Infinite-dimensional Banach Space Research Articles

Uniform large deviations for stochastic Burgers–Huxley equation driven by multiplicative noise

In this paper, we establish a Freidlin–Wentzell type large deviation principle uniformly with respect to initial condition in bounded subsets of an infinite dimensional Banach space for stochastic Burgers–Huxley equation driven by multiplicative small noise. The proof is based on the weak convergence approach obtained by Salins, Budhiraja, Dupuis (Trans. Amer. Math. Soc., 2019).

Approximate Morse-Sard type results for non-separable Banach spaces

For the Banach spaces E=c0(Γ),ℓp(Γ), where Γ is an arbitrary infinite set and 1<p<∞, we show that for every (non-zero) quotient F of E, every continuous function f:E→F can be uniformly approximated by smooth functions with no critical points, that is for every continuous function ε:E→(0,∞), there exists a Ck smooth function g:E→F such that g′(x) is surjective and ‖f(x)−g(x)‖≤ε(x) for all x∈E (k=∞ if E=c0(Γ) or p is an even integer, k=p−1 if p is an odd integer and k=[p] otherwise). Moreover, for a wide class of (not necessarily separable) infinite dimensional Banach spaces E and a suitable class of quotients F of E, every continuous function f:E→F can be uniformly approximated by Ck smooth functions with no critical points (k depending on the properties of the smoothness of the space E). In particular, for every Banach space E with Ck smooth partitions of unity (k∈N∪{∞}) and an infinite dimensional separable complemented subspace with a Ck smooth and LUR norm, we show that every continuous function f:E→Rn (n∈N) can be uniformly approximated by C1 smooth function with no critical points.

Counterexamples in rotundity of norms in Banach spaces

We study several classical concepts in the topic of strict convexity of norms in infinite dimensional Banach spaces. Specifically, and in descending order of strength, we deal with Uniform Rotundity (UR), Weak Uniform Rotundity (WUR) and Uniform Rotundity in Every Direction (URED). Our first three results show that we may distinguish between all of these three properties in every Banach space where such renormings are possible. Specifically, we show that in every infinite dimensional Banach space which admits a WUR (resp. URED) renorming, we can find a norm with the same condition and which moreover fails to be UR (resp. WUR). We prove that these norms can be constructed to be Locally Uniformly Rotund (LUR) in Banach spaces admitting such renormings. Additionally, we obtain that in every Banach space with a LUR norm we can find a LUR renorming which is not URED. These results solve three open problems posed by A.J. Guirao, V. Montesinos and V. Zizler. The norms we construct in this first part are dense.In the last part of this note, we solve a fourth question posed by the same three authors by constructing a C∞-smooth norm in c0 whose dual norm is not strictly convex.

Unconditional bases and Daugavet renormings

We prove that the space ℓ2 – and more generally every infinite dimensional Banach space with an unconditional Schauder basis – can be renormed to have a Daugavet point. As a starting point, we introduce a new diametral notion for points of the unit sphere of Banach spaces, that complements the notion of Δ-points, but is weaker than the notion of Daugavet points. We show that this notion can be used to provide another geometric characterization of the Daugavet property, as well as to recover – and even to provide new – results about Daugavet points in various contexts such as absolute sums of Banach spaces or projective tensor products. Then, we observe that the study of this notion naturally leads to new renorming ideas in the context, and combine these with previous techniques from the literature to show that Daugavet points may exist even in super-reflexive spaces.

The Existence of Solutions for w-Weighted ψ-Hilfer Fractional Differential Inclusions of Order μ ∈ (1, 2) with Non-Instantaneous Impulses in Banach Spaces

In this research, we obtain the sufficient conditions that guarantee that the set of solutions for an impulsive fractional differential inclusion involving a w-weighted ψ-Hilfer fractional derivative, D0,tσ,v,ψ,w,of order μ∈(1,2), in infinite dimensional Banach spaces that are not empty and compact. We demonstrate the exact relation between a differential equation involving D0,tσ,v,ψ,w of order μ ∈(1,2) in the presence of non-instantaneous impulses and its corresponding fractional integral equation. Then, we derive the formula for the solution for the considered problem. The desired results are achieved using the properties of the w-weighted ψ-Hilfer fractional derivative and appropriate fixed-point theorems for multivalued functions. Since the operator D0,tσ,v,ψ,w includes many types of well-known fractional differential operators, our results generalize several results recently published in the literature. We give an example that illustrates and supports our theoretical results.

The Wolff hull of a compact holomorphic self-map on an infinite dimensional ball

AbstractFor large classes of (finite and) infinite dimensional complex Banach spaces Z, B its open unit ball and $$f:B\rightarrow B$$ f : B → B a compact holomorphic fixed-point free map, we introduce and define the Wolff hull, W(f), of f in $$\partial B$$ ∂ B and prove that W(f) is proximal to the images of all subsequential limits of the sequences of iterates $$(f^n)_n$$ ( f n ) n of f. The Wolff hull generalises the concept of a Wolff point, where such a point can no longer be uniquely determined, and coincides with the Wolff point if Z is a Hilbert space. Recall that $$(f^n)_n$$ ( f n ) n does not generally converge even in finite dimensions, compactness of f (i.e. f(B) is relatively compact) is necessary for convergence in the infinite dimensional Hilbert ball and all accumulation points $$\Gamma (f)$$ Γ ( f ) of $$(f^n)_n$$ ( f n ) n map B into $$\partial B$$ ∂ B (for any topology finer than the topology of pointwise convergence on B). The target set of f is $$\begin{aligned} T(f)=\bigcup _{g \in \Gamma (f)} g(B). \end{aligned}$$ T ( f ) = ⋃ g ∈ Γ ( f ) g ( B ) . To locate T(f), we use a concept of closed convex holomorphic hull, $${\text {Ch}}(x) \subset \partial B$$ Ch ( x ) ⊂ ∂ B for each $$x \in \partial B$$ x ∈ ∂ B and define a distinguished Wolff hull W(f). We show that the Wolff hull intersects all hulls from T(f), namely $$\begin{aligned} W(f) \cap {\text {Ch}}(x)\ne \emptyset \ \ \hbox {for all}\ \ x \in T(f). \end{aligned}$$ W ( f ) ∩ Ch ( x ) ≠ ∅ for all x ∈ T ( f ) . If B is the Hilbert ball, W(f) is the Wolff point, and this is the usual Denjoy–Wolff result. Our results are for all reflexive Banach spaces having a homogeneous ball (or equivalently, for all finite rank $$JB^*$$ J B ∗ -triples). These include many well-known operator spaces, for example, L(H, K), where either H or K is finite dimensional.

Solutions and anti-periodic solutions for impulsive differential equations and inclusions containing Atangana-Baleanu fractional derivative of order $ \zeta \in (1, 2) $ in infinite dimensional Banach spaces

&lt;abstract&gt;&lt;p&gt;In this paper, we improved recent results on the existence of solutions for nonlinear fractional boundary value problems containing the Atangana-Baleanu fractional derivative of order $ \zeta \in (1, 2) $. We also derived the exact relations between these fractional boundary value problems and the corresponding fractional integral equations in infinite dimensional Banach spaces. We showed that the continuity assumption on the nonlinear term of these equations is insufficient, give the derived expression for the solution, and present two results about the existence and uniqueness of the solution. We examined the case of impulsive impact and provide some sufficiency conditions for the existence and uniqueness of the solution in these cases. We also demonstrated the existence and uniqueness of anti-periodic solution for the studied problems and considered the problem when the right-hand side was a multivalued function. Examples were given to illustrate the obtained results.&lt;/p&gt;&lt;/abstract&gt;

Asymptotically almost periodic mild solutions for some partial integrodifferential inclusions using scale of Banach spaces

Abstract We are interested in the existence of mild solutions for a class of partial integrodifferential inclusions in infinite dimensional Banach spaces. First, we show the existence of mild solutions with the help of a scale of Banach spaces, the theory of resolvent operators, and the fixed point theory for the measure of non-compactness. Moreover, we examine the existence of asymptotically almost periodic solutions for our problem. Finally, an example of the abstract results is provided.

Attractive sets of periodic integrodifference equations under discretization

We consider periodic difference equations in infinite-dimensional Banach spaces possessing compact asymptotically stable set . It is established that such persists under spatial discretizations by means of projection methods as nearby closed and bounded uniformly asymptotically stable sets, which moreover converge to in the Hausdorff metric for increasingly more accurate schemes. The proof is based on a Lyapunov function guaranteed by an ambient converse theorem and a pullback construction. As application serves integrodifference equations on spaces of continuous and p-integrable functions over a compact habitat.

Some spectral domain in approximate point-spectrum-preserving maps on mathcal {B}(mathcal {X})

Let mathcal {X} be an infinite-dimensional complex Banach space, mathcal {B}(mathcal {X}) the algebra of all bounded linear operators on mathcal {X}. Denote the spectral domain by sigma _{gamma}(T)={lambda in sigma _{a}(T): T { text{ that is semi-Fredholm and }} asc(T-lambda I)<infty }. In this paper, we characterize the structure of additive surjective maps varphi : mathcal {B}(mathcal {X})rightarrow mathcal {B}(mathcal {X}) with sigma _{gamma}(varphi (T))=sigma _{gamma}(T) for all Tin mathcal{B(X)}.

Linear bijective maps preserving spectral functions on pairs of similar operators

Let X be an infinite-dimensional complex Banach space, and let B(X) be the algebra of all linear and bounded operators on X. In this paper, we characterize linear bijective maps φ on B(X) having the property that the spectral radius (respectively, the spectrum) of φ(A) equals the spectral radius (respectively, the spectrum) of φ(B) for each pair of similar operators A,B∈B(X). As a corollary, we obtain a characterization of linear bijective maps on B(X) preserving the equality of the spectrum.

The Bloch–Nevanlinna Problem Under Lineability

The original version of the celebrated Bloch–Nevanlinna problem asked whether there exists or not a holomorphic function in the unit disc of bounded characteristic whose derivative is not of bounded characteristic. This problem has been solved in the affirmative by a number of mathematicians. Starting from a result on topological genericity of this class of functions due to Hahn, our work intends—under an algebraic as well as a topological point of view—to contribute to this topic. Specifically, the family of solutions of the Bloch–Nevanlinna problem is proved to be residual in appropriate topological spaces, and it contains, except for the zero function, dense maximal dimensional vector subspaces, large linear algebras and infinite-dimensional Banach spaces.

Computing subdifferential limits of operators on Banach spaces

Abstract Let X , Y {X,Y} be real, infinite-dimensional Banach spaces. Let ℒ ⁢ ( X , Y ) {{\mathcal{L}}(X,Y)} be the space of bounded operators. An important aspect of understanding differentiability of the operator norm at A ∈ ℒ ⁢ ( X , Y ) {A\in{\mathcal{L}}(X,Y)} is to estimate the limit (which always exists) lim t → 0 + ⁡ ∥ A + t ⁢ B ∥ - ∥ A ∥ t for ⁢ B ∈ ℒ ⁢ ( X , Y ) , \lim_{t\rightarrow 0^{+}}\frac{\lVert A+tB\rVert-\lVert A\rVert}{t}\quad\text{% for }B\in{\mathcal{L}}(X,Y), using the values of B on the state space S A = { τ ∈ ℒ ⁢ ( X , Y ) ∗ : τ ⁢ ( A ) = ∥ A ∥ , ∥ τ ∥ = 1 } . S_{A}=\bigl{\{}\tau\in{\mathcal{L}}(X,Y)^{\ast}:\tau(A)=\lVert A\rVert,\,% \lVert\tau\rVert=1\bigr{\}}. In this paper, we give several examples of Banach spaces, including the ℓ p {\ell^{p}} spaces (for 1 &lt; p &lt; ∞ {1&lt;p&lt;\infty} ) where a more tangible estimate is possible, under additional hypotheses on A. We also use the notion of norm-weak upper-semi-continuity (usc, for short) of the preduality map to achieve this. Our results also show that the operator subdifferential limit is related to the corresponding subdifferential limit of the vectors in the range space, when A ∗ ∗ {A^{\ast\ast}} attains its norm.

A Banach space with an infinite dimensional reflexive quotient algebra [formula omitted

We construct a Banach space X of Gowers-Maurey type such that the algebra of bounded operators L(X) is a direct sum of an infinite dimensional reflexive Banach space and the operator ideal of strictly singular operators SS(X).

Weak Greedy Algorithms and the Equivalence Between Semi-greedy and Almost Greedy Markushevich Bases

We introduce and study the notion of weak semi-greedy systems—which is inspired in the concepts of semi-greedy and branch semi-greedy systems and weak thresholding sets-, and prove that in infinite dimensional Banach spaces, the notions of semi-greedy, branch semi-greedy, weak semi-greedy, and almost greedy Markushevich bases are all equivalent. This completes and extends some results from (Berná in J Math Anal Appl 470:218–225, 2019; Dilworth et al. in Studia Math 159:67–101, 2003; J Funct Anal 263:3900–3921, 2012). We also exhibit an example of a semi-greedy system that is neither almost greedy nor a Markushevich basis, showing that the Markushevich condition cannot be dropped from the equivalence result. In some cases, we obtain improved upper bounds for the corresponding constants of the systems.

Diametrically Complete Sets with Empty Interior and Constant Width Sets with Empty Interior

We show that each infinite-dimensional reflexive Banach space (X,left| cdot right| _X) has an equivalent norm left| cdot right| _{X,0} such that (X,left| cdot right| _{X,0}) is LUR and contains a diametrically complete set with empty interior. We also prove that after a suitable equivalent renorming, the Banach space C([0,1],{mathbb {R}}) contains a constant width set with empty interior.

Topological properties of some multiplication operators on L(X)

A pair (u, ?) in X ? X?, where X is an infinite dimensional Banach space and X? its topological dual space, induces in a natural way two multiplication operators L?,u and R?,u on the Banach space L(X), defined by L?,u (T) (x) = ? (T(x)) u, and R?,u (T) (x) = ?(x)T(u), for all T in L(X) and x in X. In this paper, we present necessary and sufficient conditions for the compactness, demicompactness, stongly demicompactess, power compactness and Riesz property of this family of operators. We also establish sufficient conditions for the quasi-compactness and weak compactness of these operators. Finally, we show that the Dunford-Pettis property fails for the Banach space L(X) whenever either X or L(X) is reflexive.

Nonlocal impulsive differential equations and inclusions involving Atangana-Baleanu fractional derivative in infinite dimensional spaces

&lt;abstract&gt;&lt;p&gt;The aim of this paper is to derive conditions under which the solution set of a non-local impulsive differential inclusions involving Atangana-Baleanu fractional derivative is a nonempty compact set in an infinite dimensional Banach spaces. Existence results for solutions in the presence of instantaneous or non-instantaneous impulsive effect are given. We considered the case where the right hand side is either a single valued function, or a multifunction. This generalizes recent results to the case when there are impulses, the right hand side is a multifunction, and where the dimension of the space is infinite. Examples are given to illustrate the effectiveness of the established results.&lt;/p&gt;&lt;/abstract&gt;

The Josefson-Nissenzweig property for locally convex spaces

We define a locally convex space E to have the Josefson-Nissenzweig property (JNP) if the identity map (E?, ?(E?, E)) ? (E?, ?*(E?, E)) is not sequentially continuous. By the classical Josefson-Nissenzweig theorem, every infinite-dimensional Banach space has the JNP. A characterization of locally convex spaces with the JNP is given. We thoroughly study the JNP in various function spaces. Among other results we show that for a Tychonoff space X, the function space Cp(X) has the JNP iff there is a weak* null-sequence (?n)n?? of finitely supported sign-measures on X with unit norm. However, for every Tychonoff space X, neither the space B1(X) of Baire-1 functions on X nor the free locally convex space L(X) over X has the JNP.

Gap functions and global error bounds for differential variational–hemivariational inequalities

This paper concerns with the study of a differential variational–hemivariational inequality (DVHVI, for short) in infinite-dimensional Banach spaces. We first introduce the new concept of gap functions for the variational control system of (DVHVI). Then, we consider two kinds of gap functions which are regularized gap function and Moreau–Yosida regularized gap function, respectively, and examine the relevant properties of the gap functions. Moreover, two global error bounds which depend implicitly on the regularized gap function and the Moreau–Yosida regularized gap function, accordingly, are obtained. Finally, in order to illustrate the applicability of the theoretical results, we investigate a coupled dynamic system which is formulated by a nonlinear reaction–diffusion equation described by a time-dependent nonsmooth semipermeability problem.