equations of motion for variational electrodynamics abstract we extend the variational problem of wheeler feynman electrodynamics by generalizing the electromagnetic functional to a local space of absolutely continuous trajectories possessing a derivative velocities of bounded variation we show here that the gateaux derivative of the generalized functional defines two partial lagrangians for variations in our generalized local space one for each particle we prove that the critical point conditions of the generalized variational problem are i the euler lagrange equations must hold lebesgue almost everywhere and ii the momentum of each partial lagrangian and the legendre transform of each partial lagrangian must be absolutely continuous functions generalizing the weierstrass erdmann conditions

parabolic partial differential equations with discrete state dependent delay classical solutions and solution manifold abstract classical solutions to pdes with discrete state dependent delay are studied we prove the well posedness in a set x f which is analogous to the solution manifold used for ordinary differential equations with state dependent delay we prove that the evolution operators are c 1 smooth on the solution manifold

rigid analytic quantum groups and quantum arens michael envelopes abstract we introduce a rigid analytification of the quantized coordinate algebra of a semisimple algebraic group and a quantized arens michael envelope of the enveloping algebra of the corresponding lie algebra working over a non archimedean field and when q is not a root of unity we show that these analytic quantum groups are topological hopf algebras and frechet stein algebras we then introduce an analogue of the bgg category o for the quantum arens michael envelope and show that it is equivalent to the category o for the corresponding quantized enveloping algebra

superstability and rigorous asymptotics in singularly perturbed state dependent delay differential equations abstract we study the singularly perturbed state dependent delay differential equation e x t x t k x t r r r x t 1 x t which is a special case of the equation e x t g x t x t r r r x t one knows that for every sufficiently small e 0 eq possesses at least one so called slowly oscillating periodic solution and moreover the graph of every such solution approaches a specific sawtooth like shape as e 0 in this paper we obtain higher order asymptotics of the sawtooth including the location of the minimum and maximum of the solution with the form of the solution near these turning points and as well an asymptotic formula for the period using these and other asymptotic formulas we further show that the solution enjoys the property of superstability namely the nontrivial characteristic multipliers are of size o e for small e this stability property implies that this solution is unique among all slowly oscillating periodic solutions again for small e

periodic solutions of delay differential systems via hamiltonian systems abstract with a new definition of the j m index theory developed in liu 2006 we give another application of this index theory in delay differential systems via hamiltonian systems where we do not need the potential function to be c 2 continuous

a periodic solution of a differential equation with state dependent delay we study a differential equation for delayed negative feedback which models a situation where the delay depends on the present state and becomes effective in the future the main result is existence of a periodic solution in case the equilibrium is linearly unstable the proof employs the ejective fixed point principle on a compact convex set k0 c h 0 r of lipschitz continuous functions and uses that the equation generates a smooth semiflow on an infinite dimensional submanifold of the space c1 h 0 r

merging homoclinic solutions due to state dependent delay abstract for 2 5 2 the linear differential equation x t x t 1 with constant time lag is unstable and hyperbolic for sufficiently close to 5 2 we construct a state dependent delay d u 0 2 on an open set in the space c 1 2 0 with d u 1 for close to 0 so that the nonlinear equation x t x t d u x t has a one parameter family of entire solutions c r r r r r u which are homoclinic to zero and merge in finite time at some t y 0 c r t c 0 t for r r u and t t y this complements earlier results on complicated solution behaviour caused by state dependent delay and shows that there exist delay functionals for which certain transversality and regularity conditions along homoclinic solutions are violated

d modules on rigid analytic spaces i we introduce a sheaf of infinite order differential operators d on smooth rigid analytic spaces that is a rigid analytic quantisation of the cotangent bundle we show that the sections of this sheaf over sufficiently small affinoid varieties are frechet stein algebras and use this to define co admissible sheaves of d modules we prove analogues of cartan s theorems a and b for co admissible d modules

a differential equation with state dependent delay from cell population biology we analyze a differential equation describing the maturation of a stem cell population with a state dependent delay which is implicitly defined via the solution of an ode we elaborate smoothness conditions for the model ingredients in particular vital rates that guarantee the existence of a local semiflow and allow to specify the linear variational equation the proofs are based on theoretical results of hartung et al combined with implicit function arguments in infinite dimensions moreover we elaborate a criterion for global existence for differential equations with state dependent delay to prove the result we adapt a theorem by hale and lunel to the c1 topology and use a result on metric spaces from diekmann et al

differentiability of solutions with respect to parameters in neutral differential equations with state dependent delays in this paper we consider a class of nonlinear neutral differential equations with state dependent delays in both the neutral and the retarded terms we study well posedness and continuous dependence issues and differentiability of the parameter map with respect to the initial function and other possibly infinite dimensional parameters in a pointwise sense and also in the c norm

linearized stability in periodic functional differential equations with state dependent delays in this paper we study stability of periodic solutions of a class of nonlinear functional differential equations fdes with state dependent delays using the method of linearization we show that a periodic solution of the nonlinear fde is exponentially stable if the zero solution of an associated linear periodic linear homogeneous fde is exponentially stable

partial differential equations with discrete and distributed state dependent delays this work is an attempt to treat partial differential equations with discrete concentrated state dependent delay the main idea is to approximate the discrete delay term by a sequence of distributed delay terms all with state dependent delays we study local existence and long time asymptotic behavior of solutions and prove that the model with distributed delay has a global attractor while the one with discrete delay possesses the trajectory attractor

stability of periodic solutions of state dependent delay differential equations abstract we consider a class of autonomous delay differential equations z t f z t which includes equations of the form z t g z t z t r 1 z t r n r i r i z t for 1 i n with state dependent delays r i z t 0 the functions g and r i satisfy appropriate smoothness conditions we assume there exists a periodic solution z x t which is linearly asymptotically stable namely with all nontrivial characteristic multipliers satisfying 1 we prove that the appropriate nonlinear stability properties hold for x t namely that this solution is asymptotically orbitally stable with asymptotic phase and enjoys an exponential rate of attraction given in terms of the leading nontrivial characteristic multiplier a principal difficulty which distinguishes the analysis of equations such as from ones with constant delays is that even with g and r i smooth the associated function f is not smooth in function space techniques of hartung krisztin walther and wu are employed to resolve these issues

on differentiability of solutions with respect to parameters in state dependent delay equations in this paper we study differentiability of solutions with respect to parameters in state dependent delay equations in particular we give sufficient conditions for differentiability of solutions in thew1 pnorm 1 p in establishing our main results we make use of a version of the uniform contraction principle for quasi banach spaces

singular hopf bifurcation in a differential equation with large state dependent delay we study the onset of sustained oscillations in a classical state dependent delay sdd differential equation inspired by control theory owing to the large delays considered the hopf bifurcation is singular and the oscillations rapidly acquire a sawtooth profile past the instability threshold using asymptotic techniques we explicitly capture the gradual change from nearly sinusoidal to sawtooth oscillations the dependence of the delay on the solution can be either linear or nonlinear with at least quadratic dependence in the former case an asymptotic connection is made with the rayleigh oscillator in the latter van der pol s equation is derived for the small amplitude oscillations sdd differential equations are currently the subject of intense research in order to establish or amend general theorems valid for constant delay differential equation but explicit analytical construction of solutions are rare this paper illustrates the use of singular perturbation techniques and the unusual way in which solvability conditions can arise for sdd problems with large delays

global stability and wavefronts in a cooperation model with state dependent time delay abstract this article deals with a diffusive cooperative model with state dependent delay which is assumed to be an increasing function of the population density with lower and upper bounds for the cooperative dde system the positivity and boundedness of solutions are firstly given using the comparison principle of the state dependent delay equations obtained the stability criterion of model is analyzed both from local and global points of view when the diffusion is properly introduced the existence of traveling waves is obtained by constructing a pair of upper lower solutions and schauder s fixed point theorem calculating the minimum wave speed shows that the wave is slowed down by the state dependent delay finally the traveling wavefront solutions for large wave speed are also discussed and the fronts appear to be all monotone regardless of the state dependent time delay this is an interesting property since many findings are frequently reported that delay causes a loss of monotonicity with the front developing a prominent hump in some other delay models

semigroup properties and the crandall liggett approximation for a class of differential equations with state dependent delays abstract we present an approach for the resolution of a class of differential equations with state dependent delays by the theory of strongly continuous nonlinear semigroups we show that this class determines a strongly continuous semigroup in a closed subset of c 0 1 we characterize the infinitesimal generator of this semigroup through its domain finally an approximation of the crandall liggett type for the semigroup is obtained in a dense subset of c as far as we know this approach is new in the context of state dependent delay equations while it is classical in the case of constant delay differential equations

a necessary and sufficient condition for well posedness of initial value problems of retarded functional differential equations abstract we introduce the retarded functional differential equations rfdes with general delay structure to treat various delay differential equations ddes in a unified way and to clarify the delay structure in those dynamics we are interested in the question as to which space of histories is suitable for the dynamics of each dde and investigate the well posedness of the initial value problems ivps of the rfdes a main theorem is that the ivp is well posed for any admissible history functional if and only if the semigroup determined by the trivial rfde x 0 is continuous we clarify the meaning of the hale kato axiom hale kato 12 by applying this result to rfdes with infinite delay we also apply the result to ddes with unbounded time and state dependent delays

non linear partial differential equations with discrete state dependent delays in a metric space we investigate a class of non linear partial differential equations with discrete state dependent delays the existence and uniqueness of strong solutions for initial functions from a banach space are proved to get the well posed initial value problem we restrict our study to a smaller metric space construct the dynamical system and prove the existence of a compact global attractor

on locally analytic beilinson bernstein localization and the canonical dimension let mathbf g be a connected split reductive group over a p adic field in the first part of the paper we prove under certain assumptions on mathbf g and the prime p a localization theorem of beilinson bernstein type for admissible locally analytic representations of principal congruence subgroups in the rational points of mathbf g in doing so we take up and extend some recent methods and results of ardakov wadsley on completed universal enveloping algebras ardakov and wadsley ann math 2013 to a locally analytic setting as an application we prove in the second part of the paper a locally analytic version of smith s theorem on the canonical dimension

global hopf bifurcation for differential equations with state dependent delay abstract we develop a global hopf bifurcation theory for a system of functional differential equations with state dependent delay the theory is based on an application of the homotopy invariance of s 1 equivariant degree using the formal linearization of the system at a stationary state our results show that under a set of mild conditions the information about the characteristic equation of the formal linearization with frozen delay can be utilized to detect the local hopf bifurcation and to describe the global continuation of periodic solutions for such a system with state dependent delay

asymptotic theory for a class of functional differential equations with state dependent delays we characterize the asymptotic behavior of a class of functional differential equations with state dependent delays and show that this asymptotic behavior can be numerically observed by computing corresponding solutions of approximating equations with piecewise constant arguments

on state dependent delay partial neutral functional integro differential equations in this paper the existence of mild solutions for a class of abstract neutral integro differential equations with state dependent delay is studied

positive periodic solution of second order neutral functional differential equations abstract in this paper we consider two types of second order neutral functional differential equations by choosing available operators and applying krasnoselskii s fixed point theorem we obtain sufficient conditions for the existence of periodic solutions to such equations

existence of periodic solutions for delay differential equations with state dependent delay abstract in this paper we study the existence of periodic solutions of a delay differential equation with delay depending indirectly on the state a fixed point problem related to a poincare operator is constructed and solved using an ejective fixed point theorem

stability of nonlinear differential systems with state dependent delayed impulses we consider nonlinear differential systems with state dependent delayed impulses impulses which involve the delayed state of the system for which the delay is state dependent such systems arise naturally from a number of applications and the stability issue is complex due to the state dependence of the delay we establish general and applicable results for uniform stability uniform asymptotic stability and exponential stability of the systems by using the impulsive control theory and some comparison arguments we show how restrictions on the change rates of states and impulses should be imposed to achieve system s stability in comparison with general impulsive delay differential systems with state dependent delay in the nonlinearity or the differential systems with constant delays in our approach the boundedness of the state dependent delay is not required but derives from the stability result obtained examples are given to demonstrate the sharpness and applicability of our general results and the proposed approach

the solution manifold and c1 smoothness for differential equations with state dependent delay abstract let h 0 u c 1 h 0 r n open and f u r n continuously differentiable if f satisfies two mild additional smoothness conditions then the set x u 0 f is a c1 submanifold of codimension n in c 1 h 0 r n the maximal solutions x of the initial value problems x t f x t x 0 x define a continuous semiflow f on x and all operators f t are continuously differentiable their derivatives d2f t are given in the usual way by solutions v to the variational equation along x with segments vt in the tangent spaces tf t x the additional conditions on f are motivated by properties of differential equations with state dependent delay and are verified for an example

metastable periodic patterns in singularly perturbed state dependent delayed equations abstract we consider the scalar delayed differential equation x t x t f x t r where 0 r r x 1 r x and f represents either a monotone positive feedback d f d x 0 or a monotone negative feedback d f d x 0 plus some other hypotheses when the delay is a constant i e r x 1 this equation can support metastable rapidly oscillating solutions that are transients whose duration is of order exp c for some c 0 in this paper combining analytic and numerical techniques we investigate whether this metastable behavior persists when the delay r x depends non trivially on the state variable x more precisely we proceed in three steps first we explore theoretically and numerically the existence of branches of rapid periodic solutions for these equations then to predict conditions for metastability we introduce transition layer equations that depict the asymptotic shape of transient oscillations when tends to zero and finally these predictions are validated by numerical simulations numerical explorations are also performed for other scalings of with our conclusions are i when 0 0 there are no metastable transients ii when 0 0 and 0 is finite for monotone negative feedback the metastable oscillations exist irrespective of choices of f and r iii for monotone positive feedback they require that the feedback f and the delay r satisfy extra conditions such as f being an odd function and the delay r x an even function one novel result is that state dependent delays may lead to metastable dynamics in equations that cannot support such regimes when the delay is constant

boundary layer phenomena for differential delay equations with state dependent time lags i in this paper we begin a study of the differential delay equation mathtype mtef 2 1 feaafiart1ev1aaatcvaufebsjuyzl2yd9gzlbvynv2caerbulwbln hiov2dgi1btfmbaexatlxbi9gbaerbd9wdylwzybitldharqqtubsr 4rnchbgeagqivu0je9sqqrpepc0xbbl8f4rqqrffpeea0jc9yq jc9 vqaqpepm0xbba9pwe9q8fs0 yqaqpepae9pg0firpeps0he9vr0 vr 0 vqpwqaaeaabigaciaacaqabeaadaqaaqaaaoqaaiabew7aljqadi hagaqbaiaacicacawg0bgaaiykaiabg2da9iabgkhitiaadihacagg oagaamidaiaacmcacqghrawkcawgmbgaaiikaiaadihacaggoagaam idaiabgkhitiaadkhacaggpagaaiykaiaacycacaqggagaaeiiaiaa dkhacqgh9aqpcawgybgaaiikaiaadihacaggoagaamidaiaacmcaca ggpaaaaa 5192 varepsilon x t x t f x t r r r x t we prove the existence of periodic solutions for 0 0 where 0 is an optimal positive number we investigate regularity and monotonicity properties of solutions x t which are defined for all t and of associated functions like t t r x t we begin the development of a poincare bendixson theory and phase plane analysis for such equations in a companion paper these results will be used to investigate the limiting profile and corresponding boundary layer phenomena for periodic solutions as approaches zero

smoothness of semiflows for parabolic partial differential equations with state dependent delay in this paper the smoothness properties of semiflows on c1 solution submanifold of a parabolic partial differential equations with state dependent delay are investigated the problem is formulated as an abstract ordinary retarded functional differential equation of the form du t dt au t f ut with a continuously differentiable map g from an open subset u of the space c1 h 0 l2 where a is the infinitesimal generator of a compact c0 semigroup the present study is continuation of a previous work 14 that highlights the classical solutions and c1 smoothness of solution manifold here we further prove the continuous differentiability of the semiflow we finally verify all hypotheses by a biological example which describes a stage structured diffusive model where the delay which is the time taken from birth to maturity is assumed as a function of a immature species population

morse decompositions for delay differential equations abstract the attractor for the scalar delay differential equation x t f x t x t 1 is shown to possess a morse decomposition under the hypothesis of a negative feedback condition in the delay term using this decomposition we prove results on the asymptotic rate of oscillation of solutions of the initial value problem existence of both slowly and rapidly oscillating periodic solutions is also proved

stickiness in hamiltonian systems from sharply divided to hierarchical phase space we investigate the dynamics of chaotic trajectories in simple yet physically important hamiltonian systems with nonhierarchical borders between regular and chaotic regions with positive measures we show that the stickiness to the border of the regular regions in systems with such a sharply divided phase space occurs through one parameter families of marginally unstable periodic orbits and is characterized by an exponent ensuremath gamma 2 for the asymptotic power law decay of the distribution of recurrence times generic perturbations lead to systems with hierarchical phase space where the stickiness is apparently enhanced due to the presence of infinitely many regular islands and cantori in this case we show that the distribution of recurrence times can be composed of a sum of exponentials or a sum of power laws depending on the relative contribution of the primary and secondary structures of the hierarchy numerical verification of our main results are provided for area preserving maps mushroom billiards and the newly defined magnetic mushroom billiards

differential equations with locally bounded delay abstract for differential delay equations of the general form x t g x t which include equations with unbounded finite state dependent delays we construct semiflows of continuously differentiable solution operators on suitable banach manifolds and provide local stable and unstable manifolds at equilibria examples occur in feedback systems with delays caused by signal transmission

chemical principle and pde of variational electrodynamics abstract the two body problem of variational electrodynamics possesses differential delay equations of motion with state dependent delays of neutral type and solutions that can have velocity discontinuities on countable sets from a periodic orbit possessing some mild properties at breaking points we define a synchronization function in r r 3 which is further used to construct two bounded oscillatory functions vanishing at breaking points and whose first derivatives are continuous and defined everywhere but at breaking points the oscillatory functions are associated with a natural pde identity in w 2 2 r 3 we postulate an asymptotic ordering for a generalized pde identity which defines a fredholm schroedinger operator in w 2 2 r 3 with a forcing term belonging to l 2 r 3 o 1 r 2 as an application we introduce the chemical principle criterion to select orbits with asymptotically vanishing far fields and estimate the bohr radius parameter of the fredholm schroedinger pde using the boundary layers of orbits chosen according to the chemical principle criterion last working backward we derive an orbital property of the chemical principle type from the ordering conditions

meeting time distributions in bernoulli systems meeting time is defined as the time for which two orbits approach each other within the distance in phase space we show that the distribution of the meeting time is exponential in p1 pk bernoulli systems in the limit of 0 the distribution converges to exp where is the meeting time normalized by the average the exponent is shown to be kl 1pl 1 pl for the bernoulli systems

differential equations with discrete state dependent delay uniqueness and well posedness in the space of continuous functions abstract partial differential equations with discrete concentrated state dependent delays in the space of continuous functions are investigated in general the corresponding initial value problem is not well posed so we find an additional assumption on the state dependent delay function to guarantee the well posedness for the constructed dynamical system we study the long time asymptotic behavior and prove the existence of a compact global attractor

a non local pde model for population dynamics with state selective delay local theory and global attractors we propose a non local pde model for the evolution of a single species population that involves delayed feedback where the delay such as the maturation time in the delayed birth rate is selective and the selection depends on the status of the system this delay selection in contrast with the usual state dependent delay widely used in ordinary delay differential equation ensures the lipschitz continuity of the nonlinear functional in the classical phase space we also develop the local theory and the existence and upper semi continuity of the global attractor with respect to parameters

boundedness global existence and continuous dependence for nonlinear dynamical systems describing physiologically structured populations abstract the paper is aimed as a contribution to the general theory of nonlinear infinite dimensional dynamical systems describing interacting physiologically structured populations we carry out continuation of local solutions to maximal solutions in a functional analytic setting for maximal solutions we establish global existence via exponential boundedness and by a contraction argument adapted to derive uniform existence time moreover within the setting of dual banach spaces we derive results on continuous dependence with respect to time and initial state to achieve generality the paper is organized top down in the way that we first treat abstract nonlinear dynamical systems under very few but rather strong hypotheses and thereafter work our way down towards verifiable assumptions in terms of more basic biological modelling ingredients that guarantee that the high level hypotheses hold

verma modules over p adic arens michael envelopes of reductive lie algebras abstract let k be a locally compact p adic field g a split reductive lie algebra over k and u g its universal enveloping algebra we investigate the category c g of coadmissible modules over the p adic arens michael envelope u g of u g let p g be a parabolic subalgebra the main result gives a canonical equivalence between the classical parabolic bgg category of g relative to p and a certain explicitly given highest weight subcategory of c g this completely clarifies the verma module theory over u g

c1 α strict solutions and wellposedness of abstract differential equations with state dependent delay abstract we study the existence and uniqueness of c 1 strict solutions for a general class of abstract differential equations with state dependent delay we also study the local well posedness of this type of problems on subspaces of c 1 p 0 x some examples involving partial differential equations are presented

distinguishing quasiperiodic dynamics from chaos in short time series we propose a procedure to distinguish quasiperiodic from chaotic orbits in short time series which is based on the recurrence properties in phase space the histogram of the return times in a recurrence plot is introduced to disclose the recurrence property consisting of only three peaks imposed by slater s theorem noise effects on the statistics are studied our approach is demonstrated to be efficient in recognizing regular and chaotic trajectories of a hamiltonian system with mixed phase space

a condition on delay for differential equations with discrete state dependent delay abstract parabolic differential equations with discrete state dependent delay are studied the approach based on an additional condition on the delay function introduced in a v rezounenko differential equations with discrete state dependent delay uniqueness and well posedness in the space of continuous functions nonlinear anal 70 11 2009 3978 3986 is developed we propose and study an analogue of the condition which is sufficient for the well posedness of the corresponding initial value problem on the whole space of continuous functions c the dynamical system is constructed in c and the existence of a compact global attractor is proved

invariant distributions on p adic analytic groups let p be a prime number let l be a finite extension of the field qp of p adic numbers let k be a spherically complete extension field of l and let g be the group of l rational points of a split reductive group over l we derive several explicit descriptions of the center of the algebra d g k of locally analytic distributions on g with values in k the main result is a generalization of an isomorphism of harish chandra which connects the center of d g k with the algebra of weyl invariant centrally supported distributions on a maximal torus of g this isomorphism is supposed to play a role in the theory of locally analytic representations of g as studied by p schneider and j teitelbaum

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

systems of differential delay equations floquet multipliers and discrete lyapunov functions abstract we define a discrete integer valued lyapunov function v for cyclic nearest neighbor systems of differential delay equations possessing a feedback condition this extends analogous definitions for cyclic systems of ode s and for scalar differential delay equations we relate the values of v to the real parts of the floquet multipliers for such linear periodic systems and thereby prove all floquet subspaces are at most two dimensional

stable flatness of nonarchimedean hyperenveloping algebras let l be a completely valued nonarchimedean field and g a finite dimensional lie algebra over l we show that its hyperenveloping algebra f g agrees with its arens michael envelope and furthermore is a stably flat completion of its universal enveloping algebra as an application we prove that the taylor relative cohomology for the locally convex algebra f g is naturally isomorphic to the lie algebra cohomology of g

slowly oscillating periodic solutions of autonomous state dependent delay equations more than a dozen years since the peak of activity on the question of the existence of periodic slowly oscillating solutions of autonomous delay differential equations following the early work of jones l wright 2 and grafton 3 the work of nussbaum 4 51 is to be specially noted for providing several new fixed point results and a global bifurcation theorem which are particularly useful for proving the existence of periodic solutions other important works include those of hadeler and tomiuk 6 kaplan and yorke 7 chow 8 chow and hale 9 alt lo 111 and walther 12 131 see hale s book 14 for an overview to the best of our knowledge only nussbaum 5 and alt 1 l considered the question of the existence of periodic solutions of autonomous state dependent delay differential equations nussbaum considered a special equation 0 3 below in 5 but did not prove a general result alt 1 l considered more complicated integral threshold type state dependent delays which have arisen in various models in epidemiology and structured population models alt obtained a theorem for a general class of state dependent equations on which we comment below the lack of results on periodic solutions for state dependent equations is probably explained by the fact that it is not clear what kinds of state dependent delays are interesting or natural and on a lack of compelling examples of such equations arising from plausible mathematical models however some recent papers of belair and mackey 15 and belair 16 contain some interesting state dependent delay equations arising in economics and population biology state dependent delays of threshold type arise naturally in structured population models see lo 111 the motivation for this paper stems from consideration of the innocent looking equation x t kx t r x t 0 1 where r x is bell shaped e g r x 01 emxz 1 01 0 i 01 5 1 if k 7r 2 and r o 1 then the zero solution of 0 1 is unstable a formal linearization would yield z t kz t 1 which is known to be unstable for

periodic solutions for differential equations with state dependent delay and positive feedback existence of a nontrivial periodic orbit and a homoclinic orbit connecting 0 to the periodic orbit is shown for the differential equation x t x t f x t r r r x t with state dependent delay and positive feedback here 0 f and r are smooth real functions with r 0 1 and f 0

analysis of a model representing stage structured population growth with state dependent time delay a stage structured model of population growth is proposed where the time to ma turity is itself state dependent it is shown that under appropriate assumptions all solutions are positive and bounded criteria for the existence of positive equilibria and further conditions for the uniqueness of the equilibria are given the stability of the equilibria are also discussed in addition an attracting region is determined for solutions such that this region collapses to the unique positive equilibrium in the state independent case

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

existence and controllability results for second order impulsive stochastic evolution systems with state dependent delay in this paper we discuss a kind of impulsive second order stochastic evolution systems with state dependent delay in a real separable hilbert space the results concerning the existence and controllability of mild solutions have been addressed by means of the fixed point techniques some sufficient conditions are formulated as well as an application involving partial differential equation with impulses is presented

inertial and slow manifolds for delay equations with small delays abstract yu a ryabov and r d driver proved that delay equations with small delays have lipschitz inertial manifolds we prove that these manifolds are smooth in addition we show that expansion in the small delay can be used to obtain the dynamical system on the inertial manifold this justifies post newtonian approximation for delay equations

on abstract differential equations with state dependent non local conditions abstract in this paper we introduce the concept of state dependent nonlocal condition and study the existence and uniqueness of solution for a general class of abstract differential equations with state dependent delay subject to this class of conditions in this type of problem the nonlocal condition can be dependent on times or time intervals determined by the unknown solution some examples on partial differential equations with state dependent delay arising in population dynamics are presented

boundary layer phenomena for differential delay equations with state dependent time lags iii in this paper we begin a study of the differential delay equation varepsilon x t x t f x t r r r x t

abstract algebraic delay differential systems and age structured population dynamics abstract we consider the abstract algebraic delay differential system x t a x t f x t a t a t h x t a t here a is a linear operator on d a x satisfying the hille yosida conditions x t d a x and a t r n where x is a real banach space with a global lipschitz condition on f and an appropriate hypothesis on the function h we show that the corresponding initial value problem gives rise to a continuous semiflow in a subset of the space of continuous functions we establish the positivity of the x component and give some examples arising from age structured population dynamics the examples come from situations where the age of maturity of an individual at a given time is determined by whether or not the resource concentration density which depends on the immature population reaches a prescribed threshold within that time

principle of linearized stability and instability for parabolic partial differential equations with state dependent delay abstract in this paper the stability properties of a parabolic partial differential equation with state dependent delay are investigated by the heuristic approach the previous works 1 2 obtained a continuously differentiable semiflow with continuously differentiable solution operators defined by the classical solutions and resolved the problem of linearization for this equation here we clarify the relation between the spectral properties of the linearization of the semiflow at a stationary solution and the strong continuous semigroup defined by the solutions of the linearization of this equation and consider the local stable and unstable invariant manifolds of the semiflow at a stationary solution by a biological application we finally verify all hypotheses for an age structured diffusive model with state dependent delay and consider its stability behavior

statistics of return times a general framework and new applications in this paper we provide general estimates for the errors between the distribution of the first and more generally the k th return time suitably rescaled and the poisson law for measurable dynamical systems in the case that the system exhibits strong mixing properties these bounds are explicitly expressed in terms of the speed of mixing using these approximations the poisson law is finally proved to hold for a large class of non hyperbolic systems on the interval

convergence of solutions for an equation with state dependent delay abstract a result of smith and thieme shows that if a semiflow is strongly order preserving then a typical orbit converges to the set of equilibria for the equation with state dependent delay t x t f x t r x t where 0 and f and r are smooth real functions with f 0 0 and f 0 we construct a semiflow which is monotone but not strongly order preserving we prove a convergence result under a monotonicity condition different from the strong order preserving property and apply it to the above equation to obtain generic convergence

existence results for fractional neutral integro differential equations with state dependent delay in this paper we study the existence of mild solutions for a class of abstract fractional neutral integro differential equations with state dependent delay the results are obtained by using the leray schauder alternative fixed point theorem an example is provided to illustrate the main results