方程 λ(1+ ce^(-τλ)+a+ be^(-τλ)= 0所有根具有负实部的充要条件

NECESSARY AND SUFFICIENT CONDITIONS FOR ALL ROOTS FOR NEUTRAL EQUATION λ(1 +ce^(-τλ) + a + be^(-τλ)=0 HAVING NEGATIVE REAL PARTS

考虑方程λ（１＋ｃｅ^（－τλ）＋ａ＋ｂｅ^（－τλ）＝０，（２）其中ａ，ｂ和ｃ为任意常数，τ为正常数，ｃ≠０．方程（１）为中立型方程 ｘ（ｔ）＋ｃｘ（ｔ－τ）＋ａｘ（ｔ）＋ｂｘ（ｔ－τ）＝０ （２）的特征方程．方程（１）为一常见的拟多项式方程．关于拟多项式函数， Ｐｏｎｔｒｙａｇｉｎ在 １９４２年给出了判断这类函数所有零点位于左半复平面的充要条件．但对中立型方程来说，由于这些条件往往难以验证，使得人们长期以来无法用Ｐｏｎｔｒｙａｐｉｎ定理找出方程（１）所有根具有负实部的充要条件．本文在克服了上述困难后，用Ｐｏｎｔｒｇｉｎ定理找出方程（１）所有根具有负实部的充要条件．

Consider the equation λ(1+ce^(－τλ)+a+be^(- τλ）＝０where a, b and c are real constants, T is a positive constant, c ≠0. Eq.(1) is a quasi-polynomial equation. In [1], Pontryagin gave the necessary and sufficient conditions for all zeros of quasipolynomial functions to lie in the left half-plane. As for neutral equations, however, these conditions are difficult to verify. In this paper, we have succeeded in verifying conditions given by Pontryagin and thus have obtained the necessary and sufficient conditions for all roots of Eq.(1) to have negative real parts.