Three symmetric solutions of lidstone boundary value problems for difference and partial difference equations

We consider the boundary value problem $\text{δ}{}^{\mathrm{2m}}\text{y(k-m)=f(y(k),δ}{}^2\text{y(k-1),…,δ}{}^{\mathrm{2i}}\text{y(k-1),…,δ}{}^{\mathrm{2(m-1)}}\text{y(k-(m-1)))}\text{kϵ{a+1,…,b+1},}\text{δ}{}^{\mathrm{2i}}\text{y(a+1-m)=δ}{}^{\mathrm{2i}}\text{y(b+1+m-2i)=0,}\text{0≤i≤m-1,}$ where m ≥ 1 and (−1)^m f R^m → 0, ∞) is continuous. By using Amann and Leggett-Williams' fixed-point theorems, we develop growth conditions on f so that the boundary value problem has triple positive symmetric solutions. The results obtained are then applied in the investigation of radial solutions for certain partial difference equation subject to Lidstone type conditions.