Multiple fixed-sign solutions for a system of difference equations with Sturm–Liouville conditions

We consider the following system of difference equations: Δ m u i ( k ) + P i ( k , u 1 ( k ) , u 2 ( k ) , … , u n ( k ) ) = 0 , k ∈ { 0 , 1 , … , N } , i = 1 , 2 , … , n together with Sturm–Liouville boundary conditions Δ j u i ( 0 ) = 0 , 0 ⩽ j ⩽ m - 3 , ζ Δ m - 2 u i ( 0 ) - η Δ m - 1 u i ( 0 ) = 0 , ω Δ m - 2 u i ( N + 1 ) + δ Δ m - 1 u i ( N + 1 ) = 0 , where m ⩾ 2 , N ⩾ m - 1 , ζ > 0 , ω > 0 , η ⩾ 0 , δ ⩾ ω , ζ ω ( N + 1 ) + ζ δ + η ω > 0 . By using two different fixed point theorems, we develop criteria for the existence of three solutions of the system which are of fixed signs on { 0 , 1 , … , N + m } . Examples are also included to illustrate the results obtained.