Liapunov Functionals for Integral Equations

This is an introduction to integral equations from the ground up. While it is elementary in nature, being accessible to advanced undergraduate majors, it introduces a number of problems of current research interest. The real distinguishing feature is that Liapunov’s direct method is applied to the integral equation itself without first differentiating the equation. This is the first book to offer that topic and the ramifications are substantial. Basic linear and nonlinear theory is offered using elementary contraction mappings with extensive properties being derived from a substantial collection of Liapunov functionals. The linear methods advance in a natural way to nonlinear perturbations. Fixed point methods and Liapunov’s direct method are woven together throughout the book. The results are all qualitative in nature. It is shown that solutions exist, are often unique, and satisfy commonly sought properties such as boundedness, convergence to zero, periodicity, asymptotic periodicity, and various integrability properties. A wide range of Liapunov functionals are constructed to ensure these properties and these are used extensively in conjunction with fixed point theory. Many interesting, unusual, and important properties of the resolvent are given. The book should be of interest to investigators in biology, chemistry, economics, engineering, mathematics, and physics.

T. A. Burton received his Ph. D. from Washington State University in 1964. He went directly to the University of Alberta and then to Southern Illinois University. At SIU he received the American Oil Company award as Outstanding University Teacher, directed 13 doctoral dissertations, and was twice a Fulbright Senior Scholar to Eastern Europe, once as Senior Researcher, once as Senior Lecturer. He received brief research appointments at the University of Madrid and the University of Florence. For the last ten years he has resided in Port Angeles Washington, devoting full time to research, writing, lecturing, and editing math journals.