Diffusion approximation for nonlinear evolutionary equations with large interaction and fast boundary fluctuation

Approximations are derived for both nonlinear heat equations and singularly perturbed nonlinear wave equations with highly oscillating random force on boundary and strong interaction. By a diffusion approximation method, if the interaction is large and the singular perturbation is small enough, the approximation of the nonlinear wave equation is an one dimensional stochastic ordinary differential equation with white noise from the boundary which is exactly the same as that of the nonlinear heat equation.     

Invariant manifolds for stochastic wave equations

In this paper, we consider a class of stochastic wave equations with nonlinear multiplicative noise. We first show that these stochastic wave equations generate random dynamical systems (or stochastic flows) by transforming the stochastic wave equations to random wave equations through a stationary random homeomorphism. Then, we establish the existence of random invariant manifolds for the random wave equations. Due to the temperedness of the nonlinearity, we obtain only local invariant manifolds no matter how large the spectral gap is unlike the deterministic cases. Based on these random dynamical systems, we prove the existence of random invariant manifolds in a tempered neighborhood of an equilibrium. Finally, we show that the images of these invariant manifolds under the inverse stationary transformation give invariant manifolds for the stochastic wave equations.     

非线性随机积分和微分方程组的解*

在本文中我们首先对具有随机定义域的连续随机算子组证明了Darbao型不动点定理。应用此定理我们给出了非线性随机Volterra积分方程组和非线性随机微分方程组的Cauchy问题解的存在性准则。这些随机方程组的极值随机解的存在性和随机比较结果也被获得。我们的定理改进和推广Tyaughn,Lakshmikantham,Lakshmikantham-Leela,DeBlast-Myjak和第一作者的相应结果。     

Multiple solutions for nonlinear operators and applications

In this paper, some new multiple solutions theorems are obtained by introducing some new concepts such as the O-bounded cone. The abstract results obtained here are applied to ordinary differential equations and nonlinear systems of differential equations.     

随机非线性方程的几个问题

研究了一类随机非线性积分方程和随机非线性微分方程的随机解．在无限维Banach空间上举出了一个反例，得到了一些新的结果。     

Representations of the optimal filter in the context of nonlinear filtering of random fields with fractional noise

The problem of nonlinear filtering of multiparameter random fields, observed in the presence of a long-range dependent spatial noise, is considered. When the observation noise is modelled by a persistent fractional Wiener sheet, several pathwise representations of the optimal filter are derived. The representations involve series of multiple stochastic integrals of different types and are particularly important since the evolution equations, satisfied by the best mean-square estimate of the signal random field, have a complicated analytical structure and fail to be proper (measure-valued) stochastic partial differential equations. Several of the above optimal filter representations involve a new family of strong martingale transforms associated to the multiparameter fractional Brownian sheet; the latter martingale family is of independent interest in fractional stochastic calculus of multiparameter random fields.     

弱拓扑下的非线性随机积分和微分方程组的解*

在本文中,我们首先对具有随机定义域的弱连续随机算子组证明了一个Darbo型随机不动点定理.利用这一定理,我们对Banach空间中关于弱拓扑的非线性随机Volterra积分方程组给出了随机解的存在性准则.作为应用,我们得到了非线性随机微分方程组的Canchy问题弱随机解的存在定理.也得到了这些随机方程组在Banach空间中关于弱拓扑的极值随机解的存在性和随机比较结果.我们的定理改进和推广了Szep,Mitchell－Smith,Cramer－Lakshmikantham,Lakshmikantham-Leela和丁的相应结果.     

随机定向收缩及其应用*

作为Altman的定向收缩理论[4,5]和Lee,Padgett的随机收缩理论[1,2]的推广,本文对非线性集值随机算子引入了随机定向收缩概念,利用这一新概念和超限归纳法,我们证明了非线性集值随机算子方程随机解的几个存在性定理.这些定理分别改进和推广了[1,2,4,5,11]中相应的结果.其次,给出了我们的结果对非线性随机积分和微分方程的某些应用.     

随机积分和微分方程在弱拓扑下解的存在性和比较结果*

在本文中,我们对非线性随机Volterra积分方程在Banach空间的弱拓扑下的随机解证明了几个存在定理.然后作为应用,我们得到了随机微分方程的弱随机解的存在定理.还得到了这些随机方程的极值随机解的存在性和随机比较定理.我们的定理改进和推广了[4,5,10,11,12]中的相应结果.     

Interval solution of nonlinear equations using linear programming

A new computational test is proposed for nonexistence of a solution to a system of nonlinear equations in a convex polyhedral regionX. The basic idea proposed here is to formulate a linear programming problem whose feasible region contains all solutions inX. Therefore, if the feasible region is empty (which can be easily checked by Phase I of the simplex method), then the system of nonlinear equations has no solution inX. The linear programming problem is formulated by surrounding the component nonlinear functions by rectangles using interval extensions. This test is much more powerful than the conventional test if the system of nonlinear equations consists of many linear terms and a relatively small number of nonlinear terms. By introducing the proposed test to interval analysis, all solutions of nonlinear equations can be found very efficently. This work was partially supported by the Japanese Ministry of Education.     

随机1－集压缩算子的随机不动点指数和随机不动点定理

在［１］中我们建立了随机拓扑度并得到系列新的随机不动点定理，本文建立了随机１－集压缩算子的随机不动点指数理论，得到一些新的随机不动点定理，为研究各类随机方程提供一些存在性原理，给出了在随机Ｈａｍｍｅｒｓｔｅｉｎ积分方程的应用．     

A class of concave operators with applications

In this paper we provide some properties of a class of concave operators and apply these results in discussing three-point boundary value problems for differential equations and nonlinear Volterra integral equations.     

Fixed point theorems for a class of nonlinear operators in Banach spaces and applications

In this paper, we study a class of nonlinear operator equations with more extensive conditions in ordered Banach spaces. By using the cone theory and Banach contraction mapping principle, the existence and uniqueness of solutions for such equations are investigated without demanding the existence of upper and lower solutions and compactness and continuity conditions. The results in this paper are applied to a class of abstract semilinear evolution equations with noncompact semigroup in Banach spaces and the initial value problems for nonlinear second-order integro-differential equations of mixed type in Banach spaces. The results obtained here improve and generalize many known results.     

Stochastic Lyapunov Functions for a System of Nonlinear Difference Equations

We study problems related to the stability of solutions of nonlinear difference equations with random perturbations of semi-Markov type. We construct Lyapunov functions for different classes of nonlinear difference equations with semi-Markov right-hand side and establish conditions for their existence.     

On an iterative algorithm with superquadratic convergence for solving nonlinear operator equations

We study an iterative method with order for solving nonlinear operator equations in Banach spaces. Algorithms for specific operator equations are built up. We present the received new results of the local and semilocal convergence, in case when the first-order divided differences of a nonlinear operator are Hölder continuous. Moreover a quadratic nonlinear majorant for a nonlinear operator, according to the conditions laid upon it, is built. A priori and a posteriori estimations of the method’s error are received. The method needs almost the same number of computations as the classical Secant method, but has a higher order of convergence. We apply our results to the numerical solving of a nonlinear boundary value problem of second-order and to the systems of nonlinear equations of large dimension.     

Integrable solutions of a mixed type operator equation

In this paper we prove the existence of integrable solutions for a generalized mixed type operator equation, which contains many key integral and functional equations appearing frequently in Mathematical literature. Our main tool is a Krasnosel’skii type fixed point theorem recently proved by Latrach and Taoudi, the first author. An existence theory for a class of nonlinear transport equations is also developed.     

Construction of Newton-like iteration methods for solving nonlinear equations

In this paper, we present a simple, and yet powerful and easily applicable scheme in constructing the Newton-like iteration formulae for the computation of the solutions of nonlinear equations. The new scheme is based on the homotopy analysis method applied to equations in general form equivalent to the nonlinear equations. It provides a tool to develop new Newton-like iteration methods or to improve the existing iteration methods which contains the well-known Newton iteration formula in logic; those all improve the Newton method. The orders of convergence and corresponding error equations of the obtained iteration formulae are derived analytically or with the help of Maple. Some numerical tests are given to support the theory developed in this paper.     

Random nonlinear equations and monotonic nonlinearities

The question of existence of random solutions of nonlinear random operator equations involving a monotonic nonlinearity is discussed.     

A unified approach for constructing fast two-step Newton-like methods

We introduce some new very general ways of constructing fast two-step Newton-like methods to approximate a locally unique solution of a nonlinear operator equation in a Banach space setting. We provide existence-uniqueness theorems as well as an error analysis for the iterations involved using Newton-Kantorovich-type hypotheses and the majorant method. Our results depend on the existence of a Lipschitz function defined on a closed ball centered at a certain point and of a fixed radius and with values into the positive real axis. Special choices of this function lead to favorable comparisons with results already in the literature. Some applications to the solution of nonlinear integral equations appearing in radiative transfer as well as to the solution of integral equations of Uryson-type are also provided.     

Weak convergence of approximate solutions of random equations

Approximations of random operator equations are considered where the stochastic inputs and the underlying deterministic equation are approximated simultaneously. The main convergence result asserts that, under reasonable and verifiable assumptions, a sequence of weak solutions of approximate random equations converges weakly to a weak solution of the original equation. It is shown that this theorem extends and unifies results already known. We apply our theory to approximations of random differential equations involving stochastic processes with discontinuous paths and to projection methods for nonlinear random Hammerstein integral equations in spaces of integrable functions.