Optimal conditions to ensure the stability of periodic solutions of first order difference equations lying between lower and upper solutions

This paper is devoted to the study of the stability of the solutions of the first order implicit difference periodic equation Δ u ( i ) = f ( i , u ( i + 1 ) ) ; i ∈ { 0 , … , N - 1 } , u ( 0 ) = u ( N ) . We prove that if there exist α ⩽ β a pair of lower and upper solutions respectively of this problem together with a suitable condition on the function f and it has at most two solutions in the sector [ α , β ] , then one of such solutions is asymptotically stable. er, it is proved the optimality of the given results, both in the conditions imposed on f and in the number of the periodic solutions.