一类半线性微分方程的概反自守温和解

ALMOST ANTI-AUTOMORPHIC MILD SOLUTIONS TO A CLASS OF SEMI-LINEAR DIFFERENTIAL EQUATIONS

为了探讨一类半线性微分方程的概反自守温和解的存在唯一性,首先将概自守函数的概念推广到概反自守函数,其次讨论了概反自守函数的相关性质,证明了全体概反自守函数构成的集合,关于函数的加法与数乘构成一个线性空间,赋予了无穷范数之后,又成为一个Banach空间,最后利用不动点原理得到半线性微分方程概反自守温和解存在并且唯一的充分条件。由于概反自守函数是比概自守函数更精细的函数,所以对半线性微分方程概反自守温和解的存在性与唯一性的探讨,相较于对概自守温和解的存在性以及唯一性的探讨,能够更加精确地描述半线性微分方程的动力学性质,所得结论是新颖的,是现有结论的进一步完善与补充。

In order to investigate the existence and uniqueness of almost anti-automorphic mild solutions of semi-linear differential equations,the concept of almost automorphic functions is first extended to almost anti-automorphic functions.Secondly,the related properties of almost anti-automorphic functions are discussed in this paper.It is proved that the set of all almost anti-automorphic functions constitutes a linear space with respect to the addition and multiplication of functions,which is given an infinite norm and then becomes a Banach space.Finally,by using the fixed point principle,the sufficient conditions for the existence and uniqueness of almost anti-automorphic mild solutions to a kind of semi-linear differential equations are obtained.Since the almost anti-automorphic function is more precise than the almost automorphic function,the discussion on the existence and uniqueness of the almost anti-automorphic mild solution of semi-linear differential equation can more accurately describe the dynamic properties of similinear differential equation than the discussion on the existence and uniqueness of the almost automorphic mild solution,the obtained conclusion is novel and serves as a further refinement and supplement to existing conclusions.