892 results on '"Open mapping theorem (functional analysis)"'
Search Results
2. Compact operators and integral equations in the $\cal{HK}$ space
- Author
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Varayu Boonpogkrong
- Subjects
Combinatorics ,Sequence ,Functional analysis ,Fréchet space ,Closed graph theorem ,Open mapping theorem (functional analysis) ,Compact operator ,Space (mathematics) ,Mathematics ,Continuous linear operator - Abstract
The space $${\cal H}{\cal K}$$ of Henstock-Kurzweil integrable functions on [a, b] is the uncountable union of Frechet spaces $${\cal H}{\cal K}$$ (X). In this paper, on each Frechet space $${\cal H}{\cal K}$$ (X), an F-norm is defined for a continuous linear operator. Hence, many important results in functional analysis, like the Banach-Steinhaus theorem, the open mapping theorem and the closed graph theorem, hold for the $${\cal H}{\cal K}$$ (X) space. It is known that every control-convergent sequence in the $${\cal H}{\cal K}$$ space always belongs to a $${\cal H}{\cal K}$$ (X) space for some X. We illustrate how to apply results for Frechet spaces $${\cal H}{\cal K}$$ (X) to control-convergent sequences in the $${\cal H}{\cal K}$$ space. Examples of compact linear operators are given. Existence of solutions to linear and Hammerstein integral equations is proved.
- Published
- 2021
3. Some topological properties of topological rough groups
- Author
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Jinjin Li, Yujin Lin, Qianqian Sun, and Fucai Lin
- Subjects
Physics ,0209 industrial biotechnology ,Group (mathematics) ,General Topology (math.GN) ,Inverse ,Group Theory (math.GR) ,02 engineering and technology ,Topology ,Space (mathematics) ,Theoretical Computer Science ,Separation axiom ,020901 industrial engineering & automation ,Product (mathematics) ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,Equivalence relation ,020201 artificial intelligence & image processing ,Geometry and Topology ,Open mapping theorem (functional analysis) ,Identity element ,Mathematics - Group Theory ,Primary: 22A05, 54A05. Secondary: 03E25 ,Software ,Mathematics - General Topology - Abstract
Let $(U, R)$ be an approximation space with $U$ being non-empty set and $R$ being an equivalence relation on $U$, and let $\overline{G}$ and $\underline{G}$ be the upper approximation and the lower approximation of subset $G$ of $U$. A topological rough group $G$ is a rough group $G=(\underline{G}, \overline{G})$ endowed with a topology, which is induced from the upper approximation space $\overline{G}$, such that the product mapping $f: G\times G\rightarrow \overline{G}$ and the inverse mapping are continuous. In the class of topological rough groups, the relations of some separation axioms are obtained, some basic properties of the neighborhoods of the rough identity element and topological rough subgroups are investigated. In particular, some examples of topological rough groups are provided to clarify some facts about topological rough groups. Moreover, the version of open mapping theorem in the class of topological rough group is obtained. Further, some interesting open questions are posed., 19 pages
- Published
- 2021
4. Fuzzy Open Mapping and Fuzzy Closed Graph Theorems in Fuzzy Length Space
- Author
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Raghad I. Sabri
- Subjects
Mathematics::General Mathematics ,Fuzzy set ,Pharmaceutical Science ,Space (mathematics) ,Fuzzy logic ,Linear map ,Algebra ,ComputingMethodologies_PATTERNRECOGNITION ,Complementary and alternative medicine ,Metric (mathematics) ,Graph (abstract data type) ,Pharmacology (medical) ,Closed graph theorem ,ComputingMethodologies_GENERAL ,Open mapping theorem (functional analysis) ,Mathematics - Abstract
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.
- Published
- 2020
5. Passively realizable approximations of non-realizable fractional order impedance functions
- Author
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Mohammad Saleh Tavazoei
- Subjects
0209 industrial biotechnology ,Computer Networks and Communications ,Applied Mathematics ,020208 electrical & electronic engineering ,02 engineering and technology ,Function (mathematics) ,Filter (signal processing) ,020901 industrial engineering & automation ,Operator (computer programming) ,Control and Systems Engineering ,Realizability ,Signal Processing ,0202 electrical engineering, electronic engineering, information engineering ,RLC circuit ,Applied mathematics ,Open mapping theorem (functional analysis) ,Electrical impedance ,Realization (systems) ,Mathematics - Abstract
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.
- Published
- 2020
6. Applications of Baire’s category theorem in complex analysis in one and several complex variables
- Author
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Konstantinos Makridis
- Subjects
Discrete mathematics ,Uniform boundedness principle ,Several complex variables ,Baire category theorem ,Baire space ,Property of Baire ,Open mapping theorem (functional analysis) ,Baire measure ,Mathematics ,S category - Abstract
Στη συγκεκριμένη διατριβή εξετάζουμε μερικές εφαρμογές του θεωρήματος Baire στη Μιγαδική ανάλυση, τόσο στη μία όσο και σε πολλές μιγαδικές μεταβλητές. Τα αποτελέσματα μας σχετίζονται με την έννοια της υπερκυκλικότητας, των πουθενά παραγωγίσιμων (μιγαδικών) συναρτήσεων, των προσεγγιστών Pade, όπως επίσης και με τις καθολικές σειρές Laurent.
- Published
- 2021
7. Generalized Open Mapping Theorem for X-Normed Spaces
- Author
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Angel Barria Comicheo
- Subjects
Pure mathematics ,General Mathematics ,010102 general mathematics ,01 natural sciences ,Transfinite induction ,0103 physical sciences ,Countable set ,Uniform boundedness ,Baire category theorem ,Closed graph theorem ,010307 mathematical physics ,0101 mathematics ,Open mapping theorem (functional analysis) ,Quotient ,Mathematics - Abstract
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.
- Published
- 2019
8. Strong Klee–Andô Theorems through an Open Mapping Theorem for cone-valued multi-functions
- Author
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Miek Messerschmidt
- Subjects
Mathematics::Functional Analysis ,021103 operations research ,Conjecture ,010102 general mathematics ,0211 other engineering and technologies ,Banach space ,02 engineering and technology ,46B20, 46A30, 46B40, 32A12 ,Lipschitz continuity ,Mathematical proof ,01 natural sciences ,Mathematics - Functional Analysis ,Combinatorics ,Corollary ,Homogeneous ,Bounded function ,0101 mathematics ,Open mapping theorem (functional analysis) ,Analysis ,Mathematics - Abstract
A version of the classical Klee-And\^o Theorem states the following: For every Banach space $X$, ordered by a closed generating cone $C\subseteq X$, there exists some $\alpha>0$ so that, for every $x\in X$, there exist $x^{\pm}\in C$ so that $x=x^{+}-x^{-}$ and $\|x^{+}\|+\|x^{-}\|\leq\alpha\|x\|$. The conclusion of the Klee-And\^o Theorem is what is known as a conormality property. We prove stronger and somewhat more general versions of the Klee-And\^o Theorem for both conormality and coadditivity (a property that is intimately related to conormality). A corollary to our result shows that the functions $x\mapsto x^{\pm}$, 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-And\^o Theorem generalizes beyond ordered Banach spaces to Banach spaces endowed with arbitrary collections of cones. Proofs of our Klee-And\^o 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-And\^o 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 $C\subseteq X$, for which there exist no Lipschitz functions $(\cdot)^{\pm}:X\to C$ satisfying $x=x^{+}-x^{-}$ for all $x\in X$., Comment: Major rewrite. Large parts were removed which a referee pointed out can be proven through much easier methods
- Published
- 2018
9. Hyers–Ulam stability of impulsive integral equations
- Author
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Farhan Ullah Khan, Usman Riaz, and Akbar Zada
- Subjects
Mathematics::Functional Analysis ,Picard–Lindelöf theorem ,General Mathematics ,010102 general mathematics ,Mathematics::Classical Analysis and ODEs ,01 natural sciences ,Volterra integral equation ,Stability (probability) ,Integral equation ,symbols.namesake ,Nonlinear system ,Bounded function ,symbols ,Applied mathematics ,0101 mathematics ,Open mapping theorem (functional analysis) ,Mathematics ,Variable (mathematics) - Abstract
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.
- Published
- 2018
10. A Fundamental Property of Bilinear Operators
- Author
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Larry Downey
- Subjects
Bilinear operator ,Property (philosophy) ,General Mathematics ,Bilinear interpolation ,bilinear ,Algebra ,Range (mathematics) ,Open mapping ,Dimension (vector space) ,47H60 ,Simple (abstract algebra) ,Rudin ,repelling point ,Open mapping theorem (functional analysis) ,open mapping ,46G25 ,Mathematics - Abstract
In this paper, we address a fundamental question about the nature of bilinear operators first posed by Walter Rudin. While it has been known that there is not an open mapping theorem for bilinear operators in general, we show that bilinear operators enjoy an open mapping theorem, when the range has dimension three or less, and we address the simple, but crucial property that differentiates bilinear operators from their linear counterparts. We also give an example which shows that polynomials, in general do not enjoy an open mapping theorem, even when the range has dimension three.
- Published
- 2020
11. Some Fundamental Theorems of Functional Analysis with Bicomplex and Hyperbolic Scalars
- Author
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Romesh Kumar, Aditi Sharma, and Heera Saini
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,Uniform boundedness principle ,Mathematics::K-Theory and Homology ,Applied Mathematics ,Mathematics::Classical Analysis and ODEs ,Hahn–Banach theorem ,Closed graph theorem ,Mutual fund separation theorem ,Open mapping theorem (functional analysis) ,Mathematics - Abstract
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.
- Published
- 2020
12. An open mapping theorem for finitely copresented Esakia spaces
- Author
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Samuel J. van Gool, Luca Reggio, ILLC (FNWI), and Logic and Computation (ILLC, FNWI/FGw)
- Subjects
010102 general mathematics ,General Topology (math.GN) ,0102 computer and information sciences ,Mathematics - Logic ,Mathematics - Rings and Algebras ,Topological space ,Propositional calculus ,01 natural sciences ,Algebra ,Mathematics::Logic ,Rings and Algebras (math.RA) ,010201 computation theory & mathematics ,Computer Science::Logic in Computer Science ,FOS: Mathematics ,Geometry and Topology ,0101 mathematics ,Open mapping theorem (functional analysis) ,Logic (math.LO) ,Mathematics ,Mathematics - General Topology - Abstract
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., 8 pages. Minor changes in presentation. To appear in Topology and its Applications
- Published
- 2020
13. The Uniform Boundedness Principle
- Author
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Christian Clason
- Subjects
Pointwise ,Pure mathematics ,Functional analysis ,Uniform boundedness principle ,Core (graph theory) ,Closed graph theorem ,Open mapping theorem (functional analysis) ,Bounded inverse theorem ,Complete metric space ,Mathematics - Abstract
The uniform boundedness principle is one of the core principles of functional analysis: certain pointwise properties of linear operators on a complete space hold in fact uniformly. Starting from Baire’s theorem, this chapter covers some of the main theorems of functional analysis: the Banach–Steinhaus theorem, the open mapping theorem, the bounded inverse theorem, and the closed graph theorem.
- Published
- 2020
14. Algebraic Structures of Mathematical Foundations
- Author
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Robert Murray Jones
- Subjects
Algebra ,Factorization ,Functional analysis ,Algebraic structure ,Open mapping theorem (functional analysis) ,Algebra over a field - Abstract
In this paper we undertake to examine how algebra, its tools and its methods, can be used to formulate the mathematics used in applications. We give particular attention to the mathematics used in application to physics. We suggest that methods first proposed by Henry Siggins Leonard are well suited to such an examination.
- Published
- 2018
15. A remark on the Brézis–Browder theorem in finite dimensional spaces
- Author
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John Cotrina
- Subjects
Discrete mathematics ,Mathematics::Functional Analysis ,021103 operations research ,Control and Optimization ,Fundamental theorem ,Applied Mathematics ,010102 general mathematics ,0211 other engineering and technologies ,Banach space ,Fundamental theorem of linear algebra ,02 engineering and technology ,Management Science and Operations Research ,01 natural sciences ,Tychonoff's theorem ,Fréchet space ,Compactness theorem ,0101 mathematics ,Open mapping theorem (functional analysis) ,Brouwer fixed-point theorem ,Mathematics - Abstract
Brezis and Browder gave an important characterization of maximal monotonicity for linear operators, in reflexive Banach spaces. In this paper, we show that this characterization also preserves the maximal p-monotonicity, in finite dimensional spaces.
- Published
- 2017
16. A quantitative version of the Johnson–Rosenthal theorem
- Author
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Dongyang Chen
- Subjects
Unbounded operator ,Control and Optimization ,Johnson–Rosenthal theorem ,Banach space ,01 natural sciences ,Separable space ,Surjective function ,Fréchet space ,0101 mathematics ,Open mapping theorem (functional analysis) ,46B15 ,Mathematics ,Discrete mathematics ,Algebra and Number Theory ,isomorphisms ,High Energy Physics::Phenomenology ,quantitative versions ,010102 general mathematics ,010101 applied mathematics ,Banach spaces ,Bessaga–Pełczyński theorem ,High Energy Physics::Experiment ,46C05 ,Closed graph theorem ,Isomorphism ,Analysis - Abstract
Let $X,Y$ be Banach spaces. We define \begin{equation*}\alpha_{Y}(X)=\sup\{\vert T^{-1}\vert^{-1}:T:Y\rightarrow X\mbox{ is an isomorphism with }\vert T\vert \leq1\}.\end{equation*} If there is no isomorphism from $Y$ to $X$ , we set $\alpha_{Y}(X)=0$ , and ¶ \begin{equation*}\gamma_{Y}(X)=\sup\{\delta(T):T:X\rightarrow Y\mbox{ is asurjective operator with }\vert T\vert \leq1\},\end{equation*} where $\delta(T)=\sup\{\delta\gt 0:\delta B_{Y}\subseteq TB_{X}\}$ . If there is no surjective operator from $X$ onto $Y$ , we set $\gamma_{Y}(X)=0$ . We prove that for a separable space $X$ , $\alpha_{l_{1}}(X^{*})=\gamma_{c_{0}}(X)$ and $\alpha_{L_{1}}(X^{*})=\gamma_{C(\Delta)}(X)=\gamma_{C[0,1]}(X)$ .
- Published
- 2017
17. Non-self multivariate contraction mapping principle in Banach spaces
- Author
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Yongchun Xu, Yongfu Su, Jinyu Guan, and Yanxia Tang
- Subjects
Pure mathematics ,Algebra and Number Theory ,010102 general mathematics ,Mathematical analysis ,Eberlein–Šmulian theorem ,Banach space ,Banach manifold ,01 natural sciences ,010101 applied mathematics ,Fréchet space ,Interpolation space ,Contraction mapping ,0101 mathematics ,Open mapping theorem (functional analysis) ,Lp space ,Analysis ,Mathematics - Published
- 2017
18. Sequential Analogues of the Lyapunov and Krein–Milman Theorems in Fréchet Spaces
- Author
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F. S. Stonyakin
- Subjects
Statistics and Probability ,Discrete mathematics ,Mathematics::Functional Analysis ,Dual space ,Applied Mathematics ,General Mathematics ,010102 general mathematics ,Eberlein–Šmulian theorem ,Banach space ,Hahn–Banach theorem ,02 engineering and technology ,01 natural sciences ,Fréchet space ,Locally convex topological vector space ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,Closed graph theorem ,0101 mathematics ,Open mapping theorem (functional analysis) ,Mathematics - Abstract
In this paper we develop the theory of anti-compact sets we introduced earlier. We describe the class of Frechet spaces where anti-compact sets exist. They are exactly the spaces that have a countable set of continuous linear functionals. In such spaces we prove an analogue of the Hahn–Banach theorem on extension of a continuous linear functional from the original space to a space generated by some anti-compact set. We obtain an analogue of the Lyapunov theorem on convexity and compactness of the range of vector measures, which establishes convexity and a special kind of relative weak compactness of the range of an atomless vector measure with values in a Frechet space possessing an anti-compact set. Using this analogue of the Lyapunov theorem, we prove the solvability of an infinite-dimensional analogue of the problem of fair division of resources. We also obtain an analogue of the Lyapunov theorem for nonadditive analogues of measures that are vector quasi-measures valued in an infinite-dimensional Frechet space possessing an anti-compact set. In the class of Frechet spaces possessing an anti-compact set, we obtain analogues of the Krein–Milman theorem on extreme points for convex bounded sets that are not necessarily compact. A special place is occupied by analogues of the Krein–Milman theorem in terms of extreme sequences introduced in the paper (the so-called sequential analogues of the Krein–Milman theorem).
- Published
- 2017
19. The multiplier algebra of the noncommutative Schwartz space
- Author
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Krzysztof Piszczek and Tomasz Ciaś
- Subjects
47L10 (Primary) 46K10, 46H15, 46A13, 46A11 (Secondary) ,Multiplier algebra ,01 natural sciences ,46A11 ,$\mathrm{PLS}$-space ,(Fréchet) $m$-convex algebra ,FOS: Mathematics ,46K10 ,0101 mathematics ,Open mapping theorem (functional analysis) ,46A13 ,Mathematics ,Algebra and Number Theory ,Functional analysis ,46H15 ,010102 general mathematics ,multiplier algebra ,Noncommutative geometry ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,010101 applied mathematics ,Algebra ,Uniform boundedness principle ,Schwartz space ,Domain (ring theory) ,Closed graph theorem ,47L10 ,(noncommutative) Schwartz space ,Analysis - Abstract
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.
- Published
- 2017
20. Mean ergodic theorem in symmetric spaces
- Author
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Aleksandr Veksler and Fedor Sukochev
- Subjects
Discrete mathematics ,Fundamental theorem ,Picard–Lindelöf theorem ,010102 general mathematics ,Eberlein–Šmulian theorem ,Fixed-point theorem ,General Medicine ,01 natural sciences ,Fréchet space ,0103 physical sciences ,No-go theorem ,010307 mathematical physics ,0101 mathematics ,Open mapping theorem (functional analysis) ,Brouwer fixed-point theorem ,Mathematics - Abstract
We investigate the validity of the Mean Ergodic Theorem in symmetric Banach function spaces E . The assertion of that theorem always holds when E is separable, whereas the situation is more delicate when E is non-separable. To describe positive results in the latter setting, we use the connections with the theory of singular traces.
- Published
- 2017
21. A hybrid fixed-point theorem for set-valued maps
- Author
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B. D. Gel’man
- Subjects
Discrete mathematics ,General Mathematics ,010102 general mathematics ,Fixed-point theorem ,01 natural sciences ,010101 applied mathematics ,Surjective function ,Arzelà–Ascoli theorem ,Contraction mapping ,Closed graph theorem ,0101 mathematics ,Open mapping theorem (functional analysis) ,Brouwer fixed-point theorem ,Contraction (operator theory) ,Mathematics - Abstract
In 1955, M. A. Krasnosel’skii proved a fixed-point theorem for a single-valued map which is a completely continuous contraction (a hybrid theorem). Subsequently, his work was continued in various directions. In particular, it has stimulated the development of the theory of condensing maps (both single-valued and set-valued); the images of such maps are always compact. Various versions of hybrid theorems for set-valued maps with noncompact images have also been proved. The set-valued contraction in these versions was assumed to have closed images and the completely continuous perturbation, to be lower semicontinuous (in a certain sense). In this paper, a new hybrid fixed-point theorem is proved for any set-valued map which is the sum of a set-valued contraction and a compact set-valued map in the case where the compact set-valued perturbation is upper semicontinuous and pseudoacyclic. In conclusion, this hybrid theorem is used to study the solvability of operator inclusions for a new class of operators containing all surjective operators. The obtained result is applied to solve the solvability problem for a certain class of control systems determined by a singular differential equation with feedback.
- Published
- 2017
22. Meir-Keeler theorem in b-rectangular metric spaces
- Author
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Pei Wang, Dingwei Zheng, and Nada Citakovic
- Subjects
Pure mathematics ,Algebra and Number Theory ,Injective metric space ,010102 general mathematics ,Mathematical analysis ,Eberlein–Šmulian theorem ,Equivalence of metrics ,01 natural sciences ,Convex metric space ,010101 applied mathematics ,Uniform continuity ,Fréchet space ,0101 mathematics ,Open mapping theorem (functional analysis) ,Metric differential ,Analysis ,Mathematics - Published
- 2017
23. Hyers-Ulam stability of isometries and non-expansive maps between spaces of continuous functions
- Author
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Igor A. Vestfrid
- Subjects
Pure mathematics ,Uniform continuity ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Isometry ,Interpolation space ,Open mapping theorem (functional analysis) ,Expansive ,Mathematics - Published
- 2017
24. A COMMON FIXED POINT THEOREM FOR T-CONTRACTIONS ON GENERALIZED CONE b-METRIC SPACES
- Author
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M. Rangamma and Pagidi Mallikarjun Reddy
- Subjects
Pure mathematics ,Schauder fixed point theorem ,Dual cone and polar cone ,Fréchet space ,Applied Mathematics ,General Mathematics ,Mathematical analysis ,Fixed-point theorem ,Open mapping theorem (functional analysis) ,Kakutani fixed-point theorem ,Brouwer fixed-point theorem ,Fixed-point property ,Mathematics - Published
- 2017
25. Mild Solution and Constrained Local Controllability of Semilinear Boundary Control Systems
- Author
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Nutan Kumar Tomar and Suman Kumar
- Subjects
0209 industrial biotechnology ,Numerical Analysis ,Control and Optimization ,Algebra and Number Theory ,Partial differential equation ,010102 general mathematics ,Mathematical analysis ,Extrapolation ,02 engineering and technology ,01 natural sciences ,Controllability ,020901 industrial engineering & automation ,Control and Systems Engineering ,Control system ,0101 mathematics ,Open mapping theorem (functional analysis) ,Mathematics - Abstract
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.
- Published
- 2017
26. Applying G-metric Space for Cantor's Intersection and Baire's Category Theorem
- Author
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G Dhanorkar
- Subjects
Discrete mathematics ,Cantor set ,Metric space ,Uniform boundedness principle ,Baire category theorem ,General Medicine ,Baire space ,Open mapping theorem (functional analysis) ,Baire measure ,Complete metric space ,Mathematics - Published
- 2017
27. An extension of Brosowski-Meinardus theorem in modular spaces
- Author
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Karrar Emad AbdulSada and Salwa Salman Abed
- Subjects
Discrete mathematics ,Tychonoff's theorem ,Isomorphism extension theorem ,Fréchet space ,General Mathematics ,Eberlein–Šmulian theorem ,Fixed-point theorem ,Closed graph theorem ,Open mapping theorem (functional analysis) ,Brouwer fixed-point theorem ,Mathematics - Published
- 2017
28. Livšic theorem for banach rings
- Author
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Genady Grabarnik and Misha Guysinsky
- Subjects
Mathematics::Dynamical Systems ,Applied Mathematics ,010102 general mathematics ,Eberlein–Šmulian theorem ,Banach space ,Banach manifold ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Uniform boundedness principle ,Fréchet space ,Discrete Mathematics and Combinatorics ,0101 mathematics ,Open mapping theorem (functional analysis) ,Commutative algebra ,Analysis ,Banach–Mazur theorem ,Mathematics - Abstract
The Livsic Theorem for Holder continuous cocycles with values in Banach rings is proved. We consider a transitive homeomorphism ${\sigma :X\to X}$ that satisfies the Anosov Closing Lemma and a Holder continuous map ${a:X\to B^\times}$ from a compact metric space $X$ to the set of invertible elements of some Banach ring $B$. The map $a(x)$ is a coboundary with a Holder continuous transition function if and only if $a(\sigma^{n-1}p)\ldots a(\sigma p)a(p)$ is the identity for each periodic point $p=\sigma^n p$.
- Published
- 2017
29. On the Mazur-Ulam theorem in non-Archimedean fuzzy anti-2-normed spaces
- Author
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Dongseung Kang
- Subjects
Discrete mathematics ,Mazur–Ulam theorem ,General Mathematics ,Eberlein–Šmulian theorem ,Banach space ,Fuzzy number ,Closed graph theorem ,Fuzzy subalgebra ,Open mapping theorem (functional analysis) ,Brouwer fixed-point theorem ,Mathematics - Abstract
We study the notion of a non-Archimedean fuzzy anti-2-normed space over a non-Archimedean field and prove that Mazur-Ulam theorem holds under some conditions in non-Archimedean fuzzy anti-2-normed spaces.
- Published
- 2017
30. Generalizations of Caristi-Kirk theorem in partial metric spaces and applications
- Author
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Seong-Hoon Cho
- Subjects
Discrete mathematics ,Uniform continuity ,Fréchet space ,General Mathematics ,Injective metric space ,Eberlein–Šmulian theorem ,Banach space ,Open mapping theorem (functional analysis) ,Metric differential ,Convex metric space ,Mathematics - Published
- 2017
31. The Extention of Mean Value Theorem in Asplund Spaces
- Author
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H. Eshaghi kenari and A.Shahmari
- Subjects
Mathematics::Functional Analysis ,Pure mathematics ,Picard–Lindelöf theorem ,Mathematical analysis ,Eberlein–Šmulian theorem ,Limiting subdifferentials ,Banach space ,lcsh:QA299.6-433 ,lcsh:Analysis ,Asplund space ,Convex analysis ,%22">Mean value theorem"/> ,Fréchet space ,Set-valud math ,Danskin's theorem ,Lipschitz map ,Open mapping theorem (functional analysis) ,Brouwer fixed-point theorem ,Mathematics - Abstract
In this paper a nonsmooth mean value theorem in asplund spaces, under convexity, using the properties of limiting subdifferentials is established. We research on a kind of mean value theorem and prove that this theorem for set-valued mappings under convexity of domein in banach spaces. This theorem is use full to establish new results in convex analysis.
- Published
- 2017
32. An optimal nonlinear extension of Banach–Stone theorem
- Author
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André Luis Porto da Silva, Eloi Galego, and André Luiz Meleiro Porto
- Subjects
Discrete mathematics ,Banach–Stone theorem ,010102 general mathematics ,Banach space ,Hausdorff space ,Lipschitz continuity ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,Arzelà–Ascoli theorem ,Bijection ,Locally compact space ,ESPAÇOS DE BANACH ,0101 mathematics ,Open mapping theorem (functional analysis) ,Analysis ,Mathematics - Abstract
We prove that if K and S are locally compact Hausdorff spaces and there exists a bijective coarse ( M , L ) -quasi-isometry T between the Banach spaces of real continuous functions C 0 ( K ) and C 0 ( S ) with M 2 , then K and S are homeomorphic. This nonlinear extension of Banach–Stone theorem (1933/1937) is in some sense optimal and improves some results of Amir (1965), Cambern (1967), Jarosz (1989), Dutrieux and Kalton (2005) and Gorak (2011). In the Lipschitz case, that is when L = 0 , we also improve the estimations of the distance of the map T from the isometries between the spaces C 0 ( K ) and C 0 ( S ) obtained by Gorak when K and S are compact spaces or not. As a consequence, we get a linear sharp refinement of the Amir–Cambern theorem.
- Published
- 2016
33. The Uniform Boundedness Theorem and Banach’s Open Mapping Theorem
- Author
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Yau-Chuen Wong
- Subjects
Pure mathematics ,Uniform boundedness ,Open mapping theorem (functional analysis) ,Mathematics - Published
- 2019
34. An abstract approach to optimal decay of functions and operator semigroups
- Author
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Gregory Debruyne and David Seifert
- Subjects
Tauberian theorems ,Laplace transform ,General Mathematics ,0102 computer and information sciences ,rates of decay ,01 natural sciences ,Classical Analysis and ODEs (math.CA) ,FOS: Mathematics ,RATES ,0101 mathematics ,Algebra over a field ,Open mapping theorem (functional analysis) ,Bitwise operation ,Mathematics ,Mathematics::Functional Analysis ,Analytic continuation ,010102 general mathematics ,operator semigroups ,Abelian and tauberian theorems ,Functional Analysis (math.FA) ,Mathematics - Functional Analysis ,Algebra ,analytic continuation ,optimality ,Mathematics and Statistics ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Mathematics - Classical Analysis and ODEs ,010201 computation theory & mathematics - Abstract
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., Comment: To appear in the Israel Journal of Mathematics
- Published
- 2019
35. Infinite matrix method, ($\boldsymbol~{op}$) type space, invariance and demi-linear analysis
- Author
-
Zhong Shuhui and Wu Junde
- Subjects
Pure mathematics ,Generalized function ,General Mathematics ,Closed graph theorem ,Open mapping theorem (functional analysis) ,Type (model theory) ,Space (mathematics) ,Equicontinuity ,Topological vector space ,Matrix method ,Mathematics - Abstract
This paper summarizes the research work on functional space theory, including the infinite matrix method, characterization of $({op})$ type space,invariance, nonlinear open mapping theorem, nonlinear closed graph theorem, demi-linear equicontinuity theorem, demi-linear duality theory, anddemi-linear generalized function theory.
- Published
- 2020
36. Banach fixed point theorem for Contraction Mapping Principle in a Cone b-pentagonal metric spaces
- Author
-
A Anbarasan and J.Uma Maheshwari
- Subjects
Pure mathematics ,Banach fixed-point theorem ,Fréchet space ,Mathematical analysis ,Eberlein–Šmulian theorem ,Banach space ,Fixed-point theorem ,Contraction mapping ,Open mapping theorem (functional analysis) ,Fixed-point property ,Mathematics - Published
- 2016
37. Full characterizations of minimax inequality, fixed point theorem, saddle point theorem, and KKM principle in arbitrary topological spaces
- Author
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Guoqiang Tian
- Subjects
Pure mathematics ,021103 operations research ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,0211 other engineering and technologies ,Fixed-point theorem ,02 engineering and technology ,Fixed-point property ,01 natural sciences ,Schauder fixed point theorem ,Fréchet space ,Modeling and Simulation ,Closed graph theorem ,Geometry and Topology ,0101 mathematics ,Open mapping theorem (functional analysis) ,Kakutani fixed-point theorem ,Brouwer fixed-point theorem ,Mathematics - Abstract
This paper provides necessary and sufficient conditions for the existence of solutions for some important problems from optimization and non-linear analysis by replacing two typical conditions—continuity and quasiconcavity with a unique condition, weakening topological vector spaces to arbitrary topological spaces that may be discrete, continuum, non-compact or non-convex. We establish a single condition, $$\gamma $$ -recursive transfer lower semicontinuity, which fully characterizes the existence of $$\gamma $$ -equilibrium of minimax inequality without imposing any restrictions on topological space. The result is then used to provide full characterizations of fixed point theorem, saddle point theorem, and KKM principle.
- Published
- 2016
38. Cauchy sequences and a Meir-Keeler type fixed point theorem in partial metric spaces
- Author
-
Siditë Duraj
- Subjects
Discrete mathematics ,Pure mathematics ,Banach space ,Mathematics::General Topology ,Fixed-point theorem ,Open mapping theorem (functional analysis) ,Fixed-point property ,Kakutani fixed-point theorem ,Brouwer fixed-point theorem ,Cauchy sequence ,Complete metric space ,Mathematics - Abstract
In this paper we prove some new conditions for Cauchy sequences by using the diameter of orbit in partial metric spaces. A fixed point theorem for Meir-Keeler type contractions in this space is established.
- Published
- 2016
39. Extended seminorms and extended topological vector spaces
- Author
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David Salas and Sebastián Tapia-García
- Subjects
021103 operations research ,Topological tensor product ,010102 general mathematics ,0211 other engineering and technologies ,02 engineering and technology ,Topological space ,Topology ,01 natural sciences ,Topological vector space ,Fréchet space ,Locally convex topological vector space ,Compact-open topology ,Closed graph theorem ,Geometry and Topology ,0101 mathematics ,Open mapping theorem (functional analysis) ,Mathematics - Abstract
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.
- Published
- 2016
40. Banach–Stone Theorem for Quaternion- Valued Continuous Function Spaces
- Author
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Kazuhiro Kawamura
- Subjects
Discrete mathematics ,Mathematics::Functional Analysis ,Banach–Stone theorem ,General Mathematics ,010102 general mathematics ,Mathematics::General Topology ,01 natural sciences ,010101 applied mathematics ,Surjective function ,Isometry ,0101 mathematics ,Open mapping theorem (functional analysis) ,Quaternion ,Mathematics - Abstract
We prove a Banach–Stone type theorem for surjective linear isometries of quaternion-valued continuous function spaces.
- Published
- 2016
41. A fixed point theorem in S_b-metric spaces
- Author
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Nabil Mlaikib and Nizar Souayaha
- Subjects
Discrete mathematics ,Picard–Lindelöf theorem ,General Mathematics ,Eberlein–Šmulian theorem ,Computational Mechanics ,Fixed-point theorem ,Fixed-point property ,Computer Science Applications ,Computational Mathematics ,Fréchet space ,Open mapping theorem (functional analysis) ,Brouwer fixed-point theorem ,Kakutani fixed-point theorem ,Mathematics - Published
- 2016
42. A Liouville theorem for weightedp−Laplace operator on smooth metric measure spaces
- Author
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Lin Feng Wang, Ze Yu Zhang, Liang Zhao, and Yu Jie Zhou
- Subjects
Unbounded operator ,Pure mathematics ,010308 nuclear & particles physics ,General Mathematics ,Injective metric space ,010102 general mathematics ,Mathematical analysis ,General Engineering ,01 natural sciences ,Liouville number ,Convex metric space ,Von Neumann's theorem ,Fréchet space ,0103 physical sciences ,0101 mathematics ,Open mapping theorem (functional analysis) ,Laplace operator ,Mathematics - Published
- 2016
43. The Erdős-Ko-Rado theorem for finite affine spaces
- Author
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Qiuli Xu and Jun Guo
- Subjects
Discrete mathematics ,Algebra and Number Theory ,010102 general mathematics ,Fundamental theorem of linear algebra ,0102 computer and information sciences ,01 natural sciences ,010201 computation theory & mathematics ,Kelvin–Stokes theorem ,Fréchet space ,Affine hull ,Affine space ,0101 mathematics ,Open mapping theorem (functional analysis) ,Erdős–Ko–Rado theorem ,Vector space ,Mathematics - Abstract
In [W.N. Hsieh, Intersection theorems for finite vector spaces, Discrete Math. 12 (1975) 1–16], Hsieh obtained the Erdős-Ko-Rado theorem for finite vector spaces. This paper generalizes Hsieh’s result and obtains the Erdős-Ko-Rado theorem for finite affine spaces.
- Published
- 2016
44. On a global implicit function theorem and some applications to integro-differential initial value problems
- Author
-
Marek Galewski and Marcin Koniorczyk
- Subjects
Inverse function theorem ,Pure mathematics ,Picard–Lindelöf theorem ,Implicit function ,General Mathematics ,010102 general mathematics ,Mathematical analysis ,Fixed-point theorem ,01 natural sciences ,Implicit function theorem ,010101 applied mathematics ,Danskin's theorem ,0101 mathematics ,Open mapping theorem (functional analysis) ,Brouwer fixed-point theorem ,Mathematics - Abstract
We generalize a recent global implicit function theorem from [8] to the case of a mapping acting between Banach spaces. Considerations related to duality mapping and to certain auxiliary functional are used in the proof together with the local implicit function theorem and mountain pass geometry. An application to integro-differential systems is given.
- Published
- 2016
45. A Theorem for Zeros of Maximal Monotone and Bounded Maps in Certain Banach Spaces
- Author
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Emmanuel E. Otubo, Maria Amarakristi Onyido, M. I. Uzochukwu, Markjoe Olunna Uba, and P. U. Nwokoro
- Subjects
Combinatorics ,Fréchet space ,Eberlein–Šmulian theorem ,Banach space ,General Earth and Planetary Sciences ,Finite-rank operator ,Open mapping theorem (functional analysis) ,Lp space ,Bounded inverse theorem ,General Environmental Science ,Mathematics ,Bounded operator - Published
- 2016
46. Banach Limits Revisited
- Author
-
Diethard Pallaschke and Dieter Pumplün
- Subjects
Discrete mathematics ,Normed algebra ,010102 general mathematics ,Uniformly convex space ,0102 computer and information sciences ,General Medicine ,01 natural sciences ,Bounded operator ,Continuous linear operator ,Strictly convex space ,010201 computation theory & mathematics ,0101 mathematics ,Open mapping theorem (functional analysis) ,Reflexive space ,Mathematics ,Normed vector space - Abstract
Order unit normed linear spaces are a special type of regularly ordered normed linear spaces and therefore the first section is a short collection of the fundamental results on this type of normed linear spaces. The connection between order unit normed linear spaces and base normed linear spaces within the category of regularly ordered normed linear spaces is described in Section 2, and Section 3 at last, contains the results on Banach limits in an arbitrary order unit normed linear space. It is shown that the original results on Banach limits are valid for a greater range.
- Published
- 2016
47. On theorem in three complete fuzzy metric spaces
- Author
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U. Palaniyappa, P. Thirunavukarasu, and A. Thanithamil
- Subjects
Discrete mathematics ,Uniform continuity ,Fréchet space ,Injective metric space ,Eberlein–Šmulian theorem ,Banach space ,T-norm ,Open mapping theorem (functional analysis) ,Mathematics ,Convex metric space - Published
- 2016
48. Autonomous Ovsyannikov theorem and applications to nonlocal evolution equations and systems
- Author
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Rafael F. Barostichi, A. Alexandrou Himonas, and Gerson Petronilho
- Subjects
Cauchy problem ,Picard–Lindelöf theorem ,010102 general mathematics ,Mathematical analysis ,Residue theorem ,Banach space ,01 natural sciences ,010101 applied mathematics ,Applied mathematics ,Initial value problem ,Uniqueness ,0101 mathematics ,Open mapping theorem (functional analysis) ,Analysis ,Analytic function ,Mathematics - Abstract
This work presents an Ovsyannikov type theorem for an autonomous abstract Cauchy problem in a scale of decreasing Banach spaces, which in addition to existence and uniqueness of solution provides an estimate about the analytic lifespan of the solution. Then, using this theorem it studies the Cauchy problem for Camassa–Holm type equations and systems with initial data in spaces of analytic functions on both the circle and the line, which is the main goal of this paper. Finally, it studies the continuity of the data-to-solution map in spaces of analytic functions.
- Published
- 2016
49. On Fixed Point Theorem in Fuzzy Normed Space
- Author
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F S Fadhel, Rana Adnan Mohammed, and Buthainah A. A. Ahmed
- Subjects
Strictly convex space ,Discrete mathematics ,Pure mathematics ,Normed algebra ,Banach–Alaoglu theorem ,Open mapping theorem (functional analysis) ,Fixed-point property ,Complete metric space ,Mathematics ,Banach–Mazur theorem ,Normed vector space - Abstract
The formal balls in fuzzy normed space X (characterized by closed balls in X) are ordered by reverse inclusion depending on the concept of level sets. The set of formal balls in a fuzzy normed space is called a fuzzy domain normed space denoted by BX. This set is directed complete partially ordered set (dcpo), its maximal elements are the suprema. A contraction mapping principle is defined on BX. Banach fixed point theorem is studied and proved on BX.
- Published
- 2015
50. Fundamental Theorems in Fuzzy Normed Spaces
- Author
-
Reza Saadati, Themistocles M. Rassias, and Yeol Je Cho
- Subjects
Linear map ,Pure mathematics ,Mathematics::General Mathematics ,Bounded function ,Banach space ,Open set ,Closed graph theorem ,Open mapping theorem (functional analysis) ,Fuzzy logic ,MathematicsofComputing_DISCRETEMATHEMATICS ,Bounded operator ,Mathematics - Abstract
Some important theorems in this chapter are the open mapping theorem and the closed graph theorem. These are the cornerstones of the theory of fuzzy Banach spaces. Open mapping theorem states that a fuzzy bounded linear operator T from a fuzzy Banach space onto a fuzzy Banach space is an open mapping, that is, maps open sets onto open sets. Closed graph theorem gives conditions under which a closed linear operator is fuzzy bounded. Closed linear operators are of importance in physical and other applications.
- Published
- 2018
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