vector valued variational principle in fuzzy metric space and its applications a vector valued ekeland s variational principle and its equivalents in fuzzy metric space are presented in this paper our result unifies and extends some of the previous results about single valued and vector valued ekeland s variational principle as an application we obtain a fixed point theorem in fuzzy metric space under a new contractive condition which is a new generalization of well known banach contraction principle all of our results are also new even in usual metric spaces

classification of matrix product unitaries with symmetries we prove that matrix product unitaries with on site unitary symmetries are completely classified by the chiral index and the cohomology class of the symmetry group g provided that we can add trivial and symmetric ancillas with arbitrary on site representations of g if the representations in both system and ancillas are fixed to be the same we can define symmetry protected indices spis which quantify the imbalance in the transport associated to each group element and greatly refines the classification these spis are stable against disorder and measurable in interferometric experiments our results lead to a systematic construction of two dimensional floquet symmetry protected topological phases beyond the standard classification and thus shed new light on understanding nonequilibrium phases of quantum matter

function variational principles and coercivity abstract the function type extension of ekeland s variational principle j math anal appl 47 1974 324 353 due to zhong nonlinear anal 29 1997 1421 1431 is deductible in a simplified manner and in a larger functional context this is also true for his normed coercivity result based on palais smale techniques

some existence results for a periodic problem with non smooth potential abstract in this paper two existence results for a class of second order periodic boundary value problems with non smooth potential are obtained we extend the castro lazer thews reduction method to non smooth functionals the obtained result is then exploited to prove the existence of a nontrivial solution furthermore we prove the existence of multiple solutions by using a multiplicity result based on local linking

a compression type mountain pass theorem in conical shells in this paper we present a compression type version of the mountain pass lemma in a conical shell with respect to two norms an application to second order ordinary differential equations is included

non autonomous second order hamiltonian systems abstract we study the existence of periodic solutions for a second order non autonomous dynamical system containing variable kinetic energy terms our assumptions balance the interaction between the kinetic energy and the potential energy with neither one dominating the other we study sublinear problems and the existence of non constant solutions

multiple solutions for strongly resonant periodic systems abstract we considered a semilinear second order periodic system we assumed that the differential operator x x a x has zero as an eigenvalue and has no negative eigenvalues also we imposed a strong resonance condition with respect to the zero eigenvalue on the potential function f t x using the second deformation theorem we established the existence of at least two nontrivial solutions to do this we needed to conduct a detailed analysis of the cerami compactness condition which is actually of independent interest

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

generalized caristi s fixed point theorems by bae and others abstract in this paper we extend the generalized caristi s fixed point theorems proved by bae j math anal appl 284 2003 690 697 and others

critical points for nondifferentiable functions in presence of splitting a classical critical point theorem in presence of splitting established by brezis nirenberg is extended to functionals which are the sum of a locally lipschitz continuous term and of a convex proper lower semicontinuous function the obtained result is then exploited to prove a multiplicity theorem for a family of elliptic variational hemivariational eigenvalue problems

classification of quantum cellular automata there exists an index theory to classify strictly local quantum cellular automata in one dimension fidkowski et al in interacting invariants for floquet phases of fermions in two dimensions 2017 arxiv 1703 07360 gross et al in commun math phys 310 2 419 454 2012 po et al in phys rev b 96 245116 2017 we consider two classification questions first we study to what extent this index theory can be applied in higher dimensions via dimensional reduction finding a classification by the first homology group of the manifold modulo torsion second in two dimensions we show that an extension of this index theory including torsion fully classifies quantum cellular automata at least in the absence of fermionic degrees of freedom this complete classification in one and two dimensions by index theory is not expected to extend to higher dimensions due to recent evidence of a nontrivial automaton in three dimensions haah et al in nontrivial quantum cellular automata in higher dimensions 2018 arxiv 1812 01625 finally we discuss some group theoretical aspects of the classification of quantum cellular automata and consider these automata on higher dimensional real projective spaces

on a new notion of linking and application to elliptic problems at resonance abstract in this paper we introduce a new notion of linking which includes in particular the notions of homotopical linking and local linking critical point theorems for continuous functionals on metric spaces are presented finally an application to nonlinear elliptic problems at resonance is given

quantum cellular automata tensor networks and area laws quantum cellular automata are unitary maps that preserve locality and respect causality we identify them in any dimension with simple tensor networks projected entangled pair unitary whose bond dimension does not grow with the system size as a result they satisfy an area law for the entanglement entropy they can create we define other classes of nonunitary maps the so called quantum channels that either respect causality or preserve locality we show that whereas the latter obey an area law for the number of quantum correlations they can create as measured by the quantum mutual information the former may violate it we also show that neither of them can be expressed as tensor networks with a bond dimension that is independent of the system size

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

a divide and conquer algorithm for the eigendecomposition of symmetric block diagonal plus semiseparable matrices we present a fast and numerically stable algorithm for computing the eigendecomposition of a symmetric block diagonal plus semiseparable matrix we report numerical experiments that indicate that our algorithm is significantly faster than the standard method which treats the given matrix as a general symmetric dense matrix

the leray schauder boundary condition in critical point theory abstract we present a version of schechter s bounded mountain pass theorem in a hilbert space not identified to its dual an application to positive solutions of elliptic boundary value problems is included

a brezis nirenberg splitting approach for nonlocal fractional equations in this paper we consider problems modeled by the following nonlocal fractional equation su a x u f u in u 0in rn where s 0 1 is fixed is an open bounded subset of rn n 2s with lipschitz boundary s is the fractional laplace operator and is a real parameter under two different types of conditions on the functions a and f by using a famous critical point theorem in the presence of splitting established by brezis and nirenberg we obtain the existence of at least two nontrivial weak solutions for our problem

index theory of one dimensional quantum walks and cellular automata if a one dimensional quantum lattice system is subject to one step of a reversible discrete time dynamics it is intuitive that as much quantum information as moves into any given block of cells from the left has to exit that block to the right for two types of such systems namely quantum walks and cellular automata we make this intuition precise by defining an index a quantity that measures the net flow of quantum information through the system the index supplies a complete characterization of two properties of the discrete dynamics first two systems s1 s2 can be pieced together in the sense that there is a system s which acts like s1 in one region and like s2 in some other region if and only if s1 and s2 have the same index second the index labels connected components of such systems equality of the index is necessary and sufficient for the existence of a continuous deformation of s1 into s2 in the case of quantum walks the index is integer valued whereas for cellular automata it takes values in the group of positive rationals in both cases the map s mapsto rm ind s is a group homomorphism if composition of the discrete dynamics is taken as the group law of the quantum systems systems with trivial index are precisely those which can be realized by partitioned unitaries and the prototypes of systems with non trivial index are shifts

the palais smale condition versus coercivity possesses a convergent subsequence and cp is said to satisfy ps if ir satisfies ps for every c e ir recently among other results in critical point theory shujie i i showed that if a c functional p x k is bounded from below and satisfies the condition ps then p is coercive shujie s proof uses a gradient flow approach through the so called deformation theorem cf 3 io and for that he needs the notion of a pseudo gradient vector field tjassociated with the functional cp whose existence is guaranteed for c functionals by palais 8 in this paper we present some new results which relate the palais smale condition and the notion of coercivity and are based on the well known variational principle due to ekeland 5 61 in particular a new proof of the above mentioned result of shujie is given it should be pointed out that throughout the paper the given functional p could be assumed to be only gateaux differentiable rather than c and in addition to being conceptually simpler this approach could be used in more general situations where the functional is not even differentiable cf 4 the strong form of ekeland s variational principle to be repeatedly used is given in the following theorem

ekeland s variational principle minimax theorems and existence of nonconvex equilibria in complete metric spaces abstract in this paper we introduce the concept of function which generalizes the concept of w distance studied in the literature we establish a generalized ekeland s variational principle in the setting of lower semicontinuous from above and functions as applications of our ekeland s variational principle we derive generalized caristi s common fixed point theorems a generalized takahashi s nonconvex minimization theorem a nonconvex minimax theorem a nonconvex equilibrium theorem and a generalized flower petal theorem for lower semicontinuous from above functions or lower semicontinuous functions in the complete metric spaces we also prove that these theorems also imply our ekeland s variational principle

domain decomposition based ℌ lu preconditioners hierarchical matrices provide a data sparse way to approximate fully populated matrices the two basic steps in the construction of an h matrix are a the hierarchical construction of a matrix block partition and b the blockwise approxi mation of matrix data by low rank matrices in this paper we develop a new approach to construct the necessary partition based on domain decomposition compared to standard geometric bisection based h matrices this new approach yields h lu fac torizations of finite element stiffness matrices with significantly improved storage and computational complexity requirements these rigorously proven and numerically verified improvements result from an h matrix block structure which is naturally suited for parallelization and in which large subblocks of the stiffness matrix remain zero in an lu factorization we provide numerical results in which a domain decom position based h lu factorization is used as a preconditioner in the iterative solution of the discrete three dimensional convection diffusion equation

homology of saddle point reduction and applications to resonant elliptic systems abstract in the setting of saddle point reduction we prove that the critical groups of the original functional and the reduced functional are isomorphic as application we obtain two nontrivial solutions for elliptic gradient systems which may be resonant both at the origin and at infinity the difficulty that the variational functional does not satisfy the palais smale condition is overcame by taking advantage of saddle point reduction our abstract results on critical groups are crucial

a palais smale approach to lane emden equations we consider the unbounded domain problems u u u p 2 u in u 0i n a ndu 0o n w here is an unbounded domain in r n 2 2 and 2 for n 2 the existence of a ground state solution to the problems is greatly affected by the shape of the domain to determine the existence of the solutions in a general domain remains a challenge task for the flat interior flask domain that consists a strip and a ball attached to the bottom of the strip previous results have asserted the existence of a ground state solution when the diameter of the ball is greater than a positive constant however the existence of the solutions when the diameter of the ball equals to the width of the strip is still an important open question this article resolves the open question partially by considering a variation of the flat interior flask domain which is formed by attaching a stretched ball to the bottom of the strip

multiplicity of nontrivial solutions for quasilinear elliptic equation two nontrivial solutions are obtained by the reduction method for the nonautonomous semilinear elliptic dirichlet boundary value problem some well known multiplicity results are generalized

existence and multiplicity of periodic solution for non autonomous second order systems with linear nonlinearity abstract the purpose of this paper is to study the existence and multiplicity of periodic solution for the following non autonomous second order systems u t f t u t a e t 0 t u 0 u t u 0 u t 0 some new existence and multiplicity theorems are obtained by using the least action principle and the minimax method

on a class of semilinear schrodinger equations involving critical growth and discontinuous nonlinearities abstract in this paper we deal with a class of schrodinger equation in r n involving critical sobolev exponent and jump discontinuities the basic tool employed here is an approximation technique with periodic functions and variational arguments based on a linking theorem for locally lipschitz functionals

path connected yang mills moduli spaces on utilise des techniques de min max dans le calcul des variations pour demontrer que les espaces modules des connexions auto duales sur des fibres principaux su 2 ou su 3 sur s 4 sont chemin connexe

existence and multiplicity of solutions for a class of semilinear elliptic equations the existence and multiplicity results are obtained for solutions of a class of the dirichlet problem for semilinear elliptic equations by the least action principle and the minimax methods respectively

an obstacle problem for nonlinear hemivariational inequalities at resonance in this paper we examine an obstacle problem for a nonlinear hemivariational inequality at resonance driven by the p laplacian using a variational approach based on the nonsmooth critical point theory for locally lipschitz functionals defined on a closed convex set we prove two existence theorems in the second theorem we have a pointwise interpretation of the obstacle problem assuming in addition that the obstacle is also a kind of lower solution for the nonlinear elliptic differential inclusion

periodic and homoclinic solutions of extended fisher kolmogorov equations in this paper we study the existence of periodic solutions of the fourth order equations uiv pu a x u b x u3 0 and uiv pu a x u b x u3 0 where p is a positive constant and a x and b x are continuous positive 2l periodic functions the boundary value problems p1 and p2 for these equations are considered respectively with the boundary conditions u 0 u l u 0 u l 0 existence of nontrivial solutions for p1 is proved using a minimization theorem and a multiplicity result using clark s theorem existence of nontrivial solutions for p2 is proved using the symmetric mountain pass theorem we study also the homoclinic solutions for the fourth order equation uiv pu a x u b x u2 c x u3 0 where p is a constant and a x b x and c x are periodic functions the mountain pass theorem of brezis and nirenberg and concentration compactness arguments are used

on the relation between the weak palais smale condition and coercivity given by zhong abstract in this paper we discuss zhong s result that the weak palais smale condition implies coercivity under some assumption given in c k zhong a generalization of ekeland s variational principle and application to the study of the relation between the weak p s condition and coercivity nonlinear anal 29 1997 1421 1431 we also give a simple proof of zhong s result further we generalize the result of caklovic li and willem l caklovic s j li m willem a note on palais smale condition and coercivity differential integral equations 3 1990 799 800

existence and multiplicity results for classes of elliptic resonant problems in this paper we prove some new existence results of nontrivial solutions for classes of elliptic resonant problems we also establish several multiplicity results the methods used here are based on combining the minimax methods and the morse theory especially some new observations on the critical groups of a local linking type degenerate critical point

a fast direct solver for elliptic problems on general meshes in 2d we present a fast direct algorithm for solutions to linear systems arising from 2d elliptic equations we follow the approach in xia et al 2009 on combining the multifrontal method with hierarchical matrices we present a variant of that approach with additional hierarchical structure extend it to quasi uniform meshes and detail an adaptive decomposition procedure for general meshes linear time complexity is shown for a quasi regular grid and demonstrated via numerical results for the adaptive algorithm

a direct solver for variable coefficient elliptic pdes discretized via a composite spectral collocation method a numerical method for variable coefficient elliptic problems on two dimensional domains is presented the method is based on high order spectral approximations and is designed for problems with smooth solutions the resulting system of linear equations is solved using a direct solver with o n 1 5 complexity for the pre computation and o nlogn complexity for the solve the fact that the solver is direct is a principal feature of the scheme and makes it particularly well suited to solving problems for which iterative solvers struggle in particular for problems with highly oscillatory solutions numerical examples demonstrate that the scheme is fast and highly accurate for instance using a discretization with 12 points per wavelength a helmholtz problem on a domain of size 100x100 wavelengths was solved to ten correct digits the computation was executed on a standard laptop it involved 1 6m degrees of freedom and required 100 s for the pre computation and 0 3 s for the actual solve

remarks on the resonant elliptic problem we consider the semilinear elliptic equation u f x u with the dirichlet boundary value for resonant nonlinearity at infinity using the morse theory and perturbation method some existence results of solution and multiple solutions are proved

homoclinic orbits in reversible systems and their applications in mechanics fluids and optics abstract this survey article reviews the theory and application of homoclinic orbits to equilibria in reversible continuous time dynamical systems where the homoclinic orbit and the equilibrium are both reversible the focus is on even order reversible systems in four or more dimensions local theory generic argument and global existence theories are examined for each qualitatively distinct linearisation several recent results such as coalescence caused by non transversality and the reversible orbit flip bifurcation are covered a number of open problems are highlighted applications are reviewed to systems arising from a variety of disciplines with the aid of numerical methods three examples are presented in detail one of which is infinite dimensional

saddle point techniques from the beginning of the calculus of variations over 300 years ago scientists were interested in finding extrema of expressions now called functionals involving unknown functions it was discovered that functions producing extrema are solutions of differential equations called euler equations in one dimentional problems these were ordinary differential equations and the easiest procedure was to solve the corresponding euler equation and then check if the solution produces the desired maximum or minimum however as researchers began to study higher dimensional problems it became obvious that it was more difficult to solve the euler equations than obtain an extremum of the functional moreover many partial differential equations and systems that were being studied by various scientists happened to be the euler equations of functionals in this case any critical point of the corresponding functional will provide a solution of the given equation this led to the search for critical points of functionals in this paper we describe several methods of finding critical points we present applications

the strong ekeland variational principle abstract in this paper we consider the strong ekeland variational principle due to georgiev p g georgiev the strong ekeland variational principle the strong drop theorem and applications j math anal appl 131 1988 1 21 we discuss it for functions defined on banach spaces and on compact metric spaces we also prove the distance version of it

a new critical point theorem for locally lipschitz functionals with applications to differential equations abstract in the present paper a new critical point theorem for locally lipschitz functions is given and as applications one existence and two multiplicity results for zero boundary value problems are obtained several important classical theorems are improved

generalized distance and existence theorems in complete metric spaces abstract in this paper we first introduce the concept of distance on a metric space which is a generalized concept of both w distance and tataru s distance we also improve the generalizations of the banach contraction principle caristi s fixed point theorem ekeland s variational principle and the nonconvex minimization theorem according to takahashi further we discuss the relation between w distance and tataru s distance

ergodicity of quantum cellular automata we define a class of dynamical maps on the quasi local algebra of a quantum spin system which are quantum analoges of probabilistic cellular automata we develop criteria for such a system to be ergodic i e to posses a unique invariant state intuitively ergodicity obtains if the local transition operators exhibit sufficiently large disorder the ergodicity criteria also imply bounds for the exponential decay of correlations in the unique invariant state the main technical tool is a quantum version of oscillation norms defined in the classical case as the sum over all sites of the variations of an observable with respect to local spin flips

a variational approach to discontinuous problems with critical sobolev exponents abstract we employ variational techniques to study the existence and multiplicity of positive solutions of semilinear equations of the form u h x h u a uq u2 1 in rn where a 0 are parameters h x is both nonnegative and integrable on rn h is the heaviside function 2 is the critical sobolev exponent and 0 q

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

multiple solutions of neumann problem for elliptic equations abstract the multiplicity results are obtained for solutions of the neumann problem for nonlinear elliptic equations at resonance or nonresonance with unbounded nonlinearity by the reduction method and the critical point theory in particular a three critical point theorem proposed by brezis nirenberg

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

an efficient and stable method for computing multiple saddle points with symmetries in this paper an efficient and stable numerical algorithm for computing multiple saddle points with symmetries is developed by modifying the local minimax method established in y li and j zhou siam j sci comput 23 2001 pp 840 865 y li and j zhou siam j sci comput 24 2002 pp 840 865 first an invariant space is defined in a more general sense and a principle of invariant criticality is proved for the generalization then the orthogonal projection to the invariant space is used to preserve the invariance and to reduce computational error across iterations simple averaging formulas are used for the orthogonal projections numerical computations of examples with various symmetries of which some can and others cannot be characterized by a compact group of linear isomorphisms are carried out to confirm the theory and to illustrate applications the mathematical features of various problems demonstrated in these examples fall into two categories nodal solutions of saddle point type with large morse indices and nonradial positive solutions via symmetry breaking in radially symmetric elliptic problems the new numerical algorithm generates these rather unstable solutions in an efficient and stable way the existence of many unstable solutions and their behavior found in this paper remain to be investigated

quasilinear hemivariational inequalities with strong resonance at infinity we study quasilinear driven by p laplacian elliptic equations with strong resonance at infinity and with a nonsmooth potential using nonsmooth critical point theory we obtain existence and multiplicity results

periodic solutions of nonautonomous second order hamiltonian systems with even typed potentials abstract some existence and multiplicity theorems are obtained for periodic solutions of nonautonomous second order hamiltonian systems with even typed potentials by the least action principle and the minimax methods in critical point theory

on a notion of category depending on a functional part i theory and application to critical point theory abstract we introduce the notion of category depending on a fixed functional f defined on a topological space this notion permits us to obtain a better lower bound to the number of critical points of f than the bound obtained with the lusternik schnirelman category we also introduce the notion of truncated category depending on the functional f which permits us to obtain a lower bound to the number of critical points of unbounded functionals finally we extend our result after having introduced the notions of relative category and limit relative category depending on the functional f

existence and multiplicity of solutions for nonlinear second order difference boundary value problems in the present paper the following nonlinear second order difference boundary value problems d p n 1 du n 1 q n u n f n u n 0 n z 1 n u 0 u n p 0 du 0 p n du n are studied by using the critical theory and three new multiple results of solutions are obtained where z 1 n 1 2 n f z 1 n xr r is a continuous function in the second variable p q z 0 n r and p n 0

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 and multiplicity results for some elliptic systems with discontinuous nonlinearities abstract in this paper we complete two tasks we first extend the critical point theorem obtained by li and willem s j li m willem applications of local linking to critical point theory j math anal appl 189 1995 6 32 to the nonsmooth case in which the energy functional is locally lipschitz and satisfies the weaker nonsmooth cerami condition then we study some semilinear elliptic systems with discontinuous nonlinearities and obtain some existence and multiplicity results

a parallel shared memory implementation of a high order accurate solution technique for variable coefficient helmholtz problems abstract the recently developed hierarchical poincare steklov hps method is a high order discretization technique that comes with a direct solver results from previous papers demonstrate the method s ability to solve helmholtz problems to high accuracy without the so called pollution effect while the asymptotic scaling of the direct solver s computational cost is the same as the nested dissection method serial implementations of the solution technique are not practical for large scale numerical simulations this manuscript presents the first parallel implementation of the hps method specifically we introduce an approach for a shared memory implementation of the solution technique utilizing parallel linear algebra this approach is the foundation for future large scale simulations on supercomputers and clusters with large memory nodes performance results on a desktop computer resembling a large memory node are presented

remarks on existence and multiplicity of solutions for a class of semilinear elliptic equations the existence and multiplicity results are obtained for solutions of a class of the dirichlet problem for semilinear elliptic equations by the least action principle and the minimax methods respectively

a direct solver with o n complexity for variable coefficient elliptic pdes discretized via a high order composite spectral collocation method a numerical method for solving elliptic pdes with variable coefficients on two dimensional domains is presented the method is based on high order composite spectral approximations and is designed for problems with smooth solutions the resulting system of linear equations is solved using a direct as opposed to iterative solver that has optimal o n complexity for all stages of the computation when applied to problems with nonoscillatory solutions such as the laplace and the stokes equations numerical examples demonstrate that the scheme is capable of computing solutions with a relative accuracy of 10 10 or better for challenging problems such as highly oscillatory helmholtz problems and convection dominated convection diffusion equations in terms of speed it is demonstrated that a problem with a nonoscillatory solution that was discretized using 10 8 nodes can be solved in 115 minutes on a personal workstation with two quad core 3 3 ghz cpus since the solver is direct and the solution

subrahmanyam s fixed point theorem abstract we give a sufficient and necessary condition for the convergence of the sequence of successive approximations to a fixed point which is the conclusion of subrahmanyam s fixed point theorem

nonlinear resonant periodic problems with concave terms abstract we consider a nonlinear periodic problem driven by the scalar p laplacian with a concave term and a caratheodory perturbation we assume that this perturbation f t x is p 1 linear at and resonance can occur with respect to an eigenvalue m 1 m 2 of the negative periodic scalar p laplacian using a combination of variational techniques based on the critical point theory with morse theory we establish the existence of at least three nontrivial solutions useful in our considerations is an alternative minimax characterization of 1 0 the first nonzero eigenvalue that we prove in this work

the existence and multiplicity of solutions for an impulsive differential equation with two parameters via a variational method abstract in this paper we study the existence and multiplicity of classical solutions for a second order impulsive differential equation with periodic boundary conditions by using a variational method and critical point theory we give some new criteria to guarantee that the impulsive problem has at least one solution two solutions and infinitely many solutions when the parameter pair c lies in different intervals respectively some examples are given in this paper to illustrate the main results

multiplicity of solutions for resonant neumann problems with an indefinite and unbounded potential we examine semilinear neumann problems driven by the laplacian plus an unbounded and indefinite potential the reaction is a caratheodory function which exhibits linear growth near we allow for resonance to occur with respect to a nonprincipal nonnegative eigenvalue and we prove several multiplicity results our approach uses critical point theory morse theory and the reduction method the lyapunov schmidt method