1) The document presents a fixed point theorem for a pair of random generalized non-linear contraction mappings involving four points of a separable Banach space.
2) It proves that if two random operators A1(w) and A2(w) satisfy a certain inequality involving upper semi-continuous functions, then there exists a unique random variable η(w) that is the common fixed point of A1(w) and A2(w).
3) As an example, the theorem is applied to prove the existence of a solution in a Banach space to a random non-linear integral equation of the form x(t;w) = h(t;w) + integral of k
Fixed points of contractive and Geraghty contraction mappings under the influ...IJERA Editor
In this paper, we prove the existence of fixed points of contractive and Geraghty contraction maps in complete metric spaces under the influence of altering distances. Our results extend and generalize some of the known results.
(α ψ)- Construction with q- function for coupled fixed pointAlexander Decker
This document presents a theorem to prove the existence of coupled fixed points for contractive mappings in partially ordered quasi-metric spaces. It begins with definitions of key concepts such as mixed monotone mappings, coupled fixed points, quasi-metric spaces, and Q-functions. It then states and proves a coupled fixed point theorem for mappings that satisfy an (α-Ψ)-contractive condition in a partially ordered, complete quasi-metric space with a Q-function. The theorem shows that if such a mapping F has the mixed monotone property and satisfies the contractive inequality, then F has at least one coupled fixed point.
This document presents some fixed point theorems for fuzzy mappings. It begins with introducing concepts related to fuzzy mappings such as fuzzy sets, α-level sets, approximate quantities, and fuzzy mappings. It then states some preliminary lemmas. The main results proved are:
1) A fixed point theorem for a fuzzy mapping T on a complete metric space X, showing that if T satisfies a contraction-type condition, then T has a fixed point.
2) A common fixed point theorem for a sequence of fuzzy mappings {Ti} on a complete metric space X, showing that if each Ti satisfies certain rational inequality conditions, then the mappings have a common fixed point.
The Multivariate Gaussian Probability DistributionPedro222284
The document discusses the multivariate Gaussian probability distribution. It defines the distribution and provides its probability density function. It then discusses various properties including: functions of Gaussian variables such as linear transformations and addition; the characteristic function and how to calculate moments; marginalization and conditional distributions. It also provides some tips and tricks for working with Gaussian distributions including how to calculate products.
This academic article presents a unique common fixed point theorem for four maps under contractive conditions in cone metric spaces. The authors prove the existence of coincidence points and a common fixed point theorem for four self-maps on a cone metric space that satisfy a contractive condition. They show that if one of the subspaces is complete, then the maps have a coincidence point, and if the maps are commuting, they have a unique common fixed point. This generalizes and improves on previous comparable results in the literature.
Fixed points of contractive and Geraghty contraction mappings under the influ...IJERA Editor
In this paper, we prove the existence of fixed points of contractive and Geraghty contraction maps in complete metric spaces under the influence of altering distances. Our results extend and generalize some of the known results.
(α ψ)- Construction with q- function for coupled fixed pointAlexander Decker
This document presents a theorem to prove the existence of coupled fixed points for contractive mappings in partially ordered quasi-metric spaces. It begins with definitions of key concepts such as mixed monotone mappings, coupled fixed points, quasi-metric spaces, and Q-functions. It then states and proves a coupled fixed point theorem for mappings that satisfy an (α-Ψ)-contractive condition in a partially ordered, complete quasi-metric space with a Q-function. The theorem shows that if such a mapping F has the mixed monotone property and satisfies the contractive inequality, then F has at least one coupled fixed point.
This document presents some fixed point theorems for fuzzy mappings. It begins with introducing concepts related to fuzzy mappings such as fuzzy sets, α-level sets, approximate quantities, and fuzzy mappings. It then states some preliminary lemmas. The main results proved are:
1) A fixed point theorem for a fuzzy mapping T on a complete metric space X, showing that if T satisfies a contraction-type condition, then T has a fixed point.
2) A common fixed point theorem for a sequence of fuzzy mappings {Ti} on a complete metric space X, showing that if each Ti satisfies certain rational inequality conditions, then the mappings have a common fixed point.
The Multivariate Gaussian Probability DistributionPedro222284
The document discusses the multivariate Gaussian probability distribution. It defines the distribution and provides its probability density function. It then discusses various properties including: functions of Gaussian variables such as linear transformations and addition; the characteristic function and how to calculate moments; marginalization and conditional distributions. It also provides some tips and tricks for working with Gaussian distributions including how to calculate products.
This academic article presents a unique common fixed point theorem for four maps under contractive conditions in cone metric spaces. The authors prove the existence of coincidence points and a common fixed point theorem for four self-maps on a cone metric space that satisfy a contractive condition. They show that if one of the subspaces is complete, then the maps have a coincidence point, and if the maps are commuting, they have a unique common fixed point. This generalizes and improves on previous comparable results in the literature.
The document summarizes existing research on establishing the existence and uniqueness of coupled fixed points for contraction mappings on partially ordered metric spaces. It presents several key theorems:
1) Theorems by Geraghty, Amini-Harandi and Emami, and Gnana Bhaskar and Lakshmikantham establish the existence of unique fixed points for contraction mappings on complete metric spaces and partially ordered metric spaces.
2) Choudhury and Kundu extended these results to Geraghty contractions by introducing an altering distance function.
3) GVR Babu and P. Subhashini further generalized the results to coupled fixed points for Geraghty contractions using an altering distance
The document discusses deep kernel learning, which combines deep learning and Gaussian processes (GPs). It briefly reviews the predictive equations and marginal likelihood for GPs, noting their computational requirements. GPs assume datasets with input vectors and target values, modeling the values as joint Gaussian distributions based on a mean function and covariance kernel. Predictive distributions for test points are also Gaussian. The goal of deep kernel learning is to leverage recent work on efficiently representing kernel functions to produce scalable deep kernels, allowing outperformance of standalone deep learning and GPs on various datasets.
The document summarizes Andrew Hone's work on the linearizability of the Heideman-Hogan family of maps. It describes the Heideman-Hogan recurrence relations that generate integer sequences, and presents the main theorem that these recurrences satisfy a linear relation. The proof uses reversibility and calculating iterations of the map both forwards and backwards. Open questions remain about the Poisson structure and relation to bi-Hamiltonian chains.
Fixed point theorem in fuzzy metric space with e.a propertyAlexander Decker
This document presents a theorem proving the existence of a common fixed point for four self-mappings (A, B, S, T) on a fuzzy metric space under certain conditions. Specifically:
1) The mappings satisfy containment and weakly compatible conditions, as well as property (E.A).
2) There exists a contractive inequality relating the mappings.
3) The range of one mapping (T) is a closed subspace.
Under these assumptions, the theorem proves the mappings have a unique common fixed point. The proof constructs sequences to show the mappings share a single fixed point. References at the end provide background on fuzzy metric spaces and related fixed point results.
This document presents a theorem proving the existence of a common fixed point for pairs of mappings in a fuzzy metric space under certain conditions. It begins with definitions of key concepts in fuzzy set theory and fuzzy metric spaces. It then states the main theorem, which shows that if two pairs of pointwise R-weakly commuting mappings satisfy certain continuity and contractive conditions, then they have a unique common fixed point. The proof constructs Cauchy sequences that converge to the common fixed point. Continuity of one mapping is used to establish connections between the limits of the sequences.
A common fixed point theorem in cone metric spacesAlexander Decker
This academic article summarizes a common fixed point theorem for continuous and asymptotically regular self-mappings on complete cone metric spaces. The theorem extends previous results to cone metric spaces, which generalize metric spaces by replacing real numbers with an ordered Banach space. It proves that under certain contractive conditions, the self-mapping has a unique fixed point. The proof constructs a Cauchy sequence that converges to the fixed point.
Some Common Fixed Point Results for Expansive Mappings in a Cone Metric SpaceIOSR Journals
The purpose of this work is to extend and generalize some common fixed point theorems for Expansive type mappings in complete cone metric spaces. We are attempting to generalize the several well- known recent results. Mathematical subject classification; 54H25, 47H10
This document describes a method called "k-means" for partitioning multivariate data into k groups based on minimizing within-group variance. It can be used for applications like classification, prediction, and testing independence among variables. The k-means process works by starting each group with a single data point, then iteratively assigning new points to the nearest group and updating group means. The document proves that the k-means values converge almost surely to a set of distinct points that minimize within-group variance. It also describes some applications and extensions of the k-means method.
This document summarizes part of a lecture on factor analysis from Andrew Ng's CS229 course. It begins by reviewing maximum likelihood estimation of Gaussian distributions and its issues when the number of data points n is smaller than the dimension d. It then introduces the factor analysis model, which models data x as coming from a latent lower-dimensional variable z through x = μ + Λz + ε, where ε is Gaussian noise. The EM algorithm is derived for estimating the parameters of this model.
This paper presents a common coupled fixed point theorem for two pairs of w-compatible self-mappings (F, G) and (f) in a metric space (X, d). The mappings satisfy a generalized rational contractive condition. The paper proves that if f(X) is a complete subspace of X, then F, G, and f have a unique common coupled fixed point of the form (u, u) in X × X. This result generalizes and improves previous related theorems by removing the completeness assumption on the entire space X. An example is also provided to support the usability of the theorem.
The existence of common fixed point theorems of generalized contractive mappi...Alexander Decker
The document presents a common fixed point theorem for a sequence of self maps satisfying a generalized contractive condition in a non-normal cone metric space. It begins with introducing concepts such as cone metric spaces, normal and non-normal cones, and generalized contraction mappings. It then proves the main theorem: if a sequence of self maps {Tn} on a complete cone metric space X satisfies a generalized contractive condition with constants α, β, γ, δ, η, μ ∈ [0,1] such that their sum is less than 1, and x0 ∈ X with xn = Tnxn-1, then the sequence {xn} converges to a unique common fixed point v of the maps
Fast and efficient exact synthesis of single qubit unitaries generated by cli...JamesMa54
The document describes a presentation on an algorithm for exact synthesis of single qubit unitaries generated by Clifford and T gates. The algorithm reduces the problem of implementing a unitary to the problem of state preparation. It then uses a series of HT gates to iteratively decrease the smallest denominator exponent of the state entries until it reaches a base case that can be looked up. The algorithm runs in time linear in the initial smallest denominator exponent and provides an optimal sequence of H and T gates for implementing the input unitary exactly.
This document discusses subspace clustering with missing data. It summarizes two algorithms for solving this problem: 1) an EM-type algorithm that formulates the problem probabilistically and iteratively estimates the subspace parameters using an EM approach. 2) A k-means form algorithm called k-GROUSE that alternates between assigning vectors to subspaces based on projection residuals and updating each subspace using incremental gradient descent on the Grassmannian manifold. It also discusses the sampling complexity results from a recent paper, showing subspace clustering is possible without an impractically large sample size.
Fixed point theorems for four mappings in fuzzy metric space using implicit r...Alexander Decker
This document presents theorems proving the existence and uniqueness of common fixed points for four mappings (A, B, S, T) in a fuzzy metric space using an implicit relation.
It begins with definitions of key concepts like fuzzy metric spaces, Cauchy sequences, completeness, compatibility, and occasionally weak compatibility of mappings.
The main result (Theorem 3.1) proves that if the pairs of mappings (A,S) and (B,T) are occasionally weakly compatible, and an implicit relation involving the fuzzy metric of images of x and y under the mappings is satisfied, then there exists a unique common fixed point w for A and S, and a unique common fixed point z for B and T.
The document classifies all possible Uq(sl2)-module algebra structures on the quantum plane. It produces a complete list of these structures and describes the isomorphism classes. The composition series of the representations under these structures are computed to classify the structures.
Solving the energy problem of helium final reportJamesMa54
The document discusses solving the ground state energy of a helium atom. It involves computing the Hamiltonian and overlap matrices (H and S) of the system by representing the wavefunction as a linear combination of basis functions. Computing the entries of H and S requires evaluating triple integrals over the internal coordinates of the atom. The main work is to derive a general closed form for these integrals. This involves repeatedly using integration by parts to reduce the exponents in the integrands, yielding sums of terms that can be directly evaluated or fed into computational software for further analysis. Solving these integrals is the crucial step to enable determining the ground state energy by solving the eigenvalue problem Hc = λSc.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
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.
This document provides notes for an optimization models course. It begins with an introduction to vectors and functions, including definitions of vectors, vector spaces, bases, dimensions, norms, and inner products. It then discusses related concepts like orthogonality, orthogonal complements, and projections. The document contains mathematical definitions, theorems, and examples to illustrate key concepts from linear algebra that form the foundation for optimization models.
Existence of Solutions of Fractional Neutral Integrodifferential Equations wi...inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
1) The document reviews concepts from probability and statistics including discrete and continuous random variables, their distributions (e.g. binomial, Poisson, normal), and multivariate distributions.
2) It then discusses key properties of multivariate normal distributions including their probability density function and how marginal and conditional distributions can be derived from the joint distribution.
3) Concepts like independence, mean vectors, covariance matrices, and their implications are also covered as they relate to multivariate normal distributions.
The document summarizes existing research on establishing the existence and uniqueness of coupled fixed points for contraction mappings on partially ordered metric spaces. It presents several key theorems:
1) Theorems by Geraghty, Amini-Harandi and Emami, and Gnana Bhaskar and Lakshmikantham establish the existence of unique fixed points for contraction mappings on complete metric spaces and partially ordered metric spaces.
2) Choudhury and Kundu extended these results to Geraghty contractions by introducing an altering distance function.
3) GVR Babu and P. Subhashini further generalized the results to coupled fixed points for Geraghty contractions using an altering distance
The document discusses deep kernel learning, which combines deep learning and Gaussian processes (GPs). It briefly reviews the predictive equations and marginal likelihood for GPs, noting their computational requirements. GPs assume datasets with input vectors and target values, modeling the values as joint Gaussian distributions based on a mean function and covariance kernel. Predictive distributions for test points are also Gaussian. The goal of deep kernel learning is to leverage recent work on efficiently representing kernel functions to produce scalable deep kernels, allowing outperformance of standalone deep learning and GPs on various datasets.
The document summarizes Andrew Hone's work on the linearizability of the Heideman-Hogan family of maps. It describes the Heideman-Hogan recurrence relations that generate integer sequences, and presents the main theorem that these recurrences satisfy a linear relation. The proof uses reversibility and calculating iterations of the map both forwards and backwards. Open questions remain about the Poisson structure and relation to bi-Hamiltonian chains.
Fixed point theorem in fuzzy metric space with e.a propertyAlexander Decker
This document presents a theorem proving the existence of a common fixed point for four self-mappings (A, B, S, T) on a fuzzy metric space under certain conditions. Specifically:
1) The mappings satisfy containment and weakly compatible conditions, as well as property (E.A).
2) There exists a contractive inequality relating the mappings.
3) The range of one mapping (T) is a closed subspace.
Under these assumptions, the theorem proves the mappings have a unique common fixed point. The proof constructs sequences to show the mappings share a single fixed point. References at the end provide background on fuzzy metric spaces and related fixed point results.
This document presents a theorem proving the existence of a common fixed point for pairs of mappings in a fuzzy metric space under certain conditions. It begins with definitions of key concepts in fuzzy set theory and fuzzy metric spaces. It then states the main theorem, which shows that if two pairs of pointwise R-weakly commuting mappings satisfy certain continuity and contractive conditions, then they have a unique common fixed point. The proof constructs Cauchy sequences that converge to the common fixed point. Continuity of one mapping is used to establish connections between the limits of the sequences.
A common fixed point theorem in cone metric spacesAlexander Decker
This academic article summarizes a common fixed point theorem for continuous and asymptotically regular self-mappings on complete cone metric spaces. The theorem extends previous results to cone metric spaces, which generalize metric spaces by replacing real numbers with an ordered Banach space. It proves that under certain contractive conditions, the self-mapping has a unique fixed point. The proof constructs a Cauchy sequence that converges to the fixed point.
Some Common Fixed Point Results for Expansive Mappings in a Cone Metric SpaceIOSR Journals
The purpose of this work is to extend and generalize some common fixed point theorems for Expansive type mappings in complete cone metric spaces. We are attempting to generalize the several well- known recent results. Mathematical subject classification; 54H25, 47H10
This document describes a method called "k-means" for partitioning multivariate data into k groups based on minimizing within-group variance. It can be used for applications like classification, prediction, and testing independence among variables. The k-means process works by starting each group with a single data point, then iteratively assigning new points to the nearest group and updating group means. The document proves that the k-means values converge almost surely to a set of distinct points that minimize within-group variance. It also describes some applications and extensions of the k-means method.
This document summarizes part of a lecture on factor analysis from Andrew Ng's CS229 course. It begins by reviewing maximum likelihood estimation of Gaussian distributions and its issues when the number of data points n is smaller than the dimension d. It then introduces the factor analysis model, which models data x as coming from a latent lower-dimensional variable z through x = μ + Λz + ε, where ε is Gaussian noise. The EM algorithm is derived for estimating the parameters of this model.
This paper presents a common coupled fixed point theorem for two pairs of w-compatible self-mappings (F, G) and (f) in a metric space (X, d). The mappings satisfy a generalized rational contractive condition. The paper proves that if f(X) is a complete subspace of X, then F, G, and f have a unique common coupled fixed point of the form (u, u) in X × X. This result generalizes and improves previous related theorems by removing the completeness assumption on the entire space X. An example is also provided to support the usability of the theorem.
The existence of common fixed point theorems of generalized contractive mappi...Alexander Decker
The document presents a common fixed point theorem for a sequence of self maps satisfying a generalized contractive condition in a non-normal cone metric space. It begins with introducing concepts such as cone metric spaces, normal and non-normal cones, and generalized contraction mappings. It then proves the main theorem: if a sequence of self maps {Tn} on a complete cone metric space X satisfies a generalized contractive condition with constants α, β, γ, δ, η, μ ∈ [0,1] such that their sum is less than 1, and x0 ∈ X with xn = Tnxn-1, then the sequence {xn} converges to a unique common fixed point v of the maps
Fast and efficient exact synthesis of single qubit unitaries generated by cli...JamesMa54
The document describes a presentation on an algorithm for exact synthesis of single qubit unitaries generated by Clifford and T gates. The algorithm reduces the problem of implementing a unitary to the problem of state preparation. It then uses a series of HT gates to iteratively decrease the smallest denominator exponent of the state entries until it reaches a base case that can be looked up. The algorithm runs in time linear in the initial smallest denominator exponent and provides an optimal sequence of H and T gates for implementing the input unitary exactly.
This document discusses subspace clustering with missing data. It summarizes two algorithms for solving this problem: 1) an EM-type algorithm that formulates the problem probabilistically and iteratively estimates the subspace parameters using an EM approach. 2) A k-means form algorithm called k-GROUSE that alternates between assigning vectors to subspaces based on projection residuals and updating each subspace using incremental gradient descent on the Grassmannian manifold. It also discusses the sampling complexity results from a recent paper, showing subspace clustering is possible without an impractically large sample size.
Fixed point theorems for four mappings in fuzzy metric space using implicit r...Alexander Decker
This document presents theorems proving the existence and uniqueness of common fixed points for four mappings (A, B, S, T) in a fuzzy metric space using an implicit relation.
It begins with definitions of key concepts like fuzzy metric spaces, Cauchy sequences, completeness, compatibility, and occasionally weak compatibility of mappings.
The main result (Theorem 3.1) proves that if the pairs of mappings (A,S) and (B,T) are occasionally weakly compatible, and an implicit relation involving the fuzzy metric of images of x and y under the mappings is satisfied, then there exists a unique common fixed point w for A and S, and a unique common fixed point z for B and T.
The document classifies all possible Uq(sl2)-module algebra structures on the quantum plane. It produces a complete list of these structures and describes the isomorphism classes. The composition series of the representations under these structures are computed to classify the structures.
Solving the energy problem of helium final reportJamesMa54
The document discusses solving the ground state energy of a helium atom. It involves computing the Hamiltonian and overlap matrices (H and S) of the system by representing the wavefunction as a linear combination of basis functions. Computing the entries of H and S requires evaluating triple integrals over the internal coordinates of the atom. The main work is to derive a general closed form for these integrals. This involves repeatedly using integration by parts to reduce the exponents in the integrands, yielding sums of terms that can be directly evaluated or fed into computational software for further analysis. Solving these integrals is the crucial step to enable determining the ground state energy by solving the eigenvalue problem Hc = λSc.
Research Inventy : International Journal of Engineering and Science is published by the group of young academic and industrial researchers with 12 Issues per year. It is an online as well as print version open access journal that provides rapid publication (monthly) of articles in all areas of the subject such as: civil, mechanical, chemical, electronic and computer engineering as well as production and information technology. The Journal welcomes the submission of manuscripts that meet the general criteria of significance and scientific excellence. Papers will be published by rapid process within 20 days after acceptance and peer review process takes only 7 days. All articles published in Research Inventy will be peer-reviewed.
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.
This document provides notes for an optimization models course. It begins with an introduction to vectors and functions, including definitions of vectors, vector spaces, bases, dimensions, norms, and inner products. It then discusses related concepts like orthogonality, orthogonal complements, and projections. The document contains mathematical definitions, theorems, and examples to illustrate key concepts from linear algebra that form the foundation for optimization models.
Existence of Solutions of Fractional Neutral Integrodifferential Equations wi...inventionjournals
International Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
1) The document reviews concepts from probability and statistics including discrete and continuous random variables, their distributions (e.g. binomial, Poisson, normal), and multivariate distributions.
2) It then discusses key properties of multivariate normal distributions including their probability density function and how marginal and conditional distributions can be derived from the joint distribution.
3) Concepts like independence, mean vectors, covariance matrices, and their implications are also covered as they relate to multivariate normal distributions.
Common Fixed Point Theorems in Compatible Mappings of Type (P*) of Generalize...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings
under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy
metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
Mathematics subject classifications: 45H10, 54H25
Unique fixed point theorems for generalized weakly contractive condition in o...Alexander Decker
This document summarizes a research paper that proves some new fixed point theorems for generalized weakly contractive mappings in ordered partial metric spaces. The paper extends previous theorems proved by Nashine and Altun in 2017. It presents definitions of partial metric spaces and properties. It proves a new fixed point theorem (Theorem 2.1) for nondecreasing mappings on ordered partial metric spaces that satisfy a generalized contractive condition. The theorem shows the mapping has a fixed point and the partial metric of the fixed point to itself is 0. It uses properties of partial metrics, contractive conditions and continuity to prove the sequence generated by iterating the mapping is Cauchy and converges.
The document discusses quantum mechanical concepts including:
1) The time derivative of the momentum expectation value satisfies an equation involving the potential gradient.
2) For an infinite potential well, the kinetic energy expectation value is proportional to n^2/a^2 and the potential energy expectation value vanishes.
3) Eigenfunctions of an eigenvalue problem under certain boundary conditions correspond to positive eigenvalues that are sums of squares of integer multiples of pi.
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.
1) Probability is defined as a set function that satisfies three axioms: non-negativity, the probability of the sample space is 1, and countable additivity.
2) Conditional probability is the probability of an event B given that event A has occurred, defined as P(B|A)=P(A∩B)/P(A). Events A and B are independent if P(B|A)=P(B) and P(A|B)=P(A).
3) Bayes' theorem gives the probability of an event A given that event B has occurred as P(A|B)=P(A)P(B|A)/P(B).
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.
Semicompatibility and fixed point theorem in fuzzy metric space using implici...eSAT Journals
This document summarizes a research paper on proving fixed point theorems in fuzzy metric spaces using the concept of semicompatibility and implicit relations. It defines key concepts such as fuzzy metric spaces, Cauchy sequences, completeness, compatibility, semicompatibility, weak compatibility, and implicit relations. It then proves a fixed point theorem for four self-mappings on a complete fuzzy metric space where the mappings satisfy conditions related to semicompatibility, weak compatibility, and an implicit relation.
Semicompatibility and fixed point theorem in fuzzy metric space using implici...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
COMMON FIXED POINT THEOREMS IN COMPATIBLE MAPPINGS OF TYPE (P*) OF GENERALIZE...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
COMMON FIXED POINT THEOREMS IN COMPATIBLE MAPPINGS OF TYPE (P*) OF GENERALIZE...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings under the conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy metric spaces.
COMMON FIXED POINT THEOREMS IN COMPATIBLE MAPPINGS OF TYPE (P*) OF GENERALIZE...mathsjournal
In this paper, we give some new definition of Compatible mappings of type (P), type (P-1) and type (P-2) in intuitionistic generalized fuzzy metric spaces and prove Common fixed point theorems for six mappings under the
conditions of compatible mappings of type (P-1) and type (P-2) in complete intuitionistic fuzzy metric spaces. Our results intuitionistically fuzzify the result of Muthuraj and Pandiselvi [15]
Mathematics subject classifications: 45H10, 54H25
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LECTURE # 3:
ABSTRACT RITZ-GALERKIN METHOD
MATH610: NUMERICAL METHODS FOR PDES:
RAYTCHO LAZAROV
1. Variational Formulation
In the previous lecture we have introduced the following space of functions
defined on (0, 1):
(1)
V =
v :
v(x) is continuous function on (0, 1);
v′(x) exists in generalized sense and in L2(0, 1);
v(0) = v(1) = 0
:= H10 (0, 1)
and equipped it with the L2 and H1 norms
‖v‖ = (v,v)1/2 and ‖v‖V = (v,v)
1/2
V =
(∫ 1
0
(u′2 + u2)dx
)1
2
.
We also introduced the following variational and minimization problems:
(V ) find u ∈ V such that a(u,v) = L(v), ∀ v ∈ V,
(M) find u ∈ V such that F(u) ≤ F(v), ∀ v ∈ V,
where a(u,v) is a bilinear form that is symmetric, coercive and contin-
uous on V and L(v) is continuous on V and F(v) = 1
2
a(u,u) −L(v).
As an example we can take
a(u,v) ≡
∫ 1
0
(k(x)u′v′ + q(x)uv) dx and L(v) ≡
∫ 1
0
f(x)v dx.
Here we have assumed that there are positive constants k0, k1, M such that
(2) k1 ≥ k(x) ≥ k0 > 0, M ≥ q(x) ≥ 0, f ∈ L2(0, 1).
These are sufficient for the symmetry, coercivity and continuity of the
bilinear form a(., .) and the continuity of the linear form L(v).
The proof of these properties follows from the following theorem:
Theorem 1. Let u ∈ V ≡ H10 (0, 1). Then the following inequalities are
valid:
(3)
|u(x)|2 ≤ C1
∫ 1
0
(u′(x))2dx for any x ∈ (0, 1),∫ 1
0
u2(x)dx ≤ C0
∫ 1
0
(u′(x))2dx.
with constants C0 and C1 that are independent of u.
1
2 MATH610: NUMERICAL METHODS FOR PDES: RAYTCHO LAZAROV
Proof: We give two proofs. The simple one proves the above inequali-
ties with C0 = 1/2 and C1 = 1. The better proof establishes the above
inequalities with C0 = 1/6 and C1 = 1/4.
Indeed, for any x ∈ (0, 1) we have:
u(x) = u(0) +
∫ x
0
u′(s)ds.
Since u ∈ H10 (0, 1) then u(0) = 0. We square this equality and apply
Cauchy-Swartz inequality:
(4) |u(x)|2 =
∣∣∣∫ x
0
u′(s)ds
∣∣∣2 ≤ ∫ x
0
1ds
∫ x
0
(u′(s))2ds ≤ x
∫ x
0
(u′(s))2ds.
Taking the maximal value of x on the right hand side of this inequality
w ...
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Fixed points theorem on a pair of random generalized non linear contractions
1. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
326
Fixed Points Theorem on A Pair of Random Generalized Non-
Linear Contractions
R.N.Yadava
Director, Patel Group Of Institutions Bhopal (M.P.), India
(dryadava@gmail.com)
Rajesh Shrivastava
Department of Mathematics
Govt Science & commerce college Benazir, Bhopal(M.P.), India
(Rajeshraju0101@rediffmail.com)
S.K.Pandey
Department of Mathematics
Millennium Group of Institutions Bhopal(M.P.), India
(Sudeeppandey1976@gmail.com)
ABSTRACT
A fixed point theorem for a pair of random generalized non-linear contraction mappings involving four points of
the space under consideration is proven. It is shown that this result includes the result of Lee and Padgett [1].
Also an application of the result is given.
Key words : Complete probability measure space, δ-algebras, Borel subsets, random operator, separable Banach
space, upper semi-continuous functions. Bochner integral,
Mathematics Subject classification: Primary 60H99, 47H10, and Secondary 60H 20.
1. Introduction and preliminaries
The idea of fixed points plays a very important role in solving deterministic operator equations. Recently the
idea of random fixed point theorems which are the stochastic generalization of the classical fixed point theorems
has become a very important part of the theory of some operator equations which can be regarded as random
operator equations. Many interesting results have been established by various authors (see for example Bharucha
– Reid [2], Hans [3], Padgett [4], Tsokos [5], Tsokas and padgett [6], Lee and padgett [7] in this area.
Recently, a fixed point theorem for a pair of generalized non-linear contraction mappings involving four
points of the space under consideration, which includes many well known results as special cases has been
established by Achari [8] (see also Achari [9], Pittanuer [10]).
The object of this paper is to study a stochastic version of a pair of generalized non-linear contraction
mappings of Achari [8]. Also it has been shown that this result generalizes the result of Lee and Padgett [1]. It is
interesting to note that with suitable modification of the conditions of the theorem, we can easily obtain
stochastic generalizations of the results of different classical fixed points. Finally, we apply theorem 2 to prove
the existence of a solution in a Banach space of a random non-linear integral equation of the form
x (t; w) = h (t; w) + ∫s k (t, s; w) ƒ(s, x (s; w) d, µ (s) ...... (1.1)
where s is a locally compact metric space with metric d defined on S X S, µ is a complete δ-finite measure
defined on the collection of Borel subsets of S and the integral is a Bochner integral.
In this section, we state some definitions as used by Lee and padgett [1]. Let (Ω, S. P) be complete probability
measure space, and Let (X, β) and (Y, C) be two measurable spaces, where X and Y are Banach spaces and B
and C are δ-algebras of Borel subsets of X and Y, respectively. First, we state the usual definitions of a Banach
space-valued random variable and of a random operator.
Definition 1.1 : A function V : Ω → X is said to be an X – valued random variable (Random element in X, or
generalized random variable)
if { ω ∈ Ω : v (w) ∈ B} ∈ S for each B ∈ β.
Definition 1.2 : A mapping T (w) : Ω x X → Y is said to be a random operator if y (w) = T (w) x is a Y-valued
random variable for every x ε X.
Definition 1.3 : Any X-valued random variable x (w) which satisfies the condition.
P ( { W : T (w) x (w) – y (w) } ) = 1
is said to be a random solution of the random operator equation
T (w) x = y (w).
Definition 1.4 : Let yj (w), j = 1, 2, ........, n be second order real valued random variables on a probability space
(Ω, S, P), that is yj (w) ∈ L2 (Ω, S, P). The collection of all n-component random vectors y' (w) = (y1 (w), ....., yn
(w)) constitutes a linear vector space if all equivalent random vectors are identified. Define the norm of y by
2. Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.6, 2013-Selected from International Conference on Recent Trends in Applied Sciences with Engineering Applications
327
( )½|||||||||| 2max
12
max
12 ∫ Ω<<<< == dpyyy jnjLjnj
n
L
The space of all n-component random vectors y with second order components and norm given by
n
2L
||.|| above
is separable Banach space and will be denoted by
n
2L (Ω, s, p) or
n
2L . Let S be locally compact metric space
with metric d defined on S X S and let µ be a complete δ-finite measure defined on the Borel subsets of S.
Definition 1.5 : We define the space C (S,
n
2L (Ω, s, p)) to be the space of all continuous functions from S into
n
2L (Ω, s, p) with the topology of uniform convergence on compacta. It may be noted that C (S,
n
2L (Ω, s, p)) is
locally convex space whose topology is defined by a countable family of saminorms given by –
|| x (t; w) ||j =
jct
sup
∈
|| x (t; w)
n
2L
|| , j = 1, 2, ......
Definition 1.6 : Let B and D be Banach spaces. The pair (B, D) is said to be admissible with respect to a random
operator U (w) if U (w) (B) ⊂D.
Definition 1.7 : A random operator T (w) on a Banach space x with domain D (T(w)) is said to be a random
generalized nonlinear contraction if there exists non-negative real-valued upper semi-continuous functions Ψi
(w), i=1, 2, ......, 7. Satisfying Ψi (w) (r) <
7
r
for r > 0, Ψi (w) (0) and such that
|| T (w) x1 – T (w) x2 || < Ψ1 (||x1-x2||) + Ψ2 (||x1 – T(w)x1 ||)
+ Ψ3 (||x2-T(w)x2||) + Ψ4 (||x1-T (w)x2 ||)
+ Ψ5 (||x2-T (w)x1||)
+ Ψ6 [||x1-T (w) x1 ||) + (||x1-T(w)x2||)]
1+[x1-T(w)x1||) (||x1-T(w)x2||)]
+ Ψ7 [||x2-T (w) x2 ||) + (||x2-T(w)x1||)]
1+[x2-T(w)x2||) (||x2-T(w)x1||)]
For all x1, x2 ∈ D (T (w)
2.Main results
A fixed point theorem for a pair of random generalized non-linear contraction.
Theorem 2.1 :- suppose A1 (w) and A2 (w) are a pair of random operators from a separable banach space x into
itself such that
||A1(w) x1-A2 (w) x2 ||
< Ψ1 (||x1-x2||) + Ψ2 (||x1-A1 (w)x3||) +Ψ3 (||x2-A2 (w) x4||
+ Ψ4 (||x1-A2 (w)x4||) + Ψ5 (||x2-A1 (w) x3||)
+ Ψ6
||)])((||||)([(||1
||)])((||||))([(||
421311
421311
xwAxxwAx
xwAxxwAx
−−+
−+−
+ Ψ7
||)]x)w(Ax(||||x)w(Ax[(||1
||)]x)w(Ax(||||)x)w(Ax[(||
312422
312422
−−+
−+−
............(2.1)
Where Ψi (w), i = 1, 2, ........... 7, are non-negative real-valued upper semi-continuous functions
satisfying Ψi (w) (r) ,
7
r
< for r > 0, Ψi = 0 (w) (o) and for all x1, x2, x3, x4 ∈ X. Then there exists an x – valued
random variable η (w) which is the unique common fixed point of A1 (w) and A2 (w).
Proof : Let x, y ∈ x and we define.
x1 = A2 (w)y, x2 = A1 (w) x, x3 = x, x4 = y
Then (2.1) takes the form
||A1 (w) A2 (w) y – A2 (w) A1 (w) x ||
< Ψ1 A1 (w) x - A2 (w) y ||)+ Ψ2 (|| A1 (w) x – A2 (w) y ||)
+ Ψ3 (||A1 (w) x – A2 (w) y ||)+ Ψ4 (||A2 (w) y – A2 (w) y||)
+ Ψ5 (||A1 (w) x – A1 (w) x ||)
+ Ψ6
||)]y)w(Ay)w(A[(||||]y)w(Ax)w(A[(||1
||)]y)w(Ay)w(A(||||)y)w(2Ax)w(A[(||
2121
221
−−+
−+−
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328
+ Ψ7
||)]x)w(Ax)w(A[(||||]y)w(Ax)w(A[(||1
||)]x)w(Ax)w(A(||||)y)w(Ax)w(A[(||
1121
1121
−−+
−+−
< Ψ1 (||A1 (w) x–A2 (w)y|| + Ψ2 [||A1(w) x – A2 (w) y ||)
+ Ψ3 (|| A1 (w) x - A2 (w) y||)+ Ψ6 (||A1 (w) x - A2 (w) y ||)
+ Ψ7 (||A1 (w) x- A2 (w) y ||) ..... (2.2)
Let x0 ∈ X be arbitrary and construct a sequence {xn) defined by
A1(w) xn-1 = xn, A2 (w) xn = xn+1, A1 (w) xn+1 = xn+2, n=1,2....
Let us put x = xn-1 and y = xn in (2.2), then we have
A1 (w) A2 (w) xn-A2(w) A1 (w) xn-1||< Ψ1 (||A1 (w) xn-1-A2 (w) xn||)
+ Ψ2 (||A1 (w) xn-1 - A2 (w) xn ||
+ Ψ3 (||A1 (w) xn-1 - A2 (w) xn||)
+ Ψ6 (||A1 (w) xn-1 - A2 (w) xn ||
+ Ψ7 (||A1 (w) xn-1 - A2 (w) xn||)
or
||xn+2 – xn+1|| < Ψ1 (||xn – xn+1||) + Ψ2 (||xn-xn+1||)
+ Ψ3 (||xn-xn+1||) + Ψ2 (||xn-xn+1||)
+ Ψ7 (||xn – xn+1 ||) ...... (2.3)
We take n to be even and set αn = ||xn-1 – xn||.
Then αn+2 = || xn+1 – xn+2||
< Ψ1 (||xn-xn+1||) + Ψ2 (||xn-xn+1)|| + Ψ3 (||xn – xn+1) ||
+ Ψ6 (|| xn-xn+1||)+ Ψ7 (||xn – xn+1||)
< Ψ1 (αn+1) + Ψ2 (αn+1) + Ψ3 (αn+1)
+ Ψ6 (αn+1) + Ψ7 (αn+1) ......... (2.4)
From (2.4) it is clear αn decrease with n and hence αn → α as n → ∞.
Let α > 0. Then since Ψ1 is upper semi-continuous, we obtain in the
limit as n → ∞
α < Ψ1 (α) + Ψ2 (α) + Ψ3 (α) + Ψ6 (α) + Ψ7 (α) < α
7
5
which is impossible unless α = 0
Now, we shall show that {xn} is a cauchy sequence. If not, then there is an ∈ > 0 and for all positive integers K,
there exist {m (k)} and {n (k)}with m (k) > n (k) > k, such that
dk = ||xm(k) – xn(k) || > ∈, .......... (2.5)
We may assume that - ||xm(k)-1
–
xn(k) || < ∈
by choosing m (k) to be the smallest number exceeding n (k) for which (2.5) holds. Then we have
dk < || xm(k) – xm(k)-1 || + ||xm(k)-1 – xn(k) ||
< αm(k) + ∈ < αk + ∈
which implies that dk → ∈ as k → ∞. Now the following cases are to be considered.
(i) m is even and n is odd,
(ii) m and n are both odd,
(iii) m is odd and n is even,
(iv) m and n are both even
i.e. (i) :
dk = ||xm-xn|| < ||xm – xm+1||+||xm+1 – xn+1|| + || xn-xn+1||
< αn+1 + αn+1 + ||A1 (w) xm – A2 (w) xn||
By putting x1 = xn, x2 = xm, x3 = xn-1, x4 = xm-1 in (2.1), we have
< αm+1 + αn+1 + Ψ1 (||xm-xn||) + Ψ2 (||xn-A1 (w) xn-1 ||)
+ Ψ3 (||xm-A2(w) xm-1||) + Ψ4 (|| xn-A2(w) xm-1||)
+ Ψ5 (||xm-A1(w) xn-1||)
+ Ψ6
||)]x)w(Ax[(||||)]x)w(Ax[(||1
||)x)w(Ax||||)x)w(Ax[(||
1m2n1n1n
1m2n1n1n
−−
−−
−−+
−+−
+ Ψ7
||)])([(||||)])([(||1
||))((||||))([(||
1112
1112
−−
−−
−−+
−+−
nmmm
nmmm
xwAxxwAx
xwAxxwAx
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< αm+1 + αn+1 +Ψ1 (dk) +Ψ4 (dk) + Ψ5 (dk) +Ψ6 (dk) + Ψ7 (dk) letting k → ∞ we have
∈ <
7
5
∈
This is contradiction if ∈ > 0. In the case (ii), we have
dk = || xm-xn|| < || xm - xm+1|| + ||xm+2 - xm+1||+||xm+2 - xn+1||+||xn-xn+1||
< αm+2 + αm+1 + αn+1 + || A2 (w) xm+1 – A1 (w) xn ||
By putting x1 = xn, x2 = xm+1, x3 = xn-1, x4 = xm in (3.1), we get
< αm+2 + αm+1+αn+1+Ψ1 (||xm+1 – xn||) + Ψ2 (||xn-A1 (w) xn-1|| )
+ Ψ3 (||xm+1-A2(w) xm||) + Ψ4 (|| xn-A2(w) xm ||)
+ Ψ5 (||xm-A1(w) xn-1||)
+ Ψ6
||)]x)w(Ax[(||||)]x)w(Ax[(||1
||)x)w(Ax||||)x)w(Ax[(||
m2n1n1n
m2n1n1n
−−+
−+−
−
−
+ Ψ7
||)]x)w(Ax[(||||)]x)w(Ax[(||1
||)x)w(Ax(||||)x)w(Ax[(||
1n1mm21m
1n1mm21m
−+
−+
−−+
−+−
< αm+2 + αm+1 + αn+1 + Ψ1 (dk+αm+1) + Ψ4 (dk + αm+1)
+ Ψ5 (dk+αm+1) +Ψ6 (dk+αm+1) + Ψ7 (dk+αm+1).
Letting k → ∞ in the above inequality we obtain ∈ <
7
5
∈, which is a contradiction if ∈ > 0. Similarly, the
cases (iii) and (iv) may be disposed of. This Leads us to conclude that {xn} is a cachy sequence. Let η (w) be the
limit of the sequence. We shall now show that
A1 (w) η (w) = η (w) = A2 (w) η (w). Putting x1 = xn-1, x2 = η (w), x3 = xn+1, x4 = xn, in (2.1), we get
|| A1 (w) xn-1 – A2 (w) η (w) ||
< Ψ1 (||xn-1 - η (w) ||) + Ψ2 (||xn-1 – A1 (w) xn+1||)
+ Ψ3 (|| η (w) – A2(w) xn ||) + Ψ4 (||xn-1 – A2 (w) xn||)
+ Ψ5 (|| η (w)–A1 (w) xn+1||) +
Ψ6
||)]x)w(Ax[(||||)]x)w(Ax[(||1
||)x)w(Ax(||||)x)w(Ax[(||
n21n1n11n
n21n1n11n
−−+
−+−
−+−
−+−
+ Ψ7
||)]x)w(A)w((||(||)x)w(A)w([(||1
||)]x)w(A)w((||||)x)w(A)w([(||
1n1n2
1n1n2
+
+
−η−η+
−η+−η
Letting n → ∞, we get || η(w)- A2 (w) η (w) || < 0 which is a contradiction and hence η (w) = A2 (w) η (w). In
the same, way, it is possible to show that η (w) = A1 (w) η (w). Thus η (w) is a common fixed point of A1 (w)
and A2 (w). Suppose there is another fixed point ξ (w) ≠ η (w) of A1 (w) and A2 (w). Then putting x1 = x4 = η (w)
and x2 = x4 = ξ (w) in (2.1), we have
||η (w) - ξ (w)|| < Ψ1 || η(w) - ξ (w) || + Ψ2 (||η (w) - ξ (w) ||
+ Ψ3 (||η(w)-ξ (w)|| + Ψ6 (||η(w)|| - ξ (w)||) + Ψ7 || (||η (w) - ξ (w)||)
<
7
5
|| η (w) - ξ (w) ||
which is contradiction. Hence η (w) = ξ (w). This completes the proof. If in theorem 2.1, we put A1 (w) = A2 (w)
= A (w) and x1 = x3 = x, x2 = x4 = y then we have the following theorem which we only state without proof.
Theorem 2.2 : If A (w) is a random generalized non-linear contraction from a separable Banach space x into
itself, then there exists an x – valued random variable which is the unique fixed point of A (w).
We now have the following corollary of theorem 2.2
Corollary 2.3 : If Ab
(w) is a random generalized contraction from x into itself for some positive integer b, then
A (w) has a unique fixed point η (w) which is an x – valued random variable.
Proof : Since Ab
(w) is a random generalized non-linear contraction operator on X, by theorem 2.2, there exists a
unique x – valued random variable η (w) such that
Ab
(w) η (w) = η (w)
We claim that A (w) η (w) = η (w). If not, consider
||Ab+1
(w) η (w) – Ab
(w) η (w) ||
< Ψ1 (||A (w) η (w) - η (w) ||)+ Ψ2 (||A (w) η (w) – Ab+1
(w) η (w) ||)
+ Ψ3 (||η (w) – Ab
(w) η (w)||)+ Ψ4 (||A (w) η (w) – Ab
(w) η (w)||)
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+ Ψ5 (||η (w) - Ab+1
(w) η (w)||)
+ Ψ6
||])w()w(A)w()w(A[||||])w()w(A)w()w(A[(||1
||])w()w(A)w()w(A(||||))w()w(A)w()w(A[(||
b1b
b1b
η−ηη−η+
η−η+η−η
+
+
+ Ψ7
||])w()w(A)w([||||])w()w(A)w([(||1
||])w()w(A)w((||||))w()w(A)w([(||
1bb
1bb
η−ηη−η+
η−η+η−η
+
+
.....(2.31)
Moreover, the left hand side of (2.31) is
|| Ab+1
(w) η (w) – Ab
(w) η (w)|| = ||A (w) η (w) - η (w)|| .... (2.32)
From (2.31) and (2.32), we have
||A (w) η (w) - η (w) ||
< Ψ1 (|| A (w) η (w) - η (w) || + Ψ4 (||A (w) η (w) - η (w)||
+ Ψ5 (||A (w) η (w) - η (w)||) + Ψ6 (||A (w) η (w) - η (w)||)
+ Ψ7 (||A (w) η (w) - η (w) ||)
which is, a contradiction and hence A (w) η (w) = η (w)
we remark that under the conditions A1 (w) = A2 (w) = A (w) and
x1 = x3 = x, x2 = x4 = y, Ψi = Ψ, Ψj (w) (r) = 0, j = 2, 3, 4, 5, 6, 7,
the theorem 2.1 reduces to the following corollary.
Corollary 2.4 : (Lee and padgett [1]) :- If A (w) is a random non-linear contraction operator from a separable
Banach space x into itself, then there exists an x – valued random variable η (w) which is the unique random
fixed point of A (w).
2.4 Application to a Random Non-linear integral Equation :-
In this section we give an application of a random non-linear integral equation. To do so we have
followed the steps of Lee and padgett [1] with necessary modifications as required for the more general settings.
We shall assume the following conditions concerning the random kernel k (t, s; w). The function k (., .; .) : S x S
x Ω → R is such that
(i) K (t, s; w) : S x S → L∞ (Ω, S, P) such that || k (t, s; w) || . || x (s; w) ||
n
2L is µ-integrable with respect to
s∈S for each t∈S and x ∈ C (s,
n
2L (Ω, s, p) where for each (t, S) ∈ s x S
||| k (t, s; w) ||| = || k (t, s; w) || L∞ (Ω,S,P) is the norm in L∞ (Ω,S,P);
(ii) For each s ∈ S, k (t, s; w) is continuous in t ∈ S from S into L∞ (Ω,S, P); for each t ∈ S, k (t, s; w) is
continuous in s ∈ S from s into L∞ (Ω, S, P) and
(iii) There exists a positive real-valued function H on S such that H (S) || x (s; w) ||
n
2L
is µ-integrable for x
∈ c (s,
n
2L (Ω, S, P) and such that for each t, s ∈ S.
||| K (t, u; w) – K (s, u; w) ||| . || x (u; w) ||
n
2L
< H (U) || x (u; w) || L
n
2
Thus, for each (t, s) ∈ S x S, we have K (t, s; w) x (s; w) ∈ L
n
2
(Ω, S, P). we now define the random integral
operator T (w) on c (s, L
n
2
(Ω, S, P) by
[T (w) x ] (t; w) = ∫S
k (t, s; w) x (s; w) d µ (s) .......... (2.41)
where the integral is a Bochner Integral. Moreover, we have that for each t ∈ S, [T (w) x] (t; w) ∈ L
n
2
(Ω, S, P) and that is a continuous linear operator from c (s, L
n
2
(Ω, S, P)) into itself. We now have the
following theorem.
Theorem 2.5 : We consider the stochastic integral equation (1.1) subject to the following conditions :
(a) B and D are Banach spaces stronger (cf. [1]) then C (s, L
n
2
(Ω, S, P) such that (B, D) is admissible
with respect to the integral operator defined by (4.1);
(b) x ( t; w) → f (t, x (t; w)) is an operator from the set Q (ρ) = { x (t; w) : x (t; w) ∈ D, || x (t; w) ||D < ρ }
into the space B satisfying.
|| f (t, x (t; w) – f (t, y (t; w)) ||B
< Ψ1 (w) (|| x (t; w) – y (t; w) ||D )+ Ψ2 (w) (|| x (t; w) – f(t, x (t; w))||D)
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+ Ψ3 (w) (||y (t; w) – t (t, y (t; w))||D)+ Ψ4 (w) (||x (t; w) – f (t, y (t; w))||D)
+ Ψ5 (w) (||y (t; w) – f (t, x (t; w))||D)
+ Ψ6
]||))w;t(y,t(f)w;t(x(||)||))w;t(x,t(f)w;t(x[(||1
]||))w;t(y,t(f)w;t(x(||)||))w;t(x,t(f)w;t(x[(||
DD
DD
−−+
−+−
+ Ψ7
]||))w;t(x,t(f)w;t(y(||)||))w;t(x,t(f)w;t(y[(||1
]||))w;t(x,t(f)w;t(y(||)||))w;t(y,t(f)w;t(y[(||
DD
DD
−−+
−+−
for x (t; w), y (t; w) ∈ Q (ρ), where Ψi (w), i = 1, 2, ......, 7 are non-negative real-valued upper semi-
continuous functions satisfying Ψi (w) (r) <
7
r
for r > 0 and Ψi (w) (0) = O;
(c) h (t; w) ∈ D.
Then there exists a unique random solution of (1.1) in Q (ρ), provided c (w) < 1 and ||h (t; w) ||D + 2c (w) || f (t;
o)||B < ρ (1- c (w))
where c (w) is the norm of T (w).
Proof : Define the operator U (w) from Q (ρ) into D by
[U (w) x] (t; w) = h (t; w) + ∫S
k (t, s; w) f (s; w) dµ (s).
Now ||[U(w) x ] (t; w)||D < ||h (t; w) ||D + c (w)|| f (t, x (t; w) ||B
< ||(t;w)||D + c (w) ||f (t; 0)||B + c (w)|| f (t, x (t; w) – f (t; 0) ||B.
Then from the conditions of the theorem.
c (w) || f (t, x (t; w)) – f (t, 0)||B
< C (w) [Ψ1 (w) (||x (t; w)||D)]+ Ψ2 (w) (||x(t;w) – f (t, x (t; w)||D)
+ Ψ3 (w) ||f (t; o)||D)+ Ψ4 (w)(||x(t;w)||D) + Ψ5 (w) (|| f (t, x (t; w))||D)
+Ψ6
)]||)w;t(x[(||]||)w;t(x,t(f)w;t(x[(||1
)]||)w;t(x(||)||)w;t(x,t(f)w;t(x[(||
DD
DD
−+
+−
+Ψ7
)]||)w;t(x,t(f(||)||)0;t(f[(||1
)]||)w;t(x,t(f(||)||)0;t(f[(||
DD
DD
+
+
i.e.
7
5
cw|| f(t, x (t; w))–f (t; o)||B <
7
5
c (w) ρ +
7
5
c (w)||f (t; 0)||B
Hence
|| [ U (w)x ] (t; w)||D < ||h (t; w)||D + 2 c (w)|| f (t; 0)||B + c (w) ρ
< ρ (1- c(w)) + c (w) ρ
< ρ
Hence, [ U (w) x ] (t; w) ∈ Q (ρ)
Now, for x (t; w), y (t; w) ∈ Q (ρ) We have by condition (b)
|| [U (w) x ] (t; w) – [U (w)y] (t; w) ||D
= || ∫S
K (t, s; w) [f s, (x, w)) – f (s, y (s; w))] dµ (s) ||D
< c (w)|| f (t, x (t; w)) – f (t, y (t;w))||B
< c (w) [Ψ1 (|| x (t; w) – y (t; w)||D]+ Ψ2 (w) (|| x (t; w) – f (t, x(t, w)||D)
+ Ψ3 (w) (||y (t;w) – f (t, y (t; w)) ||D)+ Ψ4 (w) (||x (t; w) – f (t, y (t;w))||D)
+ Ψ5 (w) (||y (t; w) – f (t, x (t; w))||D)
+ Ψ6
)]||))w;t(y,t(f)w;t(x[(||)]||))w;t(x,t(f)w;t(x[(||1
)||))w;t(y,t(f)w;t(x(||)||))w;t(x,t(f)w;t(x[(||
DD
DD
−−+
−+−
+ Ψ7
)]||))w;t(x,t(f)w;t(y[(||)]||))w;t(y,t(f)w;t(y[(||1
)||))w;t(x,t(f)w;t(y(||)||))w;t(y,t(f)w;t(y[(||
DD
DD
−−+
−+−
< Ψ1 (w) (||x (t; w) – y (t; w))||D) + Ψ2 (w) (||x (t; w)-f (t, x (t; w))||D)
+ Ψ3 (w) (||y (t; w) – f (t, y (t;w)||D) + Ψ4 (w) (||x (t;w) – f (t, y (t;w))||D)
+ Ψ5 (w) (||y (t;w) – f (t, x (t; w))||D)
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+ Ψ6
)]||))w;t(y,t(f)w;t(x[(||)]||))w;t(x,t(f)w;t(x[(||1
)||))w;t(y,t(f)w;t(x(||)||))w;t(x,t(f)w;t(x[(||
DD
DD
−−+
−+−
+Ψ7
)]||))w;t(x,t(f)w;t(y[(||)]||))w;t(y,t(f)w;t(y[(||1
)||))w;t(x,t(f)w;t(y(||)||))w;t(y,t(f)w;t(y[(||
DD
DD
−−+
−+−
Since c (w) < 1. Thus U (w) is a random non-linear contraction operator on Q (ρ). Hence, by theorem 2.2 there
exists a unique x – valued random variable x* (t; w) ∈ Q (ρ) which is a fixed point of U (w), that is x * (t; w) is
the unique random solution of the equation (1.1).
REFERENCES
1. Lee, Arthur C.H., Padgett, W.J. “On random non-linear contraction”, math. systems theory 11 (1977),
77-84.
2. Bharucha – Reid, A.T. “Random Integral Equations”, Academic Press, New York (1972).
3. Hans, O. “Random fixed point-theorems”, Trans. First prague conf. on information theory, statist.
decision functions on random process, (1957), 105-125.
4. Padgett, W.J. “On a non-linear stochastic integral equations of Hammerstein type”, proc. Amer. Math.
Soc. 38 (1973), 625-631.
5. Tsokos, C.P. “On a stochastic integral equation of voltera type, math”, systems theory 3 (1969), 222-
231.
6. Tsokos, C.P. , Padgett, W.J. “Random Integral equations with applications in life science and
engineering”, Academic Press, New York (1974).
7. Lee, A.C.H. , Padgett, W.J. “On heavily non-linear stochastic integral equations”, utilitas mathematics 9
(1976), 128, 138.
8. Achari, J. “A result on fixed points, kyungpook math”. Journal 19 (1979), 245-248.
10. Achari, J. “On a pair of Random generalized non-linear contractions”, Internal J. Math. and math. sci.
vol. 6 No. 3 (1983) 467-475.
11. Pittnaur, F. “A fixed point theorem in complete metric spaces, to appear in periodica math”. Hungarica.
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