STIELTJES DIFFERENTIAL CALCULUS WITH APPLICATIONS

The Stieltjes derivative is a modification of the usual derivative through a nondecreasing and left-continuous map. This change in the definition allows us to study several differential problems under the same framework.

This monograph is the first published book that offers a comprehensive view of the fundamentals of Stieltjes calculus and its applications, making it approachable to newcomers and experts. It aims to provide an integrated approach to the foundations and recent developments in the area of the Stieltjes derivatives and the qualitative theory of the Stieltjes differential equations. Through 10 pedagogically organized chapters, the authors examine a wide scope of the concept of the Stieltjes derivative and its applications. Each chapter focuses on theory, and proofs, and contains sufficient examples to enrich the reader's understanding.

The Stieltjes derivative contains the Hilger delta derivative on time scales. Thus, offering a new unification and extension of continuous and discrete calculus. Further, a study of differential equations in the sense of the Stieltjes derivative allows the study of many classical problems in a unique framework. This theory has the advantage that ordinary differential equations, ordinary difference equations, quantum difference equations, impulsive differential equations, dynamic equations on time scales, and generalized differential equations can be treated as particular instances of the Stieltjes differential equations. Hence, this book serves as a basic reference for researchers to harness this powerful technique further to unlock new insights and embrace the intricacies of natural processes. Researchers and graduate students at various levels interested in learning about the Stieltjes differential calculus and related fields will find this text a valuable resource of both introductory and advanced material.

Contents:
The Stieltjes IntegralThe Stieltjes DerivativeStieltjes Elementary FunctionsThe Stieltjes–Laplace TransformFirst-Order Linear Stieltjes Differential EquationsSecond-Order Linear Stieltjes Differential EquationsStieltjes Differential SystemsQualitative Analysis of Stieltjes Differential SystemsStability Theory for Stieltjes Differential SystemsLinear Stieltjes Boundary Value Problems
Readership: Can be used as a reference book for advanced undergraduate and graduate students, professors, and researchers in the field of differential equations and related areas. It can also be used as a textbook for a 'Special Topics' graduate course.

Svetlin G Georgiev is a Professor of Mathematics in the Department of Mathematics at Sorbonne University, Paris, France. He works on various aspects of mathematics. Currently, he focuses on ordinary and partial differential equations, harmonic analysis, integral equations, Clifford and quaternion analysis, time scales calculus, differential and dynamic geometry.Sanket Tikare is an Assistant Professor in the Department of Mathematics at Ramniranjan Jhunjhunwala College, which is an empowered autonomous college affiliated with University of Mumbai, Mumbai, Maharashtra, India. He did his MSc and PhD in mathematics both from Shivaji University, Kolhapur, Maharashtra, India in the years 2008 and 2012, respectively. He specializes in Applied Analysis. His research area of interest includes differential equations, difference equations, quantum difference equations, dynamic equations on time scales, impulsive systems, and equations with delays.