Variational methods and their generalizations have been verified to be useful tools in proving the existence of solutions to a variety of boundary value problems for ordinary, impulsive, and partial differential equations as well as for difference equations. In this monograph, we look at how variational methods can be used in all these settings. In our first chapter, we gather the basic notions and fundamental theorems that will be applied in the remainder of this monograph. While many of these items are easily available in the literature, we gather them here both for the convenience of the reader and for the purpose of making this volume somewhat self-contained. Subsequent chapters deal with the Sturm-Liouville problems, multi-point boundary value problems, problems with impulses, partial differential equations, and difference equations. An extensive bibliography is also included.
Mathematical Preliminaries; Sturm-Liouville Problems, Multiple Solutions, Infinitely Many Solutions; Multi-Point Problems, Multiple Solutions, Two Parameter Systems, Existence by the Dual Action Principle; Impulsive Problems, Existence of Infinitely Many Solutions, Anti-Periodic Solutions; Partial Differential Equations, Kirchhoff-type Problems with Two Parameters, Biharmonic Systems, Elliptic Problems with a p(x)-biharmonic Operator; Difference Equations, Periodic Problems with One and Two Parameters, Periodic Problems, Multi-point Problems with Several Parameters, Homoclinic Solutions, Anti-periodic Solutions;