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The paper considers a general neural network model with impulses at a given sequence of instants, discontinuous neuron activations, delays, and time-varying data and inputs. It is shown that when the neuron interconnections satisfy an M-matrix condition, or a dominance condition, then the state solutions and the output solutions display a common asymptotic behavior as time t--+infinity. It is also shown, via a new technique based on prolonging the solutions of the delayed neural network to -infinity, that it is possible to select a unique special solution that is globally exponentially stable and can be considered as the unique global attractor for the network. Finally, this paper shows that for almost periodic data and inputs the selected solution is almost periodic; moreover, it is robust with respect to a large class of perturbations of the data. Analogous results also hold for periodic data and inputs. A by-product of the analysis is that a sequence of almost periodic impulses is able to induce in the generic case (nonstationary) almost periodic solutions in an otherwise globally convergent nonimpulsive neural network. To the authors' knowledge the results in this paper are the only available results on global exponential stability of the unique periodic or almost periodic solution for a general neural network model combining three main features, i.e., impulses, discontinuous neuron activations and delays. The results in this paper are compared with several results in the literature dealing with periodicity or almost periodicity of some subclasses of the neural network model here considered and some hints for future work are given.

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, we continue the investigations done in the literature about the so called Bohr-Neugebauer property for almost periodic differential equations. More specifically, for a class of functional heat equations, we prove that each $L^2$-bounded solution is almost periodic. This extends a result in [5] to the delay case.

An averaging result is proved for stochastic evolution equations with highly oscillating coefficients. This result applies in particular to equations with almost periodic coefficients. The convergence to the solution of the averaged equation is obtained in distribution, as in previous works by Khasminskii and Vrko{\v c}.This version corrects two minor errors from our paper published in J. Math. Anal. Appl. 427(1):336--364, 2015.

In this paper we study the two-dimensional dissipative Euler equations in a smooth and bounded domain. In the presence of a sufficiently large dissipative term (or equivalently a sufficiently small external force) precise uniform estimates on the modulus of continuity of the vorticity are proved. These allow us to show existence of Stepanov almost-periodic solutions.

In this paper, we obtain sufficient conditions for the existence of at least four positive almost periodic solutions to an impulsive delayed periodic plankton allelopathy system with multiple exploited (or harvesting) terms. This result is obtained through the use of Mawhins continuation theorem of coincidence degree theory along with some properties relating to inequalities., {"references":["A. Mukhopadhyay, J. Chattopadhyay, P.K. Tapaswi, \"A delay differential\nequations model of plankton allelopathy\", Mathematical Biosciences,,\nVol.149, pp. 167-189,1998.","A.M. Samoilenko, N.A. Perestyuk, \"Impulsive Differential\nEquations\",World Scientific, Singapore, 1995.","B.X. Yang, J.L. Li, An almost periodic solution for an impulsive\ntwo-species logarithmic population model with time -varying delay,\nMathematical and Computer Modelling, Vol.55 n0o.7-8, pp. 1963-1968,\n2012.","C.Y. 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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 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.

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].

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.

International audience; In this paper, we study almost periodic (a.p.) solutions of discrete dynamical systems. We first adapt some results on a.p. differential equations to a.p. difference equations, on the link between boundedness of solutions and existence of a.p. solutions. After, we obtain an existence result in the space of the Harmonic Synthesis for an equation $A_t (x_t,...,x_{t+p})=0$ when the dependance of $A$ on $t$ is a.p. and when $A_t$ and $D A_t$ are uniformly Lipschitz and satisfy another condition which is exactly the extension of a simple one for the basic linear system. The main tools for that are Nonlinear Functional Analysis and the Newton method.

This paper presents a couple of existence results, related to the classes of functional equations of the form $x+k\ast x=f$, or $\frac{d}{dt}[\dot{x}+k\ast x]=f$, with $f, x\in AP_r(R, \,{\mathcal{C}})=$ the space of almost periodic functions defined by \[ AP_r(R,\, {\mathcal{C}})=\left\{f : f\simeq \sum_{j=1}^{\infty} f_j\,e^{i\lambda_j t},\,f_j\in {\mathcal{C}},\lambda_j\in R,\sum_{j=1}^{\infty}|f_j|^r \lt \infty\right\}, \] the norm being given by $|f|_r= \left(\sum_{j=1}^{\infty}|f_j|^r\right)^{\frac{1}{r}}$, for each $r\in [1, 2]$. The convolution product $k\ast x$, $k\in L^1(R,\, {\mathcal{C}})$, $x\in AP_r(R,\, {\mathcal{C}})$ is defined by \[ (k\ast x)(t)= \sum_{j=1}^{\infty} x_j\left( \int_R k(s)\,e^{\lambda_j\,s}\,ds\right)\,e^{i\lambda_j\,t}, \] where $x(t)\simeq \sum_{j=1}^{\infty} x_j\,e^{i\lambda_j\,t}$.