Two‐point Boundary Value Problems at Resonance

Abstract

Consider two-point boundary value problems of resonance typeWe investigate existence and multiplicity of solutions to such problems by using the quasilinearization process.

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

“…As example (15) shows, there are solutions, which do not possess the property ( * ). The respective ( ) (the solutions of (12), (25)) are oscillatory functions, which may have multiple zeros.…”

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

“…This idea was realized first in the work by Yermachenko [25]. The Dirichlet boundary conditions were considered.…”

“…Then there exists a sequence { } such that ( ) = , where are solutions of auxiliary problems Let a solution ( ) of the problem (16) have type( ( )) = . This means that a solution ( ) of problem(12),(25) has exactly zeros in ( , ) and ( ) ̸ = 0. We wish to construct a sequence ( ; ) of solutions of auxiliary problems the same type and which converge to ( ).Consider solutions ( ; ) of the problems then ( ; ) → ( ).…”

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].…”