two classes of asymptotically different positive solutions of the equation y t f t yt abstract the equation y t f t y t is considered for t the existence of two classes of positive solutions which are asymptotically different is proved using two different techniques the method of monotone sequences and the retract method combined with razumikhin s technique with the aid of linear estimates of the right hand side of the equation considered inequalities for both types of positive solutions are given as well the results are illustrated by an example

identification of systems containing linear dynamic and static nonlinear elements identification of nonlinear systems which can be represented by combinations of linear dynamic and static nonlinear elements are considered previous results by the authors based on correlation analysis are combined to provide a unified treatment for this class of systems it is shown that systems composed of cascade feedforward feedback and multiplicative connections of linear dynamic and zero memory nonlinear elements can be identified in terms of the individual component subsystems from measurements of the system input and output only

convergence rates of the truncated euler maruyama method for stochastic differential equations influenced by higham et al 2002 several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations sdes under the local lipschitz condition these numerical methods include the tamed euler maruyama em method the tamed milstein method the stopped em the backward em the backward forward em etc recently we developed a new explicit method in mao 2015 called the truncated em method for the nonlinear sde d x t f x t d t g x t d b t and established the strong convergence theory under the local lipschitz condition plus the khasminskii type condition x t f x p 1 2 g x 2 k 1 x 2 however due to the page limit there we did not study the convergence rates for the method which is the aim of this paper we will under some additional conditions discuss the rates of l q convergence of the truncated em method for 2 q p and show that the order of l q convergence can be arbitrarily close to q 2

a positivity preserving numerical method for stochastic r d model abstract the stochastic research and development r d model plays an important role in the growth rates of technological progress and capital accumulation however we can not obtain the explicit solution of stochastic r d model to explore a numerical approximate method preserving positivity for the r d model with uncertainty from the population growth we first construct an explicit euler maruyama em method and then verify it converges to the true solution with strong order 1 2 1 1 p but the em method may lead to a negative approximation which is unrealistic so we establish the balanced explicit method bem to overcome the defect of em method and give sufficient conditions to preserve the positivity of bem then the convergence order of bem is obtained finally numerical simulations are carried out to verify our theoretical work

oscillation analysis of numerical solutions for nonlinear delay differential equations of hematopoiesis with unimodal production rate this paper is concerned with oscillation and non oscillation of numerical solutions for nonlinear delay differential equation of hematopoiesis with unimodal production rate by investigating the roots of characteristic equation some conditions under which the numerical solution is oscillatory and non oscillatory are obtained the properties of non oscillatory numerical solutions are investigated to verify our results we give numerical experiments

dominant and subdominant positive solutions to generalized dickman equation abstract the paper considers a generalized dickman equation t x t i 1 s a i x t i for t where s n ai 0 i 0 i 1 s and i 1 s a i 1 it is proved that there are two mutually disjoint sets of positive decreasing solutions such that for every two solutions from different sets the limit of their ratio for t equals 0 or the asymptotic behavior of such solutions is derived and a structure formula utilizing such solutions and describing all the solutions of a given equation is discussed in addition a criterion is proved giving sufficient conditions for initial functions to generate solutions falling into the first or the second set illustrative examples are given some open problems are suggested to be solved

bounded solutions of nonlinear discrete equations abstract the goal of this paper is to give sufficient conditions for the existence of bounded solutions of a system of discrete equations u k 1 f k u k k a a 1 a 2 with a n n 0 1 and f n a r n r n a principle giving sufficient conditions for the existence of at least one initial problem generating solution with its graph staying in a previously prescribed domain for every k a a 1 a 2 is formulated illustrative examples are considered as well

on global identifiability for arbitrary model parametrizations it is a fundamental problem of identification to be able even before the data have been analyzed to decide if all the free parameters of a model structure can be uniquely recovered from data this is the issue of global identifiability in this contribution we show how global identifiability for an arbitrary model structure basically with analytic non linearities can be analyzed using concepts and algorithms from differential algebra it is shown how the question of global structural identifiability is reduced to the question of whether the given model structure can be rearranged as a linear regression an explicit algorithm to test this is also given furthermore the question of persistent excitation for the input can also be tested explicitly is a similar fashion the algorithms involved are very well suited for implementation in computer algebra one such implementation is also described

oscillation criteria of first order linear difference equations with delay argument abstract this paper presents new sufficient conditions for the oscillation of all proper solutions of the first order linear difference equation with delay argument u k p k u k 0 k n where u k u k 1 u k p n r n n and lim k k examples illustrating the results are given it is to be pointed out that this is the first paper dealing with the oscillatory behaviour of the equation in the case of a general delay argument k

wazewski theorem on time scales with a set of egress points that is not the whole boundary abstract in the paper the celebrated wazewski retract method is developed for planar dynamic equations on an arbitrary time scale without a restrictive assumption that the whole boundary of a set of constraints where we look for solutions is a set of egress points as a consequence some recently published results are essentially generalized one transparent example illustrating the main theorem is presented the results are new and more general even for ordinary difference equations

paper zeros of sampled systems the zeros of the discrete time system obtained when sampling a continuous time system are explored theorems for the limiting zeros for large and small sampling periods are given conditions which guarantee that the sampled system has stable zeros are also presented

positive solutions of p type retarded functional differential equations abstract for systems of retarded functional differential equations with unbounded delay and with finite memory sufficient and necessary conditions of existence of positive solutions on an interval of the form t 0 are derived a general criterion is given together with corresponding applications including a linear case too examples are inserted to illustrate the results

on a problem of i gyorl abstract a necessary and sufficient condition is given for the oscillation of equation x t x t t and a problem of i gyori is also disproved

recursive prediction error identification using the nonlinear wiener model the nonlinear wiener model consisting of a linear dynamic block in cascade with a static nonlinearity is considered a recursive prediction error identification algorithm based on the wiener model is derived the linear dynamic block is modelled as a siso transfer function operator and the static nonlinearity is approximated with a piecewise linear function a theoretical analysis of the method is carried out and conditions for local convergence to the true parameter vector are given in particular the analysis shows that the input signal should be such that there is signal energy in the whole range of the piecewise linear approximation a numerical example illustrates the performance of the algorithm further practical guidelines on how to apply the algorithm are also included in the paper

convergence analysis of recursive identification algorithms based on the nonlinear wiener model recursive identification algorithms based on the nonlinear wiener model are presented a recursive identification algorithm is first derived from a general parameterization of the wiener model using a stochastic approximation framework local and global convergence of this algorithm can be tied to the stability properties of an associated differential equation since inversion is not utilized noninvertible static nonlinearities can be handled which allows a treatment of for example saturating sensors and blind adaptation problems gauss newton and stochastic gradient algorithms for the situation where the static nonlinearity is known are then suggested in the single input single output case the proposed methods can outperform conventional linearizing inversion of the nonlinearity when measurement disturbances affect the output signal for fir finite impulse response models it is also proved that global convergence of the schemes is tied to sector conditions on the static nonlinearity in particular global convergence of the stochastic gradient method is obtained provided that the nonlinearity is strictly monotone the local analysis performed for iir infinite impulse response models illustrates the importance of the amplitude contents of the exciting signals

asymptotic behaviour of solutions of systems of discrete equations via liapunov type technique abstract the second liapunov method serves as a powerful tool for the investigation of the stability of the trivial solution of ordinary differential equations systems and discrete equations systems in the presented paper a liapunov type qualitative approach is used for the investigation of asymptotic behaviour of the solutions of systems of discrete equations conditions for the existence of continuum of solutions the graphs of which remain within a given prescribed set are formulated for the general systems of discrete equations u k f k u k an additional advantage of the presented approach consists of the fact that no assumption concerning the existence of the trivial solution or the existence of an equilibrium point of systems considered is assumed moreover the asymptotic behaviour of solutions of some classes of linear difference systems is given by means of concrete asymptotic formulae several illustrative examples are considered too

construction of positivity preserving numerical method for stochastic age dependent population equations the aim of this paper is to construct a numerical method preserving positivity for stochastic age dependent population equations we use the balanced implicit numerical techniques to maintain the nonnegative path of the exact solution it is proved that the balanced implicit method bim preserves positivity and converges with strong order 1 2 under given conditions finally two examples are simulated to verify the positivity and efficiency of the proposed method

strong convergence of the stopped euler maruyama method for nonlinear stochastic differential equations in this paper numerical methods for the nonlinear stochastic differential equations sdes with non global lipschitz drift coefficient are discussed the existing known results have only so far shown that the classical explicit euler maruyama em approximate solutions converge to the true solution in probability 22 23 more recently the authors in 16 proved that the classical em method will diverge in l 2 sense for the underlying sdes in this paper and those sdes with superlinearly growing coefficients these strongly indicate that the classical em method is not good enough for the highly nonlinear sdes however in this paper we introduce a modified em method using stopping time and show successfully that the discrete version of the modified em approximate solution converges to the true solution in the strong sense namely in l 2 with a order arbitrarily close to a half

asymptotic behavior of solutions of systems of dynamic equations on time scales in a set whose boundary is a combination of strict egress and strict ingress points abstract in this paper we study the asymptotic behavior of solutions of nonlinear dynamic systems on time scales of the form y t f t y t where f t r n r n and t is a time scale for a given set t r n we formulate the conditions for function f which guarantee that at least one solution y of the above system stays in the dimension of the space of initial data generating such solutions is discussed and perturbed linear systems are considered as well a linear system with singularity at infinity is considered as an example

stability analysis of numerical schemes for stochastic differential equations stochastic differential equations sdes represent physical phenomena dominated by stochastic processes as for deterministic ordinary differential equations odes various numerical schemes are proposed for sdes in this paper we study the stability of numerical schemes for scalar sdes with respect to the mean square norm which we call ms stability we will show some figures of the ms stability domain or regions for some numerical schemes and present numerical results which confirm it this notion is an extension of absolute stability in numerical methods for odes

transformation and canonical forms of functional differential equations functional differential equations especially linear ones are considered with respect to global pointwise transformations two types of canonical forms for certain classes of these equations are introduced these transformations and the corresponding canonical forms preserve oscillatory or non oscillatory behaviour of solutions they are also suitable for studying both side solutions of equivalent functional differential equations

asymptotic convergence criteria of solutions of delayed functional differential equations abstract the convergence of all solutions of the scalar equation x f t x t is studied under the main assumption that every constant is its solution the criterion and the sufficient conditions for convergence are proved for the considered classes of equations it is among others proved that all solutions are convergent if and only if there exists at least one increasing and convergent solution as applications criteria for some classes of linear equations as well as for nonlinear equations are obtained some comparisons with already known results are given too

an existence criterion of positive solutions of p type retarded functional differential equations the conditions of existence of a positive solution i e a solntion with positive coordinates on a considered interval of systems of retarded functional equations in the case of unbounded delay with finite memory are discussed a general criterion for nonlinear case is given as well as its application to a linear system illustrative special cases are considered too

on the sharpness of a theorem by cooke and verduyn lunel abstract the sharpness of a theorem of k cooke and s m verduyn lunel differential integral equations 6 1993 1101 1117 about small solutions of linear time dependent delay equations is proved a wide class of nonautonomous functional differential equations possessing small solutions is constructed the construction is based on the existence of quickly decreasing solutions of functional differential equations with linearly expanded arguments

mean square and asymptotic stability of the stochastic theta method stability analysis of numerical methods for ordinary differential equations odes is motivated by the question for what choices of stepsize does the numerical method reproduce the characteristics of the test equation we study a linear test equation with a multiplicative noise term and consider mean square and asymptotic stability of a stochastic version of the theta method we extend some mean square stability results in saito and mitsui siam j numer anal 33 1996 pp 2254 2267 in particular we show that an extension of the deterministic a stability property holds we also plot mean square stability regions for the case where the test equation has real parameters for asymptotic stability we show that the issue reduces to finding the expected value of a parametrized random variable we combine analytical and numerical techniques to get insights into the stability properties for a variant of the method that has been proposed in the literature we obtain precise analytic expressions for the asymptotic stability region this allows us to prove a number of results the technique introduced is widely applicable and we use it to show that a fully implicit method suggested by kloeden and platen numerical solution of stochastic differential equations springer verlag 1992 has an asymptotic stability extension of the deterministic a stability property we also use the approach to explain some numerical results reported in milstein platen and schurz siam j numer anal 35 1998 pp 1010 1019

nonlinear black box models in system identification mathematical foundations we discuss several aspects of the mathematical foundations of the nonlinear black box identification problem we shall see that the quality of the identification procedure is always a result of a certain trade off between the expressive power of the model we try to identify the larger the number of parameters used to describe the model the more flexible is the approximation and the stochastic error which is proportional to the number of parameters a consequence of this trade off is the simple fact that a good approximation technique can be the basis of a good identification algorithm from this point of view we consider different approximation methods and pay special attention to spatially adaptive approximants we introduce wavelet and neuron approximations and show that they are spatially adaptive then we apply the acquired approximation experience to estimation problems finally we consider some implications of these theoretical developments for the practically implemented versions of the spatially adaptive algorithms

oscillation criteria for delay and difference equations with non monotone arguments consider the first order linear delay differential equation 1 x t p t x t t 0 t t 0 where p email protected c t 0 r t t t 0 and lim t email protected t and the discrete analogue difference equation 1 dx n p n x t n 0 n 0 1 2 where d denotes the forward difference operator dx n x n 1 x n p n is a sequence of nonnegative real numbers and t n is a sequence of integers such that t n 0 and lim n email protected n the state of the art on the oscillation of all solutions to these equations are established especially in the case of non monotone arguments examples illustrating the results are given

boundedness and asymptotic behavior results for nonlinear difference equations with positive and negative coefficients this paper is concerned with the nonlinear delay difference equations with positive and negative coefficients x n 1 x n p n f x n k q n f x n l 0 n n 0 sufficient conditions are obtained under which every solution of eq is bounded and tends to a constant as n

convergence of the solutions of the equation y t β t y t δ y t τ in the critical case we study the asymptotic behavior of the solutions of the first order differential equation containing two delays y t t y t y t with t0 r 0 the convergence of all solutions is characterized by the existence of a strictly increasing bounded solution a critical case is found for the coefficient function for coefficients below the critical function a strictly increasing and bounded solution is constructed and thus the convergence of all solutions is shown relations with known results are discussed too

asymptotic representation of solutions of equationẏ t β t y t y t τ t abstract in this paper the equation t t y t y t t is considered where c i 1 r i 1 t 1 r 0 t t is an increasing function on i 1 t t i 1 0 c i 1 r results about asymptotic structure and behavior of the solutions of this equation are proved a connection with the equation t c t x t t and the asymptotic representation of its solutions is obtained

parametric and nonparametric curve fitting we are concerned with convergence issues in the identification of a static nonlinear function our investigation focuses on properties of the input signal that ensure convergence of the estimate both parametric and nonparametric approaches are considered in the parametric case we offer sufficient conditions under which the estimated parameters converge to their true values almost surely for the nonparametric case we offer necessary and sufficient conditions under which the estimated function converges almost surely to the true nonlinearity

adaptive control of weiner type nonlinear systems abstract this study is concerned with the problem of adaptive control of a wiener type nonlinear process it is assumed that the linear dynamic part of the process can be represented by a pulse transfer function of known order with stable inverse followed by a known time delay the memoryless nonlinearity is one to one on a compact set and belongs to one of two classes 1 piecewise linear with known breakpoints or 2 smooth function in which case an approximation is made it is represented by linear splines the output signal from the linear part is not available for measurement the global stability of the proposed model reference adaptive control scheme is established subject to the assumption that the nonlinearity can be represented exactly by the linear spline function with a given set of breakpoints otherwise there will be a residual tracking error which depends on the nonlinearity approximation error but the boundedness of all signals can be assured the proposed scheme is used for adaptive control of the ph process in a continuous flow reactor

numerical analysis of the balanced implicit method for stochastic age dependent capital system with poisson jumps abstract the aim of this paper is to construct a numerical method to preserve positivity and mean square dissipativity of stochastic age dependent capital system with poisson jumps we use the balanced implicit numerical techniques to maintain the nonnegative path of the exact solution it is proved that the balanced implicit method bim preserves positivity and converges with order 1 2 under given conditions in addition some sufficient conditions are obtained for ensuring the system and the balanced implicit method bim are mean square dissipative finally a numerical example is simulated to illustrate the efficiency of theoretical results

a topological principle for retarded functional differential equations of caratheodory type abstract the paper presents a razumikhin type version of wazewski s principle for rfdes of caratheodory type thus generalizing the author s previous results j differential equations 36 117 138 1980 a few facts from the theory of rfdes are reviewed and the concept of a regular polyfacial set with respect to an rfde of caratheodory type is introduced and the results are state and proved most of the examples given in the above work can easily be adapted to the more general setting of caratheodory systems an indication how to do it is given in only one case throughout the paper standard notation is used

discrete retract principle for systems of discrete equations abstract a power tool for investigation of various problems of ordinary differential equations as well as delayed differential equations is a retraction method the development of this method is discussed here in the case of system discrete equations corresponding definition of a point of a type of strict egress for a given set with respect to the system of discrete equations u k f k u k is given in an example it is shown that this definition cannot exactly copy the analogous one known from the theory of ordinary differential equations conditions for existence of at least one solution with graph remaining in a given set are formulated the proof is based on idea of a retract principle in construction of a retract mapping the property of continuous dependence on their initial data is used illustrative applications are considered too

oscillations of delay difference equations consider the delay difference equation n 1 n pn n k 0 n 0 1 2 where pn is a sequence of real numbers and k is a positive integer new oscillation criteria for this equation are established an extension of the results to difference equations with several delays is also presented

existence of global solutions of delayed differential equations via retract approach abstract the purpose of this contribution is to give sufficient conditions for the existence of global solutions or left semi global solutions for some classes of delayed functional differential equations the topological approach known as the topological retract principle is used inequalities for coordinates of global solutions are derived as a consequence of used method examples illustrate the results

identification of non linear systems using empirical data and prior knowledge an optimization approach abstract the choice of a parametric model structure in empirical and semi empirical non linear modeling is usually viewed as an important and critical step however it is known that by augmenting the least squares identification criterion with a term that imposes a penalty on the non smoothness of the model an optimal non parametric model can be found explicitly the optimal non parametric model will depend on the particular form of the penalty which can be looked upon as a priori knowledge or the desired properties of the model in this paper these results are extended in several directions i we show how useful types of prior knowledge other than smoothness can be included as a term in the criterion or as a constraint and how this influences the optimal model ii dynamic models and a general prediction error penalty are considered iii we present a practical numerical procedure for the identification of a close to optimal semi parametric model the numerical approach is motivated by the difficulty of deriving the optimal non parametric model if there are complicated constraints or penalty terms in the criterion and finally iv we discuss determination of the appropriate model complexity through selecting weights on the different penalties on the basis of empirical data since the optimal non parametric model is identified using an augmented optimization criterion and all prior knowledge and desired model properties may be specified through the augmented optimization criterion it is the choice of this criterion that is critical in this approach this elevates the empirical and semi empirical modeling problems to a more transparent and engineering friendly level than is achieved by directly specifying a parametric model structure hence our results shed some more light on modern non linear empirical modeling approaches like radial basis functions splines and neural networks in addition to providing a fundamental insight into the role of different kinds of prior knowledge this high level formulation is also attractive from a practical point of view as it lies closer to engineering thinking and less guesswork is required the flexibility and power of this approach are illustrated with a semi realistic simulation example

exponential solutions of equation ẏ t β t y t δ y t τ abstract asymptotic behaviour of solutions of first order differential equation with two deviating arguments of the form y t t y t y t is discussed for t a criterion for representing solutions in exponential form is proved as consequences inequalities for such solutions are given connections with known results are discussed and a sufficient condition for existence of unbounded solutions generalizing previous ones is derived an illustrative example is considered too

preserving positivity in solutions of discretised stochastic differential equations we consider the euler discretisation of a scalar linear test equation with positive solutions and show for both strong and weak approximations that the probability of positivity over any finite interval of simulation tends to unity as the step size approaches zero although a s positivity in an approximation is impossible to achieve we develop for the strong maruyama approximation an asymptotic estimate of the number of mesh points required for positivity as our tolerance of non positive trajectories tends to zero and examine the effectiveness of this estimate in the context of practical numerical simulation we show how this analysis generalises to equations with a drift coefficient that may display a high level of nonlinearity but which must be linearly bounded from below i e when acting towards zero and a linearly bounded diffusion coefficient finally in the linear case we develop a refined asymptotic estimate that is more useful as an a priori guide to the number of mesh points required to produce positive approximations with a given probability

stability and asymptotic behaviour of a two dimensional differential system with delay stability and asymptotic behaviour of a real two dimensional system with delay are studied the method of investigation is based on the transformation of the considered real system to one equation with complex valued coefficients stability and asymptotic propertis of this equation are studied by means of a suitable lyapunov krasovskii functional

a necessary and sufficient condition for the oscillation of first order linear difference equations with several delay arguments abstract in this article we discuss the oscillation of all solutions of a first order linear delay difference equation with several delay arguments and obtain some new oscillation criteria under more relaxed conditions in particular we establish a necessary and sufficient condition only based on the coefficient functions and delay arguments when the coefficient functions are non negative examples and tables are given to illustrate our results

partially observed control of markov processes iv abstract a class of degenerate elliptic nonlinear second order equations is considered these equations arising from a related class of probabilistic variational problems are defined on the space of probability measures on rd the existence of c1 1 viscosity solutions is established as well as the existence uniqueness and feedback characterization of the minima of the variational problems

asymptotic behavior results for nonlinear neutral delay difference equations abstract this paper is concerned with the nonlinear neutral delay difference equation x n c n x n m p n f x n k 0 n n n 0 where is the forward difference operator defined by x n x n 1 x n c n is a sequence of real numbers p n is a positive sequence f c r r m and k are positive integers n 0 is a nonnegative integer and n n 0 n 0 n 0 1 n 0 2 sufficient conditions are obtained under which every solution of equation is bounded and tends to a constant as n our results improve and extend some known results one example is given to illustrate our results

construction of positivity preserving numerical method for jumpdiffusion option pricing models using the euler scheme to simulate the stochastic differential equations sdes models in finance often gives rise to the problem that the exact solution is positive while the numerical solution is not recently we find that this problem existed in the jumpdiffusion models as well hence this paper aims to construct a numerical method preserving positivity for jumpdiffusion option pricing models we generalize the balanced implicit method bim to the jumpdiffusion models which already turned out to be efficient for preserving positivity in sde models then the positivity of bim for jumpdiffusion models is proved under some conditions finally a numerical example is simulated to verify the positivity and efficiency of the proposed method

viscosity solutions of fully nonlinear second order equations and optimal stochastic control in infinite dimensions part i the case of bounded stochastic evolutions on etudie des equations elliptiques degenerees d ordre 2 non lineaires de la forme f d 2 u du u x 0 dans h ou h est un espace de hilbert separable x un point generique dans h u une fonction de h r

positive and oscillating solutions of differential equations with delay in critical case abstract this article is devoted to the problem of existence of positive solution for common classes of nonlinear retarded functional differential equations some criteria of its existence are proved as well as some comparison results the obtained results are applied to the linear case moreover the existence of positive and oscillating solutions of a differential equation with delay x t a t x t in the critical case is considered some comparisons with known results are given

anti lyapunov method for systems of discrete equations abstract in the present paper problems concerning the qualitative behaviour of solutions of discrete equations are considered a general principle which gives a guarantee that the graph of at least one solution stays in a prescribed domain is formulated as a consequence the auxiliary tool namely a method of regular polyfacial sets is developed and an application of the general principle is given this application is based with exceptions on the fact that points of the boundary of a domain considered are strict egress points such a situation is often in the theory of ordinary differential equations termed the anti lyapunov case illustrative applications and examples are given and open problems are formulated

stochastic stabilization and destabilization it is shown in this paper that any nonlinear systems dot x t f x t t in rd can be stabilized by brownian motion provided f hook x t k x for some k 0 on the other hand this system can also be destabilized by brownian motion if the dimension d 2 similar results are also obtained for any given stochastic differential equation dx t f hook x t t g x t t dw t

adaptive timestepping for pathwise stability and positivity of strongly discretised nonlinear stochastic differential equations abstract we consider the use of adaptive timestepping to allow a strong explicit euler maruyama discretisation to reproduce dynamical properties of a class of nonlinear stochastic differential equations with a unique equilibrium solution and non negative non globally lipschitz coefficients solutions of such equations may display a tendency towards explosive growth countered by a sufficiently intense and nonlinear diffusion we construct an adaptive timestepping strategy which closely reproduces the almost sure a s asymptotic stability and instability of the equilibrium and which can ensure the positivity of solutions with arbitrarily high probability our analysis adapts the derivation of a discrete form of the ito formula from appleby et al 2009 in order to deal with the lack of independence of the wiener increments introduced by the adaptivity of the mesh we also use results on the convergence of certain martingales and semi martingales which influence the construction of our adaptive timestepping scheme in a way proposed by liu mao 2017

stability of finite delay difference systems the stability of finite delay difference systems has recently been discussed in l in this paper we consider the delay difference systems with infinite delay of the following form x n 1 g n x s s 1 1 1 n g n x 1 where 1 is an integer or w we assume g n 0 0 forn l l l so that 1 has the zero solution x n 0 clearly for any given integer n l 1 and a given bounded initial function 4 l 1 1 no rk there is a unique solution of l denoted by x n n 4 such that it satisfies 1 for all integers n 2 n and x n no 4 444 forn f l l no for any integer m 2 co let 2 m m 1 i and 2 z the integer set if m a for any n e z define ii411 tlnl sup l s 15 s 5 n and for any h 0 let c n 4 z rk i nl h in the sequel we always assume that the variables n r and s take integer values and the corresponding intervals are discrete intervals

asymptotic formulae for a particular solution of linear nonhomogeneous discrete equations abstract the aim of this contribution is to develop the asymptotic formulae which describe the asymptotic behaviour of a particular solution of linear nonhomogeneous differential equation u k a k u k g k if k it is shown that asymptotic formulae are generated by partial sums of certain formal series that satisfy the given equation a comparison with the known explicit formula for solution of the initial problem is given and moreover some illustrative examples are discussed the used method of investigation is based on results published in 1

oscillation analysis of advertising capital model analytical and numerical studies abstract this paper mainly deals with the oscillation of nonlinear delay differential equation which is used to describe advertising capital model analytically and numerically firstly the condition of oscillation of the analytic solution is presented by the technique of the theory of characteristic secondly the asymptotic behavior of non oscillatory analytic solution is verified thirdly the methods are applied to the mentioned equation some conditions under which the numerical solution oscillates are obtained moreover it is proved that every non oscillatory numerical solution tends to the steady state of the model finally some numerical simulations for verifying the theoretical findings are provided

infinite dimensional hamilton jacobi equations with large zeroth order coefficient abstract we establish the existence and uniqueness of the c 1 1 solution of an infinite dimensional hamilton jacobi equation defined on the space of probability measures on r d when the zeroth order coefficient is sufficiently large

positive solutions of retarded functional differential equations abstract for systems of retarded functional differential equations with unbounded delay and with finite memory sufficient conditions of existence of positive solutions on an interval of the form t 0 are derived

instrumental variable methods for identification of hammerstein systems abstract consistency conditions for instrumental variable methods applied to hammerstein models are derived and analysed it is necessary to have a model that is not over parametrized and an input that is strongly persistently exciting i e the inputs and their powers are jointly persistently exciting some specific choices of the instruments are proved to give consistency those instruments are formed as filtered inputs and powers of the input consistency can be guaranteed if either the input is white noise or if a certain transfer function is positive real the vector of instruments is in general of larger dimension than the parameter vector it is shown then that a non singular linear transformation of the instruments gives a new and still consistent estimate some simulations using the iv variants proposed in this paper are also included

l2 perturbation of a linear delay differential equation abstract in this paper we present an asymptotic formula for the solutions of the nonautonomous delay differential equation t a 0 a t x t b 0 b t x t assuming that it is an l 2 perturbation of the autonomous equation t a 0 x t b 0 x t our theorem is an essential generalization of the corresponding result in the classical book by r bellman and k l cooke the proof is based on a new result on the asymptotic constancy of the solutions of a linear delay equation

explicit criteria for the existence of positive solutions for a scalar differential equation with variable delay in the critical case a scalar linear differential equation with time dependent delay x t a t x t t t is considered where t i t 0 t 0 r a i r 0 is a continuous function and t i r is a continuous function such that t t t t 0 t t 0 if t t 0 the goal of our investigation is to give sufficient conditions for the existence of positive solutions as t in the critical case in terms of inequalities on a and t a generalization of one known final in a certain sense result is given for the case of t being not a constant analysing this generalization we show e g that it differs from the original statement with a constant delay since it does not give the best possible result this is demonstrated on a suitable example

on oscillation of differential and difference equations with non monotone delays abstract for the delay differential equation x t p t x h t 0 p t 0 h t t t 0 lim t h t we demonstrate that the inequality limsup t h t t p u d u 1 is not sufficient for oscillation moreover for any a 0 the relation limsup t h t t p u d u a generally does not imply oscillation a similar result is obtained for difference equations in terms of the maximal argument new sufficient oscillation conditions are presented for differential and difference equations

sufficient conditions for the existence of global solutions of delayed differential equations the purpose of this contribution is to give sufficient conditions for the existence of global solutions or left semi global solutions for some classes of delayed functional differential equations the principle of monotone sequences is used as a main tool of investigation inequalities for coordinates of global solutions are derived as a consequence of used method examples illustrate the results

asymptotic formulae for the solutions of a linear delay difference equation consider the linear delay difference equation xn 1 xn anxnk where an s are real numbers and k is a positive integer in this paper we establish asymptotic formulae for the solutions of the above equation assuming n 0 an p for some p 1 the asymptotic formulae are given by an appropriate recursive sequence for p 1 2 3 4 we present explicit formulae we also construct an example which shows that some comparison results in a recent paper of yan and qian are incorrect

gaussian approximation in recursive estimation of multiple state of nonlinear wiener systems abstract the paper concerns recursive state estimation in wiener systems where the output is a scalar weighted sum of multiple states a gaussian approximation to the computed conditional probability distribution for the scalar sum of states is shown to lead to a practicable numerical algorithm for estimating the multiple states this algorithm mimics the kalman filter simulations show the method giving good performance when the nonlinearity is severe quantization

the asymptotic bounds of solutions of linear delay systems abstract our aim is to obtain conditions under which every solutionyof the vector functional differential equation formula can be estimated in terms of a solution of a scalar functional equation formula wherea x is a suitable real positive function andbis a suitable real negative constant moreover ifb x is diagonal matrix we derive more precise estimates ofyin terms of a solution of a vector functional equation formula where the elements ofa x are formed by the absolute values of the corresponding elements ofa x these asymptotic results enable us to obtain sharp estimates of solutions of certain linear delay systems

strong convergence of euler type methods for nonlinear stochastic differential equations traditional finite time convergence theory for numerical methods applied to stochastic differential equations sdes requires a global lipschitz assumption on the drift and diffusion coefficients in practice many important sde models satisfy only a local lipschitz property and since brownian paths can make arbitrarily large excursions the global lipschitz based theory is not directly relevant in this work we prove strong convergence results under less restrictive conditions first we give a convergence result for euler maruyama requiring only that the sde is locally lipschitz and that the pth moments of the exact and numerical solution are bounded for some p 2 as an application of this general theory we show that an implicit variant of euler maruyama converges if the diffusion coefficient is globally lipschitz but the drift coefficient satisfies only a one sided lipschitz condition this is achieved by showing that the implicit method has bounded moments and may be viewed as an euler maruyama approximation to a perturbed sde of the same form second we show that the optimal rate of convergence can be recovered if the drift coefficient is also assumed to behave like a polynomial

stabilization and destabilization of nonlinear differential equations by noise this paper considers the stabilization and destabilization by a brownian noise perturbation that preserves the equilibrium of the ordinary differential equation x t f x t in an extension of earlier work we lift the restriction that f obeys a global linear bound and show that when f is locally lipschitz a function g can always be found so that the noise perturbation g x t db t either stabilizes an unstable equilibrium or destabilizes a stable equilibrium when the equilibrium of the deterministic equation is nonhyperbolic we show that a nonhyperbolic perturbation suffices to change the stability properties of the solution

unified set membership theory for identification prediction and filtering of nonlinear systems the problem of making inferences from data measured on nonlinear systems is investigated within a set membership sm framework and it is shown that identification prediction and filtering can be treated as specific instances of the general presented theory the sm framework presents an alternative view to the parametric statistical ps framework more widely used for studying the above specific problems in particular in the sm framework a bound only on the gradient of the model regression function is assumed at difference from ps methods which assume the choice of a parametric functional form of the regression function moreover the sm theory assumes only that the noise is bounded in contrast with ps approaches which rely on noise assumptions such as stationarity uncorrelation type of distribution etc the basic notions and results of the general inference making theory are presented moreover some of the main results that can be obtained for the specific inferences of identification prediction and filtering are reviewed concluding comments on the presented results are also reported focused on the discussion of two basic questions what may be gained in identification prediction and filtering of nonlinear systems by using the presented sm framework instead of the widely diffused ps framework why sm methods could provide stronger results than the ps methods requiring weaker assumptions on system and on noise

explicit integral criteria for the existence of positive solutions of the linear delayed equation x t c t x t τ t abstract the paper analyzes the linear differential equation with single delay x t c t x t t with continuous t 0 0 r r 0 t 0 r and c t 0 r 0 new explicit integral criteria for the existence of a positive solution expressed in terms of c and are derived an overview of known relevant criteria is provided and relevant comparisons are also given it is demonstrated that the known criteria are consequences of the new results

the truncated euler maruyama method for stochastic differential equations influenced by higham et al 2003 several numerical methods have been developed to study the strong convergence of the numerical solutions to stochastic differential equations sdes under the local lipschitz condition these numerical methods include the tamed euler maruyama em method the tamed milstein method the stopped em the backward em the backward forward em etc in this paper we will develop a new explicit method called the truncated em method for the nonlinear sde d x t f x t d t g x t d b t and establish the strong convergence theory under the local lipschitz condition plus the khasminskii type condition x t f x p 1 2 g x 2 k 1 x 2 the type of convergence specifically addressed in this paper is strong l q convergence for 2 q p and p is a parameter in the khasminskii type condition

a change of variables in the asymptotic theory of differential equations with unbounded delays in this paper the author investigates the asymptotic properties of solutions of the nonhomogeneous linear differential equation x t ax t bx t f t with nonzero real scalars a b and the unbounded lag using the change of the independent and dependent variable he relates the asymptotic behaviour of solutions of this equation to the asymptotic behaviour of solutions of auxiliary functional nondifferential equations

remark on positive solutions of discrete equation δu k n p k u k abstract a delayed discrete equation u k n p k u k with positive function p k is considered sufficient conditions are formulated to guarantee the existence of a positive solution if k

asymptotic variance expressions for identified black box transfer function models identification of black box transfer function models is considered it is assumed that the transfer function models possess a certain shift property which is satisfied for example by all polynomial type models expressions for the variances of the transfer function estimates are derived that are asymptotic both in the number of observed data and in the model orders the result is that the joint covariance matrix of the transfer functions from input to output and from driving white noise source to the additive output disturbance respectively is proportional to the inverse of the joint spectrum matrix for the input and driving noise multiplied by the spectrum of the additive output noise the factor of proportionality is the ratio of model order to number of data this result is independent of the particular model structure used the result is applied to evaluate the performance degradation due to variance for a number of typical model uses some consequences for input design are also drawn

positive solutions of the equation ẋ t c t x t τ in the critical case abstract the equation t c t x t is considered in the critical case for it the asymptotic behavior of dominant and subdominant solutions is studied a generalization is made and connections with known results are discussed

oscillation of first order delay equations we study the oscillation of first order delay equations using a method that parallels the use of riccati equations in the study of the oscillation of second order ordinary differential equations without delay we give an alternative proof of a comparison theorem first established by kwong and patula obtain some asymptotic behaviors of nonoscillatory solutions prove a new criterion that falls within a gap left open by established results and confirm a conjecture raised by hunt and yorke

nonlinear black box modeling in system identification a unified overview a nonlinear black box structure for a dynamical system is a model structure that is prepared to describe virtually any nonlinear dynamics there has been considerable recent interest in this area with structures based on neural networks radial basis networks wavelet networks and hinging hyperplanes as well as wavelet transform based methods and models based on fuzzy sets and fuzzy rules this paper describes all these approaches in a common framework from a user s perspective it focuses on what are the common features in the different approaches the choices that have to be made and what considerations are relevant for a successful system identification application of these techniques it is pointed out that the nonlinear structures can be seen as a concatenation of a mapping form observed data to a regression vector and a nonlinear mapping from the regressor space to the output space these mappings are discussed separately the latter mapping is usually formed as a basis function expansion the basis functions are typically formed from one simple scalar function which is modified in terms of scale and location the expansion from the scalar argument to the regressor space is achieved by a radial or a ridge type approach basic techniques for estimating the parameters in the structures are criterion minimization as well as two step procedures where first the relevant basis functions are determined using data and then a linear least squares step to determine the coordinates of the function approximation a particular problem is to deal with the large number of potentially necessary parameters this is handled by making the number of used parameters considerably less than the number of offered parameters by regularization shrinking pruning or regressor selection

boundedness of infinite delay difference systems x n 1 g n x s s i 1 1 n g n x 1 where i is an integer or co g is a volterra functional determined by the integer n and the values of x s for s i 1 1 n and taking values in rk the stability results of 1 have been recently established in 11 however in order to explore the existence of periodic solutions of 1 we need to know not only the stability but also the boundedness of solutions for the linear case of 1 for the relevant results we refer to 2 while for the corresponding results of infinite delay differential systems see 3 41 in this paper we are going to establish theorems on boundedness for 1 in terms of both liapunov functionals and functions with the so called razumikhin techniques before doing this we give the following notations and definitions for any integer m 2 00 let z m m 1 and z z the integer set if m a trivially for any ie e z and a given bounded initial function 4 i i 1 no rk there is a unique solution of l denoted by x n n d such that it satisfies 1 for all integers 17 2 n and x n n 4 d n forn l l l n