Caristi’s Fixed Point Theorem
In 1976, Caristi  published a novel generalization of the contraction principle. Using transfinite arguments which was also later simplified by Wong . Brondstedt  provided an alternative proof by introducing an interesting partial order. On the other hand, Ekeland  established a variational principle whence deducing Caristi’s theorem. Brezis and Browder  proved an ordering principle also leading to this fixed point theorem. Subsequently Altman , 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 , Penot  and Seigel . That both Ekcland’s principle and Caristi’s theorem characterize completeness is also brought out.
- 3.Brondstedt, A.: On a lemma of Bishop and Phelps. Pac. J. Math. 55, 335–341 (1974)Google Scholar
- 5.De Figuiredo, D.G.: Lectures on the Ekeland Variational Principle with Applications and Detours. T.I.F.R (1972)Google Scholar
- 10.Penot, J.P.: A short constructive proof of the Caristi fixed point theorem. Publ. Math. Univ. Paris 10, 1–3 (1976)Google Scholar
- 13.Takahashi, W.: Existence theorems generalizing fixed point theorems for multivalued mappings. In: Baillon, J.B., Thera, M. (eds.) Fixed Point Theory and Applications, vol. 252, pp. 387–406. Longman Scientific & Technical, Harlow (1991)Google Scholar