Open Mapping Theorem Research Articles

Fuzzy Open Mapping and Fuzzy Closed Graph Theorems in Fuzzy Length Space

The theory of fuzzy set includes many aspects that regard important and significant in different fields of science and engineering in addition to there applications. Fuzzy metric and fuzzy normed spaces are essential structures in the fuzzy set theory. The concept of fuzzy length space has been given analogously and the properties of this space are studied few years ago. In this work, the definition of a fuzzy open linear operator is presented for the first time and the fuzzy Barise theorem is established to prove the fuzzy open mapping theorem in a fuzzy length space. Finally, the definition of a fuzzy closed linear operator on fuzzy length space is introduced to prove the fuzzy closed graph theorem.

Some Fundamental Theorems of Functional Analysis with Bicomplex and Hyperbolic Scalars

We discuss some properties of linear functionals on topological hyperbolic and topological bicomplex modules. The hyperbolic and bicomplex analogues of the uniform boundedness principle, the open mapping theorem, the closed graph theorem and the Hahn Banach separation theorem are proved.

Third order open mapping theorems and applications to the end-point map**The first author has been supported by University of Padova STARS Project ‘Sub-Riemannian geometry and geometric measure theory issues: old and new’.

This paper is devoted to a third order study of the end-point map in sub-Riemannian geometry. We first prove third order open mapping results for maps from a Banach space into a finite dimensional manifold. In a second step, we compute the third order term in the Taylor expansion of the end-point map and we specialize the abstract theory to the study of length-minimality of sub-Riemannian strictly singular curves. We conclude with the third order analysis of a specific strictly singular extremal that is not length-minimizing.

A Class of Analytic Starlike Functions Associated with Petal Like Region on the Positive Half of Complex Plane

In the light of Riemann open mapping theorem, if we map open unit disk U conformally onto a region then depending on the geometry of boundary of we can always extract a subclass of H[a, n] by subordinating various functionals of the function f ∈ H[a, n]. Depending upon the geometry of the range set attempts have been made to find some algebraic structure in such classes, for that Hankel determinant of coefficients of functions pertaining to these classes have been studied, bounds of various coefficients have been determined and also based on the subordination principle we have determined radius |z| < r ;z ∈ U for which f belongs to such a class. In this paper our focus would be on n−PS<sup>*</sup> defined as n − PS<sup>*</sup> = {f ∈ A : Re {zf<sup>'</sup>(z)/f(z)} > 0,|(zf<sup>'</sup>(z)/f(z))<sup>n</sup> - 1|<1}.

Passively realizable approximations of non-realizable fractional order impedance functions

Approximating the fractional order differentiation and integration operators is a common approach in implementation of fractional order dynamics. This paper aims to investigate how the procedure of approximating the fractional order operators influences on the realizability of a fractional order impedance function by passive networks. To this aim, conditions for the possibility of passive realization of the approximations of the fractional order impedance functions by using RLC components are obtained. More precisely, considering two general forms for the filters approximating the fractional order operators, the open mapping theorem in complex analysis is applied to obtain the realizability conditions on the polar plots of the approximating filters. It is found that the approximated impedance function may be realizable by a passive RLC network, whereas the original fractional order impedance function cannot be realized by passive networks composed of resistors and fractional inductors and capacitors. Furthermore, for a class of impedance functions, the realizability condition is simplified as a condition on the phase of the filter approximating the fractional order differentiation operator. Some examples are presented to verify the usefulness of the obtained conditions.

Optimality of the quantified Ingham\u2013Karamata theorem for operator semigroups with general resolvent growth

We prove that a general version of the quantified Ingham–Karamata theorem for C_0-semigroups is sharp under mild conditions on the resolvent growth, thus generalising the results contained in a recent paper by the same authors. It follows in particular that the well-known Batty–Duyckaerts theorem is optimal even for bounded C_0-semigroups whose generator has subpolynomial resolvent growth. Our proof is based on an elegant application of the open mapping theorem, which we complement by a crucial technical lemma allowing us to strengthen our earlier results.

An abstract approach to optimal decay of functions and operator semigroups

We provide a new and significantly shorter optimality proof of recent quantified Tauberian theorems, both in the setting of vector-valued functions and of $C_0$-semigroups, and in fact our results are also more general than those currently available in the literature. Our approach relies on a novel application of the open mapping theorem.

Order-Preserving Variants of the Basic Principles of Functional Analysis

We will establish order-preserving versions of the basic principles of functional analysis such as Hahn-Banach, Banach-Steinhaus, open mapping, and Banach-Alaoglu theorems.

Generalized Open Mapping Theorem for X-Normed Spaces

The theory of X-normed spaces over non-Archimedean valued fields with valuations of higher rank was introduced by H. Ochsenius and W. H. Schikhof in [9] and further developed in [10–12, 16, 17] and [13]. In order to obtain results like the Open Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem, H. Ochsenius and W. H. Schikhof used 1st countability conditions in the value group of the based field. In this article the author develops a new tool to work with transfinite induction simplifying the techniques employed in X-normed spaces, thus accomplishing a Generalized Baire Category Theorem that allows the proof of an Open Mapping Theorem for X-normed spaces without restrictions on the value group of the based field. Additionally, some contributions to the theory of X-normed spaces are presented regarding quotient spaces.

Division algebras of slice functions

AbstractThis work studies slice functions over finite-dimensional division algebras. Their zero sets are studied in detail along with their multiplicative inverses, for which some unexpected phenomena are discovered. The results are applied to prove some useful properties of the subclass of slice regular functions, previously known only over quaternions. Firstly, they are applied to derive from the maximum modulus principle a version of the minimum modulus principle, which is in turn applied to prove the open mapping theorem. Secondly, they are applied to prove, in the context of the classification of singularities, the counterpart of the Casorati-Weierstrass theorem.

Constrained controllability of second order retarded nonlinear systems with nonlocal condition

AbstractIn this paper, the constrained controllability of the second order retarded nonlinear systems with nonlocal condition has been established by using the theory of cosine families and the generalized open mapping theorem. A new set of sufficient conditions for the constrained controllability of retarded nonlinear systems is established under the assumption that the associated linear system is controllable. By using the Banach fixed point theorem, the existence of mild solution for the considered system with nonlocal delay condition has been deduced. Finally, an example is provided to verify the effectiveness of the obtained theory.

Hyers–Ulam stability of impulsive integral equations

In this manuscript, first we study the Hyers–Ulam stability of linear impulsive Volterra integral equations with the help of open mapping theorem approach. Then we give an existence and uniqueness theorem for the solutions of a class of nonlinear impulsive integral equations with a bounded variable delay. Moreover, with the help of integral inequality of Gronwall type, we investigate the Hyers–Ulam stability and Hyers–Ulam–Rassias stability of the same class of integral equations. We also present examples to support our main results.

An open mapping theorem for finitely copresented Esakia spaces

We prove an open mapping theorem for the topological spaces dual to finitely presented Heyting algebras. This yields in particular a short, self-contained semantic proof of the uniform interpolation theorem for intuitionistic propositional logic, first proved by Pitts in 1992. Our proof is based on the methods of Ghilardi & Zawadowski. However, our proof does not require sheaves nor games, only basic duality theory for Heyting algebras.

Strong Klee–Andô Theorems through an Open Mapping Theorem for cone-valued multi-functions

A version of the classical Klee–Andô Theorem states the following: For every Banach space X, ordered by a closed generating cone C⊆X, there exists some α>0 so that, for every x∈X, there exist x±∈C so that x=x+−x− and ‖x+‖+‖x−‖≤α‖x‖.The conclusion of the Klee–Andô Theorem is what is known as a conormality property.We prove stronger and somewhat more general versions of the Klee–Andô Theorem for both conormality and coadditivity (a property that is intimately related to conormality). A corollary to our result shows that the functions x↦x±, as above, may be chosen to be bounded, continuous, and positively homogeneous, with a similar conclusion yielded for coadditivity. Furthermore, we show that the Klee–Andô Theorem generalizes beyond ordered Banach spaces to Banach spaces endowed with arbitrary collections of cones. Proofs of our Klee–Andô Theorems are achieved through an Open Mapping Theorem for cone-valued multi-functions/correspondences.We very briefly discuss a potential further strengthening of The Klee–Andô Theorem beyond what is proven in this paper, and motivate a conjecture that there exists a Banach space X, ordered by a closed generating cone C⊆X, for which there exist no Lipschitz functions (⋅)±:X→C satisfying x=x+−x− for all x∈X.

Quotient operators and the open mapping theorem

Quotients of bounded operators on normed spaces have been discussed. Openmapping theorem for quotients of bounded operators and its consequences are given.

The multiplier algebra of the noncommutative Schwartz space

We describe the multiplier algebra of the noncommutative Schwartz space. This multiplier algebra can be seen as the largest ${}^*$-algebra of unbounded operators on a separable Hilbert space with the classical Schwartz space of rapidly decreasing functions as the domain. We show in particular that it is neither a $\mathcal{Q}$-algebra nor $m$-convex. On the other hand, we prove that classical tools of functional analysis, for example, the closed graph theorem, the open mapping theorem or the uniform boundedness principle, are still available.

On Geometric Aspects of Quaternionic and Octonionic Slice Regular Functions

The aim of this paper is twofold. On the one hand, we enrich from a geometrical point of view the theory of octonionic slice regular functions. We first prove a boundary Schwarz lemma for slice regular self-mappings of the open unit ball of the octonionic space. As applications, we obtain two Landau–Toeplitz type theorems for slice regular functions with respect to regular diameter and slice diameter, respectively, together with a Cauchy type estimate. Along with these results, we introduce some new and useful ideas, which also allow us to prove the minimum principle and one version of the open mapping theorem. On the other hand, we adopt a completely new approach to strengthen a version of boundary Schwarz lemma first proved in Ren and Wang (Trans Am Math Soc 369:861–885, 2017) for quaternionic slice regular functions. Our quaternionic boundary Schwarz lemma with optimal estimate improves considerably a well-known Osserman type estimate and provides additionally all the extremal functions.

Mild Solution and Constrained Local Controllability of Semilinear Boundary Control Systems

The existence of mild solution and the constrained local controllability of a retarded boundary control system with nonlocal delay condition have been established. The theory of extrapolation spaces is applied to derive the mild solution. Then, the constrained local controllability is established using the generalized open mapping theorem. In the last section, application of the result is shown through examples of control systems represented by hyperbolic partial differential equations.

Extended seminorms and extended topological vector spaces

We introduce the notions of extended topological vector spaces and extended seminormed spaces, following the main ideas of extended normed spaces, which were introduced by G. Beer and J. Vanderwerff. We provide a topological study of such structures, giving a unifying theory with main applications in the study of spaces of continuous functions. We also generalize classical results of functional analysis, as open mapping theorem and closed graph theorem.

A Banach–Dieudonné theorem for the space of bounded continuous functions on a separable metric space with the strict topology

Let X be a separable metric space and let β be the strict topology on the space of bounded continuous functions on X, which has the space of τ-additive Borel measures as a continuous dual space. We prove a Banach–Dieudonné type result for the space of bounded continuous functions equipped with β: the finest locally convex topology on the dual space that coincides with the weak topology on all weakly compact sets is a k-space. As a consequence, the space of bounded continuous functions with the strict topology is hypercomplete and a Pták space. Additionally, the closed graph, inverse mapping and open mapping theorems holds for linear maps between space of this type.