This paper is concerned with the nonlinear advanced difference equation with constant coefficients $$x_{n+1}-x_n+\sum _{i=1}^m{p}_i{f}_i\left(x_{n-k_i}\right)=0,n=0,1,\cdots$$ where $p_i\in (-\infty ,0)$ and $k_i\in \{\cdots ,-2,-1\}$ for $i=1,2,\cdots ,m$. We obtain sufficient conditions and also necessary and sufficient conditions for the oscillation of all solutions of the difference equation above by comparing with the associated linearized difference equation. Furthermore, oscillation criteria are established for the nonlinear advanced difference equation with variable coefficients $$x_{n+1}-x_n+\sum _{i=1}^m{p}_{in}{f}_i\left(x_{n-k_i}\right)=0,n=0,1,\cdots$$ where $p_{in}\le 0$ and $k_i\in \{\cdots ,-2,-1\}$ for $i=1,2,\cdots ,m$.

In order to study the impact of fishing on a grouper population, we propose in this paper to model the dynamics of a grouper population in a fishing territory by using structured models. For that purpose, we have integrated the natural population growth, the fishing, the competition for shelter and the dispersion. The dispersion was considered as a consequence of the competition. First we prove, that the grouper stocks may be less sensitive to the...

The purpose of these notes is to give a short survey of an interesting connection between partition functions of supersymmetric gauge theories and hypergeometric functions and to present the recent progress in this direction.

We consider the maximal regularity problem for the discrete time evolution equation $u_{n+1}-Tuₙ=fₙ$ for all n ∈ ℕ₀, u₀ = 0, where T is a bounded operator on a UMD space X. We characterize the discrete maximal regularity of T by two types of conditions: firstly by R-boundedness properties of the discrete time semigroup ${\left(Tⁿ\right)}_{n\in \mathbb{N}₀}$ and of the resolvent R(λ,T), secondly by the maximal regularity of the continuous time evolution equation u’(t) - Au(t) = f(t) for all t > 0, u(0) = 0, where A:= T - I. By recent results of...

By using the semi-discrete method of differential equations, a new version of discrete analogue of stochastic shunting inhibitory cellular neural networks (SICNNs) is formulated, which gives a more accurate characterization for continuous-time stochastic SICNNs than that by Euler scheme. Firstly, the existence of the 2th mean almost periodic sequence solution of the discrete-time stochastic SICNNs is investigated with the help of Minkowski inequality, Hölder inequality and Krasnoselskii's fixed...

We establish a q-shift difference analogue of the logarithmic derivative lemma. We also investigate the value distributions of q-shift difference polynomials and the growth of solutions of complex q-shift difference equations.