fixed points of asymptotic contractions abstract let r r be a contractive gauge function in the sense that is continuous s 0 and if f m m satisfies d f x f y d x y for all x y in a complete metric space m d then f always has a unique fixed point it is proved that if t m m satisfies d t n x t n y n d x y x y m where each n is continuous and n uniformly on the range of d then t has a unique fixed point and moreover all of the picard iterates of t converge to this fixed point

best proximity points for asymptotic cyclic contraction mappings abstract in this paper we prove the existence and convergence of best proximity points for asymptotic cyclic contractions in metric spaces with the property uc as well as for asymptotic proximal pointwise contractions in uniformly convex banach spaces moreover we consider a generalized cyclic contraction mapping and prove the existence of best proximity points for such a mapping in banach spaces which do not necessarily satisfy the geometric property uc

nonlinear dynamics theory of up down transitions in neocortical neural networks the neurons of the neocortex show 1 hz synchronized transitions between an active up state and a quiescent down state the up down state transitions are highly coherent over large sections of the cortex yet they are accompanied by pronounced incoherent noise we propose a simple model for the up down state oscillations that allows analysis by straightforward dynamical systems theory an essential feature is a nonuniform network geometry composed of groups of excitatory and inhibitory neurons with strong coupling inside a group and weak coupling between groups the enhanced deterministic noise of the up state appears as the natural result of the proximity of a partial synchronization transition the synchronization transition takes place as a function of the long range synaptic strength linking different groups of neurons

a rate of convergence for asymptotic contractions abstract in p gerhardy a quantitative version of kirk s fixed point theorem for asymptotic contractions j math anal appl 316 2006 339 345 p gerhardy gives a quantitative version of kirk s fixed point theorem for asymptotic contractions this involves modifying the definition of an asymptotic contraction subsuming the old definition under the new one and giving a bound expressed in the relevant moduli and a bound on the picard iteration sequence on how far one must go in the iteration sequence to at least once get close to the fixed point however since the convergence to the fixed point needs not be monotone this theorem does not provide a full rate of convergence we here give an explicit rate of convergence for the iteration sequence expressed in the relevant moduli and a bound on the sequence we furthermore give a characterization of asymptotic contractions on bounded complete metric spaces showing that they are exactly the mappings for which every picard iteration sequence converges to the same point with a rate of convergence which is uniform in the starting point

discrete and continuous time probabilistic models and algorithms for inferring neuronal up and down states up and down states the periodic fluctuations between increased and decreased spiking activity of a neuronal population are a fundamental feature of cortical circuits understanding up down state dynamics is important for understanding how these circuits represent and transmit information in the brain to date limited work has been done on characterizing the stochastic properties of up down state dynamics we present a set of markov and semi markov discrete and continuous time probability models for estimating up and down states from multiunit neural spiking activity we model multiunit neural spiking activity as a stochastic point process modulated by the hidden up and down states and the ensemble spiking history we estimate jointly the hidden states and the model parameters by maximum likelihood using an expectation maximization em algorithm and a monte carlo em algorithm that uses reversible jump markov chain monte carlo sampling in the e step we apply our models and algorithms in the analysis of both simulated multiunit spiking activity and actual multi unit spiking activity recorded from primary somatosensory cortex in a behaving rat during slow wave sleep our approach provides a statistical characterization of up down state dynamics that can serve as a basis for verifying and refining mechanistic descriptions of this process

extensions and applications of acf mappings abstract using a definition of asf sequences derived from the definition of asymptotic contractions of the final type of acf we give some new fixed point theorem for cyclic mappings and an alternating mapping which extend results from suzuki 2007 8 theorem 2 and zhang 2007 10 theorem 1

fixed point theorem for asymptotic contractions of meir keeler type in complete metric spaces abstract in this paper we introduce the notion of asymptotic contraction of meir keeler type and prove a fixed point theorem for such contractions which is a generalization of fixed point theorems of meir keeler and kirk in our discussion we use the characterization of meir keeler contraction proved by lim on characterizations of meir keeler contractive maps nonlinear anal 46 2001 113 120 we also give a simple proof of this characterization

quasi asymptotic contractions set valued dynamic systems uniqueness of endpoints and generalized pseudodistances in uniform spaces abstract for set valued dynamic systems in uniform spaces we introduce the concept of quasi asymptotic contractions with respect to some generalized pseudodistances describe a method which we use to establish general conditions guaranteeing the existence and uniqueness of endpoints stationary points of these contractions and exhibit conditions such that for each starting point each generalized sequence of iterations in particular each dynamic process converges and the limit is an endpoint the definition result ideas and techniques are new for set valued dynamic systems in uniform locally convex and metric spaces and even for single valued maps

weakly compact homomorphisms we study the structure of weakly compact homomorphisms between banach algebras in particular it is shown that between many pairs of algebras the only weakly compact homomorphisms are those of finite rank

real interpolation of banach algebras and factorization of weakly compact homomorphisms abstract we consider the problem of when the real interpolation method respects banach algebra structure using our results we show that every weakly compact homomorphism between banach algebras factors through a reflexive banach algebra

asymptotic pointwise contractions a simple and elementary without using the ultrapower technique proof for the existence of a fixed point of an asymptotic pointwise contraction is provided the existence of fixed points for asymptotic pointwise nonexpansive mappings in a uniformly convex banach space is also investigated

remarks on a paper by f browder about contraction throughout this note x denotes a complete metric space with distance function d and t is a map from x into itself powers of t are defined by tax x and t lx t t x n 2 0 occasionally we use the notation xk tkx in particular x0 x x1 tx for the sake of brevity the set x tx t x t x is called an orbit starting at x and denoted by o x the set 0 x y is the union of the two orbits starting at x and y 0 x y xk yk k 3 o the diameter diam a of a nonempty subset a of x is the supremum of the numbers d x y where x and y vary in a the letter 4 denotes a contractive gauge function i e a continuous increasing function from r the non negative reals into r which satisfies 4 s 0

a definitive result on asymptotic contractions abstract we generalize a fixed point theorem for asymptotic contractions due to kirk w a kirk fixed points of asymptotic contractions j math anal appl 277 2003 645 650 our result is the final generalization in some sense

cone uniform cone locally convex and cone metric spaces endpoints set valued dynamic systems and quasi asymptotic contractions abstract the concepts of cone pseudometrics cone uniform spaces generated by these pseudometrics cone pseudodistances set valued quasi asymptotic contractions with respect to these pseudodistances and cone closed maps are introduced and studied conditions guaranteeing the existence and uniqueness of endpoints stationary points of these contractions and conditions ensuring that for each starting point each generalized sequence of iterations of these contractions in particular each dynamic process converges and then the limit is an endpoint are all established also the concept of the cone locally convex space as a special case of the cone uniform space is introduced and examples of quasi asymptotic contractions in cone metric spaces are constructed the definitions results ideas and methods are new for set valued dynamic systems in cone uniform cone locally convex and cone metric spaces and even for single valued maps in these spaces

contractivity of leader type and fixed points in uniform spaces with generalized pseudodistances abstract recently jachymski and jo wik proved that among various classes of contractions which are introduced and studied in the metric fixed point theory the leader contractions are greatest general contractions in this article we want to show how generalized pseudodistances in uniform spaces can be used to obtain new and general results of leader type without complete graph assumptions about maps and without sequentially complete assumptions about spaces which was not done in the previous publications on this subject the definitions results and methods presented here are new for maps in uniform and locally convex spaces and even in metric spaces examples showing a difference between our results and the well known ones are given

on a fixed point theorem of kirk abstract w a kirk j math anal appl 277 2003 645 650 first introduced the notion of asymptotic contractions and proved the fixed point theorem for this class of mappings in this note we present a new short and simple proof of kirk s theorem

on characterizations of meir keeler contractive maps let x d be a metric space a map t x x is called meir keeler contractive if 0 0 such that d x y d tx ty we introduce l functions and characterize meir keeler contractive maps as maps that satisfy d tx ty d x y for some l function this characterization makes it easy to compare such maps with those satisfying the boyd wong s condition

existence and uniqueness of endpoints of closed set valued asymptotic contractions in metric spaces abstract investigations of problems of set valued asymptotic fixed point theory of set valued dynamic systems are initiated the concepts of the set valued asymptotic contractions of set valued dynamic systems t m 2 m in metric space m are introduced and conditions guaranteeing the existence and uniqueness of endpoints v m of set valued dynamic systems of set valued asymptotic contractions t m 2 m and convergence to v of all generalized sequences of iterations of t are established examples and remarks show a fundamental difference between our results and the well known ones

generalized distance and existence theorems in complete metric spaces abstract in this paper we first introduce the concept of distance on a metric space which is a generalized concept of both w distance and tataru s distance we also improve the generalizations of the banach contraction principle caristi s fixed point theorem ekeland s variational principle and the nonconvex minimization theorem according to takahashi further we discuss the relation between w distance and tataru s distance

inverse closedness of approximation algebras abstract we prove the inverse closedness of certain approximation algebras based on a quasi banach algebra x using two general theorems on the inverse closedness of subspaces of quasi banach algebras in the first theorem commutative algebras are considered while the second theorem can be applied to arbitrary x and to subspaces of x which can be obtained by a general k method of interpolation between x and an inversely closed subspace y of x having certain properties as application we present some inversely closed subalgebras of c t and c 1 1 in particular we generalize wiener s theorem i e we show that for many subalgebras s of l 1 z the property c k f s c k f being the fourier coefficients of f implies the same property for 1 f if f c t vanishes nowhere on t

chaotic balanced state in a model of cortical circuits the nature and origin of the temporal irregularity in the electrical activity of cortical neurons in vivo are not well understood we consider the hypothesis that this irregularity is due to a balance of excitatory and inhibitory currents into the cortical cells we study a network model with excitatory and inhibitory populations of simple binary units the internal feedback is mediated by relatively large synaptic strengths so that the magnitude of the total excitatory and inhibitory feedback is much larger than the neuronal threshold the connectivity is random and sparse the mean number of connections per unit is large though small compared to the total number of cells in the network the network also receives a large temporally regular input from external sources we present an analytical solution of the mean field theory of this model which is exact in the limit of large network size this theory reveals a new cooperative stationary state of large networks which we term a balanced state in thi

fixed point theorems in metric spaces abstract in this paper we establish two general theorems for equivalence between the meir keeler type contractive conditions and the contractive definitions involving gauge functions one of these theorems is an extension of a recent result of lim on characterization of meir keeler contractive maps nonlinear anal 46 2001 113 120 also we establish the following new fixed point theorem suppose r r is a contractive gauge function in the sense that for any e 0 there exists e such that e t implies t e and suppose t is a continuous and asymptotically regular selfmapping on a complete metric space x d satisfying the following i d tx ty d x y for all x y x and ii d tx ty d x y for all x y x with x y where d x y d x y d x tx d y ty with 0 then t has a unique fixed point and all of the picard iterates of t converge to this fixed point this result includes those of jachimski equivalent conditions and the meir keeler type theorems j math anal appl 194 1995 293 303 matkowski fixed point theorems for contractive mappings in metric spaces cas pest mat 105 1980 341 344 and others

endpoints of set valued dynamical systems of asymptotic contractions of meir keeler type and strict contractions in uniform spaces abstract in this paper we introduce the concepts of the set valued dynamical systems of asymptotic contractions of meir keeler type and set valued dynamical systems of strict contractions in uniform spaces and we present a method which is useful for establishing conditions guaranteeing the existence and uniqueness of endpoints of these contractions and the convergence to these endpoints of all generalized sequences of iterations of these contractions the result concerning the investigations of problems of the set valued asymptotic fixed point theory include some well known results of meir and keeler kirk and suzuki concerning the asymptotic fixed point theory of single valued maps in metric spaces the result concerning set valued strict contractions in which the contractive coefficient is not constant is different from the result of yuan concerning the existence of endpoints of tarafdar vyborny generalized contractions in which the contractive coefficient is constant in bounded metric spaces and provides some examples of tarafdar yuan topological contractions in compact uniform spaces definitions and results presented here are new for set valued dynamical systems in uniform locally convex and metric spaces and even for single valued maps examples show a fundamental difference between our results and the well known ones

a quantitative version of kirk s fixed point theorem for asymptotic contractions abstract in w a kirk fixed points of asymptotic contractions j math anal appl 277 2003 645 650 w a kirk introduced the notion of asymptotic contractions and proved a fixed point theorem for such mappings using techniques from proof mining we develop a variant of the notion of asymptotic contractions and prove a quantitative version of the corresponding fixed point theorem

subrahmanyam s fixed point theorem abstract we give a sufficient and necessary condition for the convergence of the sequence of successive approximations to a fixed point which is the conclusion of subrahmanyam s fixed point theorem

a note on a paper of i d arandelovic on asymptotic contractions abstract w a kirk w a kirk fixed points of asymptotic contractions j math anal appl 277 2003 645 650 defined the notion of an asymptotic contraction on a metric space and using ultrapower techniques he gave a nonconstructive proof of an asymptotic version of the boyd wong fixed point theorem subsequently i d arandelovic i d arandelovic on a fixed point theorem of kirk j math anal appl 301 2005 384 385 established somewhat more general version of kirk s result and he gave an elementary proof of it however our purpose is to show that there is an error in this proof and moreover arandelovic s theorem is false we also explain how to correct this result

convolution weights on l2 spaces abstract it is known that the weighted l p space on a locally compact group is stable with respect to convolution if the weight function satisfies a certain type of convolution inequality there are several counterexamples showing that this sufficient condition is not necessary however for one class of groups namely discrete abelian groups no such counterexample is known thus there remains the possibility that the convolution inequality truly characterizes stability of convolution for weighted l p spaces on these groups in this paper we investigate this inequality and in the case p 2 reinterpret it in the light of operator theory and in the context of the theory of reproducing kernel hilbert spaces

shellable weakly compact subsets of c 0 1 we show that for every weakly compact subset k of c 0 1 with finite cantor bendixson rank there is a reflexive banach lattice e and an operator t e rightarrow c 0 1 such that k subseteq t b e on the other hand we exhibit an example of a weakly compact set of c 0 1 homeomorphic to omega omega 1 for which such t and e cannot exist this answers a question of m talagrand in the 80 s

on kirk s asymptotic contractions abstract recently kirk introduced the notion of asymptotic contractions on a metric space and using ultrapower techniques he obtained an asymptotic version of the boyd wong fixed point theorem in this paper we extend this result and moreover we give a constructive proof of it furthermore we obtain a complete characterization of asymptotic contractions on a compact metric space as a by product we establish a separation theorem for upper semicontinuous functions satisfying some limit condition