The Dirichlet resonant boundary value problems are considered. If the respective nonlinear equation can be reduced to a quasilinear one with a nonresonant linear part and both equations are equivalent in some domainΩand if solutions of the quasilinear problem are inΩ, then the original problem has a solution. We say then that the original problem allows for quasilinearization. If quasilinearization is possible for essentially different linear parts, then the original problem has multiple solutions. We give conditions for Emden-Fowler type resonant boundary value problem solvability and consider examples.

We consider resonant problems of the type (i) x + p(t)x + q(t)x = f (t, x, x), (ii) x (0) = 0, x(T) = 0, where p, q, f are continuous functions and the homogeneous problem (iii) x + p(t)x + q(t)x = 0 with boundary-value conditions (ii) has a nontrivial solution. We study this problem by modifying the linear part and applying the quasilinearization technique to the modified problem. Розглянуто задачi типу (i) x + p(t)x + q(t)x = f (t, x, x), (ii) x (0) = 0, x(T) = 0, з резонансом, де p, q, f-неперервнi функцiї та однорiдна задача (iii) x + p(t)x + q(t)x = 0 разом з граничними умовами (ii) має нетривiальний розв'язок. Задача вивчається за допомогою змiни лiнiйної частини та застосування технiки квазiлiнеаризацiї до модифiкованої задачi.

Three-point boundary value problems for the second order nonlinear ordinary differential equations are considered. Existence of solutions are established by using the quasilinearization approach. As an application, the Emden-Fowler type problems with nonresonant and resonant linear parts are considered to demonstrate our results.

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 a nonlinear resonant boundary value problem. To prove the existence of a solution to a given boundary value problem we replace the linear part of a given equation by non-resonant linear part. First, to modify a resonant problem to a regular one we use the Taylor expansion for f .t; x; x 0 / with respect to x. The second way of conversion a given problem to nonresonant one is based on an appropriate choice of "good" approximation to expected solution. We provide the existence results illustrating both ways.