Existence of nonoscillatory solutions to second-order nonlinear neutral difference equations

We study a class of second-order neutral delay difference equations with positive and negative coefficients Δ ( r n ( Δ ( x n + p x n − m ) ) ) + p n f ( x n − k ) − q n g ( x n − l ) = 0 , n = n 0 , n 0 + 1 , . . . , where p ∈ R , m ; k ; l ; n 0 ∈ N , p n ; q n ; r n ∈ R + ; f ; g ∈ C ( R ; R ) with x f ( x ) 0 and x g ( x ) 0 ( x ≠ 0 ) . Some sufficient conditions for the existence of a nonoscillatory solution of the studied equation expressed in terms of ∑ ∞ R n p n 1 and ∑ ∞ R n q n 1 are obtained, where R n = ∑ n s = n 0 1 r s ; n ≥ n 0 .