Real Separable Banach Space Research Articles

Approximation of common random fixed points of finite families of N-uniformly $L_i$-Lipschitzian asymptotically hemicontractive Random maps in Banach spaces

Let $(\Omega ,\Sigma ,\mu )$ be a complete probability measure space, $E$ be a real separable Banach space, $K$ a nonempty closed convex subset of E. Let $T : \Omega \times K \to K$, such that $\{T_i\}_{i=1}^N$, be N-uniformly $L_i$-Lipschitzian asymptotically hemicontractive random maps of $K$ with $F=\displaystyle\bigcap_{i=1}^N F(T_i)\ne \emptyset$. We construct an explicit iteration scheme and prove neccessary and sufficient conditions for approximating common fixed points of finite family of asymptotically hemicontractive random maps.

On Cantrell–Rosalsky’s strong laws of large numbers

Starting from a result of almost sure (a.s.) convergence for nonnegative random variables, a simplified proof of a strong law of large numbers (SLLN) established in ([Cantrell, A., Rosalsky, A., 2003. Some strong law of large numbers for Banach space valued summands irrespective of their joint distributions. Stochastic Anal. Appl. 21, 79–95], Theorem 1) for random elements in a real separable Banach space is presented, and some other results of a.s. convergence related to SLLNs in ([Cantrell, A., Rosalsky, A., 2004. A strong law for compactly uniformly integrable sequences of independent random elements in Banach spaces. Bull. Inst. Math. Acad. Sinica 32, 15–33], Th. 3.1) and ([Cantrell, A., Rosalsky, A., 2003. Some strong law of large numbers for Banach space valued summands irrespective of their joint distributions. Stochastic Anal. Appl. 21, 79–95], Theorem 2) are derived. No conditions of independence or on the joint distribution of random elements are required. Likewise, no geometric condition on the Banach space where random elements take values is imposed. Some applications to weighted (for an array of constants) sums of random elements and to the case of random sets are also considered.

Strictly convex or “non-singular” continuous homogeneous polynomials in real or complex Banach spaces

Fix an even integer d≥2. Let X be a separable real Banach space. Here we prove the existence a closed subspace Y of X and homogeneous degree d polynomials Q, Q 1, respectively on X and on Y with the following properties: i. Q(x) ≥ 0 for all x∈X and Q(x)=0 if and only if x∈Y; ii. Q 1(y)>0 for all y∈Y; iii. Q 1 is strictly convex, i.e. for every ε > 0 the set B(Q 1,ε):= {y∈Y: Q 1(y)≤ε} is convex and Q 1(tu+(1-t)v))<ε for all u, v∈B(Q 1,ε) and all 0 <t <1; iv. the continuous homogeneous form Q˜ induced by Q on X / Y is strictly convex. In the complex case we prove a similer existence result for complete intersections of the projects spaces P(Y) and P(X/Y).

Degree theory for monotone type mappings in non-reflexive Banach spaces

Let E be a real separable Banach space, E ∗ the dual space of E , and Ω ⊂ E an open bounded subset, and let T : D ( T ) ⊆ E → 2 E ∗ be a mapping of class ( S + ) L with D ( T ) ∩ Ω ⁄ = 0̸ see Definition 1.2. A degree theory is constructed for such a mapping.

Multiple solutions for operator equations involving duality mappings on Orlicz-Sobolev spaces

Let $X$ be a real reflexive, smooth and separable Banach space having the Kadeč-Klee property and compactly imbedded in the real Banach space $Y$ and let $G:Y\rightarrow \mathbb{R} $ be a differentiable functional. By using the "fountain theorem" (Bartsch [3]), we will study the multiplicity of solutions for the operator equation \[ J_{\varphi}u=G^{\prime}(u)\text{,} \] where $J_{\varphi}$ is the duality mapping on $X$, corresponding to the gauge function $\varphi$. Equations having the above form with $J_{\varphi}$ a duality mapping on Orlicz-Sobolev spaces are considered as applications. As particular cases of the latter results, some multiplicity results concerning duality mappings on Sobolev spaces are derived.

Extension of the Leray–Schauder degree for abstract Hammerstein type mappings

We introduce a new extension of the classical Leray–Schauder topological degree in a real separable reflexive Banach space. The new class of mappings for which the degree will be constructed is obtained essentially by replacing the compact perturbation by a composition of mappings of monotone type. It turns out that the class contains the Leray–Schauder type maps as a proper subclass. The new class is not convex thus preventing the free application of affine homotopies. However, there exists a large class of admissible homotopies including subclass of affine ones so that the degree can be effectively used. We shall construct the degree and prove that it is unique. We shall generalize the Borsuk theorem of the degree for odd mappings and show that the ‘principle of omitted rays’ remains valid. To illuminate the use of the new degree we shall briefly consider the solvability of abstract Hammerstein type equations and variational inequalities.

Odd Degree Polynomials on Real Banach Spaces

A classical result of Birch claims that for given k, n integers, n-odd there exists some N = N(k, n) such that for an arbitrary n-homogeneous polynomial P on , there exists a linear subspace of dimension at least k, where the restriction of P is identically zero (we say that Y is a null space for P). Given n > 1 odd, and arbitrary real separable Banach space X (or more generally a space with w*-separable dual X*), we construct an n-homogeneous polynomial P with the property that for every point 0 ≠ x ∈ X there exists some k ∈ such that every null space containing x has dimension at most k. In particular, P has no infinite dimensional null space. For a given n odd and a cardinal τ , we obtain a cardinal N = N(τ, n) = exp n +1τ such that every n-homogeneous polynomial on a real Banach space X of density N has a null space of density τ .

Complete Moment Convergence for Sequences of Independent Random Elements in Banach Spaces

We obtain complete moment convergence results for sequences of independent identically distributed (i.i.d.) random elements taking values in a real separable Banach space, which extend and generalize some results of Chow [6] and Wang and Su [7]. In the main results, no assumptions are made concerning the geometry of the underlying Banach space.

On Complete Convergence in Mean of Normed Sums of Independent Random Elements in Banach Spaces

For a sequence of random elements {T n , n ≥ 1} in a real separable Banach space 𝒳, we study the notion of T n converging completely to 0 in mean of order p where p is a positive constant. This notion is stronger than (i) T n converging completely to 0 and (ii) T n converging to 0 in mean of order p. When 𝒳 is of Rademacher type p (1 ≤ p ≤ 2), for a sequence of independent mean 0 random elements {V n , n ≥ 1} in 𝒳 and a sequence of constants b n → ∞, conditions are provided under which the normed sum converges completely to 0 in mean of order p. Moreover, these conditions for converging completely to 0 in mean of order p are shown to provide an exact characterization of Rademacher type p Banach spaces. Illustrative examples are provided.

Asymptotic expansions for the Laplace approximations of sums of Banach space-valued random variables

Let X_i, i\in N, be i.i.d. B-valued random variables, where B is a real separable Banach space. Let \Phi be a smooth enough mapping from B into R. An asymptotic evaluation of Z_n=E(\exp (n\Phi (\sum_{i=1}^nX_i/n))), up to a factor (1+o(1)), has been gotten in Bolthausen [Probab. Theory Related Fields 72 (1986) 305-318] and Kusuoka and Liang [Probab. Theory Related Fields 116 (2000) 221-238]. In this paper, a detailed asymptotic expansion of Z_n as n\to \infty is given, valid to all orders, and with control on remainders. The results are new even in finite dimensions.

Random Mann iteration scheme and random fixed point theorems

In this paper, some concepts such as random monotone operators, random Mann iteration and so on in a separable real Banach space are introduced. Also the existence and uniqueness theorems of random fixed points for random monotone operators satisfying Condition(H) are proved.

Characterization of Gaussian semigroups on separable Banach spaces

Let E be a separable real Banach space and denote by BUC (E) the space of bounded and uniformly continuous functions on E. For a C 0-semigroup $(T(t))_{t\ge 0}$ acting on BUC(E), we obtain necessary and sufficient conditions ensuring that $(T(t))_{t \ge 0}$ is Gaussian.

A Lie-Trotter product formula for Ornstein-Uhlenbeck semigroups in infinite dimensions

We prove a Lie-Trotter product formula for the Ornstein--Uhlenbeck semigroup associated with the stochastic linear Cauchy problem \( dX(t) = AX(t)\,dt + dW(t), t \leq 0,\\ X(0) = x_0. \) Here A is the generator of a C 0-semigroup on a separable real Banach space E and \( \{W(t)\}_{t\leq 0} \) is an E-valued Brownian motion.

Some Strong and Weak Limit Theorems for Weighted Sums of i.i.d. Banach Space Valued Random Elements with Slowly Varying Weights

For a sequence of independent and identically distributed random elements {V n , n ≥ 1} in a real separable Banach space 𝒳 and a monotone (increasing or decreasing) positive slowly varying function L, it is shown that the normed weighted sum converges almost surely to an element v ∈ 𝒳 if and only if E‖V 1‖ < ∞ and E V 1 = v. No geometric conditions on 𝒳 are required for this result. Moreover, when 𝒳 is of Rademacher type p (1 < p ≤ 2), conditions are provided (which are weaker than E‖V 1‖ < ∞) which ensure that U n converges in probability to some element v ∈ 𝒳. This result can fail without the Rademacher type p hypothesis. Both of these limit theorems are new results even when 𝒳 is the real line.

A note on the imbedding theorem of Browder and Ton

The imbedding theorem of Browder and Ton states that for any real separable Banach space X there exist a real separable Hilbert space H and a compact linear injection ψ: H → X such that ψ(H) is dense in X. We shall give a short and elementary new proof to this result. We also briefly discuss the corresponding result without the completeness assumption.

On the Strong Law of Large Numbers for Weighted Sums of Random Elements in Banach Space

In this paper we present some results for the general strong law of large numbers. We consider the normed weighted sums of random elements in real separable Banach space with random and nonrandom weights. The sufficient conditions presented in the paper are formulated in terms of Chung–Teicher type conditions (cf. Chow and Teicher, Probability Theory, Independence, Interchangeability, Martingales; Springer‐Verlag: New York, 1978, Teicher, Some new conditions for the strong law. Proc. Nat. Acad. Sci. USA 1968, 59 (3), 705–707.)

Some Strong Laws of Large Numbers for Banach Space Valued Summands Irrespective of Their Joint Distributions

Conditions are provided under which a sequence of random elements in a real separable Banach space obeys a general strong law of large numbers. No conditions are imposed on the joint distributions of the random elements or on the Banach space, and the results are new even when the Banach space is the real line. Illustrative examples are provided which compare the results or which show that they are sharp.

Existence results for impulsive semilinear damped differential inclusions

In this paper we investigate the existence of mild solutions for rst and second order impulsive semilinear evolution inclusions in real separable Banach spaces. By using suitable xed point theorems, we study the case when the multivalued map has convex and nonconvex values.

On Closability of Directional Gradients

Let μ be a centred Gaussian measure on a separable real Banach space E, and let H be a Hilbert subspace of E. We provide necessary and sufficient conditions for closability in L p (E,μ) of the gradient D H in the direction of H. These conditions are further elaborated in case when the gradient D H corresponds to a bilinear form associated with a certain nonsymmetric Ornstein–Uhlenbeck operator. Some natural examples of closability and nonclosability are presented.

Transition Semigroups of Banach Space Valued Ornstein-Uhlenbeck Processes

We investigate the transition semigroup of the solution to a stochastic evolution equation dX(t)=AX(t) dt+dW H (t), t≥0, where A is the generator of a C 0-semigroup S on a separable real Banach space E and W H (t) t≥0 is cylindrical white noise with values in a real Hilbert space H which is continuously embedded in E. Various properties of these semigroups, such as the strong Feller property, the spectral gap property, and analyticity, are characterized in terms of the behaviour of S in H. In particular we investigate the interplay between analyticity of the transition semigroup, S-invariance of H, and analyticity of the restricted semigroup S H .