Georgian Mathematical Journal 1(1994), No. 3, 315-323

LIMIT BEHAVIOR OF SOLUTIONS OF ORDINARY LINEAR DIFFERENTIAL EQUATIONS

FRANTIŠEK NEUMAN

Abstract. A classification of classes of equivalent linear differential equations with respect to ω-limit sets of their canonical representatives is introduced. Some consequences of this classification to the oscillatory behavior of solution spaces are presented.

1. Introduction

Many authors dealt with the behavior of solutions of differential equations to the (mostly right) end of the interval of definition – the limit behavior (often considered for the independent variable tending to ∞). Asymptotic, oscillatory and other qualitative properties of solutions of linear differential equations were intensively studied e.g. by N.V.Azbelev and Z.B.Caljuk [1], J.H.Barrett [2], G.D.Birkhoff [3], O.Boru^{∘}vka [4], W.A.Coppel [5], M.Greguš [6], G.B.Gustafson [7], M.Hanan [8], I.T.Kiguradze and T.A.Chanturia [9], G.Sansone [14], C.A.Swanson [15], and many others.

The aim of this paper is to introduce a certain classification of the limit behavior of solutions of linear differential equations, a classification which is invariant with respect to the most general pointwise transformations of these equations. This classification has natural consequences to the oscillatory and asymptotic behavior of solutions. The main tool is based on the geometric approach introduced in [11] which enables us to convert some ”non-compact” problems into ”compact” ones. This method was applied for solving some open problems [12], and it has recently been explained systematically in detail together with other methods and results concerning linear differential equations in the monograph [13].

1991 Mathematics Subject Classification. 34A26, 34A30, 34C05, 34C10, 34C11, 34C20.

2. Background and preliminary results

Let C^{n}(I) denote the set of all functions defined on an open interval I ⊆ ℝ with continuous derivatives up to and including the order n. For n ≥ 2, let L_{n} stand for all ordinary linear differential equations of the form

P_{n} ≡ y^{(n)} + p_{n−1}(x)y^{(n−1)} + ⋯ + p_{0}(x)y = 0 on I,

I being an open interval of the reals, p_{i} are real continuous functions defined on I for i = 0, 1, …, n − 1, i.e. p_{i} ∈ C^{0}(I), p_{i} : I → ℝ.

Consider Q_{n} ∈ L_{n},

Q_{n} ≡ z^{(n)} + q_{n−1}(t)z^{(n−1)} + ⋯ + q_{0}(t)z = 0 on J.

We say that the equation P_{n} is globally equivalent to the equation Q_{n} if there exist two functions,

f ∈ C^{n}(J), f(t) ≠ 0 for each t ∈ J, and
h ∈ C^{n}(J), h′(t) ≠ 0 for each t ∈ J, and h(J) = I,

such that whenever y : I → ℝ is a solution of P_{n} then

z : J → ℝ, z(t) := f(t) · y(h(t)), t ∈ J,  (1)

is a solution of Q_{n}.

Let y(x) = (y_{1}(x), …, y_{n}(x))^{T} denote an n-tuple of linearly independent solutions of the equation P_{n} considered as a column vector function or as a curve in n-dimensional euclidean space 𝔼_{n} with the independent variable x as the parameter and y_{1}(x), …, y_{n}(x) as its coordinate functions; M^{T} denotes the transpose of the matrix M.

If z(t) = (z_{1}(t), …, z_{n}(t))^{T} denotes an n-tuple of linearly independent solutions of the equation Q_{n}, then the global transformation (1) can be equivalently written as

z(t) = f(t) · y(h(x))  (1′)

or, for an arbitrary regular constant n × n matrix A,

z(t) = Af(t) · y(h(x))  (1′′)

expressing only that another n-tuple of linearly independent solutions of the same equation Q_{n} is taken.

Denote the n-tuple v = (v_{1}, …, v_{n})^{T},

v(x) := y(x)/∥y(x)∥,

where ∥y(x)∥ := (y_{1}^{2}(x) + ⋯ + y_{n}^{2}(x))^{1/2} is the euclidean norm of y in 𝔼_{n}. It was shown (see [11] or [13]) that v ∈ C^{n}(I), v : I → 𝔼^{n}, and the Wronski determinant of v is different from zero on I. Of course, ∥v(x)∥ = 1, i.e. v ∈ 𝕊_{n−1}, where 𝕊_{n−1} is the unit sphere in 𝔼_{n}. Denote by T_{n} the differential equation from L_{n} which has this v as its n-tuple of linearly independent solutions. Evidently T_{n} is globally equivalent to P_{n}. Moreover (see again [11] or [13]), if

u(s) := v(g(s)),

where the function g satisfies

g(s) : J → I ⊆ ℝ, g(J) = I, |(g^{−1}(x))′| = ∥v′(x)∥

for the inverse g^{−1} to g, and hence g ∈ C^{n}(J), g′(s) ≠ 0 on J, we have ∥u′(s)∥ = 1, i.e. this u is the length reparametrization of the curve v. Of course, ∥u(s)∥ = ∥v(g(s))∥ = 1. If R_{n} denotes the differential equation admitting u as its n-tuple of linearly independent solutions on J ⊆ ℝ, then the above considered equation P_{n} is globally equivalent both to equation T_{n} and to R_{n}; equation R_{n} is also called the canonical equation of the whole class of equations from L_{n} globally equivalent to P_{n}. Canonical equations are characterized by admitting n-tuples of linearly independent solutions u satisfying

∥u(s)∥ = 1, ∥u′(s)∥ = 1;

for more details see [13].

The following result describes the connection between the behavior of curves y, v and u and the zeros of solutions of the corresponding equations P_{n}, T_{n} and R_{n}, see [11] or [13].

Proposition 1. Let P_{n}, T_{n} and R_{n} be equations from L_{n}, and let y, v and u denote their n-tuples of linearly independent solutions defined as above. For an arbitrary nonzero constant vector c = (c_{1}, …, c_{n})^{T}, the solution c^{T} y(x) of the equation P_{n} has the zero at x_{0} if and only if the hyperplane

H(c) ≡ c_{1}ξ_{1} + ⋯ + c_{n}ξ_{n} = 0 in 𝔼_{n}

intersects the curve y at the point of the parameter x_{0}.

Moreover, the solution c^{T} v(x) of the equation T_{n} has the zero at x_{0} if and only if the great circle H(c) ∩ 𝕊_{n−1} intersects the curve v at the point of the parameter x_{0}. And the solution c^{T} u(s) of the equation R_{n} has the zero at s_{0} = g^{−1}(x_{0}) if and only if the great circle H(c) ∩ 𝕊_{n−1} intersects the curve u at the point of the parameter s_{0}.

In each of the above cases, the order of contact corresponds to the multiplicity of zero.

3. Classification of ω-limit behavior

We have seen that a class of globally equivalent equations from L_{n} is characterized by curve v ∈ 𝕊_{n−1}, having coordinates in C^{n} with the nonvanishing wronskian. Since the sphere 𝕊_{n−1} is compact, the ω-limit set of v, denoted by ω(v), is nonempty, closed and connected, see e.g. [10]. Exactly one from the following cases occurs:

a_{1}: ω(v) is a point p ∈ 𝕊_{n−1}, i.e. a connected subset of the intersection of a 1-dimensional subspace with 𝕊_{n−1}; a_{2}: ω(v) ⊆ (𝕊_{n−1} ∩ 𝔼_{2}), where 𝔼_{2} is a 2-dimensional subspace of 𝔼_{n}, and the case a_{1} is not valid;

… a_{i}: ω(v) ⊆ (𝕊_{n−1} ∩ 𝔼_{i}), where 𝔼_{i} is an i-dimensional subspace of 𝔼_{n}, and neither from the above cases is valid;

… a_{n−1}: ω(v) ⊆ (𝕊_{n−1} ∩ 𝔼_{n−1}), and neither from the above cases holds; a_{n}: neither from the above cases is valid.

We will consider also the following subcases of the cases a_{i} for i = 1, …, n: a_{i}^{0}: if the case a_{i} is valid and ω(v) ⊆ 𝕊_{n−1}^{0}, where 𝕊_{n−1}^{0} is an open

hemisphere of 𝕊_{n−1}.

Evidently the case a_{1} coincides with a_{1}^{0}.

4. Main result

Theorem. Consider an equation P_{n} from L_{n}; let T_{n} and R_{n} be equations defined as in §2, and y, v and u denote their n-tuples of linearly independent solutions. Let ω(v) and ω(u) be the ω-limit sets of v and u, respectively. If, for some i, the case a_{i} is valid for v (or for u), then the same case holds for every equation globally equivalent to P_{n}. Moreover, if the subcase a_{i}^{0} is valid for some i, then the same subcase is true for every equation globally equivalent to P_{n}.

Proof. Suppose first that the case a_{i} is valid for P_{n} ∈ L_{n}. First it means that ω(v) ⊆ 𝕊_{n−1} ∩ 𝔼_{i} for v := y/∥y∥. Then for each z,

z(t) := Af(t) · y(h(t)),

obtained by a global transformation (1′′), we have

ω(z/∥z∥) = ω(Af · y(h)/∥Af · y(h)∥) ⊆ S_{n−1} ∩ (A𝔼_{i}),

where A𝔼_{i} is again an i-dimensional subspace of 𝔼_{n}. Moreover, if ω(z/|z|) ⊆ S_{n−1} ∩ (A𝔼_{j}) for some j < i, we would get the contradiction to our supposition. Hence the case a_{i} is valid for every equation from L_{n} globally equivalent to P_{n}.

Now suppose that the subcase a_{i}^{0} is valid for P_{n}, that means that ω(v) ⊆ 𝕊_{n−1}^{0} ∩ 𝔼_{i} for v := y/∥y∥. Then for each z, z(t) := Af(t) · y(h(t)), we have ω(z/∥z∥) ⊆ Ŝ_{n−1}^{0} ∩ (A𝔼_{i}), where Ŝ_{n−1}^{0} = {s; s = Ar/∥Ar∥, r ∈ S_{n−1}^{0}} is again an open hemisphere in 𝔼_{n} and A𝔼_{i} is an i-dimensional subspace of 𝔼_{n}. Hence the case a_{1}^{0} is valid for every equation from L_{n} globally equivalent to P_{n}. ■

Remark 1. This theorem also shows that we may speak about the above cases and subcases with respect to a given equation and not only with respect to a particular n-tuple of its solutions, because, due to an arbitrary matrix A in (1′′), these cases and subcases are characterized by the properties which are invariant with respect to a choice of an n-tuple of linearly independent solutions of the considered equation.

5. Consequences

Oscillation or nonoscillation will be always considered with respect to the right end of the definition interval of a considered equation.

Corollary 1. (Oscillatory behavior of solutions). If the case a_{1} is valid for P_{n} ∈ L_{n}, then there do not exist n linearly independent oscillatory solutions (for t → b_{−}) of P_{n}. Moreover, there exist n linearly independent nonoscillatory solutions of P_{n} as (t → b_{−}).

Proof. Let P_{n} be a given equation, and y denote an n-tuple of its linearly independent solutions. Suppose that there exist n linearly independent oscillatory solutions of P_{n}. Then, due to Proposition 1, there are n great circles on 𝕊_{n−1}, not containing a common point, each of them being intersected by v = y/∥y∥, or equivalently, by u (see notation in §2) at points with infinitely many parameters to the right end of the interval of definition. Hence on each of these great circles there is at least one point belonging to ω(v) (ω(u)). Under our assumption, the case a_{1} is valid for P_{n}, i.e. ω(v) is a single point, say p on 𝕊_{n−1}. Thus this point must be common to n considered circles, which is a contradiction to the linear independence of the solutions. Hence there do not exist n linearly independent oscillatory solutions of P_{n}.

Now choose n independent vectors c_{1}, …, c_{n} in 𝔼_{n} such that the hyperplanes H(c_{i}), i = 1, …, n do not go through the point p. Then each solution c_{i}^{T} ·y(x) is nonoscillatory. In fact, if c_{i}^{T} ·y(x) were oscillatory, then y/∥y∥∩H(c_{i}) would be an infinite sequence on the great circle 𝕊_{n−1} ∩H(c_{i}) that should have an accumulation point in ω(y/∥y∥) = p, contrary to our choice of the hyperplanes. ■

Corollary 2. (Asymptotic behavior of solutions). If the case a_{1} is valid for equation P_{n} from L_{n}, then P_{n} admits an n-tuple y^{∗} = (y_{1}^{∗}, …, y_{n}^{∗})^{T} of linearly independent solutions such that

lim_{x→b_{−}} \frac{y_{1}^{∗}}{\sqrt{(y_{1}^{∗})^{2} + ⋯ + (y_{n}^{∗})^{2}}} = 1

and

lim_{x→b_{−}} \frac{y_{i}^{∗}}{\sqrt{(y_{1}^{∗})^{2} + ⋯ + (y_{n}^{∗})^{2}}} = 0 for i = 2, …, n.

Proof. In the case a_{1} we have lim_{x→b_{−}} y(x)/∥y(x)∥ = p, p being a point on 𝕊_{n−1}. Choose an n-tuple of orthonormal vectors c_{1}, …, c_{n}, where c_{1} := p, otherwise arbitrary. Denote by C the orthogonal matrix (c_{1}, …, c_{n}). Define y_{i}^{∗} := c_{i}^{T} · y, i.e. y^{∗} = C^{T} · y. Then

lim_{x→b_{−}} y_{1}^{∗}/∥y^{∗}∥ = lim_{x→b_{−}} \frac{c_{1}^{T} y}{∥y^{T} CC^{T} y∥} = c_{1}^{T} · lim_{x→b_{−}} y/∥y∥ =

c_{1}^{T} · p = c_{1}^{T} · c_{1} = 1

and for i = 2, …, n,

lim_{x→b_{−}} y_{i}^{∗}/∥y^{∗}∥ = lim_{x→b_{−}} \frac{c_{i}^{T} y}{|y^{T} CC^{T} y|} = c_{i}^{T} · lim_{x→b_{−}} y/∥y∥ =

c_{i}^{T} · p = c_{i}^{T} · c_{1} = 0. ■

Corollary 3. If the second order equation

y′′ + p_{1}(x)y′ + p_{0}(x)y = 0 on I = (a, b), −∞ ≤ a < b ≤ ∞ (2)

is nonoscillatory (for x → b_{−}), then the case a_{1} is valid for (2). If the equation (2) is oscillatory (for x → b_{−}), then the case a_{2} holds for (2). The subcase a_{2}^{0} cannot occur.

Proof. For two linearly independent solutions y_{1}, y_{2} of equation (2), y = (y_{1}, y_{2})^{T}, the curve v = y/∥y∥ is an arc on the unit circle 𝕊_{1} in the plane 𝔼_{2}. Due to Proposition 1, if equation (2) is oscillatory for x → b_{−}, then this arc v infinitely many times encircles the origin (without turning points, see [13]), and hence ω(v) is exactly 𝕊_{1}. If equation (2) is nonoscillatory for x → b_{−}, then the arc v ends by approaching a point on 𝕊_{1}, exactly its ω-limit set, and the case a_{1} holds for (2). ■

Corollary 4. If the case a_{j} is valid for an equation P_{n} for some j > 1, then there exist n linearly independent oscillatory solutions of P_{n}.

Proof. In the case a_{j} for some j > 1, the set ω(v) contains two different points on 𝕊_{n−1}, say p_{1} and p_{2}. Evidently, there exist n hyperplanes H(c_{i}), i = 1, …, n, in 𝔼_{n} with linearly independent vectors c_{1}, …, c_{n}, each of them separating points p_{1} and p_{2} into opposite open halfspaces of 𝔼_{n}, i.e. c_{i}^{T} p_{1} > 0 and c_{i}^{T} p_{2} < 0 for each i = 1, …, n. Hence, due to Proposition 1, each solution c_{i}^{T} y(x) oscillates for x → b_{−}, because the curve v intersects infinitely many times the hyperplane H(c_{i}) as x → b_{−}. ■

Remark 1. As an immediate consequence of this corollary we may state: If equation L_{n} does not admit n linearly independent oscillatory solu-

tions, then the case a_{1} is valid for it. In particular, if each solution of equation P_{n} is nonoscillatory, then the case a_{1} takes place for L_{n}.

Corollary 5. If, for some i = 1, …, n, the case a_{i}^{0} is valid for equation P_{n}, then there exist n linearly independent nonoscillatory solutions of P_{n}.

Proof. Let y denote an n-tuple of linearly independent solutions of P_{n}. Under our assumption, ω(y/∥y∥) lies inside an open hemisphere of 𝕊_{n−1} determined by a hyperplane H(p). Evidently p^{T} y(x) is a nonoscillatory solution. Moreover, ω(y/∥y∥) is closed, and hence there exists an neighbourhood N of the point p ∈ 𝕊_{n−1} such that H(q)∩ω(y/∥y∥) = ∅ for each q ∈ N. If we take n linearly independent vectors (points) q_{1}, …, q_{n} from N, then

y_{i} := q_{i}^{T} y, i = 1, …, n,

are required nonoscillatory solutions. In fact, if one of these solutions were oscillatory, then, again due to Proposition 1, the corresponding hyperplane would intersect the curve y (or equivalently y/∥y∥) infinitely many times. Hence this hyperplane would contain at least one point in ω(y/∥y∥), contrary to our choice of the above hyperplanes. ■

Remark 2. Comparing Corollaries 4 and 5 we see that in the case a_{i}^{0} with i > 1 for L_{n}, this equation admits both an n-tuple of oscillatory solutions and, at the same time, another n-tuple of nonoscillatory solutions.

Remark 3. Also other (e.g. topological) properties of ω(v) that are invariant with respect to the centroaffine transformations can be considered for introducing other, more detailed classifications of the classes of equivalent linear differential equations from L_{n}.

6. Examples

1. The differential equation

y^{(n)} = 0 on (0, ∞)

has n linearly independent solutions: x^{n−1}, x^{n−2}, …, 1. For this equation the case a_{1} holds, no solution is oscillatory and

lim_{x→∞} \frac{x^{n−1}}{\sqrt{∑_{j=0}^{n−1} x^{2j}}} = 1,

lim_{x→∞} \frac{x^{n−2}}{\sqrt{∑_{j=0}^{n−1} x^{2j}}} = 0, …, lim_{x→∞} \frac{1}{\sqrt{∑_{j=0}^{n−1} x^{2j}}} = 0,

in accordance with Corollary 2 and Remark 1.

2. The equation

y′′′ + 2y′′ + 2y′ = 0 on (0, ∞)

admits the solutions: 1, e^{−x} sin x, e^{−x} cos x. For this equation the case a_{1} is valid. There are two linearly independent oscillatory solutions as x → ∞, there are no three linearly independent oscillatory solutions. This equation admits three linearly independent nonoscillatory solutions, and

lim_{x→∞} \frac{1}{\sqrt{1 + e^{−2x}}} = 1, lim_{x→∞} \frac{e^{−x} sin x}{\sqrt{1 + e^{−2x}}} = 0, lim_{x→∞} \frac{e^{−x} cos x}{\sqrt{1 + e^{−2x}}} = 0,

as Corollaries 1 and 2 state.

3. However, the equation

y′′′ − 2y′′ + 2y′ = 0 on (0, ∞)

admits the solutions: 1, e^{x} sin x, e^{x} cos x; the corresponding ω-limit set is a great circle on the sphere 𝕊_{2} in 𝔼_{3} and hence the case a_{2} is valid for it. However, the subcase a_{2}^{0} does not take place. Except of the constant solutions, each other solution is oscillatory (as x → ∞), see Corollaries 4,5 and Remark 2.

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(Received 06.04.1993)

Author’s address: Mathematical Institute, Academy of Sciences of the Czech Republik, Mendelovo nám. 1, CR-66 282 Brno, Czech Republik