neural networks with mixed delays on time scales. By using general Lipschitz condition, contraction mapping principle and exponential dichotomy of linear dynamic equations, we prove the existence and uniqueness and the stability of almost periodic solutions of global exponential neural networks. Finally, one example is given to illustrate the efficiency of our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we study the existence and linear stability of almost periodic solutions for a NLS equation on the circle with external parameters. Starting from the seminal result of Bourgain in [15] on the quintic NLS, we propose a novel approach allowing to prove in a unified framework the persistence of finite and infinite dimensional invariant tori, which are the support of the desired solutions. The persistence result is given through a rather abstract "counter-term theorem" à la Herman, directly in the original elliptic variables without passing to action-angle ones. Our framework allows us to find "many more" almost periodic solutions with respect to the existing literature and consider also non-translation invariant PDEs. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to investigate the existence of almost periodic solutions for the Duffing differential equation. By combining the theory of exponential dichotomies with Liapunov functions, we obtain an intersting result on the existence of almost periodic solutions. [ABSTRACT FROM AUTHOR]

In this paper, it is the first time that some important inequalities are obtained to estimate the exponential type of upper and lower bounds of solutions for the three representative classes of homogeneous impulsive dynamic systems on time scales. Based on these, some new criteria are established for admitting an exponential dichotomy of the impulsive dynamic systems. The obtained results are essentially new, even the time scale T D R or T D Z. In addition, in applications, we apply the obtained results to discuss the almost periodic problems of a class of integro-differential systems, and the numerical simulations are given to illustrate that our timescalemethods are feasible and effective. Finally, we present the conclusion and further discussion related to this topic. [ABSTRACT FROM AUTHOR]

In the paper, the problem of existence of almost periodic solutions to non-linear impulsive systems of Caputo fractional order is addressed. Sufficient conditions for stability of almost periodic solutions are also established. The investigations are carried out by using of a fractional comparison principle and the Lyapunov's function method. We apply our results to an impulsive cellular neural network model of fractional order. [ABSTRACT FROM AUTHOR]

We consider the uniqueness of bounded continuousL3, ∞-solutions on the whole time axis to the Navier-Stokes equations in 3-dimensional unbounded domains. Here,Lp, qdenotes the scale of Lorentz spaces. Thus far, uniqueness of such solutions to the Navier-Stokes equations in unbounded domain, roughly speaking, is known only for a small solution inBC(ℝ;L3, ∞) within the class of solutions which have sufficiently smallL∞(L3, ∞)-norm. In this paper, we discuss another type of uniqueness theorem for solutions inBC(ℝ;L3, ∞) using a smallness condition for one solution and a precompact range condition for the other one. The proof is based on the method of dual equations. [ABSTRACT FROM AUTHOR]

This paper deals with viscosity solutions of Hamilton-Jacobi equations in which the Hamiltonian H is weakly monotone with respect to the zero order term: this leads to non-uniqueness of solutions, even in the class of periodic or almost periodic (briefly a.p.) functions. The lack of uniqueness of a.p. solutions leads to introduce the notion of minimal (maximal) a.p. solution and to study its properties. The classes of asymptotically almost periodic (briefly a.a.p.) and pseudo almost periodic (briefly p.a.p.) functions are also considered. [ABSTRACT FROM AUTHOR]

In this paper, we first propose a concept of square-mean almost periodic function on timescales. Then, by means of the fixed point theory and differential inequality techniques on timescales, we establish some sufficient conditions on the existence and global exponential stability of square-mean almost periodic solutions for a class of stochastic Hopfield neural networks with time-varying delays on timescales. Our results are new even if the timescale $${\mathbb {T}}={\mathbb {R}}$$ or $${\mathbb {Z}}$$ . Finally, we present an example to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

The aim of this paper is to establish existence and uniqueness of time almost periodic viscosity solutions to first-order evolution equations in Carnot groups. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of almost periodic solutions for impulsive fractional evolution equations involving Caputo fractional derivative. The main results are obtained by means of the theory of operators semi-group, probability density functions, fixed point theorems and the techniques based on fractional calculus. An example is also discussed to illustrate the theory. Some known results are improved and generalized. [ABSTRACT FROM AUTHOR]

Our work is concerned with a neural network with n nodes, where the activity of the k-th cell depends on external, stochastic inputs as well as the coupling generated by the activity of the adjacent cells, transmitted through a diffusion process in the network. This paper aims to throw some light on time-varying, stochastically perturbed, neuronal networks. We show that when the coefficients oscillate around a reference value, with oscillations that are almost periodic and suitably small in percentage, then there exists a unique solution for the system, that is almost periodic and uniformly bounded in the square-mean norm for all times. [ABSTRACT FROM AUTHOR]

We present a uniqueness theorem for almost periodic-in-time solutions to the Navier-Stokes equations in 3-dimensional unbounded domains. Thus far, uniqueness of almost periodic-in-time solutions to the Navier-Stokes equations in unbounded domain, roughly speaking, is known only for a small almost periodic-in-time solution in $${BC(\mathbb {R};L^{3}_w)}$$ within the class of solutions that have sufficiently small $${L^{\infty}(L^{3}_w)}$$ -norm. In this paper, we show that a small almost periodic-in-time solution in $${BC(\mathbb {R};L^{3}_w\cap L^{6,2})}$$ is unique within the class of all almost periodic-in-time solutions in $${BC(\mathbb {R};L^{3}_w\cap L^{6,2})}$$ . The proof of the present uniqueness theorem is based on the method of dual equations. [ABSTRACT FROM AUTHOR]

In this paper we show the existence of a maximal and a minimal almost periodic solution of a non-autonomous functional differential equation using the method of lower and upper solutions. [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]