Polynomial Operator Equations in Abstract Spaces and Applications / Edition 1

Polynomial Operator Equations in Abstract Spaces and Applications / Edition 1

by Ioannis K. Argyros, Aoannis K. Argyros
     
 

Polynomial operators are a natural generalization of linear operators. Equations in such operators are the linear space analog of ordinary polynomials in one or several variables over the fields of real or complex numbers. Such equations encompass a broad spectrum of applied problems including all linear equations. Often the polynomial nature of many nonlinear

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Overview

Polynomial operators are a natural generalization of linear operators. Equations in such operators are the linear space analog of ordinary polynomials in one or several variables over the fields of real or complex numbers. Such equations encompass a broad spectrum of applied problems including all linear equations. Often the polynomial nature of many nonlinear problems goes unrecognized by researchers. This is more likely due to the fact that polynomial operators - unlike polynomials in a single variable - have received little attention. Consequently, this comprehensive presentation is needed, benefiting those working in the field as well as those seeking information about specific results or techniques.
Polynomial Operator Equations in Abstract Spaces and Applications - an outgrowth of fifteen years of the author's research work - presents new and traditional results about polynomial equations as well as analyzes current iterative methods for their numerical solution in various general space settings.
Topics include:

  • Special cases of nonlinear operator equations
  • Solution of polynomial operator equations of positive integer degree n
  • Results on global existence theorems not related with contractions
  • Galois theory
  • Polynomial integral and polynomial differential equations appearing in radiative transfer, heat transfer, neutron transport, electromechanical networks, elasticity, and other areas
  • Results on the various Chandrasekhar equations
  • Weierstrass theorem
  • Matrix representations
  • Lagrange and Hermite interpolation
  • Bounds of polynomial equations in Banach space, Banach algebra, and Hilbert space
    The materials discussed can be used for the following'studies
  • Advanced numerical analysis
  • Numerical functional analysis
  • Functional analysis
  • Approximation theory
  • Integral and differential equations
    Tables include
  • Numerical solutions for Chandrasekhar's equation I to VI
  • Error bounds comparison
  • Accelerations schemes I and II for Newton's method
  • Newton's method
  • Secant method
    The self-contained text thoroughly details results, adds exercises for each chapter, and includes several applications for the solution of integral and differential equations throughout every chapter.
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    Product Details

    ISBN-13:
    9780849387029
    Publisher:
    Taylor & Francis
    Publication date:
    03/25/1998
    Pages:
    592
    Product dimensions:
    6.25(w) x 9.50(h) x 1.50(d)

    Meet the Author

    Table of Contents

    Introduction
    Quadratic Equations and Perturbation Theory
    Algebraic Theory of Quadratic Operators
    Perturbation Theory
    Chandrasekhar's Integral Equation
    Anselone and Moore's Equation
    Other Perturbation Theorems
    More Methods for Solving Quadratic Equations
    Banach Algebras
    The Majorant Method
    Compact Quadratic Equations 83
    Finite Rank Equations
    Noncontractive Solutions
    On a Class of Quadratic Integral Equations with Perturbation
    Polynomial Equations in Banach Space
    Polynomial Equations
    Noncontractive Results
    Solving Polynomial Operator Equations in Ordered Banach Spaces
    Integral and Differential Equations
    Equations of Hammerstein Type
    Radiative Transfer Equations
    Differential Equations
    Integrals on a Separable Hilbert Space
    Approximation of Solutions of Some Quadratic Integral Equations in Transport Theory
    Multipower Equations
    Uniformly Contractive Systems and Quadratic Equations in Banach Space
    Polynomial Operators in Linear Spaces
    A Weierstrass Theorem
    Matrix Representations
    Lagrange and Hermite Interpolation
    Bounds of Polynomial Equations
    Representations of Multilinear and Polynomial Operators on Vector Spaces
    Completely Continuous and Related Multilinear Operators
    General Methods for Solving Nonlinear Equations
    Accessibility of Solutions of Equations by Newton-Like Methods and Applications
    The Super-Halley Method
    Convergence Rates for Inexact Newton-Like Methods at Singular Points
    A Newton-Mysovskii-Type Theorem with Applications to Inexact Newton-Like Methods and Their Discretizations
    Convergence Domains for Some Iterative Processes in Banach Spaces Using Outer and Generalized Inverses
    Convergence of Inexact Newton Methods on Banach Spaces with a Convergence Structure
    References
    Glossary of Symbols
    Subject Index

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