Kurzweil–Stieltjes Integral

The book is primarily devoted to the Kurzweil–Stieltjes integral and its applications in functional analysis, theory of distributions, generalized elementary functions, as well as various kinds of generalized differential equations, including dynamic equations on time scales. It continues the research that was paved out by some of the previous volumes in the Series in Real Analysis. Moreover, it presents results in a thoroughly updated form and, simultaneously, it is written in a widely understandable way, so that it can be used as a textbook for advanced university or PhD courses covering the theory of integration or differential equations.

Contents:Introduction:Areas of Planar RegionsCenter of Mass and MomentsLine IntegralsFunctions of Bounded Variation: Definition and Basic PropertiesSpace of Functions of Bounded VariationBounded Variation and ContinuityDerivatives of Bounded Variation FunctionsStep FunctionsDecomposition Into Continuous and Jump PartsPointwise ConvergenceVariation on Elementary SetsAbsolutely Continuous Functions: Definition and Basic PropertiesAbsolutely Continuous Functions and the Lebesgue IntegralLebesgue Decomposition of Functions of Bounded VariationRegulated Functions:Definition and Basic PropertiesSpace of Regulated Functions and Its SubspacesRelatively Compact Subsets of G([a,b])Riemann–Stieltjes Integral: Definition and Basic PropertiesPseudo-AdditivityAbsolute IntegrabilitySubstitutionIntegration by PartsExistence of the IntegralConvergence TheoremsConsequences of Riemann–Stieltjes IntegrabilityMean Value TheoremsOther Integrals of Stieltjes TypeKurzweil–Stieltjes Integral: IntroductionDefinition and Basic PropertiesExistence of the IntegralIntegration by PartsThe Indefinite IntegralSubstitutionAbsolute IntegrabilityConvergence TheoremsIntegration Over Elementary SetsIntegrals of Vector, Matrix and Complex FunctionsRelation to the Perron–Stieltjes IntegralRelation to the Lebesgue–Stieltjes IntegralRelation to Other Stieltjes-Type IntegralsGeneralized Linear Differential Equations: IntroductionDifferential Equations with ImpulsesLinear OperatorsExistence and Uniqueness of SolutionsA Priori Estimates of SolutionsContinuous Dependence of Solutions on ParametersFundamental MatricesVariation of Constants FormulaMiscellaneous Additional Topics:Functionals on the Space of Continuous FunctionsFunctionals on Spaces of Regulated FunctionsAdjoint Classes of Kurzweil–Stieltjes Integrable FunctionsDistributionsGeneralized Elementary FunctionsIntegration on Time ScalesDynamic Equations on Time Scales
Readership: Advanced undergraduates, graduate students and researchers in mathematical analysis and differential equations.
Key Features:Unique treatment of the Kurzweil–Stieltjes integral and its applications to differential equations and functional analysisThe exposition is written in a widely understandable form