tailieunhanh - ON A CLASS OF FOURTH-ORDER NONLINEAR DIFFERENCE EQUATIONS MAŁGORZATA MIGDA, ANNA MUSIELAK, AND EWA

ON A CLASS OF FOURTH-ORDER NONLINEAR DIFFERENCE EQUATIONS MAŁGORZATA MIGDA, ANNA MUSIELAK, AND EWA SCHMEIDEL Received 18 August 2003 and in revised form 22 October 2003 We consider a class of fourth-order nonlinear difference equations. The classification of nonoscillatory solutions is given. Next, we divide the set of solutions of these equations into two types: F+ - and F− -solutions. Relations between these types of solutions and their nonoscillatory behavior are obtained. Necessary and sufficient conditions are obtained for the difference equation to admit the existence of nonoscillatory solutions with special asymptotic properties. 1. Introduction Consider the difference equation ∆ an ∆. | ON A CLASS OF FOURTH-ORDER NONLINEAR DIFFERENCE EQUATIONS MALGORZATA MIGDA ANNA MUSIELAK AND EWA SCHMEIDEL Received 18 August 2003 and in revised form 22 October 2003 We consider a class of fourth-order nonlinear difference equations. The classification of nonoscillatory solutions is given. Next we divide the set of solutions of these equations into two types F - and F- -solutions. Relations between these types of solutions and their nonoscillatory behavior are obtained. Necessary and sufficient conditions are obtained for the difference equation to admit the existence of nonoscillatory solutions with special asymptotic properties. 1. Introduction Consider the difference equation A nA bnA cnAyn f n yn 0 n e N where N 0 1 2 . A is the forward difference operator defined by Ayn yn 1 - yn and an bn and cn are sequences of positive real numbers. Function f N X R R. By a solution of we mean a sequence yn which satisfies for n sufficiently large. We consider only such solutions which are nontrivial for all large n. A solution of is called nonoscillatory if it is eventually positive or eventually negative. Otherwise it is called oscillatory. In the last few years there has been an increasing interest in the study of oscillatory and asymptotic behavior of solutions of difference equations. Compared to second-order difference equations the study of higher-order equations and in particular fourth-order equations see . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 has received considerably less attention. An important special case of fourth-order difference equations is the discrete version of the Schrodinger equation. The purpose of this paper is to establish some necessary and sufficient conditions for the existence of solutions of with special asymptotic properties. Throughout the rest of our investigations one or several of the following assumptions will be imposed Copyright 2004 Hindawi Publishing Corporation Advances in Difference Equations 2004 1 2004 23-36 .