New ”orders of infinity”. (English) Zbl 0539.26003

In the paper the author continues the development of the theory initiated in his previous paper reviewed above. He extends the main results related to the field E (for notations see the preceding review) to the more general, perfect fields (a subfield K of the ring CB is called perfect if it coincides with the intersection of all maximal differential subfields of CB containing K). For example, Lemma 11.6. Let K be a perfect field. Then 1) \(E\subset K\); 2) K is a real closed field; 3) K is closed under integration; 4) if \(f\in K\), then \(e^ f\in K\); 5) if f is a non-zero function in K, then ln \(| f| \in K\); 6) if \(f\in K\) and f(x) tends to a finite number as \(x\to +\infty\), then sin f and cos f also belong to K; 7) if \(f\in E\), \(g\in K\) and \(\lim_{x\to +\infty} g(x)=+\infty,\) then \(f{\mathbb{O}}g\in K.\) The author gives upper bounds of the speed with which a function, satisfying an algebraic differential equation over \({\mathbb{R}}\) and more general fields of functions, may tend to infinity. For example, if f(x) is contained in some differential subfield of B and satisfies an algebraic differential equation of order k over \({\mathbb{R}}\), then \(f(x)\ll \exp_ k(x^ n)\) for some n (where \(\exp_ k(x)\) means the k-th iterate of \(e^ x)\). For the functions from E are given more precise bounds. Theorem 13.2. Let \(f\in E\) and assume that \(\lim_{x\to +\infty} f(x)=+\infty.\) Then there exists a natural number k such that \(\ln_ k(x)\ll f(x)\ll \exp_ k(x).\)
As in the previous paper the author applies the obtained results to study the growth properties of real solutions of differential equations. He gives necessary and sufficient conditions for the differential equation \(\ddot y+\phi(x)y=0(\phi(x)\in E)\) to have no oscillatory solutions at infinity. In the paper the differential fields of function at any finite point are studied also. Some connections between such fields and preceding fields are observed. The author also generalizes the main result of a paper by O. A. Gel’fond and A. G. Khovanskij [Funkts. Anal. Prilozh. 14, No.2, 52-53 (1980; Zbl 0468.30006)]. In conclusion he formulates three new conjectures. Here is one of them: Let \(\phi\) and \(k\in E\). Then the differential equation \(\ddot y+\phi(x)y=k(x)\) has a solution in E.
Reviewer: N.V.Grigorenko


26A12 Rate of growth of functions, orders of infinity, slowly varying functions
12H05 Differential algebra
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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