Asymptotic behavior of solutions of an area-preserving crystalline curvature flow equation is investigated. In this equation, the area enclosed by the solution polygon is preserved, while its total interfacial crystalline energy keeps on decreasing. In the case where the initial polygon is essentially admissible and convex, if the maximal existence time is finite, then vanishing edges are essentially admissible edges. This is a contrast to the case where the initial polygon is admissible and convex:...

The asymptotic behaviour for solutions of a difference equation $z_n=f(n,z_n)$, where the complex-valued function $f(n,z)$ is in some meaning close to a holomorphic function $h$, and of a Riccati difference equation is studied using a Lyapunov function method. The paper is motivated by papers on the asymptotic behaviour of the solutions of differential equations with complex-valued right-hand sides.

In this paper we consider the first order difference equation in a Banach space $\Delta x_n={\sum }_{i=0}^{\infty }{a}_n^{i}f\left(x_{n+i}\right)$. We show that this equation has a solution asymptotically equal to a. As an application of our result we study the difference equation $\Delta x_n={\sum }_{i=0}^{\infty }{a}_n^{i}g\left(x_{n+i}\right)+{\sum }_{i=0}^{\infty }{b}_n^{i}h\left(x_{n+i}\right)+y_n$ and give conditions when this equation has solutions. In this note we extend the results from [8,9]. For example, in [9] the function f is a real Lipschitz function. We suppose that f has values in a Banach space and satisfies some conditions with respect to the measure of noncompactness...

In this paper we investigate the asymptotic properties of all solutions of the delay differential equation y’(x)=a(x)y((x))+b(x)y(x),      xI=[x0,). We set up conditions under which every solution of this equation can be represented in terms of a solution of the differential equation z’(x)=b(x)z(x),      xI and a solution of the functional equation |a(x)|((x))=|b(x)|(x),      xI.

In the paper we consider the difference equation of neutral type $${\Delta }^3[x\left(n\right)-p\left(n\right)x\left(\sigma \left(n\right)\right)]+q\left(n\right)f\left(x\left(\tau \left(n\right)\right)\right)=0,n\in \mathbb{N}\left(n_0\right),$$ where $p,q:\mathbb{N}\left(n_0\right)\to {ℝ}_{+}$; $\sigma ,\tau :\mathbb{N}\to \mathbb{Z}$, $\sigma$ is strictly increasing and $\underset{n\to \infty }{lim}\sigma \left(n\right)=\infty ;$$\tau$ is nondecreasing and $\underset{n\to \infty }{lim}\tau \left(n\right)=\infty$, $f:ℝ\to ℝ$, $xf\left(x\right)>0$. We examine the following two cases: $$0<p\left(n\right)\le {\lambda }^{*}<1,\sigma \left(n\right)=n-k,\tau \left(n\right)=n-l,$$ and $$1<{\lambda }_{*}\le p\left(n\right),\sigma \left(n\right)=n+k,\tau \left(n\right)=n+l,$$ where $k$, $l$ are positive integers. We obtain sufficient conditions under which all nonoscillatory solutions of the above equation tend to zero as $n\to \infty$ with a weaker assumption on $q$ than the...

The purpose of this paper is to give some results on the asymptotic relationship between the solutions of a linear difference equation and its perturbed nonlinear equation.

We discuss the asymptotic behaviour of all solutions of the functional differential equation $$y^{\text{'}}\left(x\right)=\sum _{i=1}^m{a}_i\left(x\right)y\left({\tau }_i\left(x\right)\right)+b\left(x\right)y\left(x\right),$$ where $b\left(x\right)<0$. The asymptotic bounds are given in terms of a solution of the functional nondifferential equation $$\sum _{i=1}^m\left|a_i\left(x\right)\right|\omega \left({\tau }_i\left(x\right)\right)+b\left(x\right)\omega \left(x\right)=0.$$

The paper discusses the asymptotic properties of solutions of the scalar functional differential equation $$y^{\text{'}}\left(x\right)=ay\left(\tau \left(x\right)\right)+by\left(x\right),x\in [x_0,\infty )$$ of the advanced type. We show that, given a specific asymptotic behaviour, there is a (unique) solution $y\left(x\right)$ which behaves in this way.

Asymptotic properties of solutions of the difference equation of the form $${\Delta }^m{x}_n=a_n\varphi (x_{{\tau }_1\left(n\right)},\cdots ,x_{{\tau }_k\left(n\right)})+b_n$$ are studied. Conditions under which every (every bounded) solution of the equation ${\Delta }^m{y}_n=b_n$ is asymptotically equivalent to some solution of the above equation are obtained.