In this paper we prove two results. The first is an extension of the result of G. D. Jones [4]: (A) Every nontrivial solution for \[ \left\rbrace \begin{array}{ll}(-1)^n u^{(2n)} + f(t,u) = 0,\hspace{5.0pt}\text{in} \hspace{5.0pt}(\alpha , \infty ), u^{(i)}(\xi ) = 0, \quad i = 0,1,\dots , n-1, \hspace{5.0pt} \text{and} \hspace{5.0pt}\xi \in (\alpha , \infty ), \end{array}\right.\] must be unbounded, provided $f(t,z)z\ge 0$, in $E \times \mathbb R$ and for every bounded subset $I$, $f(t,z)$ is bounded in $E \times I$. (B) Every bounded solution for $(-1)^n u^{(2n)} + f(t,u) = 0$, in $\mathbb R$, must be constant, provided $f(t,z)z\ge 0$ in $\mathbb R \times \mathbb R$ and for every bounded subset $I$, $f(t,z)$ is bounded in $\mathbb R \times I$.
The nonlinear difference equation (E) xn+1 − xn = anϕn(xσ(n) ) + bn, where (an), (bn) are real sequences, ϕn : −→ , (σ(n)) is a sequence of integers and lim n−→∞ σ(n) = ∞, is investigated. Sufficient conditions for the existence of solutions of this equation asymptotically equivalent to the solutions of the equation yn+1 − yn = bn are given. Sufficient conditions under which for every real constant there exists a solution of equation (E) convergent to this constant are also obtained.
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: a solution polygon converges to the boundary of the Wulff shape without vanishing edges as time tends to infinity.
The BIPF algorithm is a Markovian algorithm with the purpose of simulating certain probability distributions supported by contingency tables belonging to hierarchical log-linear models. The updating steps of the algorithm depend only on the required expected marginal tables over the maximal terms of the hierarchical model. Usually these tables are marginals of a positive joint table, in which case it is well known that the algorithm is a blocking Gibbs Sampler. But the algorithm makes sense even when these marginals do not come from a joint table. In this case the target distribution of the algorithm is necessarily improper. In this paper we investigate the simplest non trivial case, i. e. the 2×2×2 hierarchical interaction. Our result is that the algorithm is asymptotically attracted by a limit cycle in law.
We present several results dealing with the asymptotic behaviour of a real twodimensional system x ′ (t) = A(t)x(t) + ∑ Pm k=1 Bk(t)x(θk(t)) + h(t, x(t), x(θ1(t)), . . . , x(θm(t))) with bounded nonconstant delays t − θk(t) ≥ 0 satisfying limt→∞ θk(t) = ∞, under the assumption of instability. Here A, Bk and h are supposed to be matrix functions and a vector function, respectively. The conditions for the instable properties of solutions together with the conditions for the existence of bounded solutions are given. The methods are based on the transformation of the real system considered to one equation with complex-valued coefficients. Asymptotic properties are studied by means of a suitable Lyapunov-Krasovskii functional and the Wa˙zewski topological principle. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one constant or nonconstant delay were studied.
In this paper we investigate the asymptotic properties of all solutions of the delay differential equation y'(x)=a(x)y(\tau(x))+b(x)y(x),\qquad x\in I=[x_0,\infty). 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),\qquad x\in I and a solution of the functional equation |a(x)|\varphi(\tau(x))=|b(x)|\varphi(x),\qquad x\in I.
In the paper we consider the difference equation of neutral type (E) ∆3 [x(n) − p(n)x(σ(n))] + q(n)f(x(τ (n))) = 0, n ∈ N(n0 ), where p, q : N(n0 ) → R+; σ, τ : N → Z, σ is strictly increasing and lim n→∞ σ(n) = ∞; τ is nondecreasing and lim n→∞ τ (n) = ∞, f : R → R, xf(x) > 0. We examine the following two cases: 0 < p(n) ≤ λ ∗ < 1, σ(n) = n − k, τ (n) = n − l, and 1 < λ∗ ≤ p(n), σ(n) = n + k, τ (n) = 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 → ∞ with a weaker assumption on q than the usual assumption ∑∞ i=n0 q(i) = ∞ that is used in literature.
The main goal of the paper is the presentation of several new results on the asymptotic dynamics of modes in strong global solutions to the homogeneous Navier-Stokes equations. It is proved as the main result that if w is such a solution then there exists a unique eigenvalue od the Stokes operator such that its associated eigenfunctions prevail asymptotically in the solution w for t i-> oo. and Obsahuje seznam literatury