homoclinic solutions for a class of the second order hamiltonian systems abstract we study the existence of homoclinic orbits for the second order hamiltonian system q v q t q f t where q r n and v c 1 r r n r v t q k t q w t q is t periodic in t a map k satisfies the pinching condition b 1 q 2 k t q b 2 q 2 w is superlinear at the infinity and f is sufficiently small in l 2 r r n a homoclinic orbit is obtained as a limit of 2 kt periodic solutions of a certain sequence of the second order differential equations

existence of infinitely many homoclinic orbits for fourth order difference systems containing both advance and retardation abstract by using the symmetric mountain pass theorem we establish some existence criteria to guarantee the fourth order difference system 4u n 2 q n u n f n u n 1 u n u n 1 have infinitely many homoclinic orbits where n z u r n q z r n n and f z r 3 n r are no periodic in n

positive solutions to a second order three point boundary value problem abstract the existence nonexistence and multiplicity of nonnegative solutions are established for the three point boundary value problem u t p t f u t 0 u 0 0 u 1 u where 0 1 0 1 and is a nonnegative parameter under appropriate hypotheses the key idea is that the problem of finding a nonnegative solution is transformed into the problem of finding a fixed point of a completely continuous operator the arguments involve the schauder fixed point the method of upper and lower solutions for three point boundary value problems and the leray schauder degree theory

existence of infinitely many homoclinic orbits in discrete hamiltonian systems abstract by using the symmetric mountain pass theorem we establish some existence criteria to guarantee the second order self adjoint discrete hamiltonian system p n u n 1 l n u n w n u n 0 has infinitely many homoclinic orbits where n z u r n p l z r n n and w z r n r are no periodic in n our conditions on the potential w n x are rather relaxed

eigenvalues of second order difference equations with periodic and antiperiodic boundary conditions this paper is concerned with periodic and antiperiodic boundary value problems for self adjoint second order difference equations existence of eigenvalues of these two different boundary value problems is proved numbers of their eigenvalues are calculated and their relationships are obtained in addition a representation of solutions of a nonhomogeneous linear equation with initial conditions is given

nonlinear discrete sturm liouville problems at resonance abstract in this paper we study nonlinear bounded value problems of the form p t 1 y t 1 q t y t k y t f t y t h t a 11 y a a 12 y a 0 a 21 y b 1 a 22 y b 1 0 where k is an eigenvalue of the associated linear problem f is subject to the sublinear growth condition f t s a s b t a 1 b 1 s r for some 0 1 and a b 0 we prove the existence and multiplicity of solutions by using the connectivity properties of solution sets of parameterized families of compact vector fields

existence of one signed periodic solutions of some second order differential equations via a krasnoselskii fixed point theorem this paper is devoted to study the existence of periodic solutions of the second order equation x 00 f t x where f is a caratheodory function by combining some new properties of green s function together with krasnoselskii fixed point theorem on compression and expansion of cones as applications we get new existence results for equations with jumping nonlinearities as well as equations with a repulsive or attractive singularity in this latter case our results cover equations with weak singularities and are compared with some recent results by i rachunkova m tvrdyand i vrkoc r 2002 elsevier science usa all rights reserved

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

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

homoclinic orbits for second order self adjoint difference equations abstract in this paper we discuss how to use variational methods to study the existence of nontrivial homoclinic orbits of the following nonlinear difference equations p t u t 1 q t u t f t u t t z without any periodicity assumptions on p t q t and f providing that f t x grows superlinearly both at origin and at infinity or is an odd function with respect to x r and satisfies some additional assumptions

global bifurcation of periodic solutions to ordinary differential equations abstract general homotopy continuation and bifurcation results are discussed and proved for a class of semiflows these results and further conley homotopy index methods are applied to obtain a type of global bifurcation for periodic solutions of second order ordinary differential equations

homoclinic orbits and subharmonics for nonlinear second order difference equations abstract we show the existence of a nontrivial homoclinic orbit and subharmonic solutions for a class of second order difference equations by applying the mountain pass theorem relying on ekeland s variational principle and the diagonal method and the homoclinic orbit as the limit of the subharmonics a completely new way is provided for dealing with the existence of solutions for difference equations

positive periodic solutions for nonlinear difference equations with periodic coefficients abstract we prove the existence of positive solutions of second order nonlinear difference equations on a finite discrete segment with periodic boundary conditions obtained results yield positive periodic solutions of the equation on the whole discrete axis provided that the coefficients are periodic

asymptotic and oscillatory properties of the fourth order nonlinear difference equations in this article a class of fourth order difference equations with quasi differences and a deviating argument is considered we state new oscillation theorems for the super linear and for the half linear case and we complete the existing results in the literature

homoclinic orbits for second order hamiltonian systems possessing superquadratic potentials received by the editors november 27 1990 1991 mathematics subject classification primary 34c99 58e99 58f99 the first author was supported by ministero p i gruppo 40 calcolo delle variazioni the second author s research was sponsored in part by the u s army research office under contract daal03 87 k 0043 the national science foundation under grant mcs 8110556 and the office of naval research under grant n00014 88 ko134 any reproduction for the purpose of the united states govemment is permitted

periodic and subharmonic solutions for superquadratic discrete hamiltonian systems abstract in this paper some new results are obtained for the existence of multiple periodic solutions and subharmonic solutions to discrete hamiltonian systems x 1 n h x 2 n x 1 n 1 x 2 n x 2 n h x 1 n x 1 n 1 x 2 n by using critical point theory where x1 x 2 r d h c 1 r r d r d r

homoclinic orbits for a class of hamiltonian systems consider the second order hamiltonian system where q n and v c 1 n is t periodic in t suppose v q t 0 0 0 is a local maximum for v t and v t x x under these and some additional technical assumptions we prove that hs has a homoclinic orbit q emanating from 0 the orbit q is obtained as the limit as k of 2 kt periodic solutions i e subharmonics q k of hs the subharmonics q k are obtained in turn via the mountain pass theorem

homoclinic solutions for a class of second order hamiltonian systems abstract some new results for the existence of homoclinic orbits are obtained for the second order hamiltonian systems u t v t u t f t where t r u r n and v c 1 r r n r v t x k t x w t x is t periodic with respect to t t 0 and f r r n is a continuous and bounded function our assumptions on k and f are rather loose

homoclinic solutions for a class of second order hamiltonian systems abstract a new result for existence of homoclinic orbits is obtained for the second order hamiltonian systems u t l t u t w 1 t u t w 2 t u t f t where t r u r n and w 1 w 2 c 1 r r n r and f c r r n are not necessary periodic in t this result generalizes and improves some existing results in the literature

unbounded perturbations of nonlinear discrete periodic problem at resonance abstract in this paper we study the existence of solutions of nonlinear discrete boundary value problems 2 u t 1 k u t g t u t h t u 0 u t u 1 u t 1 where t 1 t h t r k is the k th eigenvalue of the linear problem 2 u t 1 u t 0 u 0 u t u 1 u t 1 g n r r satisfies some asymptotic nonuniform resonance conditions and g t u u 0 for u r

periodic solutions of higher dimensional discrete systems consider the second order discrete system where f c r r m r m f t m z f t z for any t z r r m and m is a positive integer by making use of critical point theory the existence of m periodic solutions of is obtained

homoclinic orbits for nonlinear difference equations containing both advance and retardation in this paper we discuss how to use the critical point theory to study the existence of a nontrivial homoclinic orbit for nonlinear difference equations containing both advance and retardation without any periodic assumptions moreover if the nonlinearity is an odd function the existence of an unbounded sequence of homoclinic orbits is obtained

existence of homoclinic solution for the second order hamiltonian systems abstract an existence theorem of homoclinic solution is obtained for a class of the nonautonomous second order hamiltonian systems u t l t u t w t u t 0 t r by the minimax methods in the critical point theory specially the generalized mountain pass theorem where l t is unnecessary uniformly positively definite for all t r and w t x satisfies the superquadratic condition w t x x 2 as x uniformly in t and need not satisfy the global ambrosetti rabinowitz condition

existence of nontrivial homoclinic orbits for fourth order difference equations we obtain a sufficient condition for the existence of nontrivial homoclinic orbits for fourth order difference equations by using mountain pass theorem a weak convergence argument and a discrete version of lieb s lemma

positive solutions for nonlinear discrete periodic boundary value problems we are concerned with the nonlinear discrete periodic boundary value problem p d p t 1 du t 1 q t u t rg t f u t t 1 t z u 0 u t p 0 du 0 p t du t where r is a positive parameter optimal interval will be given to the parameter r to ensure that p has at least one positive solution

smoothing functions for second order cone complementarity problems smoothing functions have been much studied in the solution of optimization and complementarity problems with nonnegativity constraints in this paper we extend smoothing functions to problems in which the nonnegative orthant is replaced by the direct product of second order cones these smoothing functions include the chen mangasarian class and the smoothed fischer burmeister function we study the lipschitzian and differential properties of these functions and in particular we derive computable formulas for these functions and their jacobians these properties and formulas can then be used to develop and analyze noninterior continuation methods for solving the corresponding optimization and complementarity problems in particular we establish the existence and uniqueness of the newton direction when the underlying mapping is monotone

multiplicity results for periodic solutions to delay differential equations via critical point theory abstract by the critical point theory we study the existence and multiplicity of periodic solutions to the following system of delay differential equations x t f x t r where x r n f c r n r n and r 0 is a given constant a sufficient condition on the existence and multiplicity of periodic solutions for eq is obtained to the authors knowledge this is the first time the existence of periodic solutions to systems of delay differential equations is dealt with by using variational approaches directly

positive periodic solutions of functional differential equations abstract we consider the existence multiplicity and nonexistence of positive periodic solutions for the periodic equation x t a t g x x t b t f x t t where a b c r 0 are periodic 0 a t dt 0 0 b t dt 0 f g c 0 0 and f u 0 for u 0 g x is bounded t is a continuous periodic function define f 0 lim u 0 f u u f lim u f u u i0 number of zeros in the set f 0 f and i number of infinities in the set f 0 f we show that the equation has i0 or i positive periodic solution s for sufficiently large or small 0 respectively

homoclinic solutions for nonautonomous second order hamiltonian systems with a coercive potential abstract a new existence result of homoclinic orbits is obtained for the second order hamiltonian systems u t f t u t f t where f t x is periodic with respect to t this result generalizes some known results in the literature

existence of homoclinic orbits for 2nth order nonlinear difference equations containing both many advances and retardations abstract by establishing a proper variational framework and using the critical point theory we establish some new existence criteria to guarantee the 2nth order nonlinear difference equation containing both many advances and retardations n r t n n u t n q t u t f t u t n u t u t n n z 3 t z has at least one or infinitely many homoclinic orbits where r q f are nonperiodic in t our conditions on the potential are rather relaxed and some existing results in the literature are improved

existence and multiple solutions for a second order difference boundary value problem via critical point theory in this paper the critical point theory is employed to establish existence and multiple solutions for a second order difference boundary value problem

existence multiplicity and dependence on a parameter for a periodic boundary value problem the authors consider the boundary value problem y 2 y g t f y 0 0 t 2 y 0 y 2 y 0 y 2

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

existence of nonnegative and nonpositive solutions for second order periodic boundary value problems we are interested in nonnegative and nonpositive solutions of the boundary value problem u f t u u 0 u 1 u 0 u 1 where f fulfils the caratheodory conditions on 0 1 r we generalize the results reached by m n nkashama j santanilla and l sanchez and present estimates for solutions in addition we apply our existence theorems to periodic boundary value problems for nonlinear duffing equations whose right hand sides have a repulsive or attractive singularity at the origin we extend or generalize existence results by a c lazer and s solimini and other authors moreover we get some multiplicity results and in the case of a repulsive singularity we also admit a weak singularity in constrast to the previous papers on this subject our proofs are based on the method of lower and upper functions and topological degree arguments and the results are tested on examples

solitary waves with prescribed speed on infinite lattices using a variant of the mountain pass theorem we prove the existence of solitary waves with prescribed speed on infinite lattices of particles with nearest neighbor interaction the problem is to solve a second order forward backward differential difference equation c 1997 academic press

existence of a periodic solution for subquadratic second order discrete hamiltonian system some solvability conditions of periodic solutions are obtained for a class of nonautonomous subquadratic second order discrete hamiltonian system by minimax methods in critical point theory

periodic and subharmonic solutions for fourth order nonlinear difference equations abstract by using the critical point theory some new criteria are obtained for the existence and multiplicity of periodic and subharmonic solutions to fourth order nonlinear difference equations the main approach used in our paper is a variational technique and the linking theorem our results generalize and improve the existing ones

periodic solutions of second order non autonomous singular dynamical systems abstract in this paper we establish two different existence results of positive periodic solutions for second order non autonomous singular dynamical systems the first one is based on a nonlinear alternative principle of leray schauder and the result is applicable to the case of a strong singularity as well as the case of a weak singularity the second one is based on schauder s fixed point theorem and the result sheds some new light on problems with weak singularities and proves that in some situations weak singularities may help create periodic solutions recent results in the literature are generalized and significantly improved

on the existence of positive solutions for nonlinear differential equations with periodic boundary conditions we prove the existence of positive solutions of second order nonlinear differential equations on a finite interval with periodic boundary conditions and give upper and lower bounds for these positive solutions obtained results yield positive periodic solutions of the equation on the whole real axis provided that the coefficients are periodic