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Let X be a Banach space. We say that X satisfies the fixed point property (weak fixed point property) if every non-expansive mapping defined from a convex closed bounded (convex weakly compact) subset of X into itself has a fixed point. We say that X satisfies the stable fixed point property (stable weak fixed point property) if the same is true for every equivalent norm which is close enough to the original one. Denote by P ( X ) the set formed by all equivalent norms with the topology of the uniform convergence on the unit ball of X . We prove that the subset of P ( X ) formed by the norms failing the fixed point property is dense in P ( X ) when X is a non-distortable space which fails the fixed point property. In particular, no renorming of l 1 can satisfy the stable fixed point property. Furthermore, we show some examples of distortable spaces failing the weak fixed point property, which can be renormed to satisfy the stable weak fixed point property. As a consequence we prove that every separable Banach space can be renormed to satisfy the stable weak fixed point property.

This study characterizes the convex sets whose complements in the unit square exhibit the fixed point property for mappings which are jointly continuous and order preserving. Hence, one can readily construct simple sets with this fixed point property, but which neither have the fixed point property individually for continuous mappings nor for order preserving mappings. This is the first characterization of any non-trivial set with this property.

In this paper, coincidence and common fixed point results are established in a partially ordered cone metric space. An application of our results obtained to prove the existence of a common solution to integral equations is presented.

In this paper, we obtain some fixed and common fixed point theorems in finite dimensional normed Boolean vector spaces. Our results extend and unify some known results.

The purpose of this paper is to present some coupled fixed point theorems for a mixed monotone operator in a complete metric space endowed with a partial order by using altering distance functions. We also present an application to integral equations.

Following [W.A. Kirk, P.S. Srinivasan, P. Veeramany, Fixed points for mappings satisfying cyclical contractive conditions, Fixed Point Theory 4 (1) (2003) 79–89], we present a fixed point theorem for cyclic φ -contractions and following [I.A. Rus, The theory of a metrical fixed point theorem: Theoretical and applicative relevances, Fixed Point Theory 9 (2) (2008) 541–559] we construct a theory of this fixed point theorem. This theory is in connection with data dependence, well-posedness of the fixed point problem, limit shadowing property and sequences of operators and fixed points. A Maia type fixed point theorem for cyclic φ -contractions is also given.

Let ( X , ⪯ ) be a partially ordered set and d be a complete metric on X . Let F , G be two set-valued mappings on X . We obtained sufficient conditions for the existence of common fixed point of F and G satisfying an implicit relation in partially ordered set X .

In this paper, we establish some new fixed point theorems for the sum of two operators. In particular, one new theorem complements the well-known Krasnosel’skii fixed-point theorem in the sense that the mapping T is expansive rather than contractive. Some previous and related results are extended and developed. As an application to one of our main results, the existence of periodic solutions for some nonlinear integral equation is considered.

In this paper, fixed point theorems of a nonlinear operator are developed, where the domain space C [ [ a , b ] , E ] is different from the range, E , which is a metric space with partial order. An application of the fixed point theorem to a periodic boundary value problem with delay is also given.

We prove a fixed point theorem for a mixed monotone mapping in a metric space endowed with partial order, using a weak contractivity type of assumption. Besides including several recent developments, our theorem can be used to investigate a large class of problems. As an application, we discuss the existence and uniqueness of solution for a periodic boundary value problem.