Caristi’s fixed point theorem

In 1976, Caristi [4] published a novel generalization of the contraction principle. Using transfinite arguments which was also later simplified by Wong [15]. Brondstedt [3] provided an alternative proof by introducing an interesting partial order. On the other hand, Ekeland [6] established a variational principle whence deducing Caristi’s theorem. Brezis and Browder [2] proved an ordering principle also leading to this fixed point theorem. Subsequently Altman [1], Turinici [14, 15] and others have extended this principle. In this chapter, we discuss some of these as well as proofs of Caristi’s theorem by Kirk [8], Penot [10] and Seigel [11]. That both Ekcland’s principle and Caristi’s theorem characterize completeness is also brought out.