Variant Forms of Ekeland’s Variational Principle in Abstract Spaces with Generalized Distances and Their Applications

This research proposal focuses on the study of new versions of Ekelands variational principle (EVP) in different settings with applications to fixed-point theorems and optimization. We will pay our attention to different versions of EVP in generalized spaces and some other spaces, namely, pseudo-quasimetric spaces, and their equivalent fixed-point results. We shall also use generalized distances; in particular, modular distances, P-distances, Q-distances and c-distances in our research, and we shall avoid the technical requirement of limit uniqueness used in a majority of publications in this topic. It is important to emphasize that a generalized distance with the limit uniqueness assumption is equivalent to a distance and that there are several extended / generalized versions of EVP which are indeed equivalent to the original form. In addition, we shall investigate the characterization of completeness via EVP and that whether an existing version of EVP in the literature is equivalent to the original form in the sense that one implies the other. The equilibrium version of Ekelands variational principle will also be discussed under different settings and by using different distance functions mentioned above.