approximation from the exterior of a multifunction and its applications in the sweeping process abstract we approximate from the exterior an upper semicontinuous multifunction c from a metric space into the closed convex subsets of a normed space by means of globally lipschitzean multifunctions in particular when c is continuous this approximation allows us to reduce the problem of the existence of solutions of the associated evolution equation to the case in which c is lipschitzean

approachability and fixed points for non convex set valued maps this paper studies the approachability of upper semicontinuous nonconvex set valued maps by single valued continuous functions and its applications to fixed point theory the use of approximate selections was initiated by j von neumann 41 in the proof of his well known minimax principle this remarkable property was extended to the normed space setting by cellina s and cellina and lasota 9 in the context of a degree theory for upper semicontinuous convex compact valued maps it also holds for maps with compact contractible values defined on finite polyhedra of iw ma cole11 40 and more generally defined on compact anrs mclennan 39 it was recently taken up by gbrniewicz granas and kryszewski 22 241 in the context of an index theory for non convex maps defined on compact anrs the content of this paper is divided into three parts the first part is devoted to background material concerning the types of spaces and set valued maps studied herein in the second part we define the abstract class d of approachable set valued maps that is the class of maps a x y

on quadratic integral equation of fractional orders abstract we present an existence theorem for a nonlinear quadratic integral equations of fractional orders arising in the queuing theory and biology in the banach space of real functions defined and continuous on a bounded and closed interval the concept of measure of noncompactness and a fixed point theorem due to darbo are the main tool in carrying out our proof

theoretical analysis of a sinc nystrom method for volterra integro differential equations and its improvement abstract a sinc nystrom method for volterra integro differential equations was developed by zarebnia 2010 the method is quite efficient in the sense that exponential convergence can be obtained even if the given problem has endpoint singularity however its exponential convergence has not been proved theoretically in addition to implement the method the regularity of the solution is required although the solution is an unknown function in practice this paper reinforces the method by presenting two theoretical results 1 the regularity of the solution is analyzed and 2 its convergence rate is rigorously analyzed moreover this paper improves the method so that a much higher convergence rate can be attained and theoretical results similar to those listed above are provided numerical comparisons are also provided

monotone viable trajectories for functional differential inclusions this paper is a study on functional differential inclusions with memory which represent the multivalued version of retarded functional differential equations the main result gives a necessary and sufficient equations the main result gives a necessary and sufficient condition ensuring the existence of viable trajectories that means trajectories remaining in a given nonempty closed convex set defined by given constraints the system must satisfy to be viable some motivations for this paper can be found in control theory where f t f t u u u is the set of possible velocities of the system at time t depending on the past history represented by the function and on a control u ranging over a set u of controls other motivations can be found in planning procedures in microeconomics and in biological evolutions where problems with memory do effectively appear in a multivalued version all these models require viability constraints represented by a closed convex set

on a perturbed quadratic fractional integral equation of abel type we present an existence theorem for monotonic solutions of a perturbed quadratic fractional integral equation in c 0 1 our equation contains the famous chandrasekhar integral equation as a special case the concept of the measure of noncompactness related to monotonicity introduced by j banas and l olszowy and a fixed point theorem due to darbo are the main tools in carrying out our proof moreover we give an example for indicating the natural realizations of our abstract result presented in this paper

global convergence of neural networks with mixed time varying delays and discontinuous neuron activations in this paper we investigate the dynamical behavior of a class of delayed neural networks with discontinuous neuron activations and general mixed time delays involving both time varying delays and distributed delays due to the presence of time varying delays and distributed delays the step by step construction of local solutions cannot be applied this difficulty can be overcome by constructing a sequence of solutions to delayed dynamical systems with high slope activations and show that this sequence converges to a desired filippov solution of the discontinuous delayed neural networks we then derive two sets of sufficient conditions for the global exponential stability and convergence of the neural networks in terms of linear matrix inequalities lmis and m matrix properties equivalently some diagonally dominant conditions respectively convergence behavior of both the neuron state and the neuron output are discussed the obtained results extend previous work on global stability of delayed neural networks with lipschitz continuous neuron activations and neural networks with discontinuous neuron activations and only constant delays

approximating time evolution related to ginzburg landau functionals via sobolev gradient methods in a finite element setting sobolev gradients have previously 1 been used to approximate time evolution related to a model a functional in a finite difference setting in this journal here a related approach in a finite element setting is discussed

transmutation method for solving erdelyi kober fractional differintegral equations abstract by means of fractional calculus techniques we find explicit solutions of volterra integral equations of the second kind and fractional differential and differintegral equations involving erdelyi kober fractional integrals and derivatives also some hypergeometric integral equations have been considered and solved as double erdelyi kober equations of the second kind we use the transmutation method to reduce the solutions of all these equations to known solutions of simpler riemann liouville equations of the same type some examples are given

playable differential games playability conditions of differential games are studied by using viability theory first the results on payability of time independent differential games are extended to time dependent games in fact time is introduced in the dynamics of the game in the state dependent constraints bearing on controls and in state constraints second some examples of pursuit games are studied necessary and sufficient conditions of playability of the game are provided here pursuit games are directly considered as games of kind in isaacs s sense and are not considered as games of degree the viability condition does not always provide the optimal strategy to be as close as possible to a certain goal but it supplies strategies allowing the system to reach a given goal

application of sinc collocation method for solving a class of nonlinear fredholm integral equations in this paper the numerical solution of nonlinear fredholm integral equations of the second kind is considered by two methods the methods are developed by means of the sinc approximation with the single exponential se and double exponential de transformations these numerical methods combine a sinc collocation method with the newton iterative process that involves solving a nonlinear system of equations we provide an error analysis for the methods so far approximate solutions with polynomial convergence have been reported for this equation these methods improve conventional results and achieve exponential convergence some numerical examples are given to confirm the accuracy and ease of implementation of the methods

topological regularity of the set of solutions of a differential inclusion on etudie la regularite topologique de l ensemble solution de l inegalite variationnelle pour une fonction a valeur d ensemble x t tk x t f x t x 0 x ou x k k est un ensemble compact convexe non vide dans un espace de dimension finie m f est une correspondance semi continue superieurement t k x t est le cone tangent en x a k et l y est la projection de y dans l on etablit un theoreme d approximation pour la solution de l ensemble de l inegalite dans 0 t st x

error estimates with explicit constants for sinc approximation sinc quadrature and sinc indefinite integration error estimates with explicit constants are given for approximations of functions definite integrals and indefinite integrals by means of the sinc approximation although in the literature various error estimates have already been given for these approximations those estimates were basically for examining the rates of convergence and several constants were left unevaluated giving more explicit estimates i e evaluating these constants is of great practical importance since by this means we can reinforce the useful formulas with the concept of verified numerical computations in this paper we reveal the explicit form of all constants in a computable form under the same assumptions of the existing theorems the function to be approximated is analytic in a suitable region we also improve some formulas themselves to decrease their computational costs numerical examples that confirm the theory are also given

on the asymptotic stability of discontinuous systems analysed via the averaging method abstract the averaging method is one of the most powerful methods used to analyse differential equations appearing in the study of nonlinear problems the idea behind the averaging method is to replace the original equation by an averaged equation with simple structure and close solutions a large number of practical problems lead to differential equations with discontinuous right hand sides in a rigorous theory of such systems developed by filippov solutions of a differential equation with discontinuous right hand side are regarded as being solutions to a special differential inclusion with upper semi continuous right hand side the averaging method was studied for such inclusions by many authors using different and rather restrictive conditions on the regularity of the averaged inclusion in this paper we prove natural extensions of bogolyubov s first theorem and the samoilenko stanzhitskii theorem to differential inclusions with an upper semi continuous right hand side we prove that the solution set of the original differential inclusion is contained in a neighbourhood of the solution set of the averaged one the extension of bogolyubov s theorem concerns finite time intervals while the extension of the samoilenko stanzhitskii theorem deals with solutions defined on the infinite interval the averaged inclusion is defined as a special upper limit and no additional condition on its regularity is required

almost periodic dynamics of a class of delayed neural networks with discontinuous activations we use the concept of the filippov solution to study the dynamics of a class of delayed dynamical systems with discontinuous right hand side which contains the widely studied delayed neural network models with almost periodic self inhibitions interconnection weights and external inputs we prove that diagonal dominant conditions can guarantee the existence and uniqueness of an almost periodic solution as well as its global exponential stability as special cases we derive a series of results on the dynamics of delayed dynamical systems with discontinuous activations and periodic coefficients or constant coefficients respectively from the proof of the existence and uniqueness of the solution we prove that the solution of a delayed dynamical system with high slope activations approximates to the filippov solution of the dynamical system with discontinuous activations

a viability theorem for set valued states in a hilbert space abstract applications in robust control problems and shape evolution motivate the mathematical interest in control problems whose states are compact possibly non convex sets rather than vectors this leads to evolutions in a basic set which can be supplied with a metric like the well established pompeiu hausdorff distance but it does not have an obvious linear structure this article extends differential inclusions with state constraints to compact valued states in a separable hilbert space h the focus is on sufficient conditions such that a given constraint set of compact subsets is viable a k a weakly invariant our main result extends the tangential criterion in the well known viability theorem usually for differential inclusions in a vector space to the metric space of non empty compact subsets of h

on erdelyi kober type quadratic integral equation with linear modification of the argument abstract in this paper we study the existence of solutions for a nonlinear erdelyi kober fractional quadratic integral equations with linear modification of the argument the main tool used in our considerations is the technique associated to measures of noncompactness we compare our results with the ones obtained in recent papers finally we present some examples illustrating our results

directionally continuous selection in banach spaces soit e un espace de banach m 0 et on considere le cone m dans r e m t x r e x mt on dit qu une application f r e e est m continue si pour tout t 0 x 0 et e 0 il existe 0 tel que f t x f t 0 x 0 quand t 0 t t 0 x x 0 m t t 0 etant donnee une multifonction semicontinue inferieurement g r e e a valeurs fermees non vides on etudie l existence d une selection m continue f t x g t x

existence of slow monotone solutions to a differential inclusion la stabilisation des solutions des problemes de commande se pose en termes d obtention des commandes a reaction le long desquelles une fonction donnee de lyapunov decroit

chaperoning state evolutions by variable durations mckendrick partial differential equations in demography provide the state of a population depending on chronological time t and age d and involve two time evolutions the chronological one t in 0 t mapsto t geq 0 and the calendar age t in t d t mapsto t t d geq 0 both with constant velocity equal to 1 the calendar age evolution chaperons so to speak the evolution t in t d t mapsto x t of the state of a system some physical biological and economic problems motivate the introduction of variable durations with variable velocities representing the fluidity of time offering mathematical metaphors of a subjective fleeting specious time passing more or less slowly variable durations are no longer prescribed but chosen among those available and regulated the joint evolution of the variable evolution and the state is assumed to be governed by a differential inclusion or control system and provides as a byproduct the unknown temporal windows on which they evolve toge

de sinc methods have almost the same convergence property as se sinc methods even for a family of functions fitting the se sinc methods in this paper the theoretical convergence rate of the trapezoidal rule combined with the double exponential de transformation is given for a class of functions for which the single exponential se transformation is suitable it is well known that the de transformation enables the rule to achieve a much higher rate of convergence than the se transformation and the convergence rate has been analyzed and justified theoretically under a proper assumption here it should be emphasized that the assumption is more severe than the one for the se transformation and there actually exist some examples such that the trapezoidal rule with the se transformation achieves its usual rate whereas the rule with de does not such cases have been observed numerically but no theoretical analysis has been given thus far this paper reveals the theoretical rate of convergence in such cases and it turns out that the de s rate is almost the same as but slightly lower than that of the se by using the analysis technique developed here the theoretical convergence rate of the sinc approximation with the de transformation is also given for a class of functions for which the se transformation is suitable the result is quite similar to above the convergence rate in the de case is slightly lower than in the se case numerical examples which support those two theoretical results are also given

impulse capture basins of sets under impulse control systems given a target contained in a constrained set and an impulse control system governing the evolutions of runs or executions that are hybrids of continuous and discrete evolutions this paper studies and provides several characterizations of the capture basin of the target viable in the constrained set it is the subset of initial runs from which start at least one run viable in the constrained set until it reaches the target in finite time it also provides algorithms and regulation rules governing the runs that reach the targets while obeying state constraints

two formulas for numerical indefinite integration we derive two formulas for approximating the indefinite integral over a finite interval the approximation error is 0 c cv uniformly where m is the number of integrand evaluations the integrand is required to be analytic in the interior of the integration interval but may be singular at the endpoints some sample calculations indicate that the actual convergence rate accords with the error bound derived

on the approximation of upper semi continuous correspondences and the equilibriums of generalized games in section 1 we explain some of the definitions and terminology that we use in section 2 we prove several theorems concerning the approximation of upper semi continuous correspondences having for range a locally convex space theorems 1 and 2 and corollary 1 generalize certain approximation theorems by g haddad 16 pp 1352213541 g haddad and j m lasry 17 pp 299 3001 and j p aubin and a cellina 2 pp 86891 see also f s de blasi lo and j m lasry and r robert 21 although some of the details of the proofs of theorems 1 and 2 are new the basic ideas are taken from the above mentioned papers theorem 3 shows that the correspondences in the approximating families can be chosen so that they are regular see section 1 this theorem see the remark following its proof contains a classical result of m hukuhara 18 pp 56573 theorem 3 is proved using theorem 2 and propositions 1 and 4 variants of propositions 2 3 and 4 were given in 193 in section 3 we give among others theorem 5 which concerns the existence of equilibriums of generalized games abstract economies the main purpose of this result is to replace the continuity hypothesis in the w shafer h sonnenschein equilibrium theorem for generalized games by an upper semi continuity one the proof of theorem 5 is based on the results on the approximation of upper semi continuous correspondences obtained in section 2 theorem 5 us used in section 4 in section 4 we establish theorems 6 and 7 these theorems show that certain statements concerning the equilibrium of generalized games are equivalent to certain statements concerning minimax inequalities of k fan type theorems 1 2 3 5 6 and 7 are the main results of this paper 267 0022 247x 88 3 00

observability of systems under uncertainty the evolution of the state x cdot of a system under uncertainty governed by a differential inclusion text for almost all t in 0 t qquad x t in f t x t is observed through an observation map h forall t in 0 t quad y t in h x t the set valued character due to the uncertainty leads to the introduction of the following sharp input output map which is the usual product forall x 0 in x quad i x 0 h circ mathcal s x 0 mathop cup limits x cdot in mathcal s x 0 h x cdot hazy input output map which is the square product forall x 0 in x quad i x 0 h square mathcal s x 0 mathop cap limits x cdot in mathcal s x 0 h x cdot where mathcal s denotes the solution map recovering the input x 0 from the outputs i x 0 or i x 0 means that these input output maps are injective in the sense that locally x 1 ne x 2 rightarrow i x 1 cap i x 2 emptyset criteria for bo

sinc galerkin method based on the de transformation for the boundary value problem of fourth order ode in this paper a sinc galerkin method incorporated with the double exponential transformation abbreviated as the de transformation for the two point boundary value problem of fourth order ordinary differential equation is considered in this method the error bound o exp c n logn c 0 is attained as in the sinc collocation method based on the de transformation where n is a parameter representing the number of terms in the sinc approximation high efficiency of the sinc galerkin method with the de transformation is confirmed by some numerical examples and the numerical results were compared with ones obtained by sinc collocation method based on the de transformation

on erdelyi kober fractional urysohn volterra quadratic integral equations a very general nonlinear singular integral equation is introduced namely u f 1 u f 2 u 0 s 1 k s a u s s 1 d s 0 1 0 and 0 1 the above equation is called erdelyi kober fractional urysohn volterra quadratic integral equation the main goal is to show that the above equation has solutions in c0 1 and these solutions are nonnegative and nondecreasing on 0 1 by means of a measure of noncompactness and darbo fixed point theorem we prove our main results in the end of the paper we give an example to show that our assumptions of our abstract results are rather easy to verify

an upper bound estimate and stability for the global error of numerical integration using double exponential transformation abstract the double exponential formula was introduced for calculating definite integrals with singular point oscillation functions and fourier integral the double exponential transformation is not only useful for numerical computations but it is also used in different methods of sinc theory in this paper we give an upper bound estimate for the error of double exponential transformation by improving integral estimates having singular final points in theorem 1 we prove that the method is convergent and the rate of convergence is o h 2 where h is a step size our main tool in the proof is de formula in sinc theory the advantage of our method is that the time and space complexity is drastically reduced furthermore we discovered an upper bound error in de formula independent of n truncated number as a matter of fact we proved stability numerical tests are presented to verify the theoretical predictions and confirm the convergence of the numerical solution

monotonic solutions of a quadratic integral equation of volterra type abstract we study a nonlinear quadratic integral equation of volterra type in the banach space of real functions defined and continuous on a bounded and closed interval with the help of a suitable measure of noncompactness we show that the mentioned integral equation has monotonic solutions

a fixed point approach in the study of the solution sets of lipschitzian functional differential inclusions abstract this paper deals with applications of general results on fixed point sets of set valued contraction mappings to the study of solution sets of functional differential inclusions with lipschitz type right hand side in separable banach spaces

viability in a non local population model structured by size and spatial position abstract motivated conceptually by multiscale models of cancer cell migration the focus is on a broad class of set valued population models structured both physiologically and spatially for time dependent sets of constraints given sufficient conditions are provided such that every permitted density function initializes at least one admissible weak solution this result about viability a k a weak invariance concerns non local hyperbolic differential inclusions of first order with memory and with one boundary condition for the minimal value of the scalar structure parameter

the double exponential sinc collocation method for singular sturm liouville problems sturm liouville problems are abundant in the numerical treatment of scientific and engineering problems in the present contribution we present an efficient and highly accurate method for computing eigenvalues of singular sturm liouville boundary value problems the proposed method uses the double exponential formula coupled with sinc collocation method this method produces a symmetric positive definite generalized eigenvalue system and has exponential convergence rate numerical examples are presented and comparisons with single exponential sinc collocation method clearly illustrate the advantage of using the double exponential formula

generalized lyapunov approach for functional differential inclusions abstract this paper is concerned with the generalized lyapunov approach for functional differential inclusions fdi at first we prove a class of generalized halanay s inequalities then by applying the generalization of halanay s inequalities we investigate the uniformly ultimate boundedness and uniformly asymptotic stability of fdi

error estimates with explicit constants for the sinc approximation over infinite intervals abstract the sinc approximation is a function approximation formula that attains exponential convergence for rapidly decaying functions defined on the whole real axis even for other functions the sinc approximation works accurately when combined with a proper variable transformation the convergence rate has been analyzed for typical cases including finite semi infinite and infinite intervals recently for verified numerical computations a more explicit computable error bound has been given in the case of a finite interval in this paper such explicit error bounds are derived for other cases

numerical indefinite integration by double exponential sinc method we present a numerical method for approximating an indefinite integral by the double exponential sinc method the approximation error of the proposed method with n integrand function evaluations is o exp c 1 n log c 2 n for a reasonably wide class of integrands including those with endpoint singularities the proposed method compares favorably with the existing formulas based on the ordinary sinc method computational results show the accordance of the actual convergence rates with the theoretical estimate

near optimality of the sinc approximation near optimality of the sinc approximation is established in a variety of spaces of functions analytic in a strip region about the real axis each space being characterized by the decay rate of their elements functions in the neighborhood of the infinity

summary of sinc numerical methods sinc approximation methods excel for problems whose solutions may have singularities or infinite domains or boundary layers this article summarizes results obtained to date on sinc numerical methods of computation sinc methods provide procedures for function approximation over bounded or unbounded regions encompassing interpolation approximation of derivatives approximate definite and indefinite integration solving initial value ordinary differential equation problems approximation and inversion of fourier and laplace transforms approximation of hilbert transforms and approximation of indefinite convolutions the approximate solution of partial differential equations and the approximate solution of integral equations methods for constructing conformal maps and methods for analytic continuation indeed sinc are ubiquitous for approximating every operation of calculus

monotonic solutions of a quadratic integral equation of fractional order abstract in the paper we indicate an error made in the proof of the main result of the paper m a darwish on quadratic integral equation of fractional orders j math anal appl 311 2005 112 119 moreover we provide correct proof of a slightly modified version of the mentioned result the main tool used in our proof is the technique associated with the hausdorff measure of noncompactness

function classes for double exponential integration formulas the double exponential de formulas for numerical integration are known to be highly efficient more efficient than the single exponential se formulas in many cases function classes suited to the se formulas have already been investigated in the literature through rigorous mathematical analysis whereas this is not the case with the de formulas this paper identifies function classes suited to the de formulas in a way compatible with the existing theoretical results for the se formulas the de formulas are good for more restricted classes of functions but more efficient for such functions two concrete examples demonstrate the subtlety in the behavior of the de formulas that is revealed by our theoretical analysis

new exponential variable transform methods for functions with endpoint singularities the focus of this paper is the approximation of functions which are analytic on a compact interval except at the endpoints typical numerical methods for approximating such functions depend upon the use of particular conformal maps from the original interval to either a semi infinite or an infinite interval followed by an appropriate approximation procedure on the new region we first analyze the convergence of these existing methods and show that in a precisely defined sense they are suboptimal specifically they exhibit poor resolution properties by which we mean that many more degrees of freedom are required to resolve oscillatory functions than standard approximation schemes for analytic functions such as chebyshev interpolation to remedy this situation we introduce two new transforms one for each of the above settings we provide full convergence results for these new approximations and then demonstrate that for a particular choice of parameters these methods lead to substantially better re

double exponential formulas for numerical indefinite integration in this paper we derive a formula for indefinite integration of analytic functions over 1 s where 1 s 1 by means of the double exponential transformation and the sinc method the integrand must be analytic on 1 x 1 but may have a singularity at the end points x 1 the error of the formula behaves approximately as exp c1n logc2n where n is the number of function evaluations of the integrand this error term shows a much faster convergence to zero when n becomes large than that of the known formula by haber also we derive efficient double exponential formulas for numerical evaluation of indefinite integrals over 0 s 0 s and over s s several numerical examples indicate high efficiency of the formulas

slow and quasi slow solutions of differential inclusions on donne des conditions suffisantes pour l existence d une solution viable lente pour le probleme de cauchy x r x t p pent 0 x 0 x 0 k x t kt 0 ou k est un sous ensemble convexe ferme d un espace vectoriel de dimension finie x et f k x est une application multiforme semi continue bornee superieurement a images compactes convexes

explicit nearly optimal linear rational approximation with preassigned poles this paper gives explicit rational functions for interpolating and approximating functions on the intervals 1 1 0 oc and oc oc the rational functions are linear in the functions to be approximated and they have preassigned poles the error of approximation of these rationals is nearly as small as the error of best rational approximation with numerator and denominator polynomials of the same degrees regions of analyticity are described which make it possible to tell a priori the accuracy which we can expect from this type of rational approximation 1 introduction and summary in this paper we attempt to give a constructive affirmative answer to each of the following questions

recent developments of the sinc numerical methods abstract this paper gives a survey of recent developments of the sinc numerical methods a variety of sinc numerical methods have been developed by stenger and his school for a certain class of problems the sinc numerical methods have the convergence rates of o exp n with some 0 where n is the number of nodes or bases used in the methods recently it has turned out that the sinc numerical methods can achieve convergence rates of o exp n log n with some 0 for a smaller but still practically meaningful class of problems and that these convergence rates are best possible the present paper demonstrates these facts for two sinc numerical methods the sinc approximation and the sinc collocation method for two point boundary value problems

new conformal map for the sinc approximation for exponentially decaying functions over the semi infinite interval abstract the sinc approximation has shown high efficiency for numerical methods in many fields conformal maps play an important role in the success i e appropriate conformal map must be employed to elicit high performance of the sinc approximation appropriate conformal maps have been proposed for typical cases however such maps may not be optimal thus the performance of the sinc approximation may be improved by using another conformal map rather than an existing map in this paper we propose a new conformal map for the case where functions are defined over the semi infinite interval and decay exponentially then we demonstrate in both theoretical and numerical ways that the convergence rate is improved by replacing the existing conformal map with the proposed map

the double exponential transformation in numerical analysis the double exponential transformation was first proposed by takahasi and mori in 1974 for the efficient evaluation of integrals of an analytic function with end point singularity afterwards this transformation was improved for the evaluation of oscillatory functions like fourier integrals recently it turned out that the double exponential transformation is useful not only for numerical integration but also for various kinds of sinc numerical methods the purpose of the present paper is to review the double exponential transformation in numerical integration and in a variety of sinc numerical methods

numerical approximation of time evolution related to ginzburg landau functionals using weighted sobolev gradients sobolev gradients have been discussed in sial et al 2003 as a method for energy minimization related to ginzburg landau functionals in this article a weighted sobolev gradient approach for the time evolution of a ginzburg landau functional is presented for different values of k a comparison is given between the weighted and unweighted sobolev gradients in a finite element setting it is seen that for small values of k the weighted sobolev gradient method becomes more and more efficient compared to using the unweighted sobolev gradient a comparison with newton s method is given where the failure of newton s method is demonstrated for a test problem

stability theorems for delay differential inclusions this technical note addresses stability problems of delay differential inclusions a chain rule is proposed for a wide class of lyapunov krasovskii functionals which are not necessarily invariantly differentiable we also obtain an invariance like theorem where the derivative of the candidate functional is bounded above by a continuous negative semidefinite function several examples are given to show the effectiveness of the results

images of connected sets by semicontinuous multifunctions abstract the image of a connected set by an upper semicontinuous or a lower semicontinuous multifunction whose values are nonempty and connected is connected we prove this theorem in its most general setting and show its usefulness in various examples from optimization and nonlinear analysis

numerical methods based on whittaker cardinal or sinc functions this paper summarizes the results known to date for using sine functions composed with other functions as bases for approximations in numerical analysis described in this paper are methods of interpolation and approximation of functions and their derivatives quadrature the approximate evaluation of transforms hilbert fourier laplace hankel and mellin and the approximate solution of differential and integral equations the methods have many advantages over classical methods which use polynomials as bases in addition all of the methods converge at an optimal rate if singularities on the boundary of approximation are ignored

function classes for successful de sinc approximations the de sinc formulas resulting from a combination of the sinc approximation formula with the double exponential de transformation provide a highly efficient method for function approximation in many cases they are more efficient than the se sinc formulas which are the sinc approximation formulas combined with the single exponential se transformations function classes suited to the se sinc formulas have already been investigated in the literature through rigorous mathematical analysis whereas this is not the case with the de sinc formulas this paper identifies function classes suited to the de sinc formulas in a way compatible with the existing theoretical results for the se sinc formulas furthermore we identify alternative function classes for the de sinc formulas as well as for the se sinc formulas which are more useful in applications in the sense that the conditions imposed on the functions are easier to verify

optimality of the double exponential formula functional analysis approach in the light of the functional analysis theory we establish the optimality of the double exponential formula the argument consists of the following three ingredients 1 introduction of a number of spaces of functions analytic in a strip region about the real axis each space being characterized by the decay rate of their elements functions in the neighborhood of the infinity 2 proof of the near optimality of the trapezoidal formula in each space introduced in 1 by showing the near equality between an upper estimate for the error norm of the trapezoidal formula and a lower estimate for the minimum error norm of quadratures 3 nonexistence theorem for the spaces the characterizing decay rate of which is more rapid than the double exponential

dynamical behaviors of delayed neural network systems with discontinuous activation functions in this letter without assuming the boundedness of the activation functions we discuss the dynamics of a class of delayed neural networks with discontinuous activation functions a relaxed set of sufficient conditions is derived guaranteeing the existence uniqueness and global stability of the equilibrium point convergence behaviors for both state and output are discussed the constraints imposed on the feedback matrix are independent of the delay parameter and can be validated by the linear matrix inequality technique we also prove that the solution of delayed neural networks with discontinuous activation functions can be regarded as a limit of the solutions of delayed neural networks with high slope continuous activation functions

de sinc methods have almost the same convergence property as se sinc methods even for a family of functions fitting the se sinc methods part ii indefinite integration in this paper the theoretical convergence rate of the sinc indefinite integration combined with the double exponential de transformation is given for a class of functions for which the single exponential se transformation is suitable although the de transformation is considered as an enhanced version of the se transformation for sinc related methods the function space for which the de transformation is suitable is smaller than that for se and therefore there exist some examples such that the de transformation is not better than the se transformation even in such cases however some numerical observations in the literature suggest that there is almost no difference in the convergence rates of se and de in fact recently the observations have been theoretically explained for two explicit approximation formulas the sinc quadrature and the sinc approximation the conclusion is that in such cases the de s rate is slightly lower but almost the same as that of the se the contribution of this study is the derivation of the same conclusion for the sinc indefinite integration numerical examples that support the theoretical result are also provided

double exponential transformation in the sinc collocation method for two point boundary value problems in 1974 takahasi and mori proposed the double exponential transformation for efficient evaluation of integrals of an analytic function with end point singularity recently it has turned out that the double exponential transformation is also useful for various kinds of sinc methods in the present paper we consider the sinc collocation method for two point boundary value problems to show that the method can be considerably enhanced when incorporated with the double exponential transformation technique