The Method Of Getting Solution Of Almost Periodic Type Differential Equation

This paper consists of two parts. In the first part, we investigate some method to solve differential equation of the form x'(t)=Ax(t)+f(t). In history, this equation has been investigated by many mathematicians. As a corollary of the Bohr-Neugebauer Theorem, in the case the real part of all eigenvalues of matrix A does not equal to zero, then the equation has a unique solution. If f is an almost periodic function in Bohr's sense, then the solution is also almost periodic in Bohr's sense. The general method is to transform matrix A to upper triangular's form, then one can get the solution successively. Using the constant variation formula, one can get the solution. After proving the boundedness of the solution, one can get the solution's almost periodicity from the function f . That the real part of all eigenvalues of matrix A does not equal to zero is essential. But in this article we regard matrix A as a whole . We can see that matrix A is the infinitesimal generator of a semigroup. So we apply the theory of semigroup of operators to solve this equation. If the real part of all eigenvalues are greater or less than zero, then the equation has a unique solution. Moreover, we also discuss the case that the function f is almost periodic in Besicovith's sense. When the eigenvalues of matrix A and the Fourier exponents of f satisfy some conditions, we show that the equation has a unique solution. At the same time the solution belongs to a smaller almost periodic function space.The second part focus on another functional equation x(t)=px(λt)+f(t)and discusses the existence and uniqueness of the solution. We get some conclusions. When the underlying space consists of all continuous and bounded functions from R to C and the function f is almost periodic in Bohr's sense, then the solution only depends on parameter p . When p's module does not equal to one, then the unique solution is also almost periodic in Bohr's sense. When the underlying space consists of all locally integrable maps from R to C satisfying a certain bounded condition in Stepanov's sense, then the solution depends on parameter p andλ. Whenλis greater than zero and p's module does not equal toλ, then the unique solution is also almost periodic in Stepanov's sense.