A fixed point theorem for the infinite-dimensional simplex

We define the infinite-dimensional simplex to be the closure of the convex hull of the standard basis vectors in R^∞, and prove that this space has the fixed point property: any continuous function from the space into itself has a fixed point. Our proof is constructive, in the sense that it can be used to find an approximate fixed point; the proof relies on elementary analysis and Sperner's lemma. The fixed point theorem is shown to imply Schauder's fixed point theorem on infinite-dimensional compact convex subsets of normed spaces.     

Fixed point theorems for multifunctions having KKM property on almost convex sets

In this paper, for two nonempty subsets X and Y of a linear space E, we define the class KKM(X,Y) and investigate the fixed point problem for T∈KKM(X,X) with X an almost convex subset of a locally convex space. Our fixed point theorem contains Lassonde fixed point theorem for Kakutani factorizable multifunctions as special case.     

Common coupled fixed point theorems in cone metric spaces for w-compatible mappings

In this paper we introduce the concept of a w-compatible mappings to obtain coupled coincidence point and coupled point of coincidence for nonlinear contractive mappings in cone metric space with a cone having non-empty interior. Coupled common fixed point theorems for such mappings are also proved. Our results generalize, extend and unify several well known comparable results in the literature. Results are supported by three examples.     

Some fixed point results without monotone property in partially ordered metric-like spaces

The purpose of this paper is to obtain the fixed point results for F-type contractions which satisfies a weaker condition than the monotonicity of self-mapping of a partially ordered metric-like space. A fixed point result for F-expansive mapping is also proved. Therefore, several well known results are generalized. Some examples are included which illustrate the results.     

Fixed point iteration processes for asymptotic pointwise nonexpansive mappings in Banach spaces

Let X be a uniformly convex Banach space with the Opial property. Let T:C→C be an asymptotic pointwise nonexpansive mapping, where C is bounded, closed and convex subset of X. In this paper, we prove that the generalized Mann and Ishikawa processes converge weakly to a fixed point of T. In addition, we prove that for compact asymptotic pointwise nonexpansive mappings acting in uniformly convex Banach spaces, both processes converge strongly to a fixed point.     

Distortion and stability of the fixed point property for non-expansive mappings

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 ?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.     

Fixed point properties of semigroups of non-expansive mappings

In recent years, there have been considerable interests in the study of when a closed convex subset K of a Banach space has the fixed point property, i.e. whenever T is a non-expansive mapping from K into K, then K contains a fixed point for T. In this paper we shall study fixed point properties of semigroups of non-expansive mappings on weakly compact convex subsets of a Banach space (or, more generally, a locally convex space). By considering the classes of bicyclic semigroups we answer two open questions, one posted earlier by the first author in 1976 (Dalhousie) and the other posted by T. Mitchell in 1984 (Virginia). We also provide a characterization for the existence of a left invariant mean on the space of weakly almost periodic functions on separable semitopological semigroups in terms of fixed point property for non-expansive mappings related to another open problem raised by the first author in 1976.     

On coincidence point and fixed point theorems for nonlinear multivalued maps

Several characterizations of MT-functions are first given in this paper. Applying the characterizations of MT-functions, we establish some existence theorems for coincidence point and fixed point in complete metric spaces. From these results, we can obtain new generalizations of Berinde-Berinde?s fixed point theorem and Mizoguchi-Takahashi?s fixed point theorem for nonlinear multivalued contractive maps. Our results generalize and improve some main results in the literature.     

Compression-expansion fixed point theorem in two norms and applications

In this paper we present a two-norms version of Krasnoselskii's fixed point theorem in cones. The abstract result is then applied to prove the existence of positive Lp solutions of Hammerstein integral equations with better integrability properties on the kernels.     

Some fixed point results on a metric space with a graph

Combining some branches is a typical activity in different fields of science, especially in mathematics. Naturally, it is notable in fixed point theory. Over the past few decades, there have been a lot of activity in fixed point theory and another branches in mathematics such differential equations, geometry and algebraic topology. In 2006, Espinola and Kirk made a useful contribution on combining fixed point theory and graph theory. Recently, Reich and Zaslavski studied a new inexact iterative scheme for fixed points of contractive and nonexpansive multifunctions. In this paper, by using main idea of their work and the idea of combining fixed point theory and graph theory, we present some iterative scheme results for G-contractive and G-nonexpansive mappings on graphs.     

Some fixed point results on L-embedded Banach spaces

In this paper we prove that w-fixed point property and w^★-fixed point property are equivalent concepts for L-embedded Banach spaces which are duals of M-embedded spaces. Similar results will be obtained with respect to the normal structure. These equivalences will be applied to establish new fixed point results for different examples. We will also prove the existence of fixed points for both nonexpansive and asymptotically regular mappings defined on subsets of L-embedded Banach spaces which are sequentially compact for the abstract measure topology. We will check that our results do not hold in the case of the weak topology.     

Coupled fixed point results for nonlinear integral equations

In this paper, we prove some coupled coincidence point theorems for such nonlinear contraction mappings having a mixed monotone property in partially ordered metric spaces by dropping the condition of commutative. We also prove coupled common fixed point theorem for w-compatible mappings. An example of a nonlinear contraction mapping which is not applied by Lakshmikantham and ?iri?’s theorem [1] but applied by our result is given. Further, we apply our results to the existence theorem for solution of nonlinear integral equations.     

Asymptotic fixed point theory and the beer barrel theorem

In Sections 2 and 3 of this paper we refine and generalize theorems of Nussbaum (see [42]) concerning the approximate fixed point index and the fixed point index class. In Section 4 we indicate how these results imply a wide variety of asymptotic fixed point theorems. In Section 5 we prove a generalization of the mod p theorem: if p is a prime number, f belongs to the fixed point index class and f satisfies certain natural hypothesis, then the fixed point index of f ^p is congruent mod p to the fixed point index of f. In Section 6 we give a counterexample to part of an asymptotic fixed point theorem of A. Tromba [55]. Sections 2, 3, and 4 comprise both new and expository material. Sections 5 and 6 comprise new results. This paper is dedicated to Felix Browder on the occasion of his eightieth birthday and in recognition of his many contributions to nonlinear analysis     

KKM mappings in cone metric spaces and some fixed point theorems

In this paper, we define KKM mappings in cone metric spaces and define Nℜ-cone metric spaces to obtain some fixed point theorems and hence generalize the results obtained in [A. Amini, M. Fakhar, J. Zafarani, KKM mapping in metric spaces, Nonlinear Anal. 60 (2005) 1045-1052].     

An economic production quantity model with a positive resetup point under random demand

In this paper, an extended economic production quantity (EPQ) model is investigated, where demand follows a random process. This study is motivated by an industrial case for precision machine assembly in the machinery industry. Both a positive resetup point s and a fixed lot size Q are implemented in this production control policy. To cope with random demand, a resetup point, i.e., the lowest inventory level to start the production, is adapted to minimize stock shortage during the replenishment cycle. The considered cost includes setup cost, inventory carrying cost, and shortage cost, where shortage may occur at the production stage and/or at the end of one replenishment cycle. Under some mild conditions, the expected cost per unit time can be shown to be convex with respect to decision parameters s and Q. Further computational study has demonstrated that the proposed model outperforms the classical EPQ when demand is random. In particular, a positive resetup point contributes to a significant portion of this cost savings when compared with that in the classical lot sizing policy.     

The split common fixed point problem for generalized demimetric mappings in two Banach spaces

ABSTRACT

In this paper, we consider the split common fixed point problem for new demimetric mappings in two Banach spaces. Using the hybrid method, we prove a strong convergence theorem for finding a solution of the split common fixed point problem in two Banach spaces. Furthermore, using the shrinking projection method, we obtain another strong convergence theorem for finding a solution of the problem in two Banach spaces. Using these results, we obtain well-known and new strong convergence theorems in Hilbert spaces and Banach spaces.     

Fixed point and maximal element theorems with applications to abstract economies and minimax inequalities

In this paper, we establish some fixed point theorems for a family of multivalued maps under mild conditions. By using our fixed point theorems, we derive some maximal element theorems for a particular family of multivalued maps, namely the Φ-condensing multivalued maps. As applications of our results, we prove some general equilibrium existence theorems in the generalized abstract economies with preference correspondences. Further applications of our results are also given to minimax inequalities for a family of functions.     

Fixed point theorems for some generalized multivalued nonexpansive mappings

In this paper, we introduce a condition on multivalued mappings which is a multivalued version of condition (C_λ) defined by Garcia-Falset et al. (2011) [3]. It is shown here that some of the classical fixed point theorems for multivalued nonexpansive mappings can be extended to mappings satisfying this condition. Our results generalize the results in Lim (1974), Lami Dozo (1973), Kirk and Massa (1990), Garcia-Falset et al. (2011), Dhompongsa et al. (2009) and Abkar and Eslamian (2010) [4], [5], [6], [3], [7] and [8] and many others.     

The fixed point property via dual space properties

A Banach space has the weak fixed point property if its dual space has a weak^∗ sequentially compact unit ball and the dual space satisfies the weak^∗ uniform Kadec-Klee property; and it has the fixed point property if there exists ε>0 such that, for every infinite subset A of the unit sphere of the dual space, A∪(−A) fails to be (2−ε)-separated. In particular, E-convex Banach spaces, a class of spaces that includes the uniformly nonsquare spaces, have the fixed point property.     

Some new extensions of Edelstein-Suzuki-type fixed point theorem to G-metric and G-cone metric spaces

In this paper, we prove some fixed point theorems for generalized contractions in the setting of G-metric spaces. Our results extend a result of Edelstein [M. Edelstein, On fixed and periodic points under contractive mappings, J. London Math. Soc., 37 (1962), 74–79] and a result of Suzuki [T. Suzuki, A new type of fixed point theorem in metric spaces, Nonlinear Anal., 71 (2009), 5313–5317]. We prove, also, a fixed point theorem in the setting of G-cone metric spaces.