performance advantage of complex lms for controlling narrow band adaptive arrays in narrow band adaptive array applications the mean square convergence of the discrete time real least mean square lms algorithm is slowed by image frequency noises generated in the lms loops the complex lms algorithm proposed by widrow et al is shown to eliminate these noises yielding convergence of the mean squared error mse at slightly over twice the rate this paper includes a comprehensive analysis of the mse of adaptation for lms the analysis is based upon the method developed in the 1968 dissertation by k d senne and it represents the most complete treatment of the subject published to date

morse theory and asymptotic linear hamiltonian system on considere l existence de solutions 2 periodiques des systemes hamiltoniens asymptotiquement quadratiques jz h z t ou j 0 i i 0 i in la matrice identite de n n

dual morse index estimates for periodic solutions of hamiltonian systems in some nonconvex superquadratic cases in l ekeland and hofer stated a very general result for the existence of periodic solutions with any prescribed minimal period for hamiltonian systems assuming that the hamiltonian function h is strictly convex with superquadratic behaviour at the origin and at infinity the arguments given in l are based on an estimate for the morse index of a critical point of mountain pass type for a functional which is associated with the hamiltonian system via the duality principle of clarke and ekeland 2 unfortunately the functional is only c and not c then the result is achieved only through a finite dimensional reduction and as the authors themselves say through an inordinate amount of technicalities in this paper we give a direct definition of the morse index of a periodic solution which does not use the duality principle indeed it is based on a study of the morse indexes of approximating solutions obtained by a galerkin scheme such as the one introduced by rabinowitz in 3 taking into account some results about linking point theory recently stated in 4 71 we get the same index estimates as in l but in a simple way moreover some generalizations are obtained in the sense that the convexity assumption on h can be replaced either by a local convexity condition or by a condition on the second derivative of h plus eventually a symmetry condition see also 8 91 for some other results obtained in this framework in particular in the locally convex case the index estimates yield direct information on the minimal period of the corresponding solution finally let us remark that this definition of the morse index can be adapted to other nonconvex cases and one can still get some index estimates of the same type

solutions of asymptotically linear operator equations via morse theory abstract in this paper we use the classical morse theory of critical points to study the existence and multiplicity of solutions of a class of asymptotically linear operator equations as special cases we include multiplicity results for semilinear elliptic bvp s as well as existence and multiplicity of periodic solutions of semilinear wave equation and of hamiltonian systems these results simplify and complement recent work of amann zehnder and castro lazer author

iteration inequalities of the maslov type index theory with applications abstract in this paper by using the index theory introduced by y long in 1999 pacific j math 187 113 149 in particular the splitting numbers maslov type mean index and the homotopy component of symplectic matrix we establish various inequalities of the maslov type index theory for iterations of symplectic paths starting from the identity as an application these results are used to study rabinowitz conjecture on the prescribed minimal period solution problem of nonlinear hamiltonian systems

the minimal period problem of periodic solutions for autonomous superquadratic second order hamiltonian systems abstract in this paper we study the existence of periodic solutions with prescribed minimal period for superquadratic autonomous second order hamiltonian systems defined on r n with no convexity assumptions we use the direct variational approach for this problem on a w 1 2 space of even functions and prove new iteration inequalities on morse indices using these tools and the saddle point theorem we obtain results under precisely rabinowitz superquadratic condition on potential functions we show that for every t 0 the above mentioned system possesses a t periodic even solution with minimal period not smaller than t n 2

hybrid polynomial filters for gaussian and non gaussian noise environments traditional polynomial filtering theory based on linear combinations of polynomial terms is able to approximate important classes of nonlinear systems the linear combination of polynomial terms however yields poor performance in environments characterized by gaussian and heavy tailed distributions weighted median and weighted myriad filters in contrast are well known for their outlier suppression and detail preservation properties it is shown here that the weighted median and weighted myriad methodologies are naturally extended to the polynomial sample case yielding hybrid filter structures that exploits the higher order statistics of the observed samples while simultaneously being robust to outliers for both gaussian and heavy tailed distributions environments moreover the introduced hybrid polynomial filter classes are well motivated by analysis of cross and square term statistics of gaussian and heavy tailed distributions a presented asymptotic tail mass analysis shows that polynomial terms both under gaussian and heavy tailed noise statistics have heavier tails than the observed samples indicating that robust combination methods should be utilized to avoid undue influence of outliers further analysis shows weighted median processing of polynomial terms for the gaussian noise case and weighted median and weighted myriad processing of cross and square terms respectively for the heavy tailed noise case are justified from a maximum likelihood perspective filters parameter optimization procedures are also presented finally the effectiveness of hybrid filters is demonstrated through simulations that include temporal spectrum and bispectrum analysis

nonautonomous nonlinear odes nonresonance conditions and rotation numbers abstract we deal with the t periodic problem associated with a nonlinear scalar differential equation like 1 x f t x 0 where for x 0 and x the nonlinearity f is assumed behave linearly with a time dependent coefficient we prove that the landesman lazer conditions at zero and at infinity possess a rotational effect on small and large in the phase plane solutions of 1 as a consequence we are able to generalize previous multiplicity results in the resonant case i e when the linearizations at zero and or at infinity are resonant through the use of the poincare birkhoff fixed point theorem

constrained equations with impasse points abstract we give local normal forms for generic constrained equations of the form f z z h z wherefis a smooth real function defined on a manifoldmandhis a smooth vector field onm

multiple solutions for an asymptotically linear duffing equation with newmann boundary value conditions abstract we investigate multiple solutions of an asymptotically linear duffing equation satisfying newmann boundary value conditions with resonance using index theory and morse theory and obtain some new results

prediction of stable processes spectral and moving average representations for stable processes which are fourier transforms of processes with independent increments we obtain a wold decomposition we characterize their regularity and singularity and in the discrete parameter case we derive their linear predictors in sharp contrast with the gaussian case regular stable processes which are fourier transforms of processes with independent increments are not moving averages of stable motion

some results on solutions of minimal period to superquadratic hamiltonian systems then it is well known that rabinowitz s theorem see 8 guarantees the existence for every t 0 of at least one nontrivial solution to l but he left open the question to find a solution having minimal period t when h is convex some results have been recently obtained about such a problem via duality methods the problem has been completely solved by ambrosetti mancini see 2 7 with a suitable hypothesis on the second derivative of h and by clarke ekeland see 4 replacing hj ha and h5 with a subquadratic hypothesis in another framework rabinowitz determines in 9 a family of solutions to l having minimal periods which diverge in the superquadratic case see theorem 2 34 under the

linear problems in linear problems in pth order and stable processes this work extends to processes with finite moments of order p 1 p 2 and to symmetric alpha stable processes 1 alpha 2 some of the basic linear theory known for processes with finite second moments p 2 and for gaussian processes alpha 2 here the covariation plays a role analogous to the covariance specifically stochastic integrals of two types are introduced and studied for pth order processes and in particular for symmetric stable processes regression estimates and linear estimates on certain symmetric stable processes are evaluated including regression and linear filtering of signal in noise also for certain symmetric stable inputs the identification of a linear system from the input covariation and the input output cross covariation is considered and the way the distribution of the output depends on the linear system is studied

index theory nontrivial solutions and asymptotically linear second order hamiltonian systems abstract in this paper we consider the existence and multiplicity of solutions of second order hamiltonian systems we propose a generalized asymptotically linear condition on the gradient of hamiltonian function classify the linear hamiltonian systems prove the monotonicity of the index function and obtain some new conditions on the existence and multiplicity for generalized asymptotically linear hamiltonian systems by global analysis methods such as the leray schauder degree theory the morse theory the ljusternik schnirelman theory etc

symmetric alpha stable filter theory symmetric spl alpha stable s spl alpha s processes are used to model infinite variance impulsive noise in general wiener filter theory is not meaningful in s spl alpha s environments because the expectations may be unbounded to develop a theory for linear finite impulse response systems with independent identically distributed s spl alpha s inputs we propose median orthogonality as a linear filter criterion derive a generalized wiener hopf solution equation and show a sufficient condition for a filter to achieve the criterion for non gaussian s spl alpha s densities zero forcing least mean squares is the only well known filter that satisfies the criterion but others can be designed we present a second algorithm and simulations showing that both converge to the generalized wiener hopf solution

sign sign lms convergence with independent stochastic inputs the sign sign adaptive least mean square lms identifier filter is a computationally efficient variant of the lms identifier filter it involves the introduction of signum functions in the traditional lms update term consideration is given to global convergence of parameter estimates offered by this algorithm to a ball with radius proportional to the algorithm step size for white input sequences specially from gaussian and uniform distributions

periodic solutions with prescribed minimal period for convex autonomous hamiltonian systems clarke has shown that the problem of findingt periodic solutions for a convex hamiltonian system is equivalent to the problem of finding critical points to a certain functional dual to the classical action functional in this paper we relate the morse index of the critical point to the minimal period of the correspondingt periodic solution in particular we show that if the critical point is obtained by the ambrosetti rabinowitz mountain pass theorem the corresponding solution has minimal periodt that is it cannot bet k periodic withk integer k 2 as a consequence we prove that if the hamiltonian is flat near an equilibrium and superquadratic near infinity then for anyt 0 the corresponding hamiltonian system has a periodic solution with minimal periodt

desingularization of implicit analytic differential equations the question of finding solutions to a given implicit differential equation ide is an important one in part because it appears very naturally in several problems in physics engineering and many other fields in this work we show how to reduce a given analytic ide to an analytic ide of locally constant rank this can be done by using some fundamental results on subanalytic subsets and desingularization of closed subanalytic subsets an example from nonholonomic mechanics is studied using these methods

a new index theory for linear self adjoint operator equations and its applications abstract we develop an index theory for linear self adjoint operator equation with the linear self adjoint operator having no compact resolvent as applications we consider the existence and multiplicity of periodic solutions of wave equations and beam equations

alpha stable signals and adaptive filtering let d sub k x sub k be jointly stationary symmetric spl alpha stable processes with index 1 spl alpha 2 we first consider the problem of optimal fir linear and regression filters we then consider the signed error and sign sign adaptive filtering algorithms for the estimation of the desired signal d sub j on the basis of the input vector x sub j various convergence results are established for the signal and tap weights estimation errors

meridian filtering for robust signal processing a broad range of statistical processes is characterized by the generalized gaussian statistics for instance the gaussian and laplacian probability density functions are special cases of generalized gaussian statistics moreover the linear and median filtering structures are statistically related to the maximum likelihood estimates of location under gaussian and laplacian statistics respectively in this paper we investigate the well established statistical relationship between gaussian and cauchy distributions showing that the random variable formed as the ratio of two independent gaussian distributed random variables is cauchy distributed we also note that the cauchy distribution is a member of the generalized cauchy distribution family recently proposed myriad filtering is based on the maximum likelihood estimate of location under cauchy statistics an analogous relationship is formed here for the laplacian statistics as the ratio of laplacian statistics yields the distribution referred here to as the meridian interestingly the meridian distribution is also a member of the generalized cauchy family the maximum likelihood estimate under the obtained statistics is analyzed motivated by the maximum likelihood estimate under meridian statistics meridian filtering is proposed the analysis presented here indicates that the proposed filtering structure exhibits characteristics more robust than that of median and myriad filtering structures the statistical and deterministic properties essential to signal processing applications of the meridian filter are given the meridian filtering structure is extended to admit real valued weights utilizing the sign coupling approach finally simulations are performed to evaluate and compare the proposed meridian filtering structure performance to those of linear median and myriad filtering

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

implicit differential equations near a singular point abstract this paper presents general and precise results on existence and number of solutions to implicitly defined ordinary differential equations in the vicinity of a singular point where the equation is not equivalent to an explicit one unlike in most if not in all other studies devoted to this problem in the literature it is not assumed that either rank drop occurs on an entire neighborhood of the singular point or that the equation is a scalar one attention is confined to the simplest but most frequently encountered kind of singular points it is shown that such standard singular points split into two complementary classes those from which exactly two distinct solutions emanate and those at which exactly two distinct solutions terminate whether a point of the latter class is eventually encountered is unaffected by slight modifications of the nonsingular initial condition and further evolution of the system governed by the singular ode then requires using a suitable jump condition according to appropriate physical criteria points of the former class are also shown to play a more subtle but equally important role in the dynamics the phenomena described here are relevant in various problems from the sciences such as phase transitions or plasticity they should also be relevant in some aspects of the classical domain of application of differential algebraic equations singular perturbations of ode s

linear prediction of speech with a least absolute error criterion most methods for linear prediction of speech are based on a least square error criterion l 2 criterion in this paper it is shown that the robustness of a least absolute error criterion l 1 criterion is particularly well adapted to the analysis of voiced sounds the well known linear programming l 1 algorithms have two important drawbacks the stability of the synthesis filter is not guaranteed and the computation load is too heavy for real time analysis all these problems are alleviated by the new l 1 algorithm proposed in this paper experiments show comparable efficiency with classical l 2 methods

unbounded perturbations of forced second order ordinary differential equations at resonance we prove the existence of at least one solution for the differential equation x t m2x t g t x t e t with periodicity conditions x 0 x 2 x 0 x 2 0 where m 0 is an integer e is integrable and g satisfies caratheodory conditions our results are obtained for the case when there is resonance at the eigenvalue m2 of the linear second order differential equation t x t 0 r with x 0 x 2 x 0 x 2 0 the function g may be unbounded and touching of the eigenvalue m 1 2 resp m 1 2 if m 0 on a subset of positive measure is allowed our approach also works when periodicity conditions are replaced by dirichlet or neumann boundary conditions the proofs are based on topological degree mawhin s continuation theorem and leray schauder techniques

second order convergence analysis of stochastic adaptive linear filtering the convergence of an adaptive filtering vector is studied when it is governed by the mean square error gradient algorithm with constant step size we consider the mean square deviation between the optimal filter and the actual one during the steady state this quantity is known to be essentially proportional to the step size of the algorithm however previous analyses were either heuristic or based upon the assumption that successive observations were independent which is far from being realistic actually in most applications two successive observation vectors share a large number of components and thus they are strongly correlated in this work we deal with the case of correlated observations and prove that the mean square deviation is actually of the same order or less than the step size of the algorithm this result is proved without any boundedness or barrier assumption for the algorithm as it has been done previously in the literature to ensure the nondivergence our assumptions are reduced to the finite strong memory assumption and the finite moments assumption for the observation they are satisfied in a very wide class of practical applications

adaptive filtering with binary reinforcement recently there has been increased interest in high speed adaptive filtering where the usual stochastic gradient or least mean square lms algorithm is replaced with the simpler algorithm where adaptation is guided only by the polarity of the error signal in this paper the convergence of this binary reinforcement br algorithm is proved under the usual independence assumption and the surprising observation is made that unlike the lms algorithm convergence occurs for any positive value of the step size parameter while the stochastic gradient algorithm attempts to minimize a mean square error cost function the binary reinforcement algorithm in fact attempts to minimize a mean absolute error cost function it is proved for the binary reinforcement algorithm that the tap weight vector converges in distribution to a random vector that is suitably concentrated about the optimal value based on a least mean absolute error cost function for a sufficiently small step size the expected cost of the asymptotic weight vector can be made as close as desired to thc minimum cost attain ed by the optimum weight vector

comparison of the convergence of two algorithms for adaptive fir digital filters the convergence properties of two different algorithms for the updating of the coefficients of an adaptive fir digital filter are investigated and compared with one another these algorithms are the stochastic iteration algorithm and the sign algorithm in this latter algorithm a one bit gradient estimation is used which makes its implementation very simple the convergence is characterized by the residual echo variance after convergence and a parameter that indicates the speed of the convergence it is shown that the convergence of the sign algorithm can always be assured but is much slower than that of the stochastic iteration algorithm if the same variance of the residual echo is to be obtained

convergence analysis of lms filters with uncorrelated gaussian data statistical analysis of the least mean squares lms adaptive algorithm with uncorrelated gaussian data is presented exact analytical expressions for the steady state mean square error mse and the performance degradation due to weight vector misadjustment are derived necessary and sufficient conditions for the convergence of the algorithm to the optimal wiener solution within a finite variance are derived it is found that the adaptive coefficient which controls the rate of convergence of the algorithm must be restricted to an interval significantly smaller than the domain commonly stated in the literature the outcome of this paper therefore places fundamental limitations on the mse performance and rate of convergence of the lms adaptive scheme

convergence analysis of the sign algorithm for adaptive filtering we consider the convergence analysis of the sign algorithm for adaptive filtering when the input processes are uncorrelated and gaussian and a fixed step size spl mu 0 is used exact recursive equations for the covariance matrix of the deviation error are established for any step size spl mu 0 asymptotic time averaged convergence for the mean absolute deviation error mean square deviation error and for the signal mean square estimation error are established these results are shown to hold for arbitrary step size spl mu 0

polynomial weighted median filtering this paper extends weighted median wm filters to the class of polynomial weighted median pwm filters traditional polynomial filtering theory based on linear combinations of polynomial terms is able to approximate important classes of nonlinear systems the linear combination of polynomial terms however yields poor performance in environments characterized by heavy tailed distributions weighted median filters in contrast are well known for their outlier suppression and detail preservation properties the weighted median sample selection methodology is naturally extended to the polynomial sample case yielding a filter structure that exploits the higher order statistics of the observed samples while simultaneously being robust to outliers moreover the pwm filter class is well motivated by an analysis of cross and square term statistics a presented probability density function analysis shows that these terms have heavier tails than the observed samples indicating that robust combination methods should be utilized to avoid undue influence of outliers further analysis shows weighted median processing of polynomial terms is justified from a maximum likelihood perspective the established pwm filter class is statistically analyzed through the determination of the filter output distribution and breakdown probability filter parameter optimization procedures are also presented finally the effectiveness of pwm filters is demonstrated through simulations that include temporal spectrum and bispectrum analysis

the robust covariation based music roc music algorithm for bearing estimation in impulsive noise environments this paper presents a new subspace based method for bearing estimation in the presence of impulsive noise which can be modeled as a complex symmetric alpha stable s spl alpha s process we define the covariation matrix of the array sensor outputs and show that eigendecomposition based methods such as the music algorithm can be applied to the sample covariation matrix to extract the bearing information from the measurements a consistent estimator for the marginals of the covariation matrix is presented and its asymptotic performance is studied the improved performance of the proposed source localization method in the presence of a wide range of impulsive noise environments is demonstrated via monte carlo experiments

on the use of morse index and rotation numbers for multiplicity results of resonant bvps abstract motivated by a recent series of papers by k li and co workers 14 15 16 17 18 we consider the problem of the existence and multiplicity of solutions for the neumann or periodic bvps associated to a class of scalar equations of the form x f t x 0 the class considered is such that the behavior of its solutions near zero and near infinity may be compared with the behavior of the solutions of two suitable linear systems one considered near zero and the other near infinity we show how a rotation number approach together with the poincare birkhoff theorem and a recent variant of it allows to obtain multiplicity results in terms of the gap between the morse indexes of the referred linear systems at zero and at infinity these systems may also be resonant when the gaps are sufficiently large our multiplicity results improve the ones obtained by variational methods in the quoted papers also our approach allows a description of the solutions obtained in terms of their nodal properties

squared and absolute errors in optimal approximation of nonlinear systems optimal squared error and absolute error based approximation problems for static polynomial models of nonlinear discrete time systems are studied in detail these problems have many similarities with other linear in the parameters approximation problems such as with optimal approximation problems for linear time invariant models of linear and nonlinear systems nonprobabilistic signal analysis is used close connections between the studied approximation problems and certain classical topics in approximation theory such as optimal l 2 1 1 and l 1 1 1 approximation are established by analysing conditions under which sample averages of static nonlinear functions of the input converge to appropriate riemann integrals of the static functions these results should play a significant role in the analysis of corresponding system identification and model validation problems furthermore these results demonstrate that optimal modelling based on the absolute error can offer advantages over squared error based modelling especially modelling problems in which some signals possess heavy tails can benefit from absolute value based signal and error analysis

asymptotic analysis of stochastic gradient based adaptive filtering algorithms with general cost functions this paper presents an analysis of stochastic gradient based adaptive algorithms with general cost functions the analysis holds under mild assumptions on the inputs and the cost function the method of analysis is based on an asymptotic analysis of fixed stepsize adaptive algorithms and gives almost sure results regarding the behavior of the parameter estimates whereas previous stochastic analyses typically considered mean and mean square behavior the parameter estimates are shown to enter a small neighborhood about the optimum value and remain there for a finite length of time furthermore almost sure exponential bounds are given for the rate of convergence of the parameter estimates the asymptotic distribution of the parameter estimates is shown to be gaussian with mean equal to the optimum value and covariance matrix that depends on the input statistics specific adaptive algorithms that fall under the framework of this paper are signed error least mean square lms dual sign lms quantized state lms least mean fourth dead zone algorithms momentum algorithms and leaky lms

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

weak convergence and local stability properties of fixed step size recursive algorithms a recursive equation that subsumes several common adaptive filtering algorithms is analyzed for general stochastic inputs and disturbances by relating the motion of the parameter estimate errors to the behavior of an unforced deterministic ordinary differential equation ode the odes describing the motion of several common adaptive filters are examined in some simple settings including the least mean square lms algorithm and all three of its signed variants the signed regressor the signed error and the sign sign algorithms stability and instability results are presented in terms of the eigenvalues of a correlation like matrix this generalizes known results for lms signed regressor lms and signed error lms and gives new stability criteria for the sign sign algorithm the ability of the algorithms to track moving parameterizations can be analyzed in a similar manner by relating the time varying system to a forced ode the asymptotic distribution about the forced ode is an ornstein uhlenbeck process the properties of which can be described in a straightforward manner

essential critical points of linking type and solutions of minimal period to superquadratic hamiltonian systems in a previous paper 3 the authors stated a result about the existence of t periodic solutions for any t 0 of autonomous hamiltonian systems having minimal period t or t 2 in the case where the hamiltonian function h has a superquadratic behaviour of the type considered by ekeland and hofer in 2 but without a global convexity assumption which is replaced by a local convexity condition in this paper we use the notion of essentiality for critical points of linking type see i 4 71 for an exhaustive exposition in order to construct a t periodic solution having t as its minimal period we remark that this result really generalizes the main theorem by ekeland and hofer in 2 the solution is found through a riesz galerkin method as exhibited in 3 following the basic scheme introduced in 6 by rabinowitz and the use of the marino prodi perturbation method more precisely as a first step one finds a critical point of linking type 2 for the restriction f of f to a suitable finite dimensional space for any n e n then some suitable estimates for derived by the linking construction enable the use of the marino prodi theorem in order to approximate each 2 by a sequence zig in such a way that z is a nondegenerate critical point of a functional fim with ff f in the c2 norm as m oo indeed 2 can be chosen as a critical point of linking type satisfying the same estimates as 2 and verifying the further property of essentiality roughly speaking one says that a linking critical point of a functional is not essential or avoidable if by looking at its linking definition one can find a local deformation i e in the neighbourhood of w which enables the construction for any minimizing sequence of surfaces having infinitesimal distances from w of another minimizing sequence of surfaces which keep away from w the solution z of the hamiltonian system is found as the limit of a suitable diagonal subsequence 1 zi j the fact that the minimal period of z can be either t or t 2 derives from some arguments which were already exploited in 3 based on some estimates for the morse indexes of finite dimensional linking points see 7 as a final step one shows that t 2 cannot be a period for z hence the conclusion that t is the minimal period of z by showing that the property of t 2 periodicity for z should yield the avoidability of the points of a subsequence of 1 first of all one looks at the maximum negative eigenvalue uj of d2f zj

morse type index theory for flows and periodic solutions for hamiltonian equations an index theory for flows is presented which extends the classical morse theory for gradient flows on compact manifolds the theory is used to prove a morse type existence statement for periodic solutions of a time dependent periodic in time and asymptotically linear hamiltonian equation

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

multiple solutions for asymptotically linear duffing equations with neumann boundary value conditions ii abstract we study multiple solutions of asymptotically linear duffing equations satisfying neumann boundary value conditions with resonance by using index theory and lusternik schnirelmann theory and obtain some new results