uniform asymptotic stability in impulsive perturbed systems of differential equations abstract in the present paper the uniform asymptotic stability of the zero solution of perturbed impulsive systems of differential equations with unfixed moments of impulse action is investigated the direct method of lyapunov is used but the auxiliary functions possess first kind discontinuities on the impulsive hypersurfaces

stability with respect to part of the variables in systems with impulse effect abstract effective sufficient conditions are found for stability with respect to part of the variables in systems of ordinary differential equations with impulse effect the approach presented is based on the specially introduced piecewise continuous ljapunov functions

mathematical models for the control of a pest population by infected pest in this paper we propose two mathematical models concerning continuous and impulsive pest control strategies respectively therefore our models are the ordinary differential equations and the impulsive differential equations as a result the global asymptotic stability of the equilibria of the ordinary differential equations is studied in the case when an impulsive control is used it is observed that there exists a globally asymptotically stable susceptible pest eradication periodic solution when the amount of infective pests released periodically is larger than some critical value when the amount of infective pests released is less than this critical value the system is shown to be permanent which implies that the trivial susceptible pest eradication solution loses its stability finally by means of numerical simulation we obtain the critical values of the control variable under different methods of release of infected pests

sturmian comparison theorem for half linear second order differential equations let be defined by s s p 2 s with p 1 a fixed number we extend sturm comparison theorem of the linear differential equation to the nonlinear differential equation by using the wirtinger inequality a lyapunov inequality and some oscillation criteria of e are also given

bifurcation problems for the laplacian in ⁿ in this paper we consider the bifurcation problem div ivulp 2vu ag x julp 2u f a x u in rn with p 1 we show that a continuum of positive solutions bifurcates out from the principal eigenvalue a1 of the problem div ivuip 2vu ag x ulp 2u here both functions f and g may change sign

global attractors and convergence to equilibrium for degenerate ginzburg landau and parabolic equations abstract we study the dynamics of a degenerate complex ginzburg landau cgl and a real parabolic equation with a variable generally nonsmooth diffusion coefficient which may vanish at some points or be unbounded for the cgl equation there exists a global attractor in l 2 for the parabolic equation we show the existence of a global branch of nonnegative stationary states the global bifurcation result is used in order to establish in conjunction with the definition of a gradient dynamical system in the natural phase space that any solution with nonnegative initial data tends to the trivial or the nonnegative equilibrium

convergence for degenerate parabolic equations abstract we prove that any bounded global solution to a degenerate parabolic problem in one spatial dimension converges to a unique stationary state

stability criteria of impulsive differential systems the notion of eventual stability has been recently discussed we extend this notion to impulsive systems of differential equations our technique depends on liapunov s direct method

global attractors for the complex ginzburg landau equation abstract this paper concerns the long time behavior of the following complex ginzburg landau equations u t i u i u p 2 u u 0 without any restriction on p 2 under the assumptions 1 4 we first prove the well posedness of strong solutions for the complex ginzburg landau equations and then the existence of absorbing sets in l 2 h 0 1 l p and h 2 l 2 p 1 respectively for the semigroup s t t 0 generated by 1 1 1 3 is established finally we prove the existence of global attractors in l 2 and h 0 1 for the semigroup s t t 0 generated by 1 1 1 3 by the sobolev compactness embedding theorem and prove the existence of global attractor in l p for the semigroup s t t 0 generated by 1 1 1 3 using interpolation inequality

stability for impulsive differential systems in terms of two measures stability properties of impulsive differential systems are investigated using two different measures and employing piecewise continuous lyapunov functions

on the principal eigenvalue of a second order linear elliptic problem on considere le probleme de dirichlet 7b lu mu dans u 0 sur dans un domaine borne r n a frontiere lisse 7b l est un operateur differentiel uniformement elliptique sur de la forme 7b lu i n j 1 a ij 2 u x j x i i n 1 a i u x i a 0 u a coefficients a valeurs reels appartenant a c 0 1

principal eigenvalues and sturm comparison via picone s identity we characterize the principal eigenvalues and eigenfunctions in rn and present comparison results for higher dimensional p laplacian our main tool is picone s identity in this way we extend several recent results on spectral and comparison properties for differential equations

bounds for ratios of the membrane eigenvalues abstract for a bounded planar region in r 2 we obtain the ratios of lower order eigenvalues of laplace operator combining our results with the recursive formula in cheng and yang 2007 11 we can obtain better upper bound of the k 1 th k 3 membrane eigenvalues

on the principal eigenvalues of indefinite elliptic problems on considere les deux systemes d equations lu u 2 b j d j u au wu lv wv k v dans un domaine borne lisse r n on suppose que a w sont des matrices m m ueh 1 2 on donne des conditions suffisantes pour l existence de vecteurs propres positifs et des bornes superieures et inferieures sur les valeurs propres correspondantes

bounds on eigenvalues of dirichlet laplacian in this paper we investigate an eigenvalue problem of dirichlet laplacian on a bounded domain in an n dimensional euclidean space rn if k 1 is the k 1 th eigenvalue of dirichlet laplacian on then we prove that for n 41 and k geq 41 lambda k 1 leq k frac2n lambda 1 and for any n and k lambda k 1 leq c 0 n k k frac2n lambda 1 with c 0 n k leq j 2 n 2 1 j 2 n 2 1 1 where jp k denotes the k th positive zero of the standard bessel function jp x of the first kind of order p from the asymptotic formula of weyl and the partial solution of the conjecture of polya we know that our estimates are optimal in the sense of order of k

dimension of the attractors associated to the ginzburg landau partial differential equation abstract we study the long time behavior of solutions to the ginzburg landau partial differential equation it is shown that a finite dimensional attractor captures all the solutions an upper bound on this dimension is given in terms of physical quantities by estimating the lyapunov exponents on the trajectories finally using the well known side band instabilities of an exact time dependent solution stokes solution we derive lower bounds on the dimension of the universal attractor moreover the lower and upper bounds agree

on stability for impulsive perturbed systems via cone valued lyapunov function method the notion of f 0 stability recently was introduced in this paper we will extend this notion to the so called eventual f 0 stability for impulsive systems of differential equations under more relax conditions our technique depends on lyapunov s direct method

stability of linear impulsive differential equations abstract in the present paper the notion of stability of linear impulsive differential equations at fixed moments is made more precise

a sharp bound for the ratio of the first two eigenvalues of dirichlet laplacians and extensions the first author s work was partially supported by fondecyt chile project 0132 88 and by a summer research fellowship provided by the research council of the university of missouri columbia he would like to thank the physics department and others at the universidad de chile for their hospitality during his visit in april 1990 when much of this research was completed the second author was supported in part by fondecyt chile projects 0132 88 and 1238 90 both authors also thank fritz gesztesy for general comments and encouragement

on cone perturbing liapunov function for impulsive differential systems the concept of f 0 stability and f 0 boundedness recently were introduced for systems of ordinary differential equations odes perturbing liapunov function method was discussed for systems of odes and extend to systems of functional differential equations fdes in this paper we extend these notions to impulsive systems of differential equations via cone perturbing liapunov function method

the maximum principle and the existence of principal eigenvalues for some linear weighted boundary value problems abstract in this work we deal with the problem of the existence and uniqueness of principal eigenvalues for some linear weighted boundary value problems associated to a general second order uniformly elliptic operator for a large class of sign definited weights we characterize whether the boundary value problem admits a principal eigenvalue or not

structure of the set of steady state solutions and asymptotic behaviour of semilinear heat equations soit un domaine borne lisse connexe de r n on considere a la fois le probleme aux valeurs initiales u t u f u dans 0 u x t 0 sur 0 u x o u 0 x dans et le probleme stationnaire u g u dans u 0 sur ou u 0 est une condition initiale donnee et f une non linearite donnee de classe c 1

on the asymptotic stability of systems with impulses by the direct method of lyapunov abstract in the present paper the asymptotic and globally asymptotic stability of the zero solution of systems with impulses is investigated for this purpose piecewise continuous auxiliary functions are used which are an analogue of lyapunov s functions the theorem of marachkov on the asymptotic stability of systems without impulses is generalized the results obtained are formulated in six theorems

the strong maximum principle revisited in this paper we rst present the classical maximum principle due to e hopf together with an extended commentary and discussion of hopf s paper we emphasize the comparison technique invented by hopf to prove this principle which has since become a main mathematical tool for the study of second order elliptic partial dierential equations and has generated an enormous number of important applications while hopf s principle is generally understood to apply to linear equations it is in fact also crucial in nonlinear theories such as those under consideration here in particular we shall treat and discuss recent generalizations of the strong maximum principle and also the compact support principle for the case of singular quasilinear ellip tic dierential inequalities under generally weak assumptions on the quasilinear operators and the nonlinearities involved our principal interest is in necessary and sucient con ditions for the validity of both principles in exposing and simplifying earlier proofs of corresponding results and in extending the conclusions to wider classes of singular oper ators than previously considered the results have unexpected rami cations for other problems as will develop from the exposition e g i two point boundary value problems for singular quasilinear ordinary dierential equations sections 3 4 ii the exterior dirichlet boundary value problem section 5 iii the existence of dead cores and compact support solutions i e dead cores at in nity section 7 iv euler lagrange inequalities on a riemannian manifold section 9 v comparison and uniqueness theorems for solutions of singular quasilinear dierential inequalities section 10 the case of p regular elliptic inequalities is briesy considered in section 11

principal eigenvalues with indefinite weight functions both existence and non existence results for principal eigenvalues of an elliptic operator with indefinite weight function have been proved the existence of a continuous family of principal eigenvalues is demonstrated

global existence and gradient estimates for the quasilinear parabolic equations of m laplacian type with a nonlinear convection term abstract in this paper we derive precise estimates for u t including smoothing effects near t 0 and decay as t as well as global existence of the solutions u t to the initial boundary value problem in a bounded domain in r n for the quasilinear parabolic equation of the m laplacian type with a nonlinear convection term b u u for the initial data u 0 we only assume u 0 l q 1 q

the cauchy problem in local spaces for the complex ginzburg landau equation i compactness methods we continue the study of the initial value problem for the complex ginzburg landau equation with a 0 b 0 g 0 in initiated in a previous paper i we treat the case where the initial data and the solutions belong to local uniform spaces more precisely to spaces of functions satisfying local regularity conditions and uniform bounds in local norms but no decay conditions or arbitrarily weak decay conditions at infinity in in i we used compactness methods and an extended version of recent local estimates 3 and proved in particular the existence of solutions globally defined in time with local regularity of the initial data corresponding to the spaces l r for r 2 or h 1 here we treat the same problem by contraction methods this allows us in particular to prove that the solutions obtained in i are unique under suitable subcriticality conditions and to obtain for them additional regularity properties and uniform bounds the method extends some of those previously applied to the nonlinear heat equation in global spaces to the framework of local uniform spaces

a simple unified approach to some convergence theorems of l simon abstract we give a new and easier proof for the convergences result of l simon 15 concerning global and bounded solutions of a heat equation with analytic nonlinearity by constructing a new liapunov functional for the elliptic equation we are able to prove the second convergence theorem of 15 in the same way as the first theorem the results are more general and can be applied to other problems such as fourth order equation and vector valued systems of gradient type

convergence of global and bounded solutions of the wave equation with linear dissipation and analytic nonlinearity abstract we prove convergence of global bounded and smooth solutions of the wave equation with linear dissipation and analytic nonlinearity a generalization and examples of applications are given at the end of the paper