Applied Mathematics, 2016, 7, 1612-1631 Published Online August 2016 in SciRes. http://www.scirp.org/journal/am http://dx.doi.org/10.4236/am.2016.714139 How to cite this paper: Sadhasivam, V., Sundar, P. and Santhi, A. (2016) On the Asymptotic Behavior of Second Order Qua- silinear Difference Equations. Applied Mathematics, 7, 1612-1631. http://dx.doi.org/10.4236/am.2016.714139 On the Asymptotic Behavior of Second Order Quasilinear Difference Equations Vadivel Sadhasivam 1 , Pon Sundar 2 , Annamalai Santhi 1 1 PG and Research Department of Mathematics, Thiruvalluvar Government Arts College, Rasipuram, Namakkal, India 2 Om Muruga College of Arts and Science, Salem, India Received 15 June 2016; accepted 26 August 2016; published 29 August 2016 Copyright © 2016 by authors and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/ Abstract In this paper, we investigate the asymptotic behavior of the following quasilinear difference equa- tions ( ) ( ) ( ) ( ) ( ) ( ) yn yn pn yn yn 1 1 , α β − − ∆ ∆ ∆ = (E) where { } n N n n n 0 0 0 0 , 1, 2, ∈ = + +  , n N 0 ∈ . We classified the solutions into six types by means of their asymptotic behavior. We establish the necessary and/or sufficient conditions for such equa- tions to possess a solution of each of these six types. Keywords Asymptotic Behavior, Positive Solutions, Homogeneous, Quasilinear Difference Equations 1. Introduction Recently, the asymptotic properties of the solutions of second order differential equations [1] [2] difference equ- ations of the type (E) and/or related equations have been investigated by many authors, for example see, [3]-[19] and the references cited there in. Following this trend, we investigate the existence of these six types of solutions of the Equation (E) showing the necessary and/or sufficient conditions can be obtained for the existence of those solutions. For the general backward on difference equations, the reader is referred to the monographs [20]-[24]. In 1996, PJY Wang and R.P. Agarwal [25] considered the quasilinear equation ( ) ( ) ( ) 1 1 0 n n n n a y qf y σ − − ∆ ∆ + = (1)