existence of homoclinic solutions for higher order periodic difference equations with p laplacian abstract using the critical point theory in combination with periodic approximations we establish sufficient conditions on the existence of homoclinic solutions for higher order periodic difference equations with p laplacian our results provide rather weaker conditions to guarantee the existence of homoclinic solutions and considerably improve some existing ones even for some special cases

utd diffraction coefficients for higher order wedge diffracted fields an asymptotic expansion of a multiple integral for knife edge diffraction is derived the expansion expresses the field in terms of single edge diffraction by considering this expression a similar expansion for higher order utd diffracted fields is proposed by means of a set of transition functions the result removes some of the shortcomings of the original set out of the utd when the incident field is not a ray optical field

existence of uncountably many bounded nonoscillatory solutions and their iterative approximations for second order nonlinear neutral delay difference equations in this paper the banach fixed point theorem is employed to establish several existence results of uncountably many bounded nonoscillatory solutions for the second order nonlinear neutral delay difference email protected n dx n b n x email protected email protected n x h 1 n x h 2 n x h k n f n x f 1 n x f 2 n x f k n c n n n 0 where t email protected n n email protected n 0 a b c n n 0 r with a n 0 for email protected n n 0 h f n n 0xr k r and h l f l n n 0 z withlimn h l n limn f l n email protected 1 2 k a few mann type iterative approximation schemes with errors are suggested and the errors estimates between the iterative approximations and the nonoscillatory solutions are discussed seven nontrivial examples are given to illustrate the advantages of our results

new kamenev type oscillation criteria for second order neutral nonlinear differential equations abstract in this paper some new oscillation criteria for second order neutral nonlinear differential equation r t y t p t y t i 1 n q i t f i y i t 0 t t 0 are established new oscillation criteria are different from most known ones in the sense that they are based on a class of new functions t s l defined in the sequel our results are sharper than some previous results which can be seen by the example at the end of this paper

asymptotic behavior of nonlinear difference systems in this paper we consider two dimensional nonlinear difference systems of the email protected n a nf y n dy n 1 b ng x n we classify their solutions according to asymptotic behavior and give some necessary and sufficient conditions for the existence of solutions of such classes

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

oscillation criteria for second order quasi linear neutral difference equations abstract in this paper we obtain some oscillation criteria for the second order quasi linear neutral delay difference equation where 0 0 and 0 are constants a n p n q n are nonnegative sequences and f c r r

oscillation for difference equations with continuous variable consider the delay difference equation y t y t p t h y t t t 0 where 0 0 and p c r r f c r r h c r r some oscillation criteria for this equation are obtained

oscillation and nonoscillation theorems for second order nonlinear difference equations abstract some new oscillation and nonoscillation theorems are obtained for second order nonlinear difference equations a p n a x n q n a x n f n x n b x n which generalize and improve all results proved by popenda in j math anal appl 123 1987 34 48

subharmonic solutions and homoclinic orbits of second order discrete hamiltonian systems with potential changing sign in this paper a class of second order discrete hamiltonian systems with periodicity assumptions is considered based on the critical point theory some sufficient conditions for the existence of subharmonic solutions and homoclinic orbits are obtained the results obtained extend the results in 2007 by relaxing the assumptions on the sign of the potential

nonoscillatory solutions for discrete equations abstract a nonoscillatory theory is presented for discrete equations our results rely on a nonlinear alternative of leray schauder type for condensing operators

asymptotic behavior of solutions of second order differential equations with integrable coefficients the differential equation x a t f x 0 t 0 is considered under the condition that lim x fl a s ds exists and is finite and necessary and or sufficient conditions are given for this equation to have solutions which behave asymptotically like nontrivial linear functions cl c2t

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

oscillation and asymptotic behavior of second order difference equations abstract the oscillation of the second order difference equation c n y n p n y v n 1 0 n 0 1 2 1 1 is investigated where yn yn 1 yn v is a quotient of odd positive integers some necessary and sufficient conditions for oscillation of 1 1 with v 1 and v

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

existence of uncountably many bounded positive solutions for a third order nonlinear neutral delay difference equation this paper deals with the solvability of the third order nonlinear neutral delay difference equation d 2 a n d x n b nx n t d 2f n x n b 1 n x n b 2 n x n b k n dg n x n c 1 n x n c 2 n x n c k n h n x n d 1 n x n d 2 n x n d k n n n 0 relative to the sequence b n n n n 0 r using the krasnoselskii s fixed point theorem and schauder s fixed point theorem a few sufficient conditions of the existence of uncountably many bounded positive solutions for the equation are presented seven examples are included to demonstrate the advantages and effectiveness of the results presented in this paper

theoretical and experimental study of modal interaction in a two degree of freedom structure abstract fpr a two degree of freedom structure an experimental and theoretical investigation has been made of the primary resonances of the system which occur when the frequency of excitation is near one of the natural frequencies 1 and 2 the additional constraint of 2 2 1 has been included and its profound influence on the response studied the investigation has revealed some characteristics of the response that are the result of both non linearity and more than one degree of freedom the principal one of these being saturation as far as is known this is the first time that saturation has beeb observed and studied in detail in a physical structure

an existence theorem for a nonlinear difference equation where p 1 and s p is a real sequence we establish conditions under which 1 1 has a positive nondecreasing solution here a solution of 1 1 is a real sequence y y which satisfies 1 1 since 1 1 is a recurrence relation given real initial values y and y it is clear that we can inductively obtain y y since p is a real number greater than 1 the sequence yk may not be real that is not a solution note however that any constant multiple of a solution is again a solution the existence question originated in l where it was shown that hardy s inequality for a series 2 can be viewed as a necessary condition for the existence of a positive nondecreasing solution of 1 1 our existence theorem reduces to a nonoscillation criterion in the case p 2 since then 1 1 changes to a second order linear difference equation see e g 3 a brief outline of the paper is as follows section 2 introduces a riccati type transformation and then develops various necessary conditions for the existence of a positive nondecreasing solution section 3 discusses some of the functional analytic background which is needed in order to apply schauder s theorem section 4 contains the proof of our existence theorem and section 5 describes a comparison theorem which allows us to drop the assumption that the coefficients sk in 1 1 are nonnegative

set contractive mappings and difference equations in banach spaces in this paper we consider the difference equation on an arbitrary banach space x x qn xn fn xn 0 where qn is a positive sequence and fn is x valued we shall give conditions so that for a given x x there exists a solution of this equation asymptotically equal to x

oscillation theorems for certain second order nonlinear difference equations in this paper discrete inequalities are used to offer sufficient conditions for the oscillation of all solutions of the difference equation delta bigl a n delta y n sigma bigr q n 1 f y n 1 r n qquad n ge 0 where 0 p qwithp qodd integers orpeven andqodd integers several examples which dwell upon the importance of our results are also included

oscillatory and asymptotic properties of higher order nonlinear neutral difference equations abstract in this paper we study the oscillatory and asymptotic properties of solutions of higher order nonlinear neutral difference equations using krasnoselskii s fixed point theorem schauder s fixed point theorem and lebesgue s dominated convergence theorem we obtain some necessary and sufficient conditions for all bounded solutions to be oscillatory or tend to zero our results extend and improve some known results

wiener hopf analysis of the em diffraction by an impedance discontinuity in a planar surface and by an impedance half plane the wiener hopf technique is used to solve two canonical problems the first problem considered is the electromagnetic em diffraction by a planar surface with an impedance discontinuity two part surface of an arbitrarily polarized plane wave obliquely incident to the axis of the two dimensional objects the second problem considers the em diffraction by a half plane with equal impedances on both sides the solutions obtained are cast in a matrix notation which is useful for diffraction problems the exact formal solutions are expressed in terms of integrals which can be asymptotically evaluated uniform asymptotic expressions are obtained where the presence of the geometrical optics go poles as well as the surface wave poles near the saddle point are fully taken into account several numerical examples are presented and it is shown that the solutions are continuous across the shadow boundaries of the go and surface wave fields

bounded solutions and wavefronts for discrete dynamics this paper deals with the second order nonlinear difference equation dd r k dd u k q kg u k 1 0 where r k and q k are positive real sequences defined on n cup 0 and the nonlinearity g r to r is nonnegative and nontrivial sufficient and necessary conditions are given for the existence of bounded solutions starting from a fixed initial condition u 0 the same dynamic with f instead of g such that uf u 0 for u not 0 was recently extensively investigated on the contrary our nonlinearity g is of a small appearance in the discrete case its introduction is motivated by the analysis of wavefront profiles in biological and chemical models the paper emphasizes the many different dynamical behaviors caused by such a g with respect to the equation involving function f some applications in the study of wavefronts complete this work

oscillation criteria for second order quasi linear neutral delay differential equations abstract in this paper some new oscillation criteria for second order neutral nonlinear differential equation r t x t p t x t m 1 x t p t x t q t f x t 0 are established new oscillation criteria are different from most known ones in the sense that they are based on a class of new functions t s l defined in the sequel

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

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

multiple solutions for discrete boundary value problems a recent multiplicity theorem for the critical points of a functional defined on a finite dimensional hilbert space established by ricceri is extended an application to dirichlet boundary value problems for difference equations involving the discrete p laplacian operator is presented

nonlinear discrete boundary value problems for the discrete p laplacian with potential term abstract in this paper we show the existence of a positive solution of nonlinear discrete boundary value problems for the discrete p laplacian with potential term moreover the multiplicity of solutions for the problems is also investigated

bounded oscillation for second order nonlinear delay differential equation in a critical state in this paper we establish sufficient conditions for bounded oscillation of the second order nonlinear delay differential equation x t i 1m pix t i f t x t 1 t x t n t under a critical state by the second order ordinary differential equation y t 1 e e 0t f t e 0 t 1 t y t e 0 t t y t 0 at the same time necessary conditions for the above bounded oscillation equation are obtained

oscillation criteria for second order functional difference equations with summation small coefficient we consider the second order functional difference equation a n n p n x g n 0 where n p n are sequences of integers with a n 0 0 g n n limn g n p n 0 and p n 0 we obtain some necessary conditions for equation 1 exists nonoscillatory solutions and some sufficient conditions for equation 1 is oscillatory

interval oscillation criteria for second order neutral nonlinear differential equations some oscillation criteria for the second order neutral nonlinear differential equation y t p t y s t email protected i 1 nq i t f i y t i t 0 t t 0are established new oscillation criteria are different from most known ones in the sense that they are based on the information only on a sequence of subintervals of t 0 rather than on the whole half line our results are more natural according to the sturm separation theorem

simple expressions for a function occurring in diffraction theory the maliuzhinets integral function arises in connection with diffraction by a half plane two simple expressions are derived which when used in conjunction with known identities serve to approximate the function to a high degree of accuracy throughout the entire complex plane

oscillation criteria for nonlinear partial difference equations with delays abstract we offer sufficient conditions for the oscillation of all solutions of the partial difference equations y m 1 n m n y m n 1 m n y m n p m n y m k n l q m n y m k n l and y m 1 n m n y m n 1 m n y m n i 1 p i m n y m k i n l i i 1 q i m n y m k i n l i several examples which dwell upon the importance of our results are also included

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

new oscillation criteria for second order nonlinear neutral delay differential equations in this paper several new oscillation criteria for the second order nonlinear neutral delay differential equation r t x t p t x s t m 1 x t p t x s t q t f x t t 0 t t 0 are established these oscillation criteria extend and improve some known results an interesting example illustrating the importance of our results is also provided

nonoscillatory solutions for the one dimensional p laplacian abstract in this paper we study the existence of nonoscillatory solutions of the equation xk 1 f k xk 0 k 1 2 where r r is defined by s s p 2 s with p 1 and xk 1 is a nonnegative sequence with infinitely many positive terms

asymptotic properties of solutions of second order neutral difference equations abstract in this paper we study asymptotic behavior of solutions of second order neutral difference equation 2 x n px n k f n x n 0 we present conditions under which all nonoscillatory solutions of the above equation have the property x n cn o n for some c r as well as sufficient conditions under which all nonoscillatory solutions are asymptotically linear

monotone properties of certain classes of solutions of second order difference equations the authors consider the difference equations an xn qnxn 1 and an xn qnf xn 1 where an 0 qn 0 and f r r is continuous with uf u 0 for u 0 they obtain necessary and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of and sufficient conditions for the asymptotic behavior of certain types of nonoscillatory solutions of sufficient conditions for the existence of these types of nonoscillatory solutions are also presented some examples illustrating the results and suggestions for further research are included

bounded oscillation of second order delay difference equations of unstable type abstract a series of sufficient conditions are obtained for the bounded oscillation of second order delay difference equations of unstable type some of them are sharp

oscillation theorems for second order advanced functional difference equations abstract we consider the second order advanced functional difference equation a n n p n g n 0 where a n 0 s n 0 1 a s p n 0 p n 0 g n n 1 g n is a monotone increasing integer sequence we obtain some new oscillation criteria through an appropriate riccati equation

diffraction by a thin dielectric half plane a uniform diffraction coefficient is derived for a thin dielectric half plane illuminated by an h polarized incident plane wave this is accomplished by modeling the half plane using a pair of resistive and modified conductive sheets and then solving the resulting integral equations via the wiener hopf technique in contrast to a recently published solution the one presented here satisfies reciprocity and the numerical results which are included confirm its accuracy

new oscillation criteria for second order nonlinear neutral delay difference equations in this paper we will establish some new oscillation criteria for the second order nonlinear neutral delay difference equation d a n dx n p nx n t c f n x n s 0 n 0 1 2 where c 0 is a quotient of odd positive integers

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

classification schemes for positive solutions of a second order nonlinear difference equation classification schemes for eventually positive solutions of a class of second order nonlinear difference equations are given in terms of their asymptotic magnitudes and necessary as well as sufficient conditions for the existence of these solutions are also provided

comparison theorems for second order nonlinear difference equations abstract new oscillation results are obtained for the second order nonlinear difference equation r n f x n 1 g n x n 0 and its functional form r n f x n 1 g n x n 0 the role played by the argument n on the oscillation of the functional equation is explored in particular we characterize a class of sequences n which have a harmless effect on the oscillation of this type of equations some of our results generalize improve or unify known fundamental oscillation results for several particular cases of the above equations

existence of bounded positive solutions for a system of difference equations abstract this paper deals with the second order nonlinear neutral delay system of difference equations a n x n b n x n f n x h 1 n x h kn y w 1 n y w kn c n n n 0 p n y n q n y n g n x s 1 n x s kn y t 1 n y t kn r n n n 0 by virtue of the banach fixed point theorem and some natural modifications of the techniques in the literature we prove the existence results of uncountably many bounded positive solutions for the system of difference equations seven examples are constructed to explain the main results presented in this paper

fast homoclinic solutions for a class of damped vibration problems in this paper we deal with the existence and multiplicity of homoclinic solutions of the following damped vibration problemsu t q t u t l t u t w t u t 0 where l t and w t x are neither autonomous nor periodic in t our approach is variational and it is based on the critical point theory we prove existence and multiplicity results of fast homoclinic solutions under general growth conditions on the potential function moreover some open problems proposed by zhang and ruan are resolved our theorems appear to be the first such result

oscillation criteria for second order neutral delay difference equations abstract in this paper we establish some oscillation criteria of the second order delay difference equation a n 1 x n 1 p n 1 x n 1 q n f x n 0 where and are nonnegative constants an and pn qn are nonnegative sequences and f c

existence for nonoscillatory solutions of higher order nonlinear neutral difference equations abstract in this paper we consider the following forced higher order nonlinear neutral difference equation m x n c n x n k s 1 u p s n f s x n r s q n n n 0 where m u 1 k 0 and rs 0 are integers cn p s n s 1 2 u and qn are sequence of real numbers and f s c r r s 1 2 u by using krasnoselskii s fixed point theorem and some new techniques we obtain sufficient conditions for the existence of nonoscillatory solutions for general p s n s 1 2 u and qn which means that we allow oscillatory p s n s 1 2 u and qn

monotone solutions of a class of nonlinear difference equations abstract comparison theorems and existence criteria are summarized and new ones are derived for the monotone solutions of a class of nonlinear difference equations which arises from hardy s inequality

a uniform gtd formulation for the diffraction by a wedge with impedance faces a uniform high frequency solution is presented for the diffraction by a wedge with impedance faces illuminated by a plane wave perpendicularly incident on its edge arbitrary uniform isotropic impedance boundary conditions may be imposed on the faces of the wedge and both the transverse electric te and transverse magnetic tm cases are considered this solution is formulated in terms of a diffraction coefficient which has the same structure as that of the uniform geometrical theory of diffraction utd for a perfectly conducting wedge its extension to the present case is achieved by introducing suitable multiplying factors which have been derived from an asymptotic evaluation of the exact solution given by maliuzhinets when the field point is located on the surface near the edge a more accurate asymptotic evaluation is employed to obtain a high frequency expression for the diffracted field which is suitable for several specific applications the formulation described in this paper may provide a useful rigorous basis to search for a more numerically efficient but yet accurate approximation

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

oscillation and asymptotic analysis on a new generalized emden fowler equation abstract in this work we analyze the new generalized emden fowler equation with neutral type delays r t z t 1 z t q t x t 1 x t 0 where z t x t p t x t 0 by use of averaging technique and specific analytical skills some easily accessible oscillation and asymptotic criteria are established which have extended the results in the cited literature

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

nonexistence of unbounded nonoscillatory solutions of partial difference equations abstract we develop criteria for the nonexistence of eventually positive negative and nondecreasing nonincreasing solutions of the partial difference equation m n y m n p m n y m k n l q m n y m k n l and m n y m n i 1 p i m n y m k i n l i i 1 q i m n y m k i n l i several examples which dwell upon the importance of our results are also included

homoclinic orbits of a class of second order difference equations abstract in this paper we apply the variational method and the spectral theory of difference operators to investigate the existence of homoclinic orbits of the second order difference equation 2 x t 1 l t x t v x t x t 0 in the two cases that v t is superquadratic and subquadratic under the assumptions that l t is positive definite for sufficiently large t z we show that there exists at least one non trivial homoclinic orbit of the difference equation further if v t x is superquadratic and even with respect to x then it has infinitely many different non trivial homoclinic orbits at the end two illustrative examples are provided

fourt order nonlinear oscillations of difference equations abstract the authors consider the fourth order difference equation where f n u may be classified as superlinear sublinear strongly superlinear and strongly sublinear and n n 0 n r n

existence of uncountably many bounded positive solutions for second order nonlinear neutral delay difference equations this paper deals with the second order nonlinear neutral delay difference equation d a n d x n b n x n t c n f n x f 1 n x f 2 n x f k n d n n n 0 by using krasnoselskii s fix

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

amplitude modulated and chaotic dynamics in resonant motion of strings abstract the weakly non linear resonant motions of a stretched string to a harmonic planar excitation are investigated one mode truncation of the in plane and out of plane equation results in a two degree to freedom system which is examined by using the method of averaging and the method of integral manifolds it is shown that the averaged equations for small enough damping possess non planar constant solutions which become unstable by a hopf bifurcation the resulting limit cycle solutions correspond to amplitude modulated whirling or ballooning motions of the string there are two limit cycle branches for the averaged equations one arising due to the hopf bifurcation and the other due to a global saddle node bifurcation the two branches merge as the damping is further reduced also with variation in detuning the isolated branch exhibits period doubling bifurcations chaotic attractors and merging of attractors giving rise to rossler as well as lorentz type solutions the truncated string equations are also directly integrated it is shown that there are non planar periodic responses that bifurcate into amplitude modulated motions on a 2 torus changes in detuning result in torus doubling merging of tori and then destruction of the torus leading to chaotic amplitude modulations the results of asymptotic analysis via the method of averaging are found to be in qualitative agreement with the actual response

unbounded solutions of quasi linear difference equations we study positive increasing solutions of the nonlinear difference equation an p n bnf n 1 p u u p 2u p 1 where an bn are positive real sequences for n 1 fr r is continuous with uf u 0 for u 0 a full characterization of limit behavior of all these solutions in terms of an bn is established examples showing the essential role of used hypotheses are also included the tools used are the schauder fixed point theorem and a comparison method based on the reciprocity principle

oscillation of solutions for second order nonlinear difference equations with nonlinear neutral term abstract some riccati type difference inequalities are given for the second order nonlinear difference equations with nonlinear neutral term and using these inequalities we obtain some oscillation criteria for the above equation

oscillation criteria for a nonlinear difference equation abstract essential and nontrivial improvements and extensions of several recent oscillation criteria are obtained for the nonlinear difference equation 1 below

unbounded positive solutions of higher order difference equations with singular nonlinear term in this paper several necessary and or sufficient conditions are given for the existence of unbounded positive solutions of nonlinear difference equations of the form mxn 1 f n xn 0 nen0 where m 2 1 or 1 and f n0 r r is a real valued continuous function such that f n x 0 for all n x n0 0 where n0 0 1

homoclinic orbits on compact manifolds abstract under a suitable assumption on the potential energy v we prove the existence of a homoclinic orbit of the equation dt x t grad v x t 0 x m where m is a compact riemannian manifold our assumption is satisfied for instance if the function v has a unique nondegenerate maximum point

forcing induced asymmetry on dynamical systems with cubic non linearities abstract the present work investigates the dynamics of a class of two degree of freedom oscillators with cubic non linearity in the restoring forces these oscillators are under the action of an external load including constant and harmonic components initially a perturbation analysis is applied to the equations of motion demonstrating the effect of the asymmetry induced by the constant loading component on the classical 1 1 and 1 3 internal resonances as well as on the possibility of the appearance of a first order 1 2 internal resonance next sets of slow flow equations governing the amplitudes and phases of vibration are derived for the special case of no internal resonance and for the most complicated case corresponding to 1 1 internal resonance the analytical findings are then complemented by numerical results obtained by examining the dynamics of a two degree of freedom mechanical system first the effect of certain system parameters on the existence and stability of constant and periodic solutions of the slow flow equations is illustrated by presenting a sequence of response diagrams finally the dynamics of the system used as an example is investigated further by direct integration of the slow flow equations this shows the existence of a period doubling sequence culminating into a continual interchange between quasiperiodic and chaotic response it also demonstrates a new transition scenario from phase locked to phase entrained and drift response

asymptotically polynomial solutions to difference equations of neutral type asymptotic properties of solutions to difference equation of the form m x n u n x n k a n f n x n b n are studied we give sufficient conditions under which all solutions or all solutions with polynomial growth or all nonoscillatory solutions are asymptotically polynomial we use a new technique which allows us to control the degree of approximation

a uniform geometrical theory of diffraction for an imperfectly conducting half plane diffraction tensors are presented in the context of the uniform geometrical theory of diffraction utd for the high frequency scattering by an impedance half plane at normal and oblique skew incidence these are based on the exact wiener hopf solution and were derived according to the utd ansatz in addition unlike previous uniform diffraction coefficients the ones given here reduce to the known utd diffraction coefficients for the perfectly conducting case the coefficients are explicit and therefore appropriate for practical applications several scattering patterns are also presented and compared to a previous heuristic solution

oscillation criteria for a class of neutral difference equations with continuous variable in this paper we are mainly concerned with oscillatory behaviour of solutions for a class of second order nonlinear neutral difference equations with continuous variable using an integral transformation the riccati transformation and iteration some oscillation criteria are obtained

oscillatory and asymptotic behavior of second order neutral difference equations with maxima abstract in this paper the asymptotic behavior of the non oscillatory solutions of neutral difference equations with maxima is considered sufficient conditions for oscillation of all solutions are also obtained

bounded and zero convergent solutions of second order difference equations abstract in this paper we study boundedness and monotonicity properties of a homogeneous second order linear difference equation under certain conditions it is shown that every solution must eventually be monotonic the question of when such monotonic solutions diverge to infinity or converge to zero and the corresponding impact this behavior has on recessive and dominant solutions is investigated a necessary and sufficient condition for all solutions to be bounded is presented comparison theorems are also discussed

bounded oscillation for second order nonlinear neutral difference equations in critical and non critical states the sufficient and necessary conditions for the bounded oscillation of the second order nonlinear neutral delay difference email protected 2 x n px email protected email protected i 1mq ix email protected i f n x email protected 1 n x email protected l n are obtained in critical and non critical cases