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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, 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]

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]

We study perturbations of homogeneous linear difference systems over infinite fields with absolute values. The coefficient matrices of the treated systems belong to commutative groups which do not need to be bounded. We present a general limit periodic transformation of an arbitrarily given system such that the obtained system has non-almost periodic solutions. We also formulate corollaries which show how the presented construction of the perturbed system improves and extends known results. [ABSTRACT FROM AUTHOR]

We study the response of various linear and nonlinear differential equations to almost periodic forced oscillations. A special attention is given to differential equations with state-dependent and almost periodic type delays. To the best of our knowledge, there are no results in the literature that address this problem. [ABSTRACT FROM AUTHOR]