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Fixing the point: The contribution of early game theory to the tool-box of modern economics

Journal of Economic Methodology 2003年第1期10卷 1-39+97页

作者： Giocoli, Nicola Department of Economics University of Pisa Pisa Italy

The paper aims at reconstructing the sequence of works through which the fixed-point technique entered the tool-box of modern economics and at establishing a link between this sequence and the neoclassical approach to economic modeling. The focus is on the change in the demonstration techniques caused by the spread of the so-called formalist approach to mathematical economics;this change was embodied by the fixed-point technique. The main conclusions of the paper are that the formalist revolution marked a dramatic discontinuity in the history of economic theory and that early game theory - despite having been the gateway through which the fixed-point entered economics - was only partly responsible for such a discontinuity.

关键词： Demonstration techniques Equilibrium point fixed-point theorems History of game theory John Nash John von Neumann

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Teoremas de ponto fixo, teoria dos jogos e existência do Equilíbrio de Nash em jogos finitos em forma normal

Teoremas de ponto fixo, teoria dos jogos e existência do Eq...

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作者： Guarnieri, Felipe Milan

In this work, the fixed-point theorems of Kakutani and Brouwer are proved with the intention of showing the existence of Nash equilibrium in finite normal-form games. In the first chapter the needed definitions of game theory are shown, starting with finite normal-form games and ending with the concept of Nash equilibrium. In the first section of chapter two, simplex theory in Rn is developed and then the Brouwer fixer point theorem is proved. In the next section, some relations of upper hemi-continuity and closed graph in set functions are shown, then proving the theorems of Celina and von Neumann that, along with Brouwer theorem, result in Kakutani fixed-point theorem in the end of the section. As the last result, the existence of Nash equilibrium in finite normal-form games is proved through Kakutani’s theorem, relating the Nash equilibrium to the fixed-point of a set function. ...

关键词： Teorema : Ponto fixo fixed-point theorems Teoria dos jogos Finite games Nash equilibrium Jogos : Matemática Game theory Kakutani Brouwer Dissertação

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Existence and uniqueness of blow-up solution to a fully fractional thermostat model

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CHAOS SOLITONS & FRACTALS 2023年 174卷

作者： Saha, Kiran Kumar Sukavanam, N. Indian Inst Technol Roorkee Dept Math Roorkee 247667 Uttarakhand India

In this article, we deal with a fully fractional thermostat model in the settings of the Riemann-Liouville fractional derivatives. The equivalences between the fractional differential equations and the corresponding Volterra-Fredholm integral equations are rigorously derived. By choosing an appropriate weighted Banach space of continuous functions, we employ two standard fixed-point theorems, Leray-Schauder alternative and Banach contraction principle, to establish the existence of blow-up solutions - the unbounded mathematical solutions in the operational interval. Furthermore, an implicit numerical scheme based on the right product rectangle rule is presented, which provides the numerical approximation of the obtained solution. Some examples are provided to validate our theoretical findings, along with numerical simulations of the solutions.

关键词： Fractional differential equations Thermostat model Riemann-Liouville fractional derivative fixed-point theorems Product rectangle rule Numerical simulation

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Metric fixed point theory and partial impredicativity

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PHILOSOPHICAL TRANSACTIONS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES 2023年第2248期381卷 20220012页

作者： Fernandez-Duque, D. Shafer, P. Towsner, H. Yokoyama, K. Univ Ghent Dept Math WE16 Ghent Belgium Czech Acad Sci Inst Comp Sci Prague Czech Republic Univ Leeds Sch Math Leeds W Yorkshire England Univ Penn Dept Math Philadelphia PA USA Tohoku Univ Math Inst Sendai Miyagi Japan

We show that the Priess-Crampe & Ribenboim fixed point theorem is provable in RCA0. Furthermore, we show that Caristi's fixed point theorem for both Baire and Borel functions is equivalent to the transfinite leftmost path principle, which falls strictly between ATR0 and p11-CA0. We also exhibit several weakenings of Caristi's theorem that are equivalent to WKL0 and to *** article is part of the theme issue 'Modern perspectives in Proof Theory'.

关键词： computability theory reverse mathematics second-order arithmetic fixed-point theorems variational principles

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Functional impulsive differential equation of order α ε (1,2) with nonlocal initial and integral boundary conditions

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2017年第7期40卷 2409-2420页

作者： Gupta, Vidushi Dabas, Jaydev IIT Roorkee Dept Appl Sci & Engn Saharanpur Campus Saharanpur 247001 India

The main work is related to show the existence and uniqueness of solution for the fractional impulsive differential equation of order alpha epsilon (1,2) with an integral boundary condition and finite delay. Using the application of the Banach and Sadovaskii fixed-point theorems, we obtain the main results. An example is presented at the end to verify the results of the paper. Copyright (c) 2016 John Wiley & Sons, Ltd.

关键词： fractional order differential equation impulsive conditions boundary value problems fixed-point theorems

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A fractional-order hybrid system of differential equations: Existence theory and numerical solutions

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2022年第7期45卷 4024-4034页

作者： Shah, Anwar Khan, Rahmat Ali Khan, Hasib Univ Malakand Dept Math Chakdara Dir Lower Khyber Pakhtunk Pakistan Shaheed Benazir Bhutto Univ Dept Math Sheringal Dir Upper Khyber Pakhtunk Pakistan

This paper is a study about a fractional-order hybrid system of two fractional-order differential equations. We have given the existence of solution and uniqueness of solutions, and finally, the numerical solutions for an application is presented. The existence results are based on the classical fixed-point approach while the numerical results are obtained with the help of fractional-order Euler's technique. In the numerical solutions, we can see that the orders of the differential equations are playing important role. We have given related literature for the readers for their detail explanation.

关键词： Caputo's fractional derivative fixed-point theorems hybrid fractional differential equations numerical simulations

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Perturbed fixed point iterative methods based on pattern structure

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2018年第14期41卷 5582-5592页

作者： Nikazad, T. Abbasi, M. Iran Univ Sci & Technol Sch Math POB 16846-13114 Tehran Iran Inst Res Fundamental Sci IPM Sch Math POB 19395-5746 Tehran Iran Univ Qom Dept Math POB 37161-46611 Qom Iran

We introduce a new idea of algorithmic structure, called assigning algorithm, using a finite collection of a subclass of strictly quasi-nonexpansive operators. This new algorithm allows the iteration vectors to take steps on a pattern which is based on a connected directed acyclic graph. The sequential, simultaneous, and string-averaging methods for solving convex feasibility problems are the special cases of the new algorithm which may be used to reduce idle time of processors in parallel implementations. We give a convergence analysis for such algorithmic structure with perturbation. Also, we extend some existence results of the split common fixed point problem based on the new algorithm. The performance of the new algorithm is illustrated with numerical examples from computed tomography.

关键词： convex feasibility fixed-point theorems iterative procedures split common fixed point problem string-averaging quasi-nonexpansive operator

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Affine-periodic orbits for linear dynamical systems

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2023年第6期46卷 7230-7238页

作者： Yang, Xue Cheng, Ming Li, Yong Jilin Univ Coll Math Changchun 130012 Peoples R China Northeast Normal Univ Ctr Math & Interdisciplinary Sci Changchun 130024 Peoples R China

We observe the relationship between forward bounded orbits and affine periodic orbits for infinite-dimensional linear dynamical systems and prove a Massera type criterion, which asserts that a forward bounded orbit implies the existence of affine periodic orbits. Such an affine-periodic orbit might be periodic, quasi(almost)-periodic, or spiral one. We also give an analog for hybrid (switching) discrete dynamical systems.

关键词： fixed-point theorems affine-periodic orbit massera-type criterion linear dynamical system

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Probabilistic unifying relations for modelling epistemic and aleatoric uncertainty: Semantics and automated reasoning with theorem proving

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THEORETICAL COMPUTER SCIENCE 2024年 1021卷

作者： Ye, Kangfeng Woodcock, Jim Foster, Simon Univ York Dept Comp Sci Deramore Lane York YO10 5GH England

Probabilistic programming combines general computer programming, statistical inference, and formal semantics to help systems make decisions when facing uncertainty. Probabilistic programs are ubiquitous, including having a significant impact on machine intelligence. While many probabilistic algorithms have been used in practice in different domains, their automated verification based on formal semantics is still a relatively new research area. In the last two decades, it has attracted much interest. Many challenges, however, remain. The work presented in this paper, probabilistic unifying relations (ProbURel), takes a step towards our vision to tackle these challenges. Our work is based on Hehner's predicative probabilistic programming, but there are several obstacles to the broader adoption of his work. Our contributions here include (1) the formalisation of its syntax and semantics by introducing an Iverson bracket notation to separate relations from arithmetic;(2) the formalisation of relations using Unifying Theories of Programming (UTP) and probabilities outside the brackets using summation over the topological space of the real numbers;(3) the constructive semantics for probabilistic loops using Kleene's fixed-point theorem;(4) the enrichment of its semantics from distributions to subdistributions and superdistributions to deal with the constructive semantics;(5) the unique fixed-point theorem to simplify the reasoning about probabilistic loops;and (6) the mechanisation of our theory in Isabelle/UTP, an implementation of UTP in Isabelle/HOL, for automated reasoning using theorem proving. We demonstrate our work with six examples, including problems in robot localisation, classification in machine learning, and the termination of probabilistic loops.

关键词： Formal semantics fixed-point theorems Predicative programming UTP Probabilistic models Probabilistic programs Probability distributions Formal verification Quantitative verification Automated reasoning Theorem proving Isabelle/HOL Robotics Machine learning Classification

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The measure of noncompactness in a generalized coupled fixed point theorem and its application to an integro-differential system

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 2022年 413卷

作者： Tamimi, H. Saiedinezhad, S. Ghaemi, M. B. Iran Univ Sci & Technol Sch Math Tehran *** Iran

This paper discusses the existence of a solution for the system of integro-differential equations. By using a certain measure of noncompactness through the generalized Darbo fixed point theorem, we deduce conditions that guarantee the existence of a solution for the system. The advantage of the Darbo fixed point theorem is it helps us to impose weaker conditions in equations, because the contraction condition reduces to continuous and self-mapping conditions. To illustrate our results, we provide multiple examples and solve them by using the artificial small parameter method, with tables and figures that demonstrate the exactitude of the results. (c) 2022 Elsevier B.V. All rights reserved.

关键词： Measures of noncompactness and condensing mappings fixed-point theorems Systems of nonlinear integral equations

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