GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 3, 1995, 225-236

ON STRUCTURE OF SOLUTIONS OF A SYSTEM OF FOUR DIFFERENTIAL INEQUALITIES

MIROSLAV BARTUŠEK

Abstract. The aim of the paper is to study a global structure of solutions of four differential inequalities

α_{i}y′_{i}(t)y_{i+1} ≥ 0, y_{i+1}(t) = 0 ⇒ y′_{i}(t) = 0, i = 1, 2, 3, 4 ,
α_{i} ∈ {−1, 1}, α_{1}α_{2}α_{3}α_{4} = −1

with respect to their zeros. The structure of an oscillatory solution is described, and the number of points with trivial Cauchy conditions is investigated.

1. Introduction

The aim of this paper is to investigate the global structure with respect to zeros of solutions of the system of differential inequalities

\begin{matrix}α_{i}y′_{i}(t)y_{i+1} ≥ 0, \\ y_{i+1}(t) = 0 ⇒ y′_{i}(t) = 0, i ∈ N_{4}, t ∈ J, \end{matrix}  (1)

where J = (a, b), −∞ ≤ a < b ≤ ∞, y_{5} = y_{1}, N_{4} = {1, 2, 3, 4}

α_{i} ∈ {−1, 1}, α_{1}α_{2}α_{3}α_{4} = −1 .  (2)

y = (y_{1}, y_{2}, y_{3}, y_{4}) is called a solution of (1) if y_{i} : J → R, R = (−∞, ∞) is locally absolutely continuous and (1) holds for all t ∈ J such that y′_{i} exists.

Let us mention two special cases of (1) which are often studied; see, for example, [1-4] (and the references therein).

(a) A system of four differential equations

\begin{matrix}y′_{i} = f_{i}(t, y_{1}, y_{2}, y_{3}, y_{4}), i ∈ N_{4} , \\ α_{i}f_{i}(t, x_{1}, x_{2}, x_{3}, x_{4})x_{i+1} ≥ 0 , \\ x_{i+1} = 0 ⇒ f_{i}(t, x_{1}, x_{2}, x_{3}, x_{4}) = 0 on D, i ∈ N_{4}, \end{matrix}  (3)

225 where (2) holds, x_{5} = x_{1}, f_{i} : D = R^{5} → R fulfills the local Carathéodory conditions, i ∈ N_{4}.

1991 Mathematics Subject Classification. 34C10, 34A40.

Key words and phrases. Differential inequalities, global structure of solutions, oscillatory solution.

1072-947X/95/0500-0225$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

(b) A fourth-order differential equation with quasiderivatives

\begin{matrix}L_{4}x(t) = f(t, L_{0}x, L_{1}x, L_{2}x, L_{3}x) , \\ f(t, x_{1}, x_{2}, x_{3}, x_{4})x_{1} ≤ 0, f(t, 0, x_{2}, x_{3}, x_{4}) = 0, \end{matrix}  (4)

where f : R^{5} → R fulfills the local Carathéodory conditions, a_{j} : R → R is continuous and positive, j = 0, 1, 2, 3, 4, and L_{j}x is the jth quasiderivative of x : L_{0}x = a_{0}(t)x, L_{i}x = a_{i}(t)(L_{i−1}x)′, i ∈ N_{4}.

By the use of the standard transformation we can see that (4) is a special case of (3): y_{j} = L_{j−1}x, j ∈ N_{4},

y′_{i} = \frac{y_{i+1}}{a_{i}(t)}, i = 1, 2, 3, y′_{4} = \frac{1}{a_{4}(t)}f(t, y_{1}, y_{2}, y_{3}) .  (5)

In [1] the structure of oscillatory solutions (defined in the usual sense) was studied for the fourth-order differential equation (4), a_{j} ≡ 1. It was shown that two different types of them can exist and their structure was described. For example, for every type the zeros of a solution y and its derivatives y^{(i)}, i = 1, 2, 3 are uniquely ordered. This information allows a more profound study of the asymptotic behavior. In [2] it was shown that the zeros of components of a solution of (1) (under further assumptions) are simple in some neighborhood of their cluster point (the zero τ of y_{i}, i ∈ N_{4}, is simple if y_{i+1}(τ) ≠ 0 holds).

In the present paper the above-mentioned results are generalized for (1).

Notation. Let y be a solution of (1). Put Y_{1} = y_{1}, Y_{2} = α_{1}y_{2}, Y_{3} = α_{1}α_{2}y_{3}, Y_{4} = α_{1}α_{2}α_{3}y_{4}, Y_{i+4k} = Y_{i}, y_{i+4k} = y_{i}, i ∈ N_{4}, k ∈ Z = {…, −1, 0, 1, … }.

Definition 1. Let y : (a, b) → R^{4} be a solution of (1). Then y is called trivial if y_{i}(t) = 0 in (a, b), i ∈ N_{4}. Let c be a point such that c ∈ (a, b), y_{i}(c) = 0, i ∈ N_{4}, holds. Then c is called a Z-point of y.

Definition 2. Let y : (a, b) → R^{4} be a solution of (1). Then y has Property W if for every i ∈ N_{4}

(a) there exists at most one maximal bounded interval J ⊂ (a, b) such that either

y_{i}(t) = y_{i+1}(t) = y_{i+2}(t) = 0, y_{i+3}(t) ≠ 0, t ∈ J,

or

y_{i}(t) = y_{i+2}(t) = 0, y_{i+1}(t)y_{i+3}(t) ≠ 0, t ∈ J ;

(b) there exist at most two maximal bounded intervals J, J_{1} ⊂ (a, b), J ∩ J_{1} = ∅ such that y_{i}(t) = y_{i+1}(t) = 0, y_{i+2}(t)y_{i+3}(t) ≠ 0 in J ∪ J_{1}.

For the study of the structure of solutions of (1) we define the following types. Let y : J = (a, b) → R^{4}.

Type I(s, s̄): For given s ∈ Z ∪ {−∞}, s̄ ∈ Z ∪ {∞}, s̄ ≥ s − 1 there exist s_{i}, s̄_{i} and the sequences {t_{k}^{i}}, {t̄_{k}^{i}}, k ∈ {s_{i}, s_{i} + 1, …, s̄_{i}}, i ∈ N_{4} such that s_{1} = s, s̄_{1} = s̄, s_{i} ∈ {s_{1} − 1, s_{1}}, s_{j} ≥ s_{j−1}, s̄_{i} ∈ {s̄_{1} − 1, s̄_{1}}, s̄_{j} ≤ s̄_{j+1}, j = 2, 3, 4, s̄_{5} = s̄_{1} holds and for all admissible k we have

t_{k}^{1} ≤ t̄_{k}^{1} < t_{k}^{4} ≤ t̄_{k}^{4} < t_{k}^{3} ≤ t̄_{k}^{3} < t_{k}^{2} ≤ t̄_{k}^{2} < t_{k+1}^{1} ≤ t̄_{k+1}^{1} ,

Y_{i}(t) = 0 for t ∈ [t_{k}^{i}, t̄_{k}^{i}], Y_{i}(t) ≠ 0 for t ∈ J − ⋃_{k=s_{i}}^{s̄_{i}}[t_{k}^{i}, t̄_{k}^{i}] ,

\begin{aligned}Y_{j}(t)Y_{1}(t) & > 0 for t ∈ (t̄_{k}^{1}, t_{k}^{j}) , \\ & < 0 for t ∈ (t̄_{k}^{j}, t_{k+1}^{1}), j = 2, 3, 4, i ∈ N_{4} . \end{aligned}

Moreover, for i ∈ N_{4}, cd = −1, where c(d) is the sign of Y_{i} in some left (right) neighborhood of t_{k}^{i} (t̄_{k}^{i}). If s = −∞ (s̄ = ∞), then lim_{k→−∞} t_{k}^{i} = a (lim_{k→∞} t_{k}^{i} = b) holds.

Type II(s, s̄): For given s ∈ Z ∪ {−∞}, s̄ ∈ Z ∪ {∞}, s̄ ≥ s − 1 there exist s_{i}, s̄_{i} and the sequences {t_{k}^{i}}, {t̄_{k}^{i}}, k ∈ {s_{i}, s_{i} + 1, …, s̄_{i}}, i ∈ N_{4} such that s_{1} = s, s̄_{1} = s̄, s_{i} ∈ {s_{1} −1, s_{1}}, s_{j} ≤ s_{j−1}, s̄_{j} ∈ {s̄_{1} −1, s̄_{1}}, s̄_{j} ≤ s_{j−1}, j = 2, 3, 4, and for all admissible k

t_{k−1}^{1} ≤ t̄_{k−1}^{1} < t_{k}^{2} ≤ t̄_{k}^{2} < t_{k}^{3} ≤ t̄_{k}^{3} < t_{k}^{4} ≤ t̄_{k}^{4} < t_{k}^{1} ≤ t̄_{k}^{1} ,

Y_{i}(t) = 0 for t ∈ [t_{k}^{i}, t̄_{k}^{i}], Y_{i}(t) ≠ 0 for t ∈ J − ⋃_{k=s_{i}}^{s̄_{i}}[t_{k}^{i}, t̄_{k}^{i}] ,

\begin{aligned}(−1)^{j+1}Y_{j}(t)Y_{1}(t) & > 0 for t ∈ (t̄_{k−1}^{1}, t_{k}^{j}) \\ & < 0 for t ∈ (t̄_{k}^{j}, t_{k}^{1}), j = 2, 3, 4, i ∈ N_{4}, \end{aligned}

hold. Moreover, if i ∈ N_{4}, cd = −1, where c (d) is the sign of Y_{i} in some left (right) neighborhood of t_{k}^{i} (t̄_{k}^{i}). If s = −∞(s̄ = ∞), then lim_{k→−∞} t_{k}^{i} = a (lim_{k→∞} t_{k}^{i} = b) holds.

Type III. There exist j ∈ N_{4}, τ ∈ {−1, 1} such that

τY_{j}(t) ≥ 0, τc_{i}Y_{j+1}(t) > 0, t ∈ J, i = 1, 2, 3 ,

|Y_{j+k}| is nondecreasing, |Y_{j+3}| is nonincreasing on J, k = 0, 1, 2, where c_{1} = 1 for j = 1, 2, 3, c_{1} = −1 for j = 4, c_{2} = 1 for j = 1, 2, c_{2} = −1 for j = 3, 4, c_{3} = 1 for j = 1, c_{3} = −1 for j = 2, 3, 4.

Type IV. There exist j ∈ N_{4}, τ ∈ {−1, 1} such that

τY_{j}(t) ≥ 0, τc_{i}Y_{j+i}(t) > 0, t ∈ J, i = 1, 2, 3 ,

|Y_{j}| is nondecreasing, |Y_{j+k}| is nonincreasing in J, k = 1, 2, 3, where c_{1} = 1 for j = 1, 2, 3, c_{1} = −1 for j = 4, c_{2} = 1 for j = 3, 4, c_{2} = −1 for j = 1, 2, c_{3} = 1 for j = 1, c_{3} = −1 for j = 2, 3, 4.

Type V. There exist j ∈ N_{4}, τ ∈ {−1, 1} such that

Y_{j} = 0, τY_{j+1}(t) ≥ 0, τcY_{j+3}(t) > 0, sign Y_{j+2}(t) is

constant in J where c = 1 for j = 3, 4, c = −1 for j = 1, 2.

Type VI. y ≡ 0 in J.

Remark 1. The solutions of either Type I(s, ∞) or II(s, ∞) are called oscillatory. The solutions of Type III, IV, V are usually called nonoscillatory.

Definition 3. Let y : (a, b) → R^{4} and let A_{i}, i = 0, 1, …, s be one of Types I - VI. y is successively of Types A_{1}, A_{2}, …, A_{s} if numbers τ_{0}, …, τ_{s} exist such that a = τ_{0} ≤ τ_{1} ≤ ⋯ ≤ τ_{s} = b and y is of Type A_{j} on (τ_{j−1}, τ_{j}), j = 1, 2, …, s. At the same time, if y is of Type A in (τ, τ), then Type A is missing.

2. Main results

Theorem 1. Let y : J = (a, b) → R^{4}, −∞ ≤ a < b ≤ ∞ be a solution of (1).

(i) Let Z-points of y not exist in J. Then numbers s, s̄, r, r̄ exist such that s, r ∈ Z ∪ {−∞}, s̄, r̄ ∈ Z ∪ {∞} and y is successively of Types IV, II (s, s̄), V, I (r, r̄), III on J; if r = −∞ (s̄ = +∞), then Types IV, II, V (Types V, I, III) are missing, if r̄ = ∞ (s = −∞), then Type III (Type IV) is missing. Moreover, y has Property W.

(ii) Let y have the only Z-point τ in J. Then

(a) y is either of Type I(s, ∞) or of Type II(s, ∞) in some left neighborhood of τ, where s ∈ Z is a suitable number;

(b) y is either of Type I(−∞, s) or of Type II(−∞, s) in some right neighborhood of τ, where s ∈ Z is a suitable number.

(iii) Let τ, τ_{1}, a < τ < τ_{1} < b be Z-points of y and let no Z-point of y exist in (τ, τ_{1}). Then y is either of Type I(−∞, ∞) or of Type II(−∞, ∞) or numbers s, r exist such that s, r ∈ Z and y is successively of Types II(−∞, s), V, I(r, ∞) in (τ, τ_{1}). In the last case Types I, II are always present.

Remark 2. Let i ∈ N_{4}. Consider (1) with an extra condition

α_{i}y′_{i}(t)y_{i+1}(t) > 0 for y_{i+1}(t) ≠ 0 .

Let y be a solution of this problem, y : J = (a, b) → R^{4}, a < b. If y is either of Type I or II, then t_{k}^{i} = t̄_{k}^{i} for all admissible k. If y is either of Type III or IV or V, then the relation

y_{i+1} ≠ 0 on I ⇒ y_{i} ≠ 0 on I

holds. Note that in the case of the system (3) an extra condition

α_{i}f_{i}(t, x_{1}, x_{2}, x_{3}, x_{4})x_{i+1} > 0 for x_{i+1} ≠ 0

is added. The following theorem gives some conditions under which a nontrivial solution of (3) has no Z-points.

Theorem 2. Let ε > 0, ε̄ > 0, K > 0, and y : J = (a, b) → R^{4} be a nontrivial solution of (3). Let nonnegative functions a_{i} ∈ L_{loc}(R), g_{i} ∈ C^{∘}([0, ε]), i ∈ N_{4} exist such that g_{i} is nondecreasing, g_{i}(0) = 0,

|f_{i}(t, x_{1}, …, x_{4})| ≦ a_{i}(t) g_{i}(|x_{i+1}|) on R × [−ε, ε]^{4}, i ∈ N_{4}, (6)

and

g_{1}(ε̄g_{2}(ε̄g_{3}(ε̄g_{4}(z)))) ≤ Kz, z ∈ [0, ε],  (7)

hold, where x_{4i+j} = x_{i}, i ∈ Z, j ∈ N_{4}. Then y has no Z-point in J and the statement (i) of Theorem 1 holds.

Remark 3.

1. Theorem 2 generalizes the well-known condition for the nonexistence of Z-points of a nontrivial solution:

ε > 0, |f_{i}(t, x_{1}, …, x_{4})| ≤ d(t) ∑_{i=1}^{4} |x_{i}|,

t ∈ R, |x_{i}| ≤ ε, i ∈ N_{4}, d ∈ L_{loc}(R)

(i.e., the Lipschitz condition for y ≡ 0; for (4), a_{j} ≡ 1 see [3]).

2. The condition (7) cannot be replaced by

δ > 0, g_{1}(ε̄g_{2}(ε̄g_{3}(ε̄g_{4}(z)))) ≤ Kz^{1−δ}, z ∈ [0, ε] .  (8)

In [4] where sufficient conditions are given for the existence of a solution of (3) with a Z-point (when studying singular solutions of the 1st kind), the inequality (8) is fulfilled but (7) is not.

3. Let k ∈ N_{4}. Then (7) can be replaced by

g_{k}(ε̄g_{k+1}(ε̄g_{k+2}(ε̄g_{k+3}(z)))) ≤ Kz, z ∈ [0, ε],

where g_{4k+j} = g_{j}, j ∈ N_{4}, k ∈ N. The proof is similar.

The last part is devoted to equation (4). It is proved that for every solution x at most one maximal interval exists on which x is trivial.

Theorem 3. Let x : J = (a, b) → R be a nontrivial solution of (4) and let y_{i} = L_{i−1}x, i ∈ N_{4}, y = (y_{i})_{i=1}^{4}.

(i) Let the number ε > 0 and a nonnegative function d ∈ L_{loc}(R) exist such that

|f(t, x_{1}, x_{2}, x_{3}, x_{4})| ≤ d(t)|x_{1}| for t ∈ R, |x_{i}| ≤ ε, i ∈ N_{4}.

Then the statement (i) of Theorem 1 holds.

(ii) Let a_{j} ∈ C^{1}(R), j = 1, 2, \frac{a_{3}}{a_{1}} ∈ C^{2}(R). Then either the statement (i) of Theorem 1 holds or numbers s ∈ Z ∪ {−∞}, r ∈ Z ∪ {∞} exist such that y is successively of Types IV, II(s, ∞), VI, I(−∞, r), III in J; if s = −∞ (r = ∞), then Type IV (Type III) is missing; if Type I (II) is missing, then Type III (IV) is missing, too.

(iii) y has Property W.

Remark 4. Let y have a Z-point. Then the intervals from the definition of Property W do not exist.

3. Proof of the main results

We start with some lemmas.

Lemma 1. Let y be a solution of (1) defined on the interval I.

(a) Let j ∈ {2, 3, 4} and Y_{j}(t) ≥ 0 (≤ 0) on I. Then Y_{j−1} is nondecreasing (nonincreasing) in I.

(b) If Y_{1} ≥ 0 (Y_{1} ≤ 0) on I, then Y_{4} is nonincreasing (nondecreasing) in I.

(c) Let j ∈ N_{4}, Y_{j}(t) = 0 on I. Then Y_{j−1} is constant in I.

Proof. (a) Let j = 2, Y_{2}(t) = α_{1}y_{2}(t) ≥ 0 on I. As by (1) α_{1}y′_{1}(t)y_{2}(t) ≥ 0 we have y′_{1} = Y′_{1} ≥ 0 for almost all t ∈ I. In the other cases the proof is similar.

(b) Let Y_{1} ≥ 0 in I. By (1) α_{4}y′_{4}y_{1} ≥ 0 holds. Then, according to (2)

Y′_{4}(t) = α_{1}α_{2}α_{3}y′_{4}(t) = −α_{4}y′_{4}(t)y_{1}(t) ≤ 0 .

The case (c) is a consequence of (a), (b).

Remark 5. The conclusions about the monotonicity in the definitions of Types III–V follow directly from Lemma 1.

Lemma 2. Let y I = [t_{1}, t_{2}] → R^{4} be a solution of (1), i ∈ N_{4}, Y_{i}(t_{1}) = Y_{i+1}(t) = 0, Y_{i+2}(t) ≠ 0 on I. Then either

Y_{i} ≡ Y_{i+1} ≡ 0 on I  (9)

or there exists a number τ ∈ [t_{1}, t_{2}) such that Y_{i}(t) ≡ Y_{i+1} ≡ 0 in [t_{1}, τ] and c Y_{i+1}(t)Y_{i+2}(t) > 0 in (τ, t_{2}] hold, where c = 1 (c = −1) for i = 1, 2, 4 (i = 3).

Proof. Suppose that (9) is not valid and i = 1, Y_{3}(t) > 0 on I. Then by Lemma 1 the function Y_{2} is nondecreasing, Y_{2} ≥ 0 in I, and Y_{1} is nondecreasing, Y_{1} ≥ 0 in I, too. From this and according to Lemma 1(c) there exists τ ∈ [t_{1}, t_{2}) such that

Y_{1} ≡ Y_{2} ≡ 0 on [t_{1}, τ] ,  (10)
Y_{1}^{2}(t) + Y_{2}^{2}(t) > 0 on (τ, t_{2}) .  (11)

Suppose that Y_{2}(t) = 0 in some right neighborhood I_{1} of τ. Then by (1) we have y′_{1}(t) = 0 for almost all t ∈ I_{1} and according to (10) y_{1}(t) = Y_{1}(t) = 0 on I. This contradiction to (11) proves the statement for i = 1. For the other i the proof is similar.

Proof of Theorem 1. (i) Let t_{0} ∈ J be an arbitrary number. Divide all possible initial conditions at t ∈ J into 32 cases:

\begin{matrix}1^{∘} & Y_{1}Y_{4} ≥ 0, & Y_{i}Y_{4} > 0, & i=2, 3, & 17^{∘} & Y_{1} =Y_{2} =0, & Y_{3}Y_{4} > 0 \\ 2^{∘} & Y_{1}Y_{i} > 0, & Y_{4}Y_{1} ≤ 0, & i=2, 3, & 18^{∘} & Y_{1} =Y_{2} =0, & Y_{3}Y_{4} < 0 \\ 3^{∘} & Y_{i}Y_{4} < 0, & Y_{3}Y_{4} ≥ 0, & i=1, 2, & 19^{∘} & Y_{1} =Y_{3} =0, & Y_{2}Y_{4} > 0 \\ 4^{∘} & Y_{1}Y_{i} < 0, & Y_{1}Y_{2} ≤ 0, & i=3, 4, & 20^{∘} & Y_{1} =Y_{3} =0, & Y_{2}Y_{4} < 0 \\ 5^{∘} & Y_{1}Y_{3} ≤ 0, & Y_{i}Y_{3} < 0, & i=2, 4, & 21^{∘} & Y_{1} =Y_{4} =0, & Y_{2}Y_{3} > 0 \\ 6^{∘} & Y_{i}Y_{3} < 0, & Y_{2}Y_{3} ≥ 0, & i=1, 4, & 22^{∘} & Y_{1} =Y_{4} =0, & Y_{2}Y_{3} < 0 \\ 7^{∘} & Y_{i}Y_{2} < 0, & Y_{2}Y_{3} ≤ 0, & i=1, 4, & 23^{∘} & Y_{2} =Y_{3} =0, & Y_{1}Y_{4} > 0 \\ 8^{∘} & Y_{i}Y_{2} < 0, & Y_{2}Y_{4} ≥ 0, & i=1, 3, & 24^{∘} & Y_{2} =Y_{3} =0, & Y_{1}Y_{4} > 0 \\ 9^{∘} & Y_{1} =0, & Y_{i}Y_{4} < 0, & i=2, 3, & 25^{∘} & Y_{2} =Y_{4} =0, & Y_{1}Y_{3} > 0 \\ 10^{∘} & Y_{1} =0, & Y_{2}Y_{i} < 0, & i=3, 4, & 26^{∘} & Y_{2} =Y_{4} =0, & Y_{1}Y_{3} < 0 \\ 11^{∘} & Y_{i}Y_{4} > 0, & Y_{2} =0, & i=1, 3, & 27^{∘} & Y_{3} =Y_{4} =0, & Y_{1}Y_{2} > 0 \\ 12^{∘} & Y_{i}Y_{4} < 0, & Y_{2} =0, & i=1, 3, & 28^{∘} & Y_{3} =Y_{4} =0, & Y_{1}Y_{2} < 0 \\ 13^{∘} & Y_{1}Y_{i} > 0, & Y_{3} =0, & i=2, 4, & 29^{∘} & Y_{1} =Y_{2} =Y_{3} =0, & Y_{4} ≠ 0 \\ 14^{∘} & Y_{1}Y_{i} < 0, & Y_{3} =0, & i=2, 4, & 30^{∘} & Y_{1} =Y_{3} =Y_{4} =0, & Y_{2} ≠ 0 \\ 15^{∘} & Y_{i}Y_{3} < 0, & Y_{4} =0, & i=1, 2, & 31^{∘} & Y_{1} =Y_{2} =Y_{4} =0, & Y_{3} ≠ 0 \\ 16^{∘} & Y_{1}Y_{i} < 0, & Y_{4} =0, & i=2, 3, & 32^{∘} & Y_{2} =Y_{3} =Y_{4} =0, & Y_{1} ≠ 0. \end{matrix}

Note that the last case Y_{i} = 0, i ∈ N_{4}, is impossible in view of the assumptions of the theorem. Sometimes, if, for example, 1^{∘} is valid at t̄ we shall write 1^{∘}(t̄).

We shall investigate how the initial conditions vary when t increases in [t_{0}, b). First note that for J_{1} ⊂ [t_{0}, b), y is of

\begin{matrix}Type III in J_{1} iff one of the cases 1^{∘} – 4^{∘} holds in J_{1}; \\ Type IV in J_{1} iff one of the cases 5^{∘} – 8^{∘} holds in J_{1}; \\ Type V in J_{1} iff one of the cases 9^{∘} – 32^{∘} holds in J_{1} \end{matrix}\}  (12)

(see Remark 5, too).

Consider y in J_{1} = [t_{0}, b̄), b̄ ≤ b. Let j, k ∈ {1, 2, …, 32^{∘}}. The symbol j^{∘}(t_{0}) → k^{∘}(t_{1}) denotes that either j^{∘} holds in J_{1} (and y is one of Types III–V according to (12)) or t_{1}, t_{2} ∈ J_{1}, t_{0} < t_{2} < t_{1} exist such that j^{∘} holds in [t_{0}, t_{2}), k^{∘} holds in (t_{2}, t_{1}] and either j^{∘} or k^{∘} is valid at t_{2}. Generally, the notation j^{∘}(t_{0}) → {k_{1}^{∘}, …, k_{s}^{∘}}(t_{1}) denotes that j^{∘}(t_{0}) → k_{e}^{∘}(t_{1}) is valid for suitable e ∈ {1, …, s}. The following relations can be proved for y defined in [t_{0}, b)

\begin{matrix}1^{∘}(t_{0}) → 2^{∘}(t_{1}), 2^{∘}(t_{0}) → 3^{∘}(t_{1}), 3^{∘}(t_{0}) → 4^{∘}(t_{1}), 4^{∘}(t_{0}) → 1^{∘}(t_{1}), \\ 5^{∘}(t_{0}) → \{6^{∘}, 13^{∘}, 15^{∘}, 18^{∘}, 19^{∘}, 23^{∘}, 26^{∘}, 27^{∘}, 29^{∘}, 32^{∘}\}(t_{1}), \\ 6^{∘}(t_{0}) → \{7^{∘}, 9^{∘}, 16^{∘}, 20^{∘}, 21^{∘}, 23^{∘}, 26^{∘}, 28^{∘}, 30^{∘}, 32^{∘}\}(t_{1}), \\ 7^{∘}(t_{0}) → \{8^{∘}, 10^{∘}, 11^{∘}, 17^{∘}, 20^{∘}, 21^{∘}, 23^{∘}, 26^{∘}, 28^{∘}, 30^{∘}, 32^{∘}\}(t_{1}), \\ 8^{∘}(t_{0}) → \{5^{∘}, 12^{∘}, 14^{∘}, 18^{∘}, 19^{∘}, 22^{∘}, 24^{∘}, 25^{∘}, 29^{∘}, 31^{∘}\}(t_{1}), \\ 9^{∘}(t_{0}) → \{2^{∘}, 20^{∘}\}(t_{1}), 10^{∘}(t_{0}) → \{3^{∘}, 17^{∘}\}(t_{1}), \\ 11^{∘}(t_{0}) → \{1^{∘}, 25^{∘}\}(t_{1}), 12^{∘}(t_{0}) → \{2^{∘}, 24^{∘}\}(t_{1}), \\ 13^{∘}(t_{0}) → \{1^{∘}, 27^{∘}\}(t_{1}), 14^{∘}(t_{0}) → \{4^{∘}, 19^{∘}\}(t_{1}), \\ 15^{∘}(t_{0}) → \{3^{∘}, 26^{∘}\}(t_{1}), 16^{∘}(t_{0}) → \{4^{∘}, 21^{∘}\}(t_{1}), \\ 17^{∘}(t_{0}) → 1^{∘}(t_{1}), 18^{∘}(t_{0}) → \{2^{∘}, 9^{∘}, 29^{∘}\}(t_{1}), \\ 19^{∘}(t_{0}) → \{1^{∘}, 13^{∘}\}(t_{1}), 20^{∘}(t_{0}) → \{3^{∘}, 10^{∘}\}(t_{1}), \\ 21^{∘}(t_{0}) → 2^{∘}(t_{1}), 22^{∘}(t_{0}) → \{3^{∘}, 15^{∘}, 31^{∘}\}(t_{1}), \\ 23^{∘}(t_{0}) → \{1^{∘}, 11^{∘}, 32^{∘}\}(t_{1}), 24^{∘}(t_{0}) → 4^{∘}(t_{1}), \\ 25^{∘}(t_{0}) → \{2^{∘}, 12^{∘}\}(t_{1}), 26^{∘}(t_{0}) → \{4^{∘}, 16^{∘}\}(t_{1}), \\ 27^{∘}(t_{0}) → 3^{∘}(t_{1}), 28^{∘}(t_{0}) → \{4^{∘}, 14^{∘}, 30^{∘}\}(t_{1}), \\ 29^{∘}(t_{0}) → \{1^{∘}, 17^{∘}\}(t_{1}), 30^{∘}(t_{0}) → \{3^{∘}, 27^{∘}\}(t_{1}), \\ 31^{∘}(t_{0}) → \{2^{∘}, 21^{∘}\}(t_{1}), 32^{∘}(t_{0}) → \{4^{∘}, 24^{∘}\}(t_{1}), t_{1} ∈ (t_{0}, b). \end{matrix} \}(13)

We prove only the validity of

18^{∘}(t_{0}) → {2^{∘}, 9^{∘}, 29^{∘}}(t_{1}) .  (14)

The other relations can be proved similarly. Thus suppose for simplicity that

Y_{1}(t_{0}) = Y_{2}(t_{0}) = 0 , Y_{3}(t_{0}) > 0, Y_{4}(t_{0}) < 0

holds. Then, according to Lemma 1, Y_{2} is nonincreasing in some right neighborhood of t_{0} and Y_{1}, Y_{2} are nondecreasing, Y_{4} nonincreasing until Y_{2} ≥ 0. From this and according to Lemma 2 one of the following possibilities is valid:

(i) Y_{1} ≡ Y_{2} ≡ 0, Y_{3} > 0, Y_{4} < 0 in [t_{0}, b) (i.e., y is of Type V);
(ii) t_{1}, t_{1} ∈ (t_{0}, b) exists such that Y_{1} ≡ Y_{2} ≡ 0, Y_{4} < 0 in [t_{0}, t_{1}],
Y_{3}(t) > 0 in [t_{0}, t_{1}), Y_{3}(t_{1}) = 0 (i.e., 29^{∘} holds at t_{1});

(iii) t_{1}, t_{1} ∈ (t_{0}, b) exists such that Y_{1} ≡ Y_{2} ≡ 0, Y_{3} > 0, Y_{4} < 0 in [t_{0}, t_{1}], Y_{1} = 0, Y_{2} > 0 in some right neighborhood J_{2} of t_{1} (i.e., 9^{∘} holds in J_{2})

(iv) t_{1}, t_{1} ∈ (t_{0}, b) exists such that Y_{1} ≡ Y_{2} ≡ 0, Y_{3} > 0, Y_{4} < 0 on [t_{0}, t_{1}], Y_{1} = 0, Y_{2} > 0, Y_{3} > 0, Y_{4} < 0 in some right neighborhood J_{2} of t_{1} (i.e., 2^{∘} holds in J_{2}).

From this we can conclude that (14) holds. Further, note that if the cases 1^{∘}, 2^{∘}, 3^{∘}, 4^{∘} are repeated, 1^{∘} → 2^{∘} → 3^{∘} → 4^{∘} → 1^{∘}, we get just the solution of Type I; in the case 5^{∘} → 6^{∘} → 7^{∘} → 8^{∘} → 5^{∘}, we get Type II.

Let y be either of Type I(s, ∞) or of Type II(s, ∞) on the interval [t_{0}, b̄), b̄ ≤ b. We prove by an indirect method that b̄ = b. Suppose that b̄ < b. As y is oscillatory, y_{i}(b̄) = 0, i ∈ N_{4} is valid and b̄ is a Z-point of y. The contradiction to the assumptions of the theorem proves that b̄ = b.

The statement of the theorem for the interval [t_{0}, b) follows from this and from (13). The statement in (a, t_{0}] can be proved similarly, or the fact that t_{0} is arbitrary can be used.

(ii) (a) If τ, τ ∈ J, is a Z-point of y, then y_{i}(τ) = 0, i ∈ N_{4} and the only types which can fulfill these conditions are I(s, ∞), II(s, ∞), s ∈ Z. Thus the statement follows from (i). The case (b) can be proved similarly. (iii) The statement follows directly from (i), (ii).

Proof of Remark 2. It can be proved similarly to Lemma 1 that the following two statements hold:

(a) If i ∈ {2, 3, 4}, Y_{i}(t) > 0 (< 0) in I, then Y_{j−1} is increasing (decreasing) in I.

(b) If Y_{1}(t) > 0 (< 0) in I, then Y_{4} is decreasing (increasing) in I. The statement of the remark follows from this.

Proof of Theorem 2. On the contrary, suppose that a Z-point τ ∈ J exists. Without loss of generality we can suppose that τ is such that a right neighborhood of τ exists in which y is not trivial (for a left neighborhood the proof is similar).

As y_{i}(τ) = 0, i ∈ N_{4}, an interval J_{1} = [τ, τ + δ], δ > 0 exists such that

|y_{i}(t)| ≤ ε, t ∈ J_{1}, i ∈ N_{4} .  (15)

Let ε_{1}, δ_{1} and J_{2} = [τ, τ + δ_{1}] be such that 0 < ε_{1} ≤ ε̄, 0 < δ_{1} ≤ δ

\begin{matrix}ε_{1}K < 1, ε_{1} max_{j∈N_{4}^{0≤s≤ε}} g_{j}(s) ≤ ε , \\ max_{j∈N_{4}} ∫_{J_{2}} a_{j}(t) dt ≤ ε_{1} . \end{matrix} \}  (16)

Then by the use of (6), (15) we have for t ∈ J_{2} and i ∈ N_{4}

|y_{i}(t)| ≤ ∫_{τ}^{t} |f_{i}(t, y_{1}(t), …, y_{4}(t))| dt ≤

≤ ∫_{J_{2}} a_{i}(t)dtg_{i}(max_{s∈J_{2}} |y_{i+1}(s)|) .

From this, by the use of (16) we get

max_{s∈J_{2}} |y_{i}(s)| ≤ ε_{1}g_{i}(max_{s∈J_{2}} |y_{i+1}(s)|) , i ∈ N_{4} .  (17)

Denote ν = max_{s∈J_{2}}|y_{1}(s)|. As y is not trivial in J_{2} and g_{i}(0) = 0, it follows

from (17) that ν > 0 must be valid.

Further, according to (7), (16), and (17) we have

ν ≤ ε_{1}g_{1}(ε_{1}g_{2}(ε_{1}g_{3}(ε_{1}g_{4}(ν)))) ≤ ε_{1}g_{1}(ε̄g_{2}(ε̄g_{3}(ε̄g_{4}(ν)))) ≦
≦ ε_{1}Kν < ν .

The contradiction proves the theorem.

Proof of Theorem 3. (i) The statement is a consequence of Theorem 2 and (5).

(ii) Let τ ∈ J be a Z-point such that no Z-point exists in some right neighborhood J_{1} of τ.

Then according to Theorem 1, (ii) there exists a right neighborhood J_{2} of τ, J_{2} ⊂ J_{1} such that y is either of Type I(−∞, s) or of Type II(−∞, s), s ∈ {Z, ∞} in J_{2}. We prove by an indirect method that Type II is impossible. Thus, suppose that y is of Type II (−∞, s) in J. Let α > τ, α ∈ J_{2}.

Put

F = Ay_{4}y_{1} + By_{2}y_{3} + Cy_{2}^{2} + Dy_{1}y_{3} + Ey_{1}y_{2} + Gy_{1}^{2},  (18)

where

A(t) = − ∫_{τ}^{t} \frac{1}{a_{3}(s)} ∫_{s}^{α} \frac{E(v)}{a_{2}(v)} dv ds , B(t) = −\frac{a_{3}(t)}{a_{1}(t)}A(t) ,

C(t) = −\frac{a_{2}(t)}{a_{1}(t)} ∫_{t}^{α} \frac{E(v)}{a_{2}(v)} dv − \frac{a_{2}(t)}{2}(\frac{a_{3}(t)}{a_{1}(t)})′ ×

× ∫_{τ}^{t} \frac{1}{a_{3}(s)} ∫_{s}^{α} \frac{E(v)}{a_{2}(v)} dv ds ,

D(t) = ∫_{t}^{α} \frac{E(v)}{a_{2}(v)} dv; E(t) = (v − τ + σ)^{\frac{1}{2}} ; G(t) = −\frac{a_{1}(t)}{4E(t)} .

The number σ > 0 is chosen in a manner such that

\begin{matrix}G′(t) = \frac{−a′_{1}(t)}{4E(t)} + \frac{a_{1}(t)}{F (t)^{3}} ≥ 0 , \\ C′(t) + \frac{E(t)}{a_{1}(t)} = 2\frac{E(t)}{a_{1}(t)} + [\frac{a′_{2}(t)}{a_{1}(t)} + \frac{a′_{1}(t)a_{2}(t)}{2a_{3}(t)} − \frac{a_{2}(t)}{a_{1}(t)}(\frac{a_{3}(t)}{a_{1}(t)})′] × \\ × ∫_{t}^{α} \frac{E(s)}{a_{2}(s)} ds − \frac{1}{2}(a_{2}(t)(\frac{a_{3}(t)}{a_{1}(t)})′)′ ∫_{τ}^{t} \frac{1}{a_{3}(s)} ∫_{s}^{t} \frac{E(v)}{a_{2}(v)} dv ds ≥ 0 , \end{matrix} (19)

t ∈ J_{3}, holds, where J_{3}, J_{3} = [τ, t̄], τ < t̄≤ α, is a suitable interval. From this

F′ = Ay′_{4}y_{1} + \frac{B}{a_{2}} y_{3}^{2} + (C′ + \frac{E}{a_{1}} )y_{2}^{2} + G′y_{1}^{2}

and according to (4), (5), (19) and y_{i}(τ) = 0, i ∈ N_{4}, we have

F′(t) ≥ 0, F (t) ≥ 0 on J_{3} .  (20)

Further, let β, β ∈ (τ, t̄), be an arbitrary zero of y_{2}. Then according to the definition of Type II and (18) we have

A(t) < 0, D(t) > 0, G(t) < 0 on (τ, t̄) ,
y_{4}(β)y_{1}(β) > 0, y_{1}(β)y_{3}(β) < 0 .

Thus F (β) < 0 and the contradiction to (20) proves that y is of Type I (−∞, s) in J_{2}.

Let τ ∈ J be a Z-point such that no Z-point exists in some left neighborhood of τ. Then we can prove similarly that y is of Type II (s, ∞), s ∈ {Z, −∞}, in some left neighborhood of τ. Also, the transformation of the independent variable can be used for x = τ − t.

The above-mentioned results and Theorem 1, (i) show that there exists at most one maximal interval of Z-points and the statement is a consequence of Theorem 1, (i).

(iii) The intervals mentioned in the definition of Property W may occur only in Type V. Thus the statement follows from the case (ii).

Proof of Remark 4. This can be done similarly to that of Theorem 3, (iii).

References

1. M. Bartušek, The asymptotic behavior of oscillatory solutions of an equation of the fourth order. Arch. Math. 21(1985), No. 2, 93 -104.

2. —–, On oscillatory solutions of differential inequalities. Czech Math. J. 42(117)(1992), 45-52.

3. I. T. Kiguradze and T. A. Chanturia, Asymptotic properties of solutions of nonautonomous ordinary differential equations. (Russian) Nauka, Moscow, 1990.

4. T. A. Chanturia, On singular solutions of nonlinear system of ordinary differential equations. Colloquia Math. Soc. János Bolyai 15,Differential Equations, Keszethely (Hungary) 1975, 107-119.

(Received 10.02.1994)

Author’s address: Faculty of Sciences Masaryk University Janáčkovo nám. 2a 662 95 Brno, Czech Republic