block monotone iterative methods for numerical solutions of nonlinear elliptic equations two block monotone iterative schemes for a nonlinear algebraic system which is a finite difference approximation of a nonlinear elliptic boundary value problem are presented and are shown to converge monotonically either from above or from below to a solution of the system this monotone convergence result yields a computational algorithm for numerical solutions as well as an existence comparison theorem of the system including a sufficient condition for the uniqueness of the solution an advantage of the block iterative schemes is that the thomas algorithm can be used to compute numerical solutions of the sequence of iterations in the same fashion as for one dimensional problems the block iterative schemes are compared with the point monotone iterative schemes of picard jacobi and gauss seidel and various theoretical comparison results among these monotone iterative schemes are given these comparison results demonstrate that the sequence of iterations from the block iterative schemes converges faster than the corresponding sequence given by the point iterative schemes application of the iterative schemes is given to a logistic model problem in ecology and numerical ressults for a test problem with known analytical solution are given

a fast collocation method for an inverse boundary value problem in this paper we present an implementation of a fast multiscale collocation method for boundary integral equations of the second kind and its application to solving an inverse boundary value problem of recovering a coefficient function from a boundary measurement we illustrate by numerical examples the insensitive nature of the map from the coefficient to measurement and design and test a gauss newton iteration algorithm for obtaining the best estimate of the unknown coefficient from the given measurement based on a least squares formulation

combined effects of concave and convex nonlinearities in elliptic equation on rn abstract in this paper we consider the existence of two positive solutions of the elliptic equation on r n with concave and convex nonlinearities

generalized lyapunov criteria on finite time stability of stochastic nonlinear systems abstract this paper considers the problem of finite time stability for stochastic nonlinear systems a general lyapunov theorem of stochastic finite time stability is proposed and an important corollary is obtained some comparisons with the existing results are also given this theorem not only generalizes the classical stochastic finite time theorem but also somewhat shows that brownian noise takes an active role in stochastic finite time stability in addition more general conditions on stochastic finite time stability via multiple lyapunov functions are given to further relax the constraint of the differential operator l v moreover with the help of this criterion it shows that an unstable deterministic system can even be finite time stabilized by brownian noise some concrete examples are constructed to show the significant features of the proposed results and simulations are presented to verify the theoretical analysis

fast collocation methods for second kind integral equations in this paper we develop fast collocation methods for integral equations of the second kind with weakly singular kernels for this purpose we construct multiscale interpolating functions and collocation functionals having vanishing moments moreover we propose a truncation strategy for the coefficient matrix of the corresponding discrete system which forms a basis for fast algorithms an optimal order of convergence of the approximate solutions obtained from the fast algorithms is proved and the computational complexity of the algorithms is estimated the stability of the numerical method and the condition number of the truncated coefficient matrix are analyzed

historical development of the newton raphson method this expository paper traces the development of the newton raphson method for solving nonlinear algebraic equations through the extant notes letters and publications of isaac newton joseph raphson and thomas simpson it is shown how newton s formulation differed from the iterative process of raphson and that simpson was the first to give a general formulation in terms of fluxional calculus applicable to nonpolynomial equations simpson s extension of the method to systems of equations is exhibited

a minimax method for finding multiple critical points in banach spaces and its application to quasi linear elliptic pde a minimax method for finding multiple critical points in banach spaces is developed by using a modified pseudogradient the method is implemented successfully to solve several quasi linear elliptic boundary value problems for multiple solutions numerical solutions are presented by their profiles to illustrate the theory and method a unified convergence result of the algorithm is established in a subsequent paper x yao and j zhou a unified convergence result on a minimax method for finding multiple critical points in banach spaces preprint

a bifurcation method for solving multiple positive solutions to the boundary value problem of the henon equation on a unit disk three algorithms based on the bifurcation theory are proposed to compute the o 2 symmetric positive solutions to the boundary value problem of the henon equation on the unit disk taking l in the henon equation as a bifurcation parameter the symmetry breaking bifurcation point on the branch of the o 2 symmetric positive solutions is found via the extended systems finally other symmetric positive solutions are computed by the branch switching method based on the lyapunov schmidt reduction

fixed time stability theorem of stochastic nonlinear systems to solve the fixed time stability problem for stochastic systems which has not been investigated due to the difficulty caused by the existence of stochastic the fixed time stability is investigat

finite time stabilization of stochastic nonlinear systems in strict feedback form in this paper we investigate the problem of almost surely finite time stabilization of a class of stochastic nonlinear systems based on the recently proposed almost surely finite time stability theorem in yin khoo man and yu 2011 we prove that almost surely global finite time stability of stochastic nonlinear systems in strict feedback form can be guaranteed by a continuous control law a systematic design algorithm is developed for the construction of the controller simulation results are given to illustrate the theoretical analysis

gauss type quadratures for weakly singular integrals and their application to fredholm integral equations of the second kind in this paper we establish gauss type quadrature formulas for weakly singular integrals an application of the quadrature scheme is given to obtain numerical solutions of the weakly singular fredholm integral equation of the second kind we call this method a discrete product integration method since the weights involved in the standard product integration method are computed numerically

lyapunov like techniques for stochastic stability the purpose of this paper is to study the stabilizability problem for control stochastic nonlinear systems driven by a wiener process sufficient conditions for the existence of stabilizing feedback laws that are smooth except possibly at the equilibrium point of the system are provided by means of stochastic lyapunov like techniques the notion of dynamic asymptotic stability in probability of control stochastic differential systems is introduced and the stabilization by means of dynamic controllers is studied

finite time stabilization of weak solutions for a class of non local lipschitzian stochastic nonlinear systems with inverse dynamics in this paper finite time stabilization is investigated for a class of non local lipschitzian stochastic nonlinear systems with stochastic inverse dynamics different from the existing work about finite time control to guarantee the existence of the solution under mild conditions we study the stabilization in the sense of weak solution we first present a finite time stability theory under the framework of weak solution then for a class of stochastic nonlinear systems with stochastic inverse dynamics a finite time controller via state feedback is constructively designed under the assumption that the stochastic inverse dynamics is stochastic input to state stable the trivial weak solution of the closed loop system is proved to be globally finite time stable in probability finally a simulation example is given to illustrate the efficiency of the proposed design procedure

output feedback regulation of stochastic nonlinear systems with stochastic iiss inverse dynamics this paper develops a unifying framework for output feedback regulation of stochastic nonlinear systems with more general stochastic inverse dynamics the contributions of this work are characterized by the following novel features 1 motivated by the concept of integral input to state stability iiss in deterministic systems and stochastic input to state stability siss using lyapunov function in stochastic systems a concept of stochastic integral input to state stability siiss using lyapunov function is first introduced two important properties of siiss are obtained i siiss is strictly weaker than siss using lyapunov function ii siiss is stronger than the minimum phase property however only under the minimum phase assumption there is no dynamic output feedback control law for global stabilization in probability 2 almost sure boundedness a reasonable and stronger concept than boundedness in probability is introduced the purpose of introducing the concept is to prove the boundedness and convergence of some signals in the closed loop control system 3 some important mathematical tools which play an essential role in the boundedness and convergence analysis of the closed loop system are established 4 a unifying framework is proposed to design a dynamic output feedback control law which drives the states to the origin almost surely while maintaining all the closed loop signals bounded almost surely

some properties of finite time stable stochastic nonlinear systems in this paper we study some properties of finite time stable stochastic nonlinear systems we begin by showing several continuous dependence theorems of solutions on initial values under some conditions on the coefficients of stochastic systems we then derive some regular properties of its stochastic settling time for a finite time stable stochastic nonlinear system we show continuity positive definiteness and boundedness of the expected stochastic settling time under appropriate conditions finally a lyapunov function is constructed by making use of the expectation of the stochastic settling time and the infinitesimal generator of the stochastic system defined on the lyapunov function is also given and hence resulting in a converse lyapunov theorem of finite time stochastic stability

brief paper finite time stability and instability of stochastic nonlinear systems this paper presents a new definition of finite time stability for stochastic nonlinear systems this definition involves stability in probability and finite time attractiveness in probability an important lyapunov theorem on finite time stability for stochastic nonlinear systems is established a theorem extending the stochastic lyapunov theorem is also proved moreover an example and a lemma are presented to illustrate the scope of extension a useful inequality extended from bihari s inequality is derived which plays an important role in showing the lyapunov theorem finally a lyapunov theorem on finite time instability is proved which states that almost surely globally asymptotical stability is not equivalent to finite time stability for some stochastic systems two simulation examples are given to illustrate the theoretical analysis

a minimax method for finding multiple critical points and its applications to semilinear pdes most minimax theorems in critical point theory require one to solve a two level global optimization problem and therefore are not for algorithm implementation the objective of this research is to develop numerical algorithms and corresponding mathematical theory for finding multiple saddle points in a stable way in this paper inspired by the numerical works of choi mckenna and ding costa chen and the idea to define a solution submanifold some local minimax theorems are established which require us to solve only a two level local optimization problem based on the local theory a new local numerical minimax method for finding multiple saddle points is developed the local theory is applied and the numerical method is implemented successfully to solve a class of semilinear elliptic boundary value problems for multiple solutions on some nonconvex non star shaped and multiconnected domains numerical solutions are illustrated by their graphics for visualization in a subsequent paper y li and j zhou convergence results of a minimax method for finding critical points in review we establish some convergence results for the algorithm

search extension method for multiple solutions of a nonlinear problem abstract based on the eigensystem j j of multiple solutions of nonlinear problem u u 3 0 min u 0 on are approximated a new method called the search extension method sem composed of three steps is proposed numerical examples in a square and the unit circle are presented to show the efficiency of our method

combined effects of concave and convex nonlinearities in some elliptic problems abstract this paper deals with a class of semilinear elliptic dirichlet boundary value problems where the combined effects of a sublinear and a superlinear term allow us to establish some existence and multiplicity results

correspondence comments on finite time stability theorem of stochastic nonlinear systems automatica 46 2010 2105 2108 in a recent paper chen jiao 2010 a new concept of finite time stability called stochastically finite time attractiveness is defined for a class of stochastic nonlinear systems a theorem and a corollary on the finite time stability are also proved based on lyapunov functions in this comment we will point out that there exist some errors which are fundamental to the main result of the paper

convergence rates of stochastic homogeneous systems abstract this study considers stochastic homogeneous systems and focuses on the relation between their homogeneity and the convergence rates we give a definition of stochastic homogeneous systems with weighted dilations as an extension of deterministic homogeneous systems we also introduce the stability properties such as exponential rational and finite time stability with respect to homogeneous norms as a main result we show that their homogeneous degrees imply the convergence rates of asymptotically stable systems based on the result we also deal with the stabilization of stochastic homogeneous control systems we provide a design method of homogeneous feedback laws with homogeneous control lyapunov functions which guarantees the convergence rates

a construction of interpolating wavelets on invariant sets we introduce the concept of a refinable set relative to a family of contractive mappings on a metric space and demonstrate how such sets are useful to recursively construct interpolants which have a multiscale structure the notion of a refinable set parallels that of a refinable function which is the basis of wavelet construction the interpolation points we recursively generate from a refinable set by a set theoretic multiresolution are analogous to multiresolution for functions used in wavelet construction we then use this recursive structure for the points to construct multiscale interpolants several concrete examples of refinable sets which can be used for generating interpolatory wavelets are included

stabilization of stochastic nonlinear systems driven by noise of unknown covariance this paper poses and solves a new problem of stochastic nonlinear disturbance attenuation where the task is to make the system solution bounded by a monotone function of the supremum of the covariance of the noise this is a natural stochastic counterpart of the problem of input to state stabilization in the sense of sontag 1989 our development starts with a set of new global stochastic lyapunov theorems for an exemplary class of stochastic strict feedback systems with vanishing nonlinearities where the equilibrium is preserved in the presence of noise we develop an adaptive stabilization scheme based on tuning functions that requires no a priori knowledge of a bound on the covariance next we introduce a control lyapunov function formula for stochastic disturbance attenuation finally we address optimality and solve a differential game problem with the control and the noise covariance as opposing players for strict feedback systems the resulting isaacs equation has a closed form solution

global stability and stabilization of more general stochastic nonlinear systems abstract this paper considers the global stability and stabilization of more general stochastic nonlinear systems due to the absence of the conventional assumptions e g lipschitz condition the stochastic nonlinear systems under investigation may have more than one weak solution however the most associated results are only applicable to the stochastic systems having a unique strong solution and therefore it is meaningful to refine and extend the relevant concepts and methods to the more general case in this paper the concepts of stochastic stability in the more general sense are first introduced to cover the stochastic nonlinear systems having more than one weak solution then the generalized stochastic barbashin krasovskii theorem and lasalle theorem are established which present the criterions of stochastic stability for more general stochastic nonlinear systems as one of the main contributions in this paper we rigorously prove the generalized stochastic barbashin krasovskii theorem moreover based on the generalized theorems the output feedback and state feedback stabilization are accomplished respectively for two classes of high order stochastic nonlinear systems under rather weaker assumptions comparing to the existing literature

newton s method s basin of attraction for sign changing solutions of concave and convex nonlinearities abstract in this paper we will illustrate and visualize the chaotic behavior for solving the sign changing solutions of concave convex nonlinearities we propose contractor iteration algorithm and bifurcation method to improve the computation speed and avoid local convergence of newton raphson method by this method sign changing numerical solutions will be computed and visualized in various domains

boundary element monotone iteration scheme for semilinear elliptic partial differential equations the monotone iteration scheme is a constructive method for solving a wide class of semilinear elliptic boundary value problems with the availability of a supersolution and a subsolution the iterates converge monotonically to one or two solutions of the nonlinear pde however the rates of such monotone convergence cannot be determined in general in addition when the monotone iteration scheme is implemented numerically through the boundary element method error estimates have not been analyzed in earlier studies in this paper we formulate a working assumption to obtain an exponentially fast rate of convergence this allows a margin 6 for the numerical implementation of boundary elements within the range of monotone convergence we then interrelate several approximate solutions and use the aubin nitsche lemma and the triangle inequalities to derive error estimates for the galerkin boundary element iterates with respect to the h r 0 r 2 sobolev space norms such estimates are of optimal order furthermore as a peculiarity we show that for the nonlinearities that are of separable type higher than optimal order error estimates can be obtained with respect to the mesh parameter h several examples of semilinear elliptic partial differential equations featuring different situations of existence nonexistence uniqueness multiplicity and stability are discussed computed and the graphics of their numerical solutions are illustrated

a high linking algorithm for sign changing solutions of semilinear elliptic equations where n is a bounded domain with regular boundary the operator can be replaced by a more general second order uniformly elliptical operator in divergence form the solutions we are interested in nding are of minimax type and sign changing nonlinear elliptic equations of type 1 1 arise naturally in physics engineering and mathematical biology it is known that many such equations have multiple solutions their study has attracted the attention of many pure and applied mathematicians during the past two decades let us assume that f satis es the following regularity and growth conditions a1 f x t is locally lipschitz continuous in