Arbitrary Banach Space Research Articles

Octahedral norms in duals and biduals of Lipschitz-free spaces

We continue with the study of octahedral norms in the context of spaces of Lipschitz functions and in their duals. First, we prove that the norm of F(M)⁎⁎ is octahedral as soon as M is unbounded or is not uniformly discrete. Further, we prove that a concrete sequence of uniformly discrete and bounded metric spaces (Km) satisfies that the norm of F(Km)⁎⁎ is octahedral for every m. Finally, we prove that if X is an arbitrary Banach space and the norm of Lip0(M) is octahedral, then the norm of Lip0(M,X⁎) is octahedral. These results solve several open problems from the literature.

Measures of Non-Compactness and Sobolev–Lorentz Spaces

We show that the measure of non-compactness of the limiting embedding of Sobolev–Lorentz spaces is equal to the norm. This is a consequence of our general theorem for arbitrary Banach spaces.

Stability of a pair of Banach spaces for coarse Lipschitz embeddings

A pair of Banach spaces (X, Y) is said to be coarsely stable if for every coarse Lipschitz embedding $$f: X\rightarrow Y$$, there exist $$\alpha ,\gamma >0$$ and a Lipschitz mapping $$T:L(f)\rightarrow X$$ with its Lipschitz constant $$\Vert T\Vert _{Lip}\le \alpha $$ such that $$\Vert Tf(x)-x\Vert \le \gamma $$ for all $$x\in X$$, where L(f) is the closed linear span of f(X). In this paper, we study the coarse stability of a pair of Banach spaces (X, Y) when X is an absolute Lipschitz retract (resp. X is an arbitrary Banach space; $$L_2$$)and Y is an arbitrary Banach space (resp. Y is a Hilbert space; $$L_p$$ for $$2<p<+\infty $$).

Bifurcations in periodic integrodifference equations in &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ C(\Omega) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; Ⅱ: Discrete torus bifurcations

We provide a convenient Neimark-Sacker bifurcation result for time-periodic difference equations in arbitrary Banach spaces. It ensures the bifurcation of invariant tori caused by a pair of complex-conjugated Floquet multipliers crossing the complex unit circle. This criterion is made explicit for integrodifference equations, which are infinite-dimensional discrete dynamical systems popular in theoretical ecology, and are used to describe the temporal evolution and spatial dispersal of populations with nonoverlapping generations. As an application, we combine analytical and numerical tools for a detailed bifurcation analysis of a spatial predator-prey model. Since such realistic models can frequently only be studied numerically, we formulate our assumptions in such a fashion as to allow for numerically stable verification.

DENSENESS OF SETS OF CAUCHY PROBLEMS WITHHOUT SOLUTIONS AND WITH NONUNIQUE SOLUTIONS IN THE SET OF ALL CAUCHY PROBLEMS

When finding solutions of differential equations it is necessary to take into account the theorems on innovation and unity of solutions of equations. In case of non-fulfillment of the conditions of these theorems, the methods of finding solutions of the studied equations used in computational mathematics may give erroneous results. It should also be borne in mind that the Cauchy problem for differential equations may have no solutions or have an infinite number of solutions. The author presents two statements obtained by the author about the denseness of sets of the Cauchy problem without solutions (in the case of infinite-dimensional Banach space) and with many solutions (in the case of an arbitrary Banach space) in the set of all Cauchy problems. Using two examples of the Cauchy problem for differential equations, the imperfection of some methods of computational mathematics for finding solutions of the studied equations is shown.

Hypercyclic bilinear operators on Banach spaces

We study the dynamics induced by an m-linear operator. We answer a question of Bès and Conejero showing an example of an m-linear hypercyclic operator acting on a Banach space. Moreover, we prove the existence of m-linear hypercyclic operators on arbitrary infinite dimensional separable Banach spaces. We also prove an existence result about symmetric bihypercyclic bilinear operators, answering a question by Grosse-Erdmann and Kim.

Characterization of k-smooth operators between Banach spaces

We study k-smoothness of bounded linear operators defined between arbitrary Banach spaces. As an application, we characterize k-smooth operators defined from ℓ1n to an arbitrary Banach space. We also completely characterize k-smooth operators defined between arbitrary two-dimensional Banach spaces.

Hölder strong metric subregularity and its applications to convergence analysis of inexact Newton methods

<p style='text-indent:20px;'>In this paper we conduct local convergence analysis of the inexact Newton methods for solving the generalized equation <inline-formula><tex-math id="M1">\begin{document}$ 0\in f(x)+F(x) $\end{document}</tex-math></inline-formula> under the assumption of Hölder strong metric subregularity, where <inline-formula><tex-math id="M2">\begin{document}$ f : X \rightarrow Y $\end{document}</tex-math></inline-formula> is a single-valued mapping while <inline-formula><tex-math id="M3">\begin{document}$ F : X \rightrightarrows Y $\end{document}</tex-math></inline-formula> is a set-valued mapping between arbitrary Banach spaces. Our work are proceeded as twofold: we first explore fully the property of Hölder strong metric subregularity by establishing a verifiable necessary and sufficient condition as well as discussing its stability under small perturbations, and secondly, with the help of aforementioned theoretical analysis, we conclude that every sequence generated by the inexact (quasi) Newton method and staying in a neighborhood of the solution <inline-formula><tex-math id="M4">\begin{document}$ \bar x $\end{document}</tex-math></inline-formula> is convergent (superlinearly) of order <inline-formula><tex-math id="M5">\begin{document}$ p(1+q) $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M6">\begin{document}$ p $\end{document}</tex-math></inline-formula> is the order of Hölder strong metric subregularity imposed on the mapping <inline-formula><tex-math id="M7">\begin{document}$ f+F $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ q $\end{document}</tex-math></inline-formula> is the order of Hölder calmness property for the derivative <inline-formula><tex-math id="M9">\begin{document}$ Df $\end{document}</tex-math></inline-formula> while <inline-formula><tex-math id="M10">\begin{document}$ p $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M11">\begin{document}$ q $\end{document}</tex-math></inline-formula> complement each other as long as <inline-formula><tex-math id="M12">\begin{document}$ p(1+q)\geq 1 $\end{document}</tex-math></inline-formula>.

A local characterization of Kazhdan projections and applications

We give a local characterization of the existence of Kazhdan projections for arbitary families of Banach space representations of a compactly generated locally compact group G . We also define and study a natural generalization of the Fell topology to arbitrary Banach space representations of a locally compact group. We give several applications in terms of stability of rigidity under perturbations. Among them, we show a Banach-space version of the Delorme–Guichardet theorem stating that property (T) and (FH) are equivalent for \sigma -compact locally compact groups. Another kind of applications is that many forms of Banach strong property (T) are open in the space of marked groups, and more generally every group with such a property is a quotient of a compactly presented group with the same property. We also investigate the notions of central and non central Kazhdan projections, and present examples of non central Kazhdan projections coming from hyperbolic groups.

In Koenigs' footsteps: Diagonalization of composition operators

Let φ:D→D be a holomorphic map with a fixed point α∈D such that 0≤|φ′(α)|<1. We show that the spectrum of the composition operator Cφ on the Fréchet space Hol(D) is {0}∪{φ′(α)n:n=0,1,⋯} and its essential spectrum is reduced to {0}. This contrasts the situation where a restriction of Cφ to Banach spaces such as H2(D) is considered. Our proofs are based on explicit formulae for the spectral projections associated with the point spectrum found by Koenigs. Finally, as a byproduct, we obtain information on the spectrum for bounded composition operators induced by a Schröder symbol on arbitrary Banach spaces of holomorphic functions.

Nonlinear Caputo Fractional Derivative with Nonlocal Riemann-Liouville Fractional Integral Condition Via Fixed Point Theorems

In this paper, we study and investigate an interesting Caputo fractional derivative and Riemann–Liouville integral boundary value problem (BVP): c D 0 + q u ( t ) = f ( t , u ( t ) ) , t ∈ [ 0 , T ] , u ( k ) ( 0 ) = ξ k , u ( T ) = ∑ i = 1 m β i R L I 0 + p i u ( η i ) , where n − 1 &lt; q &lt; n , n ≥ 2 , m , n ∈ N , ξ k , β i ∈ R , k = 0 , 1 , … , n − 2 , i = 1 , 2 , … , m , and c D 0 + q is the Caputo fractional derivatives, f : [ 0 , T ] × C ( [ 0 , T ] , E ) → E , where E is the Banach space. The space E is chosen as an arbitrary Banach space; it can also be R (with the absolute value) or C ( [ 0 , T ] , R ) with the supremum-norm. RL I 0 + p i is the Riemann–Liouville fractional integral of order p i &gt; 0 , η i ∈ ( 0 , T ) , and ∑ i = 1 m β i η i p i + n − 1 Γ ( n ) Γ ( n + p i ) ≠ T n − 1 . Via the fixed point theorems of Krasnoselskii and Darbo, the authors study the existence of solutions to this problem. An example is included to illustrate the applicability of their results.

Controllability of impulsive fractional functional evolution equations with infinite state‐dependent delay in Banach spaces

Many evolutionary processes from various fields of physical and engineering sciences undergo abrupt changes of state at certain moments of time between intervals of continuous evolution. These processes are more suitably modeled by impulsive differential equations. In this paper, we study the controllability of an impulsive fractional differential equation with infinite state‐dependent delay in an arbitrary Banach space. We apply semigroup theory and Schaefer fixed point theorem. As an application, we include an example to illustrate the theory.

Small Pre-Quasi Banach Operator Ideals of Type Orlicz-Cesáro Mean Sequence Spaces

In this paper, we give the sufficient conditions on Orlicz-Cesáro mean sequence spaces cesφ, where φ is an Orlicz function such that the class Scesφ of all bounded linear operators between arbitrary Banach spaces with its sequence of s-numbers which belong to cesφ forms an operator ideal. The completeness and denseness of its ideal components are specified and Scesφ constructs a pre-quasi Banach operator ideal. Some inclusion relations between the pre-quasi operator ideals and the inclusion relations for their duals are explained. Moreover, we have presented the sufficient conditions on cesφ such that the pre-quasi Banach operator ideal generated by approximation number is small. The above results coincide with that known for cesp (1&lt;p&lt;∞).

The Bolzano–Poincaré–Miranda theorem in infinite-dimensional Banach spaces

We study the existence of zeroes of mappings defined in Banach spaces. We obtain, in particular, an extension of the well-known Bolzano–Poincare–Miranda theorem to infinite-dimensional Banach spaces. We also establish a result regarding the existence of periodic solutions to differential equations posed in an arbitrary Banach space.

Periodic approximation of exceptional Lyapunov exponents for semi-invertible operator cocycles

We prove that for semi-invertible and H\"older continuous linear cocycles $A$ acting on an arbitrary Banach space and defined over a base space that satisfies the Anosov Closing Property, all exceptional Lyapunov exponents of $A$ with respect to an ergodic invariant measure for base dynamics can be approximated with Lyapunov exponents of $A$ with respect to ergodic measures supported on periodic orbits. Our result is applicable to a wide class of infinite-dimensional dynamical systems.

Small operator ideals formed by s numbers on generalized Ces\xe1ro and Orlicz sequence spaces

In this article, we establish sufficient conditions on the generalized Cesáro and Orlicz sequence spaces mathbb{E} such that the class S_{mathbb{E}} of all bounded linear operators between arbitrary Banach spaces with its sequence of s-numbers belonging to mathbb{E} generates an operator ideal. The components of S_{mathbb{E}} as a pre-quasi Banach operator ideal containing finite dimensional operators as a dense subset and its completeness are proved. Some inclusion relations between the operator ideals as well as the inclusion relations for their duals are obtained. Finally, we show that the operator ideal formed by mathbb{E} and approximation numbers is small under certain conditions.

Duality and distance formulas in Banach function spaces

We consider pairs of non reflexive Banach spaces $$(E_0, E)$$ such that $$E_0$$ is defined in terms of a little-o condition and E is defined by the corresponding big-O condition. Under suitable assumptions on the pair $$(E_0, E)$$ there exists a reflexive and separable Banach space X (in which E is continuously embedded and dense) naturally associated to E which characterizes quantitatively weak compactness of bounded linear operators $$\begin{aligned} T: E_0 \rightarrow Z \end{aligned}$$ where Z is an arbitrary Banach space. Pairs include (VMO, BMO), where BMO is the space of John-Nirenberg, $$(B_0, B)$$ where B is a recently introduced space by Bourgain-Brezis-Mironescu ([6]) and some Orlicz pairs $$(L^{\psi }_0, L^{\psi })$$ where $$L^{\psi }_0$$ is the closure of $$L^\infty $$ in the Orlicz space $$L^{\psi }$$ , Marcinkiewicz pairs $$(L^{q, \infty }_0, L^{q, \infty })$$ where $$L^{q, \infty }_0$$ is the closure of $$L^\infty $$ in the Marcinkiewicz weak– $$L^q$$ denoted by $$L^{q, \infty }$$ . More generally, Banach function spaces are considered. The main results are duality formulas of the type 1 $$\begin{aligned} E_0^{**}&\simeq E\qquad \text {isometrically}\end{aligned}$$ 2 $$\begin{aligned}E^{*}&\simeq E_0^*\oplus _1 E_0^\perp\end{aligned}$$ and distance formulas.

On Selections from the Best n-Nets

The discontinuity of any selection from a best n-net for n ≥ 2 in an arbitrary not strictly convex Banach space is proved. It is also proved that there is no Lipschitz selection on an arbitrary Banach space of dimension at least 2 whose unit sphere contains an attainable point of smoothness.

Extending surjective isometries defined on the unit sphere of $$\ell _\infty (\Gamma )$$ ℓ ∞ ( Γ )

Let $$\Gamma $$ be an infinite set equipped with the discrete topology. We prove that the space $$\ell _{\infty }(\Gamma ,{\mathbb {C}}),$$ of all complex-valued bounded functions on $$\Gamma $$ , satisfies the Mazur–Ulam property, that is, every surjective isometry from the unit sphere of $$\ell _{\infty }(\Gamma , {\mathbb {C}})$$ onto the unit sphere of an arbitrary complex Banach space X admits a unique extension to a surjective real linear isometry from $$\ell _{\infty }(\Gamma ,{\mathbb {C}})$$ to X.

General Embedding Theorem

The quotient space of an arbitrary Banach space B by any subspace B1 ⊂ B equipped with the norm \(||b|B/{B_1}|| = \mathop {\inf }\limits_{f \in B/{B_1}} ||f|b||\) is considered. In the case where the infimum is a minimum, i.e., it is attained at some element, a formula for this element is presented. The proof is based on restating the original problem in dual spaces with the help of corresponding Legendre transforms. Although the original problem is nonlinear, it is found that its formulation in dual spaces is always linear and solvable. The results are applied to the general theory of boundary value problems for differential equations of mathematical physics.