Copyright © 2018 Authors. 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. International Journal of Engineering & Technology, 7 (4.10) (2018) 793-796 International Journal of Engineering & Technology Website: www.sciencepubco.com/index.php/IJET Research paper Oscillation of Generalized Second-Order Quasi Linear Difference Equations V.Srimanju 1 *, Sk.Khadar Babu 2 , V.Chandrasekar 3 1,2 Vellore Institute of Technology, Vellore – 632 014, Tamil Nadu, India 3 Thiruvalluvar University College of Arts and Science, Thennangur - 604 408, Tamil Nadu, India *Corresponding author E-mail: srimanjushc@gmail.com Abstract Authorsipresent sufficienticonditions for theioscillation of the generalizediperturbed quasilinearidifferenceiequation   1 (( 1) ) (( 1) ) (( 1) ) ( ,( )) = ( ,( ), ( )) a x i v x i v x i Fx ivx i Gx ivx i vx i                   where 0< <1  , [0, ) x   and x i x         . Examplesiillustrates the importanceiof our results are alsoiincluded. Keywords: Generalizedidifference equation; Oscillation; iQuasilinear, 1. Introduction Difference equations represent captivating mathematical field, has rich field of the applications in such diverse disciplines as popula- tion dynamics, operations research, ecology, economics, biology etc. For thelbackgroundlof differencelequations and its applicationslin diverselfields withlexamples, see [1]. The study of difference equations is based on the operator  defined as ( )= ( 1) ( ), [0, ). ux ux ux x      Thoughlmany authors [1],[16] have discussed the definition of  as ( )= ( ) ( ), (0, ), ux ux ux      (1) no notable progress have been taken on this line. Butlin [13] the authors took up the definition of  as given in (1), and given many important results and applications. They labeled the operator  defined by (1) as  and its inversely 1   , many interesting results in number theory were obtained. Qualitativelproperties like rotatory, expanding, lshrinking, spiral and web like were estab- lished by extending theory ofl  to complex function, for the solutions of difference equations involving  in [2-15, 17-21]. In the sequel, in this paper we will be considered the generalized perturbed quasi linear difference equation for [0, ) x     1 (( 1) ) (( 1) ) (( 1) ) a x i v x i v x i            ( ,( )) = ( ,( ), ( )) Fx ivx i Gx ivx i vx i        (2) where 0< <1  , ( ) ax i  is an eventually positive real valued function, land  is the generalized forward difference operator defined as ( ) = (( 1) ) ( ) vx i v x i vx i       . By alsolution of (2), we mean a nontrivial real valued function ( ) vx i  satisfying (2) for [0, ) x   . A solution ( ) vx i  is said to be oscillatory if it is neither eventually positive nor negative, and non oscillatory otherwise. 2. Main Results In this paper we assume that there exist real valued functions ( ) qx i  , ( ) px i  andla function : f R R  such that (i). ( )>0 vf v for all 0 v  ; (ii). () ( )= (, )( ) fv fw gvwv w   for , 0 vw  , where g is a nonnegativelfunction; land (iii). ( , ) ( ) ( ) Fx iv i qx i fv i      , ( , , ) ( ) ( ) Gx iv jw i px i fv i       for , 0 vw  . Thelfollowing conditionslare usedlthroughout thislpaper: 1/ 1 = , (( 1) ) a x i      (3) 1/ 1/ < , < forall >0 () () dx dx fx fx             (4) 0 = 0 inf ( ( ) ( )) 0 for all large , lim x x r x qr i pr i x       (5) 1/ 1/ 0 0 < , < for all > 0, () () dx dx fx fx           (6)