Convergence Theorems for Implicit Iteration Scheme With Errors For A Finite Family Of Generalized Asymptotically Quasi-Nonexpansive Mappings In Convex Metric Spaces
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.
A Tau Approach for Solving Fractional Diffusion Equations using Legendre-Cheb...iosrjce
In this paper, a modified numerical algorithm for solving the fractional diffusion equation is
proposed. Based on Tau idea where the shifted Legendre polynomials in time and the shifted Chebyshev
polynomials in space are utilized respectively.
The problem is reduced to the solution of a system of linear algebraic equations. From the computational point
of view, the solution obtained by this approach is tested and the efficiency of the proposed method is confirmed.
Notes on intersection theory written for a seminar in Bonn in 2010.
Following Fulton's book the following topics are covered:
- Motivation of intersection theory
- Cones and Segre Classes
- Chern Classes
- Gauss-Bonet Formula
- Segre classes under birational morphisms
- Flat pull back
Existence of Solutions of Fractional Neutral Integrodifferential Equations wi...inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, 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.
8 fixed point theorem in complete fuzzy metric space 8 megha shrivastavaBIOLOGICAL FORUM
ABSTRACT: In this paper our works establish a new fixed point theorem for a different type of mapping in complete fuzzy metric space. Here we define a mapping by using some proved results and obtain a result on the actuality of fixed points. We inspired by the concept of Hossein Piri and Poom Kumam [15]. They introduced the fixed point theorem for generalized F-suzuki -contraction mappings in complete b-metric space. Next Robert plebaniak [16] express his idea by result “New generalized fuzzy metric space and fixed point theorem in fuzzy metric space”. This paper also induces comparing of the outcome with existing result in the literature.
Keywords: Fuzzy set, Fuzzy metric space, Cauchy sequence Non- decreasing sequence, Fixed point, Mapping.
Complete l fuzzy metric spaces and common fixed point theoremsAlexander Decker
This document presents definitions and theorems related to complete L-fuzzy metric spaces and common fixed point theorems. It begins with introducing concepts such as L-fuzzy sets, L-fuzzy metric spaces, and triangular norms. It then defines Cauchy sequences and completeness in L-fuzzy metric spaces. The main result is Theorem 2.2, which establishes conditions under which four self-mappings of a complete L-fuzzy metric space have a unique common fixed point. These conditions include the mappings having compatible pairs, one mapping having a closed range, and the mappings satisfying a contractive-type inequality condition. The proof of the theorem constructs appropriate sequences to show convergence.
Generalized fixed point theorems for compatible mapping in fuzzy 2 metric spa...Alexander Decker
This document discusses generalized fixed point theorems for compatible mappings in fuzzy 2-metric spaces. It begins with introductions and preliminaries on fixed point theory, fuzzy metric spaces, and compatible mappings. It then provides new definitions of compatible mappings of types (I) and (II) in fuzzy-2 metric spaces. The main results extend, generalize, and improve previous theorems by proving common fixed point theorems for four mappings under the condition of compatible mappings of types (I) and (II) in complete fuzzy-2 metric spaces.
This document presents a research paper that proves some fixed point theorems for occasionally weakly compatible maps in fuzzy metric spaces. The paper begins with an introduction discussing the importance of fixed point theory and its applications. It then provides relevant definitions for fuzzy metric spaces and concepts like weakly compatible mappings. The main results of the paper are fixed point theorems for mappings satisfying integral type contractive conditions in fuzzy metric spaces for occasionally weakly compatible maps. The proofs of these fixed point theorems generalize existing contractive conditions to establish the existence and uniqueness of a fixed point.
A Tau Approach for Solving Fractional Diffusion Equations using Legendre-Cheb...iosrjce
In this paper, a modified numerical algorithm for solving the fractional diffusion equation is
proposed. Based on Tau idea where the shifted Legendre polynomials in time and the shifted Chebyshev
polynomials in space are utilized respectively.
The problem is reduced to the solution of a system of linear algebraic equations. From the computational point
of view, the solution obtained by this approach is tested and the efficiency of the proposed method is confirmed.
Notes on intersection theory written for a seminar in Bonn in 2010.
Following Fulton's book the following topics are covered:
- Motivation of intersection theory
- Cones and Segre Classes
- Chern Classes
- Gauss-Bonet Formula
- Segre classes under birational morphisms
- Flat pull back
Existence of Solutions of Fractional Neutral Integrodifferential Equations wi...inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, 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.
8 fixed point theorem in complete fuzzy metric space 8 megha shrivastavaBIOLOGICAL FORUM
ABSTRACT: In this paper our works establish a new fixed point theorem for a different type of mapping in complete fuzzy metric space. Here we define a mapping by using some proved results and obtain a result on the actuality of fixed points. We inspired by the concept of Hossein Piri and Poom Kumam [15]. They introduced the fixed point theorem for generalized F-suzuki -contraction mappings in complete b-metric space. Next Robert plebaniak [16] express his idea by result “New generalized fuzzy metric space and fixed point theorem in fuzzy metric space”. This paper also induces comparing of the outcome with existing result in the literature.
Keywords: Fuzzy set, Fuzzy metric space, Cauchy sequence Non- decreasing sequence, Fixed point, Mapping.
Complete l fuzzy metric spaces and common fixed point theoremsAlexander Decker
This document presents definitions and theorems related to complete L-fuzzy metric spaces and common fixed point theorems. It begins with introducing concepts such as L-fuzzy sets, L-fuzzy metric spaces, and triangular norms. It then defines Cauchy sequences and completeness in L-fuzzy metric spaces. The main result is Theorem 2.2, which establishes conditions under which four self-mappings of a complete L-fuzzy metric space have a unique common fixed point. These conditions include the mappings having compatible pairs, one mapping having a closed range, and the mappings satisfying a contractive-type inequality condition. The proof of the theorem constructs appropriate sequences to show convergence.
Generalized fixed point theorems for compatible mapping in fuzzy 2 metric spa...Alexander Decker
This document discusses generalized fixed point theorems for compatible mappings in fuzzy 2-metric spaces. It begins with introductions and preliminaries on fixed point theory, fuzzy metric spaces, and compatible mappings. It then provides new definitions of compatible mappings of types (I) and (II) in fuzzy-2 metric spaces. The main results extend, generalize, and improve previous theorems by proving common fixed point theorems for four mappings under the condition of compatible mappings of types (I) and (II) in complete fuzzy-2 metric spaces.
This document presents a research paper that proves some fixed point theorems for occasionally weakly compatible maps in fuzzy metric spaces. The paper begins with an introduction discussing the importance of fixed point theory and its applications. It then provides relevant definitions for fuzzy metric spaces and concepts like weakly compatible mappings. The main results of the paper are fixed point theorems for mappings satisfying integral type contractive conditions in fuzzy metric spaces for occasionally weakly compatible maps. The proofs of these fixed point theorems generalize existing contractive conditions to establish the existence and uniqueness of a fixed point.
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)
Fixed point theorems in random fuzzy metric space throughAlexander Decker
This document defines key concepts related to fixed point theorems in random fuzzy metric spaces. It begins by introducing fuzzy metric spaces, fuzzy 2-metric spaces, and fuzzy 3-metric spaces. It then defines random fuzzy variables and random fuzzy metric spaces. The document aims to prove some fixed point theorems in random fuzzy metric spaces, random fuzzy 2-metric spaces, and random fuzzy 3-metric spaces using rational expressions. It provides 18 definitions related to t-norms, fuzzy metric spaces, convergence of sequences, completeness, and mappings to lay the groundwork for the main results.
Generalized fixed point theorems for compatible mapping in fuzzy 3 metric spa...Alexander Decker
This document discusses generalized fixed point theorems for compatible mappings in fuzzy 3-metric spaces. It begins with introductions and preliminaries on fixed point theory, fuzzy metric spaces, and compatible mappings. It then provides new definitions of compatible mappings of types (I) and (II) in fuzzy 3-metric spaces. The main results extend, generalize, and improve previous theorems by proving common fixed point theorems for four mappings under the conditions of compatible mappings of types (I) and (II) in complete fuzzy 3-metric spaces.
Partitioning procedures for solving mixed-variables programming problemsSSA KPI
This document presents two procedures for solving mixed-variables programming problems. The key steps are:
1. The problem is partitioned into two subproblems - a programming problem defined on the set S, and a linear programming problem defined in Rp.
2. A partitioning theorem shows that solving the programming problem on S determines whether the original problem is feasible, feasible but not optimal, or determines an optimal solution and reduces the original problem to a linear programming problem.
3. To efficiently solve the programming problem on S without computing the entire constraint set, the problem is decomposed into subproblems where only constraints determining optimality need to be identified.
This document is a project report submitted by a student for a Bachelor of Science degree in Mathematics. It provides an introduction to the field of topology. The report includes a title page, certificate, declaration, acknowledgements, table of contents, and several chapters. The introduction defines topology and provides some examples. It states that the project aims to provide a thorough grounding in general topology.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
On fixed point theorems in fuzzy 2 metric spaces and fuzzy 3-metric spacesAlexander Decker
1) The document discusses fixed point theorems for mappings in fuzzy 2-metric and fuzzy 3-metric spaces.
2) It defines concepts like fuzzy metric spaces, Cauchy sequences, compatible mappings, and proves some fixed point theorems for compatible mappings.
3) The theorems show that under certain contractive conditions on the mappings, there exists a unique common fixed point for the mappings in a complete fuzzy 2-metric or fuzzy 3-metric space.
The existence of common fixed point theorems of generalized contractive mappi...Alexander Decker
The document presents a common fixed point theorem for a sequence of self maps satisfying a generalized contractive condition in a non-normal cone metric space. It begins with introducing concepts such as cone metric spaces, normal and non-normal cones, and generalized contraction mappings. It then proves the main theorem: if a sequence of self maps {Tn} on a complete cone metric space X satisfies a generalized contractive condition with constants α, β, γ, δ, η, μ ∈ [0,1] such that their sum is less than 1, and x0 ∈ X with xn = Tnxn-1, then the sequence {xn} converges to a unique common fixed point v of the maps
This document presents a modified Mann iteration method for finding common fixed points of a countable family of multivalued mappings in Banach spaces. It introduces using the best approximation operator PTn to define the iteration (1.2), where xn+1 is defined as a convex combination of the previous iterate xn and an element in PTn xn . The paper aims to establish weak and strong convergence theorems for this iterative method to find a common fixed point for a countable family of nonexpansive multivalued mappings. It provides relevant background on fixed points, nonexpansive mappings, and best approximation operators in Banach and Hilbert spaces.
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.
11.final paper -0047www.iiste.org call-for_paper-58Alexander Decker
This document discusses generating new Julia sets and Mandelbrot sets using the tangent function. It introduces using the tangent function of the form tan(zn) + c, where n ≥ 2, and applying Ishikawa iteration to generate new Relative Superior Mandelbrot sets and Relative Superior Julia sets. The results are entirely different from existing literature on transcendental functions. It describes using escape criteria for polynomials to generate the fractals and discusses the geometry of the Relative Superior Mandelbrot and Julia sets generated, which possess symmetry along the real axis.
Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...Cristiano Longo
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On Fuzzy - Semi Open Sets and Fuzzy - Semi Closed Sets in Fuzzy Topologic...IOSR Journals
Abstract: The aim of this paper is to introduce the concept of fuzzy - semi open and fuzzy - semi closed sets of a fuzzy topological space. Some characterizations are discussed, examples are given and properties are established. Also, we define fuzzy - semi interior and fuzzy - semi closure operators. And we introduce fuzzy
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MSC 2010: 54A40, 03E72.
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This document summarizes the key points of a thesis oral presentation on wavelet and frame theory. It discusses systems and frames, and how frames overcome limitations of orthonormal systems by providing stable reconstructions. It then outlines contributions made in the thesis to analyzing Gabor and wavelet systems using dual Gramian analysis. This allows constructing frames with desired properties like compact support and symmetry. It presents two papers published from the work and provides background on notions like Bessel systems, frames, Riesz sequences, and how the dual Gramian analysis connects frame properties to the adjoint system through a duality principle.
Este documento describe las características de los sustantivos o nombres en español. Explica que los sustantivos son palabras que nombran personas, animales, cosas e ideas. Luego detalla las diferentes clases de sustantivos, incluyendo si son primitivos o derivados, simples o compuestos, comunes o propios, individuales o colectivos, concretos o abstractos, contables o incontables, animados o inanimados. También explica el género y número de los sustantivos. En resumen, este documento provee una guía completa
Research Inventy : International Journal of Engineering and Scienceinventy
This document describes the design of a contactless power supply system capable of generating 300V and 10A of DC power. It is composed of a primary power converter and a secondary power converter. The primary converter uses an inverter to convert 50/60Hz power to 20kHz power, which is then transferred through magnetic coupling to the secondary converter. The secondary converter uses a boosting circuit to generate the high-voltage DC power from the induced 20kHz AC power. Simulation and experimental results showed the system was effective at contactlessly transferring power at high voltages.
El documento habla sobre los adjetivos calificativos en español. Explica que son palabras variables que expresan cualidades o estados referidos a sustantivos. Describe las funciones de los adjetivos como complemento del sustantivo o atributo, y las clases de adjetivos como especificativos y explicativos. También cubre los grados del adjetivo como positivo, comparativo y superlativo. Por último, introduce los gentilicios como una clase de adjetivos que denotan el origen geográfico.
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)
Fixed point theorems in random fuzzy metric space throughAlexander Decker
This document defines key concepts related to fixed point theorems in random fuzzy metric spaces. It begins by introducing fuzzy metric spaces, fuzzy 2-metric spaces, and fuzzy 3-metric spaces. It then defines random fuzzy variables and random fuzzy metric spaces. The document aims to prove some fixed point theorems in random fuzzy metric spaces, random fuzzy 2-metric spaces, and random fuzzy 3-metric spaces using rational expressions. It provides 18 definitions related to t-norms, fuzzy metric spaces, convergence of sequences, completeness, and mappings to lay the groundwork for the main results.
Generalized fixed point theorems for compatible mapping in fuzzy 3 metric spa...Alexander Decker
This document discusses generalized fixed point theorems for compatible mappings in fuzzy 3-metric spaces. It begins with introductions and preliminaries on fixed point theory, fuzzy metric spaces, and compatible mappings. It then provides new definitions of compatible mappings of types (I) and (II) in fuzzy 3-metric spaces. The main results extend, generalize, and improve previous theorems by proving common fixed point theorems for four mappings under the conditions of compatible mappings of types (I) and (II) in complete fuzzy 3-metric spaces.
Partitioning procedures for solving mixed-variables programming problemsSSA KPI
This document presents two procedures for solving mixed-variables programming problems. The key steps are:
1. The problem is partitioned into two subproblems - a programming problem defined on the set S, and a linear programming problem defined in Rp.
2. A partitioning theorem shows that solving the programming problem on S determines whether the original problem is feasible, feasible but not optimal, or determines an optimal solution and reduces the original problem to a linear programming problem.
3. To efficiently solve the programming problem on S without computing the entire constraint set, the problem is decomposed into subproblems where only constraints determining optimality need to be identified.
This document is a project report submitted by a student for a Bachelor of Science degree in Mathematics. It provides an introduction to the field of topology. The report includes a title page, certificate, declaration, acknowledgements, table of contents, and several chapters. The introduction defines topology and provides some examples. It states that the project aims to provide a thorough grounding in general topology.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
On fixed point theorems in fuzzy 2 metric spaces and fuzzy 3-metric spacesAlexander Decker
1) The document discusses fixed point theorems for mappings in fuzzy 2-metric and fuzzy 3-metric spaces.
2) It defines concepts like fuzzy metric spaces, Cauchy sequences, compatible mappings, and proves some fixed point theorems for compatible mappings.
3) The theorems show that under certain contractive conditions on the mappings, there exists a unique common fixed point for the mappings in a complete fuzzy 2-metric or fuzzy 3-metric space.
The existence of common fixed point theorems of generalized contractive mappi...Alexander Decker
The document presents a common fixed point theorem for a sequence of self maps satisfying a generalized contractive condition in a non-normal cone metric space. It begins with introducing concepts such as cone metric spaces, normal and non-normal cones, and generalized contraction mappings. It then proves the main theorem: if a sequence of self maps {Tn} on a complete cone metric space X satisfies a generalized contractive condition with constants α, β, γ, δ, η, μ ∈ [0,1] such that their sum is less than 1, and x0 ∈ X with xn = Tnxn-1, then the sequence {xn} converges to a unique common fixed point v of the maps
This document presents a modified Mann iteration method for finding common fixed points of a countable family of multivalued mappings in Banach spaces. It introduces using the best approximation operator PTn to define the iteration (1.2), where xn+1 is defined as a convex combination of the previous iterate xn and an element in PTn xn . The paper aims to establish weak and strong convergence theorems for this iterative method to find a common fixed point for a countable family of nonexpansive multivalued mappings. It provides relevant background on fixed points, nonexpansive mappings, and best approximation operators in Banach and Hilbert spaces.
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.
11.final paper -0047www.iiste.org call-for_paper-58Alexander Decker
This document discusses generating new Julia sets and Mandelbrot sets using the tangent function. It introduces using the tangent function of the form tan(zn) + c, where n ≥ 2, and applying Ishikawa iteration to generate new Relative Superior Mandelbrot sets and Relative Superior Julia sets. The results are entirely different from existing literature on transcendental functions. It describes using escape criteria for polynomials to generate the fractals and discusses the geometry of the Relative Superior Mandelbrot and Julia sets generated, which possess symmetry along the real axis.
Herbrand-satisfiability of a Quantified Set-theoretical Fragment (Cantone, Lo...Cristiano Longo
The document discusses the quantified fragment of set theory called ∀π0. ∀π0 allows for restricted quantification over sets and ordered pairs. A decision procedure for the satisfiability of ∀π0 formulas works by non-deterministically guessing a skeletal representation and checking if its realization is a model of the formula. The document considers encoding the conditions on skeletal representations as first-order formulas to view ∀π0 as a first-order logic and leverage tools developed for first-order logic fragments.
On Fuzzy - Semi Open Sets and Fuzzy - Semi Closed Sets in Fuzzy Topologic...IOSR Journals
Abstract: The aim of this paper is to introduce the concept of fuzzy - semi open and fuzzy - semi closed sets of a fuzzy topological space. Some characterizations are discussed, examples are given and properties are established. Also, we define fuzzy - semi interior and fuzzy - semi closure operators. And we introduce fuzzy
- t-set, -SO extremely disconnected space analyse the relations between them.
MSC 2010: 54A40, 03E72.
The paper reports on an iteration algorithm to compute asymptotic solutions at any order for a wide class of nonlinear
singularly perturbed difference equations.
This document summarizes the key points of a thesis oral presentation on wavelet and frame theory. It discusses systems and frames, and how frames overcome limitations of orthonormal systems by providing stable reconstructions. It then outlines contributions made in the thesis to analyzing Gabor and wavelet systems using dual Gramian analysis. This allows constructing frames with desired properties like compact support and symmetry. It presents two papers published from the work and provides background on notions like Bessel systems, frames, Riesz sequences, and how the dual Gramian analysis connects frame properties to the adjoint system through a duality principle.
Este documento describe las características de los sustantivos o nombres en español. Explica que los sustantivos son palabras que nombran personas, animales, cosas e ideas. Luego detalla las diferentes clases de sustantivos, incluyendo si son primitivos o derivados, simples o compuestos, comunes o propios, individuales o colectivos, concretos o abstractos, contables o incontables, animados o inanimados. También explica el género y número de los sustantivos. En resumen, este documento provee una guía completa
Research Inventy : International Journal of Engineering and Scienceinventy
This document describes the design of a contactless power supply system capable of generating 300V and 10A of DC power. It is composed of a primary power converter and a secondary power converter. The primary converter uses an inverter to convert 50/60Hz power to 20kHz power, which is then transferred through magnetic coupling to the secondary converter. The secondary converter uses a boosting circuit to generate the high-voltage DC power from the induced 20kHz AC power. Simulation and experimental results showed the system was effective at contactlessly transferring power at high voltages.
El documento habla sobre los adjetivos calificativos en español. Explica que son palabras variables que expresan cualidades o estados referidos a sustantivos. Describe las funciones de los adjetivos como complemento del sustantivo o atributo, y las clases de adjetivos como especificativos y explicativos. También cubre los grados del adjetivo como positivo, comparativo y superlativo. Por último, introduce los gentilicios como una clase de adjetivos que denotan el origen geográfico.
Este documento ofrece consejos para ser un instructor cristiano eficaz. Explica que el instructor cristiano no solo enseña conocimientos, sino que también moldea el carácter de los estudiantes a la imagen de Cristo. Para lograr esto, el instructor debe definir su propósito y objetivos, depender del Espíritu Santo y desarrollar las áreas espiritual, ética y didáctica. El proceso de enseñanza debe conducir a los estudiantes al cambio de vida, culto, conducta y servicio para lograr la madurez en
Will Selva, a sports enthusiast, takes a look at some of the NBA's 2015 draft prospects and how some carry a specific chip on their should heading into June's draft.
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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.
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This document introduces a two parameter entropy measure for uncertain variables based on two parameter probabilistic entropy. It begins by reviewing concepts of uncertainty theory such as uncertainty space, uncertain variables, and independence of uncertain variables. It then defines entropy of an uncertain variable using Shannon entropy. Previous work on one parameter entropy of uncertain variables is summarized. The document proposes a new two parameter entropy for uncertain variables, providing its definition and examples of calculating it for specific uncertainty distributions. Properties of the two parameter entropy are discussed.
Iterative procedure for uniform continuous mapping.Alexander Decker
This document presents an iterative procedure for finding a common fixed point of a finite family of self-maps on a nonempty closed convex subset of a normed linear space. Specifically:
1. It defines an m-step iterative process that generates a sequence from an initial point by applying m self-maps from the family sequentially at each step.
2. It proves that if one of the maps is uniformly continuous and hemicontractive, with bounded range, and the family has a nonempty common fixed point set, then the iterative sequence converges strongly to a common fixed point.
3. It extends previous results by allowing some maps in the family to satisfy only asymptotic conditions, rather than uniform continuity. The conditions
A common fixed point theorem in cone metric spacesAlexander Decker
This academic article summarizes a common fixed point theorem for continuous and asymptotically regular self-mappings on complete cone metric spaces. The theorem extends previous results to cone metric spaces, which generalize metric spaces by replacing real numbers with an ordered Banach space. It proves that under certain contractive conditions, the self-mapping has a unique fixed point. The proof constructs a Cauchy sequence that converges to the fixed point.
This document discusses compactness estimates for nonlinear partial differential equations (PDEs), specifically Hamilton-Jacobi equations. It provides background on Kolmogorov entropy measures of compactness and covers recent results estimating the Kolmogorov entropy of solutions to scalar conservation laws and Hamilton-Jacobi equations, showing it is on the order of 1/ε. The document outlines applications of these estimates and open questions regarding extending the estimates to non-convex fluxes and non-uniformly convex Hamiltonians.
Fixed Point Theorm In Probabilistic Analysisiosrjce
Probabilistic operator theory is the branch of probabilistic analysis which is concerned with the study of
operator-valued random variables and their properties. The development of a theory of random operators is of
interest in its own right as a probabilistic generalization of (deterministic) operator theory and just as operator
theory is of fundamental importance in the study of operator equations, the development of probabilistic operator
theory is required for the study of various classes of random equations
Similar to Convergence Theorems for Implicit Iteration Scheme With Errors For A Finite Family Of Generalized Asymptotically Quasi-Nonexpansive Mappings In Convex Metric Spaces (20)
Experimental Investigation of a Household Refrigerator Using Evaporative-Cool...inventy
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Copper Strip Corrossion Test in Various Aviation Fuelsinventy
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Additional Conservation Laws for Two-Velocity Hydrodynamics Equations with th...inventy
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Comparative Study of the Quality of Life, Quality of Work Life and Organisati...inventy
People’s lives are increasingly centred on work; they spend at least one-third of their time within the organisations that employ them. Investigating the factors that interfere with employees’ well-being and the organisational environment is becoming an increasing concern in organisations. This article identifies the criteria of the quality of life (QoL), quality of working life (QWL) and organisational climate instruments to point out their similarities. For bibliographic construction and data research, articles were sought in national and international journals, books and dissertations/articles in SciELO, Science Direct, Medline and Pub Med databases. The results show direct relationships amongst QoL, QWL and organisational climate instruments. The relationship between QoL and QWL instruments is based on fair compensation, social interaction, organisational communication, working conditions and functional capacity. QWL and organisational climate instruments are related through social interaction and interfaces. QoL and organisational climate instruments are related based on social interaction, organisational communication, and work conditions.
A Study of Automated Decision Making Systemsinventy
The decision making process of many operations are dependent on analysing very large data sets, previous decisions and their results. The information generated from the large data sets are used as an input for making decisions. Since the decisions to be taken in day to day operations are expanding, the time taken for manual decision making is also expanding. In order to reduce the time, cost and to increase the efficiency and accuracy, which are the most important things for customer satisfaction, many organisations are adopting the automated decision making systems. This paper is about the technologies used for automated decision making systems and the areas in which automated decisions systems works more efficiently and accurately.
Crystallization of L-Glutamic Acid: Mechanism of Heterogeneous β -Form Nuclea...inventy
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Evaluation of Damage by the Reliability of the Traction Test on Polymer Test ...inventy
In recent decades, polymers have undergone a remarkable historical development and their use has been greatly imposed by gradually dethroning most of the secular materials. These polymer materials have always distinguished themselves by their simple shaping and inexpensive price, their versatility, lightness, and chemical stability but despite their massive use in everyday life as well as in advanced technologies. Generally, these materials still not understood which requires a thorough knowledge of their chemical, physical, rheological and mechanical properties. This paper, we study the mechanical behavior of an amorphous polymer: Acrylonitrile Butadiene Styrene “ABS” by means of uniaxial tensile testing on pierced test pieces with different notch lengths ranging between 1 to 14mm.The proposed approach consists in analyzing the evolution of the global geometry of the obtained strain curves by taking into account the zones and characteristic points of these curves as well as the effect of the damage on the mechanical behavior of the polymer ABS, in order to visualize the evolution of the damage by a static model
Application of Kennelly’model of Running Performances to Elite Endurance Runn...inventy
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Development and Application of a Failure Monitoring System by Using the Vibra...inventy
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Removal of Chromium (VI) From Aqueous Solutions Using Discarded Solanum Tuber...inventy
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Effect of Various External and Internal Factors on the Carrier Mobility in n-...inventy
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Transient flow analysis for horizontal axial upper-wind turbineinventy
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Choice of Numerical Integration Method for Wind Time History Analysis of Tall...inventy
Wind tunnel tests are being performed routinely around the world for designing tall buildings but the advent of powerful computational tools will make time-history analysis for wind more common in near future. As the duration of wind storms ranges from tens of minutes to hours while earthquake durations are typically less than a three to four minutes, the choice of a time step size (Δt) for wind studies needs to be much larger both to reduce the computational time and to save disk space. As the error in any numerical solution of the equation of motion is dependent on step size (Δt), careful investigations on the choice of numerical integration methods for wind analyses are necessary. From a wide variety of integration methods available, it was decided to investigate three methods that seem appropriate for 3D-time history analysis of tall buildings for wind. These are modal time history analysis, the Hilber-Hughes-Taylor (HHT) method or α-method with α=- 0.1, and the Newmark method with β=0.25 and γ=0.5 ( i.e., trapezoidal rule). SAP2000, a common structural analysis software tool, and a 64-story structure are used to conduct all the analyses in this paper. A boundary layer wind tunnel (BLWT) pressure time history measured at 120 locations around the building envelope of a similar structure is used for the analyses. Analyses performed with both the HHT and Newmark-method considering P-delta effects show that second order effects have a considerable impact on both displacement and acceleration response. This result shows that it is necessary to account P-delta effect for wind analysis of tall buildings. As the direct integration time history analysis required very large computation times and very large computer physical memory for a wind duration of hours, a modal analysis with reduced stiffness is considered as a good alternative. For that purpose, a non-linear static analysis of the structure with a load combination of 1.0D + 1.0L is performed in SAP2000 and the reduced stiffness of the structure after the analysis is used to conduct an eigenvalue analysis to extract the mode shapes and frequencies of this structure. Then the first 20- modes are used to perform a modal time history analysis for wind load. The result shows that the responses from modal analysis with “20-mode (reduced stiffness)” are comparable with that from the P-Δ analyses of Newmark-method
Impacts of Demand Side Management on System Reliability Evaluationinventy
This summary provides an overview of the impacts of demand side management (DSM) techniques on power system reliability in Saudi Arabia:
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Reliability Evaluation of Riyadh System Incorporating Renewable Generationinventy
In this paper, the experience of Saudi Electricity Company (SEC) in analyzing the generation adequacy for Year 2013 is presented. This analysis is conducted by calculating several reliability indices for Riyadh system hourly load during all four seasonal periods. The reliability indices are gauged against the international utility practice. SEC also plans to introduce renewable energy into the network in order to secure the environmental standards and reduce fuel costs of conventional generation. Thus, the reliability improvement due to different integration levels of Solar and Wind generating sources has also been investigated. The capacity value provided by these variable renewable energy sources (VERs) to reliably meet the system load has been calculated using effective load carrying capability (ELCC) technique with a loss of load expectancy metric.
The effect of reduced pressure acetylene plasma treatment on physical charact...inventy
The capacitors are increasingly being used as energy storage devicesin various power systems. The scientists of the world are tryingto maximize the electrical capacity of the supercapacitors. To achieve this purpose, numerous method sare used: the surface activation of electrodes, the surface etching using the electronbeam, the electrode etching with variousgasplasma, etc. The purpose of this work is toresearch how the properties of carbon electrodes depend on the plasma parameters at whichtheywere formed. The largest surface area ofcarbonelectrodeof47.25m2 /gis obtainedat 15 ofAr/C2H2gasratio. Meanwhile, theSEMimages show that the disruption of structures with low bond energies and the formation of new onesare taking place when the carbon electrodes are etched at acetylene plasma and placed on carbon electrode. The measurements of capacitance showthat capacitors with affectedelectrodes have about10-15% highercapacity than those not treated with acetyleneplasma.
Experimental Investigation of Mini Cooler cum Freezerinventy
In general cases the refrigerator could be converted into an air conditioner by attaching a fan. Thus a cooler as well as freezer is obtained in a single set up. The freezer can be converted to an air conditioner when the outside air is allowed to flow beside the cooling coil and is forced outside by an exhaust fan. In this case a mini scale cooler cum freezer using R134a as refrigerant was fabricated and tested In our mini project work we had designed, fabricated and experimentally analysed a mini cooler cum freezer. From the observations and calculations, the results of mini cooler cum freezer are obtained and are compared.
Growth and Magnetic properties of MnGeP2 thin filmsinventy
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How information systems are built or acquired puts information, which is what they should be about, in a secondary place. Our language adapted accordingly, and we no longer talk about information systems but applications. Applications evolved in a way to break data into diverse fragments, tightly coupled with applications and expensive to integrate. The result is technical debt, which is re-paid by taking even bigger "loans", resulting in an ever-increasing technical debt. Software engineering and procurement practices work in sync with market forces to maintain this trend. This talk demonstrates how natural this situation is. The question is: can something be done to reverse the trend?
Convergence Theorems for Implicit Iteration Scheme With Errors For A Finite Family Of Generalized Asymptotically Quasi-Nonexpansive Mappings In Convex Metric Spaces
1. Research Inventy: International Journal of Engineering And Science
Vol.4, Issue 8 (August 2014), PP 49-55
Issn (e): 2278-4721, Issn (p):2319-6483, www.researchinventy.com
49
Convergence Theorems for Implicit Iteration Scheme
With Errors For A Finite Family Of Generalized Asymptotically
Quasi-Nonexpansive Mappings In Convex Metric Spaces
Renu Chugh1 and Preety2 and Sanjay Kumar 3
1,2,3Department of Mathematics, Maharshi Dayanand University, Rohtak-124001(INDIA)
ABSTRACT: In this paper, we prove the strong convergence of an implicit iterative scheme with errors to a
common fixed point for a finite family of generalized asymptotically quasi-nonexpansive mappings in convex
metric spaces. Our results refine and generalize several recent and comparable results in uniformly convex
Banach spaces. With the help of an example we compare implicit iteration used in our result and some other
implicit iteration.
KEYWORDS: Implicit iteration process with errors, Generalized asymptotically quasi-nonexpansive
mappings, Common fixed point, Strong convergence, Convex metric spaces.
I. INTRODUCTION AND PRELIMINARIES
In 1970, Takahashi [1] introduced the notion of convexity in metric space and studied some fixed
point theorems for nonexpansive mappings in such spaces.
Definition 1.1 [1] A map
2 W : X [0,1] X is a convex structure in X if
d(u,W(x, y,)) d(u, x) (1)d(u, y)
for all x, y,u X and [0,1]. A metric space (X,d) together with a convex structure W is known as
convex metric space and is denoted by (X,d,W). A nonempty subset C of a convex metric space is convex
if W(x, y,)C for all x, yC and [0,1] . All normed spaces and their subsets are the examples of
convex metric spaces.
Remark 1.2 Every normed space is a convex metric space, where a convex structure
W(x, y, z;, , ) x y z, for all x, y, z X and , , [0,1] with 1. In
fact,
d(u,W(x, y, z;, , )) u ( x y z)
u x u y u z
d(u, x) d(u, y) d(u, z),
for all uX. But there exists some convex metric spaces which cannot be embedded into normed spaces.
Throughout this paper, we assume that X is a metric space and let ( ) ( ) i F T iN be the set of all fixed
points of mappings i T respectively, that is, ( ) { : }. i i F T x X T x x The set of common fixed points of
(1, 2,...... ) iT N denoted by
1
F ( ).
N
i i
F T
Definition 1.3 [2, 3] Let T : X X be a mapping and F(T) denotes the fixed point of T. Then the mapping T
is said to be
(1) nonexpansive if
d(Tx,Ty) d(x, y), x, y X.
(2) quasi-nonexpansive if F(T) and
d(Tx, p) d(x, p), xX, pF(T).
(3) asymptotically nonexpansive [4] if there exists a sequence { } n u in [0,) with lim 0 n
n
u
such that
2. Convergence Theorems for Implicit...
50
( , ) (1 ) ( , ), , n n
n d T x T y u d x y x y X and n 1.
(4) asymptotically quasi-nonexpansive if F(T) and there exists a sequence { } n u in [0,) with
lim 0 n
n
u
such that
( , ) (1 ) ( , ), , ( ) n
n d T x p u d x p x X pF T and n 1.
(5) generalized asymptotically quasi-nonexpansive [5] if
F(T) and there exist two sequences of real numbers { } n u and { } n c with lim 0 lim n n
n n
u c
such that
( , ) (1 ) ( , ) , , ( ) n
n n d T x p u d x p c x X pF T and n 1.
(6) Uniformly L-Lipschitzian
if there exists a positive constant L such that
( , ) ( , ), , n n d T x T y Ld x y x y X and n 1.
Remark 1.4 From definition 1.2, it follows that if F(T) is nonempty, then a nonexpansive mapping is quasi-nonexpansive
and an asymptotically nonexpansive mapping is asymptotically quasi-nonexpansive. But the
converse does not hold. From (5), if 0 n c for all n 1, then T becomes asymptotically quasi-nonexpansive.
Hence the class of generalized asymptotically quasi-nonexpansive maps includes the class of asymptotically
quasi-nonexpansive maps.
The Mann and Ishikawa iteration processes have been used by a number of authors to approximate the fixed
point of nonexpansive, asymptotically nonexpansive mappings and quasi-nonexpansive mappings on Banach
spaces (see e.g., [9-16].
In 1998, Xu [9] gave the following definitions: For a nonempty subset C of a normed space E and
T :CC, the Ishikawa iteration process with errors is the iterative sequence { } n x defined by
1
1
' ' '
,
,
, 1,
n
n n n n n n n
n
n n n n n n n
x x C
x x T y u
y x T x v n
(1.1)
where { }, { } n n u v are bounded sequence in C and ' ' ' { }, { }, { }, { }, { }, { } n n n n n n are sequences in
[0,1] such that ' ' ' 1 n n n n n n for all n 1.
If ' ' 0 n n for all n 1, then (1.3) reduces to Mann iteration process with errors. The normal Ishikawa
and Mann iteration processes are special cases of the Ishikawa iteration process with errors.
In 2001, Xu and Ori [6] have introduced an implicit iteration process for a finite family of
nonexpansive mappings in a Hilbert space H. Let C be a nonempty subset of H. Let 1 2 , ,......., N T T T be
selfmappings of C and suppose that
1
F ( ) ,
N
i i
F T
the set of common fixed points of
, 1,2,..... . i T i N An implicit iteration process for a finite family of nonexpansive mappings is defined as
follows: Let { } n t a real sequence in (0, 1), 0 x C :
1 1 0 1 1 1
2 2 1 2 2 2
1
1 1 1 1 1
(1 ) ,
(1 ) ,
... ....
(1 ) ,
(1 ) ,
.... ....
N N n N N N
N N N N N
x t x t T x
x t x t T x
x t x t T x
x t x t T x
which can be written in the following compact form:
3. Convergence Theorems for Implicit...
51
1 (1 ) , 1, n n n n n n x t x t T x n (1.2)
where Tk Tk (mod N ) . (Here the mod N function takes values in the set {1,2,….N}). They proved the weak
convergence of the iterative scheme (1.2).
In 2003, Sun [7] extend the process (1.2) to a process for a finite family of asymptotically quasi-nonexpansive
mappings, with { }n a real sequence in (0,1) and an initial point 0 x C, which is defined as
follows:
1 1 0 1 1 1
2 2 1 2 2 2
1
2
1 1 1 1 1
2
2 2 2 1 2 2
3
2 1 2 1 2 2 1 1 2 1
(1 ) ,
(1 ) ,
... ....
(1 ) ,
(1 ) ,
.... ....
(1 ) ,
(1 ) ,
.... ....
N N N N N N
N N N N N
N N N N N N
N N N N N
x x T x
x x T x
x x T x
x x T x
x x T x
x x T x
which can be written in the following compact form:
1 (1 ) , 1, k
n n n n i n x x T x n (1.3)
Where n (k 1)N i, i{1,2,....N}.
Sun [7] proved the strong convergence of the process (1.3) to a common fixed point in real uniformly
convex Banach spaces, requiring only one member T in the family { , 1,2,..... } i T i N to be semi compact.
Definition 1.5 Let C be a convex subset of a convex metric space (X,d,W) with a convex structure W and
let I is the indexing set i.e. I {1,2,....N}. Let { : } i T iI be N generalized asymptotically quasi-nonexpansive
mappings on C. For any given 0 x C, the iteration process { } n x defined by
1 0 1 1 1 1 1 1
2 1 2 2 2 2 2 2
1
2
1 1 1 1 1 1 1
2
2 2 1 2 2 2 2 2
3
2 1 2 1 2 1 2 1 2
( , , ; , , ),
( , , ; , , ),
... ...
( , , ; , , ),
( , , ; , , ),
... ....
( , , ; , , ),
( , , ;
N N N N N N N N
N N N N N N N
N N N N N N N N
N N N N N
x W x T x u
x W x T x u
x W x T x u
x W x T x u
x W x T x u
x W x T x u
2 2 , , ),
... ...
N N
where { } n u is a bounded sequence in C and { }, { }, { } n n n are sequences in [0,1] such that
1. n n n
The above sequence can be written in compact form as
1 ( , , ; , , ), 1, k
n n i n n n n n x W x T x u n (1.4)
with (mod ) ( 1) , n i N i n k N i iI and T T T .
Let { : } i T iI be N uniformly L-Lipschitzian generalized asymptotically quasi-nonexpansive selfmappings of
C. Then clearly (1.5) exists.
If 0 n u in 1.5 then,
4. Convergence Theorems for Implicit...
52
1 ( , ; , ), 1, k
n n i n n n x W x T x n (1.5)
with (mod ) ( 1) , n i N i n k N i iI and T T T and { }, { } n n are sequences in [0,1] such that
1. n n
The purpose of this paper is to prove the strong convergence of implicit iteration with errors defined by
(1.5) for generalized asymptotically quasi-nonexpansive in the setting of convex metric spaces.
The following Lemma will be used to prove the main results:
Lemma 1.6 [8] Let { } n a and { } n b be sequences of nonnegative real numbers satisfying the inequality
1 , 1. n n n a a b n
If
1
, n n
b
then lim n
n
a
exists. In particular, if { } n a has a subsequence converging to zero, then
lim 0. n
n
a
Now we prove our main results:
2. Main Results
Theorem 2.1 Let C be a nonempty closed convex subset of a complete Convex metric space X. Let
: ( {1,2..... }) i T C C iI N be N uniformly L-Lipschitzian generalized quasi-nonexpansive mappings
with { }, { } [0, ) in in u c such that
1
in
n
u
and
1
. in
n
c
Suppose that
1
F ( ) .
N
i i
F T
Let { } n x
be the implicit iteration process with errors defined by (1.4) with the restrictions
1
n
n
and
{ } ( ,1 ) n s s for some
1
(0, ).
2
s Then lim ( ,F) n
n
d x
exists.
Proof: Let
1
( ),
N
i i
p F F T
using (1.5) and (5), we have
1 ( , ) ( ( , , ; , , ), ) k
n n i n n n n n d x p d W x T x u p
1 ( , ) ( , ) ( , ) k
n n n i n n n d x p d T x p d u p
1 ( , ) [(1 ) ( , ) ] ( , ) n n n ik n ik n n d x p u d x p c d u p
1 ( , ) (1 )[(1 ) ( , ) ] ( , ) n n n n ik n ik n n d x p u d x p c d u p
1 ( , ) (1 )[(1 ) ( , ) ] ( , ) n n n ik n ik n n d x p u d x p c d u p
1 ( , ) (1 )(1 ) ( , ) (1 ) ( , ) n n n ik n n ik n n d x p u d x p c d u p
1 ( , ) (1 ) ( , ) (1 ) ( , ) n n n ik n n ik n n d x p u d x p c d u p (2.1.1)
Since lim 0, n
n
there exists a natural number 1 n such that for 1n n , .
2 n
s
Hence,
1 1 (1 )
2 2 n n n
s s
s (2.1.2)
for 1n n . Thus, We have from (2.1.1) that
1 ( , ) ( , ) ( , ) (1 ) ( , ) n n n n ik n n ik n n d x p d x p u d x p c d u p (2.1.3)
and
1
1
( , ) ( , ) ( , ) 1 ( , ) ik n
n n n ik n
n n n
u
d x p d x p d x p c d u p
1
2 2 2
( , ) ( , ) 1 ( , ) ik n
n n ik n
u
d x p d x p c d u p
s s s
(2.1.4)
5. Convergence Theorems for Implicit...
53
1
2 2
( , ) ( , ) 1
2 2 2 n n ik n
ik ik ik
s s s M
d x p d x p c
s s s s s
1
2 2 2 2 2
1 ( , ) 1 1 1
2 2 2
ik ik ik
n ik n
ik ik ik
M
d x p c
s s s s s
(2.1.5)
where 1 sup { ( , )}, n n M d u p since { } n u is a bounded sequence in C. Since
1
ik
n
u
for all iI.
This gives that there exists a natural number 0 0 n (as k (n / N) 1) such that 2 0 ik s and
/ 4 ik s for all 0n n . Let
2
2
ik
ik
ik
v
s
and
2 2
1 1 .
2
ik
ik ik
ik
t c
s s
Since
k 1 ik
u
and
k 1 ik
c
for all iI , it follows that
k 1 ik
v
and
1
. k ik
t
Therefore from (2.1.5) we
get,
1
2
( , ) (1 ) ( , ) (1 ) n ik n ik ik n
M
d x p v d x p t v
s
(2.1.6)
This further implies that
1
2
( , ) (1 ) ( , ) (1 ) n ik n ik ik n
M
d x F v d x F t v
s
(2.1.7)
Since by assumptions,
1 1
, k ik k ik
v t
and
1
. k ik
Therefore, applying Lemma (1.6)
to the inequalities (2.1.6) and (2.1.7), we conclude that both lim ( , ) n
n
d x p
and lim ( ,F) n
n
d x
exists.
Theorem 2.2 Let C be a nonempty closed convex subset of a complete Convex metric space X. Let
: ( {1,2..... }) i T C C iI N be N uniformly L-Lipschitzian generalized asymptotically quasi-nonexpansive
mappings with { }, { } [0, ) in in u c such that
1
in
n
u
and
1
. in
n
c
Suppose that
1
F ( ) .
N
i i
F T
Let { } n x be the implicit iteration process with errors defined by (1.4) with the
restrictions
1
n
n
and { } ( ,1 ) n s s for some
1
(0, ).
2
s Then the sequence { } n x converges
strongly to a common fixed point p of the mapping { : } i T iI if and only if
liminf ( ,F) 0. n
n
d x
Proof: From Theorem 2.1 lim ( ,F) n
n
d x
exists. The necessity is obvious. Now we only prove the sufficiency.
Since by hypothesis liminf ( ,F) 0, n
n
d x
so by Lemma 1.6, we have
lim ( ,F) 0. n
n
d x
(2.2.1)
Next we prove that { } n x is a Cauchy sequence in C. Note that when 0,1 . x x x e It follows from (2.1.6)
that for any m 1, for all 0 n n and for any pF, we have
1 1 1 1
( , ) exp ( , )
N N
n m ik n ik
i k i k
d x p v d x p t
1 1 1
2
exp
N
ik n
i k n
M
v
s
6. Convergence Theorems for Implicit...
54
1 1 1
2
( , ) ,
N
n ik n
i k n
QM
Qd x p t
s
(2.2.2)
where,
1 1
exp 1 .
N
ik
i k
Q v
Let 0. Also lim ( , ) n
n
d x p
exists, therefore for 0, there exists a natural number 1 n such that
1 1
( , ) / 6(1 ), / 3
N
n ik
i k
d x p Q t
and
1
/ 6 n
n
s QM
for all 1n n . So we can find
* p F
such that *
1 ( , ) / 3(1 ). n d x p Q Hence, for all 1 n n and m 1, we have that
* * ( , ) ( , ) ( , ) n m n n m n d x x d x p d x p
*
1
1 1
( , )
N
n ik
i k n
Qd x p t
*
1
1
2
( , ) n n
n n
QM
d x p
s
*
1
1 1
(1 ) ( , )
N
n ik
i k n
Q d x p t
1
2
n
n n
QM
s
2
(1 ). . .
3(1 ) 3 6
QM s
Q
Q s QM
(2.2.3)
This proves that { } n x is a Cauchy sequence in C. And, the completeness of X implies that { } n x must be
convergent. Let us assume that lim . n
n
x p
Now, we show that p is common fixed point of the mappings the
mappings { : }. i T iI And, we know that
1
F ( )
N
i i
F T
is closed. From the continuity of d(x,F) 0
with lim ( ,F) 0 n
n
d x
and lim , n
n
x p
we get
d( p,F) 0, (2.2.4)
and so pF, that is p is a common fixed point of the mappings 1 { } . N
i i T
Hence the proof.
If 0 n u , in above theorem, we obtain the following result:
Theorem 2.3 Let C be a nonempty closed convex subset of a complete Convex metric space X. Let
: ( {1,2..... }) i T C C iI N be N uniformly L-Lipschitzian generalized asymptotically quasi-nonexpansive
mappings with { } [0, ) in c such that
1
. in
n
c
Suppose that
1
F ( ) .
N
i i
F T
Let
{ } n x be the implicit iteration process with errors defined by (1.5) with { } ( ,1 ) n s s for some
1
(0, ).
2
s Then the sequence { } n x converges strongly to a common fixed point p of the mapping
{ : } i T iI if and only if
liminf ( ,F) 0. n
n
d x
From Lemma 1.6 and Theorem 2.2, we can easily obtain the result.
Corollary 2.4 Let C be a nonempty closed convex subset of a complete Convex metric space X. Let
: ( {1,2..... }) i T C C iI N be N uniformly L-Lipschitzian generalized asymptotically quasi-nonexpansive
mappings with { }, { } [0, ) in in u c such that
1
in
n
u
and
1
. in
n
c
Suppose that
7. Convergence Theorems for Implicit...
55
1
F ( ) .
N
i i
F T
Let { } n x be the implicit iteration process with errors defined by (1.4) with the
restrictions
1
n
n
and { } ( ,1 ) n s s for some
1
(0, ).
2
s Then the sequence { } n x converges
strongly to a common fixed point p of the mapping { : } i T iI if and only if there exists a subsequence { } nj x
of { } n x which converges to p.
Numerical Example: Let 1 { }n
i i T be the family of generalized asymptotically quasi- non expansive mappings
defined by, , 1
2 n
x
T x n
n
, we know that for n 1, given family is non expansive and since zero is
the only common fixed point of given family, so it is quasi-nonexpansive and so generalized asymptotically
quasi-nonexpansive. Now the initial values used in C++ program for formation of our result are x0=.5 and
0 n , we have following observations about the common fixed point of family of generalized
asymptotically quasi non expansive mappings and with help of these observations we prove that the implicit
iteration (1.2) and (1.3) has same rate of convergence and converges fast as 0 n .From the table given
below we can prove the novelty of our result.
Tabular observations of Implicit Iteration (1.2) and (1.3), where initial approximation is x0==0.5.
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N 0 1 2 3 --- --- 27 28 29 30
n x
for implicit iteration
(1.2)
1e-011 1.5e-022 2e-033 2.5e-044 --- --- 1.45e-307 1.5e-318 0 0
n x
for implicit k-step
iteration
(1.3)
5.02488e-
012
5.04988e-
023
5.075e-
034
5.10025e-
045
--- --- 5.74538e-
309
5.77415e-
320
0 0