Problem 5.1 Before proceeding to the proof of the four assertions, let us observe that, in the case of the equation considered, every fundamental matrix is of type 1 1 and of the form

for every , where .

1. In view of Theorem 5.2.2, the null solution of the equation considered is stable if and only if there exists a fundamental matrix bounded on , or equivalently every fundamental matrix is bounded on . According to the remark from the beginning, this happens if and only if

   (*)

   for every , t ₀ ?t. If the mentioned inequality is satisfied, we have x(t) ?e ^{K (0)} for every , and therefore x is bounded on . Hence the null solution is stable. Conversely, if there exists M>0 such that x(t) ? M for every , then from (*), one observes that the function , which satisfies the inequality in question, can be taken

   
   

   for every .

2. By virtue of Theorem 5.2.4, the null solution of the equation is uniformly stable if and only if there exists a fundamental matrix (t) and there exists M>0 such that for every , t ₀ ? t. According to the initial remark, the null solution is uniformly stable if and only if

   
   

   for every , t ₀ ? t, or equivalently

   
   

   for every , t ₀ ? t.

3. According to Theorem 5.2.2, the null solution is asymptotically stable...