Polynomial Operator Equations in Abstract Spaces and Applications: 1st Edition (Hardback) book cover

Polynomial Operator Equations in Abstract Spaces and Applications

1st Edition

By Ioannis K. Argyros

CRC Press

592 pages

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Hardback: 9780849387029
pub: 1998-03-25

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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.

  • Reviews

    "This book provides a valuable service to those mathematicians working in the area of polynomial operator equations…The theoretical material addressed has a spectrum of applications…applications [that are] quite relevant and important…Anyone doing research in this area should have a copy of this monograph."

    Patrick J. Van Fleet, Mathematical and Information Sciences, Huntsville, Texas

    "A comprehensive presentation of this rapidly growing field…benefiting not only those working in the field but also those interested in, and in need of, information about specific results or techniques…Clear…Logical…Elegant…The author has achieved the optimum at this point."

    - Dr. George Anastassiou, University of Memphis, Tennessee

    Table of Contents


    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


    Glossary of Symbols

    Subject Index

    Subject Categories

    BISAC Subject Codes/Headings:
    MATHEMATICS / Applied
    SCIENCE / Mathematical Physics