Polynomial Operator Equations in Abstract Spaces and Applications

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Detailed Description 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 Äthat areÜ 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 Markets TOC: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.