exponential stability of nonlinear differential delay equations abstract the aim of this paper is to investigate the exponential stability of a nonlinear differential delay equation dot x t t x t x t introduce the corresponding differential equation without delay dot y t t y t y t and assume it is exponentially stable it will be shown in this paper that the differential delay equation will remain exponentially stable provided the time lag is small enough

abstract functional differential equations and reaction diffusion systems functional differential equations and reaction diffusion systems r h martin jr and h l smith abstract several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays several fundamental results on the existence and behavior of solutions to semilinear functional differential equations are developed in a banach space setting the ideas are applied to reaction diffusion systems that have time delays in the nonlinear reaction terms the techniques presented here include differential inequalities invariant sets and lyapunov functions and therefore they provide for a wide range of applicability the results on inequalities and especially strict inequalities are new even in the context of semilinear equations whose nonlinear terms do not contain delays suppose q is a bounded domain in rn with aq smooth and a is the laplacian operator on q also let m be a positive integer z a positive number and f fi m a continuous bounded function from 0 xc x q x c z 0 m into rm where c t 0 is the space of continuous functions from z 0 into r the purpose of this paper is to apply abstract results for semilinear functional differential equations in banach spaces to reactiondiffusion systems with time delays having the form a tu x t diau x t i t x ut x t a xeq i 1 m rdd ai x ui x t au i x t 81 x t t a x eaq u x a 0 x x 0 5 t 0 di 0 and ca q 0 o is c and i q x 0 oo r is c2 here an is the outward normal derivative on aq and if di 0 it is assumed that no boundary conditions are specified for this i also tu x t denotes the partial with respect to t whereas ut x denotes the member of c z 0 defined by 0 u x t 0 u x t 0 m our techniques provide basic existence criteria but the main point is that they can also be effectively applied to obtain estimates for solutions especially received by the editors october 7 1987 and in revised form october 7 1988 1980 mathematics subject classification 1985 revision primary 35r10 34k30

stabilization of second order evolution equations with unbounded feedback with time dependent delay we consider abstract second order evolution equations with unbounded feedback with time varying delay existence results are obtained under some realistic assumptions we prove the exponential decay under some conditions by introducing an abstract lyapunov functional our abstract framework is applied to the wave to the beam and to the plate equations with boundary delays

on the reducibility of the voigt functions the voigt functions which are employed in such areas as neutron physics and spectroscopy are shown to be capable of representation in terms of circular functions and confluent hypergeometric functions

lattice sums of generalized hypergeometric functions representations and convergence criteria for one dimensional 1d and two dimensional 2d lattice sums of generalized hypergeometric functions pfp 1 are derived by using apparently new integral representations for the latter functions and 1d and 2d forms of the poisson summation formula these lattice sum representations concisely unify and generalize various specializations given previously by several authors and should prove useful in analysing finite size effects in systems subjected to non periodic boundary conditions

a method of lines for a nonlinear abstract functional evolution equation let x be a real banach space with x uniformly convex a method of lines is introduced and developed for the abstract functional problem lipschitzian like function in its two variables further conditions are given for the convergence of the method to a strong solution of e recent results for perturbed abstract ordinary equations are substantially improved the method applies also to large classes of functional parabolic problems as well as problems of integral perturbations the method is straightforward because it avoids the introduction of the operators a t and the corresponding use of nonlinear evolution operator theory 1 introduction preliminaries let x be a real banach space with norm 11 let

the weak solution of a functional differential equation in a general banach space abstract the existence of a unique weak lipschitz continuous solution of the abstract functional differential equation u t a t u t g t u t t 0 t fde u t t t r 0 is obtained by a fixed point argument here x is a general banach space a t d a t x x is m accretive t 0 t and g is lipschitz continuous in both variables stability properties of this weak solution are given and a first order hyperbolic pde to which the theory can be applied is considered

on the asymptotic behavior of a semilinear functional differential equation in banach space the family a t t 3 0 is supposed to generate a linear evolution system and f t u is a nonlinear operator continuous with respect to a fractional power of a r for some r 2 0 it is known that local solutions to 1 exist under fairly general continuity assumptions on the nonlinear term f t u see 3 4 combined with condition 5 below this guarantees the existence of a global solution u t cf 3 p 5391 our results then say that if the linearized equation u t a t u t is asymptotically stable the same holds true for 1 provided that the nonlinear perturbation f is sufficiently small the main point hereby lies in the fact that the bound forfis independent of the time lag r 0

approximation schemes for nonlinear neutral optimal control systems abstract we discuss methods of approximating stable neutral functional differential equations and associated optimal control problems by sequences of optimal control problems for ordinary differential equations by introducing a class of mollified neutral functional differential equations convergence of the linear interpolating spline and the averaging approximation scheme is proved a number of numerical examples are included

on the conjectures of henkel and weston recent conjectures of henkel and weston 1992 on certain two dimensional sums are examined and some significant generalizations of these sums are developed the results so obtained should be useful in analysing finite size effects in systems subjected to non periodic boundary conditions

on the approximate k controllability of nonlinear systems in banach spaces abstract let x be a real banach space control problems of the type x a t x b t x u t 0 t x 0 0 are considered where for every t 0 t a t x d a x and b 0 t x 2 x are given operators the concept of approximate k controllability or controllability with preassigned responses is introduced for systems of the type it is shown that there exist lipschitz continuous approximating control functions u t t 0 t for a variety of response types evans responses and kato responses are considered for fully nonlinear problems as well as mild ones for semilinear problems it is also shown that the function r which determines the proximity of the response x t with x t a t x t b t x t u t t 0 t x 0 0 to the preassigned response f t x f r is of the type c where c is a positive constant independent of a more natural approximate k controllability concept approximate ek controllability is also introduced and a result is given about it using leray schauder theory an application is given in the field of partial differential equations

existence and upper semi continuity of pullback attractors of a p laplacian equation with delay in this paper we study the dynamics of a non autonomous p laplacian equation with delay we first prove the existence and uniqueness of solutions for this equation by using the galerkin approximations then applying the sobolev compactness theorem ascoli azerla theorem and energy method we show the existence of pullback attractors for the associated process in space c h 0 lp as well as in space c h 0 l2 the upper semi continuity result of the pullback attractors is also established here

on certain schlomilch type series abstract by using a form of the poisson summation formula together with a generalization due to srivastava and exton of the discontinuous integral of weber and schafheitlin and the heretofore not readily available cosine transform of the hypergeometric function 1 f 2 b 2 t 2 b 0 several new schlomilch and fourier type series are evaluated by specialization of the latter series numerous results appearing in the literature are obtained in a unified way

existence and asymptotic behavior of a functional differential equation in banach space abstract existence and asymptotic behavior of solutions are given for the equation u t a t u t f t u t t 0 and u 0 c r 0 x c the space x is a banach space the family a t 0 t t of unbounded linear operators defined on d a x x generates a linear evolution system and f c x is continuous with respect to a fractional power of a t 0 for some t 0 0 t

stability analysis of heat flow with boundary time varying delay effect abstract in this paper we investigate the stabilization of heat flow with boundary time varying delay effect under some assumptions we prove exponential stability of the solution applying variable norm technique and modified lyapunov functional approach

reaction diffusion systems with time delays monotonicity invariance comparison and convergence in a recent paper we have obtained differential inequality results developed comparison theorems and established invariance criteria for mild solutions of abstract functional differential equations in a banach space the primary application for these results is to reaction diffusion systems in which the reaction nonlinearity contains time delays in the present paper we apply these results to reaction diffusion systems with time delays of the form u t d u f x u t x x u u n 0 x where is a bounded open set in r n with smooth boundary u r n d diag d 1 d n with d i 0

asymptotic behaviour of two dimensional navier stokes equations with delays some results on the asymptotic behaviour of solutions to navier stokes equations when the external force contains some hereditary characteristics are proved we show two different approaches to prove the convergence of solutions to the stationary one when this is unique the first is a direct method while the second is based on a razumikhin type method

functional evolution equations abstract of concern is the functional evolution equation du dt bu f t u t t 0 u 0 s c 1 r 0 x where ut s u t s r 0 s r 0 and x is a real banach space it is shown by the method of lines combined with the crandall pazy theorem that for the initial data u0 in a generalized domain this equation has a limit solution which is lipschitz continuous in t and that this limit solution is a unique strong one if further assumptions on b are imposed and that the zero solution is asymptotically stable an application is given to a class of partial functional differential boundary value problems

on the mellin transform of products of bessel and generalized hypergeometric functions we deduce in an elementary way representations for the mellin transform of a product of bessel functions 0f1 a2x2 and generalized hypergeometric functions pfp 1 b2x2 for a b 0 as a corollary we obtain a transformation formula for p 1fp 1 which was discovered by wimp in 1987 by using bailey s method for the specialization 3f2 1

flow invariance for nonlinear partial differential delay equations several fundamental results on existence flow invariance regularity and linearized stability of solutions to the nonlinear partial differential delay equation u t bu t f u t t 0 uo with b c x x x w accretive are developed for a general banach space x in contrast to existing results with the history response f globally defined and at least lipschitz on bounded sets the results are tailored for situations with f defined on possibly thin subsets of the initial history space e only and are applied to place several classes of population models in their natural l 1 setting the main result solves the open problem of a subtangential condition for flow invariance of solutions in the fully nonlinear case paralleling those known for the cases of a no delay b ordinary delay equations with b 0 and c the semilinear case

on a time discretization method for a semilinear heat equation with purely integral conditions in a nonclassical function space abstract in this paper we construct a semidiscrete approximate solution to a semilinear one dimensional heat equation subject to integral boundary conditions by means of the rothe discretization in time method the convergence of the approximation scheme obtained is proved yielding the well posedness of the problem considered

evolution equations for the probabilistic generalization of the voigt profile function the spectrum profile that emerges in molecular spectroscopy and atmospheric radiative transfer as the combined effect of doppler and pressure broadenings is known as the voigt profile function because of its convolution integral representation the voigt profile can be interpreted as the probability density function of the sum of two independent random variables with gaussian density due to the doppler effect and lorentzian density due to the pressure effect since these densities belong to the class of symmetric levy stable distributions a probabilistic generalization is proposed as the convolution of two arbitrary symmetric levy densities we study the case when the widths of the distributions considered depend on a scale factor t that is representative of spatial inhomogeneity or temporal non stationarity the evolution equations for this probabilistic generalization of the voigt function are here introduced and interpreted as generalized diffusion equations containing two riesz space fractional derivatives thus classified as space fractional diffusion equations of double order

a uniform stability result for thermoelasticity of type iii with boundary distributed delay abstract in this paper we consider a thermoelastic system of type iii with boundary distributed delay under suitable assumption on the weight of the delay we prove using the energy method that the damping effect through heat conduction given by green and naghdi s theory is still strong enough to uniformly stabilize the system even in the presence of time delay

existence and continuous dependence results for nonlinear differential inclusions with infinite delay this paper is concerned with nonlinear functional differential inclusions with infinite delay in banach spaces using tools involving the measure of noncompactness and multi valued fixed point theory existence and continuous dependence results are obtained for integral solutions without the assumption of compactness on the associated nonlinear semigroup

on certain two dimensional schlomilch series by using representations for one dimensional 1d schlomilch series lommel s expansion for bessel functions of the first kind and elementary series manipulations several algebraic formulae for two dimensional 2d schlomilch series are derived one of these formulae contains as a special case an identity involving a certain trigonometric lattice sum conjectured by henkel and weston 1993 which has recently been derived by boersma and de doelder 1993 using the 2d poisson summation formula the latter method is used to give an alternative derivation for 2d schlomilch series further it is shown that representation by 2d schlomilch series of a null function is not unique in addition representations for several other 1d type series are given

asymptotic behavior of nonlinear evolution equations in banach spaces abstract in this study we show how yosida s approach for the approximation of the derivative obtains new results for evolution systems using this method we obtain multivalued time dependent perturbation results in addition we consider translation invariant subspaces y of the bounded and uniformly continuous functions to obtain criteria for the existence of solutions u y to the equation u t a t u t u t f t t r or of solutions u which are asymptotically close to y for the inhomogeneous differential equation u t a t u t u t f t t 0 u 0 u 0 in general banach spaces where a t denotes a possibly nonlinear time dependent dissipative operator particular examples of the space y are spaces of functions with various properties of almost periodicity and more general types of asymptotic behavior furthermore an application to functional differential equations is described

on some trigonometric and exponential lattice sums certain two dimensional trigonometric lattice series the ordinary convergence of which involves unresolved questions of a deep and delicate number theoretic nature are shown to be summable by a special abelian method and their abelian sums are obtained this is done by first evaluating an absolutely convergent exponential lattice series and then analytically extending its sum

some new aspects of lyapunov type theorems for stochastic differential equations of neutral type some new lyapunov type theorems for stochastic differential equations of neutral type are proved it is shown that these theorems simplify an application of kolmanovskii and shaikhet s general method of lyapunov functionals construction for stability investigation of different mathematical models

a new product integral representation for differential equations in separable banach spaces abstract letxbe a separable banach space consider the problem formula where the continuous functionf 0 x xis locally lipschitz continuous inx uniformly inton bounded intervals and continuous intuniformly w r t x the product integral formula x t lim n i 0 n i t n f i t n x 0 0 t t max for the solutionx t of d has been shown to converge we also show that iff t is lipschitz continuous onxwith constantl then the mappingx0 x t is lipschitz continuous with constanteltfor anyt 0 this formula has been recently developed for differential inclusions inrnby wolenski but the infinite dimensional case is considerably more involved

the method of lines applied to nonlinear nonlocal functional differential equations this paper discusses nonlinear functional differential equation in a real reflexive banach space with nonlocal history condition by using the method of lines the existence and uniqueness of a strong solution are established finally some applications of the abstract results are presented

method of lyapunov functionals construction in stability of delay evolution equations the investigation of stability for hereditary systems is often related to the construction of lyapunov functionals the general method of lyapunov functionals construction which was proposed by v kolmanovskii and l shaikhet and successfully used already for functional differential equations for difference equations with discrete time for difference equations with continuous time is used here to investigate the stability of delay evolution equations in particular partial differential equations

on nonautonomous functional differential equations abstract in this paper we prove a theorem on boundary perturbation of nonautonomous cauchy problems and then apply this result to show the existence and uniqueness of classical solutions of the nonautonomous banach space valued functional differential equation formula

the exponential stability region of timoshenko beam with interior delays and boundary damping abstractin this paper we investigate the exponential stability of a timoshenko beam with interior time delays and boundary damping at first we show that the system is well posed by the semigroup method next we construct an appropriate functional to study the exponential stability we transform the exponential stability into the solvability of inequality equations and then by analysing the relationship between delays parameters 1 2 and damping parameters 1 2 we describe the 1 2 region for which the system is exponentially stable

convergence of the kato approximants for evolution equations involving functional perturbations the existence of a unique strong solution of the nonlinear abstract functional differential equation u t a t u t f t ut u0 ec1 r 0 x te 0 t e is established x is a banach space with uniformly convex dual space and for t 0 t a t is m accretive and satisfies a time dependence condition suitable for applications to partial differential equations the function f satisfies a lipschitz condition the novelty of the paper is that the solution u t of e is shown to be the uniform limit as n of the sequence un t where the functions un t are continuously differentiate solutions of approximating equations involving the yosida approximants thus a straightforward approximation scheme is now available for such equations in parallel with the approach involving the use of nonlinear evolution operator theory

remarks on a note by h exton on the reducibility of the voigt functions correct expressions for integrals given in a recent letter by exton ibid vol 14 p l75 1981 are found

some connections between generalized voigt functions with the different parameters the voigt functions play an important role in several diverse fields of physics such as astrophysical spectroscopy and the theory of neutron reactions motivated by the contributions toward the unification and generalization of these functions in this paper we establish some connections between generalized voigt functions with the different parameters involving one and two variable hypergeometric functions further we derive various other interesting results as applications of these connections

stability results for 2d navier stokes equations with unbounded delay abstract some results related to 2d navier stokes equations when the external force contains hereditary characteristics involving unbounded delays are analyzed first the existence and uniqueness of solutions is proved by galerkin approximations and the energy method the existence of stationary solution is then established by means of the lax milgram theorem and the schauder fixed point theorem the local stability analysis of stationary solutions is studied by several different methods the classical lyapunov function method the razumikhin lyapunov technique and by constructing appropriate lyapunov functionals finally we also verify the polynomial stability of the stationary solution in a particular case of unbounded variable delay exponential stability in this infinite delay setting remains as an open problem

flow invariance for solutions to nonlinear nonautonomous partial differential delay equations we investigate the problem of existence and flow invariance of mild solutions to nonautonomous partial differential delay equations u t b t u t f t ut t s us where b t is a family of nonlinear multivalued accretive operators with d b t possibly depending on t and the operators f t being defined and lipschitz continuous possibly only on thin subsets of the initial history space e the results are applied to population dynamics models we also study the asymptotic behavior of solutions to this equation our analysis will be based on the evolution operator associated to the equation in the initial history space e

properties of the semigroup and the approximation of crandall liggett associated with a class of differential equations with delays depending implicitly on the state abstract in this paper we are concerned with the semigroup properties and the crandall liggett approximation associated with a class of differential equations with delays depending implicitly on the state the existence of a strongly continuous nonlinear semigroup associated with the solutions of the equations is proved various properties of this semigroup have been established such as the continuous dependence the compactness and the infinitesimal generators an approximation of crandall liggett type has also been established

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

feedback stabilisation of an euler bernoulli beam with the boundary time delay disturbance in this paper we consider the feedback stabilisation of an euler bernoulli beam with the boundary time delay disturbance due to unknown time delay coefficient the system might be exponentially i

uniform stability for thermoelastic systems with boundary time varying delay abstract in this paper we consider a thermoelastic system with boundary time varying delay using the energy method we show under suitable assumptions that the damping effect through heat conduction is still strong enough to uniformly stabilize the system even in the presence of boundary time varying delay our result improves earlier results existing in the literature

on multiindices and multivariables presentation of the voigt functions this paper aims at presenting multiindices and multivariables study of the unified or generalized voigt functions which play an important role in the several diverse field of physics such as astrophysical spectroscopy and the theory of neutron reactions some expressions representations of these functionl are given in terms of familiar special functions of multivariables further representations and series expansions involving multidimensional classical polynomials laguerre and hermite of mathematical physics are established

spatial structures and periodic travelling waves in an integro differential reaction diffusion population model an integro differential reaction diffusion equation is proposed as a model for populations where local aggregation is advantageous but intraspecific competition increases as global populations increase it is claimed that this is inherently more realistic than the usual kind of reaction diffusion model for mobile populations three kinds of bifurcation from the uniform steady state solution are considered i to steady spatially periodic structures ii to periodic standing wave solutions and iii to periodic travelling wave solutions these correspond to aggregation and motion of populations

nonlinear partial functional differential equations existence and stability existence and uniqueness of solutions for a class of nonlinear functional differential equations in hilbert spaces are established sufficient conditions which guarantee the transference of exponential stability from partial differential equations to partial functional differential equations are studied the stability results derived are applied also to ordinary differential equations with hereditary characteristics

large diffusivity and asymptotic behavior in parabolic systems abstract for systems of reaction diffusion equations with neumann boundary conditions it is shown that the solutions are asymptotic to the solutions of an ordinary differential equation if the diffusivity is large the methods apply also to reaction diffusion systems with time delays

periodic solutions to functional evolution equations the object of the present paper is the study of a class of functional evolution equations of the type u t a t ut u t 0 t r where a t are nonlinear operators acting on a banach space x we establish under certain additional assumptions on x and the operator a the existence of a periodic generalized solution

on the existence of solutions for functional evolution equations soit x un espace de banach de norme soit pc l espace des fonctions continues par morceaux r 0 x pour un r fixe 0 pour pc on pose pc sup r 0 on considere l equation d evolution fonctionnelle non lineaire abstraite g t u t u t a t u t u 0 t 0 t

nonlinear functional differential equations of monotone type in hilbert spaces abstract in this paper we study a class of semilinear functional evolution equations with finite delay in which the nonlinearity satisfies the weak condition of demicontinuity with respect to functional variable and also a semimonotone condition we will prove the existence uniqueness and measurability of the mild solutions based on an extension of the analogous results for non delay initial value problems on banach spaces and a version of recently developed random schauder s fixed point theorem this will be done first for the generalized solutions and then we follow an approximating procedure to obtain the same results for the mild solutions

a class of generalized hypergeometric summations abstract summations over the positive integers n of the generalized hypergeometric expressions 1 n p f p 1 n 2 x 2 x 0 are derived in closed form the specialization p 0 for example reduces to known results for schlomilch series in addition we record the apparently not readily available sine and cosine transforms of p f p 1 b 2 x 2 b 0 the latter of which is used together with a form of the poisson summation formula to deduce the aforementioned results

global existence asymptotic stability and uniform attractors for non autonomous thermoelastic systems with constant time delay in this paper we first consider a non autonomous thermoelastic system with a boundary delay and obtain the asymptotic behavior then using the contractive function method and multiplier techniques we establish the asymptotic behavior and the existence of a uniform attractor of the system with a boundary time delay the main advantage of this method is that we only need to verify compactness condition with the type of energy estimates same as those for establishing the absorbing set moreover we investigate an alternative result of solutions to semilinear thermoelastic systems by virtue of the semigroup method

functional differential equations of delay type and nonlinear evolution in lp spaces abstract nonlinear nonautonomous functional differential equations of delay type with lp initial functions is studied by means of the theory of evolution operators in lp spaces using the evolution operator in c it is shown that the evolution operator constructed in lp rn determines the solutions of the initial value problem in the exact sense a variation of parameters formula is developed for the nonlinear perturbed equation

existence results for partial neutral functional differential equations with unbounded delay abstract we prove the existence of mild solutions for a partial neutral functional integrodifferential equation with unbounded delay using the leray schauder alternative

general decay of the solution for a viscoelastic wave equation with a time varying delay term in the internal feedback in this paper we consider a viscoelastic wave equation with a time varying delay term the coefficient of which is not necessarily positive by introducing suitable energy and lyapunov functionals under suitable assumptions we establish a general energy decay result from which the exponential and polynomial types of decay are only special cases

navier stokes equations with delays some results on the existence and uniqueness of solutions to navier stokes equations when the external force contains some hereditary characteristics are proved