Showing papers on "C0-semigroup 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.

Nonlinear Integral Equations in Abstract Spaces

Abstract: Preface. 1. Preliminaries. 2. Nonlinear Integral Equations in Banach Spaces. 3. Nonlinear Integro-Differential Equations in Banach Spaces. 4. Nonlinear Impulsive Integral Equations in Banach Spaces. References.

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

An algorithm for complete enumeration of all factorizations of a linear ordinary differential operator

Abstract: We discuss the problem of exhaustive enumeration of all possible factorization for a given linem ordinary differential operator. A theoretical investigation of topological and combinatorial obstacles to uniform description of factors which include arbitrary parameters and a complete algorithm for enumeration of all (discrete and parameterized) factorization are given.

Local null controllability of nonlinear functional differential systems in Banach space

Abstract: Sufficient conditions for the local null controllability of non-linear functional differential systems with unbounded linear operators in Banach space are established. The results are obtained using the semigroup of linear operators, fractional powers of operators, and the Schauder fixed-point theorem. Applications to parabolic differential systems are given.

A few remarks on ordinary differential equations

Abstract: We present in this note existence and uniqueness results for solutions of ordinary differential equations and linear transport equations with discontinuous coefficients in a bounded open subset Ωof...

Iterative solutions of nonlinear equations in smooth Banach spaces

Abstract: Let E be a smooth Banach space over the real field, ^ K C E closed convex and bounded, T : K —* A' uniformly continuous and strongly pseudo-contractive. It is proved that the Ishikawa iteration process converges strongly to the unique fixed point of T. Applications of this result to the operator equations Au = f or u + Av. = f where A is a strongly accretive mapping of E into itself and under various continuity assumptions on A ate also given. M1RAMARE TRIESTE May 1994

Extensions Of Subdifferential Calculus Rules in Banach Spaces

Abstract: This paper is devoted to extending formulas for the geometric approximate subdifferential and the Clarke subdifferential of extended-real-valued functions on Banach spaces. The results are strong enough to include completely the finite dimensional setting.

Integral equations in reflexive Banach spaces and weak topologies

Abstract: The Schauder Tychonoff theorem in a locally convex topological space is used to establish existence results for Volterra-Hammerstein and Hammerstein integral equations in a reflexive Banach space.

E-Subdifferentials in Terms of Subdifferentials *

Abstract: In this note, we give a formula which expresses the e-subdifferential operator of a lower semicontinuous convex proper function on a given Banach space in terms of its subdifferential.

D'Alembertian solutions of inhomogeneous linear equations (differential, difference, and some other)

Abstract: Let an Ore polynomial ring k[X; a, 6] and a nonzero pseudolinear map 19: K + K, where K is a O, &compatible extension of the field k, be given. Then we have the ring k[O] of operators K ~ K. It is assumed that if a first-order equation Fg = O, F ~ k[9], has a nonzero solution in a u, b-compatible extension of the field k, then the equation has a nonzero solution in K. These solutions form the set %,t C K of hyperexponential elements. An equation Py = O, P ~ k[O], is called completely factorable if P can be decomposed in the product of first-order operators over k. Solutions of all completely factorable equations form the linear space dk C K of d’Alembertian elements. The order of minimal operator over k which annihilates a G ~k is called the height of a. It is easy to see that %k C J& and the height of any a C %k is equal to 1. It is known ([12, 4]) that if L E ,k[@] and ~ G ‘?-l~ then all the hyperexponential solutions of the equation

C0-groups andC0-semigroups of linear operators on hereditarily indecomposable Banach spaces

Abstract: A Banach space X is called hereditarily indecomposable (briefly, H.I3 if, whenever Y and Z are closed, infinite dimensional subspaces of X and ~5 > 0, then there exist unit vectors y ~ Y and z ~ Z such that II Y z < 3. This is equivalent to the following property: whenever Y and Z are closed, infinite dimensional subspaces of X satisfying Yc~ Z = {0}, then Y + Z is non-closed. Examples of H.I. spaces were recently exhibited by Gowers and Maurey [9], where it was also shown that bounded linear operators T in such spaces are somewhat special, e.g. there is a unique point 2 r in the spectrum e(T) of T such that T 2 r I is strictly singular, [9; w This is further exemplified in the recent articles [8], [22]. In this note we consider the class of (closed) operators T on H.I. spaces which generate Co-groups or Co-semigroups. One of the main results is that generators of Co-groups are always bounded operators. Moreover, if the group is of polynomial growth, then ( T 2 ~ I ) k is compact for some k ~ N qsee Theorems 2.3 and 3.2), and ( T ~rI1 k is a finite rank operator iff a(T) is a finite set tsee Proposition 3.3). For the generator T of a uniformly bounded Co-grou p, the H.I. property places a severe (and somewhat curious~ restriction on or(T); namely, there is a finite set H ~ iN. such that ~(T) is contained in the rational span of H (cf. Proposition 3.4). There are also analogues of these results for discrete groups. For the case of Co-semigroups m H.I. spaces the situation changes somewhat. Firstly, the generator T need not be bounded; see Example 2.4. However. a(T) is always either a convergent sequence in tI2~ = G w {oo} or a finite set (possibly empty); see Proposition 2.2. If ~(T) ~ ~ is a bounded, infinite set, then T e L(X); see Proposition 1.3. Most importantly, the generator T of any Co-semigroup satisfies the spectral mapping theorem, i.e. e t~(r) = cr(e tr) \\ {0} ; see Proposition 2.5. As for the case of Co-groups, if (Z D(Tt\"t is the generator of a uniformly bounded Co-semigrou p, then a(T) c~ ilR cannot contain an infinite, rationally independent set (cf. Remark 2). If, in addition, the H.I. space is reflexive and or(T)c~ iN is an infinite set, then ~r(T)\\ iN is necessarily finite, T is bounded and T )~rI is compact; see Proposition 4.2.

Bounded Solutions of Nonlinear Ordinary Differential Equations

Abstract: A classical result states that if A is a n × n real matrix and T > 0, then the system Open image in new window has a T-periodic solution for each T-periodic continuous forcing term p if and only if no eigenvalue of A has the form ikw with k ∈ ℤ and ω = 2π/T. The homogeneous part of equation (1) is then said to be non-resonant.

Asymptotic behavior of c0-semigroups in banach spaces

Abstract: We present optimal estimates for the asymptotic behavior of strongly continuous semigroups UA : [0,∞[→ L(X) in terms of growth abscissas of the resolvent function R(·, A) of the generator A. In particular we give Ljapunov’s classical stability condition a definite form for (infinite dimensional) abstract Cauchy problems: The abscissa of boundedness of R(·, A) equals the growth bound of the classical solutions of y′ = Ay.

Self-induced compactness in Banach spaces

Abstract: The question which led to the title of this note is the following: {\it If $X$ is a Banach space and $K$ is a compact subset of $X$, is it possible to find a compact, or even approximable, operator $v:X\to X$ such that $K\subset\ol{v(B_X)}$?} This question was first posed by P.G.Dixon [6] in connection with investigating the problem of the existence of approximate identities in certain operator algebras. We shall provide a couple of observations related to the above question and give in particular a negative answer in case of approximable operators. We shall also provide the first examples of Banach spaces having the approximation property but failing the bounded compact approximation property though all of their duals do even have the metric compact approximation property.

An inverse-free Jarratt type approximation in a Banach space

Abstract: In this study, we introduce a new iterative method for solving nonlinear operator equations in Banach spaces. We establish a sufficient as well as a local convergence theorem. We also provide the best possible practical error bound for this method.

Resolution of quadratic equations in Banach spaces

Abstract: A new convergence theorem is established for the super-Halley method. This method has, in general, order three, but when it is applied to quadratic equations, its order is four.

Bi-shadowing and delay equations

Abstract: Bi-shadowing is an extension to the concept of shadowing and is usually used in the context of comparing computed trajectories with the true trajectories of a dynamical system in Rn. Here the concept is defined in a Banach space and is applied to delay equations to give an apparently new result on nonlinear perturbations of linear delay equations. This is essentially a form of robustness with respect to small nonlinear disturbances.

Faddeev differential equations as a spectral problem for a nonsymmetric operator

Abstract: We consider a nonsymmetric matrix operator whose eigenvalue problem is the system of Faddeev differential equations for a three-particle system. For this operator and its adjoint, the resolvents are represented in terms of Faddeev T-matrix components of the three-particle Schrodinger operator. On the basis of these representations, the invariant spaces of the operators under consideration are investigated and their eigenfunctions are determined. The biorthogonality and completeness of the eigenfunction system are proved.