Research On Qualitative Theory Of Functional Differential Equations

In this thesis, we discuss several problems which exist to qualitative theory of the functional differential equation, within which we develop further two important theories, so-called the 3/2 - stability theory and the distribution of zeros of solutions of the functional differential equation.Based on the foundational theories of these two topics, we bring out several new front problems. In this thesis, we mainly refer to the similar inferential reasoning method on the classical theories to solve these problems in method of the similar inferential reasoning to achieved the improvement of the theories.In the second chapter, considering a certain class of neutral differential equation with unbounded delay, we achieve such optimized conditions that the 3/2 - stability theorem is improved. First of all, based on the idea of expanding the applicable scope of the stability theory, we improve the structure of the conventional neutral differential equation and also develop its delay item from a constant to a variable. Secondly, based on the idea of improving the 3/2 - conditions, we mainly refer to the idea of X. H. Thang who achieved the optimized 3/2 - conditions in the neutral differential equation with a constant delay in 2001. We transfer his idea into the certain class of neutral differential equation with unbounded delay to improve its 3/2 - conditions and prove the theories we obtain. Thus, the stability theorem for a certain class of neutral differential equationwith unbounded delay contains that of the conventional neutral differential equation and rich the stability theories as well.In the third chapter, after reading many references on the distribution of zeros of solution of the functional differential equation, we emphatically analyze the literature by B. G. Zhang in 2000, who established the theory for the functional differential equation with unbounded delay instead of that with constant delay, which reduced the request on delay item to expand the research scope. His theory involves to solve the distribution of zeros of solutions of the functional differential equation with constant delay as well as more variable items. Therefore, in this thesis, we transfer this idea into advanced equations. We define the advanced variable similar to what B. G. Zhang did for unbounded delay and improve the lemmas. Finally, we establish the distribution of zeros of solutions of functional differential equation with an advanced variable.Wang bin (Applied mathematics). Supervised by Ye haiping.