Georgian Mathematical Journal 1(1994), No. 1, 31-45

ON TWO-POINT BOUNDARY VALUE PROBLEMS FOR

SYSTEMS OF HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS WITH SINGULARITIES

I. KIGURADZE AND G. TSKHOVREBADZE

Abstract. The sufficient conditions of solvability and unique solvability of the two-point boundary value problems of Vallèe-Poussin and Cauchy-Niccoletti have been found for a system of ordinary differential equations of the form

u^{(n)} = f(t, u, u′, …, u^{(n−1)}), where the vector function f :]a, b[×ℝ^{nl} → ℝ^{l} has nonintegrable singularities with respect to the first argument at the points a and b.

§ 1. Statement of the main results

In this paper for an l-dimensional system of differential equations

u^{(n)} = f(t, u, u′, …, u^{(n−1)})  (1.1)

we consider the boundary value problem of Vallèe-Poussin

\begin{matrix}u(a+) = ⋯ = u^{(m−1)}(a+) = 0, \\ u(b−) = ⋯ = u^{(n−m−1)}(b−) = 0 \end{matrix}  (1.2)

and that of Cauchy-Niccoletti

\begin{matrix}u(a+) = ⋯ = u^{(m−1)}(a+) = 0, \\ u^{(m)}(b−) = ⋯ = u^{(n−1)}(b−) = 0, \end{matrix}  (1.3)

where l ≥ 1, n ≥ 2, m is an integer part of the number \frac{n}{2}, −∞ < a < b < +∞, and the vector function f :]a, b[×ℝ^{nl} → ℝ^{l} satisfies the Caratheodory conditions on each compact contained in ]a, b[×ℝ^{nl}. We are interested mainly in the singular case when f is nonintegrable with respect to the first argument on [a, b], having singularities at the ends of this interval. The above problems were investigated for l = 1 in [2-6].

1991 Mathematics Subject Classification. 34B15.

The following notations will be used:

I_{n}(a, b) = \{ \begin{matrix}]a, b[ for n = 2m \\ ]a, b] for n = 2m + 1 \end{matrix} ;

μ_{n} = \{ \begin{matrix}1 for n = 2m \\ \frac{n}{2} for n = 2m + 1 \end{matrix} ;

λ_{im}(a, b; t) = \frac{min\{(t − a)^{2m−i}, (b − t)^{2m−i}\}}{(m − 1)!(m − i)!\sqrt{(2m − 1)(2m − 2i + 1)}}

(i = 1, …, m);

ℝ is a set of real numbers, ℝ_{+} = [0, +∞[; ξ = (ξ_{j})_{j=1}^{l} ∈ ℝ^{l} and A = (a_{kj})_{k,j=1}^{l} ∈ ℝ^{l×l} are respectively an l -

dimensional column vector and an l×l matrix with real components ξ_{j} (j = 1, …, l) and a_{kj} (k, j = 1, …, l),

|ξ| = (|ξ_{j}|)_{j=1}^{l}, ∥ξ∥ = ∑_{j=1}^{l} |ξ_{j}|, ∥A∥ = ∑_{k,j=1}^{l} |a_{kj}|,

S(ξ) = ( \begin{matrix}sign ξ_{1} & 0 & … & 0 \\ 0 & sign ξ_{2} & … & 0 \\ … & … & … & … \\ 0 & 0 & … & sign ξ_{l} \end{matrix} );

r(A) is the spectral radius of the matrix A; ℝ_{+}^{l} and ℝ_{+}^{l×l} are sets of l-dimensional vectors and l × l matrices with

nonnegative components;

the inequalities ξ ≤ ξ̄ and A ≤ Ā, where ξ and ξ̄ ∈ ℝ^{l} and A and Ā ∈ ℝ^{l×l}, imply respectively ξ̄− ξ ∈ ℝ_{+}^{l} and Ā − A ∈ ℝ_{+}^{l×l};

L_{loc}(I; ℝ_{+}), where I ⊂ ℝ is an interval, is a set of functions x : I → ℝ_{+} which are Lebesgue integrable on each segment contained in I;

K_{loc}(I × ℝ^{p}; ℝ^{l}), where p is a natural number, is a set of vector functions mapping I × ℝ^{p} into ℝ^{l} and satisfying the Caratheodory conditions on each compact contained in I × ℝ^{p};

C̃_{loc}^{p}(I; ℝ^{l}) is a set of vector functions u : I → ℝ^{l} which are absolutely continuous together with all their derivatives up to order p inclusive on each segment contained in I;

C̃^{n−1,m}(I; ℝ^{l}) is a set of vector functions u ∈ C̃_{loc}^{n−1}(I; ℝ^{l}) satisfying the condition

∫_{I} ∥u^{(m)}(τ)∥^{2}dτ < +∞.

As mentioned above, throughout this paper it is assumed that

f ∈ K_{loc}(]a, b[×ℝ^{nl}; ℝ^{l}).

Theorem 1.1. Let the following inequalities be fulfilled on ]a, b[×ℝ^{nl}:

(−1)^{n−m−1}S(x_{1})f(t, x_{1}, …, x_{n}) ≥−∑_{i=1}^{m} H_{i}(t)|x_{i}| − h(t)  (1.4)

and

∥f(t, x_{1}, …, x_{n})∥≤q(t, x_{1}, …, x_{m}) ∑_{i=m+1}^{n}(1+∥x_{i}∥)^{\frac{2n−2m−1}{2i−2m−1}} , (1.5)

where

q ∈ K_{loc}(I_{n}(a, b) × ℝ^{ml}; ℝ_{+}),  (1.6)

and H_{i} :]a, b[→ ℝ_{+}^{l×l} (i = 1, …, m) and h :]a, b[→ ℝ_{+}^{l} are respectively measurable matrix and vector functions satisfying the conditions

∫_{a}^{b}(τ − a)^{n−m− \frac{1}{2}} (b − τ)^{m− \frac{1}{2}} ∥h(τ)∥dτ < +∞,  (1.7)
∫_{a}^{b}(τ −a)^{n−i}(b−τ)^{2m−i}∥H_{i}(τ)∥dτ <+∞ (i=1, …, m),  (1.8)
r( ∑_{i=1}^{m} ∫_{a}^{b}(τ − a)^{n−2m}λ_{im}(a, b; τ)H_{i}(τ)dτ) < μ_{n}.  (1.9)

Then the problem (1.1),(1.2) is solvable in the class C̃^{n−1,m}(I_{n}(a,b);ℝ^{l}).

Theorem 1.2. Let on ]a, b[×ℝ^{nl} the inequalities (1.4) and (1.5) be fulfilled, where q ∈ K_{loc}(]a, b] × ℝ^{ml}; ℝ_{+}), and H_{i} :]a, b] → ℝ_{+}^{l×l} (i = 1, …, m) and h :]a, b] → ℝ_{+}^{l} are respectively measurable matrix and vector functions satisfying the conditions

∫_{a}^{b}(τ − a)^{n−m− \frac{1}{2}} ∥h(τ)∥dτ < +∞,  (1.10)
∫_{a}^{b}(τ − a)^{n−i}∥H_{i}(τ)∥dτ < +∞ (i = 1, …, m),  (1.11)

r( ∑_{i=1}^{m} \frac{1}{(m − 1)!(m − i)!\sqrt{(2m − 1)(2m − 2i + 1)}} ×

× ∫_{a}^{b}(τ − a)^{n−i}H_{i}(τ)dτ) < μ_{n}.  (1.12)

Then the problem (1.1), (1.3) is solvable in the class C̃^{n−1,m}(]a, b]; ℝ^{l}).

For a differential system

u^{(n)} = f(t, u, u′, …, u^{(m−1)}),  (1.1′)

not containing intermediate derivatives of order higher than (m − 1), Theorems 1.1 and 1.2 can be formulated as follows:

Theorem 1.1′. Let f ∈ K_{loc}(I_{n}(a, b) × ℝ^{ml}; ℝ^{l}) and on ]a, b[×ℝ^{ml}

(−1)^{n−m−1}S(x_{1})f(t, x_{1}, …, x_{m})≥−∑_{i=1}^{m} H_{i}(t)|x_{i}|−h(t), (1.4′)

where H_{i} :]a, b[→ ℝ_{+}^{l×l} (i = 1, …, m) and h :]a, b[→ ℝ_{+}^{l} are measurable matrix and vector functions satisfying the conditions (1.7)-(1.9). Then the problem (1.1′), (1.2) is solvable in the class C̃^{n−1,m}(I_{n}(a, b);ℝ^{l}).

Theorem 1.2′. Let f ∈ K_{loc}(]a, b] × ℝ^{ml}; ℝ^{l}) and on ]a, b[×ℝ^{ml} the inequality (1.4′) be fulfilled, where H_{i} :]a, b] → ℝ_{+}^{l×l} (i = 1, …, m) and h :]a, b] → ℝ_{+}^{l} are measurable matrix and vector functions satisfying the conditions (1.10)-(1.12).Then the problem (1.1′), (1.3)is solvable in the class C̃^{n−1,m}(]a, b]; ℝ^{l}).

Theorem 1.3. Let

\begin{matrix}f ∈ K_{loc}(I_{n}(a, b) × ℝ^{ml}; ℝ^{l}), \\ ∫_{a}^{b}(τ − a)^{n−m− \frac{1}{2}} (b − τ)^{m− \frac{1}{2}} ∥f(τ, 0, …, 0)∥dτ < +∞ \end{matrix} (1.13)

and on ]a, b[×ℝ^{ml}

\begin{matrix}(−1)^{n−m−1}S(x_{1} − y_{1})[f(t, x_{1}, …, x_{m}) − f(t, y_{1}, …, y_{m})] ≥ \\ ≥ − ∑_{i=1}^{m} H_{i}(t)|x_{i} − y_{i}|, \end{matrix} (1.14)

where H_{i} :]a, b[→ ℝ_{+}^{l×l} (i = 1, …, m) are measurable matrix functions satisfying the conditions (1.8) and (1.9). Then the problem (1.1′), (1.2) is uniquely solvable in the class C̃^{n−1,m}(I_{n}(a, b); ℝ^{l}).

Theorem 1.4. Let

f ∈ K_{loc}(]a, b] × ℝ^{ml}; ℝ^{l}), ∫_{a}^{b}(τ − a)^{n−m− \frac{1}{2}} ∥f(τ, 0, …, 0)∥dτ < +∞

and on ]a, b[×ℝ^{ml} the inequality (1.14) be fulfilled, where H_{i} :]a, b] → ℝ_{+}^{l×l} (i = 1, …, m) are measurable matrix functions satisfying the conditions (1.11) and (1.12). Then the problem (1.1′), (1.3) is uniquely solvable in the class C̃^{n−1,m}(]a, b]; ℝ^{l}).

§ 2. Auxiliary propositions

Lemma 2.1. Let I ⊂ ℝ be some interval, k be a natural number, ρ_{0} ∈ ]0, +∞[ and

φ ∈ L_{loc}(I; ℝ_{+}).  (2.1)

Then there exists a continuous function ρ : I → ℝ_{+} such that for any vector function v ∈ C̃_{loc}^{k}(I; ℝ^{l}) satisfying almost everywhere on I the differential inequality

∥v^{(k+1)}(t)∥ ≤ φ(t) [1 + ∑_{i=0}^{k} ∥v^{(i)}(t)∥^{\frac{2k+1}{2i+1}} ]  (2.2)

and the condition

∫_{I} ∥v(τ)∥^{2} dτ ≤ ρ_{0}^{2},  (2.3)

the estimates

∥v^{(i)}(t)∥ < ρ(t) for t ∈ I (i = 0, …, k)  (2.4)

hold.

Proof. In the case I = [a, b] it is not difficult to verify by Lemma 2.2 from [6] that there exists a positive constant ρ̃ such that the estimates ∥v^{(i)}(t)∥ < ρ̃ for a ≤ t ≤ b (i = 0, …, k) hold for any vector function v ∈ C̃_{loc}^{k}(I; ℝ^{l}) satisfying the conditions (2.2) and (2.3); in other words, we have (2.4), where ρ(t) ≡ ρ̃.

Now consider the case I =]a, b]. Choose any decreasing sequence a_{j} ∈ ]a, b] (j = 0, 1, 2, …) such that a_{0} = b and lim_{j→+∞} a_{j} = a. Then, by virtue

of the above reasoning, for any natural number j there exists a positive constant ρ_{j} such that any vector function v ∈ C̃_{loc}^{k}(I; ℝ^{l}) satisfying the conditions (2.2) and (2.3) admits the estimates

∥v^{(i)}(t)∥ < ρ_{j} for a_{j} ≤ t ≤ b (i = 0, …, k).  (2.5)

Without loss of generality the sequence (ρ_{j})_{j=1}^{+∞} can be assumed to be nondecreasing. Then (2.5) yields the estimates (2.4), where

ρ(t) = ρ_{j} + \frac{t − a_{j−1}}{a_{j} − a_{j−1}} (ρ_{j+1} − ρ_{j}) for a_{j} < t ≤ a_{j−1} (j = 1, 2, …)

with ρ : I → ℝ_{+} being continuous and independent of v.

The cases I = [a, b[ and I =]a, b[ are considered similarly.

Lemma 2.2. Let H_{i} :]a, b[→ ℝ_{+}^{l×l} (i = 1, …, m) and h :]a, b[→ ℝ_{+}^{l} be measurable matrix and vector functions satisfying the conditions (1.7)-(1.9) and

H = ∑_{i=1}^{m} ∫_{a}^{b}(τ − a)^{n−2m}λ_{im}(a, b; τ)H_{i}(τ) dτ.  (2.6)

Then for any vector function u ∈ C̃^{n−1,m}(]a, b[; ℝ^{l}) satisfying a system of differential inequalities

(−1)^{n−m−1}S(u(t))u^{(n)}(t)≥− ∑_{i=1}^{m} H_{i}(t)|u^{(i−1)}(t)|−h(t)  (2.7)
for a < t < b

and the boundary conditions (1.2) we have the estimates

∫_{a}^{b} ∥u^{(m)}(τ)∥^{2} dτ ≤ ρ_{0}^{2}  (2.8)

and

∥u^{(i−1)}(t)∥ ≤ ρ_{0}σ_{im}(a, b; t) for a < t < b (i = 1, …, m), (2.9)

where

σ_{im}(a, b; t) = \frac{min\{(t − a)^{m−i+ \frac{1}{2}} , (b − t)^{m−i+ \frac{1}{2}} \}}{(m − i)!\sqrt{2m − 2i + 1}},

ρ_{0} = \sqrt{l}∥(μ_{n}E − H)^{−1}∥ ×
× ∫_{a}^{b}(τ − a)^{n−2m}σ_{1m}(a, b; τ)∥h(τ)∥ dτ  (2.10)

and E is the unit l × l matrix.

To prove this lemma we need

Lemma 2.3. Let

w(t) = ∑_{i=1}^{n−m} ∑_{k=i}^{n−m} c_{ik}(t)v^{(n−k)}(t)v^{(i−1)}(t),

where

\begin{matrix}v ∈ C̃^{n−1,m}(]a, b[; ℝ), v^{(i−1)}(a+) = 0 (i = 1, …, m), \\ v^{(j−1)}(b−) = 0 (j = 1, …, n − m) \end{matrix}  (2.11)

and each c_{ik} : [a, b] → ℝ is a (n−k −i+1)-times continuously differentiable function; in that case there exists a positive constant c_{0} such that

|c_{ii}(t)| ≤ c_{0}(t − a)^{n−2m} for a ≤ t ≤ b  (2.12)
(i = 1, …, n − m).

Then

lim_{t→a+} inf |w(t)| = 0, lim_{t→b−} inf |w(t)| = 0.

Proof. In the first place it will be shown that

lim_{t→a+} inf |w(t)| = 0.  (2.13)

Let the opposite be true. Then without loss of generality one may assume that the inequality w(t) ≥ δ for a < t ≤ a + 2ε_{0} is fulfilled for some δ ∈]0, +∞[ and ε_{0} ∈ ]0, \frac{b−a}{4}[∩]0, 1[.

Therefore

∑_{i=1}^{n−m} ∑_{k=i}^{n−m} q_{ik}(t; ε)v^{(n−k)}(t)v^{(i−1)}(t) ≥ δ(t − a − ε)^{n}(a + 2ε − t)^{n} (2.14)
for a + ε ≤ t ≤ a + 2ε, 0 < ε ≤ ε_{0},

where q_{ik}(t; ε) = (t − a − ε)^{n}(a + 2ε − t)^{n}c_{ik}(t). After integrating the latter inequality from a + ε to a + 2ε according to Lemma 4.1 from [7], we obtain

∑_{i=1}^{n−m} ∑_{k=i}^{n−m} ∑_{j=0}^{m_{ik}} ν_{ikj}∫_{a+ε}^{a+2ε}q_{ik}^{(n−k−i−2j+1)}(τ; ε)[v^{(i+j−1)}(τ)]^{2}dτ ≥

≥ δ ∫_{a+ε}^{a+2ε}(τ − a − ε)^{n}(a + 2ε − τ)^{n} dτ,  (2.15)

where m_{ik} is the integer part of the number \frac{1}{2}(n − k − i + 1) and ν_{ikj} (i = 1, …, n − m; k = i, …, n − m; j = 0, …, m_{ik}) are positive constants independent of a, ε and v.

If k ∈ {i + 1, …, n − m}, then we have i + j − 1 ≤ m − 1, 2n − (n − k − i − 2j + 1) ≥ 2i + 2j + n for any j ∈ {0, …, m_{ik}}.

Therefore, taking into account (2.11) and (2.14), we find

[v^{(i+j−1)}(t)]^{2} = [\frac{1}{(m − i − j)!} ∫_{a}^{t}(t − τ)^{m−i−j}v^{(m)}(τ) dτ]^{2} ≤

≤ α(ε)ε^{2m−2i−2j+1} for a < t ≤ a + 2ε  (2.16)

and |q_{ik}^{(n−k−i−2j+1)}(t; ε)| ≤ α_{1}ε^{2i+2j+n} for a ≤ t ≤ a + 2ε, where

α(ε) = 2^{2m−1} ∫_{a}^{a+2ε}[v^{(m)}(τ)]^{2} dτ → 0 for ε → 0  (2.17)

and α_{1} is a positive constant independent of ε. Therefore

|∫_{a+ε}^{a+2ε} q_{ik}^{(n−k−i−2j+1)}(τ; ε)[v^{(i+j−1)}(τ)]^{2} dτ| ≤ α_{1}α(ε)ε^{2m+2+n} ≤

≤ α_{1}α(ε)ε^{2n+1}.

Consider now the case k = i. By virtue of (2.12) and (2.14) we have

|q_{ik}^{(n−k−i−2j+1)}(t; ε)| = |q_{ii}^{(n−2i−2j+1)}(t; ε)| ≤ α_{2}ε^{2n−2m+2i+2j−1}

for a ≤ t ≤ a + 2ε,

where α_{2} is a positive constant independent of ε. Therefore if i+j −1 = m, then

|∫_{a+ε}^{a+2ε} q_{ii}^{(n−2i−2j+1)}(τ; ε)[v^{(i+j−1)}(τ)]^{2} dτ| =
= |∫_{a+ε}^{a+2ε} q_{ii}(τ; ε)[v^{(m)}(τ)]^{2} dτ| ≤ α_{2}α(ε)ε^{2n+1},

if however i + j − 1 < m, then, taking into account (2.16), we obtain

|∫_{a+ε}^{a+2ε} q_{ii}^{(n−2i−2j+1)}(τ; ε)[v^{(i+j−1)}(τ)]^{2} dτ| ≤ α_{2}α(ε)ε^{2n+1}.

Thus

|∫_{a+ε}^{a+2ε} q_{ik}^{(n−k−i−2j+1)}(τ; ε)[v^{(i+j−1)}(τ)]^{2} dτ| ≤ α_{0}α(ε)ε^{2n+1}, (2.18)
(i = 1, …, n − m; k = i, …, n − m, j = 0, …, m_{ik}),

where α_{0} = max{α_{1}, α_{2}}.

On the other hand,

∫_{a+ε}^{a+2ε}(τ − a − ε)^{n}(a + 2ε − τ)^{n} dτ ≥ \frac{ε^{n}}{2^{n}} ∫_{a+ε}^{a+ \frac{3ε}{2}} (τ − a − ε)^{n}dτ =

= \frac{1}{2^{2n+1}(n + 1)}ε^{2n+1}.

Due to (2.18) and the latter inequality we find from (2.15) that α(ε) ≥ δ_{0} for 0 < ε ≤ ε_{0}, where δ_{0} is a positive constant independent of ε. But the latter inequality contradicts the condition (2.17). This contradiction proves that (2.13) holds.

The equality lim_{t→b−} inf |w(t)| = 0 is proved similarly, the only difference

being that for n = 2m + 1 instead of (2.12) the condition v^{(m)}(b−) = 0 is used.

Proof of Lemma 2.2. For each component u_{j} (j = 1, …, l) of the solution u of the problem (2.7), (1.2) we have

|u_{j}^{(i−1)}(t)| = |\frac{1}{(m − i)!} ∫_{a}^{t}(t − τ)^{m−i}u_{j}^{(m)}(τ) dτ| ≤

≤ \frac{1}{(m − i)!\sqrt{2m − 2i + 1}}(t − a)^{m−i+ \frac{1}{2}} (∫_{a}^{b}[u_{j}^{(m)}(τ)]^{2} dτ)^{\frac{1}{2}}

for a < t < b (i = 1, …, m)

and

|u_{j}^{(i−1)}(t)| = |\frac{1}{(m − i)!} ∫_{t}^{b}(τ − t)^{m−i}u_{j}^{(m)}(τ) dτ| ≤

≤ \frac{1}{(m − i)!\sqrt{2m − 2i + 1}}(b − t)^{m−i+ \frac{1}{2}} (∫_{a}^{b}[u_{j}^{(m)}(τ)]^{2} dτ)^{\frac{1}{2}}

for a < t < b (i = 1, …, m).

Therefore

|u_{j}^{(i−1)}(t)| ≤ σ_{im}(a, b; t)ρ_{j} for a < t < b (i = 1, …, m), (2.19)

where

ρ_{j} = (∫_{a}^{b}[u_{j}^{(m)}(τ)]^{2} dτ)^{\frac{1}{2}} .

Let H_{i}(t) = (h_{ijk}(t))_{j,k=1}^{l} (i = 1, …, m), h(t) = (h_{j}(t))_{j=1}^{l}. Rewrite (2.7) in terms of components as

(−1)^{n−m−1}u_{j}^{(n)}(t) sign u_{j}(t) ≥
≥−∑_{i=1}^{m} ∑_{k=1}^{l} h_{ijk}(t)|u_{k}^{(i−1)}(t)|−h_{j}(t) (j =1, …, l).  (2.7′)

40 I. KIGURADZE AND G. TSKHOVREBADZE

After multiplying both sides of (2.7′) by (t−a)^{n−2m}|u_{j}(t)| and integrating from s to t, we obtain

(−1)^{n−m} ∫_{s}^{t}(τ − a)^{n−2m}u_{j}^{(n)}(τ)u_{j}(τ) dτ ≤

≤ ∑_{i=1}^{m} ∑_{k=1}^{l} ∫_{s}^{t}(τ − a)^{n−2m}h_{ijk}(τ)|u_{k}^{(i−1)}(τ)||u_{j}(τ)| dτ +

+ ∫_{s}^{t}(τ − a)^{n−2m}h_{j}(τ)|u_{j}(τ)| dτ for a < s ≤ t < b.  (2.20)

By virtue of (2.19)

∑_{k=1}^{l} ∫_{s}^{t}(τ − a)^{n−2m}h_{ijk}(τ)|u_{k}^{(i−1)}(τ)||u_{j}(τ)| dτ ≤

≤ ρ_{j} ∑_{k=1}^{l} ρ_{k} ∫_{s}^{t}(τ − a)^{n−2m}σ_{1m}(a, b; τ)σ_{im}(a, b; τ)h_{ijk}(τ) dτ =

= ρ_{j} ∑_{k=1}^{l} ρ_{k} ∫_{s}^{t}(τ − a)^{n−2m}λ_{im}(a, b; τ)h_{ijk}(τ) dτ  (2.21)
(i = 1, …, m),
∫_{s}^{t}(τ − a)^{n−2m}h_{i}(τ)|u_{j}(τ)| dτ ≤
≤ ρ_{j} ∫_{s}^{t}(τ − a)^{n−2m}σ_{1m}(a, b; τ)h_{j}(τ) dτ.  (2.22)

On the other hand, by Lemma 4.1 from [7]

∫_{s}^{t}(τ − a)^{n−2m}u_{j}^{(n)}(τ)u_{j}(τ) dτ =

= w_{j}(t) − w_{j}(s) + (−1)^{n−m}μ_{n} ∫_{s}^{t}[u_{j}^{(m)}(τ)]^{2} dτ,  (2.23)

where

w_{j}(t) = \begin{cases}∑_{p=1}^{n−m}(−1)^{p−1}u_{j}^{(n−p)}(t)u_{j}^{(p−1)}(t) & for n = 2m, \\ ∑_{p=1}^{n−m−1}(−1)^{p−1}[(t − a)u_{j}^{(n−p)}(t)− \\ −pu_{j}^{(n−p−1)}(t)]u_{j}^{(p−1)}(t)+(−1)^{m} \frac{t−a}{2}[u_{j}^{(m)}(t)]^{2} & for n=2m+1. \end{cases}

As one may readily verify, the functions w_{j} (j = 1, …, l) satisfy the conditions of Lemma 2.3 and therefore

lim_{s→a+} inf |w_{j}(s)| = 0, lim_{t→b−} inf |w_{j}(t)| = 0 (j = 1, …, l).

Taking into account the latter equalities and conditions (1.7) and (1.8) from (2.20)-(2.23) we obtain

μ_{n}ρ_{j}^{2} ≤ ρ_{j} ∑_{i=1}^{m} ∑_{k=1}^{l} ρ_{k} ∫_{a}^{b}(τ − a)^{n−2m}λ_{im}(a, b; τ)h_{ijk}(τ)dτ +

+ρ_{j} ∫_{a}^{b}(τ − a)^{n−2m}σ_{1m}(a, b; τ)h_{j}(τ)dτ (j = 1, …, l).

Hence by virtue of (2.6) we have

μ_{n}ρ ≤ Hρ + ∫_{a}^{b}(τ − a)^{n−2m}σ_{1m}(a, b; τ)h(τ)dτ,

where ρ = (ρ_{j})_{j=1}^{l}. In view of (1.9) and the notation (2.10) from the latter inequality we find

ρ ≤ (μ_{n}E − H)^{−1} ∫_{a}^{b}(τ − a)^{n−2m}σ_{1m}(a, b; τ)h(τ)dτ

and

∥ρ∥ ≤ ∥(μ_{n}E − H)^{−1}∥ ∫_{a}^{b}(τ − a)^{n−2m}σ_{1m}(a, b; τ)∥h(τ)∥dτ = l^{− \frac{1}{2}} ρ_{0}.

Hence

∫_{a}^{b} ∥u^{(m)}(τ)∥^{2}dτ ≤ l∥ρ∥^{2} ≤ ρ_{0}^{2}.

On the other hand, in view of (2.19)

∥u^{(i−1)}(t)∥ ≤ σ_{im}(a, b; t)∥ρ∥ ≤ ρ_{0}σ_{im}(a, b; t) for a < t < b
(i = 1, …, m).

Therefore, the estimates (2.8) and (2.9) hold.

In a similar manner we prove

Lemma 2.4. Let H_{i} :]a, b] → ℝ_{+}^{l×l} (i = 1, …, m), and let h :]a, b] → ℝ_{+}^{l} be measurable matrix and vector functions satisfying the conditions (1.10)- (1.12). Then for any solution u ∈ C̃^{n−1,m}(]a, b]; ℝ^{l}) of the problem (2.7), (1.3) we have the estimates

∫_{a}^{b} ∥u^{(m)}(τ)∥^{2}dτ ≤ ρ_{0}^{2}

and ∥u^{(i−1)}(t)∥ ≤ ρ_{0}(t − a)^{m−i+ \frac{1}{2}} for a < t < b (i = 1, …, m), where

ρ_{0} = \frac{\sqrt{l}}{(m − 1)!\sqrt{2m − 1}}∥(μ_{n}E − H)^{−1}∥ ∫_{a}^{b}(τ − a)^{n−m− \frac{1}{2}} ∥h(τ)∥ dτ,

H =∑_{i=1}^{m} \frac{1}{(m−1)!(m−i)!\sqrt{(2m−1)(2m−2i+1)}} ∫_{a}^{b}(τ −a)^{n−i}H_{i}(τ)dτ,

and E is the unit l × l matrix.

§ 3. Proof of the main results

Proof of Theorem 1.1. Let ρ_{0} and σ_{im}(a, b; t) (i = 1, …, m) be respectively the number and functions from Lemma 2.2 and

φ(t) = 4^{n} sup{q(t, x_{1}, …, x_{m}) : ∥x_{i}∥ ≤ ρ_{0}σ_{im}(a, b; t)
(i = 1, …, m)}.  (3.1)

Then due to (1.6), (2.1) holds with I = I_{n}(a, b).

For k = n − m − 1, ρ_{0}, and φ, by virtue of Lemma 2.1 there exists a continuous function ρ : I_{n}(a, b) → ℝ_{+} such that estimates (2.4) are valid for any vector function v ∈ C̃_{loc}^{k}(I_{n}(a, b); ℝ^{l}) satisfying the conditions (2.2) and (2.3).

Let

ρ_{i}(t) = \begin{cases}ρ_{0}σ_{im}(a, b; t) & for i ∈ \{1, …, m\} \\ ρ(t) & for i ∈ \{m + 1, …, n\} \end{cases}  (3.2)

and f^{∗}(t) = sup{∥f(t, x_{1}, …, x_{n})∥ : ∥x_{i}∥ ≤ ρ_{i}(t) (i = 1, …, n)}. For any i ∈ {1, …, n} and ξ = (ξ_{p})_{p=1}^{l} we set

χ_{ip}(t, ξ) = \begin{cases}ξ_{p} & for |ξ_{p}| ≤ ρ_{i}(t) \\ ρ_{i}(t) sign ξ_{p} & for |ξ_{p}| > ρ_{i}(t) \end{cases} ,

χ_{i}(t, ξ) = (χ_{ip}(t, ξ))_{p=1}^{l}.

(3.3)

Let j be an arbitrary natural number,

I_{nj}(a, b) = \begin{cases}[a + \frac{b−a}{3j}, b − \frac{b−a}{3j}] & for n = 2m \\ [a + \frac{b−a}{3j}, b] & for n = 2m + 1 \end{cases} ,

f_{j}(t, x_{1}, …, x_{n}) =
= \begin{cases}f(t, χ_{1}(t, x_{1}), …, χ_{n}(t, x_{n})) & for t ∈ I_{nj}(a, b) \\ 0 & for t ∈ [a, b]\I_{nj}(a, b) \end{cases} , (3.4)

f_{j}^{∗}(t) = \begin{cases}f^{∗}(t) & for t ∈ I_{nj}(a, b) \\ 0 & for t ∈ [a, b]\I_{nj}(a, b) \end{cases} .

Clearly, that f_{j}^{∗} : [a, b] → ℝ_{+} is the Lebesgue integrable function and on the [a, b] × ℝ^{nl} the inequality ∥f_{j}(t, x_{1}, …, x_{n})∥ ≤ f_{j}^{∗}(t) holds. On the other hand the homogeneous differential system u^{(n)} = 0 by boundary conditions (1.2) has only the trivial solution. Therefore by virtue of the Conti theorem [1] ^{1} the differential system u^{(n)} = f_{j}(t, u, …, u^{(n−1)}) has a solution u_{j} ∈ C̃_{loc}^{n−1}([a, b]; ℝ^{l}) satisfying the boundary conditions (1.2). It is obvious that u_{j} ∈ C̃^{n−1,m}([a, b]; ℝ^{l}). Simultaneously, from (1.4), (3.3), and (3.4) it follows that u_{j} is the solution of the system of the differential inequalities (2.7). Therefore by virtue of Lemma 2.2

∫_{a}^{b} ∥u_{j}^{(m)}(τ)∥^{2} dτ ≤ ρ_{0}^{2}  (3.5)

and

∥u_{j}^{(i−1)}(t)∥ ≤ ρ_{0}σ_{im}(a, b; t) for a < t < b (i = 1, …, m). (3.6)

From conditions (1.5) and (3.1)-(3.6) it is clear that the vector function v_{j}(t) = u_{j}^{(m)}(t) satifies the inequalities (2.2) and (2.3). Hence by Lemma 2.1 ∥u_{j}^{(i−1)}(t)∥ < ρ(t) for t ∈ I_{n}(a, b) (i = m + 1, …, n). Therefore

∥u_{j}^{(i−1)}(t)∥ < ρ_{i}(t) for a < t < b (i = 1, …, n)  (3.7)

and

∥u_{j}^{(n)}(t)∥ ≤ f^{∗}(t) for a < t < b.  (3.8)

Moreover, in view of (3.3),(3.4) and (3.7) it is clear that

u_{j}^{(n)}(t) = f(t, u_{j}(t), …, u_{j}^{(n−1)}(t)) for t ∈ I_{nj}(a, b).  (3.9)

Since f^{∗} ∈ L_{loc}(I_{n}(a, b); ℝ_{+}), the estimates (3.7) and (3.8) imply that the sequences (u_{j}^{(i−1)})_{j=1}^{+∞} (i = 1, …, n) are uniformly bounded and equicontinuous on each segment contained in I_{n}(a, b). Therefore, by virtue of the Arcela-Ascoli lemma these sequences can be regarded without loss of generality as uniformly converging on each segment from I_{n}(a, b).

If we set lim_{j→+∞} u_{j}(t) = u(t) for t ∈ I_{n}(a, b), then

lim_{j→+∞} u_{j}^{(i−1)}(t) = u^{(i−1)}(t) for t ∈ I_{n}(a, b) (i = 1, …, n) (3.10)

1See also [8], Corollary 2.1.

uniformly on each segment contained in I_{n}(a, b). Therefore from (3.5) and (3.6) we obtain

∫_{a}^{b} ∥u^{(m)}(τ)∥^{2} dτ ≤ ρ_{0}^{2},  (3.11)
∥u^{(i−1)}(t)∥ ≤ ρ_{0}σ_{im}(a, b; t) for t ∈ I_{n}(a, b)  (3.12)
(i = 1, …, m).

In view of (3.9) for arbitrary fixed s and t ∈ I_{n}(a, b) there exists a natural number j_{0} such that

u_{j}^{(n−1)}(t) − u_{j}^{(n−1)}(s) = ∫_{s}^{t} f(τ, u_{j}(τ), …, u_{j}^{(n−1)}(τ))dτ
(j = j_{0}, j_{0} + 1, …)

and s, t ∈ I_{nj}(a, b) for j ≥ j_{0}. Passing to the limit in the latter equality by j → +∞, we obtain

u^{(n−1)}(t) − u^{(n−1)}(s) = ∫_{s}^{t} f(τ, u(τ), …, u^{(n−1)}(τ))dτ.

Therefore u is the solution of the system (1.1). Simultaneously, (3.10)-(3.12) imply that u ∈ C̃^{n−1,m}(I_{n}(a, b); ℝ^{l}) and satisfies the boundary conditions (1.2).

Theorem 1.1′ immediately follows from Theorem 1.1, since in the case where f(t, x_{1}, …, x_{n}) ≡ f(t, x_{1}, …, x_{m}) and f ∈ K_{loc}(I_{n}(a, b) × ℝ^{ml}; ℝ^{l}), inequality (1.5) is fulfilled automatically and the function q(t, x_{1}, …, x_{m}) ≡ ∥f(t, x_{1}, …, x_{m})∥ satisfies the condition (1.6).

Proof of Theorem 1.3. (1.13) and (1.14) yield the conditions (1.4′) and (1.7), where h(t) = |f(t, 0, …, 0)|. Therefore by virtue of Theorem 1.1′ the problem (1.1′), (1.2) is solvable in the class C̃^{n−1,m}(I_{n}(a, b); ℝ^{l}).

To complete the proof of the theorem it remains for us to verify that the problem under consideration has at most one solution in the class C̃^{n−1,m}(I_{n}(a, b); ℝ^{l}).

Let u, ū ∈ C̃^{n−1,m}(I_{n}(a, b); ℝ^{l}) be two arbitrary solutions of the problem (1.1′), (1.2). We set v(t) = u(t) − ū(t) for t ∈ I_{n}(a, b).

It is clear that v ∈ C̃^{n−1,m}(I_{n}(a, b); ℝ^{l}) and v(a+) = ⋯ = v^{(m−1)}(a+) = 0 , v(b−) = ⋯ = v^{(n−m−1)}(b−) = 0.

On the other hand, by the condition (1.14) from the equality

v^{(n)}(t) = f(t, u(t), …, u^{(n−1)}(t)) − f(t, ū(t), …, ū^{(n−1)}(t))

we have

(−1)^{n−m−1}S(v(t))v^{(n)}(t) ≥ − ∑_{i=1}^{m} H_{i}(t)|v^{(i−1)}(t)|.

Therefore due to Lemma 2.2 v(t) ≡ 0, i.e., u(t) ≡ ū(t).

Theorems 1.2 and 1.2′ are proved similarly to Theorems 1.1 and 1.1′, while Theorem 1.4 is proved similarly to Theorem 1.3 with the only difference being that Lemma 2.4 is used instead of Lemma 2.2.

References

1. R. Conti, Equazioni differenziali oridinarie quasilineari con condizioni lineari. Ann. Mat. Pura Appl. 57(1962), 49-61.

2. I. T. Kiguradze, Some singular boundary value problems for ordinary differential equations. (Russian) Tbilisi University Press, Tbilisi, 1975.

3. —–, On some singular boundary value problems for ordinary differential equations. Equadiff, 5. Proc. Czech. Conf. Diff. Equations and Applications, 174-178, Teubner Verlag, Leipzig, 1982.

4. I. T. Kiguradze, On solvability of boundary value problems of de la Vallèe Poussin. (Russian) Differentsial’nie Uravneniya 21(1985), No. 3, 391-398.

5. —–, On boundary value problems for high order ordinary differential equations with singularities. (Russian) Uspekhi Mat. Nauk 41(1986), No. 4, 166-167.

6. —–, On two-point singular boundary value problems for nonlinear ordinary differential equations. (Russian) Proceedings of All-Union symposium on current problems of mathematical physics, Vol. 1 (Russian), 276-279, Tbilisi University Press, Tbilisi, 1987.

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

8. I. T. Kiguradze, Boundary value problems for systems of ordinary differential equations. (Russian) Current problems in mathematics. Newest results, vol. 30 (Russian), 3-103, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vses. Inst. Nauchn. i Tekhn. Inform., Moscow, 1987.

(Received 23.09.1992)

Authors’ address:

A.Razmadze Mathematical Institute

Georgian Academy of Sciences

1 Rukhadze St., Tbilisi 380093

Republic of Georgia