Refinements of Caristis fixed point theorem

Abstract

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.

A Proposal for Revisiting Ćirić and Caristi Type Theorems in Metric Spaces

“…Let (E, d, < ) be a partially ordered complete metric space satisfying the (OSC) property and T: E ⟶ E be a monotonically increasing map such that there exists a function φ: E ⟶ [0, +∞[ satisfying ∀x, y ∈ E, x < y ⟹ d(x, Tx)d(Tx, Ty) ≤ (φ(x) − φ(Tx))d(x, y). (6) en, for any x 0 ∈ E such that Tx 0 < x 0 , the sequence (x n ) n∈N defined by x n+1 � Tx n converges to a fixed point of T.…”

“…It has been successfully applied in many topics such as differential equations, convex minimization, operator theory, variational inequalities, and control theory. For known Caristi-type fixed point results in the literature, see [5][6][7][8][9][10][11][12][13]. Recall that this theorem states that any map T: E ⟶ E has a fixed point provided that E is complete and there exists a lower semicontinuous map φ: E ⟶ [0, +∞[ such that d(x, Tx) ≤ φ(x) − φ(Tx), for every x ∈ E. e proofs given to Caristi's result vary and use different techniques (see [14,15]).…”