Bernstein approximations of nonlinear Sturm–Liouville problems

Abstract

In this paper we deal with Sturm–Liouville boundary value problems (∗) { u ″ ( t ) + φ ( t , u ( t ) , u ′ ( t ) , λ ) = 0 t ∈ ( 0 , 1 ) l ( u ) = 0 and their finitely dimensional Bernstein approximations (∗∗) { u ″ ( t ) + ∑ k = 0 n n k φ ( k n , u ( k n ) , u ′ ( k n ) , λ ) t k ( 1 − t ) n − k = 0 t ∈ ( 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.