solution of multi delay systems using hybrid of block pulse functions and taylor series a method for finding the solution of linear time varying multi delay systems using a hybrid function is proposed the properties of the hybrid functions which consist of block pulse functions plus taylor series are presented the method is based upon expanding various time functions in the system as their truncated hybrid functions operational matrices of integration delay and product are presented and are utilized to reduce the solution of multi delay systems to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the technique

stability criteria for certain high even order delay differential equations abstract in this paper we study the asymptotic stability of the zero solution of even order linear delay differential equations of the form y 2 m t j 0 2 m 1 a j y j t j 0 2 m 1 b j y j t where a j and b j are certain constants and m 1 here 0 is a constant delay in proving our results we make use of pontryagin s theory for quasi polynomials

integral criteria for a classification of solutions of linear differential equations are positive continuous function on the interval 0 co by solution we always mean a continuously differentiable function such that r t x t has a continuous derivative satisfying e when the func tion r is continuous but does not have a continuous derivative e can be interpreted as the first order system for the vector x y x rx 381

necessary and sufficient conditions for oscillations of higher order delay differential equations we obtain necessary and sufficient conditions under which ail solutions of certain n th order delay differential equations oscillate when n is odd and necessary and sufficient conditions under which all bounded solutions oscillate when n is even the conditions derived involve the characteristic equation of the delay differential equation

stability criteria for certain third order delay differential equations in this paper we study the asymptotic stability of the zero solution of third order linear delay differential equations of the form y t p1y t p2y t q1y t q2y t v1y t v2y t where p1 p1 q1 q2 v1 and v2 are certain constants here 0 is a constant delay in proving our results we make use of pontryagin s theory for quasi polynomials

asymptotic properties of solutions of third order nonlinear differential equations with deviating argument abstract the aim of this paper is to establish comparison principles on property a between a nonlinear differential equation of the third order with deviating argument with delay advanced or mixed argument and the corresponding linear equation without deviating argument on the basis of these comparison principles the sufficient conditions for delay advanced and mixed equations to have property a are presented the results obtained are compared with existing ones in the framework of the papers

properties of kneser solutions for third order differential equations abstract we study asymptotic behavior of nonoscillatory solutions for the third order differential equations of the form r t y t p t y t 0 for property a of studied equations we extend the information about the asymptotic properties of positive decreasing solutions

on stability of systems of delay differential equations abstract in this paper we give necessary and sufficient conditions for the asymptotic stability of the zero solution of the system of linear delay differential equations of the form x t ax t 1 ax t where a is an n n matrix 0 is constant and 0 1 we reduce this to systems of first and second order problems our stability results are given in terms of the eigenvalues of a the proof of our results are carried out by an application of pontryagin s criterion for quasi polynomials to the characteristic functions of subsystems of the delay differential equations we also provide four algorithmic stability tests and include several examples

asymptotic and exponential stability of certain third order non linear delayed differential equations frequency domain method we use the frequency domain method to prove that the zero solution of certain third order nonlinear delayed differential equations is asymptotically stable when there is no forcing term we also prove the existence of a bounded solution which is exponentially stable when there is a bounded forcing term the situation for which the non linear term is delayed is also proved

critical delays and polynomial eigenvalue problems in this work we present a new method to compute the delays of delay differential equations ddes such that the dde has a purely imaginary eigenvalue for delay differential equations with multiple delays the critical curves or critical surfaces in delay space that is the set of delays where the dde has a purely imaginary eigenvalue are parameterized we show how the method is related to other works in the field by treating the case where the delays are integer multiples of some delay value i e commensurate delays the parameterization is done by solving a quadratic eigenvalue problem which is constructed from the vectorization of a matrix equation and hence typically of large size for commensurate delay differential equations the corresponding equation is a polynomial eigenvalue problem as a special case of the proposed method we find a closed form for a parameterization of the critical surface for the scalar case we provide several examples with visualizations where the computation is done with some exploitation of the structure of eigenvalue problems

taylor series analysis of time varying multi delay systems a method for finding an approximate solution of a multi delay system with time varying coefficients is proposed the method is based upon expanding various time functions in the system as their truncated taylor series using the operational matrices for integration and delay and hence reducing the system into a set of algebraic simultaneous equations the properties of taylor series are first briefly reviewed and associated operational matrices are utilized to solve multi delay systems the method is computationally very attractive and applications are demonstrated through illustrative examples

analysis of time delay systems using an alternative technique the taylor delayed operational matrix is first developed with the introduction of the taylor product matrix the taylor series approximation can be extended to the analysis of time varying systems the method is better than that using walsh series as demonstrated by numerical examples the result is quite satisfactory

stability criteria for certain second order delay differential equations with mixed coefficients in this paper we study asymptotic stability of the zero solution of the second order linear delay differential equation y t p1y t p2y t q1y t q2y t where p1p2 q1 and q2 are constants with p1p2 0 and q1q2 0 is a constant in proving our results we make use of pontryagin s theory of quasi polynomials

asymptotic stability of a mechanical robotics model with damping and delay in this paper we study the asymptotic stability of a mechanical robotics model with damping and delay this model yields a certain linear third order delay differential equation in proving our results we make use of pontryagin s theory for quasi polynomials

oscillation of functional trinomial differential equations with positive and negative term abstract in the paper we establish new technique for investigation of properties of trinomial differential equations with positive and negative term b t a t x t p t f x t q t h x t 0 we offer new criteria for asymptotic properties of nonoscillatory solutions for the studied equations we support our results with illustrative examples

stability criteria for certain high odd order delay differential equations in this paper we study the asymptotic stability of the zero solution of odd order linear delay differential equations of the formy 2 m 1 t j 02ma jy j t j 02mb jy j t t where a j and b j are certain constants and m 1 here t 0 is a constant delay in proving our results we make use of pontryagin s theory for quasi polynomials

taylor series approach for function approximation using estimated higher derivatives the paper proposes a new approach for function approximation by estimating the second derivative using taylor series where the samples of the function and its first derivatives at the sample points are known without such estimation these initial data can approximate the function traditionally using the first order taylor approximation but with more error if we desire to improve the approximation via second order taylor series then we can estimate the pseudo second derivatives of the function in three different ways all these three ways are investigated in this paper the pseudo second derivatives help in computing many more sample points of the function within each sampling interval thus the approach acts like a mathematical magnifying glass two examples are treated to compare the efficiencies of the methods also a qualitative study for upper bound of error of the approximations is studied in detail

a class of nonlinear optimal control problems governed by fredholm integro differential equations with delay this paper introduces an efficient numerical scheme for solving a significant class of nonlinear delay optimal control problem governed by fredholm integro differential equations a direct approach

a novel computational hybrid approach in solving hankel transform in this paper we use a combination of taylor and block pulse functions on the interval 0 1 that is called hybrid functions to estimate fast and stable solution of hankel transform first hybrid of block pulse and taylor polynomial basis is obtained and orthonormalized using gram schmidt process which are used as basis to expand a part of the integrand rf r appearing in the hankel transform integral thus transforming the integral into a fourier bessel series truncating the series an efficient stable algorithm is obtained for the numerical evaluation of the hankel transforms of order 1 the novelty of our method is that we give error analysis and stability of the hybrid algorithm and corroborate our theoretical findings by various numerical experiments for the first time the solutions obtained by projected method indicate that the approach is easy to implement and computationally very attractive

analysis and parameter identification of bilinear systems via shifted legendre polynomials the problems of analysis and parameter identification of lime delay systems have been recently studied using rectangular functions such as walsh and block pulse functions in this paper the continuous shifted legendre polynomials are first used to solve the above mentioned problems the approach followed is that of converting the delay differential equation to an algebraic form through using the operational matrices of integration and delay the key point is the derivation of a delay operational matrix d which relates the coefficient vector x of the shifted legendre series of a given function x t with the coefficient vector x of its delay form x t in the form x dx where the elements of a are obtained from the initial function of x t two computational examples are given to illustrate the utility of the method the availability of recursive algorithms for constructing the delay operational matrix d makes the method particularly attractive to practise

stability analysis of the zero solution for two class evolution equations abstract by constructing some suitable lyapunov type functionals and applying the theory of the definite quadratic form we obtain the stability of the zero solution of two classes of evolution equations with delays including reaction diffusion equations and damped wave equations our criteria depend on the derivatives of delays consequently when the delays are constants these criteria are independent of the magnitudes of the delays so the delays are harmless for the stability of the zero solution

equilibrium problems in hadamard manifolds abstract an equilibrium theory is developed in hadamard manifolds the existence of equilibrium points for a bifunction is proved under suitable conditions and applications to variational inequality fixed point and nash equilibrium problems are provided the convergence of picard iteration for firmly nonexpansive mappings along with the definition of resolvents for bifunctions in this setting is used to devise an algorithm to approximate equilibrium points

on kneser solutions of nonlinear third order differential equations sufficient conditions are given under which the third order differential equation with the middle term y q t y r t f y 0 q positive has a kneser solution y the derivatives of which change their signs

oscillation of third order trinomial delay differential equations the purpose of this paper is to study oscillation and asymptotic behavior of solutions to a third order linear delay differential equation y t p t y t q t y tau t 0 new comparison th

properties of the third order trinomial differential equations with delay argument abstract in this paper we study properties of the third order trinomial delay differential equation y t p t y t g t y t 0 by transforming this equation to binomial second third order differential equation employing suitable comparison theorems we establish new results on asymptotic behavior of solutions of the studied equation

nonexpansive iterations in hyperbolic spaces one of the most active research areas in nonlinear functional analysis is the asymptotics of nonexpansive mappings most of the results however have been obtained in normed linear spaces it is natural therefore to try to develop a theory of nonexpansive iterations in more general infinite dimensional manifolds this is the purpose of the present paper more specifically we propose the class of hyperbolic spaces as an appropriate background for the study of operator theory in general and of iterative processes for nonexpansive mappings in particular this class of metric spaces which is defined in section 2 includes all normed linear spaces and hadamard manifolds as well as the hilbert ball and the cartesian product of hilbert balls in section 3 we introduce co accretive operators and their resolvents and present some of their properties in the fourth section we discuss the concept of uniform convexity for hyperbolic spaces section 5 is devoted to two new geometric properties of infinite dimensional banach spaces theorem 5 6 provides a characterization of banach spaces having these properties in terms of nonlinear accretive operators in sections 6 7 and 8 we study explicit implict and continuous iterations repectively using the same approach in all three sections we illustrate this common approach with the following special case let c be a closed convex subset of a hyperbolic space x p let t c f c be a nonexpansive mapping and let x be a point in c in order to study the iteration t x n 0 1 2 we set z 1 l n x 0 l n t x k clco zj j i l and d inf p y ty y e c the first step is to show that p x k lim p x t x n d this leads to the convergence m of lz when x is uniformly convex and to the weak convergence of z when x is a banach space which is reflexive and strictly convex when t is an averaged mapping we are also able to establish the following triple equality for all k 2 1

even order half linear advanced differential equations we establish some new criteria for oscillation and asymptotic behavior of solutions of even order half linear advanced differential equations we study the case of canonical and the case of noncanonical equations subject to various conditions

parameter identification of linear multi delay systems via a hybrid of block pulse functions and taylor s polynomials abstractin this paper an efficient and effective procedure is successfully developed for parameter identification of linear time invariant multi delay systems the proposed framework is based on a hybrid of block pulse functions and taylor s polynomials two upper error bounds corresponding to hybrid functions are established the excellent properties of these functions together with the associated operational matrices of integration and delay are utilised to transform the original problem into a system of linear algebraic equations the least squares method is then implemented for estimation of the unknown parameters several numerical experiments are investigated to demonstrate the usefulness and effectiveness of the proposed procedure easy implementation simple operations and accurate solutions are the main features of the suggested approximation scheme

a new prediction scheme for input delay compensation in restricted feedback linearizable systems the input output inversion of a system under the effect of input delays typically relies on the ability to predict the future of the system s state indeed if the latter is known ahead of time one can cope with the input delay by using a prediction of the state instead of the state itself such methods are efficient when the plant is stable but become numerically unstable otherwise we present a new method to compensate input delays our approach relies on imposing a desired error dynamics which is designed to be linear and asymptotically stable at the origin then the state prediction is computed from the state reference trajectory and the predicted error dynamics in this technical note we concentrate on the case study of systems in strict feedback form and present a simple backstepping procedure

a hybrid approach to parameter identification of linear delay differential equations involving multiple delays abstractin this paper we are concerned with the parameter identification of linear time invariant systems containing multiple delays the approach is based upon a hybrid of block pulse functions and legendre s polynomials the convergence of the proposed procedure is established and an upper error bound with respect to the l2 norm associated with the hybrid functions is derived the problem under consideration is first transformed into a system of algebraic equations the least squares technique is then employed for identification of the desired parameters several multi delay systems of varying complexity are investigated to evaluate the performance and capability of the proposed approximation method it is shown that the proposed approach is also applicable to a class of nonlinear multi delay systems it is demonstrated that the suggested procedure provides accurate results for the desired parameters

analysis parameter estimation and optimal control of time delay systems via chebyshev series the chebyshev delay operational matrix is introduced first and then applied to approximate the solutions of linear time invariant and time varying delay systems with arbitrary time delay the parameter identification problem of the delay control system is also studied furthermore an approximate solution of the optimal control problem with quadratic performance measure is then discussed four examples are given and the results are shown to be very accurate and satisfactory

monotonicity resolvents and yosida approximations of operators on hilbert manifolds abstract monotonicity resolvents and yosida approximations of operators on manifolds of class c 3 are considered strong convergence of difference approximations to differential equations is shown by using resolvents our motivation is to give an elementary step toward constructing nonlinear semigroup theory on infinite dimensional manifolds

the asymptotic behavior of a class of nonlinear semigroups in the hilbert ball abstract we study the asymptotic behavior of the nonlinear semigroup generated by a nonexpansive self mapping of a closed convex subset of the hilbert ball equipped with the hyperbolic metric

parameters estimation of systems with delayed and structured entries this paper deals with on line identification of continuous time systems with structured entries such entries which may consist of inputs perturbations or piecewise polynomial time varying parameters can be defined as signals that can be easily annihilated the proposed cancellation method allows to obtain non asymptotic estimators for the unknown coefficients application to delayed and switching hybrid systems are proposed numerical simulations with noisy data but also experimental results on a delay process are provided

oscillation criteria of a certain class of third order nonlinear delay differential equations with damping abstract in this paper we are concerned with the oscillation of third order nonlinear delay differential equations of the form r 2 t r 1 t y p t y q t f y g t 0 by using a generalized riccati transformation and integral averaging technique we establish some new sufficient conditions which insure that any solution of this equation oscillates or converges to zero in particular several examples are given to illustrate the importance of our results

optimal control method with time delay in control abstract optimal control method for active vibration control of linear time delay systems is investigated in this paper in terms of two cases that time delay is integer and non integer times of sampling period motion equation with time delay is transformed as standard discrete forms which contain no time delay by using zero order holder respectively discrete quadratic function is used as objective function in design of controller to guarantee good control efficiency on sampling points in every step of computation of the deduced controller it contains not only current step of state feedback but also linear combination of some former steps of control because the controller is deduced directly from time delay differential equation system stability can be guaranteed easily thus this method is generally applicable to ordinary control systems the performance of the control method proposed and system stability when using this method are all demonstrated by numerical simulation results simulation results demonstrate that the presented method is a viable and attractive control strategy for applications to active vibration control instability in responses occurs possibly if the systems with time delay are controlled using controller designed in case of no time delay

analysis of time varying delay systems by hybrid of block pulse functions and biorthogonal multiscaling functions this paper proposes new effective methods for analysis of time varying delay systems using new hybrid functions the excellent properties of the hybrid functions which consist of block pulse functions and biorthogonal multiscaling functions are presented we utilise operational matrices of product delay and integration to reduce the solution of delay systems to the solution of algebraic equations illustrative examples are included to demonstrate the validity and applicability of the presented technique

analysis and parameter identification of time delay systems via shifted jacobi polynomials the shifted jacobi delay operational matrix is introduced and applied to the approximation of the solutions of linear systems with arbitrary time delay the parameter identification problem for time delay systems is also dealt with the new method is algebraic and computer oriented two examples are given and the results are very accurate and satisfactory

analysis and parameter identification of time delay systems via polynomial series time delay frequently occurs in electrical and mechanical systems in addition systems such as population and economic growth epidemic growth and neural networks are commonly modelled by delay differential equations although it is difficult to solve time delay systems we can apply the delay matrix l s to simplify the problem as has been shown taylor series have the simplest operational properties therefore we can apply the integration and delay matrix of taylor series via transformation as a result we can derive all of the polynomial delay matrices with arbitrary time delay

oscillation of third order nonlinear damped delay differential equations this paper is concerned with the oscillation of certain third order nonlinear delay differential equations with damping we give new characterizations of oscillation of the third order equation in terms of oscillation of a related well studied second order linear differential equation without damping we also establish new oscillation results for the third order equation by using the integral averaging technique due to philos numerous examples are given throughout

coaccretive operators and firmly nonexpansive mappings in the hilbert ball recently there has been an extensive research concerning p nonexpansive mappings of b see 5 7 14 161 in fact the more general nonexpansive mappings of b the cartesian product of n hilbert balls with respect to its hyperbolic metric had also been studied see lo 11 171 in this paper we shall present a new approach to the study of p nonexpansive mappings and more generally coaccretive operators see the definition below this approach involves the structure of b p as a hilbert manifold and can be used to simplify and unify a large part of the known basic results of the theory which can be found in the book 7 however we prefer to concentrate on the new results that our methods lead to relying on the known results when needed these new results include a study of a new class of generally multivalued operators namely coaccretive operators which are closely related to accretive operators in banach spaces and of the asymptotic behavior both at the origin and at infinity of certain families of firmly nonexpansive mappings of both kinds in particular we answer a question raised by goebel and reich 7 p 135 6 13 problem 161 our methods lead also to results of ergodic nature for p nonexpansive mappings which will appear elsewhere

on third order differential equations with property a and b we present some new results concerning relationships between oscillation and property a oscillation and property b the obtained results have necessary and sufficient character they are applied to the nonlinear third order differential equations too

positive solutions of functional differential equations with singular nonlinear terms soit p i a 0 0 i n des fonctions continues on definit les operateurs differentiels l i 0 i n par l 0 y t y t p 0 t l i y t 1 p i t dl i 1 y t dt 1 i n et on considere les equations l n y t f t y g t 0 l n y t f t y g t 0 ou f a 0 0 et g a 0 sont des fonctions continues et lim t g t on etudie la structure des solutions de ces equations definies et positives au voisinage de l infini

oscillatory and asymptotic behaviour of solutions of a class of linear ordinary differential equations the equation to be considered iswhere pi t 0 i n and q t are continuous and positive on some half line a it is known that always has strictly monotone nonoscillatory solutions defined on a so that of particular interest is the extreme situation in which such strictly monotone solutions are the only possible nonoscillatory solutions of in this paper sufficient conditions are given for this situation to hold for the structure of the solution space of is also studied

on functional inequalities and their applications in the oscillation theory in the paper we present several functional inequalities and offer their application for higher order advanced differential equations of the formx n t q t x s t 0 to be oscillatory the conditions obtained essentially improve complements and extends many other known results and involves unimprovable constants

integral criteria for oscillation of third order nonlinear differential equations abstract in this paper we are concerned with the oscillation of third order nonlinear differential equations of the form r 2 t r 1 t y p t y q t f y 0 t t 0 where t 0 0 is a fixed real number r 1 r 2 p q i t 0 r are nonnegative continuous functions with r 1 0 r 2 0 f r r is a continuous function by making use of a generalized riccati transformation we establish some new sufficient integral conditions under which the equation has at least one oscillatory solution examples are given to illustrate the importance of our results