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

摘要

We consider the following system of difference equations:Δmui(k)+Pi(k,u1(k),u2(k),…,un(k))=0,k∈{0,1,…,N},i=1,2,…,ntogether with Sturm–Liouville boundary conditionsΔjui(0)=0,0⩽j⩽m-3,ζΔm-2ui(0)-ηΔm-1ui(0)=0,ωΔm-2ui(N+1)+δΔm-1ui(N+1)=0,where m⩾2,N⩾m-1,ζ>0,ω>0,η⩾0,δ⩾ω,ζω(N+1)+ζδ+ηω>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.