Random Attractors for Stochastic Reaction-Diffusion Equations with Distribution Derivatives on Unbounded Domains

In this paper, we prove the existence of random attractors for a stochastic reaction-diffusion equation with distribution derivatives on unbounded domains. The nonlinearity is dissipative for large values of the state and the stochastic nature of the equation appears spatially distributed temporal white noise. The stochastic reaction-diffusion equation is recast as a continuous random dynamical system and asymptotic compactness for this demonstrated by using uniform estimates far-field values of solutions. The results are new and appear to be optimal.

ATTRACTORS FOR THE GINZBURG-LANDAU-BBM EQUATIONS IN AN UNBOUNDED DOMAIN

By the interpolating inequality and a priori estimates in the weighted space,the existence of the global solutions for the Ginzburg-Landau equation coupled with the BBM equation in an unbounded domain is considered, and the existence of the maximal attractor is obtained.

INFINITE ELEMENT METHOD FOR PROBLEMS ON UNBOUNDED AND MULTIPLY CONNECTED DOMAINS

The author studies the infinite element method for the boundary value problems of second order elliptic equations on unbounded and multiply connected domains. The author makes a partition of the domain into infinite number of elements. Without dividing the domain, as usual, into a bounded one and an exterior one, he derives an initial value problem of an ordinary differential equation for the combined stiffness matrix, then obtains the approximate solution with a small amount of computer work. Numerical examples are given.

ON SOLUTIONS OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS,WITH UNBOUNDED STOPPING TIMES AS TERMINAL AND WITH NON-LIPSCHITZ COEFFICIENTS,AND PROBABILISTIC INTERPRETATION OF QUASI-LINEAR ELLIPTIC TYPE INTEGRO-DIFFERENTIAL EQUATIO

The existence and uniqueness of solutions to backward stochastic differential equations with jumps and with unbounded stopping time as terminal under the non_Lipschitz condition are obtained. The convergence of solutions and the continuous dependence of solutions on parameters are also derived. Then the probabilistic interpretation of solutions to some kinds of quasi_linear elliptic type integro_differential equations is obtained.

Global Topological Linearization with Unbounded Nonlinear Term

In 1960s, Hartman and Grobman pointed out that if all eigenvalues of a matrix A have no zero real part and f(x) is small Lipchitzian, then x′=Ax+f(x) can be locally linearized on a neighborhood of the origin. Later, the above result was generalized to global under the condition that f(x) is a bounded function. In this paper, we delete the condition that f(x) is a bounded function, and prove that if f(x) has suitable structure, then x′=Ax+f(x) can be linearized.

Homotopy Continuous Method for Weak Efficient Solution of Multiobjective Optimization Problem with Feasible Set Unbounded Condition

In this paper, we propose a homotopy continuous method (HCM) for solving a weak efficient solution of multiobjective optimization problem (MOP) with feasible set unbounded condition, which is arising in Economical Distributions, Engineering Decisions, Resource Allocations and other field of mathematical economics and engineering problems. Under the suitable assumption, it is proved to globally converge to a weak efficient solution of (MOP), if its x-branch has no weak infinite solution.

On the Stable Method Computing Values of Unbounded Operators

Unbounded operators can transform arbitrarily small vectors into arbitrarily large vectors—a phenomenon known as instability. Stabilization methods strive to approximate a value of an unbounded operator by applying a family of bounded operators to rough approximate data that do not necessarily lie within the domain of unbounded operator. In this paper we shall be concerned with the stable method of computing values of unbounded operators having perturbations and the stability is established for this method.

On the Method of Multiplier-enlargement and Approximation of Unbounded Continuous Functions

By combining the classical appropriate functions “1, x, x 2” with the method of multiplier enlargement, this paper establishes a theorem to approximate any unbounded continuous functions with modified positive linear operators. As an example, Hermite Fejér interpolation polynomial operators are analysed and studied, and a general conclusion is obtained.

THE EXISTENCE OF INFINITELY MANY SO UTIONS OF QUASILINEAR PARTIAL DIFFERENTIAL EQUATIONS IN UNBOUNDED DOMAINS

In this paper, we use the concentration-compactness principle together with the Mountain Pass Lemma to get the existence of nontrivial solutions and the existence of infinitely many solutions of the problem need not be compact operators from E to R~1.

Error Analysis of ERM Algorithm with Unbounded and Non-Identical Sampling

A standard assumption in the literature of learning theory is the samples which are drawn independently from an identical distribution with a uniform bounded output. This excludes the common case with Gaussian distribution. In this paper we extend these assumptions to a general case. To be precise, samples are drawn from a sequence of unbounded and non-identical probability distributions. By drift error analysis and Bennett inequality for the unbounded random variables, we derive a satisfactory learning rate for the ERM algorithm.

BOUNDARY VALUE PROBLEMS FOR FULLY NONLINEAR UNIFORMLY ELLIPTIC EQUATIONS OF SECOND IN UNBOUNDED DOMAINS

We consider fully nonlinear equations of the formF（z,u,Du,D2u） = F（x,y,u,uz,uy,uzz,uzy,uyy） = 0 （1）in unbounded open subset G = R2\Ωof the plane R2,where F is a real continuous function on U = G×R×R2×R3 and Ω= Ωi,Ωi is a simply connected region （i=1,2,"",N） . We assume the function F hascontinuous partial der ivatives Fuzz, Fuzy, Fuyy. on U.For a real function r C（ G） a real function u（x,y） is called a solution of （1） satisfyingu = r on G,（2）if there exists a constant P0】2 such that u C1 （ ） W2.pLoc0 （G） satisfies （1） almost everywhere and （2）in the common sense.The method for treating the above exterior Dirichlet problem in a given unbounded region is as fol-

EXISTENCE AND STABILITY OF PERIODIC AND ALMOST PERIODIC SOLUTIONS TO THE BOUSSINESQ SYSTEM IN UNBOUNDED DOMAINS

In this paper we investigate the existence and stability of periodic solutions(on a half-line R_(+))and almost periodic solutions on the whole line time-axis R to the Boussinesq system on several classes of unbounded domains of R^(n) in the framework of interpolation spaces.For the linear Boussinesq system we combine the L^(p)—L^(q)-smoothing estimates and interpolation functors to prove the existence of bounded mild solutions.Then,we prove the existence of periodic solutions by invoking Massera’s principle.We also prove the existence of almost periodic solutions.Then we use the results of the linear Boussinesq system to establish the existence,uniqueness and stability of the small periodic and almost periodic solutions to the Boussinesq system using fixed point arguments and interpolation spaces.Our results cover and extend the previous ones obtained in[13,34,38].

ON NONLINEAR HYPERBOLIC EQUATION IN UNBOUNDED DOMAIN

The following nonlinear hyperbolic equation is discussed in this paper:where A= -? + Iandx∈Rn. The model comes from the transverse deflection equation of an extensible beam. We prove that there exists a unique local solution of the above equation as M depends on x.

THE EXPONENTIAL PROPERTY OF SOLUTIONS BOUNDED FROM BELOW TO DEGENERATE EQUATIONS IN UNBOUNDED DOMAINS

This paper is focused on studying the structure of solutions bounded from below to degenerate elliptic equations with Neumann and Dirichlet boundary conditions in unbounded domains.After establishing the weak maximum principles,the global boundary Holder estimates and the boundary Harnack inequalities of the equations,we show that all solutions bounded from below are linear combinations of two special solutions(exponential growth at one end and exponential decay at the other)with a bounded solution to the degenerate equations.

Asymptotic Stability of Neutral Differential Equations with Unbounded Delay

Consider the neutral differential equation with positive and negative coefficients and unbounded delay ddt[x(t)-P(t)x(h(t))]+Q(t)x(q(t))-R(t)x(r(t))=0, t≥t 0, where P(t)∈C([t 0, ∞), R), Q(t), R(t)∈C([t 0, ∞), [WTHZ]R +), and h, q, r: [t 0, ∞)→R are continuously differentiable and strictly increasing, h(t)<t, q(t)<t, r(t)<t for all t≥t 0. In this paper, the authors obtain sufficient conditions for the zero solution of this equation with unbounded delay to be uniformly stable as well as asymptotically stable. [WTH1X]

NORM SUMMABILITY OF NRLUND LOGARITHMIC MEANS ON UNBOUNDED VILENKIN GROUPS

The （Noerlund） logarithmic means of the Fourier series is：tnf=1/ln ^n-1∑k=1 Skf/n-k, where ln=^n-1∑k=1 1/k In general, the Fej6r （C, 1） means have better properties than the logarithmic ones. We compare them and show that in the case of some unbounded Vilenkin systems the situation changes.

PROBABILITY INEQUALITIES FOR SUMS OF INDEPENDENT UNBOUNDED RANDOM VARIABLES

The tail probability inequalities for the sum of independent unbounded random variables on a probability space (Omega, T, P) were studied and a new method was proposed to treat the sum of independent unbounded random variables by truncating the original probability space (Omega, T, P). The probability exponential inequalities for sums of the results, some independent unbounded random variables were given. As applications of interesting examples were given. The examples show that the method proposed in the paper and the results of the paper are quite useful in the study of the large sample properties of the sums of independent unbounded random variables.

Equivalence Relation between Initial Values and Solutions for Evolution p-Laplacian Equation in Unbounded Space

In this paper,an equivalence relation between the ω-limit set of initial values and the ω-limit set of solutions is established for the Cauchy problem of evolution p-Laplacian equation in the unbounded space Yσ(RN).To overcome the difficulties caused by the nonlinearity of the equation and the unbounded solutions,we establish the propagation estimate and the growth estimate for the solutions.It will be demonstrated that the equivalence relation can be used to study the asymptotic behavior of solutions.

Approximate Method of Riemann-Hilbert Problem for Elliptic Complex Equations of First Order in Multiply Connected Unbounded Domains

In this article, we discuss the approximate method of solving the Riemann-Hilbert boundary value problem for nonlinear uniformly elliptic complex equation of first order (0.1) with the boundary conditions (0.2) in a multiply connected unbounded domain D, the above boundary value problem will be called Problem A. If the complex Equation (0.1) satisfies the conditions similar to Condition C of (1.1), and the boundary condition (0.2) satisfies the conditions similar to (1.5), then we can obtain approximate solutions of the boundary value problems (0.1) and (0.2). Moreover the error estimates of approximate solutions for the boundary value problem is also given. The boundary value problem possesses many applications in mechanics and physics etc., for instance from (5.114) and (5.115), Chapter VI, [1], we see that Problem A of (0.1) possesses the important application to the shell and elasticity.

Asymptotic Behavior for A Class of Non-autonomous Nonclassical Parabolic Equations with Delay on Unbounded Domain

In this article,we investigate the longtime behavior for the following nonautonomous nonclassical parabolic equations on unbounded domain ut−∆ut−∆u+λu=f(x,u(x,t−ρ(t)))+g(x,t).Under some suitable conditions on the delay term f and the non-autonomous forcing term g,we prove the existence of uniform attractors in Banach space CH1(RN)for the multivalued process generated by non-autonomous nonclassical parabolic equations with delays in unbounded domain.