extended kronecker summation for cluster treatment of lti systems with multiple delays a new procedure is presented for determining the kernel and the offspring hypersurfaces for general linear time invariant lti dynamics with multiple delays these hypersurfaces as they have very recently been introduced in a concept paper r sipahi and n olgac automatica 41 2005 pp 1413 1422 form the basis of the overriding paradigm which is called the cluster treatment of characteristic roots ctcr in fact these two sets of hypersurfaces exhaustively represent the locations in the domain of the delays where the system possesses at least one pair of imaginary characteristic roots to determine the kernel and offspring we use the extraordinary features of the extended kronecker summation operation in this paper the end result is that the infinite dimensional problem reduces to a finite dimensional one and preferably into an eigenvalue problem following the procedure described in this paper we are able to shorten the computational time considerably in determining these hypersurfaces we demonstrate these concepts via some example case studies one of the examples treats a 3 delay system for this case another interesting perspective called the building block is also utilized to display the kernel in three dimensional space in the domain of spectral delays

bounds for the spectrum of a two parameter matrix eigenvalue problem abstract we consider the two parameter eigenvalue problem t j v j 1 a j 1 v j 2 a j 2 v j 0 where j c t j a j k j k 1 2 are matrices bounds for the spectral radius of that problem are suggested our main tool is a norm estimate for the operator inverse to the operator a 11 a 22 a 12 a 21 where means the tensor product in addition by virtue of that norm estimate and the ostrowsky schneider theorem we establish a condition that provides the conservation of the number of the eigenvalues of the considered problem in a half plane

critical delays and polynomial eigenvalue problems in this work we present a new method to compute the delays of delay differential equations ddes such that the dde has a purely imaginary eigenvalue for delay differential equations with multiple delays the critical curves or critical surfaces in delay space that is the set of delays where the dde has a purely imaginary eigenvalue are parameterized we show how the method is related to other works in the field by treating the case where the delays are integer multiples of some delay value i e commensurate delays the parameterization is done by solving a quadratic eigenvalue problem which is constructed from the vectorization of a matrix equation and hence typically of large size for commensurate delay differential equations the corresponding equation is a polynomial eigenvalue problem as a special case of the proposed method we find a closed form for a parameterization of the critical surface for the scalar case we provide several examples with visualizations where the computation is done with some exploitation of the structure of eigenvalue problems

on ground states of superlinear p laplacian equations in rn abstract we consider superlinear p laplacian equations in r n with a potential which is periodic or has a bounded potential well without assuming the ambrosetti rabinowitz type condition and the monotonicity of the function t f x t t p 1 we prove the existence of ground states even for the case p 2 our results extend the recent results of li wang and zeng ann inst h poincare anal non lineaire 23 2006 829 837

b weyl spectrum and poles of the resolvent abstract let t be a bounded linear operator acting on a banach space and let bw t c such that t i is not a b fredholm operator of index 0 be the b weyl spectrum of t define also e t to be the set of all isolated eigenvalues in the spectrum t of t and t to be the set of the poles of the resolvent of t in this paper two new generalized versions of the classical weyl s theorem are considered more precisely we seek for conditions under which an operator t satisfies the generalized weyl s theorem bw t t e t or the version ii of the generalized weyl s theorem bw t t t

a new method for computing delay margins for stability of linear delay systems this note is concerned with stability properties of linear time invariant delay systems we consider delay systems of retarded type modeled both as a high order scalar differential difference equation and as a set of first order differential difference equations expressed in state space form we provide a computational method that can be used to compute a delay interval such that the system under consideration is stable for all delay values that lie in the computed interval this method requires computing only the eigenvalues and generalized eigenvalues of certain constant matrices and it can be implemented efficiently based on this method we further state a simple necessary and sufficient condition concerning stability independent of delay for each of the two types of the models

a note on generalized weyl s theorem abstract we prove that if either t or t has the single valued extension property then the spectral mapping theorem holds for b weyl spectrum if moreover t is isoloid and generalized weyl s theorem holds for t then generalized weyl s theorem holds for f t for every f h t an application is given for algebraically paranormal operators

abstract critical point theorems and applications to some nonlinear problems with strong resonance at infinity critical point theorems and applications to some nonlinear problems with strong resonance at infinity p bartolo and v benci istituto di matematica applicata via re david x0 bari ital and istituto di analisi matematica palazzo ateneo l ia scolai 2 bari italy received in revised form 20 december 1952

on superlinear p x laplacian equations in rn abstract we consider the p x laplacian equations in r n the nonlinearity is superlinear but does not satisfy the ambrosetti rabinowitz type condition we obtain ground states of the equations improving a recent result of fan x l fan p x laplacian equations in r n with periodic data and nonperiodic perturbations j math anal appl 341 2008 103 119 we also establish a bartsch wang type compact embedding theorem for variable exponent spaces then a multiplicity result for the equations is proved for odd nonlinearity

ground state solutions for a periodic schrodinger equation with superlinear nonlinearities abstract in this paper we consider the following schrodinger equation u v x u g x u for x r n u x 0 as x where v is a periodic continuous real function with 0 in a gap of the spectrum a a v and the classical ambrosetti rabinowitz superlinear condition on g is replaced by a general super quadratic condition

on linearizations of the quadratic two parameter eigenvalue problems abstract we present several transformations that can be used to solve the quadratic two parameter eigenvalue problem qmep by formulating an associated linear multiparameter eigenvalue problem two of these transformations are generalizations of the well known linearization of the quadratic eigenvalue problem and linearize the qmep as a singular two parameter eigenvalue problem the third replaces all nonlinear terms by new variables and adds new equations for their relations the qmep is thus transformed into a nonsingular five parameter eigenvalue problem the advantage of these transformations is that they enable one to solve the qmep using existing numerical methods for multiparameter eigenvalue problems we also consider several special cases of the qmep where some matrix coefficients are zero

linear systems with commensurate time delays stability and stabilization independent of delay notions of exponential stability independent of delay and stabilizability independent of delay are developed for the class of delay differential systems of the retarded type with commensurate time delays various criteria for exponential stability independent of delay with a given order are specified in terms of matrices whose entries are functions of a single real parameter and polynomials in one variable whose coefficients are functions of a single real parameter sufficient conditions and a necessary condition based on local stabilizability are given for stabilizability independent of delay using state feedback with commensurate time delays constructive methods for determining a stabilizing feedback are also presented the last part of the paper deals with a standard type of observer and regulator with the requirement that the closed loop system be stable independent of delay

ground state solutions for some indefinite variational problems abstract we consider the nonlinear stationary schrodinger equation u v x u f x u in r n here f is a superlinear subcritical nonlinearity and we mainly study the case where both v and f are periodic in x and 0 belongs to a spectral gap of v inspired by previous work of li et al 2006 11 and pankov 2005 13 we develop an approach to find ground state solutions i e nontrivial solutions with least possible energy the approach is based on a direct and simple reduction of the indefinite variational problem to a definite one and gives rise to a new minimax characterization of the corresponding critical value our method works for merely continuous nonlinearities f which are allowed to have weaker asymptotic growth than usually assumed for odd f we obtain infinitely many geometrically distinct solutions the approach also yields new existence and multiplicity results for the dirichlet problem for the same type of equations in a bounded domain

multiple solutions of schrodinger equations with indefinite linear part and super or asymptotically linear terms abstract based on new information concerning strongly indefinite functionals without palais smale conditions we study existence and multiplicity of solutions of the schrodinger equation u v x u g x u for x r n u x 0 as x where v and g are periodic with respect to x and 0 lies in a gap of v supposing g is asymptotically linear as u and symmetric in u we obtain infinitely many geometrically distinct solutions we also consider the situation where g is super linear with mild assumptions different from those studied previously and establish the existence and multiplicity

solutions of superlinear elliptic equations and their morse indices we consider here solutions of semilinear second order elliptic equations with superlinear nonlinearities and we present some relationships between their morse indices and some qualitative properties in particular we show that for uo subcritical nonlinearities bounds on solutions are equivalent to bounds on their morse indices

asymptotically linear schrodinger equation with potential vanishing at infinity we are concerned with the existence of bound states and ground states of the following nonlinear schrodinger equation

jacobi davidson methods for polynomial two parameter eigenvalue problems we propose jacobi davidson type methods for polynomial two parameter eigenvalue problems pmep such problems can be linearized as singular two parameter eigenvalue problems whose matrices are of dimension k k 1 n 2 where k is the degree of the polynomial and n is the size of the matrix coefficients in the pmep when k 2 n is relatively small the problem can be solved numerically by computing the common regular part of the related pair of singular pencils for large k 2 n computing all solutions is not feasible and iterative methods are required when k is large we propose to linearize the problem first and then apply jacobi davidson to the obtained singular two parameter eigenvalue problem the resulting method may for instance be used for computing zeros of a system of scalar bivariate polynomials close to a given target on the other hand when k is small we can apply a jacobi davidson type approach directly to the original matrices the original matrices are projected onto a low dimensional subspace and the projected polynomial two parameter eigenvalue problems are solved by a linearization

existence of infinitely many solutions for p laplacian equations in rn abstract based on the cohomological linking method for cones and a new direct sum decomposition of w 1 p r n we study the existence and multiplicity of solutions of a p laplacian equation in r n with sign changing potential and subcritical p superlinear nonlinearity our assumptions are mild and different from those studied previously

property gb and perturbations abstract an operator t acting on a banach space x possesses property gb if a t sbf t t where a t is the approximate point spectrum of t sbf t is the essential semi b fredholm spectrum of t and t is the set of all poles of the resolvent of t in this paper we study property gb in connection with weyl type theorems which is analogous to generalized browder s theorem several sufficient and necessary conditions for which property gb holds are given we also study the stability of property gb for a polaroid operator t acting on a banach space under perturbations by finite rank operators by nilpotent operators and more generally by algebraic and riesz operators commuting with t

projective spectrum and cyclic cohomology abstract for a tuple a a 1 a 2 a n of elements in a unital algebra b over c its projective spectrum p a or p a is the collection of z c n or respectively z p n 1 such that the multi parameter pencil a z z 1 a 1 z 2 a 2 z n a n is not invertible in b b valued 1 form a 1 z d a z contains much topological information about p c a c n p a in commutative cases invariant multi linear functionals are effective tools to extract that information this paper shows that in non commutative cases the cyclic cohomology of b does a similar job in fact a chen weil type map from the cyclic cohomology of b to the de rham cohomology h d p c a c is established as an example we prove a closed high order form of the classical jacobi s formula

on computing the maximal delay intervals for stability of linear delay systems this note is concerned with stability properties of linear time invariant delay systems the authors consider delay systems of both retarded and neutral types expressed in state space forms the author s main goal is to provide a computation oriented method for computing the maximal delay intervals over which the systems under consideration maintain stability the author s results show that this can be accomplished by computing the generalized eigenvalues of certain frequency dependent matrices based on these results the author also states a necessary and sufficient condition concerning stability independent of delay for each of the retarded and neutral systems the author s results can be readily implemented and appear suitable for analyzing systems with high dimensions and many delay units

technical communique invariance properties in the root sensitivity of time delay systems with double imaginary roots if i w ir is an eigenvalue of a time delay system for the delay t 0 then i w is also an eigenvalue for the delays t k t 0 k2 p w for any k z we investigate the sensitivity periodicity and invariance properties of the root i w for the case that i w is a double eigenvalue for some t k it turns out that under natural conditions the condition that the root exhibits the completely regular splitting property if the delay is perturbed the presence of a double imaginary root i w for some delay t 0 implies that i w is a simple root for the other delays t k k 0 moreover we show how to characterize the root locus around i w the entire local root locus picture can be completely determined from the square root splitting of the double root we separate the general picture into two cases depending on the sign of a single scalar constant the imaginary part of the first coefficient in the square root expansion of the double eigenvalue

nontrivial solutions of superlinear p laplacian equations we consider p laplacian equations on a bounded domain where the nonlinearity is superlinear but dose not satisfy the usual ambrosetti rabinowitz condition near infinity or its dual version near zero nontrivial solutions are obtained by computing the critical groups and morse theory

property w and perturbations ii abstract a bounded linear operator t l x defined on a banach space x satisfies property w a variant of weyl s theorem if the complement in the approximate point spectrum a t of the weyl essential approximate spectrum wa t coincides with the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity in this note we study the stability of property w for a bounded operator t acting on a banach space under perturbations by finite rank operators by nilpotent operator and quasi nilpotent operators commuting with t

roots of bivariate polynomial systems via determinantal representations we give two determinantal representations for a bivariate polynomial they may be used to compute the zeros of a system of two of these polynomials via the eigenvalues of a two parameter eigenvalue problem the first determinantal representation is suitable for polynomials with scalar or matrix coefficients and consists of matrices with asymptotic order n 2 4 where n is the degree of the polynomial the second representation is useful for scalar polynomials and has asymptotic order n 2 6 the resulting method to compute the roots of a system of two bivariate polynomials is very competitive with some existing methods for polynomials up to degree 10 as well as for polynomials with a small number of terms

existence of weak solutions for a quasilinear equation in rn abstract this paper studies the p laplacian equation p u v x u p 2 u f x u in r n where 1 p n 1 and v x is a nonnegative continuous function under some conditions on f x u and v x we prove the existence of nontrivial solutions for sufficiently large

existence of nontrivial solutions for p laplacian equations in rn abstract in this paper we consider a p laplacian equation in r n with sign changing potential and subcritical p superlinear nonlinearity by using the cohomological linking method for cones an existence result is obtained we also give a result on the existence of periodic solutions for one dimensional p laplacian equations which can be proved by the same method

on the quadratic two parameter eigenvalue problem and its linearization we introduce the quadratic two parameter eigenvalue problem and linearize it as a singular two parameter eigenvalue problem this together with an example from model updating shows the need for numerical methods for singular two parameter eigenvalue problems and for a better understanding of such problems there are various numerical methods for two parameter eigenvalue problems but only few for nonsingular ones we present a method that can be applied to singular two parameter eigenvalue problems including the linearization of the quadratic two parameter eigenvalue problem it is based on the staircase algorithm for the extraction of the common regular part of two singular matrix pencils

solutions in spectral gaps for a nonlinear equation of schrodinger type abstract in this paper we study the existence of a nontrivial h2 r n solution for an equation of the form u x p x u x f x u x u x x r n r where p l r n is a periodic function we assume that the operator p h2 r n l2 r n l2 r n is strongly indefinite and invertible and that x r r is odd and satisfies some superlinear but subcritical growth conditions we extend the class of nonlinearities which has been studied up to now in particular under standard technical restrictions the existence of a solution is derived when lim x f x s f s 0 exists for all s r if we assume that f x s f s for all s r and a e on r n

some results on connecting orbits for a class of hamiltonian systems abstract the existence of various kinds of connecting orbits is established for a certain hamiltonian system as well as its time dependent analogue for the autonomous case our main assumption is that v has a global maximum e g at x o and we find a various kinds of orbits terminating at o for the time dependent case v has a local but not global maximum at x o and we find a homoclinic orbit emanating from and terminating at o keywords periodic solution

stability of linear neutral time delay systems exact conditions via matrix pencil solutions in this note we study the stability properties of linear neutral delay systems we consider systems described by both neutral differential difference and state space equations and we seek to determine the delay margin of such systems that is the largest range of delay values for which a neutral delay system may preserve its stability in both cases we show that the delay margin can be found by computing the eigenvalues and generalized eigenvalues of certain constant matrices which can be executed efficiently and with high precision the results extend previously known work on retarded systems and demonstrate that similar stability tests exist for neutral systems in particular the tests require essentially the same amount of computation required for retarded systems

existence of multiple solutions for modified schrdingerkirchhoffpoisson type systems via perturbation method with sign changing potential in this paper we prove the existence of positive solutions and negative solutions for the following modified schrdingerkirchhoffpoisson type systems a br3u2 u v x u u12u u2 f x u inr3 u2 inr3 where a 0 b0 v and f are continuous functions and v x is allowed to be sign changing under some certain assumptions on v and f we prove the existence of nontrivial nonnegative solutions nontrivial nonpositive solutions and sequence of high energy solutions via the perturbation method

variations on weyl s theorem in this note we study the property w a variant of weyl s theorem introduced by rakocevic by means of the localized single valued extension property svep we establish for a bounded linear operator defined on a banach space several sufficient and necessary conditions for which property w holds we also relate this property with weyl s theorem and with another variant of it a weyl s theorem we show that weyl s theorem a weyl s theorem and property w for t respectively t coincide whenever t respectively t satisfies svep as a consequence of these results we obtain that several classes of commonly considered operators have property w

bifurcation in lp rn for a semilinear elliptic equation keywords lsp p bifurcation constrained minimization sobolev spaces nonlinear eigenvalue problem semilinear spectrum linearisation variational structure reference ana article 1988 001doi 10 1112 plms s3 57 3 511 record created on 2008 12 10 modified on 2016 08 08

on multiplicity and concentration of positive solutions for a class of quasilinear problems with critical exponential growth in rn abstract in this work we study the existence multiplicity and concentration of positive solutions for the following class of quasilinear problems n n u v x u n 2 u f u in r n u x 0 x r n where n u is the n laplacian operator is a positive parameter n 2 v r n r is a continuous function and f r r is a c 1 function having critical exponential growth

nonstationary homoclinic orbits for an infinite dimensional hamiltonian system we study the following system of diffusion equations on r rn tu xu v x u g t x v tv xv v x v f t x u where z u v r rn r2 v c rn r g t x v f t x u are periodic in t x and superlinear at infinity by a variant linking theorem and concentration compactness arguments we establish the existence of nonstationary homoclinic type solutions of least energy for this systems

stability in linear delay equations abstract for linear autonomous differential difference equations of retarded or neutral type necessary and sufficient conditions are given for the zero solution to be stable hyperbolic for all values of the delays

semiclassical solutions of perturbed p laplacian equations with critical nonlinearity abstract we consider the perturbed p laplacian equation e p p u v x u p 2 u k x u p 2 u f x u u w 1 p r n where p u div u p 2 u is the p laplacian operator with 1 p n p p n n p denotes the sobolev critical exponent k x is a bounded positive function under some mild conditions on v and f we show that the equation has at least one nontrivial solution provided that e e 0 where the bound e 0 is formulated in terms of p n v k and f

a method for computing the interval of delay values for which a differential delay system is stable a test for determining the interval of delay values for which delay differential system is asymptotically stable is presented i e an explicit method is given for calculating h such that for 0

polynomial two parameter eigenvalue problems and matrix pencil methods for stability of delay differential equations several recent methods used to analyze asymptotic stability of delay differential equations ddes involve determining the eigenvalues of a matrix a matrix pencil or a matrix polynomial constructed by kronecker products despite some similarities between the different types of these so called matrix pencil methods the general ideas used as well as the proofs differ considerably moreover the available theory hardly reveals the relations between the different methods in this work a different derivation of various matrix pencil methods is presented using a unifying framework of a new type of eigenvalue problem the polynomial two parameter eigenvalue problem of which the quadratic two parameter eigenvalue problem is a special case this framework makes it possible to establish relations between various seemingly different methods and provides further insight in the theory of matrix pencil methods we also recognize a few new matrix pencil variants to determine dde stability finally the recognition of the new types of eigenvalue problem opens a door to efficient computation of dde stability

on a class of superlinear p q laplacian type equations on rn starting from a new sum decomposition of w1 p rn w1 q rn and using a variational approach we investigate the existence of multiple weak solutions of a p q laplacian equation on rn for 1 q p n with a sign changing potential and a caratheodory reaction term satisfying the celebrated ambrosetti rabinowitz condition our assumptions are mild and different from those used in related papers and moreover our results improve or complement previous ones for the single p laplacian

on ground state solutions for quasilinear elliptic equations with a general nonlinearity in the critical growth abstract in this paper we consider the following problem p u v u p 2 u f u u w 1 p r n where 1 p n and f has critical growth the purpose of this paper is to study the existence of the ground state solution for a class of quasilinear elliptic equations some recent results are extended

compactness results for schrodinger equations with asymptotically linear terms abstract we study the nonlinear problem u v x f x u x r n lim x u x 0 where the schrodinger operator v is semibounded and the nonlinearity f is linearly bounded we establish compactness of palais smale sequences and cerami sequences for the associated energy functional under general spectral theoretic assumptions applying these results we obtain existence of three nontrivial solutions if the energy functional has a mountain pass geometry

property w for perturbations of polaroid operators abstract a bounded linear operator t l x acting on a banach space satisfies property w a variant of weyl s theorem if the complement in the approximate point spectrum a t of the weyl essential approximate point spectrum wa t is the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity in this note we study the stability of property w for a polaroid operator t acting on a banach space under perturbations by finite rank operators by nilpotent operators and more generally by algebraic operators commuting with t