15. Prove that the system of differential equations dx = y² - xy, dt dy = x²-x³y dt has no nonconstant periodic solution. 16. A function f satisfies the Lipschitz condition in a neighborhood of the origin in R" and f(0)=0. Denote by x(t, to, xo), t≥ to, the solution to Cauchy problem for the system d = f(x) under initial condition x (to) = xo. Prove that: (a) If zero solution x(t, to, d), t≥ to, is stable in the sense of Lyapunov for some to € R, then it is stable in the sense of Lyapunov for every to R and uniformly in to- (b) If zero solution x(t, to, ŏ), t≥ to, is asymptotically stable in the sense of Lyapunov then it holds lim ||x(t, to, xo) || = 0 uniformly in xo from some neigh- borhood of the origin in R". 1+00 2000 17 17. A function f: [1, +00) → [0, +∞) is Lebesgue measurable, and so f(x) dλ(x) < ∞ (here à denotes the Lebesgue measure). Prove that: ∞ (a) the series f(nx) converges for A-almost all x = [1, +∞0). n=1 (b) lim xf(x)dx(x) = 0. T→+∞ 18. Let & be a nonnegative random variable. Suppose that for every x ≥ 0, the expectations f(x) = E(x)+ ≤ ∞ are known. Evaluate the expectation Ees. (Here y denotes max(y, 0).)