symplectic lie algebras with degenerate center abstract every symplectic lie algebra with degenerate including non abelian nilpotent symplectic lie algebras has the structure of a quadratic extension we give a standard model and describe the equivalence classes on the level of corresponding quadratic cohomology sets finally we give a scheme to classify the isomorphism classes of symplectic lie algebras with degenerate center we also give a classification scheme for nilpotent symplectic lie algebras and compute a complete list of all 6 dimensional nilpotent symplectic lie algebras which removes inaccuracies in an older list

hybrid iterative scheme for fixed point problem infinite systems of equilibrium and variational inequality problems the purpose of this paper is to construct a new iterative scheme by the hybrid method and prove a strong convergence theorem for the approximation of a common fixed point of two countable families of weak relatively nonexpansive mappings which is also a common solution to infinite systems of equilibrium and variational inequality problems in a uniformly smooth and strictly convex real banach space with kadec klee property using the properties of a generalized f projection operator using this result we give some applications our results extend many known recent results in the literature

on uniform decay of the solution for a damped nonlinear coupled system of wave equations with nonlinear boundary damping and memory term in this paper we are concerned with the existence and energy decay of the solution to the initial boundary value problem for the nonlinear coupled wave equations with nonlinear boundary damping and memory term

flat affine or projective geometries on lie groups abstract this paper deals essentially with affine or projective transformations of lie groups endowed with a flat left invariant affine or projective structure these groups are called flat affine or flat projective lie groups we give necessary and sufficient conditions for the existence of flat left invariant projective structures on lie groups we also determine lie groups admitting flat bi invariant affine or projective structures these groups could play an essential role in the study of homogeneous spaces m g h having a flat affine or flat projective structures invariant under the natural action of g on m a medina asked several years ago if the group of affine transformations of a flat affine lie group is a flat projective lie group in this work we provide a partial positive answer to this question

some problems and results in the study of nonlinear analysis this paper is presented as a survey of the latest and new results on some topics in the study of nonlinear analysis which were obtained by the author and some chinese mathematicians

long time asymptotic expansion for the damped semilinear wave equation the first initial boundary value problem is considered for the damped semilinear wave equation with the quadratic nonlinearity for small initial data and homogeneous boundary conditions its solution is constructed in the form of a series in the eigenfunctions of the laplace operator the long time asymptotic expansion is obtained which shows the nonlinear effects of amplitude and frequency multiplication the same results hold for the admissible initial data that are not small

a general iterative method for equilibrium problems and fixed point problems in hilbert spaces abstract in this paper we introduce two iterative schemes by the general iterative method for finding a common element of the set of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a hilbert space then we prove two strong convergence theorems for nonexpansive mappings to solve a unique solution of the variational inequality which is the optimality condition for the minimization problem these results extended and improved the corresponding results of marino and xu g marino h k xu a general iterative method for nonexpansive mapping in hilbert spaces j math anal appl 318 2006 43 52 s takahashi and w takahashi s takahashi w takahashi viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 331 1 2007 506 515 and many others

convergence theorems of fixed points for κ strict pseudo contractions in hilbert spaces abstract let c be a closed convex subset of a real hilbert space h and assume that t c h is a strict pseudo contraction such that f t x c x t x 0 consider the normal mann s iterative algorithm given by x 1 c x n 1 n x n 1 n p c s x n n 1 where s c h is defined by s x x 1 t x p c is the metric projection of h onto c and n n 1 for all n 1 it is proved that if the control parameter sequence n is chosen so that n 1 and n 1 n 1 n then x n converges weakly to a fixed point of t in order to get a strong convergence theorem we modify the normal mann s iterative algorithm by using a suitable convex combination of a fixed vector and a sequence in c the results presented in this article respectively improve and extend the recent results of marino and xu g marino h k xu weak and strong convergence theorems for strict pseudo contractions in hilbert spaces j math anal appl 329 2007 336 349 from strictly pseudo contractive self mappings to nonself mappings and of kim and xu t h kim h k xu strong convergence of modified mann iterations nonlinear anal 61 2005 51 60 from nonexpansive mappings to strict pseudo contractions

bifurcation of periodics and subharmonics in abstract nonlinear undamped wave equations abstract periodic and subharmonic bifurcation is investigated for certain abstract wave differential equations modelled after a nonlinear beam equation

a general iterative method for nonexpansive mappings in hilbert spaces let h be a real hilbert space consider on h a nonexpansive mapping t with a fixed point a contraction f with coefficient 0 0 let 0

periodic solutions to one dimensional wave equation with piece wise constant coefficients abstract this work is concerned with analysis and optimization in coefficients of the 1 d t periodic wave equation with piece wise constant coefficients inx fort rational the system is in resonant case i e the corresponding homogeneous problem has nontrivial solutions free oscillations or breathers the case oft irrational remains an outstanding open problem

double extension symplectique d un groupe de lie symplectique resume lie groups carrying a left invariant symplectic form symplectic groups are described in terms of semi direct product of lie groups or symplectic reduction and principal fiber bundles with affine fiber we give a generalization of medina and revoy s symplectic double extension which realizes a symplectic group as the reduction of another symplectic group we show that every group obtained by this process carries an invariant lagrangian foliation such that the affine structure defined by the simplectic form over each leaf is complete

viscosity approximation methods for nonexpansive mappings abstract viscosity approximation methods for nonexpansive mappings are studied consider a nonexpansive self mapping t of a closed convex subset c of a banach space x suppose that the set fix t of fixed points of t is nonempty for a contraction f on c and t 0 1 let x t c be the unique fixed point of the contraction x tf x 1 t tx consider also the iteration process x n where x 0 c is arbitrary and x n 1 n f x n 1 n tx n for n 1 where n 0 1 if x is either hilbert or uniformly smooth then it is shown that x t and under certain appropriate conditions on n x n converge strongly to a fixed point of t which solves some variational inequality

symplectic structures on quadratic lie algebras abstract we study quadratic lie algebras over a field k of null characteristic which admit at the same time a symplectic structure we see that if k is algebraically closed every such lie algebra may be constructed as the t extension of a nilpotent lie algebra admitting an invertible derivation and also as the double extension of another quadratic symplectic lie algebra by the one dimensional lie algebra finally we prove that every symplectic quadratic lie algebra is a special symplectic manin algebra and we give an inductive description in terms of symplectic quadratic double extensions

strong convergence theorems for nonexpansive mappings and inverse strongly monotone mappings abstract in this paper we introduce an iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly monotone mapping in a hilbert space then we show that the sequence converges strongly to a common element of two sets using this result we consider the problem of finding a common fixed point of a nonexpansive mapping and a strictly pseudocontractive mapping and the problem of finding a common element of the set of fixed points of a nonexpansive mapping and the set of zeros of an inverse strongly monotone mapping

viscosity approximation schemes for fixed point problems and equilibrium problems and variational inequality problems abstract in this paper we first introduce a general iterative scheme by halpern approximation method for finding a common element of the set of solutions of an equilibrium problem the set of solutions of the variational inequality for an infinite family of i inverse strongly monotone mappings and the set of common fixed points for a family of infinitely nonexpansive mappings in a hilbert space then we extend the halpern iterative scheme to a general viscosity approximation scheme by a simple and different method as applications we utilize our results to investigate some convergence problems for strictly pseudocontractive mappings and maximal monotone operators the results obtained in this paper extend and improve the recent ones announced by many others

a general iterative algorithm for nonexpansive mappings in hilbert spaces abstract let h be a real hilbert space suppose that t is a nonexpansive mapping on h with a fixed point f is a contraction on h with coefficient 0 1 and f h h is a k lipschitzian and strongly monotone operator with k 0 0 let 0 2 k 2 0 k 2 2 we proved that the sequence x n generated by the iterative method x n 1 n f x n i n f t x n converges strongly to a fixed point x f i x t which solves the variational inequality f f x x x 0 for x f i x t

viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces abstract in this paper we introduce an iterative scheme by the viscosity approximation method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a nonexpansive mapping in a hilbert space then we prove a strong convergence theorem which is connected with combettes and hirstoaga s result p l combettes s a hirstoaga equilibrium programming in hilbert spaces j nonlinear convex anal 6 2005 117 136 and wittmann s result r wittmann approximation of fixed points of nonexpansive mappings arch math 58 1992 486 491 using this result we obtain two corollaries which improve and extend their results

a new variational principle and duality for periodic solutions of hamilton s equations we are dealing with a classical boundary value problem concerning periodic solutions of hamiltonian equations p f k f 4th p f i f h x t p t 1 where h r x r x r r is a given function called the hamiltonian in mechanics 1 and also in mathematics 171 with a given hamiltonian the lagrangian l defined by the legendre transform l 4 x a sup k p wc x pi d 2

a unified algebraic approach to the classical yang baxter equation in this paper the different operator forms of the classical yang baxter equation are given in the tensor expression through a unified algebraic method it is closely related to left symmetric algebras which play an important role in many fields in mathematics and mathematical physics by studying the relations between left symmetric algebras and the classical yang baxter equation we can construct left symmetric algebras from certain classical r matrices and conversely there is a natural classical r matrix constructed from a left symmetric algebra which corresponds to a parakahler structure in geometry moreover the former in a special case gives an algebraic interpretation of the left symmetry as a lie bracket left twisted by a classical r matrix

a general iterative method for equilibrium problems and strict pseudo contractions in hilbert spaces abstract in this paper we introduce two iterative schemes by the general iterative method for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a k strictly pseudo contractive non self mapping in the setting of real hilbert spaces our results improve and extend the corresponding results given by many others

an iterative scheme for equilibrium problems and fixed point problems of strict pseudo contraction mappings in this paper we propose an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and the set of fixed points of a strict pseudo contraction mapping in the setting of real hilbert spaces we establish some weak and strong convergence theorems of the sequences generated by our proposed scheme our results combine the ideas of marino and xu s result g marino h k xu weak and strong convergence theorems for strict pseudo contractions in hilbert spaces j math anal appl 329 2007 336 346 and takahashi and takahashi s result s takahashi w takahashi viscosity approximation methods for equilibrium problems and fixed point problems in hilbert spaces j math anal appl 331 2007 506 515 in particular necessary and sufficient conditions for strong convergence of our iterative scheme are obtained

existence of periodic solutions for nonlinear differential equations abstract in this paper we shall study the existence of a periodic solution to the nonlinear differential equationxtt axt a xt a ax b t x f t x t 0 in some complex hilbert space using duality and variational methods

strong convergence theorems for infinite family of relatively quasi nonexpansive mappings and systems of equilibrium problems abstract in this paper we construct a new iterative scheme by hybrid method for approximation of common element of set of common fixed points of countably infinite family of relatively quasi nonexpansive mappings and set of common solutions to a system of equilibrium problems in a uniformly convex and uniformly smooth real banach space using the properties of generalized f projection operator then we prove strong convergence of the scheme to a common element of the two sets furthermore we apply our results to solve convex minimization problem our results extend important recent results

on periodic solutions of a damped wave equation in a thin domain using degree theoretic methods abstract we combine the methods of the topological degree with techniques developed by j k hale and g raugel to study the periodic solutions of a damped nonlinear hyperbolic equation in a thin domain

hybrid algorithm for generalized mixed equilibrium problems and variational inequality problems and fixed point problems in this paper we introduce a new iterative algorithm by hybrid method for finding a common element of the set of solutions of finite general mixed equilibrium problems and the set of solutions of a general variational inequality problem for finite inverse strongly monotone mappings and the set of common fixed points of infinite family of strictly pseudocontractive mappings in a real hilbert space then we prove strong convergence of the scheme to a common element of the three above described sets our result improves and extends the corresponding results announced by many others

strong convergence theorems of modified mann iterative process for strict pseudo contractions in hilbert spaces abstract in this paper we modify the normal mann s iterative process to have strong convergence for a k strictly pseudo contractive non self mapping in the framework of hilbert spaces our results improve and extend the corresponding results announced by many others

periodic solutions of hamilton s equations and local minima of the dual action the dual action is a functional whose extremals lead to solutions of hamilton s equations up to now extremals of the dual action have been obtained either through its global minimization or through application of crit ical point theory a new methodology is introduced in which local minima of the dual action are found to exist applications are then made to the existence of hamiltonian trajectories having prescribed period 1 introduction a classical and very interesting boundary value problem is concerned with periodic solutions of hamilton s equations 1 1 p i vxtf x i p i x t vph x t p t where the function h rn x rn r is called the hamiltonian any such hamil tonian trajectory x p is such that h x t p t is constant this constant is called the energy associated with x p if x 0 x t p 0 p t for some t 0 then x p is said to have period t one can ask for what energies and periods are their associated hamiltonian trajectories this and related issues have both a long history and an active present we refer to desolneux and moulis 16 mancini 20 and rabinowitz 23 for surveys and a more extensive bibliography than would be appropriate here in this article attention will be focused upon the case in which the hamiltonian is a convex function a situation which has drawn perhaps the most attention of late we begin by reviewing briefly the basis for much of this recent work a dual action principle introduced by clarke in 1978 6 let g denote the conjugate of h in the sense of convex analysis i e the function

on the existence of periodic solutions for a certain type of nonlinear differential equations abstract bifurcations of periodic solutions are studied for certain types of weakly perturbed partial differential equations it is shown that a bifurcation occurs for almost all in the sense of the lebesque measure periodic small perturbations a generalized implicit function theorem is applied

on a quasilinear wave equation with memory equation 1 l with a 0 and n 1 has its roots in a model for the small amplitude vibrations of a string in which the dependence of the tension on the deformation cannot be ignored see for example bernstein 2 carrier 3 41 dickey s and nishida 13 when a 0 1 1 may be used to describe the dynamics of an extensible string with fading memory this equation states that the dynamic equilibrium of a body depends not only on the present state of deformation but also on the previous history of the deformation gradient since the publication of the aforementioned papers equation 1 1 with a 0 has been the subject of intense scrutiny see for example arosio and spagnolo 11 lions 9 medeiros lo menzala ll nishihara 14 151 pohoiaev 16 and yamada 17 181 to name but a few among the facts thus far proved pohoiaev s results and their extension to the degenerate case of arosio and spagnolo are of particular significance they show that if the data satisfy an analytic type condition then solutions are global in time in spite of the abundant literature on the memoryless form of equation l l to our best knowledge equation 1 1 with a f 0 appears to be untouched our main results theorem 5 1 theorem 5 2 and theorem 5 4 will show that under appropriate assumptions on the kernel

pseudo kahler lie algebras with abelian complex structures we study lie algebras endowed with an abelian complex structure which admit a symplectic form compatible with the complex structure we prove that each of those lie algebras is completely determined by a pair u h where u is a complex commutative associative algebra and h is a sesquilinear hermitian form on u which verifies certain compatibility conditions with respect to the associative product on u the riemannian and ricci curvatures of the associated pseudo kahler metric are studied and a characterization of those lie algebras which are einstein but not ricci flat is given it is seen that all pseudo kahler lie algebras can be inductively described by a certain method of double extensions applied to the associated complex associative commutative algebras

a general composite iterative algorithm for nonexpansive mappings in hilbert spaces let h be a real hilbert space suppose that t is a nonexpansive mapping on h with a fixed point f is a contraction on h with coefficient email protected 0 1 f h h is a k lipschitzian and h strongly monotone operator with k 0 h 0 and a h h is a strongly positive bounded linear operator with coefficient email protected email protected 1 2 let 0 0 where a n 0 1 and b n 0 1 converges strongly to a fixed point email protected email protected fix t which solves the variational inequality 0 email protected fix t

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

some iterative methods for finding fixed points and for solving constrained convex minimization problems the present paper is divided into two parts in the first part we introduce implicit and explicit iterative schemes for finding the fixed point of a nonexpansive mapping defined on the closed convex subset of a real hilbert space we establish results on the strong convergence of the sequences generated by the proposed schemes to a fixed point of a nonexpansive mapping such a fixed point is also a solution of a variational inequality defined on the set of fixed points in the second part we propose implicit and explicit iterative schemes for finding the approximate minimizer of a constrained convex minimization problem and prove that the sequences generated by our schemes converge strongly to a solution of the constrained convex minimization problem such a solution is also a solution of a variational inequality defined over the set of fixed points of a nonexpansive mapping the results of this paper extend and improve several results presented in the literature in the recent past