Boundary Value Problems for Second-Order Finite Difference Equations and Systems

The mathematical modeling of many problems from economics, computer science, engineering, biological neural networks and others leads to the consideration of nonlinear difference equations. Many authors have studied such problems by using various methods, such as: the fixed point theory, the fixed point index theory, variational methods, the critical point theory, different transformations, extensions of Perron's second theorem, diversified criteria for the stability of solutions, and so on. 

This monograph studies the existence of positive solutions for some classes of second-order nonlinear finite difference equations, and systems of second-order nonlinear finite difference equations, subject to various multi-point boundary conditions. In the case of systems, these boundary conditions may be uncoupled or coupled. It also investigates a class of nonlinear 𝜈th order Atıcı-Eloe fractional difference equations supplemented with varied boundary conditions, and some systems of generalized second-order difference equations in Hilbert spaces with multi-point boundary conditions. The book draws together our results that have been obtained in the last years. Chapter 1 deals with the existence of positive solutions for two second-order finite difference equations which contain a linear term and a sign-changing nonlinearity, with or without parameters, subject to multi-point boundary conditions. Chapter 2 is focused on the existence and multiplicity of positive solutions for two systems of nonlinear second-order difference equations with uncoupled multi-point boundary conditions. The nonlinearities from the systems are nonnegative functions and satisfy some assumptions containing concave functions, or they are sign-changing functions. Chapter 3 studies the existence and nonexistence of positive solutions for two systems of nonlinear second-order difference equations supplemented with coupled multi-point boundary conditions, with positive parameters in the systems or in the boundary conditions. The nonlinearities of the systems are nonnegative functions and satisfy various assumptions. Chapter 4 is concerned with the existence and multiplicity of positive solutions for two systems of nonlinear second-order difference equations subject to coupled multi-point boundary conditions, without parameters. The nonlinearities of the systems are nonnegative functions and satisfy various assumptions. Chapter 5 is devoted to the existence of positive solutions for a system of nonlinear second-order difference equations with parameters and sign-changing nonlinearities, supplemented with multi-point coupled boundary conditions. Chapter 6 deals with the existence of nontrivial solutions, nonnegative solutions and positive solutions for a class of nonlinear 𝜈th order Atıcı-Eloe fractional difference equations with left focal boundary conditions or Dirichlet boundary conditions. In each chapter, various examples are presented which support the main results.

Finally, the new Chapter 7 investigates the existence and uniqueness of solutions for some nonlinear systems of generalized second-order difference equations in Hilbert spaces, subject to multi-point boundary conditions containing monotone operators. Some applications to initial-boundary value problems for nonlinear first-order differential systems with monotone operators are also addressed. The methods used in the proof of our theorems include results from the fixed point theory, the fixed point index theory, the theory of monotone operators and nonlinear evolution equations of monotone type in Hilbert spaces.

This monograph can serve as a good resource for the mathematical and scientific researchers, and for the graduate students in mathematics and science interested in the existence of solutions and positive solutions for finite difference equations and systems.

Johnny Henderson has focused on these topics for over 45-years, having  published over 700 refereed articles (many in highly indexed journals), along with publication of some 11 research monographs.  His research is highly known and cited (MathSciNet Citation database shows 5359 citations of his research by 2150 authors with an h-index = 32, and he is listed as a World’s Top 2% Researcher on the Stanford University Database), and he has given hundreds of invited research talks in colloquia and conferences  (often as a plenary  speaker).  He served on editorial boards of over 30 mathematical journals. He has received several awards of distinction for both teaching and research.  For example, in 2002, he received the Mathematical Association of America Southeastern Section Award for Distinguished Teaching, and in 2023, the Mathematical Association of America Texas Section Award for Distinguished Teaching. In 2016, he was named the (Baylor University) Cornelia Marschall Smith Professor of the Year. In addition, he is an Inaugural  Fellow of the American Mathematical Society.  He served on the faculties at the Missouri University of Technology, at Auburn University (17 years as an endowed Alumni Professor, and 2 years as the endowed Scharnagel Professor of Mathematics), and he retired in 2023 following his 21st year as the endowed Distinguished Professor of Mathematics at Baylor University. He also held fully funded research fellowships at Tamkang University (Taiwan), University of Queensland (Australia) and the University of New South Wales (Australia). 

Rodica Luca is Full Professor of Mathematics at “Gheorghe Asachi” Technical University of
Iasi, Romania. She received her PhD in mathematics from “Alexandru Ioan Cuza” University
of Iasi in 1996, and her Habilitation Certificate in mathematics from the School of Advanced
Studies of the Romanian Academy, Bucharest in 2017. She received the "Spiru Haret" Prize
of the Romanian Academy in 2018 for her book published by Elsevier in 2016 written in
collaboration with professor Johnny Henderson, the "Award for the researcher with the best performances in scientific research" and the "Gheorghe Asachi" Medal in 2020 from the management of the "Gheorghe Asachi" Technical University of Iasi for the scientific activity carried out during 2017-2019, and 40 prizes in the "National Research, Development and Innovation Plan - Awards for Research Results" program for her papers. She has published more than 190 papers. Her research interests are boundary value problems for nonlinear systems of ordinary differential equations, finite difference equations, and fractional differential equations, and initial-boundary value problems for nonlinear hyperbolic systems.