New results for the non-oscillatory asymptotic behavior of high order di ﬀ erential equations of Poincar´e type Poincar´e-Perron problem;

: This paper discusses the study of asymptotic behavior of non-oscillatory solutions for high order di ﬀ erential equations of Poincar´e type. We present two new and weaker hypotheses on the coe ﬃ cients, which implies a well posedness result and a characterization of asymptotic behavior for the solution of the Poincar´e equation. In our discussion, we use the scalar method: we deﬁne a change of variable to reduce the order of the Poincar´e equation and thus demonstrate that a new variable can satisﬁes a nonlinear di ﬀ erential equation; we apply the method of variation of parameters and the Banach ﬁxed-point theorem to obtain the well posedness and asymptotic behavior of the nonlinear equation; and we establish the existence of a fundamental system of solutions and formulas for the asymptotic behavior of the Poincar´e type equation by rewriting the results in terms of the original variable. Moreover we present an example to show that the results introduced in this paper can be used in class of functions where classical theorems fail to be applied.


Introduction
In this paper we are interested in the non-oscillatory asymptotic behavior of the following differential equation: [a i + r i (t)]y (i) = 0, n ∈ N, n ≥ 2, (1.1) where a i ∈ R are given constants and r i are real-valued functions. The Eq (1.1) is called of a Poincaré type, since its analysis was motivated and introduced by Poincaré [17], where the author obtained conditions guaranteeing that all nonzero solutions of (1.1) are such that lim t→∞ (y /y)(t) exists as a finite number. Perron [16] completed the result of Poincaré in the sense of existence. Later on, the asymptotic behavior of (1.1) have been investigated by several authors with a long and rich history of results [3,8,12,13]. A recent short review of the most relevant landmarks of the evolution of the Poincaré problem, is included in [5,18,19]. The purpose of this paper is to solve the Poincaré problem by introducing hypotheses, on perturbation functions r i , that are weaker than the classical ones. Our analysis is based on the modified scalar method which consists of three big steps, for details on cases n = 3, 4 see [10] and [5,6], respectively. In the first step, let us consider that y is a nontrivial solution of (1.1), we introduce the change of variable z(t) = y (t) y(t) − µ or equivalently y(t) = exp t t 0 (z(s) + µ)ds , (1.2) to reduce the order of (1.1). If µ ∈ R is a simple characteristic root of (1.1) when r i = 0 then z satisfies a non-linear differential equation of the following type z (n−1) (t) + n−2 i=0 b i (µ)z (i) (t) = P(µ, t, r 0 (t), . . . , r n−1 (t), z(t), z (t), . . . , z (n−2) (t)), (1.3) where b i are real-valued functions, µ ∈ R is a given (fix) parameter, and P : R 2n+1 → R is a polynomial of n degree in the n − 1 last variables and the coefficients depends on the first n + 2 variables, i.e., P admits the representation P(e, x) = Here, we have used the standard multindex notation for |α| and x α , i.e. |α| = α 1 + . . . + α n−1 and x α = x α 1 1 . . . x α n−1 n−1 . We observe the inclusion of r 0 (t), . . . , r n−1 (t) as arguments of P is only by notational convenience and the Eq (1.3) is the abstract form of a more particular equation obtained in the specific case of change of variable (1.2). For instance, in the case n = 3, 4 there is several coefficients such that Ω α = 0 [5,10]. In the second step, we analyze the existence and asymptotic behavior of (1.3). Finally, we particularize the results of (1.3) to the specific equation obtained for µ characteristic root of (1.1) and using the definition of y in (1.3) we deduce the well posedness and asymptotic behavior of (1.1).
The main aims of the paper are the following The rest of the paper is organized as follows: we introduce some notation in section 2, we present the main results on section 3, we develop the proofs on section 4, and we present an example on section 5.
2. Assumptions on coefficients of (1.3) and (1.1) 2.1. Assumptions for the coefficients of (1.3) Let us introduce some notation. Throughout the paper unless otherwise stated we assume the notation t 0 is a given (fix) real number. Given the Green function g (see (4.1)), we consider that the functions R, L k and Υ are defined as follows Hereinafter we consider that the integrals ∞ t 0 defining R and L k exists as improper Riemann integrals. Then, we assume the following hypotheses on b i (µ) and P: (R1) The functions b i (µ) for i = 0, n − 2 are such that the set of characteristic roots for (1.3) when P = 0 is given by Γ µ = γ i (µ), i = 1, n − 1 : γ 1 > γ 2 > . . . > γ n−1 ⊂ R. (R2) The functions Ω α given on (1.4a) satisfy the following requirements lim t→∞ R(t) = lim t→∞ L 1 (t) = 0 and lim t→∞ n k=2 L k (t) < 1, (2.4) where R and L k are defined on (2.2). (R3) The functions Ω α on (1.4a) satisfy the following property: given Φ 1 and γ i with and The assumptions (R1) and (R2) are considered to study the existence of a solution for (1.3). Since we are interested in the non-oscillatory behavior, we considered the hypothesis (R1). Meanwhile, (R2) is the natural condition in order to get the application of Banach fixed point theorem. Now, the hypothesis (R3) is used in order to get the asymptotic behavior of the solution for (1.3).

2.2.
Assumptions for the coefficients of (1.1) Let us consider γ j (λ i ) and the function H defined by whereŜ 0, j (t) = j + µ( j − 1) and for m > 1 we have that Then, related with the coefficients of (1.1), we consider that: (H1) The constants a i are selected such that the set of characteristic roots for (1.1) when r i = 0 is given by Λ = λ i , i = 1, n : λ 1 > λ 2 > . . . > λ n ⊂ R. (H2) The perturbation functions satisfy the asymptotic behavior: for a Green function g defined on (4.1). (H3) The perturbation functions r i and the characteristic set Λ satisfy the following property: given Φ Λ i and γ k (λ i ) with

A brief discussion of the assumptions.
Related with the constant coefficients part of (1.1), i.e. when r i = 0, there is a coincidence or common hypothesis used by the different researchers, in order to deduce the non-oscillatory asymptotic behavior. All of them consider the fact that There is a simple characteristic root µ such that Re(µ) Re(µ 0 ) for any other characteristic root µ 0 . (2.10) Then, (H1) is an extension of the condition (2.10) for all characteristic roots of (1.1). Now, with respect to the regularity and the asymptotic behavior of perturbation functions r i we have different assumptions. For instance, the seminal work of Poincaré [17] consider that r i are rational functions, Perron [16] assumes that r i are continuous functions, Levinson [15] considers that r i ∈ L 1 ([t 0 , ∞[), and Hartman and Wintner in [14] select r i ∈ L p ([t 0 , ∞[) for some p ∈]1, 2]. There is a coincidence of the authors with respect to the asymptotic behavior of r i , by considering that: r i (t) → 0 when t → ∞. Then (H2) and (H3) are new and more weak than the previous assumptions. Indeed, we refer to the recent work [5], in the case n = 4, for an example of perturbation functions r i which satisfy (H2) and (H3) and neither satisfy the assumptions of L p regularity and nor satisfy the asymptotic behavior r i (t) → 0 when t → ∞.
In principle, the asymptotic behavior of (1.1) can be obtained by application of the typical existing results for the asymptotic behavior for systems. However, it has some drawbacks as is specified below by synthesizing the system methodology in three steps: (a) Rewrite (1.1) as a diagonal system. We consider the change of variable and introduce the notation with a = [a 1 . . . a n ], r(t) = [r 0 (t) . . . r n−1 (t)] and I n−1 the identity matrix of order n − 1. Then, we observe that the Eq (1.1) is equivalently to the following system We note that the eigenvalues of A are the characteristic roots of (1.1) when r = 0. Then, if we assume that (H1) holds, we deduce that we can diagonalize the system (2.12). More precisely, under (H1) the system (2.12) can be rewritten in a diagonal form as follows where M n is the Vandermonde matrix of order n asocieted with (λ 1 , . . . , λ n ). then we deduce that the system (2.13) has n solutions Y i for i = 1, . . . , n with the following asymptotic behavior The notation e 1 , . . . , e n is used to denote the vectors of the standard canonical base of R n . The Hartman-Wintner Theorem (see [8,Theorem 1.5.1]) consider that the perturbation matrix satisfies 16) and deduce that the system (2.13) has n solutions Y i for i = 1, . . . , n with the following asymptotic behavior n B(t)M n satisfies the dichotomy Levinson condition, then deduce the system (2.13) has n solutions Y i for i = 1, . . . , n with the following asymptotic behavior (2.18) (c) Translation of the results for the behavior of Y to the variable X. By recalling that X = M n Y M −1 n an using (2.11), we deduce the asymptotic behavior of (1.1).
Theoretically, the process (a)-(c) can be rigorously applied to study the asymptotic behavior of any specific equation of the form (1.1). However, it has some clear and practical computation disadvantages for higher order differential equations, among them we have the the conditions (2.14), (2.16) and dichotomy Levinson condition for µ k (t) are hard to verify and the change of variable for the diagonalization M −1 n XM n is expensive. Note also that the analytic computation of µ k (t) is not always possible since, most of the time, M −1 n B(t)M n is a full matrix. Consequently, in order to overcome this disadvantages, in this paper we apply the scalar method [1-3, 5, 6, 10, 11].
In section 5, we present an example to compare the application of the results in this paper in comparison with the classical results.

Main results
The main results are given by the following four theorems: which is a Banach space with the norm z 0 = sup t≥t 0 Assume that the coefficients of the Eq (1.3) satisfy (R1) and (R2). Then, exists z ∈ C n−2 0 ([t 0 , ∞[) a solution of (1.3). Theorem 3.2. Let us introduce the notation Consider that the hypotheses of Theorem 3.1 and the assumption (R3) are valid. Then, z µ the solution of (1.3) has the following asymptotic behavior when G µ ⊂ E n−1 n and β ∈]0, γ n−1 ], when t → ∞, for all j ∈ {0, . . . , n − 2}.
Theorem 3.4. Let us consider that (H1) and (H2) are satisfied. Then, the Eq (1.1) has a fundamental system of solutions given by Moreover, the following properties about the asymptotic behavior is satisfied when t → ∞. Furthermore, if π i = 1≤i< j≤n, j i (λ j − λ i ) and (H3) is satisfied, then holds, when t → ∞ with z ( j) λ i , j = 0, n − 2 given asymptotically by (3.9) and F defined on (3.5).

Proof of Theorem 3.1
The proof is mainly based on variation of parameters technique and Banach fixed point Theorem. More specifically we proceed as follows: we introduce the precise notation of Green functions for (1.3) when P = 0, we apply the method of variation of parameters to get the operator equation, and we deduce that the operator satisfies the hypotheses of Banach fixed point Theorem.
The Green function g for (1.3) when P = 0 is defined by where Υ 0 is the notation on (2.3), G µ is the set defined on (3.1), the notation E n−1 i is given on (3.2), and g µ : R 2 → R are the functions defined as follows where G = (−1) Υ (γ 1 , . . . , γ n−1 ) with Υ defined on (2.3) and H is the Heaviside function, i.e., H(x) = 1 for x ≥ 0 and H(x) = 0 for x < 0. For further details on Green functions the reader may be consult [3] (see also [9] for n = 2, [10] for n = 3 and [5] for n = 4). We apply the method of variation of parameters to get that (1.3) is equivalent to the following integral equation where g is the Green function defined on (4.1). Thus, if we define the operator T from C n−2 we note that (4.2) can be rewritten as the operator equation where η ∈ R + will be selected in order to apply the Banach fixed point theorem. Indeed, we have that (a) T is well defined from C n−2 0 ([t 0 , ∞[) to C n−2 0 ([t 0 , ∞[). Let us consider an arbitrary z ∈ C n−2 0 ([t 0 , ∞[), by the definition of the operator T we deduce that ∂ j g ∂t j (t, s)P µ, s, r 0 (s), . . . , r n−1 (s), z(s), . . . , z (n−2) (s) ds, j = 0, n − 2.
Then, considering the hypothesis (R2) and by using the notation (1.4), we can deduce the following estimates |α|=k Ω α µ, s, r 0 (s), . . . , r n−1 (s) ds Now, by (2.4) and the fact that z ∈ C n−2 0 ([t 0 , ∞[), we have that the right hand side of (4.6) tend to 0 when t → ∞. Thus, for all j = 0, . . . , n − 2 we have that T ( j) z(t) → 0 when t → ∞ or equivalently For all η ∈]0, 1[, the set D η is invariant under T . Let us consider z ∈ D η . From (4.5), we can deduce the following estimate in a right neighborhood of η = 0. Here we notice that L 2 (t) + n k=3 η k−2 L k (t) ≤ η in a right neighborhood of η = 0. Now, by (2.4) we deduce that the first two terms on the right side of (4.7) tend to 0 when t → ∞. Hence, by (4.7) and (R2) we have that T z 0 ≤ η 3 ≤ η, or equivalently T z ∈ D η for all z ∈ D η .
Hence, from (a)-(c) and by the application of Banach fixed point theorem, we deduce that there is a unique z ∈ D η ⊂ C n−2 0 ([t 0 , ∞[) solution of (4.4).

Proof of Theorem 3.2
The proof is based on the operator Eq (1.3) and the invariant and contraction properties of T . Indeed, let us first introduce some notation. We denote by z µ the solution of the Eq (1.3) given by Theorem 3.1. Moreover, on D η with η ∈]0, 1/n[, we define recursively the sequence {ω m } m∈N by assuming that ω m+1 = T ω m with ω 0 = 0. We note that ω m → z µ when m → ∞ as a consequence of the contraction property of T .
Proof of (4.10). From (4.14), using recursively the definition of Φ h−1 , . . . , Φ 2 , we can rewrite Φ h as the sum of the terms of a geometric progression where the common ratio is given by σ γ i Φ 1 . Then, the hypothesis (R3) implies that σ γ i Φ 1 ∈]0, 1[, and we can deduce that or equivalently Φ m converges to Φ γ i .

Proof of Theorem 3.3
[(a)] In order to prove (3.4), we first construct a formula for y ( j) and then we use that formula in (1.1). Indeed, we have that the j-th derivative for y is given by The proof of (4.15) for j = 1, 2 is given by direct differentiation of y in (1.2). Now, for j ≥ 3, we proceed inductively by using the Leibniz formula for differentiation. Indeed, by using the relation for y (1) we have that (4.16) The order of derivatives for y is strictly decreasing with respect to 1 , since Then, if j − 1 > 2 we can again apply the Leibniz formula to compute y ( j− 1 −1) = (y ) ( j− 1 −2) and from (4.16) we get (4.17) Similarly, by observing the order of derivatives for y, we deduce that if j − 2 > 2 we can use the Leibniz formula to compute y ( j− 1 − 2 −2) . We note that, we can apply this strategy j − 2 times to obtain the desired result given in (4.15) for j ≥ 3 and conclude the proof of item (a).
Remark 4.1. We observe that the derivatives of y(t) = exp( t t 0 (z(s) + µ)ds) can be deduced by other methodologies instead of Leibniz formula, for instance by using the Faà di Bruno's formula and complete Bell polynomials [4,7].
[(b)] The proof of (3.8) is a consequence of the following claim: If λ i , λ j ∈ Λ for i j, then λ j − λ i is a root of the characteristic polynomial associated with the differential Eq (3.4) when µ = λ i and the right hand side is zero. Indeed, using the identity we deduce that and by selecting u = λ j and v = λ i we can prove the claim.

We cannot apply Levinson, Hartman-Wintner and Eastham theorems.
In order to apply the Levinson, Hartman-Wintner and Eastham theorems, we rewrite (5.1) as a system of the form (2.13), i.e.,  Then the process of analytic calculus of the eigenvalues of the matrixB(t) is not possible, resulting that the verification of assumptions for Eastham Theorem (see [8, Theorem 1.6.1]) are not simple and consequently the result is not straightforward to apply in this particular case.

The hypothesis (H2) is satisfied.
To verify the hypothesis (H2) we proceed in three steps: construction of Green functions using the definition given (4.1), calculus of integrals and limits in (2.8), and calculus of integrals and limits in (2.9).
Step 1. Construction of Green functions using the definition given (4.1). We calculate G µ and Υ i for i = 0, . . . , 4 defined on (3.1) and (2.3). For instance, if we fix µ = λ 1 = 3, by application of theorem 3.3(b), we deduce that Proceeding similarly, we deduce that the Green functions for other cases of selection of µ are given by   By follow a similar process to step 2 in section 5.3 we deduce that ∞ t exp(t − s) H(s, λ 1 ) ds → 0 when t → ∞. Then for all > 0, there is N > 0 such that t > N implies that ∞ t exp(t − s) H(s, λ 1 ) < . Particularly, by selecting σ γ 1 = ∈ [0, 21/209] we deduce that the hypothesis (H3) is satisfied. We proceed analogously to prove the other cases.

Conclusions
In this paper, we have introduced and proved a new result on the asymptotic behavior of nonoscillatory solutions for high order differential equations of Poincaré type. The construction of the proof is based on the scalar method which was developed in [5,9,10] for differential equations with orders 2-4, respectively. The scalar method consist in three big steps: (i) a change of variable to reduce the order of the Poincaré equation and demonstrate that the new variable is a solution of a nonlinear differential equation; (ii) the application of the method of variation of parameters and the Banach fixedpoint theorem to obtain the well posedness and asymptotic behavior of the non-linear equation; (iii) the proof of existence of a fundamental system of solutions and formulas for the asymptotic behavior of the Poincaré type equation by rewriting the results in terms of the original variable. Comparing with the classical results, the major contribution of the new result is the fact that the perturbation functions, appearing as coefficients of the equation, are more weak.