This document presents a common fixed point theorem involving six self-mappings of a complete metric space that satisfy certain contractive conditions and compatibility properties. The theorem is proved over multiple steps: 1) a Cauchy sequence is constructed from the mappings and shown to converge to a point z, 2) z is shown to be a common fixed point of the mappings, 3) uniqueness of the common fixed point is proved using the contractive conditions. The theorem generalizes several existing common fixed point results.
Field Induced Josephson Junction (FIJJ) is defined as the physical system made by placement of ferromagnetic strip directly or indirectly [insulator layer in-between] on the top of superconducting strip [3, 4, 7]. The analysis conducted in extended Ginzburg-Landau, Bogoliubov-de Gennes and RCSJ [11] models essentially points that the system is in most case a weak-link Josephson junction [2] and sometimes has features of tunneling Josephson junction [1]. Generalization of Field Induced Josephson junctions leads to the case of network of robust coupled field induced Josephson junctions [4] that interact in inductive way. Also the scheme of superconducting Random Access Memory (RAM) for Rapid Single Flux [8, 9] quantum (RSFQ) computer is drawn [6, 10] using the concept of tunneling Josephson junction [1] and Field Induced Josephson junction [3, 4].
The given presentation is also available by YouTube (https://www.youtube.com/watch?v=uIqXqiwDsSM).
Literature
[1]. B.D.Josephson, Possible new effects in superconductive tunnelling, PL, Vol.1, No. 251, 1962
[2]. K.Likharev, Josephson junctions Superconducting weak links, RMP, Vol. 51, No. 101, 1979
[3]. K.Pomorski and P.Prokopow, Possible existence of field induced Josephson junctions, PSS B, Vol.249, No.9, 2012
[4]. K.Pomorski, PhD thesis: Physical description of unconventional Josephson junction, Jagiellonian University, 2015
[4]. K.Pomorski, H.Akaike, A.Fujimaki, Towards robust coupled field induced Josephson junctions, arxiv:1607.05013, 2016
[6]. K.Pomorski, H.Akaike, A.Fujimaki, Relaxation method in description of RAM memory cell in RSFQ computer, Procedings of Applied Conference 2016 (in progress)
[7]. J.Gelhausen and M.Eschrig, Theory of a weak-link superconductor-ferromagnet Josephson structure, PRB, Vol.94, 2016
[8]. K.K. Likharev, Rapid Single Flux Quantum Logic (http://pavel.physics.sunysb.edu/RSFQ/Research/WhatIs/rsfqre2m.html)
[9]. Proceedings of Applied Superconductivity Confence 2016, plenary talk by N.Yoshikawa, Low-energy high-performance computing based on superconducting technology (http://ieeecsc.org/pages/plenary-series-applied-superconductivity-conference-2016-asc-2016#Plenary7)
[10]. A.Y.Herr and Q.P.Herr, Josephson magnetic random access memory system and method, International patent nr:8 270 209 B2, 2012
[11]. J.A.Blackburn, M.Cirillo, N.Gronbech-Jensen, A survey of classical and quantum interpretations of experiments on Josephson junctions at very low temperatures, arXiv:1602.05316v1, 2016
Field Induced Josephson Junction (FIJJ) is defined as the physical system made by placement of ferromagnetic strip directly or indirectly [insulator layer in-between] on the top of superconducting strip [3, 4, 7]. The analysis conducted in extended Ginzburg-Landau, Bogoliubov-de Gennes and RCSJ [11] models essentially points that the system is in most case a weak-link Josephson junction [2] and sometimes has features of tunneling Josephson junction [1]. Generalization of Field Induced Josephson junctions leads to the case of network of robust coupled field induced Josephson junctions [4] that interact in inductive way. Also the scheme of superconducting Random Access Memory (RAM) for Rapid Single Flux [8, 9] quantum (RSFQ) computer is drawn [6, 10] using the concept of tunneling Josephson junction [1] and Field Induced Josephson junction [3, 4].
The given presentation is also available by YouTube (https://www.youtube.com/watch?v=uIqXqiwDsSM).
Literature
[1]. B.D.Josephson, Possible new effects in superconductive tunnelling, PL, Vol.1, No. 251, 1962
[2]. K.Likharev, Josephson junctions Superconducting weak links, RMP, Vol. 51, No. 101, 1979
[3]. K.Pomorski and P.Prokopow, Possible existence of field induced Josephson junctions, PSS B, Vol.249, No.9, 2012
[4]. K.Pomorski, PhD thesis: Physical description of unconventional Josephson junction, Jagiellonian University, 2015
[4]. K.Pomorski, H.Akaike, A.Fujimaki, Towards robust coupled field induced Josephson junctions, arxiv:1607.05013, 2016
[6]. K.Pomorski, H.Akaike, A.Fujimaki, Relaxation method in description of RAM memory cell in RSFQ computer, Procedings of Applied Conference 2016 (in progress)
[7]. J.Gelhausen and M.Eschrig, Theory of a weak-link superconductor-ferromagnet Josephson structure, PRB, Vol.94, 2016
[8]. K.K. Likharev, Rapid Single Flux Quantum Logic (http://pavel.physics.sunysb.edu/RSFQ/Research/WhatIs/rsfqre2m.html)
[9]. Proceedings of Applied Superconductivity Confence 2016, plenary talk by N.Yoshikawa, Low-energy high-performance computing based on superconducting technology (http://ieeecsc.org/pages/plenary-series-applied-superconductivity-conference-2016-asc-2016#Plenary7)
[10]. A.Y.Herr and Q.P.Herr, Josephson magnetic random access memory system and method, International patent nr:8 270 209 B2, 2012
[11]. J.A.Blackburn, M.Cirillo, N.Gronbech-Jensen, A survey of classical and quantum interpretations of experiments on Josephson junctions at very low temperatures, arXiv:1602.05316v1, 2016
How to Solve a Partial Differential Equation on a surfacetr1987
Familiar techniques of separation of variables and Fourier series can be used to solve a variety of pde based on domains in the plane, however these techniques do not extend naturally to surface problems. Instead we look to take a computational approach. The talk will cover the basics of finite difference and finite element approximations of the one dimensional heat equation and show how to extend these ideas on to surfaces. If time allows, we will show numerical results of an optimal partition problem based on a sphere. No background knowledge of pde or computation is required.
A Common Fixed Point Theorem on Fuzzy Metric Space Using Weakly Compatible an...inventionjournals
The aim of this paper is to prove a fixed point theorem in a complete fuzzy metric space using six self maps. We prove our theorem with the concept of weakly compatible mappings and semi-compatible mappings in complete fuzzy metric space.
It is a new theory based on an algorithmic approach. Its only element
is called nokton. These rules are precise. The innities are completely
absent whatever the system studied. It is a theory with discrete space
and time. The theory is only at these beginnings.
Dyadics algebra.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Describes the simulation model of the backlash effect in gear mechanisms. For undergraduate students in engineering. In the download process a lot of figures are missing.
I recommend to visit my website in the Simulation Folder for a better view of this presentation.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
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.
Connected Total Dominating Sets and Connected Total Domination Polynomials of...iosrjce
Let G = (V, E) be a simple graph. A set S of vertices in a graph G is said to be a total dominating set
if every vertex v V is adjacent to an element of S. A total dominating set S of G is called a connected total
dominating set if the induced subgraph <s> is connected. In this paper, we study the concept of connected total
domination polynomials of the star graph Sn and wheel graph Wn. The connected total domination polynomial of
a graph G of order n is the polynomial Dct(G, x) =
ct
n
i=γ (G)
ct
i
d (G, i) x , where dct(G, i) is the number of
connected total dominating set of G of size i and ct(G) is the connected total domination number of G. We
obtain some properties of Dct(Sn, x) and Dct(Wn, x) and their coefficients. Also, we obtain the recursive formula
to derive the connected total dominating sets of the star graph Sn and the Wheel graph Wn
How to Solve a Partial Differential Equation on a surfacetr1987
Familiar techniques of separation of variables and Fourier series can be used to solve a variety of pde based on domains in the plane, however these techniques do not extend naturally to surface problems. Instead we look to take a computational approach. The talk will cover the basics of finite difference and finite element approximations of the one dimensional heat equation and show how to extend these ideas on to surfaces. If time allows, we will show numerical results of an optimal partition problem based on a sphere. No background knowledge of pde or computation is required.
A Common Fixed Point Theorem on Fuzzy Metric Space Using Weakly Compatible an...inventionjournals
The aim of this paper is to prove a fixed point theorem in a complete fuzzy metric space using six self maps. We prove our theorem with the concept of weakly compatible mappings and semi-compatible mappings in complete fuzzy metric space.
It is a new theory based on an algorithmic approach. Its only element
is called nokton. These rules are precise. The innities are completely
absent whatever the system studied. It is a theory with discrete space
and time. The theory is only at these beginnings.
Dyadics algebra.
Please send comments and suggestions to solo.hermelin@gmail.com. Thanks.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
Describes the simulation model of the backlash effect in gear mechanisms. For undergraduate students in engineering. In the download process a lot of figures are missing.
I recommend to visit my website in the Simulation Folder for a better view of this presentation.
Please send comments to solo.hermelin@gmail.com.
For more presentations on different subjects visit my website at http://www.solohermelin.com.
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.
Connected Total Dominating Sets and Connected Total Domination Polynomials of...iosrjce
Let G = (V, E) be a simple graph. A set S of vertices in a graph G is said to be a total dominating set
if every vertex v V is adjacent to an element of S. A total dominating set S of G is called a connected total
dominating set if the induced subgraph <s> is connected. In this paper, we study the concept of connected total
domination polynomials of the star graph Sn and wheel graph Wn. The connected total domination polynomial of
a graph G of order n is the polynomial Dct(G, x) =
ct
n
i=γ (G)
ct
i
d (G, i) x , where dct(G, i) is the number of
connected total dominating set of G of size i and ct(G) is the connected total domination number of G. We
obtain some properties of Dct(Sn, x) and Dct(Wn, x) and their coefficients. Also, we obtain the recursive formula
to derive the connected total dominating sets of the star graph Sn and the Wheel graph Wn
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 ].
Some Common Fixed Point Theorems for Multivalued Mappings in Two Metric SpacesIJMER
In this paper we prove some common fixed point theorems for multivalued mappings in two
complete metric spaces.
AMS Mathematics Subject Classification: 47H10, 54H25
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 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.
On Convergence of Jungck Type Iteration for Certain Contractive Conditionsresearchinventy
In this article we prove the strong convergence result for a pair of nonself mappings using Jungck S- iterative scheme in Convex metric spaces satisfying certain contractive condition. The results are the generalization of some existing results in the literature
Key Trends Shaping the Future of Infrastructure.pdfCheryl Hung
Keynote at DIGIT West Expo, Glasgow on 29 May 2024.
Cheryl Hung, ochery.com
Sr Director, Infrastructure Ecosystem, Arm.
The key trends across hardware, cloud and open-source; exploring how these areas are likely to mature and develop over the short and long-term, and then considering how organisations can position themselves to adapt and thrive.
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
Le nuove frontiere dell'AI nell'RPA con UiPath Autopilot™UiPathCommunity
In questo evento online gratuito, organizzato dalla Community Italiana di UiPath, potrai esplorare le nuove funzionalità di Autopilot, il tool che integra l'Intelligenza Artificiale nei processi di sviluppo e utilizzo delle Automazioni.
📕 Vedremo insieme alcuni esempi dell'utilizzo di Autopilot in diversi tool della Suite UiPath:
Autopilot per Studio Web
Autopilot per Studio
Autopilot per Apps
Clipboard AI
GenAI applicata alla Document Understanding
👨🏫👨💻 Speakers:
Stefano Negro, UiPath MVPx3, RPA Tech Lead @ BSP Consultant
Flavio Martinelli, UiPath MVP 2023, Technical Account Manager @UiPath
Andrei Tasca, RPA Solutions Team Lead @NTT Data
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Generative AI Deep Dive: Advancing from Proof of Concept to ProductionAggregage
Join Maher Hanafi, VP of Engineering at Betterworks, in this new session where he'll share a practical framework to transform Gen AI prototypes into impactful products! He'll delve into the complexities of data collection and management, model selection and optimization, and ensuring security, scalability, and responsible use.
Encryption in Microsoft 365 - ExpertsLive Netherlands 2024Albert Hoitingh
In this session I delve into the encryption technology used in Microsoft 365 and Microsoft Purview. Including the concepts of Customer Key and Double Key Encryption.
DevOps and Testing slides at DASA ConnectKari Kakkonen
My and Rik Marselis slides at 30.5.2024 DASA Connect conference. We discuss about what is testing, then what is agile testing and finally what is Testing in DevOps. Finally we had lovely workshop with the participants trying to find out different ways to think about quality and testing in different parts of the DevOps infinity loop.
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
PHP Frameworks: I want to break free (IPC Berlin 2024)Ralf Eggert
In this presentation, we examine the challenges and limitations of relying too heavily on PHP frameworks in web development. We discuss the history of PHP and its frameworks to understand how this dependence has evolved. The focus will be on providing concrete tips and strategies to reduce reliance on these frameworks, based on real-world examples and practical considerations. The goal is to equip developers with the skills and knowledge to create more flexible and future-proof web applications. We'll explore the importance of maintaining autonomy in a rapidly changing tech landscape and how to make informed decisions in PHP development.
This talk is aimed at encouraging a more independent approach to using PHP frameworks, moving towards a more flexible and future-proof approach to PHP development.
Welocme to ViralQR, your best QR code generator.ViralQR
Welcome to ViralQR, your best QR code generator available on the market!
At ViralQR, we design static and dynamic QR codes. Our mission is to make business operations easier and customer engagement more powerful through the use of QR technology. Be it a small-scale business or a huge enterprise, our easy-to-use platform provides multiple choices that can be tailored according to your company's branding and marketing strategies.
Our Vision
We are here to make the process of creating QR codes easy and smooth, thus enhancing customer interaction and making business more fluid. We very strongly believe in the ability of QR codes to change the world for businesses in their interaction with customers and are set on making that technology accessible and usable far and wide.
Our Achievements
Ever since its inception, we have successfully served many clients by offering QR codes in their marketing, service delivery, and collection of feedback across various industries. Our platform has been recognized for its ease of use and amazing features, which helped a business to make QR codes.
Our Services
At ViralQR, here is a comprehensive suite of services that caters to your very needs:
Static QR Codes: Create free static QR codes. These QR codes are able to store significant information such as URLs, vCards, plain text, emails and SMS, Wi-Fi credentials, and Bitcoin addresses.
Dynamic QR codes: These also have all the advanced features but are subscription-based. They can directly link to PDF files, images, micro-landing pages, social accounts, review forms, business pages, and applications. In addition, they can be branded with CTAs, frames, patterns, colors, and logos to enhance your branding.
Pricing and Packages
Additionally, there is a 14-day free offer to ViralQR, which is an exceptional opportunity for new users to take a feel of this platform. One can easily subscribe from there and experience the full dynamic of using QR codes. The subscription plans are not only meant for business; they are priced very flexibly so that literally every business could afford to benefit from our service.
Why choose us?
ViralQR will provide services for marketing, advertising, catering, retail, and the like. The QR codes can be posted on fliers, packaging, merchandise, and banners, as well as to substitute for cash and cards in a restaurant or coffee shop. With QR codes integrated into your business, improve customer engagement and streamline operations.
Comprehensive Analytics
Subscribers of ViralQR receive detailed analytics and tracking tools in light of having a view of the core values of QR code performance. Our analytics dashboard shows aggregate views and unique views, as well as detailed information about each impression, including time, device, browser, and estimated location by city and country.
So, thank you for choosing ViralQR; we have an offer of nothing but the best in terms of QR code services to meet business diversity!
UiPath Test Automation using UiPath Test Suite series, part 3DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 3. In this session, we will cover desktop automation along with UI automation.
Topics covered:
UI automation Introduction,
UI automation Sample
Desktop automation flow
Pradeep Chinnala, Senior Consultant Automation Developer @WonderBotz and UiPath MVP
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Essentials of Automations: Optimizing FME Workflows with ParametersSafe Software
Are you looking to streamline your workflows and boost your projects’ efficiency? Do you find yourself searching for ways to add flexibility and control over your FME workflows? If so, you’re in the right place.
Join us for an insightful dive into the world of FME parameters, a critical element in optimizing workflow efficiency. This webinar marks the beginning of our three-part “Essentials of Automation” series. This first webinar is designed to equip you with the knowledge and skills to utilize parameters effectively: enhancing the flexibility, maintainability, and user control of your FME projects.
Here’s what you’ll gain:
- Essentials of FME Parameters: Understand the pivotal role of parameters, including Reader/Writer, Transformer, User, and FME Flow categories. Discover how they are the key to unlocking automation and optimization within your workflows.
- Practical Applications in FME Form: Delve into key user parameter types including choice, connections, and file URLs. Allow users to control how a workflow runs, making your workflows more reusable. Learn to import values and deliver the best user experience for your workflows while enhancing accuracy.
- Optimization Strategies in FME Flow: Explore the creation and strategic deployment of parameters in FME Flow, including the use of deployment and geometry parameters, to maximize workflow efficiency.
- Pro Tips for Success: Gain insights on parameterizing connections and leveraging new features like Conditional Visibility for clarity and simplicity.
We’ll wrap up with a glimpse into future webinars, followed by a Q&A session to address your specific questions surrounding this topic.
Don’t miss this opportunity to elevate your FME expertise and drive your projects to new heights of efficiency.
Essentials of Automations: Optimizing FME Workflows with Parameters
3.common fixed point theorem for compatible mapping of type a -21-24
1. Computer Engineering and Intelligent Systems www.iiste.org
ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online)
Vol 2, No.8, 2011
Common Fixed Point Theorem for Compatible Mapping
of Type (A)
Vishal Gupta
Department pf Mathematics,
Maharishi Markandeshwar University,
Mullana, Ambala, Haryana, India.
vishal.gmn@gmail.com, vkgupta09@rediffmail.com
Received: 2011-10-20
Accepted: 2011-10-29
Published:2011-11-04
Abstract
The purpose of this paper is to prove a common fixed point theorem involving two pairs
of compatible mappings of type (A) using six maps using a contractive condition. This
article represents a useful generalization of several results announced in the literature.
Key Words: Complete metric space, Compatible mapping of type (A), Commuting
mapping, Cauchy Sequence, Fixed points.
1. Introduction
The study of common fixed point of mappings satisfying contractive type conditions has
been studied by many mathematicians.Seesa(1982) introduce the concept of weakly
commuting mapping and proved some theorem of commutativity by useing the condition
to weakly commutativity, Jungck(1988) gave more generalized commuting and weakly
commuting maps called compatible maps and use it for compatibility of two mappings.
After that Jungck Muthy and Cho(1993) made another generalization of weak commuting
mapping by defining the concept of compatible map of type (A).
We proposed to re-analysis the theorems of Aage C.T (2009) on common fixed point
theorem compatibility of type (A)
2.Preliminaries
Definition 2.1. Self maps S and T of metric space (X,d) are said to be weakly
commuting pair
iff d(STx,TSx)≤d(Sx,Tx) for all x in X.
Definition 2.2. Self maps S and T of a metric space (X,d) are said to be compatible of
type (A) if
21 | P a g e
www.iiste.org
2. Computer Engineering and Intelligent Systems www.iiste.org
ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online)
Vol 2, No.8, 2011
lim d(TSxn,SSxn) =0 and lim d(STxn, TTxn)=0 as n→∞ whenever
{xn} is a
sequence in X such that lim Sxn=lim Txn =t as n→∞ for some t in
X.
Definition 2.3. A function Φ: [0, ∞) → [0, ∞) is said to be a contractive modulus if Φ (0)
=0 and
Φ (t) <t for t > 0.
3.Main Result
Theorem 3.1. Let S, R, T, U, I and J are self mapping of a complete metric space (X,d)
into itself satisfying the conditions
(i) SR(X) ⊂ J(X) , TU(X) ⊂ I(X)
(ii) d(SRx,TUy)≤ α d(Ix,Jy) +β [d(Ix,SRx) + d(Jy,TUy)] +γ [d(Ix,TUy)+
d(Jy,SRx)]
for all x,y Є X and α,β and γ are non-negative reals such that
α+2β+2γ<1
(iii) One of S,R,T,U,I and J is continuous.
(iv) (SR,I) and (TU.J) are compatible of type (A).Then SR,TU,I,J have a unique
common
fixed point. Further if the pairs (S,R) , (S,I) , (R,I) , T,U) , (T,I) , (U.J)
are commuting
pairs then S,R,T,U,I and J have a unique common fixed point.
Proof: Let x0 Є X be arbitrary. Choose a point x1 in X such that SRx0 = Jx1.
This can be done since SR(X) ) ⊂ J(X).
Let x2 be a point in X such that TUx1 = Ix2. This can be done since TU(X) ⊂ I(X).
In general we can choose x2n , x2n+1, x2n+2 …, such that SRx2n =Jx2n+1 and TUx2n+1 = Ix2n+2.
So that we obtain a sequence SRx0, TUx1, SRx2, TUx3 …..
Using condition (ii) we have
d(SRx2n,TUx2n+1)≤ α d(I2n,Jx2n+1) +β [d(Ix2n, SRx2n) + d( Jx2n+1, TUx2n+1)] +γ [d(Ix2n,
TUx2n+1) +
d(Jx2n+1,SRx2n)]
= α d (TUx2n-1, SRx2n) + β [d(TUx2n-1,SRx2n)+d(SRx2n, TUx2n+1)] +
γ[d(TUx2n-1,TUx2n+1) + d(SRx2n, SRx2n)]
≤ α d(TUx2n-1, SRx2n) + β [d(TUx2n-1, SRx2n) +d(SRx2n, TUx2n+1)] +
γ [d(TUx2n-1 ,SRx2n) + d(SRx2n,TUx2n+1)]
= (α+β+γ) d (TUx2n-1, SRx2n) + (β+γ) (SRx2n, TUx2n+1)
Hence d(SRx2n, TUx2n+1) ≤ kd(SRx2n, TUx2n-1) where k=(α+β+γ)/ 1-(β+γ) < 1 ,
Similarly we can show d(SRx2n, TUx2n-1) ≤ k d(SRx2n-2, TUx2n-1)
Therefore d (SRx2n , TUx2n+1) ≤ k2 d(SRx2n-2 , TUx2n-1)
≤ k2n d(SRx0, TUx1)
Which implies that the sequence is a Cauchy sequence and since (X,d) is complete so the
sequence has a limit point z in X. Hence the subsequences {SRx2n} ={Jx2n-1}
and {TUx2n-1} ={Ix2n} also converges to the point z in X.
Suppose that the mapping I is continuous. Then I2x2n→ Iz and ISRx2n → Iz as n→ ∞.
Since the pair (SR,I ) is compatible of type (A). we get SRIx2n→ Iz as n→∞.
22 | P a g e
www.iiste.org
3. Computer Engineering and Intelligent Systems www.iiste.org
ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online)
Vol 2, No.8, 2011
Now by (ii)
d(SRIx2n, TUx2n+1) ≤ α d( I2x2n, Jx2n+1) + β [d( I2x2n, SRIx2n) +d( Jx2n+1,TUx2n+1)] +
γ [d(I2x2n,TUx2n+1) +d(Jx2n+1, SRIx2n)]
letting n→∞ , we get
d(Iz,z) ≤ α d(Iz,z) + β [d(Iz,z) +d(z,z)]+γ [d(Iz,z) + d(z,Iz) ]
=(α+2γ) d(Iz,z)
This gives d(Iz,z)=0 since 0≤ α+2γ<1, Hence Iz=z.
Further d(SRz,TUx2n+1) ≤ αd(Iz,Jx2n+1) +β[d(Iz,SRz) + d(Jx2n+1,TUx2n+1)] +
γ [d(Iz,TUx2n+1) + d(Jx2n+1 , SRz)]
Letting Jx2n+1, TUx2n+1 → z as n→∞ and Iz=z we get
d(SRz,z) ≤ α d(z,z) +β [d(z,SRz) + d(z,z)] +γ [d(z,z) + d(z,SRz)
= (β+γ) d(SRz,z)
Hence d(SRz,z) =0 i.e SRz=z , since 0≤ β+γ <1. Thus SRz=Iz=z
Since SR(X) ⊂ J(X) ,there is a point z1 in X such that z=SRz=-Jz1
Now by (ii)
d(z,TUz1) = d(SRz,TU z1)
≤ α d(Iz,J z1) + β [d(Iz,SRz) + d(J z1,TUz1)] +γ [d(Iz,TU z1) +d(Jz1, SRz)]
= α d(z,z) + β [ d(z,z) +d(z,TUz1)] + γ [d(z,TU z1) + d(z,z) ]
=(β+γ) d(z,TU z1)
Hence d(z,TUz1) =0 i.e TU z1 =z =Jz1 , since 0≤ β+γ <1, Take yn = z1 for n≥ 1
Then TUyn→ Tz1 =z and Jyn→ J z1=z as n→∞
Since the pair (TU,J) is compatible of type (A) , we get
Lim d( TUJyn , JJyn) =0 as n→∞ implies d(TUz,Jz)=0 since Jyn =z for all n≥ 1. Hence
TUz=Jz.
Now d(z,TUz) = d(SRz,TUz)
≤ α d(Iz,Jz) + β [d(Iz,SRz) + d( Jz,TUz)] + γ [d(Iz, TUz) + d( Jz,
SRz )
= α d(z,TUz) + β [ d(z,z) +d( TUz,TUz)]+ γ [d(z,TUz) + d(TUz,z) ]
=(α+2γ) d( z, TUz)
Since α+2γ<1, we get TUz=z, hence z=TUz=Jz therefore z is common fixed point of
SR,TU,I,J when the continuity of I is assumed .
Now suppose that SR is continuous then S2R x2n →SRz , SRIx2n → SRz as n→∞ .
By condition (ii) ,we have
d(S2Rx2n, TUx2n+1) ≤ α d( ISRx2n Jx2n+1) + β [d( ISRx2n , S2Rx2n) + d( Jx2n+1 , TUx2n+1) +
γ[d( ISRx2n,TUx2n+1) + d( Jx2n+1, S2Rx2n)]
letting n→∞ and using the compatibility of type (A) of the pair (SR,I), we get
d(SRz,z) ≤ αd(SRz,z) +β [d(SRz,SRz) +d(z,z)] +γ [d(SRz,z) +d(z,SRz)]
=(α+2γ) d(SRz,z)
Since α+2γ<1 we get SRz=z. But SR(X) ⊂ J(X) there is a point p in X such that
z=SRz=Jp ,Now by (ii)
d(S2Rx2n, TUp) ≤ α d(ISRx2n , Jp) +β [d(ISRx2n,S2Rx2n) +d(Jp,TUp)] +
γ[d( ISRx2n ,TUp)+d(Jp,S2Rx2n)
letting n→∞ we have
d(z,TUp) =d(SRz,TUp)
≤αd(z,z) +β[d(z,z) +d(z,TUp)] +γ[d(z,TUp)+ d(z,z)]
=(β+γ) d(z,TUp)
Since β+γ<1 , we get TUp=z.Thus z=Jp=TUp.
Let yn=p then TUyn → TUp=z and Jyn→ TUp =z
Since (TU,J) is compatible of type (A), we have
Lim d(TUJyn ,JJyn) =0 as n→∞
This gives TUJp=JTUp or TUz=Jz
Further
23 | P a g e
www.iiste.org
4. Computer Engineering and Intelligent Systems www.iiste.org
ISSN 2222-1719 (Paper) ISSN 2222-2863 (Online)
Vol 2, No.8, 2011
d(SRx2n, TUz) ≤ αd(Ix2n,Jz) +β[d(Ix2n,SRz) +d(Jz,TUz)] +γ [d(Ix2n, TUz)+d(Jz,SRx2n)]
Letting n→∞, we get
d(z, TUz) ≤αd(z,TUz) +β [d(z,z) +d(TUz,TUz)] +γ[d(z,TUz) +d(TUz,z)]
=(α+2γ) d(z,TUz)
Since 0≤ α+2γ <1 we get z=TUz
Again we have TU(X) ⊂ I(X) there is a point q in X such that z=TUz=Iq
Now d(SRq,z) =d(SRq,TUz) ≤ α d(Iq,Jz) +β [d(Iq,SRq) +
d(Jz,TUz)]+γ[d(Iq,TUz)+d(Jz,SRq)]
=αd(z,z) +β [d(z,SRq) + d(z,z)]+γ [d(z,TUz) +d(z,SRq)]
=(β+γ) d(z,SRq)
Since 0≤ β+γ<1 we get SRq=z, take yn =q then SRyn→ SRq =z , Iyn→Iq=z
Since (SR,I) is compatible of type (A) , we get
Lim d( ISRyn , IIyn) =0 as n→∞
This implies that SRIq=ISRq or SRz=Iz.
Thus we have z=SRz=Iz=Jz=TUz Hence z is a common fixed point of SR,TU,I and J,
when S is continuous
The proof is similar that z is common fixed point of SR,TU,I and J when I is
continuous,R and U is continuous.
For uniqueness let z and w be two common fixed point os SR,TU,I and J , then by
condition (ii)
d(z,w)= d(SRz,TUw) ≤ αd(Iz,Jw) +β[d(Iz,SRz) +d(Jw,TUw)] +γ [d(Iz,TUw)
+d(Jw,SRz)]
=αd(z,w) +β [d(z,z) +d(w,w)]+γ [d(z,w) +d(w,z)]
= (α+2γ) d(z,w)
Since α+2γ<1 we have z=w.
Again let z be the unique common fixed point of both the pairs (SR,I) , (TU,J) then
Sz=S(SRz) = S(RSz) =SR (Sz)
Sz=S(Iz)=I(Sz)
Rz=R(SRz)=(RS)(RS) =(SR)(Rz)
Rz=R(Iz)=I(Rz)
Which shows that Sz and Rz is the common fixed point of (SR,I) yielding thereby
Sz=z=Rz=Iz=SRz
In view of uniqueness of the common fixed point of the pair (SR,I).
Similarly using the commutativity of (T,U), (T,J), (U,J) it can be shown that
Tz=z=Uz=Jz=TUz.Thus z is the unique common fixed point of S,R,T,U,I and J.
Hence the proof.
4. References
Aage C.T., Salunke J.N.(2009), “On Common fixed point theorem in complete metric
space,” IMF,Vol.3,pp.151-159.
Jungck.G.(1986), “Compatible mappings and common fixed points” Internat. J. Math.
Math. Sci., Vol. 9, pp.771-779.
Jungck.G., P.P.Murthy and Y.J.Cho,(1993), “Compatible mapping of type (A) and
common fixed points” Math.Japonica Vol.38,issue 2, pp. 381-390.
S.Sessa(1986), “On a weak commutativity condition in a fixed point consideration.”Publ.
Inst. Math. 32(46), pp. 149-153.
24 | P a g e
www.iiste.org