Qualitative Study of Nonlinear Problems in Analysis.

Abstract

Abstract : The objective of the work was to use qualitative methods, especially fixed point theorems and topological degree, to study nonlinear problems in analysis. Existence of solutions of wide classes of abstract nonlinear functional equations are established and the existence of recurrent and period solutions of nonlinear ordinary differential equations are obtained. Upper bounds for the number of such solutions are also obtained. (Such upper bounds for the number of periodic solutions of nonlinear ordinary differential equations have become significant in mathematical studies of chemical reactions.) Several results in the stability of periodic solutions of nonlinear ordinary differential equations were also established. Several mathematical problems which arise in physiology were studied. The most specific conclusion obtained is a clinical recommendation for the treatment of patients suffering from a rupture of an aneurysm of the circle of Willis. Finally, two books have been written: a textbook on qualitative theory of ordinary differential equations and a set of lecture notes on the mathematics of cell electrophysiology. (Author)