The document presents some fixed point theorems for expansion mappings in complete metric spaces. It begins with definitions of terms like metric spaces, complete metric spaces, Cauchy sequences, and expansion mappings. It then summarizes several existing fixed point theorems for expansion mappings established by other mathematicians. The main result proved in this document is Theorem 3.1, which establishes a new fixed point theorem for expansion mappings under certain conditions on the metric space and mapping. It shows that if the mapping satisfies the given inequality, then it has a fixed point. The proof of this theorem constructs a sequence to show that it converges to a fixed point.
This document presents a research paper that proves some fixed point theorems for occasionally weakly compatible maps in fuzzy metric spaces. The paper begins with an introduction discussing the importance of fixed point theory and its applications. It then provides relevant definitions for fuzzy metric spaces and concepts like weakly compatible mappings. The main results of the paper are fixed point theorems for mappings satisfying integral type contractive conditions in fuzzy metric spaces for occasionally weakly compatible maps. The proofs of these fixed point theorems generalize existing contractive conditions to establish the existence and uniqueness of a fixed point.
A Tau Approach for Solving Fractional Diffusion Equations using Legendre-Cheb...iosrjce
In this paper, a modified numerical algorithm for solving the fractional diffusion equation is
proposed. Based on Tau idea where the shifted Legendre polynomials in time and the shifted Chebyshev
polynomials in space are utilized respectively.
The problem is reduced to the solution of a system of linear algebraic equations. From the computational point
of view, the solution obtained by this approach is tested and the efficiency of the proposed method is confirmed.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
This document summarizes a presentation on matrix computations in machine learning. It discusses how matrix computations are used in traditional machine learning algorithms like regression, PCA, and spectral graph partitioning. It also covers specialized machine learning algorithms that involve numerical optimization and matrix computations, such as Lasso, non-negative matrix factorization, sparse PCA, matrix completion, and co-clustering. Finally, it discusses how machine learning can be applied to improve graph clustering algorithms.
Intuitionistic Fuzzy W- Closed Sets and Intuitionistic Fuzzy W -ContinuityWaqas Tariq
The aim of this paper is to introduce and study the concepts of intuitionistic fuzzy w- closed sets, intuitionistic fuzzy w-continuity and inttuitionistic fuzzy w-open & intuitionistic fuzzy w-closed mappings in intuitionistic fuzzy topological spaces.
This document outlines the plan for a seminar on kernel methods. The seminar will cover four main topics: kernel methods, random projections, deep learning, and more about kernels. The overall goal is to provide enough theoretical background to understand several related papers on convolutional kernel networks, fastfood kernel approximations, and deep fried convolutional neural networks. The document includes an agenda, definitions, theorems, and references to support understanding kernel methods and their applications.
This document presents a research paper that proves some fixed point theorems for occasionally weakly compatible maps in fuzzy metric spaces. The paper begins with an introduction discussing the importance of fixed point theory and its applications. It then provides relevant definitions for fuzzy metric spaces and concepts like weakly compatible mappings. The main results of the paper are fixed point theorems for mappings satisfying integral type contractive conditions in fuzzy metric spaces for occasionally weakly compatible maps. The proofs of these fixed point theorems generalize existing contractive conditions to establish the existence and uniqueness of a fixed point.
A Tau Approach for Solving Fractional Diffusion Equations using Legendre-Cheb...iosrjce
In this paper, a modified numerical algorithm for solving the fractional diffusion equation is
proposed. Based on Tau idea where the shifted Legendre polynomials in time and the shifted Chebyshev
polynomials in space are utilized respectively.
The problem is reduced to the solution of a system of linear algebraic equations. From the computational point
of view, the solution obtained by this approach is tested and the efficiency of the proposed method is confirmed.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
This document summarizes a presentation on matrix computations in machine learning. It discusses how matrix computations are used in traditional machine learning algorithms like regression, PCA, and spectral graph partitioning. It also covers specialized machine learning algorithms that involve numerical optimization and matrix computations, such as Lasso, non-negative matrix factorization, sparse PCA, matrix completion, and co-clustering. Finally, it discusses how machine learning can be applied to improve graph clustering algorithms.
Intuitionistic Fuzzy W- Closed Sets and Intuitionistic Fuzzy W -ContinuityWaqas Tariq
The aim of this paper is to introduce and study the concepts of intuitionistic fuzzy w- closed sets, intuitionistic fuzzy w-continuity and inttuitionistic fuzzy w-open & intuitionistic fuzzy w-closed mappings in intuitionistic fuzzy topological spaces.
This document outlines the plan for a seminar on kernel methods. The seminar will cover four main topics: kernel methods, random projections, deep learning, and more about kernels. The overall goal is to provide enough theoretical background to understand several related papers on convolutional kernel networks, fastfood kernel approximations, and deep fried convolutional neural networks. The document includes an agenda, definitions, theorems, and references to support understanding kernel methods and their applications.
This document summarizes a research article that introduces a new iterative scheme for finding a common element of the set of common fixed points of a countable family of nonexpansive multivalued maps, the set of solutions to a variational inequality problem, and the set of solutions to an equilibrium problem in a Hilbert space. The authors provide background on existing methods for related problems and note the need to construct an iteration process that can solve this more general problem. They then introduce their new monotone hybrid iterative scheme and state that they will establish strong convergence theorems for their proposed iteration method.
This document contains the solutions to 8 questions from a homework assignment for a math class. The solutions include:
1) Deriving the gradient of a cost function and expressing it in matrix form.
2) Explaining how the number of data points and degree of a polynomial model relate to whether an equation has a unique solution.
3) Plotting data points and a fitted curve.
4) Fitting curves with different polynomial degrees to different data sets to determine the best fit.
Research Inventy : International Journal of Engineering and Scienceinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Green's theorem in classical mechanics and electrodynamicsCorneliu Sochichiu
The document discusses applications of Green's theorem in classical mechanics and electrodynamics. It begins by explaining Green's theorem and related theorems. In classical mechanics, Green's theorem can be used to calculate properties like mass, moments of inertia, and determine whether a force is conservative. It also shows how Green's theorem derives Kepler's second law about equal areas being swept in equal times for planetary orbits.
Stability of Iteration for Some General Operators in b-MetricKomal Goyal
This document discusses the stability of the Jungck-Mann iteration scheme for approximating fixed points of operators in b-metric spaces. It defines the Jungck-Mann iterative procedure and establishes conditions for its stability. Specifically, it proves that if two operators S and T satisfy a contractive condition in a b-metric space, and Sx converges to a coincidence point p of S and T using the Jungck-Mann scheme, then any perturbed sequence Sy will also converge to p as the perturbations go to zero. The stability result relies on deriving inequalities relating the distances between iterates.
Here are the key points about g given f:
- g represents the area under the curve of f over successive intervals of the x-axis
- As x increases over an interval, g will increase if f is positive over that interval and decrease if f is negative
- The concavity (convexity or concavity) of g will match the concavity of f over each interval
In summary, the area function g, as defined by the integral of f, will have properties that correspond directly to the sign and concavity of f over successive intervals of integration.
An implicit partial pivoting gauss elimination algorithm for linear system of...Alexander Decker
This document proposes a new method for solving fully fuzzy linear systems of equations (FFLS) using Gauss elimination with implicit partial pivoting. It begins by introducing concepts of fuzzy sets theory and arithmetic operations on fuzzy numbers. It then presents the FFLS problem and shows how it can be reduced to a crisp linear system. The key steps of the proposed Gauss elimination method with implicit partial pivoting are outlined. Finally, the method is illustrated on a numerical example, showing the steps to obtain solutions for the variables x, y and z.
This document discusses optimal weighted distributions and their applications to financial time series. It addresses the problems of minimizing distances between weighted sample distributions and target distributions for 1) a single population, 2) two populations, and 3) N populations. Explicit formulas are derived for the optimal weights in each case. An iterative algorithm is presented to compute the target distribution as a fixed point. Applications to Google stock data are shown. Possible developments include extending the analysis to more populations, relaxing assumptions, and considering alternative distances.
11.common fixed points of weakly reciprocally continuous maps using a gauge f...Alexander Decker
The document presents a common fixed point theorem for weakly reciprocally continuous self-mappings on a complete metric space. It begins with definitions of various types of compatible mappings and introduces the concept of weak reciprocal continuity. The main result, Theorem 2.1, proves that if two self-mappings satisfy conditions (i) and (ii) and are either compatible, A-compatible, or T-compatible, then the mappings have a unique common fixed point. Condition (ii) is a contractive condition involving an upper semi-continuous function. The proof constructs Cauchy sequences to show the existence of the common fixed point.
Common fixed points of weakly reciprocally continuous maps using a gauge func...Alexander Decker
The document summarizes a mathematical research paper that proves a common fixed point theorem for weakly reciprocally continuous self-mappings on a complete metric space. The theorem establishes that if two self-mappings satisfy a contractive condition and are either compatible, A-compatible, or T-compatible, then they have a unique common fixed point. The proof constructs Cauchy sequences from the mappings and uses properties like weak reciprocal continuity, compatibility, and the contractive condition to show the sequences converge to a common fixed point.
This document discusses rectangular coordinate systems in three dimensions. It begins by introducing the 3D coordinate system with three perpendicular axes (x, y, z) that intersect at the origin. Points in 3D space are located using ordered triples (x, y, z). It then discusses visualizing points and objects in 3D, finding distances between points, and graphing simple objects like lines and planes. Examples are provided to illustrate key concepts like plotting points, finding distances, and graphing lines and planes defined by equations.
CVPR2010: Advanced ITinCVPR in a Nutshell: part 5: Shape, Matching and Diverg...zukun
The document discusses using divergence measures like the Jensen-Shannon divergence to align multiple point sets represented as probability density functions. It motivates using the JS divergence by modeling point sets as mixtures of density functions, and shows how the likelihood ratio between models leads to the JS divergence. It then formulates the problem of group-wise point set registration as minimizing the JS divergence between density functions, combined with a regularization term. Experimental results on aligning multiple 3D hippocampus point sets are also presented.
Tensor Decomposition and its ApplicationsKeisuke OTAKI
This document discusses tensor factorizations and decompositions and their applications in data mining. It introduces tensors as multi-dimensional arrays and covers 2nd order tensors (matrices) and 3rd order tensors. It describes how tensor decompositions like the Tucker model and CANDECOMP/PARAFAC (CP) model can be used to decompose tensors into core elements to interpret data. It also discusses singular value decomposition (SVD) as a way to decompose matrices and reduce dimensions while approximating the original matrix.
A new class of a stable implicit schemes for treatment of stiffAlexander Decker
The document presents a new class of implicit rational Runge-Kutta schemes for solving stiff systems of ordinary differential equations. It describes the development of the schemes, which are motivated by implicit conventional Runge-Kutta schemes and rational function approximation. Taylor series expansion and Pade approximation techniques are used in the analysis. The schemes are convergent and A-stable, meaning they are stable for large step sizes. Examples of one-stage and multi-stage schemes are presented.
A common fixed point theorem in cone metric spacesAlexander Decker
This academic article summarizes a common fixed point theorem for continuous and asymptotically regular self-mappings on complete cone metric spaces. The theorem extends previous results to cone metric spaces, which generalize metric spaces by replacing real numbers with an ordered Banach space. It proves that under certain contractive conditions, the self-mapping has a unique fixed point. The proof constructs a Cauchy sequence that converges to the fixed point.
Fixed points of contractive and Geraghty contraction mappings under the influ...IJERA Editor
In this paper, we prove the existence of fixed points of contractive and Geraghty contraction maps in complete metric spaces under the influence of altering distances. Our results extend and generalize some of the known results.
Fixed point theorems in random fuzzy metric space throughAlexander Decker
This document defines key concepts related to fixed point theorems in random fuzzy metric spaces. It begins by introducing fuzzy metric spaces, fuzzy 2-metric spaces, and fuzzy 3-metric spaces. It then defines random fuzzy variables and random fuzzy metric spaces. The document aims to prove some fixed point theorems in random fuzzy metric spaces, random fuzzy 2-metric spaces, and random fuzzy 3-metric spaces using rational expressions. It provides 18 definitions related to t-norms, fuzzy metric spaces, convergence of sequences, completeness, and mappings to lay the groundwork for the main results.
We use stochastic methods to present mathematically correct representation of the wave function. Informal construction was developed by R. Feynman. This approach were introduced first by H. Doss Sur une Resolution Stochastique de l'Equation de Schrödinger à Coefficients Analytiques. Communications in Mathematical Physics
October 1980, Volume 73, Issue 3, pp 247–264.
Primary intention is to discuss formal stochastic representation of the Schrodinger equation solution with its applications to the theory of demolition quantum measurements.
I will appreciate your comments.
New Contraction Mappings in Dislocated Quasi - Metric SpacesIJERA Editor
In this paper the concept of new contraction mappings has been used in proving fixed point theorems. We
establish some common fixed point theorems in complete dislocated quasi metric spaces using new contraction
mappings.
This document summarizes a research article that introduces a new iterative scheme for finding a common element of the set of common fixed points of a countable family of nonexpansive multivalued maps, the set of solutions to a variational inequality problem, and the set of solutions to an equilibrium problem in a Hilbert space. The authors provide background on existing methods for related problems and note the need to construct an iteration process that can solve this more general problem. They then introduce their new monotone hybrid iterative scheme and state that they will establish strong convergence theorems for their proposed iteration method.
This document contains the solutions to 8 questions from a homework assignment for a math class. The solutions include:
1) Deriving the gradient of a cost function and expressing it in matrix form.
2) Explaining how the number of data points and degree of a polynomial model relate to whether an equation has a unique solution.
3) Plotting data points and a fitted curve.
4) Fitting curves with different polynomial degrees to different data sets to determine the best fit.
Research Inventy : International Journal of Engineering and Scienceinventy
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
Green's theorem in classical mechanics and electrodynamicsCorneliu Sochichiu
The document discusses applications of Green's theorem in classical mechanics and electrodynamics. It begins by explaining Green's theorem and related theorems. In classical mechanics, Green's theorem can be used to calculate properties like mass, moments of inertia, and determine whether a force is conservative. It also shows how Green's theorem derives Kepler's second law about equal areas being swept in equal times for planetary orbits.
Stability of Iteration for Some General Operators in b-MetricKomal Goyal
This document discusses the stability of the Jungck-Mann iteration scheme for approximating fixed points of operators in b-metric spaces. It defines the Jungck-Mann iterative procedure and establishes conditions for its stability. Specifically, it proves that if two operators S and T satisfy a contractive condition in a b-metric space, and Sx converges to a coincidence point p of S and T using the Jungck-Mann scheme, then any perturbed sequence Sy will also converge to p as the perturbations go to zero. The stability result relies on deriving inequalities relating the distances between iterates.
Here are the key points about g given f:
- g represents the area under the curve of f over successive intervals of the x-axis
- As x increases over an interval, g will increase if f is positive over that interval and decrease if f is negative
- The concavity (convexity or concavity) of g will match the concavity of f over each interval
In summary, the area function g, as defined by the integral of f, will have properties that correspond directly to the sign and concavity of f over successive intervals of integration.
An implicit partial pivoting gauss elimination algorithm for linear system of...Alexander Decker
This document proposes a new method for solving fully fuzzy linear systems of equations (FFLS) using Gauss elimination with implicit partial pivoting. It begins by introducing concepts of fuzzy sets theory and arithmetic operations on fuzzy numbers. It then presents the FFLS problem and shows how it can be reduced to a crisp linear system. The key steps of the proposed Gauss elimination method with implicit partial pivoting are outlined. Finally, the method is illustrated on a numerical example, showing the steps to obtain solutions for the variables x, y and z.
This document discusses optimal weighted distributions and their applications to financial time series. It addresses the problems of minimizing distances between weighted sample distributions and target distributions for 1) a single population, 2) two populations, and 3) N populations. Explicit formulas are derived for the optimal weights in each case. An iterative algorithm is presented to compute the target distribution as a fixed point. Applications to Google stock data are shown. Possible developments include extending the analysis to more populations, relaxing assumptions, and considering alternative distances.
11.common fixed points of weakly reciprocally continuous maps using a gauge f...Alexander Decker
The document presents a common fixed point theorem for weakly reciprocally continuous self-mappings on a complete metric space. It begins with definitions of various types of compatible mappings and introduces the concept of weak reciprocal continuity. The main result, Theorem 2.1, proves that if two self-mappings satisfy conditions (i) and (ii) and are either compatible, A-compatible, or T-compatible, then the mappings have a unique common fixed point. Condition (ii) is a contractive condition involving an upper semi-continuous function. The proof constructs Cauchy sequences to show the existence of the common fixed point.
Common fixed points of weakly reciprocally continuous maps using a gauge func...Alexander Decker
The document summarizes a mathematical research paper that proves a common fixed point theorem for weakly reciprocally continuous self-mappings on a complete metric space. The theorem establishes that if two self-mappings satisfy a contractive condition and are either compatible, A-compatible, or T-compatible, then they have a unique common fixed point. The proof constructs Cauchy sequences from the mappings and uses properties like weak reciprocal continuity, compatibility, and the contractive condition to show the sequences converge to a common fixed point.
This document discusses rectangular coordinate systems in three dimensions. It begins by introducing the 3D coordinate system with three perpendicular axes (x, y, z) that intersect at the origin. Points in 3D space are located using ordered triples (x, y, z). It then discusses visualizing points and objects in 3D, finding distances between points, and graphing simple objects like lines and planes. Examples are provided to illustrate key concepts like plotting points, finding distances, and graphing lines and planes defined by equations.
CVPR2010: Advanced ITinCVPR in a Nutshell: part 5: Shape, Matching and Diverg...zukun
The document discusses using divergence measures like the Jensen-Shannon divergence to align multiple point sets represented as probability density functions. It motivates using the JS divergence by modeling point sets as mixtures of density functions, and shows how the likelihood ratio between models leads to the JS divergence. It then formulates the problem of group-wise point set registration as minimizing the JS divergence between density functions, combined with a regularization term. Experimental results on aligning multiple 3D hippocampus point sets are also presented.
Tensor Decomposition and its ApplicationsKeisuke OTAKI
This document discusses tensor factorizations and decompositions and their applications in data mining. It introduces tensors as multi-dimensional arrays and covers 2nd order tensors (matrices) and 3rd order tensors. It describes how tensor decompositions like the Tucker model and CANDECOMP/PARAFAC (CP) model can be used to decompose tensors into core elements to interpret data. It also discusses singular value decomposition (SVD) as a way to decompose matrices and reduce dimensions while approximating the original matrix.
A new class of a stable implicit schemes for treatment of stiffAlexander Decker
The document presents a new class of implicit rational Runge-Kutta schemes for solving stiff systems of ordinary differential equations. It describes the development of the schemes, which are motivated by implicit conventional Runge-Kutta schemes and rational function approximation. Taylor series expansion and Pade approximation techniques are used in the analysis. The schemes are convergent and A-stable, meaning they are stable for large step sizes. Examples of one-stage and multi-stage schemes are presented.
A common fixed point theorem in cone metric spacesAlexander Decker
This academic article summarizes a common fixed point theorem for continuous and asymptotically regular self-mappings on complete cone metric spaces. The theorem extends previous results to cone metric spaces, which generalize metric spaces by replacing real numbers with an ordered Banach space. It proves that under certain contractive conditions, the self-mapping has a unique fixed point. The proof constructs a Cauchy sequence that converges to the fixed point.
Fixed points of contractive and Geraghty contraction mappings under the influ...IJERA Editor
In this paper, we prove the existence of fixed points of contractive and Geraghty contraction maps in complete metric spaces under the influence of altering distances. Our results extend and generalize some of the known results.
Fixed point theorems in random fuzzy metric space throughAlexander Decker
This document defines key concepts related to fixed point theorems in random fuzzy metric spaces. It begins by introducing fuzzy metric spaces, fuzzy 2-metric spaces, and fuzzy 3-metric spaces. It then defines random fuzzy variables and random fuzzy metric spaces. The document aims to prove some fixed point theorems in random fuzzy metric spaces, random fuzzy 2-metric spaces, and random fuzzy 3-metric spaces using rational expressions. It provides 18 definitions related to t-norms, fuzzy metric spaces, convergence of sequences, completeness, and mappings to lay the groundwork for the main results.
We use stochastic methods to present mathematically correct representation of the wave function. Informal construction was developed by R. Feynman. This approach were introduced first by H. Doss Sur une Resolution Stochastique de l'Equation de Schrödinger à Coefficients Analytiques. Communications in Mathematical Physics
October 1980, Volume 73, Issue 3, pp 247–264.
Primary intention is to discuss formal stochastic representation of the Schrodinger equation solution with its applications to the theory of demolition quantum measurements.
I will appreciate your comments.
New Contraction Mappings in Dislocated Quasi - Metric SpacesIJERA Editor
In this paper the concept of new contraction mappings has been used in proving fixed point theorems. We
establish some common fixed point theorems in complete dislocated quasi metric spaces using new contraction
mappings.
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
International Journal of Mathematics and Statistics Invention (IJMSI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJMSI publishes research articles and reviews within the whole field Mathematics and Statistics, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
Fixed Point Theorems for Weak K-Quasi Contractions on a Generalized Metric Sp...IJERA Editor
In this paper we obtain conditions for a k- quasi contraction on a generalized metric space with a partial order to have a fixed point. Using this, we derive certain known results as corollaries.
This document defines metric spaces and discusses their basic properties. It begins by defining what a metric is and what constitutes a metric space. It provides some basic examples of metrics, such as the discrete metric and p-norm metrics. It then discusses metric topologies, defining open and closed balls and showing that the collection of open sets forms a topology. It also introduces the concept of topologically equivalent metrics.
On fixed point theorems in fuzzy 2 metric spaces and fuzzy 3-metric spacesAlexander Decker
1) The document discusses fixed point theorems for mappings in fuzzy 2-metric and fuzzy 3-metric spaces.
2) It defines concepts like fuzzy metric spaces, Cauchy sequences, compatible mappings, and proves some fixed point theorems for compatible mappings.
3) The theorems show that under certain contractive conditions on the mappings, there exists a unique common fixed point for the mappings in a complete fuzzy 2-metric or fuzzy 3-metric space.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
This document presents a common fixed point theorem for two pairs of weakly compatible mappings in a dislocated metric space. Some key points:
- It defines dislocated metric spaces, where the self-distance of a point need not be zero.
- It establishes the existence and uniqueness of a common fixed point for four mappings (A, B, S, T) under certain contractive conditions.
- The mappings must satisfy three conditions: T(X) subset A(X), S(X) subset B(X), and the pairs (S,A) and (T,B) are weakly compatible.
- It constructs sequences to prove the mappings have a unique common fixed point z in the
A Generalized Metric Space and Related Fixed Point TheoremsIRJET Journal
This document presents a new concept of generalized metric spaces and establishes some fixed point theorems in these spaces. It begins with defining generalized metric spaces, which generalize standard metric spaces, b-metric spaces, dislocated metric spaces, and modular spaces with the Fatou property. It then proves some properties of generalized metric spaces, including conditions for convergence. Finally, it establishes an extension of the Banach contraction principle to generalized metric spaces, proving the existence and uniqueness of a fixed point under certain assumptions.
The document presents a fixed point theorem for non-expansive mappings in 2-Banach spaces. It defines key concepts such as 2-norms, 2-Banach spaces, and non-expansive mappings. The main theorem proves that if mappings F and G satisfy certain conditions, including FG=G=I and a contraction-like inequality, then F and G have a common fixed point. The proof involves showing the sequence defined by an average of F and I converges to a fixed point shared by F and G. This generalized previous fixed point results for mappings on 2-Banach spaces.
This document introduces the finite element method for solving partial differential equations. It discusses using a "master element" to perform calculations that then get transformed to individual mesh elements. The method is described for a general diffusion equation, integrating it by parts and discretizing it using basis functions defined on mesh elements. This leads to a system of equations relating the unknown values at different nodes in the mesh at each time step. Transforming between the master element coordinates and the actual mesh coordinates completes the description of how the finite element method sets up and solves the discrete system of equations approximating the original PDE.
The Probability that a Matrix of Integers Is DiagonalizableJay Liew
The Probability that a
Matrix of Integers Is Diagonalizable
Andrew J. Hetzel, Jay S. Liew, and Kent E. Morrison
1. INTRODUCTION. It is natural to use integer matrices for examples and exercises
when teaching a linear algebra course, or, for that matter, when writing a textbook in
the subject. After all, integer matrices offer a great deal of algebraic simplicity for particular
problems. This, in turn, lets students focus on the concepts. Of course, to insist
on integer matrices exclusively would certainly give the wrong idea about many important
concepts. For example, integer matrices with integer matrix inverses are quite
rare, although invertible integer matrices (over the rational numbers) are relatively
common. In this article, we focus on the property of diagonalizability for integer matrices
and pose the question of the likelihood that an integer matrix is diagonalizable.
Specifically, we ask: What is the probability that an n × n matrix with integer entries is
diagonalizable over the complex numbers, the real numbers, and the rational numbers,
respectively?
Common Fixed Point Theorems in Compatible Mappings of Type (P*) of Generalize...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings
under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy
metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
Mathematics subject classifications: 45H10, 54H25
Generalized fixed point theorems for compatible mapping in fuzzy 2 metric spa...Alexander Decker
This document discusses generalized fixed point theorems for compatible mappings in fuzzy 2-metric spaces. It begins with introductions and preliminaries on fixed point theory, fuzzy metric spaces, and compatible mappings. It then provides new definitions of compatible mappings of types (I) and (II) in fuzzy-2 metric spaces. The main results extend, generalize, and improve previous theorems by proving common fixed point theorems for four mappings under the condition of compatible mappings of types (I) and (II) in complete fuzzy-2 metric spaces.
Generalized fixed point theorems for compatible mapping in fuzzy 3 metric spa...Alexander Decker
This document discusses generalized fixed point theorems for compatible mappings in fuzzy 3-metric spaces. It begins with introductions and preliminaries on fixed point theory, fuzzy metric spaces, and compatible mappings. It then provides new definitions of compatible mappings of types (I) and (II) in fuzzy 3-metric spaces. The main results extend, generalize, and improve previous theorems by proving common fixed point theorems for four mappings under the conditions of compatible mappings of types (I) and (II) in complete fuzzy 3-metric spaces.
IJERD (www.ijerd.com) International Journal of Engineering Research and Devel...IJERD Editor
This document presents a theorem that establishes the existence of a fixed point for a mapping under a general contractive condition of integral type. The mapping considered generalizes various types of contractive mappings in an integral setting. The theorem proves that if a self-mapping on a complete metric space satisfies the given integral inequality involving the distance between images of points, where the integral involves a non-negative, summable function, then the mapping has a unique fixed point. Furthermore, the sequence of repeated applications of the mapping to any starting point will converge to this fixed point. The proof involves showing the distance between successive terms in the sequence decreases according to the integral inequality.
Similar to International Journal of Computational Engineering Research(IJCER) (20)
IJERD (www.ijerd.com) International Journal of Engineering Research and Devel...
International Journal of Computational Engineering Research(IJCER)
1. International Journal Of Computational Engineering Research (ijceronline.com) Vol. 3 Issue. 2
Some Fixed Point Theorems for Expansion Mappings
1,
A.S.Saluja, 2, Alkesh Kumar Dhakde 3, Devkrishna Magarde
1,
Department of Mathematics J.H.Govt.P.G.College Betul (M.P.)
2,
IES Group of institutions Bhopal (M.P.)
Abstract
In The Present Paper We Shall Establish Some Fixed Point Theorems For Expansion Mappings In Complete Metric
Spaces. Our Results Are Generalization Of Some Well Known Results.
Keywords: Fixed Point, Complete Metric spaces, Expansion mappings.
1. Introdution & Preliminary:
This paper is divided into two parts
Section I: Some fixed point theorems for expansion mappings in complete metric spaces.
Section II: Some fixed point theorems for expansion mappings in complete 2-Metric spaces before starting main result some
definitions.
Definition 2.1: (Metric spaces) A metric space is an ordered pair X , d where X is a set and d a function on X X
with the properties of a metric, namely:
1. d x, y 0 . (non-negative) ,
2. d x, y d y, x (symmetry),
3. d x, y 0 if and only if x y (identity of indiscernible)
4. The triangle inequality holds:
d x, y d x, z d z, y , for all x, y, z in X
Example 2.1: Let E n or R x ( x1 , x 2 , x3 ......... x n ), xi R, R the set of real numbers and let d be defined
n
as follows:
1
y y1 , y 2 , y3 ........ n then d x, y | xi yi | p d p x, y where p is a fixed number in 0, .
n p
If y
1
The fact that d is metric follows from the well-known Minkowski inequality. Also another metric on S considered above
can be defined as follows
d x, y sup| xi yi | d x, y
i
x xi 1 such
Example 2.2: Let S be the set of all sequence of real numbers that for some fixed
p 0, , | xi | p In this case if y y i is another point in S , we define
1
| x
1
d x, y i yi | p p d q x, y , and from Minkowski inequality it follows that this is a metric space on S .
Example 2.3: For x, y R , define d x, y | x y | . Then R, d is a metric space .In general, for
x x1 , x 2 , x3 ........ x n and y y1 , y 2 , y 3 ....... y n R n , define
d x, y x1 y1 2 x2 y 2 2 .......... xn y n 2
.
||Issn 2250-3005(online)|| ||February|| 2013 Page 1
2. International Journal Of Computational Engineering Research (ijceronline.com) Vol. 3 Issue. 2
n
Then R , d is a metric space .As this d is usually used, we called it the usual metric.
Definition 2.2: (Convergent sequence in metric space)
A sequence in metric space X , d is convergent to
x X if lim n d x n , x 0
Definition 2.3: (Cauchy sequence in Metric space) Let M X , d be a metric space, let x n be a sequence if and only
if R : 0 : N : m, n N : m, n N : d x n , x m
Definition 2.4: (Complete Metric space) A metric space X , d is complete if every Cauchy sequence is convergent.
Definition 2.5: A 2-metric space is a space X in which for each triple of points x, y, z there exists a real function
d x, y, z such that:
[ M1 ] To each pair of distinct points x, y, z d x, y, z 0
M 2 d x, y, z 0 When at least two of x, y, z are equal
[ M 3 ] d x, y , z d y , z , x d x, z , y
[M 4 ] d x, y, z d x, y, v d x, v, z d v, y, z for all x, y, z, v in X.
Definition 2.6: A sequence xn in a 2-metric space X , d is said to be convergent at x if
lim d xn , x, z 0 for all z in X .
n
Definition 2.7: A sequence xn in a 2-metric space X , d is said to be Cauchy sequence if lim d xn , x, z 0 for all
m,n
z in X .
Definition 2.8: A 2-metric space X , d is said to be complete if every Cauchy sequence in X is convergent.
2. Basic Theorems
In 1975, Fisher [4], proved the following results:
Theorem (A): Let T be a self mapping of a metric spaces X such that,
d Tx, Ty d x, Tx d y, Ty x, y X , T is an identity mappings.
1
2
Theorem (B): Let X be a compact metric space and T : X X satisfies 4. (A) and x y and x, y X .Then T has
r
a fixed point for some positive integer r , and T is invertible.
In 1984 the first known result for expansion mappings was proved by Wang, Li, Gao and Iseki [13].
Theorem (C): “Let T be a self map of complete metric space X into itself and if there is a constant 1 such
that, d Tx, Ty d x, y for all x, y X .
Then T has a unique fixed point in X .
Theorem (D): If there exist non negative real numbers 1 and 1 such that
d Tx ,Ty min d x,Tx , d y,Ty , d x, y x, y X ,
T is continuous on X onto itself, and then T has a fixed point.
Theorem (F): If there exist non negative real numbers 1 such that d Tx , T x d x, Tx x X , T is onto and
2
continuous then T has a fixed point.
In 1988, Park and Rhoades [8] shows that the above theorems are consequence of a theorem of Park [7] .In 1991, Rhoades
[10] generalized the result of Iseki and others for pair of mappings.
Theorem (G): If there exist non negative real numbers 1 and T , S be surjective self-map on a complete metric space
X , d such that;
d Tx , Sy d x, y x X , Then T and S have a unique common fixed point.
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In 1989 Taniguchi [12] extended some results of Iseki .Later, the results of expansion mappings were extended to 2-metric
spaces, introduced by Sharma, Sharma and Iseki [11] for contractive mappings. Many other Mathematicians worked on this
way.
Rhoades [10] summarized contractive mapping of different types and discussed on their fixed point theorems. He considered
many types of mappings and analyzed the relationship amongst them, where d Tx , Ty is governed by,
d x, y , d x, Tx , d y, Ty , d x, Ty , d y, Tx , d x, y , d x, Tx , d y, Ty , d x, Ty , d y, Tx
Many other mathematicians like Wang,Gao , Isekey [13] , Popa [9] , Jain and Jain [5] , Jain and Yadav [6] worked on
expansion mappings Recently , Agrawal and Chouhan [1,2] , Bhardwaj , Rajput and Yadava [1] worked for common fixed
point for expansion mapping.
Our object in this paper is, to obtain some result on fixed point theorems of expansion type’s maps on complete metric space.
Now In Section I, We will find Some Fixed Point Theorems For Expansion Mappings In Complete Metric Spaces.
3. Main Results
Theorem (3.1): Let X denotes the complete metric space with metric d and f is a mapping of X into itself .If there
exist non negative real’s , , , 1 with 2 2 1 such that
d fx , fy
1 d y, fy d x, fx d x, fx d y, fy d x, fy d y, fx d x, y
1 d x, y
For each x, y X with x y and f is onto then f has a fixed point.
Proof: Let x0 X .since f is onto, there is an element x1 satisfying x1 f x0 .similarly we can write
1
xn f 1 xn 1 , n 1,2,3.........
From the hypothesis
d xn 1 , xn d fx n , fx n 1
1 d xn 1 , fx n 1 d xn , fx n d x , fx d x , fx d x , fx d x , fx d x , x
1 d xn , xn 1
n n n 1 n 1 n n 1 n 1 n n n 1
1 d xn 1, xn d xn , xn 1 d x , x d x , x d x , x d x , x d x , x
1 d xn , xn 1
n n 1 n 1 n n n n 1 n 1 n n 1
1 d xn , xn 1 d xn , xn 1
1
d xn , xn 1 d xn , xn 1
Therefore xn converges to x in X .Let y f x for infinitely many n , xn x for some n .
1
d xn , x d fx n 1 , fy d fy , fx n 1
1 d y, fy d xn 1 , fx n 1 d x , fx d y, fy d x , fy d y, fx d x , y
1 d xn 1 , y
n 1 n 1 n 1 n 1 n 1
1 d y, x d xn 1 , xn d x , x d y, x d x , x d y, x d x , y
1 d xn 1 , y
n 1 n n 1 n n 1
Since d xn , x as n we have d xn 1 , xn d xn 1 , x 0
Therefore d x, y 0 when x y .
i.e y f x x
This completes the proof of the theorem.
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Theorem (3.2): Let X denotes the complete metric space with metric d and S and T is a mapping of X into itself .If
there exists non negative real’s , , , 1 with 2 2 1 such that
d Sx, Ty
1 d y,Ty d x, Sx d x, Sx d y,Ty d x,Ty d y, Sx d x, y
1 d x, y
For each x, y in X with x y and S
T has a fixed point.
&
Proof: Let x 0 be an arbitrary point in X . Since S & T maps itself there exist points x1 , x2 in X such that
x1 S 1 x0 , x2 T 1 x1
Continuing the process, we get a sequence xn in X such that
x2 n 1 S 1 x2 n & T 1 x2 n 1 x2 n 2
We see that if x2 n x2 n 1 for some
n then x2 n is a common fixed point of S & T .
Therefore we suppose that x2 n x2 n 1 for all n 0
From the hypothesis
d x2 n , x2 n 1 d Sx2 n 1 , Tx 2 n 2
1 d x2n 2 ,Tx2n 2 d x2n 1, Sx2n 1 d x , Sx d x ,Tx
1 d x2 n 1 , x2 n 2
2 n 1 2 n 1 2n 2 2n 2
d x2 n 1 , Tx2 n 2 d x2 n 2 , Sx2 n 1 d x2 n 1 , x2 n 2
d x2 n 1 , x2 n d x2 n 1 , x2 n d x2 n 2 , x2 n 1 d x2 n 1 , x2 n 1 d x2 n 2 , x2 n d x2 n 1 , x2 n 2
1 d x2 n , x2 n 1 d x2 n 1 , x2 n 2
1
d x2 n 1, x2 n 2 d x2n , x2n 1
1
Where k 1 .similarly it can be shown that
d x2 n 1 , x2 n 2 kdx2 n , x2 n 1
Therefore, for all n ,
d xn 1 , xn 2 kdxn , xn 1
....... k d x0 , x1
n1
Now, for any mn ,
d xn , xm d xn , xn 1 d xn 1 , xn 2 ...... d xm 1 , xm
k k
n n 1
.......... k
m1
d x , x
1 0
n
d x1 , x0
k
1 k
Which shows that xn is a Cauchy sequence in X and so it has a limit z .Since X is complete metric space, we have
z X , there exist v, w X such that S v z, Tw z
From the hypothesis we have
d x2 n , z d Sx2 n 1 , Tw
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1 d w,Twd x2n 1, Sx2n 1 d x , Sx d w,Tw
1 d x2 n 1 , w
2 n 1 2 n 1
d x2 n 1 , Tw d w, Sx2 n 1 d x2 n 1 , w
1 d w, z d x2 n 1 , x2 n d x , x d w, Tw d x , Tw d w, x d x , w
1 d x2 n 1 , w
2 n 1 2n 2 n 1 2n 2 n 1
Since d xn , z as n we have d x2 n 1 , x2 n d x2 n 1 , z 0
Therefore w z i.e Sw Tw z
This completes the proof of the theorem.
Section II: Some Fixed Point Theorems In 2-Metric Spaces For Expansion Mappings
Theorem 3.3: Let X denotes the complete 2-metric space with metric d and f is a mapping of X into itself. If there
exist non negative real’s , , , 1 with 2 2 1 such that
d fx, fy, a
1 d y, fy, a d x, fx, a d x, fx, a d y, fy, a d x, fy, a d y, fx, a
1 d x, y, a
d x, y, a
For each x, y in X with x y and f is onto then f has a fixed point.
Proof: Let x0 X .since f is onto, there is an element x1 f 1 x0 .similarly we can write
xn f 1 xn 1 n 1,2,3.........
From the hypothesis
d xn 1 , xn , a d fx n , fx n 1 , a
1 d xn 1, fxn 1, a d xn , fxn , a d x , fx , a d x , fx , a d x , fx , a d x , fx , a
1 d xn , xn 1 , a
n n n 1 n 1 n n 1 n 1 n
d xn , xn 1 , a
1 d xn 1 , xn , a d xn , xn 1 , a d x , x , a d x , x , a d x , x , a d x , x , a
1 d xn , xn 1 , a
n n 1 n 1 n n n n 1 n 1
d xn , xn 1 , a
d xn , xn 1 , a d xn , xn 1 , a d xn 1 , xn , a d xn 1 , xn , a d xn , xn 1 , a d xn , xn 1 , a
1 d xn , xn 1 , a d xn , xn 1 , a
1
d xn , xn 1 , a d xn , xn 1 , a
Therefore xn converges to x in X .Let y f x for infinitely many n, xn x for such n
1
d xn , x, a d fx n 1 , fy , a d fy , fx n 1 , a
1 d y, fy, a d xn 1, fxn 1, a d x , fx , a d y, fy, a d x , fy, a d y, fx , a
1 d xn 1 , y, a
n 1 n 1 n 1 n 1
d xn 1 , y, a
1 d y, x, a d xn 1 , xn , a d x , x , a d y, x, a d x , x, a d y, x , a d x , y, a
1 d xn 1 , y, a
n 1 n n 1 n n 1
Since d xn , x, a as n we have d xn 1 , xn , a d xn 1 , x, a 0
Therefore d x, y, a 0 x y
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i.e y f x x .
Theorem 3.4: Let X denotes the complete metric space with metric d and S & T is a mappings of X into itself .If
there exists non negative real’s , , , 1 with 2 2 1 such that
d Sx, Ty, a
1 d y,Ty, a d x, Sx, a d x, Sx, a d y,Ty, a d x,Ty, a d y, Sx, a
1 d x, y, a
dx, y, a
For each x, y in X with x y and S & T has a fixed point.
Proof: Let x 0 be an arbitrary point in X . Since S & T maps into itself there exist points x1 , x2 in X such that
x1 S 1
x0 x2 T 1
x1
Continuing the process, we get a sequence xn in C such that
x2 n 1 S 1 x2 n & T 1 x2 n 1 x2 n 2
Sx2 n 1 x2 n Tx2 n 2 x2 n 1
We see that if x2 n x2 n 1 for some n then x2 n is a common fixed point of S & T .
Therefore we suppose that x2 n x2 n 1 for all n 0
From the hypothesis
d x2 n , x2 n 1 , a d Sx2 n 1 , Tx 2 n 2 , a
1 d x2n 2 ,Tx2n 2 , a d x2n 1, Sx2n 1, a d x , Sx , a d x ,Tx , a
1 d x2 n 1 , x2 n 2 , a
2 n 1 2 n 1 2n 2 2n 2
d x2 n 1 , Tx2 n 2 , a d x2 n 2 , Sx2 n 1 , a d x2 n 1 , x2 n 2 , a
1 d x2n 2 , x2n 1, a d x2n 1, x2n , a d x , x , a d x , x , a
1 d x2 n 1, x2 n 2 , a
2 n 1 2n 2n 2 2 n 1
d x2 n 1, x2 n 1, a d x2 n 2 , x2 n , a d x2 n 1, x2 n 2 , a
1 d x2 n , x2 n 1 , a d x2 n 1 , x2 n 2 , a
1
d x2 n 1 , x2 n 2 , a d x2 n , x2 n 1 , a
1
where k 1
d x2 n 1 , x2 n 2 , a kdx2 n , x2 n 1 , a
Similarly d x2 n 2 , x2 n 3 , a kdx2 n 1 , x2 n 2 , a
Therefore we obtain d xn 1 , xn 2 , a kdxn , xn 1 , a
Which shows that xn is a Cauchy sequence in X and so it has a limit is complete metric space, we have X ,
there exists v, w X such that S v , T w
From the hypothesis we have
d x2 n , , a d Sx2 n 1 , Tw, a
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1 d w,Tw, a d w, Sx2n 1, a d x , Sx , a d w,Tw, a
1 d x2 n 1 , w, a
2 n 1 2 n 1
d x2 n 1 , Tw, a d w, Sx2 n 1 , a d x2 n 1 , w, a
1 d w, , a d w, x2n , a d x , x , a d w, Tw, a
1 d x2 n1 , w, a
2 n1 2n
d x2 n1 , Tw, a d w, x2 n , a d x2 n1 , w, a
Since d x2 n , , a as n we have d x2 n 1 , x2 n , a d x2 n 1 , w, a 0 . Therefore d w, , a 0
implies w i.e Sw Tw
This completes the proof of the theorem.
References
[1]. Agrawal, A.K. and Chouhan, P. “Some fixed point theorems for expansion mappings” Jnanabha 35 (2005) 197-199.
[2]. Agrawal, A.K. and Chouhan, P. “Some fixed point theorems for expansion mappings” Jnanabha 36 (2006) 197-199.
[3]. Bhardwaj, R.K. Rajput, S.S. and Yadava, R.N. “Some fixed point theorems in complete Metric spaces” International
J.of Math.Sci. & Engg.Appls.2 (2007) 193-198.
[4]. Fisher, B. “Mapping on a metric space” Bull. V.M.I. (4), 12(1975) 147-151.
[5]. Jain, R.K. and Jain, R. “Some fixed point theorems on expansion mappings” Acta Ciencia Indica 20(1994) 217-220.
[6]. Jain, R. and Yadav, V. “A common fixed point theorem for compatible mappings in metric spaces” The Mathematics
Education (1994) 183-188.
[7]. Park, S. “On extensions of the Caristi-Kirk fixed point theorem” J.Korean Math.Soc.19(1983) 223-228.
[8]. Park, S. and Rhoades, B.E. “Some fixed point theorems for expansion mappings” Math.Japonica 33(1) (1988) 129-
133.
[9]. Popa, V. “Fixed point theorem for expansion mappings” Babes Bolyal University, Faculty of Mathematics and physics
Research Seminor 3 (1987) 25-30.
[10]. Rhoades B.E. “A comparison of various definitions of contractive mappings” Trans. Amer Math.Soc.226 (1976) 257-
290.
[11]. Sharma.P.L., Sharma.B.K. and Iseki, K.“Contractive type mappings on 2-metric spaces” Math.Japonica 21 (1976) 67-
70.
[12]. Taniguchi, T. “Common fixed point theorems on expansion type mappings on complete metric spaces” Math.Japonica
34(1989) 139-142.
[13]. Wang, S.Z. Gao, Z.M. and Iseki, K. “Fixed point theorems on expansion mappings” Math.Japonica .29 (1984) 631-
636.
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