Janusz Migda

Asymptotically polynomial solutions of difference equations

AbstractAsymptotic properties of solutions of a difference equation of the form Δmxn=anf(n,xσ(n))+bn are studied. We present sufficient conditions under which, for any polynomial φ(n) of degree at most m−1 and for any real s≤0, there exists a solution x of the above equation such that xn=φ(n)+o(ns). We give also sufficient conditions under which, for given real s≤m−1, all solutions x of the equation satisfy the condition xn=φ(n)+o(ns) for some polynomial φ(n) of degree at most m−1.MSC:39A10.

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.

Iterated remainder operator, tests for multiple convergence of series, and solutions of difference equations

We establish some properties of iterations of the remainder operator which assigns to any convergent series the sequence of its remainders. Moreover, we introduce the spaces of multiple absolute summable sequences. We also present some tests for multiple absolute convergence of series. These tests extend the well-known classical tests for absolute convergence of series. For example we generalize the Raabe, Gauss, and Bertrand tests. Next we present some applications of our results to the study of asymptotic properties of solutions of difference equations. We use the spaces of multiple absolute summable sequences as the measure of approximation.MSC:39A10.

Approximative solutions to difference equations of neutral type

Asymptotic properties of solutions to difference equations of the form Δ m ( x n - u n x n - k ) = a n f ( x n ) + b n are studied. Replacing the sequence u by its limit and the right side of the equation by zero we obtain an equation which we call the fundamental equation. First we investigate the space of all solutions of the fundamental equation. We show that any such solution is a sum of a polynomial sequence and a product of a geometric sequence and a periodic sequence. Next, using a new version of the Krasnoselski fixed point theorem and the iterated remainder operator, we establish sufficient conditions under which a given solution of the fundamental equation is an approximative solution to the above equation. Our approach, based on the iterated remainder operator, allows us to control the degree of approximation. In this paper we use o(ns), for a given nonpositive real s, as a measure of approximation.

Asymptotically polynomial solutions to difference equations of neutral type

Asymptotic properties of solutions of difference equation of the form \[ \Delta^m(x_n+u_nx_{n+k})=a_nf(n,x_{\sigma(n)})+b_n \] are studied. We give sufficient conditions under which all solutions, or all solutions with polynomial growth, or all nonoscillatory solutions are asymptotically polynomial. We use a new technique which allows us to control the degree of approximation.

Bounded solutions of nonlinear discrete volterra equations

Abstract We give sufficient conditions, under which for every real constant, there exists a solution of the nonlinear discrete Volterra equation Δx(n)=b(n)+∑i=0nK(n,i)f(x(i)),

Nonoscillatory Solutions to Second-Order Neutral Difference Equations

\Delta x(n) = b(n) + \sum\limits_{i=0}^{n}K(n,i)f(x(i)),

On the asymptotic behavior of solutions of higher order nonlinear difference equations

convergent to this constant. We give also conditions under which all solutions are asymptotically constant. Sufficient conditions for the existence of asymptotically periodic solutions of the above equation are also derived.