tions and almost periodic solutions by using Liapunov functions. Seifert [11 ...
almost periodic function, and hence we are not required to assume the
uniqueness.
Funkcialaj Ekvacioj, 12 (1969), 23-40
Asymptotically Almost Periodic Solutions
of an Almost Periodic System By Taro YOSHIZAWA
(Tohoku University)
1.
Introduction. Assuming the existence of a bounded solution with some stability property, several authors discussed the existence theorem for almost periodic solutions. In this direction, there are many interesting results for ordinary differential equations and for functional-differential equations. Fixed point theorems, Liapunov’s second method and the theory of dynamical systems etc. are applied to . obtain existence theorems for periodic solutions and almost periodic Hale [8] and the author [16, 17] have discussed the existence of periodic solutions and almost periodic solutions by using Liapunov functions. Seifert [11, 12] has applied Amerio’ result [1]. In these papers, they have assumed that a bounded solution is asymptotically stable. Under a weaker condition, properties of dynamical systems have been applied to the existence of almost periodic solutions. For a periodic system, Deysach and Sell [4] have shown the existence of an almost periodic solution under the assumption that the system has a bounded uniformly stable solution. MiUer [9] has considered almost periodic differential equations as dynamical systems and assumed that the system has a bounded solution which is totally stable. Seifert [13] also has applied a result of Deysach and Sell under the assumption that a bounded solution has a kind of stability property called -stability which is weaker than total stability. Sell [14, 15] has extended a result of Deysach and Sell to the existence of periodic solutions for a periodic system and to a more general nonautonomous system. All of them required the uniqueness of solutions for the initial value problem, because of dynamical systems. Recently, Coppel [3] has used properties of an asymptotically almost periodic function introduced by Frechet [5] and proved Miller’s result without the assumption that the solution is unique. To obtain existence theorems for an almost periodic solution, properties of an asymptotically almost periodic function were utilized by Reuter [10] for a second order differential equation and by Halanay [6, 7] for a quasi-linear system. In this paper, more generally, we shall discuss functional-differential equations and we shall see that all of results mentioned above will be obtained by using properties of an asymptotically $¥mathrm{s}¥mathrm{o}¥mathrm{l}¥mathrm{u}¥mathrm{t}¥mathrm{i}¥mathrm{o}¥dot{¥mathrm{n}}¥mathrm{s}$
