multiperiodicity and exponential attractivity evoked by periodic external inputs in delayed cellular neural networks we show that an n neuron cellular neural network with time varying delay can have 2n periodic orbits located in saturation regions and these periodic orbits are locally exponentially attractive in addition we give some conditions for ascertaining periodic orbits to be locally or globally exponentially attractive and allow them to locate in any designated region as a special case of exponential periodicity exponential stability of delayed cellular neural networks is also characterized these conditions improve and extend the existing results in the literature to illustrate and compare the results simulation results are discussed in three numerical examples

periodic solutions of integro differential equations in vector valued function spaces abstract operator valued fourier multipliers are used to study well posedness of integro differential equations in banach spaces both strong and mild periodic solutions are considered strong well posedness corresponds to maximal regularity which has proved very efficient in the handling of nonlinear problems we are concerned with a large array of vector valued function spaces lebesgue bochner spaces l p the besov spaces b p q s and related spaces such as the holder zygmund spaces c s and the triebel lizorkin spaces f p q s we note that the multiplier results in these last two scales of spaces involve only boundedness conditions on the resolvents and are therefore applicable to arbitrary banach spaces the results are applied to various classes of nonlinear integral and integro differential equations

existence learning and replication of periodic motions in recurrent neural networks a class of recurrent neural networks is shown to possess a stable limit cycle a gradient type algorithm is used to modify the parameters of the network so that it learns and replicates autonomously a time varying periodic signal the results are applied to controlling the repetitive motion of a two link robot manipulator

a note on convergence under dynamical thresholds with delays we complement the study of the asymptotic behaviour of the dynamical threshold neuron model with delay introduced by gopalsamy and leung 1997 by providing a description of the dynamics of the system in the remaining parameters range we characterize the regions of harmless delays and those in which delay induced oscillations appear

on uniqueness for the navier stokes system in 3d bounded lipschitz domains in this paper we prove uniqueness of solutions of the navier stokes system in cb 0 t l3 3 l 0 t l3 2 where is a bounded lipschitz domain in r3

global exponential periodicity of a class of recurrent neural networks with oscillating parameters and time varying delays in this paper we present the analytical results on the global exponential periodicity of a class of recurrent neural networks with oscillating parameters and time varying delays sufficient conditions are derived for ascertaining the existence uniqueness and global exponential periodicity of the oscillatory solution of such recurrent neural networks by using the comparison principle and mixed monotone operator method the periodicity results extend or improve existing stability results for the class of recurrent neural networks with and without time delays

quasilinear evolutionary equations and continuous interpolation spaces in this paper we analyze the abstract parabolic evolutionary equations dt u x a u u f u h t u 0 x in continuous interpolation spaces allowing a singularity as t 0 here dt denotes the time derivative of order 0 2 we first give a treatment of fractional derivatives in the spaces lp 0 t x and then consider these derivatives in spaces of continuous functions having at most a prescribed singularity as t 0 the corresponding trace spaces are characterized and the dependence on is demonstrated via maximal regularity results on the linear equation dt u x au f u 0 x we arrive at results on existence uniqueness and continuation on the quasilinear equation finally an example is presented

maximal regularity of second order equations with delay abstract we characterize existence and uniqueness of solutions of an inhomogeneous abstract delay equation in holder spaces the method is based on the theory of operator valued fourier multipliers

regularity theory for a nonlinear fractional reaction diffusion equation abstract this paper is dedicated to the study of a nonlinear fractional reaction diffusion equation we analyze the behavior of the resolvent family associated with the problem in the scale of fractional power spaces associated with the laplace operator we ensure the existence and uniqueness of regular mild solutions to the problem in the l q setting furthermore we consider global existence or non continuation by a blow up of such solutions

pointwise growth and uniqueness of positive solutions for a class of sublinear elliptic problems where bifurcation from infinity occurs in this paper we analyze the uniqueness and the pointwise growth of the positive solutions of a nonlinear elliptic boundary value problem of general sublinear type with a weight function multiplying the nonlinearity when this function vanishes on some subdomain the problem exhibits a bifurcation from infinity in this case almost nothing is known about the pointwise growth of the positive solutions as the parameter approaches the critical value where the bifurcation from infinity occurs in this work we show that the positive solutions grow to infinity in the region where the weight function vanishes and that on its support they stabilize to the minimal positive solution of the original equation subject to infinite dirichlet boundary conditions this behavior provides us with the uniqueness of the positive solution near the value of the parameter where the bifurcation from infinity occurs also we solve the problem using spectral collocation methods coupled with path following techniques to show how the main uniqueness result is optimal throughout the paper the mathematical analysis aids the numerical study and the numerical study confirms and illuminates the analysis

lmi approach for stationary oscillation of interval neural networks with discrete and distributed time varying delays under impulsive perturbations in this paper a class of impulsive interval neural networks with discrete and distributed time varying delays is discussed several new sufficient conditions are obtained ensuring the existence uniqueness and global exponential stability of periodic solution i e stationary oscillation for the addressed models based on inequality analysis techniques the obtained results can be checked easily by the linear matrix inequality control toolbox in matlab finally two numerical examples are given to show the effectiveness of our results

the poincare bendixson theorem for monotone cyclic feedback systems with delay abstract we consider cyclic nearest neighbor systems of differential delay equations in which the coupling between neighbors possesses a monotonicity property using a discrete integer valued lyapunov function we prove that the poincare bendixson theorem holds for such systems we also obtain results on piecewise monotonicity and stability of periodic solutions of such systems

multiplicity results of periodic solutions for a class of second order delay differential systems abstract this paper is concerned with the periodic boundary value problem 1 u t u t f t u t a e t 0 2 u 0 u 2 u 0 u 2 0 where 0 is a given constant r is a parameter the variational principle is given and some multiplicity results of periodic solutions of 1 are obtained directly via variational methods

compact kernel sections for lattice systems with delays abstract in this paper we consider the existence upper semi continuity and singular limiting behavior of compact kernel sections for the process associated to non autonomous lattice systems with delays

oscillations in i o monotone systems under negative feedback oscillatory behavior is a key property of many biological systems the small gain theorem sgt for input output monotone systems provides a sufficient condition for global asymptotic stability of an equilibrium and hence its violation is a necessary condition for the existence of periodic solutions one advantage of the use of the monotone sgt technique is its robustness with respect to all perturbations that preserve monotonicity and stability properties of a very low dimensional in many interesting examples just one dimensional model reduction this robustness makes the technique useful in the analysis of molecular biological models in which there is large uncertainty regarding the values of kinetic and other parameters however verifying the conditions needed in order to apply the sgt is not always easy this paper provides an approach to the verification of the needed properties and illustrates the approach through an application to a classical model of circadian oscillations as a nontrivial case study and provides a theorem in the converse direction of predicting oscillations when the sgt conditions fail

on some necessary and sufficient conditions for a recurrent neural network model with time delays to generate oscillations in this paper the existence of oscillations for a class of recurrent neural networks with time delays between neural interconnections is investigated by using the fixed point theory and liapunov functional we prove that a recurrent neural network might have a unique equilibrium point which is unstable this particular type of instability combined with the boundedness of the solutions of the system will force the network to generate a permanent oscillation some necessary and sufficient conditions for these oscillations are obtained simple and practical criteria for fixing the range of parameters in this network are also derived typical simulation examples are presented

effect of delay on the boundary of the basin of attraction in a self excited single graded response neuron little attention has been paid in the past to the effects of interunit transmission delays representing a xonal and synaptic delays ontheboundary of the basin of attraction of stable equilibrium points in neural networks as a first step toward a better understanding of the influence of delay we study the dynamics of a single graded response neuron with a delayed excitatory self connection the behavior of this system is representative of that of a family of networks composed of graded response neurons in which most trajectories converge to stable equilibrium points for any delay value it is shown that changing the delay modifies the location of the boundary of the basin of attraction of the stable equilibrium points without affecting the stability of the equilibria the dynamics of trajectories on the boundary are also delay dependent and influence the transient regime of trajectories within the adjacent basins our results suggest that when dealing with networks with delay it is important to stud

domain decomposition method for the global dynamics of delay differential equations with unimodal feedback x t zkmx t c f x tkt with unimodal feedback is studied the existence of the global attractor is shown and bounds of the attractor are given we find attractive invariant intervals and give sufficient conditions that guarantee that all solutions enter the domain where f 0 is negative with respect to a positive equilibrium so the results for delayed monotone feedback can be applied to describe the asymptotic behaviour of solutions in particular the existence of heteroclinic orbits from the trivial equilibrium to a periodic orbit oscillating around the positive equilibrium is established numerical examples using nicholson s blowflies equation and the mackey glass equation are provided to illustrate the main results

estimating the ultimate bound and positively invariant set for a class of hopfield networks in this paper we investigate the ultimate bound and positively invariant set for a class of hopfield neural networks hnns based on the lyapunov stability criterion and lagrange multiplier method it is shown that a hyperelliptic estimate of the ultimate bound and positively invariant set for the hnns can be calculated by solving a linear matrix inequality lmi furthermore the global stability of the unique equilibrium and the instability region of the hnns are analyzed respectively finally the most accurate estimate of the ultimate bound and positively invariant set can be derived by solving the corresponding optimization problems involving the lmi constraints some numerical examples are given to illustrate the effectiveness of the proposed results

nontrivial periodic solutions for delay differential systems via morse theory abstract the existence of the nontrivial periodic solutions to the system of delay differential equations 1 1 x t f x t is investigated where x r n f c r n r n we prove several new existence results via morse theory and the galerkin methods

on maximal regularity for the cauchy dirichlet parabolic problem with fractional time derivative abstract we prove two maximal regularity results in spaces of continuous and holder continuous functions for a linear cauchy dirichlet problem with a fractional time derivative d t this derivative is intended in the sense of caputo and is taken in 0 2 in case 1 we obtain maximal regularity results for parabolic problems already known in mathematical literature

elliptic eigenvalue problems and unbounded continua of positive solutions of a semilinear elliptic equation abstract we derive a result on the limit of certain sequences of principal eigenvalues associated with some elliptic eigenvalue problems this result is then used to give a complete description of the global structure of the curves of positive steady states of a parameter dependent diffusive version of the classical logistic equation in particular we characterize the bifurcation values from infinity to positive steady states the stability of the positive steady states as well as the asymptotic behaviour of positive solutions is also discussed

slowly oscillating wave solutions of a single species reaction diffusion equation with delay we study positive bounded wave solutions u t x x ct 0 of equation ut t x u t x u t x g u t h x x rm this equation is assumed to have two non negative equilibria u1 0 and u2 0 the birth function g c r r is unimodal and differentiable at 0 and some results also require the feedback condition g s s c our results are based on a detailed geometric description of the wave profile in particular the monotonicity of its leading edge is established we also discuss the uniqueness problem indicating a subclass g of asymmetric tent maps such that given g g there exists exactly one positive travelling front for each fixed admissible speed

connecting orbits from synchronous periodic solutions to phase locked periodic solutions in a delay differential system we consider a system of delay differential equations modelling the excitatory interaction of two identical neurons assuming the delay is sufficiently large we show that the closure of the forward extension w5 of a 5 dimensional leading unstable manifold of the trivial solution contains a phase locked periodic orbit and a synchronized periodic orbit and we classify the dynamics of the semiflow restricted to w5 we also obtain the precise information about the floquet multipliers of the synchronized periodic orbit which enables us to establish the existence of heteroclinic orbits from the synchronized periodic orbit to the phase locked orbit

multiplicity results of periodic solutions for systems of first order delay differential equation abstract this paper is concerned with the periodic solutions of the following delay non autonomous systems 1 u t f t u t r where r 0 f c r 1 r n r n satisfies f t r z f t z for all z r n some multiplicity results of periodic solutions of 1 are obtained via variational methods

a sufficient condition for additive d stability and application to reaction diffusion models abstract the matrix a is said to be additively d stable if a d remains hurwitz for all non negative diagonal matrices d in reaction diffusion models additive d stability of the matrix describing the reaction dynamics guarantees the stability of the homogeneous steady state thus ruling out the possibility of diffusion driven instabilities we present a new criterion for additive d stability using the concept of compound matrices we first give conditions under which the second additive compound matrix has non negative off diagonal entries we then use this metzler property of the compound matrix to prove additive d stability with the help of an additional determinant condition this result is then applied to investigate the stability of cyclic reaction networks in the presence of diffusion finally a reaction network structure that fails to achieve additive d stability is exhibited

pullback attractors of a multi valued process generated by parabolic differential equations with unbounded delays abstract the theory of nonautonomous multi valued dynamical systems and a method of asymptotic compactness based on the concept of the kuratowski measure of the noncompactness of a bounded set are used to prove the existence of pullback attractors in the weighted space c v for the multi valued process associated with the nonautonomous parabolic differential equation with infinite delay for which the uniqueness of solutions need not hold

monostability and multistability for almost periodic solutions of fractional order neural networks with unsaturating piecewise linear activation functions since the unsaturating activation function is unbounded more complex dynamics may exist in neural networks with this kind of activation function in this article monostability and multistability results of almost periodic solutions are developed for fractional order neural networks with unsaturating piecewise linear activation functions some globally mittag leffler attractive sets are given and the existence of globally mittag leffler stable almost periodic solution is demonstrated by using ascoli arzela theorem in particular some sufficient conditions are provided to ascertain the multistability of almost periodic solutions based on locally positively invariant set it shows that there exists an almost periodic solution in each positively invariant set and all trajectories converge to this periodic trajectory in that rectangular area two illustrative examples are provided to demonstrate the effectiveness of the proposed sufficient criteria

undamped oscillations generated by hopf bifurcations in fractional order recurrent neural networks with caputo derivative in this paper a fractional order recurrent neural network is proposed and several topics related to the dynamics of such a network are investigated such as the stability hopf bifurcations and undamped oscillations the stability domain of the trivial steady state is completely characterized with respect to network parameters and orders of the commensurate order neural network based on the stability analysis the critical values of the fractional order are identified where hopf bifurcations occur and a family of oscillations bifurcate from the trivial steady state then the parametric range of undamped oscillations is also estimated and the frequency and amplitude of oscillations are determined analytically and numerically for such commensurate order networks meanwhile it is shown that the incommensurate order neural network can also exhibit a hopf bifurcation as the network parameter passes through a critical value which can be determined exactly the frequency and amplitude of bifurcated oscillations are determined

traveling waves in the nonlocal kpp fisher equation different roles of the right and the left interactions abstract we consider the nonlocal kpp fisher equation u t t x u x x t x u t x 1 k u t x which describes the evolution of population density u t x with respect to time t and location x the non locality is expressed in terms of the convolution of u t with kernel k 0 r k s d s 1 the restrictions k s s 0 and k s s 0 are responsible for interactions of an individual with his left and right neighbors respectively we show that these two parts of k play quite different roles as for the existence and uniqueness of traveling fronts to the kpp fisher equation in particular if the left interaction is dominant the uniqueness of fronts can be proved while the dominance of the right interaction can induce the co existence of monotone and oscillating fronts we also present a short proof of the existence of traveling waves without assuming various technical restrictions usually imposed on k

finite horizon h_ infty state estimation for periodic neural networks over fading channels the problem of finite horizon h infty state estimator design for periodic neural networks over multiple fading channels is studied in this paper to characterize the measurement signals transmitted through different channels experiencing channel fading a multiple fading channels model is considered for investigating the situation of correlated fading channels a set of correlated random variables is introduced specifically the channel coefficients are described by white noise processes and are assumed to be correlated two sufficient criteria are provided by utilizing a stochastic analysis approach to guarantee that the estimation error system is stochastically stable and achieves the prescribed h infty performance then the parameters of the estimator are derived by solving recursive linear matrix inequalities finally some simulation results are shown to illustrate the effectiveness of the proposed method

impulsive control for existence uniqueness and global stability of periodic solutions of recurrent neural networks with discrete and continuously distributed delays in this paper a class of recurrent neural networks with discrete and continuously distributed delays is considered sufficient conditions for the existence uniqueness and global exponential stability of a periodic solution are obtained by using contraction mapping theorem and stability theory on impulsive functional differential equations the proposed method which differs from the existing results in the literature shows that network models may admit a periodic solution which is globally exponentially stable via proper impulsive control strategies even if it is originally unstable or divergent two numerical examples and their computer simulations are offered to show the effectiveness of our new results

holder continuous solutions for fractional differential equations and maximal regularity abstract we characterize the existence and uniqueness of solutions of an abstract fractional differential equation with infinite delay in holder spaces

absence of small solutions and existence of morse decomposition for a cyclic system of delay differential equations abstract we consider the unidirectional cyclic system of delay differential equations x i t g i x i t x i 1 t i t 0 i n where the indexes are taken modulo n 1 with n n 0 i 0 i 0 n i 0 and for all 0 i n the feedback functions g i u v t are continuous in t r and c 1 in u v r 2 and each of them satisfies either a positive or a negative feedback condition in the delayed term we show that all components of a superexponential solution i e nonzero solutions that converge to zero faster than any exponential function must have infinitely many sign changes on any interval of length as a corollary we obtain that if a backwards bounded global pullback attractor exists then it does not contain any superexponential solutions in the autonomous case we also prove that the global attractor possesses a morse decomposition that is based on a discrete lyapunov function this generalizes former results by mallet paret 1988 28 and polner 2002 37 in which the scalar case was studied

instabilities in threshold diffusion equations with delay the introduction of delays into ordinary or partial differential equation models is well known to facilitate the production of rich dynamics ranging from periodic solutions through to spatio temporal chaos in this paper we consider a class of scalar partial differential equations with a delayed threshold nonlinearity which admits exact solutions for equilibria periodic orbits and travelling waves importantly we show how the spectra of periodic and travelling wave solutions can be determined in terms of the zeros of a complex analytic function using this as a computational tool to determine stability we show that delays can have very different effects on threshold systems with negative as opposed to positive feedback direct numerical simulations are used to confirm our bifurcation analysis and to probe some of the rich behaviour possible for mixed feedback

multiplicity results for periodic solutions to delay differential equations via critical point theory abstract by the critical point theory we study the existence and multiplicity of periodic solutions to the following system of delay differential equations x t f x t r where x r n f c r n r n and r 0 is a given constant a sufficient condition on the existence and multiplicity of periodic solutions for eq is obtained to the authors knowledge this is the first time the existence of periodic solutions to systems of delay differential equations is dealt with by using variational approaches directly

the poincare bendixson theorem for a class of delay equations with state dependent delay and monotonic feedback abstract we consider the real valued differential equation x t f x t x t d x t with state dependent delay where f is strictly monotonic in its second argument we describe a class of such equations for which a version of the poincare bendixson theorem holds

a general family of morphed nonlinear phase oscillators with arbitrary limit cycle shape we present a general family of nonlinear phase oscillators which can exhibit arbitrary limit cycle shapes and infinitely large basins of attraction this general family is the superset of familiar control methods like pd control over a periodic reference and rhythmic dynamical movement primitives the general methodology is based on morphing the limit cycle of an existing phase oscillator with phase based scaling functions to obtain a desired limit cycle behavior the introduced methodology can be represented as first second or n th order dynamical systems the elegance of the formulation provides the possibility to define explicit arbitrary convergence behavior for simple cases we analyze the stability properties of the methodology with the poincare bendixson theorem and the contraction theory and use numerical simulations to show the properties of some oscillators that are a subset of this general family c 2013 elsevier b v all rights reserved

operator valued fourier multiplier theorems and maximal l_p regularity we prove a mihlin type multiplier theorem for operator valued multiplier functions on umd spaces the essential assumption is r boundedness of the multiplier function as an application we give a characterization of maximal l p regularity for the generator of an analytic semigroup t t in terms of the r boundedness of the resolvent of a or the semigroup t t

configurations of periodic orbits for equations with delayed positive feedback abstract we consider scalar delay differential equations of the form x t x t f x t 1 where 0 and f is a nondecreasing c 1 function if is a fixed point of f r u f u r with f 1 then 1 0 s r is an unstable equilibrium a periodic solution is said to have large amplitude if it oscillates about at least two fixed points of f with f 1 and f 1 we investigate what type of large amplitude periodic solutions may exist at the same time when the number of such fixed points and hence the number of unstable equilibria is an arbitrary integer n 2 it is shown that the number of different configurations equals the number of ways in which n symbols can be parenthesized the location of the floquet multipliers of the corresponding periodic orbits is also discussed

mild solutions to integro differential equations in banach spaces abstract in this paper we study the existence and uniqueness of mild solutions to integro differential equations in terms of a resolvent operator on the interval 0 2 and on the real line moreover we characterize the spectrum of the resolvent family that solves the volterra equation u a u a a u f in terms of their mild periodic solutions

asymptotic constancy and periodicity for a single neuron model with delay abstract we study the single neuron model equation x t d x t a f x t b f x t 1 i t 0 where d 0 b 0 and f x 1 2 x 1 x 1 we examine the asymptotic behaviour of the solutions and confirm that although most of them are convergent there exists a periodic solution if d a b i

stable phase locked periodic solutions in a delay differential system abstract for a delay differential system where the nonlinearity is motivated by applications of neural networks to spatiotemporal pattern association and can be regarded as a perturbation of a step function we obtain the existence stability and limiting profile of a phase locked periodic solution using an approach very much similar to the asymptotic expansion of inner and outer layers in the analytic method of singular perturbation theory

on the periodic dynamics of memristor based neural networks with time varying delays abstract in this paper we study the existence uniqueness and stability of periodic solution for a wide class of memristor based neural networks with time varying delays by employing the topological degree theory in set valued analysis differential inclusions theory and a new lyapunov function method we prove that the neural network has a unique periodic solution which is globally exponentially stable moreover we prove the existence uniqueness and global exponential stability of equilibrium point for time varying delayed memristor based neural networks with constant coefficients the obtained results improve and extend previous works on memristor based or usual neural network dynamical systems with continuous or discontinuous right hand side finally two numerical examples are provided to show the applicability and effectiveness of our main results

multiplicity of large solutions for quasi monotone pulse type nonlinearities abstract this paper discusses the keller osserman condition from a dynamical perspective in order to obtain a rather astonishing multiplicity result of large solutions it turns out that for any given increasing positive function f u that satisfies the keller osserman condition destroying the monotonicity of f u on a compact set with arbitrarily small measure can originate an arbitrarily large number of explosive solutions moreover some counterexamples to an important result of 6 are given

existence of periodic solutions for a system of delay differential equations abstract in this paper we mainly study the existence of periodic solutions for a system of delay differential equations representing a simple two neuron network model of hopfield type with time delayed connections between the neurons we first examine the local stability of the trivial solution propose some sufficient conditions for the uniqueness of equilibria and then apply the poincare bendixson theorem for monotone cyclic feedback delayed systems to establish the existence of periodic solutions in addition a sufficient condition that ensures the trivial solution to be globally exponentially stable is also given numerical examples are provided to support the theoretical analysis

equilibria and their bifurcations in a recurrent neural network involving iterates of a transcendental function some practical models contain so complicated mathematical expressions that it is hard to determine the number and distribution of all equilibria not mentioning the qualitative properties and bifurcations of those equilibria the three node recurrent neural network system with two free weight parameters originally introduced by ruiz owens and townley in 1997 is such a system for which the equation of equilibria involves transcendental function and its iterates not computing coordinates of its equilibria in this paper we display an effective technique to determine the number and distribution of its equilibria without full information about equilibria our method enables to further study qualitative properties of those equilibria and discuss their saddle node pitchfork and hopf bifurcations by approximating center manifolds

a passivity based approach to stability of spatially distributed systems with a cyclic interconnection structure a class of distributed systems with a cyclic interconnection structure is considered these systems arise in several biochemical applications and they can undergo diffusion driven instability which leads to a formation of spatially heterogeneous patterns in this paper a class of cyclic systems in which addition of diffusion does not have a destabilizing effect is identified for these systems global stability results hold if the ldquosecantrdquo criterion is satisfied in the linear case it is shown that the secant condition is necessary and sufficient for the existence of a decoupled quadratic lyapunov function which extends a recent diagonal stability result to partial differential equations for reaction diffusion equations with nondecreasing coupling nonlinearities global asymptotic stability of the origin is established all of the derived results remain true for both linear and nonlinear positive diffusion terms similar results are shown for compartmental systems

minimal instability and unstable set of a phase locked periodic orbit in a delayed neural network abstract the existence of a non trivial phase locked periodic orbit is established for a system of delay differential equations describing the dynamics of networks of two identical saturating neurons we discuss the instability and the unstable manifold of this phase locked orbit in particular we give detailed information about the spectrum of the related monodromy operator and establish the connection between the unstable manifold of the phase locked orbit and the boundary of the global extension of a three dimensional c 1 submanifold of the origin we obtain a smooth solid spindle contained in the global attractor separated by a disk bordered by the phase locked orbit major technical tools include a discrete lyapunov functional invariant manifolds and other recently developed geometric theory of delay differential equations

existence of solutions for neutral integro differential equations with state dependent delay in this paper we study the existence and regularity of solutions for a neutral functional integro differential equation with state dependent delay in banach space the mild solutions are investigated by sadovskii fixed point theorem under compactness condition for the resolvent operator and the theory of fractional power and a norm are also used in the discussion since the nonlinear terms of the system involve spacial derivatives the strict solutions are obtained under holder continuity condition hence this work extends and develops the existed results on this topic in the end an example is provided to illustrate the application of the obtained results

a proof of jones conjecture abstract in this paper we prove that wright s equation y t y t 1 1 y t has a unique slowly oscillating periodic solution for parameter values 2 1 9 up to time translation this result proves jones conjecture formulated in 1962 that there is a unique slowly oscillating periodic orbit for all 2 furthermore there are no isolas of periodic solutions to wright s equation all periodic orbits arise from hopf bifurcations

attractors for 2d navier stokes models with delays abstract the existence of an attractor for a 2d navier stokes system with delay is proved the theory of pullback attractors is successfully applied to obtain the results since the abstract functional framework considered turns out to be nonautonomous however on some occasions the attractors may attract not only in the pullback sense but in the forward one as well also this formulation allows to treat in a unified way terms containing various classes of delay features constant variable distributed delays etc as a consequence some results for the autonomous model are deduced as particular cases of our general formulation

approximate controllability of semilinear neutral evolution systems with delay this paper considers the approximate controllability for a class of neutral control systems governed by semilinear neutral delayed equations sufficient conditions for approximate controllability are established by constructing fundamental solutions and using resolvent condition banach contraction mapping principle and techniques on fractional power operators the obtained results improve some existing analogous ones on this topic to some extent an example is presented finally to illustrate the applications of the obtained results

multi valued characteristics and morse decompositions abstract a rapid growth of molecular and systems biology in recent years challenges mathematicians to develop robust modeling and analytical tools for this area we combine a theory of monotone input output systems with a classical theory of morse decompositions in the context of ordinary differential equations models of biochemical reactions we show that a multi valued input output characteristic can be used to define non trivial morse decompositions which provide information about a global structure of the attractor the previous work on input output characteristics is shown to apply locally to individual morse sets and is seamlessly incorporated into our global theory we apply our tools to a model of cell cycle maintenance we show that changing the strength of the negative feedback loop can lead to cessation of cell cycle in two different ways it can either lead to globally attracting equilibrium or to a pair of equilibria that attract almost all solutions

biochemical oscillations in delayed negative cyclic feedback existence and profiles in this paper we propose a theoretical framework to systematically analyze the existence and the profiles of chemical oscillations in gene regulatory networks with negative cyclic feedback in particular we analytically derive the existence conditions and the profiles of oscillations in terms of reaction kinetic parameters and reveal dimensionless quantities that essentially characterize the oscillatory dynamics these discoveries then allow us to provide general biological insights that are useful for the design of synthetic gene circuits and wet lab experiments we point out that time delays due to splicing and transport play an important role for both of the existence and the profiles of oscillations to this end we first show that local instability leads to oscillations in cyclic gene regulatory networks and we derive the existence conditions based on local instability analysis then we analyze the period phase and amplitude of oscillations using multivariable harmonic balance analysis these results are demonstrated with two existing biochemical networks the pentilator and a self repression network of a hes protein

exponential stability and oscillation of hopfield graded response neural network both exponential and stochastic stabilities of the hopfield neural network are analyzed the results are especially useful for analyzing the stabilities of asymmetric neural networks a constraint on the connection matrix has been found under which the neural network has a unique and exponentially stable equilibrium given any connection matrix this constraint can be satisfied through the adjustment of the gains of the amplifiers and the resistances in the neural net circuit a one to one and smooth map between input currents and the equilibria of the neural network can be set up the above results can be applied to the master slave net to prove that the master net can find the best connection matrix for the slave net for the neural network disturbed by some noise the stochastic stability of the network is also analyzed a special asymmetric neural network formed by a closed chain of formal neurons is also studied for its stability and oscillation both stable and oscillatory dynamics are obtained in the closed chain network through the adjustment of the gains and resistances of the amplifiers

holder continuous solutions for integro differential equations and maximal regularity we characterize existence and uniqueness of solutions for a linear integro differential equation in holder spaces our method is based on operator valued fourier multipliers the solutions we consider may be unbounded concrete equations of the type we study arise in the modeling of heat conduction in materials with memory

multiple periodic solutions with minimal period 4 of the delay differential equation x t f t x t 1 abstract in this paper some sufficient conditions are obtained to guarantee the existence of multiple periodic solutions with minimal period 4 of the delay differential equation x t f t x t 1 by using relative morse index and some considerate analysis techniques

producing stable periodic solutions of switched impulsive delayed neural networks using a matrix based cubic convex combination approach this article is dedicated to designing a novel periodic impulsive control strategy for producing globally exponentially stable periodic solutions for switched neural networks with discrete and finite distributed time varying delays first tunable parameters and cubic convex combination approach are proposed to study the globally exponential convergence of switched neural networks second a sufficient criterion for the existence uniqueness and globally exponential stability of a periodic solution is demonstrated by using contraction mapping theorem and the impulse delay dependent lyapunov krasovskii functional method it is worth emphasizing that the addressed lyapunov krasovskii functional covers both triple integral terms and novel quadruple integral terms which makes the conservatism of the above criteria decrease even if the original neural network models are unstable or the impulsive effects are strong the addressed neural network model can produce a globally exponentially stable periodic solution these results here which include boundedness globally uniformly exponential convergence and globally exponentially stability of the periodic solution generalize and improve the earlier publications finally two numerical examples and their computer simulations are given to show the effectiveness of theoretical results

cones of rank 2 and the poincare bendixson property for a new class of monotone systems abstract we study systems that are monotone in a generalized sense with respect to cones of rank 2 the main result of the paper is the existence of a poincare bendixson property for some solutions of those systems

the structure of the set of 1 node solutions of a class of degenerate bvp s abstract a recent study 12 13 of a class of degenerate bvp s showed that there were two possibilities for the structure of the set of 1 node solutions left open was the question of whether each one was possible using some elementary phase plane techniques combined with some comparison arguments obtained from the maximum principle this paper constructs examples showing that indeed each can occur