Theorem For Banach Spaces Research Articles

Generalized projections, decompositions, and the Pythagorean-type theorem in Banach spaces

In this paper, we investigate new properties of the generalized projection operators on convex closed cones in uniformly convex and uniformly smooth Banach spaces; establish decompositions theorems for arbitrary elements both in primary and dual spaces; and prove the Banach space analogue of the Pythagorean-type theorem. Earlier, all these results were known only in Hilbert spaces.

Boundary control of semilinear elliptic equations with discontinuous leading coefficients and unbounded controls

We consider optimal control problems governed by semilinear elliptic equations with pointwise constraints on the state variable. The main difference with previous papers is that we consider nonlinear boundary conditions, elliptic operators with discontinuous leading coefficients and unbounded controls. We can deal with problems with integral control constraints and the control may be a coefficient of order zero in the equation. We derive optimality conditions by means of a new Lagrange multiplier theorem in Banach spaces.

Nonlinear boundary control of semilinear parabolic problems with pointwise state constraints

We consider optimal control problems governed by semilinear par- abolic equations with nonlinear boundary conditions and pointwise constraints on the state variable. In Robin boundary conditions considered here, the nonlinear term is neither necessarily monotone nor Lipschitz with respect to the state variable. We derive optimality conditions by means of a Lagrange multiplier theorem in Banach spaces. The adjoint state must satisfy a parabolic equation with Radon measures in Robin boundary conditions, in the terminal condition and in the distributed term. We give a precise meaning to the adjoint equation with measures as data and we prove the existence of a unique weak solution for this equation in an appropriate space.

Perturbed Optimization in Banach Spaces I: A General Theory Based on a Weak Directional Constraint Qualification

Using a directional form of constraint qualification weaker than Robinson's, we derive an implicit function theorem for inclusions and use it for first- and second-order sensitivity analyses of the value function in perturbed constrained optimization. We obtain Holder and Lipschitz properties and, under a no-gap condition, first-order expansions for exact and approximate solutions. As an application, differentiability properties of metric projections in Hilbert spaces are obtained, using a condition generalizing polyhedricity. We also present in the appendix a short proof of a generalization of the convex duality theorem in Banach spaces.

On the uniform ergodic theorem in Banach spaces that do not contain duals

Let T be a power-bounded linear operator in a real Banach space X. We study the equality (*) $(I-T)X = {z ∈ X: sup_{n} ∥∑_{k=0}^{n} T^{k}z∥ < ∞}$. For X separable, we show that if T satisfies and is not uniformly ergodic, then $\overline{(I-T)X}$ contains

Evolutionary Semigroups and Lyapunov Theorems in Banach Spaces

We present a spectral mapping theorem for continuous semigroups of operators on any Banach space E. The condition for the hyperbolicity of a semigroup on E is given in terms of the generator of an evolutionary semigroup acting in the space of E-valued functions. The evolutionary semigroup generated by the propagator of a nonautonomous differential equation in E is also studied. A "discrete" technique for investigating the evolutionary semigroup is developed and applied in describing the hyperbolicity (exponential dichotomy) of the nonautonomous equation.

The RAGE theorem in Banach spaces

A generalization of the so called RAGE theorems from scattering theory in Hilbėrt spaces to a more general class of uniformly boundedC0-semigroups on Banach spaces is given. The extension is based on the notion of weak almost periodicity (in the sense of Eberlein).

Asymptotic expansions in the integral and local limit theorems in banach spaces with applications to ?-statistics

LetB be a real separable Banach space and letX,X1,X2,...∈B denote a sequence of independent identically distributed random variables taking values inB. DenoteS n =n−1/2(X1+...X n ). Let π:B→R be a polynomial. We consider (truncated) Edgeworth expansions and other asymptotic expansions for the distribution function of the r.v. π(S n ) with uniform and nonuniform bounds for the remainder terms. Expansions for the density of π(S n ) and its higher order derivatives are derived as well. As an application of the general results we get expansions in the integral and local limit theorems for ω-statistics $$\omega _n^p (q)\mathop { = n^{{p \mathord{\left/ {\vphantom {p 2}} \right. \kern-\nulldelimiterspace} 2}} }\limits^\Delta \smallint _{(0,1)} \{ F_n (x) - x\} ^p q(x)dx$$ and investigate smoothness properties of their distribution functions. Herep≥2 is an even number,q: [0, 1]→[0, ∞] is a measurable weight function, andF n denotes the empirical distribution function. Roughly speaking, we show that in order to get an asymptotic expansion with remainder termO(n−α), α 0. Expansions of arbitrary length are available provided the weight functionq is absolutely continuous and positive on an nonempty subinterval of (0, 1). Similar results hold for the density of the distribution function and its derivatives providedq satisfies certain very mild smoothness condition and is bounded away from zero. The last condition is essential since the distribution function of the ω-statistic has no density whenq is vanishing on an nonempty subinterval of (0, 1).

Nonuniform estimates in the central limit theorem in Banach spaces

Nonuniform estimates in the central limit theorem in Banach spaces

Some fixed point theorems in convex metric spaces

The concept of a convex metric space was introduced by Takahashi [10]. He observed that it is possible to generalize fixed point theorems in Banach spaces. Subsequently, Machado [8], Itoh [5], Naimpally, Singh and Whitfield [9] and Beg and Azam [2], among others have studied fixed point theorems in convex metric spaces. This paper is a continuation of these investigations.

The differentiability of convex functions on topological linear spaces

In this paper topological linear spaces are categorised according to the differentiability properties of their continuous convex functions. Mazur's Theorem for Banach spaces is generalised: all separable Baire topological linear spaces are weak Asplund. A class of spaces is given for which Gateaux and Fréchet differentiability of a continuous convex function coincide, which with Mazur's theorem, implies that all Montel Fréchet spaces are Asplund spaces. The effect of weakening the topology of a given space is studied in terms of the space's classification. Any topological linear space with its weak topology is an Asplund space; at the opposite end of the topological spectrum, an example is given of the inductive limit of Asplund spaces which is not even a Gateaux differentiability space.

Local mean ergodic theorems in Banach spaces

Local mean ergodic theorems in Banach spaces

Nonexpansive retractions and nonlinear ergodic theorems in Banach spaces

On donne des theoremes ergodiques non lineaires pour des semigroupes commutatifs d'applications non dilatation sans utiliser le concept de moyenne compacte

Two Banach space methods and dual operator algebras

In this paper we present several new slightly nonlinear variants of the bipolar and the open mapping theorems in Banach spaces, which we abstracted from the recent developments in the theory of dual operator algebras. A new application of our techniques to the theory of operator algebras is also given.

Global Surjection and Global Inverse Mapping Theorems in Banach Spaces

Global Surjection and Global Inverse Mapping Theorems in Banach Spaces

A mean ergodic theorem in Banach spaces

Let Γ = { U t : t ∈ Λ } ( Λ = Z + − { 0 } or R + − { 0 } ) \Gamma = \left \{ {{U_t}:t \in \Lambda } \right \}\left ( {\Lambda = {{\mathbf {Z}}^ + } - \left \{ 0 \right \}{\text {or }}{{\mathbf {R}}^ + } - \left \{ 0 \right \}} \right ) be a commuting family of nonexpansive affine operators in a Banach space X X satisfying the following conditions: (i) there is a function M ( x | Γ ) ≥ 0 M\left ( {x\left | \Gamma \right .} \right ) \geq 0 defined on X X such that \[ ‖ U t + s x − U t U s x ‖ ≤ M ( x | Γ ) ( s , t ∈ Λ , x ∈ X ) , \left \| {{U_{t + s}}x - {U_t}{U_s}x} \right \| \leq M\left ( {x\left | \Gamma \right .} \right )\quad \left ( {s,t \in \Lambda ,x \in X} \right ), \] (ii) \[ sup { t − 1 ‖ U t x ‖ : t ∈ Λ , t ≥ 1 } = K ( x | Γ ) &gt; ∞ ( x ∈ X ) . \sup \left \{ {{t^{ - 1}}\left \| {{U_t}x} \right \|:t \in \Lambda ,t \geq 1} \right \} = K\left ( {x\left | \Gamma \right .} \right ) &gt; \infty \left ( {x \in X} \right ). \] Then it is proved that if { t − 1 U t x : t ∈ Λ } \left \{ {{t^{ - 1}}{U_t}x:t \in \Lambda } \right \} is relatively compact for x ∈ X x \in X , the limit X − lim t → ∞ t − 1 U t x = x ¯ X -\lim _{t \to \infty }{t^{ - 1}}{U_t}x = \bar x exists in X X and U t x ¯ = x ¯ ( t ∈ Λ ) \overline {{U_t}x} = \overline x \left ( {t \in \Lambda } \right ) .

On the Rate of Convergence in the Central Limit Theorem in Banach Spaces

Let $E$ denote a separable Banach space and let $X_i, i \in \mathbb{N}$, be a sequence of i.i.d. $E$-valued random vectors having finite third moment such that the central limit theorem holds. We prove that the convergence rate in the central limit theorem is $O(n^{-1/2})$ for regions $\{x \in E: F(x) < r\}$ which are defined by means of a smooth real valued function $F$ on $E$, provided that the limiting distribution of the gradient of $F$ fulfills a variance condition. Using this result we prove that the rate of convergence in the functional limit theorem for empirical processes is of order $O(n^{-1/2})$.

Lower bounds for the rate of convergence in the central limit theorem in Banach spaces

Let E c B be a pair of real separable Banach spaces, connected by the identity continuous linear imbedding, with norms ['[E and ['[B, respectively. Let X, XI, X2, ... e E be a sequence of independent identically distributed random elements, assuming values in the space E and having mean zero. We set Sn = Sn(X) = n -I/= (X~ +...+ Xn). We shall assume that the centered Gaussian random element Y e E and the random element X have identical covariance.

Asymptotic expansions for moments in the central limit theorem in banach spaces

Asymptotic expansions for moments in the central limit theorem in banach spaces

Sur une inegalite de H.P. Rosenthal et le theoreme limite central dans les espaces de Banach

This paper studies an inequality of H. P. Rosenthal for vector valued random variables, its relations with some geometric properties of Banach spaces and its applications to the study of the central limit theorem in Banach spaces.