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.
The materials discussed can be used for the following studies
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.
"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
Quadratic Equations and Perturbation Theory
Algebraic Theory of Quadratic Operators
Chandrasekhar's Integral Equation
Anselone and Moore's Equation
Other Perturbation Theorems
More Methods for Solving Quadratic Equations
The Majorant Method
Compact Quadratic Equations 83
Finite Rank Equations
On a Class of Quadratic Integral Equations with Perturbation
Polynomial Equations in Banach Space
Solving Polynomial Operator Equations in Ordered Banach Spaces
Integral and Differential Equations
Equations of Hammerstein Type
Radiative Transfer Equations
Integrals on a Separable Hilbert Space
Approximation of Solutions of Some Quadratic Integral Equations in Transport Theory
Uniformly Contractive Systems and Quadratic Equations in Banach Space
Polynomial Operators in Linear Spaces
A Weierstrass Theorem
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