Singular discrete and continuous mixed boundary value problems

For each n∈Nn∈N, n≥2n≥2, we prove the existence of a solution (u0,…,un)∈Rn+1(u0,…,un)∈Rn+1 of the singular discrete problem 1h2Δ2uk−1+f(tk,uk)=0,k=1,…,n−1,Δu0=0,un=0, where uk>0uk>0 for k=0,…,n−1k=0,…,n−1. Here T∈(0,∞)T∈(0,∞), h=Tn, tk=hktk=hk, f(t,x):[0,T]×(0,∞)→Rf(t,x):[0,T]×(0,∞)→R is continuous and has a singularity at x=0x=0. We prove that for n→∞n→∞ the sequence of solutions of the above discrete problems converges to a solution yy of the corresponding continuous boundary value problem y″(t)+f(t,y(t))=0,y″(t)+f(t,y(t))=0,y′(0)=0,y(T)=0.