Nonlinear discrete Sturm–Liouville problems at resonance

In this paper, we study nonlinear bounded value problems of the form Δ[p(t−1)Δy(t−1)]+q(t)y(t)+λky(t)=f(t,y(t))+h(t),Δ[p(t−1)Δy(t−1)]+q(t)y(t)+λky(t)=f(t,y(t))+h(t),a11y(a)+a12Δy(a)=0,a21y(b+1)+a22Δy(b+1)=0, where λkλk is an eigenvalue of the associated linear problem, ff is subject to the sublinear growth condition |f(t,s)|≤A|s|α+B,t∈{a+1,…,b+1},s∈R for some 0≤α<10≤α<1 and A,B∈(0,∞). We prove the existence and multiplicity of solutions by using the connectivity properties of solution sets of parameterized families of compact vector fields.