International Journal of Mathematics and Mathematical Sciences
Volume 15 (1992), Issue 1, Pages 129-142
doi:10.1155/S0161171292000152
Comparison results and linearized oscillations for higher-order difference equations
1Department of Mathematics, The University of Rhode Island, Kingston 02881-0816, R.I., USA
2Department of Mathematics, Yangzhou Teacher's College, Yangzhou, Jiangsu, China
Received 8 May 1989; Revised 20 February 1990
Copyright © 1992 G. Ladas and C. Qian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Consider the difference equationsΔmxn+(−1)m+1pnf(xn−k)=0, n=0,1,… (1)andΔmyn+(−1)m+1qng(yn−ℓ)=0, n=0,1,…. (2)We establish a comparison result according to which, when m is odd, every solution of Eq.(1) oscillates provided that every solution of Eq.(2) oscillates and, when m is even, every bounded solution of Eq.(1) oscillates provided that every bounded solution of Eq.(2) oscillates. We also establish a linearized oscillation theorem according to which, when m is odd, every solution of Eq.(1) oscillates if and only if every solution of an associated linear equationΔmzn+(−1)m+1pzn−k=0, n=0,1,… (*)oscillates and, when m is even, every bounded solution of Eq.(1) oscillates if and only if every bounded solution of (*) oscillates.