on the global convergence of a class of functional differential equations with applications in neural network theory abstract we study a system of retarded functional differential equations which generalise both the hopfield neural network model as well as hybrid network models of the cellular neural network type our main results give sufficient conditions for the global asymptotic stability of such systems and are milder than previously known conditions for the hybrid models when specialised to neural networks our models allow us to consider several different types of activation functions including piecewise linear sigmoids and unbounded activations as well as the usualc smooth sigmoids these issues are vital in the applications we also study neural network models with nonconstant delaysr t

spaces of holomorphic functions in regular domains let be a regular domain in the complex plane c c let gb be the linear space over c of the holomorphic functions f in such that f n is bounded in and is continuously extendible to the closure of n 0 1 2 we endow gb in a natural manner with a structure of frechet space and we obtain dense subspaces f of gb with good topological linear properties also satisfying that each function f of f distinct from zero does not extend holomorphically outside

a fourth order finite difference method based on non uniform mesh for a class of singular two point boundary value problems abstract we present a three point finite difference method based on non uniform mesh for a class of singular two point boundary value problem x y f x y y 0 a y 1 b 0 we show that the method based on non uniform mesh provides o h4 convergent approximations this method is illustrated by two numerical examples one is linear and the other is non linear

a uniform mesh finite difference method for a class of singular two point boundary value problems a new finite difference method based on uniform mesh is given for the weakly singular two point boundary value problem x alpha y f x y y 0 a y 1 b 0 alpha 1 under quite general conditions on f and f we show that our present method based on uniform mesh provides o h 2 convergent approximations for all alpha in 0 1 our method is based on one evaluation of f and for alpha 0 it reduces to the classical second order method for y f x y

a novel numerical approach and its convergence for numerical solution of nonlinear doubly singular boundary value problems this paper deals with two new recursive numerical schemes for the solution of a class of doubly singular two point boundary value problems with different types of boundary conditions we first transform the original problem into an equivalent integral equation to overcome the singular behaviour at the origin and then establish recursive schemes by employing homotopy perturbation method for the solution of the resultant equation we establish the convergence results of the new numerical methods four numerical examples are considered to demonstrate the efficiency and accuracy of the proposed methods the present methods are shown their advantage over some existing methods chawla and katti 1982 pandey and singh 2007 pandey and singh 2004 caglar et al 2009 kumar and aziz 2004 khuri and sayfy 2010 developed for the solution of singular two point boundary value problems moreover the new methods are used to obtain the approximate solutions of the singular boundary value problems arising in various physical problems

lineability criteria with applications lineability is a property enjoyed by some subsets within a vector space x a subset a of x is called lineable whenever a contains except for zero an infinite dimensional vector subspace if additionally x is endowed with richer structures then the more stringent notions of dense lineability maximal dense lineability and spaceability arise naturally in this paper several lineability criteria are provided and applied to specific topological vector spaces mainly function spaces sometimes such criteria furnish unified proofs of a number of scattered results in the related literature families of strict order integrable functions hypercyclic vectors non extendable holomorphic mappings riemann non lebesgue integrable functions sequences not satisfying the lebesgue dominated convergence theorem nowhere analytic functions bounded variation functions entire functions with fast growth and peano curves among others are analyzed from the point of view of lineability

estimate of exponential convergence rate and exponential stability for neural networks estimates of exponential convergence rate and exponential stability are studied for a class of neural networks which includes hopfield neural networks and cellular neural networks both local and global exponential convergence are discussed theorems for estimation of the exponential convergence rate are established and the bounds on the rate of convergence are given the domains of attraction in the case of local exponential convergence are obtained simple conditions are presented for checking exponential stability of the neural networks

a note on existence uniqueness results for a class of doubly singular boundary value problems abstract in this paper we establish the existence and uniqueness of solution for nonlinear singular boundary value problem p x y x q x f x y for 0 x b and lim x 0 p x y x 0 1 y b 1 p b y b 1 here p 0 0 and also q is not assumed to be continuous at 0 so the problem may be doubly singular in this note we discuss the cases when y 0 0 at the singular end is not possible but lim x 0 p x y x 0 regions for multiple solutions have also been determined

a fourth order spline method for singular two point boundary value problems this paper describes a new fourth order spline approximation method for the class of singular two point boundary value problems y frac alpha x y f x y 0 0 x 1 y 0 0 y 1 a for all alpha geqslant 1

on the convergence of finite difference methods for weakly regular singular boundary value problems the second order finite difference methods m 1 based on a non uniform mesh and m 2 based on an uniform mesh developed by chawla and katti finite difference methods and their convergence for a class of singular two point boundary value problems numer math 39 1982 341 350 for weakly regular singular boundary value problems p x y f x y 0

neurocomputing with time delay analysis for solving convex quadratic programming problems this paper presents a neural network computational scheme with time delay consideration for solving convex quadratic programming problems based on some known results a delay margin is explicitly determined for the stability of the neural dynamics under which the states of the neural network does not oscillate the configuration of the proposed neural network is provided operational characteristics of the neural network are demonstrated via numerical examples

haar wavelet collocation method for lane emden equations with dirichlet neumann and neumann robin boundary conditions abstract in this paper we present a numerically stable algorithm based on the haar wavelet collocation method hwcm for numerical solution of a class of lane emden equation with dirichlet neumann and neumann robin type boundary conditions arising in various physical models the haar wavelet collocation method uses simple box functions and hence the formulation of the numerical algorithm is straightforward unlike other methods the hwcm can be applied easily and accurately to the neumann type boundary conditions the advantage of the haar wavelet method is its efficiency and simple applicability for a variety of boundary conditions the haar wavelet collocation method is utilized to reduce the lane emden equation with boundary conditions to a system of algebraic equations then the newton s method is employed for numerical solutions nine examples of the lane emden equation with various types of boundary conditions are provided to demonstrate the accuracy and efficiency of the hwcm the numerical results are compared with the results obtained by other techniques and the exact solution

global exponential convergence of recurrent neural networks with variable delays convergence analysis of recurrent neural networks is an important research direction in the field of neural networks novel methods to study the global exponential convergence of recurrent neural networks with variable delays are proposed a condition for global exponential stability which is independent of the delays is derived by the method of delayed inequalities analysis another condition for global exponential stability which depends on the delays is obtained via the method of constructing a suitable and interesting lyapunov functional

on the convergence of a finite difference method for a class of singular boundary value problems arising in physiology using chawla s identity bit 29 1989 566 a finite difference method based on uniform mesh is described for a class of singular boundary value problems p x y p x f x y 0 x 1 y 0 a y 1 y 1 or y 0 0 y 1 y 1 with p x xb0 g x b0 0 and it is shown that the method is of second order accuracy under quite general conditions on p x and f x y this work also extends the method developed by chawla et al bit 26 1986 326 for p x xb0 b0 1 to a general class of function p x xb0 g x b0 0 numerical examples for general function p x verify the order of the convergence of the method and two physiological problems have also been solved

a uniform mesh finite difference method for a class of singular two point boundary value problems abstract a method is described based on a uniform mesh for the class of singular two point boundary value problems x y f x y 0 x 1 y 0 a y 1 b it is shown to be order h 4 convergent for all 0 1

a fourth order finite difference method based on uniform mesh for singular two point boundary value problems we present a new fourth order finite difference method based on uniform mesh for the weakly singular two point boundary value problem x y f x y 0 x 1 y 0 a y 1 b 0 1 our method provides o h4 convergent approximations for all 0 1 for 0 it reduces to the well known fourth order method of numerov for y x y

existence uniqueness results for a class of singular boundary value problems ii abstract in this paper we establish existence uniqueness of solution of a class of singular boundary value problem p x y x q x f x y for 0 x b and y 0 a 1 y b 1 y b 1 where p x satisfies i p x 0 in 0 b ii p x c 1 0 r and for some r b iii x p x p x is analytic in z z r and q x satisfies i q x 0 in 0 b ii q x l 1 0 b and for some r b iii x 2 q x p x is analytic in z z r with quite general conditions on f x y region for multiple solutions have also been determined

the space h ω zj of holomorphic functions abstract let be a domain in c n let h be the linear space over c of the holomorphic functions in endowed with the compact open topology let z j be a sequence in without adherent points in in this paper we define the space h z j and some of its linear topological properties are studied we also show that for some domains of holomorphy and some sequences z j the non zero elements of h z j cannot be extended holomorphically outside as a consequence we obtain some characterizations of the domains of holomorphy in c n

fundamental study global dynamics for non autonomous reaction diffusion neural networks with time varying delays in this paper a class of non autonomous reaction diffusion neural networks with time varying delays is considered novel methods to study the global dynamical behavior of these systems are proposed employing the properties of diffusion operator and the method of delayed inequalities analysis we investigate global exponential stability positive invariant sets and global attracting sets of the neural networks under consideration furthermore conditions sufficient for the existence and uniqueness of periodic attractors for periodic neural networks are derived and the existence range of the attractors is estimated finally two examples are given to demonstrate the effectiveness of these results

spline finite difference methods for singular two point boundary value problems in this paper we discuss the construction of a spline function for a class of singular two point boundary value problemx x u f x u u 0 a u 1 b 0 1 or 1 2 the boundary conditions may also be of the formu 0 0 u 1 b three point finite difference methods using the above splines are obtained for the solution of the boundary value problem these methods are of second order and are illustrated by four numerical examples

robust stability for interval hopfield neural networks with time delay the conventional hopfield neural network with time delay is intervalized to consider the bounded effect of deviation of network parameters and perturbations yielding a novel interval dynamic hopfield neural network idhnn model a sufficient condition related to the existence of unique equilibrium point and its robust stability is derived

wavelet galerkin method for solving nonlinear singular boundary value problems arising in physiology abstract in this study wavelet galerkin method has been developed for solving nonlinear singular boundary value problems associated with physiology science the quasilinearization technique is applied to reduce the given nonlinear problem to a sequence of linear problems we modify the resulting set of differential equations at the singular point then treat this set of boundary value problems by using compactly supported cubic b spline wavelets and galerkin method these wavelets are used as testing and weighting functions the method is computationally attractive and applications are demonstrated through illustrative examples

how delays affect neural dynamics and learning we investigate the effects of delays on the dynamics and in particular on the oscillatory properties of simple neural network models we extend previously known results regarding the effects of delays on stability and convergence properties we treat in detail the case of ring networks for which we derive simple conditions for oscillating behavior and several formulas to predict the regions of bifurcation the periods of the limit cycles and the phases of the different neurons these results in turn can readily be applied to more complex and more biologically motivated architectures such as layered networks in general the main result is that delays tend to increase the period of oscillations and broaden the spectrum of possible frequencies in a quantifiable way simulations show that the theoretically predicted values are in excellent agreement with the numerically observed behavior adaptable delays are then proposed as one additional mechanism through which neural systems could tailor their own dynamics accordingly we derive recurrent backpropagation learning formulas for the adjustment of delays and other parameters in networks with delayed interactions and discuss some possible applications

a difference method for singular two point boundary value problems a difference method based on uniform mesh for solving singular two point boundary value problems of the form x ay f x y 0

absolute exponential stability of a class of recurrent neural networks with multiple and variable delays in this paper we derive some new conditions for absolute exponential stability aest of a class of recurrent neural networks with multiple and variable delays by using the holder s inequality and the young s inequality to estimate the derivatives of the lyapunov functionals we are able to establish more general results than some existing ones the first type of conditions established involves the convex combinations of column sum and row sum dominance of the neural network weight matrices while the second type involves the p norm of the weight matrices with p 1

a difference scheme based on non uniform mesh for singular two point boundary value problems a difference scheme based on non uniform mesh for singular two point boundary value problems of the form y 1xy f x y 0 0

on a class of regular singular two point boundary value problems abstract existence of a unique solution for a class of regular singular two point boundary value problems p x y p x f x y 0 x b lim x 0 y x 0 y b b has been established wherep x satisfies i p x 0 on 0 b ii p x c1 0 r and for somer b iii xp x p x is analytic in z z xp x p x b 0 b 1 x b 0 1 and with quite general conditions onf x y these conditions onf x y are sharp which is seen through one example regions for multiple solutions have also been determined

the numerical solution of non linear singular boundary value problems arising in physiology a class of non linear singular ordinary differential equations is solved by a new method based on non polynomial cubic spline we use the quesilinearization technique to reduce the given non linear problem to a sequence of linear problems we modify the resulting set of differential equations at the singular point then treat this set of boundary value problems by using non polynomial cubic spline approximation the resulting system of algebraic equations is solved by using a tri diagonal solver computational results are provided to demonstrate the viability of the new method

multistability analysis for recurrent neural networks with unsaturating piecewise linear transfer functions multistability is a property necessary in neural networks in order to enable certain applications e g decision making where monostable networks can be computationally restrictive this article focuses on the analysis of multistability for a class of recurrent neural networks with unsaturating piecewise linear transfer functions it deals fully with the three basic properties of a multistable network boundedness global attractivity and complete convergence this article makes the following contributions conditions based on local inhibition are derived that guarantee boundedness of some multistable networks conditions are established for global attractivity bounds on global attractive sets are obtained complete convergence conditions for the network are developed using novel energy like functions and simulation examples are employed to illustrate the theory thus developed

a new finite difference method for a class of singular two point boundary value problems we present a new three point finite difference method based on uniform mesh for a class of singular two point boundary value problem x ay f x y y 0 a y 1 b 0 a 1 we show that the method based on uniform mesh provides o h 4 convergent approximations this method is illustrated by two numerical examples one is linear and another is non linear

infinite dimensional holomorphic non extendability and algebraic genericity abstract in this note the linear structure of the family h e g of holomorphic functions in a domain g of a complex banach space that are not holomorphically continuable beyond the boundary of g is analyzed more particularly we prove that h e g contains except for zero a closed and a dense vector space having maximal dimension as well as a maximally generated free algebra the results obtained complete a number of previous ones by several authors

a second order spline finite difference method for singular two point boundary value problems a second order spline finite difference method based on a non uniform mesh for singular two point boundary value problems of the form x a x au f x u 0

finite difference methods and their convergence for a class of singular two point boundary value problems we discuss the construction of three point finite difference approximations and their convergence for the class of singular two point boundary value problems x y f x y y 0 a y 1 b 0 1 we first establish a certain identity based on general non uniform mesh from which various methods can be derived to obtain a method having order two for all 0 1 we investigate three possibilities by employing an appropriate non uniform mesh over 0 1 we obtain a methodm1 based on just one evaluation off for uniform mesh we obtain two methodsm2 andm3 each based on three evaluations off for 0 m1 andm2 both reduce to the classical second order method based on one evaluation off these three methods are investigated theiro h2 convergence established and illustrated by numerical examples

stability in asymmetric hopfield nets with transmission delays sufficient conditions are derived for the delay independent stability of the equilibria in hopfield s graded response networks of the type dxi t dt bixi t nj 1aijfj jxj t ij fi t i 1 2 n when the external inputs fi are held temporally uniform a generalization to continuously distributed delays is briefly indicated several illustrative examples are numerically simulated and the results of simulations are graphically displayed