invariant manifolds of partial functional differential equations this paper is concerned with the existence smoothness and attractivity of invariant manifolds for evolutionary processes on general banach spaces when the nonlinear perturbation has a small global lipschitz constant and locally ck smooth near the trivial solution such a nonlinear perturbation arises in many applications through the usual cut off procedure but the requirement in the existing literature that the nonlinear perturbation is globally ck smooth and has a globally small lipschitz constant is hardly met in those systems for which the phase space does not allow a smooth cut off function our general results are illustrated by and applied to partial functional differential equations for which the phase space c r 0 x where r 0 and x being a banach space has no smooth inner product structure and for which the validity of variation of constants formula is still an interesting open problem

almost automorphic mild solutions to some partial neutral functional differential equations and applications abstract the paper considers the existence and uniqueness of almost automorphic mild solutions to some classes of first order partial neutral functional differential equations sufficient conditions for the existence and uniqueness of almost automorphic mild solutions to the above mentioned equations are obtained as an application a first order boundary value problem arising in control systems is considered

transient policies in discrete dynamic programming linear programming including suboptimality tests and additional constraints this paper investigates the computation of transient optimal policies in discrete dynamic programming the model is quite general it may contain transient as well as nontransient policies and the transition matrices are not necessarily substochastic a functional equation for the so called transient value vector is derived and the concept of superharmonicity is introduced this concept provides the linear program to compute the transientvalue vector and a transient optimal policy we also discuss the elimination of suboptimal actions the solution of problems with additional constraints and the computation of an efficient policy for a multiple objective dynamic programming problem

bohr neugebauer type theorem for some partial neutral functional differential equations in this work we study the existence of almost periodic solutions for some partial neutral functional dierential equations using the variation of constants formula and the spectral decomposition of the phase space developed in 6 we prove that the existence of an almost periodic solution is equivalent to the existence of a bounded solution on r

finding the k best policies in a finite horizon markov decision process abstract directed hypergraphs represent a general modelling and algorithmic tool which have been successfully used in many different research areas such as artificial intelligence database systems fuzzy systems propositional logic and transportation networks however modelling markov decision processes using directed hypergraphs has not yet been considered in this paper we consider finite horizon markov decision processes mdps with finite state and action space and present an algorithm for finding the k best deterministic markov policies that is we are interested in ranking the first k deterministic markov policies in non decreasing order using an additive criterion of optimality the algorithm uses a directed hypergraph to model the finite horizon mdp it is shown that the problem of finding the optimal policy can be formulated as a minimum weight hyperpath problem and be solved in linear time with respect to the input data representing the mdp using different additive optimality criteria

approximation by smooth functions with no critical points on separable banach spaces abstract we characterize the class of separable banach spaces x such that for every continuous function f x r and for every continuous function e x 0 there exists a c 1 smooth function g x r for which f x g x e x and g x 0 for all x x that is g has no critical points as those infinite dimensional banach spaces x with separable dual x we also state sufficient conditions on a separable banach space so that the function g can be taken to be of class c p for p 1 2 in particular we obtain the optimal order of smoothness of the approximating functions with no critical points on the classical spaces l p n and l p r n some important consequences of the above results are 1 the existence of a non linear hahn banach theorem and the smooth approximation of closed sets on the classes of spaces considered above and 2 versions of all these results for a wide class of infinite dimensional banach manifolds

rate of convergence of empirical measures and costs in controlled markov chains and transient optimality the purpose of this paper is two fold first bounds on the rate of convergence of empirical measures in controlled markov chains are obtained under some recurrence conditions these include bounds obtained through large deviations and central limit theorem arguments these results are then applied to optimal control problems bounds on the rate of convergence of the empirical measures that are uniform over different sets of policies are derived resulting in bounds on the rate of convergence of the costs finally new optimal control problems that involve not only average cost criteria but also measures on the transient behavior of the cost namely the rate of convergence are introduced and applied to a problem in telecommunications the solution to these problems rely on the bounds introduced in previous sections

denumerable constrained markov decision processes and finite approximations the purpose of this paper is two fold first to establish the theory of discounted constrained markov decision processes with a countable state and action spaces with general multi chain structure second to introduce finite approximation methods we define the occupation measures and obtain properties of the set of all achievable occupation measures under the different admissible policies we establish the optimality of stationary policies for the constrained control problem and obtain an lp with a countable number of decision variables through which stationary optimal policies are computed since for such an lp one cannot expect to find an optimal solution in a finite number of operations we present two schemes for finite approximations and establish the convergence of optimal values and policies for both the discounted and the expected average cost with unbounded cost sometimes it turns out to be easier to solve the problem with infinite state space than the problem with finite yet large state space based on the optimal policy for the problem with infinite state space we construct policies which are almost optimal for the problem with truncated state space this method is applied to obtain an e optimal policy for a problem of optimal priority assignment under constraints for a system of k finite queues

constrained markov decision models with weighted discounted rewards this paper deals with constrained optimization of markov decision processes both objective function and constraints are sums of standard discounted rewards but each with a different discount factor such models arise e g in production and in applications involving multiple time scales we prove that it a feasible policy exists then there exists an optimal policy which is i stationary nonrandomized from some step onward ii randomized markov before this step but the total number of actions which are added by randomization is bounded by the number of constraints optimality of such policies for multi criteria problems is also established these new policies have the pleasing aesthetic property that the amount of randomization they require over any trajectory is restricted by the number of constraints this result is new even for constrained optimization with a single discount factor where the optimality of randomized stationary policies is known however a randomized stationary policy may req

continuity of optimal values and solutions for control of markov chains with constraints we consider in this paper constrained markov decision processes this type of control model has many applications in telecommunications and other fields e altman and a shwartz ieee trans automat control 34 1989 pp 1089 1102 e a feinberg and m i reiman probab engrg inform sci 8 1994 pp 463 489 a hordijk and f spieksma adv in appl probab 21 1989 pp 409 431 a lazar ieee trans automat control 28 1983 pp 1001 1007 p nain and k w ross ieee trans automat control 31 1986 pp 883 888 k w ross and b chen ieee trans automat control 33 1988 pp 261 267 we address the issue of the convergence of the value and optimal policies of the problem with discounted costs to the ones for the problem with expected average cost we consider the general multichain ergodic structure we present two stability results in this paper we establish the continuity of optimal values and solutions of as well as some type of robustness of some suboptimal solutions in the discount factor our proof relies on same general theory on continuity of values and solutions in convex optimization that relies on well known notions of gamma convergence

an algorithm to identify and compute average optimal policies in multichain markov decision processes this paper concerns discrete time finite state multichain mdps with compact action sets the optimality criterion is long run average cost simple examples illustrate that optimal stationary markov policies do not always exist we establish the existence of e optimal policies that are stationary markovian and develop an algorithm that computes these approximate optimal policies we establish a necessary and sufficient condition for the existence of an optimal policy that is stationary markovian and in case that such an optimal policy exists the algorithm computes it

control of a hybrid stochastic system abstract we consider in this paper a continuous time stochastic hybrid control system with a finite time horizon the objective is to minimize a linear function of the expected state trajectory the state evolves according to a linear dynamics however the parameters of the state evolution equation may change at discrete times according to a controlled markov chain which has finite state and action spaces we use a procedure similar in form to the maximum principle this determines a control strategy which is asymptotically optimal as the number of transitions during the finite time horizon grows to infinity

on the empirical state action frequencies in markov decision processes under general policies we consider the empirical state action frequencies and the empirical reward in weakly communicating finite state markov decision processes under general policies we define a certain polytope and establish that every element of this polytope is the limit of the empirical frequency vector under some policy in a strong sense furthermore we show that the probability of exceeding a given distance between the empirical frequency vector and the polytope decays exponentially with time under every policy we provide similar results for vector valued empirical rewards

a massera type theorem for almost automorphic solutions of differential equations abstract in this note we present a massera type theorem for the existence of almost automorphic solutions of periodic linear evolution equations of the form x t a t x t f t where a t is unbounded linear operator depending on t periodically and generates a periodic evolutionary process f is almost automorphic the main results are stated in terms of the almost automorphy of solutions and their carleman spectra

multichain markov decision processes with a sample path constraint a decomposition approach we consider finite state finite action markov decision processes which accumulate both a reward and a cost at each decision epoch we study the problem of finding a policy that maximizes the expected long run average reward subject to the constraint that the long run average cost be no greater than a given value with probability one we establish that if there exists a policy that meets the constraint then there exists an e optimal stationary policy furthermore an algorithm is outlined to locate the e optimal stationary policy the proof of the result hinges on a decomposition of the state space into maximal recurrent classes and a set of transient states

adaptive control of constrained markov chains adaptive control of finite state markov chains is discussed the optimal performance is characterized through the minimization of a long run average cost functional subject to constraints on several other such functionals under mild structural and feasibility conditions two explicit adaptive control policies are exhibited for the case where the transition probabilities are unknown the policies are optimal under the constrained optimization criterion they rely on a powerful estimation scheme which provides consistent estimators for the transition probabilities this scheme is of independent interest as it provides strong consistency under a large number of adaptive schemes and is independent of any identifiability conditions as an application an optimal adaptive policy is derived for a system of k competing queues with countable state space for which the constrained criteria arise naturally in the context of communication networks

a variation of constants formula for abstract functional differential equations in the phase space abstract for linear functional differential equations with infinite delay in a banach space a variation of constants formula is established in the phase space as an application one applies it to study the admissibility of some spaces of functions whose spectra are contained in a closed subset of the real line

constrained undiscounted stochastic dynamic programming in this paper we investigate the computation of optimal policies in constrained discrete stochastic dynamic programming with the average reward as utility function the state space and action sets are assumed to be finite constraints which are linear functions of the state action frequencies are allowed in the general multichain case an optimal policy will be a randomized nonstationary policy an algorithm to compute such an optimal policy is presented furthermore sufficient conditions for optimal policies to be stationary are derived there are many applications for constrained undiscounted stochastic dynamic programming e g in multiple objective markovian decision models

optimal policies for controlled markov chains with a constraint abstract the time average reward for a discrete time controlled markov process subject to a time average cost constraint is maximized over the class of al causal policies each epoch a reward depending on the state and action is earned and a similarly constituted cost is assessed the time average of the former is maximized subject to a hard limit on the time average of the latter it is assumed that the state space is finite and the action space compact metric an accessibility hypothesis makes it possible to utilize a lagrange multiplier formulation involving the dynamic programming equation thus reducing the optimization problem to an unconstrained optimization parametrized by the multiplier the parametrized dynamic programming equation possesses compactness and convergence properties that lead to the following if the constraint can be satisfied by any causal policy the supremum over time average rewards respective to all causal policies is attained by either a simple or a mixed policy the latter is equivalent to choosing independently at each epoch between two specified simple policies by the throw of a biased coin

achieving target state action frequencies in multichain average reward markov decision processes in this paper we address a basic problem that arises naturally in average reward markov decision processes with constraints and or nonstandard payoff criteria given a feasible state action frequency vector the target construct a policy whose state action frequencies match those of the target vector while it is well known that the solution to this problem cannot in general be found in the space of stationary randomized policies we construct a solution that has ultimately stationary structure it consists of two stationary policies where the first one is used initially and then the switch to the second one is made at a certain random switching time the computational effort required to construct this solution is minimal we also show that our problem can always be solved by a stationary policy if the original mdp is extended by adding certain states and actions the solution in the original mdp is obtained by mapping the solution in the extended mdp back to the original process

extraction of critical points of smooth functions on banach spaces abstract let e be an infinite dimensional separable hilbert space we show that for every c 1 function f e r d every open set u with c f x e d f x is not surjective u and every continuous function e e 0 there exists a c 1 mapping e r d such that f x x e x for every x e f outside u and has no critical points c this result can be generalized to the case where e c 0 or e l p 1 p in the case e c 0 it is also possible to get that d f x d x e x for every x e

almost automorphic solutions for some partial functional differential equations abstract in this work we study the existence of almost automorphic solutions for some partial functional differential equations we prove that the existence of a bounded solution on r implies the existence of an almost automorphic solution our results extend the classical known theorem by bohr and neugebauer on the existence of almost periodic solutions for inhomegeneous linear almost periodic differential equations we give some applications to hyperbolic equations and lotka volterra type equations used to describe the evolution of a single diffusive animal species

on the energy delay tradeoff in streaming data finite blocklength analysis this paper investigates basic trade offs between energy and delay in wireless communication systems using finite blocklength theory we first assume that data arrive in constant stream of bits which are put into packets and transmitted over a communications link our results show that depending on exactly how energy is measured in general energy depends on sqrt d 1 or sqrt d 1 log d where d is the delay this means that the energy decreases quite slowly with increasing delay furthermore to approach the absolute minimum of 1 59 db on energy bandwidth has to increase very rapidly much more than what is predicted by infinite blocklength theory we then consider the scenario when data arrive stochastically in packets and can be queued we devise a scheduling algorithm based on finite blocklength theory and develop bounds for the energy delay performance our results again show that the energy decreases quite slowly with increasing delay

asymptotic optimization of a nonlinear hybrid system governed by a markov decision process we consider in this paper a continuous time stochastic hybrid control system with finite time horizon the objective is to minimize a nonlinear function of the state trajectory the state evolves according to a nonlinear dynamics the parameters of the dynamics of the system may change at discrete times l epsilon l 0 1 according to a controlled markov chain which has finite state and action spaces under the assumption that epsilon is a small parameter we justify an averaging procedure allowing us to establish that our problem can be approximated by the solution of some deterministic optimal control problem

smooth approximations without critical points of continuous mappings between banach spaces and diffeomorphic extractions of sets abstract let e f be separable hilbert spaces and assume that e is infinite dimensional we show that for every continuous mapping f e f and every continuous function e e 0 there exists a c mapping g e f such that f x g x e x and d g x e f is a surjective linear operator for every x e we also provide a version of this result where e can be replaced with a banach space from a large class including all the classical spaces with smooth norms such as c 0 l p or l p 1 p and f can be taken to be any banach space such that there exists a bounded linear operator from e onto f in particular for such e f every continuous mapping f e f can be uniformly approximated by smooth open mappings part of the proof provides results of independent interest that improve some known theorems about diffeomorphic extractions of closed sets from banach spaces or hilbert manifolds

the failure of rolle s theorem in infinite dimensional banach spaces we prove the following new characterization of cp lipschitz smoothness in banach spaces an infinite dimensional banach space x has a cp smooth lipschitz bump function if and only if it has another cp smooth lipschitz bump function f such that its derivative does not vanish at any point in the interior of the support of f that is f does not satisfy rolle s theorem moreover the support of this bump can be assumed to be a smooth starlike body the twisted tube method we use in the proof is interesting in itself as it provides other useful characterizations of cp smoothness related to the existence of a certain kind of deleting diffeomorphisms as well as to the failure of brouwer s fixed point theorem even for smooth self mappings of starlike bodies in all infinite dimensional spaces

randomized and past dependent policies for markov decision processes with multiple constraints the markov decision problem of locating a policy to maximize the long run average reward subject to k long run average cost constraints is considered it is assumed that the state and action spaces are finite and the law of motion is unichain that is every pure policy gives rise to a markov chain with one recurrent class it is first proved that there exists an optimal stationary policy with a degree of randomization no greater than k consequently it is never necessary to randomize in more than k states a linear program produces the optimal policy with limited randomization for the special case of a single constraint we also address the problem of finding optimal nonrandomized but nonstationary policies we show that a round robin type policy is optimal and conjecture the same for a steering policy that depends on the entire past history of the process but whose implementation requires essentially no more storage than that of a pure policy

markov decision processes with sample path constraints the communicating case we consider time average markov decision processes mdps which accumulate a reward and cost at each decision epoch a policy meets the sample path constraint if the time average cost is below a specified value with probability one the optimization problem is to maximize the expected average reward over all policies that meet the sample path constraint the sample path constraint is compared with the more commonly studied constraint of requiring the average expected cost to be less than a specified value although the two criteria are equivalent for certain classes of mdps their feasible and optimal policies differ for many nontrivial problems in general there does not exist optimal or nearly optimal stationary policies when the expected average cost constraint is employed assuming that a policy exists that meets the sample path constraint we establish that there exist nearly optimal stationary policies for communicating mdps a parametric linear programming algorithm is given to construct nearly optimal stationary policies the discussion relies on well known results from the theory of stochastic processes and linear programming the techniques lead to simple proofs of the existence of optimal and nearly optimal stationary policies for unichain and deterministic mdps respectively

smooth negligibility of compact sets in infinite dimensional banach spaces with applications this article deals with smooth removability of compact sets in infinite dimensional banach spaces the main result states that ifx is an infinite dimensional banach space which has a not necessarily equivalent cp smooth norm and k is a compact subset of x then x and x r k are cp diffeomorphic the proof relies on the construction of a deleting path through a nontrivial refinement of bessaga s incomplete norm technique however norms are not at present available and the construction requires the use of asymmetric functionals the noncompleteness of such functionals relies in turn on james theorem on existence of linear functionals which do not attain their norm on every nonreflexive space applications are given which show that several important theorems on finite dimensional spaces completely fail in the infinite dimensional case for instance on any banach space isomorphic to its cartesian square and for any natural number n 2 there exists a c1 diffeomorphism of pure period n with no fixed point this work opens the way to several interesting open questions on nonseparable banach spaces does every banach space with a c1 smooth norm admit a nonequivalent c1 smooth norm in which banach spaces is every compact subset the set where a certain c1 real valued function vanishes

boundedness and almost periodicity of solutions of partial functional differential equations abstract we study necessary and sufficient conditions for the abstract functional differential equation x ax fx t f t to have almost periodic quasi periodic solutions with the same structure of spectrum as f the main conditions are stated in terms of the imaginary solutions of the associated characteristic equations and the spectrum of the forcing term f the obtained results extend recent results to abstract functional differential equations

behavior near hyperbolic stationary solutions for partial functional differential equations with infinite delay abstract the aim of this work is to investigate the asymptotic behavior of solutions near hyperbolic stationary solutions for partial functional differential equations with infinite delay we suppose that the linear part satisfies the hille yosida condition on a banach space and it is not necessarily densely defined firstly we establish a new variation of constants formula for the nonhomogeneous linear equations secondly we use this formula and the spectral decomposition of the phase space to show the existence of stable and unstable manifolds the estimations of solutions on these manifolds are obtained for illustration we propose to study the stability of stationary solutions for the lotka volterra model with diffusion

reduction principle and dynamic behaviors for a class of partial functional differential equations abstract we investigate the dynamic of solutions in the norm for some nonhomogeneous linear partial functional differential equations we suppose that the undelayed homogeneous part is the infinitesimal generator of an analytic semigroup and the delayed part is continuous with respect to fractional powers of the generator we establish a reduction principle for the infinite dimensional system in order to reduce its qualitative analysis to a finite dimensional one our reduction method is based on a new variation of constants formula in application the reduced system is used to prove the existence of almost automorphic almost periodic and periodic solutions for the whole infinite dimensional system

markov decision problems and state action frequencies consider a controlled markov chain with countable state and action spaces basic quantities that determine the values of average cost functionals are identified under some regularity conditions these turn out to be a collection of numbers one for each state action pair describing for each state the relative number of uses of each action these conditional frequencies which are defined pathwise are shown to determine the state action frequencies that in the finite case are known to determine the costs this is extended to the countable case allowing for unbounded costs the space of frequencies is shown to be compact and convex and the extreme points are identified with stationary deterministic policies conditions under which the search for optimality in several optimization problems may be restricted to stationary policies are given these problems include the standard markov decision process as well as constrained optimization both in terms of average cost functionals and variability sensitive optimization an application to a queueing problem is given where these results imply the existence and explicit computation of optimal policies in constrained optimization problems the pathwise definition of the conditional frequencies implies that their values can be controlled directly moreover they depend only on the limiting behavior of the control this has immediate application to adaptive control of markov chains including adaptive control under constraints

variability sensitive markov decision processes considered are time average markov decision processes mdps with finite state and action spaces two definitions of variability are introduced namely the expected time average variability and time average expected variability the two criteria are in general different although they can both be employed to penalize for variance in the stream of rewards for communicating mdps we construct a randomized stationary policy that is e optimal for both criteria the policy is optimal and pure for a specific variability function for general multichain mdps a state space decomposition leads to a similar result for the expected time average variability we also consider the problem of the decision maker choosing the initial state along with the policy

simulation based optimization of markov decision processes an empirical process theory approach we generalize and build on the pac learning framework for markov decision processes developed in jain and varaiya 2006 we consider the reward function to depend on both the state and the action both the state and action spaces can potentially be countably infinite we obtain an estimate for the value function of a markov decision process which assigns to each policy its expected discounted reward this expected reward can be estimated as the empirical average of the reward over many independent simulation runs we derive bounds on the number of runs needed for the convergence of the empirical average to the expected reward uniformly for a class of policies in terms of the v c or pseudo dimension of the policy class we then propose a framework to obtain an e optimal policy from simulation we provide sample complexity of such an approach

time sharing policies for controlled markov chains we propose a class of nonstationary policies called policy time sharing pts which possesses several desirable properties for problems where the criteria are of the average cost type an optimal policy exists within this class the computation of optimal policies is straightforward and the implementation of this policy is easy while in the finite state case stationary policies are also known to share these properties the new policies are much more flexible in the sense that they can be applied to solve adaptive problems and they suggest new ways to incorporate the particular structure of the problem at hand into the derivation of optimal policies in addition they provide insight into the pathwise structure of controlled markov chains to use pts policies one alternates between the use of several stationary deterministic policies switching when reaching some predetermined state in some countable state cases optimal solutions of the pts type are available and easy to compute whereas optimal stationary policies are not available examples that illustrate the last point and the usefulness of the new approach are discussed involving constrained optimization problems with countable state space or compact action space

variance penalized markov decision processes we consider a markov decision process with both the expected limiting average and the discounted total return criteria appropriately modified to include a penalty for the variability in the stream of rewards in both cases we formulate appropriate nonlinear programs in the space of state action frequencies averaged or discounted whose optimal solutions are shown to be related to the optimal policies in the corresponding variance penalized mdp the analysis of one of the discounted cases is facilitated by the introduction of a cartesian product of two independent mdps

simulation based uniform value function estimates of markov decision processes the value function of a markov decision process mdp assigns to each policy its expected discounted reward this expected reward can be estimated as the empirical average of the reward over many independent simulation runs we derive bounds on the number of runs needed for the uniform convergence of the empirical average to the expected reward for a class of policies in terms of the vapnik chervonenkis or p dimension of the policy class further we show through a counterexample that whether we get uniform convergence or not for an mdp depends on the simulation method used uniform convergence results are also obtained for the average reward case for partially observed markov decision processes and can be easily extended to markov games the results can be viewed as a contribution to empirical process theory and as an extension of the probably approximately correct pac learning theory for partially observable mdps and markov games

variation of constants formula and reduction principle for a class of partial functional differential equations with infinite delay abstract in this work the dynamic behavior of solutions is investigated for a class of partial functional differential equations with infinite delay we suppose that the undelayed homogeneous part generates an analytic semigroup and the delayed part is continuous with respect to fractional powers of the generator firstly a variation of constants formula is obtained in the corresponding norm space which is mainly used to establish a reduction principle of complexity of the considered equation the reduction principle proves that the dynamics of the considered equation is governed by an ordinary differential equation in finite dimensional space as an application we investigate the existence of periodic almost periodic and almost automorphic solutions for the original equation

uniform approximation of continuous mappings by smooth mappings with no critical points on hilbert manifolds we prove that every continuous mapping from a separable infinite dimensional hilbert space x into r m can be uniformly approximated by c infinity smooth mappings with no critical points this kind of result can be regarded as a sort of strong approximate version of the morse sard theorem some consequences of the main theorem are as follows every two disjoint closed subsets of x can be separated by a one codimensional smooth manifold that is a level set of a smooth function with no critical points in particular every closed set in x can be uniformly approximated by open sets whose boundaries are c infinity smooth one codimensional submanifolds of x finally since every hilbert manifold is diffeomorphic to an open subset of the hilbert space all of these results still hold if one replaces the hilbert space x with any smooth manifold m modeled on x

hamiltonian cycles and singularly perturbed markov chains we consider the hamiltonian cycle problem embedded in a singularly perturbed markov decision process we also consider a functional on the space of deterministic policies of the process that consists of the 1 1 entry of the fundamental matrices of the markov chains induced by the same policies we show that when the perturbation parameter e is less than or equal to 1 n2 the hamiltonian cycles of the directed graph are precisely the minimizers of our functional over the space of deterministic policies in the process we derive analytical expressions for the possiblen distinct values of the functional over the typically much larger space of deterministic policies

variation of constants formula and almost periodic solutions for some partial functional differential equations with infinite delay abstract in this work we give a variation of constants formula for partial functional differential equations with infinite delay we assume that the linear part is not necessarily densely defined and its resolvent operator satisfies the hille yosida condition we establish a reduction of the problem to a finite dimensional space which allows us to prove the existence of almost periodic solutions

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