THE STUDY in fixed point theory has generally developed in three main directions: generalization of conditions which ensure existence, and, if possible, uniqueness, of fixed points; investigation of the character of the sequence of iterates (T”x)~~~, where T: X + X, X a complete metric space, is the map under consideration; study of the topoligical properties of the set of fixed points, wheneve...

We look for geometric structures which underlie both the 'classical' geometric and more modern 'order theoretic' conditions for a Banach space (dual space) to have the weak (weak*) fixed point property; that is, for every nonexpansive self-mapping of a nonempty weak (weak*) compact convex subset to have a fixed point.

In this paper, we give some definitions and conclusions in dislocated quasimetric spaces. Also, we establish two fixed point theorems for multivalued mappings in these spaces.

Some results regarding the existence of a fixed point for a multivalued non-expansive mapping defined on an unbounded subset of a reflexive Banach space are established.

The alternation hierarchy problem asks whether every μ-term φ, that is, a term built up also using a least fixed point constructor as well as a greatest fixed point constructor, is equivalent to a μ-term where the number of nested fixed points of a different type is bounded by a constant independent of φ. In this paper we give a proof that the alternation hierarchy for the theory of μ-lattices ...

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 me...

We prove some fixed point theorems for nonexpansive selfand non-self-mappings in product spaces; in particular, we provide a constructive proof of a result of Kirk and Martinez and a partial answer to a question of Khamsi. Our proofs are elementary in the sense that we do not use any universal (or ultra) nets.

We prove the following result. Let C be a convex compact subset in a topological vector space, and T : C → C a convex continuous mapping. (See Definition 1.1.) Then T has a fixed point. Moreover, continuous mappings that can be approximated by convex continuous mappings also have the fixed point property.

In this paper we use degree and index theory to present new applicable fixed point theory for permissible maps.

Very recently, Jleli and Samet [53] and Samet et. al. [52] reported that some fixed point result in G-metric spaces can be derived from the fixed point theorems in the setting of usual metric space. In this paper, we prove the existence and uniqueness of fixed points of certain cyclic mappings in the context of G-metric spaces that can not be obtained by usual fixed point results via techniques...