We consider the Dirichlet problemunder 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.

We consider the second order BVP x″ = f (t, x, x′), x′(a) = A, x′(b) = B provided that there exist α and β (lower and upper functions) such that: β′ (α) < A < α′(a) and β′(b) < B < α′ (b). We consider monotone and non-monotone approximations of solutions to the Neumann problem. The results and examples are provided.

We review the results concerning types of solutions of boundary value problems for the second order nonlinear equation(l2x)(t)=f(t,x,x′),where(l2x)(t)is the second order linear differential form. The existence results and the multiplicity results are stated in terms of types of solutions.

Abstract. We consider the second-order nonlinear boundary value problems (BVPs) with Sturm-Liouville boundary conditions. We define types of solutions and show that if there exist solutions of different types then there exist intermediate solutions also.Keywords: nonlinear boundary value problem, multiplicity of solutions, Sturm-Liouville problem.

We consider boundary value problems of the type x'' = f(t, x, x'), (∗) x(a) = A, x(b) = B. A solution ξ(t) of the above BVP is said to be of type i if a solution y(t) of the respective equation of variations y'' = fx(t, ξ(t), ξ' (t))y + fx' (t, ξ(t), ξ' (t))y' , y(a) = 0, y' (a) = 1, has exactly i zeros in the interval (a, b) and y(b) 6= 0. Suppose there exist two solutions x1(t) and x2(t) of the BVP. We study properties of the set S of all solutions x(t) of the equation (∗) such that x(a) = A, x'1(a) ≤ x' (a) ≤ x'2(a) provided that solutions extend to the interval [a, b].