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.
The 1741 Goldbach [1] made his most famous contribution to mathematics with the conjecture that all even numbers can be expressed as the sum of the two primes (currently Conjecture) referred to as “all even numbers greater than 2 can be expressed as the sum-two primes” (DOI:10.13140/RG.2.2.32893.69600/1)
This document discusses solving linear homogeneous recurrence relations with constant coefficients. It begins by defining such a recurrence relation as one where the terms are expressed as a linear combination of previous terms. It then explains that these types of relations can be solved by finding the characteristic roots of the characteristic equation. The document provides an example of solving a degree two recurrence relation and outlines the basic approach of finding a solution of the form an = rn. It also discusses solving coupled recurrence relations by eliminating variables to obtain a single recurrence relation that can be solved. Finally, it revisits the Martian DNA problem and shows its solution is a Fibonacci number.
A Proof of Hypothesis Riemann and it is proven that apply only for equation ζ(z)=0.Also here it turns out that it does not apply in General Case.(DOI:10.13140/RG.2.2.10888.34563/5)
I am Rachael W. I am a Statistical Physics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, Massachusetts Institute of Technology, USA
I have been helping students with their homework for the past 6 years. I solve assignments related to Statistical.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Physics Assignments.
The document discusses recurrence relations and the Master Theorem for solving recurrences that arise from divide-and-conquer algorithms. It introduces recurrence relations and examples. It then explains the substitution method, iteration method, and Master Theorem for solving recurrences. The Master Theorem provides a "cookbook" for determining the running time of a divide-and-conquer algorithm where the problem of size n is divided into a subproblems of size n/b and the cost is f(n). It presents the three cases of the Master Theorem and works through examples of its application.
We use stochastic methods to present mathematically correct representation of the wave function. Informal construction was developed by R. Feynman. This approach were introduced first by H. Doss Sur une Resolution Stochastique de l'Equation de Schrödinger à Coefficients Analytiques. Communications in Mathematical Physics
October 1980, Volume 73, Issue 3, pp 247–264.
Primary intention is to discuss formal stochastic representation of the Schrodinger equation solution with its applications to the theory of demolition quantum measurements.
I will appreciate your comments.
The document outlines a stochastic approach to modeling warm inflation that accounts for both quantum and thermal fluctuations. It presents the Langevin-like equation of motion for the inflaton field that includes dissipation and thermal noise terms. Solutions to this equation are used to derive explicit expressions for the power spectra of density fluctuations due to thermal and quantum contributions. Results show that the model recovers cold inflation in the zero-temperature limit and warm inflation in the strong dissipation limit. Thermal fluctuations are also shown to make inflation models with steeper potentials compatible with observational constraints.
The 1741 Goldbach [1] made his most famous contribution to mathematics with the conjecture that all even numbers can be expressed as the sum of the two primes (currently Conjecture) referred to as “all even numbers greater than 2 can be expressed as the sum-two primes” (DOI:10.13140/RG.2.2.32893.69600/1)
This document discusses solving linear homogeneous recurrence relations with constant coefficients. It begins by defining such a recurrence relation as one where the terms are expressed as a linear combination of previous terms. It then explains that these types of relations can be solved by finding the characteristic roots of the characteristic equation. The document provides an example of solving a degree two recurrence relation and outlines the basic approach of finding a solution of the form an = rn. It also discusses solving coupled recurrence relations by eliminating variables to obtain a single recurrence relation that can be solved. Finally, it revisits the Martian DNA problem and shows its solution is a Fibonacci number.
A Proof of Hypothesis Riemann and it is proven that apply only for equation ζ(z)=0.Also here it turns out that it does not apply in General Case.(DOI:10.13140/RG.2.2.10888.34563/5)
I am Rachael W. I am a Statistical Physics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, Massachusetts Institute of Technology, USA
I have been helping students with their homework for the past 6 years. I solve assignments related to Statistical.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Physics Assignments.
The document discusses recurrence relations and the Master Theorem for solving recurrences that arise from divide-and-conquer algorithms. It introduces recurrence relations and examples. It then explains the substitution method, iteration method, and Master Theorem for solving recurrences. The Master Theorem provides a "cookbook" for determining the running time of a divide-and-conquer algorithm where the problem of size n is divided into a subproblems of size n/b and the cost is f(n). It presents the three cases of the Master Theorem and works through examples of its application.
We use stochastic methods to present mathematically correct representation of the wave function. Informal construction was developed by R. Feynman. This approach were introduced first by H. Doss Sur une Resolution Stochastique de l'Equation de Schrödinger à Coefficients Analytiques. Communications in Mathematical Physics
October 1980, Volume 73, Issue 3, pp 247–264.
Primary intention is to discuss formal stochastic representation of the Schrodinger equation solution with its applications to the theory of demolition quantum measurements.
I will appreciate your comments.
The document outlines a stochastic approach to modeling warm inflation that accounts for both quantum and thermal fluctuations. It presents the Langevin-like equation of motion for the inflaton field that includes dissipation and thermal noise terms. Solutions to this equation are used to derive explicit expressions for the power spectra of density fluctuations due to thermal and quantum contributions. Results show that the model recovers cold inflation in the zero-temperature limit and warm inflation in the strong dissipation limit. Thermal fluctuations are also shown to make inflation models with steeper potentials compatible with observational constraints.
This document provides an overview of key topics in mathematics including trigonometry, coordinate geometry, calculus, algebra, sequences and series, and permutations and combinations. It discusses important formulas and concepts for each topic, as well as strategies for understanding and solving problems. Key areas covered include trigonometric functions and their inverses, equations of circles, parabolas, ellipses and hyperbolas, limits, derivatives, integrals, complex numbers, and series.
Abstract: The Riemann zeta function is one of the most Leonhard Euler important and fascinating functions in mathematics. Analyzing the matter of conjecture of Riemann divide our analysis in the zeta function and in the proof of conjecture, which has consequences on the distribution of prime numbers.
This document discusses algorithms analysis and recurrence relations. It begins by defining recurrences as equations that describe a function in terms of its value on smaller inputs. Solving recurrences is important for determining an algorithm's actual running time. Several methods for solving recurrences are presented, including iteration, substitution, recursion trees, and the master method. Examples are provided to demonstrate each technique. Overall, the document provides an overview of recurrences and their analysis to determine algorithmic efficiency.
This document presents research on semi-periodic vectors for n-tuples (sets) of operators on Banach spaces. It introduces concepts such as the orbit of a vector under an n-tuple of operators and defines hypercyclic, fixed, and semi-periodic vectors. Theorems are presented to establish conditions under which an n-tuple satisfies the hypercyclicity criterion, including if it has a subset of semi-periodic vectors or a dense generalized kernel. The research was supported by a grant from the Islamic Azad University.
1. The document presents a new approach to proving comparison theorems for stochastic differential equations (SDEs) using differentiation of solutions with respect to initial data.
2. It proves that if the drift term of one SDE is always greater than or equal to the other, and their initial values satisfy the same relation, then the solutions will also satisfy this relation for all time.
3. Two methods are provided: the first uses explicit solutions, the second avoids this by showing the difference process cannot reach zero in finite time based on its behavior.
The document discusses recurrences and methods for solving them. It covers:
1) Divide-and-conquer algorithms can often be modeled with recurrences. Examples include merge-sort and matrix multiplication.
2) Common methods for solving recurrences are substitution, iteration/recursion trees, and the master method. The master method provides a general solution for recurrences of the form T(n) = aT(n/b) + nc.
3) Strassen's matrix multiplication algorithm improves on the naive O(n^3) time by using a recurrence with a=7 to achieve O(n^2.81) time via the master method. Changing variables can sometimes simplify recurrences.
A recurrence relation defines a sequence based on a rule that gives the next term as a function of previous terms. There are three main methods to solve recurrence relations: 1) repeated substitution, 2) recursion trees, and 3) the master method. Repeated substitution repeatedly substitutes the recursive function into itself until it is reduced to a non-recursive form. Recursion trees show the successive expansions of a recurrence using a tree structure. The master method provides rules to determine the time complexity of divide and conquer recurrences.
The document summarizes key concepts of quantum mechanics from chapters 3 and 4 of McQuarrie, including:
1) The Schrodinger equation and its solutions in 1D and 3D.
2) Solving the time-independent Schrodinger equation involves finding the general solution, applying boundary conditions, and normalizing the wavefunction.
3) Wavefunctions represent probability distributions, and the probability of finding a particle in a region is calculated by integrating the wavefunction.
The document discusses recurrence relations and methods for solving them. It defines a recurrence relation as an equation that expresses the terms of a sequence in terms of previous terms. It provides examples of homogeneous and non-homogeneous recurrence relations. For homogeneous relations, it describes guessing a solution of the form T(n)=x^n and finding the characteristic equation. For non-homogeneous relations, it explains adding the non-recursive term to the characteristic equation. It then works through examples of solving both homogeneous and non-homogeneous recurrence relations.
The document discusses solving recurrence relations through the following steps:
1) Write the recurrence relation and initial conditions as a characteristic equation.
2) Find the solutions to the characteristic equation.
3) Use the solutions to write the general solution as a linear combination with constants.
4) Determine the constants using the initial conditions.
Two examples are provided to demonstrate solving both regular and similar solutions to linear homogeneous recurrence relations.
Quantum mechanics and the square root of the Brownian motionMarco Frasca
The document discusses taking the square root of Brownian motion and how it relates to quantum mechanics. It shows that defining the square root through stochastic integration reproduces the heat kernel and Schrodinger's equation. This indicates the process is doing quantum mechanics. The approach is generalized to include potentials, deriving the harmonic oscillator case. Finally, using Dirac's algebra trick and introducing additional Brownian motions, the formalism reproduces the Dirac equation and introduces spin naturally through stochastic behavior.
A short remark on Feller’s square root condition.Ilya Gikhman
This document presents a proof of Feller's square root condition for the Cox-Ingersoll-Ross model of short interest rates.
The CIR model describes the dynamics of the short rate r(t) as a scalar SDE with parameters k, θ, and σ.
The theorem states that if the Feller condition 2kθ > σ^2 is satisfied, then there exists a unique positive solution r(t) on each finite time interval t ∈ [0, ∞).
The proof uses Ito's formula and Gronwall's inequality to show that as ε approaches 0, the probability that the solution falls below ε approaches 0 as well.
The one-dimensional wave equation governs vibrations of an elastic string. It is solved by separating variables, yielding solutions of the form F(x)G(t) where F and G satisfy ordinary differential equations. Boundary conditions require F(x) to be sinusoidal, with wavelengths that are integer multiples of the string length. The general solution is a superposition of these sinusoidal modes, with coefficients determined by the initial conditions. Motions of strings with different initial displacements are expressed as solutions to the one-dimensional wave equation.
This paper investigates conditions under which a pair of operators (T1, T2) acting on a Banach space is hereditarily hypercyclic. It proves three main theorems:
1) A pair (T1, T2) is hereditarily hypercyclic with respect to sequences {m(k,1)}, {m(k,2)} if and only if for any open sets U,V there exist M,N such that Tm1Tn2(U) intersects V for all m>M and n>N.
2) If a pair satisfies the hypercyclicity criterion and the differences between terms in the sequences {mk}, {nj} are bounded,
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
This document presents a common fixed point theorem for six self-maps (A, B, S, T, L, M) on a Menger space using the concept of weak compatibility. It proves that if the maps satisfy certain conditions, including being weakly compatible and their images being complete subspaces, then the maps have a unique common fixed point. The proof constructs sequences to show the maps have a coincidence point, then uses weak compatibility and lemmas to show this point is the unique common fixed point.
I am Stacy W. I am a Statistical Physics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, University of McGill, Canada
I have been helping students with their homework for the past 7years. I solve assignments related to Statistical.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Physics Assignments.
The document defines unit tangent and normal vectors for vector-valued functions. Specifically, it defines the unit tangent vector T(t) as the normalized derivative of the vector-valued function C(t). It also defines the principal normal vector N as the derivative of the unit tangent vector normalized. Finally, it introduces the binormal vector B as the cross product of T and N, forming the Frenet-Serret frame TNB at each point to describe the curve's geometry.
Brocard conjectured in 1904 that the only solutions to the diophantine equation 1! 2 - = mn are n = 4, 5, 7. The document provides the initial solutions for n = 4 and 5 by putting m = 2x+1 into the equation. It then generalizes the approach for n > 5 by putting m = 2k+1 and factorizing n! into terms with integers p and q. This leads to the equation 1 = q5 - p6, which has solutions for k = 1 (n = 7, m = 71) but no other values of k. Therefore, the only solutions to Brocard's conjecture are n = 4, 5, 7.
This document discusses applications of differential equations to model various engineering systems. It begins by introducing Newton's law of cooling and shows how to derive and solve the first-order differential equation that models it. Next, it examines modeling of electrical circuits using first-order differential equations. It then discusses modeling the dissolution of a spherical pill using a first-order equation. Finally, it considers second-order differential equations for modeling free and forced mechanical oscillations, deriving the general solution for different cases of damping.
This document discusses using recurrence relations to model problems involving counting techniques. It provides examples of modeling problems related to bacteria population growth, rabbit population growth, the Tower of Hanoi puzzle, and valid codeword enumeration. For each problem, it defines the recurrence relation and initial conditions, derives a closed-form solution, and proves its correctness using mathematical induction. Recurrence relations provide a way to define sequences and solve problems recursively by relating terms to previous terms in the sequence.
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.
This document defines and provides necessary and sufficient conditions for a pair of operators (T1, T2) on a topological vector space to be syndetically hypercyclic. It begins by introducing key concepts such as hypercyclic pairs and sequences. The main result is that a pair (T1, T2) is syndetically hypercyclic if and only if it satisfies the Hypercyclicity Criterion for syndetic sequences. Additionally, it is shown that if a pair satisfies the Hypercyclicity Criterion for a syndetic sequence, then it is topologically mixing. The proof of the main theorem utilizes the Hypercyclicity Criterion and shows topological weak mixing is equivalent to
This document provides an overview of key topics in mathematics including trigonometry, coordinate geometry, calculus, algebra, sequences and series, and permutations and combinations. It discusses important formulas and concepts for each topic, as well as strategies for understanding and solving problems. Key areas covered include trigonometric functions and their inverses, equations of circles, parabolas, ellipses and hyperbolas, limits, derivatives, integrals, complex numbers, and series.
Abstract: The Riemann zeta function is one of the most Leonhard Euler important and fascinating functions in mathematics. Analyzing the matter of conjecture of Riemann divide our analysis in the zeta function and in the proof of conjecture, which has consequences on the distribution of prime numbers.
This document discusses algorithms analysis and recurrence relations. It begins by defining recurrences as equations that describe a function in terms of its value on smaller inputs. Solving recurrences is important for determining an algorithm's actual running time. Several methods for solving recurrences are presented, including iteration, substitution, recursion trees, and the master method. Examples are provided to demonstrate each technique. Overall, the document provides an overview of recurrences and their analysis to determine algorithmic efficiency.
This document presents research on semi-periodic vectors for n-tuples (sets) of operators on Banach spaces. It introduces concepts such as the orbit of a vector under an n-tuple of operators and defines hypercyclic, fixed, and semi-periodic vectors. Theorems are presented to establish conditions under which an n-tuple satisfies the hypercyclicity criterion, including if it has a subset of semi-periodic vectors or a dense generalized kernel. The research was supported by a grant from the Islamic Azad University.
1. The document presents a new approach to proving comparison theorems for stochastic differential equations (SDEs) using differentiation of solutions with respect to initial data.
2. It proves that if the drift term of one SDE is always greater than or equal to the other, and their initial values satisfy the same relation, then the solutions will also satisfy this relation for all time.
3. Two methods are provided: the first uses explicit solutions, the second avoids this by showing the difference process cannot reach zero in finite time based on its behavior.
The document discusses recurrences and methods for solving them. It covers:
1) Divide-and-conquer algorithms can often be modeled with recurrences. Examples include merge-sort and matrix multiplication.
2) Common methods for solving recurrences are substitution, iteration/recursion trees, and the master method. The master method provides a general solution for recurrences of the form T(n) = aT(n/b) + nc.
3) Strassen's matrix multiplication algorithm improves on the naive O(n^3) time by using a recurrence with a=7 to achieve O(n^2.81) time via the master method. Changing variables can sometimes simplify recurrences.
A recurrence relation defines a sequence based on a rule that gives the next term as a function of previous terms. There are three main methods to solve recurrence relations: 1) repeated substitution, 2) recursion trees, and 3) the master method. Repeated substitution repeatedly substitutes the recursive function into itself until it is reduced to a non-recursive form. Recursion trees show the successive expansions of a recurrence using a tree structure. The master method provides rules to determine the time complexity of divide and conquer recurrences.
The document summarizes key concepts of quantum mechanics from chapters 3 and 4 of McQuarrie, including:
1) The Schrodinger equation and its solutions in 1D and 3D.
2) Solving the time-independent Schrodinger equation involves finding the general solution, applying boundary conditions, and normalizing the wavefunction.
3) Wavefunctions represent probability distributions, and the probability of finding a particle in a region is calculated by integrating the wavefunction.
The document discusses recurrence relations and methods for solving them. It defines a recurrence relation as an equation that expresses the terms of a sequence in terms of previous terms. It provides examples of homogeneous and non-homogeneous recurrence relations. For homogeneous relations, it describes guessing a solution of the form T(n)=x^n and finding the characteristic equation. For non-homogeneous relations, it explains adding the non-recursive term to the characteristic equation. It then works through examples of solving both homogeneous and non-homogeneous recurrence relations.
The document discusses solving recurrence relations through the following steps:
1) Write the recurrence relation and initial conditions as a characteristic equation.
2) Find the solutions to the characteristic equation.
3) Use the solutions to write the general solution as a linear combination with constants.
4) Determine the constants using the initial conditions.
Two examples are provided to demonstrate solving both regular and similar solutions to linear homogeneous recurrence relations.
Quantum mechanics and the square root of the Brownian motionMarco Frasca
The document discusses taking the square root of Brownian motion and how it relates to quantum mechanics. It shows that defining the square root through stochastic integration reproduces the heat kernel and Schrodinger's equation. This indicates the process is doing quantum mechanics. The approach is generalized to include potentials, deriving the harmonic oscillator case. Finally, using Dirac's algebra trick and introducing additional Brownian motions, the formalism reproduces the Dirac equation and introduces spin naturally through stochastic behavior.
A short remark on Feller’s square root condition.Ilya Gikhman
This document presents a proof of Feller's square root condition for the Cox-Ingersoll-Ross model of short interest rates.
The CIR model describes the dynamics of the short rate r(t) as a scalar SDE with parameters k, θ, and σ.
The theorem states that if the Feller condition 2kθ > σ^2 is satisfied, then there exists a unique positive solution r(t) on each finite time interval t ∈ [0, ∞).
The proof uses Ito's formula and Gronwall's inequality to show that as ε approaches 0, the probability that the solution falls below ε approaches 0 as well.
The one-dimensional wave equation governs vibrations of an elastic string. It is solved by separating variables, yielding solutions of the form F(x)G(t) where F and G satisfy ordinary differential equations. Boundary conditions require F(x) to be sinusoidal, with wavelengths that are integer multiples of the string length. The general solution is a superposition of these sinusoidal modes, with coefficients determined by the initial conditions. Motions of strings with different initial displacements are expressed as solutions to the one-dimensional wave equation.
This paper investigates conditions under which a pair of operators (T1, T2) acting on a Banach space is hereditarily hypercyclic. It proves three main theorems:
1) A pair (T1, T2) is hereditarily hypercyclic with respect to sequences {m(k,1)}, {m(k,2)} if and only if for any open sets U,V there exist M,N such that Tm1Tn2(U) intersects V for all m>M and n>N.
2) If a pair satisfies the hypercyclicity criterion and the differences between terms in the sequences {mk}, {nj} are bounded,
International Journal of Mathematics and Statistics Invention (IJMSI) inventionjournals
This document presents a common fixed point theorem for six self-maps (A, B, S, T, L, M) on a Menger space using the concept of weak compatibility. It proves that if the maps satisfy certain conditions, including being weakly compatible and their images being complete subspaces, then the maps have a unique common fixed point. The proof constructs sequences to show the maps have a coincidence point, then uses weak compatibility and lemmas to show this point is the unique common fixed point.
I am Stacy W. I am a Statistical Physics Assignment Expert at statisticsassignmenthelp.com. I hold a Masters in Statistics from, University of McGill, Canada
I have been helping students with their homework for the past 7years. I solve assignments related to Statistical.
Visit statisticsassignmenthelp.com or email info@statisticsassignmenthelp.com.
You can also call on +1 678 648 4277 for any assistance with Statistical Physics Assignments.
The document defines unit tangent and normal vectors for vector-valued functions. Specifically, it defines the unit tangent vector T(t) as the normalized derivative of the vector-valued function C(t). It also defines the principal normal vector N as the derivative of the unit tangent vector normalized. Finally, it introduces the binormal vector B as the cross product of T and N, forming the Frenet-Serret frame TNB at each point to describe the curve's geometry.
Brocard conjectured in 1904 that the only solutions to the diophantine equation 1! 2 - = mn are n = 4, 5, 7. The document provides the initial solutions for n = 4 and 5 by putting m = 2x+1 into the equation. It then generalizes the approach for n > 5 by putting m = 2k+1 and factorizing n! into terms with integers p and q. This leads to the equation 1 = q5 - p6, which has solutions for k = 1 (n = 7, m = 71) but no other values of k. Therefore, the only solutions to Brocard's conjecture are n = 4, 5, 7.
This document discusses applications of differential equations to model various engineering systems. It begins by introducing Newton's law of cooling and shows how to derive and solve the first-order differential equation that models it. Next, it examines modeling of electrical circuits using first-order differential equations. It then discusses modeling the dissolution of a spherical pill using a first-order equation. Finally, it considers second-order differential equations for modeling free and forced mechanical oscillations, deriving the general solution for different cases of damping.
This document discusses using recurrence relations to model problems involving counting techniques. It provides examples of modeling problems related to bacteria population growth, rabbit population growth, the Tower of Hanoi puzzle, and valid codeword enumeration. For each problem, it defines the recurrence relation and initial conditions, derives a closed-form solution, and proves its correctness using mathematical induction. Recurrence relations provide a way to define sequences and solve problems recursively by relating terms to previous terms in the sequence.
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.
This document defines and provides necessary and sufficient conditions for a pair of operators (T1, T2) on a topological vector space to be syndetically hypercyclic. It begins by introducing key concepts such as hypercyclic pairs and sequences. The main result is that a pair (T1, T2) is syndetically hypercyclic if and only if it satisfies the Hypercyclicity Criterion for syndetic sequences. Additionally, it is shown that if a pair satisfies the Hypercyclicity Criterion for a syndetic sequence, then it is topologically mixing. The proof of the main theorem utilizes the Hypercyclicity Criterion and shows topological weak mixing is equivalent to
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
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.
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if you are struggling with your Multiple Linear Regression homework, do not hesitate to seek help from our statistics homework help experts. We are here to guide you through the process and ensure that you understand the concept and the steps involved in performing the analysis. Contact us today and let us help you ace your Multiple Linear Regression homework!
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The document discusses recursive algorithms and recurrence relations. It provides examples of solving recurrence relations for different algorithms like Towers of Hanoi, selection sort, and merge sort. Recurrence relations define algorithms recursively in terms of smaller inputs. They are solved to find closed-form formulas for the running time of algorithms.
PaperNo14-Habibi-IJMA-n-Tuples and ChaoticityMezban Habibi
This document presents theorems and definitions related to n-tuples of operators on a Frechet space and conditions for chaoticity. It begins with definitions of key concepts such as the orbit of a vector under an n-tuple of operators and what it means for an n-tuple to be hypercyclic or for a vector to be periodic. The main results section presents two theorems, the first characterizing when an n-tuple satisfies the hypercyclicity criterion and the second proving conditions under which an n-tuple of weighted backward shifts is chaotic. The second theorem shows the equivalence of an n-tuple being chaotic, hypercyclic with a non-trivial periodic point, having a non-trivial periodic point, and a
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.
- The document summarizes Stirling's formula, which approximates n! asymptotically as n approaches infinity.
- It proves a weaker version showing nlogn is the right order of magnitude for log(n!).
- It then proves Stirling's formula precisely by using Euler's integral representation of n!, applying a change of variables to center the integrand at n, and showing the integrand converges to a Gaussian.
https://utilitasmathematica.com/index.php/Index
Utilitas Mathematica journal that publishes original research. This journal publishes mainly in areas of pure and applied mathematics, statistics and others like algebra, analysis, geometry, topology, number theory, diffrential equations, operations research, mathematical physics, computer science,mathematical economics.And it is official publication of Utilitas Mathematica Academy, Canada.
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.
PaperNo8-HabibiSafari-IJAM-CHAOTICITY OF A PAIR OF OPERATORSMezban Habibi
This document summarizes research on chaotic pairs of operators on Frechet spaces. It defines key concepts such as hypercyclic pairs and chaotic pairs of operators. The main results proven are:
1) A hypercyclicity criterion is presented to characterize when a pair of operators is hypercyclic.
2) Conditions are identified under which a pair of backward weighted shifts on an F-sequence space are chaotic, involving the convergence of certain series.
3) It is shown that for such pairs of backward weighted shifts, being chaotic is equivalent to being hypercyclic with a non-trivial periodic point.
linear transformation and rank nullity theorem Manthan Chavda
In these notes, I will present everything we know so far about linear transformations.
This material comes from sections in the book, and supplemental that
I talk about in class.
The document provides an outline for a course on quantum mechanics. It discusses key topics like the time-dependent Schrodinger equation, eigenvalues and eigenfunctions, boundary conditions for wave functions, and applications like the particle in a box model. Specific solutions to the Schrodinger equation are explored for stationary states with definite energy, including the wave function for a free particle and the quantization of energy for a particle confined to a one-dimensional box.
The lattice Boltzmann equation: background and boundary conditionsTim Reis
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Some Results on Common Fixed Point Theorems in Hilbert Space
1. www.ajms.com 66
RESEARCH ARTICLE
Some Results on Common Fixed Point Theorems in Hilbert Space
N. Seshagiri Rao1
, K. Kalyani2
1
Department of Applied Mathematics, School of Applied Natural Science, Adama Science and Technology
University, Post Box No. 1888, Adama, Ethiopia, 2
Department of Science and Humanities, Vignan’s Foundation
for Science, Technology and Research, Guntur, Andhra Pradesh, India
Received: 25-12-2019; Revised: 17-01-2020; Accepted: 31-01-2020
ABSTRACT
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.
Key words: Hilbert space, closed subset, Cauchy sequence, completeness.
Mathematics Subject Classification: 40A05, 47H10, 54H25.
INTRODUCTION
From the well-known Banach’s contraction principle, every contraction mapping of a complete metric
space into itself has a unique fixed point. This celebrated principle has played an important role in the
development of different results related to fixed points and approximation theory. Later, it has been
generalized by many authors[1-14]
in metric space by involving either ordinary or rational terms in the
inequalities presented there. Again, these results were extended by other or the same researchers[15-21]
in
different spaces. Thereafter, these results were again extended/developed for getting a unique fixed point
to a pair of continuous self/different mappings[22-26]
or more than two or a sequence of mappings, which
may be continues or non-continues or imposing some weaker conditions on the mappings in the same or
different spaces.[27-33]
As already stated, many authors have generalized Banach’s contraction principle through rational
expressions in different spaces such as metric, convex Banach’s space, and normed spaces. But still
here are some extensions/generalizations of this principle paying an attention in terms of taking rational
expression in Hilbert spaces.
In this paper, we have considered a pair of self-mappings T1
and T2
of a closed subset X of a Hilbert
space, satisfying certain rational inequalities and obtained a unique fixed point for both T1
and T2
. We
have developed this result to the pair T1
p
and T2
q
where p and q are some positive integers and then also
further extended the same results to a sequence of mappings. In each of the above three cases, we have
obtained common fixed point in X. These results are generalizations of Koparde and Waghmode[29]
in
Hilbert space.
MAIN RESULTS
Theorem 1: Let T1
and T2
be two self-mappings of a closed subset X of a Hilbert space satisfying the
inequality
Address for correspondence:
N. Seshagiri Rao
E-mail: seshu.namana@gmail.com
ISSN 2581-3463
2. Rao and Kalyani: Some common fixed theorems in Hilbert space
AJMS/Jan-Mar-2020/Vol 4/Issue 1 67
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 X
, ∈ and ≠
x y, where α β γ δ
, , , are positive real’s with 4 4 1
α β γ δ
+ + + . Then, T1
and T2
have a unique common fixed point in X.
Proof: Let us start with an arbitrary point x X
0 ∈ , define a sequence { }
xn as
x T x
1 1 0
= , x T x
2 2 1
= , x T x
3 1 2
= ,.........
In general,
x T x
n n
2 1 1 2
+ = , x T x
n n
2 2 2 2 1
+ +
= , for n = 0 1 2 3
, , , ,........
Next, we have to show that this sequence { }
xn is a Cauchy sequence in X. For this consider
x x T x T x
x T x x T x
n n n n
n n n n
2 1 2
2
1 2 2 2 1
2
2 2 2 1
2
2 1 1 2
2
1
1
+ −
− −
− = −
≤
− + −
α
+
+ −
+
− + −
+ −
+
−
− −
−
x x
x x x T x
x x
x
n n
n n n n
n n
2 2 1
2
2 2 1
2
2 2 2 1
2
2 2 1
2
2
1
1
β
γ n
n n n n n n
T x x T x x x
− − −
− + −
+ −
1 1 2
2
2 2 2 1
2
2 2 1
2
δ
⇒ − ≤
+ +
−
−
+ −
x x x x
n n n n
2 1 2
2
2 2 1
2
2
1 2
β γ δ
γ
A similar calculation indicates that
x x
x x
x x
n n
n n
n
2 2 2 1
2 2 1 2
2
2 1
2 2
1 2 1 2
+ +
+
+
− ≤
+ + + + −
− + − −
( ) ( )
( ) ( )
α β γ β δ
α β 2
2
2 2 1 2
2
n
n n
x x
+ −
Let k =
+ +
−
β γ δ
γ
2
1 2
and s n
x x
x x
n n
n n
( )
( ) ( )
( ) ( )
=
+ + + + −
− + − −
+
+
2 2
1 2 1 2
2 1 2
2
2 1 2
2
α β γ β δ
α β
, where s n
( ) depends n. Since
4 4 1
α β γ δ
+ + + , we see that k 1 and s n
( ) 1, for all n.
Let S s n n
= =
{ }
sup ( ) : , , ,......
0 1 2 and let 2
= { }
max ,
k S so that 0 1
, as a result of which we get
x x x x
n n n n
+ −
− ≤ −
1 1
Repeating the above process in a similar manner, we get
x x x x n
n n
n
+ − ≤ − ≥
1 1 0 1
,
On taking n → ∞, we obtain x x
n n
+ − →
1 0. Hence, it follows that the sequence{ }
xn is a Cauchy sequence.
However, X is a closed subset of Hilbert space and so by the completeness of X , there is a point ∈X
such that
→
n
x as n → ∞.
3. Rao and Kalyani: Some common fixed theorems in Hilbert space
AJMS/Jan-Mar-2020/Vol 4/Issue 1 68
Consequently, the sequences { } { }
x T x
n n
2 1 1 2
+ = and { } { }
x T x
n n
2 2 2 2 1
+ +
= converge to the same limit . Now,
we shall show that this is a common fixed point of both T1 and T2 .
For this, in view of the hypothesis note that
µ µ µ µ
µ α
µ
− = − + −
≤ − +
− + −
+ +
+
+ +
T x x T
x
T x x
n n
n
n n
1
2
2 2 2 2 1
2
2 2
2 2 2 1
2
2 1
1
( ) ( )
T
T
x
x T x
x
x
n
n n
n
1
2
2 1
2
2 1
2
2 2 1
2
2 1
2
1
1
1
µ
µ
β
µ µ
µ
γ
+ −
+
− + −
+ −
+
+
+ +
+
2
2 1 1
2
2 2 1
2
2 1
2
2 2 2 2 1
2
n n
n n n
T T x
x x x T
+ +
+ + +
− + −
+ − + − −
µ µ
δ µ µ µ
Letting n → ∞, we obtain
µ µ γ δ µ µ
− ≤ + −
T T
1
2
1
2
( ) ,
⇒ T1
μ =μ, since γ δ
+ 1,
Similarly, by considering
2
2
2 2 1 2 1 2
( ) ( )
+ +
− = − + −
n n
T x x T ,
we get 2
=
T . Thus, is a common fixed point of T1 and T2 .
Finally, to prove the uniqueness of a fixed point , let ( )
≠
v v be another fixed point of T1 and T2 . Then,
from the inequality, we obtain
µ µ α
µ µ
µ
β
µ µ
− = − ≤
− + −
+ −
+
− + −
v T T v
T v v T
v
v T v
2
1 2
2 2
2
1
2
2
2
2
2
1
1
1
1+
+ −
+ − + −
+ −
µ
γ µ µ δ µ
v
v T T v v
2
1
2
2
2 2
⇒ − ≤ + + + −
µ α β γ δ µ
v v
2 2
2
( )
v
⇒ =
, since α β γ δ
+ + +
2 1.
Hence, is a unique common fixed point of both T1 and T2 in X.
This completes the proof of the theorem.
Theorem 2: Let X be a closed subset of a Hilbert space and T1, T2 be two mappings on X itself satisfying
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 X
, ∈ and ≠
x y, where α β γ δ
, , , are positive real’s with 4 4 1
α β γ δ
+ + + and p q
, are
positive integers. Then, T1 and T2 have a unique common fixed point in X.
4. Rao and Kalyani: Some common fixed theorems in Hilbert space
AJMS/Jan-Mar-2020/Vol 4/Issue 1 69
Proof: From above Theorem-1, T p
1 and T q
2 have a unique common fixed point ∈X so that
1
=
p
T and 2
=
q
T .
Now, ( ) ( )
1 1 1 1 1
= =
p p
T T T T T
1
⇒ T is a fixed point of T p
1 .
But is a unique fixed point of T p
1 .
1
∴ =
T
Similarly, we get 2
=
T
∴ is a common fixed point of T1 and T2 .
For uniqueness, let v be another fixed point of T1 and T2 so that Tv T v v
1 2
= = . Then,
µ µ α
µ µ
µ
β
µ µ
− = − ≤
− + −
+ −
+
− + −
v T T v
T v v T
v
v T
p q
q p q
2
1 2
2 2
2
1
2
2
2
2
1
1
1 v
v
v
v T T v v
p q
2
2
1
2
2
2 2
1
+ −
+ − + −
+ −
µ
γ µ µ δ µ
⇒ − ≤ + + + −
µ α β γ δ µ
v v
2 2
2
( )
⇒ =
v, since α β γ δ
+ + +
2 1
Hence, is a unique common fixed point of T1 and T2 in X.
Complete the proof of the theorem.
In the following theorem, we have taken a sequence of mappings on a closed subset of a Hilbert space
converges pointwise to a limit mapping and show that if this limit mapping has a fixed point; then, this
fixed point is also the limit of fixed points of the mappings of the sequence.
Theorem 3: Let X be a closed subset of a Hilbert space and let { }
Ti be a sequence of mappings on X
into itself converging pointwise to T satisfying the following condition
T x T y
x T y y T x
x y
x y x T y
x y
i i
i i i
− ≤
− + −
+ −
+
− + −
+ −
2
2 2
2
2 2
2
1
1
1
1
α β
+
+ − + −
+ −
γ δ
y T x x T y x y
i i
2 2 2
for all x y X
, ∈ and ≠
x y, where α β γ δ
, , , are positive real’s with 4 4 1
α β γ δ
+ + + . If each Ti has a
fixed point i and T has a fixed point , then the sequence { }
n converges to μ.
Proof: Since i is a fixed point of Ti , then we have
µ µ µ µ µ µ µ µ
µ µ µ µ µ
− = − = − + −
≤ − + − +
n n n n n n n
n n n n
T T T T T T
T T T T T
2 2 2
2 2
2
( ) ( )
) −
− −
= − +
− + −
+ −
+
−
T T T
T T
T T
n n n n
n
n n n n
n
n
µ µ µ
µ µ α
µ µ µ µ
µ µ
β
µ µ
)
2
2 2
2
2
1
1
1
1
1
2
2
2
2 2 2
+ −
+ −
+ − + −
+ − + −
µ µ
µ µ
γ µ µ µ µ δ µ µ µ µ
T
T T T T T
n n
n
n n n n n n n
n n n
T
µ µ
−
5. Rao and Kalyani: Some common fixed theorems in Hilbert space
AJMS/Jan-Mar-2020/Vol 4/Issue 1 70
which implies that
µ µ µ µ α
µ µ µ µ
µ µ
β
µ µ µ µ
− = − +
− + −
+ −
+
− + −
n n
n n n n
n
n n n
T T
T T
T
2 2
2 2
2
2 2
1
1
1
+ −
+ − + −
+ − + − −
1
2
2
2 2
2
µ µ
γ µ µ µ µ
δ µ µ µ µ µ µ
n
n n n n
n n n n n
T T
T T T T
Letting n → ∞ so that
→
n
T T , 2 0
− − →
n n n n
T T T T , we get
lim lim
n
n
n
n
→∞ →∞
− ≤ + + +
( ) −
µ µ α β γ δ µ µ
2 2
2
lim 0
→∞
⇒ − =
n
n
, since α β γ δ
+ + +
2 1
⇒ →
n as n → ∞
This completes the proof.
The following theorem is the refinement of the above Theorem-1, by involving three rational terms in
the inequality.
Theorem 4: Let X be a closed subset of a Hilbert space and T1, T2 be two self-mappings on X
satisfying the following condition, then T1 and T2 have a unique common fixed point in X .
T x T y
x T y y T x
x y
y T x x T y
x
1 2
2 2
2
1
2
2
1
2
2
2
1
1
1
1
− ≤
− + −
+ −
+
− + −
+ −
α β
y
y
x y x T x
y T y
x y
2
2
1
2
2
2
2
1
1
+
− + −
+ −
+ −
γ δ
for all x y X
, ∈ and ≠
x y, where α β γ δ
, , , are positive real’s with 4 1
( )
α β γ δ
+ + + .
Proof: Let us construct a sequence { }
xn for an arbitrary point x X
0 ∈ as follows:
x T x
n n
2 1 1 2
+ = , x T x
n n
2 2 2 2 1
+ +
= for n = 0 1 2 3
, , , ,........
Then
x x T x T x
x T x x T x
n n n n
n n n n
2 1 2
2
1 2 2 2 1
2
2 2 2 1
2
2 1 1 2
2
1
1
+ −
− −
− = −
≤
− + −
α
+
+ −
+
− + −
+ −
+
−
− −
−
x x
x T x x T x
x x
n n
n n n n
n n
2 2 1
2
2 1 1 2
2
2 2 2 1
2
2 2 1
2
1
1
β
γ
x
x x x T x
x T x
x x
n n n n
n n
n n
2 2 1
2
2 1 2
2
2 1 2 2 1
2 2 2 1
2
1
1
− + −
+ −
+ −
−
− −
−
δ
which suggests that
x x s n x x
n n n n
2 1 2
2
2 2 1
2
+ −
− ≤ −
( ) , where s n
x x
x x
n n
n n
( )
( )
( ) ( )
=
+ + + −
− + − −
−
−
2
1 2 1
2 2 1
2
2 2 1
2
β γ δ δ
β γ
Similarly, we get
x x t n x x
n n n n
2 2 2 1
2
2 1 2
2
+ + +
− ≤ −
( ) , where t n
x x
x x
n n
n n
( )
( ) ( )
( )
=
+ + + + −
− + −
+
+
2
1 2
2 1 2
2
2 1 2
2
α γ δ γ δ
α
6. Rao and Kalyani: Some common fixed theorems in Hilbert space
AJMS/Jan-Mar-2020/Vol 4/Issue 1 71
Since 4 1
( )
α β γ δ
+ + + then both s n
( ) 1 and t n
( ) 1, for all n.
Let S s n n
= =
{ }
sup ( ) : , ,......
1 2 and T t n n
= =
{ }
sup ( ) : , ,......
1 2 .
Let 2
= { }
max ,
S T so that 0 1
. Hence, from above, we get
x x x x
n n n n
+ −
− ≤ −
1 1
Continuing the above process, we obtained
x x x x n
n n
n
+ − ≤ − ≥
1 1 0 1
,
Hence, the sequence { }
xn is a Cauchy sequence in X and so it converges to a limit in X . Since the
sequences { } { }
x T x
n n
2 1 1 2
+ = and { } { }
x T x
n n
2 2 2 2 1
+ +
= are subsequences of { }
xn , then { }
T x n
1 2 and { }
T x n
2 2 1
+
converges to the same . Now to see that this is a common fixed point of T1 and T2 . For this using the
inequality, we arrive at
µ µ µ µ
µ α
µ
− = − + −
≤ − +
− + −
+ +
+
+ +
T x x T
x
T x x
n n
n
n n
1
2
2 2 2 2 1
2
2 2
2 2 2 1
2
2 1
1
( ) ( )
T
T
x
x T T x
x
n
n n
n
1
2
2 1
2
2 1 1
2
2 2 1
2
2 1
2
1
1
1
µ
µ
β
µ µ
µ
+ −
+
− + −
+ −
+
+
+ +
+
γ
γ
µ µ µ
δ µ µ
− + −
+ −
+ − + −
+
+ +
+ +
x T
x T x
x x x
n
n n
n n
2 1
2
1
2
2 1 2 2 1
2
2 1
2
2 2
1
1
2 2
2 2 1
n T
+ − µ
On taking n → ∞, we obtained µ µ β µ µ
− ≤ −
T T
1
2
1
2
, because 0 1
, it follows immediately that
1
=
T .
Similarly, we get 2
=
T by considering,
2
2
2 2 1 2 1 2
( ) ( )
+ +
− = − + −
n n
T x x T
Thus, μ is a common fixed point of T1 and T2 .
Next, we want to show that is a unique fixed point of T1 and T2 . Let us suppose that ( )
≠
v v is also a
common fixed point of T1 and T2 . Then, in view of hypothesis, we have
µ α
µ µ
µ
β
µ µ
µ
γ
− ≤
− + −
+ −
+
− + −
+ −
+
v
T v v T
v
v T T v
v
2 2
2
1
2
2
1
2
2
2
2
1
1
1
1
µ
µ µ µ
δ µ
− + −
+ −
+ −
v T
v T v
v
2
1
2
2
2
2
1
1
⇒ − ≤ + + + −
µ α β γ δ µ
v v
2 2
( )
⇒ =
v, since α β γ δ
+ + + 1
It follows that = v
and so the common fixed point is unique.
7. Rao and Kalyani: Some common fixed theorems in Hilbert space
AJMS/Jan-Mar-2020/Vol 4/Issue 1 72
This completes the proof of the theorem.
Theorem 5: Let T1 and T2 be the self-mappings on a closed subset X of a Hilbert space satisfying
T x T y
x T y y T x
x y
y T x x T y
p q
q p p q
1 2
2 2
2
1
2
2
1
2
2
2
1
1
1
− ≤
− + −
+ −
+
− + −
α β
+ −
+
− + −
+ −
+ −
1
1
1
2
2
1
2
2
2
2
x y
x y x T x
y T y
x y
p
q
γ δ
for all x y X
, ∈ and ≠
x y, where α β γ δ
, , , are positive real’s with 4 1
( )
α β γ δ
+ + + and p q
, are
positive integers. Then, T1 and T2 have a unique common fixed point in X .
Proof: From above Theorem-3, T p
1 and T q
2 have a unique common fixed point ∈X so that 1
=
p
T
and 2
=
q
T .
Now, ( ) ( )
1 1 1 1 1
= =
p p
T T T T T
⇒ T1μ is a fixed point of T p
1 .
But is a unique fixed point of T p
1 .
1
∴ =
T
Similarly, we get 2
=
T
∴ is a common fixed point of T1 and T2 .
For uniqueness, let v be another fixed point of T1 andT2 , so we have 1 2
= =
Tv T v v. Then,
µ µ α
µ µ
µ
β
µ µ
− = − ≤
− + −
+ −
+
− + −
v T T v
T v v T
v
v T
p q
q p p
2
1 2
2 2
2
1
2
2
1
2
1
1
1 T
T v
v
v T
v T v
v
q
p
q
2
2
2
2
1
2
2
2
2
1
1
1
+ −
+
− + −
+ −
+ −
µ
γ
µ µ µ
δ µ
⇒ − ≤ + + + −
µ α β γ δ µ
v v
2 2
( )
⇒ =
v, since α β γ δ
+ + + 1
This completes the proof of the theorem.
In the following last theorem, we consider a sequence of mappings, which converges pointwise to a limit
mapping and shows that if this limit mapping has a fixed point, then this fixed point is also the limit of
fixed points of the mappings of the sequence.
Theorem 6: Let { }
Ti be a sequence of mappings of X into itself converging pointwise to T and let
T x T y
x T y y T x
x y
y T x x T y
x
i i
i i i i
− ≤
− + −
+ −
+
− + −
+ −
2
2 2
2
2 2
1
1
1
1
α β
y
y
x y x T x
y T y
x y
i
i
2
2 2
2
2
1
1
+
− + −
+ −
+ −
γ δ
8. Rao and Kalyani: Some common fixed theorems in Hilbert space
AJMS/Jan-Mar-2020/Vol 4/Issue 1 73
for all x y X
, ∈ and ≠
x y, where α β γ δ
, , , are positive real’s with 4 1
( )
α β γ δ
+ + + . If each Ti has a
fixed point i and T has a fixed point , then the sequence { }
n converges to in X .
Proof: We know that i is a fixed point of Ti then
µ µ µ µ
µ µ µ µ
µ µ µ µ µ
− = −
= − + −
≤ − + − +
n n n
n n n n
n n n n
T T
T T T T
T T T T T
2 2
2
2 2
2
( ) ( )
) −
− −
= − +
− + −
+ −
+
−
T T T
T T
T T
T
n n n n
n
n n n n
n
n n
µ µ µ
µ µ α
µ µ µ µ
µ µ
β
µ
)
2
2 2
2
1
1
µ
µ µ µ
µ µ
γ
µ µ µ µ
µ µ
δ µ µ
2 2
2
2 2
2
1
1
1
1
+ −
+ −
+
− + −
+ −
+ −
T T
T
n n
n
n n
n n n
n
2
2
2
+ − −
T T T T
n n n n
µ µ µ µ
Letting n → ∞ so that
→
n
T T , 2 0
− − →
n n n n
T T T T , we get
lim lim
n
n
n
n
→∞ →∞
− ≤ + + +
( ) −
µ µ α β γ δ µ µ
2 2
lim 0
→∞
⇒ − =
n
n
, since α β γ δ
+ + + 1
⇒ →
n as n → ∞
This completes the proof.
CONCLUSIONS
In this paper, Banach’s contraction principle has been generalized to obtain a common fixed point for
two, positive powers of two, and a sequence of self-mappings defined over a closed subset of a Hilbert
space. The existence and uniqueness of a common fixed point for three different types of self-mappings
are carried out in all six theorems with various inequalities involving square rational terms.
ACKNOWLEDGMENT
I am very thankful to my Professor G. M. Deheri for his valuable suggestions and constant support for
making me to get these fruitful results.
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