By M. H. Lantsman (auth.)

The asymptotic concept bargains with the problern of picking the behaviour of a functionality in a local of its singular element. The functionality is changed by means of one other identified functionality ( named the asymptotic functionality) shut (in a feeling) to the functionality into account. Many difficulties of arithmetic, physics, and different divisions of ordinary sci ence convey out the need of fixing such difficulties. this day asymptotic conception has develop into an incredible and self reliant department of mathematical research. the current attention is especially in keeping with the speculation of asymp totic areas. each one asymptotic house is a set of asymptotics united by way of an linked genuine functionality which determines their progress close to the given element and (perhaps) another analytic houses. the most contents of this booklet is the asymptotic concept of standard linear differential equations with variable coefficients. The equations with energy order development coefficients are thought of intimately. because the software of the idea of differential asymptotic fields, we additionally think about the subsequent asymptotic difficulties: the behaviour of particular and implicit features, wrong integrals, integrals depending on a wide parameter, linear differential and distinction equations, and so on .. The received effects have an self sustaining that means. The reader is thought to be acquainted with a entire process the mathematical research studied, for example at mathematical departments of universities. additional useful details is given during this ebook in summarized shape with proofs of the most aspects.

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Then G is called a multiplicative abelian group. Instead of the zero element (}, we have the unity element I with the property ai = a for any a E G. Instead of the opposite element -a, we have the inverse element a-I such that a-Ia = I. 26 7. ASYMPTOTICS OF LINEAR DIFFERENTIAL EQUATIONS METRIC GROUPS. 30. Let G be an additive abelian group. We say that G is a metric group if for any (fixed) a E G it is assigned a non-negative number p( a) satisfying the following conditions (axioms): (1) p(a) = 0 if and only if a = 0; (2) for any a, b E G, p(a + b) :S p(a) + p(b) (triangle axiom).

And supm[h{f(m)(t)} + m]=p. PROOF. Properties (1) and (2) are obvious. Prove the inequality II{f'(t)} ::; p- 1. Choose a number r > p. Clearly, j(t)cr E Ct. Therefore (by the definition) (f(t)cr)'t = f'(t)cr+l- r f(t)cr E Ct. Hence f'(t)cr+l E Ct. This leads to the relation II{f'(t)}::; r- 1. Taking into account the arbitrariness of r, we obtain II{f'(t)} ::; p- 1. Prove property (4). Let p ~ -1. Then f(t) = o(tP+c) fort-+ +oo. Hence ll: f(s)dsl ::; l: sP+cds = O(tP+l+c) for t-+ +oo. If p < -1, in the same way (for any c < - p - 1) litoo f(s)dsl ::; loo sP+cds = O(tP+l+c).

45. 24) 32 ASYMPTOTICS OF LINEAR DIFFERENTIAL EQUATIONS PROOF. The function lif(t)ll is also continuous on [a,ß], and for a T-partition P = [to, t1, ... , tn], we have n-1 IIS(P, f(t))ll = L f(tj)(tj+l- tj) j=O n-1 ~ 2: IIJ(tj)ll(tj+l- tj) j=O = S(P, llf(t)ll). 25). 46. Let c E [a, ß], then 1: f(s)ds = ic f(t)dt + lß 0 f(t)dt. 26) Choose a sequence of Tn-partitions S(Pn, f(t)) on the segment [a, ß] such that any partition Pn contains the point c. Then Pn = P~ + P:: where P~ and P:: are Tn-partitions of the segments [a,c) and [c,ß) respectively.