periodic solutions with prescribed energy for singular conservative systems involving strong forces this paper establishes under appropriate conditions which are stated below the existence of periodic solutions of the equation ii t vv q t 0 1 1 where q t e rn n 1 2 and v i is the gradient of a real function i which has a singularity at the origin in the literature the problem of the existence of periodic solutions of 1 1 has been addressed following two different points of view i given t 0 find periodic solutions of 1 1 having t as period fixed period problem ii given h e r find periodic solutions of 1 1 such that for all t e r

a chebyshev method for the solution of boundary value problems abstract an expansion procedure using the chebyshev polynomials as base functions is proposed the method yields more accurate results than either of the galerkin or tau methods as indicated from solving the orr sommerfeld equation for both the plane poiseuille flow and the blasius velocity profile the chebyshev approximation is also applied to resolve the radial dependence of the flow field for a circular cylinder or a sphere in a uniform flow

multiple brake orbits for some classes of singular hamiltonian systems from which one can easily construct a periodic solution of period 2t by reflecting q around t 0 t if q x e e 1 v x hj is homeomorphic to a ball then the existence of one brake orbit is a classical result of seifert 11 generalizations of that result as well as alternative proofs were given in 2 6 141 in the case in which the topology of sz is richer then the existence of multiple solutions has been proved by bolotin and kozlov in 7 see also 8 in this paper we are mainly concerned with the case in which the potential v has singularities inside 2 in the sense that v x co as x x e ai and the topology of q is nontrivial more precisely in section 2 we shall assume that sz has a certain number of holes k and that v behaves like lxi o1 with 0 cx 2 near x 0 and we shall show that 1 3 has at least k classical brake orbits let us point out that the main interest of this result lies in the fact that we are able to prove that the solutions we find are classical ones that is they do not collide with the singularity while most of the papers dealing with existence of fixed energy 9 or fixed period lo solutions for singular potentials only prove existence of generalized solutions that is solutions which can collide with the singularity existence of a noncollision periodic solution for a singular potential

homoclinic solutions for singular hamiltonian systems without the strong force condition abstract we consider the existence of homoclinic orbits at the origin of a hamiltonian system q v q 0 in r n n 3 where v has a strict global maximum at q 0 and a singularity at a point e 0 namely v q as q e we establish via variational methods the existence of a generalized homoclinic orbit q that may enter the singularity e without assuming the strong force condition of gordon moreover when v 1 q e 0 2 near e we give a bound for the number of collisions of q based on the morse index of approximated solutions as a consequence we obtain that q is classical non collision orbit for 1 2 and enters the singularity e at most one time in r if 0 1

forced vibrations of superquadratic hamiltonian systems on etudie l existence des solutions t periodiques ter t 0 du systeme hamiltonien suivant z gh z f t ou g est la matrice asymetrique standard

new characterisations of positive realness and static output feedback control of discrete time systems this article is concerned with the problems of positive real analysis and control synthesis for discrete time systems new linear matrix inequality lmi characterisations of positive realness are derived which enable one to check the positive realness by using parameter dependent lyapunov function the relationship between the proposed characterisations and the existing ones are clarified which shows that our new results are of less conservatism for characterising the positive realness of discrete time systems with polytopic uncertainty in addition sufficient conditions for static output feedback positive real controller design are given in terms of solutions to a set of linear matrix inequalities numerical examples are included for illustration

integration preconditioning matrix for ultraspherical pseudospectral operators in this paper we give an integration rectangular preconditioner for ultraspherical pseudospectral operators we present an explicit solution of matrix equations with the structure provided by the pseudospectral discretization of the nth order differential operator subject to various types of boundary conditions this occurs in two steps in the first step we introduce an integration rectangular preconditioner in the second step we use a new strategy to impose the boundary conditions on the condition that the coefficients matrix has a known inverse computational examples are given

morse index of some min max critical points i application to multiplicity results on donne un resultat general sur des bornes inferieures pour des indices de morse de points critiques obtenus par des principes de min max en combinant cette information avec une inegalite semiclassique on obtient des estimations pointues sur la croissance de certaines valeurs critiques a partir desquelles on deduit de nouveaux resultats de multiplicite pour des solutions d equations elliptiques semi lineaires d ordre 2

existence and multiplicity of homoclinic orbits for potentials on unbounded domains we study the system in r n where v is a potential with a strict local maximum at 0 and possibly with a singularity first using a minimising argument we can prove the existence of a homoclinic orbit when the component of x r n v x containing 0 is an arbitrary open set in the case unbounded we allow v x to go to 0 at infinity although at a slow enough rate then we show that the presence of a singularity in implies that a homoclinic solution can also be found via a minimax procedure and comparing the critical levels of the functional associated to the system we see that the two solutions are distinct whenever the singularity is not too far from 0

representation theorems for sobolev spaces on intervals and multiplicity results for nonlinear odes abstract we present some topological isomorphisms between w m p 0 1 and l p 0 1 r m combined with some arguments from fourier analysis we obtain explicit schauder bases for w m p 0 1 and some of its subspaces in addition we apply such results to treat some boundary value problems involving nonlinear elliptic differential equations

positive solutions of superlinear elliptic equations abstract in this paper we study the existence of two positive solutions of superlinear elliptic equations without assuming the conditions which have been used in the literature to deduce either the p s condition or a priori bounds of positive solutions the first solution is proved as the minimal positive solution while the second one is obtained as the limit of a gradient flow whose starting point is properly chosen the dependence of the minimal solution upon a parameter is also considered

extended lmi characterizations for stability and performance of linear systems over the past ten years extensive research has been devoted to extended lmi characterizations for stability and performance of linear systems these characterizations constitute a valuable tool for reducing conservatism in hard problems like multi objective control and robust stability and performance analysis the present paper proposes a general projection lemma based methodology for deriving such extended lmis and hereby provides a straightforward and unified proof for all known literature results as well as some currently missing extended lmis

multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces abstract the existence of multiple positive solutions to superlinear periodic boundary value problems with repulsive singular forces is discussed in this paper our nonlinearity may be singular in its dependent variable and our analysis relies on a nonlinear alternative of leray schauder type and on a fixed point theorem in cones

noncollision periodic solutions to some singular dynamical systems with very weak forces abstract we prove a double variational characterization of the set of all the periodic solutions of the kepler problem we provide a sufficient condition for the existence of a periodic solution with prescribed period or energy for autonomous second order differential systems with keplerian type potentials the proofs are based on linking arguments

extensions of the mountain pass theorem on considere une fonctionnelle c 1 i x r ou x est un espace de banach reel et ou i satisfait la condition de compacite de palais smale on presente une serie de theoremes qui generalisent le theoreme du col

forced oscillations of singular dynamical systems with an application to the restricted three body problem abstract we consider forced dynamical systems with two degrees of freedom having singular potentials and we prove existence of infinitely many classical noncollision periodic solutions these solutions have a prescribed rotation behavior with respect to the singularities and a prescribed period the same of the systems they are obtained variationally as minima of a suitable functional on open subsets of a hubert space this investigation was motivated by the elliptic restricted three body problem with arbitrary masses of the two primaries for that problem we obtain infinitely many distinct generalized periodic solutions i e solutions which possibly experience collisions

periodic solutions of lienard equations with singular forces of repulsive type in this paper we will use the coincidence degree to give an existence result of periodic solutions for the scalar lienard equations with singular forces of repulsive type the main result of this paper clearly describes what balance conditions between the singular force at the singularity and at the infinity should be imposed if the equation has a periodic solution as an application we will improve the results of ding ye and wang for the brillouin electron beam focusing system

periodic solutions of a class of singular hamiltonian systems soit r n 0 n n n 3 et soit v c 1 rx r on etudie l existence des solutions periodiques t u t avec une periode prescrite du systeme suivant de n equations differentielles d ordre 2 u v t u

topological results on a certain class of functionals and application abstract we study in this paper problems of the type u u p 1 u x bounded r n u 0 i where x is given and where p 1 n 2 n 2 p 1 if n 2 our main result is that i has an infinite number of solutions for a residual set of in h 1 in particular for many n n there exists an open and dense subset of in h 1 such that i has n distinct solutions for such an this result is to be related to the conjecture developed in 1 of the existence of an infinite number of solutions to i the proof relies on a general characterization of level sets for a certain class of functionals when there are no critical value in a large enough interval in addition to the study of problem i we apply this characterization to give another proof using e g brouwer s fixed point theorem for some classical results about even functionals and saddle points

periodic solutions of a class of hamiltonian systems with singularities this paper deals with a class of time periodic hamiltonian systems obtained by a time dependent perturbation from an autonomous system with a singularity at q 0 in configuration space it is shown that t being the period of the perturbation nondegenerate families of t periodic orbits in the unperturbed problem branch off into a certain number of t periodic orbits for the perturbed problem

the variational contribution of the periodic orbits provided by the birkhoff lewis theorem with applications to convex hamiltonians and to three body type problems

a bahri lions theorem revisited abstract in 1988 a bahri and p l lions a bahri p l lions morse index of some min max critical points i application to multiplicity results comm pure appl math 41 1988 1027 1037 studied the following elliptic problem u u p 2 u f x u u h 0 1 where is a bounded smooth domain of r n 2 p 2 n 2 n 2 and f x u is not assumed to be odd in u they proved the existence of infinitely many solutions under an appropriate growth restriction on f in the present paper we improve this result by showing that under the same growth assumption on f the problem admits in fact infinitely many sign changing solutions in addition we derive an estimate on the number of their nodal domains we also deal with the corresponding fourth order equation 2 u u p 2 u f x u with both dirichlet and navier boundary conditions as well as with strongly coupled elliptic systems

multiplicity results for some perturbed elliptic problems in unbounded domains with non homogeneous boundary conditions abstract we study the following nonlinear elliptic problem u m u g x u f x in r n b r u h on b r u 0 as x with n 3 m r under suitable conditions on the radial functions g f and h multiplicity results are proved in the perturbed case f 0 and h 0 by using bolle s perturbation method and suitable growth estimates on min max critical levels

on the existence of infinitely many periodic solutions to some problems of n body type we prove the existence of infinitely many periodic solutions with prescribed period to a class of problems of n body type

new characterization of positive realness and control of a class of uncertain polytopic discrete time systems abstract this paper deals with the problems of positive real analysis and control synthesis for a class of discrete time polytopic systems with uncertainties the systems under consideration are modelled in a polytopic form with linear fractional uncertainties in its vertices a new linear matrix inequality lmi characterization of positive realness for this class of systems is given it enables one to check the positive realness by using parameter dependent lyapunov function this new characterization exhibits a kind of decoupling between the lyapunov matrix and the system matrices which is subsequently exploited for control design based on the new result sufficient conditions with reduced conservatism are obtained a numerical example is also included to demonstrate the applicability of the proposed results

perturbation of hamiltonian systems with keplerian potentials on considere l existence des solutions periodiques pour une classe de systemes hamiltoniens obtenus comme des perturbations de l hamiltonien keplerien k p q 1 2 p 2 q 0

a relationship between the periodic and the dirichlet bvps of singular differential equations in this paper a relationship between the periodic and the dirichlet boundary value problems for second order ordinary differential equations with singularities is established this relationship may be useful in explaining the difference between the nonresonance of singular and nonsingular differential equations using this relationship we give in this paper an existence result of positive periodic solutions to singular differential equations when the singular forces satisfy some strong force condition at the singularity 0 and some linear growth condition at infinity

multiple solutions for prescribed boundary value problem we in this paper consider the semilinear elliptic boundary value problem u f x u in and u g on and the corresponding bolza problem x v t x 0 x 0 x0 x t x1 where is

a minimax method for a class of hamiltonian systems with singular potentials abstract this paper presents a minimax method which gives existence and multiplicity results for time periodic solutions of a class of hamiltonian systems when a singular potential is present the singularity satisfies the strong force condition of gordon when milder singularities are permitted a notion of generalized t periodic solution is introduced and we get existence and multiplicity results for such solutions

on a class of singular second order hamiltonian systems with infinitely many homoclinic solutions abstract we show existence of infinitely many homoclinic orbits at the origin for a class of singular second order hamiltonian systems u v u t u 0 t we use variational methods under the assumption that v t u satisfies the so called strong force condition

topologically distinct collision free periodic solutions for the n center problem this work concerns the planar n center problem with homogeneous potential of degree alpha alpha in 1 2 the existence of infinitely many topologically distinct non collision periodic solutions with a prescribed energy is proved a notion of admissibility in the space of loops on the punctured plane is introduced so that in any admissible class and for any positive h the existence of a classical periodic solution with energy h for the n center problem with alpha in 1 2 is proven in case alpha 1 a slightly different result is shown it is the case that there is either a non collision periodic solution or a collision reflection solution the results hold for any position of the centres and it is possible to prescribe in advance the shape of the periodic solutions the proof combines the topological properties of the space of loops in the punctured plane with variational and geometrical arguments

the fixed energy problem for a class of nonconvex singular hamiltonian systems we consider a noncompact hypersurface h in r2n which is the energy level of a singular hamiltonian of strong force type under global geometric assumptions on h we prove that it carries a closed characteristic as a consequence of a result by hofer and viterbo on the weinstein conjecture in cotangent bundles of compact manifolds our theorem contains as particular cases earlier results on the fixed energy problem for singular lagrangian systems of strong force type

collisionless periodic solutions to some three body problems we provide sufficient conditions for the existence of periodic solutions to some three body problems periodic solutions are found as minima of the associated action integral and are shown to be free of double and triple collisions

periodic solutions of prescribed energy for a class of hamiltonian systems with singular potentials abstract we study the existence of periodic solutions of prescribed energy for a class of hamiltonian systems with singular potentials we found periodic solutions lying between two prescribed spheres

all controllers for the general h control problem lmi existence conditions and state space formulas this paper presents all controllers for the general h control problem with no assumptions on the plant matrices necessary and sufficient conditions for the existence of an h controller of any order are given in terms of three linear matrix inequalities lmis our existence conditions are equivalent to scherer s results but with a more elementary derivation furthermore we provide the set of all h controllers explicitly parametrized in the state space using the positive definite solutions to the lmis even under standard assumptions full rank etc our controller parametrization has an advantage over the q parametrization the freedom q a real rational stable transfer matrix with the h norm bounded above by a specified number is replaced by a constant matrix l of fixed dimension with a norm bound and the solutions x y to the lmis the inequality formulation converts the existence conditions to a convex feasibility problem and also the free matrix l and the pair x y define a finite dimensional design space as opposed to the infinite dimensional space associated with the q parametrization

positive real control problem for uncertain linear time invariant systems abstract this paper focuses on positive real control of linear time invariant systems which are subjected to norm bounded uncertainties in the state equation we address the problem of designing a linear dynamic output feedback controller that robustly stabilizes the uncertain system and achieves the extended strict positive realness property for a given closed loop transfer function it is shown that a solution to the above problem can be obtained by solving a scaled strict positive real control problem for which no parameter uncertainty occurs

application of a multiplier method to uncertain lur e like systems abstract this paper investigates the conditions under which an abstract matrix multiplier method can be applied to determine guaranteeing cost controls for systems containing nonlinear uncertain elements via linear matrix inequalities lmis quadratically constrained uncertainties and nonlinearities are considered which comprehend the cases of norm bounded positive real and sector bounded uncertainties nonlinearities both the discrete time and the continuous time cases are discussed necessary and sufficient conditions are formulated in the case of unstructured uncertainty the conditions are sufficient in the structured case the cost guaranteeing controls can be determined by solving lmis numerical examples illustrate the effectiveness of the proposed method

solution to the positive real control problem for linear time invariant systems in this paper we study the problem of synthesizing an internally stabilizing linear time invariant controller for a linear time invariant plant such that a given closed loop transfer function is extended strictly positive real necessary and sufficient conditions for the existence of a controller are obtained state space formulas for the controller design are given in terms of solutions to algebraic riccati equations or inequalities the order of the constructed controller does not exceed that of the plant

new proof and generalizations of the demidowitsch schneider criterion a new proof of the demidowitsch schneider criterion on the absence of closed trajectories of r3 is given the new proof is generalized and applied in several directions rn vector fields n 3

periodic orbits of radially symmetric keplerian like systems a topological degree approach abstract we are concerned with non autonomous radially symmetric systems with a singularity which are t periodic in time by the use of topological degree theory we prove the existence of large amplitude periodic solutions whose minimal period is an integer multiple of t precise estimates are then given in the case of keplerian like systems showing some resemblance between the orbits of those solutions and the circular orbits of the corresponding classical autonomous system

chebyshev expansion methods for the solution of the extended graetz problem abstract the extended graetz problem a classical problem described by an equation with anon self adjoint second order elliptic differential operator is solved by two numerical methods based on chebyshev expansions a chebyshev finite difference method and a chebyshev finite element method the latter relies entirely on chebyshev expansions global in the bounded radial coordinate direction and local in the unbounded axial direction and proves to be more accurate than the chebyshev finite difference method in resolving singularities both methods can be applied to general type boundary value problems involving second order elliptic operators have accuracy comparable to high order finite difference schemes and are operation cost efficient

sign changing solutions for coupled nonlinear schrodinger equations with critical growth abstract we consider the following nonlinear schrodinger system with critical growth u j j u j i 1 k i j u i 2 2 u j 2 2 2 u j in u j 0 on j 1 k where is a bounded smooth domain in r n 2 2 n n 2 0 j 1 j 1 k 1 is the first eigenvalue of with zero dirichlet boundary condition we consider the repulsive case namely j j 0 j 1 k i j j i 0 i j i j 1 k the existence of infinitely many sign changing solutions as bound states is proved provided n 7 by approximations of systems with subcritical growth and by the concentration analysis on approximating solutions

transitive decomposition of symmetry groups for the n body problem periodic and quasi periodic solutions of the n body problem are critical points of the action functional constrained to the sobolev space of symmetric loops variational methods yield collisionless orbits provided the group of symmetries fulfills certain conditions such as the rotating circle property here we generalize such conditions to more general group types and show how to constructively classify all groups satisfying such hypothesis by a decomposition into irreducible transitive components as examples we show approximate trajectories of some of the resulting symmetric minimizers

category of loop spaces of open subsets in euclidean space has been studied in several recent papers e g l 2 5 6 91 in situations where the potential function v is singular in the sense that it is defined and c only on an open subset u of euclidean k space rk and not on the whole of rk if e wgt2 r rk is the sobolev space of t periodic functions from r to rk and e u is the subset of e of functions which map into u then the topology of e u plays a crucial role in these studies in particular the knowledge that the lusternik schnirelmann category of e u is infinite when used in conjunction with min max methods implies the existence of infinitely many critical points another subset e u which consists of those functions in e u which take the origin to a base point in u plays a dual role first of all in problems concerning the existence of geodesics in u from one subset a of u to another b of u see e g 3 the category of e u plays a role secondly the general inequality cat e u 2 cat e u see 6 when u is connected and simply connected can be applied to show that cat e u cat e u thus providing a tool for computing cat e u the topological analogue of e u is the space of free loops au c s u of continuous maps from the circle s to u and the analogue of e u is the space of based loops ru which consists of maps in au which take a base point in s to a base point in u au has the same homotopy type as e u and s2u has the same homotopy type as e u thus homotopy type invariants e g category of e u and e u may be studied through au and szu respectively the following classical theorem of serre l l is useful in this regard and is stated in terms of a general space x

study of the generalized solutions of n body type problems with weak forces we show that the generalized solutions of n body type problems with weak forces obtained as limits of classical solutions of problems with strong force potentials have at most a finite number of collisions we also estimate the number of collisions using the morse index of appropriate fuctionals

closed orbits of fixed energy for singular hamiltonian systems this paper deals with the existence of periodic solutions of q v q 0 such that 1 2 q 2 v q h where q r n h is a given number v c 2 r n 0 r has a singularity at x 0 and v denotes the gradient of v

periodic solutions on hypersurfaces and a result by c viterbo on considere un champ vectoriel hamiltonien x j h x x h x sur x r 2n h etant une fonction lisse dont le gradient h est defini par rapport a la metrique euclidienne on cherche des solutions periodiques sur une surface d energie donnee s x r 2n h x const supposee reguliere

a mountain pass theorem 143 in other words the mountain separating 0 and e may even be assumed of zero altitude and the conclusion is still true moreover as we shall show if c a the critical point can be chosen with llx ll r it is interesting to ask whether this extension of the ambrosetti rabinowitz theorem remains true in the infinite dimensional case we prove here that if 1 is strengthened a little to the form 1 there exist real numbers u r r such that 0 a for every x e a x e x y a moreover if c a the critical point can be chosen with r i x0 i r roughly speaking in brief the mountain pass theorem continues to hold for a mountain of zero altitude provided it also has non zero thickness in addition if c a the pass itself occurs precisely on the mountain i e satisfying r llroll r there are two interesting and immediate corollaries the first one says that a c function which has iwo local minimum points also has a third critical point the second states that a u periodic c function with a local minimum e has a critical point x e ku k 0 1 2 the precise statements of the above results will be given in the next sec tions the proof depends upon a lemma of clark 3 formulated here in a version suitable to our purpose in section 3 some applications are presen ted for the forced pendulum equation 2

an integral chebyshev expansion method for boundary value problems of o d e type abstract a chebyshev expansion method for the solution of boundary value problems of o d e type is presented the method employs the pseudospectral collocation approximation and generates approximations to the lower order derivatives of the function through successive integrations of the chebyshev polynomial approximation to the highest order derivative the method is easier to implement than spectral methods employing the galerkin and tau approximations and yields results of comparable accuracy to these methods with reduced computing requirements applications to the linear stability problems for plane poiseuille and the blasius boundary layer flows are presented

an homotopical index and multiplicity of periodic solutions to dynamical systems with singular potential this paper has two main purposes the first one is to define a tool homotopical index theory which provides critical points for functionals having a singularity and a lack of compactness the second one is to apply the abstract results in order to find a multiplicity of periodic solutions to a class of dynamical systems with singular potential let us fix up some notations in the following t 0 will be fixed and we shall denote hj h r tz w and en the subspace of hf of all the constant functions we consider

the structure of the critical set in the mountain pass theorem we show that the critical set generated by the mountain pass theorem of ambrosetti and rabinowitz must have a well defined structure in particular if the underlying banach space is infinite dimensional then either the critical set contains a saddle point of mountain pass type or the set of local minima intersects at least two components of the set of saddle points related conclusions are also established for the finite dimensional case and when other special conditions are assumed throughout the paper no hypotheses of nondegeneracy are required on the critical set

non simultaneous collision periodic solutions forn body problems abstract in this paper we prove the existence of a non simultaneous collision t periodic solution for any given t for a class of n body type problems

positive solutions to superlinear semipositone periodic boundary value problems with repulsive weak singular forces this paper is devoted to study the existence of positive solutions to the second order semipositone periodic boundary value problem x a t x f t x x 0 x 1 x 0 x 1 here f t x may be singular at x 0 and may be superlinear at x our analysis relies on a fixed point theorem in cones

periodic orbits of n body type problems the fixed period case this paper gives a proof of the existence and multiplicity of periodic solutions to hamiltonian systems of the form a v t q o q t t q t vt e r where qi e e 3 1 i n q q1 qn and n v e vlj t qi qj i i j with vij t t periodic in t and singular in 4 at 0 under additional hypotheses on v when a is posed as a variational problem the corresponding functional i is shown to have an unbounded sequence of critical values if the singularity of v at 0 is strong enough the critical points of i are classical t periodic solutions of a then assuming that i has only non degenerate critical points up to translations morse type inequalities are proved and used to show that the number of critical points with a fixed morse index k grows exponentially with k at least when k 0 1 mod e 2 the proof is based on the study of the critical points at infinity done by the author in a previous paper and generalizes the topological arguments used by a bahri and p rabinowitz in their study of the 3 body problem it uses a recent result of e fadell and s husseini on the homology of free loop spaces on configuration spaces the detailed proof is given for the 4 body problem then generalized to the n body problem

a new discrete time robust stability condition a new robust stability condition for uncertain discrete time systems with convex polytopic uncertainty is given it enables to check stability using parameter dependent lyapunov functions which are derived from lmi conditions it is shown that this new condition provides better results than the classical quadratic stability besides the use of a parameter dependent lyapunov function this condition exhibits a kind of decoupling between the lyapunov and the system matrices which may be explored for control synthesis purposes a numerical example illustrates the results

a note on periodic solutions of prescribed energy for singular hamiltonian systems abstract in this note we show under a less restrictive setting than previously that there exists a collision brake orbit of prescribed energy to a certain second order hamiltonian system

nontrivial solutions of operator equations and morse indices of critical points of min max type on montre que si les hypotheses du theoreme de point selle sont satisfaites si dim y 2 et si f 1 c contient seulement des points critiques non degeneres alors f 1 c contient au moins un point critique u 0 tel que 2 indice de morse u 0 dim y

instability analysis of saddle points by a local minimax method the objective of this work is to develop some tools for local instability analysis of multiple critical points which can be computationally carried out the morse index can be used to measure local instability of a nondegenerate saddle point however it is very expensive to compute numerically and is ineffective for degenerate critical points a local weak linking index can also be defined to measure local instability of a degenerate saddle point but it is still too difficult to compute in this paper a local instability index called a local minimax index is defined by using a local minimax method this new instability index is known beforehand and can help in finding a saddle point numerically relations between the local minimax index and other local instability indices are established those relations also provide ways to numerically compute the morse local linking indices in particular the local minimax index can be used to define a local instability index of a saddle point relative to a reference trivial critical point even in a banach space while others failed to do so

existence of infinitely many critical values of some nonsymmetric functionals abstract this paper studies the existence and multiplicity results for semi linear elliptic equations u q x u p 1 u h x in u 0 on here is a smooth bounded domain in r n q c 1 gw is positive somewhere we show that for some p and h given the above equation has infinitely many solutions

morse type information on palais smale sequences obtained by min max principles in the context of the min max approach to critical point theory but without the usual compactness assumptions a la palais smale and the nondegeneracy conditions a la fredholm we construct almost critical points with two sided estimates on their approximate morse indices which are arbitrarily close to certain a priori given dual sets this additional topological and analytical information about almost critical sequences can sometimes be crucial in the proof of their convergence and therefore in solving the corresponding variational problem 1994 john wiley sons inc

a prescribed energy problem for a singular hamiltonian system with a weak force we consider the existence of periodic solutions of a hamiltonian system q v q 0 hs such that 12 q t 2 v q t h for all t where q rn n 3 h 0 we prove the existence of a generalized solution that may enter the singularity 0 moreover under the assumption v q 1 q w q where 0 2 and q w q q 1 w q q 2 2w q 0 as q 0 we estimate the number of collisions of generalized solutions in particular we get the existence of a classical non collision solution of hs for l 2 when n 4 and for 4 3 2 when n 3

on a class of dynamical systems with singular potential we are concerned with the problem of searching for t periodic solutions t prescribed for the equation x v x where x t c 2 r r n and v c 1 r 0 r in the case when v is singular i e v behaves like k x near the origin

multiple closed orbits of fixed energy forn body type problems with gravitational potentials abstract using the equivariant ljusternik schnirelmann theory and the estimates of the upper bound and lower bound of the critical values for the variational functional we obtain results in general concerning multiple geometrically distinct periodic solutions of fixed energy for a class of planarn body type problems

periodic solutions of singular hamiltonian systems with prescribed energy abstract we study the existence of non collision periodic solutions for some class of singular hamiltonian systems in particular in the case that the weak force potential has a set of singular points which is a compact subset of r n we prove the existence of a non collision periodic solution for all positive prescribed energy

periodic solutions to some problems of n body type we prove the existence of at least one t periodic solution to a dynamical system of the type m i ddot u i sum limits j 1 j ne i n triangledown v ij u i u j text t 1 where the potentials v ij are t periodic in t and singular at the origin u i e r k i 1 n and k 3 we also provide estimates on the h 1 norm of this solution the proofs are based on a variant of the ljusternik schnirelman method the results here generalize to the n body problem some results obtained by bahri rabinowitz on the 3 body problem in 6