The Fredholm Alternative for Functional Dierential Equations of Mixed Typ e John MalletParet Division of Applied Mathematics Brown University Providence RI August Abstract We prove a Fredholm alternative theorem for a class of asymptotically hyp erb olic linear dierential dierence equations of mixed typ e We also establish the co cycle prop erty and the sp ectral ow prop erty for such equations providing an eective means of calculating the Fredholm index Such systems can arise from equations which describ e traveling waves in a spatial lattice Intro duction Our interest in this pap er is the linear functional dierential equation of mixed typ e N X x A x r h j j j Generally nonlinear nonlo cal dierential equations arise in the study of traveling waves in domains with nonlo cal interactions such as on a spatial lattice see for example and equations such as arise as their linearizations ab out particular solutions The p erturbation stability and bifurcation theory of such solutions via the LyapunovSchmidt pro cedure relies up on a Fredholm alternative for equation to which the present pap er is devoted With the exception of early work of Rustichini not much is known ab out general prop erties of such linear equations In Sacker and Sell consider a general class of innite dimensional systems and study prop erties related to exp onential dichotomies Although our results are somewhat in this spirit equation diers in that it is not an evolutionary system that is initial value problems are not wellp osed In a companion pap er our results are used to study the global structure of the set of traveling wave solutions on a spatial lattice We employ there a Melnikov metho d much in the spirit of earlier work involving global continuation techniques See the survey articles for a broader outline of the problems and for additional references We make as a standing assumption for that the d d complex matrix co ecients A j dd J C are measurable and uniformly b ounded on some usually innite interval J and that the d p inhomogeneity h J C is lo cally integrable We shall later imp ose other conditions in particular L b ounds on h The quantities r the socalled shifts can b e of either sign As a technical convenience j we assume that r r r j k N N j k that is the shifts are distinct with r and hence with r for j N We take these as j standing hyp otheses throughout this pap er In fact the conditions are not restrictions on the form of the equation but merely a matter of notation since any co ecient A is p ermitted to j vanish identically on J Denoting r min fr g r max fr g min j max j j N j N we observe that r r and r r min max min max We may write equation as x L x h where L for almost every J denotes the linear functional N X d L A r C r r C j j min max j d d from C r r C into C When the function h is absent then we have the homogeneous min max system x L x A sp ecial case of o ccurs with a constant co ecient op erator N X A r L j j j where each A is a xed matrix Here equation takes the form j N X x L x h A x r h j j j or simply x L x in the homogeneous case Our notation parallels very closely the standard notation of delay dierential equations for exam ple as set forth in the b o ok of Hale and Verduyn Lunel where we denote x x for r r Observe that for almost every the norm of the linear functional in is min max N X kL k jA j j j d and also that for any b ounded function x L L R C on the line we have the p ointwise estimate 1 jL x j kL kkxk L Asso ciated to equation we have the linear op erator dened by L N X x x A x r L j j j d for appropriately dierentiable x R C when we take J R As we are assuming a uniform p p d p b ound for the co ecients A it follows that h x b elongs to L L R C whenever x W j L p d W R C that is p p W L L is a b ounded op erator for p Thus equation on the real line R takes the form x h L when b oth the inhomogeneous term h and the solution x b elong to the appropriate function spaces Understanding the op erator is the main concern of this pap er In particular we shall prove L as a main result a Fredholm alternative for in the case that the asso ciated homogeneous equation L is asymptotically hyp erb olic We also establish related results such as the co cycle prop erty and the sp ectral ow prop erty by which the Fredholm index of can b e eectively calculated We have the L following theorems Theorem A The Fredholm Alternative Assume the homogeneous equation is asymptot p p ical ly hyperbolic Then for each p with p the operator from W to L is a Fredholm L p p p W of is independent of p so we denote K K and similarly operator The kernel K L L L L p p p L of associated to the adjoint L The range R for the kernel of the operator K K L L L L L p in L is given by Z p p fh L j R g y h d for al l y K L L In particular p p ind ind dim K co dim R co dim R dim K L L L L L L where ind denotes the Fredholm index Final ly when L L is a hyperbolic constant coecient operator we have p ind dim K co dim R L L 0 0 L 0 and so is an isomorphism L 0 We shall discuss and dene several of the concepts o ccurring in the statements of Theorem A and also the theorems b elow more formally in the next sections In particular the adjoint L of L will b e dened it is related to the adjoint of the op erator but with a sign change L p p p W L for each p and similarly In the statement of Theorem A we have that K K L L q L where p q and this implies that the integral in In particular K for K L L decay exists In fact we show in a remark following Prop osition that the elements of K and K L L exp onentially at Theorem B The Co cycle Prop erty Assume the homogeneous equation is asymptotical ly hyperbolic Then the Fredholm index of depends only on the limiting operators L namely the L limits of L as Denoting ind L L L we have that L L L L L L for any triple L L L of hyperbolic constant coecient operators The ab ove result in particular equation is termed the co cycle prop erty For an ordinary dierential equation x A x it is the case that u u L L dim W L dim W L u where dim W L denotes the numb er of unstable eigenvalues of the related constant co ecient system and thus the co cycle prop erty holds The same formula holds for delay dierential equations that is when all the shifts r are nonp ositive Formula thus allows direct j calculation of the Fredholm index ind in these cases L Unfortunately in the general case of arbitrary shifts r R considered here is not valid in j u u general in particular b ecause dim W L is p ossible even if one of the quantities dim W L is nite and the other is innite must fail Nevertheless the Fredholm index can b e eectively calculated using the sp ectral ow formula given as in the following result Roughly this gives the index as the net numb er of eigenvalues which cross the imaginary axis along a homotopy b etween the constant co ecient op erators L and L Theorem C The Sp ectral Flow Prop erty Let L for be a continuously varying oneparameter family of constant coecient operators and suppose the operators L L are hyperbolic Suppose further there are only nitely many values f g J of for which L is not hyperbolic Then L L crossL is the net number of eigenvalues of which cross the imaginary axis from left to right as increases from to To dene crossL more precisely x any as in in the statement of Theorem C and j K j j let f g denote those eigenvalues of the corresp onding equation with L L on the jk k imaginary axis Re We list these eigenvalues with rep etitions according to their multiplici ty jk as ro ots of the characteristic equation Let M denote the sum of their multiplicitie s For near j with this equation has exactly M eigenvalues counting multiplicity near the j j j L R L R imaginary axis M with Re and M with Re where M M M The net j j j j j R R crossing numb er of eigenvalues at is thus M M We dene j j j J X R R M M crossL j j j This pap er is organized as follows Following Sections and in which notation and prelimi nary results are develop ed we construct in Section the Greens function for a hyp erb olic constant co ecient system This result given in Theorem is used in Section to provide fundamental estimates on solutions of the general asymptotically hyp erb olic system and ultimately to prove Theorem A in that section Section is devoted to the pro of of Theorem B Following Section which provides additional asymptotic information ab out solutions of we prove Theorem C in Section The pro of of this result the sp ectral ow prop erty involves making generic approximations to one parameter families of systems and in the App endix we prove the technical result Prop osition that such approximations are p ossible Acknowledgment This work was partially supp orted by NSF Grant DMS and by ONR Contract NJ Preliminaries and Notation d By a solution of on an interval J we mean a continuous function x J C dened on the larger interval J f j J and r r g min max such that x is absolutely continuous on J and satises for almost every J Thus for such x d we have that x C r r C for every J We shall always take J R unless otherwise min max noted We shall o ccasionally use subscripts