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Journal of Function Spaces
Volume 2015, Article ID 737420, 9 pages
Research Article

Some Discussions on the Difference Equation

1Department of Mathematics, Faculty of Science and Arts, University of Jeddah (UJ), P.O. Box 355, Khulais 21921, Saudi Arabia
2Department of Mathematics, Faculty of Science, Ain Shams University, P.O. Box 1156, Abbassia, Cairo 11566, Egypt

Received 6 November 2014; Revised 8 April 2015; Accepted 15 April 2015

Academic Editor: Jaeyoung Chung

Copyright © 2015 Awad A. Bakery. 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.


We give in this work the sufficient conditions on the positive solutions of the difference equation ,  , where α, k, and under positive initial conditions ,   to be bounded, α-convergent, the equilibrium point to be globally asymptotically stable and that every positive solution converges to a prime two-periodic solution. Our results coincide with that known for the cases of Amleh et al. (1999) and of Hamza and Morsy (2009). We offer improving conditions in the case of of Gümüs and Öcalan (2012) and explain our results by some numerical examples with figures.