Asymptotic Properties Of Solutions Research Articles

The asymptotic behaviour of q-difference equations with multiple delays

Abstract We investigate asymptotic properties of solutions of a class of linear q-difference equations. We relate their behaviour to the asymptotics of some q-exponential functions and mention possible connections with the corresponding continuous case.

Oscillatory and asymptotic properties of solutions of even order neutral difference equations

We study the neutral difference equation of the form Explicit sufficient conditions which guarantee that all solutions of the above equation are oscillatory are obtained.

Asymptotic properties of solutions of systems of nonlinear functional differential equations of neutral type

We establish sufficient conditions for systems of nonlinear functional differential equations of neutral type to have solutions that are continuously differentiable and bounded for t ∈ ℝ (together with their first derivatives) and investigate the asymptotic properties of these solutions.

Existence of positive solutions of deay diffetential equations

The article is concerned with the existence of positive solutions of the delay differential equations. The asymptotic properties of solutions are also treated.

Random walks with random velocities

We consider a random walk model that takes into account the velocity distribution of random walkers. Random motion with alternating velocities is inherent to various physical and biological systems. Moreover, the velocity distribution is often the first characteristic that is experimentally accessible. Here, we derive transport equations describing the dispersal process in the model and solve them analytically. The asymptotic properties of solutions are presented in the form of a phase diagram that shows all possible scaling regimes, including superdiffusive, ballistic, and superballistic motion. The theoretical results of this work are in excellent agreement with accompanying numerical simulations.

On unilateral contact of two plates aligned at an angle to each other

The contact problem for two elastic plates aligned at a prescribed angle to each other is considered. The set of contact points is assumed to be unknown in advance and to be determined only after the problem is solved. Various formulations of the problem are given, and their equivalence is proved. A complete set of boundary conditions fulfilled on the contact domain is found, and the character of satisfaction of these conditions is described. The asymptotic properties of solutions are studied for rigidity parameters of the contacting plates tending to infinity.

Approximate solution to a boundary value problem for the Lyapunov differential equation with a parameter

A special boundary value problem is studied for the Lyapunov differential equation which is used for investigation of the asymptotic properties of solutions to systems of periodic differential equations with a parameter. An algorithm is proposed for constructing an approximate solution to this boundary value problem, and conditions on the parameter are found under which the zero solution to the system is asymptotically stable.

On the asymptotic properties of solutions of linear functional differential equations with linearly transformed argument

We establish new properties of solutions of the functional differential equation {fx153-01} in the neighborhood of the singular point t = +∞.

On asymptotic properties of solutions of functional differential equations of neutral type

In this paper, we obtain sharp estimates for strong solutions of functional differential equations of neutral type. Our result is closely connected with our previous results devoted to the initial-value problem for above-mentioned equations in the scale of Sobolev spaces. To obtain our estimates of the solutions, we essentially use Riesz basis properties of the system of exponential solutions. The fact that they form a Riesz basis is one of the main results of this article.

On an approach to the analysis of asymptotic properties of solutions of first-order ordinary delay differential equations

For the first-order ordinary delay differential equation $$ u'(t) + p(t)u(r(t)) = 0, $$ where p ∈ Lloc(ℝ+; ℝ+), τ ∈ C(ℝ+; ℝ+), τ(t) ≤ t for t ∈ ℝ+, limt→+∞τ(t) = +∞, and ℝ+:= [0, ∞), we obtain new criteria for the existence of sign-definite and oscillating solutions, thus generalizing some earlier-known results.

Unilateral contact problem for two inclined elastic bodies

The paper concerns an analysis of unilateral contact problems between two inclined elastic plates and between a plate and a beam. Considered problems are characterized by a contact set having a dimension less than one of that of a domain. This property leads to a new class of free boundary problems with inequality type boundary conditions. The main attention is paid to a suitable description of boundary conditions along the contact zone. Asymptotic properties of solutions are established provided that elasticity parameters of the contacting bodies are changing, in particular, in the case when a stiffness of the elastic body goes to infinity.

Asymptotic properties of solutions to some nonlinear integral equations of convolution type

For a class of nonlinear integral equations of convolution type we give necessary and sufficient conditions for the boundedness of nonnegative solutions. Moreover, conditions for the solution to converge asymptotically to a determined limit are obtained.

On the asymptotic properties of solutions of nonlinear parabolic equations

Let Πa,b = {(x, t) : x ∈ Ω, a < t < b}, σa,b = {(x, t) : x ∈ ∂Ω, a < t < b}, Πa = Πa,∞, σa = σa,∞, and ∂Ω be the boundary of Ω. Assume that the coefficients aij(x), ai(x), i = 1, . . . , n, are Holder continuous functions in Ω. The function f(x, t, u) satisfies the Holder condition in x and t in each of the domains Π0,T , is nondecreasing in u, and has the derivative ∂f ∂u ; moreover, f(x, t, 0) ≡ ∂f ∂u (x, t, 0) = 0 and ∂f ∂u is continuous in u uniformly in Π0. Consider the boundary conditions

Asymptotic properties of solutions of the Cauchy problem for a degenerate singularly perturbed system of differential equations in the case of the multiple spectrum of the principal operator

We prove the asymptotic character of a solution of the Cauchy problem for a singularly perturbed linear system of differential equations with degenerate matrix of the coefficients of derivatives in the case where the limit matrix pencil is regular and has multiple “finite” and “infinite” elementary divisors. We establish conditions under which the constructed formal solutions are asymptotic expansions of the corresponding exact solutions.

On asymptotic properties of solutions of functional differential equations with linearly transformed argument

We establish new properties of solutions of the functional differential equation x′(t) = ax(t) + bx(t − r) + cx′(t − r) + px(qt) + hx′(qt) + f1(x(t), x(t − r), x′(t − r), x(qt), x′(qt)) in the neighborhood of the singular point t = +∞.

On the (x,t) asymptotic properties of solutions of the Navier-Stokes equations in the half-space

We study the space-time asymptotic behavior of classical solutions of the initial-boundary value problem for the Navier-Stokes system in the half-space. We construct a (local in time) solution corresponding to an initial data that is only assumed to be continuous and decreasing at infinity as |x|−µ, µ ∈ (1/2,n). We prove pointwise estimates in the space variable. Moreover, if μ ∈ [1, n) and the initial data is suitably small, then the above solutions are global (in time), and we prove space-time pointwise estimates. Bibliography: 19 titles.

Asymptotic properties of solutions of semilinear second-order elliptic equations in cylindrical domains

The equations under consideration have the following structure: $$\frac{{\partial ^2 u}}{{\partial x_n^2 }} + \sum\limits_{i,j = 1}^{n - 1} {\frac{\partial }{{\partial x_i }}\left( {a_{ij} (x)\frac{{\partial u}}{{\partial x_j }}} \right)} + \sum\limits_{i = 1}^{n - 1} {a_i (x)\frac{{\partial u}}{{\partial x_i }}} - f(u,x_n ) = 0,$$ where 0 < xn < ∞, (x1, …, xn−1) ∈ Ω, Ω is a bounded Lipschitz domain, \(f(0,x_n ) \equiv 0,\tfrac{{\partial f}}{{\partial u}}(0,x_n ) \equiv 0\) is a function that is continuous and monotonic with respect to u, and all coefficients are bounded measurable functions. Asymptotic formulas are established for solutions of such equations as xn → + ∞; the solutions are assumed to satisfy zero Dirichlet or Neumann boundary conditions on ∂Ω. Previously, such formulas were obtained in the case of aij, ai depending only on (x1, …, xn−1).

ASYMPTOTICS FOR NONLINEAR NONLOCAL EQUATIONS ON A HALF-LINE

We study the initial-boundary value problem for a general class of nonlinear pseudo-differential equations on a half-line [Formula: see text] where the number M depends on the order of the pseudo-differential operator [Formula: see text] on a half-line. The nonlinear term [Formula: see text] is such that [Formula: see text] as u, v → 0, with ρ, σ &gt; 0. Pseudo-differential operator [Formula: see text] is defined by the inverse Laplace transform. The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem (0.1) and to find the main term of the asymptotic representation of solutions taking into account the influence of inhomogeneous boundary data and a source on the asymptotic properties of solutions.

On asymptotic properties of solutions of linear differential functional equations with linearly transformed argument

We establish new properties of C 1[−1, +∞)-solutions of the linear functional differential equation ẋ(t) = ax(t) + bx(qt) + hx(t−1) + cẋ(qt) + rẋ(t−1) in the neighborhood of the singular point t = +∞.

Asymptotic properties of solutions of an equation of order n with almost constant coefficients and pulse action

We investigate the asymptotic behavior of solutions of linear differential equations with almost constant coefficients and pulse action at fixed times as t tends to infinity. We establish conditions for the times of pulse action under which there exist values of pulse action for which the solution of the considered Cauchy problem with initial conditions that coincide with the initial conditions for a certain (arbitrary but fixed) solution of the original equation without pulse action is bounded, unbounded, or tending to infinity.