By I. T. Kiguradze, T. A. Chanturia (auth.)

This quantity offers a accomplished evaluate of the advancements that have taken position over the past thirty years in regards to the asymptotic houses of options of nonautonomous traditional differential equations. The stipulations of oscillation of suggestions are verified, and a few normal theorems at the category of equations in keeping with their oscillatory homes are proved. moreover, the stipulations are discovered below which nonlinear equations would not have singular, right, oscillatory and monotone options.

The booklet has 5 chapters: bankruptcy I bargains with linear differential equations; bankruptcy II with quasilinear equations; bankruptcy III with basic nonlinear differential equations; and bankruptcy IV and V deal, respectively, with higher-order and second-order differential equations of the Emden-Fowler sort.

every one part comprises difficulties, together with a few which almost immediately stay unsolved. the amount concludes with an intensive checklist of references.

For researchers and graduate scholars drawn to the qualitative concept of differential equations.

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71) hold. Moreover, C1i) = WI = Ul, and (1. 70) is fulfilled. €2 for t E [a, b). 10. 79) (1 - €)n+lc8(1 - eo)n-l Cl ~ (n - I)! 80) to. 21. 82) Iw'(t)1 ::; € for t ~ to. 62), u(t) -=I- 0 for t ~ to, a E [to, +oo[ and b = a + w(a). 82), for t E [a, b) we have w(t) (b - a)n-k lb = w(a) + w'(e)(t - Ipk(t)1 dt ::; (b _ + (1- ~)n-k lb a) ~ (b - a)(1 - E), a)(~ _ €)n-k lb n k w - (t)lpk2(t)ldt::; 8 (b - at- 1 lb wn-k+l(t)lpkl(t)! dt (k = 2, ... ,n), Ip12(t)l dt ::; 8. 21 there exist functions ak that E C([a,b)) (k = 0, ...

Either n = 5 or n E {7,8, ... }). Then m~ < M~ (m... 10)). Now if c E ] - M~, -m~[ (c E ]m*n, M~n[), then the equation P~(x) + c = 0 (P*n(x) - c = 0) has real roots belonging to the interval ]0, n - 1[, as well as complex roots. 39) does not have property A (property B), but possesses an oscillatory solution. So, for higher order equations the problem on whether the equation has property A or B does not coincide with the problem on the existence of an oscillatory solution. 1. 1) is said to be nonoscillatory if its all solutions are nonoscillatory, and oscillatory if at least one solution is oscillatory.

16 1. 35) with u = VI + Wo is valid. Since to - 8 > Tn(to; p), we have u(t) > 0 for t E ]to, +00[. Besides, U(io-l) (to) = V~io-1) (to) + (OV~io-l) (to) = -(0 < O. Hence there exists t* > to such that U(io-l)(t*) < O. 38) Put V = u + WI. 38) we can find ( > 0 so that v(i-l)(t*) ~ (i = 1, ... ,n), 0 ftn(v;t*) > O. Thus, t* E E~*(to;p). But t* > T~*(to;p). This contradiction completes the proof in the case under consideration. Now let T~*(to; p) = +00. Assume Tn(tO; p) < +00. Then there exists t 1 E E~*(to; p) such that t 1 > Tn(tO;p).