This document summarizes research on generalized classical Fréchet derivatives in real Banach spaces. It begins by reviewing basic definitions and results for Fréchet derivatives, including the chain rule, mean value theorem, and Taylor's formula. It then proves several theorems on generalized Fréchet derivatives, including a generalized chain rule, mean value theorem, and implicit function theorem. It also presents a generalized Taylor's formula for nth order Fréchet differentiable functions. The proofs of the main results on generalized Fréchet derivatives are provided.
On Generalized Classical Fréchet Derivatives in the Real Banach SpaceBRNSS Publication Hub
This document summarizes research on generalized classical Fréchet derivatives in real Banach spaces. It begins by reviewing basic definitions and results for Fréchet derivatives, including the chain rule, mean value theorem, and Taylor's formula. It then proves that Fréchet derivatives exist and are continuous in real Banach spaces. The main results generalize the chain rule, mean value theorem, and Taylor's formula to higher order Fréchet derivatives in real Banach spaces. Proofs are provided for the generalized chain rule and other theorems.
This document summarizes a research article that reviews integration in Banach spaces, with a focus on the Bochner integral, generalized derivatives, and generalized gradients. It presents key definitions related to strongly measurable functions, the Bochner integral, Lp spaces of functions from an interval to a Banach space, generalized derivatives, and monotone operators. It also states theorems regarding Banach and Hilbert space properties, generalized derivative properties under weak convergence, and finite extensions of the Bochner integral for sums and products. The main results develop these finite extensions of the Bochner integral.
This document presents several theorems regarding the existence, uniqueness, and extendibility of solutions to ordinary differential equation systems. It begins by introducing Peano's theorem on existence of solutions and Picard-Lindelof theorem on uniqueness of solutions locally. It then discusses extension theorems which aim to extend local solutions to open connected domains. The main results section discusses using extension theorems to show existence of solutions on the entire positive real line by extending solutions from finite intervals to infinity.
This document discusses a theory solver for the theory of uninterpreted functions (UF) in satisfiability modulo theories (SMT). It presents the key components of a UF solver, including union-find algorithms to handle equalities, congruence closure to handle functions, and computing theory conflicts. The solver decides satisfiability of UF formulas in incremental, backtrackable, and theory-propagating manner. It can also be used as a base layer for other theory solvers like LRA.
On The Generalized Topological Set Extension Results Using The Cluster Point ...BRNSS Publication Hub
In this work, we seek generalized finite extensions for a set of real numbers in the topological space through the cluster point approach. Basically, we know that in the topological space, a point is said to be a cluster point of a subset X if and only if every open set containing the point say x contains another point of x1 different from x. This concept with the aid basic known ideas on set theory was carefully used in the definition of linear, radial, and circular types of operators which played the major roles in realizing generalized extension results as in our main results of section three.
This document provides an overview of preliminary topological concepts needed for applied mathematics. It defines topological spaces and metric spaces, and introduces key topological notions like open and closed sets, bases for topologies, convergence of sequences, accumulation points, interior and closure of sets, and dense sets. Metric spaces are shown to induce a natural topological structure, though not all topologies come from a metric. Examples are provided to illustrate various definitions and properties.
1) The document presents results on generalized topological set extensions using the cluster point approach. It defines key concepts like cluster points, interior points, and closure.
2) It introduces three types of topological operators - linear, radial, and circular - and proves extension theorems using these operators.
3) The main result (Theorem 3.1) proves that if the cluster points of a set X form a nonempty set, then the closure of X is equal to the union of X and its cluster points.
The document discusses concepts related to partial differentiation and its applications. It covers topics like tangent planes, linear approximations, differentials, Taylor expansions, maxima and minima problems, and the Lagrange method. Specifically, it defines the tangent plane to a surface at a point using partial derivatives, describes how to find the linear approximation of functions, and explains how to find maximum and minimum values of functions using critical points and the second derivative test.
On Generalized Classical Fréchet Derivatives in the Real Banach SpaceBRNSS Publication Hub
This document summarizes research on generalized classical Fréchet derivatives in real Banach spaces. It begins by reviewing basic definitions and results for Fréchet derivatives, including the chain rule, mean value theorem, and Taylor's formula. It then proves that Fréchet derivatives exist and are continuous in real Banach spaces. The main results generalize the chain rule, mean value theorem, and Taylor's formula to higher order Fréchet derivatives in real Banach spaces. Proofs are provided for the generalized chain rule and other theorems.
This document summarizes a research article that reviews integration in Banach spaces, with a focus on the Bochner integral, generalized derivatives, and generalized gradients. It presents key definitions related to strongly measurable functions, the Bochner integral, Lp spaces of functions from an interval to a Banach space, generalized derivatives, and monotone operators. It also states theorems regarding Banach and Hilbert space properties, generalized derivative properties under weak convergence, and finite extensions of the Bochner integral for sums and products. The main results develop these finite extensions of the Bochner integral.
This document presents several theorems regarding the existence, uniqueness, and extendibility of solutions to ordinary differential equation systems. It begins by introducing Peano's theorem on existence of solutions and Picard-Lindelof theorem on uniqueness of solutions locally. It then discusses extension theorems which aim to extend local solutions to open connected domains. The main results section discusses using extension theorems to show existence of solutions on the entire positive real line by extending solutions from finite intervals to infinity.
This document discusses a theory solver for the theory of uninterpreted functions (UF) in satisfiability modulo theories (SMT). It presents the key components of a UF solver, including union-find algorithms to handle equalities, congruence closure to handle functions, and computing theory conflicts. The solver decides satisfiability of UF formulas in incremental, backtrackable, and theory-propagating manner. It can also be used as a base layer for other theory solvers like LRA.
On The Generalized Topological Set Extension Results Using The Cluster Point ...BRNSS Publication Hub
In this work, we seek generalized finite extensions for a set of real numbers in the topological space through the cluster point approach. Basically, we know that in the topological space, a point is said to be a cluster point of a subset X if and only if every open set containing the point say x contains another point of x1 different from x. This concept with the aid basic known ideas on set theory was carefully used in the definition of linear, radial, and circular types of operators which played the major roles in realizing generalized extension results as in our main results of section three.
This document provides an overview of preliminary topological concepts needed for applied mathematics. It defines topological spaces and metric spaces, and introduces key topological notions like open and closed sets, bases for topologies, convergence of sequences, accumulation points, interior and closure of sets, and dense sets. Metric spaces are shown to induce a natural topological structure, though not all topologies come from a metric. Examples are provided to illustrate various definitions and properties.
1) The document presents results on generalized topological set extensions using the cluster point approach. It defines key concepts like cluster points, interior points, and closure.
2) It introduces three types of topological operators - linear, radial, and circular - and proves extension theorems using these operators.
3) The main result (Theorem 3.1) proves that if the cluster points of a set X form a nonempty set, then the closure of X is equal to the union of X and its cluster points.
The document discusses concepts related to partial differentiation and its applications. It covers topics like tangent planes, linear approximations, differentials, Taylor expansions, maxima and minima problems, and the Lagrange method. Specifically, it defines the tangent plane to a surface at a point using partial derivatives, describes how to find the linear approximation of functions, and explains how to find maximum and minimum values of functions using critical points and the second derivative test.
This document is a project report submitted by a student for a Bachelor of Science degree in Mathematics. It provides an introduction to the field of topology. The report includes a title page, certificate, declaration, acknowledgements, table of contents, and several chapters. The introduction defines topology and provides some examples. It states that the project aims to provide a thorough grounding in general topology.
A Solution Manual And Notes For Kalman Filtering Theory And Practice Using ...Daniel Wachtel
This document contains notes and solutions for a book on Kalman filtering. It includes notes on chapters explaining linear dynamic systems and derivations of fundamental solutions and state transition matrices for differential equations like dy/dt=0 and d2y/dt2=0. The author provides complete solutions to problems at the end of chapters and welcomes feedback to improve the notes.
On Analytic Review of Hahn–Banach Extension Results with Some GeneralizationsBRNSS Publication Hub
The useful Hahn–Banach theorem in functional analysis has significantly been in use for many years ago. At this point in time, we discover that its domain and range of existence can be extended point wisely so as to secure a wider range of extendibility. In achieving this, we initially reviewed the existing traditional Hahn–Banach extension theorem, before we carefully and successfully used it to generate the finite extension form as in main results of section three.
Existance Theory for First Order Nonlinear Random Dfferential Equartioninventionjournals
In this paper, the existence of a solution of nonlinear random differential equation of first order is proved under Caratheodory condition by using suitable fixed point theorem. 2000 Mathematics Subject Classification: 34F05, 47H10, 47H4
1. The chain rule describes how to take the derivative of a function composed of other functions. It states that if z = f(x, y) and x and y are functions of t, then the derivative of z with respect to t is the sum of the partial derivatives multiplied by the derivatives of x and y with respect to t.
2. The chain rule is applied to find the derivative of two example functions, w = xy and w = xy + z, with respect to t along given paths for x, y, and z in terms of t.
3. The chain rule is generalized to the case of a function u of n variables, where each variable is a function of m other variables
This document summarizes research on extending the domain of fixed point solutions to ordinary differential equations. It begins with definitions of fixed points and extendability. The Peano and Picard-Lindelof theorems are used to prove local existence and uniqueness of fixed points. Several theorems are presented to establish conditions under which a fixed point defined on a finite interval can be extended to the endpoints of that interval or beyond. Specifically, if the vector field is bounded and limits exist at the endpoints, the fixed point can be extended. The results ensure extendability to infinite intervals under boundedness assumptions. Extendability is key to studying existence of fixed points on the half line R+.
This document summarizes research on extending the domain of fixed point solutions to ordinary differential equations. It begins with definitions of fixed points and extendability. The Peano and Picard-Lindelof theorems are used to prove local existence and uniqueness of fixed points. Several theorems are presented to establish conditions under which a fixed point defined on a finite interval can be extended to the endpoints of the interval. Specifically, if the vector field is bounded and limits exist at the endpoints, the fixed point can be extended. The results are applied to extend fixed points defined on finite intervals to the entire positive real line.
1) The document discusses the existence of solutions for a nonlinear fractional differential equation with an integral boundary condition. It considers equations defined on the interval [0,T] with values in the space of continuous functions C[0,T].
2) It introduces relevant definitions, lemmas, and theorems for fractional calculus and fixed point theory. It then proves that solutions exist based on the Ascoli-Arzela theorem and Schauder-Tychonoff fixed point theorem.
3) The main result provides an explicit formula for the solution in terms of integrals involving the functions defining the fractional differential equation and the boundary conditions. Existence is established by showing the solution map is a contraction.
This document provides an overview and proofs of several theorems related to the Hahn-Banach theorem. It begins with an introduction to linear functionals and the Hahn-Banach theorem. It then presents two main theorems - the Hahn-Banach theorem and the topological Hahn-Banach theorem. The document provides proofs of these theorems and several related theorems using the Hahn-Banach extension lemma. It also discusses consequences of the Hahn-Banach extension form and provides proofs of the theorems using the lemma.
On the Fixed Point Extension Results in the Differential Systems of Ordinary ...BRNSS Publication Hub
This document summarizes research on extending the domain of fixed points for ordinary differential equations. It begins with definitions of fixed points and extendability. It then establishes several theorems on extending fixed points, including using Peano's theorem on existence and Picard-Lindelof theorem on uniqueness to extend fixed points over open connected domains where the vector field is continuous. The document proves that if a fixed point is bounded on its domain and the limits at the endpoints exist, then the fixed point can be extended to those endpoints. It concludes by discussing extending fixed points defined on intervals to the whole positive real line using boundedness conditions.
1) Fourier series and integrals are used to represent periodic functions as an infinite sum or integral of sines and cosines. They are useful for solving differential equations.
2) The Fourier series of a periodic function f(x) with period T is the sum of its coefficients multiplied by sines and cosines of integer multiples of x/T. The coefficients are calculated using integrals involving f(x).
3) Half range expansions like half range cosines (HRC) and half range sines (HRS) are used when f(x) is defined on a finite interval, by extending it to a periodic function on the entire real line.
Existence of Extremal Solutions of Second Order Initial Value Problemsijtsrd
In this paper existence of extremal solutions of second order initial value problems with discontinuous right hand side is obtained under certain monotonicity conditions and without assuming the existence of upper and lower solutions. Two basic differential inequalities corresponding to these initial value problems are obtained in the form of extremal solutions. And also we prove uniqueness of solutions of given initial value problems under certain conditions. A. Sreenivas ""Existence of Extremal Solutions of Second Order Initial Value Problems"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-4 , June 2019,
URL: https://www.ijtsrd.com/papers/ijtsrd25192.pdf
Paper URL: https://www.ijtsrd.com/mathemetics/other/25192/existence-of-extremal-solutions-of-second-order-initial-value-problems/a-sreenivas
The document discusses Fourier series and two of their applications. Fourier series can be used to represent periodic functions as an infinite series of sines and cosines. This allows approximating functions that are not smooth using trigonometric polynomials. Two key applications are representing forced oscillations, where a periodic driving force can be modeled as a Fourier series, and solving the heat equation, where the method of separation of variables results in a Fourier series representation of temperature over space and time.
Complete l fuzzy metric spaces and common fixed point theoremsAlexander Decker
This document presents definitions and theorems related to complete L-fuzzy metric spaces and common fixed point theorems. It begins with introducing concepts such as L-fuzzy sets, L-fuzzy metric spaces, and triangular norms. It then defines Cauchy sequences and completeness in L-fuzzy metric spaces. The main result is Theorem 2.2, which establishes conditions under which four self-mappings of a complete L-fuzzy metric space have a unique common fixed point. These conditions include the mappings having compatible pairs, one mapping having a closed range, and the mappings satisfying a contractive-type inequality condition. The proof of the theorem constructs appropriate sequences to show convergence.
This document discusses the chain rule for functions of multiple variables. It begins by reviewing the chain rule for single-variable functions, then extends it to functions of more variables. The chain rule is presented for cases where the dependent variable z is a function of intermediate variables x and y, which are themselves functions of independent variables s and t. General formulas are given using partial derivatives. Examples are worked out, such as finding the derivative of a function defined implicitly by an equation. Diagrams are used to illustrate the relationships between variables.
This document discusses principles of communication and representation of signals. It begins with an introduction to the communication process and challenges involved. Signals exist in the time and frequency domains, and Fourier analysis using the Fourier series and Fourier transform helps characterize signals in the frequency domain. Periodic signals can be represented by a Fourier series which decomposes the signal into a sum of complex exponentials at discrete frequencies that are integer multiples of the fundamental frequency. Examples are provided to illustrate calculation of Fourier coefficients and representation of periodic signals in the exponential and trigonometric forms of the Fourier series. Spectral plots from a spectrum analyzer are also presented for various waveforms.
This document discusses differential equations. It defines ordinary and partial differential equations, and explains that ordinary differential equations involve functions of a single variable while partial differential equations involve functions of multiple variables. It also defines linear and non-linear differential equations, and discusses methods for solving different types of differential equations including separable, exact, and linear equations with constant or variable coefficients. Examples are provided for population growth models, price dynamics, and economic models involving differential equations.
1. The document provides an introduction to Fourier analysis and Fourier series. It discusses how periodic functions can be represented as the sum of infinite trigonometric terms.
2. Examples are given of arbitrary functions being approximated by Fourier series of increasing lengths. As the length of the series increases, the ability to mimic the behavior of the original function also increases.
3. The Fourier transform is introduced as a method to represent functions in terms of sine and cosine terms. It allows problems involving differential equations to be transformed into an algebraic form and then transformed back to find the solution.
This document discusses whether and when the entropy of a spatially homogeneous Boltzmann equation system with infinite initial entropy will become finite. It presents examples showing that for some initial distributions, the entropy remains infinite over time, while for others it becomes finite after a finite time. The main tool used to estimate the entropy is the Duhamel formula for the solution. For hard potential and hard sphere models, rules are given for determining whether the entropy will appear finite based on properties of the initial distribution and collision kernel.
Common Fixed Theorems Using Random Implicit Iterative Schemesinventy
This document summarizes research on common fixed point theorems using random implicit iterative schemes. It defines random Mann, Ishikawa, and SP iterative schemes. It also defines modified implicit random iterative schemes associated with families of random asymptotically nonexpansive operators. The paper proves the convergence of two random implicit iterative schemes to a random common fixed point. This generalizes previous results and provides new convergence theorems for random operators in Banach spaces.
This document is a project report submitted by a student for a Bachelor of Science degree in Mathematics. It provides an introduction to the field of topology. The report includes a title page, certificate, declaration, acknowledgements, table of contents, and several chapters. The introduction defines topology and provides some examples. It states that the project aims to provide a thorough grounding in general topology.
A Solution Manual And Notes For Kalman Filtering Theory And Practice Using ...Daniel Wachtel
This document contains notes and solutions for a book on Kalman filtering. It includes notes on chapters explaining linear dynamic systems and derivations of fundamental solutions and state transition matrices for differential equations like dy/dt=0 and d2y/dt2=0. The author provides complete solutions to problems at the end of chapters and welcomes feedback to improve the notes.
On Analytic Review of Hahn–Banach Extension Results with Some GeneralizationsBRNSS Publication Hub
The useful Hahn–Banach theorem in functional analysis has significantly been in use for many years ago. At this point in time, we discover that its domain and range of existence can be extended point wisely so as to secure a wider range of extendibility. In achieving this, we initially reviewed the existing traditional Hahn–Banach extension theorem, before we carefully and successfully used it to generate the finite extension form as in main results of section three.
Existance Theory for First Order Nonlinear Random Dfferential Equartioninventionjournals
In this paper, the existence of a solution of nonlinear random differential equation of first order is proved under Caratheodory condition by using suitable fixed point theorem. 2000 Mathematics Subject Classification: 34F05, 47H10, 47H4
1. The chain rule describes how to take the derivative of a function composed of other functions. It states that if z = f(x, y) and x and y are functions of t, then the derivative of z with respect to t is the sum of the partial derivatives multiplied by the derivatives of x and y with respect to t.
2. The chain rule is applied to find the derivative of two example functions, w = xy and w = xy + z, with respect to t along given paths for x, y, and z in terms of t.
3. The chain rule is generalized to the case of a function u of n variables, where each variable is a function of m other variables
This document summarizes research on extending the domain of fixed point solutions to ordinary differential equations. It begins with definitions of fixed points and extendability. The Peano and Picard-Lindelof theorems are used to prove local existence and uniqueness of fixed points. Several theorems are presented to establish conditions under which a fixed point defined on a finite interval can be extended to the endpoints of that interval or beyond. Specifically, if the vector field is bounded and limits exist at the endpoints, the fixed point can be extended. The results ensure extendability to infinite intervals under boundedness assumptions. Extendability is key to studying existence of fixed points on the half line R+.
This document summarizes research on extending the domain of fixed point solutions to ordinary differential equations. It begins with definitions of fixed points and extendability. The Peano and Picard-Lindelof theorems are used to prove local existence and uniqueness of fixed points. Several theorems are presented to establish conditions under which a fixed point defined on a finite interval can be extended to the endpoints of the interval. Specifically, if the vector field is bounded and limits exist at the endpoints, the fixed point can be extended. The results are applied to extend fixed points defined on finite intervals to the entire positive real line.
1) The document discusses the existence of solutions for a nonlinear fractional differential equation with an integral boundary condition. It considers equations defined on the interval [0,T] with values in the space of continuous functions C[0,T].
2) It introduces relevant definitions, lemmas, and theorems for fractional calculus and fixed point theory. It then proves that solutions exist based on the Ascoli-Arzela theorem and Schauder-Tychonoff fixed point theorem.
3) The main result provides an explicit formula for the solution in terms of integrals involving the functions defining the fractional differential equation and the boundary conditions. Existence is established by showing the solution map is a contraction.
This document provides an overview and proofs of several theorems related to the Hahn-Banach theorem. It begins with an introduction to linear functionals and the Hahn-Banach theorem. It then presents two main theorems - the Hahn-Banach theorem and the topological Hahn-Banach theorem. The document provides proofs of these theorems and several related theorems using the Hahn-Banach extension lemma. It also discusses consequences of the Hahn-Banach extension form and provides proofs of the theorems using the lemma.
On the Fixed Point Extension Results in the Differential Systems of Ordinary ...BRNSS Publication Hub
This document summarizes research on extending the domain of fixed points for ordinary differential equations. It begins with definitions of fixed points and extendability. It then establishes several theorems on extending fixed points, including using Peano's theorem on existence and Picard-Lindelof theorem on uniqueness to extend fixed points over open connected domains where the vector field is continuous. The document proves that if a fixed point is bounded on its domain and the limits at the endpoints exist, then the fixed point can be extended to those endpoints. It concludes by discussing extending fixed points defined on intervals to the whole positive real line using boundedness conditions.
1) Fourier series and integrals are used to represent periodic functions as an infinite sum or integral of sines and cosines. They are useful for solving differential equations.
2) The Fourier series of a periodic function f(x) with period T is the sum of its coefficients multiplied by sines and cosines of integer multiples of x/T. The coefficients are calculated using integrals involving f(x).
3) Half range expansions like half range cosines (HRC) and half range sines (HRS) are used when f(x) is defined on a finite interval, by extending it to a periodic function on the entire real line.
Existence of Extremal Solutions of Second Order Initial Value Problemsijtsrd
In this paper existence of extremal solutions of second order initial value problems with discontinuous right hand side is obtained under certain monotonicity conditions and without assuming the existence of upper and lower solutions. Two basic differential inequalities corresponding to these initial value problems are obtained in the form of extremal solutions. And also we prove uniqueness of solutions of given initial value problems under certain conditions. A. Sreenivas ""Existence of Extremal Solutions of Second Order Initial Value Problems"" Published in International Journal of Trend in Scientific Research and Development (ijtsrd), ISSN: 2456-6470, Volume-3 | Issue-4 , June 2019,
URL: https://www.ijtsrd.com/papers/ijtsrd25192.pdf
Paper URL: https://www.ijtsrd.com/mathemetics/other/25192/existence-of-extremal-solutions-of-second-order-initial-value-problems/a-sreenivas
The document discusses Fourier series and two of their applications. Fourier series can be used to represent periodic functions as an infinite series of sines and cosines. This allows approximating functions that are not smooth using trigonometric polynomials. Two key applications are representing forced oscillations, where a periodic driving force can be modeled as a Fourier series, and solving the heat equation, where the method of separation of variables results in a Fourier series representation of temperature over space and time.
Complete l fuzzy metric spaces and common fixed point theoremsAlexander Decker
This document presents definitions and theorems related to complete L-fuzzy metric spaces and common fixed point theorems. It begins with introducing concepts such as L-fuzzy sets, L-fuzzy metric spaces, and triangular norms. It then defines Cauchy sequences and completeness in L-fuzzy metric spaces. The main result is Theorem 2.2, which establishes conditions under which four self-mappings of a complete L-fuzzy metric space have a unique common fixed point. These conditions include the mappings having compatible pairs, one mapping having a closed range, and the mappings satisfying a contractive-type inequality condition. The proof of the theorem constructs appropriate sequences to show convergence.
This document discusses the chain rule for functions of multiple variables. It begins by reviewing the chain rule for single-variable functions, then extends it to functions of more variables. The chain rule is presented for cases where the dependent variable z is a function of intermediate variables x and y, which are themselves functions of independent variables s and t. General formulas are given using partial derivatives. Examples are worked out, such as finding the derivative of a function defined implicitly by an equation. Diagrams are used to illustrate the relationships between variables.
This document discusses principles of communication and representation of signals. It begins with an introduction to the communication process and challenges involved. Signals exist in the time and frequency domains, and Fourier analysis using the Fourier series and Fourier transform helps characterize signals in the frequency domain. Periodic signals can be represented by a Fourier series which decomposes the signal into a sum of complex exponentials at discrete frequencies that are integer multiples of the fundamental frequency. Examples are provided to illustrate calculation of Fourier coefficients and representation of periodic signals in the exponential and trigonometric forms of the Fourier series. Spectral plots from a spectrum analyzer are also presented for various waveforms.
This document discusses differential equations. It defines ordinary and partial differential equations, and explains that ordinary differential equations involve functions of a single variable while partial differential equations involve functions of multiple variables. It also defines linear and non-linear differential equations, and discusses methods for solving different types of differential equations including separable, exact, and linear equations with constant or variable coefficients. Examples are provided for population growth models, price dynamics, and economic models involving differential equations.
1. The document provides an introduction to Fourier analysis and Fourier series. It discusses how periodic functions can be represented as the sum of infinite trigonometric terms.
2. Examples are given of arbitrary functions being approximated by Fourier series of increasing lengths. As the length of the series increases, the ability to mimic the behavior of the original function also increases.
3. The Fourier transform is introduced as a method to represent functions in terms of sine and cosine terms. It allows problems involving differential equations to be transformed into an algebraic form and then transformed back to find the solution.
This document discusses whether and when the entropy of a spatially homogeneous Boltzmann equation system with infinite initial entropy will become finite. It presents examples showing that for some initial distributions, the entropy remains infinite over time, while for others it becomes finite after a finite time. The main tool used to estimate the entropy is the Duhamel formula for the solution. For hard potential and hard sphere models, rules are given for determining whether the entropy will appear finite based on properties of the initial distribution and collision kernel.
Common Fixed Theorems Using Random Implicit Iterative Schemesinventy
This document summarizes research on common fixed point theorems using random implicit iterative schemes. It defines random Mann, Ishikawa, and SP iterative schemes. It also defines modified implicit random iterative schemes associated with families of random asymptotically nonexpansive operators. The paper proves the convergence of two random implicit iterative schemes to a random common fixed point. This generalizes previous results and provides new convergence theorems for random operators in Banach spaces.
ALPHA LOGARITHM TRANSFORMED SEMI LOGISTIC DISTRIBUTION USING MAXIMUM LIKELIH...BRNSS Publication Hub
The document discusses the alpha logarithm transformed semi-logistic distribution and its maximum likelihood estimation method. It introduces the distribution, provides its probability density function and cumulative distribution function. It then describes generating random numbers from the distribution and outlines the maximum likelihood estimation method to estimate the distribution's unknown parameters. This involves deriving the likelihood function and taking its partial derivatives to obtain equations that are set to zero and solved to find maximum likelihood estimates of the location, scale, and shape parameters.
AN ASSESSMENT ON THE SPLIT AND NON-SPLIT DOMINATION NUMBER OF TENEMENT GRAPHSBRNSS Publication Hub
This document summarizes research on the split and non-split domination numbers of tenement graphs. It defines tenement graphs and provides basic definitions of domination, split domination, and non-split domination. Formulas for the split and non-split domination numbers of tenement graphs are presented based on the number of vertices. Theorems are presented stating that the mid vertex set of a tenement graph is always a split dominating set, but its size is not always equal to the split domination number.
This document summarizes research on generalized Cantor sets and functions where the standard construction is modified. It introduces Cantor sets defined by an arbitrary base where the intervals removed at each stage are not all the same length. It also defines irregular or transcendental Cantor sets generated by transcendental numbers like e. The key findings are:
1) There exists a unique probability measure for generalized Cantor sets that generates the cumulative distribution function.
2) The Holder exponent of generalized Cantor sets is shown to be logn/s where n is the base and s is the number of subintervals.
3) Lower and upper densities are defined for the measure on generalized Cantor functions and their properties are
SYMMETRIC BILINEAR CRYPTOGRAPHY ON ELLIPTIC CURVE AND LIE ALGEBRABRNSS Publication Hub
1) The document discusses symmetric bilinear pairings on elliptic curves and Lie algebras in the context of cryptography. It provides an overview of the theoretical foundations and applications of combining these areas.
2) Key concepts covered include the Weil pairing as a symmetric bilinear pairing on elliptic curves, its properties of bilinearity and non-degeneracy, and efficient computation. Applications of elliptic curves in cryptography like ECDH and ECDSA are also summarized.
3) The security of protocols like ECDH and ECDSA relies on the assumed difficulty of solving the elliptic curve discrete logarithm problem (ECDLP). The document proves various mathematical aspects behind symmetric bilinear pairings and their use in elliptic curve cryptography.
SUITABILITY OF COINTEGRATION TESTS ON DATA STRUCTURE OF DIFFERENT ORDERSBRNSS Publication Hub
This document summarizes research investigating the suitability of cointegration tests on time series data of different orders. The researchers used simulated time series data from normal and gamma distributions at sample sizes of 30, 60, and 90. Three cointegration tests (Engle-Granger, Johansen, and Phillips-Ouliaris) were applied to the data. The tests were assessed based on type 1 error rates and power to determine which test was most robust for different distributions and sample sizes. The results indicated the Phillips-Ouliaris test was generally the most effective at determining cointegration across different sample sizes and distributions.
Artificial Intelligence: A Manifested Leap in Psychiatric RehabilitationBRNSS Publication Hub
Artificial intelligence shows promise in improving psychiatric rehabilitation in 3 key ways:
1) AI can help diagnose and treat mental health issues through virtual therapists and chatbots, improving access and reducing stigma.
2) Technologies like machine learning and big data allow personalized interventions and more accurate diagnoses.
3) The COVID-19 pandemic has increased need for mental health support, and AI may help address gaps by providing remote services.
A Review on Polyherbal Formulations and Herbal Medicine for Management of Ul...BRNSS Publication Hub
This document provides a review of polyherbal formulations and herbal medicines for treating peptic ulcers. It discusses how peptic ulcers occur due to an imbalance between aggressive and protective factors in the gastrointestinal tract. Common causes include H. pylori infection and NSAID use. While synthetic medications are available, herbal supplements are more affordable and have fewer side effects. The review examines various herbs that have traditionally been used to treat ulcers, including their active chemical constituents. It defines polyherbal formulations as combinations of two or more herbs, which can enhance therapeutic effects while reducing toxicity. The document aims to summarize recent research on herb and polyherbal formulation treatments for peptic ulcers.
Current Trends in Treatments and Targets of Neglected Tropical DiseaseBRNSS Publication Hub
This document summarizes current trends in treatments and targets of neglected tropical diseases. It begins by stating that neglected tropical diseases affect over 1.7 billion people globally each year and are caused by a variety of microbes. The World Health Organization is working to eliminate 30 neglected tropical diseases by 2030. The document then discusses several specific neglected tropical diseases in more detail, including human African trypanosomiasis, Chagas disease, leishmaniasis, soil-transmitted helminths, and schistosomiasis. It describes the causative agents, transmission methods, symptoms, affected populations, and current treatment options for each of these diseases. Overall, the document aims to briefly discuss neglected infectious diseases and treatment
Evaluation of Cordia Dichotoma gum as A Potent Excipient for the Formulation ...BRNSS Publication Hub
This document summarizes a study that evaluated Cordia dichotoma gum as an excipient for oral thin film drug delivery. Films were prepared with varying ratios of the gum, plasticizers (methyl paraben and glycerine), and the model drug diclofenac sodium. The films were evaluated for properties like thickness, folding endurance, tensile strength, water uptake, and drug release kinetics. The results found that a film with 10% gum, 0.2% methyl paraben and 2.5% glycerine (CDF3) exhibited the best results among the formulations tested. Stability studies showed the films were stable for 30 days at different temperatures. Overall, the study demonstrated that C.
Assessment of Medication Adherence Pattern for Patients with Chronic Diseases...BRNSS Publication Hub
This study assessed medication adherence and knowledge among rural patients with chronic diseases in South Indian hospitals. 1500 hypertensive patients were divided into intervention and control groups. The intervention group received education from pharmacists at various times, while the control group did not. A questionnaire evaluated patients' medication knowledge at baseline and several follow-ups. The intervention group showed improved medication knowledge scores after education compared to the control group. Female gender, lower education, and income were linked to lower knowledge. The study highlights the need to educate rural patients to improve medication understanding and adherence.
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Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
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Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
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Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
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01_AJMS_277_20_20210128_V1.pdf
1. www.ajms.com 1
ISSN 2581-3463
RESEARCH ARTICLE
On Generalized Classical Fréchet Derivatives in the Real Banach Space
Chigozie Emmanuel Eziokwu
Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria
Received: 01-08-2020; Revised: 20-09-2020; Accepted: 10-10-2020
ABSTRACT
In this work, we reviewed the Fréchet derivatives beginning with the basic definitions and touching most
of the important basic results. These results include among others the chain rule, mean value theorem, and
Taylor’s formula for differentiation. Obviously, having clarified that the Fréchet differential operators
exist in the real Banach domain and that the operators are clearly continuous, we then in the last section
for main results developed generalized results for the Fréchet derivatives of the chain rule, mean value
theorem, and Taylor’s formula among others which become highly useful in the analysis of generalized
Banach space problems and their solutions in Rn
.
Key words: Banach space, continuity, Fréchet derivatives, mean value theorem, Taylor’s formula
2010 Mathematics Subject Classification: 46BXX, 46B25
Address for correspondence:
Chigozie Emmanuel Eziokwu
E-mail: okereemm@yahoo.com
THE USUAL FRECHET DERIVATIVES
Given x a fixed point in a Banach space X and Y another Banach space, a continuous linear operator
S: X→Y is called the Frechet derivative of the operator T: X→Y at x if
T x x T x S x x x
,
and
lim
,
x
x x
x
0
0
Or equivalently,
lim
x
T x x T x S x
x
0
0
This derivative is usually denoted by dT (x) or
T x and T is Frechet differentiable on its domain if
T x exists at every point of the domain as in Abdul[1]
and Argyros[2]
.
Remark: If X = R, Y = R, then the classical derivative
f x of real function f: R→R at x
f x
f x x f x
x
x
lim
0
Is a number representing the slope of the graph of the function f at x where the Frechet derivative of f is
not a number but a linear operator on R into R. Existence of
f x implies the existence of the Frechet
derivative[3]
as the two are related by
f x x f x f x x x g x
2. Eziokwu: Generalized classical fréchet derivatives
AJMS/Oct-Dec-2020/Vol 4/Issue 4 2
While S is the operator which multiplies every δx by the number
f x . In elementary calculus the
derivative at x is a local approximation of f in the neighborhood of x while the Frechet derivative is
interpreted as the best local linear approximation. It is clear from definition that if T is linear, then the
Frechet derivative is linear as well, that is,
dT x T x
THEOREM 1.1:[4]
If an operator has the Frechet derivative at a point, then it has the Gateaux derivative at that point and
both derivatives have equal values.
THEOREM 1.2:[5]
Let Ω be an open subset of X and T Y
: have Frechet derivative at an arbitrary point a of Ω. Then
T is continuous at a. This means that every Frechet differentiable operator defined on an open subset of
a Banach space is continuous.
THEOREM 1.3(CHAIN RULE):[1,6]
Let A, B, and C be real Banach spaces. If S: A→B and T: B→C are Frechet differentiable at x and
U x T S x S x . Then, the higher order Frechet derivatives for real U = To
S can successively be
generated iteratively such that
U x T S x S x
n n n
For n ≥ 2 and integer.
THEOREM 1.5 (IMPLICIT FUNCTION THEOREM)[1,7,8]
Suppose that X, Y, and Z are Banach spaces, C an open subset of X×Y and T: C→Z is continuous, suppose
further that for some x y C
1 1
,
i. T x y
1 1 0
,
ii. The Frechet derivative of T (.,.) when x is fixed is denoted by Ty
(x, y) called the partial Frechet derivative
with respect to y, exists at each point in a neighborhood of (x1
, y1
) and is continuous at (x, y).
iii. T x y B z y
y 1 1
1
, ,
then there is an open subset of X containing x and a unique continuous
mapping y: D→Y such that T(x, y (x)) = 0 and y(x1
)=y1
Corollary 1.6: If in addition to theorem 1.5 Tx
(x, y) also exists in the open set, and is continuous at
(x1
, y1
). Then, F: x→y (x) has Frechet derivative at x1
given by
F x T x y T x y
y x
1 1
1
1 1
, ,
THEOREM 1.7 (Taylor’s Formula for differentiation)[1,9,10]
Let T X Y
: and let a a x
,
be any closed segment lying in Ω. If T is Frechet differentiable at
a, then
T a x T a x x x
x
x
lim
0
0
and
T a h T a T a x T a x x x
x
x
1
2
0
2
0
lim
For twice differentiable functions.
3. Eziokwu: Generalized classical fréchet derivatives
AJMS/Oct-Dec-2020/Vol 4/Issue 4 3
MAIN RESULTS ON GENERALIZED FRECHET DERIVATIVES
Let x be a fixed point in the real Banach space. Also let the continuous linear operator S: X→Y be a real
Frechet derivative of the operator T: X→Y such that
lim
x
T x x T x S x
x
0
0
Then, the higher order Frechet derivative successively can be generated in an iterative manner such that
lim
x
i
i
n
i
i
n
i
i
n
T x x T x x S x
0
1 1
1
1
i
i
i
n
i
i
n
x
1
1
n≥2 and an integer.
THEOREM 2.1 (CHAIN RULE): Let A, B, and C be a unitary spaces, if S: A→B and T: B→C are
Frechet differentiable at z and
u x u s x s x
. Then, the higher order Frechet derivative for
U x
n
can be generated with U S T
= generating n n
z U z
if and only if
lim
z
i
i
n
i
i
n
i
i
i
i
n
U z z U z z z
1
0
1 1
1
1
n
n
i
i
n
x
1
lim
x
i
i
n
i
i
n
i
i
n
S T x x S T x x
1
0
1 1
1
x
x
i
i
n
i
i
n
1
1
THEOREM 2.2 [Generalized Frechet Mean Value theorem]: Let T: A→B where A is an open convex
set containing a, b, and c is a normed space. T n x
exists for each a a b
, and T x
n
1
is continuous
on [a, b], then
T b T a T a T b T a
n n
x a b
n n n
1 1 2 2
,
sup
THEOREM 2.3 [Generalized Implicit function theorem]
Suppose that A, B, and C are real Banach spaces, D is an open subset of A×B and T: D→C is continuous.
Suppose further that for some a b D
,
, then
i. T a b
n
, 0
ii. The nthFrechet derivative of T (.,.) where x is fixed and denoted by T a b
b
n
1 1 1
, called the nth
partial
derivative with respect to b exists at each point in a neighborhood of (a1
, b1
) and is continuous at a1
, b1
iii. T a b B C B
x
n
1 1
1
, , then there is a subset E of A containing a1
and a unique continuous
mapping S: E→C such that T a b a
n
1 1 1 0
, and S a b
n
1 1
4. Eziokwu: Generalized classical fréchet derivatives
AJMS/Oct-Dec-2020/Vol 4/Issue 4 4
Corollary 2.4: If the addition to conditions of theorem 2.3, T a b
a
n
1
, also exists on the open set, and
is continuous at (a1
, b1
), then F: a→b (a) has the nth Frechet derivative at a1
given by
F a T a b T a b
n
n
n
a
n
1
1
1
1
1 1
, ,
THEOREM 2.5 [Taylors formula for nth Frechet differentiable functions]
Let T X Y
: and a a n x
,
be any closed segment lying in Ω. If T is differentiable in Ω and nth
differentiable at a, then
T a n x T a T a x T a x x
n
T x x
n n n n
1
2
1
!
x
where
lim
x
x
0
PROOF OF MAIN RESULTS
Proof of Theorem 2.1 (chain rule)
Let x x X
, and suppose Un
(x) can be generated with U S T
= such that the generalized Frechet
derivative
n n
i
i
n
i
i
n
x U x U x x U x
T S x
1 1
0 x
x T S x T x y
i
i
n
i
i
n
i
i
n
1 1 1
T yi
i
n
1
where
x S x x S x
i
i
n
i
i
n
i
i
n
1 1 1
Thus
U x x U x T x x z
i i
i
n
n
1
0
since
x S x x x
i
i
n
n
i
i
n
1 1
We get
U x x U x T x S x x
U x x
i
i
n
i
i
n
n n
i
i
n
1 1
0
1
U x T y x T y S x x
i
i
n
n n n
1
In view of the fact that S is continuous at x, we obtain
x x
i
i
n
1
5. Eziokwu: Generalized classical fréchet derivatives
AJMS/Oct-Dec-2020/Vol 4/Issue 4 5
therefore
n n
x
i
i
n
i
i
n
x x
S T x x S T x x
i
i
n
lim
1
0
1 1
1
1
1
1
x
x
i
i
n
i
i
n
conversely
lim
x
i
i
n
i
i
n
i
i
n
S T x x S T x x x
1
0
1 1
1
i
i
i
n
i
i
n
x
1
1
implies
T S x x T S x
T S x x x
i
i
n
i
i
n
i i
i
n
1
1 1
T S x x
i
i
n
i
i
n
1 1
and
T S x x S T x
T S x x x
i
i
n
i
i
n
i
i
n
1
1
1
1
T S x x
i
i
n
i
i
n
1 1
lim
x
i i
i
n
i
i
n
i
T S x x T S x x x
0
1
1
1
x
x
x x
i
i
n
x
i
i
n
i
x
i
i n
1
1
0
1
1
0
0
0
lim lim By L'H
Hospital Rule
Hence, U x x
n n
is Frechet differentiable and the proof is complete.
PROOF OF THE GENERALIZED FRECHET MEAN VALUE THEOREM
Let T: K→B where K an open convex set containing is a and b. B is a normal space and T x
n
exists
for each x in [a, b] and
T x is continuous in [a, b] such that
T b T a T x b a
x a b
,
sup
Then by induction for the nth complex iterative Frechet derivative of T, the mean value theorem becomes
T b T a T x T b T a
n n
x a b
n n n
1 1 2 2
,
sup
Proof of Theorem 2.3 [generalized implicit function theorem]
For the sake of convenience, we may take x1
= 0 and x2 0
*
= . let
A T B C B
x
n
2
1
0 0
, ,
6. Eziokwu: Generalized classical fréchet derivatives
AJMS/Oct-Dec-2020/Vol 4/Issue 4 6
Since D is an open set containing 0 0
,
, we find that
D D x C x x D
x
1 2 1 2
,
For all sufficiently small say x1 . For each x1 having this property, we define a function S x x
n
1
1 2
,
D C
x1
→ by S x x x FT x x
n n
1
1 2 2
2
1 2
, , . To prove the theorem, we must prove the existence of a
fixed point for S x x
n
1
1 2
, under the condition that x1 is sufficiently small. Continuity of the mapping
x x x
1 2 1
and x x x
2 1 2
*
. Now,
S x x U U T x x U
x
n
x
2 2
1
1 2 1 2
, ,
and
FF FTx
n
1 1
2
0 0
,
Therefore, assumptions on T(n−1)
guarantees the existence of S(n−1)
(x1
, x2
) for sufficiently small x1 and
x2 and
S x x U F T T x x U
n
x
n
x
n
1
1 2
1 1
1 2
2 2
0 0
, , ,
hence
S x x F T T x x
n
x
n
x
n
1
1 2
1 1
1 2
2 2
0 0
, , ,
Since, Tx
n
2
1
is continuous at (0, 0) there exists a constant L 0 such that
S x x L
n
1
1 2
, (3.3.1)
Or sufficiently small x1 and x2 , we say that x1 1
and x2 2
. Since T n
1
is continuous at
(0, 0), there exists an ε ≤ ε1
such that
S x FT x L
n n
1
1 0 1 0 2 1
, , (3.3.2)
For all x1
with x1 . We now show that S x
n
1
1,. maps the closed ball S x B x
n
1
2 2 2
0
into itself. For this let x1 and x2 2
. Then by the Mean Value theorem and (3.3.1), (3.3.2), we
have
S x x S x x S x S x
n n n n
1
1 2
1
1 2
1
1
1
0 1
0 0
, , , ,
s
sup *
, ,
S x x x S x L L
x
n n
2 2
1 2 2
1
1 2 2
0 1
Therefore, for x S x S
n n
1
1
1
1
2
0
, ,. : . Also for x x S
2 2 2
0
* **
, ;
we obtain by the mean value
theorem of section 2.2 and equation (3.3.1)
S x x S x x S x x x x
n n
x x
n
1
1 2
1
1 2
1
1 2 2
2 2 2
, , ,
* ** sup *
2
2 2 2
** * **
L x x
The Banach contraction mapping theorem guarantees that for each x1
with x1 there exists a unique
x x S n
2 1
1
2
0
such that
x x S x x x x x FT x x x
n n
2 1
1
1 2 1 2 1
1
1 2 1
, ,
That is,
T x x x
n
1
1 2 1 0
,
7. Eziokwu: Generalized classical fréchet derivatives
AJMS/Oct-Dec-2020/Vol 4/Issue 4 7
By the uniqueness of x2
, we have that x2
(0) = 0 since
T n
1
0 0 0
,
Finally, we show that x x x
1 2 1
is continuous for if x1
*
and x1
**
, and then selecting
x x x x
2
0
1 2 1
, **
and x S x x
n
2
1
1 2
0
* *
,
. We have by the error bound for fixed point iteration on the
mapping S x
n
1
1,.
x x x x
L
x x
2 1 2 1 2
0
2
1
1
** *
We can write
x x x x S x x x
S x x x
n
n
2
0
2 2 1
1
1 2 1
1
1 2 1
* ** * **
* **
,
,
T x x x
F T x x x T x x x
n
n n
1
1 2 1
1
1 2 1
1
1 2 1
* **
* **
,
, , *
**
Therefore, by continuity of T x x x x
n
1
2 1 2 1
, ** **
can be made arbitrary small for x x
1 1
** *
− sufficiently
small and hence the proof.
Proof of Corollary 2.4
We set x x x
1 1 1
*
and G x F x x
n n
1 1 1
*
. Then G n
0 0 and
G x T x x x
T x x
n
x
n
x
n
1
1
1
1 2
1
1
1
1 2
2
2
* *
* *
,
1
1
1 2 1
1
1
1
1 2 1
2 1
T x x x G x T x x x
x
n n
x
n
, ,
* * *
and
T x x x G x T x x x
T
x
n n
x
n
n
2 1
1
1 2 1
1
1
1
1 2 1
1
, ,
* * *
x x x G x x T x x
T G
n n
x
n
1 1 2
1
1 1
1
1 2
1
2
* * * * * *
, ,
n
n
x
n
x x x x
1
1
1
1 2 1
1
* *
,
If O1
, O2
are numbers in (0,1), then
T x x x G x T x x x
T
x
n n
x
n
x
2 1
1
1
1 2 1
1
1
1
1 2 1
, ,
sup
* * *
n
n n
x
n
x O x x O G x T x x x
1
1 1 1 2 2
1
1
1
1 2 1
1
* * * * * *
, ,
O O x
n n
x
n
T x O x x O G x T x
1 2 1 2
1
1 1 1 2 2
1
1
1
1
sup * * * *
, ,
x
x G x
n
2
2
1
* *
Thus applying continuity od Tx
n
1
1
, Tx
n
2
1
for 0, we find that
such that on x x
1 1
*
, we
have
8. Eziokwu: Generalized classical fréchet derivatives
AJMS/Oct-Dec-2020/Vol 4/Issue 4 8
G x T x x T x x x
T
n
x
n
x
n
x
1
1
1
1 2
1
1
1 2 1
2 1
* * * *
, ,
2
2 2 1
1
1 2
1
1
1 2
1
1
1
1
n
x
n
x
n
x x T x x T x
* * * *
, ,
*
* * *
,
,
x x
T x x
x
n
2 1
1
1 2
1
1 2
The coefficient of
*
1
∆x can be as small as required as x1 0
*
. Thus,
F x F x T x x T x x x
n n
x
n
x
n
1 1
1
1 2
1
1
1 1 1
2 1
** *
, ,
x x x
2 1 1
* *
Hence,
F x T x x T x x
n n
x
n
1
1
1 2
1
1
1 2
2
* * *
, ,
Proof of Taylor’s formula for nth Frechet differentiable function
The proof of this theorem can be generated as in Carton[11]
and Nasheed.[12]
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Centre; 2010.
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non-linear functional analysis In: Rall LB, editor. Non Linear Functional Analysis and Applications. New York, London:
Academic Press; 1971. p. 103-309.