on the existence and multiplicity of positive periodic solutions for first order vector differential equation abstract in this paper we consider the existence and multiplicity of positive periodic solutions for first order vector differential equation x t f t x t 0 a e t 0 under the periodic boundary value condition x 0 x here is a positive constant and f 0 r n r n is a caratheodory function some existence and multiplicity results of positive periodic solutions are derived by using a fixed point theorem in cones

remarks on the periodic boundary value problems for first order differential equations this paper is concerned with the existence of solutions and the monotone method of first order periodic boundary value problems when the lower and upper solutions and violate the boundary conditions 0 2 and 0 2 using the topological degree theory two existence theorems are established under weaker conditions than the one side lipschitz conditions an example is given which illustrates that pbvp may not have solutions between and without further restrictions to f t u the monotone method is also discussed with some new results

remarks on first and second order periodic boundary value problems on etudie les problemes aux valeurs limites periodiques d ordre 1 et 2 suivants u f t u u 0 u 2 et u f t u u 0 u 2 u 0 u 2 on obtient l existence de solutions extremales comme limites d iterees monotones

the method of upper lower solutions and hyperbolic partial differential equations on presente l etude de la methode des solutions superieures et inferieures pour des equations differentielles hyperboliques on considere la theorie des inegalites differentielles hyperboliques d une facon generale et on demontre un resultat d existence

on the structure of the solution set for first order differential equations we study the structure of the solution set for the periodic boundary value problem for first order differential equations we present some existence results and give conditions under which the solution set is compact connected or acyclic if any of the assumptions is slightly modified then it is shown that the solution set may be chaotic

monotone method for second order delayed differential equations with boundary value conditions in this paper we investigate the existence of extremal solutions of boundary value problems for second order differential equations with delayed arguments we show that a lower and upper solution method has been applied successfully to such problems an one sided lipschitz condition with corresponding constants is important assumption in our research

generalization of flanders theorem to matrix triples abstract this paper deals with several ways to generalize the flanders theorem to matrix triples we consider six invertible matrices and try to write them as the possible products of three matrices initially we describe a wide set of necessary conditions so that this system be solvable showing that they are not sufficient next we study the simultaneous solvability of two equations selected appropriately among the matrix system the rest of the paper is devoted to the study of a particular case in which the six given matrices are simultaneously diagonalizable with distinct nonzero eigenvalues in this case we obtain a necessary and sufficient condition for the solvability of the full matrix system moreover an explicit solution to it is constructed certain technical results necessary for this work may be of independent interest

the method of lower and upper solutions for systems of boundary value problems the method of lower and upper solutions for second order differential system x f t x with dirichlet boundary condition x 0 x 1 0 is used to give sufficient conditions for the existence of at least one solution

complex reflection groups braid groups hecke algebras presentations a la coxeter are given for all irreducible nite com plex re ection groups they provide presentations for the corresponding generalized braid groups for all but six cases which allow us to generalize some of the known properties of nite coxeter groups and their associated braid groups such as the computation of the center of the braid group and the construction of deformations of the nite group algebra hecke algebras we introduce monodromy representations of the braid groups which factorize through the hecke algebras extending results of cherednik opdam kohno and others summary introduction complex re ection groups and their presentations a background from complex re ection groups b presentations braid groups and their diagrams a generalities about hyperplane complements b generalities about the braid groups c the braid diagrams proofs of the main theorems for the braid groups b de e r a notation and prerequisites b computation of b de e r and of its center for d c computation of b e e r and of its center hecke algebras a background from di erential equations and monodromy b a family of monodromy representations of the braid group c hecke algebras diagrams and tables appendix generators of the monodromy around a divisor appendix tables to mathematics subject classi cation primary g we thank jean michel peter orlik pierre vogel for useful conversations and the isaac newton institute for its hospitality while the last version of this manuscript was written up the second named author gratefully acknowledges nancial support by the fondation alexander von humboldt for his stays in paris michel brou e gunter malle raphael rouquier

ordinary differential equations with nonlinear boundary conditions of antiperiodic type abstract the quasilinearization method is successfully applied to differential problems with nonlinear boundary conditions of antiperiodic type given are sufficient conditions under which the monotone iterations converge quadratically to a unique solution of our problem the advantage of this approach lies in the fact that the approximate solutions are obtained iteratively as solutions of the above mentioned iterations

positive solution for system of nonlinear first order pbvps on time scales abstract in this paper we are concerned with the following system of nonlinear first order periodic boundary value problems on time scale t x i t f i t x 1 t x 2 t x n t 0 t 0 t x i 0 x i t i 1 2 n where f i 0 t 0 n r is continuous and there exists a constant m i 0 such that m i x i f i t x 1 x 2 x n 0 for x 1 x 2 x n 0 n t 0 t some existence criteria of positive solution are established by using a fixed point theorem for operators on cone

periodic boundary value problems for second order impulsive differential systems on etudie le probleme aux valeurs limites periodiques pour un systeme d equations differentielles d ordre 2 faiblement couplees avec impulsion

on the similarity classes among products of m nonsingular matrices in various orders abstract it is shown that for any k 1 2 m 1 there exist m invertible complex matrices such that among the m products a a 1 a 2 a m s m exactly k different similarity classes occur the cases in which the matrices a i are upper triangular or are 2 by 2 are considered in detail in the former case it is shown that any m 1 unispectral matrices with common eigenvalue may be found among the a and in the latter case it is shown explicitly how to achieve m 1 and m 1 2 similarity classes as well as any number from 1 to 6 when m 4 other particular results are given as well as a discussion of further natural questions

periodic n dimensional first order coupled systems and periodic schizophrenia phenomena abstract this paper introduces some existence and location results for periodic solutions of first order coupled nonlinear systems of n ordinary differential equations the arguments used are based on the arzela ascoli theorem and schauder s fixed point theorem combined with the upper and lower solution technique to illustrate the applicability of these problems an application to a mathematical model of the thyroid pituitary interaction and their homeostatic mechanism is included for specific cases of periodic events such as relapsing catatonia

waveform relaxation methods for periodic differential functional systems the convergence of waveform relaxation methods applied to a periodic differential functional system is studied the continuous dependence is also discussed

a generalized upper and lower solutions method and multiplicity results for nonlinear first order ordinary differential equations abstract we introduce a generalized upper and lower solutions method for the solvability of first order ordinary differential equations u t t u t u 0 u 1 in order to cover the case when the function satisfies caratheodory conditions this method is then applied to get multiplicity results when the nonlinearity interacts with the real eigenvalue of the linearized problem our proofs are based on differential inequalities and classical leray schauder degree

clifford algebra analogue of the hopf koszul samelson theorem theρ decompositionc g end vρ c p and the g module structure of g

extremal solutions of periodic boundary value problems for first order integro differential equations of mixed type this paper investigates the maximal and minimal solutions of periodic boundary value problems for first order nonlinear integro differential equations of mixed type by establishing a comparison result and using the monotone iterative technique

monotone method for second order periodic boundary value problems in recent years there has been an extensive study of the existence of periodic solutions l s 11 14 151 in 8 111 the existence of solutions of first and second order pbvp period boundary value problems has been obtained by a novel approach of combining the classical method of lower and upper solutions and the method of alternative problems lyapunovschmidt method which provide conditions that are easily verifiable and which covers previous known results of other authors as a constructive method for obtaining extremal solutions of initial and boundary value problems the monotone iterative procedure has been employed by several researchers 5 7 11 13 151 the objective of this paper is to employ this useful technique for second order pbvp to obtain the minimal and maximal solutions as limits of monotone iterates our method can be used to study semilinear parabolic initial boundary value problems and other problems at resonance

dunkl operators for complex reflection groups dunkl operators for complex reflection groups are defined in this paper these commuting operators give rise to a parameterized family of deformations of the polynomial de rham complex this leads to the study of the polynomial ring as a module over the rational cherednik algebra and a natural contravariant form on this module in the case of the imprimitive complex reflection groups g m p n the set of singular parameters in the parameterized family of these structures is described explicitly using the theory of non symmetric jack polynomials

nonlinear second order periodic boundary value problems abstract in this paper we study a periodic boundary value problem pbvp for brevity of second order it is well known that if and are upper and lower solutions respectively with then there exists a solution u such that u when nothing was known the main purpose of this paper is to consider this last case also we study the structure of the set of solutions of the pbvp in the sector when and we apply the results to the case

monotone method for a system of second order periodic boundary value problems the existence of coupled maximal and minimal quasi solutions of a system of second order periodic boundary value problems is discussed when the right hand side possesses a mixed monotone property

quadratic approximation of solutions for differential equations with nonlinear boundary conditions the quasilinearization method is used for nonlinear ordinary differential equations with nonlinear boundary conditions given are sufficient conditions when corresponding monotone sequences converge to the unique solution and this convergence is quadratic

the monotone method for first order problems with linear and nonlinear boundary conditions in this paper we develop the monotone method for the first order problem u t f t u t for a e t a b when f is a caratheodory function and u w 1 1 a b we consider the nonlinear boundary conditions l u a u b 0 with l c r 2 r nondecreasing in x or nonincreasing in y and the linear boundary conditions a 0u 0 b 0u t l 0 with a 0 b 0 and l 0 r we prove the existence of solutions and the validity of the monotone method if there exists a lower solution a and an upper solution b with either a b for the linear conditions we obtain eight new concepts of lower and upper solutions which generalize previous known cases

existence of closed solutions for a polynomial first order differential equation we find new criteria for the existence of closed solutions in a first order polynomial differential equation which contains the abel equation as a particular case such results are applied to the problem of the existence of limit cycles in planar polynomial vector fields

explicit jordan form for certain block triangular matrices ii abstract let a mn b mm and c be given for x mn m we seek to determine the jordan structure of m a x o b associated with in terms of the jordan structure of a and b associated with and properties of x the principal result is an explicit prescription for the jordan structure of m associated with in the case that the geometric multiplicity of in b is one

existence of solutions of differential equations with nonlinear multipoint boundary conditions abstract we establish sufficient conditions for the existence of solutions unique or extremal of differential equations with nonlinear multipoint boundary conditions using the monotone iterative technique

existence of periodic solutions for first order differential equations in this paper we study the periodic boundary value problem for first order differential equations by combining techniques of the theory of differential inequalities namely the method of upper and lower solutions and the alternative method for nonlinear problems at resonance the results obtained are in terms of the behavior of the nonlinear part at infinity

iterative approximation of limit cycles for a class of abel equations this article considers the analytical approximation of limit cycles that may appear in abel equations written in the normal form the procedure uses an iterative approach that takes advantage of the contraction mapping theorem thus the obtained sequence exhibits uniform convergence to the target periodic solution the effectiveness of the technique is illustrated through the approximation of an unstable limit cycle that appears in an abel equation arising in a tracking control problem that affects an elementary second order bilinear power converter

periodic solutions of some lienard and duffing equations on etudie certains aspects du probleme periodique u f u u g t u 0 u 0 u 2 u 0 u 2 ou f est une fonction continue sur r et g t u est une fonction de caratheodory soumise a certaines restrictions de croissances on donne deux resultats d existence

existence and approximation of solutions for discontinuous functional differential equations in this paper existence of solutions of initial value problems for discontinuous functional differential equations is investigated firstly by applying the method of upper and lower solutions which may be discontinuous some existence results of extremal solutions are obtained furthermore we also develop a monotone iterative technique for obtaining extremal solutions which are obtained as limits of monotone sequences

order invariant spectral properties for several matrices abstract the collections of m n by n matrices over a field such that the products in any of the m orders share a common similarity class resp spectrum trace are studied the spectral and trace order invariant properties are characterized and the similarity invariant one is related to them in several cases a complete explicit description is given in case m 3 and n 2

on the quotient ring by diagonal invariants for a finite coxeter group w and its reflection representation we find the character and hilbert series for a quotient ring of by an ideal containing the w invariant polynomials without constant term this confirms conjectures of haiman

rational cherednik algebras and hilbert schemes let h c be the rational cherednik algebra of type a n 1 with spherical subalgebra u c e h c e then u c is filtered by order of differential operators with associated graded ring mbox gr u c mathbb c mathfrak h oplus mathfrak h where w is the n th symmetric group we construct a filtered mathbb z algebra b such that under mild conditions on c the category b qgr of graded noetherian b modules modulo torsion is equivalent to u c mod the associated graded mathbb z algebra mbox gr b has mbox gr b lqgr simeq coh hilb n this can be regarded as saying that u c simultaneously gives a non commutative deformation of mathfrak h oplus mathfrak h w and of its resolution of singularities hilb n rightarrow mathfrak h oplus mathfrak h as we show elsewhere this result is a powerful tool for studying the representation theory of h c and its relationship to hilb n

generalized quasilinearization and first order periodic boundary value problem the method of quasilinearization was extended and generalized to include functions which are neither convex nor concave this is known as generalized quasilinearization the generalized method yields both sided bounds as in the monotone method but the rate of convergence is quadratic as in the quasilinearization method in this paper we extend the generalization method to a first order periodic boundary value problem

periodic solutions of nonlinear parabolic equations abstract in this paper we study the periodic neumann boundary value problem for semilinear parabolic equations we present some existence results and study the structure of the set of solutions giving sufficient conditions under which that solution set is compact connected or acyclic

a monotone method for the equation uxyz f comparison theorems for the nonlinear boundary value problem u x y z f x y z u u x u y u z u x y u y z u z x with the boundary conditions u o y z a y z u x 0 z b z x u x y 0 c x y a y 0 c 0 y a 0 z b z 0 b 0 x c x 0 are established under certain conditions then it is shown that there exist sequences of lower and upper solutions which converge uniformly and monotonically to minimal and maximal solutions respectively

periodic solutions of nonlinear boundary value problems one of the well known techniques in the theory of nonlinear boundary value problems bvp is the method of differential inequalities or the method of upper and lower solutions the method of alternative problems a global variant of the lyapunov schmidt method has been used in the study of problems at resonance the investigation of periodic bvp s forms an important subclass of problems at resonance our aim is to combine the two approaches to discuss nonlinear problems at resonance we restrict ourselves in this paper to the discussion of periodic boundary value problems in section 1 we shall indicate the method of upper and lower solutions the results for systems are given in a general way so as to include known results and also offer new directions section 2 deals with the abstract existence results at resonance in the desired framework in section 3 we consider the scalar periodic bvp s by combining the two techniques namely alternative methods and the method of differential inequalities

monotone iterative techniques of v lakshmikantham for a boundary value problem for systems of impulsive differential difference equations by means of the monotone iterative techniques of v lakshmikantham a couple of minimal and maximal quasisolutions of a two point boundary value problem for a system of impulsive differential difference equations are constructed

positive solutions for first order periodic boundary value problem this paper discusses the existence and multiplicity of positive solutions for the following first order periodic boundary value problemx t f t x t 0 0

periodic boundary value problem for first order differential equations with impulses at variable times in this paper we show the validity of the method of upper and lower solutions to obtain an existence result for a periodic boundary value problem of first order impulsive differential equations at variable times

upper and lower solutions with jumps abstract we study a periodic boundary value problem for a first order differential equation from a new point of view which permits us to obtain an existence result involving upper and lower solutions with jump discontinuities we present some applications of our result with illustrative examples

representations of the rational cherednik algebras of dihedral type abstract in this paper we describe the jordan holder series of the standard modules over the rational cherednik algebras associated with the dihedral group in particular we compute the characters of the irreducible representations from the category o and classify the finite dimensional representations

comparison results for parabolic differential equations at resonance the objective of this paper is to develop systematically general comparison techniques for semilinear parabolic equations with periodic and homogeneous neumann boundary conditions these results are expected to play a significant role in the existence theory for parabolic boundary value problems at resonance

comparison results for first and second order boundary value problems at resonance various types of comparison results for first and second order periodic boundary value problems are developed it is hoped that these comparison results play an important role in the existence theory of boundary value problems at resonance

existence of solutions of boundary value problems for differential equations with delayed arguments we study the existence and approximation of solution of boundary value problems for differential equations with delayed arguments sufficient conditions are established for the existence of a unique solution or extremal ones of the given problem a monotone iterative technique is applied

existence of periodic solutions for fully nonlinear first order differential equations in this paper we establish some existence results for periodic solutions of fully nonlinear first order scalar differential equation f t x x 0 the approach here uses the viscosity solutions method developed in recent decades and the classical upper lower solutions method

rational cherednik algebras and hilbert schemes ii representations and sheaves let h c be the rational cherednik algebra of type a n 1 with spherical subalgebra u c e h c e then u c is filtered by order of differential operators with associated graded ring gr u c mathbb c mathfrac h mathfrac h w where is the n th symmetric group using the z algebra construction from gs it is also possible to associate to a filtered h c or u c module hat m on the hilbert scheme hilb n using this technique we study the representation theory of u c and h c and we relate it to hilb n and to the resolution of singularities hilb n mathfrac h mathfrac h w for example we prove the following if c 1 n so that l c triv is the unique one dimensional simple h c module then hat e l c triv mathcal o z n where z n 1 0 is the punctual hilbert scheme if c 1 n k for k in mathbb n then under a canonical filtration on the finite dimensional module l c triv gr e l c triv has a natural bigraded structure that coincides with that on h 0 z n mathscr l k where mathscr l mathcal o hilb n 1 this confirms conjectures of berest etingof and ginzburg beg2 conjectures 7 2 7 3 under mild restrictions on c the characteristic cycle of hat e c equals sum k z where k are kostka numbers and the z are known irreducible components of 1 mathfrak h w

singular polynomials for finite reflection groups the dunkl operators involve a multiplicity function as parameter for generic values of this function the simultaneous kernel of these operators acting on polynomials is equal to the constants for special values however this kernel is larger we determine these singular values completely and give partial results on the representations of g that occur in this kernel

existence of solutions for some nonlinear problems with p laplacian like operators abstract the purpose of this paper is to obtain some existence results of solutions for the nonlinear boundary value problems with p laplacian like operators

differential difference operators associated to reflection groups there is a theory of spherical harmonics for measures invariant under a finite reflection group the measures are products of powers of linear functions whose zero sets are the mirrors of the reflections in the group times the rotation invariant measure on the unit sphere in rn a commutative set of differential difference operators each homogeneous of degree 1 is the analogue of the set of first order partial derivatives in the ordinary theory of spherical harmonics in the case of r2 and dihedral groups there are analogues of the cauchy riemann equations which apply to gegenbauer and jacobi polynomial expansions the analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions this paper is part of an ongoing program to establish a workable theory for one particular class the underlying structure is based on finite coxeter groups these are finite groups acting on euclidean space generated by reflections in the zero sets of a collection of linear functions the roots the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere in addition the weight function is required to be invariant under the action of the group the resulting theory has strong similarities to the theory of spherical harmonics this was established in previous papers of the author 3 4 5 most notably a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second order differential difference operator ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials in this paper we construct a commutative set of first order differentialdifference operators associated to the second order operator previously mentioned received by the editors june 1 1987 1980 mathematics subject classification 1985 revision primary 33a45 33a65 20h15 secondary 20c30 42c10 51f15

approximation of solutions for nonlinear periodic boundary value problems with discontinuous upper and lower solutions we prove that it is possible to approximate the solutions of a nonlinear periodic boundary value problem via monotone sequences starting at piecewise continuous upper and lower solutions in fact we present here two different iterative methods explicit expression of each iteration is obtained by solving some linear initial value problems

rapid convergence of the iterative technique for first order initial value problems in this paper we study the convergence of the approximate solutions of the first order problem u t f t w t u o u o we prove that if there exists kf u k k 1 and it is a continuous function then it is possible to construct two sequences of approximate solutions converging to a solution with rate of convergence of order k

periodic solutions of nonlinear hyperbolic problems abstract we discuss two different approaches to study the problem of existence of periodic solutions of nonlinear hyperbolic partial differential equations these approaches are motivated by the recent results for existence of periodic solutions of systems of ordinary differential equations

double affine hecke algebras and macdonald s conjectures in this paper we prove the main results about the structure of double affine hecke algebras announced in c1 c2 the technique is based on the realization of these algebras in terms of demazure lusztig operators bgg d l2 ls c3 and rather standard facts from the theory of affine weyl groups in particular it completes the proof partially published in c2 of the macdonald scalar product conjecture see ml 12 6 including the famous macdonald constant term conjecture the q t case we mainly follow the opdam paper 0 where the macdonald mehta conjectures in the degenerate differential case were deduced from certain properties of the heckman opdam operators ho and the existence of the shift operators heckman s interpretation of these operators via the so called dunkl operators see he and also c5 was important to our approach we note that the ho operators are closely related to the so called quantum many body problem calogero sutherland moser olshanetsky perelomov the conformal field theory knizhnik zamolodchikov equations the harmonic analysis on symmetric spaces harish chandra helgason etc and last but not the least the classic theory of the hypergeometric functions establishing the connection between the difference counterparts of heckman opdam operators introduced in c4 and the macdonald theory ml m2 including the construction of the difference shift operators is the main thrust of this paper once the connection is established it is not very difficult to calculate the scalar squares of the macdonald polynomials and to prove the constant term conjecture from his fundamental paper m3 to simplify the exposition we discuss the reduced root systems only and impose the relation q tk for k e z to avoid infinite products in the definition of macdonald s pairing the purpose of this work is to present a concrete application of the new technique arbitrary q t can be handled in a

flandersʼ theorem for many matrices under commutativity assumptions we analyze the relationship between the jordan canonical form of products in different orders of k square matrices a 1 a k our results extend some classical results by h flanders motivated by a generalization of fiedler matrices we study permuted products of a 1 a k under the assumption that the graph of non commutativity relations of a 1 a k is a forest under this condition we show that the jordan structure of all nonzero eigenvalues is the same for all permuted products for the eigenvalue zero we obtain an upper bound on the dierence between the sizes of jordan blocks for any two permuted products and we show that this bound is attainable for k 3 we show that moreover the bound is exhaustive

periodic solutions of hyperbolic partial differential equations abstract constructive methods by obtaining multiple solutions of first and second order nonlinear hyperbolic periodic boundary value problems are given appropriate partial differential inequalities are developed and are then employed to derive the iterative schemes which yield the desired solutions the advantage of these methods is that the solutions of the given nonlinear problems are constructed as uniform limits of monotone iterates these iterates being solutions of linear problems are explicitly computable

time periodic boundary layer solutions to singularly perturbed parabolic problems abstract in this paper we present a result of implicit function theorem type which was designed for applications to singularly perturbed problems this result is based on fixed point iterations for contractive mappings in particular no monotonicity or sign preservation properties are needed then we apply our abstract result to time periodic boundary layer solutions which are allowed to be non monotone with respect to the space variable in semilinear parabolic problems with two independent singular perturbation parameters we prove existence and local uniqueness of those solutions and estimate their distance to certain approximate solutions

quasiharmonic polynomials for coxeter groups and representations of cherednik algebras we introduce and study deformations of finite dimensional modules over rational cherednik algebras our main tool is a generalization of usual harmonic polynomials for each coxeter group the so called quasiharmonic polynomials a surprising application of this approach is the construction of canonical elementary symmetric polynomials and their deformations for all coxeter groups