Georgian Mathematical Journal 1(94), No. 4, 429-458

ON THE MODIFIED BOUNDARY VALUE PROBLEM OF DE LA VALLÉE-POUSSIN FOR NONLINEAR ORDINARY

DIFFERENTIAL EQUATIONS

G. TSKHOVREBADZE

Abstract. The sufficient conditions of the existence, uniqueness, and correctness of the solution of the modified boundary value problem of de la Vallée-Poussin have been found for a nonlinear ordinary differential equation

u^{(n)} = f(t, u, u′, …, u^{(n−1)}),

where the function f has nonitegrable singularities with respect to the first argument.

§ 1. Statement of the Main Results

In this paper for an ordinary differential equation

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

we shall consider the boundary value problem

u^{(k−1)}(t_{i}) = 0 (k = 1, …, n_{i}; i = 1, …, m),^{1}  (1.2_{1})
sup {(t − a)^{l−1−λ_{1}}(b − t)^{l−1−λ_{2}}|u^{(l−1)}(t)| : a < t < b} < +∞, (1.2_{2})

where n ≥ 2, l ∈ {1, …, n}, m ∈ {2, …, n}, −∞ < a = t_{1} < ⋯ < t_{m} = b < +∞, n_{i} ∈ {1, …, n − 1} (i = 1, …, m), ∑_{i=1}^{m} n_{i} = n, λ_{1} ∈]n_{1} − 1, n_{1}[, λ_{2} ∈]n_{m} − 1, n_{m}[, I_{m} = [a, b]\{t_{1}, …, t_{m}} and the function f : I_{m} × ℝ^{n} → ℝ satisfies the Caratheodory conditions on each compactum contained in I_{m} × ℝ^{n}.

Problem (1.1), (1.2_{1}) is the well-known boundary value problem of de la Vallée-Poussin and has been studied with sufficient thoroughness both when f is integrable with respect to the first argument on [a, b] (see, for example,

[2] and references from [5]) and when f has nonintegrable singularities at the 429 points t_{1}, …, t_{m} (see, for example, [5]–[7]). However, in the works devoted to the study of problem (1.1), (1.2_{1}) it is assumed that

1991 Mathematics Subject Classification. 34B15.

1Here u^{(i)}(a) (u^{(i)}(b)) denotes the right (left) limit of the function u^{(i)} at the point a (b).

1072-947X/94/0700-0429$12.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1994 Plenum Publishing Corporation

∫_{a}^{b}(t − a)^{n−n_{1}−1}(b − t)^{n−n_{m}−1}f_{r}^{∗}(t)dt < +∞ for r > 0, (1.3)

where

f_{r}^{∗}(t)=max {|f(t, ∏_{i=1}^{m} |t − t_{i}|^{n_{i1}}x_{1}, …, ∏_{i=1}^{m} |t − t_{i}|^{n_{in}}x_{n})|:∑_{k=1}^{n} |x_{k}| ≤ r},

n_{ik} = \begin{cases}n_{i} − k + 1 & for k ≤ n_{i}, \\ 0 & for k > n_{i}. \end{cases}

This assumption is not casual. The fact is that if condition (1.3) is not fulfilled, then problem (1.1), (1.2_{1}) is not, generally speaking, uniquely solvable even in the simplest case. For example, given the boundary condition (1.2_{1}), the equation

u^{(n)} = \frac{(−1)^{n}δ}{(t − a)^{n}} u

has an infinite number of solutions for n_{1} = 1 and any sufficiently small δ > 0.

Therefore, for the solution to be unique, we have to introduce an additional and, of course, natural condition such as, for example, (1.2_{2}). This condition is natural because if (1.3) is fulfilled, then (1.2_{1}) yields (1.2_{2}), i.e., problem (1.1), (1.2_{1}), (1.2_{2}) coincides with the problem of de la Vallée-Poussin (1.1), (1.2_{1}). However, if (1.3) is not fulfilled, then, as follows from the above example, this is not so.

Problem (1.1), (1.2_{1}), (1.2_{2}) is the generalization of the boundary value problem of de la Vallée-Poussin (1.1), (1.2_{1}) and has been studied in [14]^{2} for the linear differential equation

u^{(n)} = ∑_{k=1}^{l} p_{k}(t)u^{(k−1)} + q(t),  (1.4)

where p_{k} : I_{m} → ℝ (k = 1, …, l), q :]a, b[→ ℝ.

In this paper we shall establish the sufficient conditions for the existence, uniqueness, and correctness of the solution of problem (1.1), (1.2_{1}), (1.2_{2}). Note that the solution of this problem is sought for in the class of functions u :]a, b[→ ℝ absolutely continuous with u^{(k)} (k = 1, …, n − 1) inside ]a, b[.^{3} The following notation will be used:

2See also [10]–[13].

3That is, on each segment contained in ]a, b[.

ν_{kl} = |(k − 1 − λ) ⋯ (l − 2 − λ)| (k = 1, …, l − 1), ν_{ll}(λ) = 1;

σ_{k,λ_{1},λ_{2}}(t) = (t − a)^{λ_{1}−k+1}(b − t)^{λ_{2}−k+1} ∏_{i=2}^{m−1} |t − t_{i}|^{n_{ik}};^{4}

σ_{k,n}(t) = (t − a)^{n−k}(b − t)^{n−k} ∏_{i=2}^{m−1} |t − t_{i}|^{n_{ik}};

ℝ =] − ∞, +∞[; ℝ_{+} = [0, +∞[;

ℝ^{p}, where p is a natural number, is a p-dimensional Euclidean space; ℂ_{loc}^{n−1}(]a, b[; ℝ) is a set of functions v :]a, b[→ ℝ which are continuous,

with v^{(k)} (k = 1, …, n − 1), inside ]a, b[;

ℂ̃_{loc}^{n−1}(]a, b[; ℝ) is a set of functions v :]a, b[→ ℝ which are absolutely continuous, with v^{(k)} (k = 1, …, n − 1), inside ]a, b[;

L([a, b]; I), where I ⊂ ℝ, is a set of functions v : [a, b] → I which are Lebesgue integrable on [a, b];

L_{loc}(]a, b[; I) is a set of functions v :]a, b[→ I which are Lebesgue integrable inside ]a, b[;

L_{α,β}(]a, b[; I), where α ≥ 0, β ≥ 0, is a set of measurable functions v :]a, b[→ I satisfying the condition

|v(·)|_{α,β} = sup \{(t − a)^{α}(b − t)^{β}| ∫_{\frac{a+b}{2}}^{t} v(τ)dτ| : a < t < b\} < +∞;

𝕂_{loc}(I × ℝ^{p}; ℝ), where I ∈]a, b[ is a measurable set and p a natural number, is a set of functions g : I × ℝ^{p} → ℝ satisfying the Caratheodory conditions on each compactum contained in I × ℝ^{p};

𝕂_{loc}^{0}(I × ℝ^{p}; ℝ) is a set of functions g : I × ℝ^{p} → ℝ such that g(·, x_{1}(·), …, x_{p}(·)) : I → ℝ is a measurable function for any continuous vector function (x_{1}, …, x_{p}) : I → ℝ^{p};

D_{λ_{1},λ_{2}}(u_{0}; r), where r ∈ ℝ_{+}, u_{0} ∈ ℂ_{loc}^{n−1}(]a, b[; ℝ) is a set of vectors (x_{1}, …, x_{l}) ∈ ℝ^{l} satisfying the condition

inf \{ ∑_{k=1}^{l} |x_{k} − \frac{u_{0}^{(k−1)}(t)}{σ_{k,λ_{1},λ_{2}}(t)}| : t ∈ I_{m}\} ≤ r;

4∏_{i=2}^{m−1} |t − t_{i}|^{n_{ik}} will denote unity when m = 2.

W_{λ_{1},λ_{2}}(u_{0}; r) is a set of functions u ∈ ℂ_{loc}^{n−1}(]a, b[; ℝ) satisfying the condition

∑_{k=1}^{l} \frac{|u^{(k−1)}(t) − u_{0}^{(k−1)}(t)|}{σ_{k,λ_{1},λ_{2}}(t)} ≤ r for a < t < b;

M([α, β] × ℝ_{+}; ℝ_{+}), where a ≤ α < β ≤ b, is a set of functions ω ∈ 𝕂_{loc}([α, β] × ℝ_{+}; ℝ_{+}) which are nondecreasing with respect to the second argument and satisfy the condition ω(t, 0) = 0 for α ≤ t ≤ β.

Throughout the paper it will be assumed that f ∈ 𝕂_{loc}(I_{m} × ℝ^{n}; ℝ) and the solution of problem (1.1), (1.2_{1}), (1.2_{2}) will be sought for in the class ℂ̃_{loc}^{n−1}(]a, b[; ℝ).

Some definitions will be given.

Definition 1.1. Let n_{0} ∈ {1, …, n − 1} and λ ∈]n_{0} − 1, n_{0}[. A vector function (h_{1}, …, h_{l}) with measurable components h_{k} :]a, b[→ ℝ (k = 1, …, l) is said to belong to the set S^{+}(a, b; n, n_{0}; λ) (S^{−}(a, b; n, n_{0}; λ)) if there exists α ∈]a, b[ such that we have the inequality

lim sup_{t→a} \frac{(t − a)^{l−1−λ}}{(n − l)!} ∑_{k=1}^{l} \frac{1}{ν_{kl}(λ)} ∫_{t}^{α}(τ − t)^{n−l}(τ − a)^{λ−k+1}|h_{k}(τ)|dτ < 1

[ lim sup_{t→b} \frac{(b − t)^{l−1−λ}}{(n − l)!} ∑_{k=1}^{l} \frac{1}{ν_{kl}(λ)} ∫_{α}^{t}(t − τ)^{n−l}(b − τ)^{λ−k+1}|h_{k}(τ)|dτ < 1]

in the case l ∈ {n_{0} + 1, …, n} and the inequality

lim sup_{t→a} \frac{(t − a)^{l−1−λ}}{(n − n_{0} − 1)!(n_{0} − l)!} ×

×∑_{k=1}^{l} \frac{1}{ν_{kl}(λ)}∫_{a}^{t}(t−s)^{n_{0}−l} ∫_{s}^{α}(τ −s)^{n−n_{0}−1}(τ −a)^{λ−k+1}|h_{k}(τ)|dτds<1

[ lim sup_{t→b} \frac{(b − t)^{l−1−λ}}{(n − n_{0} − 1)!(n_{0} − l)!} ×

×∑_{k=1}^{l} \frac{1}{ν_{kl}(λ)} ∫_{t}^{b}(s−t)^{n_{0}−l}∫_{α}^{s}(s−τ)^{n−n_{0}−1}(b−τ)^{λ−k+1}|h_{k}(τ)|dτds<1]

in the case l ∈ {1, …, n_{0}}.

Definition 1.2. Let

σ_{k,λ_{1},λ_{2}}(·)p_{jk}(·) ∈ L_{loc}(]a, b[; ℝ) (j = 1, 2; k = 1, …, l),
p_{1k}(t) ≤ p_{2k}(t) for a < t < b (k = 1, …, l),
(p_{1}^{∗}, …, p_{l}^{∗}) ∈ S^{+}(a, b; n, n_{1}; λ_{1}) ∩ S^{−}(a, b; n, n_{m}; λ_{2}),

where p_{k}^{∗}(t) = max{|p_{1k}(t)|, |p_{2k}(t)|} (k = 1, …, l) and, moreover, under the boundary conditions (1.2_{1}), (1.2_{2}) the differential equation

u^{(n)} = ∑_{k=1}^{l} p_{k}(t)u^{(k−1)}  (1.4_{0})

haa no nontrivial solution no matter what the measurable functions p_{k}: ]a, b[→ ℝ (k = 1, …, l) satisfying the inequalities

p_{1k}(t) ≤ p_{k}(t) ≤ p_{2k}(t) for a < t < b (k = 1, …, l)  (1.5)

are. Then the vector function (p_{11}, …, p_{1l}; p_{21}, …, p_{2l}) is said to belong to the class V (t_{1}, …, t_{m}; n_{1}, …, n_{m}; λ_{1}, λ_{2}).

1.1.Existence and Uniqueness Theorems

The General Case.

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

|f(t, x_{1}, …, x_{n}) − ∑_{k=1}^{l} p_{k}(t, x_{1}, …, x_{n})x_{k} − q_{0}(t)| ≤

≤ q(t, ∑_{k=1}^{l} \frac{|x_{k}|}{σ_{k,λ_{1},λ_{2}}(t)})  (1.6)

and

p_{1k}(t) ≤ p_{k}(t, x_{1}, …, x_{n}) ≤ p_{2k}(t) (k = 1, …, l),  (1.7)

where

(p_{11}, …, p_{1l}; p_{21}, …, p_{2l}) ∈ V (t_{1}, …, t_{m}; n_{1}, …, n_{m}; λ_{1}, λ_{2}),  (1.8)

p_{k} ∈ 𝕂_{loc}^{0}(I_{m} × ℝ^{n}; ℝ) (k = 1, …, l), q_{0} ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ),

the function q :]a, b[×ℝ_{+} → ℝ_{+} is nondecreasing with respect to the second argument, q(·, ϱ) ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ_{+}) for any ϱ ≥ 0, and

lim_{ϱ→+∞} \frac{|q(·, ϱ)|_{n−1−λ_{1},n_{1}−1−λ_{2}}}{ϱ} = 0.  (1.9)

Then problem (1.1), (1.2_{1}), (1.2_{2}) is solvable.

Corollary 1.1. Let the following inequality be fulfilled:

|f(t, x_{1}, …, x_{n}) − q_{0}(t)| ≤ ∑_{k=1}^{l} p_{k}(t)|x_{k}| +

+q(t, ∑_{k=1}^{l} \frac{|x_{k}|}{σ_{k,λ_{1},λ_{2}}(t)}),  (1.10)

where

q_{0} ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ),
(−p_{1}, …, −p_{l}; p_{1}, …, p_{l}) ∈ V (t_{1}, …, t_{m}; n_{1}, …, n_{m}; λ_{1}, λ_{2}), (1.11)

and the function q :]a, b[×ℝ_{+} → ℝ_{+} satisfy the conditions of Theorem 1.1. Then problem (1.1), (1.2_{1}), (1.2_{2}) is solvable.

For the case where the right-hand side of equation (1.1) is independent of the last n − l arguments, i.e.,

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

we have

Theorem 1.2. Let f(·, 0, …, 0) ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ) and the function f have partial derivatives with respect to phase arguments belonging to 𝕂_{loc}(I_{m} × ℝ^{l}; ℝ). Let, moreover, the following inequalities be fulfilled on ]a, b[×ℝ^{l}:

p_{1k}(t) ≤ \frac{∂f(t, x_{1}, …, x_{l})}{∂x_{k}} ≤ p_{2k}(t) (k = 1, …, l),  (1.12)

and (p_{11}, …, p_{1l}; p_{21}, …, p_{2l}) ∈ V (t_{1}, …, t_{m}; n_{1}, …, n_{m}; λ_{1}, λ_{2}). Then problem (1.1′), (1.2_{1}), (1.2_{2}) has unique solution.

The case l = 1.

In this subsection we shall consider the problem

u^{(n)} = f(t, u),  (1.13)
u^{(k−1)}(t_{i}) = 0 (k = 1, …, n_{i}; i = 1, …, m),  (1.14_{1})
sup{(t − a)^{−λ_{1}}(b − t)^{−λ_{2}}|u(t)| : a < t < b} < +∞,  (1.14_{2})

assuming that f ∈ 𝕂_{loc}(I_{m} × ℝ; ℝ). For any r ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ) we introduce the notation

ϱ_{λ_{1},λ_{2}}(r) = vrai max \{\frac{|u_{0}(r)(t)|}{σ_{1,λ_{1},λ_{2}}(t)} : a < t < b\},

where u_{0}(r)(·) is the unique solution of the equation u^{(n)} = r(t) satisfying the boundary conditions (1.14_{1}), (1.14_{2}) (see, for example, Proposition 2.3).

Theorem 1.3. Let the following inequality be fulfilled on ]a, b[×ℝ:

|f(t, x) − q_{0}(t)| ≤ \frac{r(t)}{σ_{1,λ_{1},λ_{2}}(t)}|x| + q(t, \frac{|x|}{σ_{1,λ_{1},λ_{2}}(t)}),  (1.15)

where

q_{0}, r ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ),

and let the function q :]a, b[×ℝ_{+} → ℝ_{+} be nondecreasing with respect to the second argument, q(·, ϱ) ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ_{+}) for any ϱ ≥ 0 and

lim_{ϱ→+∞} \frac{|q(·, ϱ)|_{n−1−λ_{1},n−1−λ_{2}}}{ϱ} = 0.

Let, moreover,

ϱ_{λ_{1},λ_{2}}(r) < 1.  (1.16)

Then problem (1.13), (1.14_{1}), (1.14_{2}) is solvable.

Corollary 1.2. Let m = 2 and the following inequality be fulfilled on ]a, b[×ℝ:

|f(t, x) − q_{0}(t)| ≤ \frac{r_{0}}{(t − a)^{n}(b − t)^{n}} |x| + q(t, \frac{|x|}{(t − a)^{λ_{1}}(b − t)^{λ_{2}}} ),

where the functions q_{0} and q satisfy the conditions of Theorem 1.3, and the number r_{0} the inequality

r_{0} < [n_{1}!n_{2}!(λ_{1} − n_{1} + 1)(λ_{2} − n_{2} + 1)(n_{1} − λ_{1})(n_{2} − λ_{2})(b − a)^{n}] × × [2n_{1}(λ_{2} − n_{2} + 1)(n_{1} − λ_{1})(λ_{1} − n_{1} + 1 + n_{2} − λ_{2}) +

+ 2n_{2}(λ_{1} − n_{1} + 1)(n_{2} − λ_{2})(λ_{2} − n_{2} + 1 + n_{1} − λ_{1})]^{−1}.  (1.17)

Then problem (1.13), (1.14_{1}), (1.14_{2}) is solvable.

For the two-point boundary value problem

u′′ = f(t, u),  (1.18)
u(a) = u(b) = 0,  (1.19_{1})
sup{(t − a)^{−λ_{1}}(b − t)^{−λ_{2}}|u(t)| : a < t < b} < +∞,  (1.19_{2})

where 0 < λ_{1}, λ_{2} < 1, and for the three-point boundary value problem

u′′′ = f(t, u),  (1.20)
u(a) = u(t_{0}) = u(b) = 0,  (1.21_{1})
sup{(t − a)^{−λ_{1}}(b − t)^{−λ_{2}}|u(t)| : a < t < b} < +∞,  (1.21_{2})

where a < t_{0} < b, 0 < λ_{1}, λ_{2} < 1, from Theorem 1.3 we obtain

Corollary 1.3. Let the following inequality be fulfilled on ]a, b[×ℝ:

|f(t, x) − q_{0}(t)| ≤ r_{0}(\frac{λ_{1}(1 − λ_{1})}{(t − a)^{2}} + \frac{2λ_{1}λ_{2}}{(t − a)(b − t)} + \frac{λ_{2}(1 − λ_{2})}{(b − t)^{2}})|x| +

+q(t, \frac{|x|}{(t − a)^{λ_{1}}(b − t)^{λ_{2}}} ),

where r_{0} ∈]0, 1[ and the functions q_{0} and q satisfy the conditions of Theorem 1.3 for n = m = 2, n_{1} = n_{2} = 1. Then problem (1.18), (1.19_{1}), (1.19_{2}) is solvable.

Corollary 1.4. Let the following inequality be fulfilled on ]a, b[×ℝ:

|f(t, x) − q_{0}(t)| ≤ r_{0}|\frac{λ_{1}(1 − λ_{1})(2 − λ_{1})}{(t − a)^{3}} − \frac{3λ_{1}(1 − λ_{1})}{(t − a)^{2}(t − t_{0})} +

+\frac{3λ_{1}λ_{2}(1 − λ_{1})}{(t − a)^{2}(b − t)} − \frac{6λ_{1}λ_{2}}{(t − a)(t − t_{0})(b − t)} − \frac{3λ_{1}λ_{2}(1 − λ_{2})}{(t − a)(b − t)^{2}} −

−\frac{3λ_{2}(1 − λ_{2})}{(t − t_{0})(b − t)^{2}} − \frac{λ_{2}(1 − λ_{2})(2 − λ_{2})}{(b − t)^{3}} | |x| +

+q(t, \frac{|x|}{(t − a)^{λ_{1}}|t − t_{0}|(b − t)^{λ_{2}}} ),

where r_{0} ∈]0, 1[ and the functions q_{0} and q satisfy the conditions of Theorem 1.3 for n = m = 3, n_{1} = n_{2} = n_{3} = 1. Then problem (1.20), (1.21_{1}), (1.21_{2}) is solvable.

Theorem 1.4. Let f(·, 0) ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ) and the following inequality be fulfilled on ]a, b[×ℝ:

|f(t, x) − f(t, y)| ≤ \frac{r(t)}{σ_{1,λ_{1},λ_{2}}(t)}|x − y|,  (1.22)

where the function r ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ) satisfies condition (1.16). Then problem (1.13), (1.14_{1}), (1.14_{2}) has the unique solution u and

vrai max \{\frac{|u_{j}(t) − u(t)|}{σ_{1,λ_{1},λ_{2}}(t)} : a < t < b\} → 0 for j → +∞, (1.23)

where u_{0}(t) ≡ 0, and for each natural number j the function u_{j} is a solution of the equation

u_{j}^{(n)}(t) = f(t, u_{j−1}(t)),  (1.24)

satisfying the boundary conditions (1.14_{1}), (1.14_{2}).

Corollary 1.5. Let m = 2, f(·, 0) ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ) and the following inequality be fulfilled on ]a, b[×ℝ:

|f(t, x) − f(t, y)| ≤ \frac{r_{0}}{(t − a)^{n}(b − t)^{n}} |x − y|,

where r_{0} is a number satisfying (1.17). Then problem (1.13), (1.14_{1}), (1.14_{2}) has the unique solution u and

vrai max \{\frac{|u_{j}(t) − u(t)|}{(t − a)^{λ_{1}}(b − t)^{λ_{2}}} : a < t < b\} → 0 for j → +∞,

where u_{0}(t) ≡ 0, and for each natural number j the function u_{j} is a solution of problem (1.24), (1.14_{1}), (1.14_{2}).

Corollary 1.6. Let f(·, 0) ∈ L_{1−λ_{1},1−λ_{2}}(]a, b[; ℝ) and the following inequality be fulfilled on ]a, b[×ℝ:

|f(t, x) − f(t, y)| ≤ r_{0}(\frac{λ_{1}(1 − λ_{1})}{(t − a)^{2}} + \frac{2λ_{1}λ_{2}}{(t − a)(b − t)} + \frac{λ_{2}(1 − λ_{2})}{(b − t)^{2}})|x − y|,

where r_{0} ∈]0, 1[. Then problem (1.18), (1.19_{1}), (1.19_{2}) has the unique solution u and

vrai max \{\frac{|u_{j}(t) − u(t)|}{(t − a)^{λ_{1}}(b − t)^{λ_{2}}} : a < t < b\} → 0 for j → +∞,

where u_{0}(t) ≡ 0, and for each natural number j the function u_{j} is a solution of the equation u′′_{j}(t) = f(t, u_{j−1}(t)), satisfying the boundary conditions (1.19_{1}), (1.19_{2}).

Corollary 1.7. Let f(·, 0) ∈ L_{2−λ_{1},2−λ_{2}}(]a, b[; ℝ) and the following inequality be fulfilled on ]a, b[×ℝ:

|f(t, x) − f(t, y)| ≤ r_{0}|\frac{λ_{1}(1 − λ_{1})(2 − λ_{1})}{(t − a)^{3}} − \frac{3λ_{1}(1 − λ_{1})}{(t − a)^{2}(t − t_{0})} +

+\frac{3λ_{1}λ_{2}(1 − λ_{1})}{(t − a)^{2}(b − t)} − \frac{6λ_{1}λ_{2}}{(t − a)(t − t_{0})(b − t)} − \frac{3λ_{1}λ_{2}(1 − λ_{2})}{(t − a)(b − t)^{2}} −

−\frac{3λ_{2}(1 − λ_{2})}{(t − t_{0})(b − t)^{2}} − \frac{λ_{2}(1 − λ_{2})(2 − λ_{2})}{(b − t)^{3}} | |x − y|,

where r_{0} ∈]0, 1[. Then problem (1.20), (1.21_{1}), (1.21_{2}) has the unique solution u and

vrai max \{\frac{|u_{j}(t) − u(t)|}{(t − a)^{λ_{1}}|t − t_{0}|(b − t)^{λ_{2}}} : a < t < b\} → 0 for j → +∞,

where u_{0}(t) ≡ 0, and for each natural number j the function u_{j} is a solution of the equation u′′′_{j}(t) = f(t, u_{j−1}(t)), satisfying the boundary conditions (1.21_{1}), (1.21_{2}).

Remark 1.1. In Corollaries 1.3, 1.4, 1.6, and 1.7 the condition

r_{0} ∈]0, 1[  (1.25)

cannot be replaced by the equality

r_{0} = 1.  (1.26)

1.2.Continuous Dependence of Solutions of the Right-Hand Side of the Equation

In this subsection we shall consider the boundary value problem

u^{(n)} = f(t, u, u′, …, u^{(l−1)}),  (1.1′)
u^{(k−1)}(t_{i}) = 0 (k = 1, …, n_{i}; i = 1, …, m),  (1.1′)
sup {(t − a)^{l−1−λ_{1}}(b − t)^{l−1−λ_{2}}|u^{(l−1)}(t)| : a < t < b} < +∞, (1.2_{2})

assuming that f, \frac{∂f}{∂x_{k}} ∈ 𝕂_{loc}(I_{m} × ℝ^{l}; ℝ) (k = 1, …, l), and give the sufficient conditions for its solutions to be stable with respect to small perturbations of the right-hand side of equation (1.1′).

Definition 1.3. Let u_{0} be a solution of problem (1.1′), (1.2_{1}), (1.2_{2}) and r be a positive number. It will be said that u_{0} is r-stable with respect to small perturbations of the right-hand side of equation (1.1′) if for any ε ∈]0, r[, α ∈]a, t_{2}[, β ∈]t_{m−1}, b[, (x_{10}, …, x_{l0}) ∈ D_{λ_{1},λ_{2}}(u_{0}; r) and ω ∈ M([α, β] × ℝ_{+}; ℝ_{+}) there exists δ > 0 such that for any function η ∈ 𝕂_{loc}(I_{m} × ℝ^{l}; ℝ) satisfying the conditions

| ∫_{α}^{t} η(τ, σ_{1,λ_{1},λ_{2}}(τ)x_{1}, …, σ_{l,λ_{1},λ_{2}}(τ)x_{l})dτ| ≤ δ

for α ≤ t ≤ β, (x_{1}, …, x_{l}) ∈ D_{λ_{1},λ_{2}}(u_{0}; r),

|η(t, σ_{1,λ_{1},λ_{2}}(t)x_{1}, …, σ_{l,λ_{1},λ_{2}}(t)x_{l}) −

−η(t, σ_{1,λ_{1},λ_{2}}(t)y_{1}, …, σ_{l,λ_{1},λ_{2}}(t)y_{l})| ≤ ω(t, ∑_{k=1}^{l} |x_{k} − y_{k}|)

for α ≤ t ≤ β, (x_{1}, …, x_{l}) and (y_{1}, …, y_{l}) ∈ D_{λ_{1},λ_{2}}(u_{0}; r),

(t − a)^{n−1−λ_{1}}[ ∫_{t}^{α} η_{λ_{1},λ_{2}}^{∗}(τ; u_{0}; r) dτ + | ∫_{t}^{α} η_{0}(τ)dτ|] ≤ δ

for a < t ≤ α,

(b − t)^{n−1−λ_{2}}[ ∫_{β}^{t} η_{λ_{1},λ_{2}}^{∗}(τ; u_{0}; r) dτ + | ∫_{β}^{t} η_{0}(τ)dτ|] ≤ δ

for β ≤ t < b,

where

η_{0}(t) = η(t, σ_{1,λ_{1},λ_{2}}(t)x_{10}, …, σ_{l,λ_{1},λ_{2}}(t)x_{l0}),
η_{λ_{1},λ_{2}}^{∗}(t; u_{0}; r)=sup{|η(t, σ_{1,λ_{1},λ_{2}}(t)x_{1}, …, σ_{l,λ_{1},λ_{2}}(t)x_{l})−η_{0}(t)| :
(x_{1}, …, x_{l})D_{λ_{1},λ_{2}}(u_{0}, r)},

the equation u^{(n)} = f(t, u, …, u^{(l−1)}) + η(t, u, …, u^{(l−1)}) has at least one solution in W_{λ_{1},λ_{2}}(u_{0}; r), and every such solution is also contained in W_{λ_{1},λ_{2}}(u_{0}; ε).

Definition 1.4. It will be said that the solution u_{0} of problem (1.1′), (1.2_{1}), (1.2_{2}) is stable with respect to small perturbations of the right-hand side of equation (1.1′) if it is r-stable for any r > 0.

Theorem 1.5. Let u_{0} be a solution of problem (1.1′), (1.2_{1}), (1.2_{2}), r > 0, and let the following inequalities be fulfilled on ]a, b[×D_{λ_{1},λ_{2}}(u_{0}; r):

\begin{matrix}p_{1k}(t) ≤ \frac{∂f(t, σ_{1,λ_{1},λ_{2}}(t)x_{1}, …, σ_{l,λ_{1},λ_{2}}(t)x_{l})}{∂x_{k}} ≤ p_{2k}(t) \\ (k = 1, …, l), \end{matrix} (1.27)

where

(p_{11}, …, p_{1l}; p_{21}, …, p_{2l}) ∈ V (t_{1}, …, t_{m}; n_{1}, …, n_{m}; λ_{1}, λ_{2}).  (1.8)

Then u_{0} is r-stable with respect to small perturbations of the right-hand side of equation (1.1′).

Theorem 1.6. Let

f(·, 0, …, 0) ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ)

and the following inequalities be fulfilled on ]a, b[×ℝ^{l}:

p_{1k}(t) ≤ \frac{∂f(t, x_{1}, …, x_{l})}{∂x_{k}} ≤ p_{2k}(t) (k = 1, …, l),

where p_{jk} (j = 1, 2; k = 1, …, l) satisfy condition (1.8). Then problem (1.1′), (1.2_{1}), (1.2_{2}) has the unique solution u_{0} and, moreover, this solution is stable with respect to small perturbations of the right-hand side of equation (1.1′).

§ 2. Auxiliary Propositions

Proposition 2.1. Let

(p_{11}, …, p_{1l}; p_{21}, …, p_{2l}) ∈ V (t_{1}, …, t_{m}; n_{1}, …, n_{m}; λ_{1}, λ_{2}).

Then there exists a positive number ϱ_{0} such that for any q ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ) and measurable functions p_{k} :]a, b[→ ℝ (k = 1, …, l) satisfying inequalities (1.5) an arbitrary solution u of problem (1.4), (1.2_{1}), (1.2_{2}) admits the estimate

|u^{(k−1)}(t)| ≤ ϱ_{0}σ_{k,λ_{1},λ_{2}}(t)|q(·)|_{n−1−λ_{1},n−1−λ_{2}}
for a < t < b (k = 1, …, l).

Proposition 2.2. Let

σ_{k,λ_{1},λ_{2}}(·)p_{k}(·) ∈ L_{loc}(]a, b[; ℝ) (k = 1, …, l)  (2.1)

and

(p_{1}, …, p_{l}) ∈ S^{+}(a, b; n, n_{1}; λ_{1}) ∩ S^{−}(a, b; n, n_{m}; λ_{2}).  (2.2)

Then for problem (1.4), (1.2_{1}), (1.2_{2}) to be uniquely solvable for each q ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ) it is necessary and sufficient that the corresponding homogeneous problem (1.4_{0}), (1.2_{1}), (1.2_{2}) have the trivial solution only.

Proposition 2.3. Let

p_{k}(t) = \frac{g_{1k}(t)}{(t − a)^{n−k+1}} + \frac{g_{2k}(t)}{(b − t)^{n−k+1}} + p_{0k}(t) (k = 1, …, l),

where σ_{k,n}(·)p_{0k}(·) ∈ L([a, b]; ℝ) (k = 1, …, l), and g_{1k}, g_{2k} : [a, b] → ℝ (k = 1, …, l) are continuous functions satisfying the inequalities

∑_{k=1}^{l} \frac{|g_{1k}(a)|}{ν_{kl}(λ_{1})ν_{ln+1}(λ_{1})} < 1, ∑_{k=1}^{l} \frac{|g_{2k}(b)|}{ν_{kl}(λ_{2})ν_{ln+1}(λ_{2})} < 1.

Then for problem (1.4), (1.2_{1}), (1.2_{2}) to be uniquely solvable for each q ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ) it is necessary and sufficient that the corresponding homogeneous problem (1.4_{0}), (1.2_{1}), (1.2_{2}) have the trivial solution only.

In the rest of this paragraph it is assumed that p_{k} :]a, b[→ ℝ (k = 1, …, l) are the fixed functions satisfying conditions (2.1), (2.2) and problem (1.4_{0}), (1.2_{1}), (1.2_{2}) has the trivial solution only. Then by Proposition 2.2 problem (1.4), (1.2_{1}), (1.2_{2}) has the unique solution for each q ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ). The operator G : Ł_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ) → ℂ̃_{loc}^{n−1}(]a, b[; ℝ) that puts the solution u(t) = G(q)(t) of problem (1.4), (1.2_{1}), (1.2_{2}) into correspondence with each q ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ) will be called the Green operator of problem (1.4_{0}), (1.2_{1}), (1.2_{2}).

Proposition 2.4. There exists a positive number ϱ_{0} such that for any q ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ) we have the inequalities

|\frac{d^{k−1}G(q)(t)}{dt^{k−1}} | ≤ ϱ_{0}σ_{k,λ_{1},λ_{2}}(t)|q(·)|_{n−1−λ_{1},n−1−λ_{2}}

for a < t < b (k = 1, …, l)

and

|\frac{d^{n−1}G(q)(t)}{dt^{n−1}} − \frac{d^{n−1}G(q)(s)}{ds^{n−1}} | ≤ ∫_{s}^{t} p^{∗}(τ)dτ + | ∫_{s}^{t} q(τ)dτ|

for a < s ≤ t < b,

where

p^{∗}(t) = ϱ_{0}|q(·)|_{n−1−λ_{1},n−1−λ_{2}} ∑_{k=1}^{l} |p_{k}(t)|σ_{k,λ_{1},λ_{2}}(t).

Proposition 2.5. Let

q, q_{j} ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ) (j = 1, 2, …),

lim_{j→+∞} ∫_{\frac{a+b}{2}}^{t} q_{j}(τ)dτ = ∫_{\frac{a+b}{2}}^{t} q(τ)dτ uniformly inside ]a, b[

and

lim sup_{j→+∞} |q_{j}(·) − q(·)|_{n−1−λ_{1},n−1−λ_{2}} < +∞.

Then

lim_{j→+∞} \frac{d^{k−1}G(q_{j})(t)}{dt^{k−1}} = \frac{d^{k−1}G(q)(t)}{dt^{k−1}} uniformly inside ]a, b[.

The proofs of Propositions 2.1-2.5 are given in [14] ^{5} (see Lemma 1, Theorems 1 and 5, Corollaries 1 and 4).

To conclude this paragraph let us consider a quasilinear differential equation

u^{(n)} = ∑_{k=1}^{l} p_{k}(t)u^{(k−1)} + q(t, u, …, u^{(n−1)}),  (2.3)

where p_{k} : I_{m} → ℝ (k = 1, …, l) are measurable functions satisfying (2.1), (2.2),

q ∈ K_{loc}(]a, b[×ℝ^{n}; ℝ).  (2.4)

5See also [12] and [13].

Proposition 2.6. Let problem (1.4_{0}), (1.2_{1}), (1.2_{2}) have the trivial solution only. Let, moreover, there exist functions q_{0} ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ) and q^{∗} ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ_{+}) such that the inequality

|q(t, x_{1}, …, x_{n}) − q_{0}(t)| ≤ q^{∗}(t)  (2.5)

is fulfilled on ]a, b[×ℝ^{n}. Then problem (2.3), (1.2_{1}), (1.2_{2}) is solvable.

Proof. Let G be the Green operator of problem (1.4_{0}), (1.2_{1}), (1.2_{2}). By Proposition 2.4 there exists ϱ_{0} > 0 such that the inequalities

\begin{matrix}|\frac{d^{k−1}G(q̃)(t)}{dt^{k−1}} | ≤ ϱ_{0}σ_{k,λ_{1},λ_{2}}(t)|q̃(·)|_{n−1−λ_{1},n−1−λ_{2}} \\ for a < t < b (k = 1, …, l) \end{matrix}  (2.6)

and

|\frac{d^{n−1}G(q̃)(t)}{dt^{n−1}} − \frac{d^{n−1}G(q̃)(s)}{ds^{n−1}} | ≤

≤ϱ_{0}|q̃(·)|_{n−1−λ_{1},n−1−λ_{2}} ∑_{k=1}^{l} ∫_{s}^{t} |p_{k}(τ)|σ_{k,λ_{1},λ_{2}}(τ)dτ+| ∫_{s}^{t} q̃(τ)dτ| (2.7)
for a < s ≤ t < b

are fulfilled for any q̃ ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ).

Let ℂ^{n−1}(]a, b[; ℝ) be a topologic space of n − 1 times continuously differentiable functions u :]a, b[→ ℝ, where by the convergence of the sequence (u_{i})_{i=1}^{+∞} we mean the uniform convergence of the sequences (u_{i}^{(k)})_{i=1}^{+∞} (k = 0, …, n − 1) inside ]a, b[.

It is assumed that

ϱ = ϱ_{0}[|q_{0}(·)|_{n−1−λ_{1},n−1−λ_{2}} + |q^{∗}(·)|_{n−1−λ_{1},n−1−λ_{2}}],

p^{∗}(t) = ϱ ∑_{k=1}^{l} |p_{k}(t)|σ_{k,λ_{1},λ_{2}}(t) + |q_{0}(t)| + q^{∗}(t),

and A is the set of all elements u of the space ℂ^{n−1}(]a, b[; ℝ) satisfying the inequalities

|u^{(k−1)}(t)| ≤ ϱσ_{k,λ_{1},λ_{2}}(t) for a < t < b (k = 1, …, l),

|u^{(n−1)}(t) − u^{(n−1)}(s)| ≤ ∫_{s}^{t} p^{∗}(τ)dτ for a < s ≤ t < b.

It is obvious that A is a convex set. On the other hand, by the Arcela–Askoli lemma it immediately follows that A is a compactum.

The operator G̃ is given on A as follows:

G̃(u)(t) = G(q(·, u(·), …, u^{(n−1)}(·)))(t) for u ∈ A.

According to (2.5)–(2.7), for any u ∈ A the function ũ(·) = G̃(u)(·) satisfies the inequalities

|u^{(k−1)}(t)| ≤ ϱ_{0}|q(·, u(·), …, u^{(n−1)}(·))|_{n−1−λ_{1},n−1−λ_{2}}σ_{k,λ_{1},λ_{2}}(t) ≤

≤ ϱσ_{k,λ_{1},λ_{2}}(t) a < t < b for a < t < b (k = 1, …, l),

|u^{(n−1)}(t) − u^{(n−1)}(s)| ≤ ϱ ∑_{k=1}^{l} ∫_{s}^{t} |p_{k}(τ)|σ_{k,λ_{1},λ_{2}}(τ) dτ +

+| ∫_{s}^{t} q(τ, u(τ), …, u^{(n−1)}(τ))dτ| ≤ ∫_{s}^{t} p^{∗}(τ)dτ for a < s ≤ t < b.

Therefore the operator G̃ maps A into itself. On the other hand, by Proposition 2.5 condition (2.4) guarantees the continuity of the operator G̃. According to the Chauder–Tikhonov theorem [4], G̃ has at least one fixed point. Therefore there exists a function u such that u(t) = G(q(·, u(·), …, u^{(n−1)}(·)))(t) for a < t < b. Hence it is clear that u is the solution of problem (2.3), (1.2_{1}), (1.2_{2}).

§ 3. Proof of the Main Results

Proof of Theorem 1.1. Let ρ_{0} be a positive number for which Proposition 2.1 is valid. According to (1.9) there exists ρ^{∗} > 0 such that

lρ_{0}[|q(·, ρ)|_{n−1−λ_{1},n−1−λ_{2}} +|q(·)|_{n−1−λ_{1},n−1−λ_{2}}] < ρ for ρ≥ρ^{∗}. (3.1)

Let

χ(s) = \begin{cases}1 & for 0 ≤ s ≤ ρ^{∗}, \\ 2 − \frac{s}{ρ^{∗}} & for ρ^{∗} ≤ s ≤ 2ρ^{∗}, \\ 0 & for s ≥ 2ρ^{∗} \end{cases}  (3.2)

and ε_{0} = \frac{a+b}{4}. Assume that for each j

ε_{j}(t) = \begin{cases}0 & for t ∈]a, a + \frac{ε_{0}}{j} [∪]b − \frac{ε_{0}}{j} , b[, \\ 1 & for t ∈ [a + \frac{ε_{0}}{j} , b − \frac{ε_{0}}{j}], \end{cases}  (3.3)

q_{j}(t, x_{1}, …, x_{n}) = ε_{j}(t)χ( ∑_{i=1}^{l} \frac{|x_{i}|}{σ_{i,λ_{1},λ_{2}}(t)})[f(t, x_{1}, …, x_{n}) −

− ∑_{k=1}^{l} p_{1k}(t)x_{k} − q_{0}(t)] + q_{0}(t)  (3.4)

and consider the differential equation

u^{(n)} = ∑_{k=1}^{l} p_{1k}(t)u^{(k−1)} + q_{j}(t, u, …, u^{(n−1)})  (3.5)

for an arbitrary natural number j.

Taking into account inequalities (1.6), (1.7), from (3.2)–(3.4) we obtain on ]a, b[×ℝ^{n} |q_{j}(t, x_{1}, …, x_{n}) − q_{0}(t)| ≤ q_{j}^{∗}(t), where

q_{j}^{∗}(t) = ε_{j}(t)q(t, 2ρ^{∗}) + 2ρ^{∗}ε_{j}(t) ∑_{k=1}^{l} |p_{2k}(t) − p_{1k}(t)|σ_{k,λ_{1},λ_{2}}(t).

By (1.8) and (3.3) it is easy to ascertain that q_{j}^{∗} ∈ L([a, b]; ℝ_{+}). Therefore, according to Proposition 2.6, Problem (3.5), (1.2_{1}), (1.2_{2}) has a solution u_{j}.

By (3.4) and (3.5) it is obvious that

u_{j}^{(n)}(t) = ∑_{k=1}^{l} p̃_{kj}(t)u_{j}^{(k−1)}(t) + q̃_{j}(t),  (3.6)

where

p̃_{kj}(t) = p_{1k}(t) + ε_{j}(t)χ( ∑_{i=1}^{l} \frac{|u_{j}^{(i−1)}(t)|}{σ_{i,λ_{1},λ_{2}}(t)}) ×

× [p_{k}(t, u_{j}(t), …, u_{j}^{(n−1)}(t)) − p_{1k}(t)],

q̃_{kj}(t) = ε_{j}(t)χ( ∑_{i=1}^{l} \frac{|u_{j}^{(i−1)}(t)|}{σ_{i,λ_{1},λ_{2}}(t)})[f(t, u_{j}(t), …, u_{j}^{(n−1)}(t)) −

− ∑_{k=1}^{l} p_{k}(t, u_{j}(t), …, u_{j}^{(n−1)}(t))u_{j}^{(k−1)}(t) − q_{0}(t)] + q_{0}(t).

On the other hand, (1.6) and (1.7) imply

p_{1k}(t) ≤ p̃_{kj}(t) ≤ p_{2k}(t) for a < t < b (k = 1, …, l)  (3.7)

and

|q̃_{j}(t) − q_{0}(t)| ≤ q(t, ρ_{j}) for a < t < b,  (3.8)

where

ρ_{j} = vrai max \{ ∑_{i=1}^{l} \frac{|u_{j}^{(i−1)}(t)|}{σ_{i,λ_{1},λ_{2}}(t)} : a < t < b\}.  (3.9)

In view of conditions (1.8), (3.7) and the choice of the number ρ_{0} we have

|u_{j}^{(i−1)}(t)| ≤ ρ_{0}|q̃_{j}(·)|_{n−1−λ_{1},n−1−λ_{2}}σ_{i,λ_{1},λ_{2}}(t) for a < t < b (i = 1, …, l).

Hence, taking into account (3.8) and (3.9), we find

ρ_{j} ≤ lρ_{0}[|q_{0}(·)|_{n−1−λ_{1},n−1−λ_{2}} + |q(·, ρ_{j})|_{n−1−λ_{1},n−1−λ_{2}}].

Consequently, by (3.1) ρ_{j} ≤ ρ^{∗}. According to this inequality, for each j (3.2), (3.6)–(3.9) imply

χ( ∑_{i=1}^{l} \frac{|u_{j}^{(i−1)}(t)|}{σ_{i,λ_{1},λ_{2}}(t)}) = 1 for a<t<b,  (3.10)
|q̃_{j}(t) − q_{0}(t)| ≤ q(t, ρ^{∗}) for a<t<b,  (3.11)
|u_{j}^{(k−1)}(t)| ≤ ρ^{∗}σ_{k,λ_{1},λ_{2}}(t) for a<t<b (k =1,…, l)  (3.12)

and

|u_{j}^{(n)}(t) − q_{0}(t)| ≤ p^{∗}(t) for a < t < b,  (3.13)

where

p^{∗}(t) = ρ^{∗} ∑_{k=1}^{l} (|p_{1k}(t)| + |p_{2k}(t)|)σ_{k,λ_{1},λ_{2}}(t) + q(t, ρ^{∗})

belongs to the class L_{loc}(]a, b[; ℝ_{+}).

By virtue of (3.12) and (3.13) the sequences (u_{j}^{(k−1)})_{j=1}^{+∞} (k = 1, …, n) are uniformly bounded and equicontinuous inside ]a, b[. Therefore by the Arcela–Askoli lemma it can be assumed without loss of generality that they converge uniformly inside ]a, b[.

Let u(t) = lim_{j→+∞} u_{j}(t) for a < t < b. Then

u^{(k−1)}(t) = lim_{j→+∞} u_{j}^{(k−1)}(t) for a < t < b (k = 1, …, n). (3.14)

Taking into account (3.3), (3.6), (3.10), and (3.13), it readily follows from (3.14) that u is a solution of equation (1.1).

On the other hand, since the sequences (u_{j}^{(k−1)})_{j=1}^{+∞} (k = 1, …, n) are uniformly bounded inside ]a, b[ and on account of (1.8), (3.11), (3.12) and the equalities u_{j}^{(k−1)}(a) = 0 (k = 1, …, n_{1}), u_{j}^{(k−1)}(b) = 0 (k = 1, …, n_{m}), from (3.6) we find

|u_{j}^{(k−1)}(t)|≤r_{1}(t − a)^{λ_{1}−k+1}+∫_{a}^{t}(t − s)^{n_{1}−k} ∫_{s}^{\frac{a+b}{2}}(τ − s)^{n−n_{1}−1}p̃(τ) dτds

for a < t ≤ \frac{a + b}{2}, l ∈ \{1, …, n_{1}\} (k = 1, …, n_{1})

and

|u_{j}^{(k−1)}(t)|≤r_{2}(b − t)^{λ_{2}−k+1}+∫_{t}^{b}(s − t)^{n_{m}−k}∫_{\frac{a+b}{2}}^{s} (s − τ)^{n−n_{m}−1}p̃(τ) dτds

for \frac{a + b}{2} ≤ t < b, l ∈ \{1, …, n_{m}\} (k = 1, …, n_{m}),

where p̃(t) = p^{∗}(t) − q(t, ρ^{∗}), and r_{1} and r_{2} are positive numbers not depending on j. Taking into account these estimates together with (3.12) and (3.14), we ascertain that u satisfies the boundary conditions (1.2_{1}),(1.2_{2}).

Thus u is the solution of problem (1.1),(1.2_{1}),(1.2_{2}).

Proof of Corollary 1.1. We set

γ(t, x_{1}, …, x_{n}) = [f(t, x_{1}, …, x_{n}) − q_{0}(t)] ×

×[1 + ∑_{k=1}^{l} p_{k}(t)|x_{k}| + q(t, ∑_{i=1}^{l} \frac{|x_{i}|}{σ_{i,λ_{1},λ_{2}}(t)})]^{−1}

and p̃_{k}(t, x_{1}, …, x_{n}) = γ(t, x_{1}, …, x_{n})p_{k}(t) sign x_{k} (k = 1, …, l). Then

f(t, x_{1}, …, x_{n}) = ∑_{k=1}^{l} p̃_{k}(t, x_{1}, …, x_{n})x_{k} +

+[1 + q(t, ∑_{i=1}^{l} \frac{|x_{i}|}{σ_{i,λ_{1},λ_{2}}(t)})γ(t, x_{1}, …, x_{n})] + q_{0}(t)

and

p̃_{k} ∈ K_{loc}^{0}(I_{m} × ℝ^{n}; ℝ).  (3.15)

On the other hand, according to (1.9) and (1.10), on ]a, b[×ℝ^{n} we have

\begin{matrix}|γ(t, x_{1}, …, x_{n})| < 1, \\ −p_{k}(t) ≤ p̃_{k}(t, x_{1}, …, x_{n}) ≤ p_{k}(t) (k = 1, …, l) \end{matrix}  (3.16)

and

|f(t, x_{1}, …, x_{n}) − ∑_{k=1}^{l} p̃_{k}(t, x_{1}, …, x_{n})x_{k} − q_{0}(t)| ≤

≤ q̃(t, ∑_{i=1}^{l} \frac{|x_{i}|}{σ_{i,λ_{1},λ_{2}}(t)}),  (3.17)

where the function q̃(t, ρ) = 1 + q(t, ρ) satisfies the condition

lim_{ρ→+∞} \frac{|q̃(·, ρ)|_{n−1−λ_{1},n−1−λ_{2}}}{ρ} = 0.  (3.18)

By Theorem 1.1 from conditions (1.11), (3.15)–(3.18) we conclude that problem (1.1),(1.2_{1}),(1.2_{2}) is solvable.

Proof of Theorem 1.2. By the inequalities (1.12), the equality

f(t, x_{1}, …, x_{l}) = ∑_{k=1}^{l} p_{k}(t, x_{1}, …, x_{l})x_{k} + q_{0}(t),

where

p_{k}(t, x_{1}, …, x_{l}) = ∫_{0}^{1} \frac{∂f(t, ξx_{1}, …, ξx_{l})}{∂x_{k}} dξ (k = 1, …, l),

q_{0}(t) = f(t, 0, …, 0),

yields

|f(t, x_{1}, …, x_{l}) − ∑_{k=1}^{l} p_{k}(t, x_{1}, …, x_{l})x_{k} − q_{0}(t)| = 0

and p_{1k}(t) ≤ p_{k}(t, x_{1}, …, x_{l}) ≤ p_{2k}(t) (k = 1, …, l). Therefore by Theorem 1.1 problem (1.1′),(1.2_{1}),(1.2_{2}) has a solution u_{0}.

Let us show that u_{0} is the unique solution of this problem. Assume that u_{1} is an arbitrary solution of problem (1.1′),(1.2_{1}),(1.2_{2}). Set

u(t) = u_{0}(t) − u_{1}(t), p̃_{k}(t; x_{1}, …, x_{l}; y_{1}, …, y_{l}) =

= ∫_{0}^{1} \frac{∂f(t, ξx_{1} + (1 − ξ)y_{1}, …, ξx_{l} + (1 − ξ)y_{l})}{∂x_{k}} dξ

and p_{k}(t) = p̃_{k}(t; u_{0}(t), …, u_{0}^{(l−1)}(t); u_{1}(t), …, u_{1}^{(l−1)}(t)).

By virtue of the equality

f(t, x_{1}, …, x_{l}) − f(t, y_{1}, …, y_{l}) = ∑_{k=1}^{l} p̃_{k}(t; x_{1}, …, x_{l}; y_{1}, …, y_{l})(x_{k} − y_{k})

and the inequalities (1.12), u is a solution of the equation

u^{(n)} = ∑_{k=1}^{l} p_{k}(t)u^{(k−1)}  (3.19)

satisfying the boundary conditions (1.2_{1}),(1.2_{2}) and

p_{1k}(t) ≤ p_{k}(t) ≤ p_{2k}(t) for a < t < b (k = 1, …, l).  (3.20)

But by (1.8) and (3.20) problem (3.19),(1.2_{1}),(1.2_{2}) has the trivial solution only, i.e., u_{1}(t) ≡ u_{0}(t).

Proof of Theorem 1.3. Set

p(t) = \frac{r(t)}{σ_{1,λ_{1},λ_{2}}(t)} for a < t < b.

Let us show that

(p) ∈ S^{+}(a, b; n, n_{1}; λ_{1}) ∩ S^{−}(a, b; n, n_{m}; λ_{2}).  (3.21)

By virtue of (1.14_{1}) there exist points t_{0n_{1}}, t_{0 n_{1}+1}, …, t_{0 n−1} such that

a < t_{0n_{1}} ≤ ⋯ ≤ t_{0 n−1} < b  (3.22)

and

u_{0}^{(k)}(r)(t_{0k}) = 0 (k = n_{1}, …, n − 1).  (3.23)

At the same time, u_{0}^{(n)}(r)(t) = r(t) ≥ 0 for a < t < b. Therefore by (3.22) it is obvious that for each k ∈ {n_{1}, …, n − 1} the function u_{0}^{(k)}(r)(·) does not change its sign on ]a, t_{0k}[ and, taking into account (3.23), we readily obtain

\begin{matrix}|u_{0}^{(n_{1})}(r)(t)| ≥ \frac{1}{(n − n_{1} − 1)!} ∫_{t}^{t_{0n_{1}}}(τ − t)^{n−n_{1}−1}r(τ) dτ \\ for a < t ≤ t_{0n_{1}}. \end{matrix}  (3.24)

At the same time, by (1.14_{1}) we have

|u_{0}(r)(t)| = \frac{1}{(n_{1} − 1)!} ∫_{a}^{t}(τ − t)^{n_{1}−1}|u^{(n_{1})}(r)(τ)| dτ

for a < t ≤ min{t_{2}, t_{0n_{1}}}.

Using (1.16), choose α ∈]a, t_{2}[ such that α ≤ t_{0n_{1}} and

ρ_{λ_{1},λ_{2}}(r) < \frac{(b − α)^{λ_{2}} ∏_{i=2}^{m−1} |α − t_{i}|^{n_{i}}}{(b − a)^{λ_{2}} ∏_{i=2}^{m−1} |a − t_{i}|^{n_{i}}} .  (3.25)

ON THE MODIFIED BOUNDARY VALUE PROBLEM 449

By (3.24) the latter equality yields

|u_{0}(r)(t)| ≥ \frac{(b − α)^{λ_{2}} ∏_{i=2}^{m−1} |α − t_{i}|^{n_{i}}}{(n_{1} − 1)!(n − n_{1} − 1)!} ×

× ∫_{a}^{t}(t − s)^{n_{1}−1} ∫_{s}^{α}(τ − t)^{n−n_{1}−1}(τ − a)^{λ_{1}}|p(τ)| dτds

for a < t ≤ α,

whence

ρ_{λ_{1},λ_{2}}(r) ≥ lim sup_{t→a} \frac{|u_{0}(r)(t)|}{σ_{1,λ_{1},λ_{2}}(t)} ≥

≥ \frac{(b − α)^{λ_{2}} ∏_{i=2}^{m−1} |α − t_{i}|^{n_{i}}}{(b − a)^{λ_{2}} ∏_{i=2}^{m−1} |a − t_{i}|^{n_{i}}} lim sup_{t→a} \frac{(t − a)^{−λ_{1}}}{(n_{1} − 1)!(n − n_{1} − 1)!} ×

× ∫_{a}^{t}(t − s)^{n_{1}−1} ∫_{s}^{α}(τ − t)^{n−n_{1}−1}(τ − a)^{λ_{1}}|p(τ)| dτds.

Therefore by virtue of (3.25) we obtain

lim sup_{t→a} \frac{(t − a)^{−λ_{1}}}{(n_{1} − 1)!(n − n_{1} − 1)!} ×

× ∫_{a}^{t}(t − s)^{n_{1}−1} ∫_{s}^{α}(τ − t)^{n−n_{1}−1}(τ − a)^{λ_{1}}|p(τ)| dτds < 1.

The inequality

lim sup_{t→b} \frac{(b − t)^{−λ_{2}}}{(n_{m} − 1)!(n − n_{m} − 1)!} ×

× ∫_{t}^{b}(s − t)^{n_{m}−1} ∫_{β}^{s}(s − τ)^{n−n_{m}−1}(b − τ)^{λ_{2}}|p(τ)| dτds < 1,

where β ∈]t_{m−1}, b[, is proved in a similar manner. Thus (3.21) is valid.

Now let us show that

(−p, p) ∈ V (t_{1}, …, t_{m}; n_{1}, …, n_{m}; λ_{1}, λ_{2}).  (3.26)

Due to (3.21) it it is sufficient to verify that under the boundary conditions (1.14_{1}),(1.14_{2}) the equation

u^{(n)} = p̃(t)u  (3.27)

has the trivial solution only for any function p̃ :]a, b[→ ℝ satisfying the conditions σ_{1,λ_{1},λ_{2}}(·)p̃(·) ∈ L_{loc}(]a, b[; ℝ) and

−p(t) ≤ p̃(t) ≤ p(t) for a < t < b.  (3.28)

Let u be an arbitrary solution of problem (3.27),(1.14_{1}),(1.14_{2}) and g(·, ·) be the Green function of the equation

u^{(n)} = 0  (3.29)

by the boundary conditions of de la Vallée-Poussin (1.14_{1}). According to the Chichkin theorem [3]^{6} g(t, τ)(t−t_{1})^{n_{1}} ⋯ (t−t_{m})^{n_{m}} ≥ 0 for a ≤ t, τ ≤ b. Therefore

|u_{0}(r)(t)| = ∫_{a}^{b} |g(t, τ)|r(τ) dτ.  (3.30)

On account of (3.28), (3.30) the equality

u(t) = ∫_{a}^{b} g(t, τ)p̃(τ)u(τ) dτ

yields γ ≤ γρ_{λ_{1},λ_{2}}(r), where

γ = vrai max \{\frac{|u(t)|}{σ_{1,λ_{1},λ_{2}}(t)} : a < t < b\}.

Hence, taking into account (1.16), we obtain γ = 0, i.e., u(t) ≡ 0. Therefore (3.26) holds. Now by virtue of (1.15) we conclude that all the conditions of Corollary 1.1 are fulfilled. Therefore problem (1.13),(1.14_{1}),(1.14_{2}) is solvable. Proof of Corollary 1.2. We introduce the notation

η(a, b; n_{1}, n_{2}; λ_{1}, λ_{2}) =
= [n_{1}!n_{2}!(λ_{1} − n_{1} + 1)(λ_{2} − n_{2} + 1)(n_{1} − λ_{1})(n_{2} − λ_{2})(b − a)^{n}] ×
×[2n_{1}(λ_{2} − n_{2} + 1)(n_{1} − λ_{1})(λ_{1} − n_{1} + 1 + n_{2} − λ_{2}) +
+2n_{2}(λ_{1} − n_{1} + 1)(n_{2} − λ_{2})(λ_{2} − n_{2} + 1 + n_{1} − λ_{1})]^{−1},

r(t) = \frac{r_{0}}{(t − a)^{n−λ_{1}}(b − t)^{n−λ_{2}}} .

Now condition (1.17) can be written as

r_{0} < η(a, b; n_{1}, n_{2}; λ_{1}, λ_{2}).  (3.31)

6See also [9], Lemma 4.2.

By Theorem 1.3, to prove the corollary it is sufficient to show that

ρ_{λ_{1},λ_{2}}(r) = sup \{\frac{|u_{0}(r)(t)|}{(t − a)^{λ_{1}}(b − t)^{λ_{2}}} : a < t < b\} < 1,

where u_{0}(r)(·) is the solution of the equation u^{(n)} = r(t), satisfying the boundary conditions (1.14_{1}) for m = 2.

As shown in [1], for m = 2 the Green function g(·, ·) of problem (3.29), (1.14_{1}) admits the estimate

|g(t, τ)| ≤ \frac{(t − a)^{n_{1}−1}(b − t)^{n_{2}−1}(τ − a)^{n_{2}−1}(b − τ)^{n_{1}−1}}{(n_{1} − 1)!(n_{2} − 1)!(b − a)^{n−1}} ×

× \begin{cases}\frac{(t−a)(b−τ)}{n_{1}} & for t ≤ τ \\ \frac{(b−t)(τ−a)}{n_{2}} & for t < τ \end{cases} .

Therefore from the equality

u_{0}(r)(t) = ∫_{a}^{b} g(t, τ)r(τ) dτ

we find

|u_{0}(r)(t)| ≤ \frac{r_{0}}{η(a, b; n_{1}, n_{2}; λ_{1}, λ_{2})}(t − a)^{λ_{1}}(b − t)^{λ_{2}},

whence by (3.31) we obtain

ρ_{λ_{1},λ_{2}}(r) < 1.

Proof of Corollary 1.3. Let

r(t) = r_{0}(t − a)^{λ_{1}}(b − t)^{λ_{2}}[\frac{λ_{1}(1 − λ_{1})}{(t − a)^{2}} +

+ \frac{2λ_{1}λ_{2}}{(t − a)(b − t)} + \frac{λ_{2}(1 − λ_{2})}{(b − t)^{2}}].

Clearly, r ∈ L_{1−λ_{1},1−λ_{2}}(]a, b[; ℝ_{+}). On the other hand, it is not difficult to verify that the function u_{0}(r)(t) = −r_{0}(t − a)^{λ_{1}}(b − t)^{λ_{2}} is the solution of the problem

u′′ = r(t), u(a) = u(b) = 0.

By condition (1.25) we obtain ρ_{λ_{1},λ_{2}}(r) = r_{0} < 1. Thus all the conditions of Theorem 1.3 are fulfilled.

Proof of Corollary 1.4. Let

r(t)=r_{0}(t−a)^{λ_{1}}(t_{0}−t)(b−t)^{λ_{2}}[\frac{λ_{1}(1−λ_{1})(2−λ_{1})}{(t−a)^{3}} − \frac{3λ_{1}(1−λ_{1})}{(t−a)^{2}(t−t_{0})} +

+ \frac{3λ_{1}λ_{2}(1 − λ_{1})}{(t − a)^{2}(b − t)} − \frac{6λ_{1}λ_{2}}{(t − a)(t − t_{0})(b − t)} − \frac{3λ_{1}λ_{2}(1 − λ_{2})}{(t − a)(b − t)^{2}} −

− \frac{3λ_{2}(1 − λ_{2})}{(t − t_{0})(b − t)^{2}} − \frac{λ_{2}(1 − λ_{2})(2 − λ_{2})}{(b − t)^{3}}].

Clearly, r ∈ L_{2−λ_{1},2−λ_{2}}(]a, b[; ℝ). On the other hand, it is not difficult to verify that the function u_{0}(r)(t) = −r_{0}(t−a)^{λ_{1}}(t−t_{0})(b−t)^{λ_{2}} is the solution of the problem

u′′′ = r(t), u(a) = u(t_{0}) = u(b) = 0.

By condition (1.25) we obtain ρ_{λ_{1},λ_{2}}(r) = r_{0} < 1. Thus all the conditions of Theorem 1.3 are fulfilled.

Proof of Theorem 1.4. As we have ascertained in proving Theorem 1.3, condition (1.16) guarantees the fulfilment of condition (3.26), where

p(t) = \frac{r(t)}{σ_{1,λ_{1},λ_{2}}(t)}.

Therefore, according to Theorem 1.2, problem (1.13),(1.14_{1}),(1.14_{2}) has the unique solution u.

From the conditions of the theorem we have

f(·, ũ(·)) ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ)

for any ũ ∈ ℂ_{loc}^{n−1}(]a, b[; ℝ) satisfying the boundary conditions (1.14_{1}), (1.14_{2}). Therefore, by Proposition 2.3, for each natural number j problem (1.24), (1.14_{1}), (1.14_{2}) has the unique solution u_{j}.

It is assumed that for each j

v_{j}(t) = u_{j}(t) − u(t).  (3.32)

Clearly, v_{j} satisfies the boundary conditions (1.14_{1}), (1.14_{2}),

γ_{j} = vrai max \{\frac{|v_{j}(t)|}{σ_{1,λ_{1},λ_{2}}(t)} : a < t < b\} < +∞  (3.33)

and

v_{j}^{(n)} = f(t, u_{j−1}(t)) − f(t, u(t)).  (3.34)

Repeating the reasoning from the proof of Theorem 1.3 and using the Chichkin theorem, by virtue of (1.22), (3.32)–(3.34) we obtain

γ_{j} ≤ γ_{j−1}ρ_{λ_{1},λ_{2}}(r),

Hence γ_{j} ≤ γ_{1}ρ_{λ_{1},λ_{2}}^{j−1}(r) (j = 1, 2, …). Therefore by (1.16), (3.32), (3.33) we obtain (1.23).

Corollaries 1.5-1.7 are proved similarly to Corollaries 1.2–1.4, the only difference being that Theorem 1.4 is used instead of Theorem 1.3.

Remark 3.1. In Corollaries 1.3, 1.4, 1.6, and 1.7 condition (1.25) cannot be replaced by equality (1.26), since for λ_{1} ∈]\frac{1}{2}, 1[ problem (1.18), (1.19_{1}), (1.19_{2}), where

f(t, x) = −\frac{λ_{1}(1 − λ_{1})}{(t − a)^{2}} x + (t − a)^{λ_{1}−2},

and problem (1.20),(1.21_{1}),(1.21_{2}), where

f(t, x) = \frac{λ_{1}(1 − λ_{1})(2 − λ_{1})}{(t − a)^{3}} x + (t − a)^{λ_{1}−3},

have no solutions though all the conditions of these corollaries are fulfilled with the exception of condition (1.25) which is replaced by (1.26).

Proof of Theorem 1.5. Let us assume that the theorem is not true. Then there exist ε ∈]0, r[, α ∈]a, t_{2}[, β ∈]t_{m−1}, b[, (x_{10}, …, x_{l0}) ∈ D_{λ_{1},λ_{2}}(u_{0}; r), ω ∈ M([α, β] × ℝ_{+}; ℝ_{+}) and a sequence of functions η_{i} ∈ K_{loc}(I_{m} × ℝ^{l}; ℝ) such that

\begin{matrix}| ∫_{α}^{t} η_{i}(τ, σ_{1,λ_{1},λ_{2}}(τ)x_{1}, …, σ_{1,λ_{1},λ_{2}}(τ)x_{l}) dτ| ≤ \frac{1}{i} \\ for α ≤ t ≤ β, (x_{1}, …, x_{l}) ∈ D_{λ_{1},λ_{2}}(u_{0}; r), \end{matrix}  (3.35)

|η_{i}(t, σ_{1,λ_{1},λ_{2}}(t)x_{1}, …, σ_{l,λ_{1},λ_{2}}(t)x_{l}) −
−η_{i}(t, σ_{1,λ_{1},λ_{2}}(t)y_{1}, …, σ_{l,λ_{1},λ_{2}}(t)y_{l}| ≤ ω( ∑_{k=1}^{l} |x_{k} − y_{k}|) (3.36)
for α ≤ t ≤ β, (x_{1}, …, x_{l}) and (y_{1}, …, y_{l}) ∈ D_{λ_{1},λ_{2}}(u_{0}; r),

\begin{matrix}(t − a)^{n−1−λ_{1}}[ ∫_{t}^{α} η_{λ_{1},λ_{2},i}^{∗}(τ; u_{0}; r) dτ + | ∫_{t}^{α} η_{0i}(τ), dτ|] ≤ \frac{1}{i} \\ for a < t ≤ α, \end{matrix} (3.37)

\begin{matrix}(b − t)^{n−1−λ_{2}}[ ∫_{β}^{t} η_{λ_{1},λ_{2},i}^{∗}(τ; u_{0}; r) dτ + | ∫_{β}^{t} η_{0i}(τ), dτ|] ≤ \frac{1}{i} \\ for β ≤ t < b, \end{matrix} (3.38)

where

η_{0i}(t) = η_{i}(t, σ_{1,λ_{1},λ_{2}}(t)x_{10}, …, σ_{l,λ_{1},λ_{2}}(t)x_{l0}),
η_{λ_{1},λ_{2},i}^{∗}(t; u_{0}; r) =
= sup {|η_{i}(t, σ_{1,λ_{1},λ_{2}}(t)x_{1}, …, σ_{l,λ_{1},λ_{2}}(t)x_{l}) − η_{0i}| :
(x_{1}, …, x_{l}) ∈ D_{λ_{1},λ_{2}}(u_{0}; r)},

and for each i the equation

u^{(n)} = f(t, u, …, u^{(l−1)}) + η_{i}(t, u, …, u^{(l−1)})  (3.39)

either has no solution contained in W_{λ_{1},λ_{2}}(u_{0}; r) or has at least one solution contained in W_{λ_{1},λ_{2}}(u_{0}; r)\W_{λ_{1},λ_{2}}(u_{0}; ε).

It is assumed that

\begin{matrix}p_{k}(t, x_{1}, …, x_{l}) = ∫_{0}^{1} \frac{∂f(t, u_{0}(t) + ξx_{1}, …, u_{0}^{(l−1)}(t) + ξx_{l})}{∂x_{k}} dξ \\ (k = 1, …, l), \end{matrix} (3.40)

χ(s) = \begin{cases}1 & for 0 ≤ s ≤ r, \\ 2 − \frac{s}{r} & for r < s < 2r, \\ 0 & for s ≥ 2r, \end{cases} ,  (3.41)

\begin{matrix}p̃_{k}(t, x_{1}, …, x_{l})=p_{k}(t, x_{1}χ( ∑_{j=1}^{l} \frac{|x_{j}|}{σ_{j,λ_{1},λ_{2}} (t)}), …, x_{l}χ( ∑_{j=1}^{l} \frac{|x_{j}|}{σ_{j,λ_{1},λ_{2}} (t)})) \\ (k = 1, …, l), \end{matrix}(3.42)

η̃_{i}(t, x_{1}, …, x_{l}) =η_{i}(t, u_{0}(t)+x_{1}χ(∑_{j=1}^{l} \frac{|x_{j}|}{σ_{j,λ_{1},λ_{2}}(t)}), …, u_{0}^{(l−1)}(t)+

+x_{l}χ(∑_{j=1}^{l} \frac{|x_{j}|}{σ_{j,λ_{1},λ_{2}}(t)})) (k = 1, …, l).  (3.43)

Let us consider the equation

v^{(n)} = ∑_{k=1}^{l} p̃_{k}(t, v, …, v^{(l−1)})v^{(k−1)} + η̃_{i}(t, v, …, v^{(l−1)}) (3.44)

for each i. By (1.37), (3.40)–(3.43) ascertain that the following inequalities are fulfilled on ]a, b[×ℝ^{l}:

p_{1k}(t) ≤ p̃_{k}(t, x_{1}, …, x_{l}) ≤ p_{2k}(t) (k = 1, …, l)  (3.45)

and

|η̃_{i}(t, x_{1}, …, x_{l}) − η_{0i}(t)| ≤ q_{i}(t),  (3.46)

where

q_{i}(t) = \begin{cases}ω(t, r + ∑_{j=1}^{l} \frac{|u_{0}^{(j−1)}(t)|}{σ_{j,λ_{1},λ_{2}} (t)} + |x_{k0}|) & for α ≤ t ≤ β, \\ η_{λ_{1},λ_{2}}^{∗}(t; u_{0}; r) & for ]a, b[\[α, β]. \end{cases}

From (3.37), (3.38) we readily obtain η_{0i}, q_{i} ∈ L_{n−1−λ_{1},n−1−λ_{2}}(]a, b[; ℝ) (i = 1, 2, …). However, since condition (1.8)is fulfilled, by virtue of Theorem 1.1 equation (3.44) has, for each natural number i, at least one solution satisfying (1.2_{1}),(1.2_{2}).

Let v_{0} be an arbitrary solution of problem (3.44),(1.2_{1}),(1.2_{2}). Then either

vrai max \{ ∑_{k=1}^{l} \frac{|v_{0}^{(k−1)}(t)|}{σ_{k,λ_{1},λ_{2}}(t)} : a < t < b\} > r,

or

vrai max \{ ∑_{k=1}^{l} \frac{|v_{0}^{(k−1)}(t)|}{σ_{k,λ_{1},λ_{2}}(t)} : a < t < b\} ≤ r.

If the latter inequality is fulfilled, then in view of (3.40)-(3.43) and the equality

f(t, u_{0}(t) + x_{1}, …, u_{0}^{(l−1)}(t) + x_{l}) − f(t, u_{0}(t), …, u_{0}^{(l−1)}(t)) =

= ∑_{k=1}^{l} p_{k}(t, x_{1}, …, x_{l})x_{k}

it is obvious that the function u(t) = u_{0}(t) + v_{0}(t) is a solution of problem (3.39),(1.2_{1}),(1.2_{2}) contained in W_{λ_{1},λ_{2}}(u_{0}; r). However, by our assumption, in the case under consideration problem (3.39),(1.2_{1}),(1.2_{2}) has at least one solution ũ such that ũ ∈ W_{λ_{1},λ_{2}}(u_{0}; r)\W_{λ_{1},λ_{2}}(u_{0}; ε).

It is obvious that the function v(t) = ũ(t)−u_{0}(t) is a solution of problem (3.44),(1.2_{1}),(1.2_{2}) satisfying the inequality

vrai max \{ ∑_{k=1}^{l} \frac{|v^{(k−1)}(t)|}{σ_{k,λ_{1},λ_{2}}(t)} : a < t < b\} > ε.

Thus, for each natural number i problem (3.44),(1.2_{1}),(1.2_{2}) has the solution v_{i} satisfying the condition

vrai max \{ ∑_{k=1}^{l} \frac{|v_{i}^{(k−1)}(t)|}{σ_{k,λ_{1},λ_{2}}(t)} : a < t < b\} > ε.  (3.47)

Let ρ_{0} be the number from Proposition 2.1. Then in view of (1.8) and (3.45) the inequality

\begin{matrix}|v_{i}^{(k−1)}(t)| ≤ ρ_{0}|η̃_{i}(·, v_{i}(·), …, v_{i}^{(l−1)}(·))|_{n−1−λ_{1},n−1−λ_{2}}σ_{k,λ_{1},λ_{2}}(t) \\ for a < t < b (k = 1, …, l) \end{matrix}(3.48)

is fulfilled for each i.

By (3.37), (3.38) and (3.46) we have

η^{∗} = sup {|η̃(·, v_{i}(·), …, v_{i}^{(l−1)}(·))|_{n−1−λ_{1},n−1−λ_{2}} : i = 1, 2, … } ≤

≤sup {|η_{0i}(·)|_{n−1−λ_{1},n−1−λ_{2}} +|q_{i}(·)|_{n−1−λ_{1},n−1−λ_{2}} : i=1, 2, … }<+∞.

Using (3.45), (3.46), (3.48) and the latter inequality, from (3.44) we have

\begin{matrix}∑_{k=1}^{l} |v_{i}^{(k−1)}(t)| ≤ ρ^{∗}, |v_{i}^{(n)} − η_{0i}(t)| ≤ p^{∗}(t) \\ for α ≤ t ≤ β (i = 1, 2, …), \end{matrix}  (3.49)

where

ρ^{∗} = η^{∗}ρ_{0} max { ∑_{k=1}^{l} σ_{k,λ_{1},λ_{2}}(t) : α ≤ t ≤ β},

p^{∗}(t) = η^{∗}ρ_{0} ∑_{k=1}^{l} (|p_{1k}(t)| + |p_{2k}(t)|)σ_{k,λ_{1},λ_{2}}(t) +

+ ω(t, r + ∑_{j=1}^{l} (\frac{|u_{0}^{(j−1)}(t)|}{σ_{j,λ_{1},λ_{2}}(t)} + |x_{j0}|))

and p^{∗} ∈ L([α, β]; ℝ). Moreover, since

max {| ∫_{s}^{t} η_{0i}(τ) dτ| : α ≤ s ≤ t ≤ β} → 0 for i → +∞,

it follows from (3.49) that the sequences (v_{i}^{(k−1)})_{i=1}^{+∞} (k = 1, …, l) are uniformly bounded and equicontinuous on [α, β]. Therefore, by Lemma 3.1 [8], conditions (3.35), (3.36), (3.41), and (3.43) imply

\begin{matrix}γ_{i} = max \{| ∫_{α}^{t} η̃_{i}(τ, v_{i}(τ), …, v_{i}^{(l−1)}(τ)) dτ| : α ≤ t ≤ β\} → 0 \\ for i → +∞. \end{matrix} (3.50)

In view of (3.37),(3.38),(3.46), and (3.50)

|η̃_{i}(·, v_{i}(·), …, v_{i}^{(l−1)}(·))|_{n−1−λ_{1},n−1−λ_{2}} ≤

≤ (2γ_{j}(b − a)^{2n−2−λ_{1}−λ_{2}} + \frac{1}{i}[(b − a)^{n−1−λ_{1}} + (b − a)^{n−1−λ_{2}}]) → 0

for i → +∞

and now from (3.48) we find

vrai max \{ ∑_{k=1}^{l} \frac{|v_{i}^{(k−1)}(t)|}{σ_{k,λ_{1},λ_{2}}(t)} : a < t < b\} → 0 for i → +∞.

But this contradicts (3.47), which proves the theorem.

Theorem 1.6 immediately follows from Theorems 1.2 and 1.5.

References

1. P.W. Bates and G.B. Gustafson, Green’s function inequalities for twopoint boundary value problem. Pacif. J. Math. 59(1975), No. 2, 327-343.

2. Ch.-J. de la Vallée-Poussin, Sur l’equation differentielle lineaire du second ordre. Determination d’une integrale par deux valeurs assignees. Extension aux equation d’ordre n. J. Math. Pures et Appl. 8(1929), No. 2, 125-144.

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11. —–, On the correctness of a singular problem of the de la Vallée-Poussin type. (Russian) Bull. Acad. Sci. Georgian SSR 135(1989), No. 3, 489-492.

12. —–, On a class V (t_{1}, …, t_{m}; n_{1}, …, n_{m}; λ_{1}, λ_{2}). Preprint. A. Razmadze Math. Inst. Acad. Sci. Rep. of Georgia, Tbilisi, 1991.

13. —–, On the existence, uniqueness, and correctness of the solution of a Vallée-Poussin type linear boundary value problem.Preprint. A. Razmadze Math. Inst. Acad. Sci. Rep. of Georgia, Tbilisi, 1991.

14. —–, On a multipoint boundary value problem for linear ordinary differential equations with singularities. Arch. Math. (to appear).

(Received 7.07.1993)

Author’s address: A.Razmadze Mathematical Institute Georgian Academy of Sciences 1, Z.Rukhadze St., 380093 Tbilisi Republic of Georgia