uniqueness theorem for quasilinear 2nth order equations abstract we are concerned with existence and uniqueness of the solution of initial value problems for quasilinear 2nth order equations of the type 1 n u n p 2 u n n u q 2 u where n n r and p q 1 we show that there exists a global solution for p q while the solution can blow up for p q we give an example of nonuniqueness we prove the uniqueness theorem for a general equation involving nonconstant coefficients and jumping nonlinearity

operator means and range inclusion we prove the range inclusions rana12 ranb12 ran amb 1 2 rana1 2 ranb1 2 for an operator mean m an equality condition in each inclusion is given by using the representing function x 1 mx and f x xm 1 further we show that an operator mean m is majorized by a geometric mean if and only if ran amb rana12 ranb12

algorithms associated with arithmetic geometric and harmonic means and integrable systems abstract gauss algorithm for arithmetic geometric mean agm can be regarded as a discrete time integrable dynamical system having an elliptic theta function solution and a conserved quantity in this paper we consider algorithms associated with arithmetic geometric and harmonic means from a viewpoint of integrable systems first a max plus limit and its inverse limit of the agm algorithm are discussed these mean operations are shown to be connected to each other by the max plus limit secondly continous time dynamical systems associated with the arithmetic harmonic mean ahm algorithm are found thirdly it is shown that the ahm algorithm in indefinite case has a chaotic dynamics and is a generator of numbers which obey the cauchy distribution finally an extension of the ahm algorithm to the space of positive definite symmetric matrices is considered

comparisons between the pendulum with varying length and the pendulum with oscillating support we consider two forced dissipative pendulum systems the pendulum with vertically oscillating support and the pendulum with periodically varying length with a view to draw comparisons between their behaviour we study the two systems for values of the parameters for which the dynamics are non chaotic we focus our investigation on the persisting attractive periodic orbits and their basins of attraction utilising both analytical and numerical techniques although in some respect the two systems have similar behaviour we find that even within the perturbation regime they may exhibit different dynamics in particular for the same value of the amplitude of the forcing the pendulum with varying length turns out to be perturbed to a greater extent furthermore the periodic attractors persist under larger values of the damping coefficient in the pendulum with varying length finally unlike the pendulum with oscillating support the pendulum with varying length cannot be stabilised around the upward position for any values of the parameters

global attraction to the origin in a parametrically driven nonlinear oscillator we consider a parametrically driven nonlinear ode which encompasses a simple model of an electronic circuit known as a parametric amplifier whose linearisation has a zero eigenvalue by adopting two different approaches we obtain conditions for the origin to be a global attractor which is approached a non monotonically and b monotonically in case b we obtain an asymptotic expression for the convergence to the origin some further numerical results are reported

operator means and matrix functions abstract operator means are nonlinear matrix functions that arise in the study of interconnection of n ports hilbert ports we survey some of the relations among operator means shorted operators dini s theorem and norm convergence matrix valued functions are used to present new examples of many diverse convergence behaviors we also give a new result on the geometric mean

riemannian geometry and matrix geometric means the geometric mean of two positive definite matrices has been defined in several ways and studied by several authors including pusz and woronowicz and ando the characterizations by these authors do not readily extend to three matrices and it has been a long standing problem to define a natural geometric mean of three positive definite matrices in some recent papers new understanding of the geometric mean of two positive definite matrices has been achieved by identifying the geometric mean of a and b as the midpoint of the geodesic with respect to a natural riemannian metric joining a and b this suggests some natural definitions for a geometric mean of three positive definite matrices we explain the necessary geometric background and explore the properties of some of these candidates

nonlinear eigenvalue problems for a class of ordinary differential equations ani 1 x a1 x a0 x u p0 0 p1 0 pni 1 0 b x u r where y p jyj p sgny pi 0 and p0p1 pni 1 r with various boundary conditions we prove the existence of eigenvalues and study the zero properties and structure of the corresponding eigenfunctions

the coexistence of quasi periodic and blow up solutions in a class of hamiltonian systems abstract the coexistence of quasi periodic solutions and blow up phenomena in a class of higher dimensional duffing type equations is proved in this paper moreover we show that the initial point sets for both kinds of solutions are of infinite lebesgue measure in the phase space for the part of quasi periodic solutions the tool we used is the small twist theorem for higher dimensional cases

boundedness for solutions of nonlinear hill s equations with periodic forcing terms via moser s twist theorem we consider the nonlinear hill s equation with periodic forcing term x x2n 1 a1 ea t x p t n 1 where a t and p t are continuous and 1 periodic a1 and are positive constants and e is a small parameter the purpose of this paper is to prove that all solutions of the above mentioned equation are bounded for t r and that there are infinitely many quasi periodic solutions and an infinity of periodic solutions of minimal period m for each positive integer m

matrix power means and the karcher mean we define a new family of matrix means pt a t 1 1 where and a vary over all positive probability vectors in rn and n tuples of positive definite matrices resp each of these means except t 0 arises as a unique positive definite solution of a non linear matrix equation satisfies all desirable properties of power means of positive real numbers and interpolates between the weighted harmonic and arithmetic means the main result is that the karcher mean coincides with the limit of power means as t 0 this provides not only a sequence of matrix means converging to the karcher mean but also a simple proof of the monotonicity of the karcher mean conjectured by bhatia and holbrook and other new properties which have recently been established by lawson and lim and also bhatia and karandikar using probabilistic methods on the metric structure of positive definite matrices equipped with the trace metric

globally and locally attractive solutions for quasi periodically forced systems abstract we consider a class of differential equations x x g x f t with r d describing one dimensional dissipative systems subject to a periodic or quasi periodic diophantine forcing we study existence and properties of trajectories with the same quasi periodicity as the forcing for g x x 2 p 1 p n we show that when the dissipation coefficient is large enough there is only one such trajectory and that it describes a global attractor in the case of more general nonlinearities including g x x 2 describing the varactor equation we find that there is at least one trajectory which describes a local attractor

bifurcation phenomena and attractive periodic solutions in the saturating inductor circuit in this paper we investigate bifurcation phenomena such as those of the periodic solutions for the unperturbed nonlinear system g x cbxz 0 with g x z ac x 2 1c x 2 and ao1 bo0 when we add the two competing termskf t cg x with f t a time periodic analytic forcing function and go0 the dissipative parameter the resulting differential equation g x xcbxcg xkf t z 0 describes approximately an electronic system known as the saturating inductor circuit for any periodic orbit of the unperturbed system we provide conditions which give rise to the appearance of subharmonic solutions furthermore we show that other bifurcation phenomena arise as there is a periodic solution with the same period as the forcing function f t which branches off the origin when the perturbation is switched on we also show that such a solution which encircles the origin attracts the entire phase space when the dissipative parameter becomes large enough we then compute numerically the basins of attraction of the attractive periodic solutions by choosing specific examples of the forcing function f t which are dictated by experience we provide evidence showing that all the dynamics of the saturating inductor circuit is organized by the persistent subharmonic solutions and by the periodic solution around the origin

quasiperiodic motions in superquadratic time periodic potentials it is shown that for a large class of potentials on the line with superquadratic growth at infinity and with the additional time periodic dependence all possible motions under the influence of such potentials are bounded for all time and that most in a precise sense motions are in fact quasiperiodic the class of potentials includes as very particular examples the exponential polynomial and much more this extends earlier results and gives an answer to a problem posed by littlewood in the mid 1960 s along the way machinery is developed for estimating the action angle transformation directly in terms of the potential and also some apparently new identities involving singular integrals are derived

operator power means due to lawson lim palfia for 1 t 2 abstract for 1 t 1 lim palfia defined a new family of operator power means of positive definite matrices and subsequently by lawson lim their notion and most of their results extend to the setting of positive invertible operators on a hilbert space each of these means except t 0 arises as a unique positive invertible solution of a non linear operator equation and satisfies all desirable properties of power arithmetic means of positive real numbers the purpose of this paper is to extend the range in which operator power means due to lawson lim palfia are defined we investigate some properties of operator power means for t 2 2 1 1

boundedness of solutions of duffing s equations abstract j littlewood l markus and j moser proposed independently the boundedness problem for solutions of duffing s equation x g x p t where p t is continuous and periodic and g x is superlinear at infinity the purpose of this paper is to prove that all solutions of the above mentioned duffing s equation are bounded for t r when p t is even or when p t is odd and g x is odd

on simplicity of spectra of p biharmonic equations abstract in this paper we deal with the nonlinear fourth order boundary value problem with general robin type boundary conditions u p 2 u u q 2 u in 0 1 md u 0 o nd u 1 o where r p q 1 m n are 2 4 matrices rank m rank n 2 and d u u p 2 u u p 2 u u p 2 u u p 2 u t under some restrictions on m and n we show that every positive eigenvalue is simple on the other hand we give a counterexample showing that without these restrictions it need not be true we also discuss the question of positivity of all eigenvalues

the relative operator entropy and the karcher mean abstract the karcher mean for positive invertible operators on a hilbert space which is the unique solution of the karcher operator equation was established by lawson lim based on the geometric considerations for positive invertible operators in this note we extend both this mean and the equation to the non invertible case the key concept is the relative operator entropy for operators whose ranges are closed we show that the karcher mean for the non invertible case is the unique solution of the extended karcher equation as a byproduct we can show the uniqueness of the karcher solution for the invertible case based only on that of the power means