Two-point nonlinear boundary value problems for ordinary differential equations: existence, multiple solutions, properties

In our talk we discuss some basic issues of the theory of nonlinear boundary value problems (BVP), such as the existence of a solution, multiplicity of solutions and properties of solutions. In the first part we consider the second order BVP. Brief historical report and overview of the method of a priori bounds and the method of the upper and lower solutions (functions) are given. We devote special attention to applicability of the second method to problems which have oscillatory solutions. Two alternative approaches are discussed, the first one, based on the analysis of a phase plane of similar autonomous equations and then generalizing the results to non-autonomous equations; and the second one, based on the so called quasi-linearization of an equation and distinguishing solutions by their types. Examples and applications are provided. In the second part we pass to the third and fourth order equations. We start with fundamentals of the linear theory and introduce some basic concepts, such as classes of equations, oscillation, conjugate points. Then we pass to the existence and multiplicity results. The existence results are formulated in terms of the upper and lower solutions (functions), where the boundary conditions may be essentially nonlinear (fully nonlinear, in terminology of some authors). Multiplicity results are provided of the similar nature, as in the case of the second order equations.