Remarks on Caristi’s fixed point theorem

Abstract

Abstract. In this work, we give a characterization of the existence of minimal elements in partially ordered sets in terms of fixed point of multivalued maps. This characterization shows that the assumptions in Caristi's fixed point theorem can, a priori, be weakened. Finally, we discuss Kirk's problem on an extension of Caristi's theorem and prove a new positive result which illustrates the weakening mentioned before.

Maximal and minimal point theorems and Caristi's fixed point theorem

“…In the past decades, Caristi's fixed point theorem has been generalized and extended in several directions, and the proofs given for Caristi's result varied and used different techniques, we refer the readers to [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15].…”

“…However, if h is not subadditive, then the relationship ≼ defined by (1) may not be a partial order on X, and consequently the method used there becomes invalid. Recently, Khamsi [13] removed the additivity of h by introducing a partial order on Q as follows…”

“…He showed that (Q, ≼*) has a maximal point which is exactly the maximal point of (X, ≼) and hence each Caristi-type mapping has a fixed point. Very recently, the results of [9,12,13] were improved by Li [14] in which the continuity, subadditivity and nondecreasing property of h are removed at the expense that (H) there exists c >0 and ε >0 such that h(t) ≥ ct for each…”

“…From [14, Theorem 2 and Remark 2] we know that the assumptions made on h in [12,13] force that (H) is satisfied. In other words, (H) is necessarily assumed in [12][13][14].…”

“…In other words, (H) is necessarily assumed in [12][13][14]. Meanwhile, is always assumed to be lower semicontinuous there.…”