We investigate which points in the unit sphere of the Besicovitch--Orlicz space of almost periodic functions, equipped with the Luxemburg norm, are extreme points. Sufficient conditions for the strict convexity of this space are also given.

In the beginning of the twentieth century, Plemelj introduced the notion of factorization of matrix functions. The matrix factorization finds applications in many fields such as in the diffraction theory, in the theory of differential equations and in the theory of singular integral operators. However, the explicit formulas for the factors of the factorization are known only in a few classes of matrices. In the present paper we consider a new approach to obtain the factorization of a rational matrix...

On commence par présenter une méthode de résolution d’une famille de systèmes fuchsiens d’opérateurs de pseudo-dérivations associées à une famille à deux paramètres d’homographies, qui unifie et généralise les cas connus des systèmes différentiels, aux différences ou aux $q$-différences. Nous traitons ensuite dans cette famille des problèmes de confluence que l’on peut voir comme des problèmes de continuité en ces deux paramètres.

Firstly we study the growth of meromorphic solutions of linear difference equation of the form$$A_k\left(z\right)f(z+c_k)+\cdots +A_1\left(z\right)f(z+c_1)+A_0\left(z\right)f\left(z\right)=F\left(z\right),$$ where $A_k\left(z\right),...,A_0\left(z\right)$ and $F\left(z\right)$ are meromorphic functions of finite logarithmic order, $c_i$$(i=1,......$

Let f(z) be a finite order transcendental meromorphic function such that λ(1/f(z)) < σ(f(z)), and let c ∈ ℂ∖0 be a constant such that f(z+c) ≢ f(z) + c. We mainly prove that $max\tau \left(f\left(z\right)\right),\tau \left({\Delta }_{c}f\left(z\right)\right)=max\tau \left(f\right(z\left)\right),\tau \left(f\right(z+c\left)\right)=max\tau \left({\Delta }_{c}f\left(z\right)\right),\tau \left(f(z+c)\right)=\sigma \left(f\left(z\right)\right)$, where τ(g(z)) denotes the exponent of convergence of fixed points of the meromorphic function g(z), and σ(g(z)) denotes the order of growth of g(z).

Consider the third order nonlinear dynamic equation $x^{\Delta \Delta \Delta }\left(t\right)+p\left(t\right)f\left(x\right)=g\left(t\right)$, (*) on a time scale which is unbounded above. The function f ∈ C(,) is assumed to satisfy xf(x) > 0 for x ≠ 0 and be nondecreasing. We study the oscillatory behaviour of solutions of (*). As an application, we find that the nonlinear difference equation $\Delta ³x\left(n\right)+n^{\alpha }{\left|x\right|}^{\gamma }sgn\left(n\right)=(-1)ⁿn^c$, where α ≥ -1, γ > 0, c > 3, is oscillatory.

We develop a difference equations analogue of recent results by F. Gesztesy, K. A. Makarov, and the second author relating the Evans function and Fredholm determinants of operators with semi-separable kernels.

This paper is concerned with a class of nonlinear delay partial difference equations with variable coefficients, which may change sign. By making use of frequency measures, some new oscillatory criteria are established. This is the first time oscillation of these partial difference equations is discussed by employing frequency measures.