First, existence criteria for at least three nonnegative solutions to the following boundary value problem of fourth-order difference equation Δ 4 x ( t − 2 ) = a ( t ) f ( x ( t ) ) , t ∈ [ 2 , T ] , x ( 0 ) = x ( T + 2 ) = 0 , Δ 2 x ( 0 ) = Δ 2 x ( T ) = 0 are established by using the well-known Leggett-Williams fixed point theorem, and then, for arbitrary positive integer m , existence results for at least 2 m - 1 nonnegative solutions are obtained.