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Laijiu Lin - One of the best experts on this subject based on the ideXlab platform.
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common fixed point Theorems for a finite family of discontinuous and noncommutative maps
Fixed Point Theory and Applications, 2011Co-Authors: Laijiu Lin, Sungyu WangAbstract:We study common fixed point Theorems for a finite family of discontinuous and noncommutative single-valued functions defined in complete metric spaces. We also study a common fixed point theorem for two multivalued self-mappings and a stationary point theorem in complete metric spaces. Throughout this paper, we establish common fixed point Theorems without commuting and continuity assumptions. In contrast, commuting or continuity assumptions are often assumed in common fixed point Theorems. We also give examples to show our results. Results in this paper except those that generalized Banach contraction principle and those improve and generalize recent results in fixed point theorem are original and different from any existence result in the literature. The results in this paper will have some applications in nonlinear analysis and fixed point theory.
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from an abstract maximal element principle to optimization problems stationary point Theorems and common fixed point Theorems
Journal of Global Optimization, 2010Co-Authors: Laijiu LinAbstract:In this paper, we first establish an existence theorem related with intersection theorem, maximal element theorem and common fixed point theorem for multivalued maps by applying an abstract maximal element principle proved by Lin and Du. Some new stationary point Theorems, minimization problems, new fixed point Theorems and a system of nonconvex equilibrium theorem are also given.
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systems of equilibrium problems with applications to new variants of ekeland s variational principle fixed point Theorems and parametric optimization problems
Journal of Global Optimization, 2008Co-Authors: Laijiu LinAbstract:In this paper, we first establish some existence Theorems of systems of generalized vector equilibrium problems. From these results, we obtain new variants of Ekeland's variational principle in a Hausdorff t.v.s., a minimax theorem and minimization Theorems. Some applications to the existence theorem of systems of semi-infinite problem, a variant of flower petal theorem and a generalization of Schauder's fixed point theorem are also given.
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on maximal element Theorems variants of ekeland s variational principle and their applications
Nonlinear Analysis-theory Methods & Applications, 2008Co-Authors: Laijiu LinAbstract:Abstract In this paper, we establish several different versions of generalized Ekeland’s variational principle and maximal element theorem for τ -functions in ≲ complete metric spaces. The equivalence relations between maximal element Theorems, generalized Ekeland’s variational principle, generalized Caristi’s (common) fixed point Theorems and nonconvex maximal element Theorems for maps are also proved. Moreover, we obtain some applications to a nonconvex minimax theorem, nonconvex vectorial equilibrium Theorems and convergence Theorems in complete metric spaces.
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System of Generalized Vector Quasi-Equilibrium Problems with Applications to Fixed Point Theorems for a Family of Nonexpansive Multivalued Mappings
Journal of Global Optimization, 2006Co-Authors: Laijiu LinAbstract:In this paper, we establish the existence Theorems of the generalized vector quasi-equilibrium problems. From some existence theorem, we establish fixed point Theorems for a family of lower semicontinuous or nonexpansive multivalued mappings. We also obtain the existence Theorems of system of mixed generalized vector variational-like inequalities and existence Theorems of the Debreu vector equilibrium problems and the Nash vector equilibrium problems.
Sungyu Wang - One of the best experts on this subject based on the ideXlab platform.
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common fixed point Theorems for a finite family of discontinuous and noncommutative maps
Fixed Point Theory and Applications, 2011Co-Authors: Laijiu Lin, Sungyu WangAbstract:We study common fixed point Theorems for a finite family of discontinuous and noncommutative single-valued functions defined in complete metric spaces. We also study a common fixed point theorem for two multivalued self-mappings and a stationary point theorem in complete metric spaces. Throughout this paper, we establish common fixed point Theorems without commuting and continuity assumptions. In contrast, commuting or continuity assumptions are often assumed in common fixed point Theorems. We also give examples to show our results. Results in this paper except those that generalized Banach contraction principle and those improve and generalize recent results in fixed point theorem are original and different from any existence result in the literature. The results in this paper will have some applications in nonlinear analysis and fixed point theory.
Daniel Cordero Grau - One of the best experts on this subject based on the ideXlab platform.
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The continuum hypothesis
2015Co-Authors: Daniel Cordero GrauAbstract:In this paper we prove the continuum hypothesis with categorical logic, by proving that the theory of initial ordinals and the theory of cardinals are isomorphic. To prove that cardinals and initial ordinals are isomorphic structures, that the Theorems of the theory of cardinals are Theorems of the theory of initial ordinals, and that, conversely, the Theorems of the theory of initial ordinals are Theorems of the theory of cardinals, we use the de\u85nition of an isomorphism of theories, in its equivalent form, the de\u85nition of an isomorphism of categories, the de\u85nition of a functor, the de\u85nition of a category, the axioms of mathematical logic and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms for classes and sets. And so, applying both the theorem of comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal arithmetic to the theory of ordinals, we prove the theorem. Theorem "generalized continuum hypothesis": For every trans\u85nite cardinal number , there is no cardinal number between and 2
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The continuum hypothesis
2015Co-Authors: Daniel Cordero GrauAbstract:In this paper we prove the continuum hypothesis with categorical logic by proving that the theory of initial ordinals and the theory of cardinals are isomorphic. To prove that cardinals and initial ordinals are isomorphic structures, and so, since isomorphic structures are isomorphic theories by the fundamental theorem of mathematical logic, the Theorems of the theory of cardinals are Theorems of the theory of initial ordinals, and conversely the Theorems of the theory of initial ordinals are Theorems of the theory of cardinals, we use the the de\u85nition of an isomorphism of structures, in its equivalent form, the de\u85nition of an isomorphism of categories, the de\u85nition of a functor, the de\u85nition of a category, the axioms of mathematical logic and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms for classes and sets. And thus, applying both the theorem on the comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal arithmetic to the theory of ordinals, we prove the theorem. Theorem "generalized continuum hypothesis": For every trans\u85nite cardina
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The continuum hypothesis
2015Co-Authors: Daniel Cordero GrauAbstract:In this paper we prove of the continuum hypothesis, by proving that the theory of initial ordinals and the theory of cardinals are equivalent. To prove that the Theorems of the theory of cardinals are Theorems of the theory of initial ordinals, and that, conversely, the Theorems of the theory of initial ordinals are Theorems of the theory of cardinals, we use the de\u85nition of an isomorphism of theories in mathematical logic, in its equivalent form, the de nition of an isomorphism of categories from the theory of categories, and also, we use the de\u85nition of a functor, the de\u85nition of a category, the axioms of mathematical logic and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms for classes and sets, and so, applying both the theorem of comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal arithmetic to the theory of ordinals, we prove the theorem. Theorem "generalized continuum hypothesis": For every trans\u85nite cardinal number , there is no cardinal number between and 2
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The continuum hypothesis
2014Co-Authors: Daniel Cordero GrauAbstract:In this paper we prove the continuum hypothesis with categorical logic, by proving that the theory of initial ordinals and the theory of cardinals are equivalent. To prove that the Theorems of the theory of cardinals are Theorems of the theory of initial ordinals, and that, conversely, the Theorems of the theory of initial ordinals are Theorems of the theory of cardinals, we use the de nition of an isomorphism of theories in mathematical logic, in its equivalent form, the de\u85nition of an isomorphism of categories from the theory of categories, and also, we use the de\u85nition of a functor, the de\u85nition of a category, the axioms of mathematical logic and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms for classes and sets, and so, applying both the theorem of comparablity of ordinals to the theory of cardinals, and the fundamental theorem of cardinal arithmetic to the theory of ordinals, we prove the theorem. Theorem "generalized continuum hypothesis": For every trans\u85nite cardinal number , there is no cardinal number between and 2
Guido Gherardi - One of the best experts on this subject based on the ideXlab platform.
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G.: Effective choice and boundedness principles in computable analysis. http://arxiv.org/abs/0905.4685 (preliminary version
2012Co-Authors: Vasco Brattka, Guido GherardiAbstract:Abstract. In this paper we study a new approach to classify mathematical Theorems according to their computational content. Basically, we are asking the question which Theorems can be continuously or computably transferred into each other? For this purpose Theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice principles such as co-finite choice, discrete choice, interval choice, compact choice and closed choice, which are cornerstones among Weihrauch degrees and it turns out that certain core Theorems in analysis can be classified naturally in this structure. In particular, we study Theorems such as the Intermediate Value Theorem, the Baire Category Theorem, the Banach Inverse Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem. We also explore how existing classifications of the Hahn-Banach Theorem and Weak Kőnig’s Lemma fit into this picture. Well-known omniscience principles from constructive mathematics such as LPO and LLPO can also naturally be considered as Weihrauch degrees and they play an importan
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Effective choice and boundedness principles in computable analysis
'Association for Symbolic Logic', 2011Co-Authors: Vasco Brattka, Guido GherardiAbstract:In this paper we study a new approach to classify mathematical Theorems ac- cording to their computational content. Basically, we are asking the question which Theorems can be continuously or computably transferred into each other? For this purpose Theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice principles such as co-finite choice, discrete choice, interval choice, compact choice and closed choice, which are cornerstones among Weihrauch degrees and it turns out that certain core Theorems in analysis can be classified naturally in this structure. In particular, we study Theorems such as the Intermediate Value Theorem, the Baire Cate- gory Theorem, the Banach Inverse Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem. We also explore how existing classifications of the Hahn\u2013 Banach Theorem and Weak K\uf6nig\u2019s Lemma fit into this picture. Well-known omniscience principles from constructive mathematics such as LPO and LLPO can also naturally be con- sidered as Weihrauch degrees and they play an important role in our classification. Based on thiswe compare the results of our classificationwith existing classifications in constructive and reverse mathematics and we claim that in a certain sense our classification is finer and sheds some new light on the computational content of the respective Theorems. Our classification scheme does not require any particular logical framework or axiomatic setting, but it can be carried out in the framework of classical mathematics using tools of topology, computability theory and computable analysis. We develop a number of separation techniques based on a new parallelization principle, on certain invariance properties of Weihrauch reducibility, on the Low Basis Theorem of Jockusch and Soare and based on the Baire Category Theorem. Finally, we present a number of metaTheorems that allow to derive upper bounds for the classification of the Weihrauch degree of many Theorems and we discuss the Brouwer Fixed Point Theorem as an example
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effective choice and boundedness principles in computable analysis
6th International Conference on Computability and Complexity in Analysis (CCA'09), 2009Co-Authors: Guido GherardiAbstract:In this paper we study a new approach to classify mathematical Theorems according to their computational content. Basically, we are asking the question which Theorems can be continuously or computably transferred into each other? For this purpose Theorems are considered via their realizers which are operations with certain input and output data. The technical tool to express continuous or computable relations between such operations is Weihrauch reducibility and the partially ordered degree structure induced by it. We have identified certain choice principles on closed sets which are cornerstones among Weihrauch degrees and it turns out that certain core Theorems in analysis can be classified naturally in this structure. In particular, we study Theorems such as the Intermediate Value Theorem, the Baire Category Theorem, the Banach Inverse Mapping Theorem, the Closed Graph Theorem and the Uniform Boundedness Theorem. Well-known omniscience principles from constructive mathematics such as $\LPO$ and $\LLPO$ can naturally be considered as Weihrauch degrees and they play an important role in our classification. Our classification scheme does not require any particular logical framework or axiomatic setting, but it can be carried out in the framework of classical mathematics using tools of topology, computability theory and computable analysis. Finally, we present a number of metaTheorems that allow to derive upper bounds for the classification of the Weihrauch degree of many Theorems and we discuss the Brouwer Fixed Point Theorem as an example.
Skrettingland Eirik - One of the best experts on this subject based on the ideXlab platform.
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A Wiener Tauberian theorem for operators and functions
2020Co-Authors: Luef Franz, Skrettingland EirikAbstract:We prove variants of Wiener's Tauberian theorem in the framework of quantum harmonic analysis, i.e. for convolutions between an absolutely integrable function and a trace class operator, or of two trace class operators. Our results include Wiener's Tauberian theorem as a special case. Applications of our Tauberian Theorems are related to localization operators, Toeplitz operators, isomorphism Theorems between Bargmann-Fock spaces and quantization schemes with consequences for Shubin's pseudodifferential operator calculus and Born-Jordan quantization. Based on the links between localization operators and Tauberian Theorems we note that the analogue of Pitt's Tauberian theorem in our setting implies compactness results for Toeplitz operators in terms of the Berezin transform. In addition, we extend the results on Toeplitz operators to other reproducing kernel Hilbert spaces induced by the short-time Fourier transform, known as Gabor spaces. Finally, we establish the equivalence of Wiener's Tauberian theorem and the condition in the characterization of compactness of localization operators due to Fern\'andez and Galbis.Comment: 39 page
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A Wiener Tauberian theorem for operators and functions
2020Co-Authors: Luef Franz, Skrettingland EirikAbstract:We prove variants of Wiener's Tauberian theorem in the framework of quantum harmonic analysis, i.e. for convolutions between an absolutely integrable function and a trace class operator, or of two trace class operators. Our results include Wiener's Tauberian theorem as a special case. Applications of our Tauberian Theorems are related to localization operators, Toeplitz operators, isomorphism Theorems between Bargmann-Fock spaces and quantization schemes with consequences for Shubin's pseudodifferential operator calculus and Born-Jordan quantization. Based on the links between localization operators and Tauberian Theorems we note that the analogue of Pitt's Tauberian theorem in our setting implies compactness results for Toeplitz operators in terms of the Berezin transform. In addition, we extend the results on Toeplitz operators to other reproducing kernel Hilbert spaces induced by the short-time Fourier transform, known as Gabor spaces. Finally, we establish the equivalence of Wiener's Tauberian theorem and the condition in the characterization of compactness of localization operators due to Fern\'andez and Galbis.Comment: 39 pages v2) Accepted for publication in Journal of Functional Analysis. Smaller changes and references added based on helpful feedback from refere
