Asymptotic Behaviour of Solutions of n-order Differential Equations with Regularly Varying Nonlinearities

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1 International Workshop QUALITDE 206, December 24 26, 206, Tbilisi, Georgi4 Asymptotic Behaviour of Solutions of n-order Differential Equations with Regularly Varying Nonlinearities K. S. Korepanova Odessa National University, Odessa, Ukraine Consider the differential equation y (n = αp(t φ j (y (j, ( where n 2, α {, }, p : [a, [ ]0, [ is a continuous function, a R, the φ j : Y j ]0; [ are continuous functions regularly varying as y (j Y j of order σ j, j = 0, n, Y j is a one-sided neighborhood of the point Y j, Y j {0, ± }. The equation ( is a particular case of the equation, comprehensively studied by V. M. Evtukhov and A. M. Klopot [3] m y (n = α k p k (t φ kj (y (j, k= where n 2, α k {, } (k =, m, the p k : [a, ω[ ]0, [ (k =, m are continuous functions, < a < ω, the φ kj : Y j ]0; [ (k =, m, j = 0, n are continuous functions regularly varying as y (j Y j of order σ j, Y j is a one-sided neighborhood of the point Y j, Y j is equal to either 0 or ±. From mentioned results necessary and sufficient existence conditions of the so-called P (Y 0,..., Y, λ 0 -solutions of equation ( can be obtained for all λ 0 ( λ 0. Moreover, asymptotic representations as t of such solutions and their derivatives of order up to n can be established. It follows from the definition of these solutions that j=0 j=0 t y(j (t = Y j {0, ± } (j = 0, n, t [y ( (t] 2 y ( (ty (n (t = λ 0. However, the set of the monotonous solutions of equation (, defined in some neighborhood of, also can have the solutions such that for each of them there exists a number k {,..., n} so that y (n k (t = c + o( (c 0 as t. (2 When k =, 2 or the functions φ i (y (i (i = n k +, n 2 tend to positive constants, as y (i Y i, a question on the existence of the solutions of type (2 of equation ( can be solved without any assumption on the its. Otherwise, we can not get asymptotic formulas of these solutions and their derivatives of order up to n directly from equation (. Some results concerning existence of solutions of type (2 were obtained in corollary 8.2 of the monograph by I. T. Kiguradze and T. A. Chanturia [2, Ch. II, 8, p. 207] for general type equations. But these results provide for considerably strict restriction on the (n k+-st derivative of solution. For Y j = ± here and in the following all numbers in the neighborhood Y j are assumed to have constant sign.

2 42 International Workshop QUALITDE 206, December 24 26, 206, Tbilisi, Georgia In order to receive new results with less strict restrictions on behaviour of this and following derivatives of order n in case, when k {3,..., n} and not all φ i (y (i (i = n k +, n 2 tend to positive constant as y (i Y i, let us introduce the following definition. Definition. The solution y of the differential equation ( is referred (for k {3,..., n} to as a P k (λ 0 -solution, where λ 0, if it is defined on the interval [t 0, [ [a, [ and satisfies the conditions t y(n k (t = c (c 0, t [y ( (t] 2 y ( (ty (n (t = λ 0. (3 It is obvious that by virtue of the first relative (3 for these solutions the following representations hold y (l ct n l k+ (t = [ + o(] (l =, n k as t (4 (n l k +! and c Y n k. It readily follows from the form of equation ( that y (n (t has constant sign in some neighborhood of. Then y (n l (t (l =, k are strictly monotone functions in neighborhood of and by virtue of (2 can tend only to zero as t. Therefore it is necessary that Y j = 0 for j = n k + 2, n. (5 Let us introduce the numbers µ j (j = 0, n if Y j =, or Y j = 0, and Y j is a right neighborhood of the point 0, µ j = if Y j =, or Y j = 0 and Y j is a left neighborhood of the point 0, such that µ j µ j+ > 0 for j = 0, n k, µ j µ j+ < 0 for j = n k +, n 2. (6 Besides, note that in some neighborhood of sign y (j (t = µ j (j = 0, n, sign y (n (t = α. (7 In this case along with (6 the following inequality hold αµ < 0. (8 Moreover, it follows from (4 that { if µ n k > 0, Y j = if µ n k < 0, for j =, n k. (9 These conditions on µ j (j = 0, n and α are necessary for existence of P(λ k 0 -solutions of equation (. The aim of the present paper is to obtain necessary and sufficient existence conditions of P(λ k 0 -solutions (k {3,..., n} of equation ( for λ 0 R \ {0, 2,..., k 3 k 2, }, and establish asymptotic as t formulas of their derivatives of order n. Moreover, the question on the quantity of studied solutions will be solved. It is significant that by virtue of the obtained results by V. M. Evtukhov [] studied solutions of equation ( hold the following a priori asymptotic conditions.

3 International Workshop QUALITDE 206, December 24 26, 206, Tbilisi, Georgi43 Lemm. Let k {3,..., n} and λ 0 R \ {0, 2,..., k 3 k 2, }. Then for each Pk (λ 0 -solution of equation ( the following asymptotic as t relations hold y (l (t [(λ 0 t] n l y ( (t (l = n k + 2, n, (0 i=l where y : [t 0, [ R is an arbitrary P k (λ 0 -solution of equation (, = (n iλ 0 (n i (i =, n. We say that a continuous function L : Y 0 ]0, [ slowly varying as y Y 0 satisfies condition S 0 if L(µe [+o(] ln y = L(y[ + o(] as y Y 0 (y Y 0, where µ = sign y. Condition S 0 is necessarily satisfied for functions L that have a nonzero finite it as y Y 0, for functions of the form where, 2 0, and for many other functions. We need the following auxiliary notations: L(y = ln y, L(y = ln y ln ln y 2, = σ j, ν = a 0j = (n jλ 0 (n j (j =, n, C = t I(t = φ n k (cm(c A σ j (n j, (λ 0 n j i=j+ p(ττ ν φ 0 (µ 0 τ n k φ n k (µ n k τ dτ, σ j, where A = if if p(ττ ν φ 0 (µ 0 τ n k φ n k (µ n k τ dτ = ±, p(ττ ν φ 0 (µ 0 τ n k φ n k (µ n k τ dτ <, a such that µ j t n k j+ Y j (j =, n k for t, M(c = n k j= c σ j. (n j k +! Theorem. Let 0, k {3,..., n} and λ 0 R \ {0, 2,..., k 3 k 2, }. Then for existence of P k (λ 0 -solutions of equation (, it is necessary that c Y n k, along with (5, (6, (8, (9 inequalities λ 0 <, a 0j+ > 0 (j = n k +, n 2 (

4 44 International Workshop QUALITDE 206, December 24 26, 206, Tbilisi, Georgia hold and the following condition be satisfied: ti (t t I(t = λ 0. (2 Moreover, each solution of that kind admits along with (2 and (4 the asymptotic representations (0 as t and L (y ( (t y ( (t ( L [(λ0 t] n j j i=j+ = αµ CI(t[ + o(]. y ( (t Theorem 2. Let 0, k {3,..., n}, λ 0 R \ {0, 2,..., k 3 k 2, } and functions L j (j = n k +, n slowly varying as y (j Y j satisfy condition S 0. In addition, let c Y n k and conditions (5, (6, (8, (9, ( and (2 hold. Then, in case of existence of P(λ k 0 -solutions of equation (, a 2 τ k 2 I(τL (µ τ a 0j λ 0 a 0j+ L j (µ j τ λ 0 dτ <, (3 where a 2 such that µ j t λ 0 Y j (j = n k + 2, n, µ t λ 0 Y for t a 2, and each solution of that kind admits along with (4 the following asymptotic representations as t where y (n k (t = c + µ (λ 0 k 2 W (t[ + o(], i=n k+2 y (l (t = µ (λ 0 n l t n l k+2 W (t[ + o(] (l = n k + 2, n, i=l y ( (t = µ W (t t k 2 W (t = t [ + o(], τ k 2 CI(τL (µ τ λ 0 a 0j+ L j (µ j τ λ 0 If the inequality σ holds along with mentioned conditions, then equation ( has at least one P(λ k 0 -solution that admits such representations. Moreover, for each c Y n k in case λ 0 ], k 2 k [ \{0, 2,..., k 3 k 2 } (λ 0 [ k 2 k ; [ there exists an (n k + -parameter ((n k- parameter, respectively family of solutions with such representations if σ >, and (n k- parameter ((n k -parameter if σ <. References [] V. M. Evtukhov, Asymptotic representations of solutions of nonautonomous differential equations. Doctoral (Phys.-Math. Dissertation, , Kiev, 998, 295 pp. dτ. (4

5 International Workshop QUALITDE 206, December 24 26, 206, Tbilisi, Georgi45 [2] I. T. Kiguradze and T. A. Chanturia, Asymptotic properties of solutions of non-autonomous ordinary differential equations. (Russian Nauka, Moskow, 990. [3] A. M. Klopot, Asymptotic representations of solutions of n-order differential equations with regularly varying nonlinearitites. Dissertation Cand. Phys.-Math. Nauk, 205, 48 pp.