This document presents several theorems related to establishing the existence and uniqueness of a common fixed point for nonlinear contractive mappings in Hilbert spaces. It begins by introducing the background and motivation for studying fixed point theory and various generalizations of the Banach contraction principle. It then lists several existing theorems that establish fixed point results for mappings satisfying different contraction conditions in complete metric and Hilbert spaces. The main part of the document presents new fixed point theorems for continuous self-mappings on closed subsets of a Hilbert space, proving the existence and uniqueness of a fixed point when the mappings satisfy certain rational-type contraction conditions.
The document presents three theorems regarding common fixed point theorems in Hilbert spaces:
1. The first theorem proves that if two self-mappings T1 and T2 of a closed subset X of a Hilbert space satisfy a certain contraction condition involving rational terms, then T1 and T2 have a unique common fixed point in X.
2. The second theorem extends the first by proving the result for the mappings T1p and T2q where p and q are positive integers.
3. The third theorem considers a sequence of mappings {Ti} on a closed subset of a Hilbert space converging to a limit mapping T, and proves if T has a fixed point, then it is the limit of
This article provides the existence and uniqueness of a common fixed point for a pair of self-mappings, positive integers powers of a pair, and a sequence of self-mappings over a closed subset of a Hilbert space satisfying various contraction conditions involving rational expressions.
This document provides estimates for several number theory functions without assuming the Riemann Hypothesis (RH), including bounds for ψ(x), θ(x), and the kth prime number pk. The following estimates are derived:
1) θ(x) - x < 1/36,260x for x > 0.
2) |θ(x) - x| < ηk x/lnkx for certain values of x, where ηk decreases as k increases.
3) Estimates are obtained for θ(pk), the value of θ at the kth prime number pk, showing θ(pk) is approximately k ln k + ln2 k - 1.
This document summarizes key concepts about linear time-invariant (LTI) systems including:
1. LTI systems exhibit superposition and the output is the summation of individual impulse responses.
2. Inputs can be represented as a linear combination of shifted unit impulses. The output is the convolution sum of the input and impulse response.
3. Convolution represents the output of an LTI system and can be computed using a summation or integral. Common properties like commutativity and distributivity apply.
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
I am Frank P. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Malacca, Malaysia. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
The document provides solutions to questions from an IIT-JEE mathematics exam. It includes 8 questions worth 2 marks each, 8 questions worth 4 marks each, and 2 questions worth 6 marks each. The solutions solve problems related to probability, trigonometry, geometry, calculus, and loci. The summary focuses on the high-level structure and content of the document.
The document presents three theorems regarding common fixed point theorems in Hilbert spaces:
1. The first theorem proves that if two self-mappings T1 and T2 of a closed subset X of a Hilbert space satisfy a certain contraction condition involving rational terms, then T1 and T2 have a unique common fixed point in X.
2. The second theorem extends the first by proving the result for the mappings T1p and T2q where p and q are positive integers.
3. The third theorem considers a sequence of mappings {Ti} on a closed subset of a Hilbert space converging to a limit mapping T, and proves if T has a fixed point, then it is the limit of
This article provides the existence and uniqueness of a common fixed point for a pair of self-mappings, positive integers powers of a pair, and a sequence of self-mappings over a closed subset of a Hilbert space satisfying various contraction conditions involving rational expressions.
This document provides estimates for several number theory functions without assuming the Riemann Hypothesis (RH), including bounds for ψ(x), θ(x), and the kth prime number pk. The following estimates are derived:
1) θ(x) - x < 1/36,260x for x > 0.
2) |θ(x) - x| < ηk x/lnkx for certain values of x, where ηk decreases as k increases.
3) Estimates are obtained for θ(pk), the value of θ at the kth prime number pk, showing θ(pk) is approximately k ln k + ln2 k - 1.
This document summarizes key concepts about linear time-invariant (LTI) systems including:
1. LTI systems exhibit superposition and the output is the summation of individual impulse responses.
2. Inputs can be represented as a linear combination of shifted unit impulses. The output is the convolution sum of the input and impulse response.
3. Convolution represents the output of an LTI system and can be computed using a summation or integral. Common properties like commutativity and distributivity apply.
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
I am Frank P. I am a Physical Chemistry Assignment Expert at eduassignmenthelp.com. I hold a Ph.D. in Physical Chemistry, from Malacca, Malaysia. I have been helping students with their homework for the past 6 years. I solve assignments related to Physical Chemistry.
Visit eduassignmenthelp.com or email info@eduassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Physical Chemistry Assignments.
The document provides solutions to questions from an IIT-JEE mathematics exam. It includes 8 questions worth 2 marks each, 8 questions worth 4 marks each, and 2 questions worth 6 marks each. The solutions solve problems related to probability, trigonometry, geometry, calculus, and loci. The summary focuses on the high-level structure and content of the document.
This document summarizes research on the consistency and stability of linear multistep methods for solving initial value differential problems. It discusses the local truncation error and consistency conditions for convergence. The consistency condition requires that the truncation error approaches zero as the step size decreases. Stability conditions like relative and weak stability are also analyzed. It is shown that linear multistep methods satisfy the conditions of the Banach fixed point theorem, ensuring a unique solution. Specifically, a two-step predictor-corrector method is presented where the predictor provides an initial estimate that is corrected.
(1) The document discusses various integration techniques including: review of integral formulas, integration by parts, trigonometric integrals involving products of sines and cosines, trigonometric substitutions, and integration of rational functions using partial fractions.
(2) Examples are provided to demonstrate each technique, such as using integration by parts to evaluate integrals of the form ∫udv, using trigonometric identities to reduce powers of trigonometric functions, and using partial fractions to break down rational functions into simpler fractions.
(3) The key techniques discussed are integration by parts, trigonometric substitutions to transform integrals involving quadratic expressions into simpler forms, and partial fractions to decompose rational functions for integration. Various examples illustrate the
This document summarizes a research article about Seidel's method, an iterative method for solving systems of linear equations. Seidel's method ensures convergence if the system satisfies diagonal dominance after each iteration. The document outlines the Seidel's method iteration formula and proves that it generates a Cauchy sequence that converges to a unique fixed point, providing a contraction mapping. It also discusses conditions for the convergence of Seidel's method, relating it to the eigenvalues of matrices used in the iteration formula.
This document summarizes a research article about Seidel's method, an iterative method for solving systems of linear equations. Seidel's method ensures convergence if the system satisfies diagonal dominance after each iteration. The document outlines the Seidel's method iteration formula and proves that it generates a Cauchy sequence that converges to a unique fixed point, providing a contraction mapping. It also discusses conditions for the convergence of Seidel's method, relating it to the eigenvalues of matrices derived from the system's coefficients.
On the Seidel’s Method, a Stronger Contraction Fixed Point Iterative Method o...BRNSS Publication Hub
In the solution of a system of linear equations, there exist many methods most of which are not fixed point iterative methods. However, this method of Sidel’s iteration ensures that the given system of the equation must be contractive after satisfying diagonal dominance. The theory behind this was discussed in sections one and two and the end; the application was extensively discussed in the last section.
1. The document provides definitions and identities relating to theoretical computer science topics like asymptotic analysis, series, recurrences, and discrete structures.
2. It includes definitions for big O, Omega, and Theta notation used to describe the asymptotic behavior of functions.
3. Recurrences, like the master method, are presented for analyzing the runtime of divide-and-conquer algorithms.
4. Discrete structures like combinations, Stirling numbers, and trees are also covered, along with their properties and relationships.
The document provides solutions to problems from an IIT-JEE 2004 mathematics exam. Problem 1 asks the student to find the center and radius of a circle defined by a complex number relation. The solution shows that the center is the midpoint of points dividing the join of the constants in the ratio k:1, and gives the radius. Problem 2 asks the student to prove an inequality relating dot products of four vectors satisfying certain conditions. The solution shows that the vectors must be parallel or antiparallel.
This document provides a formula booklet covering topics in mathematics. It contains 24 sections with over 100 formulas related to topics like straight lines, circles, parabolas, ellipses, hyperbolas, limits, differentiation, integration, equations, sequences, series, vectors, and more. For each topic, relevant geometric definitions and properties are stated along with the key formulas.
11.a focus on a common fixed point theorem using weakly compatible mappingsAlexander Decker
The document presents a common fixed point theorem that generalizes an earlier theorem by Bijendra Singh and M.S. Chauhan. It replaces the conditions of compatibility and completeness with weaker conditions of weakly compatible mappings and an associated convergent sequence. The theorem proves that if self-maps A, B, S, and T of a metric space satisfy certain conditions, including (1) A(X) ⊆ T(X) and B(X) ⊆ S(X), (2) the pairs (A,S) and (B,T) are weakly compatible, and (3) the associated sequence converges, then the maps have a unique common fixed point. An example is given where the
A focus on a common fixed point theorem using weakly compatible mappingsAlexander Decker
The document presents a theorem that generalizes an existing fixed point theorem using weaker conditions. Specifically, it replaces the conditions of compatibility and completeness with weakly compatible mappings and an associated convergent sequence. The theorem proves that if four self-maps satisfy certain conditions, including being weakly compatible and having an associated sequence that converges, then the maps have a unique common fixed point. The conditions are shown to be weaker using an example where the associated sequence converges even though the space is not complete.
Calculus Early Transcendentals 10th Edition Anton Solutions Manualnodyligomi
This document contains a table of contents for a calculus textbook, outlining 15 chapters that cover topics from limits and continuity to vector calculus and partial derivatives. It also includes 3 appendix sections on graphing functions, trigonometry review, and solving polynomial equations.
The document provides an introduction to the binomial theorem. It defines binomial coefficients through the Pascal triangle and gives an explicit formula for computing them using factorials. The binomial theorem is then derived and stated, providing a formula for expanding expressions of the form (a + b)^n in terms of binomial coefficients. Several examples are worked out to demonstrate expanding expressions and finding coefficients using the binomial theorem. Applications to estimating interest calculations are also briefly discussed.
The document discusses solving multiple non-linear equations using the multidimensional Newton-Raphson method. It provides an example of solving for the equilibrium conversion of two coupled chemical reactions. The key steps are: (1) writing the reaction equations in root-finding form as two non-linear equations f1 and f2; (2) defining the Jacobian matrix J with the partial derivatives of f1 and f2; and (3) using the multidimensional Taylor series expansion and Jacobian matrix to linearize the system and iteratively solve for the root where f1 and f2 equal zero.
This document discusses randomized algorithms. It begins by listing different categories of algorithms, including randomized algorithms. Randomized algorithms introduce randomness into the algorithm to avoid worst-case behavior and find efficient approximate solutions. Quicksort is presented as an example randomized algorithm, where randomness improves its average runtime from quadratic to linear. The document also discusses the randomized closest pair algorithm and a randomized algorithm for primality testing. Both introduce randomness to improve efficiency compared to deterministic algorithms for the same problems.
The document discusses statistical representation of random inputs in continuum models. It provides examples of representing random fields using the Karhunen-Loeve expansion, which expresses a random field as the sum of orthogonal deterministic basis functions and random variables. Common choices for the covariance function in the expansion include the radial basis function and limiting cases of fully correlated and uncorrelated fields. The covariance function can be approximated from samples of the random field to enable representation in applications.
The document provides information on various integration techniques including the midpoint rule, trapezoidal rule, Simpson's rule, integration by parts, trigonometric substitutions, and applications of integrals such as finding the area between curves, arc length, surface area of revolution, and volume of revolution. It also covers integrals of common functions, properties of integrals, and techniques for parametric and polar coordinates.
This document presents a summary of a talk on building a harmonic analytic theory for the Gaussian measure and the Ornstein-Uhlenbeck operator. It discusses how the Gaussian measure is non-doubling but satisfies a local doubling property. It introduces Gaussian cones and shows how they allow proving maximal function estimates for the Ornstein-Uhlenbeck semigroup in a similar way as for the heat semigroup. The talk outlines estimates for the Mehler kernel of the Ornstein-Uhlenbeck semigroup and combines them to obtain boundedness of the maximal function.
This document discusses the application of fixed-point theorems to solve ordinary differential equations. It begins by introducing the Banach contraction principle and proving it. It then states two other important fixed-point theorems - the Schauder-Tychonoff theorem and the Leray-Schauder theorem. The rest of the document focuses on proving the Schauder-Tychonoff theorem, which characterizes compact subsets of function spaces and shows that if an operator maps into a relatively compact subset, it has a fixed point. This allows the fixed-point theorems to be applied to finding solutions to differential equations.
This document discusses the application of fixed-point theorems to solve ordinary differential equations. It begins by introducing the Banach contraction principle and proving it. It then states two other important fixed-point theorems - the Schauder-Tychonoff theorem and the Leray-Schauder theorem. The rest of the document focuses on proving the Schauder-Tychonoff theorem, which characterizes compact subsets of function spaces and shows that if an operator maps into a relatively compact subset, it has a fixed point. This allows the fixed-point theorems to be applied to finding solutions to differential equations.
This document summarizes research on the consistency and stability of linear multistep methods for solving initial value differential problems. It discusses the local truncation error and consistency conditions for convergence. The consistency condition requires that the truncation error approaches zero as the step size decreases. Stability conditions like relative and weak stability are also analyzed. It is shown that linear multistep methods satisfy the conditions of the Banach fixed point theorem, ensuring a unique solution. Specifically, a two-step predictor-corrector method is presented where the predictor provides an initial estimate that is corrected.
(1) The document discusses various integration techniques including: review of integral formulas, integration by parts, trigonometric integrals involving products of sines and cosines, trigonometric substitutions, and integration of rational functions using partial fractions.
(2) Examples are provided to demonstrate each technique, such as using integration by parts to evaluate integrals of the form ∫udv, using trigonometric identities to reduce powers of trigonometric functions, and using partial fractions to break down rational functions into simpler fractions.
(3) The key techniques discussed are integration by parts, trigonometric substitutions to transform integrals involving quadratic expressions into simpler forms, and partial fractions to decompose rational functions for integration. Various examples illustrate the
This document summarizes a research article about Seidel's method, an iterative method for solving systems of linear equations. Seidel's method ensures convergence if the system satisfies diagonal dominance after each iteration. The document outlines the Seidel's method iteration formula and proves that it generates a Cauchy sequence that converges to a unique fixed point, providing a contraction mapping. It also discusses conditions for the convergence of Seidel's method, relating it to the eigenvalues of matrices used in the iteration formula.
This document summarizes a research article about Seidel's method, an iterative method for solving systems of linear equations. Seidel's method ensures convergence if the system satisfies diagonal dominance after each iteration. The document outlines the Seidel's method iteration formula and proves that it generates a Cauchy sequence that converges to a unique fixed point, providing a contraction mapping. It also discusses conditions for the convergence of Seidel's method, relating it to the eigenvalues of matrices derived from the system's coefficients.
On the Seidel’s Method, a Stronger Contraction Fixed Point Iterative Method o...BRNSS Publication Hub
In the solution of a system of linear equations, there exist many methods most of which are not fixed point iterative methods. However, this method of Sidel’s iteration ensures that the given system of the equation must be contractive after satisfying diagonal dominance. The theory behind this was discussed in sections one and two and the end; the application was extensively discussed in the last section.
1. The document provides definitions and identities relating to theoretical computer science topics like asymptotic analysis, series, recurrences, and discrete structures.
2. It includes definitions for big O, Omega, and Theta notation used to describe the asymptotic behavior of functions.
3. Recurrences, like the master method, are presented for analyzing the runtime of divide-and-conquer algorithms.
4. Discrete structures like combinations, Stirling numbers, and trees are also covered, along with their properties and relationships.
The document provides solutions to problems from an IIT-JEE 2004 mathematics exam. Problem 1 asks the student to find the center and radius of a circle defined by a complex number relation. The solution shows that the center is the midpoint of points dividing the join of the constants in the ratio k:1, and gives the radius. Problem 2 asks the student to prove an inequality relating dot products of four vectors satisfying certain conditions. The solution shows that the vectors must be parallel or antiparallel.
This document provides a formula booklet covering topics in mathematics. It contains 24 sections with over 100 formulas related to topics like straight lines, circles, parabolas, ellipses, hyperbolas, limits, differentiation, integration, equations, sequences, series, vectors, and more. For each topic, relevant geometric definitions and properties are stated along with the key formulas.
11.a focus on a common fixed point theorem using weakly compatible mappingsAlexander Decker
The document presents a common fixed point theorem that generalizes an earlier theorem by Bijendra Singh and M.S. Chauhan. It replaces the conditions of compatibility and completeness with weaker conditions of weakly compatible mappings and an associated convergent sequence. The theorem proves that if self-maps A, B, S, and T of a metric space satisfy certain conditions, including (1) A(X) ⊆ T(X) and B(X) ⊆ S(X), (2) the pairs (A,S) and (B,T) are weakly compatible, and (3) the associated sequence converges, then the maps have a unique common fixed point. An example is given where the
A focus on a common fixed point theorem using weakly compatible mappingsAlexander Decker
The document presents a theorem that generalizes an existing fixed point theorem using weaker conditions. Specifically, it replaces the conditions of compatibility and completeness with weakly compatible mappings and an associated convergent sequence. The theorem proves that if four self-maps satisfy certain conditions, including being weakly compatible and having an associated sequence that converges, then the maps have a unique common fixed point. The conditions are shown to be weaker using an example where the associated sequence converges even though the space is not complete.
Calculus Early Transcendentals 10th Edition Anton Solutions Manualnodyligomi
This document contains a table of contents for a calculus textbook, outlining 15 chapters that cover topics from limits and continuity to vector calculus and partial derivatives. It also includes 3 appendix sections on graphing functions, trigonometry review, and solving polynomial equations.
The document provides an introduction to the binomial theorem. It defines binomial coefficients through the Pascal triangle and gives an explicit formula for computing them using factorials. The binomial theorem is then derived and stated, providing a formula for expanding expressions of the form (a + b)^n in terms of binomial coefficients. Several examples are worked out to demonstrate expanding expressions and finding coefficients using the binomial theorem. Applications to estimating interest calculations are also briefly discussed.
The document discusses solving multiple non-linear equations using the multidimensional Newton-Raphson method. It provides an example of solving for the equilibrium conversion of two coupled chemical reactions. The key steps are: (1) writing the reaction equations in root-finding form as two non-linear equations f1 and f2; (2) defining the Jacobian matrix J with the partial derivatives of f1 and f2; and (3) using the multidimensional Taylor series expansion and Jacobian matrix to linearize the system and iteratively solve for the root where f1 and f2 equal zero.
This document discusses randomized algorithms. It begins by listing different categories of algorithms, including randomized algorithms. Randomized algorithms introduce randomness into the algorithm to avoid worst-case behavior and find efficient approximate solutions. Quicksort is presented as an example randomized algorithm, where randomness improves its average runtime from quadratic to linear. The document also discusses the randomized closest pair algorithm and a randomized algorithm for primality testing. Both introduce randomness to improve efficiency compared to deterministic algorithms for the same problems.
The document discusses statistical representation of random inputs in continuum models. It provides examples of representing random fields using the Karhunen-Loeve expansion, which expresses a random field as the sum of orthogonal deterministic basis functions and random variables. Common choices for the covariance function in the expansion include the radial basis function and limiting cases of fully correlated and uncorrelated fields. The covariance function can be approximated from samples of the random field to enable representation in applications.
The document provides information on various integration techniques including the midpoint rule, trapezoidal rule, Simpson's rule, integration by parts, trigonometric substitutions, and applications of integrals such as finding the area between curves, arc length, surface area of revolution, and volume of revolution. It also covers integrals of common functions, properties of integrals, and techniques for parametric and polar coordinates.
This document presents a summary of a talk on building a harmonic analytic theory for the Gaussian measure and the Ornstein-Uhlenbeck operator. It discusses how the Gaussian measure is non-doubling but satisfies a local doubling property. It introduces Gaussian cones and shows how they allow proving maximal function estimates for the Ornstein-Uhlenbeck semigroup in a similar way as for the heat semigroup. The talk outlines estimates for the Mehler kernel of the Ornstein-Uhlenbeck semigroup and combines them to obtain boundedness of the maximal function.
This document discusses the application of fixed-point theorems to solve ordinary differential equations. It begins by introducing the Banach contraction principle and proving it. It then states two other important fixed-point theorems - the Schauder-Tychonoff theorem and the Leray-Schauder theorem. The rest of the document focuses on proving the Schauder-Tychonoff theorem, which characterizes compact subsets of function spaces and shows that if an operator maps into a relatively compact subset, it has a fixed point. This allows the fixed-point theorems to be applied to finding solutions to differential equations.
This document discusses the application of fixed-point theorems to solve ordinary differential equations. It begins by introducing the Banach contraction principle and proving it. It then states two other important fixed-point theorems - the Schauder-Tychonoff theorem and the Leray-Schauder theorem. The rest of the document focuses on proving the Schauder-Tychonoff theorem, which characterizes compact subsets of function spaces and shows that if an operator maps into a relatively compact subset, it has a fixed point. This allows the fixed-point theorems to be applied to finding solutions to differential equations.
ALPHA LOGARITHM TRANSFORMED SEMI LOGISTIC DISTRIBUTION USING MAXIMUM LIKELIH...BRNSS Publication Hub
The document discusses the alpha logarithm transformed semi-logistic distribution and its maximum likelihood estimation method. It introduces the distribution, provides its probability density function and cumulative distribution function. It then describes generating random numbers from the distribution and outlines the maximum likelihood estimation method to estimate the distribution's unknown parameters. This involves deriving the likelihood function and taking its partial derivatives to obtain equations that are set to zero and solved to find maximum likelihood estimates of the location, scale, and shape parameters.
AN ASSESSMENT ON THE SPLIT AND NON-SPLIT DOMINATION NUMBER OF TENEMENT GRAPHSBRNSS Publication Hub
This document summarizes research on the split and non-split domination numbers of tenement graphs. It defines tenement graphs and provides basic definitions of domination, split domination, and non-split domination. Formulas for the split and non-split domination numbers of tenement graphs are presented based on the number of vertices. Theorems are presented stating that the mid vertex set of a tenement graph is always a split dominating set, but its size is not always equal to the split domination number.
This document summarizes research on generalized Cantor sets and functions where the standard construction is modified. It introduces Cantor sets defined by an arbitrary base where the intervals removed at each stage are not all the same length. It also defines irregular or transcendental Cantor sets generated by transcendental numbers like e. The key findings are:
1) There exists a unique probability measure for generalized Cantor sets that generates the cumulative distribution function.
2) The Holder exponent of generalized Cantor sets is shown to be logn/s where n is the base and s is the number of subintervals.
3) Lower and upper densities are defined for the measure on generalized Cantor functions and their properties are
SYMMETRIC BILINEAR CRYPTOGRAPHY ON ELLIPTIC CURVE AND LIE ALGEBRABRNSS Publication Hub
1) The document discusses symmetric bilinear pairings on elliptic curves and Lie algebras in the context of cryptography. It provides an overview of the theoretical foundations and applications of combining these areas.
2) Key concepts covered include the Weil pairing as a symmetric bilinear pairing on elliptic curves, its properties of bilinearity and non-degeneracy, and efficient computation. Applications of elliptic curves in cryptography like ECDH and ECDSA are also summarized.
3) The security of protocols like ECDH and ECDSA relies on the assumed difficulty of solving the elliptic curve discrete logarithm problem (ECDLP). The document proves various mathematical aspects behind symmetric bilinear pairings and their use in elliptic curve cryptography.
SUITABILITY OF COINTEGRATION TESTS ON DATA STRUCTURE OF DIFFERENT ORDERSBRNSS Publication Hub
This document summarizes research investigating the suitability of cointegration tests on time series data of different orders. The researchers used simulated time series data from normal and gamma distributions at sample sizes of 30, 60, and 90. Three cointegration tests (Engle-Granger, Johansen, and Phillips-Ouliaris) were applied to the data. The tests were assessed based on type 1 error rates and power to determine which test was most robust for different distributions and sample sizes. The results indicated the Phillips-Ouliaris test was generally the most effective at determining cointegration across different sample sizes and distributions.
Artificial Intelligence: A Manifested Leap in Psychiatric RehabilitationBRNSS Publication Hub
Artificial intelligence shows promise in improving psychiatric rehabilitation in 3 key ways:
1) AI can help diagnose and treat mental health issues through virtual therapists and chatbots, improving access and reducing stigma.
2) Technologies like machine learning and big data allow personalized interventions and more accurate diagnoses.
3) The COVID-19 pandemic has increased need for mental health support, and AI may help address gaps by providing remote services.
A Review on Polyherbal Formulations and Herbal Medicine for Management of Ul...BRNSS Publication Hub
This document provides a review of polyherbal formulations and herbal medicines for treating peptic ulcers. It discusses how peptic ulcers occur due to an imbalance between aggressive and protective factors in the gastrointestinal tract. Common causes include H. pylori infection and NSAID use. While synthetic medications are available, herbal supplements are more affordable and have fewer side effects. The review examines various herbs that have traditionally been used to treat ulcers, including their active chemical constituents. It defines polyherbal formulations as combinations of two or more herbs, which can enhance therapeutic effects while reducing toxicity. The document aims to summarize recent research on herb and polyherbal formulation treatments for peptic ulcers.
Current Trends in Treatments and Targets of Neglected Tropical DiseaseBRNSS Publication Hub
This document summarizes current trends in treatments and targets of neglected tropical diseases. It begins by stating that neglected tropical diseases affect over 1.7 billion people globally each year and are caused by a variety of microbes. The World Health Organization is working to eliminate 30 neglected tropical diseases by 2030. The document then discusses several specific neglected tropical diseases in more detail, including human African trypanosomiasis, Chagas disease, leishmaniasis, soil-transmitted helminths, and schistosomiasis. It describes the causative agents, transmission methods, symptoms, affected populations, and current treatment options for each of these diseases. Overall, the document aims to briefly discuss neglected infectious diseases and treatment
Evaluation of Cordia Dichotoma gum as A Potent Excipient for the Formulation ...BRNSS Publication Hub
This document summarizes a study that evaluated Cordia dichotoma gum as an excipient for oral thin film drug delivery. Films were prepared with varying ratios of the gum, plasticizers (methyl paraben and glycerine), and the model drug diclofenac sodium. The films were evaluated for properties like thickness, folding endurance, tensile strength, water uptake, and drug release kinetics. The results found that a film with 10% gum, 0.2% methyl paraben and 2.5% glycerine (CDF3) exhibited the best results among the formulations tested. Stability studies showed the films were stable for 30 days at different temperatures. Overall, the study demonstrated that C.
Assessment of Medication Adherence Pattern for Patients with Chronic Diseases...BRNSS Publication Hub
This study assessed medication adherence and knowledge among rural patients with chronic diseases in South Indian hospitals. 1500 hypertensive patients were divided into intervention and control groups. The intervention group received education from pharmacists at various times, while the control group did not. A questionnaire evaluated patients' medication knowledge at baseline and several follow-ups. The intervention group showed improved medication knowledge scores after education compared to the control group. Female gender, lower education, and income were linked to lower knowledge. The study highlights the need to educate rural patients to improve medication understanding and adherence.
This document proposes a system to hide information using four algorithms for image steganography. The system first encrypts data using a modified AES algorithm. It then encrypts the encrypted data using a modified RSA algorithm. Next, it uses a fuzzy stream algorithm to add ambiguity. Finally, it hides the encrypted data in the least significant bits of cover images using LSB steganography. The document evaluates the proposed system using metrics like PSNR, MSE, and SSIM to analyze image quality and the ability to hide data imperceptibly compared to other techniques. It selects four color images as cover files and tests the system on them.
The document discusses Goldbach's problems and their solutions. It summarizes that the ternary Goldbach problem, which states that every odd number greater than 7 can be represented as the sum of three odd primes, was solved in 2013. It also discusses Ramare's 1995 proof that any even number can be represented as the sum of no more than 6 primes. The document then provides proofs for theorems related to representing numbers as sums of primes and concludes there are an infinite number of twin primes.
The document summarizes research on k-super contra harmonic mean labeling of graphs. It defines k-super Lehmer-3 mean labeling of a graph as an injective vertex labeling such that the induced edge labels satisfy certain properties. It proves that several families of graphs admit k-super Lehmer-3 mean labeling for any positive integer k, including triangular snakes, double triangular snakes, alternative triangular snakes, quadrilateral snakes, and alternative quadrilateral snakes. The document introduces the concept of k-super Lehmer-3 mean labeling and investigates this property for these families of graphs.
The document summarizes research on using various iterative schemes to solve fixed-point problems and inequalities involving self-mappings and contractions in Banach spaces. It defines concepts like non-expansive mappings, mean non-expansive mappings, and rates of convergence. The paper presents two theorems: 1) an iterative scheme for a sequence involving a self-mapping T is shown to converge to a fixed point of T, and 2) an iterative process involving a self-contraction mapping T is defined and shown to converge. Limiting cases are considered to prove convergence as the number of iterations approaches infinity.
This document summarizes research on analyzing and simulating the accuracy and stability of closed-loop control systems. It discusses various techniques for evaluating accuracy and stability, including steady-state error analysis, stability analysis, and simulation. Factors that can affect accuracy and stability are also identified, such as sensor noise, model inaccuracies, and environmental disturbances. The paper provides an overview of closed-loop control systems and their uses in various engineering fields like manufacturing, chemical processes, vehicles, aircraft, and power systems.
Leveraging Generative AI to Drive Nonprofit InnovationTechSoup
In this webinar, participants learned how to utilize Generative AI to streamline operations and elevate member engagement. Amazon Web Service experts provided a customer specific use cases and dived into low/no-code tools that are quick and easy to deploy through Amazon Web Service (AWS.)
Andreas Schleicher presents PISA 2022 Volume III - Creative Thinking - 18 Jun...EduSkills OECD
Andreas Schleicher, Director of Education and Skills at the OECD presents at the launch of PISA 2022 Volume III - Creative Minds, Creative Schools on 18 June 2024.
Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.pptHenry Hollis
The History of NZ 1870-1900.
Making of a Nation.
From the NZ Wars to Liberals,
Richard Seddon, George Grey,
Social Laboratory, New Zealand,
Confiscations, Kotahitanga, Kingitanga, Parliament, Suffrage, Repudiation, Economic Change, Agriculture, Gold Mining, Timber, Flax, Sheep, Dairying,
Gender and Mental Health - Counselling and Family Therapy Applications and In...PsychoTech Services
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1. www.ajms.com 1
ISSN 2581-3463
RESEARCH ARTICLE
Common Fixed Point Theorems for Non-linear Contractive Mapping in Hilbert
Space
M. Raji, F. Adegboye
Department of Mathematics, Confluence University of Science and Technology, Osara, Kogi State, Nigeria
Received: 25-02-2021; Revised: 18-03-2022; Accepted: 10-04-2022
ABSTRACT
In this manuscript, we prove the existence and uniqueness of a common fixed point for a continuous self-
mapping over a closed subset of Hilbert space satisfying non-linear rational type contraction condition.
Key words: Hilbert space, Fixed point, Closed subset, Completeness
Address for correspondence:
Muhammed Raji,
Email: rajimuhammed11@gmail.com
INTRODUCTION
Fixed point theory is an important tool used in
solving a variety of problems in control theory,
economic theory, global analysis, and non-
linear analysis. Several authors worked on the
applications, generalizations, and extensions of
the Banach contraction principle[1]
in different
directions by either weakening the hypothesis,
using different setups or considering different
mappings.
Banachcontractionprinciplehasbeenextendedand
generalized by several authors such as Kannan[2]
who investigated the extension of Banach fixed
point theorem by removing the completeness of
the space with different conditions. Chatterji[3]
considered various contractive conditions for
self-mappings in metric space. Dass and Gupta[5]
also investigated the rational type of contractions
to obtain a unique fixed point in complete metric
space. Fisher[10]
developed the approach of
Khann[2]
and proved analogous results involving
two mappings on a complete metric space.
PRELIMINARIES
The main aim of this paper is to prove the
existence and uniqueness of a common fixed point
for a continuous self-mapping T, some positive
integers r,s of a pair of continuous self-mapping
Tr
,Ts
. These results generalized and extend the
results of[2-13]
and[10]
here are the lists of some of
the results that motivated our results.
Theorem 2.1.[2]
proved that “If f is self-mapping
of a complete metric space X into itself satisfying
d(x,y)≤α[d(Tx,x)+d(Ty,y)], for all x,y∈X and
α∈[0,1/2), then f has a unique fixed point in X”.
Theorem 2.2.[10]
Prove the result with
d(x,y)≤α[d(Tx,x)+d(Ty,y)]+βd(x,y), for all x,y ∈ X
and α,β ∈[0,1/2), then f has a unique fixed point
in X”.
Theorem 2.3.[8]
Proved the common fixed point
theorems in Hilbert space with the following
condition.
Tx Ty
x Tx y Ty
x Tx y Ty
x y
− ≤
− + −
− + −
+ −
α β
2 2
for
all x,y ∈S,x ≠y, α ∈[0,1/2), β≥0 and 2α + β<1.
Theorem 2.4.[9]
Proved the fixed point theorems
in Hilbert space with the following condition.
Tx Ty
x Tx y Ty
x Tx y Ty
x Ty y Tx
x Ty y T
− ≤
− + −
− + −
+
− + −
− + −
α
β
2 2
2 2
x
x
x y
+ −
γ
for all x,y ∈S,x ≠y, 0≤ α,β < 1/2, 0 ≤ 𝛾 2α and 2β
+ 𝛾 < 1.
Theorem 2.5.[14]
Proved the unique fixed point
theorems in Hilbert space with the following
condition.
2. Raji and Adegboye: Common Fixed Point Theorems for Non-linear Contractive Mapping in Hilbert Space
AJMS/Apr-Jun-2022/Vol 6/Issue 2 2
Tx Ty a
x Tx y Ty x Ty y Tx
x Tx y Ty x Ty y Tx
− ≤
− + − + − + −
− + − + − + −
1
2 2 2 2
+
− + −
− + −
+
− + −
− + −
a
x Tx y Ty
x Tx y Ty
a
x Ty y Tx
x Ty y T
2
2 2
3
2 2
x
x
a x y
+ −
4
For all x,y ∈S and 4α1
,α2
,α3,
α4
≥ 0and 4α1
+ 2α2
+
2α3
+α4
1.
Theorem 2.6.[7]
Proved the unique common fixed
point theorems for two self-mappings T1
,T2
of closed
subset X in Hilbert space satisfying the inequality.
T x T y a
x T x y T y
x y
a
1 2 1
1 2
2
1
1
− ≤
− + −
+ −
+
y T y y T x
x y
− + −
+ −
+
2 1
1
1
a
x T y y T x
x y
3
2 1
1
1
− + −
+ −
+
a
x y T x T y
x y
4
1 2
1
1
− + −
+ −
+
− + −
+ −
+
a
x y x T x
y T y
5
1
2
1
1
a
x T x x T y
y T y
a
x T y T x T y
x y
a
6
1 2
2
7
2 1 2
8
1
1
1
1
− + −
+ −
+
− + −
+ −
+
x y x T y
x y
a
x T x y T y x y
x T x x T y y T y x y
− + −
+ −
+
− + − + −
+ − − − −
+
1
1
1
2
9
1 2
1 2 2
a
x T y y T x
x T y y T x
a x T x y T y
a x T y y
10
2
2
1
2
2 1
11 1 2
12 2
− + −
− + −
+ − + −
+ − + −T
T x a x y
1 13
+ − .
For all x,y ∈ X andx ≠ y, where ai
(i=1,2,3,…,13) are
non-negative reals with 0 ≤ ∑8
i=1
ai
+3a9
+2∑12
i=10
ai
+a13
1.
Theorem 2.7.[6]
Proved the unique common fixed
point theorems for two self-mappings T1
T2
of
closed subset X in Hilbert space satisfying the
inequality.
T x T y
x T y y T x
x y
x y x T y
x y
1 2
2 2
2
1
2
2
2
2
2
2
1
1
1
1
− ≤
− + −
+ −
+
− + −
+ −
α
β +
+ − + −
+ −
γ
δ
y T x x T y
x y
1
2
2
2
2
for all x y S x y
, ,�
∈ ≠ where α,β,γ,δ and 4α + β +
4γ + δ 1.
Theorem 2.8.[6]
Proved the unique common fixed
point theorems for two self-mappings T1
,T2
and p,q
(positive integers) of closed subset X in Hilbert
space satisfying the inequality.
T x T y
x T y y T x
x y
x y x T y
p q
q p
q
1 2
2 2
2
1
2
2
2
2
2
1
1
1
− ≤
− + −
+ −
+
− + −
α
β
+ −
+ − + −
+ −
1
2 1
2
2
2
2
x y
y T x x T y
x y
p q
γ
δ
for all 𝑥,𝑦 ∈ 𝑆,𝑥≠𝑦, where 𝛼,𝛽,𝛾,𝛿 and 4𝛼 +𝛽+
4𝛾+𝛿 1.
MAIN RESULTS
Theorem 3.1. Let X be a closed subset of a Hilbert
space and T: X→X be a continuous self-mapping
satisfying the following inequality.
Tx Ty c
x Tx y Ty
x y
c
y Ty y Tx
x y
− ≤
− + −
+ −
+
− + −
+ −
+
1
2
1
1
1
1
c
x Ty y Tx
x y
c
x y Tx Ty
x y
3 4
1
1
1
1
− + −
+ −
+
− + −
+ −
3. Raji and Adegboye: Common Fixed Point Theorems for Non-linear Contractive Mapping in Hilbert Space
AJMS/Apr-Jun-2022/Vol 6/Issue 2 3
+
− + −
+ −
+
− + −
+ −
+
c
x y x Tx
y Ty
c
x Tx x Ty
y Ty
5 6
1
1
1
1
c
x Ty Tx Ty
x y
c
x y x Ty
x y
7 8
1
1
1
1
− + −
+ −
+
− + −
+ −
+
− + − + −
+ − − − −
c
x Tx y Ty x y
x Tx x Ty y Ty x y
9
1
+
− + −
− + −
+ − + −
+ − + −
c
x Ty y Tx
x Ty y Tx
c x Tx y Ty
c x Ty y Tx
10
2 2
11
12 +
+ −
c x y
13
for all x,y∈X and x≠y, where ci
(i=1,2,3,…,13) are
non-negative real numbers with 0≤ ci
i=
∑ 1
8
+ 3c9
+2
ci
i=
∑ 10
12
+ c13
1. Then, T has a unique fixed point
in X.
Proof. We construct a sequence {xn
} for an
arbitrary point x0
∈ X defined as follows
x2n+1
= Tx2n
, x2n+2
=Tx2n
+1, for 𝑛=1,2,3.,
We show that the sequence {xn
} is a Cauchy
sequence in X
x x Tx Tx
n n n n
2 1 2 2 2 1
+ −
− = −
≤
− + −
+ −
+
− +
− −
−
− −
c
x Tx x Tx
x x
c
x Tx x
n n n n
n n
n n
1
2 2 2 1 2 1
2 2 1
2
2 1 2 1
1
1
1 2
2 1 2
2 2 1
1
n n
n n
Tx
x x
−
−
−
+ −
+
− + −
+ −
− −
−
c
x Tx x Tx
x x
n n n n
n n
3
2 2 1 2 1 2
2 2 1
1
1
+
− + −
+ −
+
− + −
− −
−
−
c
x x Tx Tx
x x
c
x x x
n n n n
n n
n n n
4
2 2 1 2 2 1
2 2 1
5
2 2 1 2
1
1
1 T
Tx
x Tx
n
n n
2
2 1 2 1
1
+ −
+
− −
c
x Tx x Tx
x Tx
n n n n
n n
6
2 2 2 2 1
2 1 2 1
1
1
− + −
+ −
−
− −
+
− + −
+ −
+
− +
− −
−
−
c
x Tx Tx Tx
x x
c
x x x
n n n n
n n
n n n
7
2 2 1 2 2 1
2 2 1
8
2 2 1 2
1
1
1 −
−
+ −
−
−
Tx
x x
n
n n
2 1
2 2 1
1
+
− + −
+ −
+ − −
− −
−
−
−
c
x Tx x Tx
x x
x Tx x Tx
x
n n n n
n n
n n n n
n
9
2 2 2 1 2 1
2 2 1
2 2 2 2 1
2
1
1
1 2 1 2 2 1
− −
− −
Tx x x
n n n
+
− + −
− + −
− −
− −
c
x Tx x Tx
x Tx x Tx
n n n n
n n n n
10
2 2 1
2
2 1 2
2
2 2 1 2 1 2
+ − + −
+ − + −
− −
− −
c x Tx x Tx
c x Tx x Tx
n n n n
n n n n
11 2 2 2 1 2 1
12 2 2 1 2 1 2 +
+ − −
c x x
n n
13 2 2 1
This implies
x x a n Tx Tx
n n n n
2 1 2 2 2 1
+ −
− = ( ) − ,
where
a n
B c c c c c
x x
B c c c c c c
n n
( )=
+ + + + +
( )
−
+ − − − − − −
−
1 2 9 10 11 12
2 2 1
2 1 2 4 5 9
2
1 1
10 11 12
2 2 1
− −
( )
− −
c c
x x
n n
,
1 2 4 5 8 9 10 11 12 13
2
B c c c c c c c c c
= + + + + + + + + and
B2
=1-c1
-c6
-c9
-c10
-c11
-c12 S
Clearly, δ=a (n)1,∀n=1,2,3,.., we get
x x x x
n n n n
+ −
− = −
1 1
,
Recursively, we have
x x x x n
n n
n
+ − = − ≥
1 1 0 1
, ,
Takingn→∞, we have x x
n n
+ − →
1 0.
Hence, {xn
} is a Cauchy sequence in X and so it has
a limit λ in X. Since the sequences {x2n+1
}={Tx2n
}
and {x2n+2
}={Tx2n+1
}are subsequences of {xn
}, and
also these subsequences have the same limit λ in X.
We now show that λ is a common fixed point of T
Now consider the following inequality
4. Raji and Adegboye: Common Fixed Point Theorems for Non-linear Contractive Mapping in Hilbert Space
AJMS/Apr-Jun-2022/Vol 6/Issue 2 4
− = − + − = −
( )
+ −
( )≤ − + −
+ + +
+ +
T x x T x
x T x T Tx
n n n
n n n
2 2 2 2 2 2
2 2 2 2 2 +
+1
≤
− + −
+ −
+
− + −
+ +
+
+ + +
c
T x Tx
x
c
x Tx x T
n n
n
n n n
1
2 1 2 1
2 1
2
2 1 2 1 2 1
1
1
1
+ − +
1 2 1
x n
c
Tx x T
x
c
x T Tx
n n
n
n n
3
2 1 2 1
2 1
4
2 1 2 1
1
1
1
− + −
+ −
+
− + −
+ +
+
+ +
1
1 2 1
+ − +
x n
+
− + −
+ −
+
− + −
+
+
+ +
+
c
x T
x Tx
c
T Tx
x
n
n n
n
5
2 1
2 1 2 1
6
2 1
1
1
1
1
2
2 1 2 1
n n
Tx
+ +
−
+
− + −
+ −
+
− + −
+ +
+
+ +
c
Tx T Tx
x
c
x Tx
n n
n
n n
7
2 1 2 1
2 1
8
2 1 2 1
1
1
1
+ − +
1 2 1
x n
+
− + − + −
+ − − − −
+ + +
+ + +
c
T x Tx x
T Tx x Tx x
n n n
n n n
9
2 1 2 1 2 1
2 1 2 1 2 1 2
1
n
n
n n
n n
c
Tx x T
Tx x T
+
+ +
+ +
+
− + −
− + −
1
10
2 1
2
2 1
2
2 1 2 1
+ − + −
+ − + −
+ −
+ +
+ +
c T x Tx
c Tx x T c x
n n
n n
11 2 1 2 1
12 2 1 2 1 13 2
n
n+1 .
Letting n→∞, we get ‖λ-Tλ‖ ≤
(c1
+c6
+c9
+c10
+c11
+c12
)‖λ-Tλ‖, since
c1
+c6
+c9
+c10
+c11
+c12
1, hence λ = Tλ.
Similarly, from the hypothesis, we get λ = Tλ by
considering the following
− = −
( )+ −
( )
+ +
T x x T
n n
2 1 2 1 .
We now show that λ is a unique fixed point of T.
Suppose that t (λ≠t) is also a common fixed point
of T.
Then, by the hypothesis, we get
− = − ≤
− + −
+ −
+
− + −
+ −
+
t T Tt c
T t Tt
t
c
t Tt t T
t
c
1
2 3
1
1
1
1
− + −
+ −
+
− + −
+ −
+
− + −
Tt t T
t
c
t T Tt
t
c
t T
1
1
1
1
1
1
4
5
+
+ −
t Tt
+
− + −
+ −
+
− + −
+ −
+
− + −
c
T Tt
t Tt
c
Tt T Tt
t
c
t Tt
6 7
8
1
1
1
1
1
+ −
1 t
+
− + − + −
+ − − − −
+
− + −
− + −
c
T t Tt t
T Tt t Tt t
c
Tt t T
Tt t T
9
10
2 2
1
+ − + −
+ − + −
+ −
c T t Tt c Tt t T
c t
11 12
13
.
Thus,
− ≤ + + + + + + + +
( )
− −
t c c c c c c c c c
t t
3 4 5 7 8 9 10 12 13
2
a contradiction. Hence λ=t (common fixed point λ
is unique in X).
Theorem 3.2. Let X be a closed subset of a Hilbert
space and T: X→X be a continuous self mapping
satisfying
T x T y c
x T x y T y
x y
c
y T y y T x
x y
r s
r s
s r
− ≤
− + −
+ −
+
− + −
+ −
+
1
2
1
1
1
1
c
x T y y T x
x y
c
x y T x T y
x y
s r
r s
3
4
1
1
1
1
− + −
+ −
+
− + −
+ −
5. Raji and Adegboye: Common Fixed Point Theorems for Non-linear Contractive Mapping in Hilbert Space
AJMS/Apr-Jun-2022/Vol 6/Issue 2 5
+
− + −
+ −
+
− + −
+ −
+
−
c
x y x T x
y T y
c
x T x x T y
y T y
c
x T y
r
s
r s
s
s
5 6
7
1
1
1
1
1
1
1
1
1
8
+ −
+ −
+
− + −
+ −
T x T y
x y
c
x y x T y
x y
r s s
+
− + − + −
+ − − − −
+
− + −
−
c
x T x y T y x y
x T x x T y y T y x y
c
x T y y T x
x T
r s
r s s
s r
9
10
2 2
1
s
s r
y y T x
+ −
+ − + −
+ − + −
+ −
c x T x y T y
c x T y y T x c x y
r s
s r
11
12 13
for all x, y∈ X and x≠y, where ci
(i=1,2,3,…,13)
are non-negative real numbers with
0 3 2 1
1
8
9
10
12
13
≤ + + +
= =
∑ ∑
i
i
i
i
c c c c and r,s are two
positive integers. Then T has a unique fixed point
in X.
Proof. From theorem 3.1.Tr
and Ts
have a unique
common fixed point λ∈X, so that Tr
λ=λ and Ts
λ=λ.
From Tr
(Tλ)=T(Tr
λ)=Tλ, it follows that Tλ is a
fixed point of Tr
. But λ is a unique fixed point of
Tr
. Therefore Tλ=λ.
Similarly,wecangetTλ=λfromTs
(Tλ)=T(Ts
λ)=Tλ.
Hence λ is a unique fixed point of T.
Now, we show uniqueness. Let t be another fixed
point of T, so that Tt=t. then from the hypothesis,
we have
− = − ≤
− + −
+ −
+
− + −
+ −
t T T t c
T t T t
t
c
t T t t T
r s
r r
s r
1
2
1
1
1
1 t
t
+
c
T t t T
t
c
t T T t
t
c
t T
s r r s
3 4
5
1
1
1
1
1
− + −
+ −
+
− + −
+ −
+
− + − r
r
s
t T t
+ −
1
+
− + −
+ −
+
− + −
+ −
+
−
c
T T t
t T t
c
T t T T t
t
c
t
r s
s
s r s
6
7 8
1
1
1
1
1
1
1
1
9
+ −
+ −
+
− + − + −
+ − − − −
T t
t
c
T t T t t
T T t t T t t
s
r s
r s s
+
− + −
− + −
+ − + −
+ − +
c
T t t T
T t t T
c T t T t
c T t t
s r
s r
r s
s
10
2 2
11
12
−
−
+ −
T c t
r
13 .
Therefore, we have
− ≤ + + + + + + + +
( )
− −
t c c c c c c c c c
t t
3 4 5 7 8 9 10 12 13
2
Implies λ=t, since c3
+c4
+c5
+c7
+c8
+c9
+c10
+2c12
+c13
1.
Hence, λ is a unique common fixed point of
Tin X.
Example 3.3. Let T: [0,1]→[0,1] be a mapping
defined by Tx=x3
/6, for all x∈[0,1]with usual norm
‖x-y‖=|x-y|, for all x∈[0,1].
Proof. From theorem 3.1, we have
Tx Ty
x y
c
x
x
y
y
x y
c
y
y
y
x
− = − ≤
− + −
+ −
+
− + −
3 3
1
3 3
2
3 3
6 6
6
1
6
1
6
1
6
+ −
+
1 x y
c
x
y
y
x
x y
c
x y
x y
x y
c
x y
3
3 3
4
3 3
5
6
1
6
1
1
6 6
1
1
− + −
+ −
+
− + −
+ −
+
− +
+ −
+ −
x
x
y
y
3
3
6
1
6
6. Raji and Adegboye: Common Fixed Point Theorems for Non-linear Contractive Mapping in Hilbert Space
AJMS/Apr-Jun-2022/Vol 6/Issue 2 6
+
− + −
+ −
+
− + −
+ −
+
c
x
x
x
y
y
y
c
x
y x y
x y
6
3 3
3
7
3 3 3
6
1
6
1
6
6
1
6 6
1
c
x
y
y
x
x y
c
x y
x y
x y
c
x y
3
3 3
4
3 3
5
6
1
6
1
1
6 6
1
1
− + −
+ −
+
− + −
+ −
+
− +
+ −
+ −
x
x
y
y
3
3
6
1
6
+
− + −
+ −
+
− + −
+ −
+
c
x
x
x
y
y
y
c
x
y x y
x y
6
3 3
3
7
3 3 3
6
1
6
1
6
6
1
6 6
1
c
x y x
y
x y
c
x
x
y
y
x y
x
x
x
y
y
y
8
3
9
3 3
3 3 3
1
6
1
6 6
1
6 6 6
− + −
+ −
+
− + − + −
+ − − − x
x y
−
+
− + −
− + −
+ − + −
+ −
c
x
y
y
x
x
y
y
x
c x
x
y
y
c x
y
10
3 2 3 2
3 3 11
3 3
12
3
6 6
6 6
6 6
6
+
+ −
+ −
y
x
c x y
3
13
6
Continuewiththeprocedureintheproofoftheorem
3.1, we get that 0 is the only fixed point of T.
CONCLUSION
In this paper, we proved the existence and
uniqueness of a common fixed point for a
continuous self mapping,T some positive integers
r,s of a pair of continuous self mapping ,
r s
T T
of Hilbert space. These results generalized and
extend the results of some literatures.
ACKNOWLEDGMENT
The authors would like to extend their sincere thanks
to all that have contributed in one way or the other
for the actualization of the review of this work.
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