the laplacian spectral radius of some graphs the laplacian spectral radius of a graph is the largest eigenvalue of the associated laplacian matrix in this paper we determine those graphs which maximize the laplacian spectral radius among all bipartite graphs with edge connectivity at most k we also characterize graphs of order n with k cut edges having laplacian spectral radius equal to n

periodic solutions of a class of dynamical systems of second order abstract we give some sufficient conditions for the existence of infinitely many periodic solutions with prescribed period for a class of dynamical systems of second order variational methods and finite dimensional approximations are used

a symplectic fixed point theorem for cpn two symplectic diffeomorphisms 0 1 of a symplectic manifold x are said to be homologous if there exists a smooth homotopy 1 t 0 1 of symplectic diffeomorphisms between them such that the timedependent vector field t defined byd dt t t o t is a globally hamiltonian vector field for allt i e there exists a smooth real valued timedependent hamiltonian functionh x t onx x 0 1 such that t dh t whereht h x t

the spectral radius of unicyclic and bicyclic graphs with n vertices and k pendant vertices in this paper we determine graphs with the largest spectral radius among all the unicyclic and all the bicyclic graphs with n vertices and k pendant vertices respectively

subharmonic oscillations of forced pendulum type equations where f is periodic with minimal period t and mean value zero we have in mind as a particular case the pendulum equation where g x a sin x first results on the existence of subharmonic orbits in a neighborhood of a given periodic motion were obtained by birkhoff and lewis cf 3 and 143 by perturbation type techniques rabinowitz 151 was able to prove the existence of subharmonic solutions for hamiltonian systems by the use of variational methods his approach is not of local type like the one in 3 and enables one to obtain a sequence of solutions whose minimal period tends toward infinity in the case when the hamiltonian function has subquadratic or superquadratic growth these results have been extended in various directions cf 2 5 6 8 13 16 181 local results on subharmonies for the forced pendulum equation can be found in 19 hamiltonian systems with periodic nonlinearity were studied by conley and zehnder 6 they proved the existence of subharmonic solutions under some assumptions on the nondegenerateness of the solutions by the use of morse conley theory in this paper we will prove the existence of subharmonic oscillations of a pendulum type equation by the use of classical morse theory together with an iteration formula for the index due to bott 4 and developed in 7 and l

periodic solutions for nonlinear second order difference equations abstract this paper is devoted to introduce a new approach to investigate the existence of periodic solutions for nonlinear difference equations transforming a second order nonlinear difference equation on a finite discrete segment with periodic boundary conditions into a continuous system we obtain the existence conditions of periodic solutions of the equation the proof relies on coincidence degree theory and matrix theory without using green s function

on the dimension and the index of the solution set of nonlinear equations we study the covering dimension and the index of the solution set to multiparameter nonlinear and semilinear operator equations involving fredholm maps of positive index the classes of maps under consideration are pseudo a proper and either approximation essential or equivariant approximation essential applications are given to semilinear elliptic bvp s

on the periodic fucik spectrum and a superlinear sturm liouville equation in the first part of the paper a variational characterisation of the periodic eigenvalues the so called fucik spectrum of a semilinear positive homogeneous sturm liouville equation is given the proof relies on the s 1 invariance of the equation in the second part a nonlinear sturm liouville equation with typically an exponential nonlinearity is considered it is proved that under certain conditions this equation is solvable for arbitrary forcing terms the proof uses a comparison of the minimax levels of the functional associated to this equation with suitable values in the fucik spectrum

equivariant morse theory for flows and an application to the n body problem a l aide de l indice de conley et de la cohomologie equivariante on deduit certaines inegalites de type morse pour un flot equivariant par rapport a l action d un groupe topologique compact

periodic and subharmonic solutions for a class of superquadratic hamiltonian systems abstract some existence theorems are obtained for periodic and subharmonic solutions of a class of superquadratic hamiltonian systems by the minimax methods in critical point theory specially a local link theorem due to li and willem and the generalized mountain pass theorem respectively

multiple solutions of hamiltonian systems via limit relative category by means of an extension of relative category we prove by a linking type theorem the existence of two distinct critical levels for a strongly indefinite functional applications to asymptotically linear hamiltonian systems are given

nonlinear problems with strong resonance at infinity an abstract theorem and applications on considere l equation au n u u e ou a est un operateur lineaire n avec c 1 e r et e est un espace de hilbert on suppose que n a une derivee a l infini n et que o appartient au spectre de a n on demontre un theoreme pour la multiplicite des solutions

periodic solutions of symmetric autonomous newtonian systems abstract in this paper we show the existence and bifurcation of t periodic solutions of a special form for an autonomous newtonian system with symmetry if the phase space r 2 n is equipped with the structure of an orthogonal representation w w and the potential v w r is invariant then for every such a solution the set of indices of nonvanishing fourier coefficients is finite and depends on w only if the potential v depends on the squares of complex coordinates then for every such a solution t is the minimal period

multiple critical points of invariant functionals and applications abstract this paper deals with some multiplicity results of periodic orbits of hamiltonian systems and for solution of a non linear dirichlet problem these results follow from an abstract theorem of lusternik schnirelman type as applied to an invariant equation of the form lu delta f u 0 in a hilbert space x l sub 2 omega r sub n where l is an unbounded self adjoint operator and f is a c sub 1 strictly convex function author

a borsuk ulam theorem for orthogonal tk and zpr actions and applications abstract a version if the borsuk ulam theorem is proved for the torus tk and p torus zpr we deal with the case of a nonempty fixed point set of groups on the sphere of representation as an application a geometrical index theory is introduced for these groups the index is used to obtain the existence of multiple critical points for functions invariant under tk or zpr action

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

infinitely many solutions of a symmetric dirichlet problem here q is a bounded domain in r with smooth boundary and f i r is c and satisfies certain growth conditions if m 1 and f is an even function hence f r r is an odd function then ambrosetti and rabinowitz l 21 proved the existence of infinitely many weak solutions of d using a symmetric version of the mountain pass theorem a similar result is due to michalek 3 who considered the case where m 21 rm s c and f is invariant under a free action of the cyclic group z p on c here z p is considered as a subgroup of s c c acting essentially via scalar multiplication on c in this paper we want to investigate which kind of symmetries guarantee the existence of infinitely many solutions of d let g be a compact lie group and p g o m an orthogonal representation of g on lrm then g acts on irm via the map g x ir ir g u y gv p g v we write v for the space rm considered as a g space do there exist infinitely many solutions of d if f is invariant under this symmetry i e f gu f u for all g e g and u e v without further assumptions on p the answer to this question is no in general for example if p g id for all g e g then there is no symmetry at all and one cannot expect infinitely many solutions more generally one should exclude the existence of fixed points vg u e vi gu u for all g e g 0 our first result says that there exist infinitely many solutions of d if the following condition is satisfied

lusternik schnirelman method for functionals invariant with respect to a finite group action has at least g i n 1 distinct critical points see kl ma we refer also to corollary 1 5 which is a generalization of this result consequently the existence of a symmetry group g for the func tionalf in general defined on a banach manifold has a significant impact on the number of critical points the problem of finding the best possible estimates for the number of orbits of critical points of an invariant func tional was studied by many authors for a i action with p prime number see el for s action see bl b2 cw2 cw3 fr2 r2 for a non free action where morse theory is applied see pl cwl w see also da fh frl this work is an attempt to give a general and explicit formula cf theorem 1 3 for the number of critical points of an invariant func tional

periodic oscillations for a nonlinear suspension bridge model abstract we look for time periodic solutions of the suspension bridge equation lazer and mckenna showed that for a certain configuration of the parameters one may expect the existence of large amplitude periodic solutions having the same period as the forcing term we prove the existence of large scale subharmonic solutions

periodic solutions of dynamical systems with bounded potential where ve c2 r x rn r v t t x y t x v t x jjbxiw y is bounded and v v t x 0 as 1x1 co for example we prove that if 0 d v r x m and v r x m as 1x1 co there exists at least one t periodic solution of v moreover if v j t 0 0 under suitable assumptions on v y t 0 this solution is not identically zero we also give existence results for the forced system

subharmonics for convex nonautonomous hamiltonian systems on etudie des solutions periodiques du systeme hamiltonien jx h t x ou j est la matrice symplectique standard et h c 2 r r 2n r est t periodique en la premiere variable pour tout t x h t x note le gradient partiel par rapport a la seconde variable

on critical point theory for indefinite functionals in the presence of symmetries we consider functionals which are not bounded from above or from below even modulo compact perturbations and which exhibit certain symmetries with respect to the action of a compact lie group we develop a method which permits us to prove the existence of multiple critical points for such functionals the proofs are carried out directly in an infinite dimensional hilbert space and they are based on minimax arguments the applications given here are to hamiltonian systems of ordinary differential equations where the existence of multiple time periodic solutions is established for several classes of hamiltonians symmetry properties of these hamiltonians such as time translation invariancy or evenness are exploited introduction many variational problems which arise from physics or mathematical physics are indefinite in the sense that the functionals involved are not bounded from below or from above however some of these functionals defined in an appropriate function space are semidefinite in the sense that they are bounded from below or from above if perturbed with a weakly continuous functional this paper deals with functionals which are not semidefinite usually problems involving indefinite functionals are more difficult to handle and only very recently a method has been developed which permits us to treat such problems directly in an infinite dimensional function space 7 in 7 some theorems have been proved which establish the existence of at least one nontrivial critical point for such functionals in many physical situations there are problems which have symmetries with respect to the action of some lie group in this case we expect the existence of many critical points this has been established for semidefinite functionals cf e g 8 and 3 for even functionals 9 for a zp action with p prime numbers 15 16 10 and 6 for an s action in this paper we develop a method which allows us to estimate the number of critical points of indefinite functionals in the presence of symmetries more precisely received by the editors july 16 1980 and in revised form june 26 1981 1980 mathematics subject classification primary 58e05 70h05 secondary 34c25

on the spectral radius of graphs with cut edges we study the spectral radius of graphs with n vertices and k cut edges in this paper we show that of all the connected graphs with n vertices and k cut edges the maximal spectral radius is obtained uniquely at k k where k k n is a graph obtained by joining k independent vertices to one vertex of kn k we also discuss the limit point of the maximal spectral radius

on the laplacian spectral radii of bipartite graphs the laplacian spectral radius of a graph is the largest eigenvalue of the associated laplacian matrix in this paper we provide structural and behavioral details of graphs with maximum laplacian spectral radius among all bipartite connected graphs of given order and size using these results we provide a unified approach to determine the graphs with maximum laplacian spectral radii among all trees and all bipartite unicyclic bicyclic tricyclic and quasi tree graphs respectively

on the ring of invariants of ordinary quartic curves in characteristic 2 abstract the moduli space of the ordinary non singular quartic curves over fields of characteristic 2 is isomorphic to a certain open subset of an affine variety whose coordinate ring in turn is given as the invariant algebra of a certain module of the finite group gl 3 f 2 we derive a complete description of this invariant algebra by combining theoretical analysis with application of specially tailored computational techniques

existence of multiple periodic orbits of hamiltonian systems on positive type hypersurfaces in r2n abstract this paper deals with the fixed energy problem of hamiltonian systems in r 2n the main result is an existence theorem of multiple periodic orbits for hamiltonian vector fields on a class of contact type but not necessarily star shaped hypersurfaces this is the first multiplicity result for hypersurfaces that are more general than star shaped ones the main result is achieved by using critical point theory for a special hamiltonian function and central gravity of the proof relies on a delicate construction of such hamiltonian function

applications of a zp index theory to periodic solutions for a class of functional differential equations abstract by means of a new geometrical index with z p group actions multiplicity results for a certain class of nonautonomous time periodic functional differential systems are obtained

on the fucik spectrum of the laplacian on a torus abstract we study the fucik spectrum of the laplacian on a two dimensional torus t 2 exploiting the invariance properties of the domain t 2 with respect to translations we obtain a good description of large parts of the spectrum in particular for each eigenvalue of the laplacian we will find an explicit global curve in the fucik spectrum which passes through this eigenvalue these curves are ordered and we will show that their asymptotic limits are positive on the other hand using a topological index based on the mentioned group invariance we will obtain a variational characterization of global curves in the fucik spectrum also these curves emanate from the eigenvalues of the laplacian and we will show that they tend asymptotically to zero thus we infer that the variational and the explicit curves cannot coincide globally and that in fact many curve crossings must occur we will give a bifurcation result which partially explains these phenomena

multiple periodic solutions for a semilinear wave equation with nonmonotone nonlinearity on cherche des solutions t periodiques de l equation d onde non lineaire u tt u xx g u u t x u t t x pour t r x 0 u t 0 u t 0 pour t r ou t est un multiple rationnel de et g est une application continument differentiable

solutions of minimal period for a class of nonconvex hamiltonian systems and applications to the fixed energy problem on considere le systeme hamiltonien jz h z ou j x y y x x y r n r n h est non convexe a un bon comportement superquadratique et satisfait une condition simple reliant ses coefficients de superquadraticite et l exposant dans ce cas on demontre que pour t 0 l existence de solutions pour le systeme hamiltonien de periode minimale t

multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree abstract we study the periodic boundary value problem associated with the second order nonlinear differential equation u c u a t a t g u 0 where g u has superlinear growth at zero and at infinity a t is a periodic sign changing weight c r and 0 is a real parameter our model includes for c 0 the so called nonlinear hill s equation we prove the existence of 2 m 1 positive solutions when a t has m positive humps separated by m negative ones in a periodicity interval and is sufficiently large thus giving a complete solution to a problem raised by g j butler in 1976 the proof is based on mawhin s coincidence degree defined in open possibly unbounded sets and applies also to neumann boundary conditions our method also provides a topological approach to detect subharmonic solutions

subharmonics for not uniformly coercive hamiltonian systems abstract in this paper we prove the existence of subharmonic solutions with prescribed minimal period for the hamiltonian system u t j h t u t when the hamiltonian h is subquadratic with unbounded gradient and not uniformly coercive

subharmonic solutions with prescribed minimal period for nonautonomous hamiltonian systems on etablit un resultat de multiplicite pour une certaine classe de systemes hamiltoniens periodiques en temps

interlacing eigenvalues on some operations of graphs in this paper we focus on some operations of graphs and give a kind of eigenvalue interlacing in terms of the adjacency matrix standard laplacian and normalized laplacian also we explore some applications of this interlacing

existence of multiple normal mode trajectories on convex energy surfaces of even classical hamiltonian systems hamiltonian systems of n degrees of freedom for which the hamiltonian is a function that is even both in its joint n coordinate variables as well as in its joint n momentum variables are discussed for such systems the number of distinct trajectories which correspond to particular periodic solutions normal modes with the same energy is investigated to that end a constrained dual action principle is introduced applying min max methods to this variational problem several results are obtained among which the existence of at least n distinct trajectories if specific conditions are satisfied

morse indices of critical manifolds generated by min max methods with compact lie group actions and applications we derive some upper and lower bounds for morse indices of critical manifolds generated by min max principles for functionals invariant under a general compact lie group or a finite group action the results generalize the similar results in the nonequivariant no group action case in doing so we also generalize the extension theorem of dugundji type in the nonequivariant case to the equivariant group action case as an application we obtain a precise growth estimate for the whole sequence of critical values given by the min max procedure for some superquadratic second order differential equations it is well known that this growth estimate is crucial in showing the existence of multiple solutions of some superquadratic perturbed hamiltonian systems and equations

the laplacian spectral radius of some bipartite graphs in this paper we study the largest laplacian spectral radius of the bipartite graphs with n vertices and k cut edges and the bicyclic bipartite graphs respectively identifying the center of a star k1 k and one vertex of degree n of km n we denote by km nk the resulting graph we show that the graph k2 n k 2k 1 k n 4 is the unique graph with the largest laplacian spectral radius among the bipartite graphs with n vertices and k cut edges and k2 3n 5 n 7 is the unique graph with the largest laplacian spectral radius among all the bicyclic bipartite graphs

periodic solutions to second order hamiltonian systems in an unbounded potential well in this paper we give some results on the existence of periodic solutions to the second order hamiltonian system where and is an open set of n with non empty bounded complement n we suppose v t x is periodic in t v t x as x and v is super or sub quadratic as x

atlas of finite groups maximal subgroups and ordinary characters for simple groups this atlas covers groups from the families of the classification of finite simple groups recently updated incorporating corrections

periodic and subharmonic solutions of a class of superquadratic hamiltonian systems abstract periodic solutions and infinitely distinct subharmonic solutions are obtained for a class of nonconvex and nonautonomous superquadratic hamiltonian systems z j h z z t by using the minimax methods in critical point theory

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

multiple periodic solutions for superquadratic second order discrete hamiltonian systems abstract some multiplicity results are obtained for periodic solutions of the nonautonomous superquadratic second order discrete hamiltonian systems 2 u t 1 f t u t 0 t z by using critical point theory especially a three critical points theorem proposed by brezis and nirenberg

a z p borsuk ulam theorem and index theory with a multiplicity result in partial differential equations on demontre un theoreme de borsuk ulam z p generalise c est le resultat topologique essentiel pour developper un theoreme d indice z p on demontre un resultat de multiplicite pour un systeme d equations aux derivees partielles elliptiques non lineaires

morse theory for symmetric functionals on the sphere and an application to a bifurcation problem abstract this paper uses conley s index to study the cortical points of a function f on a finite dimensional sphere in presence of a symmetry group the authors prove a theorem which leads to a lower bound on the number of critical points of f when the group is finite even if the action is not free this investigation has been motivated by the following bifurcation problem au lambda where a is a variational g equivariant operator an estimated on the number of branches bifurcating from an eigenvalue of a 0 is given author

periodic and subharmonic solutions of a class of subquadratic second order hamiltonian systems some solvability conditions of periodic and subharmonic solutions are obtained for a class of subquadratic nonautonomous second order hamiltonian systems by the minimax methods in critical point theory

analytical mini max methods for hamiltonian brake orbits of prescribed energy on considere des caracterisations explicites de solutions periodiques de systemes hamiltoniens autonomes d ordre 2 de la forme q v q q t r n ou le potentiel v satisfait les conditions suivantes v c 2 r n v x v o 0 v est strictement convexe sur r n

periodic solutions to hamiltonian inclusions abstract we prove the existence of periodic trajectories of hamiltonian inclusions which reduce to the usual hamiltonian equations in the presence of smoothness this is accomplished via a direct variational principle involving a new action integral in an extended sense

nonlinear oscillations and boundary value problems for hamiltonian systems we prove the existence of solutions of various boundary value problems for nonautonomous hamiltonian systems with forcing terms begin gathered dot x t h p t x t p t g t hfill dot p t h x t x t p t f t hfill end gathered

existence of multiple periodic orbits on star shaped hamiltonian surfaces consider the hamiltonian system hs i 1 n here h c2 2n in this paper we investigate the existence of periodic orbits of hs on a given energy surface z 2n h z c c o is a constant the surface is required to verify certain geometric assumptions bounds a star shaped compact region and for some ellipsoid 2n o o depending in an explicit fashion on the lengths of the main axes of and one other geometrical parameter of such that if furthermore 2 2 1 then hs has at least n distinct geometric orbits on this result is shown to extend and unify several earlier works on this subject among them works by weinstein rabinowitz ekeland lasry and ekeland in proving this result we construct index theories for an s1 action from which we derive abstract critical point theorems for s1 invariant functionals we also derive an estimate for the minimal period of solutions to differential equatious