Memoirs on Differential Equations and Mathematical Physics

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1 Memoirs on Differential Equations and Mathematical Physics Volume 6, 04, 37 6 V. M. Evtukhov and A. M. Klopot ASYMPTOTIC BEHAVIOR OF SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS OF n-th ORDER WITH REGULARLY VARYING NONLINEARITIES

2 Abstract. The conditions of existence and asymptotic for t ω ω + ) representations of one class of monotonic solutions of n-th order differential equations containing in the right-hand side a sum of terms with regularly varying nonlinearities, are established. 00 Mathematics Subject Classification. 34D05, 34C. Key words and phrases. Ordinary differential equations, regularly varying nonlinearities, asymptotics of solutions. ÒÄÆÉÖÌÄ. n-öòé ÒÉÂÉÓ ÜÅÄÖËÄÁÒÉÅÉ ÃÉ ÄÒÄÍÝÉÀËÖÒÉ ÂÀÍÔÏ- ËÄÁÄÁÉÓÀÈÅÉÓ, ÒÏÌÄËÈÀ ÌÀÒãÅÄÍÀ ÌáÀÒÄÄÁÉ ßÀÒÌÏÀÃÂÄÍÄÍ ÒÄÂÖËÀ- ÒÖËÀà ÝÅÀËÄÁÀÃÉ ÀÒÀßÒ ÉÅÏÁÉÓ ÌØÏÍÄ ßÄÅÒÈÀ ãàìó, ÃÀÃÂÄÍÉËÉÀ ÂÀÒÊÅÄÖËÉ ÊËÀÓÉÓ ÌÏÍÏÔÏÍÖÒ ÀÌÏÍÀáÓÍÈÀ ÀÒÓÄÁÏÁÉÓ ÓÀÊÌÀÒÉÓÉ ÐÉÒÏÁÄÁÉ ÃÀ ÌÉÙÄÁÖËÉ ÌÀÈÉ ÀÓÉÌÐÔÏÔÖÒÉ ßÀÒÌÏÃÂÄÍÄÁÉ.

3 Asymptotic Behavior of Solutions of ODEs Introduction The theory of regularly varying functions created by J. Karamata in 930 has been later see, for example, monographs [], []) extensively developed and widely used in various mathematical researches. Particularly, the last decades of the past century is mentioned by a great interest in studying regularly and slowly varying solutions of various differential equations and in equations of the type y = α 0 pt)φy), where α 0 {, }, p : [a, + [ ]0, + [ is a continuous function and φ : Y0 ]0, + [ is a regularly varying continuous function of order σ as y Y 0 ; here Y 0 equals either zero or ±, and Y0 is a one-sided neighborhood of Y 0. Among the researches carried out within that period and dedicated to determination of asymptotics as t + of monotonic solutions for such equations, of special mention are the works [3], [4] and the monograph [5]. Here, according to the definition of regularly varying function see E. Seneta [, Ch., Sect.., pp. 9 0]), φy) = y σ Ly), where L is slowly varying as y Y 0 function, i.e., the condition y Y 0 y Y0 Lλy) Ly) = with any λ > 0 is satisfied. Considering such representation for φ, such class of equations is a natural extension of the class of generalized second order Emden Fowler equations y = α 0 pt) y σ sign y. The basic results dealing with asymptotic properties of solutions for the second- and n-th order Emden Fowler equations, obtained before 990, can be found in the monograph due to I.T. Kiguradze and T.A. Chanturiya [6, Ch. IV, V, pp ]. The works [7] [6], dedicated to the determination of asymptotics of monotonic differential equations of second and higher orders with power nonlinearities are also worth mentioning. For the last decade, the results obtained in [7] [] and also those obtained in [] [6] were applied to differential equations m y = α 0 pt)φ 0 y)φ y ), y = α k p k t)φ k0 y)φ k y ), k= y n) = α 0 pt)φy) n ) with nonlinearities, regularly varying as y Y 0 and y Y, where Y i {0; ± } i = 0, ), and with some additional restrictions to nonlinearity for the first two equations.

4 40 V. M. Evtukhov and A. M. Klopot In the present paper we consider the following differential equation: m y n) = α k p k t) φ kj y j) ),.) k= where n, α k { ; } k =, m), p k : [a, ω[ ]0, + [ k =, m) are continuous functions, φ kj : Yj ]0, + [ k =, m; j = 0, n ) are continuous and regularly varying as y j) Y j functions of orders σ kj, < a < ω +, * Yj is one-sided neighborhood of Y j, Y j equal either to 0 or to ±. It is assumed that numbers ν j j = 0, n ) determined by, if either Y j = +, or Y j = 0 and Yj -right neighborhood of 0, ν j =.), if either Y j =, or Y j = 0 and Yj -left neighborhood of 0, are such that ν j ν j+ > 0 with Y j = ± and ν j ν j+ < 0 with Y j = 0 j = 0, n )..3) Such conditions for ν j j = 0, n ) are necessary for the equation.) to have solutions defined in the left neighborhood of ω, each of which satisfying the conditions y j) t) Yj with t [t 0, ω[, y j) t) = Y j j = 0, n )..4) Among strictly monotonic, with derivatives up to the n order inclusive, in some left neighborhood of ω, solutions of equation.) these ones are of special academic interest, because each of the rest ones admits only one representation of the type yt) = πω k t)[c k + o)] k =, n), where c k k =, n) are the non-zero real constants and { t, if ω = +, π ω t) = t ω, if ω < +..5) The question on the existence of solutions of.) with similar representations may be solved, in a whole, in a rather simple way by applying, for example, Corollary 8. for ω = + from the monograph of I. T. Kiguradze and T. A. Chanturiya [, Ch. II, p. 8, p. 07] and the schemes from the works [0], [] as ω +. As for the solutions with properties.4), for lack of particular representations for them, there arises the necessity to single out a class of solutions admitting one to get such representations. * if a >, then ω = +, and ω < a < ω if ω < +.

5 Asymptotic Behavior of Solutions of ODEs... 4 One of such rather wide classes of solutions has been introduced in [4] [6] dedicated to generalized Emden Fowler type equations of n-th order, y n) = α 0 pt) y j) σ j. For the equation.), this class is determined as follows. Definition.. A solution y of the equation.) defined on the interval [t 0, ω[ [a, ω[, is called a P ω Y 0,..., Y, λ 0 )-solution, where λ 0 +, if along with.4) the condition [y ) t)] y n ) t)y = λ n) 0.6) is satisfied. If y is a solution of the differential equation.) with properties.4) and the functions ln y ) t) and ln π ω t) are comparable with order one see [3, Ch. 5, Sect. 4,5, pp ]) as t ω, then it is easy to check that this solution is the P ω Y 0,..., Y, λ 0 )-solution for some λ 0 depending on the value of π ωt)y n) t). y ) t) Moreover, using assertions,, 5 and 9 on the properties of regularly varying functions) from the monograph [5, Appendix, pp. 5 7], it can be verified that in the case of regularly varying as t ω coefficients p k k =, m) of the equation.), each of its regularly varying as t ω solutions with properties.4) is a P ω Y 0,..., Y, λ 0 )-solution for some final or equal to ± value λ 0. The aim of this note is to determine the conditions for existence of P ω Y 0,..., Y, λ 0 )-solutions of.) in special cases, where λ 0 = n i n i as i {,..., n }, and also asymptotic representations as t ω for such solutions and their derivatives up to and including n order. By virtues of the results from [6], these solutions of the equation.) possess the following a priori asymptotic properties. Lemma.. Let y : [t 0, ω[ Y0 be an arbitrary P ω Y 0,..., Y, λ 0 )- solution of the equation.). Then: ) if n > and λ 0 = n i n i for some i {,..., n }, then for t ω, y k ) t) [π ωt)] i k y i ) t) k =,..., i ) *, i k)! y y i) t) ) t) = o, π ω t).7) y k) t) ) k i k i)! [π ω t)] k i yi) t) k = i +,..., n);.8) At i = these relationships do not exist.

6 4 V. M. Evtukhov and A. M. Klopot ) if n and λ 0 = 0, then for t ω, y k ) t) [π ωt)] n k n k )! yn ) t) k =,..., n ) *, y y ) n ) t) ).9) t) = o π ω t) and, in the case of existence of finite or equal to ± ) it π ω t)y n) t), y ) t) y n) t) π ω t) y ) t). Statement of the Main Results with t ω..0) In order to formulate the theorems, we will need some auxiliary notation and one definition. By virtue of the definition of regularly varying function, the nonlinearity in.) is representable in the form φ kj y j) ) = y j) σ kj L kj y j) ) k =, m; j = 0, n ),.) where L kj : Yj ]0, + [ are continuous and slowly varying as y j Y j functions, for which with any λ > 0 y j) Y j y j) Yj L kj λy j) ) L kj y j) ) = k =, m; j = 0, n )..) It is also known see [, Ch., Sect.., pp. 0 5]) that the its.) are uniformly fulfilled with respect to λ on any interval [c, d] ]0, + [ property M ) and there exist continuously differentiable slowly varying as y j) Y j functions L 0kj : Yj ]0, + [ property M ) such that y j) Y j y j) Yj L kj y j) ) L 0kj y j) ) = and y j) Y j y j) Yj y j) L 0kj yj) ) = 0.3) L 0kj y j) ) k =, m; j = 0, n ). Definition.. We say that a slowly varying as z Z 0 function L : Z0 ]0, + [, where Z 0 either equals zero, or ±, and Z0 is one-sided neighborhood of Z 0, satisfies condition S 0, if L νe [+o)] ln z ) = Lz)[ + o)] with z Z 0 z Z0 ), where ν = sign z. At n = these relationships do not exist.

7 Asymptotic Behavior of Solutions of ODEs Remark.. If the slowly varying as z Z 0 function L : Z0 ]0, + [ satisfies the condition S 0, then for every slowly varying as z Z 0 function l : Z0 ]0, + [, Lzlz)) = Lz)[ + o)] when z Z 0 z Z0 ). The validity of this statement follows from the theorem of representation see [, Ch., Sect.., p. 0]) of slowly varying function l and property M of function L. Remark. see []). If slowly varying as z Z 0 function L : Z0 ]0, + [ satisfies condition S 0, then the function y : [t 0, ω[ Y0 is continuously differentiable and such that yt) = Y 0, y t) yt) = ξ t) [r + o)] when t ω, ξt) where r is the non-zero real constant, ξ is continuously differentiable in some left neighborhood of ω real function, for which ξ t) 0, then Lyt)) = L ν ξt) r) [ + o)] when t ω, where ν = sign yt) in the left neighborhood of ω. Remark.3. If slowly varying as z Z 0 function L : Z0 ]0, + [ satisfies condition S 0 and the function r : Z0 K R, where K is compact in R m, is such that then rz, v) = 0 unifornly with respect to v K, z Z 0 z Z0 Lνe[+rz,v)] ln z ) = z Z 0 Lz) z Z0 uniformly with respect to v K, where ν = sign z. Indeed, if it shouldn t be true, then there would exist a sequence {v n } K and a sequence {z n } Z0 converging to Z 0 such that the inequality inf Lνe[+rz n,v n )] ln z n ) > 0.5) n + Lz n ) is fulfilled. Thus it is clear that there is the function v : Z0 K such that vz n ) = v n. For this function it is obvious that z Z 0 rz, vz)) = 0 and hence z Z0 Lνe[+rz,vz))] ln z ) =, z Z 0 Lz) z Z0 which contradicts the inequality.5).

8 44 V. M. Evtukhov and A. M. Klopot Finally, let us introduce auxiliary definitions assuming i µ ki =n i + σ kj i j ) γ k = σ kj, C ki = i [i j )!] σ kj n i)! t J ki t)= A ki p k s) π ω s) µki J kii t) = t A kii j=i+ σ kj j i) k =, m; i=, n), γ ki = σ kj k =, m; i =, n ), J ki s) j=i j=i+ [j i)!] σ kj k =, m; i=, n ), L kj νj π ω s) i j ) ds k =, m; i=, n), γ ki ds k =, m; i =, n), where each of the its of integration A km, A kmm m {0, }) is chosen equal to the point a 0 [a, ω[ on the right of which, i.e., as t [a 0, ω[, the integrand function is continuous) if under this value of its of integration the corresponding integral tends to ± as t ω, and equal to ω if at such value of its of integration it tends to zero as t ω. Theorem.. Let n >, i {,..., n } and for some s {,..., m} the inequalities sup < β ln p k t) ln p s t) β ln π ω t) < σ sj σ kj )i j ) at all k {,..., m} \ {s},.6 i ) be fulfilled, where β = sign π ω t) for t [a, ω[. Moreover, let γ s 0 and the functions L sj for all j {0,..., n } \ {i } satisfy condition S 0. Then for the existence of P ω Y 0,..., Y, n i n i )-solutions of the equation.) it is necessary, and if algebraic equation j=i+ σ sj j i)! j i m ρ) + σ si = m= n i n i)! m= m ρ).7) has no roots with zero real part it is sufficient that along with.3)) the inequalities ν j ν j i j)π ω t) > 0 at all j {,..., n } \ {i}, ν i ν i γ s J sii t) > 0,.8 i ) ν i α s ) n i π n i ω t) J si t) > 0.9 i )

9 Asymptotic Behavior of Solutions of ODEs be fulfilled in some left neighborhood of ω, as well as the conditions ν j π ω t) i j = Y j at all j {,..., n} \ {i}, ν i J sii t) γsi γs = Y i,.0 i ) π ω t)j si t) π ω t)j sii =, t) = 0.. i ) J si t) J sii t) Moreover, each solution of that kind admits as t ω the asymptotic representations y j ) t) = [π ωt)] i j y i ) t)[ + o)] j =,..., i ), i j)!. i ) y j) t) = ) j i j i)! [π ω t)] j i γsij sii t) γ s J sii t) yi ) t)[ + o)].3 i ) j = i,..., n ), y i ) t) γ s L si y i ) t)) = C si γ s J sii t) [ + o)] with t ω,.4i ) and in case ω = + there is i + -parameter family of solutions if the inequality ν i ν i γ s > 0 is valid, and i + l-parameter family if the inequality ν i ν i γ s < 0 is valid, in case ω < +, there is r+-parameter family if the inequality ν i ν i γ s > 0 is valid, and r-parameter family if the inequality ν i ν i γ s < 0 is valid, where l is a number of roots of the equation.7) with negative real part and r is a number of its roots with positive real part. Remark.4. Algebraic equation.7) has a fortiori no roots with zero real part, if n σ sj < σ s. j=i In Theorem., asymptotic representation for y i ) is written implicitly. The following theorem shows an additional restriction under which this representation may be presented explicitly. Theorem.. If the conditions of Theorem. are fulfilled and a slowly varying at y i ) Y i function L si satisfies condition S 0, then for each P ω Y 0,..., Y, n i n i )-solution of the equation.), asymptotic representations. i ),.3 i ) and hold when t ω. y i ) γsi t) = ν i C si L si ν i J sii t) γs ) γs γ s γs J sii t) [ + o)].5 i )

10 46 V. M. Evtukhov and A. M. Klopot 3. Proof of Theorems Proof of Theorem.. Necessity. Let y : [t 0, ω[ Y0 be an arbitrary P ω Y 0,..., Y, n i n i -solution of the equation.). Then the conditions.4) are satisfied, there is t [a, ω[ such that ν j y j) t) > 0 j = 0, n ) for t [t, ω[ and by Lemma., the asymptotic relations.7),.8) hold. From.7) and.8) we obtain the relations y j) t) y j ) t) = i j + o) π ω t) j =, n) when t ω 3. i ) and therefore ln y j ) t) = [ i j + o) ] ln π ω t) j =, n) when t ω. 3. i ) By virtue of 3. i ), the first of inequalities.8 i ) are fulfilled, and by virtue of 3. i ), the first of conditions.0 i ) are satisfied. Taking into account 3. i ), the representations.) and the conditions y j) Y j y j) y j ln L kj y j) ) ln y j) = 0 k =, m, j = 0, n ), 3.3) which are satisfied due to the properties of slowly varying functions see [, Ch., p..5, p. 4]), we find that ln φ kj y j) t)) = σ kj ln y j) t) + ln L kj y j) t)) = = [ σ kj + o) ] ln y j) t) = [ σ kj i j ) + o) ] ln π ω t) k =, m, j = 0, n ) when t ω. That is why for each k {,..., m} \ {s}, [p k t) φ kj y j) t))] ln =ln p kt) p s t) p φ sj y j) s t) + [ ] ln φ kj y j) t) ln φ sj y j) t) = t)) = ln p kt) p s t) + ln π [ ωt) σkj σ sj )i j ) + o) ] = [ ln pk t) ln p s t) = β ln π ω t) + β β ln π ω t) ] σ kj σ sj )i j ) + o) as t ω. Since the expression, appearing on the right of this correlation, by virtue of.6 i ) and the type of the function π ω from.5), tends to when t ω,

11 Asymptotic Behavior of Solutions of ODEs therefore p k t) φ kj y j) t)) = 0 at all k {,..., m} \ {s}. 3.4) p s t) φ sj y j) t)) Then from.) it follows that this solution implies asymptotic relation y n) t) = α s p s t)[ + o)] φ sj y j) t)) when t ω. 3.5) Here, for all j {0,..., n } \ {i }, the functions L sj in the representations.) of functions φ sj satisfy the condition S 0. Therefore, by virtue of 3. i ) and Remark., for them we have L sj y j) t)) = L sj νj π ω t) i j ) [ + o)] when t ω. Taking into account.) and the above representations, we can rewrite 3.5) in the form y n) t) = α s p s t)y i ) t) σ si L si y i ) t)) y j) t) σ sj L sj νj π ω t) i j )) [ + o)] at t ω. Hence, using.7),.8) and bearing in mind the fact that according to 3. i ), y n) t) = yn) t) y ) t) yi+) t) y i+) t) yi+) t) )n i n i)! π n i y i+) t) at t ω, t) ω and the notation introduced before formulation of theorems, we get the following relation: y i+) t) y i) t) y i ) t) γ slsi y i ) t)) = = α s ) n i sign[π ω t)] n i ) C si pt) π ω t) µsi L sj νj π ω t) i j ) [ + o)] at t ω. 3.6) By virtue of property M of slowly varying functions, there is a continuously differentiated function L 0si : Yi ]0, + [ satisfying the

12 48 V. M. Evtukhov and A. M. Klopot conditions.3) for k = s and j = i. Using these conditions and 3. i ), we find that y i) t) y i ) t) γ sl0si y i ) t)) γ s ) yi) t) y i+) t) ) = ν i y i+) t) y i) t) y i ) t) γ sl0si y i ) t)) y i) t) y i ) t) yi) t) y i+) t) y i) t) y i ) t) yi ) t)l 0si yi ) t)) = L 0si y i ) t)) y i+) t) y i) t) [ = νi γ y i ) t) γ sl0si y i ) si + o) ] at t ω. t)) Therefore 3.6) can be rewritten in the form y i) t) γsi ) = y i ) t) γsi γs L 0si y i ) t)) = ν i α s ) n i sign[πω t)] n i ) C si pt) π ω t) µ si L sj νj π ω t) i j ) [ + o)] at t ω. Integrating this relation on the interval between t and t and taking into account that the fraction under the derivative sign due to the condition 0 tends either to zero, or to ± as t ω, we get y i) t) y i ) t) γ sl0si y i ) t)) = = ν i α s ) n i sign[πω t)] n i ) C si J si t)[ + o)] at t ω. From here first of all follows that the inequality.9 i ) is fulfilled. Moreover, from this and 3.6), due to the equivalence of functions L si and L 0si as y i ) Y i, we have y i+) t) y i) t) = J si t) [ + o)] at t ω, J si t) whence, according to 3. i ) for j = i +, it follows that the first condition of. i ) is valid. From the obtained relation we also have y i ) t) γ s y i) t) L 0si yi ) t)) = = ν i C si J si t) [ + o)] at t ω. 3.7) )

13 Asymptotic Behavior of Solutions of ODEs By virtue of the fact that y i ) t) γ s ) = L = ν i y i) t) y i ) t) γs γsi L from 3.7) it follows y i ) t) γs L 0si yi ) t)) 0si yi ) t)) = ν i y i) t) y i ) t) γs γsi L 0si yi ) t)) 0si yi ) t)) [ γs y i ) t)l 0si yi ) t)) L 0si y i ) t)) [ γs ] + o) at t ω, ] = ) = ν i ν i γ s C si J si t) [ + o)] when t ω. Here the fraction appearing under the derivative sign tends either to zero or to ± as t ω, since by virtue of.4) and properties of slowly varying functions see 3.3)), y i ) t) γ s [ ln = ln y i ) γs t) ln L 0si y i ) ] t)) = γ L si γ 0si yi ) t)) si ln y i ) t) [ = ln y i ) γs ] t) + o) ± at t ω. That is why, by integrating this correlation on the interval from t to t, we get L y i ) t) γ s 0si yi ) t)) = ν iν i γ s C si J sii t)[ + o)] at t ω. 3.8) From here it follows the validity of the second inequality of.8 i ) and also, in view of the equivalence of functions L si and L 0si as y i ) Y i, the validity of the asymptotic representation.4 i ). Besides, 3.7) and 3.8) yield y i) t) y i ) t) = J sii t) γ s J sii t) [ + o)] at t ω. 3.9 i) By virtue of the last relation and Lemma., the second conditions of.0 i ) and. i ) are fulfilled, and asymptotic representations. i ) and.3 i ) hold. Sufficiency. Let the conditions.8 i ). i ) be satisfied, and the algebraic equation.7) have no roots with zero real part. Let us show that in this case the equation.) has solutions admitting asymptotic relations. i ).4 i ) as t ω.

14 50 V. M. Evtukhov and A. M. Klopot Towards this end, we consider first the relation Y γ s γ = C si si γ s J sii t) [ + v n ], 3.0) γ L si 0si Y ) where L 0si : Yi ]0, + [ are continuously differentiated slowly varying as Y Y i functions, satisfying the conditions.3) for k = s and j = i ) and existing due to the property M of slowly varying functions. Having chosen an arbitrary number d ]0, γ s [, let us show that for some t 0 ]a, ω[ the relation 3.0) defined uniquely, on the set [t 0, ω[ R, where R = {v R : v }, a continuously differentiated implicit function Y = Y t, v n ) of the type where z is the function such that Y t, v n ) = ν i J sii t) γs +zt,v n), 3.) zt, v n ) d for t, v n ) [t 0, ω[ R Assuming in 3.0) and zt, v n ) = 0 uniformly with respect to v n R. 3.) Y = ν i J sii t) γs +z 3.3) and then taking the logarithm of the obtained relation, after elementary manipulations, we find that where at) = ln γ s + ln C si γ s ln J sii t) z = at) + bt, v n ) + Zt, z), 3.4), bt, v n ) = γ s ln[ + v n] ln J sii t), Zt, z) = ln L 0si ν i J sii t) γsi γs +z ). γ s ln J sii t) Here, by virtue of the second condition of.0 i ), by the choice of the it of integration in J sii and by the property 3.3) of slowly varying functions, Since ν i J sii t) γis γs +z = Y i uniformly with respect to z [ d, d], at) = 0, 3.5) bt, v n ) = 0 uniformly with respect to v n R Zt, z) = 0 uniformly with respect to z [ d, d]. 3.6) Zt, z) z = νi J sii t) γsi γs +z L osi ν i J sii t) γs +z ), γ s L 0si ν i J sii t) γs +z )

15 Asymptotic Behavior of Solutions of ODEs... 5 by virtue of.3) and the first of the above-stated conditions, we likewise have Zt, z) = 0 uniformly with respect to z [ d, d]. z According to these conditions, there is a number t [a, ω[ such that ν i J sii t) γ is γs +z Yi at t, z) [t, ω[ R d, where R d = { z R : z d }, at) + bt, v, v ) + Zt, z) 3.7) d at t, v n, z) [t, ω[ R R d and Zt, z ) Zt, z ) z z at t [t, ω[ and z, z R d. 3.8) Having chosen in this way the number t, we denote by B the Banach space of continuous and bounded on set Ω = [t, ω[ R functions z : Ω R with the norm z = sup { zt, v n ) : t, v n ) Ω }. We distinguish from it the subspace B 0 of those functions from B, for which z d, and consider on B 0, choosing a fortiori an arbitrary number ν 0, ), the operator and Φz)t, v n ) = zt, v n ) ν [ zt, v n ) at) bt, v n ) Zt, zt, v n ) ]. 3.9) By virtue of 3.7) and 3.8), for any z B 0 and z, z B 0, we have Φz)t, v n ) ν) zt, v n ) + νd d and t, v n ) Ω Φz )t, v n ) Φz )t, v n ) ν) z t, v n ) z t, v n ) + ν Zt, z t, v n )) Zt, z t, v n ) ν) z t, v n ) z t, v n ) + ν z t, v n ) z t, v n ) ν ) z z at t, v n ) Ω. This implies that ΦB 0 ) B 0 and Φz ) Φz ) ν ) z z. It means that the operator Φ maps the space B 0 into itself and is a contractor operator on it. Then, by the contraction mapping principle, there is a unique function z B 0 such that z = Φz). By virtue of 3.9), this continuous on set Ω function is a unique solution of the equation 3.4) satisfying the condition z d. From 3.4), with regard for 3.5), 3.6), it follows that the given solution tends to zero as t ω uniformly with respect to v n R. Continuous differentiability of this solution on the set [t 0, ω[ R, where t 0 is some number from [t, ω[, follows directly from the well-known local theorem on the existence of an implicit function defined by the relation 3.4). In virtue of replacement 3.3), the obtained function z corresponds to a continuously differentiated on set [t 0, ω[ R function Y of

16 5 V. M. Evtukhov and A. M. Klopot type 3.), where z possesses the properties 3.) and which is a solution of the equation 3.0) and satisfies the conditions Y t, v n ) Yi for t, v n ) [t 0, ω[ R, Y t, v n ) = Y i uniformly with respect to v n R. 3.0) Now, applying to differential equation.) the transformation y j ) t) = [π ωt)] i j y i ) t)[ + v j τ)] j =,..., i ), i j)! y j) t) = ) j i j i)! [π ω t)] j i γsij sii t) γ s J sii t) yi ) t)[ + v j τ)] y i ) t) = Y t, v n τ)), τt) = β ln π ω t), j = i,..., n ), 3. i ) where β is defined in.6 i ), and bearing in mind that the function y i ) t) = Y t, v n τ)) for t [t 0, ω[ and v n τ) R satisfies equation y i ) t) γ s = C si γ s J sii t) [ + v n τ)], γ L si 0si yi ) t)) with the use of sign conditions.8 i ),.9 i ), we get a system of differential equations of the form [ v j = β i j)v j+ v j ) γ ] si h τ) + v j ) + v i ) j =,..., i ), γ [ s v i = β v i γ ] si h τ) + v i ) + v i ), γ [ s v j = β j i) + v j ) j + i) + v j+ ) h τ) + v j )+ + ] h τ) + v j )γ s v i ) j = i,..., n ), γ s v = β [ n i h τ) i + v j+ σ sj j=i + v n + v j σ sj + n i ) + v ) h τ) + v )+ ] + γ s h τ) + v )γ s v i ), Gτ, v,..., v n )+ [ v n = βh τ) + v n ) + v i ) + v n ) ] Hτ, v n ) + v n ) + v i ), γ s in which h τ) = h τt)) = π ωt)j sii t) J sii t), h τ) = h τt)) = π ωt)j si t) J si t),

17 Asymptotic Behavior of Solutions of ODEs Gτt), v,..., v n ) = L si Y t, v n )) L 0si Y t, v n )) m k= α k p k t)φ ki Y t, v n )) L sj Y [j] t, v j, v j+, v n )) L sj ν j π ω t) i j ) φ kj Y [j] t, v j, v j+, v n ) α s p s t)φ si Y t, v n )) φ sj Y [j] t, v j, v j+, v n )) πω j i t) Hτt), v n ) = Y t, v n)l 0si Y t, v n)) L 0si Y t, v n )) γ s Y [j] t, v j, v j+, v n ) = πω i j t) i j )! Y t, v n) + v j+ ) when j = 0, i, = j i)! J sii t) J sii t) Y t, v n) + v j ) when j = i, n. Here, the function τt) = β ln π ω t) possesses the properties,, τ t) > 0 at t [t 0, ω[, τt) = + and that is why, according to conditions. i ), h τ) = h τt)) = 0, τ + h τ) = h τt)) =. τ + 3.) By virtue of 3.0) and.3) for k = s and j = i ) the function H tends to zero as τ + uniformly with respect to v n R, and first fraction in the representation of the function G tends to unity as τ + uniformly with respect to v n R. Let us show that the second and third fractions in the representation of function G likewise tends to unity as τ + uniformly with respect to v,..., v n ) R n. By virtue of. i ) and using the l Hospital s rule, we have ln J sii t) π ω t)j sii = t) = 0, ln π ω t) J sii t) J ln sii t) J sii t) ln π ω t) = [ πω t)j si t) J si t) π ωt)j sii t) ] =. J sii t)

18 54 V. M. Evtukhov and A. M. Klopot Taking into account the type of functions Y and Y [j] j = 0, n, j i ), we find ln Y t, v n ) [ γs ] ln J sii t) = + zt, v n ) = 0 ln π ω t) ln π ω t) uniformly with respect to v n R, and ln Y [j] t, v j, v j+, v n ) = ln π ω t) ln Y t, v n ) = i j + + ln π ω t) ln +v j+ i j )! ln π ω t) uniformly with respect to v j+, v n ) R ln Y [j] t, v j, v j+, v n ) ln Y t, v n ) = i j + + ln π ω t) ln π ω t) J ln sii t) J + sii t) ln π ω t) + ln j i)! γsi+vj) γ s ln π ω t) = i j for j = 0, i = i j uniformly with respect to v j, v n ) R for j = i, n. In view of these marginal ratios and using inequalities.6 i ) we find, repeating the reasoning in proving the necessity, that for any k {,..., m} \ {s} p k t)φ ki Y t, v n )) φ kj Y [j] t, v j, v j+, v n )) = 0 p s t)φ si Y t, v n )) φ sj Y [j] t, v j, v j+, v n )) uniformly with respect to v,..., v n ) R n. Owing to these conditions, the last fraction in the representation of function G tends to unity as τ + uniformly with respect to v,..., v n ) R n Moreover, taking into account marginal ratios stated above, we obtain the following representations: Y [j] t, v j, v j+, v n ) = ν j e ln Y [j] t,v j,v j+,v n ) = where = ν j e [+r jt,v j,v j+,v n )] ln π ω t) i j as j {0,..., n } \ {i }, r j t, v j, v j+, v n ) = 0 uniformly with respect to v j, v j+, v n ) R 3 for all j {0,..., n } \ {i }. Since the functions L sj j =, n, j i ) satisfy the condition S 0, by Remark.3, it follows that.

19 Asymptotic Behavior of Solutions of ODEs L sj Y [j] t, v j, v j+, v n )) L sj ν j π ω t) i j ) = uniformly with respect to v,..., v n ) R n. Therefore, the second fraction in the representation of function G tends to unity as τ + uniformly with respect to v,..., v n ) R n. Due to above stated, the obtained system of differential equations can be written in form [ v j = β f i τ, v,..., v n ) + n ] p jk v k k =, n ), k= [ v =β f τ, v,..., v n )+ n k= [ v n = βh τ) f n τ, v,..., v n ) + ] p k v k +V v,..., v n ), n k= ] p nk v k + V n v,..., v n ), where the functions f i i =, n) are continuous on a set [τ, + [ R n some τ β ln π ω t 0 ) and are such that f iτ, v,..., v n ) = 0 i =, n) τ i ) uniformly with respect to v,..., v n ) R n, 3.4) p jj = j i, p jj+ = i j, jk = 0 at k {,..., n} \ {j, j + } j =, i ), * p i i =, p i k = 0 at k {,..., n} \ {i }, p jj = j i +, p jj+ = i j, p jk = 0 at k {,..., n} \ {j, j + } j = i, n ), p k = n i)σ sk k =, i ), p k = n i)σ sk k = i, n ), p = n i) σ s ), p n = n i), p ni = i, p nk = 0 at k {,..., n} \ {i}, V n v,..., v n ) = v i v n, V v,..., v n ) = i n) [ + n i) + i k= i + v j+ σ sj σ sk v k + k=i j=i + v j σ sj + v n γsi + σ sk v k v n ]. for

20 56 V. M. Evtukhov and A. M. Klopot Since conditions 3.4) are satisfied and V j v,..., v n ) = 0 j = n, n; k =, n), v + + v n 0 v k this system belongs to the class of systems of differential equations, for which the criteria for the existence of vanishing at infinity solutions were obtained in [4]. Let us show that for this system the conditions of Theorem.6 are fulfilled based on this paper). First of all, taking into account the conditions 3.) and the type of integral J sii t), we notice that the function h possesses the properties h τ) = 0, τ + + τ h τ) dτ = β ω t J sii t) J sii t) dt = = β ln J sii t) ω t = ± τ = β ln π ω t ) ), h τ) τ + h τ) = [ πω t)j sii = β t) + π ω t)j si t) J sii t) J si t) h τt))) t τ t)h τt)) = π ω t)j sii t) J sii t) πω t)j sii t) ) ] =0. J sii t) Next, consider the matrices P n = p jk ) n j,k= and P = p jk ) j,k=, for which we have det P = ) i i )!n i)!, det P n = ) i i )!n i)!, det [ i ] P ρe = ) i k + ρ) n i)! j=i+ k= σ sj j i)! [ n i m ρ) m= j i m ρ) n i)!σ si ], m= where E is the unit matrix of dimension n ) n ). Since algebraic equation.7), according to the conditions of Theorem, has no roots with zero real part, the characteristic equation of the matrix P has likewise no such roots, and the given characteristic equation has i roots if i > ) of the type ρ k = k k =, i ). Thus, for the system 3.3 i ), all the conditions of Theorem.6 of [4] are satisfied. According to this theorem, the system 3.3 i ) has at least one solution v j ) n j= : [τ, + [ R n τ τ ) tending to zero as τ +. Moreover, if l is a number of roots of the equation.7) with negative real part, and r is a number of roots with positive real part, then according to the same Theorem, in case β =, this system has i + - parametric family of such solutions if the inequality ν i ν i γ s > 0 is fulfilled, and has i + l- parametric family if the inequality ν i ν i γ s < 0 is fulfilled, whereas, in case β =, there is r + - parameter family of such solutions

21 Asymptotic Behavior of Solutions of ODEs if there is the inequality ν i ν i γ s > 0 and r - parametric family if there is the inequality ν i ν i γ s < 0. To every such solution of the system 3.3 i ) there corresponds, due to the replacements 3. i ) and the first condition of.3), the solution y : [t, ω[ R t [a, ω[) of the equation.) admitting as t ω asymptotic representations. i ).4 i ). Using these representations and conditions.6 i ),.8 i ). i ), it can be easily seen that it is a P ω Y 0,..., Y, n i n i )-solution. Proof of Theorem.. Let the equation.) have P ω Y 0,..., Y, n i n i )- solution y : [t 0, ω[ Y0. Then, according to Theorem., the conditions.8 i ). i ) are satisfied and for this solution the asymptotic representations. i ).4 i ) hold as t ω. Furthermore, from the proof of necessity of that theorem it is clear that the condition 3.9 i ) is satisfied. Since the functions L si satisfy the condition S 0, by virtue of 3.9 i ) and Remark., L si y i ) t)) = L sj νi J sii t) γs Therefore it follows from.4 i ) that ) [ + o)] at t ω. y i ) t) γs = = C si L si νi J sii t) ) γ s γs J sii t) [ + o)] at t ω, which results in the presentation.5 i ). 4. Example of Equation with Regularly Varying as t ω Coefficients Suppose that in the differential equation.), the continuous functions p k : [a, ω[ ]0, + [ k =, m) are regularly varying, as t ω, of orders ϱ k k =, m), and, moreover, the conditions of Theorem. as i {,..., n } are satisfied. In this case and the conditions.6 i ) take the form βϱ k ϱ s ) < β ln p k t) ln π ω t) = ϱ k 4.) σ sj σ kj )i j ) at all k {,..., m} \ {s}. 4. i ) Since as t ω the functions L sj ν j π ω t) i j ) j {0,..., n } \ {i }) are slowly varying, and the function p s is regularly varying of order ϱ s, therefore the function J si is regularly varying of order + ϱ s + µ si, and the

22 58 V. M. Evtukhov and A. M. Klopot function J sii t) is regularly varying of order + + ϱ s + µ si ) as t ω. This implies that π ω t)j si t) = + ϱ s + µ si, J si t) π ω t)j sii t) = + + ϱ s + µ si ). J sii t) Therefore the conditions. i ) will be of the following form: + ϱ s + + µ si = i ) Taking into account this condition, the function J sii t) should be slowly varying as t ω. In order to get asymptotic representation for this integral we have to know the type of a slowly varying component of the integrand equation. Suppose that the functions p s and φ sj j = 0, n ) are of the form p s t) = π ω t) ϱs ln π ω t) r s, φ sj y j) ) = y j) σ sj ln y j) λ sj j = 0, n ). 4.4) In this case, L sj y j) ) = ln y j) λ sj j = 0, n ) and hence all of them satisfy the conditions of S 0. Additionally, we get as t ω the following asymptotic relations J si t) β i j λ sj π ω t) ln π ω t) r s + λ sj, 4.5 i ) J sii t) i j λ sj rs + ) λ sj + β if r s + ln πω t) + λ sj, j i λ sj ln ln π ω t), if r s + λ sj =, r s + ) λ sj, 4.6 i )

23 Asymptotic Behavior of Solutions of ODEs J sii t) J sii t) π ω t) ln π ω t), if r s+ r s + λ sj + λ sj, π ω t) ln π ω t) ln ln π ω t), if r s+ λ sj =. 4.7 i ) From the above relations it, in particular, follows that the inequalities.8 i ),.9 i ) and the conditions.0 i ) take the form ν j ν j i j)π ω t) > 0 ν i α s ) n i π n i ω t) > 0, ν i ν i γ s > 0 < 0), if + r s + at all j {,..., n } \ {i}, 4.8 i ) λ sj ) 0 < 0), 4.9 i ) ν j π ω t) i j = Y j at j {,..., n} \ {i}, 4.0 i ) ν i Y i = = 0), if γ s r s + λ sj + ) 0 < 0). 4. i ) By virtue of above-said, from Theorem. follows the following statement. Corollary 4.. Let in the equation.) n >, the functions p k k =, m) be regularly varying of orders ϱ k at t ω, i {,..., n } and for some s {,..., m}, the inequalities 4. i ) be fulfilled. Let, moreover, the equation γ s 0 be fulfilled and the representations 4.4) hold. Then for the equation.) to have P ω Y 0,..., Y, n i n i )-solutions, it is necessary, and if algebraic equation.7) has no roots with zero real part, then it is sufficient that the conditions 4.3 i ), 4.8 i ) 4. i ) along with.3)) are satisfied. Moreover, for each such solution there exist, as t ω, the following asymptotic representations: y j ) t) = [π ωt)] i j y i ) t)[ + o)] j =,..., i ), i j)! 4. i ) y j) t) = ) j i j i)! [π ω t)] j i γsij sii t) γ s J sii t) yi ) t)[ + o)] 4.3 i ) j = i,..., n ), y i ) γsi γ si λsi γsi γs t) = ν i C si γ s J sii t) γs ln J sii t) λ si γs [ + o)],.5 i )

24 60 V. M. Evtukhov and A. M. Klopot where the functions J sii t) and J sii t) J sii t) are defined by 4.6 i) and 4.7 i ), respectively, and for such solutions in case ω = + there exists an i+-parametric family if the inequality ν i ν i γ s > 0 is fulfilled, and an i +l-parameter family if there is the inequality ν i ν i γ s < 0, while in case ω < + there exists an r + -parametric family of such solutions if the inequality ν i ν i γ s > 0 is fulfilled, and an r- parametric family if there is the inequality ν i ν i γ s < 0, where l is a number of roots of the equation.7) with negative real part and r is a number of its roots with positive real part. References. E. Seneta, Regularly varying functions. Lecture Notes in Mathematics, Vol Springer-Verlag, Berlin New York, 976; translation in Nauka, Moscow, N. H. Bingham, C. M. Goldie, and J. L. Teugels, Regular variation. Encyclopedia of Mathematics and its Applications, 7. Cambridge University Press, Cambridge, V. Marić and M. Tomić, Asymptotic properties of solutions of the equation y = fx)ϕy). Math. Z ), No. 3, S. D. Taliaferro, Asymptotic behavior of solutions of y = φt)fy). SIAM J. Math. Anal. 98), No. 6, V. Marić, Regular variation and differential equations. Lecture Notes in Mathematics, 76. Springer-Verlag, Berlin, I. T. Kiguradze and T. A. Chanturia, Asymptotic properties of solutions of nonautonomous ordinary differential equations. Translated from the 985 Russian original. Mathematics and its Applications Soviet Series), 89. Kluwer Academic Publishers Group, Dordrecht, A. V. Kostin, The asymptotic behavior of the extendable solutions of equations of Emden Fowler type. Russian) Dokl. Akad. Nauk SSSR 00 97), V. M. Evtukhov, The asymptotic behavior of solutions of one nonlinear second order differential equation of Emdem Fowler type. Diss. Kand. Fiz.-Mat. Nauk Odessa, 980, 9 p. 9. V. M. Evtukhov, Asymptotic representations of solutions of a class of second-order nonlinear differential equations. Russian) Soobshch. Akad. Nauk Gruzin. SSR 06 98), No. 3, V. M. Evtukhov, Asymptotic properties of the solutions of a certain class of secondorder differential equations. Russian) Math. Nachr ), V. M. Evtukhov, Asymptotic behavior of the solutions of a second-order semilinear differential equation. Russian) Differentsial nye Uravneniya 6 990), No. 5, , 95 96; translation in Differential Equations 6 990), Nno. 5, A. V. Kostin, Asymptotics of the regular solutions of nonlinear ordinary differential equations. Russian) Differentsial nye Uravneniya 3 987), No. 3, 54 56, V. M. Evtukhov, Asymptotic properties of monotone solutions of a class of nonlinear differential equations of the nth-order. Dokl. Rasshir. Zased. Sem. Inst. Prikl. Mat ), No. 3, V. M. Evtukhov, Asymptotic representations of monotone solutions of an nth-order nonlinear differential equation of Emden Fowler type. Russian) Dokl. Akad. Nauk 34 99), No., 58 60; translation in Russian Acad. Sci. Dokl. Math ), No. 3, ). 5. V. M. Evtukhov, A class of monotone solutions of an nth-order nonlinear differential equation of Emden Fowler type. Russian) Soobshch. Akad. Nauk Gruzii 45 99), No.,

25 Asymptotic Behavior of Solutions of ODEs V. M. Evtukhov, Asymptotic representations of the solutions of nonautonomous ordinary differential equations. Diss. Doctor. Fiz.-Mat. Nauk., Kiev, 998, 95 p. 7. V. M. Evtukhov and M. A. Belozerova, Asymptotic representations of solutions of second-order essentially nonlinear nonautonomous differential equations. Russian) Ukrain. Mat. Zh ), No. 3, 30 33; translation in Ukrainian Math. J ), No. 3, M. A. Belozerova, Asymptotic properties of a class of solutions of essentially nonlinear second-order differential equations. Russian) Mat. Stud ), No., M. A. Belozerova, Asymptotic representations of solutions of second-order nonautonomous differential equations with nonlinearities close to power type. Russian) Nelīnīĭnī Koliv. 009), No., 3 5; translation in Nonlinear Oscil. N. Y.) 009), No., V. M. Evtukhov and A. A. Koz ma, Existence criteria and asymptotics for some classes of solutions of essentially nonlinear second-order differential equations. Ukrainian Math. J. 63 0), No. 7, A. A. Koz ma, Asymptotic behavior of a class of solutions of second-order nonlinear nonautonomous differential equations. Russian) Nelīnīĭnī Koliv. 4 0), No. 4, V. M. Evtukhov and A. M. Samoilenko, Asymptotic representations of solutions of nonautonomous ordinary differential equations with regularly varying nonlinearities. Russian) Differ. Uravn. 47 0), No. 5, ; translation in Differ. Equ. 47 0), No. 5, N. Bourbaki, The functions of a real variable. Translated from the French) Nauka, Moscow, V. M. Evtukhov and A. M. Samoilenko, Conditions for the existence of solutions of real nonautonomous systems of quasilinear differential equations vanishing at a singular point. Ukrainian Math. J. 6 00), No., Authors address: Received December 4, 04) Odessa I. I. Mechnikov National University, Dvoryanska St., Odessa 6508, Ukraine. emden@farlep.net; mrtark@gmail.com