Impact of Kirk's Results on the Development of Fixed Point Theory

Citation: Fixed Point Theory and Applications 2011 2010:821961

Moduli and Characteristics of Monotonicity in Some Banach Lattices
First the characteristic of monotonicity of any Banach lattice is expressed in terms of the left limit of the modulus of monotonicity of
Citation: Fixed Point Theory and Applications 2010 2010:852346 
A Set of Axioms for the Degree of a Tangent Vector Field on Differentiable Manifolds
Given a tangent vector field on a finitedimensional real smooth manifold, its degree (also known as characteristic or rotation) is, in some sense, an algebraic count of its zeros and gives useful information for...
Citation: Fixed Point Theory and Applications 2010 2010:845631 
Ergodic Retractions for Families of Asymptotically Nonexpansive Mappings
We prove some theorems for the existence of ergodic retractions onto the set of common fixed points of a family of asymptotically nonexpansive mappings. Our results extend corresponding results of Benavides an...
Citation: Fixed Point Theory and Applications 2010 2010:281362 
Fixed Points of Discontinuous Multivalued Operators in Ordered Spaces with Applications
Existence theorems of fixed points for multivalued increasing operators in partially ordered spaces are presented. Here neither the continuity nor compactness is assumed for multivalued operators. As an applic...
Citation: Fixed Point Theory and Applications 2010 2010:745769 
Normal Structure and Common Fixed Point Properties for Semigroups of Nonexpansive Mappings in Banach Spaces
In 1965, Kirk proved that if is a nonempty weakly compact convex subset of a Banach space with normal structure, then every nonexpansiv...
Citation: Fixed Point Theory and Applications 2010 2010:580956 
On Some Properties of Hyperconvex Spaces
We are going to answer some open questions in the theory of hyperconvex metric spaces. We prove that in complete trees hyperconvex hull...
Citation: Fixed Point Theory and Applications 2010 2010:213812 
Some Variational Results Using Generalizations of Sequential Lower Semicontinuity
Kirk and Saliga and then Chen et al. introduced lower semicontinuity from above, a generalization of sequential lower semicontinuity, and they showed that wellknown results, such as Ekeland's variational prin...
Citation: Fixed Point Theory and Applications 2010 2010:323487 
Ordered NonArchimedean Fuzzy Metric Spaces and Some Fixed Point Results
In the present paper we provide two different kinds of fixed point theorems on ordered nonArchimedean fuzzy metric spaces. First, two fixed point theorems are proved for fuzzy order ...
Citation: Fixed Point Theory and Applications 2010 2010:782680 
Fixed Point Theorems for SetValued Contraction Type Maps in Metric Spaces
We first give some fixed point results for setvalued selfmap contractions in complete metric spaces. Then we derive a fixed point theorem for nonself setvalued contractions which are metrically inward. Our ...
Citation: Fixed Point Theory and Applications 2010 2010:390183 
Convergence of the Sequence of Successive Approximations to a Fixed Point
If is a complete metric space and is a contraction on
Citation: Fixed Point Theory and Applications 2010 2010:716971 
Common Fixed Point of Multivalued Generalized Weak Contractive Mappings
Fixed point and coincidence results are presented for multivalued generalized weak contractive mappings on complete metric spaces, wher...
Citation: Fixed Point Theory and Applications 2010 2010:708984 
Approximating Fixed Points of Some Maps in Uniformly Convex Metric Spaces
We study strong convergence of the Ishikawa iterates of qasinonexpansive (generalized nonexpansive) maps and some related results in uniformly convex metric spaces. Our work improves and generalizes the corre...
Citation: Fixed Point Theory and Applications 2010 2010:385986 
A Continuation Method for Weakly Kannan Maps
The first continuation method for contractive maps in the setting of a metric space was given by Granas. Later, Frigon extended Granas theorem to the class of weakly contractive maps, and recently Agarwal and ...
Citation: Fixed Point Theory and Applications 2010 2010:321594 
Generalized IFSs on Noncompact Spaces
The aim of this paper is to continue the research work that we have done in a previous paper published in this journal (see Mihail and Miculescu, 2008). We introduce the notion of GIFS, which is a family of fu...
Citation: Fixed Point Theory and Applications 2010 2010:584215 
Demiclosed Principle for Asymptotically Nonexpansive Mappings in CAT(0) Spaces
We prove the demiclosed principle for asymptotically nonexpansive mappings in CAT(0) spaces. As a consequence, we obtain a convergence ...
Citation: Fixed Point Theory and Applications 2010 2010:268780 
Convergence of Inexact Iterative Schemes for Nonexpansive SetValued Mappings
Taking into account possibly inexact data, we study iterative schemes for approximating fixed points and attractors of contractive and nonexpansive setvalued mappings, respectively. More precisely, we are con...
Citation: Fixed Point Theory and Applications 2010 2010:518243 
Does Kirk's Theorem Hold for Multivalued Nonexpansive Mappings?
Fixed Point Theory for multivalued mappings has many useful applications in Applied Sciences, in particular, in Game Theory and Mathematical Economics. Thus, it is natural to try of extending the known fixed p...
Citation: Fixed Point Theory and Applications 2010 2010:546761 
Fixed Point Theorems on Spaces Endowed with VectorValued Metrics
The purpose of this work is to present some (local and global) fixed point results for singlevalued and multivalued generalized contractions on spaces endowed with vectorvalued metrics. The results are extens...
Citation: Fixed Point Theory and Applications 2010 2010:281381 
Fixed Point Theorems for Nonlinear Operators with and without Monotonicity in Partially Ordered Banach Spaces
We establish two fixed point theorems for nonlinear operators on Banach spaces partially ordered by a cone. The first fixed point theorem is concerned with a class of mixed monotone operators. In the second fi...
Citation: Fixed Point Theory and Applications 2010 2010:108343 
Fixed Points for Discontinuous Monotone Operators
We obtain some new existence theorems of the maximal and minimal fixed points for discontinuous monotone operator on an order interval in an ordered normed space. Moreover, the maximal and minimal fixed points...
Citation: Fixed Point Theory and Applications 2009 2010:926209 
Equivalent Extensions to CaristiKirk's Fixed Point Theorem, Ekeland's Variational Principle, and Takahashi's Minimization Theorem
With a recent result of Suzuki (2001) we extend CaristiKirk's fixed point theorem, Ekeland's variational principle, and Takahashi's minimization theorem in a complete metric space by replacing the distance wi...
Citation: Fixed Point Theory and Applications 2009 2010:970579 
Weak and Strong Convergence of an Implicit Iteration Process for an Asymptotically Quasi Nonexpansive Mapping in Banach Space
We prove the weak and strong convergence of the implicit iterative process to a common fixed point of an asymptotically quasi nonexpans...
Citation: Fixed Point Theory and Applications 2009 2010:719631 
Regularization and Iterative Methods for Monotone Variational Inequalities
We provide a general regularization method for monotone variational inequalities, where the regularizer is a Lipschitz continuous and strongly monotone operator. We also introduce an iterative method as discre...
Citation: Fixed Point Theory and Applications 2009 2010:765206 
A Kirk Type Characterization of Completeness for Partial Metric Spaces
We extend the celebrated result of W. A. Kirk that a metric space is complete if and only if every Caristi selfmapping for
Citation: Fixed Point Theory and Applications 2009 2010:493298 
Halpern's Iteration in CAT(0) Spaces
Motivated by Halpern's result, we prove strong convergence theorem of an iterative sequence in CAT(0) spaces. We apply our result to find a common fixed point of a family of nonexpansive mappings. A convergenc...
Citation: Fixed Point Theory and Applications 2009 2010:471781 
Generalized Asymptotic Pointwise Contractions and Nonexpansive Mappings Involving Orbits
We give fixed point results for classes of mappings that generalize pointwise contractions, asymptotic contractions, asymptotic pointwise contractions, and nonexpansive and asymptotic nonexpansive mappings. We...
Citation: Fixed Point Theory and Applications 2009 2010:458265 
Periodic Point, Endpoint, and Convergence Theorems for Dissipative SetValued Dynamic Systems with Generalized Pseudodistances in Cone Uniform and Uniform Spaces
In cone uniform and uniform spaces, we introduce the three kinds of dissipative setvalued dynamic systems with generalized pseudodistances and not necessarily lower semicontinuous entropies, we study the conv...
Citation: Fixed Point Theory and Applications 2009 2010:864536 
On Fixed Points of Maximalizing Mappings in Posets
We use chain methods to prove fixed point results for maximalizing mappings in posets. Concrete examples are also presented.
Citation: Fixed Point Theory and Applications 2009 2010:634109 
Some Weak Convergence Theorems for a Family of Asymptotically Nonexpansive Nonself Mappings
A onestep iteration with errors is considered for a family of asymptotically nonexpansive nonself mappings. Weak convergence of the purposed iteration is obtained in a Banach space.
Citation: Fixed Point Theory and Applications 2009 2010:218573 
Properties and , Embeddings in Banach Spaces with 1Unconditional Basis and
We will use GarcÃaFalset and LlorÃ©ns Fuster's paper on the AMCproperty to prove that a Banach space that ...
Citation: Fixed Point Theory and Applications 2009 2010:342691 
A Hybrid Projection Algorithm for Finding Solutions of Mixed Equilibrium Problem and Variational Inequality Problem
We propose a modified hybrid projection algorithm to approximate a common fixed point of a strict pseudocontraction and of two sequences of non...
Citation: Fixed Point Theory and Applications 2009 2010:383740 
Nonexpansive Matrices with Applications to Solutions of Linear Systems by Fixed Point Iterations
We characterize (i) matrices which are nonexpansive with respect to some matrix norms, and (ii) matrices whose average iterates approach zero or are bounded. Then we apply these results to iterative solutions ...
Citation: Fixed Point Theory and Applications 2009 2010:821928 
Fixed Points for Pseudocontractive Mappings on Unbounded Domains
We give some fixed point results for pseudocontractive mappings on nonbounded domains which allow us to obtain generalizations of recent fixed point theorems of Penot, Isac, and NÃ©meth. An application to integ...
Citation: Fixed Point Theory and Applications 2009 2010:769858 
Browder's Convergence for Uniformly Asymptotically Regular Nonexpansive Semigroups in Hilbert Spaces
We give a sufficient and necessary condition concerning a Browder's convergence type theorem for uniformly asymptotically regular oneparameter nonexpansive semigroups in Hilbert spaces.
Citation: Fixed Point Theory and Applications 2009 2010:418030 
Coincidence Theorems for Certain Classes of Hybrid Contractions
Coincidence and fixed point theorems for a new class of hybrid contractions consisting of a pair of singlevalued and multivalued maps on an arbitrary nonempty set with values in a metric space are proved. In ...
Citation: Fixed Point Theory and Applications 2009 2010:898109 
A New System of Generalized Nonlinear Mixed Variational Inclusions in Banach Spaces
We introduce and study a new system of generalized nonlinear mixed variational inclusions in real uniformly smooth Banach spaces. We pr...
Citation: Fixed Point Theory and Applications 2009 2010:908490 
Some Convergence Theorems of a Sequence in Complete Metric Spaces and Its Applications
The concept of weakly quasinonexpansive mappings with respect to a sequence is introduced. This concept generalizes the concept of quasinonexpansive mappings with respect to a sequence due to Ahmed and Zeyad...
Citation: Fixed Point Theory and Applications 2009 2010:647085 
Fixed Points of Single and SetValued Mappings in Uniformly Convex Metric Spaces with No Metric Convexity
We study the existence of fixed points and convergence of iterates for asymptotic pointwise contractions in uniformly convex metric spaces. We also study the existence of fixed points for setvalued nonexpansi...
Citation: Fixed Point Theory and Applications 2009 2010:169837