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.
Dobkevich, M. (2010). On construction of converging sequences to solutions of boundary value problems. Mathematical Modelling and Analysis, 15(2), 189-197. https://doi.org/10.3846/1392-6292.2010.15.189-197
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