On construction of converging sequences to solutions of boundary value problems

Abstract

We consider the Dirichlet problem x″ = f(t,x), x(a) = A, x(b) = B under the assumption that there exist the upper and lower functions. We distinguish between two types of solutions, the first one, which can be approximated by monotone sequences of solutions (the so called Jackson—Schrader's solutions) and those solutions of the problem, which cannot be approximated by monotone sequences. We discuss the conditions under which this second type solutions of the Dirichlet problem can be approximated.

Keywords:

• nonlinear boundary value problems
• types of solutions
• monotone iterations
• multiplicity of solutions
• non‐monotone iterations