Abstract

The central theme of this dissertation is to present a new aspect in the study of critical point theory. A theory and its application are discussed on a closed subset D in a Hilbert space H, while the traditional approach takes place in the entire space H. This setting naturally entails certain boundary conditions on functionals. For this reason, we introduce a new concept called (BC) condition. This is basically a localized (PS) condition with some boundary restrictions for the functionals. It is demonstrated in this dissertation that we develop a theory leading to results parallel to the traditional ones, such as the deformation theorem, the mountain pass theorem, the saddle point theorem, as well as some minimax theorems. We also consider the equivariant critical point theory under our setting. By applying these results, the existence of periodic solutions of certain types of Hamiltonian Systems, and the existence of periodic solutions of some singular Hamiltonian Systems are studied. Finally, some infinite dimensional problems, to which the ordinary (PS) condition does not seem to apply readily, are solved using our results based on the notion of (BC) condition.