Differential Equations In Hilbert Spaces Research Articles

Stability and Convergence Results for Itô-Type Parabolic Partial Differential Equations in Hilbert Spaces

In this article, we consider the following Itô-type stochastic parabolic partial differential equation where A and C are families of nonlinear operators in Hilbert spaces and w t is a Hilbert valued Wiener process. Under certain regularity conditions of operators A and C to guarantee a strong solution process of this partial differential equation, the concept of vector Lyapunov-like functional technique coupled with partial differential inequalities are utilized to develop a comparison principle to investigate various types of stability and convergence in the pth moment and in probability of the solution process of the system. An example is provided to demonstrate the significance of the results developed in this article.

Stability analysis for a class of functional differential equations and applications

The problem of Lyapunov stability for functional differential equations in Hilbert spaces is studied. The system to be considered is non-autonomous and the delay is time-varying. Known results on this problem are based on the Gronwall inequality yielding relative conservative bounds on nonlinear perturbations. In this paper, using more general Lyapunov–Krasovskii functional, neither model variable transformation nor bounding restriction on nonlinear perturbations is required to obtain improved conditions for the global exponential stability of the system. The conditions given in terms of the solution of standard Riccati differential equations allow to compute simultaneously the two bounds that characterize the stability rate of the solution. The proposed method can be easily applied to some control problems of nonlinear non-autonomous control time-delay systems.

Approximation of evolution differential equations in scales of Hilbert spaces

Evolution differential equations in Hilbert spaces arise in numerous problems in different fields of natural sciences. In these problems, it is necessary to approximate infinite-dimensional systems by finite-dimensional equations (or by systems of ordinary differential equations). In the approximation of the evolution equations in scales of Hilbert spaces, high efficiency can be attained by using computers. In the present paper, we consider the problem of estimating the time of existence of solutions and approximations in the chosen scales of spaces. The results obtained allow us to develop constructive methods for obtaining such estimates in specific cases. We note that the problem of estimating the solution existence time in scales of spaces is actual both in the case of linear equations, where global solvability is possible, and in the case of nonlinear systems, where the existence time of the solution is finite. Let H be a Hilbert space. Throughout the paper, the Hilbert spaces are assumed to be separable and infinite-dimensional. We let ek denote an orthonormal basis in H. We introduce a continuous generally nonlinear operator A : D → H ,w hereD is a Hilbert space densely and continuously embedded in H and satisfying the condition {ek }⊂ D. We consider the Cauchy problem u � (t )= Au(t) ,t ∈ (0 ,T ) (1) u(0) = ϕ, ϕ ∈ H.

Nonlinear maps of convex sets in Hilbert spaces with application to kinetic equations

Let \( \mathcal{H} \) be a separable Hilbert space, \( \mathcal{U} \subseteq \mathcal{H} \) an open convex subset, and f: \( \mathcal{U} \to \mathcal{H} \) a smooth map. Let Ω be an open convex set in \( \mathcal{H} \) with \( \bar \Omega \subseteq \mathcal{U} \), where \( \bar \Omega \) denotes the closure of Ω in \( \mathcal{H} \). We consider the following questions. First, in case f is Lipschitz, find sufficient conditions such that for ɛ > 0 sufficiently small, depending only on Lip(f), the image of Ω by I + ɛf, (I + ɛf)(Ω), is convex. Second, suppose df(u): \( \mathcal{H} \to \mathcal{H} \) is symmetrizable with σ(df(u)) ⊆ (0,∞), for all u ∈ \( \mathcal{U} \), where σ(df(u)) denotes the spectrum of df(u). Find sufficient conditions so that the image f(Ω) is convex. We establish results addressing both questions illustrating our assumptions and results with simple examples. We also show how our first main result immediately apply to provide an invariance principle for finite difference schemes for nonlinear ordinary differential equations in Hilbert spaces. The main application of the theory developed in this paper concerns our second result and provides an invariance principle for certain convex sets in an L2-space under the flow of a class of kinetic transport equations so called BGK model.

Stochastic Retarded Evolution Equations: Green Operators, Convolutions, and Solutions

In this work, we shall investigate solution (strong, weak and mild) processes and relevant properties of stochastic convolutions for a class of stochastic retarded differential equations in Hilbert spaces. We introduce a strongly continuous one-parameter family of bounded linear operators which will completely describe the corresponding deterministic systematical dynamics with time delays. This family, which constitutes the fundamental solutions (Green's operators) of our stochastic retarded systems, is applied subsequently to define mild solutions of the stochastic retarded differential equations considered. The relations among strong, weak and mild solutions are explored. By virtue of a strong solution approximation method, Burkholder–Davis–Gundy's type of inequalities for stochastic convolutions are established.

Stochastic Differential Equations with Non-Lipschitz Coefficients in Hilbert Spaces

In this article, we discuss the successive approximations problem for the solutions of the semilinear stochastic differential equations in Hilbert spaces with cylindrical Wiener processes under some conditions which are weaker than the Lipschitz one. We establish the existence and the uniqueness of the solution and additionally, in our framework we consider a limiting problem for the mild solution. It is shown that the mild solution tends to the solution of the stochastic differential equation of Itô type in finite dimensional space.

Quadratic BSDEs with Random Terminal Time and Elliptic PDEs in Infinite Dimension

In this paper we study one dimensional backward stochastic differential equations (BSDEs) with random terminal time not necessarily bounded or finite when the generator $F(t,Y,Z)$ has a quadratic growth in $Z$. We provide existence and uniqueness of a bounded solution of such BSDEs and, in the case of infinite horizon, regular dependence on parameters. The obtained results are then applied to prove existence and uniqueness of a mild solution to elliptic partial differential equations in Hilbert spaces. Finally we show an application to a control problem.

Differentiability of Backward Stochastic Differential Equations in Hilbert Spaces with Monotone Generators

The aim of the present paper is to study the regularity properties of the solution of a backward stochastic differential equation with a monotone generator in infinite dimension. We show some applications to the nonlinear Kolmogorov equation and to stochastic optimal control.

Approximate controllability of nonlinear impulsive differential systems

Many practical systems in physical and biological sciences have impulsive dynamical be- haviours during the evolution process which can be modeled by impulsive differential equations. This paper studies the approximate controllability issue for nonlinear impulsive differential and neutral functional differential equations in Hilbert spaces. Based on the semigroup theory and fixed point approach, sufficient conditions for approximate controllability of impulsive differential and neutral functional differential equations are established. Finally, two examples are presented to illustrate the utility of the proposed result. The results improve some recent results.

A System of Differential-Operator Equations of Different Orders in Hilbert Spaces

We consider an abstract Cauchy problem for a system of nonhomogeneous abstract differential equations in Hilbert spaces. The “main” equation is of the second order and “boundary” equations are of the first order. Existence of a solution is proved. Application to mixed (initial boundary-value) problems for one-dimensional second order hyperbolic equations and for fourth order PDEs with the time derivative in boundary conditions has been shown.

BSDEs with stochastic Lipschitz condition and quadratic PDEs in Hilbert spaces

This paper is devoted to real valued backward stochastic differential equations (BSDEs for short) with generators which satisfy a stochastic Lipschitz condition involving BMO martingales. This framework arises naturally when looking at the BSDE satisfied by the gradient of the solution to a BSDE with quadratic growth in Z . We first prove an existence and uniqueness result from which we deduce the differentiability with respect to parameters of solutions to quadratic BSDEs. Finally, we apply these results to prove the existence and uniqueness of a mild solution to a parabolic partial differential equation in Hilbert space with nonlinearity having quadratic growth in the gradient of the solution.

Approximation Schemes for Stochastic Differential Equations in Hilbert Space

For solutions of Itô–Volterra equations and semilinear evolution-type equations we consider approximations via the Milstein scheme, approximations by finite-dimensional processes, and approximations by solutions of stochastic differential equations (SDEs) with bounded coefficients. We prove mean-square convergence for finite-dimensional approximations and establish results on the rate of mean-square convergence for two remaining types of approximation.

On nondensely defined semilinear stochastic functional differential equations with nonlocal conditions

The nonlinear alternative of Leray-Schauder type is used to investigate the existence of solutions for first-order semilinear stochastic functional differential equations in Hilbert spaces.

AN INFINITE FACTOR MODEL FOR CREDIT RISK

The defaultable term structure is modeled using stochastic differential equations in Hilbert spaces. This leads to an infinite dimensional model, which is free of arbitrage under a certain drift condition. Furthermore, the model is extended to incorporate ratings based on a Markov chain.

Perron's method and the method of relaxed limits for ``unbounded'' PDE in Hilbert spaces

We prove that Perron's method and the method of half-relaxed limits of Barles-Perthame works for the so called B-continuous viscosity solutions of a large class of fully nonlinear unbounded partiel differential equations in Hilbert spaces. Perron's method extends the existence of B-continuous viscosity solutions to many new equations that are not of Bellman type. The method of half-relaxed limits allows limiting operations with viscosity solutions without any a priori estimates. Possible applications of the method of half relaxed limits to large derivations, singular perturbation problems, and convergence of finite-dimensional approximations are discussed

On the strong stability of differential equations and difference schemes

AbstractPaper deals with the construction of a priori estimates for the solution of Cauchy problem for an abstract linear differential equation in Hilbert space under perturbations of the initial condition, right‐hand side, and operators of the problem. It is shown that a priori estimates of strong stability can be obtained directly on the basis of various a priori estimates for the solution of the Cauchy problem. The perturbations of the operators of the problem are estimated in the corresponding operator norms. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Stability of nonautonomous differential equations in Hilbert spaces

We introduce a large class of nonautonomous linear differential equations v ′ = A ( t ) v in Hilbert spaces, for which the asymptotic stability of the zero solution, with all Lyapunov exponents of the linear equation negative, persists in v ′ = A ( t ) v + f ( t , v ) under sufficiently small perturbations f. This class of equations, which we call Lyapunov regular, is introduced here inspired in the classical regularity theory of Lyapunov developed for finite-dimensional spaces, that is nowadays apparently overlooked in the theory of differential equations. Our study is based on a detailed analysis of the Lyapunov exponents. Essentially, the equation v ′ = A ( t ) v is Lyapunov regular if for every k the limit of Γ ( t ) 1 / t as t → ∞ exists, where Γ ( t ) is any k-volume defined by solutions v 1 ( t ) , … , v k ( t ) . We note that the class of Lyapunov regular linear equations is much larger than the class of uniformly asymptotically stable equations.

Asymptotic Equivalence of Triangular Differential Equations in Hilbert Spaces

In this article, we study conditions for the asymptotic equivalence of differential equations in Hilbert spaces. We also discuss the relationship between the properties of solutions of differential equations of triangular form and those of truncated differential equations.

A Series Approach to Stochastic Differential Equations with Infinite Dimensional Noise

This paper considers semilinear stochastic differential equations in Hilbert spaces with Lipschitz nonlinearities and with the noise terms driven by sequences of independent scalar Wiener processes (Brownian motions). The interpretation of such equations requires a stochastic integral. By means of a series of Ito integrals, an elementary and direct construction of a Hilbert space valued stochastic integral with respect to a sequence of independent scalar Wiener processes is given. As an application, existence and strong and weak uniqueness for the stochastic differential equation are shown by exploiting the series construction of the integral.

Multiple solutions for impulsive semilinear functional and neutral functional differential equations in Hilbert space

The well-known Krasnoselskii twin fixed point theorem is used to investigate the existence of mild solutions for first- and second-order impulsive semilinear functional and neutral functional differential equations in Hilbert spaces.