Atlantis Briefs in Differential Equations – serie

The book aims to provide an unifying view of a variety (a 'zoo') of mathematical models with some kind of singular nonlinearity, in the sense that it becomes infinite when the state variable approaches a certain point. Up to 11 different concrete models are analyzed in separate chapters. Each chapter starts with a discussion of the basic model and its physical significance. Then the main results and typical proofs are outlined, followed by open problems. Each chapter is closed by a suitable list of references. The book may serve as a guide for researchers interested in the modelling of real world processes.

This monograph covers the existing results regarding Green’s functions for differential equations with involutions (DEI).The first part of the book is devoted to the study of the most useful aspects of involutions from an analytical point of view and the associated algebras of differential operators.The work combines the state of the art regarding the existence and uniqueness results for DEI and new theorems describing how to obtain Green’s functions, proving that the theory can be extended to operators (not necessarily involutions) of a similar nature, such as the Hilbert transform or projections, due to their analogous algebraic properties.Obtaining a Green’s function for these operators leads to new results on the qualitative properties of the solutions, in particular maximum and antimaximum principles.

This book offers the reader a new approach to the solvability of boundary value problems with state-dependent impulses and provides recently obtained existence results for state dependent impulsive problems with general linear boundary conditions. It covers fixed-time impulsive boundary value problems both regular and singular and deals with higher order differential equations or with systems that are subject to general linear boundary conditions.  We treat state-dependent impulsive boundary value problems, including a new approach giving effective conditions for the solvability of the  Dirichlet problem with one state-dependent impulse condition and we show that the depicted approach can be  extended to problems with a finite number of state-dependent impulses. We  investigate  the Sturm–Liouville boundary value problem for a more general right-hand side of a differential equation. Finally, we offer  generalizations to higher order differential equations or differential systems subject to general linear boundary conditions.