Sixth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference 15 (2007), pp. ?–?. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp)

EXISTENCE OF NONCONTINUABLE SOLUTIONS

MIROSLAV BARTUŠEK

Abstract. This paper presents necessary and sufficient conditions for an n-th order differential equation to have a non-continuable solution with finite limits of its derivatives up to the orders n − 2 at the right-hand end point of the definition interval.

1. Introduction

Consider the n-th order differential equation

y^{(n)} = f(t, y, y′, …, y^{(n−2)})g(y^{(n−1)})  (1.1)

where n ≥ 2, f ∈ C^{0}(R_{+} × R^{n−1}), g ∈ C^{0}(R), R_{+} = [0, ∞), R = (−∞, ∞) and M > 0 exists such that

g(x) > 0 for |x| ≥ M.  (1.2)

This inequality will be assumed throughout the paper.

So we study equations for which g is nonzero in neighbourhoods of ∞ and −∞; this case can be easily transformed into (1.1) and (1.2).

A solution y of (1.1) defined on [T, τ) ⊂ R_{+} is called noncontinuable if τ < ∞ and y cannot be defined at t = τ. Sometimes such solutions are called singular of the second kind [1, 3, 10]. A noncontinuable solution y is called nonoscillatory if y ≠ 0 in a left neighbourhood of τ.

Sufficient conditions for the existence of noncontinuable solutions for the Cauchy problem can be found in [10]. For f(t, x_{1}, …, x_{n−1}) ≡ r(t)|x_{1}|^{λ}×sgn x_{1}, r ≠ 0 in [3]. For n = 2 in [2, 4, 8]. Sufficient conditions for the nonexistence of noncontinuable solutions of (1.1) and of its special cases be found in [5, 6, 7, 10].

Jaroš and Kusano [9] investigated the differential equation

y′′ = r(t)|y|^{σ}|y′|^{λ} sgn y  (1.3)

with σ > 0, r < 0 on R_{+}. They proved that there exists a noncontinuable solution y of (1.3) fulfilling lim_{t→τ_{−}} y(t) ∈ [0, ∞), lim_{t→τ_{−}} y′(t) = −∞ if, and only if λ > 2; they call it a black hole solution.

2000 Mathematics Subject Classification. 34C11.

Key words and phrases. Noncontinuable solutions; singular solutions of the second kind;

black-hole solutions.

\begin{matrix}c \\ ◯ \end{matrix}2007 Texas State University - San Marcos.

Published ??.

In [1], a problem is formulated for (1.1): To find conditions under which (1.1) has a noncontinuable solution y fulfilling the conditions

\begin{matrix}τ ∈ (0, ∞), c_{i} ∈ R, lim_{t→τ_{−}} y(t) = c_{i} i = 0, 1, 2, …, n − 2, \\ lim_{t→τ_{−}} |y^{(n−1)}(t)| = ∞ \end{matrix}  (1.4)

and y is defined in a left neighbourhood of τ.

Note that (1.4) is a boundary-value problem and a solution y fulfilling (1.4) is nonoscillatory. The obtained results are summed up in the following theorem.

Theorem 1.1 ([1]). Let τ ∈ (0, ∞), f(t, x_{1}, …, x_{n−1})x_{1} ≠ 0 for x_{1} ≠ 0 and g(x) ≥ 0 for x ∈ R.

(i) If M_{1} ∈ (0, ∞) is such that g(x) ≤ x^{2} for |x| ≥ M_{1}, then (1.1) has no solution y fulfilling (1.4).

(ii) Let τ ∈ (0, ∞), c_{0} ≠ 0, λ > 2, M_{1} ∈ (0, ∞) and g(x) ≥ |x|^{λ} for |x| ≥ M_{1}, then (1.1) has a solution y fulfilling (1.4) that is defined in a left neighbourhood of τ.

In the present paper, we generalize Theorem 1.1 and the necessary and sufficient condition for the existence of a noncontinuable solution y fulfilling (1.4) will be stated if f(τ, c_{0}, …, c_{n−2}) ≠ 0. Sufficient conditions for the existence of a noncontinuable solution y fulfilling (1.4) are given in case f(τ, c_{0}, …, c_{n−2}) = 0.

Notation. Let ∫_{M}^{∞} \frac{dσ}{g(σ)} < ∞. Then we put

F (z) = ∫_{z}^{∞} \frac{dσ}{g(σ)}, z ≥ M

and F^{−1} : (0, F (M)] → [M, ∞) denotes the inverse function to F. Similarly, if ∫_{−∞}^{−M} \frac{dσ}{g(σ)} < ∞, put

G(z) = ∫_{−∞}^{z} \frac{dσ}{g(σ)}, z ≤ −M

and G^{−1} : (0, G(−M)] → (−∞, −M] is the inverse function to G.

The next lemma follows from the definitions of F and G.

Lemma 1.2. (i) Let ∫_{M}^{∞} \frac{dσ}{g(σ)} < ∞. Then functions F and F^{−1} are decreas-

ing, lim_{z→∞} F (z) = 0 and lim_{z→0_{+}} F^{−1}(z) = ∞. (ii) Let ∫_{−∞}^{−M} \frac{dσ}{g(σ)} < ∞. Then functions G and G^{−1} are increasing, G >

0, G^{−1} < 0, lim_{z→−∞} G(z) = 0 and lim_{z→0_{+}} G^{−1}(z) = −∞.

Denote by [[a]] the entire part of the number a.

2. Main results

The next theorem gives conditions for the nonexistence of a solution y fulfilling (1.4).

Theorem 2.1. Let the following two assumptions hold.

(i) Let either

∫_{M}^{∞} \frac{dσ}{g(σ)} = ∞  (2.1)

or

∫_{M}^{∞} \frac{dσ}{g(σ)} < ∞ and ∫_{0}^{F (M)} F^{−1}(σ)dσ = ∞;  (2.2)

(ii) Let either

∫_{−∞}^{−M} \frac{dσ}{g(σ)} = ∞  (2.3)

or

∫_{−∞}^{−M} \frac{dσ}{g(σ)} < ∞ and ∫_{0}^{G(−M)} |G^{−1}(σ)|dσ = ∞.  (2.4)

Then (1.1) has no noncontinuable solution y fulfilling (1.4) that is defined in a left neighbourhood of τ.

Proof. Suppose, contrarily, that y is a noncontinuable solution of (1.1) fulfilling (1.4) that is defined on [T, τ) ⊂ R_{+}. Furthermore, suppose that lim_{t→τ_{−}} y^{(n−1)}(t) = ∞; the opposite case, if lim_{t→τ_{−}} y^{(n−1)}(t) = −∞, can be studied similarly using (2.3) and (2.4). From this, T_{1} ∈ [T, τ) and M_{1} > 0 exist such that

f(t, y(t), …, y^{(n−2)}(t)) ≤ M_{1}, y^{(n−1)}(t) ≥ M for t ∈ [T_{1}, τ).

Hence, the integration of (1.1) and (1.2) yields

\begin{aligned}∫_{y^{(n−1)}(t)}^{∞} \frac{dσ}{g(σ)} & = ∫_{t}^{τ} \frac{y^{(n)}(σ)dσ}{g(y^{(n−1)}(σ))} \\ & = ∫_{t}^{τ} f(σ, y(σ), …, y^{(n−2)}(σ))dσ \\ & ≤ M_{1}(τ − t) ≤ M_{1}τ, t ∈ [T_{1}, τ). \end{aligned}  (2.5)

It follows from this that (2.1) is not valid and hence (2.2) holds.

Let T_{2} ∈ [T_{1}, τ) be such that τ − T ≤ F (M)M_{1}^{−1}. From this and from (2.5)

F (y^{(n−1)}(t)) ≤ M_{1}(τ − t) ∈ (0, F (M)] for t ∈ [T_{2}, τ);

hence, Lemma 1.2 yields

y^{(n−1)}(t) ≥ F^{−1} (M_{1}(τ − t)) , t ∈ [T_{2}, τ)

and an integration on [T_{2}, τ) and (2.2) yield

∞ > c_{n−2} − y^{(n−2)}(T_{2}) = y^{(n−2)}(τ) − y^{(n−2)}(T_{2}) = ∫_{T_{2}}^{τ} y^{(n−1)}(σ)dσ

≥ ∫_{T_{2}}^{τ} F^{−1}(M_{1}(τ − s)) ds = \frac{1}{M_{1}} ∫_{0}^{M_{1}(τ−T_{2})} F^{−1}(σ)dσ = ∞.

This contradiction proves that a noncontinuable solution y fulfilling (1.4) does not exist. □

The following theorem formulates necessary and sufficient conditions for the existence of a solution y fulfilling (1.4) in case f(τ, c_{0}, …, c_{n−2}) ≠ 0.

Theorem 2.2. Let τ > 0 and c_{i} ∈ R, i = 0, 1, …, n − 2 be such that

f(τ, c_{0}, c_{1}, …, c_{n−2}) ≠ 0.  (2.6)

Then (1.1) has a noncontinuable solution y fulfilling (1.4) if and only if one of the following two conditions holds:

∫_{M}^{∞} \frac{dσ}{g(σ)} < ∞ and ∫_{0}^{F (M)} F^{−1}(σ) dσ < ∞;  (2.7)
∫_{−∞}^{−M} \frac{dσ}{g(σ)} < ∞ and ∫_{0}^{G(−M)} |G^{−1}(σ)| dσ < ∞.  (2.8)

In this case y is defined in a left neighbourhood of τ.

Moreover, let f(0, c_{0}, c_{1}, …, c_{n−2}) ≠ 0 and either (2.7) or (2.8) holds. Then there exists τ_{0} > 0 such that for every 0 < τ ≤ τ_{0}, a noncontinuable solution y fulfilling (1.4) exists and is defined on [0, τ).

Proof. Necessity: This follows from Theorem 2.1.

Sufficiency: We prove the statement in case f(τ, c_{0}, …, c_{n−2}) > 0; the opposite case can be studied similarly. There exist N > 0 and τ̄ ∈ [0, τ) such that

f(t, x_{1}, …, x_{n−1}) > 0 for t ∈ [τ̄, τ], |x_{i} − c_{i−1}| ≤ N, i = 1, 2, …, n − 1. (2.9)

From this, positive constants M_{1} and M_{2} exist such that

\begin{matrix}0 < M_{1} = min\{f(t, x_{1}, …, x_{n−1}) : t ∈ [τ̄, τ], |x_{i} − c_{i−1}| ≤ N, i = 1, 2, …, n − 1\}, \\ ∞ > M_{2} = max\{f(t, x_{1}, …, x_{n−1}) : t ∈ [τ̄, τ], |x_{i} − c_{i−1}| ≤ N, i = 1, 2, …, n − 1\}. \end{matrix}

(2.10)

Consider the auxiliary problem

\begin{matrix}y^{(n)} = f (t, χ_{0}(y), χ_{1}(y′), …, χ_{n−2}(y^{(n−2)})) g(y^{(n−1)}), \\ y^{(i)}(τ) = c_{i}, i = 0, 1, …, n − 2, y^{(n−1)}(τ) = k, \end{matrix}  (2.11)

where k ∈ {k_{0}, k_{0} + 1, … }, k_{0} ≥ [[2M]],

χ_{i}(s) = \begin{cases}s & for |s − c_{i}| ≤ N \\ c_{i} + N & for s > c_{i} + N \\ c_{i} − N & for s < c_{i} − N, \end{cases}  (2.12)

where i = 0, 1, …, n − 2. Furthermore, let J = [T, τ) ⊂ [τ̄, τ) be such that 0 < τ − T ≤ 1,

(τ − T) ∑_{j=1}^{n−2} |c_{j}| + \frac{1}{M_{1}} ∫_{0}^{M_{1}(τ−T)} F^{−1}(z) dz ≤ N,  (2.13)

and

M_{2}(τ − T) < ∫_{M}^{2M} \frac{ds}{g(s)};  (2.14)

this choice is possible due to the second inequality in (2.7), (1.2) and Lemma 1.2; it does not depend on k.

Denote by y_{k} a solution of (2.11) and by J_{k} the intersection of its maximal definition interval and [T, τ). We prove that

y_{k}^{(n−1)}(t) > M for t ∈ J_{k}.  (2.15)

As k ≥ k_{0} ≥ [[2M]], (2.11) yields (2.15) is valid in a left neighbourhood of τ. Suppose, contrarily, that T_{1} ∈ J_{k} exists such that y_{k}^{(n−1)}(T_{1}) = M and y_{k}^{(n−1)}(t) > M on (T_{1}, τ]. Then (1.2), (2.10), (2.11) and (2.12) yield y_{k}^{(n)}(t) > 0 on [T_{1}, τ] and

y_{k}^{(n)}(t) ≤ M_{2} g(y_{k}^{(n−1)}(t)), g(y_{k}^{(n−1)}(t)) > 0, t ∈ [T_{1}, τ].

From this, and an integration on [T_{1}, τ], we obtain

∫_{M}^{2M} \frac{ds}{g(s)} ≤ ∫_{M}^{k} \frac{ds}{g(s)} ≤ M_{2}(τ − T_{1}) ≤ M_{2}(τ − T).

The contradiction with (2.14) proves that (2.15) holds. According to (2.10), (2.11) and (2.12), we have

y_{k}^{(n)}(t) ≥ M_{1} g(y_{k}^{(n−1)}(t)), t ∈ J_{k}

and an integration on [t, τ], (1.2), and (2.15) yield

F (y_{k}^{(n−1)}(t)) ≥ ∫_{y_{k}^{(n−1)}(t)}^{k} \frac{ds}{g(s)} ≥ M_{1}(τ − t), t ∈ J_{k}.

As, according to (2.14), M_{1}(τ − T) ≤ M_{2}(τ − T) < F (M), we have M_{1}(τ − t) ∈ (0, f(M)] and

y_{k}^{(n−1)}(t) ≤ F^{−1}(M_{1}(τ − t)), t ∈ J_{k}.  (2.16)

From this and from Lemma 1.2, y_{k}^{(n−1)} is bounded on J_{k} and hence J_{k} = [T, τ]; moreover, J_{k} is defined on the same interval [T, τ] for every k = k_{0}, k_{0} + 1, . . ..

We estimate the functions y_{k}^{(i)}. Taylor’s formula, T − τ ≤ 1, (2.13), and (2.16) yield

|y_{k}^{(i)}(t) − c_{i}| ≤ ∑_{j=i+1}^{n−2} \frac{|c_{j}|}{(j − i)!}(τ − t)^{j−i} + | ∫_{τ}^{t} \frac{|(t − s)^{n−i−2}|}{(n − i − 2)!}F^{−1}(M_{1}(τ − s)) ds|

≤ (τ − T) ∑_{j=1}^{n−2} |c_{j}| + | ∫_{τ}^{t} F^{−1}(M_{1}(τ − s)) ds|

≤ (τ − T) ∑_{j=1}^{n−2} |c_{j}| + \frac{1}{M_{1}} ∫_{0}^{M_{1}(τ−T)} F^{−1}(z) dz

≤ N, t ∈ [T, τ], i = 0, 1, …, n − 2.

(2.17)

From this and from (2.12), y_{k} is a solution of (1.1), as well.

As the estimations (2.15), (2.16) and (2.17) and the definition interval of y_{k} do not depend on k, then according to the Arzel-Ascoli Theorem (see [3, Lemma 10.2]) there exists a subsequence of {y_{k}}_{k=k_{0}}^{∞} that converges locally uniformly to a solution y of (1.1) on [T, τ) together with all derivatives up to the order n − 1. Evidently conditions (1.4) hold for i = 0, 1, …, n − 2 and we prove lim_{t→τ_{−}} y^{(n−1)}(t) = ∞. As y^{(n−1)} is increasing on [T, τ), there exists a limit as t → τ_{−}. Suppose, contrarily, that lim_{t→τ_{−}} y^{(n−1)}(t) = Q < ∞. Then Lemma 1.2 yields the existence of T_{2} ∈ [T, τ) such that

Q < F^{−1}(M_{2}(τ − T_{2})), M(τ − T_{2}) ≤ F (M).  (2.18)

Moreover, there exists a subsequence of {y_{k}^{(n−1)}}_{k=k_{0}}^{∞}, we denote it {y_{k}^{(n−1)}}_{k=k_{0}}^{∞} for simplicity, that converges to y^{(n−1)} on [T, T_{2}]. From this, k̄ exists such that

y_{k}^{(n−1)}(T_{2}) ≤ 2Q for k = k̄, k̄ + 1, ….

According to (1.1) and (2.17), y_{k}^{(n)}(t) ≤ M_{2}g(y_{k}^{(n−1)}(t)), we obtain, by integration on [T_{2}, τ),

M_{2}(τ − T_{2}) ≥ ∫_{y_{k}^{(n−1)}(T_{2})}^{k} \frac{ds}{g(s)} ≥ ∫_{2Q}^{k} \frac{ds}{g(s)}, k ≥ k̄.

Thus,

M_{2}(τ − T_{2}) ≥ ∫_{2Q}^{∞} \frac{ds}{g(s)} = F (2Q),  (2.19)

so 2Q ≥ F^{−1}(M_{2}(τ −T_{2})), which contradicts (2.18). Hence, lim_{t→τ_{−}} y(t) = ∞. Let f(0, c_{0}, c_{1}, …, c_{n−2}) > 0. Then there exist N > 0 and τ̄_{0} ≤ 1 such that

f(t, x_{1}, …, x_{n−1}) > 0 for t ∈ [0, τ̄_{0}], [x_{i} − c_{i−1}| ≤ N, i = 1, …, n − 1.

Define

M_{1} = min{f(t, x_{1}, …, x_{n−1}) : t ∈ [0, τ̄_{0}], |x_{i} − c_{i−1}| ≤ N, i = 1, 2, …, n − 1},

M_{2} = max{f(t, x_{1}, …, x_{n−1}) : t ∈ [0, τ̄_{0}], |x_{i} − c_{i−1}| ≤ N, i = 1, 2, …, n − 1}.

Constants N, M_{1} and M_{2} are given by (2.9) and (2.10), but for [0, τ̄_{0}] instead of [τ̄, τ]. Let 0 < τ_{0} ≤ τ̄_{0} be a number such that (2.13) and (2.14) hold with T = 0 and τ = τ_{0}. It is clear that (2.13) and (2.14) are valid for τ ≤ τ_{0} and T = 0 and a noncontinuable solution y fulfilling (1.4) exists according to the first part of the proof, and it is defined on [0, τ). □

Next, we prove a comparison theorem.

Theorem 2.3. Let τ > 0 and c_{i} ∈ R_{+}, be such that f(τ, c_{0}, …, c_{n−2}) ≠ 0. Let f_{1} ∈ C^{0}(R_{+} × R^{n−1}), f_{1}(τ, c_{0}, …, c_{n−1}) ≠ 0 and let ḡ ∈ C^{0}(R) exist such that

ḡ(x) ≥ g(x) > 0 for |x| ≥ M.  (2.20)

(i) If (1.1) has a solution fulfilling (1.4), then the equation

y^{(n)} = f_{1}(t, y, …, y^{(n−2)})ḡ(y^{(n−1)})  (2.21)

has the same property. (ii) If (2.21) has no solution fulfilling (1.4), then (1.1) has the same property.

Proof. (i) According to Theorem 2.2 either (2.7) or (2.8) holds. Suppose that (2.7) is valid; if (2.8) holds the proof is similar. Then (2.20) yields

∫_{M}^{z} \frac{dσ}{ḡ(σ)} ≤ ∫_{M}^{z} \frac{dσ}{g(σ)}, z ≤ M.  (2.22)

According to (2.7) and (2.22), ∫_{M}^{∞} \frac{dσ}{ḡ(σ)} < ∞. Denote F_{1}(z) = ∫_{z}^{∞} \frac{dσ}{ḡ(σ)}, z ≥ M and let F_{1}^{−1} be the inverse function to F_{1}. As F_{1}(z) ≤ F (z), z ≥ M, and as F and F_{1} are nonincreasing, then F_{1}^{−1}(z) ≤ F^{−1}(z), z ≥ F_{1}(M) and, hence, (2.7) yields

∫_{0}^{F_{1}(M)} F_{1}^{−1}(σ)dσ ≤ ∫_{0}^{F_{1}(M)} F^{−1}(σ) ds < ∞.  (2.23)

Hence, Theorem 2.2 applied to (2.21) proves that it has a noncontinuable solution y fulfilling (1.4).

(ii) Suppose, contrarily, that (1.1) has a solution y fulfilling (1.4). Then Theorem 2.2 yields either (2.7) or (2.8) holds. Suppose that (2.7) holds. (2.21) has no solution fulfilling (1.4); hence according to Theorem 2.2 (i) (applied to (2.21) ) either

∫_{M}^{∞} \frac{dσ}{ḡ(σ)} = ∞  (2.24)

or

∫_{M}^{∞} \frac{dσ}{ḡ(σ)} = ∞ and ∫_{0}^{F_{1}(M)} F_{1}^{−1}(σ) dσ = ∞.  (2.25)

As (2.7) and (2.20) yield (2.22), (2.24) is in a contradiction with (2.7) and (2.22). As (2.7) yields (2.23), the inequality (2.23) contradicts (2.25).

If (2.8) holds, the proof is similar. □

Example. Consider problem (1.1), (1.2) with g(x) = |x|^{λ} for |x| ≥ M, λ ∈ R. Let τ > 0, c_{i}, i = 0, …, n − 2 be such that f(τ, c_{0}, …, c_{n−2}) ≠ 0. Then, according to Theorem 2.2, (1.1) has a noncontinuable solution y fulfilling (1.4) if and only if λ > 2.

Remark. Theorem 1.1 (ii) follows from Theorem 2.3 and the Example.

Let us turn our attention to the case when (2.6) does not hold.

Theorem 2.4. Let β ∈ {−1, 1}, δ > 0, ε > 0, τ ∈ (0, ∞), α ∈ {−1, 1}, s ∈ {0, 1, …, n − 2} and c_{i} ∈ R, i = 0, 1, …, n − 2 be such that τ > ε,

λ > δ(n − s − 2) + 2,  (2.26)
c_{s} = 0, (−1)^{i−s} β c_{i} ≥ 0 for i = s + 1, …, n − 2,  (2.27)
n − s + \frac{1 − α}{2} be odd,  (2.28)
g(x) ≥ |x|^{λ} for βx ≥ M.  (2.29)

Let, moreover, a positive function r exist such that

\begin{matrix}αf(t, x_{1}, …, x_{n−1}) sgn x_{s+1} ≥ r(t)|x_{s+1}|^{δ} \\ for t ∈ [τ − ε, τ] ∩ R_{+}, |x_{i} − c_{i−1}| ≤ ε, i = 1, 2, …, n − 1. \end{matrix}  (2.30)

Then there exists a solution y of (1.1) fulfilling (1.4) that is defined in a left neighbourhood of τ.

Proof. Let α = 1 and β = 1; thus n − s is odd. For the other cases the proof is similar. Note that (2.30) and c_{s} = 0 yield f(τ, c_{0}, …, c_{n−2}) = 0. Consider problem (2.11) and (2.12) with N = ε and τ̄ = max(0, τ − ε). Put

M_{1} = ((n − s − 1)!)^{−δ} min_{t∈[τ̄,τ]} r(t) > 0,

δ_{1} = \frac{δ(n − s − 1) + 1}{λ + δ − 1},

M_{2} = max{|f(t, x_{1}, …, x_{n−1})| : t ∈ [τ̄, τ], |x_{i} − c_{i−1}| ≤ ε, i = 1, 2, …, n − 1},

M_{3} = (\frac{M_{1}(λ + δ − 1)}{δ(n − s − 1) + 1})^{−1/(λ+δ−1)},

M_{4} = (λ − 1)ε^{δ} min_{t∈[0,τ]} r(t),

and M_{5} = M_{4}^{−1/(λ−1)}. Note that due to (2.26), δ_{1} ∈ (0, 1).

Furthermore, let J = [T, τ) ⊂ [τ̄, τ) be such that 0 < τ − T ≤ 1,

(τ − T) ∑_{j=0}^{n−2} |c_{j}| + \frac{M_{3}}{1 − δ_{1}} (τ − T)^{1−δ_{1}} + \frac{λ − 1}{λ − 2}M_{5}(τ − T)^{\frac{λ−2}{λ−1}} ≤ ε,  (2.31)

and

M_{2}(τ − T) < ∫_{M}^{2M} \frac{ds}{g(s)}.

Denote by y_{k} a solution of (2.11) and by J_{k} the intersection of its maximal definition interval and [T, τ]. We prove, similarly as in the proof of Theorem 2.2, (see (2.15)) that

y_{k}^{(n−1)}(t) > M for t ∈ J_{k};  (2.32)

hence (2.27), (2.30) and (2.32) yield

c_{s+1} ≤ 0, c_{s+2} ≥ 0, …, c_{n−2} ≤ 0,  (2.33)
(−1)^{j−s}y_{k}^{(j)}(t) > 0 for j = s + 1, s + 2, …, n − 2,
sgn y_{k}^{(s)}(t) = 1, t ∈ J_{k} − {τ}.

From this, (2.11), (2.12) and (2.30),

y_{k}^{(n)}(t) ≥ 0 and y_{k}^{(n−1)} is nondecreasing on J_{k}.  (2.34)

The Taylor formula at t = τ, (2.33), (2.34), and n − s being odd yield

y_{k}^{(s)}(t) = ∑_{j=s}^{n−2} c_{j} \frac{(t − τ)^{j−s}}{(j − s)!} + ∫_{τ}^{t} \frac{(t − σ)^{n−s−2}}{(n − s − 2)!} y_{k}^{(n−1)}(σ) dσ

≥ ∫_{τ}^{t} \frac{(t − σ)^{n−s−2}}{(n − s − 2)!} y_{k}^{(n−1)}(σ) dσ

≥ \frac{(τ − t)^{n−s−1}}{(n − s − 1)!} y_{k}^{(n−1)}(t), t ∈ J_{k}.

Let T^{∗} ∈ [T, τ) be a number such that

0 ≤ y_{k}^{(s)}(T) ≤ ε for t ∈ [T^{∗}, τ),

and, if T^{∗} > T,

y_{k}^{(s)}(T) > ε for t ∈ [T, T^{∗});

this choice is possible due to (2.34).

Let T^{∗} > T and t ∈ [T, T^{∗}). Then (2.11), (2.12), (2.29), (2.30) and (2.32) yield

y_{k}^{(n)}(t) ≥ r(t) ε^{δ}(y_{k}^{(n−1)}(t))^{λ},

and since λ > 1, an integration on [t, T^{∗}] shows

(y_{k}^{(n−1)}(t))^{1−λ} ≥ (y_{k}^{(n−1)}(t))^{1−λ} − (y_{k}^{(n−1)}(T^{∗}))^{1−λ} ≥ M_{4}(T^{∗} − t)

and

y_{k}^{(n−1)}(t) ≤ M_{5}(T^{∗} − t)^{− \frac{1}{λ−1}} , t ∈ [T, T^{∗}).  (2.35)

Similarly, for t ∈ [T^{∗}, τ), we have

y_{k}^{(n)}(t) ≥ r(t)(y_{k}^{(s)}(t))^{δ}(y_{k}^{(n−1)}(t))^{λ} ≥ M_{1}(τ − t)^{δ(n−s−1)}(y_{k}^{(n−1)}(t))^{λ+δ}. (2.36)

Hence, as λ + δ > 1, an integration on [t, τ] yields

y_{k}^{(n−1)}(t) ≤ M_{3}(τ − t)^{−δ_{1}}, t ∈ J_{k}, k = k_{0}, k_{0+1}, …  (2.37)

From this and from (2.32) and (2.35) we have J_{k} = [T, τ]. Moreover, as τ − T ≤ 1 and δ_{1} < 1, Taylor’s theorem, (2.35), (2.31) and (2.37) yield

|y_{k}^{(i)}(T) − c_{i}| ≤ ∑_{j=i+1}^{n−2} \frac{|c_{j}|}{(j − i)!}(τ − T)^{j−i} + | ∫_{τ}^{T^{∗}} \frac{(T − σ)^{n−i−2}}{(n − i − 2)!} y_{k}^{(n−1)}(σ) dσ|

+ | ∫_{T^{∗}}^{T} \frac{(T − σ)^{n−i−2}}{(n − i − 2)!} y_{k}^{(n−1)}(σ) dσ|

≤ (τ − T) ∑_{j=0}^{n−2} |c_{j}| + \frac{M_{3}}{1 − δ_{1}} (τ − T)^{1−δ_{1}} + \frac{λ − 1}{λ − 2}M_{5}(τ − T)^{\frac{λ−2}{λ−1}}

≤ ε, i = 0, 1, …, n − 2.

From this and from (2.12) and (2.34), χ_{i}(y^{(i)}(t)) = y^{(i)}(t), t ∈ [T, τ] and y_{k} is the solution of (1.1) fulfilling y_{k}^{(i)}(τ) = c_{i}, i = 0, 1, …, n − 2 and y_{k}^{(n−1)} = k. Moreover, as χ_{s}(y^{(s)}(t)) = y^{(s)}(t), the estimations (2.36) and (2.37) holds on [T, τ). The statement of the theorem follows from this and from the Arzèl-Ascoli Theorem similarly as in the proof of Theorem 2.2; when prooving lim_{t→τ_{−}} y(t) = ∞, T_{2} has to be defined such that M_{2}(τ − T_{2}) < ∫_{2Q}^{∞} \frac{ds}{g(s)} (this is possible due to (1.2)) and the inequality in (2.19) is in contradiction with the choice of T_{2}. □

The following Corollary shows that conditions (2.26) and (2.28) cannot be weakened.

Corollary 2.5. Let c_{i} = 0, i = 0, 1, …, n − 2, δ > 0, s ∈ {0, 1, …, n − 2}, α ∈ {−1, 1}, τ ∈ (0, ∞), r ∈ C^{0}(R_{+}) and r > 0 on [0, τ]. Then the equation

y^{(n)} = αr (t)|y^{(s)}|^{δ}|y^{(n−1)}|^{λ} sgn y^{(s)}  (2.38)

has a noncontinuable solution y fulfilling (1.4) if and only if

λ > δ(n − s − 2) + 2 and n − s + \frac{1 − α}{2} is odd.  (2.39)

Proof. If (2.39) holds the statement follows from Theorem 2.4. Let (2.39) be not valid. Let, contrarily, (2.38) have a solution y fulfilling (1.4) defined on [τ̄, τ) ⊂ R_{+}. Suppose, for simplicity, that α = 1 and lim_{t→τ_{−}} y^{(n−1)}(t) = ∞. In the other cases the proof is similar.

As c_{i} = 0 for i = 0, 1, …, n − 2, there exists t_{0} ∈ [τ̄, τ) such that

\begin{matrix}(−1)^{i−s+β}y^{(i)}(t) > 0, i = s, s + 1, …, n − 2, \\ y^{(n−1)}(t) ≥ 1, y^{(n)}(t) > 0 on J = [t_{0}, τ), \end{matrix}  (2.40)

where β = 0 (β = 1) if n − s is odd (is even).

Let n − s be even. Then (2.40) yields y^{(s)}(t) < 0 on J and according to (2.38) y^{(n)}(t) < 0 on J which contradicts (2.40).

Let n − s be odd and λ ≤ δ(n − s − 2) + 2. From this, from (1.4), (2.40) and Taylor’s theorem, we get

0 < y^{(s)}(t) = ∫_{τ}^{t} \frac{(t − σ)^{n−s−2}}{(n − s − 2)!} y^{(n−1)}(σ) dσ ≤ (τ − t)^{n−s−2}|y^{(n−2)}(t)|, (2.41)

with t ∈ J. Furthermore, using (2.40), we have

|y^{n−2}(t)| = ∫_{t}^{τ} y^{(n−1)}(s) ds ≥ y^{(n−1)}(t)(τ − t), t ∈ J.

From this and from (2.41)

0 < y^{(s)}(t)[y^{(n−1)}(t)]^{n−s−2} ≤ |y^{(n−2)}(t)|^{n−s−1} ≤ |y^{(n−2)}(t_{0})|^{n−s−1} = M_{1}, t ∈ J.

Thus, from λ ≤ δ(n − s − 2) + 2 and from y^{(n−1)}(t) ≥ 1 (see (2.40)), we have

∞ = log \frac{y^{(n−1)}(τ)}{y^{(n−1)}(t_{0})} = ∫_{t_{0}}^{τ} \frac{y^{(n)}(σ)}{y^{(n−1)}(σ)} dσ

= ∫_{t_{0}}^{τ} r(σ)(y^{(s)}(σ))^{δ}[y^{(n−1)}(σ)]^{λ−1} ds

≤ M_{1}^{δ} ∫_{t_{0}}^{τ} r(σ)[y^{(n−1)}(σ)]^{λ−1−δ(n−s−2)}dσ

≤ M_{1}^{δ} ∫_{t_{0}}^{τ} r(σ)y^{(n−1)}(σ) dσ

≤ M_{1}^{δ} max_{z∈[T,τ]} r(z) |y^{(n−2)}(t_{0})| < ∞.

This contradiction proves the statement. □

Remark. Theorem 2.4 is proved in [1] in case s = 0 and under further assumptions.

Remark. Let the assumptions of Theorems 2.4 hold with the exception of (2.26). If λ ≤ 2, Theorem 2.1 with g = x^{λ} and Theorem 2.3 (ii) yield (1.1) has no solution y fulfilling (1.4). So there is a problem what happens if 2 < λ < δ(n − s − 2) + 2. In special cases, see e.g. Corollary 2.5, no noncontinuable solution y fulfilling (1.4) exists.

Acknowledgments. This work was supported by the Grant No A1163401/04 of the Grant Agency of the Academy of Sciences of the Czech Republic and by the Ministry of Education of the Czech Republic under the project MSM 0021622409.

References

[1] M. Bartušek, On Existence of Singular Solutions of N-th Order Differential Equations, Arch. Math. (Brno), 36 (2000), 395–404.

[2] M. Bartušek, M. Cecchi, Z. Došlá, M. Marini, Global Monotonicity and Oscillation for Second Order Differential Equation, Czech Math. J. 55 (130) (2005), 209–222.

[3] T. Chanturia, On Existence of Singular and Unbounded Oscillatory Solutions of Differential Equations of Emden-Fowler Type, Diff. Urav. 28 (1992), 1009–1022 (in Russian).

[4] C. V. Coffman, D. F. Ullrich, On the Continuation of Solutions of a Certain Non-linear Differential Equation, Monatsh. Math., B 71, (1967), 385-392.

[5] C. V. Coffman, J. S. W. Wong, Oscillation and Nonoscillation Theorems for Second Order Differential Equations, Funkcial. Ekvac. 15 (1972), 119-130.

[6] J. R. Graef, P. W. Spikes, Asymptotic Behaviour of Solutions of a Second Order Nonlinear Differential Equation, J. Differential Equations 17 (1975), 461–476.

[7] J. R. Graef, P. W. Spikes, On the Nonlinear Limit-point/Limit-circle Problem, Nonlinear Anal. 7 (1983), 851–871.

[8] J. W. Heidel, Uniqueness, Continuation and Nonoscillation for a Second Order Differential Equation, Pacif. J. Math. 32 (1970), 715-721.

[9] J. Jaroš, T. Kusano, On Black Hole Solutions of Second Order Differential Equations with a Singularity in the Differential Operator, Funkcial. Ekvac. 43 (2000) 491–509.

[10] I. Kiguradze, T. Chanturia, Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations, Kluwer Acad., Dordrecht, 1993.

Miroslav Bartušek Department of Mathematics, Masaryk University, Janáčkovo nám. 2a, 602 00 Brno, Czech Republic

   E-mail address: bartusek@math.muni.cz