Showing papers on "Banach space published in 1996"

Monotone operators in Banach space and nonlinear partial differential equations

Abstract: PDE examples by type Linear problems...An introduction Nonlinear stationary problems Nonlinear evolution problems Accretive operators and nonlinear Cauchy problems Appendix Bibliography Index.

The Asymptotic Behaviour of Semigroups of Linear Operators

Abstract: 1 Spectral bound and growth bound- 11 C0-semigroups and the abstract Cauchy problem- 12 The spectral bound and growth bound of a semigroup- 13 The Laplace transform and its complex inversion- 14 Positive semigroups- Notes- 2 Spectral mapping theorems- 21 The spectral mapping theorem for the point spectrum- 22 The spectral mapping theorems of Greiner and Gearhart- 23 Eventually uniformly continuous semigroups- 24 Groups of non-quasianalytic growth- 25 Latushkin - Montgomery-Smith theory- Notes- 3 Uniform exponential stability- 31 The theorem of Datko and Pazy- 32 The theorem of Rolewicz- 33 Characterization by convolutions- 34 Characterization by almost periodic functions- 35 Positive semigroups on Lp-spaces- 36 The essential spectrum- Notes Ill- 4 Boundedness of the resolvent- 41 The convexity theorem of Weis and Wrobel- 42 Stability and boundedness of the resolvent- 43 Individual stability in B-convex Banach spaces- 44 Individual stability in spaces with the analytic RNP- 45 Individual stability in arbitrary Banach spaces- 46 Scalarly integrable semigroups- Notes- 5 Countability of the unitary spectrum- 51 The stability theorem of Arendt, Batty, Lyubich, and V?- 52 The Katznelson-Tzafriri theorem- 53 The unbounded case- 54 Sets of spectral synthesis- 55 A quantitative stability theorem- 56 A Tauberian theorem for the Laplace transform- 57 The splitting theorem of Glicksberg and DeLeeuw- Notes- Append- Al Fractional powers- A2 Interpolation theory- A3 Banach lattices- A4 Banach function spaces- References- Symbols

Theory and applications of nonlinear operators of accretive and monotone type

Abstract: Periodic solutions for a second order semilinear Volterra equation metric and generalized projection operators in Banach spaces - properties and applications the rate of asymptotic regularity is 0(1/square root of n) global existence for second order functional differential equations iterative process for finding common fixed points of nonlinear mappings regularity for semilinear abstract Cauchy problems the KdV equation via semigroups a degree for maximal monotone operators on subjectivity of perturbed nonlinear m-accretive operators the fixed point property and mappings which are eventually nonexpansive approximation-solvability of semilinear equations and applications on the approximation of zeros for locally accretive operators quasimonotonicity and the Leray-Lions condition on nonlinear ill-posed problems II - monotone operator equations and monotone variational inequalities a classical hypergeometric proof of a transformation found by Ronald Bruck periodic solutions for nonlinear 2-D wave equations the existence of resolvents of holomorphic generators in Banach spaces existence of solutions to partial functional differential equations with delay zeros of weakly inward accretive mappings via A-proper maps nonlinear wave equations with asymptotically monotone damping.

Nonsmooth sequential analysis in Asplund spaces

Abstract: We develop a generalized differentiation theory for nonsmooth functions and sets with nonsmooth boundaries defined in Asplund spaces. This broad subclass of Banach spaces provides a convenient framework for many important applications to optimization, sensitivity, variational inequalities, etc. Our basic normal and subdifferential constructions are related to sequential weak-star limits of Frechet normals and subdifferentials. Using a variational approach, we establish a rich calculus for these nonconvex limiting objects which turn out to be minimal among other set-valued di erential constructions with natural properties. The results obtained provide new developments in infinite dimensional nonsmooth analysis and have useful applications to optimization and the geometry of Banach spaces.

Banach spaces of analytic functions

Abstract: In this paper, we explore certain Banach spaces of analytic functions. In particular, we study the space A -I, demonstrating some of its basic properties including non-separability. We ask the question: Given a class C of analytic functions on the unit disk ID> and a sequence { Zn} of points in the disk, is there an non-zero analytic function f E C with f(zn) = 0 for all n? Finally, we explore the Mz invariant subspaces of A-t, demonstrating that they may possess the codimension-2 property. This paper is part of the requirements for honors in mathematics. The signatures below, by the advisor, a departmental reader, and a representative of the departmental honors committee, demonstrate that Michael T. Nimchek has met all the requirements needed to receive honors in mathematics.

A generalized Drazin inverse

Abstract: The main theme of this paper can be described as a study of the Drazin inverse for bounded linear operators in a Banach space X when 0 is an isolated spectral point ofthe operator. This inverse is useful for instance in the solution of differential equations formulated in a Banach space X. Since the elements of X rarely enter into our considerations, the exposition seems to gain in clarity when the operators are regarded as elements of the Banach algebra L(X).

Competitive exclusion and coexistence for competitive systems on ordered Banach spaces

Abstract: The dynamics of competitive maps and semiflows defined on the product of two cones in respective Banach spaces is studied. It is shown that exactly one of three outcomes is possible for two viable competitors. Either one or the other population becomes extinct while the surviving population approaches a steady state, or there exists a positive steady state representing the coexistence of both populations.

Partial differential equations : basic theory

Abstract: Basic Theory of ODE and Vector Fields.- 1 The derivative.- 2 Fundamental local existence theorem for ODE.- 3 Inverse function and implicit function theorems.- 4 Constant-coefficient linear systems exponentiation of matrices.- 5 Variable-coefficient linear systems of ODE: Duhamel's principle.- 6 Dependence of solutions on initial data and on other parameters.- 7 Flows and vector fields.- 8 Lie brackets.- 9 Commuting flows Frobenius's theorem.- 10 Hamiltonian systems.- 11 Geodesies.- 12 Variational problems and the stationary action principle.- 13 Differential forms.- 14 The symplectic form and canonical transformations.- 15 First-order, scalar, nonlinear PDE.- 16 Completely integrable Hamiltonian systems.- 17 Examples of integrable systems central force problems.- 18 Relativistic motion.- 19 Topological applications of differential forms.- 20 Critical points and index of a vector field.- A Nonsmooth vector fields.- References.- 2 The Laplace Equation and Wave Equation.- 1 Vibrating strings and membranes.- 2 The divergence of a vector field.- 3 The covariant derivative and divergence of tensor fields.- 4 The Laplace operator on a Riemannian manifold.- 5 The wave equation on a product manifold and energy conservation.- 6 Uniqueness and finite propagation speed.- 7 Lorentz manifolds and stress-energy tensors.- 8 More general hyperbolic equations energy estimates.- 9 The symbol of a differential operator and a general Green-Stokes formula.- 10 The Hodge Laplacian on k-forms.- 11 Maxwell's equations.- References.- 3 Fourier Analysis, Distributions, and Constant-Coefficient Linear PDE.- 1 Fourier series.- 2 Harmonic functions and holomorphic functions in the plane.- 3 The Fourier transform.- 4 Distributions and tempered distributions.- 5 The classical evolution equations.- 6 Radial distributions, polar coordinates, and Bessel functions.- 7 The method of images and Poisson's summation formula.- 8 Homogeneous distributions and principal value distributions.- 9 Elliptic operators.- 10 Local solvability of constant-coefficient PDE.- 11 The discrete Fourier transform.- 12 The fast Fourier transform.- The mighty Gaussian and the sublime gamma function.- References.- 4 Sobolev Spaces.- 1 Sobolev spaces on ?n.- 2 The complex interpolation method.- 3 Sobolev spaces on compact manifolds.- 4 Sobolev spaces on bounded domains.- 5 The Sobolev spaces Hs0(?).- 6 The Schwartz kernel theorem.- References.- 5 Linear Elliptic Equations.- 1 Existence and regularity of solutions to the Dirichlet problem.- 2 The weak and strong maximum principles.- 3 The Dirichlet problem on the ball in ?n.- 4 The Riemann mapping theorem (smooth boundary).- 5 The Dirichlet problem on a domain with a rough boundary.- 6 The Riemann mapping theorem (rough boundary).- 7 The Neumann boundary problem.- 8 The Hodge decomposition and harmonic forms.- 9 Natural boundary problems for the Hodge Laplacian.- 10 Isothermal coordinates and conformal structures on surfaces.- 11 General elliptic boundary problems.- 12 Operator properties of regular boundary problems.- Spaces of generalized functions on manifolds with boundary.- The Mayer-Vietoris sequence in deRham cohomology.- References.- 6 Linear Evolution Equations.- 1 The heat equation and the wave equation on bounded domains.- 2 The heat equation and wave equation on unbounded domains.- 3 Maxwell's equations.- 4 The Cauchy-Kowalewsky theorem.- 5 Hyperbolic systems.- 6 Geometrical optics.- 7 The formation of caustics.- Some Banach spaces of harmonic functions.- The stationary phase method.- References.- A Outline of Functional Analysis.- 1 Banach spaces.- 2 Hilbert spaces.- 3 Frechet spaces locally convex spaces.- 4 Duality.- 5 Linear operators.- 6 Compact operators.- 7 Fredholm operators.- 8 Unbounded operators.- 9 Semigroups.- References.- B Manifolds, Vector Bundles, and Lie Groups.- 1 Metric spaces and topological spaces.- 2 Manifolds.- 3 Vector bundles.- 4 Sard's theorem.- 5 Lie groups.- 6 The Campbell-Hausdorff formula.- 7 Representations of Lie groups and Lie algebras.- 8 Representations of compact Lie groups.- 9 Representations of SU(2) and related groups.- References.

On vector variational inequalities

Abstract: In this paper, we study vector variational inequalities. The concept of weaklyC-pseudomonotone operator is introduced. By employing the Fan lemma, we establish several existence results. The new results extend and unify existence results of vector variational inequalities for monotone operators under a Banach space setting. In particular, existence results for the generalized vector complementarity problem with weaklyC-pseudomonotone operators in Banach space are obtained.

Quasimonotone variational inequalities in Banach spaces

Abstract: Various existence results for variational inequalities in Banach spaces are derived, extending some recent results by Cottle and Yao. Generalized monotonicity as well as continuity assumptions on the operatorf are weakened and, in some results, the regularity assumptions on the domain off are relaxed significantly. The concept of inner point for subsets of Banach spaces proves to be useful.

Exponential stability of breathers in Hamiltonian networks of weakly coupled oscillators

Abstract: We prove existence and practical stability of breathers in chains of weakly coupled anharmonic oscillators. Precisely, for a large class of chains, we prove that there exist periodic solutions exponentially localized in space, with the property that, given an initial datum (with ) close to the phase space trajectory of the breather, then the corresponding solution remains at a distance from the above trajectory, up to times growing exponentially with the inverse of , being a parameter measuring the size of the interaction among the particles. This result is deduced from a general normal form theorem for abstract Hamiltonian systems in Banach spaces, which we think could be interesting in itself.

Critical points for multiple integrals of the calculus of variations

Abstract: In this paper we deal with the existence of critical points of functional defined on the Sobolev space W01,p(Ω), p>1, by $$J(u) = \int\limits_\Omega {\vartheta (x,u,Du)dx,} {\text{ }}$$ where Ω is a bounded, open subset of ℝN. Even for very simple examples in ℝN the differentiability of J(u) can fail. To overcome this difficulty we prove a suitable version of the Ambrosetti-Rabinowitz Mountain Pass Theorem applicable to functionals which are not differentiable in all directions. Existence and multiplicity of nonnegative critical points are also studied through the use of this theorem.

Generation theory for semigroups of holomorphic mappings in Banach spaces

Abstract: We study nonlinear semigroups ofholomorphic mappings in Banach spaces and their infinitesimal generators. Using resolvents, we char- acterize, in particular, bounded holomorphic generators on bounded convex domains and obtain an analog ofthe Hille exponential f ormula. We then apply our results to the null point theory ofsemi-plus complete vector fields. We study the structure ofnull point sets and the spectral characteristics of null points, as well as their existence and uniqueness. A global version of the implicit function theorem and a discussion of some open problems are also included.

Norm or Numerical Radius Attaining Multilinear Mappings and Polynomials

Abstract: We study the denseness or norm of numerical radius attaining multilinear mappings and polynomials between Banach spaces, and examine the relations between norms and numerical radii of such mappings.

Integral manifolds of Carathéodory type differential equations in Banach spaces

Abstract: Integral manifolds of Caratheodory type differential equations in Banach spaces / Bernd Aulbach ; Thomas Wanner. - In: Six lectures on dynamical systems / eds. B. Aulbach ... - Singapore u.a. : World Scientific, 1996. - S. 45-119

Viscosity Solutions and Viscosity Subderivatives in Smooth Banach Spaces with Applications to Metric Regularity

Abstract: In Gâteaux or bornologically differentiable spaces there are two natural generalizations of the concept of a Frechet subderivative. In this paper we study the viscosity subderivative (which is the more robust of the two) and establish refined fuzzy sum rules for it in a smooth Banach space. These rules are applied to obtain comparison results for viscosity solutions of Hamilton--Jacobi equations in smooth spaces. A unified treatment of metric regularity in smooth spaces completes the paper. This illustrates the flexibility of viscosity subderivatives as a tool for analysis.

Compact tripotents in bi-dual JB*-triples

Abstract: The set consisting of the partially ordered set of tripotents in a JBW*-triple C with a greatest element adjoined forms a complete lattice. This paper is mainly concerned with the situation in which C is the second dual A** of a complex Banach space A and, more particularly, when A is itself a JB*-triple. A subset of consisting of the set of tripotents compact relative to A (denned in Section 4) with a greatest element adjoined is studied. It is shown to be an atomic complete lattice with the properties that the infimum of an arbitrary family of elements of is the same whether taken in or in and that every decreasing net of non-zero elements of has a non-zero infimum. The relationship between the complete lattice and the complete lattice where B is a Banach space such that B** is a weak*-closed subtriple of A** is also investigated. When applied to the special case in which A is a C*-algebra the results provide information about the set of compact partial isometries relative to A and are closely related to those recently obtained by Akemann and Pedersen. In particular it is shown that a partial isometry is compact relative to A if and only if, in their terminology, it belongs locally to A. The main results are applied to this and other examples.

Nonconvex differential calculus for infinite-dimensional multifunctions

Abstract: The paper is concerned with generalized differentiation of set-valued mappings between Banach spaces. Our basic object is the so-called coderivative of multifunctions that was introduced earlier by the first author and has had a number of useful applications to nonlinear analysis, optimization, and control. This coderivative is a nonconvex-valued mapping which is related to sequential limits of Frechet-like graphical normals but is not dual to any tangentially generated derivative of multifunctions. Using a variational approach, we develop a full calculus for the coderivative in the framework of Asplund spaces. The latter class is sufficiently broad and convenient for many important applications. Some useful calculus results are also obtained in general Banach spaces.

Weak-type weights and normable Lorentz spaces

Abstract: We show that the Lorentz space A1(w) is a Banach space if and only if the Hardy-Littlewood maximal operator M satisfies a certain weak-type estimate. We also consider the case of general measures. Finally, we study some properties of several indices associated to these spaces.

The problem of convexity of Chebyshev sets

Abstract: Contents Introduction §1. Definitions and notation §2. Reference theorems §3. Some results Chapter I. Characterization of Banach spaces by means of the relations between approximation properties of sets §1. Existence, uniqueness §2. Prom approximate compactness to 'sun'-property §3. From 'sun'-property to approximate compactness §4. Differentiability in the direction of the gradient is sufficient for Frechet and Gâteaux differentiability §5. Sets with convex complement Chapter II. The structure of Chebyshev and related sets §1. The isolated point method §2. Restrictions of the type §3. The case where M is locally compact §4. The case where W lies in a hyperplane §5. Other cases Chapter III. Selected results §1. Some applications of the theory of monotone operators §2. A non-convex Chebyshev set in pre-Hilbert space §3. The example of Klee (discrete Chebyshev set) §4. A survey of some other results Conclusion Bibliography

The spectral mapping theorem for evolution semigroups on spaces of vector-valued functions

Abstract: (see e.g. [Da-K], [Fat], [Paz], [Tan]). In the following a family (U(t, s))(t,s)∈D in L(X) satisfying (E1)–(E3) is called an evolution family. It has been noticed by several authors (see [LM1], [LM2], [LRa], [Na2], [RaS], [Ra1], [Ra2], [Ra3], [Rha] and the references therein) that asymptotic properties of the evolution family (U(t, s))(t,s)∈D are strongly related to the asymptotic behaviour of an associated evolution semigroup (TE(t))t≥0 of operators on a Banach space E(X) of X –valued functions (see Section 1). For a large class of these function spaces this evolution semigroup is strongly continuous and hence has a generator GE . It has been shown by R. Rau [Ra1, Prop. 1.7] and Y. Latushkin and S. Montgomery–Smith [LM1, Thm. 3.1], [LM2, Thm. 4] that on the function spaces C0(IR, X) and L (IR, X), 1 ≤ p <∞ , these semigroups always satisfy the spectral mapping theorem

Stationary distributions under mutation- selection balance: structure and properties

Abstract: A general model for the evolution of the frequency distribution of types in a population under mutation and selection is derived and investigated. The approach is sufficiently general to subsume classical models with a finite number of alleles, as well as models with a continuum of possible alleles as used in quantitative genetics. The dynamics of the corresponding probability distributions is governed by an integrodifferential equation in the Banach space of Borel measures on a locally compact space. Existence and uniqueness of the solutions of the initial value problem is proved using basic semigroup theory. A complete characterization of the structure of stationary distributions is presented. Then, existence and uniqueness of stationary distributions is proved under mild conditions by applying operator theoretic generalizations of Perron-Frobenius theory. For an extension of Kingman's original house-of-cards model, a classification of possible stationary distributions is obtained. PROBABILITY MEASURES; PERRON-FROBENIUS THEORY; INTEGRO-DIFFERENTIAL EQUATIONS; POSITIVE OPERATORS; CONTINUUM-OF-ALLELES

Convergence of Waveform Relaxation Methods for Differential-Algebraic Systems

Abstract: This paper gives sufficient conditions for existence and uniqueness of solutions and for the convergence of Picard iterations and more general waveform relaxation methods for differential-algebraic systems of neutral type. The results are obtained by the contraction mapping principle on Banach spaces with weighted norms and by the use of the Perron--Frobenius theory of nonnegative and nonreducible matrices. It is demonstrated that waveform relaxation methods are convergent faster than the classical Picard iterations.

Two definitions of exponential dichotomy for skew-product semiflow in Banach spaces

Abstract: In this paper we introduce a concept of exponential dichotomy for linear skew-product semiflows (LSPS) in infinite dimensional Banach spaces, which is an extension of the classical concept of exponential dichotomy for time dependent linear differential equations in Banach spaces. We prove that the concept of exponential dichotomy used by Sacker-Sell and Magalhães in recent years is stronger than this one, but they are equivalent under suitable conditions. Using this concept we where able to find a formula for all the bounded negative continuations. After that, we characterize the stable and unstable subbundles in terms of the boundedness of the corresponding projector along (forward/backward) the LSPS and in terms of the exponential decay of the semiflow. The linear theory presented here provides a foundation for studying the nonlinear theory. Also, this concept can be used to study the existence of exponential dichotomy and the roughness property for LSPS.

Regularity and Algebras of Analytic Functions in Infinite Dimensions

Abstract: A Banach space E is known to be Arens regular if every continuous linear mapping from E to E′ is weakly compact Let U be an open subset of E, and letHb(U) denote the algebra of analytic functions on U which are bounded on bounded subsets of U lying at a positive distance from the boundary of U We endow Hb(U) with the usual Frechet topology Mb(U) denotes the set of continuous homomorphisms φ : Hb(U) → C We study the relation between the Arens regularity of the space E and the structure of Mb(U)

Abstract Perron-Stieltjes integral

Abstract: Fundamental results concerning Stieltjes integrals for functions with values in Banach spaces are presented. The background of the theory is the Kurzweil approach to integration, based on Riemann type integral sums (see e.g. \cite4). It is known that the Kurzweil theory leads to the (non-absolutely convergent) Perron-Stieltjes integral in the finite dimensional case. In \cite3 Ch. S. Honig presented a Stieltjes integral for Banach space valued functions. For Honig's integral the Dushnik interior integral presents the background. \endgraf It should be mentioned that abstract Stieltjes integration was recently used by O. Diekmann, M. Gyllenberg and H. R. Thieme in \cite1 and \cite2 for describing the behaviour of some evolutionary systems originating in problems concerning structured population dynamics.