The dissertation by MSc M. Amine Zemirni is concerned with ordinary differential equations (ODE's) in a complex domain, i.e., differential equations with solutions and coefficients that are functions of a complex variable. The dissertation contains several new results that improve several old ones in the cases of the complex plane and the unit disc. These results aim at a more comprehensive understanding of the theory of complex ODE's. One of the principal aims is to find the asymptotic properties of solutions of complex ODE's and, in particular, to find the inherent asymptotic relationship between solutions and coefficients. In such a study, the former proof methods must be adjusted to conform to the research objectives. In this dissertation, the general tools and methods used to study complex ODE's lie under several theories, such as Nevanlinna's theory and Wiman-Valiron’s theory.
For linear ODE's, several results improve the well-known Frei's theorem on the number of solutions with infinite order in both cases of the complex plane and the unit disc. Furthermore, an accurate asymptotic growth for the logarithm of the maximum modulus of the solutions is obtained for second-order ODE's. The latter result is used to study nonlinear differential equations of a specific form. Another problem addressed in this dissertation concerns the effects of the coefficients' asymptotic growth in a second-order ODE on the growth of the solutions. New conditions are introduced based on Petrenko's deviations with respect to infinity and the linear measure of angles on which coefficient functions behave similarly as polynomials. The asymptotic integration theory of Hille takes place in this dissertation, where the missing details and defects in the original theory are covered.
The doctoral dissertation of Master of Science Mohamed Amine Zemirni, entitled Results on the asymptotic growth of solutions of complex ODE's, will be examined at the Faculty of Science and Forestry on November 20th at 12 o’clock. The opponent will be Professor Shamil Makhmutov, Sultan Qaboos University, Oman, and the custos will be Associate Professor Janne Heittokangas, University of Eastern Finland. The public examination will be held in English and it can be followed online.