Quasilinearization of Resonant Boundary-value Problems with Mixed Boundary Conditions

Abstract

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.

On Types of Solutions of the Second Order Nonlinear Boundary Value Problems

“…The Dirichlet boundary conditions were considered. Later the results of the same spirit were obtained in the works [26,27]. In the work [26] the mixed boundary conditions…”

“…There are numerous papers devoted to resonant boundary value problems. Some of them can be found in the bibliography of papers [25][26][27].…”

On Three-Point Boundary Value Problem

“…We use a quasilinearization approach. Previously this approach was used for the study of different BVPs in [18,19,20].…”

Resonant Problems by Quasilinearization

“…There is an intensive literature on resonant problems. We mention several papers [5][6][7][8][9][10][11][12][13].…”