Jerzy Popenda

Oscillation and nonoscillation theorems for second-order difference equations

In this paper we are concerned with the oscillatory and nonoscillatory behavior of the solutions of some second-order difference equations. Let R be the set of real numbers, N := {n,, n, + l,...} where n, is given nonnegative integer. For the function x: N--t R, the difference operator d L1 (a is fixed real constant) will be defined as follows: d,x, = x, + , ax,, (n E N). Instead of A, we shall write A. We define inductively A:x, = A,(AtIx,) for k > 1. We shall study difference equations of the form

ON THE OSCILLATION OF SOLUTIONS OF CERTAIN DIFFERENCE EQUATIONS

For some class of the f ini te difference equations o s c i l l a tion c r i t e r i a are given. Let R be the set of a l l r e a l numbers. We le t N(nQ) = = | n Q , n 0 + 1 , . . . } , where nQ i s a natural number or zero. Let a > 0 be a real constant. The difference operator A^ will be defined in the following way AQ*n = x Q + 1 axQ (neN(O)) , where j x Q j i s the sequence of r e a l numbers. Instead of  we shall write A. In this paper we study osci l la t ion of solutions of the difference equations m

ON THE DISCRETE ANALOGY OF GRONWALL LEMMA

The present paper contains some theorems which are modifications of the well known Gronwall-Bellman inequality to a d i f ference equation theory, Gronwall inequal i t ies of considerable in teres t are associated with the names of Demidovic [ 2 ] , Pachpatte [3]-[ i>], Wil let t and Wong [ 9 ] . The integra l and discrete inequal i t ies of th is type can be used in the study of existence, boundedness, •stability and other asymptotic properties of the solutions of d i f f e r e n t i a l and f i n i t e difference equations. In th i s note we wish to establ ish some new difference inequal i t i es , similar to the d i f f e r e n t i a l inequal i t ies considered in [ 1 ] , which can be applied to the study of nonlinear f i n i t e difference equat ions of more general type. Let N = | n 0 , n Q + 1 , . . . J , where nQ i s a given integer , R the set of r e a l number. For the function y:N—«-R we denote f i n i t e difference Ay(n) = y(n+1) y (n) . S denotes the fixed set j l , . . . , s | . We assume the following hypotheses: ( i ) g,u:N—» < 0 , <*>), f :N — R, ( i i ) dk:K N,nQ ^ dk(n) « n, lim dk(n) = 00 for n e K . k e S ,

Discrete Gronwall inequalities in many variables

Abstract We offer a decisive generalization of linear Gronwall discrete inequalities in several independent variables.

The periodic solutions of the second order nonlinear difference equation

Periodic and asymptotically periodic solutions of the nonlinear equation ?2xn + anf(xn) = 0, n I N, are studied.

On the asymptotic behavior of solutions of linear difference equations

In the paper sufficient conditions for the difference equation

Remark on Gronwall’s inequality

\Delta x_n=\sum^r_{i=0}a_n^{(i)}x_{n+i}

On the Solutions of Fourth Order Difference Equations

to have a solution which tends to a constant, are given. Applying these conditions, an asymptotic formula for a solution of an