Selected methods for nonlinear boundary value problems

Abstract

The following book deals with various boundary value problems for differential equations. As Juliusz Schauder, one of the pioneers and unsurpassed masters (at least for the author), used to say, the most important thing is to know the methods, not the theorems. Thus, we are interested in a set of methods of Nonlinear Analysis applied to such boundary value problems. Since we want to avoid the difficulties associated with partial equations (already the theory of linear partial differential equations requires the use of subtle concepts and tools of Functional Analysis), we choose examples showing applications of the above-mentioned methods among ordinary differential equations. We are interested in nonlinear equations, but the boundary conditions we discuss are usually linear. By boundary conditions, we mean here any additional equations that the solutions of the differential equation are expected to satisfy. Such additional conditions are necessary if we want to have one (or more) solutions - after all, a given differential equation has infinitely many solutions. These additional conditions may be initial conditions, conditions to be satisfied by the function at the extremes of the domain (boundary conditions), but they may also be multipoint or, more broadly, nonlocal e.g. when there is an integral of the solution in the equation. On the other hand, the wealth of methods of nonlinear analysis is so great that we emphasize a certain set of methods (mainly topological) preferred by the author. The examples on which we present applications of these methods are in majority taken from the work of the research team that consists of the author and his former PhD students. Thus, this survey is not a monograph of the subject in a strict sense, which is reflected in the first word of the title "Selected".