Asymptotic behavior of solutions of functional difference equations

For linear functional difference equations, we obtain some results on the asymptotic behavior of solutions, which correspond to a Perron-type theorem for linear ordinary difference equations. We also apply our results to Volterra difference equations with infinite delay.     

Asymptotic behavior of the solutions of certain parabolic equations

Some laws in physics describe the change of a flux and are represented by parabolic equations of the form (*) \documentclass{article}\pagestyle{empty}\begin{document}$$\frac{{\partial u}}{{\partial t}}=\frac{\partial}{{\partial x_j }}(\eta \frac{{\partial u}}{{ax_j}}-vju),$$\end{document} j≤m, where η and v_j are functions of both space and time. We show under quite general assumptions that the solutions of equation (*) with homogeneous Dirichlet boundary conditions and initial condition u(x, 0) = u_o(x) satisfy The decay rate d > 0 only depends on bounds for η, v and G § R^m the spatial domain, while the constant c depends additionally on which norm is considered. For the solutions of equation (*) with homogeneous Neumann boundary conditions and initial condition u₀(x) ≥ 0 we derive bounds d₁u₁ ≤ u(x, t) ≤ d₂u₂, Where d_i, i = 1, 2, depend on bounds for η, v and G, and the u_i are bounds on the initial condition u₀.     

Asymptotic behavior of solutions of linear homogeneous second-order difference equations

We analyze and study the asymptotic behavior (asn→∞) of the general solutionx _n of the equationx _n+2 =Ax _n +Bx _n+1 ,A≠0,n=0,1,2,..., for various possible values of coefficients and initial data. Translated fromMatematicheskie Zametki, Vol. 66, No. 2, pp. 211–215, August, 99.     

Asymptotic behavior of positive solutions of fourth-order nonlinear difference equations

We consider a class of fourth-order nonlinear difference equations of the form {fx006-01} where α, β are ratios of odd positive integers and {p _n}, {q _n} are positive real sequences defined for all n ∈ ℕ(n ₀). We establish necessary and sufficient conditions for the existence of nonoscillatory solutions with specific asymptotic behavior under suitable combinations of convergence or divergence conditions for the sums {fx006-02}. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 1, pp. 8–27, January, 2008.     

Asymptotic behavior of solutions of difference equations in Banach spaces

In this paper we consider the first order difference equation

and give necessary and sufficient conditions so that there exist solutions which are asymptotically constant. These results generalize those given earlier by Popenda and Schmeidel. As an application we give necessary and sufficient conditions for the second order difference equation

to have asymptotically constant solutions.

     

Asymptotic behavior of solutions of forced nonlinear neutral difference equations

In this paper, we consider the asymptotic behavior of solutions of the forced nonlinear neutral difference equation $$\Delta \left[ {x(n) - \sum\limits_{i - 1}^m {p_i (n)x(n - k_i )} } \right] + \sum\limits_{j = 1}^s {q_j (n)f(x(n - l_j )) = r(n)} $$ with sign changing coefficients. Some sufficient conditions for every solution of (*) to tend to zero are established. The results extend and improve some known theorems in literature.     

Asymptotic behavior of solutions for a class of retarded difference equations

Consider the retarded difference equationx _n −x _n−1 =F(−f(x _n )+g(x _n−k )), wherek is a positive integer,F,f,g:R→R are continuous,F andf are increasing onR, anduF(u)>0 for allu≠0. We show that whenf(y)≥g(y) (resp. f(y)≤g(y)) fory∈R, every solution of (*) tends to either a constant or −∞ (resp. ∞) asn→∞. Furthermore, iff(y)≡g(y) fory∈R, then every solution of (*) tends to a constant asn→∞. Project supported by NNSF (19601016) of China and NSF (97-37-42) of Hunan     

Asymptotic properties of solutions of difference equations

Sufficient conditions for somem-th order finite difference equations are presented which have a solution behaving in a precisely specified way like a given polynomial.     

Asymptotic properties of solutions of nonlinear difference equations

Using discrete inequalities and Schauder's fixed point theorem we study the problem of asymptotic equilibrium for difference equations.     

Asymptotic behavior of solutions of variational equations

In questo articolo si considerano equazioni differenziali ordinarie del secondo ordine della forma $$\{ A(u')u'\} ' + \delta (r)A(u')u' + f(r,u) = 0,$$ dove cioè la nonlinearità è presente sia nella variabile soluzioneu che nella sua derivata. Si forniscono proprietà di monotonia, oscillazione e un accurato studio del comportamento asintotico all’ infinito delle soluzioni, quandoA, δ,f hanno crescite asintotiche di tipo algebrico.     

Asymptotic properties of solutions of certain third-order dynamic equations

In this paper, the well known oscillation criteria due to Hille and Nehari for second-order linear differential equations will be generalized and extended to the third-order nonlinear dynamic equation

(r₂(t)((r₁(t)x^Δ(t))^Δ)^γ)^Δ+q(t)f(x(t))=0     

Asymptotic behavior of solutions of nonlinear Volterra equations

An energy decay rate is obtained for solutions of wave type equations in a bounded region in Rⁿ whose boundary consists partly of a nontrapping reflecting surface and partly of an energy absorbing surface.     

Asymptotic behavior of solutions of linear delay equations

The delay differential equations of the formx′(t)=?a(t)x(t?1),t≥0 are considered, wherea(t)≥0 is locally integrable on [0,∞). The main result: Let 0<c(t)≤a(t)≤k(t) for large ∫^∞, andc(t)≤Mc(t′) fort, t′≥T,|t?t′|≤l with some constantsl>0,M>1,T≥0. Then the condition \(k(t) \leqslant \frac{3}{2} + \alpha c(t), t \geqslant T\) with some constant α>0 dependent onl, M, ensures that all solutions of (*) tend to zero ast→∞.     

Meromorphic solutions of certain difference equations of first order

In this article, we investigate the growth and value distribution of meromorphic solutions of a first order difference equation with small coefficients in the complex plane.