Showing papers on "C0-semigroup published in 2000"

Backward stochastic differential equations and partial differential equations with quadratic growth

Abstract: We provide existence, comparison and stability results for one- dimensional backward stochastic differential equations (BSDEs) when the coefficient (or generator) $F(t,Y, Z)$ is continuous and has a quadratic growth in $Z$ and the terminal condition is boundede also give, in this framework, the links between the solutions of BSDEs set on a diffusion and viscosity or Sobolev solutions of the corresponding semilinear partial differential equations

On the Stability of Functional Equations and a Problem of Ulam

Abstract: In this paper, we study the stability of functional equations that has its origins with S. M. Ulam, who posed the fundamental problem 60 years ago and with D. H. Hyers, who gave the first significant partial solution in 1941. In particular, during the last two decades, the notion of stability of functional equations has evolved into an area of continuing research from both pure and applied viewpoints. Both classical results and current research are presented in a unified and self-contained fashion. In addition, related problems are investigated. Some of the applications deal with nonlinear equations in Banach spaces and complementarity theory.

Exponential attractors for a nonlinear reaction-diffusion system in ?

Abstract: We give in this Note a construction of exponential attractors for a class of operators in Banach spaces (and not in Hilbert spaces only as it is the case for the classical constructions). We then apply this result to a reaction-diffusion system in R 3 .

Spectrum-preserving linear mappings between banach algebras or jordan–banach algebras

Abstract: Spectrum-preserving linear mappings were studied for the first time by G. Frobenius [18]. He proved that a linear mapping Φ from Mn([Copf ]) onto Mn([Copf ]) which preserves the spectrum has one of the forms Φ(x) = axa−1 or Φ(x) = atxa−1, for some invertible matrix a. (Incidentally the hypothesis that Φ is onto is superfluous; see Proposition 2.1(i).) This result was extended by J. Dieudonne [17] supposing Φ onto and satisfying SpΦ(x) ⊂ Sp x, for every n × n matrix x.Several results of M. Nagasawa, S. Banach and M. Stone, R. V. Kadison, A. Gleason and J. P. Kahane and W. Żelazko led I. Kaplansky in [22] to the following problem: given two Banach algebras with unit and Φ a linear mapping from A into B such that Φ(1) = 1 and SpΦ(x) ⊂ Sp x, for every x ∈ A, is it true that Φ is a Jordan morphism? With this general formulation, this question cannot be true (see [2], p. 28).

Stochastic convolution in separable Banach spaces and the stochastic linear Cauchy problem

Abstract: Let H be a separable real Hilbert space and let E be a separable real Banach space. In this paper we develop a general theory of stochastic convolution of L(H,E)− valued functions with respect to a cylindrical Wiener process {WH t }t∈[0,T ] with CameronMartin space H. This theory is applied to obtain necessary and sufficient conditions for the existence of a weak solution of the stochastic abstract Cauchy problem (ACP ) dXt = AXt dt+B dW H t (t ∈ [0, T ]), X0 = 0 almost surely, where A is the generator of a C0−semigroup {S(t)}t≥0 of bounded linear operators on E and B ∈ L(H,E) is a bounded linear operator. We further show that whenever a weak solution exists, it is unique, and given by a stochastic convolution

Stochastic functional partial differential equations: existence, uniqueness and asymptotic decay property

Abstract: Existence and uniqueness of strong solutions for a class of stochastic functional differential equations in Hilbert spaces are established. Sufficient conditions which guarantee the transference of mean–square and pathwise exponential stability from stochastic partial differential equations to stochastic functional partial differential equations are studied. The stability results derived are also applied to stochastic ordinary differential equations with hereditary characteristics. In particular, as a direct consequence our main results improve some of those by Mao & Shah in which it was proved that under certain conditions pathwise exponential stability is transferred from non–delay equations to delay equations if the constant time–lag appearing in the problem is sufficiently small, while in our treatment the transference actually holds for arbitrary bounded delay variables not only in finite but in infinite dimensions.

Normal forms for semilinear functional differential equations in Banach spaces and applications. Part II

Abstract: A normal form theory for functional differential equations in Banach spaces of retarded type is addressed. The theory is based on a formal adjoint theory for the linearized equation at an equilibrium and on the existence of center manifolds for perturbed inhomogeneous equations, established in the first part of this work under weaker hypotheses than those that usually appear in the literature. Based on these results, an algorithm to compute normal forms on finite dimensional invariant manifolds of the origin is presented. Such normal forms are important in obtaining the ordinary differential equation giving the flow on center manifolds explicitly in terms of the original functional differential equation. Applications to Bogdanov-Takens and Hopf bifurcations are presented.

The Accuracy of Gaussian Approximation in Banach Spaces

Abstract: Let B be a real separable Banach space with norm || · || = || · || B . Suppose that X, X 1, X 2, … ∈ B are independent and identically distributed (i.i.d.) random elements (r.e.’s) taking values in B. Furthermore, assume that EX = 0 and that there exists a zero-mean Gaussian r.e. Y ∈ B such that the covariances of X and Y coincide.

The Banach Envelopes of Besov and Triebel-Lizorkin Spaces and Applications to Partial Differential Equations

Abstract: With “hat” denoting the Banach envelope (of a quasi-Banach space) we prove that Open image in new window if 0

Analytical solution of a system of homogeneous delay differential equations via the Lambert function

Abstract: A new approach to obtain an analytic solution for delay differential equations (DDE) based on the concept of Lambert functions is presented in this paper. The similarity of the results with the concept of the state transition matrix in ordinary differential equations enables the approach to be used for general classes of linear delay differential equations including the matrix form of DDE. Stability criteria for delay equations in different cases are studied and the results are presented in the paper.

Iterative method for solutions and coupled quasi-solutions of nonlinear integro-differential equations of mixed type in Banach spaces

Abstract: We consider, in this paper, the existence of solutions, coupled minimal and maximal quasi-solutions and iterative approximation of the unique solution for the following initial value problems (IVP) of the nonlinear integro-di erential equations of mixed type in ordered Banach spaces E: u′ = f(t; u; u; Tu; Su); t ∈ I; u(t0) = x0; (1) where I = [t0; t0 + a] (a? 0), x0 ∈E, f∈C[I × E × E × E × E; E] and

Invertibility and the fredholm property of difference operators

Abstract: We obtain conditions for the invertibility and the Fredholm property of the difference operator (Dx)(n)=x(n) -U(n)x(n − 1),n e ℤ, in the Banach space l p (ℤ, X),p e [1, ∞], of vector sequences, whereX is a Banach space andU is a bounded operator function.

An inverse problem for differential operator pencils

Abstract: An inverse spectral problem of recovering a second-order differential operator pencil on the half-line is investigated. A uniqueness theorem is proved; necessary and sufficient conditions for the solubility, and an algorithm for the solution of the inverse problem are obtained. Relations with inverse problems for partial differential equations are pointed out.

Transfer Operators for Coupled Analytic Maps

Abstract: We consider analytically coupled circle maps (uniformly expanding and analytic) on the ${\mathbb Z}^d$-lattice with exponentially decaying interaction. We introduce Banach spaces for the infinite-dimensional system that include measures whose finite-dimensional marginals have analytic, exponentially bounded densities. Using residue calculus and ‘cluster expansion’-like techniques we define transfer operators on these Banach spaces. We get a unique (in the considered Banach spaces) probability measure that exhibits exponential decay of correlations.

Controllability of second-order differential inclusions in Banach spaces with nonlocal conditions

Abstract: In this paper, we establish sufficient conditions for the controllability ofsecond-order differential inclusions in Banach spaces with nonlocalconditions. We rely on a fixed-point theorem for condensing maps due toMartelli.

Spectral value sets of closed linear operators

Abstract: We study how the spectrum of a closed linear operator on a complex Banach space changes under affine perturbations of the form A ↝ AΔ = A + DΔE. Here A, D and E are given linear operators, whereas Δ is an unknown bounded linear operator that parametrizes the possibly unbounded perturbation DΔE. The union of the spectra of the perturbed operators AΔ, with the norm of Δ smaller than a given δ > 0, is called the spectral value set of A at level δ. In this paper we extend a known characterization of these sets for the matrix case to infinite dimensions, and in so doing present a framework that allows for unbounded perturbations of closed linear operators on Banach spaces. The results will be illustrated by applying them to a delay system with uncertain parameters and to a partial differential equation with a perturbed boundary condition.

Spectral Analysis of Fokker--Planck and Related Operators Arising From Linear Stochastic Differential Equations

Abstract: We study spectral properties of certain families of linear second-order differential operators arising from linear stochastic differential equations. We construct a basis in the Hilbert space of square-integrable functions using modified Hermite polynomials, and obtain a representation for these operators from which their eigenvalues and eigenfunctions can be computed. In particular, we completely describe the spectrum of the Fokker--Planck operator on an appropriate invariant subspace of rapidly decaying functions. The eigenvalues of the Fokker--Planck operator provide information about the speed of convergence of the corresponding probability distribution to steady state, which is important for stochastic estimation and control applications. We show that the operator families under consideration can be realized as solutions of differential equations in the double bracket form on an operator Lie algebra, which leads to a simple expression for the flow of their eigenfunctions.

Implicit operator differential equations and applications to electrodynamics

Abstract: A cylindrical waveguide with layered dispersive medium is considered. An implicit operator differential equation of second order is obtained by separating variables in the corresponding boundary-value problem for Maxwell's equations. Existence and uniqueness theorems are proved and explicit formulae for solutions are given. The spectral theory of operators and operator sheaves are used.

Continuity and Representation of Gaussian Mehler Semigroups

Abstract: We present sufficient conditions on a Gaussian Mehler semigroup on a reflexive Banach space Eto be induced by a single positive symmetric operator Q \in \(Q \in \mathcal{L}(E^* ,E)\), and give a counterexample which shows that this representation theorem is false in every nonreflexive Banach space with a Schauder basis. We also show that the transition semigroup of a Gaussian Mehler semigroup on a separable Banach space Eacts in a pointwise continuous way, uniformly on compact subsets of E, in the space BUC(E) of bounded uniformly continuous real-valued funtions on E. The transition semigroup is shown to be strongly continuous on BUC(E) if and only if S(t) = Ifor all t⩽ 0

The Cauchy problem for nonlinear abstract functional differential equations with infinite delay

Abstract: In this paper, we use existence families and admissible phase spaces to study the Cauchy problem for a class of nonlinear abstract functional differential equations with infinite delay in a unified way. Our main results concern the existence and uniqueness of mild solutions, strong solutions, and classical solutions.

The Essential Spectrum of a System of Singular Ordinary Differential Operators of Mixed Order. Part II: The Generalization of Kako's Problem

Abstract: A system of ordinary differential equations of mixed order on an interval (0, r0) is considered, where some coefficients are singular at 0. Special cases have been dealt with by Kako, where the essential spectrum of an operator associated with a linearized MHD model was calculated, and more recently by Hardt, Mennicken and Naboko. In both papers this operator is a selfadjoint extension of an operator on sufficiently smooth functions. The approach in the present paper is different in that a suitable operator associated with the given system of ordinary differential equations is explicitly defined as the closure of an operator defined on sufficiently smooth functions. This closed operator can be written as a sum of a selfadjoint operator and a bounded operator. It is shown that its essential spectrum is a nonempty compact subset of ℂ, and formulas for the calculation of the essential spectrum in terms of the coefficients are given.

Sobolev type spaces associated with jacobi differential operators

Abstract: Sobolev type spaces , associated with the Jacobi differential operators are studied. Some properties including completeness and inclusion are proved. Next, the Jacobi potential is defined as a pseudodifferential operator associated with a precise symbol. The operator is extended to a space of distributions. The Lp -space of all such Jacobi potential is defined. It is proved that this space is a Banach space, for under some conditions on s t and α. Also, it is shown that solutions of certain equations involving the Jacobi differential operator belong to these spaces.

Multivariational Inequalities and Operator Inclusions in Banach Spaces with Mappings of the Class (S)

Abstract: We prove theorems on the existence of solutions of variational inequalities and operator inclusions in Banach spaces with multivalued mappings of the class (S)+. We justify the method of penalty operators for variational inequalities.