THE PATHWISE SOLUTION FOR A CLASS OF QUASILINEAR STOCHASTIC EQUATIONS OF EVOLUTION IN BANACH SPACE Ⅲ

This is the third part of the papers with the same title. We will discuss the problem of convergence of the semi-implicit difference scheme for a class of quasilinear SEE, which generalize the Crandall's work to the stochastic case.     

THE PATHWISE SOLUTION FOR A CLASS OF QUASILINEAR STOCHASTIC DIFFERENTIAL EQUATION IN BANACH SPACES I

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UNIQUENESS OF THE MILD SOLUTION OF SEMILINEAR STOCHASTIC EVOLUTION EQUATION IN HILBERT SPACE

In this paper, we will consider following initial value problem of semilinear stochastic evolution equation in Hilbert Space: where W(t) is a wiener process in H, H and Y are two real separable Hilbert Spaces, A is an infinitesimal generator of a strongly continuous semigroup s(t) on Y,f(t, y): [0, T]×Y→Y, and G(t, y): [0, T]×Y→L(H, Y), y_0: Ω→Y is a ramdom variable of square integrable. We apply theory of the semigroup and obtain two conclusions of uniqueness of the mild solution of (1) which include the corresponding results in [4].     

THE EXISTENCE OF THE SOLUTIONS FOR A CLASS OF SEMILINEAR EVOLUTION EQUATIONS

The author obtains the existence of the local and global solution for a class of semilinear evolution equations in a Hilbert spcae, and uses the results to prove the existence of the solution for semilinear parabolic partial differential equations in R~n.     

GLOBAL SOLVABILITY IN THE WHOLE SPACE FOR A CLASS OF FIRST ORDER QUASILINEAR HYPERBOLIC SYSTEMS

For reducible quasilinear hyperbolic systems and for general quasilinear hyperbolicsystems of diagonal form in two independent variables,some sufficient conditions satisfied bythe system itself are given tc guarantee the existence of nontrivial global smooth solutionsin the whole(t,x)space.     

STABILITY OF THE MILD SOLUTION OF STOCHASTIC NONLINEAR EVOLUTION DIFFERENTIAL EQUATIONS

In this paper, we consider the stability problem associated with the mild solutions of stochastic nonlinear evolution differential equations in Hilbert space under hypothesis which is weaker than Lipschitz condition. And the result is established by employing the Ito-type inequality and the extension of the Bihari's inequality.     

EXISTENCE OF POSITIVE SOLUTION FOR A CLASS OF QUASILINEAR ELLIPTIC EQUATION

In this paper, we investigates the existence of positive solutions for a class of quasilinear elliptic equations by using the topological degree argument.     

ITERATIVE SOLUTIONS FOR SYSTEMS OF NONLINEAR OPERATOR EQUATIONS IN BANACH SPACE

By using partial order method, the existence, uniqueness and iterative approximation of solutions for a class of systems of nonlinear operator equations in Banach space are discussed. The results obtained in this paper extend and improve recent results.     

EXISTENCE OF LOCAL SOLUTION OF INTEGRAL EQUATION IN A BANACH SPACE

In this paper, by using the Daher fixed point theorem, we prove the existence of local solutions of integral equations of Volterra type, which improves or extendes the results in [1].     

THE FIRST BOUNDARY VALUE PROBLEM FOR A CLASS OF QUASILINEAR DEGENERATE ELLIPTIC EQUATIONS

In this paper, the first boundary value problem for quasilinear equation of the form △A（u,x）＋∑i=1^m δb^i（u,x）/δxi＋c（u,x）=0,Au（u,x） ≥0is studied. By using the compensated compactness theory, some results on the existence of weak solution are established. In addition, under certain condition the uniqueness of solution is proved.     

ASYMPTOTIC BEHAVIOR OF SOLUTION FOR A CLASS OF REACTION DIFFUSION EQUATIONS

A class of initial boundary value problems for the reaction diffusion equations are considered. The asymptotic behavior of solution for the problem is obtained using the theory of differential inequality.     

REMARKS OF DIFFERENTIAL EQUATIONS ON CLOSED SUBSETS IN A BANACH SPACE

1IntroductionTheabstractCauchyproblemsfordifferentialequationsonclosedsetshavebeenstudiedbymanyauthors(see[2],[3]).In[1],H.MonchandG.H.VOnHartonestablishedinequalitiesofmeasuresofnoncolnpactnessforthesequencesofcontinuouslydifferentiablefunctiofls,thed,us…     

A PHRAGMEN－LINDELOF ALTERNATIVE FOR A CLASS OF SECOND ORDER QUASILINEAR EQUATIONS IN R~3

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ON DIRECT METHOD OF SOLUTION FOR A CLASS OF SINGULAR INTEGRAL EQUATIONS

In this article, by introducing characteristic singular integral operator and associate singular integral equations (SIEs), the authors discuss the direct method of solution for a class of singular integral equations with certain analytic inputs. They obtain both the conditions of solvability and the solutions in closed form. It is noteworthy that the method is different from the classical one that is due to Lu.     

NONLOCAL INITIAL PROBLEM FOR NONLINEAR NONAUTONOMOUS DIFFERENTIAL EQUATIONS IN A BANACH SPACE

The nonlocal initial problem for nonlinear nonautonomous evolution equati-ons in a Banach space is considered. It is assumed that the nonlinearities havethe local Lipschitz properties. The existence and uniqueness of mild solutionsare proved. Applications to integro-differential equations are discussed.The main tool in the paper is the normalizing mapping (the generalizednorm).     

GLOBAL SMOOTH SOLUTION OF CAUCHY PROBLEMS FOR A CLASS OF QUASILINEAR HYPERBOLIC SYSTEMS

In this article we prove that certain kinds of inhomogeneous terms can smoothen thesolution,for first order quasilinear hyperbolic systems globally in time provided that theinitial data are small.     

GLOBAL EXISTENCE OF SOLUTION TO NONLINEAR INTEGRODIFFERENTIAL EQUATION IN A BANACH SPACE

Using Daher's fixed point theorem, we obtain a local existence theorem, in which the assumption is weaker than That in the Theorem 2.1 in [2]. Based on this theorem, we get a global existence theorem which is an extension of certain results for ordinary differential equations.     

A NEW REGULARITY CLASS FOR THE NAVIER-STOKES EQUATIONS IN IR~n

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THE FIRST BOUNDARY VALUE PROBLEM FOR QUASILINEAR DEGENERATE PARABOLIC EQUATIONS OF SECOND ORDER IN SEVERAL SPACE VARIABLES

In this paper,we study the first boundary value problem for quasilinear equations ofthe formUnder certain conditions,the existence of generalized solutions in BV is proved by meansof the method of parabolic regularization.To do this,we need some estimates on the family{u_ε}of solutions of regularized problems and the most difficult step is to estimate|grad u_ε|L~(1+)In addition,some results on the uniqueness and stability of generalized solutions areestablished.     

L_p-DICHOTOMY OF LINEAR DIFFERENTIAL EQUATIONS IN AN ARBITRARY BANACH SPACE

The notion of Lp-dichotomy for linear differential equations with possibly unbounded operator is introduced. By help of Banach fixed point theorem sufficient conditions for the existence of bounded solutions of nonlinear differential equations with an Lp-dichotomous linear part are obtained.