Refinements of Caristi’s fixed point theorem

In this paper, we introduce new types of Caristi fixed point theorem and Caristi-type cyclic maps in a metric space with a partial order or a directed graph. These types of mappings are more general than that of Du and Karapinar [W.-S. Du, E. Karapinar, Fixed Point Theory Appl., 2013 (2013), 13 pages]. We obtain some fixed point results for such Caristi-type maps and prove some convergence theorems and best proximity results for such Caristi-type cyclic maps. It should be mentioned that in our results, all the optional conditions for the dominated functions are presented and discussed to our knowledge, and the replacing of d(x, T x) by min{d(x, T x), d(T x, T y)} endowed with a graph makes our results strictly more general. Many recent results involving Caristi fixed point or best proximity point can be deduced immediately from our theory. Serval applications and examples are presented making effective the new concepts and results. Two analogues for Banach-type contraction are also provided.