This document presents a theorem that generalizes previous results on finding a common fixed point for weakly compatible self-maps. It proves that if two pairs of self-maps (S,A) and (B,T) satisfy certain conditions, including one map being non-vacuously compatible and one space being complete, then the maps have a unique common fixed point. The conditions are: 1) the maps satisfy inclusions and an inequality relating their distances, 2) the pairs are weakly compatible, 3) one pair is non-vacuously compatible or satisfies property E.A., and 4) one of the ranges is complete. The proof considers four cases based on which range is complete and shows the maps have a coincidence point and
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.
This document introduces eigenvalues and eigenvectors through examples. It begins by discussing how linear algebra can be used to solve systems of equations, moving from the "row picture" of intersecting lines to the "column picture" of linear combinations of vectors. It introduces key concepts like vectors, scalars, and linear combinations. The document then discusses how to represent systems of equations using matrices by choosing appropriate reference vectors. This allows solving problems algebraically by finding linear combinations of vectors rather than geometrically finding intersections of lines.
An approach to Fuzzy clustering of the iris petals by using Ac-meansijsc
This paper proposes a definition of a fuzzy partition element based on the homomorphism between type-1 fuzzy sets and the three-valued Kleene algebra. A new clustering method
based on the C-means algorithm, using the defined partition, is presented in this paper, which will
be validated with the traditional iris clustering problem by measuring its petals.
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.
This document summarizes a directed research report on using singular value decomposition (SVD) to reconstruct images with missing pixel values. It describes how images can be represented as matrices and SVD is commonly used for matrix completion problems. The report explores using an alternating least squares (ALS) algorithm based on SVD to fill in missing pixel values by finding feature matrices that approximate the rank k reconstruction of an image matrix. The ALS algorithm works by alternating between optimizing one feature matrix while holding the other fixed, minimizing the reconstruction error between the known pixel values and predicted values from multiplying the feature matrices.
The document classifies all possible Uq(sl2)-module algebra structures on the quantum plane. It produces a complete list of these structures and describes the isomorphism classes. The composition series of the representations under these structures are computed to classify the structures.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
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.
This document introduces eigenvalues and eigenvectors through examples. It begins by discussing how linear algebra can be used to solve systems of equations, moving from the "row picture" of intersecting lines to the "column picture" of linear combinations of vectors. It introduces key concepts like vectors, scalars, and linear combinations. The document then discusses how to represent systems of equations using matrices by choosing appropriate reference vectors. This allows solving problems algebraically by finding linear combinations of vectors rather than geometrically finding intersections of lines.
An approach to Fuzzy clustering of the iris petals by using Ac-meansijsc
This paper proposes a definition of a fuzzy partition element based on the homomorphism between type-1 fuzzy sets and the three-valued Kleene algebra. A new clustering method
based on the C-means algorithm, using the defined partition, is presented in this paper, which will
be validated with the traditional iris clustering problem by measuring its petals.
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.
This document summarizes a directed research report on using singular value decomposition (SVD) to reconstruct images with missing pixel values. It describes how images can be represented as matrices and SVD is commonly used for matrix completion problems. The report explores using an alternating least squares (ALS) algorithm based on SVD to fill in missing pixel values by finding feature matrices that approximate the rank k reconstruction of an image matrix. The ALS algorithm works by alternating between optimizing one feature matrix while holding the other fixed, minimizing the reconstruction error between the known pixel values and predicted values from multiplying the feature matrices.
The document classifies all possible Uq(sl2)-module algebra structures on the quantum plane. It produces a complete list of these structures and describes the isomorphism classes. The composition series of the representations under these structures are computed to classify the structures.
International Journal of Computational Engineering Research(IJCER)ijceronline
International Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
The document discusses pseudospectra as an alternative to eigenvalues for analyzing non-normal matrices and operators. It defines three equivalent definitions of pseudospectra: (1) the set of points where the resolvent is larger than ε-1, (2) the set of points that are eigenvalues of a perturbed matrix with perturbation smaller than ε, and (3) the set of points where the resolvent applied to a unit vector is larger than ε. It also shows that pseudospectra are nested sets and their intersection is the spectrum. The definitions extend to operators on Hilbert spaces using singular values.
This document provides an overview of linear models for classification. It discusses discriminant functions including linear discriminant analysis and the perceptron algorithm. It also covers probabilistic generative models that model class-conditional densities and priors to estimate posterior probabilities. Probabilistic discriminative models like logistic regression directly model posterior probabilities using maximum likelihood. Iterative reweighted least squares is used to optimize logistic regression since there is no closed-form solution.
4. [22 25]characterization of connected vertex magic total labeling graphs in...Alexander Decker
The document summarizes research characterizing connected vertex magic total labeling (VMTL) graphs through ideals in topological spaces. Some key results:
1. A connected VMTL graph G is Eulerian if and only if all vertices of G have even degree.
2. For a connected VMTL graph G, ideal I of even degree vertices, and topology τ on vertices, the local function {v}* is empty if and only if the degree of v is even.
3. G is Eulerian if and only if the local function is empty for all vertices, characterizing when G has only even degree vertices.
The paper presents theorems relating the topological concept of local functions of
11.characterization of connected vertex magic total labeling graphs in topolo...Alexander Decker
The document summarizes research characterizing connected vertex magic total labeling (VMTL) graphs through ideals in topological spaces. Some key results:
1. A connected VMTL graph G is Eulerian if and only if all vertices of G have even degree.
2. For a connected VMTL graph G, topology τ on vertices, and ideal I of even degree vertices, a vertex v has empty local function if and only if its degree is even.
3. G is Eulerian if and only if all local functions are empty, characterizing even degree vertices through ideals in topological spaces.
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.
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.
Density theorems for anisotropic point configurationsVjekoslavKovac1
The document summarizes recent work on density theorems for anisotropic point configurations. It discusses classical results on Euclidean density theorems and their study of "large" measurable sets. It then outlines the general approach taken, involving decomposing counting forms into structured, error, and uniform parts. Finally, it presents some of the author's recent results on anisotropic dilates of simplices, boxes, and trees, proving the existence of such configurations in sets with positive upper Banach density.
Fixed points theorem on a pair of random generalized non linear contractionsAlexander Decker
1) The document presents a fixed point theorem for a pair of random generalized non-linear contraction mappings involving four points of a separable Banach space.
2) It proves that if two random operators A1(w) and A2(w) satisfy a certain inequality involving upper semi-continuous functions, then there exists a unique random variable η(w) that is the common fixed point of A1(w) and A2(w).
3) As an example, the theorem is applied to prove the existence of a solution in a Banach space to a random non-linear integral equation of the form x(t;w) = h(t;w) + integral of k
Abstract Quadripartitioned single valued neutrosophic (QSVN) set is a powerful structure where we have four components Truth-T, Falsity-F, Unknown-U and Contradiction-C. And also it generalizes the concept of fuzzy, initutionstic and single valued neutrosophic set. In this paper we have proposed the concept of K-algebras on QSVN, level subset of QSVN and studied some of the results. In addition to this we have also investigated the characteristics of QSVN Ksubalgebras under homomorphism.
4. Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagona...Ceni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the fourth part which is discussing eigenvalues, eigenvectors and diagonalization.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Here are the slides of the second part which was discussing basis and dimension:
https://www.slideshare.net/CeniBabaogluPhDinMat/2-linear-algebra-for-machine-learning-basis-and-dimension
Here are the slides of the third part which is discussing factorization and linear transformations.
https://www.slideshare.net/CeniBabaogluPhDinMat/3-linear-algebra-for-machine-learning-factorization-and-linear-transformations-130813437
Lecture 8 nul col bases dim & rank - section 4-2, 4-3, 4-5 & 4-6njit-ronbrown
The document discusses null spaces, column spaces, and bases of matrices. It begins by defining the null space of a matrix A as the set of all solutions to the homogeneous equation Ax = 0. It then proves that the null space of any matrix is a subspace. Similarly, it defines the column space of A as the set of all linear combinations of A's columns, and proves the column space is always a subspace. The document contrasts the properties of null spaces and column spaces. It also discusses finding bases for null spaces and column spaces. Finally, it covers linear independence, spanning sets, and using pivots to determine bases.
The document provides examples to illustrate how to find the eigenvalues and eigenvectors of a matrix.
1) For a 2x2 matrix, the characteristic polynomial is computed by taking the determinant of the matrix minus the identity matrix. The roots of the characteristic polynomial are the eigenvalues. The corresponding eigenvectors are found by solving the original eigenvalue equation.
2) For a triangular matrix, the eigenvalues are the diagonal elements. The eigenvectors are found by setting rows corresponding to non-diagonal elements to zero.
3) The document provides a numerical example to demonstrate finding the eigenvalues (3, 1, -2) and eigenvectors of a 3x3 matrix.
Computer Science
Active and Programmable Networks
Active safety systems
Ad Hoc & Sensor Network
Ad hoc networks for pervasive communications
Adaptive, autonomic and context-aware computing
Advance Computing technology and their application
Advanced Computing Architectures and New Programming Models
Advanced control and measurement
Aeronautical Engineering,
Agent-based middleware
Alert applications
Automotive, marine and aero-space control and all other control applications
Autonomic and self-managing middleware
Autonomous vehicle
Biochemistry
Bioinformatics
BioTechnology(Chemistry, Mathematics, Statistics, Geology)
Broadband and intelligent networks
Broadband wireless technologies
CAD/CAM/CAT/CIM
Call admission and flow/congestion control
Capacity planning and dimensioning
Changing Access to Patient Information
Channel capacity modelling and analysis
Civil Engineering,
Cloud Computing and Applications
Collaborative applications
Communication application
Communication architectures for pervasive computing
Communication systems
Computational intelligence
Computer and microprocessor-based control
Computer Architecture and Embedded Systems
Computer Business
Computer Sciences and Applications
Computer Vision
Computer-based information systems in health care
Computing Ethics
Computing Practices & Applications
Congestion and/or Flow Control
Content Distribution
Context-awareness and middleware
Creativity in Internet management and retailing
Cross-layer design and Physical layer based issue
Cryptography
Data Base Management
Data fusion
Data Mining
Data retrieval
Data Storage Management
Decision analysis methods
Decision making
Digital Economy and Digital Divide
Digital signal processing theory
Distributed Sensor Networks
Drives automation
Drug Design,
Drug Development
DSP implementation
E-Business
E-Commerce
E-Government
Electronic transceiver device for Retail Marketing Industries
Electronics Engineering,
Embeded Computer System
Emerging advances in business and its applications
Emerging signal processing areas
Enabling technologies for pervasive systems
Energy-efficient and green pervasive computing
Environmental Engineering,
Estimation and identification techniques
Evaluation techniques for middleware solutions
Event-based, publish/subscribe, and message-oriented middleware
Evolutionary computing and intelligent systems
Expert approaches
Facilities planning and management
Flexible manufacturing systems
Formal methods and tools for designing
Fuzzy algorithms
Fuzzy logics
GPS and location-based app
This document summarizes a presentation given on April 21, 2020. It discusses models of words, trees and graphs, first-order logic, monadic second-order logic, and the relationship between automata and monadic second-order logic on finite words and trees. Specifically, it summarizes theorems showing that languages of finite words/trees are recognizable by finite automata if and only if they are definable in monadic second-order logic. Conversion between automata and logical formulas is also effective.
1. The orthogonal decomposition theorem states that any vector y in Rn can be written uniquely as the sum of a vector ŷ in a subspace W and a vector z orthogonal to W.
2. The vector ŷ is called the orthogonal projection of y onto W. It is the closest vector to y that lies in W.
3. The best approximation theorem states that the orthogonal projection ŷ provides the best or closest approximation of y using only vectors that lie in the subspace W. The distance from y to ŷ is less than the distance from y to any other vector in
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.
11.a focus on a common fixed point theorem using weakly compatible mappingsAlexander Decker
The document presents a common fixed point theorem that generalizes an earlier theorem by Bijendra Singh and M.S. Chauhan. It replaces the conditions of compatibility and completeness with weaker conditions of weakly compatible mappings and an associated convergent sequence. The theorem proves that if self-maps A, B, S, and T of a metric space satisfy certain conditions, including (1) A(X) ⊆ T(X) and B(X) ⊆ S(X), (2) the pairs (A,S) and (B,T) are weakly compatible, and (3) the associated sequence converges, then the maps have a unique common fixed point. An example is given where the
A focus on a common fixed point theorem using weakly compatible mappingsAlexander Decker
The document presents a theorem that generalizes an existing fixed point theorem using weaker conditions. Specifically, it replaces the conditions of compatibility and completeness with weakly compatible mappings and an associated convergent sequence. The theorem proves that if four self-maps satisfy certain conditions, including being weakly compatible and having an associated sequence that converges, then the maps have a unique common fixed point. The conditions are shown to be weaker using an example where the associated sequence converges even though the space is not complete.
A weaker version of continuity and a common fixed point theoremAlexander Decker
This article presents a generalization of previous theorems on common fixed points of self-maps. It introduces the concept of property E.A. and weak compatibility between self-maps. A new theorem (Theorem B) is proved which finds a unique common fixed point for three self-maps under weaker conditions than previous results, including relaxing orbital completeness and removing the requirement of orbital continuity. The proof of Theorem B is provided. It is shown that this new theorem generalizes an earlier result from the literature.
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
Fixed Point Theorem in Fuzzy Metric Space Using (CLRg) Propertyinventionjournals
The object of this paper is to establish a common fixed point theorem for semi-compatible pair of self maps by using CLRg Property in fuzzy metric space.
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]
The document discusses pseudospectra as an alternative to eigenvalues for analyzing non-normal matrices and operators. It defines three equivalent definitions of pseudospectra: (1) the set of points where the resolvent is larger than ε-1, (2) the set of points that are eigenvalues of a perturbed matrix with perturbation smaller than ε, and (3) the set of points where the resolvent applied to a unit vector is larger than ε. It also shows that pseudospectra are nested sets and their intersection is the spectrum. The definitions extend to operators on Hilbert spaces using singular values.
This document provides an overview of linear models for classification. It discusses discriminant functions including linear discriminant analysis and the perceptron algorithm. It also covers probabilistic generative models that model class-conditional densities and priors to estimate posterior probabilities. Probabilistic discriminative models like logistic regression directly model posterior probabilities using maximum likelihood. Iterative reweighted least squares is used to optimize logistic regression since there is no closed-form solution.
4. [22 25]characterization of connected vertex magic total labeling graphs in...Alexander Decker
The document summarizes research characterizing connected vertex magic total labeling (VMTL) graphs through ideals in topological spaces. Some key results:
1. A connected VMTL graph G is Eulerian if and only if all vertices of G have even degree.
2. For a connected VMTL graph G, ideal I of even degree vertices, and topology τ on vertices, the local function {v}* is empty if and only if the degree of v is even.
3. G is Eulerian if and only if the local function is empty for all vertices, characterizing when G has only even degree vertices.
The paper presents theorems relating the topological concept of local functions of
11.characterization of connected vertex magic total labeling graphs in topolo...Alexander Decker
The document summarizes research characterizing connected vertex magic total labeling (VMTL) graphs through ideals in topological spaces. Some key results:
1. A connected VMTL graph G is Eulerian if and only if all vertices of G have even degree.
2. For a connected VMTL graph G, topology τ on vertices, and ideal I of even degree vertices, a vertex v has empty local function if and only if its degree is even.
3. G is Eulerian if and only if all local functions are empty, characterizing even degree vertices through ideals in topological spaces.
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.
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.
Density theorems for anisotropic point configurationsVjekoslavKovac1
The document summarizes recent work on density theorems for anisotropic point configurations. It discusses classical results on Euclidean density theorems and their study of "large" measurable sets. It then outlines the general approach taken, involving decomposing counting forms into structured, error, and uniform parts. Finally, it presents some of the author's recent results on anisotropic dilates of simplices, boxes, and trees, proving the existence of such configurations in sets with positive upper Banach density.
Fixed points theorem on a pair of random generalized non linear contractionsAlexander Decker
1) The document presents a fixed point theorem for a pair of random generalized non-linear contraction mappings involving four points of a separable Banach space.
2) It proves that if two random operators A1(w) and A2(w) satisfy a certain inequality involving upper semi-continuous functions, then there exists a unique random variable η(w) that is the common fixed point of A1(w) and A2(w).
3) As an example, the theorem is applied to prove the existence of a solution in a Banach space to a random non-linear integral equation of the form x(t;w) = h(t;w) + integral of k
Abstract Quadripartitioned single valued neutrosophic (QSVN) set is a powerful structure where we have four components Truth-T, Falsity-F, Unknown-U and Contradiction-C. And also it generalizes the concept of fuzzy, initutionstic and single valued neutrosophic set. In this paper we have proposed the concept of K-algebras on QSVN, level subset of QSVN and studied some of the results. In addition to this we have also investigated the characteristics of QSVN Ksubalgebras under homomorphism.
4. Linear Algebra for Machine Learning: Eigenvalues, Eigenvectors and Diagona...Ceni Babaoglu, PhD
The seminar series will focus on the mathematical background needed for machine learning. The first set of the seminars will be on "Linear Algebra for Machine Learning". Here are the slides of the fourth part which is discussing eigenvalues, eigenvectors and diagonalization.
Here is the link of the first part which was discussing linear systems: https://www.slideshare.net/CeniBabaogluPhDinMat/linear-algebra-for-machine-learning-linear-systems/1
Here are the slides of the second part which was discussing basis and dimension:
https://www.slideshare.net/CeniBabaogluPhDinMat/2-linear-algebra-for-machine-learning-basis-and-dimension
Here are the slides of the third part which is discussing factorization and linear transformations.
https://www.slideshare.net/CeniBabaogluPhDinMat/3-linear-algebra-for-machine-learning-factorization-and-linear-transformations-130813437
Lecture 8 nul col bases dim & rank - section 4-2, 4-3, 4-5 & 4-6njit-ronbrown
The document discusses null spaces, column spaces, and bases of matrices. It begins by defining the null space of a matrix A as the set of all solutions to the homogeneous equation Ax = 0. It then proves that the null space of any matrix is a subspace. Similarly, it defines the column space of A as the set of all linear combinations of A's columns, and proves the column space is always a subspace. The document contrasts the properties of null spaces and column spaces. It also discusses finding bases for null spaces and column spaces. Finally, it covers linear independence, spanning sets, and using pivots to determine bases.
The document provides examples to illustrate how to find the eigenvalues and eigenvectors of a matrix.
1) For a 2x2 matrix, the characteristic polynomial is computed by taking the determinant of the matrix minus the identity matrix. The roots of the characteristic polynomial are the eigenvalues. The corresponding eigenvectors are found by solving the original eigenvalue equation.
2) For a triangular matrix, the eigenvalues are the diagonal elements. The eigenvectors are found by setting rows corresponding to non-diagonal elements to zero.
3) The document provides a numerical example to demonstrate finding the eigenvalues (3, 1, -2) and eigenvectors of a 3x3 matrix.
Computer Science
Active and Programmable Networks
Active safety systems
Ad Hoc & Sensor Network
Ad hoc networks for pervasive communications
Adaptive, autonomic and context-aware computing
Advance Computing technology and their application
Advanced Computing Architectures and New Programming Models
Advanced control and measurement
Aeronautical Engineering,
Agent-based middleware
Alert applications
Automotive, marine and aero-space control and all other control applications
Autonomic and self-managing middleware
Autonomous vehicle
Biochemistry
Bioinformatics
BioTechnology(Chemistry, Mathematics, Statistics, Geology)
Broadband and intelligent networks
Broadband wireless technologies
CAD/CAM/CAT/CIM
Call admission and flow/congestion control
Capacity planning and dimensioning
Changing Access to Patient Information
Channel capacity modelling and analysis
Civil Engineering,
Cloud Computing and Applications
Collaborative applications
Communication application
Communication architectures for pervasive computing
Communication systems
Computational intelligence
Computer and microprocessor-based control
Computer Architecture and Embedded Systems
Computer Business
Computer Sciences and Applications
Computer Vision
Computer-based information systems in health care
Computing Ethics
Computing Practices & Applications
Congestion and/or Flow Control
Content Distribution
Context-awareness and middleware
Creativity in Internet management and retailing
Cross-layer design and Physical layer based issue
Cryptography
Data Base Management
Data fusion
Data Mining
Data retrieval
Data Storage Management
Decision analysis methods
Decision making
Digital Economy and Digital Divide
Digital signal processing theory
Distributed Sensor Networks
Drives automation
Drug Design,
Drug Development
DSP implementation
E-Business
E-Commerce
E-Government
Electronic transceiver device for Retail Marketing Industries
Electronics Engineering,
Embeded Computer System
Emerging advances in business and its applications
Emerging signal processing areas
Enabling technologies for pervasive systems
Energy-efficient and green pervasive computing
Environmental Engineering,
Estimation and identification techniques
Evaluation techniques for middleware solutions
Event-based, publish/subscribe, and message-oriented middleware
Evolutionary computing and intelligent systems
Expert approaches
Facilities planning and management
Flexible manufacturing systems
Formal methods and tools for designing
Fuzzy algorithms
Fuzzy logics
GPS and location-based app
This document summarizes a presentation given on April 21, 2020. It discusses models of words, trees and graphs, first-order logic, monadic second-order logic, and the relationship between automata and monadic second-order logic on finite words and trees. Specifically, it summarizes theorems showing that languages of finite words/trees are recognizable by finite automata if and only if they are definable in monadic second-order logic. Conversion between automata and logical formulas is also effective.
1. The orthogonal decomposition theorem states that any vector y in Rn can be written uniquely as the sum of a vector ŷ in a subspace W and a vector z orthogonal to W.
2. The vector ŷ is called the orthogonal projection of y onto W. It is the closest vector to y that lies in W.
3. The best approximation theorem states that the orthogonal projection ŷ provides the best or closest approximation of y using only vectors that lie in the subspace W. The distance from y to ŷ is less than the distance from y to any other vector in
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.
11.a focus on a common fixed point theorem using weakly compatible mappingsAlexander Decker
The document presents a common fixed point theorem that generalizes an earlier theorem by Bijendra Singh and M.S. Chauhan. It replaces the conditions of compatibility and completeness with weaker conditions of weakly compatible mappings and an associated convergent sequence. The theorem proves that if self-maps A, B, S, and T of a metric space satisfy certain conditions, including (1) A(X) ⊆ T(X) and B(X) ⊆ S(X), (2) the pairs (A,S) and (B,T) are weakly compatible, and (3) the associated sequence converges, then the maps have a unique common fixed point. An example is given where the
A focus on a common fixed point theorem using weakly compatible mappingsAlexander Decker
The document presents a theorem that generalizes an existing fixed point theorem using weaker conditions. Specifically, it replaces the conditions of compatibility and completeness with weakly compatible mappings and an associated convergent sequence. The theorem proves that if four self-maps satisfy certain conditions, including being weakly compatible and having an associated sequence that converges, then the maps have a unique common fixed point. The conditions are shown to be weaker using an example where the associated sequence converges even though the space is not complete.
A weaker version of continuity and a common fixed point theoremAlexander Decker
This article presents a generalization of previous theorems on common fixed points of self-maps. It introduces the concept of property E.A. and weak compatibility between self-maps. A new theorem (Theorem B) is proved which finds a unique common fixed point for three self-maps under weaker conditions than previous results, including relaxing orbital completeness and removing the requirement of orbital continuity. The proof of Theorem B is provided. It is shown that this new theorem generalizes an earlier result from the literature.
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
Fixed Point Theorem in Fuzzy Metric Space Using (CLRg) Propertyinventionjournals
The object of this paper is to establish a common fixed point theorem for semi-compatible pair of self maps by using CLRg Property in fuzzy metric space.
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]
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.
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
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Common fixed point for two weakly compatible pairs of self maps through property e. a. under an implicit relation
1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.6, 2012
Common fixed point for two weakly compatible pairs
of self-maps through property E. A. under an implicit relation
T. Phaneendra
Applied Analysis Division
School of Advanced Sciences
VIT-University, Vellore - 632 014, Tamil Nadu, INDIA
E–mail: drtp.indra@gmail.com
Swatmaram
Department of Mathematics
Chatanya Bharathi Institute of Technology
Hyderabad – 500 075 (A. P.), INDIA
E–mail: ramuswatma@yahoo.com
Abstract
We prove some generalizations of the results of Renu Chug-Sanjay Kumar, the author with necessary corrections, and
of Rhoades et al. for weakly compatible self-maps satisfying the property E. A. under an implicit relation.
Key Words: Compatible and Weakly Compatible Self-maps, Property E. A., Contractive Mmodulus, Common Fixed
Point
1 INTRODUCTION
Throughout this paper, ( X , d ) denotes a metric space. If x ∈ X and A is a self-map on X, we write Ax for the A-image
of x, A(X) for the range of A and AS for the composition of self-maps A and S. Also IR + will denote the set of all non
negative real numbers. A point p of the space X is a fixed point of a self-map A if and only if Ap = p. A self-map A on
X with the choice d ( Ax, Ay) ≤ a d ( x, y) for all x, y ∈ X is known as a contraction, provided 0 ≤ a < 1 . According to the
celebrated Banach contraction principle (BCP), every contraction on a complete metric space has a unique fixed
point. It is easy to see that a discontinuous self-map cannot be a contraction and hence contraction principle cannot
ensure a fixed point for it even if X is complete. However, various generalizations of BCP have been established by
weakening the contraction condition, relaxing the completeness of the underlying space and/or extending to two or
more self-maps under additional assumptions. To mention a few are the works of Fisher ([2]-[3]), Fisher and Khan
[4], Chang [5], Ciric [6], Das and Naik [7], Jungck [8], Pant [11] and Singh and Singh [18].
Self-maps S and A on X are said to be weakly commuting [17] if d ( ASx, SAx) ≤ a d (Sx, Ax) for all x, y ∈ X . As a
further generalization for commuting maps, Gerald Jungck [9] proposed the compatibility as in the following lines:
Definition 1.1 Self-maps S and A on X are compatible if
lim d (SAxn , ASxn ) = 0 (1)
n∞ ∞
→
whenever xn n =1
⊂X such that
lim Axn = lim Sxn = t for some t ∈ X. (2)
n→∞ n→∞
∞
If there is no xn n = 1 in X with the choice (2), S and T are called vacuously compatible.
∞
Remark 1.1 Self-maps S and A on X are not (non-vacuously) compatible if there is a sequence xn n =1
in
1
2. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.6, 2012
X with choice (2) but lim d (SAxn , ASxn ) ≠ 0 or + ∞.
n →∞
Obviously every commuting pair of self-maps is a weakly commuting one, and a weakly commuting pair is
necessarily compatible. But neither reverse implication is true [9].
The following notion is due to Aamri and Moutawakil [1]:
∞
Definition 1.2 Self-maps S and T satisfy the property E. A. if there is an xn n =1
⊂ X with the choice (2).
In this paper CX denotes the class of all non-compatible pairs of self-maps on X, while C* , the class of all pairs of
X
self-maps on X which satisfy the property E. A.
∞
In view of Remark 1.1, non-compatibility implies the existence of the sequence xn n =1
with the choice (2).
Hence the class C*
X is potentially wider than CX. We also note that vacuously compatible self-maps do not satisfy
the property E. A.
We call a point x ∈ X , a coincidence point of self-maps S and A on X if Tx = Sx, while y ∈ X is a point of their
coincidence w. r. t. x if Tx = Sx = y. Taking xn = x for all n, from (1) and (2), it follows that STx = TSx
whenever x ∈ X is such that Tx = Sx. Hence we have
Definition 1.3 Self-maps which commute at their coincidence points are weakly compatible maps [10] which are
also called coincidentally commuting or partially commuting [16].
Being weakly compatible and possessing property E. A. are independent conditions of each other (Pathak et al.
[12])
In this paper φ : IR5 → IR+ is an upper semi-continuous (written shortly as usc) non decreasing in each coordinate
+
variable, and for ξ > 0 satisfy the conditions:
(i) max{φ(ξ, ξ,0,2ξ,0), φ(ξ, ξ,0, ,0,2ξ)} ≤ ξ
(ii) max{φ(ξ, ξ,0, αξ,0), φ(ξ, ξ,0, ,0, αξ)} < ξ when α < 2
(iii) γ(t ) = φ(ξ, ξ, α1ξ, α 2ξ, α3ξ) < ξ if α1 + α 2 + α3 = 4 .
Remark 1.2 ξ ≤ φ(ξ, ξ, ξ, ξ, ξ) implies that ξ = 0 .
With this notion Renu Chug and Sanjay Kumar [14] proved the following result for two weakly compatible pairs of
self-maps:
Theorem 1.1 Let A, B, S and T be self-maps on X satisfying the inclusions
A( X ) ⊂ T ( X ) , (3-a)
B( X ) ⊂ S ( X ) (3-b)
and the inequality
d ( Ax, By) ≤ φ(d (Sx,Ty), d ( Ax, Sx), d (By,Ty), d ( Ax,Ty), d ( By, Sx)) for all x, y ∈ X . (4)
Suppose that:
(a) X is complete
2
3. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.2, No.6, 2012
(b) The pairs ( A, S ) and (B,T ) are weakly compatible.
Then all the four maps A, B, S and T have a unique common fixed point.
The authors of Theorem 1.1 asserted that there is a point u ∈ X such that z = Su under the inclusion
(3-b) (Line 4 from the bottom, page 244). We point out that their assertion is not true and that the inclusion (3-b)
plays no role to obtain the point u. However it will be true if we assume that the map B (and hence S) is onto.
In view of this suggestion we restate Theorem 1.1 as follows:
Theorem 1.2 Let A, B, S and T be self-maps on X satisfying the inclusions (3-a), (3-b), the inequality (4), (a) and (b)
of Theorem 1.1. If
(c) either A or B is onto,
then all the four maps A, B, S and T have a unique common fixed point.
In this paper we obtain the conclusion of Theorem 1.1, by relaxing the completeness of the space X and using
the property E. A. (see below). Our result will be a generalization of those some of the authors.
2 MAIN RESULT
First we prove
Theorem 2.1 Let A, B, S and T be self-maps on X satisfying (3-a), (3-b), and the inequality (4).
Suppose that
(d) either (S , A) ∈ C* or (B,T ) ∈ C* ,
X X
(e) one of S ( X ), A( X ), T ( X ) and B(X ) is a complete subspace of X.
If the condition (b) of Theorem 1.2 holds good, then A, B, S and T will have a unique common fixed point.
∞
Proof. First suppose that (S , A) ∈ C* . Then there is a xn
X n =1
⊂ X such that lim Axn = lim Sxn = p for
n→∞ n→∞
∞
some p ∈ X. By virtue of the inclusion (3-a), we can find another sequence yn n =1
of points in X such that
Axn = Tyn for all n so that
lim Axn = lim Sxn = lim Tyn = p. (5)
n→∞ n→∞ n→∞
Observe that q = lim Byn also equals p. In fact, writing x = xn and y = yn in the inequality (4), we see that
n→∞
d ( Axn , Byn ) ≤ φ ( d (Sxn ,Tyn ), d ( Axn , Sxn ), d (Byn , Tyn ), d ( Axn ,Tyn ), d (Byn , Sxn ))
Applying the limit as n → ∞ in this, and using (5) and the usc of φ,
d ( p, q) ≤ φ (0,0, d (q, p),0, d (q, p)) ≤ φ (d ( p, q), d ( p, q), d ( p, q), d ( p, q), d ( p, q))
so that or d ( p, q) = 0 or p = q , in view of Remark 1.2. Thus
lim Axn = lim Sxn = lim Byn = lim Tyn = p. (6)
n→∞ n→∞ n→∞ n→∞
We first show that p is a common coincidence point for (S , A) and (B,T ) . That is
Ap = Sp = Bp = Tp .
(7)
Case (a): Suppose that T (X ) is complete.
Then p ∈ T (X ) so that Tu = p for some u ∈ X. But then from (4), we see that
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d ( Axn , Bu) ≤ φ (d (Sxn ,Tu ), d ( Axn , Sxn ), d (Bu,Tu ), d ( Axn ,Tu ), d (Bu, Sxn ))
Applying the limit as n → ∞ in this, and using Tu = p and upper semi-continuity of φ followed by its non
decreasing nature in each coordinate variable, we get
d ( p, Bu) ≤ φ (0,0, d (Bu, p),0, d (Bu, p)) ≤ φ (d ( Bu, p), d ( Bu, p), d (Bu, p), d (Bu, p), d (Bu, p))
or d ( p, Bu) = 0 Bu = p . Thus u is a coincidence point, and hence p is a point of coincidence for B and T . That is
Bu = Tu = p. (8)
Again from the inclusion (3-b), we see that p = Bu = Sr for some r ∈ X .
Since φ is non decreasing, from (4) and (8) we get
d ( Ar, Sr) = d ( Ar, Bu) ≤ φ (d (Sr,Tu), d ( Ar, Sr), d (Bu,Tu), d ( Ar,Tu), d (Bu, Sr))
= φ (0, d ( Ar , p), 0, d ( Ar , p), 0) ≤ φ (d ( Ar , p), d ( Ar, p), d ( Ar, p), d ( Ar, p), d ( Ar, p))
so that Ar = Sr = p , in view of Remark 1.2. That is p is a point of coincidence for A and S w. r. t. r. Thus
Ar = Sr = Bu = Tu = p. (9)
Since (A, S) and (B, T) are weakly compatible pairs, (7) follows from (9).
Case (b): Suppose that A(X ) is complete.
Then p ∈ A( X ) ⊂ T ( X ) , in view of (3-a) and hence (7) follows from Case (a).
Case (c): Suppose that S (X ) is complete.
Then p∈ S ( X ) so that Sw = p for some w ∈ X. But then from (4), we see that
d ( Aw, Byn ) ≤ φ (d (Sw,Tyn ), d ( Aw, Sw), d (Byn ,Tyn ), d ( Aw,Tyn ), d (Byn , Sw))
Applying the limit as n → ∞ , and using (6), Sw = p and upper semi-continuity of φ, followed by its non decreasing
nature in each coordinate variable, this gives
d ( Aw, p) ≤ φ (0, d ( Aw, p), 0, d ( Aw, p), 0) ≤ φ (d ( Aw, p), d ( Aw, p), d ( Aw, p), d ( Aw, p), d ( Aw, p))
or d ( p, Aw) = 0 or Aw = p . Thus w is a coincidence point and p is a point of coincidence for A and S
w. r. t. w. That is
Aw = Sw = p . (10)
Again from the inclusion (3-a), we see that p = Aw = Tq for some q ∈ X .
Since φ is non decreasing, from (4) and (10) we get
d ( p, Bq) = d ( Aw, Bq) ≤ φ (d (Sw, Tq), d ( Aw, Sw), d ( Bq, Tq), d ( Aw, Tq), d (Bq, Sw))
= φ (0,0, d ( Bq, p), 0, d ( Bq, p)) ≤ φ (d (Bq, p), d ( Bq, p), d (Bq, p), d (Bq, p), d ( Bq, p))
so that p = Bq in view of Remark 1.2.
Thus Aw = Sw = Bq = Tq = p. (11)
Now (7) will follow from (b) and (11).
Case (d): Suppose that B( X ) is complete.
Then p ∈ B( X ) ⊂ S ( X ) , in view of (3-b) and hence (7) follows from Case (c).
To establish that p is a common fixed point for all the four maps, we again use (4) with x = r and y = p so that
d ( Ar, Bp) ≤ φ (d (Sr,Tp), d ( Ar, Sr ), d (Bp,Tp), d ( Ar ,Tp), d (Bp, Sr )) .
Again since φ is non decreasing, this together with (7) and (8) gives
d ( p, Bp) ≤ φ (d ( p, Bp),0, 0, d ( p, Bp), d (Bp, p)) ≤ φ (d ( p, Bp), d ( p, Bp), d ( p, Bp), d ( p, Bp), d ( p, Bp))
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so that from Remark 1.2, it follows that Bp = p and hence p is a common fixed point for A, B, S and T, which in
fact is a point of their common coincidence.
Now suppose that (B,T ) ∈ C* . Then exchanging the roles of (S , A) and (B,T ) ; of (3-a) and (3-b); and of
X
T (X ) and S (X ) in the above proof, we can similarly obtain the conclusion.
Uniqueness of the common fixed point follows easily from the choice of φ and (4).
Remark 2.1 If A is onto then A( X ) = X . Therefore the completeness of X implies the completeness of A(X ) .
Similarly the completeness of X implies the completeness of B(X ) whenever B is onto.
Remark 2.2 We can prove that if self-maps A, B, S and T on X satisfy the inclusions (3-a), (3-b) and the inequality
(4), and X is complete, then both ( A, S ) ∈ C* and (B,T ) ∈ C* .
X X
∞
In fact for any x0 ∈ X , the inclusions (3 a-b) generate a sequence of points xn n =1
in X with the choice
Ax2n−2 = Tx2n −1 , Bx2n −1 = Sx2n for all n ≥ 1. (12)
∞ ∞
From the proof of Theorem 1.2, it follows that Ax2n n = 1 and Ax2n n = 1 are Cauchy sequences.
If X is complete, these will converge to some z ∈ X. That is
lim Ax2n−2 = lim Tx2n−1 = lim Bx2n−1 = lim Sx2n = p. (13)
n→∞ n→∞ n→∞ n→∞
With x2n = xn and x2n−1 = yn from (13), we see that
* *
* *
lim Axn = lim Sxn = p * *
and lim Byn = lim Tyn = p,
n→∞ n→∞ n→∞ n→∞
proving that the pairs (S , A) and (B,T ) satisfy the property E. A.
Remark 2.3 In view of Remarks 2.1 and 2.2, we see that Theorem 1.2 follows as a particular case of our result
(Theorem 2.1).
It is possible to relax the condition (b), and drop the inclusions in Theorem 2.1 for three self-maps
A, B and T.
In deed we prove the following
Theorem 2.2 Let A, B and T be self-maps on X satisfying the inequality
d ( Ax, By) ≤ φ(d (Tx, Ty), d ( Ax, Tx), d ( By,Ty), d ( Ax,Ty), d (By, Tx))
for all x, y ∈ X . (14)
Suppose that T ( X ) is complete subspace of X and
(f) either of the pairs ( A,T ) and (B,T ) is both (weakly compatible and belongs to the class C* ).
X
Then A, B and T will have a unique common fixed point.
∞
Proof. Suppose that the pair ( A,T ) ∈ C* . Then there exists a sequence
X xn n =1
⊂ X such that
lim Axn = lim Txn = p for some p ∈ X. (15)
n→∞ n→∞
Let q = lim Bxn . Then q = p. In fact, writing x = y = xn in the inequality (14),
n→∞
d ( Axn , Bxn ) ≤ φ ( d (Txn ,Txn ), d ( Axn ,Txn ), d (Bxn ,Txn ), d ( Axn, Txn ), d (Bxn, Txn ))
Applying the limit as n → ∞ in this and using (15) and upper semi-continuity of φ,
d ( p, q) ≤ φ (0,0, d (q, p),0, d (q, p)) ≤ φ (d ( p, q), d ( p, q), d ( p, q), d ( p, q), d ( p, q))
so that d ( p, q) = 0 or p = q , in view of Remark 1.2. Thus
lim Axn = lim Txn = lim Bxn = p. (16)
n→∞ n→∞ n→∞
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Suppose that T ( X ) is complete. Then p ∈ T ( X ) so that Tu = p for some u ∈ X. But then from (14),
d ( Axn , Bu) ≤ φ (d (Txn ,Tu), d ( Axn , Txn ), d (Bu, Tu), d ( Axn ,Tu), d ( Bu,Txn )) .
Applying the limit as n → ∞ in this, and using Tu = p and upper semi-continuity of φ followed by its non decreasing
nature in each coordinate variable, we get
d ( p, Bu) ≤ φ (0,0, d ( Bu, p),0, d ( Bu, p)) ≤ φ (d (Bu, p), d (Bu, p), d (Bu, p), d (Bu, p), d (Bu, p)) or d ( p, Bu) = 0
or Bu = p . Thus u is a coincidence point of B and T. That is Bu = Tu = p .
Since φ is non decreasing, from (14), we get
d ( Au, Bu ) ≤ φ (d (Tu,Tu ), d ( Au, Su ), d ( Bu,Tu), d ( Au,Tu ), d ( Bu,Tu))
= φ (0, d ( Au, p), 0), d ( Au, p), 0) ≤ φ (d ( Au, p), d ( Au, p), d ( Au, p), d ( Au, p), d ( Au, p))
so that Au = Tu in view of Remark 1.2.
Thus Au = Bu = Tu = p . (17)
Since (A, T) is weakly compatible pair, from (17) we get that Ap = Tp .
Again since φ is non decreasing, from (14) and using Ap = Tp , we get
d ( Ap, Bp) ≤ φ (d (Tp, Tp), d ( Ap,Tp), d (Bp, Tp), d ( Ap,Tp), d (Bp, Tp))
= φ (0,0, d ( Ap, Bp), 0, d ( Ap, Bp)) ≤ φ (d ( Ap, Bp), d ( Ap, Bp), d ( Ap, Bp), d ( Ap, Bp), d ( Ap, Bp))
so that Ap = Bp in view of Remark 1.2.
Thus Ap = Bp = Tp . (18)
Finally p will be a common fixed point for the three maps. For, writing x = u and y = p in (14) and then using
(17) and (18) we get
d ( p, Tp) = d ( Au, Bp) ≤ φ (d (Tu, Tp), d ( Au, Tp), d (Bp, Tp), d ( Au, Tp), d ( Bp, Tu))
= φ (d ( p, Tp), d ( p, Tp), 0, d ( p, Tp), d (Tp, p)) ≤ φ (d ( p, Tp), d ( p, Tp), d ( p, Tp), d (Tp, p), d ( p, Tp))
Again since φ is non decreasing, this with Remark 1.2, implies that Bp = p and hence p is a common fixed point
for A, B, and T.
Similarly if ( A,T ) is a weakly compatible pair satisfying the property E.A., it follows that A, B, and T have a
common fixed point. Uniqueness of the common fixed point follows easily from the choice of φ and (14).
Taking A = B in Theorem 2.2 we have
Corollary 2.1 Let A and T be self-maps on X satisfying the inequality
d ( Ax, Ay) ≤ φ(d (Tx, Ty), d ( Ax,Tx), d ( Ay,Ty ), d ( Ax, Ay), d ( Ay,Tx)) for all x, y ∈ X . (19)
If T ( X ) is complete subspace of X, ( A,T ) is weakly compatible and ( A,T ) ∈ C* , then A and T will have a unique
X
common fixed point.
Given x0 ∈ X and self-maps A and T on X, if there exist points x1, x2, x3,... in X with Axn −1 = Txn for n ≥ 1 , the
∞
sequence Axn n =1 defines a ( A,T ) -orbit or simply an orbit at x0. The space X is ( A,T ) -orbitally complete [13]
at x0 ∈ X if every Cauchy sequence in some orbit at x0 converges in X .
Note that every complete metric space is orbitally complete at each of its points. However there are incomplete
metric spaces which are orbitally complete at some point of it [13].
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We give the following generalization of Corollary 2.1, the proof of which can be obtained on similar lines of that of
Theorem 2.2.
Theorem 2.3 Let A and T be self-maps on X satisfying the inequality (14) and ( A,T ) ∈ C* . Suppose that any one of
X
the following conditions holds good:
(g) the subspace A( X ) is orbitally complete at x0 ∈ X
(h) T ( X ) is orbitally complete at x0 ∈ X .
Then there is a coincidence point z for A and T. Further if A and T are weakly compatible, then the point of
coincidence of A and T w. r. t. z will be a unique common fixed point for them.
In the remainder of the paper ψ : IR+ → IR+ represents a contractive modulus, due to Solomon Leader [19] with the
choice ψ(t ) < t for t > 0.
Corollary 2.2 (Theorem 2, [13]) Let A and T be self-maps on X satisfying the inclusion
A( X ) ⊂ T ( X ) (20)
and the inequality
d ( Ax, Ay) ≤ ψ(max d (Tx, Ty), d ( Ax, Tx), d ( Ay, Ty) 1 [ d ( Ax, Ty) + d ( Ay, Tx)]) for all x, y ∈ X ,
2
(21)
where ψ is non decreasing and upper semi continuous contractive modulus.
Suppose that one of the conditions (g), (h) and (i) holds good, where
(i) the space X is orbitally complete at some x0 ∈ X and T is onto
Then there is a coincidence point z for A and T. Further if A and T are weakly compatible, then the point of
coincidence of A and T w. r. t. z will be a unique common fixed point for them.
Proof. We note that the condition (i) implies (g). We set
φ(ξ1, ξ2, ξ3, ξ4, ξ5) = ψ(max{ξ1, ξ2, ξ3, 1 [ξ4 + ξ5]}) for all ξi , i = 1,2, ..., 5 .
2
Then (14) reduces to (21).
Let x0 ∈ X be arbitrary. By virtue of the inclusion (20), we can construct an ( A,T ) - orbit x0 with choice
∞
Axn −1 = Txn for n ≥ 1 . Then the sequence Axn n =1
is a Cauchy sequence in this orbit at X.
In view of Remark 2.2, A and T satisfy the property E. A. Hence the conclusion follows from Theorem 2.3.
Now write B = A in Theorem 2.1, we get
Theorem 2.4 Let A, S and T be self-maps on X satisfying the inclusion
A( X ) ⊂ [S ( X ) I T ( X )] (22)
and the inequality
d ( Ax, Ay) ≤ φ(d (Sx,Ty), d ( Ax, Sx), d ( Ay, Ty), d ( Ax,Ty), d ( Ay, Sx)) for all x, y ∈ X . (23)
Suppose that
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(j) either of the pairs ( A, S ) and ( A,T ) satisfies the property E. A.
(k) one of A( X ), B( X ) and T ( X ) is a compete subspace of X.
If ( A, S ) and ( A,T ) are weakly compatible, then A, B, S and T will have a unique common fixed point.
Since every compatible pair is weakly compatible, we have the following sufficiency part of Main theorem of [15].
Corollary 2.3 Let A, S and T be self-maps on X satisfying (20) and the inequality
d ( Ax, Ay) ≤ max{d (Sx,Ty), d ( Ax, Sx), d ( Ay,Ty), 1 [d ( Ax,Ty) + d ( Ay, Sx)]}
2
− ω(max{d (Sx,Ty), d ( Ax, Sx), d ( Ay,Ty), 1 [d ( Ax,Ty) + d ( Ay, Sx)]}) for all x, y ∈ X ,
2
(24)
where ω : IR+ → IR+ is continuous and 0 < ω(t ) < t for t > 0 . Suppose that X is complete, A is continuous and
( A, S ) and ( A,T ) are compatible. Then A, S and T will have a unique common fixed point.
Note that (23) reduces to (24) with
φ(ξ1, ξ2 , ξ3, ξ4 , ξ5 ) = maxξ1, ξ2 , ξ3, [ξ4 + ξ5 ] − ω maxξ1, ξ2 , ξ3, [ξ4 + ξ5 ] for all ξi , i = 1,2, ..., 5 .
1 1
2 2
∞
Given x0 ∈ X , due to the inclusion (22), we can construct the sequence Axn n =1
with choice
Ax2n − 2 = Tx2n−1, Ax2n −1 = Tx2n for n ≥ 1 ,
which is a Cauchy sequence in the complete space X and hence converges in it.
In view of Remark 2.2, A and T satisfy the property E. A.
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