International Journal of Mathematics Research. ISSN 0976-5840 Volume 6, Number 1 (2014), pp. 49-56 © International Research Publication House http://www.irphouse.com Oscillatory Behavior of Fractional Difference Equations M.Reni Sagayaraj 1 , A.George Maria Selvam 2 and M.Paul Loganathan 3 1, 2 Sacred Heart College, Tirupattur - 635 601, S.India, 3 Department of Mathematics, Dravidian University, Kuppam. e− mail : agmshc@gmail.com ABSTRACT In this paper, we study oscillatory behavior of the fractional difference equations of the following form 0 0 1 ( ) 1 ( ()( ( ))) () ( 1) () 0, , t t st ptg xt qtf t s xs t N α α α α −+ − +− = ⎛ ⎞ Δ Δ + − − = ∈ ⎜ ⎟ ⎝ ⎠ ∑ where ∆ α denotes the Riemann-Liouville difference operator of order α, 0 < α ≤ 1. We establish some oscillation criteria for the equation using Riccati transformation technique and Hardy inequality. Examples are provided to illustrate our main results. 1. INTRODUCTION Oscillatory behavior of fractional differential equations have been investigated by few authors, see papers [2]-[8] and the theory of fractional differential equations are presented in the books, see [13]-[15]. But the fractional difference equations are studied by very few authors, see [9]-[12]. Motivated by [3] and [8], we study the following fractional difference equation of the form 0 0 1 ( ) 1 ( ()( ( ))) () ( 1) () 0, , t t s t ptg xt qtf t s xs t N α α α α −+ − +− = ⎛ ⎞ Δ Δ + − − = ∈ ⎜ ⎟ ⎝ ⎠ ∑ (1) where ∆ α denotes the Riemann-Liouville difference operator of order 0 < α ≤ 1. In this paper, we make the following assumptions. (H 1 ). () p t and () qt are positive sequences and , : f gR R → are continuous functions with () 0, () 0 for 0 xf x xg x x > > ≠ and there exist positive constants 1 2 , k k such that 1 2 () , () fx x k k x gx ≥ ≥ for all 0 x ≠ . (H 2 ). 1 ( , ) g CRR − ∈ is a continuous function with 1 () 0 for 0 xg x x − > ≠ and there