In this work, we study the existence of almost periodic solutions for some partial neutral functional differential equations. Using the variation of constants formula and the spectral decomposition of the phase space developed in [6], we prove that the existence of an almost periodic solution is equivalent to the existence of a bounded solution on R+.

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))+ ∑n s=−∞ 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-separati...

A group is called D-quasirandom if all of its nontrivial representations over the complex numbers have dimensions at least D. In this paper, we investigate the extent to which quasirandomness is a local property; more precisely, we study the question of whether a non-principal ultraproduct of a given sequence of quasirandom groups remains quasirandom, and whether the ultraproduct of a sequence ...

By using some new analytical techniques, modified inequalities and Mawhin’s continuation theorem of coincidence degree theory, some sufficient conditions for the existence of at least one positive almost periodic solution of a kind of two-species model of facultative mutualism with time delays are obtained. Further, the global asymptotic stability of the positive almost periodic solution of thi...

For every point χ in the spectrum of the operator (h(δ)ξ) = ξn+1 + ξn−1 + β ( δe + δe ) ξn on l(Z) there exists a complex number x of modulus one such that the equation ξn+1 + ξn−1 + β ( xδe + xδe ) ξn = χξn has a non-trivial solution satisfying the condition lim |n|→∞ |ξn| 1/|n| ≤ δβ provided that β, δ > 1 and α satisfies the diophantine condition lim n→∞ | sinπαn| 1 n = 1 . The parameters xδ ...

In the present paper the problems of existence and uniqueness of almost periodic solutions for impulsive cellular neural networks with delay are considered. 2006 Elsevier Inc. All rights reserved.

for each t ∈R. When convergence in (1.1) and (1.2) is uniform in t ∈ R, f is said to be (Bochner) almost periodic. Almost periodic functions are characterized by the following so-called Bochner’s criterion. f : R→ X is almost periodic if and only if for every sequence of real numbers (s′ n), there exists a subsequence (sn) such that (f (t+sn)) is uniformly convergent in t ∈R. For more informati...

In this paper, we introduce and discuss the concept of a Π-semigroup for invariant under translations time scales and the concept of abstract weighted pseudo almost periodic functions in Banach spaces. As an application, we obtain conditions for the existence of weighted pseudo almost periodic solutions for a class of neutral functional differential equations on time scales. AMS (MOS) Subject C...

By using the frequency domain method, sufficient conditions which guarantee asymptotic stability of the null solution of a certain class of third order nonlinear delay differential equation are established. Furthermore, effective criteria for the existence of a bounded solution which is exponentially stable, periodic or almost periodic according as the forcing term is periodic or almost periodi...

The method of upper and lower solutions is a classical tool in the theory of periodic differential equations of the second order. We show that this method does not have a direct extension to almost periodic equations. To do this we construct equations of this type without almost periodic solutions but having two constants as ordered upper and lower solutions. r 2003 Elsevier Science (USA). All ...