Bernstein approximations of nonlinear Sturm–Liouville problems

In this paper we deal with Sturm–Liouville boundary value problems equation(∗){u″(t)+φ(t,u(t),u′(t),λ)=0t∈(0,1)l(u)=0 and their finitely dimensional Bernstein approximations equation(∗∗){u″(t)+∑k=0nnkφ(kn,u(kn),u′(kn),λ)tk(1−t)n−k=0t∈(0,1)l(u)=0. We prove that branches of nontrivial solutions of (∗) bifurcating from trivial solutions are approximated by branches of solutions of (∗∗). Additionally we apply the global bifurcation theorem to obtain the existence results for nonlinear Sturm–Liouville problems.