Juan J. Nieto & Donal O'Regan
talk with ScienceWatch.com and answer a few questions
about this month's New Hot Papers in the field of
Mathematics.

Juan J. Nieto with Peter D. Lax,
Professor Emeritus of Mathematics at
the Courant Institute of Mathematical
Sciences at New York University.

Article Title: Variational approach to impulsive
differential equations Authors: Nieto,
JJ;O'Regan, D
Journal: NONLINEAR ANAL-REAL WORLD APP
Volume: 10
Issue: 2
Page: 680-690
Year: APR 2009
* Univ Santiago de Compostela, Fac Matemat, Dept Anal Matemat,
Santiago De Compostela 15782, Spain.
* Univ Santiago de Compostela, Fac Matemat, Dept Anal Matemat,
Santiago De Compostela 15782, Spain.
* Natl Univ Ireland, Dept Math, Galway, Ireland.

Why do you think your paper is highly
cited?

The paper combines both linear and nonlinear functional analysis. In
particular, we use the celebrated Lax-Milgram theorem with critical point
theory and the theory of impulsive dynamical systems. See photo of Peter D.
Lax pictured above with Juan. J. Nietro.

The paper shows that the solution for some discontinuous or impulsive
differential equations is given by minimizing an action functional
associated with the system.

Coauthor Donal O'Regan

Although the ideas in the paper are simple, they can be applied in a
variety of situations to the study of theoretical and applied problems.
Some of the authors citing our paper examine new problems using our
methodology.

Does it describe a new discovery, methodology, or
synthesis of knowledge?

We combined different settings to present a new approach to the study of
dynamical systems having an impulsive, dynamical behavior. It permits us to
consider discontinuous phenomena, which is difficult using classical tools.
In particular, we reveal the variational structure underlying some
impulsive differential equations.

Would you summarize the significance of your paper in layman's
terms?

Our paper uses the classical theory of functional analysis to show how
simply one can examine dynamical systems with discontinuous phenomena which
arise in real-life applications. In a sense, it unifies and harmonizes both
linear and nonlinear theory.

How did you become involved in this research, and were there any
problems along the way? Where do you see your research leading in the
future?

Functional analysis is one of the great contributions of mathematics in the
20^{th} century and the Lax-Milgram theorem is one of the
cornerstones in the study of nonlinear partial differential equations.

We both work in the area of nonlinear functional analysis. Over the years
we've had a special interest in the theory of discontinuous differential
equations which arise in real-world applications. In our paper we fused
both themes to obtain a new approach to the examination of impulsive
differential equations.

We feel our future research will continue to promote classical analysis in
the study of real-world applications. Research in mathematics usually has
some unexpected implications for the future and is really quite hard to
predict, but we are hopeful.

Professor Juan J. Nieto
Department of Mathematical Analysis
Faculty of Mathematics
University of Santiago de Compostela
Santiago de Compostela, Spain

Professor Donal O'Regan
Department of Mathematics
National University of Ireland
Galway, Ireland