unbounded perturbation of the exponential dichotomy for evolution equations abstract in this paper we prove that the exponential dichotomy for evolution equations in banach spaces is not destroyed if we perturb the equation by small unbounded linear operator this is done by employing a skew product semiflow technique and a perturbation principle from linear operator theory finally we apply these results to partial parabolic equations and functional differential equations

existence and roughness of exponential dichotomies of linear dynamic equations on time scales in this paper we define the exponential dichotomy of linear dynamic equations on time scales then we present perturbation theorems on the roughness of exponential dichotomy and develop several explicit sufficient criteria for linear dynamic equations to have an exponential dichotomy as applications of the criteria of exponential dichotomy we derive some new sufficient conditions for the existence of periodic solutions of semi linear dynamic equations and nonlinear dynamic equations on time scales

an ergodic theorem in the qualitative behavior of non linear semiflows abstract take to be a continuous semiflow on the locally compact metric space and let a be a family of possibly unbounded densely defined closed operators on the banach space x we show that the solutions of the variational equation u a t u t r have a uniform exponential decay if and only if there exist p 1 and a sequence of positive real numbers n n n with n satisfying lim n n n 0 n x p d 0 uniformly w r t for all x x where t t is a strongly continuous cocycle solving in the classical sense the above variational equation thus we generalize some known results obtained by datko and pazy

existence and roughness of the exponential dichotomy for skew product semiflow in banach spaces abstract in this paper we introduce a concept of exponential dichotomy for skew product semiflow in infinite dimensional banach spaces which is an extension of the classic concept for evolution operators this concept is used to study the roughness property of the skew product semiflow also we introduce the concept of discrete skew product and give a necessary and sufficient condition for this discrete skew product to have a discrete dichotomy after that we give necessary and sufficient conditions for the existence of exponential dichotomy for skew product semiflow moreover we prove that the exponential dichotomy for skew product semiflow is not destroyed by small perturbation finally we apply these results to parabolic partial differential equations and functional differential equations

robustness of discrete dynamics via lyapunov sequences for a nonautonomous dynamics with discrete time defined by a sequence of matrices we give a complete characterization of nonuniform exponential contractions and nonuniform exponential dichotomies in terms of lyapunov sequences we note that these include as very special cases uniform exponential contractions and uniform exponential dichotomies due to the central role played by these properties in a substantial part of the theory of dynamical systems in particular in connection with the study of stable and unstable invariant manifolds it is important to have available optimal characterizations that are more amenable to check whether a given dynamics has such a property we also obtain inverse theorems that give explicitly lyapunov sequences for a given contraction or dichotomy as a nontrivial application we establish the robustness under sufficiently small linear perturbations both of nonuniform exponential contractions and nonuniform exponential dichotomies we emphasize that when compared to former work our proof of the robustness property is much simpler

admissibility and exponential trichotomy of dynamical systems described by skew product flows abstract the aim of this paper is to present a new and very general method for the detection of the uniform exponential trichotomy of dynamical systems the investigation is done in several constructive stages that correspond to three admissibility properties that are progressively introduced with respect to an associated input output system we prove that the uniform admissibility of the pair c b r x l 1 r x for the associated system is a sufficient condition for the existence of a uniform trichotomic behavior of the initial dynamical system if p 1 and the pair c b r x l 1 r x is uniformly p admissible then we obtain the uniform exponential trichotomy next we study whether the admissibility conditions are also necessary for the uniform exponential trichotomy supposing that a dynamical system has a uniform exponential trichotomy we prove that the associated input output system has unique bounded solutions in certain subspaces finally we obtain that the uniform p admissibility of the pair c b r x l 1 r x is a necessary and sufficient condition for uniform exponential trichotomy

lyapunov functions for trichotomies with growth rates abstract we consider linear equations x a t x that may exhibit stable unstable and central behaviors in different directions with respect to arbitrary asymptotic rates e c t determined by a function t for example the usual exponential behavior with t t is included as a very special case and when t log t we obtain a polynomial behavior we emphasize that we also consider the general case of nonuniform exponential behavior which corresponds to the existence of what we call a nonuniform exponential trichotomy this is known to occur in a large class of nonautonomous linear equations our main objective is to give a complete characterization in terms of strict lyapunov functions of the linear equations admitting a nonuniform exponential trichotomy this includes criteria for the existence of a nonuniform exponential trichotomy as well as inverse theorems providing explicit strict lyapunov functions for each given exponential trichotomy in the particular case of quadratic lyapunov functions we show that the existence of strict lyapunov sequences can be deduced from more algebraic relations between the quadratic forms defining the lyapunov functions as an application of the characterization of nonuniform exponential trichotomies in terms of strict lyapunov functions we establish the robustness of nonuniform exponential trichotomies under sufficiently small linear perturbations we emphasize that in comparison with former works our proof of the robustness is much simpler even when t t

skew product semi flows and non autonomous control systems in this paper we give a necessary and sufficient conditions for exact and approximate controllability of a wide class of linear infinite dimensional non autonomous control systems this is done by employing skew product semi flow technique finally we apply these results to prove the controllability of a broad class of non autonomous reaction diffusion equations in hilbert spaces

robust nonuniform dichotomies and parameter dependence abstract we establish the robustness of linear cocycles with an exponential dichotomy under sufficiently small lipschitz perturbations in the sense that the existence of an exponential dichotomy for a given cocycle persists under these perturbations we consider cocycles in banach spaces as well as the general case of nonuniform exponential dichotomies and also the general case of an exponential behavior e c n given by an arbitrary sequence n including the usual exponential behavior n n as a very special case moreover we show that the projections of the exponential dichotomies obtained from the perturbation vary continuously with the parameter and in fact that they are locally lipschitz on finite dimensional parameters

nonuniform h k μ ν dichotomy with applications to nonautonomous dynamical systems the paper develops and studies a very general notion of dichotomy referred to as nonuniform h k dichotomy the new notion contains as special cases most versions of dichotomy existing in the literature the paper then provides corresponding new versions of robustness hartman grobman theorem and stable manifold theorem for nonautonomous dynamical systems in banach spaces in term of the nonuniform h k dichotomy

robustness via lyapunov functions for nonautonomous linear equations x a t x we give a complete characterization of nonuniform exponential dichotomies in terms of strict quadratic lyapunov functions nonuniform exponential dichotomies include as a very special case uniform exponential dichotomies in particular we construct explicitly strict lyapunov functions for each exponential dichotomy as a nontrivial application we establish in a simple and direct manner the robustness of nonuniform exponential dichotomies under sufficiently small linear perturbations this represents a considerable simplification of former work

on the scaling limits at zeros of solutions of parabolic equations abstract we are concerned with the local scaling limits near zero points of solutions of general parabolic equations we show that at each zero point of finite vanishing order the solution is asymptotic to a homogeneous polynomial solution of the osculating equation with constant coefficients this provides useful information about the local structure of zero sets of solutions in particular we give upper bounds for hausdorff dimensions of zero sets of weak solutions our analysis combines a homothety scaling argument with the theory of nonautonomous dynamical systems

lp r x lq r x admissibility and exponential dichotomies of cocycles we analyze the existence of eventually no past exponential and ordinary dichotomies of an exponentially bounded strongly continuous cocycle t t r over a continuous semiflow our main tool is the lp r x lq r x admissibility condition i e there exist p q 1 p q 1 such that for each input f lp r x and there exists x x such that the output u x f r x u t x f t x 0t s t s f s ds belongs to lq r x we prove that the above admissibility condition implies that the output is bounded above by the input but nonuniformly with respect to requiring that the boundedness to be uniform with respect to we prove that the above admissibility condition assures the existence of a no past exponential dichotomy for t t r variants for ordinary dichotomy and also complete characterizations for the exponential dichotomy of cocycles are obtained it is worth to note that we involve a concept of a no past exponential dichotomy for cocycles weaker than the well known concept defined by sacker and sell 1994 33 our definition of exponential dichotomy follows partially the definition given in chow and leiva 1996 2 in the sense that we allow the unstable subspace to have infinite dimension the main difference is that we do not assume a priori that the cocycle is invertible on the unstable space actually we do not even assume that the unstable space is invariant under the cocycle thus we generalize some known results due to perron 1930 23 daleckij and krein 1974 6 massera and schaffer 1966 14 nguyen van minh rabiger and r schnaubelt 1998 18

robustness of nonuniform μ ν dichotomies in banach spaces abstract this paper focuses on the linear perturbation of dynamical systems in banach spaces which corresponds to a nonautonomous dynamical system admitting a nonuniform dichotomy we establish the robustness or roughness of nonuniform dichotomies in the sense that the linear dynamical system persists under a sufficiently small linear perturbation

approximating lyapunov exponents and sacker sell spectrum for retarded functional differential equations we consider lyapunov exponents and sacker sell spectrum for linear nonautonomous retarded functional differential equations posed on an appropriate hilbert space a numerical method is proposed to approximate such quantities based on the reduction to finite dimension of the evolution family associated to the system to which a classic discrete qr method is then applied the discretization of the evolution family is accomplished by a combination of collocation and generalized fourier projection a rigorous error analysis is developed to bound the difference between the computed stability spectra and the exact stability spectra the efficacy of the results is illustrated with some numerical examples

globally positive solutions of linear parabolic pdes of second order with robin boundary conditions it is shown that globally positive solutions of a linear second order parabolic partial differential equation on a bounded domain with robin boundary conditions are unique up to multiplication by a positive constant

necessary and sufficient criteria for the existence of exponential dichotomy on time scales necessary and sufficient criteria are established for the existence of exponential dichotomies for linear dynamic equations on time scales and are applied to derive some perturbation theorems on the roughness of exponential dichotomy

global attractivity in concave or sublinear monotone infinite delay differential equations abstract we study the dynamical behavior of the trajectories defined by a recurrent family of monotone functional differential equations with infinite delay and concave or sublinear nonlinearities we analyze different sceneries which require the existence of a lower solution and of a bounded trajectory ordered in an appropriate way for which we prove the existence of a globally asymptotically stable minimal set given by a 1 cover of the base flow we apply these results to the description of the long term dynamics of a nonautonomous model representing a stage structured population growth without irreducibility assumptions on the coefficient matrices

transversality of homoclinic orbits and exponential dichotomies for parabolic equations abstract in this paper we discuss the existence of exponential dichotomies on r of linear parabolic equations depending on small parameters and provide a tool of proving the transversality of the homoclinic orbits of parabolic equations and by making use of the results on exponential dichotomies of this paper we investigate the transversality of homoclinic orbits for parabolic equations

existence and robustness of nonuniform h k μ ν dichotomies for nonautonomous impulsive differential equations abstract for the linear nonautonomous impulsive differential equation we propose a new notion called the nonuniform h k dichotomy which extends the notion of dichotomy of uniform and nonuniform types and is related to the theory of nonuniform hyperbolicity we establish a sufficient criterion for the existence of the nonuniform h k dichotomy in terms of appropriate lyapunov exponents for the linear perturbation we develop the robustness or roughness of nonuniform h k dichotomies in the sense that a nonuniform h k dichotomy persists under a sufficiently small linear perturbation

spectral theory for general nonautonomous random dispersal evolution operators abstract we investigate the spectral theory of the following general nonautonomous evolution equation t u t x a u t x h t x u t x x d where d is a bounded subset of r n which can be a smooth domain or a discrete set a is a general linear dispersal operator for example a laplacian operator an integral operator with positive kernel or a cooperative discrete operator and h t x is a smooth function on r d we first study the influence of time dependence on the principal spectrum of dispersal equations and show that the principal lyapunov exponent of a time dependent dispersal equation is always greater than or equal to that of the time averaged one several results about the principal eigenvalue of time periodic parabolic equations are extended to general time periodic dispersal ones finally the investigation is generalized to random time dependent dispersal equations

exponential trichotomy and p admissibility for evolution families on the real line the aim of this paper is to obtain necessary and sufficient conditions for uniform exponential trichotomy of evolution families on the real line we prove that if p 1 and the pair c b r x c c r x is uniformly p admissible for an evolution family open image in new window u t s t s then open image in new window is uniformly exponentially trichotomic after that we analyze when the uniform p admissibility of the pair c b r x c c r x becomes a necessary condition for uniform exponential trichotomy as applications of these results we study the uniform exponential dichotomy of evolution families we obtain that in certain conditions the admissibility of the pair c b r x l p r x for an evolution family open image in new window u t s t s is equivalent with its uniform exponential dichotomy

on the asymptotic behavior of discrete dynamical systems an ergodic theory approach abstract the aim of this paper is to present a new and general method for the study of stability expansiveness and dichotomy of dynamical systems based on arguments proceeding from ergodic theory and on admissibility properties we consider a discrete variational system and we associate to it three different families of input output systems corresponding to the investigations for stability expansiveness and dichotomy respectively we introduce some input output conditions relative to ergodic measures and we show that certain nonuniform admissibility properties imply the existence of the uniform asymptotic behaviors first we obtain characterizations for uniform exponential stability as an application we prove that in order to preserve the uniform exponential stability subject to perturbations it is sufficient for the perturbed system to be close enough to the initial system on a measurable set next we deduce criteria for uniform exponential expansiveness based on some exact admissibilities relative to ergodic measures finally we present an interesting application to the study of the uniform exponential dichotomy starting from a nonuniform admissibility property and using stability criteria

asymptotic stability of harmonic maps under the schrodinger flow for schr odinger maps from r 2 r to the 2 sphere s 2 it is not known if finite energy solutions can form singularities blow up in finite time we consider equivariant solutions with energy near the energy of the two parameter family of equivariant harmonic maps we prove that if the topological degree of the map is at least four blowup does not occur and global solutions converge in a dispersive sense i e scatter to a fixed harmonic map as time tends to infinity the proof uses among other things a time dependent splitting of the solution the generalized hasimoto transform and strichartz dispersive estimates for a certain two space dimensional linear schr odinger equation whose potential has critical power spatial singularity and decay along the way we establish an energy space local well posedness result for which the existence time is determined by the length scale of a nearby harmonic map

nonlinear criteria for the existence of the exponential trichotomy in infinite dimensional spaces abstract in this paper we obtain for the first time nonlinear conditions for the existence of the exponential trichotomy of skew product flows in infinite dimensional spaces we treat the most general case without any additional assumptions concerning the cocycle and without assuming a priori the existence of the projection families we show that an inedit assembly of integral conditions imply the existence of the exponential trichotomy with all of its properties and we prove that the imposed conditions are also necessary our results generalize the previous studies on this topic and provide as particular cases many interesting situations among which we mention the detection of the exponential trichotomy of general non autonomous systems

input output admissibility and exponential trichotomy of difference equations abstract we propose a new method for the study of the asymptotic behavior of difference equations in infinite dimensional spaces providing characterizations for the property of uniform exponential trichotomy we deduce the structure of the stable unstable and bounded subspace and prove the uniqueness of the projection families we introduce a new admissibility concept with respect to a discrete input output system and prove that this is a necessary and sufficient condition for the existence of uniform exponential trichotomy throughout the paper there is no assumption on the coefficients and the obtained results are applicable to any class of difference equations

robustness of general dichotomies abstract for a linear equation v a t v we consider general dichotomies that may exhibit stable and unstable behaviors with respect to arbitrary asymptotic rates e c t for some function t this includes as a special case the usual exponential behavior when t t we also consider the general case of nonuniform exponential dichotomies we establish the robustness of the exponential dichotomies in banach spaces in the sense that the existence of an exponential dichotomy for a given linear equation persists under sufficiently small linear perturbations we also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy

exponential forward splitting for noninvertible difference equations abstract while the basic theory of exponential dichotomies deals with differential and difference equations with uniquely determined forward and backward solutions nowadays applications require to have a corresponding theory for equations whose backward solutions are not guaranteed to exist or to be unique in this paper we study the problem of how to generalize the notion of exponential dichotomy to difference equations whose solutions are only assumed to exist in forward time it turns out that for this kind of equation a straightforward generalization of the notion of exponential dichotomy does not provide satisfactory results but that on the other hand with an additional assumption that desired goal can be reached

uniform and strict persistence in monotone skew product semiflows with applications to non autonomous nicholson systems abstract we determine sufficient conditions for uniform and strict persistence in the case of skew product semiflows generated by solutions of non autonomous families of cooperative systems of odes or delay fdes in terms of the principal spectrums of some associated linear skew product semiflows which admit a continuous separation our conditions are also necessary in the linear case we apply our results to a noncooperative almost periodic nicholson system with a patch structure whose persistence turns out to be equivalent to the persistence of the linearized system along the null solution

robustness of nonuniform exponential trichotomies in banach spaces abstract we establish the robustness under sufficiently small linear perturbations of nonuniform exponential trichotomies defined by linear equations x a t x in banach spaces we also establish the continuous dependence on the perturbation of the constants in the notion of trichotomy we consider both trichotomies in semi infinite intervals and trichotomies in r

centered and staggered fourier derivatives and hilbert transforms when ringing derivatives are used in seismic modeling migration and other wave propagation problems in heterogeneous media they interact with boundaries of blocks or layers of different physical properties generating noncausal and nonphysical waves these spurious waves can be as strong as reflected or converted waves and other legitimate physical events leading to errors and misinterpretations therefore compact i e short and nonringing derivatives should be used

nonuniform mean square exponential dichotomies and mean square exponential stability abstract in this paper the existence conditions of nonuniform mean square exponential dichotomy nms ed for a linear stochastic differential equation sde are established the difference of the conditions for the existence of a nonuniform dichotomy between an sde and an ordinary differential equation ode is that the first one needs an additional assumption nonuniform lyapunov matrix to guarantee that the linear sde can be transformed into a decoupled one while the second does not therefore the first main novelty of our work is that we establish some preliminary results to tackle the stochasticity this paper is also concerned with the mean square exponential stability of nonlinear perturbation of a linear sde under the condition of nonuniform mean square exponential contraction nms ec for this purpose the concept of second moment regularity coefficient is introduced this concept is essential in determining the stability of the perturbed equation and hence we deduce the lower and upper bounds of this coefficient our results imply that the lower and upper bounds of the second moment regularity coefficient can be expressed solely by the drift term of the linear sde

uniformly rd piecewise almost periodic functions with applications to the analysis of impulsive δ dynamic system on time scales we introduce some new concepts of uniformly rd piecewise almost periodic functions on time scales and obtain some properties some new estimation inequalities and completely new theorems are established on time scales for impulsive dynamic systems several applications are given to illustrate our feasible results in the present work we introduce two equivalent concepts of uniformly rd piecewise almost periodic functions on time scales and their equivalence is proved based on this some basic properties of them are obtained then some new criteria of exponential dichotomy are established for homogeneous dynamic system on time scales also some completely new theorems are established on time scales for impulsive almost periodic dynamic systems such as favard s theorem and exponential dichotomy theorem as applications we provide a method to obtain an almost periodic solution for a given nonhomogeneous impulsive dynamic system furthermore we introduce an impulsive non autonomous nicholson s blowflies system model with patch structure and multiple nonlinear harvesting terms for which the existence and exponential stability of almost periodic solutions are studied which shows that our results can be applied feasibly and effectively

robustness of nonuniform exponential dichotomies in banach spaces we give conditions for the robustness of nonuniform exponential dichotomies in banach spaces in the sense that the existence of an exponential dichotomy for a given linear equation x a t x persists under a sufficiently small linear perturbation we also establish the continuous dependence with the perturbation of the constants in the notion of dichotomy and of the angles between the stable and unstable subspaces our proofs exhibit implicitly the exponential dichotomies of the perturbed equations in terms of fixed points of appropriate contractions we emphasize that we do not need the notion of admissibility of bounded nonlinear perturbations we also obtain related robustness results in the case of nonuniform exponential contractions in addition we establish an appropriate version of robustness for nonautonomous dynamical systems with discrete time

existence and robustness of exponential dichotomy of linear skew product semiflows over semiflows in this paper we investigate the exponential dichotomy of linear skew product semiflows over semiflows by considering the operators generated by the integral equation related to strongly continuous cocycles over metric spaces acting on banach bundles we characterize the existence of exponential dichotomy by properties of these operators and use this characterization to prove the robustness of exponential dichotomy

a version of a theorem of r datko for stability in average in this note we obtain a version of the well known theorem of r datko for the notion of the exponential stability in average we consider both cocycles over flows as well as cocycles over maps

nonuniform behavior and robustness we establish the robustness of linear cocycles in banach spaces admitting a nonuniform exponential dichotomy we first obtain robustness results for positive and negative time by establishing exponential behavior along certain subspaces and showing that the associated sequences of projections have bounded exponential growth we then establish a robustness result in z by constructing explicitly appropriate projections on the stable and unstable subspaces we emphasize that in general these projections may be different from those obtained separately from the robustness for positive and negative time we also consider the case of strong nonuniform exponential dichotomies

global and cocycle attractors for non autonomous reaction diffusion equations the case of null upper lyapunov exponent abstract in this paper we obtain a detailed description of the global and cocycle attractors for the skew product semiflows induced by the mild solutions of a family of scalar linear dissipative parabolic problems over a minimal and uniquely ergodic flow we consider the case of null upper lyapunov exponent for the linear part of the problem then two different types of attractors can appear depending on whether the linear equations have a bounded or an unbounded associated real cocycle in the first case e g in periodic equations the structure of the attractor is simple whereas in the second case which occurs in aperiodic equations the attractor is a pinched set with a complicated structure we describe situations when the attractor is chaotic in measure in the sense of li yorke besides we obtain a non autonomous discontinuous pitchfork bifurcation scenario for concave equations applicable for instance to a linear dissipative version of the chafee infante equation

discretized characterizations of exponential dichotomy of linear skew product semiflows over semiflows using the method of discretization we investigate the necessary and sufficient conditions for the existence of exponential dichotomy of linear skew product semiflows over semiflows through the existence of discrete exponential dichotomy of the discretized linear skew product semiflows we then apply the obtained results to consider the roughness of exponential dichotomy of linear skew product semiflows

criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations abstract we prove that the admissibility of any pair of vector valued schaffer function spaces satisfying a very general technical condition implies the existence of a no past exponential dichotomy for an exponentially bounded strongly continuous cocycle over a semiflow roughly speaking the class of schaffer function spaces consists in all function spaces which are invariant under the right shift and therefore our approach addresses most of the possible pairs of admissible spaces complete characterizations for the exponential dichotomy of cocycles are also obtained moreover we involve a concept of a no past exponential dichotomy for cocycles weaker than the classical concept defined by sacker and sell 1994 in 23 our definition of exponential dichotomy follows partially the definition given by chow and leiva 1996 in 4 in the sense that we allow the unstable subspace to have infinite dimension the main difference is that we do not assume a priori that the cocycle is invertible on the unstable space actually we do not even assume that the unstable space is invariant under the cocycle thus we generalize some known results due to o perron 1930 14 j daleckij and m krein 1974 7 j l massera and j j schaffer 1966 11 n van minh f rabiger and r schnaubelt 1998 26

an a posteriori kam theorem for whiskered tori in hamiltonian partial differential equations with applications to some ill posed equations the goal of this paper is to develop a kam theory for tori with hyperbolic directions which applies to hamiltonian partial differential equations even to some ill posed ones the main result has an a posteriori format that is we show that if there is an approximate solution of an invariance equation which also satisfies some non degeneracy conditions then there is a true solution nearby this allows besides dealing with the quasi integrable case for the validation of numerical computations or formal perturbative expansions as well as for obtaining quasi periodic solutions in degenerate situations the a posteriori format also has other automatic consequences smooth dependence on parameters bootstrap of regularity etc we emphasize that the non degeneracy conditions required are just quantities evaluated on the approximate solution no global assumptions on the system such as twist hence they are readily verifiable in perturbation expansions we will pay attention to the quantitative relations between the sizes of the approximation and the non degeneracy conditions this will allow us to prove what experts call small twist theorems the non degeneracy conditions vanishes as the perturbation goes to zero but much slower than the error of the approximation the method of proof is based on an iterative method for solving a functional equation for the parameterization of the torus expressing that the range of the parameterization admits an evolution and is invariant we also solve functional equations for bundles which imply that are invariant under the linearization the iterative method does not use transformation theory nor action angle variables the main result does not assume that the system is close to integrable more surprisingly we do not need that the equations we study define an evolution for all initial conditions and are well posed even if the systems we study do not admit solutions for all initial conditions we show that there is a systematic way to choose initial conditions on which one can define an evolution which is quasi periodic we first develop an abstract theorem then we show how this abstract result applies to some concrete examples the examples considered in this paper are the scalar boussinesq equation and the boussinesq system both are pde models that aim to describe water waves in the long wave limit for these equations we construct small amplitude time quasi periodic solutions which are even in the spatial variable the strategy for the abstract theorem is inspired by that in fontich et al electron res announc math sci 16 9 22 2009 j differ equ 246 8 3136 3213 2009 the main part of the paper is to study infinite dimensional analogues of dichotomies which applies even to ill posed equations and which is stable under addition of unbounded perturbations this requires that we assume smoothing properties we also present very detailed bounds on the change of the splittings under perturbations

schaffer spaces and uniform exponential stability of linear skew product semiflows we study the exponential stability of linear skew product semiflows on locally compact metric space with banach fibers our main tool is the admissibility of a pair of the so called schaffer spaces this characterization is a very general one it includes as particular cases many interesting situations among them we can mention some results due to clark datko latushkin van minh montgomery smith randolph rabiger schnaubelt

minimal sets in monotone and concave skew product semiflows i a general theory abstract the long term dynamics of a general monotone and concave skew product semiflow is analyzed paying special attention to the region delimited from below by the graph of a semicontinuous subequilibrium or by a minimal set admitting a flow extension different possibilities arise depending on the existence and number or absence of minimal sets strongly above the initially fixed one as well as on the coexistence or not of bounded and unbounded semiorbits on the region previous results are unified and extended and significative differences with the sublinear case are established some scenarios which are impossible in the autonomous or periodic cases may occur in this setting

smooth robustness of parameterized perturbations of exponential dichotomies we establish the robustness of nonuniform exponential dichotomies in banach spaces under sufficiently small c1 parameterized perturbations moreover we show that the stable and unstable subspaces of the exponential dichotomies obtained from the perturbation are also of class c1 on the parameter thus yielding an optimal smoothness