Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 19 (2000), No. 1, 279–284

Extension of the Bernstein Condition to Systems of

Ordinary Differential Equations of General Form

A. Ja. Lepin and A. D. Myshkis

Abstract. The Bernstein condition of boundedness of the derivatives of an a priori bounded solution of a 2nd order ordinary differential equation is extended to systems in which each equation has its own order.

Keywords: Ordinary differential equations, continuation of solutions, Bernstein condition

AMS subject classification: 34 A 15

1. Introduction

The Bernstein theorem for the equation

x′′(t) = f(t, x(t), x′(t))

is well-known [1: Section 1.2]. According to it, the inequality

|f(t, x, x_{1})| ≤ A x_{1}^{2} + B (A, B constants)

guarantees the boundedness of x′, if the solution x of the equation above is bounded. This theorem was extended in several directions. So, the vector equation

x^{(n)}(t) = f(t, x(t), x′(t), ..., x^{(n−1)}(t))  (x(t) ∈ ℝ^{m} (m ≥ 1), n ≥ 2)  (1)

was considered in [2] with f continuous. There was proven that, if the function f satisfies the estimation

|f(t, x, x_{1}, …, x_{n−1})| ≤ A (|x_{1}|^{n} + |x_{2}|^{\frac{n}{2}} + … + |x_{n−1}|^{\frac{n}{n−1}} ) + B  (2)

for |x| ≤ a (a > 0) and A, B > 0, then any solution x : [t_{0}, T] → ℝ^{m} of (1) which satisfies the a priori estimation |x(t)| ≤ α with sufficiently small α depending only on A, B and m, n can be continued onto the whole semiaxis [t_{0}, ∞) and has bounded derivatives x′, …, x^{(n−1)} on it. But if condition (2) is replaced by

A. Ja. Lepin: Latvian Univ., Inst. Math. & Inf. Sci., bul. Rainis 29, Riga, Latvia A. D. Myshkis: Moscow State Univ. of Comm., ul. Obraztsova 15, Moscow 101475, Russia; with a support of the Russian Foundation for Fundamental Research (grant 97-01-00909a), Foundation of Basic Research of the Ministry of Railways of RF and Foundation of MH & BSE RF (grant 97-0-1.8-53).

ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

sup_{t∈[t_{0},∞)} max_{|x|≤a} |f(t, x, x_{1}, …, x_{n−1})| = o(|x_{1}|^{n} + … + |x_{n−1}|^{\frac{n}{n−1}} )  (3)

as

|x_{1}| ; ⋮ ; |x_{n−1}| } → ∞,

for any fixed a > 0, then the condition of sufficient smallness of α is eliminated, i.e. any a priory bounded solution x has bounded derivatives x′, …, x^{(n−1)} (this statement holds under estimation (2) only if n = 2 and m = 1, i.e. in the case covered by the Bernstein theorem).

The transition to a right-hand side of equation (1) which satisfies the Carathéodory conditions (see, e.g., [3: Section 18.4]), the replacement of boundedness of the solution x on its uniform L_{p}-boundedness on segments of fixed length, and some other generalizations are contained in [4, 5]. The results of [5] can be applied especially to the system of scalar equations

x_{i}^{(n_{i})}(t) = f_{i}(t, …, x_{j}^{(k)}(t), …)  ({i,j=1,...,m \atop k=0,...,n_{j}−1}).  (4)

The aim of the present paper is to give effective sufficiency conditions on the functions f_{i} for the possibility of a continuation onto the whole semi-axis of any a priory bounded solution of system (4) and the boundedness of all its derivatives x_{j}^{(k)}(t) (k ≤ n_{j} − 1)

2. General plan of the estimation of derivatives

2.1. We consider solutions of the system of scalar equations (4), whose right-hand sides are given for t ∈ [0, ∞) and arbitrary values of other arguments and satisfy the Carathéodory condition. Uniqueness of the solution of any Cauchy problem is not supposed. Let the solution

t ↦ x(t) = (x_{1}(t), …, x_{m}(t))

of system (4) be built starting from t = 0 in the direction of growth of t, and let be known that the values of this solution, being arbitrarily continued, cannot leave some domain

Q = [−α_{1}, α_{1}] × ⋯ × [−α_{m}, α_{m}] (α_{1}, …, α_{m} ∈ (0, ∞)).

The problem is to find conditions on the functions f_{i} under which all derivatives of the solution of system (4) indicated in the right-hand sides of that system remain bounded. In particular, it follows from here that any such solution can be continued on the whole semi-axis [0, ∞).

We shall use the Kolmogorov-Gorny inequality (see, e.g., [6: Supplement 37]) for any function ψ ∈ C^{s}([a, b]; ℝ)

∥ψ^{(k)}∥ ≤ a_{s,k} ∥ψ∥^{\frac{s−k}{s}} [ max \{∥ψ^{(s)}∥, \frac{s!}{(b − a)^{s}} ∥ψ∥\}]^{\frac{k}{s}} (k = 0, …, s − 1) (5)

where ∥ · ∥ = max_{[a,b]} | · | while a_{s,k} > 0 are absolute constants with a_{s,0} = 1. The following simple lemma will be needed for us: Lemma 1. For any s ∈ ℕ there exists r_{s} > 0 such that the implication

a ∈ ℝ, b ∈ (a, ∞), φ ∈ C^{s}([a, b], ℝ) ⟹ (b − a)^{s} min |φ^{(s)}| ≤ r_{s} max |φ| holds.

2.2. Let x : [0, t] → Q (0 < t < ∞) be a solution of system (4) and denote

M_{i}(t) = max_{τ∈[0,t]} |x_{i}^{(n_{i}−1)}(τ)| (i = 1, …, m).

We find conditions under which all functions M_{i}(t) remain bounded in the continuation process of any such solution of system (4). Then the boundedness of its derivatives of lower orders will follow from (5).

Consider the i-th equation of system (4). If

|x_{i}^{(n_{i}−1)}(τ)| > \frac{1}{2}M_{i}(t) (∀ τ ∈ [0, t]),

then from Lemma 1

\frac{1}{2}t^{n_{i}−1}M_{i}(t) ≤ r_{n_{i}−1}α_{i},  i.e. M_{i}(t) ≤ 2\frac{r_{n_{i}−1}α_{i}}{t^{n_{i}−1}}  (6)

follows. Let now be

min_{τ∈[0,t]} |x_{i}^{(n_{i}−1)}(τ)| ≤ \frac{1}{2}M_{i}(t).

Then values t_{i1}, t_{i2} ∈ [0, t] depending on t exist such that

|x_{i}^{(n_{i}−1)}(t_{i1})| = M_{i}(t)
|x_{i}^{(n_{i}−1)}(t_{i2})| = \frac{1}{2}M_{i}(t)
|x_{i}^{(n_{i}−1)}(τ)| ∈ (\frac{1}{2}M_{i}(t), M_{i}(t)) ∀ τ between t_{i1} and t_{i2}.

Integrating both parts of equation (4) from t_{i1} up to t_{i2}, we obtain

M_{i}(t) = 2 |∫_{t_{i1}}^{t_{i2}} f_{i}(τ, …, x_{j}^{(k)}(τ), …) dτ| .  (7)

Moreover, from Lemma 1

\frac{1}{2}|t_{i2} − t_{i1}|^{n_{i}−1}M_{i}(t) ≤ r_{n_{i}−1}α_{i}, i.e. |t_{i2} − t_{i1}| ≤ (2\frac{r_{n_{i}−1}α_{i}}{M_{i}(t)} )^{\frac{1}{n_{i}−1}}  (8)

follows.

In order to estimate the right-hand side of (7), denote

\begin{matrix}Φ_{i}(…, b_{jk}, …; δ, t) = \\ sup & \{|∫_{t_{1}}^{t_{1}+h} f_{i}(τ, …, φ_{jk}(τ), …) dτ| : \begin{matrix}0 ≤ t_{1} ≤ t_{1} + h ≤ t, h ≤ δ \\ φ_{jk} ∈ C([0, t], [−b_{jk}, b_{jk}]) \end{matrix} \} \end{matrix} (9)

for b_{jk} > 0 (j = 1, …, m; k = 0, …, n_{i} − 1). Then we obtain from (5) - (7) (with s = n_{j} − 1) and (8) that

\begin{aligned}M_{i}(t) & ≤ 2 max \{\frac{r_{n_{i}−1}α_{i}}{t^{n_{i}−1}} , \\ & Φ_{i}( …, a_{n_{j}−1,k}αj^{\frac{n_{j} −1−k}{n_{j} −1}} [ max \{M_{j}(t), (n_{j} − 1)! \frac{α_{j}}{t^{n_{j}−1}} \}]^{\frac{k}{n_{j} −1}} , …; \\ & [2\frac{r_{n_{i}−1}α_{i}}{M_{i}(t)} ]^{\frac{1}{n_{i}−1}} , t)\} \end{aligned}  (10)

(i = 1, …, m). Here one must take M_{j}(t) instead of the inner maximum if k = n_{j} − 1. If some n_{i} = 1, then the corresponding equation (10) is not considered and M_{i}(t) is replaced by α_{i} in all other equations.

Thus we have obtained system (10) which contains m inequalities connecting m non-decreasing non-negative functions t ↦ M_{i}(t). According to that what has been said we obtain the following

Theorem 1. If from the inequality system (10) the boundedness of all functions M_{i} for any {α_{j}} or any sufficiently small {α_{j}} follows, then by continuation of any a priori bounded solution or respectively any bounded with sufficiently small constants solution of system (4), all derivatives indicated in the right-hand sides of system (4) remain bounded and the continuation is possible for arbitrary large values of t.

3. Examples

3.1. Consider equation (1) with m = 1, i.e. the scalar case, where condition (2) holds. Then we obtain from (9)

Φ(b_{0}, …, b_{n−1}; δ, t) ≤ A δ(b_{1}^{n} + … + bn−1^{\frac{n}{n−1}}) + Bδ

for all 0 ≤ t < ∞ and |b_{0}| ≤ a. We can assume that t ≥ t_{0} for some t_{0} > 0 as it is possible to apply the existence theorem for the Cauchy problem on the interval [0, t_{0}] for sufficiently small t_{0}. Then inequality (10) for M(t) > 0 takes the form

\begin{aligned}M(t) & ≤ 2 max \{\frac{r_{n−1}α}{t_{0}^{n−1}}, [2\frac{r_{n−1}α}{M(t)}]^{\frac{1}{n−1}} \\ & × [B + A ∑_{k=1}^{n−1} (a_{n−1,k}α^{\frac{n−1−k}{n−1}} [ max \{M(t), (n − 1)! \frac{α}{t_{0}^{n−1}} \}]^{\frac{k}{n−1}} )^{\frac{n}{k}} ]\} \end{aligned}

From here

M(t) ≤ max \{C_{1}α, C_{2}[\frac{α}{M(t)}]^{\frac{1}{n−1}} + C_{3} max \{ ∑_{k=1}^{n−1} α^{\frac{n−k}{k}} M(t), α^{\frac{n^{2}−n−1}{k(n−1)}} \}\}  (11)

follows where the constants C_{1}, C_{2}, C_{3} do not depend on α and M(t). We see that the function t ↦ M(t) is bounded for all t ≥ t_{0} and sufficiently small α, and therefore Theorem 1 is applicable in the 2nd variant.

If condition (2) is repalced by (3), then the boundedness of M(t) for any α follows from the fact that the value A and therefore C_{3} in estimate (11) can be chosen arbitrarily small for sufficiently large M(t). Therefore Theorem 1 is applicable in the 1st variant in this case.

3.2. Let be m = 1 and let the right-hand side of equation (1) admit the estimate

|f(t, x, x_{1}, …, x_{n−1})| ≤ ∑_{r=1}^{p} g_{r}(t) |x_{1}|^{α_{r,1}} ⋯ |x_{n−1}|^{α_{r,n−1}}

∀ t ∈ [0, ∞), x ∈ [−α, α], x_{1}, …, x_{n−1} ∈ ℝ

for some α_{i,j} ≥ 0 where ∫_{t_{1}}^{t_{2}} g_{i}(t) dt = o(|t_{2}−t_{1}|^{γ_{i}}) as t_{1}, t_{2} ∈ [0, ∞) with 0 < |t_{2}−t_{1}| → 0 and γ_{i} ∈ [0, 1] (i = 1, …, p). Then

Φ(b_{0}, …, b_{n−1}; δ, t) = o (∑_{r=1}^{p} δ^{γ_{r}}b_{1}^{α_{r,1}} ⋯ b_{n−1}^{α_{r,n−1}}) (δ → 0).

Hence we obtain, arguing as in Example 3.1, that if

∑_{k=1}^{n−1} kα_{r,k} − γ_{r} ≤ n − 1 (r = 1, …, p),

then the function t ↦ M(t) is bounded for all t ≥ 0 and any α > 0, i.e. Theorem 1 is applicable in its 1-st variant.

3.3. Consider the system of scalar equations with bounded functions g_{i}

\begin{matrix}x′′_{1}(t) & = g_{1}(t, x(t), x′(t)) |x′_{1}(t)|^{β_{11}}|x′_{2}(t)|^{β_{12}} \\ x′′_{2}(t) & = g_{2}(t, x(t), x′(t)) |x′_{1}(t)|^{β_{21}}|x′_{2}(t)|^{β_{22}} \end{matrix} \}  (0 ≤ t < ∞)  (12)

for some β_{ij} > 0 (1 ≤ i, j ≤ 2) and x = (x_{1}, x_{2}). Here

Φ_{i}(b_{11}, b_{21}; δ, t) ≤ G b_{11}^{β_{i1}}b_{21}^{β_{i2}}δ (i = 1, 2; G > 0),

therefore the system of inequalities (10) has the form

M_{i}(t) ≤ max {C_{1}α_{i}, C_{2}M_{1}(t)^{β_{i1}}M_{2}(t)^{β_{i2}}α_{i}M_{i}(t)^{−1}}  (i = 1, 2)  (13)

for t ≥ t_{0} > 0 where C_{1} > 0 and C_{2} > 0 are certain constants. After taking the logarithm of both sides of (13) and denoting y_{i} = ln M_{i}(t) we obtain the inequality system

\begin{matrix}y_{1} & ≤ max \{ ln(C_{1}α_{1}), ln(C_{2}α_{1}) + (β_{11} − 1)y_{1} + β_{12}y_{2}\} \\ y_{2} & ≤ max \{ ln(C_{1}α_{2}), ln(C_{2}α_{2}) + β_{21}y_{1} + (β_{22} − 1)y_{2}\} \end{matrix} \} .  (14)

Inequality (14)_{1} defines an angle in the (y_{1}, y_{2})-plane which is larger than π and bounded by two rays given as

\begin{matrix}y_{1} & = ln(C_{1}α_{1}) \\ y_{2} & = \frac{2 − β_{11}}{β_{12}} y_{1} − \frac{ln(C_{2}α_{1})}{β_{12}} \end{matrix} \}

where the first goes downwards and the second one to the right. Further, inequality (14)_{2} defines an angle which is larger than π and bounded by two rays given as

\begin{matrix}y_{2} & = ln(C_{1}α_{2}) \\ y_{1} & = \frac{2 − β_{22}}{β_{21}} y_{2} − \frac{ln(C_{2}α_{2})}{β_{21}} \end{matrix} \}

where the first goes to the left and the second one upwards. The direct consideration of the intersection of these angles shows that the conditions

β_{11} < 2, \frac{β_{12}}{2 − β_{11}} < \frac{2 − β_{22}}{β_{21}}  or  β_{11} < 2, \frac{β_{12}}{2 − β_{11}} ≤ \frac{2 − β_{22}}{β_{21}}  (15)

are necessary and sufficient for the boundedness from above both coordinates of its points for any, or respectively any sufficiently small values of α_{1} and α_{2}.

Thus Theorem 1 is applicable to system (12) in the 1-st or 2-nd variant, if inequalities (15)_{1} or equality in (15)_{2} hold.

Acknowledgement. We express our thanks to the referee for his helpful criticism.

References

[1] Bernstein, S.: Sur les equations du calcul des variations. Ann. Ec. Norm. 29 (1912), 431 – 485.

[2] Lepin, A. Ja. and A. D. Myshkis: On the boundedness of the derivatives of the bounded solutions of ordinary differential equations (in Russian). Diff. Uravn. 1 (1965), 1260 – 1263.

[3] Kurzweil, J.: Ordinary Differential Equations. Amsterdam: Elsevier Sci. Publ. 1986.

[4] Klokov, Ju. A.: On a priori estimates of the solutions of ordinary differential equations (in Russian). Diff. Uravn. 23 (1987), 611 – 618.

[5] Lepin, A. Ja.: A priori boundedness of the derivatives for a system of differential inequalities (in Russian). Diff. Uravn. 25 (1989), 1129 – 1135.

[6] Hardy, G. H., Littlewood, J. E. and G. Polya: Inequalities. Cambridge: Univ. Press 1934.

Received 22.06.1999