Caristi’s Fixed Point Theorem and Selections of Set-Valued Contractions

We show without the axiom of choice.that the Zermelo theorem implies directly a restriction of the Caristi fixed point theorem to continuous functions. Under the axiom of choice, this restriction is proved to be equivalent to Caristi’s theorem. We also discuss Kirk’s problem on an extension of the Caristi theorem and we establish two selection theorems for set-valued contractions