This document presents some definitions and results related to generalized D*-metric spaces and establishes a unique common fixed point theorem for such spaces. It begins by introducing D*-metric spaces and generalized D*-metric spaces, along with some key definitions like convergence and Cauchy sequences. It then presents several lemmas about convergence and uniqueness of limits in generalized D*-metric spaces. The document concludes by stating that it obtains a unique common fixed point theorem for generalized D*-metric spaces.
This document presents a mathematical theory on weak contractions in cone metric spaces. It begins by introducing concepts such as cone metric spaces, C-contractions, weak C-contractions, and f-contractions. It then defines a new concept of a C-f weak contraction, which generalizes previous weak contraction definitions. The main result proved is a coincidence point and common fixed point theorem for C-f weak contractions in complete cone metric spaces under certain conditions on the mappings. Examples and remarks are provided to show how the new C-f weak contraction definition generalizes previous contraction definitions.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
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 summarizes several theorems related to optimization and differentiation in Banach spaces. It begins by reinterpreting some previous results of Smulian to characterize subsets having the Radon-Nikodym property in terms of optimization results. It then proves that the Minkowski functional is Frechet differentiable if and only if a certain condition is met. Finally, it proves Bourgain's theorem that the set of points strongly exposing a bounded, convex set is a dense Gdelta set, and that the set is the closed convex hull of its strongly exposed points.
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.
This document introduces the concept of fuzzy compact-open topology. Some key points:
- The fuzzy compact-open topology is defined on the class of fuzzy continuous functions between two fuzzy topological spaces.
- An evaluation map from the product of this function class and the domain space into the range space is shown to be fuzzy continuous.
- For fuzzy locally compact Hausdorff domain and range spaces, there is an exponential law isomorphism between the function class with the compact-open topology and the product of this class with the domain space.
On Extendable Sets in the Reals (R) With Application to the Lyapunov Stabilit...BRNSS Publication Hub
This document summarizes a research article that defines extendable sets in the real numbers (R) and applies this concept to the Lyapunov stability comparison principle of ordinary differential equations. It begins with the author's own definition of extension on R and a basic result called the basic extension fact for R. It then reviews existing definitions and theorems on extension, including Urysohn's lemma and Tietze's extension theorem. The document concludes by extensively applying these results to prove some important results relating to the comparison principle of Lyapunov stability theory in ordinary differential equations.
Fixed Point Theorems for Weak K-Quasi Contractions on a Generalized Metric Sp...IJERA Editor
In this paper we obtain conditions for a k- quasi contraction on a generalized metric space with a partial order to have a fixed point. Using this, we derive certain known results as corollaries.
This document presents a mathematical theory on weak contractions in cone metric spaces. It begins by introducing concepts such as cone metric spaces, C-contractions, weak C-contractions, and f-contractions. It then defines a new concept of a C-f weak contraction, which generalizes previous weak contraction definitions. The main result proved is a coincidence point and common fixed point theorem for C-f weak contractions in complete cone metric spaces under certain conditions on the mappings. Examples and remarks are provided to show how the new C-f weak contraction definition generalizes previous contraction definitions.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
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 summarizes several theorems related to optimization and differentiation in Banach spaces. It begins by reinterpreting some previous results of Smulian to characterize subsets having the Radon-Nikodym property in terms of optimization results. It then proves that the Minkowski functional is Frechet differentiable if and only if a certain condition is met. Finally, it proves Bourgain's theorem that the set of points strongly exposing a bounded, convex set is a dense Gdelta set, and that the set is the closed convex hull of its strongly exposed points.
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.
This document introduces the concept of fuzzy compact-open topology. Some key points:
- The fuzzy compact-open topology is defined on the class of fuzzy continuous functions between two fuzzy topological spaces.
- An evaluation map from the product of this function class and the domain space into the range space is shown to be fuzzy continuous.
- For fuzzy locally compact Hausdorff domain and range spaces, there is an exponential law isomorphism between the function class with the compact-open topology and the product of this class with the domain space.
On Extendable Sets in the Reals (R) With Application to the Lyapunov Stabilit...BRNSS Publication Hub
This document summarizes a research article that defines extendable sets in the real numbers (R) and applies this concept to the Lyapunov stability comparison principle of ordinary differential equations. It begins with the author's own definition of extension on R and a basic result called the basic extension fact for R. It then reviews existing definitions and theorems on extension, including Urysohn's lemma and Tietze's extension theorem. The document concludes by extensively applying these results to prove some important results relating to the comparison principle of Lyapunov stability theory in ordinary differential equations.
Fixed Point Theorems for Weak K-Quasi Contractions on a Generalized Metric Sp...IJERA Editor
In this paper we obtain conditions for a k- quasi contraction on a generalized metric space with a partial order to have a fixed point. Using this, we derive certain known results as corollaries.
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.
On Analytic Review of Hahn–Banach Extension Results with Some GeneralizationsBRNSS Publication Hub
The useful Hahn–Banach theorem in functional analysis has significantly been in use for many years ago. At this point in time, we discover that its domain and range of existence can be extended point wisely so as to secure a wider range of extendibility. In achieving this, we initially reviewed the existing traditional Hahn–Banach extension theorem, before we carefully and successfully used it to generate the finite extension form as in main results of section three.
Fixed point result in probabilistic metric spaceAlexander Decker
This document presents a fixed point theorem for four self-mappings in a probabilistic metric space. It begins with definitions related to probabilistic metric spaces, including Menger spaces. It then proves a common fixed point theorem for four self-mappings defined on a complete Menger space, where the mappings satisfy certain conditions. Specifically, the mappings have nested range properties and some pairs are weakly or semi-compatible. Additionally, there is a contraction condition involving a rational function. The theorem guarantees the existence and uniqueness of a common fixed point for the four mappings. An example is also provided to support the theorem.
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.
The document discusses generalizing the tropical semiring to higher dimensions. Specifically, it considers closed convex sets in Rn as elements, with the union operation defined as the convex hull of the union, and the sum operation defined as the Minkowski sum.
It first provides background on polyhedra, recession cones, and asymptotic cones. It then shows that the set of convex polytopes in Rn forms a semiring under these operations. The proof establishes that the union operation yields a closed convex set, and the sum and other axioms hold.
Subsequent sections will consider semirings of other closed convex sets in Rn, such as compact sets and those with a fixed recession cone
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
Best Approximation in Real Linear 2-Normed SpacesIOSR Journals
This pape r d e l i n e a t e s existence, characterizations and st rong unicity of best uniform
approximations in real linear 2-normed spaces.
AMS Su ject Classification: 41A50, 41A52, 41A99, 41A28.
This document presents a common fixed point theorem for two pairs of weakly compatible mappings in a dislocated metric space. Some key points:
- It defines dislocated metric spaces, where the self-distance of a point need not be zero.
- It establishes the existence and uniqueness of a common fixed point for four mappings (A, B, S, T) under certain contractive conditions.
- The mappings must satisfy three conditions: T(X) subset A(X), S(X) subset B(X), and the pairs (S,A) and (T,B) are weakly compatible.
- It constructs sequences to prove the mappings have a unique common fixed point z in the
Convergence Theorems for Implicit Iteration Scheme With Errors For A Finite F...inventy
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.
This document provides an overview and definitions related to functional analysis and Banach spaces. It discusses:
1) The definition of a Banach space as a complete normed linear space where every Cauchy sequence converges.
2) Examples of Banach spaces including l^p spaces, C(X) for compact X, C_b(X) for any topological space X, and C^k([a,b]).
3) Measure spaces and the definition of measurable functions on a measure space. It notes the closure properties of measurable functions under scalar multiplication and (sometimes) addition.
Continuous And Irresolute Functions Via Star Generalised Closed SetsIJMERJOURNAL
ABSTRACT: In this paper, we introduce a new class of continuous functions called semi*δ-continuous function and semi* δ-irresolute functions in topological spaces by utilizing semi* δ-open sets and to investigate their properties.
This document summarizes an algorithm to computationally prove Rota's basis conjecture for matroids of rank 4 or less. The algorithm starts with the given bases and uses basis exchange properties to infer new bases. It continues inferring bases until either finding n disjoint rainbow bases, proving the conjecture, or being unable to infer any more bases, providing a counterexample. Rank 3 is solved with one case, while rank 4 requires around 100,000 cases. The algorithm chooses 3-tuples of bases judiciously to minimize the number of cases needed at each step.
Fractional Newton-Raphson Method and Some Variants for the Solution of Nonlin...mathsjournal
The following document presents some novel numerical methods valid for one and several variables, which
using the fractional derivative, allow us to find solutions for some nonlinear systems in the complex space using
real initial conditions. The origin of these methods is the fractional Newton-Raphson method, but unlike the
latter, the orders proposed here for the fractional derivatives are functions. In the first method, a function is
used to guarantee an order of convergence (at least) quadratic, and in the other, a function is used to avoid the
discontinuity that is generated when the fractional derivative of the constants is used, and with this, it is possible
that the method has at most an order of convergence (at least) linear.
Wallin Education Partners is an organization that provides scholarships and advising to help students stay in college and graduate. It focuses on students who are first-generation college students, students of color, and those with financial need, as these groups have lower national persistence and graduation rates. Wallin assigns advisors to work closely with scholarship recipients to help them navigate the transition from high school to college and ensure they utilize resources to support their progress to graduation. Recent data shows the Wallin approach has helped over 85% of its 600 college student scholars graduate within 6 years, exceeding the national average of 59%.
El blog “arsphotographicagranada.blogspot.com” forma parte de una iniciativa de los módulos de Técnicas Fotográficas y Técnicas Fotográficas Aplicadas, impartidos por la profesora Paloma Brinkmann, en la Escuela de Arte de Granada, para implementar el desarrollo didáctico y generar sinergias que permitan profundizar en la utilización de las plataformas digitales en el campo de la creación artística.
05-11-2014 Entrega Estado recursos de Todo Pavimentado a Nogales, Oquitoa, Im...Guillermo Padrés Elías
El Gobernador de Sonora entregó más de 130 millones de pesos del programa Todo Pavimentado a los municipios de Nogales, Oquitoa, Santa Cruz e Imuris. Nogales recibió 90 millones, Imuris 22.7 millones, Oquitoa 8.9 millones y Santa Cruz 8.9 millones. El dinero será utilizado para pavimentar calles y reparar daños causados por las lluvias, mejorando la infraestructura vial.
A trends of salmonella and antibiotic resistanceAlexander Decker
This document discusses trends in Salmonella and antibiotic resistance. It provides background on Salmonella, including its classification and the historical perspectives of its discovery. It then discusses definitions of antimicrobials and antibiotics, and the mechanisms by which Salmonella develops resistance to various classes of antibiotics, including aminoglycosides, beta-lactams, chloramphenicol, quinolones, tetracyclines, sulfonamides and trimethoprim. Resistance occurs through modification or destruction of antibiotics, efflux pumps, modification of antibiotic targets, and decreased membrane permeability.
A transformational generative approach towards understanding al-istifhamAlexander Decker
This document discusses a transformational-generative approach to understanding Al-Istifham, which refers to interrogative sentences in Arabic. It begins with an introduction to the origin and development of Arabic grammar. The paper then explains the theoretical framework of transformational-generative grammar that is used. Basic linguistic concepts and terms related to Arabic grammar are defined. The document analyzes how interrogative sentences in Arabic can be derived and transformed via tools from transformational-generative grammar, categorizing Al-Istifham into linguistic and literary questions.
A therapy for physical and mental fitness of school childrenAlexander Decker
This document discusses the importance of physical and mental fitness for school children. It notes that physical fitness involves more than just the absence of disease, but the ability to study efficiently without undue fatigue. Regular graded physical exercises are needed to develop and maintain fitness, but many teachers no longer prioritize this. The physical and mental fitness of pupils has declined as a result. The document argues that schools must make developing and maintaining pupil fitness a priority through organized physical education programs.
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.
On Analytic Review of Hahn–Banach Extension Results with Some GeneralizationsBRNSS Publication Hub
The useful Hahn–Banach theorem in functional analysis has significantly been in use for many years ago. At this point in time, we discover that its domain and range of existence can be extended point wisely so as to secure a wider range of extendibility. In achieving this, we initially reviewed the existing traditional Hahn–Banach extension theorem, before we carefully and successfully used it to generate the finite extension form as in main results of section three.
Fixed point result in probabilistic metric spaceAlexander Decker
This document presents a fixed point theorem for four self-mappings in a probabilistic metric space. It begins with definitions related to probabilistic metric spaces, including Menger spaces. It then proves a common fixed point theorem for four self-mappings defined on a complete Menger space, where the mappings satisfy certain conditions. Specifically, the mappings have nested range properties and some pairs are weakly or semi-compatible. Additionally, there is a contraction condition involving a rational function. The theorem guarantees the existence and uniqueness of a common fixed point for the four mappings. An example is also provided to support the theorem.
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.
The document discusses generalizing the tropical semiring to higher dimensions. Specifically, it considers closed convex sets in Rn as elements, with the union operation defined as the convex hull of the union, and the sum operation defined as the Minkowski sum.
It first provides background on polyhedra, recession cones, and asymptotic cones. It then shows that the set of convex polytopes in Rn forms a semiring under these operations. The proof establishes that the union operation yields a closed convex set, and the sum and other axioms hold.
Subsequent sections will consider semirings of other closed convex sets in Rn, such as compact sets and those with a fixed recession cone
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
Best Approximation in Real Linear 2-Normed SpacesIOSR Journals
This pape r d e l i n e a t e s existence, characterizations and st rong unicity of best uniform
approximations in real linear 2-normed spaces.
AMS Su ject Classification: 41A50, 41A52, 41A99, 41A28.
This document presents a common fixed point theorem for two pairs of weakly compatible mappings in a dislocated metric space. Some key points:
- It defines dislocated metric spaces, where the self-distance of a point need not be zero.
- It establishes the existence and uniqueness of a common fixed point for four mappings (A, B, S, T) under certain contractive conditions.
- The mappings must satisfy three conditions: T(X) subset A(X), S(X) subset B(X), and the pairs (S,A) and (T,B) are weakly compatible.
- It constructs sequences to prove the mappings have a unique common fixed point z in the
Convergence Theorems for Implicit Iteration Scheme With Errors For A Finite F...inventy
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.
This document provides an overview and definitions related to functional analysis and Banach spaces. It discusses:
1) The definition of a Banach space as a complete normed linear space where every Cauchy sequence converges.
2) Examples of Banach spaces including l^p spaces, C(X) for compact X, C_b(X) for any topological space X, and C^k([a,b]).
3) Measure spaces and the definition of measurable functions on a measure space. It notes the closure properties of measurable functions under scalar multiplication and (sometimes) addition.
Continuous And Irresolute Functions Via Star Generalised Closed SetsIJMERJOURNAL
ABSTRACT: In this paper, we introduce a new class of continuous functions called semi*δ-continuous function and semi* δ-irresolute functions in topological spaces by utilizing semi* δ-open sets and to investigate their properties.
This document summarizes an algorithm to computationally prove Rota's basis conjecture for matroids of rank 4 or less. The algorithm starts with the given bases and uses basis exchange properties to infer new bases. It continues inferring bases until either finding n disjoint rainbow bases, proving the conjecture, or being unable to infer any more bases, providing a counterexample. Rank 3 is solved with one case, while rank 4 requires around 100,000 cases. The algorithm chooses 3-tuples of bases judiciously to minimize the number of cases needed at each step.
Fractional Newton-Raphson Method and Some Variants for the Solution of Nonlin...mathsjournal
The following document presents some novel numerical methods valid for one and several variables, which
using the fractional derivative, allow us to find solutions for some nonlinear systems in the complex space using
real initial conditions. The origin of these methods is the fractional Newton-Raphson method, but unlike the
latter, the orders proposed here for the fractional derivatives are functions. In the first method, a function is
used to guarantee an order of convergence (at least) quadratic, and in the other, a function is used to avoid the
discontinuity that is generated when the fractional derivative of the constants is used, and with this, it is possible
that the method has at most an order of convergence (at least) linear.
Wallin Education Partners is an organization that provides scholarships and advising to help students stay in college and graduate. It focuses on students who are first-generation college students, students of color, and those with financial need, as these groups have lower national persistence and graduation rates. Wallin assigns advisors to work closely with scholarship recipients to help them navigate the transition from high school to college and ensure they utilize resources to support their progress to graduation. Recent data shows the Wallin approach has helped over 85% of its 600 college student scholars graduate within 6 years, exceeding the national average of 59%.
El blog “arsphotographicagranada.blogspot.com” forma parte de una iniciativa de los módulos de Técnicas Fotográficas y Técnicas Fotográficas Aplicadas, impartidos por la profesora Paloma Brinkmann, en la Escuela de Arte de Granada, para implementar el desarrollo didáctico y generar sinergias que permitan profundizar en la utilización de las plataformas digitales en el campo de la creación artística.
05-11-2014 Entrega Estado recursos de Todo Pavimentado a Nogales, Oquitoa, Im...Guillermo Padrés Elías
El Gobernador de Sonora entregó más de 130 millones de pesos del programa Todo Pavimentado a los municipios de Nogales, Oquitoa, Santa Cruz e Imuris. Nogales recibió 90 millones, Imuris 22.7 millones, Oquitoa 8.9 millones y Santa Cruz 8.9 millones. El dinero será utilizado para pavimentar calles y reparar daños causados por las lluvias, mejorando la infraestructura vial.
A trends of salmonella and antibiotic resistanceAlexander Decker
This document discusses trends in Salmonella and antibiotic resistance. It provides background on Salmonella, including its classification and the historical perspectives of its discovery. It then discusses definitions of antimicrobials and antibiotics, and the mechanisms by which Salmonella develops resistance to various classes of antibiotics, including aminoglycosides, beta-lactams, chloramphenicol, quinolones, tetracyclines, sulfonamides and trimethoprim. Resistance occurs through modification or destruction of antibiotics, efflux pumps, modification of antibiotic targets, and decreased membrane permeability.
A transformational generative approach towards understanding al-istifhamAlexander Decker
This document discusses a transformational-generative approach to understanding Al-Istifham, which refers to interrogative sentences in Arabic. It begins with an introduction to the origin and development of Arabic grammar. The paper then explains the theoretical framework of transformational-generative grammar that is used. Basic linguistic concepts and terms related to Arabic grammar are defined. The document analyzes how interrogative sentences in Arabic can be derived and transformed via tools from transformational-generative grammar, categorizing Al-Istifham into linguistic and literary questions.
A therapy for physical and mental fitness of school childrenAlexander Decker
This document discusses the importance of physical and mental fitness for school children. It notes that physical fitness involves more than just the absence of disease, but the ability to study efficiently without undue fatigue. Regular graded physical exercises are needed to develop and maintain fitness, but many teachers no longer prioritize this. The physical and mental fitness of pupils has declined as a result. The document argues that schools must make developing and maintaining pupil fitness a priority through organized physical education programs.
A universal model for managing the marketing executives in nigerian banksAlexander Decker
This document summarizes a study on developing a universal model for motivating marketing executives in Nigerian banks. The study synthesized theories of Maslow and McGregor to develop a model. A sample of 303 marketing executives was used to test the model. The results showed that managers will most effectively motivate marketing executives by considering their individual needs and creating challenging but attainable goals. The proposed universal model suggests managers take individual characteristics and abilities into account to provide satisfaction and challenge.
A time series analysis of the determinants of savings in namibiaAlexander Decker
This document summarizes a study on the determinants of savings in Namibia from 1991 to 2012. It reviews previous literature on savings determinants in developing countries. The study uses time series analysis including unit root tests, cointegration, and error correction models to analyze the relationship between savings and variables like income, inflation, population growth, deposit rates, and financial deepening in Namibia. The results found inflation and income have a positive impact on savings, while population growth negatively impacts savings. Deposit rates and financial deepening were found to have no significant impact. The study reinforces previous work and emphasizes the importance of improving income levels to achieve higher savings rates in Namibia.
This document describes the parameters and models used to evaluate the link budget for 900MHz GSM communication services. It considers factors such as transmitter power, antenna gains, losses, propagation path loss, and rain attenuation. Path loss models discussed include free space, Okumura, and Hata models. Rain attenuation is calculated using the ITU model. An example link budget is provided for an urban area in Lagos, Nigeria that incorporates the calculated rain attenuation to improve communication during rainfall.
El documento resume la situación económica reciente en España y la UE. El PIB español creció un 0,5% en el tercer trimestre de 2014 impulsado por la demanda doméstica. El desempleo en España disminuyó en octubre, aunque aumentó el número de parados registrados. La Comisión Europea revisó a la baja sus previsiones de crecimiento para la UE y la zona euro debido al debilitamiento en Alemania, Francia e Italia.
A theory of efficiency for managing the marketing executives in nigerian banksAlexander Decker
This document discusses a theory of efficiency for managing marketing executives in Nigerian banks. It examines the theory in light of Kaizen (continuous improvement) and efficiency principles. Kaizen involves small, incremental improvements through identifying and reducing waste. The document recommends adopting a 3H strategy (head, heart, hand) to improve knowledge, attitudes, and tools of management and marketing executives. This would work on structural, voluntary, and induced inefficiencies. The marketing structure can guide efficiency goals, so its design is important. Top management must be transformed first through the 3H strategy for any efficiency drive to succeed.
A validation of the adverse childhood experiences scale inAlexander Decker
This document discusses the validation of the Adverse Childhood Experiences Scale in Nigeria. It describes a study that examined the concurrent validity of the Adverse Childhood Experience International Questionnaire (ACE-IQ) and the Childhood Trauma Questionnaire (CTQ) using responses from 253 prison inmates in Nigeria. The study found significant correlations between the total scores on the ACE-IQ and CTQ, as well as between their subscales, indicating the two scales have concurrent validity in measuring adverse childhood experiences. It also found that male, young, low education, and divorced inmates reported more adverse childhood experiences.
A usability evaluation framework for b2 c e commerce websitesAlexander Decker
This document presents a framework for evaluating the usability of B2C e-commerce websites. It involves user testing methods like usability testing and interviews to identify usability problems in areas like navigation, design, purchasing processes, and customer service. The framework specifies goals for the evaluation, determines which website aspects to evaluate, and identifies target users. It then describes collecting data through user testing and analyzing the results to identify usability problems and suggest improvements.
Ontología, Epistemologia y Metodologia.mariana veliz
Este documento define tres conceptos clave: ontología, epistemología y metodología. La ontología estudia lo que existe, la epistemología analiza cómo se genera y valida el conocimiento científico, y la metodología se refiere a los procedimientos usados para llevar a cabo investigaciones científicas. Luego, describe brevemente la evolución histórica de cada concepto y cómo han influido en la elaboración de investigaciones.
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.
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.
International Journal of Engineering Research and DevelopmentIJERD Editor
Electrical, Electronics and Computer Engineering,
Information Engineering and Technology,
Mechanical, Industrial and Manufacturing Engineering,
Automation and Mechatronics Engineering,
Material and Chemical Engineering,
Civil and Architecture Engineering,
Biotechnology and Bio Engineering,
Environmental Engineering,
Petroleum and Mining Engineering,
Marine and Agriculture engineering,
Aerospace Engineering.
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1) It defines a superficial element as one where amv+s = mv+s for large v, generalizing the definition to allow zero divisors.
2) It proves three properties are equivalent for an element a to be superficial: the form ideal no:(φ) is the isolated component no, amv = mv+s for large v, and nv+s:(a) = mv for large v.
3) It shows
Fixed point theorems for four mappings in fuzzy metric space using implicit r...Alexander Decker
This document presents theorems proving the existence and uniqueness of common fixed points for four mappings (A, B, S, T) in a fuzzy metric space using an implicit relation.
It begins with definitions of key concepts like fuzzy metric spaces, Cauchy sequences, completeness, compatibility, and occasionally weak compatibility of mappings.
The main result (Theorem 3.1) proves that if the pairs of mappings (A,S) and (B,T) are occasionally weakly compatible, and an implicit relation involving the fuzzy metric of images of x and y under the mappings is satisfied, then there exists a unique common fixed point w for A and S, and a unique common fixed point z for B and T.
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Fixed point and common fixed point theorems in vector metric spacesAlexander Decker
This document presents three theorems about the existence and uniqueness of fixed points and common fixed points of mappings in vector metric spaces.
The first theorem proves that if two self-mappings S and T on a vector space satisfy a quasi-contraction condition, then they have a unique fixed point of coincidence. If the mappings are also weakly compatible, then they have a unique common fixed point.
The second theorem proves a similar result for mappings that satisfy a different quasi-contraction condition involving vector metric spaces.
The third theorem generalizes the contraction conditions to prove the existence of a unique point of coincidence and common fixed point for mappings S and T under more general assumptions. Proofs of the theorems use properties
Fixed Point Theorem in Fuzzy Metric SpaceIJERA Editor
In this present paper on fixed point theorems in fuzzy metric space . we extended to Fuzzy Metric space
generalisation of main theorem .
Mathematics Subject Classification: 47H10, 54A40
Compatible Mapping and Common Fixed Point TheoremIOSR Journals
This document presents a common fixed point theorem for compatible mappings. It begins with definitions of commuting, weakly commuting, and compatible mappings. It then states the main theorem - that if mappings P, Q, S, T satisfy certain conditions, including being compatible and a contraction-type inequality, then they have a unique common fixed point. The proof of the theorem is presented, showing that the mappings converge to a single point z, which is proven to be the unique common fixed point. A corollary is also presented as an extension of the main result.
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.
This document presents three new theorems on the existence of a unique common fixed point for occasionally weakly compatible mappings on a complete fuzzy metric space. The theorems introduce integral type inequalities involving the mappings that generalize several known fixed point results. Theorem 1 establishes a unique common fixed point for four self-mappings where two pairs of mappings are occasionally weakly compatible and satisfy a particular integral inequality. Theorem 2 and 3 prove similar results but replace the integral inequality with one involving a function Φ satisfying certain properties. The theorems reduce the minimum value of the integral inequality compared to previous results.
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.
On fixed point theorems in fuzzy metric spaces in integral typeAlexander Decker
This document presents several common fixed point theorems for occasionally weakly compatible mappings in fuzzy metric spaces. It begins with definitions of key concepts such as fuzzy sets, fuzzy metric spaces, occasionally weakly compatible mappings, and Cauchy sequences in fuzzy metric spaces. It then presents four main theorems that establish the existence and uniqueness of a common fixed point for self-mappings under certain contractive conditions on the mappings and using the concept of occasionally weakly compatible pairs. The proofs of the theorems are also provided.
International Journal of Computational Engineering Research(IJCER)ijceronline
The document presents some fixed point theorems for expansion mappings in complete metric spaces. It begins with definitions of terms like metric spaces, complete metric spaces, Cauchy sequences, and expansion mappings. It then summarizes several existing fixed point theorems for expansion mappings established by other mathematicians. The main result proved in this document is Theorem 3.1, which establishes a new fixed point theorem for expansion mappings under certain conditions on the metric space and mapping. It shows that if the mapping satisfies the given inequality, then it has a fixed point. The proof of this theorem constructs a sequence to show that it converges to a fixed point.
This academic article presents a unique common fixed point theorem for four maps under contractive conditions in cone metric spaces. The authors prove the existence of coincidence points and a common fixed point theorem for four self-maps on a cone metric space that satisfy a contractive condition. They show that if one of the subspaces is complete, then the maps have a coincidence point, and if the maps are commuting, they have a unique common fixed point. This generalizes and improves on previous comparable results in the literature.
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.
This document presents several theorems regarding common fixed points of mappings in fuzzy metric spaces. It begins with definitions of key concepts such as fuzzy sets, continuous t-norms, and fuzzy metric spaces. It then states four fixed point theorems for occasionally weakly compatible mappings in fuzzy metric spaces. The theorems provide conditions in terms of inequalities involving the fuzzy metric that guarantee the existence and uniqueness of a common fixed point. The proofs of the theorems demonstrate that the inequalities imply the mappings have a unique point of coincidence, which must then be their common fixed point.
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A unique common fixed point theorems in generalized d
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A Unique Common Fixed Point Theorems in generalized D*-
Metric Spaces
Salih Yousuf Mohammed Salih
Mathematics Department, University of Bakht Er-Ruda, Sudan.
Nuraddeen Gafai Sayyadi
Mathematics Department, Umaru Musa Yar’adua University, Nigeria.
Abstract
In this paper we establish some common fixed point theorem for contractive type mapping in
cone metric spaces and prove some generalized complete D*- metric spaces.
Keywords: D*- metric space, common fixed points, normal cones.
1 Introduction and Preliminaries
Fixed point theorems play a major role in mathematics such as optimization, mathematical
models, economy, military and medicine. So, the metric fixed point theory has been
investigated extensively in the past two decades by numerous mathematicians. Some
generalizations of a metric space concept have been studied by several authors. These
different generalizations have been improved by Gahler [7, 10], by introducing 2-metric
spaces, and Dhage [1] by studying the theory of D- metric spaces.
In 2005, Mustafa and Sims [11] introduced a new structure of generalized metric spaces
which are called G-metric spaces as a generalization of metric spaces. Later, Mustafa et al.
[12–14] obtained several fixed point theorems for mappings satisfying different contractive
conditions in G-metric spaces. Later in 2007 Shaban Sedghi et.al [8] modified the D-metric
space and defined D*-metric spaces and then C.T.Aage and J.N.Salunke [3] generalized the
D*-metric spaces by replacing the real numbers by an ordered Banach space and defined
D*-cone metric spaces and prove the topological properties.
Further, Huang and Zhang [6] generalized the notion of metric spaces by replacing the
real numbers by ordered Banach space and defined the cone metric spaces. They have
investigated the convergence in cone metric spaces, introduced the completeness of cone
metric spaces and have proved Banach contraction mapping theorem, some other fixed point
theorems of contractive type mappings in cone metric spaces using the normality condition.
Afterwards, Rezapour and Hamlbarani [9], Ilic and Rakocevic [5], contributed some fixed
point theorems for contractive type mappings in cone metric spaces.
In this paper, we obtain a unique common fixed point theorem for generalized D*-metric
spaces.
First, we present some known definitions and propositions in D*-metric spaces.
Let E be a real Banach space and P a subset of E . P is called a cone if and only if:
i P is closed, non-empty and 0 ,P
ii ax by P for all ,x y P and non-negative real numbers , ,a b
0 .iii P P
Given a cone P E , we define a partial ordering on E with respect to P by x y if
and only if y x P . We shall write x y if x y and ;x y we shall write
if int ,x y y x P where int P denotes the interior of P . The cone P is called normal if
there is a number 0K such that for all ,x y E
0 x y implies .x K y (1)
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The least positive number K satisfying the above is called the normal constant of P [12].
The cone P is called regular if every increasing sequence which is bounded from above is
convergent. That is, if 1n n
x
is a sequence such that 1 2x x y for some y E , then
there is x E such that lim 0.n nx x
The cone P is regular if and only if every decreasing sequence which is bounded from
below is convergent. Rezapour and Hamlbarani [9] proved every regular cone is normal and
there are normal cone with normal constant 1.M
Definition (1.1) [4]. Let X be a non empty set. A generalized D *-metric on X is a function,
3
*:D X E that satisfies the following conditions for all , , ,x y z a X :
(1) * , , 0,D x y z
(2) * , , 0 if and only if ,D x y z x y z
(3) * , , * , , ,D x y z D p x y z (Symmetry) where p is a permutation function,
(4) * , , * , , * , , ,D x y z D x y a D a z z
Then the function *D is called a generalized *D -metric and the pair , *X D is called a
generalized *D -metric space
Example (1.2) [4]. Let 2
, , : , 0 ,E R P x y E x y X R and *:D X X X E
defined by
* , , , ,D x y z x y y z x z x y y z x z where 0 is a constant.
Then , *X D is a generalized *D - metric space.
Proposition (1.3) [4]. If , *X D be generalized *D - metric space, then for all , ,x y z X ,
we have * , , * , , .D x x y D x y y
Proof. Let * , , * , , * , , * , ,D x x y D x x x D x y y D x y y and similarly
* , , * , , * , , * , , .D y y x D y y y D y x x D y x x Hence we have
* , , * , , .D x x y D x y y
Definition (1.4) [4]. Let , *X D be a generalized *D - metric space. Let nx be a sequence
in X and x X . If for every c E with 0 c there is N such that for all
, , * , , ,m nm n N D x x x c then nx is said to be convergent and nx converges to x , and
x is the limit of nx . We denote this by .nx x n
Lemma (1.5) [4]. Let , *X D be a generalized *D -metric space, P be a normal cone with
normal constant K . Let nx be a sequence in X . Then nx converges to x if and only if
* , , , .m nD x x x m n
Proof. Let nx be a sequence in generalized *D -metric space X converge to x X and
0 be any number. Then for any c E , with 0 c there is a positive integer N such that
,m n N implies
* , , * , , ,m n m nD x x x c D x x x K c
since 0 * , ,m nD x x x c and K is normal constant. Choose c such that .K c Then
* , ,m nD x x x for all ,m n N and hence * , , 0 as , .m nD x x x m n
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Conversely let * , , 0 as ,m nD x x x m n (in P E ). So for any c E with 0 c
(i.e. intc P ), we have 0c , let , inf : ,r dist c P c t t P where P denotes the
boundary of P , for this given ,0 ,r r c there exist a positive integer N such that
,m n N implies that , ,
2
m
r
G x x x c and for any ,t P
* , , * , ,
2 2
m n m n
r r
c D x x x t c t D x x x r
which proves that * , , int i.e. , , .m n m nc D x x x P G x x x c
Remark. If nu is a sequence in P E and nu u in E the u P as P is a closed subset
of E . From this 0 0.nu u Thus if n nu v in P then lim lim ,n nu v provided limit
exist.
Lemma (1.6) [4]. Let , *X D be a generalized *D -metric space then the following are
equivalent.
is * to .ni x D convergent x
* , , 0 asn nii D x x x n
* , , 0 as .niii D x x x n
Proof. i ii by Lemma (1.5).
ii i . Assume ii , i.e. * , , 0 asnD x x x n i.e. for every c E with 0 c
there is N such that for all , * , , / 2,n nn N D x x x c
* , , * , , * , ,m n m n nD x x x D x x x D x x x
* , , * , ,m m n nD x x x D x x x
for all , .c m n N
Hence nx is *D -convergent to x .
.ii iii Assume ii , i.e. * , , 0 asn nD x x x n i.e. for every c E with 0 c
there is N such that for all n N , * , , ,n nD x x x c
* , , * , , * , , .n n n n nD x x x D x x x D x x x c
Hence * , , 0 as ,nD x x x n since c is arbitrary.
.iii ii Assume that iii i.e. * , , 0 asnD x x x n . Then for any c E with
0 c , there is an N such that n N implies * , , .
2
n
c
D x x x Hence ,m n N gives
* , , * , , * , ,m n m n nD x x x D x x x D x x x c . Thus nx is *D - convergent to x .
Lemma (1.7) [4]. Let , *X D be a generalized *D - metric space, P be a normal cone with
normal constant K . Let nx be a sequence in X . If nx converges to x and nx
converges to y , then x y . That is the limit of nx , if exists, is unique.
Proof. For any c E with 0 ,c there is N such that for all ,m n N , * , , .m nD x x x c
We have
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0 * , , * , , * , ,n nD x x y D x x x D x y y
* , , * , ,n n n nD x x x D x x y
. for all .c n N
Hence * , , 2 .D x x y K c Since c is arbitrary, * , , 0,D x x y therefore x y .
Definition (1.8) [4]. Let , *X D be a generalized *D - metric space, nx be a sequence in
X . If for any c E with 0 ,c there is N such that for all , , ,m n l N * , , ,m n lD x x x c
then nx is called a Cauchy sequence in X .
Definition (1.9) [4]. Let , *X D be a generalized *D -metric space. If every Cauchy
sequence in X is convergent in X , then X is called a complete generalized *D -metric
space.
Lemma (1.10) [4]. Let , *X D be a generalized *D -metric space, nx be a sequence in
X . If nx converges to x , then nx is a Cauchy sequence.
Proof. For any c E with 0 c , there is N such that for all , ,m n l N ,
* , , / 2m nD x x x c and * , , / 2.l lD x x x c Hence
* , , * , , * , , .m n l m n l lD x x x D x x x D x x x c
Therefore nx is a Cauchy sequence.
Lemma (1.11) [4]. Let , *X D be a generalized *D -metric space, P be a normal cone with
normal constant K . Let nx be a sequence in X . Then nx is a Cauchy sequence if and
only if * , , 0 , , .m n lD x x x m n l
Proof. Let nx be a Cauchy sequence in generalized *D -metric space , *X D and 0
be any real number. Then for any c E with 0 ,c there exist a positive integer N such
that , ,m n l N implies * , , 0 * , ,m n l m n lD x x x c D x x x c i.e.
* , , ,m n lD x x x K c where K is a normal constant of P in E . Choose c such that
.K c Then * , ,m n lD x x x for all , ,m n l N , showing that
* , , 0 as , , .m n lD x x x m n l
Conversely let * , , 0 as , , .m n lD x x x m n l For any c E with 0 c we have
0 0, as 0 and 1 .K c c c c IntP K For given K c there is a positive integer N
such that , , * , , .m n lm n l N D x x x K c This proves that * , ,m n lD x x x c for all
, ,m n l N and hence nx is a Cauchy sequence.
Definition (1.12) [4]. Let , *X D , ', '*X D be generalized *D -metric spaces, then a
function : 'f X X is said to be *D -continuous at a point x X if and only if it is
*D -sequentially continuous at x , that is, whenever nx is *D -convergent to x we have
nfx is *D -convergent to fx .
Lemma (1.13) [4]. Let , *X D be a generalized *D -metric space, P be a normal cone with
normal constant K . Let nx , ,ny and nz be three sequences in X and , ,n nx x y y
nz z .n Then * , , * , , .n n nD x y z D x y z n
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Proof. For every 0 , choose c E with 0 c and .
6 3
c
K
From ,n nx x y y
and nz z , there is N such that for all , * , , , * , ,n n n nn N D x x x c D y y y c and
* , , .n nD z z z c We have
* , , * , , * , , * , , * , ,n n n n n n n n n n nD x y z D x y z D z z z D z x y D z z z
* , , * , , * , ,n n n n nD z x y D y y y D z z z
* , , * , , * , ,n n n n nD y z x D y y y D z z z
* , , * , , * , , * , ,n n n n n nD y z x D x x x D y y y D z z z
3 * , ,c D x y z
Similarly, we infer * , , * , , 3 .n n nD x y z D x y z c Hence
0 * , , 3 * , , 6n n nD x y z c D x y z c
and
* , , * , , * , , 3 * , , 3n n n n n nD x y z D x y z D x y z c D x y z c
6 3 , for all .K c n N
Therefore * , , * , , .n n nD x y z D x y z n
Remark. If nx x in generalized *D -metric space X , then every subsequence of nx
converges to x in X . Let nkx be any subsequence of nx and nx x in X then
* , , 0 as ,m nD x x x m n
and also * , , 0 as , since for all .m nk k nD x x x m n k n n
Definition (1.14) [4]. Let f and g be self maps of a set X . If w fx gx for some x in
X , then x is called a coincidence point of f and g , and w is called a point of coincidence
of f and g .
Proposition (1.15) [4]. Let f and g be weakly compatible self maps of a set X . If f and g
have a unique point of coincidence w fx gx , then w is the unique common fixed point of
f and g .
2. Main Results
The first main result is
Theorem (2.1). Let , *X D be complete generalized *D -metric spaces, P be a normal
cone with normal constant K and let :T X X , be a mapping satisfies the following
conditions
* , , * , , * , ,D Tx Ty Tz aD x y z bD x Tx Tx
* , , * , ,cD y Ty Ty dD z Tz Tz (2)
for all , ,x y z X , where , , , 0, 1.a b c d a b c d Then T have a unique fixed point in X
.
Proof. Let 0x X be arbitrary, there exist 1x X such that 0 1Tx x , in this way we have a
sequence nx with 1n nTx x . Then from the above inequality we have
1 1 1* , , * , ,n n n n n nD x x x D Tx Tx Tx
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1 1 1 1* , , * , ,n n n n n naD x x x bD x Tx Tx
* , , * , ,n n n n n ncD x Tx Tx dD x Tx Tx
1 1* , , * , ,n n n n n naD x x x bD x x x
1 1 1 1* , , * , ,n n n n n ncD x x x dD x x x
1 1 1* , , * , ,n n n n n na b D x x x c d D x x x
This implies
1 1 11 * , , * , ,n n n n n nc d D x x x a b D x x x
1 1 1* , , * , ,
1
n n n n n n
a b
D x x x D x x x
c d
1 1 1* , , * , ,n n n n n nD x x x qD x x x
where
1
a b
q
c d
, then0 1q . By repeated the application of the above inequality we
have
1 1 0 1 1* , , * , ,n
n n nD x x x q D x x x
(3)
Then for all , ,n m n m we have by repeated use the triangle inequality and equality (3)
that
1 1 1 2* , , * , , * , ,n m m n n n n n nD x x x D x x x D x x x
2 2 3 1 1* , , * , ,n n n m m mD x x x D x x x
1 1 1 2 2* , , * , ,n n n n n nD x x x D x x x
2 3 3 1* , , * , ,n n n m m mD x x x D x x x
1 1
0 1 1* , ,n n m
q q q D x x x
0 1 1* , ,
1
n
q
D x x x
q
.
From (1) we infer
0 1 1* , , * , ,
1
n
n m m
q
D x x x K D x x x
q
which implies that * , , 0n m mD x x x as ,n m , since 0 1 1* , ,
1
n
q
K D x x x
q
as
,n m .
For , ,n m l , and
* , , * , , * , ,n m l n m m m l lD Tx x x D x x x D x x x ,
from (1)
* , , * , , * , ,n m l n m m m l lD Tx x x K D x x x D x x x
taking limit as , ,n m l , we get * , , 0n m lD x x x . So nx is *D -Cauchy sequence,
since X is *D -complete, there exists u X such that nx u as n , there exist
p X such that p u . If T X is complete, then there exist u T X such that nx u ,
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as T X X , we have u X . Then there exist p X such that p u . We claim that
Tp u ,
* , , * , ,D Tp u u D Tp Tp u
* , , * , ,n nD Tp Tp Tx D Tx u u
* , , * , , * , ,naD p p x bD p Tp Tp cD p Tp Tp
1* , , * , ,n n n ndD x Tx Tx D x u u
* , , * , , * , ,naD u u x bD u Tp Tp cD u Tp Tp
1 1 1* , , * , ,n n n ndD x x x D x u u
This implies that
1 1 1
1
* , , * , , * , , * , ,
1
n n n n nD Tp Tp u aD u u x dD x x x D x u u
b c
from (1)
1 1 1
1
* , , * , , * , , * , ,
1
n n n n nD Tp Tp u K a D u u x d D x x x D x u u
b c
as n , right hand side approaches to zero. Hence * , , 0D Tp Tp u and Tp u . i.e.
Tp p . Now we show T has a unique fixed point. For this, assume that there exists a point q
in X such that q Tq . Now
* , , * , , * , , * , ,D Tp Tp Tq aD p p q bD p Tp Tp cD p Tp Tp
* , ,dD q Tq Tq
* , , * , , * , ,aD Tp Tp Tq bD Tp Tp Tp cD Tp Tp Tp
* , ,dD Tq Tq Tq
* , ,aD Tp Tp Tq
we have * , , * , ,D Tp Tp Tq aD Tp Tp Tq , i.e. 1 * , ,a D Tp Tp Tq P , but
1 * , ,a D Tp Tp Tq P , since 1 0k . Thus p is a unique common fixed point of T .
Corollary (2.2). Let , *X D be a complete generalized *D -metric space, P be a normal
cone with normal constant K and let :T X X be a mappings satisfy the condition
* , , * , , * , ,D Tx Ty Tzq a D x Ty Ty D y Tx Tx
* , , * , ,b D y Tz Tz D z Ty Ty
* , , * , ,c D x Tz Tz D z Tx Tx (4)
for all , ,x y z X , where , , 0,2 2 2 1.a b c a b c Then T has a unique fixed point in X .
Proof. Let 0x X be arbitrary, there exist 1x X such that 0 1,Tx x in this way we have
sequence nTx with 1.n nTx x Then from inequality (4), we have
1 1 1* , , * , ,n n n n n nD x x x D Tx Tx Tx
1 1 1* , , * , ,n n n n n na D x Tx Tx D x Tx Tx
* , , * , ,n n n n n nb D x Tx Tx D x Tx Tx
1 1 1* , , * , ,n n n n n nc D x Tx Tx D x Tx Tx
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1 1 1* , , * , ,n n n n n na D x x x D x x x
1 1 1 1* , , * , ,n n n n n nb D x x x D x x x
1 1 1* , , * , ,n n n n n nc D x x x D x x x
1 1 1* , , * , ,n n n n n na c D x x x D x x x
1 12 * , , .n n nbD x x x
This implies that
1 1 1* , , * , ,n
n n n n n nD x x x q D x x x
where
,
1 2
a c
q
a b c
then 0 1q and by repeated application of above inequality,
we have,
1 1 0 1 1* , , * , ,n
n n nD x x x q D x x x (5)
Then, for all , , ,n m m n we have, by repeated use of the rectangle inequality,
1 1 1* , , * , , * , ,m n n m m m m n nD x x x D x x x D x x x
1 1 1 2 2* , , * , ,m m m m m mD x x x D x x x
2* , ,m n nD x x x
1 1 1 2 2* , , * , ,m m m m m mD x x x D x x x
1* , ,n n nD x x x
1 1
0 1 1* , ,m m n
q q q D x x x
0 1 1* , , .
1
m
q
D x x x
q
from (1)
0 1 1* , , * , , 0 as , .
1
m
m n n
q
D x x x K D x x x m n
q
since 0 1.q So nx is *D -Cauchy sequence. By the completeness of X , there exists
u X such that nx is *D -convergent to u . Then there is p X , such that p u . If
T X is complete, then there exist u T X such that ,nx u as ,T X X we have
u X .Then there exist p X such that .p u We claim that Tp u ,
* , , * , , * , ,n nD Tp Tp u D Tp Tp Tx D Tx u u
* , , * , ,a D p Tp Tp D p Tp Tp
* , , * , ,n n nb D p Tx Tx D x Tp Tp
* , , * , , * , ,n n n nc D p Tx Tx D x Tp Tp D Tx u u
* , , * , ,a D u Tp Tp D u Tp Tu
1 1* , , * , , * , ,n n n nb D u x x D Tp Tp u D u x x
1 1* , , * , , * , ,n n n nc D u x x D Tp Tp u D u x x
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* , ,nD x u u
This implies that
1 1
1
* , , * , , * , ,
1 2
n n n nD Tp Tp u b c D u x x D u x x
a b c
1* , ,nD x u u
from (1)
1 1
1
* , , * , , * , ,
1 2
n n n nD Tp Tp u b c D u x x D u x x
a b c
1* , ,nD x u u
the right hand side approaches to zero as .n Hence * , , 0D Tp Tp u and Tp u .
Hence Tp p . Now we show that T has a unique fixed point. For this, assume that there
exists a point q in X such that q Tq . Now
* , , * , , * , ,D Tp Tp Tq a D p Tp Tp D p Tp Tp
* , , * , ,b D p Tq Tq D q Tp Tp
* , , * , ,c D p Tq Tq D q Tp Tp
* , , * , ,a D Tp Tp Tp D Tp Tp Tp
* , , * , ,b D Tp Tq Tq D Tq Tp Tp
* , , * , ,c D Tp Tq Tq D Tq Tp Tp
* , , * , ,b D Tp Tp Tq D Tp Tp Tq
* , , * , ,c D Tp Tp Tq D Tp Tp Tq
2 2 * , ,b c D Tp Tp Tq
* , , 2 2 * , ,D Tp Tp Tq b c D Tp Tp Tq .
This implies 2 2 1 * , ,b c D Tp Tp Tq P and 2 2 1 * , ,b c D Tp Tp Tq P , since
2 2 1 0b c . As 0P P , we have 2 2 1 * , , 0,b c D Tp Tp Tq i.e.
* , , 0D Tp Tp Tq . Hence Tp Tq . Also p q , since Tp p . Hence p is a unique fixed
point of T in X .
Corollary (2.3). Let , *X D be a complete generalized *D -metric space, P be a normal
cone with normal constant K and let :T X X be a mapping which satisfies the following
condition,
* , , * , , * , ,D Tx Ty Ty a D y Ty Ty D x Ty Ty
* , ,bD y Tx Tx (6)
for all ,x y X , where , 0,3 1,a b a b Then T has a unique fixed point in X .
Proof. Let 0x X be arbitrary, there exist 1x X such that 0 1Tx x , in this way we have
sequence nTx with 1.n nTx x Then from inequality (6), we have
1 1 1* , , * , ,n n n n n nD x x x D Tx Tx Tx
1* , , * , ,n n n n n na D x Tx Tx D x Tx Tx
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1 1* , ,n n nbD x Tx Tx
1 1 1 1 1* , , * , ,n n n n n na D x x x D x x x
* , ,n n nbD x x x
1 1 1* , , * , ,n n n n n na D x x x D x x x
* , ,n n nbD x x x
This implies that
1 1 1* , , * , ,n n n n n nD x x x rD x x x (7)
where ,
1 2
a
r
a
then 0 1.r Then repeating application of (7), we get
1 1 0 1 1* , , * , ,n
n n nD x x x r D x x x
Then, for all , ,n m n m we have, by repeated use of the rectangle inequality,
1 1 1 2 2* , , * , , * , ,m n n m m m m m mD x x x D x x x D x x x
1* , ,n n nD x x x
1 1
0 1 1* , ,m m n
r r r D x x x
0 1 1* , , .
1
m
r
D x x x
r
From (1)
0 1 1* , , * , , 0 as , .
1
m
m n n
r
D x x x K D x x x m n
r
since 0 1.r So nx is *D -Cauchy sequence. By the completeness of X , there exists
u X such that nx is *D -convergent to u . Then there is p X , such that p u . If
T X is complete, then there exist u T X such that nx u , as T X X , we have
u X .Then there exists p X such that p u . We claim that Tp u ,
1 1* , , * , , * , ,n nD Tp Tu u D Tp Tp x D x u u
* , , * , ,a D p Tp Tp D p Tp Tp
1* , , * , ,nbD p Tp Tp D x u u
* , , * , ,a D u Tp Tp D u Tp Tp
1* , , * , ,nbD u Tp Tp D x u u
* , , * , ,a D Tp Tp u D Tp Tp u
1* , , * , ,nbD Tp Tp u D x u u
This implies that
1
1
* , , * , ,
1 2
nD Tp u u D x u u
a b
from (1)
1
1
* , , * , ,
1 2
nD Tp u u K D x u u
a b
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right hand side approaches to zero as n . Hence * , , 0D Tp u u and Tp u and
p Tp . Now we show that T has a unique fixed point. For this, assume that there exists a
point q in X such that q Tq . Now
* , , * , , * , , * , ,D Tp Tq Tq a D q Tq Tq D p Tq Tq bD q Tp Tp
* , , * , , * , ,a D Tq Tq Tq D Tp Tq Tq bD Tq Tp Tp
* , , * , , * , , .aD Tp Tq Tq bD Tp Tq Tq a b D Tp Tq Tq
This implies 1 * , , and 1 * , , ,a b D Tp Tq Tq P a b D Tp Tq Tq P since
* , ,D Tp Tq Tq P and 1 0a b . As 0 ,P P we have
1 * , , 0a b D Tp Tq Tq , i.e. * , , 0D Tp Tq Tq . Hence Tp Tq . Also p q , since
p Tp . Hence p is a unique fixed point of T in X .
Theorem (2.4). Let , *X D be a generalized *D -metric space, P be a normal cone with
normal constant K and let : ,T X X be a mappings which satisfies the following condition
* , , * , , * , ,D Tx Ty Tz a D z Tx Tx D y Tx Tx
* , , * , ,b D y Tz Tz D x Tz Tz
* , , * , ,c D x Ty Ty D z Ty Ty (8)
for all , ,x y z X , where , , 0,3 2 3 1.a b c a b c Then T has a unique common fixed point
in X .
Proof. Setting z y in condition (8), reduces it to condition (6), and the proof follows from
Corollary (2.3).
References
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