GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 2, 1995, 165-188

ON SOME BOUNDARY VALUE PROBLEMS WITH INTEGRAL CONDITIONS FOR FUNCTIONAL

DIFFERENTIAL EQUATIONS

I. KIGURADZE AND D. CHICHUA

Abstract. For the functional differential equation u^{(n)}(t) = f(u)(t) we have established the sufficient conditions for solvability and unique solvability of the boundary value problems

u^{(i)}(0) = c_{i} (i = 0, …, m − 1), ∫_{0}^{+∞} |u^{(m)}(t)|^{2}dt < +∞

and

u^{(i)}(0) = c_{i} (i = 0, …, m − 1),
∫_{0}^{+∞} t^{2j}|u^{(j)}(t)|^{2}dt < +∞ (j = 0, …, m),

where n ≥ 2, m is the integer part of \frac{n}{2} , c_{i} ∈ R, and f is the continuous operator acting from the space of (n − 1)-times continuously differentiable functions given on an interval [0, +∞[ into the space of locally Lebesgue integrable functions.

§ 1. FORMULATION OF THE EXISTENCE AND UNIQUENESS THEOREMS

Let n ≥ 2 and f be a continuous operator acting from the space of (n−1)-times continuously differentiable functions given on an interval R_{+} = [0, +∞[ into the space of locally Lebesgue integrable functions given on the same interval. Consider the functional differential equation

u^{(n)}(t) = f(u)(t)  (1.1)

by whose solution we shall understand a function u : R_{+} → R which is locally absolutely continuous with its derivatives up to order n − 1 inclusive

1991 Mathematics Subject Classification. 34K10. Key words and phrases. Functional differential equation, boundary value problem,

integral condition.

165

1072-947X/95/0300-0165$07.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1995 Plenum Publishing Corporation

and satisfies (1.1) almost everywhere on R_{+}. In this paper we shall be concerned with the problems of the existence and uniqueness of a solution of equation (1.1) satisfying either of the two boundary conditions

u^{(i)}(0) = c_{i} (i = 0, …, m − 1), ∫_{0}^{+∞} |u^{(m)}(t)|^{2}dt < +∞ (1.2)

and

\begin{matrix}u^{(i)}(0) = c_{i} (i = 0, …, m − 1), \\ ∫_{0}^{+∞} t^{2j}|u^{(j)}(t)|^{2}dt < +∞ (j = 0, …, m). \end{matrix}  (1.3)

For the case f(u)(t) = g(t, u(t), …, u^{(n−1)}(t)) problems of type (1.1), (1.2) and (1.1), (1.3), as well as their closely related problems of the existence of so-called proper oscillatory and vanishing-at-infinity solutions of the equation

u^{(n)} = g(t, u(t), …, u^{(n−1)}(t)),

have been studied with a sufficient thoroughness (see [1,2,3] and §§4 and 14 in the monograph [4]). As to the general case, the above-mentioned problems were not previously investigated.

In this paper we establish the sufficient conditions for the solvability and unique solvability of problems (1.1), (1.2) and (1.1), (1.3). In [5] these results are specified for a differential equation with deviating arguments of the form

u^{(n)}(t) = g(t, u(τ_{0}(t)), …, u^{(m−1)}(τ_{m−1}(t))),

and criteria are found for the existence of a multiparameter family of vanishing-at-infinity proper oscillatory solutions of the above equation.

The following notation will be used throughout the paper.

C^{n−1}([t_{1}, t_{2}]) and L([t_{1}, t_{2}]) are respectively the spaces of (n − 1)-times continuously differentiable and Lebesgue integrable real functions given on the segment [t_{1}, t_{2}].

C̃^{n−1} is the space of functions u : R_{+} → R which are locally absolutely continuous (i.e., absolutely continuous on each finite interval from R_{+}) together with their derivatives up to order n − 1 inclusive.

C^{n−1} is a topological space of (n−1)-times continuously differentiable real functions given on R_{+}, where by the convergence of the sequence (u_{k})_{k=1}^{+∞} we understand the uniform convergence of sequences (u_{k}^{(i)})_{k=1}^{+∞} (i = 0, …, n−1) on each finite interval from R_{+}.

C_{0}^{n−1,m} = {u ∈ C^{n−1} : ∫_{0}^{+∞} |u^{(m)}(t)|^{2}dt < +∞},

C^{n−1,m} = {u ∈ C^{n−1} : ∫_{0}^{+∞} t^{2j}|u^{(i)}|^{2}dt < +∞ (i = 0, …, m)}.

If u ∈ C_{0}^{n−1,m}, then

∥u∥_{0,m} = [ ∑_{i=0}^{m−1} |u^{(i)}(0)|^{2} + ∫_{0}^{+∞} |u^{(m)}(s)|^{2}ds]^{1/2};

if, however, u ∈ C^{n−1,m}, then

∥u∥_{m} = [ ∫_{0}^{+∞}(1 + s)^{2m}|u^{(m)}(s)|^{2}ds]^{1/2}.

L is the space of locally Lebesgue integrable functions v : R_{+} → R with the topology of convergence in the mean on each finite interval from R_{+}.

μ_{i}^{k} (i = 0, 1, …; k = 2i, 2i + 1, …) are the numbers given by the recurrent relations

μ_{0}^{i+1} = \frac{1}{2}, μ_{i}^{2i} = 1, μ_{i+1}^{k} = μ_{i+1}^{k−1} + μ_{i}^{k−2} (k = 2i + 3, …).

γ_{n} = 0 for n ≤ 3,

γ_{n} = ∑_{j=0}^{m_{0}−1} \frac{n!}{(2m − 2 − 4j)!}μ_{m−1−2j}^{n} for n ≥ 4.

where m_{0} is the integer part of the number \frac{n}{4}.

In the sequel it will always be assumed that f : C^{n−1} → L is a continuous operator.

Theorem 1.1. Let for any u ∈ C_{0}^{n−1,m} the inequalities

(−1)^{n−m−1}u(t)f(u)(t) ≥ −a_{1}(t)∥u∥_{0,m}^{2} − a_{2}(t),  (1.4)
|f(u)(t)| ≤ b(t, |u(t)|, ∥u∥_{0,m})  (1.5)

hold almost everywhere on R_{+}, where a_{i} : R_{+} → R_{+} (i = 1, 2) are measurable functions such that

\begin{matrix}∫_{0}^{+∞}(1 + t)^{n−2m}a_{1}(t)dt < μ_{m}^{n}, \\ ∫_{0}^{+∞}(1 + t)^{n−2m}a_{2}(t)dt < +∞, \end{matrix}  (1.6)

and the function b : R_{+}^{3} → R_{+} is locally summable with respect to the first argument, nondecreasing with respect to the last two arguments and

lim_{\begin{matrix}t→0 \\ y→+∞ \end{matrix}} (y^{−2} ∫_{0}^{t} b(s, x, y)ds) = 0 for x ∈ R_{+}.  (1.7)

Then problem (1.1), (1.2) has at least one solution.

Theorem 1.2. Let for any u and \overline{u} ∈ C_{0}^{n−1,m} the inequality

(−1)^{n−m−1}(u(t) − \overline{u}(t))(f(u)(t) − f(\overline{u})(t)) ≥ −a(t)∥u − \overline{u}∥_{0,m}^{2} (1.8)

hold almost everywhere on R_{+}, where a : R_{+} → R_{+} is a measurable function such that

∫_{0}^{+∞}(1 + t)^{n−2m}a(t)dt < μ_{m}^{n}.  (1.9)

Then problem (1.1), (1.2) has at most one solution.

Theorem 1.3. Let for any u ∈ C^{n−1,m} the inequalities

(−1)^{n−m−1}u(t)f(u)(t) ≥
≥ γ(1 + t)^{−n}|u(t)|^{2} − a_{1}(t)∥u∥_{m}^{2} − a_{2}(t),  (1.10)
|f(u)(t)| ≤ b(t, |u(t)|, ∥u∥_{m})  (1.11)

hold almost everywhere on R_{+}, where γ and a_{i} : R_{+} → R_{+} (i = 1, 2) are a positive number and measurable functions such that

\begin{matrix}∫_{0}^{+∞}(1 + t)^{n}a_{i}(t)dt < +∞ (i = 1, 2), \\ δ = \frac{n!}{(2m)!}μ_{m}^{n} − ∫_{0}^{+∞}(1 + t)^{n}a_{1}(t)dt > 0, \end{matrix}  (1.12)

γ + (−1)^{m} \frac{n!}{2} >

> \frac{m − 1}{4}γ_{n}[\frac{γ_{n}}{δ} + \frac{(m − 2)(4m^{2} − m + 3)}{3} + 4]^{m−1},  (1.13)

and the function b : R_{+}^{3} → R_{+} is locally summable with respect to the first argument, nondecreasing with respect to the last two arguments, and satisfies condition (1.7). Then problem (1.1), (1.3) has at least one solution.

Theorem 1.4. Let for any u and \overline{u} ∈ C^{n−1,m} the inequality

(−1)^{n−m−1}(u(t) − \overline{u}(t))(f(u)(t) − f(\overline{u})(t)) ≥
≥ γ(1 + t)^{−n}(u(t) − \overline{u}(t))^{2} − a(t)∥u − \overline{u}∥_{m}^{2}  (1.14)

hold almost everywhere on R_{+}, where γ and a : R_{+} → R_{+} are a positive number and a measurable function such that

δ = \frac{n!}{(2m)!}μ_{m}^{n} − ∫_{0}^{+∞}(1 + t)^{n}a(t)dt > 0  (1.15)

and inequality (1.13) is fulfilled. Then problem (1.1), (1.3) has at most one solution.

§ 2. AUXILIARY STATEMENTS

2.1. Operator σ_{k}. For an arbitrary i ∈ {0, …, n − 1} we denote by v_{i} the polynomial of degree not higher than 2n − 1, satisfying the boundary conditions

v_{i}^{(j)}(0) = δ_{ij}, v_{i}^{(j)}(1) = 0 (j = 0, …, n − 1),

where δ_{ij} is the Kronecker symbol. Let v^{∗} be the maximal value among maxima of functions |v_{i}^{(j)}| (i, j = 0, …, n − 1) on the segment [0, 1]. For any natural k and function u ∈ C^{n−1}([0, k]) we set

ε_{k}(u) = (k + 2)^{−n}[1 + ∑_{j=0}^{n−1} |u^{(j)}(k)|]^{−2}(v^{∗})^{−2},

σ_{k}(u)(t) = \{ \begin{matrix}u(t) for 0 ≤ t ≤ k \\ ∑_{i=0}^{n−1} ε_{k}^{i}(u)v_{i}(\frac{t − k}{ε_{k}(u)})u^{(i)}(k) for k < t < k + ε_{k}(u) \\ 0 for t ≥ k + ε_{k}(u) \end{matrix} .

Lemma 2.1. For any natural k the operator σ_{k} : C^{n−1}([0, k]) → C^{n−1} is continuous and for any function u ∈ C^{n−1}([0, k]) satisfying the conditions

u^{(i)}(k) = 0 (i = 0, …, m − 1)  (2.1)

we have

0 ≤ ∥σ_{k}(u)∥_{0,m}^{2} − ∑_{i=0}^{m−1} |u^{(i)}(0)|^{2} − ∫_{0}^{k} |u^{(m)}(s)|^{2}ds ≤ 1,  (2.2)

0 ≤ ∥σ_{k}(u)∥_{m}^{2} − ∫_{0}^{k}(1 + s)^{2m}|u^{(m)}(s)|^{2}ds ≤ 1.  (2.3)

Proof. The continuity of the operator σ_{k} is obvious. We shall prove the validity of inequality (2.2). In view of (2.1)

0 ≤ ∥σ_{k}(u)∥_{0,m}^{2} − ( ∑_{i=0}^{m−1} |u^{(i)}(0)|^{2}) − ∫_{0}^{k} |u^{(m)}(s)|^{2}ds =

= ∫_{k}^{k+ε_{k}(u)} [ ∑_{i=m}^{n−1} ε_{k}^{i−m}(u)v_{i}^{(m)}(\frac{s − k}{ε_{k}(u)})u^{(i)}(k)]^{2}ds ≤

≤ ε_{k}(u)(v^{∗})^{2}[ ∑_{i=m}^{n−1} |u^{(i)}(k)|]^{2} < 1.

Inequality (2.3) is proved likewise.

2.2. Lemmas on A Priori Estimates. In this subsection we shall derive a priori estimates of the function u ∈ C̃^{n−1} which for some natural k satisfy either of the two systems of differential inequalities

|u^{(n)}(t)| ≤ b_{0}(t, |u(t)|, ∥σ_{k}(u)∥_{0,m}), (−1)^{n−m−1}u^{(n)}(t)u(t) ≥
≥ −a_{1}(t)∥σ_{k}(u)∥_{0,m}^{2} − a_{2}(t) for 0 ≤ t ≤ k  (2.4)

and

|u^{(n)}(t)| ≤ b_{0}(t, |u(t)|, ∥σ_{k}(u)∥_{m}), (−1)^{n−m−1}u^{(n)}(t)u(t) ≥
≥ γ(1 + t)^{−n}|u(t)|^{2} − a_{1}(t)∥σ_{k}(u)∥_{m}^{2} − a_{2}(t) for 0 ≤ t ≤ k (2.5)

and the boundary conditions

\begin{matrix}|u^{(i)}(0)| ≤ ρ_{0} (i = 0, …, m − 1), \\ u^{(j)}(k) = 0 (j = 0, …, n − m − 1), \end{matrix}  (2.6)

where ρ_{0} > 0 and m is the integer part of the number \frac{n}{2}.

Along with the above-mentioned numbers μ_{i}^{k} we shall introduce the numbers μ_{ij}^{k} (i = 0, 1, …; j = i, i + 1, …; k = i + j + 1, i + j + 2, …) using the following recurrent relations:

μ_{00}^{k} = \frac{1}{2}, μ_{0j}^{k} = 1 (k = 1, 2, …; j = 1, …, k − 1),

μ_{ii}^{2i+1} = \frac{1}{2}, μk_{i k−i−1} = 1 (i = 1, 2, …; k = 2i + 2, 2i + 3, …),

μ_{ij}^{k} = μ_{ij}^{k−1} + μ_{i−1 j−1}^{k−2}

(i = 1, 2, …; j = i, i + 1, …; k = i + j + 2, i + j + 3, …).

The following three lemmas are proved in [4] (see Lemmas 4.1-4.3 and 5.1).

Lemma 2.2. If the functions u and w : [0, t_{0}] → R are absolutely continuous together with their derivatives up to order n − 1 inclusive, then

∫_{0}^{t_{0}} w(t)u(t)u^{(n)}(t)dt =

= ∑_{i=0}^{n−m−1} ∑_{j=i}^{n−1−i}(−1)^{n−1−j}μ_{ij}^{n}(w^{(n−1−i−j)}(t_{0})u^{(i)}(t_{0})u^{(j)}(t_{0}) −

−w^{(n−1−i−j)}(0)u^{(i)}(0)u^{(j)}(0)) +

+ ∑_{i=0}^{m}(−1)^{n−i}μ_{i}^{n} ∫_{0}^{t_{0}} w^{(n−2i)}(t)|u^{(i)}(t)|^{2}dt.

Lemma 2.3. Let m ≥ 2, t_{0} > 0, r_{0} ≥ 0 and the function u : [0, t_{0}] → R be m-times continuously differentiable and

(1 + t_{0})^{2i}u^{(i)}(t_{0})u^{(i−1)}(t_{0}) − i(1 + t_{0})^{2i−1}|u^{(i−1)}(t_{0})|^{2} −
−u^{(i)}(0)u^{(i−1)}(0) + i|u^{(i−1)}(0)|^{2} ≤ r_{0} (i = 1, …, m − 1).

Then for any η > \frac{1}{3}(m − 2)(4m^{2} − m + 3) the estimates

∫_{0}^{t_{0}}(1 + t)^{2i}|u^{(i)}(t)|^{2}dt ≤ 2η^{m−2}r_{0} + α_{i}(η) ∫_{0}^{t_{0}} |u(t)|^{2}dt +
+β_{i}(η) ∫_{0}^{t_{0}}(1 + t)^{2m}|u^{(m)}(t)|^{2}dt (i = 1, …, m − 1),

hold, where

\begin{matrix}α_{1}(η) = (m − 1)(1 + \frac{η}{4} ), \\ α_{i}(η) = (m − i)(1 + \frac{η}{4} ) ∏_{j=1}^{i−1} (η − \frac{(j − 1)(4j^{2} + 7j + 6)}{3}) \\ (i = 2, …, m − 1), \\ β_{i}(η) = ∏_{j=i}^{m−1} (η − \frac{(j − 1)(4j^{2} + 7j + 6)}{3})^{−1} (i = 1, …, m − 1). \end{matrix} (2.7)

Lemma 2.4. Let δ > 0 and the function u : [0, δ] → R be i-times continuously differentiable. Then there exists a point t^{∗} ∈ [0, δ] such that

|u^{(i)}(t^{∗})| ≤ (i + 1)!(2i + 1)^{i+ \frac{1}{2}} δ^{− \frac{1}{2} −i}( ∫_{0}^{δ} |u(t)|^{2}dt)^{\frac{1}{2}} .

Lemma 2.5. Let r_{1} > 0 and the function b_{0} : [0, 1] × R_{+}^{2} → R_{+} be summable with respect to the first argument, nondecreasing with respect to the last two arguments and

lim_{\begin{matrix}t→0 \\ y→+∞ \end{matrix}} (y^{−2} ∫_{0}^{t} b_{0}(s, x, y)ds) = 0 for x ∈ R_{+}.  (2.8)

Then there exists a positive number r_{2} such that any function u : [0, 1] → R, absolutely continuous together with its derivatives up to order n−1 inclusive and satisfying the inequalities

∫_{0}^{1} |u^{(n)}(s)u(s)|ds + ∫_{0}^{1} |u^{(m)}(s)|^{2}ds ≤ r_{1}(1 + ∑_{i=m}^{n−1} |u^{(i)}(0)|), (2.9)

\begin{matrix}|u^{(n)}(t)| ≤ b_{0}(t, |u(t)|, r1^{\frac{1}{2}} (1 + ∑_{i=m}^{n−1} |u^{(i)}(0)|)^{\frac{1}{2}} ) \\ for 0 ≤ t ≤ 1, \end{matrix}  (2.10)

admits the estimate

|u^{(i)}(0)| ≤ r_{2} (i = m, …, n − 1).  (2.11)

Proof. By virtue of condition (2.8) there exist numbers δ ∈]0, 1] and y_{0} ∈ R_{+} such that

∫_{0}^{δ} b_{0}(s, 8nr_{1}, y)ds ≤ \frac{1}{8nr_{1}} y^{2} for y ≥ y_{0},

and therefore

∫_{0}^{δ} b_{0}(s, 8nr_{1}, y)ds ≤
≤ ∫_{0}^{δ} b_{0}(s, 8nr_{1}, y_{0})ds + \frac{1}{8nr_{1}} y^{2} for y ≥ 0.  (2.12)

According to Lemma 2.4 there exist point t_{i} ∈ [0, δ] (i = m, …, n − 1) such that

|u^{(i)}(t_{i})| ≤ (i − m + 1)!(2i − 2m + 1)^{i−m+ \frac{1}{2}} δ^{− \frac{1}{2} −i+m}( ∫_{0}^{δ} |u^{(m)}(s)|^{2}ds)^{\frac{1}{2}}

(i = m, …, n − 1).

Therefore from the equalities

u^{(i)}(t) = u^{(i)}(t_{i}) + ∫_{t_{i}}^{t} u^{(i+1)}(s)ds (i = m, …, n − 1)

we find

|u^{(i)}(t)| ≤ ∑_{j=i}^{n−1} δ^{j−i}|u^{(j)}(t_{j})| + δ^{n−1−i} ∫_{0}^{δ} |u^{(n)}(s)|ds ≤

≤ n^{n}δ^{−n}( ∫_{0}^{1} |u^{(m)}(s)|^{2}ds)^{\frac{1}{2}} + ∫_{0}^{δ} |u^{(n)}(s)|ds
for 0 ≤ t ≤ δ (i = m, …, n − 1).

Hence by (2.9) it follows that

∑_{i=m}^{n−1} |u^{(i)}(0)| ≤ n^{1+n}δ^{−n}( ∫_{0}^{1} |u^{(m)}(s)|^{2}ds)^{\frac{1}{2}} + n ∫_{0}^{δ} |u^{(n)}(s)|ds ≤

≤ n^{2+2n}δ^{−2n}r_{1} + \frac{1}{4r_{1}} ∫_{0}^{1} |u^{(m)}(s)|^{2}ds + n ∫_{0}^{δ} |u^{(n)}(s)|ds ≤

≤ n^{2+2n}δ^{−2n}r_{1} + \frac{1}{4}(1 + ∑_{i=m}^{n−1} |u^{(i)}(0)|) + n ∫_{0}^{δ} |u^{(n)}(s)|ds. (2.13)

Let I_{1} and I_{2} be respectively the set of points of the interval [0, δ] at which the inequalities |u(t)| ≤ 8nr_{1} and |u(t)| > 8nr_{1} hold. Then on account of inequalities (2.9), (2.10), and (2.12) we have

∫_{0}^{δ} |u^{(n)}(s)|ds = ∫_{I_{1}} |u^{(n)}(s)|ds + ∫_{I_{2}} |u^{(n)}(s)|ds ≤

≤ ∫_{I_{1}} b_{0}(s, 8nr_{1}, r1^{\frac{1}{2}} (1 + ∑_{i=m}^{n−1} |u^{(i)}(0)|)^{\frac{1}{2}} )ds + \frac{1}{8nr_{1}} ∫_{I_{2}} |u^{(n)}(s)u(s)|ds ≤

≤ ∫_{0}^{δ} b_{0}(s, 8nr_{1}, y_{0})ds + \frac{1}{4n}(1 + ∑_{i=m}^{n−1} |u^{(i)}(0)|).

Taking the latter inequality into account, from (2.13) we obtain

∑_{i=m}^{n−1} |u^{(i)}(0)| ≤ n^{2+2n}δ^{−2n}r_{1} + n ∫_{0}^{δ} b_{0}(s, 8nr_{1}, y_{0})ds + \frac{1}{2}(1 + ∑_{i=m}^{n−1} |u^{(i)}(0)|).

This immediately implies estimates (2.11), where

r_{2} = 2n^{2+2n}δ^{−2n}r_{1} + 2n ∫_{0}^{δ} b_{0}(s, 8nr_{1}, y_{0})ds + 1

is the positive number independent of u.

Lemma 2.6. Let ρ_{0} > 0, the functions a_{i} : R_{+} → R_{+} (i = 1, 2) be measurable and satisfy conditions (1.6), and the function b_{0} : R_{+}^{3} → R_{+} be locally summable with respect to the first argument, nondecreasing with respect to the last two arguments, and satisfies condition (2.8). Then there exists a positive number r such that any function u ∈ C̃^{n−1} satisfying for some natural k conditions (2.4) and (2.6) admits the estimates

∫_{0}^{k} |u^{(m)}(s)|^{2}ds + ∫_{0}^{k}(1 + s)^{n−2m}|u^{(n)}(s)u(s)|ds ≤ r,  (2.14)
|u^{(i)}(t)| ≤ r(1 + t)^{m−i− \frac{1}{2}} for 0 ≤ t ≤ k (i = 0, …, m − 1), (2.15)

|u^{(i)}(t)| ≤ r(1 + t)^{n−i−1} + ∫_{0}^{t}(t − s)^{n−i−1}b_{0}(s, r(1 + s)^{m− \frac{1}{2}} , r)ds
for 0 ≤ t ≤ k (i = m, …, n − 1).  (2.16)

Proof. By virtue of (1.6)

ε = μ_{m}^{n} − ∫_{0}^{+∞}(1 + s)^{n−2m}a_{1}(s)ds > 0,  (2.17)
ρ_{1} = (1 + mρ_{0}^{2})μ_{m}^{n} + ∫_{0}^{+∞}(1 + s)^{n−2m}a_{2}(s)ds < +∞. (2.18)

For

r_{1} = (ε^{−1} + ε^{−1}μ_{m}^{n} + 2)ρ_{1} + (ε^{−1} + ε^{−1}μ_{m}^{n} + 1)mρ_{0} + 1 + m^{2}ρ_{0}^{2}

we shall choose r_{2} > 0 such that the conclusion of Lemma 2.5 is valid and put r = r_{1}(1 + r_{2} + mr_{2}) + mρ_{0}.

Let u ∈ C̃^{n−1} be an arbitrary function satisfying, for some natural k, inequalities (2.4) and (2.6). Then

(−1)^{n−m}(1 + t)^{n−2m}u^{(n)}(t)u(t) + w_{k}(t) =
= (1 + t)^{n−2m}a_{1}(t)∥σ_{k}(u)∥_{0,m}^{2} + (1 + t)^{n−2m}a_{2}(t),

where

w_{k}(t) = (1 + t)^{n−2m}|(−1)^{n−m}u^{(n)}(t)u(t) −
−a_{1}(t)∥σ_{k}(u)∥_{0,m}^{2} − a_{2}(t)|.  (2.19)

Hence due to Lemma 2.2 we find

μ_{m}^{n} ∫_{0}^{k} |u^{(m)}(s)|^{2}ds + ∫_{0}^{k} w_{k}(s)ds = l(u) +
+∥σ_{k}(u)∥_{0,m}^{2}∫_{0}^{k}(1 + s)^{n−2m}a_{1}(s)ds+∫_{0}^{k}(1 + s)^{n−2m}a_{2}(s)ds, (2.20)

where

l(u) = ∑_{i=0}^{m−1}(−1)^{m−i}u^{(i)}(0)u^{(n−1−i)}(0) for n = 2m

and

l(u) = −\frac{1}{2}|u^{(m)}(0)|^{2} + ∑_{i=0}^{m−1}(−1)^{m−i}[(i + 1)u^{(n−2−i)}(0) −

−u^{(n−1−i)}(0)]u^{(i)}(0) for n = 2m + 1.

ON SOME BOUNDARY VALUE PROBLEMS 175

By conditions (2.2) and (2.6)

∥σ_{k}(u)∥_{0,m}^{2} ≤ 1 + mρ_{0}^{2} + ∫_{0}^{k} |u^{(m)}(s)|^{2}ds  (2.21)

and

l(u) ≤ mρ_{0} ∑_{i=m}^{n−1} |u^{(i)}(0)|.

Taking into account these inequalities and conditions (1.6), (2.17), and (2.18), from (2.19) and (2.20) we obtain

∫_{0}^{k} w_{k}(s)ds ≥ ∫_{0}^{k}(1 + s)^{n−2m}|u^{(n)}(s)u(s)|ds − −∥σ_{k}(u)∥_{0,m}^{2} ∫_{0}^{k}(1 + s)^{n−2m}a_{1}(s)ds − ∫_{0}^{k}(1 + s)^{n−2m}a_{2}(s)ds ≥

≥ ∫_{0}^{k}(1 + s)^{n−2m}|u^{(n)}(s)u(s)|ds − μ_{m}^{n} ∫_{0}^{k} |u^{(m)}(s)|^{2}ds − ρ_{1}, (2.22)

μ_{m}^{n} ∫_{0}^{k} |u^{(m)}(s)|^{2}ds + ∫_{0}^{k} w_{k}(s)ds ≤ (μ_{m}^{n} − ε) ∫_{0}^{k} |u^{(m)}(s)|^{2}ds + ρ_{1} +

+mρ_{0} ∑_{i=m}^{n−1} |u^{(i)}(0)|

and

ε ∫_{0}^{k} |u^{(m)}(s)|^{2}ds + ∫_{0}^{k} w_{k}(s)ds ≤ ρ_{1} + mρ_{0} ∑_{i=m}^{n−1} |u^{(i)}(0)|. (2.23)

Since w_{k} is nonnegative, from (2.22) and (2.23) we obtain

∫_{0}^{k} |u^{(m)}(s)|^{2}ds ≤ ε^{−1}ρ_{1} + ε^{−1}mρ_{0} ∑_{i=m}^{n−1} |u^{(i)}(0)|,  (2.24)
∫_{0}^{k}(1 + s)^{n−2m}|u^{(n)}(s)u(s)|ds ≤ (ε^{−1}μ_{m}^{n} + 2)ρ_{1} +

+(1 + ε^{−1}μ_{m}^{n})mρ_{0} ∑_{i=m}^{n−1} |u^{(i)}(0)|,

∫_{0}^{k} |u^{(m)}(s)|^{2}ds + ∫_{0}^{k}(1 + s)^{n−2m}|u^{(n)}(s)u(s)|ds ≤
≤ r_{1}(1 + ∑_{i=m}^{n−1} |u^{(i)}(0)|).  (2.25)

On the other hand, using (2.21) and (2.24), from (2.4) we have

|u^{(n)}(t)|≤b_{0}(t, |u(t)|, r1^{\frac{1}{2}} (1+ ∑_{i=m}^{n−1} |u^{(i)}(0)|)^{\frac{1}{2}} ) for 0≤t≤k. (2.26)

Therefore inequalities (2.9) and (2.10) are fulfilled. Thus estimates (2.11) hold by virtue of the choice of r_{2}.

With (2.11) taken into account, inequalities (2.25) and (2.26) imply estimates (2.14) and (2.16). On the other hand, using (2.6) and (2.11), from (2.25) we find

|u^{(i)}(t)| = | ∑_{j=i}^{m−1} \frac{t^{j−i}}{(j − i)!}u^{(j)}(0) + \frac{1}{(m − 1 − i)!} ∫_{0}^{t}(t − s)^{m−1−i}u^{(m)}(s)ds| ≤

≤ mρ_{0}(1 + t)^{m−1−i} + [ ∫_{0}^{t}(t − s)^{2m−2−2i}ds]^{\frac{1}{2}} [ ∫_{0}^{t} |u^{(m)}(s)|^{2}ds]^{\frac{1}{2}} ≤

≤ r(1 + t)^{m−i− \frac{1}{2}} for 0 ≤ t ≤ k (i = 0, …, m − 1).

Therefore estimates (2.15) hold as well.

Lemma 2.7. Let γ and a_{i} : R_{+} → R_{+} (i = 1, 2) be a positive constant and measurable functions, satisfying conditions (1.12) and (1.13), ρ_{0} > 0, and b_{0} : R_{+}^{3} → R_{+} be locally summable with respect to the first argument, nondecreasing with respect to the last two arguments, and satisfies condition (2.8). Then there is a positive number r such that any function u ∈ C̃^{n−1}, satisfying for some natural k conditions (2.5) and (2.6), admits estimates

∑_{i=0}^{m} ∫_{0}^{k}(1 + s)^{2i}|u^{(i)}(s)|^{2}ds + ∫_{0}^{k}(1 + s)^{n}|u^{(n)}(s)u(s)|ds ≤ r, (2.27)
|u^{(i)}(t)| ≤ r(1 + t)^{−i− \frac{1}{2}} for 0 ≤ t ≤ k (i = 0, …, m − 1), (2.28)
|u^{(i)}(t)|≤r(1 + t)^{n−i−1}+∫_{0}^{t}(t − s)^{n−i−1}b_{0}(s, r(1 + s)^{− \frac{1}{2}} , r)ds
for 0 ≤ t ≤ k (i = m, …, n − 1).  (2.29)

Proof. According to (1.13) there exists ε ∈]0, δ[ such that

γ + (−1)^{m} \frac{n!}{2} > \frac{m − 1}{4}γ_{n}(η + 4)^{m−1} + ε,  (2.30)

where

η = \frac{γ_{n}}{δ − ε} + \frac{(m − 2)(4m^{2} − m + 3)}{3}.

ON SOME BOUNDARY VALUE PROBLEMS 177

Let

ρ_{1} = ∑_{i=1}^{2} ∫_{0}^{+∞}(1 + s)^{n}a_{i}(s)ds, ρ_{2} = (1 + ρ_{0}^{2}) ∑_{i=0}^{m−1} ∑_{j=i}^{n−1−i} \frac{n!}{(1 + i + j)!}μ_{ij}^{n},

ρ_{3} = 2mρ_{0}^{2}η^{m−2}γ_{n}, r_{1} = [ε^{−1}(1 + γ + \frac{n!}{(2m)!}μ_{m}^{n}) + 2](ρ_{1} + ρ_{2} + ρ_{3}) + 1.

Choose r_{2} > 0 such that the conclusion of Lemma 2.5 is valid and put

r = 2(m − 1)mρ_{0}^{2}η^{m−2} + r_{1}[(m − 1)^{2}(η + 4)^{m−1} + m](1 + nr_{2}). Let u ∈ C̃^{n−1} be an arbitrary function satisfying for some natural k

conditions (2.5) and (2.6). Then

(−1)^{n−m}(1 + s)^{n}u^{(n)}(s)u(s) + γ|u(s)|^{2} + w_{k}(s) =

(1 + s)^{n}a_{1}(s)∥σ_{k}(u)∥_{m}^{2} + (1 + s)^{n}a_{2}(s) for 0 ≤ s ≤ k,

where

w_{k}(s) = |(−1)^{n−m}(1 + s)^{n}u^{(n)}(s)u(s) + γ|u(s)|^{2} −
−(1 + s)^{n}a_{1}(s)∥σ_{k}(u)∥_{m}^{2} − (1 + s)^{n}a_{2}(s)|.

Hence by virtue of Lemma 2.2 we find

∑_{i=0}^{m}(−1)^{m−i} \frac{n!}{(2i)!}μ_{i}^{n} ∫_{0}^{k}(1 + s)^{2i}|u^{(i)}(s)|^{2}ds + γ ∫_{0}^{k} |u(s)|^{2}ds +

+ ∫_{0}^{k} w_{k}(s)ds = l(u) + ∥σ_{k}(u)∥_{m}^{2} ∫_{0}^{k}(1 + s)^{n}a_{1}(s)ds + ∫_{0}^{k}(1 + s)^{n}a_{2}(s)ds, where

l(u) = ∑_{i=0}^{n−m−1} ∑_{j=i}^{n−1−i}(−1)^{m−1−j} \frac{n!}{(i + j + 1)!} μ_{ij}^{n}u^{(i)}(0)u^{(j)}(0).

If we set c_{m}(u) = 0 for m = 1,

c_{m}(u) = ∑_{i=1}^{m−1}(−1)^{m−i−1} \frac{n!}{(2i)!}μ_{i}^{n} ∫_{0}^{k}(1 + s)^{2i}|u^{(i)}(s)|^{2}ds for m > 1,

then the latter equality can be rewritten as

\frac{n!}{(2m)!}μ_{m}^{n} ∫_{0}^{k}(1 + s)^{2m}|u^{(m)}(s)|^{2}ds + [γ + (−1)^{m} \frac{n!}{2}] ∫_{0}^{k} |u(s)|^{2}ds +

+ ∫_{0}^{k} w_{k}(s)ds = c_{m}(u) + l(u) + ∥σ_{k}(u)∥_{m}^{2} ∫_{0}^{k}(1 + s)^{n}a_{1}(s)ds +

178 I. KIGURADZE AND D. CHICHUA

+ ∫_{0}^{k}(1 + s)^{n}a_{2}(s)ds.  (2.31)

By conditions (1.12), (2.3), and (2.6)

∥σ_{k}(u)∥_{m}^{2} ∫_{0}^{k}(1 + s)^{n}a_{1}(s)ds + ∫_{0}^{k}(1 + s)^{n}a_{2}(s)ds ≤

≤ (\frac{n!}{(2m)!}μ_{m}^{n} − δ) ∫_{0}^{k}(1 + s)^{2m}|u^{(m)}(s)|^{2}ds + ρ_{1},  (2.32)

l(u) ≤ ∑_{i=0}^{m−1} ∑_{j=i}^{n−1−i}(−1)^{m−1−j} \frac{n!}{(1 + i + j)!} μ_{ij}^{n}u^{(i)}(0)u^{(j)}(0) ≤

≤ ∑_{i=0}^{m−1} [ρ_{0}^{2} ∑_{j=i}^{m−1} \frac{n!}{(1 + i + j)!} μ_{ij}^{n} + ρ_{0} ∑_{j=m}^{n−1−i} \frac{n!}{(1 + i + j)!}μ_{ij}^{n}|u^{(j)}(0)|] ≤

≤ ρ_{2}[1 + ∑_{i=m}^{n−1} |u^{(i)}(0)|].  (2.33)

If m > 1, then on account of Lemma 2.3 we have

c_{m}(u)≤ ∑_{j=0}^{m_{0}−1} \frac{n!}{(2m − 2 − 4j)!}μ_{m−1−2j}^{n}∫_{0}^{k}(1+s)^{2m−2−4j}|u^{(m−1−2j)}(s)|^{2}ds≤

≤ ρ_{3} + ∑_{j=0}^{m_{0}−1} \frac{n!}{(2m − 2 − 4j)!}μ_{m−1−2j}^{n}α_{m−1−2j}(η) ∫_{0}^{k} |u(s)|^{2}ds +

+ ∑_{j=0}^{m_{0}−1} \frac{n!}{(2m − 2 − 4j)!}μ_{m−1−2j}^{n}β_{m−1−2j}(η) ∫_{0}^{k}(1 + s)^{2m}|u^{(m)}(s)|^{2}ds.

On the other hand, by virtue of the inequality

η > 1 + \frac{(m − 2)(4m^{2} − m + 3)}{3}

(2.7) implies clearly that

α_{i}(η) ≤ \frac{m − 1}{4}(η + 4)^{m−1},

β_{i}(η) ≤ [η − \frac{(m − 2)(4m^{2} − m + 3)}{3}]^{−1} = \frac{δ − ε}{γ_{n}}  (2.34)
(i = 1, …, m − 1).

Therefore

c_{m}(u) ≤ρ_{3}+\frac{m − 1}{4}(η +4)^{m−1}γ_{n} ∫_{0}^{k} |u(s)|^{2}ds+(δ−ε)∫_{0}^{k}(1+s)^{2m}|u^{(m)}(s)|^{2}ds.

If along with this inequality we take into consideration inequalities (2.30), (2.32), and (2.33), then from (2.31) we have

ε[ ∫_{0}^{k}(1 + s)^{2m}|u^{(m)}(s)|^{2}ds + ∫_{0}^{k} |u(s)|^{2}ds] + ∫_{0}^{k} w_{k}(s)ds ≤

≤ (ρ_{1} + ρ_{2} + ρ_{3})[1 + ∑_{i=m}^{n−1} |u^{(i)}(0)|].

Therefore

∫_{0}^{k}(1 + s)^{2m}|u^{(m)}(s)|^{2}ds + ∫_{0}^{k} |u(s)|^{2}ds ≤

≤ ε^{−1}(ρ_{1} + ρ_{2} + ρ_{3})[1 + ∑_{i=m}^{n−1} |u^{(i)}(0)|],  (2.35)

∫_{0}^{k}(1 + s)^{n}|u^{(n)}(s)u(s)|ds ≤ ∫_{0}^{k} w_{k}(s)ds + γ ∫_{0}^{k} |u(s)|^{2}ds +

+[1+∫_{0}^{k}(1+s)^{2m}|u^{(m)}(s)|^{2}ds] ∫_{0}^{k}(1+s)^{n}a_{1}(s)ds+∫_{0}^{k}(1+s)^{n}a_{2}(s)ds≤

≤ ρ_{1} + [ε^{−1}(γ + \frac{n!}{(2m)!}μ_{m}^{n}) + 1](ρ_{1} + ρ_{2} + ρ_{3})[1 + ∑_{i=m}^{n−1} |u^{(i)}(0)|],

∫_{0}^{k}(1 + s)^{2m}|u^{(m)}(s)|^{2}ds + ∫_{0}^{k} |u(s)|^{2}ds +

+ ∫_{0}^{k}(1 + s)^{n}|u^{(n)}(s)u(s)|ds ≤ r_{1}[1 + ∑_{i=m}^{n−1} |u^{(i)}(0)|].  (2.36)

On the other hand, using (2.3) and (2.35), from (2.5) we obtain inequality (2.26). Therefore inequalities (2.9) and (2.10) are fulfilled. Thus estimates (2.11) hold by virtue of the choice of r_{2}.

By inequalities (2.11), (2.34), (2.36) and Lemma 2.3 we have

∫_{0}^{k}(1 + s)^{2m}|u^{(m)}(s)|^{2}ds + ∫_{0}^{k} |u(s)|^{2}ds +
+ ∫_{0}^{k}(1 + s)^{n}|u^{(n)}(s)u(s)|ds ≤ r_{1}(1 + nr_{2}),

∑_{i=1}^{m−1} ∫_{0}^{k}(1 + s)^{2i}|u^{(i)}(s)|^{2}ds ≤ 2m(m − 1)ρ_{0}^{2}η^{m−2} +

+ ∑_{i=1}^{m−1} α_{i}(η) ∫_{0}^{k} |u(s)|^{2}ds + ∑_{i=1}^{m−1} β_{i}(η) ∫_{0}^{k}(1 + s)^{2m}|u^{(m)}(s)|^{2}ds ≤

≤ 2m(m − 1)ρ_{0}^{2}η^{m−2} + (m − 1)[(m − 1)(η + 4)^{m−1} + 1] ×

×[ ∫_{0}^{k}(1 + s)^{2m}|u^{(m)}(s)|^{2}ds + ∫_{0}^{k} |u(s)|^{2}ds] ≤

≤ 2m(m − 1)ρ_{0}^{2}η^{m−2} + r_{1}[(m − 1)^{2}(η + 4)^{m−1} + m − 1](1 + nr_{2}).

Therefore estimate (2.27) is valid. In view of (2.6)

|u^{(i)}(t)| = | ∫_{t}^{k} u^{(i+1)}(s)ds| ≤
≤ | ∫_{t}^{k}(1 + s)^{−2i−2}ds|^{\frac{1}{2}} [ ∫_{t}^{k}(1 + s)^{2i+2}|u^{(i+1)}(s)|^{2}ds]^{\frac{1}{2}} ≤
≤ (1 + t)^{−i− \frac{1}{2}} [ ∫_{t}^{k}(1 + s)^{2i+2}|u^{(i+1)}(s)|^{2}ds]^{\frac{1}{2}}
for 0 ≤ t ≤ k (i = 0, …, m − 1).

Hence, with (2.27) taken into account, we obtain estimates (2.28). As to estimates (2.29), they follow from inequalities (2.10), (2.11), and (2.28).

2.3. Some Properties of Functions from the Classes C_{0}^{n−1,m} and C^{n−1,m}.

Lemma 2.8. If u ∈ C_{0}^{n−1,m}, then

lim_{t→+∞} t^{i+ \frac{1}{2} −m}u^{(i)}(t) = 0 (i = 0, …, m − 1)  (2.37)

and for any constants c_{ij} (i = 0, …, n − m − 1; j = i, …, n − 1 − i) the function

w(t) = ∑_{i=0}^{n−m−1} ∑_{j=i}^{n−1−i} c_{ij}t^{i+j+1−2m}u^{(i)}(t)u^{(j)}(t)

satisfies the condition

lim inf_{t→+∞} |w(t)| = 0.  (2.38)

Lemma 2.9. If u ∈ C^{n−1,m}, then

lim_{t→+∞} t^{i+ \frac{1}{2}} u^{(i)}(t) = 0 (i = 0, …, m − 1)  (2.39)

and for any constants c_{ij} (i = 0, …, n − m − 1; j = i, …, n − 1 − i) the function

w(t) = ∑_{i=0}^{n−m−1} ∑_{j=i}^{n−1−i} c_{ij}t^{i+j+1}u^{(i)}(t)u^{(j)}(t)

satisfies condition (2.38).

Lemma 2.10. If m ≥ 2, r_{0} ≥ 0, u ∈ C^{n−1,m},

i|u^{(i−1)}(0)|^{2} − u^{(i)}(0)u^{(i−1)}(0) ≤ r_{0} (i = 1, …, m − 1),

then for any η > \frac{1}{3}(m − 2)(4m^{2} − m + 3) we have the estimates

∫_{0}^{+∞}(1 + t)^{2i}|u^{(i)}(t)|^{2}dt ≤ 2η^{m−2}r_{0} + α_{i}(η) ∫_{0}^{+∞} |u(t)|^{2}dt +
+β_{i}(η) ∫_{0}^{+∞}(1 + t)^{2m}|u^{(m)}(t)|^{2}dt (i = 1, …, m − 1).

These lemmas follow immediately from Lemmas 4.3-4.5 in the monograph [4].

2.4. Lemma on the Solvability of an Auxiliary Two-Point Boundary Value Problem. Let t_{0} ∈ ]0, +∞[, c_{i} ∈ R (i = 0, …, m − 1), \overline{c}_{j} ∈ R (j = 0, …, n − m − 1), p : [0, t_{0}] → R be a summable function and q : C^{n−1}([0, t_{0}]) → L([0, t_{0}]) be a continuous operator. Consider the boundary value problem

u^{(n)}(t) = p(t)u(t) + q(u)(t),  (2.40)
\begin{matrix}u^{(i)}(0) = c_{i} (i = 0, …, m − 1), \\ u^{(j)}(t_{0}) = \overline{c}_{j} (j = 0, …, n − m − 1). \end{matrix}  (2.41)

Lemma 2.11. Let (−1)^{n−m−1}p(t) ≥ 0 for 0 ≤ t ≤ t_{0}, and let there exist a summable function q^{∗} : [0, t_{0}] → R_{+} such that the inequality

|q(u)(t)| ≤ q^{∗}(t) for 0 ≤ t ≤ t_{0}  (2.42)

holds for any function u ∈ C^{n−1}([0, t_{0}]) satisfying the boundary conditions (2.41). Then problem (2.40), (2.41) is solvable.

Proof. We shall show in the first place that the homogeneous problem

u^{(n)}(t) = p(t)u(t),  (2.40_{0})
\begin{matrix}u^{(i)}(0) = 0 (i = 0, …, m − 1), \\ u^{(j)}(t_{0}) = 0 (j = 0, …, n − m − 1) \end{matrix}  (2.41_{0})

has only a trivial solution.

Let u be an arbitrary solution of problem (2.40_{0}), (2.41_{0}). Then, since the function (−1)^{n−m}p is nonpositive, we have

(−1)^{n−m}u^{(n)}(t)u(t) + |p(t)||u(t)|^{2} = 0.

On integrating both sides of this equality from 0 to t_{0}, by virtue of Lemma 2.2 and conditions (2.41_{0}) we obtain

μ_{m}^{n} ∫_{0}^{t_{0}} |u^{(m)}(t)|^{2}dt + ∫_{0}^{t_{0}} t^{n−2m}|p(t)||u(t)|^{2}dt + l_{0}(u) = 0,

where l_{0}(u) = 0 for n = 2m and l_{0}(u) = \frac{1}{2}|u^{(m)}(0)|^{2} for n = 2m + 1. Hence it is clear that u(t) ≡ 0.

Since problem (2.40_{0}), (2.41_{0}) has only a trivial solution, problem (2.40), (2.41) is equivalent to the integral equation

u(t) = u_{0}(t) + ∫_{0}^{t_{0}} G(t, s)q(u)(s)ds,  (2.43)

where u_{0} is the solution of the homogeneous equation (2.40_{0}) under the boundary conditions (2.41) and G is the Green function of problem (2.40_{0}), (2.41_{0}).

By Shauder’s principle [6] the continuity of the operator q : C^{n−1}([0, t_{0}]) → L([0, t_{0}]) and condition (2.42) guarantees the existence of at least one solution of the integral equation (2.43).

§ 3. PROOFS OF THE EXISTENCE AND UNIQUENESS THEOREMS

Proof of Theorem 1.1. Let

ρ_{0} = max{|c_{i}| i = 0, …, m − 1}, b_{0}(t, x, y) = b(t, x, y),

r be the number from Lemma 2.6, r_{0} = 2m^{2}ρ_{0}^{2} + 2r + 1, and

χ(s) = \{ \begin{matrix}1 for 0 ≤ s ≤ r_{0} \\ 2 − \frac{s}{r_{0}} for r_{0} < s < 2r_{0} \\ 0 for s ≥ 2r_{0} \end{matrix} .  (3.1)

For any natural k put q_{k}(u)(t) = χ(∥σ_{k}(u)∥_{0,m})f(σ_{k}(u))(t) and consider the boundary value problem

u^{(n)}(t) = q_{k}(u)(t),  (3.2)
\begin{matrix}u^{(i)}(0) = c_{i} (i = 0, …, m − 1), \\ u^{(j)}(k) = 0 (j = 0, …, n − m − 1). \end{matrix}  (3.3)

By Lemma 2.1 the continuity of the operator f : C^{n−1} → L implies the continuity of the operator q_{k} : C([0, k]) → L([0, k]). Let u ∈ C^{n−1}([0, k]) be an arbitrary function satisfying the boundary conditions (3.3). Then in view of (1.5)

|q_{k}(u)(t)| ≤ χ(∥σ_{k}(u)∥_{0,m})b(t, |u(t)|, ∥σ_{k}(u)∥_{0,m}) for 0 ≤ t ≤ k.

On the other hand,

|u(t)| = | ∑_{i=0}^{m−1} \frac{t^{i}}{i!} u^{(i)}(0) + \frac{1}{(m − 1)!} ∫_{0}^{t}(t − s)^{m−1}u^{(m)}(s)ds| ≤

≤ mρ_{0}(1 + t)^{m−1} + t^{m− \frac{1}{2}} [ ∫_{0}^{t} |u^{(m)}(s)|^{2}ds]^{\frac{1}{2}} ≤
≤ (1 + t)^{m− \frac{1}{2}} (mρ_{0} + ∥σ_{k}(u)∥_{0,m}) for 0 ≤ t ≤ k.

Therefore

|q_{k}(u)(t)| ≤ q^{∗}(t) for 0 ≤ t ≤ k,

where q^{∗}(t) = b(t, mρ_{0}+2r_{0})(1+t)^{m− \frac{1}{2}} , 2r_{0}) and q^{∗} ∈ L([0, k]). Since all the conditions of Lemma 2.11 are fulfilled for problem (3.2), (3.3), it is solvable. Let u_{k} be some solution of this problem. By inequalities (1.4) and (1.5)

(−1)^{n−m−1}u_{k}^{(n)}(t)u_{k}(t) =
= (−1)^{n−m−1}χ(∥σ_{k}(u_{k})∥_{0,m})σ_{k}(u_{k})(t)f(σ_{k}(u_{k}))(t) ≥
≥ −a_{1}(t)∥σ_{k}(u_{k})∥_{0,m}^{2} − a_{2}(t) for 0 ≤ t ≤ k,
|u_{k}^{(n)}(t)| ≤ b_{0}(t, |u_{k}(t)|, ∥σ_{k}(u_{k})∥_{0,m}) for 0 ≤ t ≤ k.  (3.4)

Therefore by virtue of Lemmas 2.1 and 2.6 we have the estimates

|u_{k}^{(i)}(t)| ≤ r(1 + t)^{m−i− \frac{1}{2}} (i = 0, …, m − 1),
|u_{k}^{(i)}(t)| ≤ r(1 + t)^{n−1−i} + ∫_{0}^{t}(t − s)^{n−1−i}b_{0}(s, r(1 + s)^{m− \frac{1}{2}} , r)ds
(i = m, …, n − 1) for 0 ≤ t ≤ k,  (3.5)
∫_{0}^{k} |u_{k}^{(m)}(s)|^{2}ds ≤ r, ∥σ_{k}(u_{k})∥_{0,m} ≤ r_{0}.  (3.6)

Let us extend u_{k} to the entire R_{+} using the equality

u_{k}(t) = σ_{k}(u_{k})(t) for t ≥ k.  (3.7)

Then due to (3.1) and (3.6) we have

u_{k}^{(n)}(t) = f(u_{k})(t) for 0 ≤ t ≤ k.  (3.8)

From (3.4)–(3.6) it is clear that the sequences (u_{k}^{(i)})_{k=1}^{+∞} (i = 0, …, n−1) are uniformly bounded and equicontinuous on each finite segment of R_{+}. Therefore by the Arzella-Ascoli lemma there exists a subsequence (u_{k_{j}} )_{j=1}^{+∞}

of (u_{k})_{k=1}^{+∞} such that (u(i)_{k_{j}})_{j=1}^{+∞} (i = 0, …, n − 1) uniformly converges on each finite segment of R_{+}.

By the continuity of the operator f : C^{n−1} → L and equality (3.8) it is clear that the function u(t) = lim_{j→+∞} u_{k_{j}} (t) for t ∈ R_{+} is a solution of equation (1.1). On the other hand, from (3.3) and (3.6) it follows that u satisfies conditions (1.2).

Proof of Theorem 1.2. Let u and \overline{u} be two arbitrary solutions of problem (1.1), (1.2). Putting u_{0}(t) = u(t) − \overline{u}(t), we obtain

u_{0}^{(i)}(0) = 0 (i = 0, …, m − 1), ∫_{0}^{+∞} |u_{0}^{(m)}(s)|^{2}ds < +∞.

On the other hand, by condition (1.8) we have

(−1)^{n−m}(1 + t)^{n−2m}u_{0}(t)u_{0}^{(n)}(t) ≤ (1 + t)^{n−2m}a(t)∥u_{0}∥_{0,m}^{2} for t ∈ R_{+}.

After integrating this inequality from 0 to t, by Lemma 2.2 we obtain

μ_{m}^{n} ∫_{0}^{t} |u_{0}^{(m)}(s)|^{2}ds ≤ w(t) + ∥u_{0}∥_{0,m}^{2} ∫_{0}^{t}(1 + s)^{n−2m}a(s)ds,

where

w(t) = (n − 2m) ∑_{i=0}^{n−m−1}(−1)^{n−m−i}(i + 1)u_{0}^{(i)}(t)u_{0}^{(n−2−i)}(t) −

−(1 + t)^{n−2m} ∑_{i=0}^{n−m−1}(−1)^{n−m−i}u_{0}^{(i)}(t)u_{0}^{(n−1−i)}(t).

However, since by Lemma 2.8 the function w satisfies condition (2.38), from the latter inequality we find

μ_{m}^{n}∥u_{0}∥_{0,m}^{2} = μ_{m}^{n} ∫_{0}^{+∞} |u_{0}^{(m)}(s)|^{2}ds ≤ ∥u_{0}∥_{0,m}^{2} ∫_{0}^{+∞}(1 + s)^{n−2m}a(s)ds.

Hence on account of (1.9) we obtain ∥u_{0}∥_{0,m} = 0, i.e., u(t) ≡ \overline{u}(t).

Proof of Theorem 1.3. Let

ρ_{0} = max{|c_{i}| : i = 0, …, m − 1}, b_{0}(t, x, y) = b(t, x, y) + γ(1 + t)^{−n}x,

r be the number from Lemma 2.7, r_{0} = r + 1, and χ be the function given by equality (3.1). For any natural k we put

q_{k}(u)(t) = χ(∥σ_{k}(u)∥_{m})[f(σ_{k}(u))(t) − (−1)^{n−m−1}γ(1 + t)^{−n}u(t)]

and consider the equation

u^{(n)}(t) = (−1)^{n−m−1}γ(1 + t)^{−n}u(t) + q_{k}(u)(t)  (3.9)

with the boundary conditions (3.3).

According to Lemma 2.1 the operator q_{k} : C^{n−1}([0, k]) → L([0, k]) is continuous. On the other hand, by (1.11), for any u ∈ C^{n−1}([0, k]) we have

|q_{k}(u)(t)| ≤ χ(∥σ_{k}(u)∥_{m})b_{0}(t, |u(t)|, ∥σ_{k}(u)∥_{m}) for 0 ≤ t ≤ k.

But

|u(t)| = |σ_{k}(u)(t)| = \frac{1}{(m − 1)!}| ∫_{t}^{+∞}(t − s)^{m−1}[σ_{k}(u)(s)]^{(m)}ds| ≤

≤ \frac{1}{(m − 1)!}| ∫_{t}^{+∞}(t − s)^{2m−2}(1 + s)^{−2m}ds|^{\frac{1}{2}} ∥σ_{k}(u)∥_{m} ≤

≤ (1 + t)^{− \frac{1}{2}} ∥σ_{k}(u)∥_{m} for 0 ≤ t ≤ k.

Therefore

|q_{k}(u)(t)| ≤ q^{∗}(t) for 0 ≤ t ≤ k,

where q^{∗}(t) = b_{0}(t, 2r_{0}(1 + t)^{− \frac{1}{2}} , 2r_{0}) and q^{∗} ∈ L([0, k]). Thus all the conditions of Lemma 2.11 are fulfilled for problem (3.9), (3.3). Therefore it has at least one solution. Let u_{k} be some solution of this problem. Then due to inequalities (1.10) and (1.11) we shall have

(−1)^{n−m−1}u_{k}(t)u_{k}^{(n)}(t) = γ(1 + t)^{−n}[1 − χ(∥σ_{k}(u_{k})∥_{m})]|u_{k}(t)|^{2} +

+(−1)^{n−m−1}χ(∥σ_{k}(u_{k})∥_{m})σ_{k}(u_{k})(t)f(σ_{k}(u_{k})(t)) ≥
≥ γ(1 + t)^{−n}|u_{k}(t)|^{2} − a_{1}(t)∥σ_{k}(u_{k})∥_{m}^{2} − a_{2}(t) for 0 ≤ t ≤ k,
|u_{k}^{(n)}(t)| ≤ γ(1 + t)^{−n}|u_{k}(t)| + b(t, |u_{k}(t)|, ∥σ_{k}(u_{k})∥_{m}) =
= b_{0}(t, |u_{k}(t)|, ∥σ_{k}(u_{k})∥_{m}) for 0 ≤ t ≤ k.  (3.10)

Thus by Lemmas 2.1 and 2.7 we have the estimates

|u_{k}^{(i)}(t)| ≤ r(1 + t)^{−i− \frac{1}{2}} (i = 0, …, m − 1),

|u_{k}^{(i)}(t)|≤r(1 + t)^{n−1−i}+∫_{0}^{t}(t − s)^{n−i−1}b_{0}(s, r(1+s)^{− \frac{1}{2}} , r)ds
(i = m, …, n − 1) for 0 ≤ t ≤ k,  (3.11)
∑_{i=0}^{m} ∫_{0}^{k}(1 + s)^{2i}|u_{k}^{(i)}(s)|^{2}ds ≤ r, ∥σ_{k}(u_{k})∥_{m} ≤ r_{0}.  (3.12)

Let us extend u_{k} to the entire R_{+} using equality (3.7). Then identity (3.8) will be fulfilled by (3.1) and (3.12).

According to estimates (3.10)–(3.12) the sequences (u_{k}^{(i)})_{k=1}^{+∞} (i=0, ..., n− 1) are uniformly bounded and equicontinuous on each finite segment of R_{+}. Therefore there exists a subsequence (u_{k_{j}} )_{j=1}^{+∞} of (u_{k})_{k=1}^{+∞} such that (u(i)_{k_{j}})_{j=1}^{+∞} (i = 0, …, n − 1) uniformly converges on each finite segment of R_{+}.

Since the operator f : C^{n−1} → L is continuous, from conditions (3.3), (3.8), and (3.12) it follows that the function

u(t) = lim_{j→+∞} u_{k_{j}} (t) for t ∈ R_{+}

is a solution of problem (1.1), (1.3).

Proof of Theorem 1.4. Let u and \overline{u} be two arbitrary solutions of problem (1.1), (1.3). If we put u_{0}(t) = u(t) − \overline{u}(t), then

u_{0}^{(i)}(0) = 0 (i = 0, …, m−1), ∫_{0}^{+∞} t^{2j}|u_{0}^{(j)}(t)|^{2}dt < +∞ (j = 0, …, m).

On the other hand by condition (1.14) we have

(−1)^{n−m}(1 + t)^{n}u_{0}(t)u_{0}^{(n)}(t) + γ|u_{0}(t)|^{2} ≤ (1 + t)^{n}a(t)∥u_{0}∥_{m}^{2} for t ∈ R_{+}.

After integrating this inequality from 0 to t, by Lemma 2.2 we obtain

∑_{i=0}^{m}(−1)^{m−i} \frac{n!}{(2i)!}μ_{i}^{n} ∫_{0}^{t}(1 + s)^{2i}|u_{0}^{(i)}(s)|^{2}ds +

+γ ∫_{0}^{t} |u_{0}(s)|^{2}ds ≤ w(t) + ∥u_{0}∥_{m}^{2} ∫_{0}^{t}(1 + s)^{n}a(s)ds for t ∈ R_{+},

where

w(t) = ∑_{i=0}^{n−m−1} ∑_{j=i}^{n−1−i}(−1)^{m−j} \frac{n!}{(1 + i + j)!}μ_{ij}^{n}(1 + t)^{1+i+j}u^{(i)}(t)u^{(j)}(t).

But by Lemma 2.9 the function w satisfies condition (2.38). Taking this fact and condition (1.15) into account, from the latter inequality we find

δ∥u_{0}∥_{m}^{2} + [γ + (−1)^{m} \frac{n!}{2}] ∫_{0}^{+∞} |u_{0}(s)|^{2}ds ≤ c_{m}(u_{0}),  (3.13)

where

c_{m}(u_{0}) = 0 for m = 1,

c_{m}(u_{0}) = ∑_{i=1}^{m−1}(−1)^{m−i−1} \frac{n!}{(2i)!}μ_{i}^{n} ∫_{0}^{+∞}(1 + s)^{2i}|u_{0}^{(i)}(s)|^{2}ds for m > 1.

In view of (1.13), for a sufficiently small ε ∈ ]0, δ[ inequality (2.30) is fulfilled, where

η = \frac{γ_{n}}{δ − ε} + \frac{(m − 2)(4m^{2} − m + 3)}{3}.

If m > 1, then by virtue of Lemma 2.10 we have

c_{m}(u_{0}) ≤

≤ ∑_{j=0}^{m_{0}−1} \frac{n!}{(2m − 2 − 4j)!}μ_{m−1−2j}^{n} ∫_{0}^{+∞}(1 + s)^{2m−2−4j}|u_{0}^{(m−1−2j)}(s)|^{2}ds ≤

≤ ∑_{j=0}^{m_{0}−1} \frac{n!}{(2m − 2 − 4j)!}μ_{m−1−2j}^{n}α_{m−1−2j}(η) ∫_{0}^{+∞} |u_{0}(s)|^{2}ds +

+ ∑_{j=0}^{m_{0}−1} \frac{n!}{(2m − 2 − 4j)!}μ_{m−1−2j}^{n}β_{m−1−2j}(η)∥u_{0}∥_{m}^{2} ≤

≤ \frac{m − 1}{4}(η + 4)^{m−1}γ_{n} ∫_{0}^{+∞} |u_{0}(s)|^{2}ds + (δ − ε)∥u_{0}∥_{m}^{2}.

If along with this estimate we take into account inequality (2.30), then from (3.13) we obtain

∥u_{0}∥_{m}^{2} + ∫_{0}^{+∞} |u_{0}(s)|^{2}ds ≤ 0.

Hence it is clear that u_{0}(t) ≡ 0, i.e., u(t) ≡ \overline{u}(t).

References

1. I. T. Kiguradze, On some singular boundary value problems for ordinary differential equations. Equadiff 5. Proc. 5 Czech. Conf. Differ. Equations and Appl., 174-178, Teubner Verlagsgesellschaft, Leipzig, 1982.

2. I. T. Kiguradze, On vanishing-at-infinity of solutions of ordinary differential equations. Czechosl. Math. J. 33(1983), No.4, 613-646.

3. I. T. Kiguradze, On a boundary value problem with the condition at infinity for ordinary differential equations of higher orders. (Russian) Partial Differential Equations and their Applications. Proc. All-Union. Symp. in Tbilisi, April 2 − 23, 1982, 91-105. Tbilisi University Press, Tbilisi, 1986.

4. I. T. Kiguradze and T. A. Chanturia, Asymptotic properties of solutions of nonautonomous ordinary differential equations. Kluwer Academie Publishers, Dordrecht, Boston, London, 1993.

5. I. Kiguradze and D. Chichua, On proper oscillatory and vanishing-at-infinity solutions of differential equations with derivating argument. Georgian Math. J. 2(1995), No. 4 (to appear).

6. L. V. Kantorovich and G. P. Akilov, Functional analysis. (Russian) Nauka, Moscow, 1977.

(Received 08.12.1993)

Authors’ addresses:

I. Kiguradze A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, Rukhadze St., Tbilisi 380093 Republic of Georgia

D. Chichua I. Vekua Institute of Applied Mathematics Tbilisi State University 2, University St., Tbilisi 380043 Republic of Georgia