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
Some Common Fixed Point Results for Expansive Mappings in a Cone Metric SpaceIOSR Journals
The purpose of this work is to extend and generalize some common fixed point theorems for Expansive type mappings in complete cone metric spaces. We are attempting to generalize the several well- known recent results. Mathematical subject classification; 54H25, 47H10
Common Fixed Point Theorems For Occasionally Weakely Compatible Mappingsiosrjce
Som [11 ] establishes a common fixed point theorem for R-weakly Commuting mappings in a Fuzzy
metric space.The object of this Paper is to prove some fixed point theorems for occasionally Weakly compatible
mappings by improving the condition of Som[11 ].
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.
Some Common Fixed Point Results for Expansive Mappings in a Cone Metric SpaceIOSR Journals
The purpose of this work is to extend and generalize some common fixed point theorems for Expansive type mappings in complete cone metric spaces. We are attempting to generalize the several well- known recent results. Mathematical subject classification; 54H25, 47H10
Common Fixed Point Theorems For Occasionally Weakely Compatible Mappingsiosrjce
Som [11 ] establishes a common fixed point theorem for R-weakly Commuting mappings in a Fuzzy
metric space.The object of this Paper is to prove some fixed point theorems for occasionally Weakly compatible
mappings by improving the condition of Som[11 ].
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 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 Uniform SpacesIJLT EMAS
In the process of generalization of metric spaces to
Topological spaces, a few aspects of metric spaces are lost.
Therefore, the requirement of generalization of metric spaces
leads to the theory of uniform spaces. Uniform spaces stand
somewhere in between metric spaces and general topological
spaces. Khan[6] extended fixed point theorems due to Hardy and
Rogers[2], Jungck[4] and Acharya[1] in uniform space by
obtaining some results on common fixed points for a pair of
commuting mappings defined on a sequentially complete
Hausdorff uniform space. Rhoades et. al.[7] generalized the
result of Khan[6] by establishing a general fixed point theorem
for four compatible maps in uniform space .
In this paper, a common fixed point theorem in
uniform spaces is proved which generalizes the result of Khan[6]
and Rhoades et al.[7] by employing the less restrictive condition
of weak compatibility for one pair and the condition of
compatibility for second pair, the result is proved for six selfmappings.
The paper reports on an iteration algorithm to compute asymptotic solutions at any order for a wide class of nonlinear
singularly perturbed difference equations.
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 Results for Weakly Compatible Mappings in Convex G-Metric Spaceinventionjournals
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.
Pseudo Bipolar Fuzzy Cosets of Bipolar Fuzzy and Bipolar Anti-Fuzzy HX Subgroupsmathsjournal
In this paper, we introduce the concept of pseudo bipolar fuzzy cosets, pseudo bipolar fuzzy double cosets of a bipolar fuzzy and bipolar anti-fuzzy subgroups. We also establish these concepts to bipolar fuzzy and bipolar anti-fuzzy HX subgroups of a HX group with suitable examples. Also we discuss some of their relative properties.
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 Results In Fuzzy Menger Space With Common Property (E.A.)IJERA Editor
This paper presents some common fixed point theorems for weakly compatible mappings via an implicit relation in Fuzzy Menger spaces satisfying the common property (E.A)
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.
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 Uniform SpacesIJLT EMAS
In the process of generalization of metric spaces to
Topological spaces, a few aspects of metric spaces are lost.
Therefore, the requirement of generalization of metric spaces
leads to the theory of uniform spaces. Uniform spaces stand
somewhere in between metric spaces and general topological
spaces. Khan[6] extended fixed point theorems due to Hardy and
Rogers[2], Jungck[4] and Acharya[1] in uniform space by
obtaining some results on common fixed points for a pair of
commuting mappings defined on a sequentially complete
Hausdorff uniform space. Rhoades et. al.[7] generalized the
result of Khan[6] by establishing a general fixed point theorem
for four compatible maps in uniform space .
In this paper, a common fixed point theorem in
uniform spaces is proved which generalizes the result of Khan[6]
and Rhoades et al.[7] by employing the less restrictive condition
of weak compatibility for one pair and the condition of
compatibility for second pair, the result is proved for six selfmappings.
The paper reports on an iteration algorithm to compute asymptotic solutions at any order for a wide class of nonlinear
singularly perturbed difference equations.
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 Results for Weakly Compatible Mappings in Convex G-Metric Spaceinventionjournals
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.
Pseudo Bipolar Fuzzy Cosets of Bipolar Fuzzy and Bipolar Anti-Fuzzy HX Subgroupsmathsjournal
In this paper, we introduce the concept of pseudo bipolar fuzzy cosets, pseudo bipolar fuzzy double cosets of a bipolar fuzzy and bipolar anti-fuzzy subgroups. We also establish these concepts to bipolar fuzzy and bipolar anti-fuzzy HX subgroups of a HX group with suitable examples. Also we discuss some of their relative properties.
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 Results In Fuzzy Menger Space With Common Property (E.A.)IJERA Editor
This paper presents some common fixed point theorems for weakly compatible mappings via an implicit relation in Fuzzy Menger spaces satisfying the common property (E.A)
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.
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.
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.
Existance Theory for First Order Nonlinear Random Dfferential Equartioninventionjournals
In this paper, the existence of a solution of nonlinear random differential equation of first order is proved under Caratheodory condition by using suitable fixed point theorem. 2000 Mathematics Subject Classification: 34F05, 47H10, 47H4
A Fixed Point Theorem Using Common Property (E. A.) In PM Spacesinventionjournals
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.
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.
https://utilitasmathematica.com/index.php/Index
Utilitas Mathematica journal that publishes original research. This journal publishes mainly in areas of pure and applied mathematics, statistics and others like algebra, analysis, geometry, topology, number theory, diffrential equations, operations research, mathematical physics, computer science,mathematical economics.And it is official publication of Utilitas Mathematica Academy, Canada.
Semicompatibility and fixed point theorem in fuzzy metric space using implici...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Semicompatibility and fixed point theorem in fuzzy metric space using implici...eSAT Journals
Abstract In this paper we proved fixed point theorem of four mapping on fuzzy metric space based on the concept of semi copatibility using implicit relation. These results generalize several corresponding relations in fuzzy metric space. All the results of this paper are new. Keywords: fuzzy metric space, compatibility, semi compatibility, implicit relation. 2000 AMS Mathematics Subject Classification: 47H10, 54H25
Some properties of two-fuzzy Nor med spacesIOSR Journals
The study sheds light on the two-fuzzy normed space concentrating on some of their properties like convergence, continuity and the in order to study the relationship between these spaces
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
including those with tonal differences. The proposed method offers a practical and efficient solution for
speaker diarization in real-world scenarios where there are labeling time and cost constraints.
A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,
a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We
will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal
argument in its original form to establish the cardinality of K between that of (N,R) respectively
A Positive Integer 𝑵 Such That 𝒑𝒏 + 𝒑𝒏+𝟑 ~ 𝒑𝒏+𝟏 + 𝒑𝒏+𝟐 For All 𝒏 ≥ 𝑵mathsjournal
According to Bertrand's postulate, we have 𝑝𝑛 + 𝑝𝑛 ≥ 𝑝𝑛+1. Is it true that for all 𝑛 > 1 then 𝑝𝑛−1 + 𝑝𝑛 ≥𝑝𝑛+1? Then 𝑝𝑛 + 𝑝𝑛+3 > 𝑝𝑛+1 + 𝑝𝑛+2where 𝑛 ≥ 𝑁, 𝑁 is a large enough value?
A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,
a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We
will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal
argument in its original form to establish the cardinality of K between that of (N,R) respectively.
Moving Target Detection Using CA, SO and GO-CFAR detectors in Nonhomogeneous ...mathsjournal
systems in complex situations. A fundamental problem in radar systems is to automatically detect targets while maintaining a
desired constant false alarm probability. This work studies two detection approaches, the first with a fixed threshold and the
other with an adaptive one. In the latter, we have learned the three types of detectors CA, SO, and GO-CFAR. This research
aims to apply intelligent techniques to improve detection performance in a nonhomogeneous environment using standard
CFAR detectors. The objective is to maintain the false alarm probability and enhance target detection by combining
intelligent techniques. With these objectives in mind, implementing standard CFAR detectors is applied to nonhomogeneous
environment data. The primary focus is understanding the reason for the false detection when applying standard CFAR
detectors in a nonhomogeneous environment and how to avoid it using intelligent approaches.
OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
including those with tonal differences. The proposed method offers a practical and efficient solution for
speaker diarization in real-world scenarios where there are labeling time and cost constraints.
The Impact of Allee Effect on a Predator-Prey Model with Holling Type II Func...mathsjournal
There is currently much interest in predator–prey models across a variety of bioscientific disciplines. The focus is on quantifying predator–prey interactions, and this quantification is being formulated especially as regards climate change. In this article, a stability analysis is used to analyse the behaviour of a general two-species model with respect to the Allee effect (on the growth rate and nutrient limitation level of the prey population). We present a description of the local and non-local interaction stability of the model and detail the types of bifurcation which arise, proving that there is a Hopf bifurcation in the Allee effect module. A stable periodic oscillation was encountered which was due to the Allee effect on the
prey species. As a result of this, the positive equilibrium of the model could change from stable to unstable and then back to stable, as the strength of the Allee effect (or the ‘handling’ time taken by predators when predating) increased continuously from zero. Hopf bifurcation has arose yield some complex patterns that have not been observed previously in predator-prey models, and these, at the same time, reflect long term behaviours. These findings have significant implications for ecological studies, not least with respect to examining the mobility of the two species involved in the non-local domain using Turing instability. A spiral generated by local interaction (reflecting the instability that forms even when an infinitely large
carrying capacity is assumed) is used in the model.
A POSSIBLE RESOLUTION TO HILBERT’S FIRST PROBLEM BY APPLYING CANTOR’S DIAGONA...mathsjournal
We present herein a new approach to the Continuum hypothesis CH. We will employ a string conditioning,a technique that limits the range of a string over some of its sub-domains for forming subsets K of R. We will prove that these are well defined and in fact proper subsets of R by making use of Cantor’s Diagonal argument in its original form to establish the cardinality of K between that of (N,R) respectively.
Moving Target Detection Using CA, SO and GO-CFAR detectors in Nonhomogeneous ...mathsjournal
Modernization of radar technology and improved signal processing techniques are necessary to improve detection systems in complex situations. A fundamental problem in radar systems is to automatically detect targets while maintaining a
desired constant false alarm probability. This work studies two detection approaches, the first with a fixed threshold and the
other with an adaptive one. In the latter, we have learned the three types of detectors CA, SO, and GO-CFAR. This research
aims to apply intelligent techniques to improve detection performance in a nonhomogeneous environment using standard
CFAR detectors. The objective is to maintain the false alarm probability and enhance target detection by combining
intelligent techniques. With these objectives in mind, implementing standard CFAR detectors is applied to nonhomogeneous
environment data. The primary focus is understanding the reason for the false detection when applying standard CFAR
detectors in a nonhomogeneous environment and how to avoid it using intelligent approaches
OPTIMIZING SIMILARITY THRESHOLD FOR ABSTRACT SIMILARITY METRIC IN SPEECH DIAR...mathsjournal
Speaker diarization is a critical task in speech processing that aims to identify "who spoke when?" in an
audio or video recording that contains unknown amounts of speech from unknown speakers and unknown
number of speakers. Diarization has numerous applications in speech recognition, speaker identification,
and automatic captioning. Supervised and unsupervised algorithms are used to address speaker diarization
problems, but providing exhaustive labeling for the training dataset can become costly in supervised
learning, while accuracy can be compromised when using unsupervised approaches. This paper presents a
novel approach to speaker diarization, which defines loosely labeled data and employs x-vector embedding
and a formalized approach for threshold searching with a given abstract similarity metric to cluster
temporal segments into unique user segments. The proposed algorithm uses concepts of graph theory,
matrix algebra, and genetic algorithm to formulate and solve the optimization problem. Additionally, the
algorithm is applied to English, Spanish, and Chinese audios, and the performance is evaluated using wellknown similarity metrics. The results demonstrate that the robustness of the proposed approach. The
findings of this research have significant implications for speech processing, speaker identification
including those with tonal differences. The proposed method offers a practical and efficient solution for
speaker diarization in real-world scenarios where there are labeling time and cost constraints
Modified Alpha-Rooting Color Image Enhancement Method on the Two Side 2-D Qua...mathsjournal
Color in an image is resolved to 3 or 4 color components and 2-Dimages of these components are stored in separate channels. Most of the color image enhancement algorithms are applied channel-by-channel on each image. But such a system of color image processing is not processing the original color. When a color image is represented as a quaternion image, processing is done in original colors. This paper proposes an implementation of the quaternion approach of enhancement algorithm for enhancing color images and is referred as the modified alpha-rooting by the two-dimensional quaternion discrete Fourier transform (2-D QDFT). Enhancement results of this proposed method are compared with the channel-by-channel image enhancement by the 2-D DFT. Enhancements in color images are quantitatively measured by the color enhancement measure estimation (CEME), which allows for selecting optimum parameters for processing by thegenetic algorithm. Enhancement of color images by the quaternion based method allows for obtaining images which are closer to the genuine representation of the real original color.
An Application of Assignment Problem in Laptop Selection Problem Using MATLABmathsjournal
The assignment – selection problem used to find one-to- one match of given “Users” to “Laptops”, the main objective is to minimize the cost as per user requirement. This paper presents satisfactory solution for real assignment – Laptop selection problem using MATLAB coding.
The aim of this paper is to study the class of β-normal spaces. The relationships among s-normal spaces, pnormal spaces and β-normal spaces are investigated. Moreover, we study the forms of generalized β-closed
functions. We obtain characterizations of β-normal spaces, properties of the forms of generalized β-closed
functions and preservation theorems.
Cubic Response Surface Designs Using Bibd in Four Dimensionsmathsjournal
Response Surface Methodology (RSM) has applications in Chemical, Physical, Meteorological, Industrial and Biological fields. The estimation of slope response surface occurs frequently in practical situations for the experimenter. The rates of change of the response surface, like rates of change in the yield of crop to various fertilizers, to estimate the rates of change in chemical experiments etc. are of
interest. If the fit of second order response is inadequate for the design points, we continue the
experiment so as to fit a third order response surface. Higher order response surface designs are sometimes needed in Industrial and Meteorological applications. Gardiner et al (1959) introduced third order rotatable designs for exploring response surface. Anjaneyulu et al (1994-1995) constructed third order slope rotatable designs using doubly balanced incomplete block designs. Anjaneyulu et al (2001)
introduced third order slope rotatable designs using central composite type design points. Seshu babu et al (2011) studied modified construction of third order slope rotatable designs using central composite
designs. Seshu babu et al (2014) constructed TOSRD using BIBD. In view of wide applicability of third
order models in RSM and importance of slope rotatability, we introduce A Cubic Slope Rotatable Designs Using BIBD in four dimensions.
The caustic that occur in geodesics in space-times which are solutions to the gravitational field equations with the energy-momentum tensor satisfying the dominant energy condition can be circumvented if quantum variations are allowed. An action is developed such that the variation yields the field equations
and the geodesic condition, and its quantization provides a method for determining the extent of the wave packet around the classical path.
Approximate Analytical Solution of Non-Linear Boussinesq Equation for the Uns...mathsjournal
For one dimensional homogeneous, isotropic aquifer, without accretion the governing Boussinesq equation under Dupuit assumptions is a nonlinear partial differential equation. In the present paper approximate analytical solution of nonlinear Boussinesq equation is obtained using Homotopy perturbation transform method(HPTM). The solution is compared with the exact solution. The comparison shows that the HPTM is efficient, accurate and reliable. The analysis of two important aquifer
parameters namely viz. specific yield and hydraulic conductivity is studied to see the effects on the height of water table. The results resemble well with the physical phenomena.
A Probabilistic Algorithm for Computation of Polynomial Greatest Common with ...mathsjournal
In the earlier work, Knuth present an algorithm to decrease the coefficient growth in the Euclidean algorithm of polynomials called subresultant algorithm. However, the output polynomials may have a small factor which can be removed. Then later, Brown of Bell Telephone Laboratories showed the subresultant in another way by adding a variant called 𝜏 and gave a way to compute the variant. Nevertheless, the way failed to determine every 𝜏 correctly.
In this paper, we will give a probabilistic algorithm to determine the variant 𝜏 correctly in most cases by adding a few steps instead of computing 𝑡(𝑥) when given 𝑓(𝑥) and𝑔(𝑥) ∈ ℤ[𝑥], where 𝑡(𝑥) satisfies that 𝑠(𝑥)𝑓(𝑥) + 𝑡(𝑥)𝑔(𝑥) = 𝑟(𝑥), here 𝑡(𝑥), 𝑠(𝑥) ∈ ℤ[𝑥]
Table of Contents - September 2022, Volume 9, Number 2/3mathsjournal
Applied Mathematics and Sciences: An International Journal (MathSJ ) aims to publish original research papers and survey articles on all areas of pure mathematics, theoretical applied mathematics, mathematical physics, theoretical mechanics, probability and mathematical statistics, and theoretical biology. All articles are fully refereed and are judged by their contribution to advancing the state of the science of mathematics.
Code of the multidimensional fractional pseudo-Newton method using recursive ...mathsjournal
The following paper presents one way to define and classify the fractional pseudo-Newton method through a group of fractional matrix operators, as well as a code written in recursive programming to implement this
method, which through minor modifications, can be implemented in any fractional fixed-point method that allows
solving nonlinear algebraic equation systems.
6th International Conference on Machine Learning & Applications (CMLA 2024)ClaraZara1
6th International Conference on Machine Learning & Applications (CMLA 2024) will provide an excellent international forum for sharing knowledge and results in theory, methodology and applications of on Machine Learning & Applications.
Final project report on grocery store management system..pdfKamal Acharya
In today’s fast-changing business environment, it’s extremely important to be able to respond to client needs in the most effective and timely manner. If your customers wish to see your business online and have instant access to your products or services.
Online Grocery Store is an e-commerce website, which retails various grocery products. This project allows viewing various products available enables registered users to purchase desired products instantly using Paytm, UPI payment processor (Instant Pay) and also can place order by using Cash on Delivery (Pay Later) option. This project provides an easy access to Administrators and Managers to view orders placed using Pay Later and Instant Pay options.
In order to develop an e-commerce website, a number of Technologies must be studied and understood. These include multi-tiered architecture, server and client-side scripting techniques, implementation technologies, programming language (such as PHP, HTML, CSS, JavaScript) and MySQL relational databases. This is a project with the objective to develop a basic website where a consumer is provided with a shopping cart website and also to know about the technologies used to develop such a website.
This document will discuss each of the underlying technologies to create and implement an e- commerce website.
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
Understanding Inductive Bias in Machine LearningSUTEJAS
This presentation explores the concept of inductive bias in machine learning. It explains how algorithms come with built-in assumptions and preferences that guide the learning process. You'll learn about the different types of inductive bias and how they can impact the performance and generalizability of machine learning models.
The presentation also covers the positive and negative aspects of inductive bias, along with strategies for mitigating potential drawbacks. We'll explore examples of how bias manifests in algorithms like neural networks and decision trees.
By understanding inductive bias, you can gain valuable insights into how machine learning models work and make informed decisions when building and deploying them.
Student information management system project report ii.pdfKamal Acharya
Our project explains about the student management. This project mainly explains the various actions related to student details. This project shows some ease in adding, editing and deleting the student details. It also provides a less time consuming process for viewing, adding, editing and deleting the marks of the students.
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
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COMMON FIXED POINT THEOREMS IN COMPATIBLE MAPPINGS OF TYPE (P*) OF GENERALIZED INTUITIONISTIC FUZZY METRIC SPACES
1. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 3, September 2015
DOI : 10.5121/mathsj.2015.2301 1
COMMON FIXED POINT THEOREMS IN
COMPATIBLE MAPPINGS OF TYPE (P*) OF
GENERALIZED INTUITIONISTIC FUZZY
METRIC SPACES
R.Muthuraj1
& R.Pandiselvi2
1
PG and Research Department of Mathematics, H.H.The Rajah’s College,
Pudukkottai – 622 001, India.
2
Department of Mathematics, The Madura college, Madurai – 625 011, India.
ABSTRACT
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
KEYWORDS
Intuitionistic fuzzy metric spaces, compatible mappings of type ( P ), type (P-1) and type (P-2) , common
fixed point .
1.INTRODUCTION
The Concept of fuzzy set was introduced by Zadeh [23] in 1965 .Following the concept of fuzzy
sets, Deng [6] Kaleva and Seikalla [12] and kramosil and Michalek [13] introduced the concept of
fuzzy metric space, George and Veeramani [7] modified the concept of fuzzy metric space
introduced by kramosil and Michalek [13] .
Further, Sedghi and Shobe [19] defined ℳ-fuzzy metric space and proved a common fixed point
theorem in it. Jong Seo Park [15] introduced the concept of semi compatible and Weak
Compatible maps in fuzzy metric space and proved some fixed point theorems satisfying certain
conditions in ℳ-fuzzy metric spaces.
As a generalization of fuzzy sets, Atanassov [1] introduced and studied the concept of
intuitionistic fuzzy sets. Using the idea of intuitionistic fuzzy sets Park [16] defined the notion of
intuitionistic fuzzy metric space with the help of continuous t- norm and continuous t- conorm as
a generalization of fuzzy metric space, George and Veeramani [8] had showed that every metric
induces an intuitionistic fuzzy metric and found a necessary and sufficient conditions for an
intuitionistic fuzzy metric space to be complete. Choudhary [4] introduced mutually contractive
sequence of self maps and proved a fixed point theorem. Kramaosil and Michalek [13] introduced
the notion of Cauchy sequences in an intuitionistic fuzzy metric space and proved the well known
fixed point theorem of Banach[2]. Turkoglu et al [22] gave the generalization of Jungck’s[11]
Common fixed point theorem to intuitionistic fuzzy metric spaces.
2. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 3, September 2015
2
In this paper, we extend the result of common fixed point theorem for compatible mappings of
type (P-1) and type (P-2) in intuitionistic fuzzy metric space and prove common fixed point
theorem of type (P-1) and type (P-2) in intuitionistic fuzzy metric spaces, we also give an
example to validate our main theorem. Our results intuitionistically fuzzify the result of Muthuraj
and Pandiselvi [15].
2. PRELIMINARIES
We start with the following definitions.
Definition 2.1
A binary operation ∗ : [0,1] × [0,1] → [0,1] is said to be a continuous t-norm if * is satisfies the
following conditions.
(i) ∗ is commutative and associative,
(ii) ∗ is continuous,
(iii) a∗1 = a for all a∈ [0,1],
(iv) a∗b ≤ c∗d whenever a ≤ c and b ≤ d for all a,b,c,d ∈ [0,1].
Definition 2.2
A binary operation ◊ : [0,1] × [0,1] → [0,1] is said to be a continuous t-conorm if ◊ satisfies the
following conditions :
(i) ◊ is commutative and associative,
(ii) ◊ is continuous,
(iii) a ◊ 0 = a for all a ∈ [0,1],
(iv) a ◊ b ≤ c ◊ d whenever a ≤ c and b ≤ d for all a,b,c,d ∈ [0,1].
Definition 2.3
A 5-tuple (X, ℳ, ࣨ, ∗, ◊) is called an intuitionistic fuzzy metric space if X is an arbitrary set, ∗
is a continuous t-norm, ◊ a continuous t-conorm and ℳ, ࣨ are fuzzy sets on X3
× (0, ∞),
satisfying the following conditions, for each x, y, z, a∈X and
t, s > 0,
a) ℳ( x, y, z, t ) + ࣨ( x, y, z, t ) ≤ 1.
b) ℳ( x, y, z, t ) > 0.
c) ℳ( x, y, z, t ) = 1 if and only if x = y = z.
d) ℳ( x, y, z, t ) = ℳ ( p{ x, y, z}, t) where p is a permutation function,
e) ℳ( x, y, a, t ) ∗ ℳ( a, z, z, s ) ≤ ℳ( x, y, z, t + s )
f) ℳ( x, y, z ) : ( 0, ∞) → [0, 1] is continuous
g) ࣨ( x, y, z, t ) > 0
h) ࣨ( x, y, z, t ) = 0, if and only if x = y = z,
i) ࣨ( x, y, z, t = ࣨ( p{ x, y, z}, t) where p is a permutation function,
j) ࣨ( x, y, a, t ) ◊ ࣨ( a, z, z, s ) ≥ ࣨ( x, y, z, t + s ),
k) ࣨ( x, y, z, ⋅) : ( 0, ∞) → [0, 1] is continuous.
Then (ℳ, ࣨ) is called an intuitionistic fuzzy metric on X.
3. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 3, September 2015
3
Example 2.4
Let X = R, and ℳ(x, y, z, t ) =
୲
୲ା|୶ି୷|ା|୷ି|ା|ି୶|
, ࣨ( x, y, z, t ) =
|୶ି୷|ା|୷ି|ା|ି୶|
୲ା|୶ି୷|ା|୷ି|ା|ି୶|
for every x,
y, z and t > 0, let A and B defined as Ax = 2x + 1, Bx = x + 2, consider the sequence xn =
ଵ
୬
+1, n
= 1 , 2,… Thus we have
lim
୬→∞
ℳ(Axn, 3, 3, t) = lim
୬→∞
ℳ(Bxn, 3, 3, t) =1 and
lim
୬→∞
ࣨ( Axn, 3, 3, t) = lim
୬→∞
ࣨ( Bxn, 3, 3, t) = 0, for every t > 0.
Then A and B satisfying the property (E).
Definition 2.5
Let (X, ℳ, ࣨ, ∗,◊ ) be an intuitionistic fuzzy metric space and {xn} be a sequence in X.
a) {xn} is said to be converges to a point x∈X, if lim
୬→∞
ℳ( x, x, xn, t ) = 1 and
lim
୬→∞
ࣨ( x, x, xn, t ) = 0, for all t > 0.
b) {xn} is called Cauchy sequence if lim
୬→∞
ℳ(xn+p, xn+p, xn, t) = 1 and
lim
୬→∞
ࣨ(xn+p, xn+p, xn, t) = 0 for all t > 0 and p > 0.
c) An intuitionistic fuzzy metric space in which every Cauchy sequence is convergent is
said to be complete.
Lemma 2.6
Let (X, ℳ, ࣨ, ∗, ◊) be an intuitionistic fuzzy metric space. Then ℳ(x, y, z, t) and ࣨ(x, y, z, t)
are non-decreasing with respect to t, for all x, y, z in X.
Proof
By definition 2.3, for each x, y, z, a ∈X and t, s > 0
we have ℳ(x, y, a, t ) ∗ ℳ(a, z, z, s ) ≤ ℳ(x, y, z, t + s ). If we set a = z,
we get ℳ(z, y, z, t ) ∗ ℳ(z, z, z, s ) ≤ ℳ(x, y, z, t + s ), that is
ℳ(x, y, z, t + s ) ≥ ℳ(x, y, z, t ).
Similarly, ࣨ(x, y, a, t) ◊ ࣨ(a, z, z, s ) ≥ ࣨ(x, y, z, t + s ), for each x, y, z, a∈X and
t, s > 0, by definition of (X, ࣨ, ◊ ). If we set a = z, we get
ࣨ(x, y, z, t ) ◊ ࣨ(z, z, z, s ) ≥ ࣨ(x, y, z, t + s )
that is ࣨ(x, y, z, t + s ) ≤ ࣨ(x, y, z, t) .Hence in IFMS (X, ℳ, ࣨ, ∗, ◊ ),
ℳ(x, y, z, t ) and ࣨ(x, y, z, t ) are non-decreasing with respect to t, for all x, y, z in X.
4. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 3, September 2015
4
3.COMPATIBLE MAPPINGS OF TYPE
Definition 3.1
Let A and S be self mappings from an intuitionistic fuzzy metric space (X, ℳ, ࣨ, ∗,◊) into itself.
Then the mappings are said to be compatible if
lim
୬→∞
ℳ(ASxn, SAxn, SAxn, t) = 1 and
lim
୬→∞
ࣨ(ASxn, SAxn, SAxn, t) = 0, for all t > 0 whenever {xn} is a sequence in X such that lim
୬→∞
Axn
= lim
୬→∞
Sxn = z for some z∈X.
Definition 3.2
Let A and S be self mappings from an intuitionistic fuzzy metric space (X, ℳ, ࣨ, ∗, ◊ ) into
itself. Then the mappings are said to be compatible of type (P), if
lim
୬→∞
ℳ(AAxn, SSxn, SSxn, t ) = 1 and lim
୬→∞
ࣨ(AAxn, SSxn, SSxn, t ) = 0 for all t > 0, whenever
{xn} is a sequence in X such that lim
୬→∞
Axn = lim
୬→∞
Sxn = z for some z∈X.
Definition 3.3
Let A and S be self mappings from an intuitionistic fuzzy metric space (X, ℳ, ࣨ, ∗, ◊) into itself.
Then the mappings are said to be R-Weakly commuting of type (P), if there exists some R > 0,
such that ℳ(AAx, SSx, SSx, t) ≥ ℳ( Ax, Sx, Sx,
୲
ୖ
),
ࣨ(AAx, SSx, SSx, t) ≤ ࣨ(Ax, Sx, Sx,
୲
ୖ
), for all x in X and t > 0.
Definition 3.4
Let A and S be self mappings from an intuitionistic fuzzy metric space (X, ℳ, ࣨ, ∗, ◊) into
itself. Then the mappings are said to be compatible of type (P-1) if
lim
୬→∞
ℳ(SAxn, AAxn, AAxn, t ) = 1 and lim
୬→∞
ࣨ( SAxn, AAxn, AAxn, t ) = 0 for all t > 0, whenever
{xn} is a sequence in X such that lim
୬→∞
Axn = lim
୬→∞
S xn = z for some z∈X.
Definition 3.5
Let A and S be self mappings from an intuitionistic fuzzy metric space (X, ℳ, ࣨ, ∗,◊ ) into itself.
Then the mappings are said to be compatible of type (P-2) if
lim
୬→∞
ℳ (AAxn, SSxn, SSxn, t) = 1 and lim
୬→∞
ࣨ(AAxn, SSxn, SSxn, t) = 0 for all t > 0 whenever {xn}
is a sequence in X such that lim
୬→∞
Axn = lim
୬→∞
Sxn = z for some z∈X.
5. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 3, September 2015
5
Proposition 3.6
Let A and S be self mappings from an intuitionistic fuzzy metric space (X, ℳ, ࣨ, ∗, ◊ ) into
itself.
a) If A is continuous map then the pair of mappings (A, S) is compatible of type (P-1) if and
only if A and S are compatible.
b) If S is a continuous map then the pair of mappings (A, S) is compatible of
type (P-2) if and only if A and S are compatible.
Proof
a) Let lim
୬→∞
Axn = lim
୬→∞
Sxn = z for some z ∈ X, and let the pair (A, S) be compatible of type
(P-1). Since A is continuous, we have lim
୬→∞
ASxn = Az and lim
୬→∞
AAxn = Az. Therefore it
follows that
ℳሺ SAx୬, ASx୬, ASx୬, t ሻ ≥ ℳ ቀ SAx୬, AAx୬, AAx୬,
୲
ଶ
ቁ
∗ ℳሺ AAx୬, ASx୬, ASx୬,
୲
ଶ
ሻ and
ࣨሺ SAx୬, ASx୬, ASx୬, t ሻ ≤ ࣨ ቀ SAx୬, AAx୬, AAx୬,
୲
ଶ
ቁ
◊ ࣨሺ AAx୬, ASx୬, ASx୬,
୲
ଶ
ሻ
yields lim
୬→∞
ℳ ( SAxn, ASxn, ASxn, t ) ≥ 1 ∗ 1 = 1 and
lim
୬→∞
ࣨ( SAxn, ASxn, ASxn, t ) ≤ 0 ◊ 0 = 0 and so the mappings A and S are compatible.
Now, let A and S be compatible. Therefore it follows that
ℳሺSAx୬, AAx୬, AAx୬, t ሻ ≥ ℳ ൬SAx୬, ASx୬, ASx୬,
t
2
൰
∗ ℳሺASx୬, AAx୬, AAx୬,
t
2
ሻ
ࣨሺSAx୬, AAx୬, AAx୬, t ሻ ≤ ࣨ ൬SAx୬, ASx୬, ASx୬,
t
2
൰
◊ ࣨሺASx୬, AAx୬, AAx୬,
t
2
ሻ
yields lim
୬→∞
ℳ(SAxn, AAxn, AAxn, t ) ≥ 1 ∗ 1 = 1 and
lim
୬→∞
ࣨ( SAxn, AAxn, AAxn, t ) ≤ 0 ◊ 0 = 0 and
so that pair of mappings (A,S) are compatible of type (P-1).
6. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 3, September 2015
6
b) Let lim
୬→∞
Sxn = lim
୬→∞
Axn = z for some z in X and let the pair (A, S) be compatible of type
(P-2). Since S is continuous, we have lim
୬→∞
SAxn = Sz and
lim
୬→∞
SSxn = Sz. Therefore it follows that
ℳሺSAx୬, ASx୬, ASx୬, tሻ ≥ ℳ ൬SAx୬, SSx୬, SSx୬,
t
2
൰
∗ ℳ ቀSSx୬, ASx୬, ASx୬,
୲
ଶ
ቁ and
ࣨሺSAx୬, ASx୬, ASx୬, tሻ ≤ ࣨ ൬SAx୬, SSx୬, SSx୬,
t
2
൰
◊ ࣨ ൬SSx୬, ASx୬, ASx୬,
t
2
൰
yields lim
୬→∞
ℳ( SAxn, ASxn, ASxn, t ) ≥ 1 ∗ 1 = 1 and
lim
୬→∞
ࣨ( SAxn, ASxn, ASxn, t ) ≤ 0 ◊ 0 = 0 and so the mappings A and S are compatible.
Now let A and S be compatible. Then we have
ℳሺASx୬, SSx୬, SSx୬, tሻ ≥ ℳ ൬ASx୬, SAx୬, SAx୬,
t
2
൰
∗ ℳ ቀSAx୬, SSx୬, SSx୬,
୲
ଶ
ቁ and
ࣨሺASx୬, SSx୬, SSx୬, tሻ ≤ ࣨ ൬ASx୬, SAx୬, SAx୬,
t
2
൰
◊ ࣨ ൬SAx୬, SSx୬, SSx୬,
t
2
൰
yields lim
୬→∞
ℳ(ASxn, SSxn, SSxn, t ) ≥ 1 ∗ 1 = 1 and
lim
୬→∞
ࣨ(ASxn, SSxn, SSxn, t) ≤ 0 ◊ 0 = 0 and so the pair of mappings (A, S) are
compatible of type (P-2).
Proposition 3.7
Let A and S be self mappings from an intuitionistic fuzzy metric space (X, ℳ, ࣨ, ∗, ◊) into itself.
If the pair (A, S) is compatible of type (P-2) and Sz = Az for some z∈X. Then ASz = SSz.
Proof:
Let { xn} be a sequence in X defined by xn = z for n=1,2,… and let Az = Sz.
Then we have lim
୬→∞
Sxn = Sz and lim
୬→∞
Axn = Az. Since the pair (A, S) is compatible of type (P-2),
we have
ℳ( ASz, SSz, SSz, t ) = lim
୬→∞
ℳሺASxn, SSxn, SSxn, t ) = 1 and
ࣨ( ASz, SSz, SSz, t ) = lim
୬→∞
ࣨ(ASxn, SSxn, SSxn, t ) = 0.
Hence ASz = SSz.
7. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 3, September 2015
7
Proposition 3.8
Let A and S self mappings from an intuitionistic fuzzy metric space (X, ℳ, ࣨ, ∗, ◊) with t ∗ t ≥
t and (1- t) ◊ (1- t) ≤ 1- t for all t ∈ [0, 1] if the pair (A, S) are compatible of type (p -1) and Axn,
Sxn → z for some z in X and a sequence {xn} in X.
Then AAxn → Sz, if S is continuous at z.
Proof
Since S is continuous at z, we have SAxn → Sz. Since the pair (A, S) are compatible of type (P-1),
we have ℳ(SAxn, AAxn, AAxn, t) → 1 as n → ∞. It follows that
ℳ( Sz, AAxn, AAxn, t) ≥ ℳ( Sz, SAxn, SAxn,
୲
ଶ
) ∗ ℳ( SAxn, AAxn, AAxn,
୲
ଶ
) and
ࣨ (Sz, AAxn, AAxn, t ) ≤ ࣨ (Sz, SAxn, SAxn,
୲
ଶ
) ∗ ࣨ( SAxn, AAxn, AAxn,
୲
ଶ
) yield
ℳ (Sz, AAxn, AAxn, t) ≥ 1 ∗1 = 1 and
ࣨ(Sz, AAxn, AAxn, t) ≤ 0 ◊ 0 = 0 and so
we have AAxn → Sz as n → ∞.
Proposition 3.9
Let A and S be self mappings from an intuitionistic fuzzy metric space (X, ℳ, ࣨ, ∗, ◊) with t ∗t ≥
t and (1- t) ◊ (1- t) ≤ 1- t for t ∈ [0, 1]. If the pair (A, S) are compatible of type (P - 2) and Axn,
Sxn→z for some z in X and sequence {xn} in X. Then SSxn → Az if A is continuous at z.
Proof
Since A is continuous at z, we have ASxn → Az. Since the pair (A, S) are compatible of type (P -
2), we have ℳ( ASxn, SSxn, SSxn, t ) →1 as n→∞, it follows that
ℳ(Az, SSxn, SSxn, t ) ≥ ℳ(Az, ASxn, ASxn,
୲
ଶ
) ∗ ℳ( ASxn, SSxn, SSxn,
୲
ଶ
) and
ࣨ(Az, SSxn, SSxn, t ) ≤ ࣨ(Az, ASxn, ASxn,
୲
ଶ
) ◊ ࣨ(ASxn, SSxn, SSxn,
୲
ଶ
) yield
lim
୬→∞
ℳ(Az, SSxn, SSxn, t ) ≥ 1 ∗ 1 = 1 and
lim
୬→∞
ࣨ( Az, SSxn, SSxn, t ) ≤ 0 ◊ 0 = 0 and so
we have SSxn → Az as n → ∞.
8. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 3, September 2015
8
4. MAIN RESULTS
Theorem 4.1
Let (X, ℳ, ࣨ, ∗,◊) be a complete generalized intuitionistic fuzzy metric space and let A, B, P,Q,
S and T be self mappings of X satisfying the following conditions.
(i) P(X) ⊆ ST (X), Q(X) ⊆ AB(X)
(ii) The pair (P, AB) and (Q, ST) are compatible mappings of type (P)
(iii) ST is continuous
(iv) ℳ( Px, Qz, Qz, qt) ≥ min {ℳ(ABx, Py, Qy, t), ℳ(ABx, Py, STz, t),
ℳ(Qy, STz, Py, t), ℳ(ABx, Qy, STz, t)} and
ࣨ( Px, Qz, Qz, qt) ≤ max{ ࣨ(ABx, Py,Qy, t), ࣨ(ABx, Py, STz, t),
ࣨ(Qy, STz, Py, t), ࣨ(ABx, Qy, STz, t)}
then the mappings P, Q, AB and ST have a unique common fixed point in X.
Proof
Let x0 be any arbitrary point in X. Thus we construct a sequence {yn} in X such that
y2n-1 = STx2n-1 = Px2n-2 and y2n = ABx2n = Qx2n-1. Put x = x2n-1, y = x2n-1, z = x2n.
ℳሺ Pxଶ୬ିଵ, Qxଶ୬, Qxଶ୬, qt ሻ ≥ min
ە
۔
ۓ
ℳሺ ABxଶ୬ିଵ, Pxଶ୬ିଵ, Qxଶ୬ିଵ, t ሻ,
ℳሺABxଶ୬ିଵ, Pxଶ୬ିଵ, STxଶ୬, t ሻ,
ℳሺ Qxଶ୬ିଵ, STxଶ୬, Pxଶ୬ିଵ, t ሻ,
ℳሺ ABxଶ୬ିଵ, Qxଶ୬ିଵ, STxଶ୬, t ሻ ۙ
ۘ
ۗ
ℳሺ yଶ୬, yଶ୬ାଵ, yଶ୬ାଵ, qt ሻ ≥ min
ە
۔
ۓ
ℳሺ yଶ୬ିଵ, yଶ୬, yଶ୬, t ሻ,
ℳሺ yଶ୬ିଵ, yଶ୬, yଶ୬, t ሻ,
ℳሺ yଶ୬, yଶ୬, yଶ୬, t ሻ,
ℳሺyଶ୬ିଵ, yଶ୬, yଶ୬, t ሻ ۙ
ۘ
ۗ
ℳሺ yଶ୬, yଶ୬ାଵ, yଶ୬ାଵ, qtሻ ≥ ℳሺyଶ୬ିଵ, yଶ୬, yଶ୬, tሻ
This implies that ℳሺ yଶ୬, yଶ୬ାଵ, yଶ୬ାଵ, tሻis an increasing sequence of positive real numbers.
ࣨሺ Pxଶ୬ିଵ, Qxଶ୬, Qxଶ୬, qt ሻ ≤ max
ە
۔
ۓ
ࣨሺ ABxଶ୬ିଵ, Pxଶ୬ିଵ, Qxଶ୬ିଵ, t ሻ,
ࣨሺABxଶ୬ିଵ, Pxଶ୬ିଵ, STxଶ୬, t ሻ,
ࣨሺ Qxଶ୬ିଵ, STxଶ୬, Pxଶ୬ିଵ, t ሻ,
ࣨሺ ABxଶ୬ିଵ, Qxଶ୬ିଵ, STxଶ୬, t ሻ ۙ
ۘ
ۗ
ࣨሺ yଶ୬, yଶ୬ାଵ, yଶ୬ାଵ, qt ሻ ≤ max
ە
۔
ۓ
ࣨሺ yଶ୬ିଵ, yଶ୬, yଶ୬, t ሻ,
ࣨሺ yଶ୬ିଵ, yଶ୬, yଶ୬, t ሻ,
ࣨሺ yଶ୬, yଶ୬, yଶ୬, t ሻ,
ࣨሺyଶ୬ିଵ, yଶ୬, yଶ୬, t ሻ ۙ
ۘ
ۗ
ࣨሺ yଶ୬, yଶ୬ାଵ, yଶ୬ାଵ, qtሻ ≤ ࣨሺyଶ୬ିଵ, yଶ୬, yଶ୬, tሻ
This implies that ࣨሺ yଶ୬, yଶ୬ାଵ, yଶ୬ାଵ, tሻ is an decreasing sequence of positive real numbers.
Now to prove that ℳሺ y୬, y୬ାଵ, y୬ାଵ, tሻ converges to 1 as n → ∞ and
ࣨሺ yଶ୬, yଶ୬ାଵ, yଶ୬ାଵ, tሻ converges to 0 as n → ∞. By lemma 2.6,
9. Applied Mathematics and Sciences: An International Journal (MathSJ ), Vol. 2, No. 3, September 2015
9
ℳሺ y୬, y୬ାଵ, y୬ାଵ, t ሻ ≥ ℳ ൬ y୬ିଵ, y୬, y୬,
t
q
൰ ≥ ℳ ൬ y୬ିଶ, y୬ିଵ, y୬ିଵ,
t
qଶ ൰
. . . ≥ ℳሺ y, yଵ, yଵ,
t
q୬
ሻ
Thus ℳሺ y୬, y୬ାଵ, y୬ାଵ, t ሻ ≥ ℳ ቀ y, yଵ, yଵ,
୲
୯ ቁ and
ࣨሺ y୬, y୬ାଵ, y୬ାଵ, t ሻ ≤ ࣨ ൬ y୬ିଵ, y୬, y୬,
t
q
൰ ≤ ࣨ ൬ y୬ିଶ, y୬ିଵ, y୬ିଵ,
t
qଶ ൰
. . . ≤ ࣨሺ y, yଵ, yଵ,
t
q୬ ሻ
Then by the definition of IFMS,
ℳ( yn, yn+p, yn+p, t ) ≥ ℳ( yn, yn+1, yn+1,
୲
୮
) ∗… p times …∗ ℳ( yn+p-1, yn+p, yn+p,
୲
୮
)
≥ ℳ( y0, y1, y1,
୲
୯ ) ∗… p times …∗ ℳ( y0, y1, y1,
୲
୮୯శ౦షభ )
Thus by the definition of IFMS,
ࣨ(yn, yn+p, yn+p, t) ≤ ࣨ( yn, yn+1, yn+1,
୲
୮
) ∗… p times …∗ ࣨ( yn+p-1, yn+p-1, yn+p,
୲
୮
)
≤ ࣨ( y0, y1, y1,
୲
୯ ) ∗ … p times … ∗ ࣨ( y0, y1, y1,
୲
୮୯శ౦షభ ).
lim
୬→∞
ℳ( yn, yn+p, yn+p, t ) ≥ 1 ∗ 1∗… p times …∗ 1. lim
୬→∞
ℳ( yn, yn+p, yn+p, t ) = 1 and
lim
୬→∞
ࣨ( yn, yn+p, yn+p, t ) ≤ 0 ∗ 0∗…∗ p times …∗ 0. lim
୬→∞
ࣨ(yn, yn+p, yn+p, t ) = 0.
Thus {yn} is a Cauchy sequence in intuitionistic fuzzy metric space X.
Since X is complete, there exists a point u∈X such that yn → u.
Thus {ABx2n}, {Qx2n-1}, {STx2n-1}, {Px2n-2} are Cauchy sequence converge to u.
Put x = ABx2n, y = u, z = STx2n-1 in (iv), we get
ℳሺ PABxଶ୬, QSTxଶ୬ିଵ, QSTxଶ୬ିଵ, qt ሻ ≥ min
ە
۔
ۓ
ℳሺABABxଶ୬, Pu, Qu, tሻ,
ℳሺABABxଶ୬, Pu, STSTxଶ୬ିଵ, t ሻ,
ℳሺQu, STSTxଶ୬ିଵ, Pu, tሻ,
ℳሺABABxଶ୬, Qu, STSTxଶ୬ିଵ, t ሻ,ۙ
ۘ
ۗ
and
ࣨሺ PABxଶ୬, QSTxଶ୬ିଵ, QSTxଶ୬ିଵ, qt ሻ ≤ max
ە
۔
ۓ
ࣨሺABABxଶ୬, Pu, Qu, tሻ,
ࣨሺABABxଶ୬, Pu, STSTxଶ୬ିଵ, t ሻ,
ࣨሺQu, STSTxଶ୬ିଵ, Pu, tሻ,
ࣨሺABABxଶ୬, Qu, STSTxଶ୬ିଵ, t ሻ,ۙ
ۘ
ۗ
.
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10
Now take the limit as n → ∞ and using (ii), we get,
ℳሺPu, Qu, Qu, qtሻ ≥ min ൜
ℳሺ Pu, Pu, Qu, tሻ, ℳሺ Pu, Pu, Qu, tሻ
ℳሺ Qu, Qu, Pu, tሻ, ℳሺ Pu, Qu, Qu, tሻ
ൠ and
ࣨሺPu, Qu, Qu, qtሻ ≤ max ൜
ࣨሺ Pu, Pu, Qu, tሻ, ࣨሺ Pu, Pu, Qu, tሻ
ࣨሺ Qu, Qu, Pu, tሻ, ࣨሺ Pu, Qu, Qu, tሻ
ൠ.
Then by lemma 2.6, we get
ℳሺPu, Qu, Qu, qtሻ ≥ ℳ ሺPu, Qu, Qu, tሻ and
ࣨሺPu, Qu, Qu, qtሻ ≤ ࣨ ሺPu, Qu, Qu, tሻ.
Therefore Pu = Qu. Now put x = ABx2n, y = x2n-1, z = x2n-1, in (iv), we get
ℳሺ PABxଶ୬, Qxଶ୬ିଵ, Qxଶ୬ିଵ, qt ሻ≥ min
ە
۔
ۓ
ℳሺ ABABxଶ୬, Pxଶ୬ିଵ, Qxଶ୬ିଵ, tሻ,
ℳሺABABxଶ୬, Pxଶ୬ିଵ, STxଶ୬ିଵ, t ሻ,
ℳሺ Qxଶ୬ିଵ, STxଶ୬ିଵ, Pxଶ୬ିଵ, t ሻ,
ℳሺABABxଶ୬, Qxଶ୬ିଵ, STxଶ୬ିଵ, t ሻۙ
ۘ
ۗ
and
ࣨሺ PABxଶ୬, Qxଶ୬ିଵ, Qxଶ୬ିଵ, qt ሻ ≤ max
ە
۔
ۓ
ࣨሺ ABABxଶ୬, Pxଶ୬ିଵ, Qxଶ୬ିଵ, tሻ,
ࣨሺABABxଶ୬, Pxଶ୬ିଵ, STxଶ୬ିଵ, t ሻ,
ࣨሺ Qxଶ୬ିଵ, STxଶ୬ିଵ, Pxଶ୬ିଵ, t ሻ,
ࣨሺABABxଶ୬, Qxଶ୬ିଵ, STxଶ୬ିଵ, t ሻۙ
ۘ
ۗ
Thus we have ℳሺ Pu, u, u, qtሻ ≥ ℳሺ Pu, u, u, tሻ and
ࣨሺ Pu, u, u, qtሻ ≥ ࣨሺ Pu, u, u, tሻ.
Therefore Pu = u. This implies Pu = Qu = u.
Now put x = Px2n-2, y = Px2n-2, z = u in (iv), we get
ℳሺPPxଶ୬ିଶ, Qu, Qu, qtሻ ≥ min
ە
۔
ۓ
ℳሺ ABPxଶ୬ିଶ, PPxଶ୬ିଶ, QPxଶ୬ିଶ, t ሻ,
ℳሺABPxଶ୬ିଶ, PPxଶ୬ିଶ, STu, t ሻ,
ℳሺ QPxଶ୬ିଶ, STu, PPxଶ୬ିଶ, t ሻ,
ℳሺABPxଶ୬ିଶ, QPxଶ୬ିଶ, STu, t ሻ ۙ
ۘ
ۗ
and
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11
ࣨሺPPxଶ୬ିଶ, Qu, Qu, qtሻ ≤ max
ە
۔
ۓ
ℳሺ ABPxଶ୬ିଶ, PPxଶ୬ିଶ, QPxଶ୬ିଶ, t ሻ,
ℳሺABPxଶ୬ିଶ, PPxଶ୬ିଶ, STu, t ሻ,
ℳሺ QPxଶ୬ିଶ, STu, PPxଶ୬ିଶ, t ሻ,
ℳሺABPxଶ୬ିଶ, QPxଶ୬ିଶ, STu, t ሻ ۙ
ۘ
ۗ
.
Now taking the limit as n → ∞ and on using (ii) and (iii), we get
ℳሺABu, u, u, qtሻ ≥ min ൜
ℳሺABu, ABu, u, t ሻ, ℳሺABu, ABu, u, t ሻ,
ℳሺ Qu, u, ABu, t ሻ, ℳሺABu, Qu, u, t ሻ
ൠ
ࣨሺABu, u, u, qtሻ ≤ max ൜
ࣨሺABu, ABu, u, t ሻ, ࣨሺABu, ABu, u, t ሻ,
ࣨሺ Qu, u, ABu, t ሻ, ࣨሺABu, Qu, u, t ሻ
ൠ.
This implies
ℳሺABu, u, u, qtሻ ≥ min ൜
ℳሺABu, ABu, u, t ሻ, ℳሺABu, ABu, u, t ሻ,
ℳሺu, u, ABu, t ሻ, ℳሺABu, u, u, t ሻ
ൠ
ࣨሺABu, u, u, qtሻ ≤ max ൜
ࣨሺABu, ABu, u, t ሻ, ࣨሺABu, ABu, u, t ሻ,
ࣨሺu, u, ABu, t ሻ, ࣨሺABu, u, u, t ሻ
ൠ .
Therefore by lemma (2.6) we have ABu = u. Thus Pu = Qu = ABu = u.
Put x = u, y = u, z = Qx2n-1, in (iv) we get
ℳሺPu, QQxଶ୬ିଵ, QQxଶ୬ିଵ, qt ሻ ≥ min ൜
ℳሺu, u, u, t ሻ, ℳሺ u, u, STu, t ሻ,
ℳሺu, STu, u, t ሻ, ℳሺ u, u, , STu, t ሻ
ൠ
ࣨሺPu, QQxଶ୬ିଵ, QQxଶ୬ିଵ, qt ሻ ≤ max ൜
ࣨሺu, u, u, t ሻ, ࣨሺ u, u, STu, t ሻ,
ࣨሺu, STu, u, t ሻ, ࣨሺ u, u, , STu, t ሻ
ൠ,
On using lemma, (2.6) we have
ℳሺSTu, STu, u, qt ሻ ≥ ℳሺ STu, STu, u, t ሻ and
ℳሺSTu, STu, u, qt ሻ ≥ ℳሺ STu, STu, u, t ሻ
ࣨ(STu, STu, u, qt ) ≤ ࣨ( STu, STu, u, t ).
Thus STu = u. We get Pu = Qu = ABu = STu = u.
Uniqueness
Let w be another common fixed point of A, B, P, Q, S and T. Then
ℳሺ Pu, Qw, Qw, qt ሻ ≥ min ൜
ℳሺABu, Pw, Qw, t ሻ, ℳሺ ABu, Pw, STw, t ሻ,
ℳሺ Qw, STw, Pw, t ሻ, ℳሺ ABu, Qw, STw, t ሻ
ൠ
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12
ℳሺ u, w, w, qt ሻ ≥ min ൜
ℳሺ u, w, w, t ሻ, ℳሺ u, w, w, t ሻ,
ℳሺ w, w, w, t ሻ, ℳሺ u, w, w, t ሻ
ൠ
ℳሺ u, w, w, qt ሻ ≥ ℳሺ u, w, w, t ሻ and
ࣨሺ Pu, Qw, Qw, qt ሻ ≤ max ൜
ࣨሺABu, Pw, Qw, t ሻ, ࣨ ሺ ABu, Pw, STw, t ሻ,
ࣨሺ Qw, STw, Pw, t ሻ, ࣨሺ ABu, Qw, STw, t ሻ
ൠ
ࣨሺ u, w, w, qt ሻ ≤ max ൜
ࣨሺ u, w, w, t ሻ, ࣨሺ u, w, w, t ሻ,
ࣨሺ w, w, w, t ሻ, ࣨሺ u, w, w, t ሻ
ൠ ࣨሺ u, w, w, qt ሻ ≤ ࣨሺ u, w, w, t ሻ,
which is a contradiction. Therefore u = w.
Hence the common fixed point is unique.
Corollary 4.2
Let (X, ℳ, ࣨ, ∗,◊) be a complete generalized intuitionistic fuzzy metric space and let A, P,Q and
S be self mappings of X satisfying the following conditions.
(i) P(X) ⊆ S(X), Q(X) ⊆ A(X)
(ii) The pair (P,A) and (Q,S) are compatible mappings of type (P)
(iii) S is continuous
(iv) ℳ( Px, Qz, Qz, qt ) ≥ min { ℳ( Ax, Py, Qy, t ), ℳ( Ax, Py, Sz, t ),
ℳ( Qy, Sz, Py, t ), ℳ( Ax, Qy, Sz, t )} and
ࣨ( Px, Qz, Qz, qt ) ≤ max {ࣨ( Ax, Py, Qy, t ), ࣨ( Ax, Py, Sz, t ),
ࣨ( Qy, Sz, Py, t ), ࣨ( Ax, Qy, Sz, t )}.
Then the mappings P, Q, A and S have a unique common fixed point in X.
Corollary 4.3
Let (X, ℳ, ࣨ, ∗,◊) be a complete generalized intuitionistic fuzzy metric space and let B,P,Q and
T be self mappings of X satisfying the conditions (i), (ii), (iii), & (iv) with S = I and A = I;
Then the mappings B, P,Q and T have a unique common fixed point.
Corollary 4.4
Let ( X, ℳ, ࣨ, ∗, ◊ ) be a complete generalized intuitionistic fuzzy metric space and let
A,B,P,Q,S and T be self mappings of X satisfying the following conditions:
(i) P(X) ⊆ ST(X), Q(X) ⊆ AB(X)
(ii) The pair (P, AB) and (Q, ST) are compatible mappings of type (P)
(iii) ST is continuous
(iv) ℳ( Px, Qz, Qz, qt) ≥ ℳ( ABx, Py, Qy, t) ∗ ℳ(ABx, Py, STz, t) ∗
ℳ( Qy, STz, Py, t) ∗ ℳ(ABx,Qy,STz,t) and
ࣨ( Px, Qz, Qz, qt) ≤ ࣨ( ABx, Py, Qy, t) ◊ ࣨ( ABx, Py, STz, t) ◊
ࣨ( Qy, STz, Py, t) ◊ ࣨ( ABx, Qy, STz, t)
Then the mappings P,Q,AB and ST have a unique common fixed point in X.
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Authors
Dr.R.Muthuraj received his Ph.D degree in Mathematics from Alagappa University
Karaikudi, Tamil nadu, India in April 2010. Presently he is an Assistant Professor ,
PG & Research Department of Mathematics, H.H.The Rajah’s College,
Pudukkottai Tamilnadu ,India. He has published over 80 papers in refereed
National and International Journals. He is the reviewer and Editor of the reputed
International Journals.Eight members are doing research work under his guidance.
His research interests are Fuzzy Algebra, Lattice Theory, Discrete Mathematics,
Fuzzy Topology, Fixed point theory and Fuzzy Graph Theory.
R. Pandiselvi received her M.Phil degree from School of Mathematics, Madurai
Kamaraj University, Madurai, Tamilnadu, India. Now she is doing Ph.D at
Bharathidasan University Tiruchirappalli,Tamilnadu, India. Presently she is
working as an Associate Professor in Mathematics , The Madura college, Madurai,
Tamilnadu, India. She has published over 10 papers in reputed National and
International journals. Her research area is Fixed Point Theory.