Types of solutions and multiplicity results for two-point nonlinear boundary value problems

Abstract

Two-point boundary value problems for the second-order ordinary nonlinear differential equations are considered. If the respective nonlinear equation can be reduced to a quasi-linear one with a non-resonant linear part and both equations are equivalent in some domain D , and if solutions of the quasi-linear problem lie in D , then the original problem has a solution. We then say that the original problem allows for quasilinearization. We show that a quasi-linear problem has a solution of definite type which corresponds to the type of the linear part. If quasilinearization is possible for essentially different linear parts, then the original problem has multiple solutions.