Georgian Mathematical Journal Volume 8 (2001), Number 4, 791–814

ON SINGULAR BOUNDARY VALUE PROBLEMS FOR FUNCTIONAL DIFFERENTIAL EQUATIONS OF HIGHER

ORDER

I. KIGURADZE, B. PŮŽA, AND I. P. STAVROULAKIS

Dedicated to the memory of Professor N. Muskhelishvili on the occasion of his 110th birthday

Abstract. Sufficient conditions are established for the solvability of the boundary value problem

x^{(n)}(t) = f(x)(t), h_{i}(x) = 0 (i = 1, …, n), where f is an operator (h_{i} (i = 1, …, n) are operators) acting from some subspace of the space of (n − 1)-times differentiable on the interval ]a, b[ m-dimensional vector functions into the space of locally integrable on ]a, b[ m-dimensional vector functions (into the space ℝ^{m}).

2000 Mathematics Subject Classification: 34K10.

Key words and phrases: singular functional differential equation, boundary value problem, Fredholm property, a priori boundedness principle.

1. Formulation of the Main Results

1.1. Formulation of the problem and a brief survey of literature. Consider the functional differential equation of n-th order

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

with the boundary conditions

h_{i}(x) = 0 (i = 1, …, n).  (1.2)

When the operators f : C^{n−1}([a, b]; ℝ^{m}) → L([a, b]; ℝ^{m}) and h_{i} : C^{n−1}([a, b]; ℝ^{m}) → ℝ^{m} (i = 1, …, n) are continuous, problem (1.1), (1.2) is called regular. If the operator f (operators h_{i} (i = 1, …, n)) acts from some subspace of the space C^{n−1}(]a, b[ ; ℝ^{m}) into the space L_{loc}(]a, b[ ; ℝ^{m}) (into the space ℝ^{m}), problem (1.1), (1.2) is called singular.

The basic principles of the theory of a wide enough class of regular problems of form (1.1), (1.2) are constructed in the monographs [4], [5], [43]. Optimal sufficient conditions for such problems to be solvable and uniquely solvable are given in [7], [8], [10]–[12], [22], [24], [26]–[28], [39].

ISSN 1072-947X / $8.00 / \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag www.heldermann.de

As to singular problems of form (1.1), (1.2), they have been studied with sufficient completeness in the case with the operator f having the form

f(x)(t) = g(t, x(t), …, x^{(n−1)}(t))

(see [1], [2], [14]–[21], [32]–[35], [37], [45] and the references cited therein). For the singular functional differential equation (1.1), the weighted initial problem is studied in [30], [31], two-point problems in [3], [6], [23], [36], [38], [40]–[42], whereas the multi-point Vallée-Poussin problem in [25]. In the general case the singular problem (1.1), (1.2) remains studied but little. An attempt is made in this paper to fill up this gap to some extent.

Throughout the paper the following notation will be used.

ℝ = ] − ∞, +∞[ , ℝ_{+} = [0, +∞[ .

ℝ^{m} is the space of m-dimensional column vectors x = (x_{i})_{i=1}^{m} with the components x_{i} ∈ ℝ (i = 1, …, m) and the norm

∥x∥ = ∑_{i=1}^{m} |x_{i}|.

ℝ_{+}^{m} = {x = (x_{i})_{i=1}^{m} : x_{i} ∈ ℝ_{+} (i = 1, …, m)}.

ℝ^{m×m} is the space of m × m matrices X = (x_{ik})_{i,k=1}^{m} with the components x_{ik} ∈ ℝ (i, k = 1, …, m) and the norm

∥X∥ = ∑_{i,k=1}^{m} |x_{ik}|.

If x = (x_{i})_{i=1}^{m} ∈ ℝ^{m} and X = (x_{ik})_{i,k=1}^{m} ∈ ℝ^{m×m}, then

|x| = (|x_{i}|)_{i=1}^{m} and |X| = (|x_{ik}|)_{i,k=1}^{m}.

ℝ_{+}^{m×m} = {X = (x_{ik})_{i,k=1}^{m} : x_{ik} ∈ ℝ_{+} (i, k = 1, …, m)}.

r(X) is the spectral radius of the matrix X ∈ ℝ^{m×m}.

Inequalities between matrices and vectors are understood componentwise, i.e., for x = (x_{i})_{i=1}^{m}, y = (y_{i})_{i=1}^{m}, X = (x_{ik})_{i,k=1}^{m} and Y = (y_{ik})_{i,k=1}^{m} we have

x ≤ y ⟺ x_{i} ≤ y_{i} (i = 1, …, m)

and

X ≤ Y ⟺ x_{ik} ≤ y_{ik} (i, k = 1, …, m).

If k is a natural number and ε ∈]0, 1[ , then

(k − ε)! = ∏_{i=1}^{k}(i − ε).

If m and n are natural numbers, −∞ < a < b < +∞, α ∈ ℝ and β ∈ ℝ, then C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) is the Banach space of (n−1)-times continuously differentiable vector functions x : ]a, b[ → ℝ^{m} having limits

lim_{t→a}(t − a)^{α_{i}}x^{(i−1)}(t), lim_{t→b}(b − t)^{β_{i}}x^{(i−1)}(t) (i = 1, …, n),  (1.3)

where

α_{i} = \frac{α + i − n + |α + i − n|}{2} , β_{i} = \frac{β + i − n + |β + i − n|}{2}  (1.4)
(i = 1, …, n).

The norm of an arbitrary element x of this space is defined by the equality

∥x∥_{C_{α,β}^{n−1}} = sup { ∑_{k=1}^{n}(t − a)^{α_{i}}(b − t)^{β_{i}}∥x^{(i−1)}(t)∥ : a < t < b}.

C̃_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) is the set of x ∈ C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) for which x^{(n−1)} is locally absolutely continuous on ]a, b[ , i.e., absolutely continuous on [a + ε, b − ε] for arbitrarily small positive ε.

L_{α,β}(]a, b[ ; ℝ^{m}) and L_{α,β}(]a, b[ ; ℝ^{m×m}) are respectively the Banach space of vector functions y : ]a, b[ → ℝ^{m} and the Banach space of matrix functions Y : ]a, b[ → ℝ^{m×m} whose components are summable with weight (t − a)^{α}(b − t)^{β}. The norms in these spaces are defined by the equalities

∥y∥_{L_{α,β}} = ∫_{a}^{b}(t − a)^{α}(b − t)^{β}∥y(t)∥ dt, ∥Y ∥_{L_{α,β}} = ∫_{a}^{b}(t − a)^{α}(b − t)^{β}∥Y (t)∥ dt.

L_{α,β}(]a, b[ ; ℝ_{+}^{m}) = {y ∈ L_{α,β}(]a, b[ ; ℝ^{m}) : y(t) ∈ ℝ_{+}^{m} for t ∈]a, b[ }.

L_{α,β}(]a, b[ ; ℝ_{+}^{m×m}) = {Y ∈ L_{α,β}(]a, b[ ; ℝ^{m×m}) : Y (t) ∈ ℝ_{+}^{m×m} for t ∈]a, b[ }. In the sequel it will always be assumed that −∞ < a < b < +∞,

α ∈ [0, n − 1], β ∈ [0, n − 1],  (1.5)

whereas f : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → L_{α,β}(]a, b[ ; ℝ^{m}) and h_{i} : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → ℝ^{m} (i = 1, …, n) are continuous operators which, for each ρ ∈]0, +∞[, satisfy the conditions

sup {∥f(x)(·)∥ : ∥x∥_{C_{α,β}^{n−1}} ≤ ρ} ∈ L_{α,β}(]a, b[ ; ℝ_{+}),  (1.6)
sup {∥h_{i}(x)∥ : ∥x∥_{C_{α,β}^{n−1}} ≤ ρ} < +∞ (i = 1, …, n).  (1.7)

By a solution of the functional differential equation (1.1) is understood a vector function x ∈ C̃_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) satisfying (1.1) almost everywhere on ]a, b[ . A solution of (1.1) satisfying (1.2) is called a solution of problem (1.1), (1.2).

1.2. Theorem on the Fredholm property of a linear boundary value problem. We begin by introducing

Definition 1.1. A linear operator p : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → ℝ^{m} is called strongly bounded if there exists ζ ∈ L_{α,β}(]a, b[ ; ℝ_{+}) such that

∥p(x)(t)∥ ≤ ζ(t)∥x∥_{C_{α,β}^{n−1}} for a < t < b, x ∈ C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}). (1.8)

Consider the boundary value problem

x^{(n)}(t) = p(x)(t) + q(t),  (1.9)
ℓ_{i}(x) = c_{0i} (i = 1, …, n),  (1.10)

where p : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → L_{α,β}(]a, b[ ; ℝ^{m}) is a linear, strongly bounded operator, ℓ_{i} : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → ℝ^{m} (i = 1, …, m) are linear bounded operators,

q ∈ L_{α,β}(]a, b[ ; ℝ^{m}), c_{0i} ∈ ℝ^{m} (i = 1, …, m).

Theorem 1.1. For problem (1.9), (1.10) to be uniquely solvable it is necessary and sufficient that the corresponding homogeneous problem

x^{(n)}(t) = p(x)(t),  (1.9_{0})
ℓ_{i}(x) = 0 (i = 1, …, n)  (1.10_{0})

have only a trivial solution. Moreover, if problem (1.9_{0}) (1.10_{0}) has only a trivial solution, then there exists a positive constant γ such that for any q ∈ L_{α,β}(]a, b[ ; ℝ^{m}) and c_{0i} ∈ ℝ^{m} (i = 1, …, m), a solution x of problem (1.9), (1.10) admits the estimate

∥x∥_{C_{α,β}^{n−1}} ≤ γ( ∑_{i=1}^{n} ∥c_{0i}∥ + ∥q∥_{L_{α,β}} ).  (1.11)

The vector differential equation with deviating arguments

x^{(n)}(t) = ∑_{i=1}^{n} P_{i}(t)x^{(i−1)}(τ_{i}(t)) + q(t),  (1.12)

where τ_{i} : [a, b] → [a, b] (i = 1, …, n) are measurable functions, P_{i} : ]a, b[ → ℝ^{m×m} (i = 1, …, n) are matrix functions with measurable components and q ∈ L_{α,β}(]a, b[ ; ℝ^{m}), is a particular case of equation (1.9). Along with (1.12), consider the corresponding homogeneous equation

x^{(n)}(t) = ∑_{i=1}^{n} P_{i}(t)x^{(i−1)}(τ_{i}(t)).  (1.12_{0})

From Theorem 1.1 follows

Corollary 1.1. Let almost everywhere on ]a, b[ the inequalities

τ_{i}(t) > a for i > n − α, τ_{j}(t) < b for j > n − β  (1.13)

be fulfilled. Moreover,

∫_{a}^{b}(t − a)^{α}(b − t)^{β}(τ_{i}(t) − a)^{−α_{i}}(b − τ_{i}(t))^{−β_{i}}∥P_{i}(t)∥ dt

< +∞ (i = 1, …, n).^{∗)}  (1.14)

∗) Here and in the sequel it will be assumed that if α_{i} = 0 (β_{i} = 0), then (τ_{i}(t) − a)^{−α_{i}} ≡ 1 ((τ_{i}(t) − b)^{−β_{i}} ≡ 1).

Then for problem (1.12), (1.2) to be uniquely solvable, it is necessary and sufficient that the corresponding homogeneous problem (1.12_{0}), (1.2_{0}) have only a trivial solution. Moreover, if problem (1.12_{0}) (1.2_{0}) has only a trivial solution, then there exists a positive constant γ such that for any q ∈ L_{α,β}(]a, b[ ; ℝ^{m}) and c_{0i} ∈ ℝ^{m} (i = 1, …, m), a solution x of problem (1.12), (1.2) admits estimate (1.11).

1.3. A priori boundedness principle for the nonlinear problem (1.1), (1.2). To formulate this principle we have to introduce

Definition 1.2. Let γ be a positive number. The pair (p, (ℓ_{i})_{i=1}^{n}) of continuous operators p : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) × C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → L_{α,β}(]a, b[ ; ℝ^{m}) and (ℓ_{i})_{i=1}^{n} : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) × C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → ℝ^{mn} is said to be γ-consistent if:

(i) the operators p(x, ·) : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → L_{α,β}(]a, b[ ; ℝ^{m}) and ℓ_{i}(x, ·) : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → ℝ^{m} are linear for any fixed x ∈ C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) and i ∈ {1, …, n};

(ii) for any x and y ∈ C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) and for almost all t ∈]a, b[ we have inequalities

∥p(x, y)(t)∥ ≤ δ(t, ∥x∥_{C_{α,β}^{n−1}})∥y∥_{C_{α,β}^{n−1}}, ∑_{i=1}^{n} ∥ℓ_{i}(x, y)∥ ≤ δ_{0}(∥x∥_{C_{α,β}^{n−1}})∥y∥_{C_{α,β}^{n−1}},

where δ_{0} : ℝ_{+} → ℝ_{+} is nondecreasing, δ(·, ρ) ∈ L_{α,β}(]a, b[ ; ℝ_{+}) for every ρ ∈ ℝ_{+}, and δ(t, ·) : ℝ_{+} → ℝ_{+} is nondecreasing for every t ∈]a, b[ ;

(iii) for any x ∈ C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}), q ∈ L_{α,β}(]a, b[ ; ℝ^{m}) and c_{i} ∈ ℝ^{m} (i = 1, …, n), an arbitrary solution y of the boundary value problem

y^{(n)}(t) = p(x, y)(t) + q(t), ℓ_{i}(x, y) = c_{i} (i = 1, …, n)  (1.15)

admits the estimate

∥y∥_{C_{α,β}^{n−1}} ≤ γ( ∑_{i=1}^{n} ∥c_{i}∥ + ∥q∥_{L_{α,β}} ).  (1.16)

Definition 1.2′. The pair (p, (ℓ_{i})_{i=1}^{n}) of continuous operators p : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) × C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → L_{α,β}(]a, b[ ; ℝ^{m}) and (ℓ_{i})_{i=1}^{n} : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m})×C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → ℝ^{mn} is said to be consistent if there exists γ > 0 such that this pair is γ-consistent.

Theorem 1.2. Let there exist a positive number ρ_{0} and a consistent pair (p, (ℓ_{i})_{i=1}^{n}) of continuous operators p : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) × C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → ℝ^{mn} and (ℓ_{i})_{i=1}^{n} : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) × C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → ℝ^{mn} such that for any λ ∈]0, 1[ an arbitrary solution of the problem

x^{(n)}(t) = (1 − λ)p(x, x)(t) + λf(x)(t),  (1.17)
(λ − 1)ℓ_{i}(x, x) = λh_{i}(x) (i = 1, …, n)  (1.18)

admits the estimate

∥x∥_{C_{α,β}^{n−1}} ≤ ρ_{0}.  (1.19)

Then problem (1.1), (1.2) is solvable.

For n = 1 and α = β = 0, Theorem 1.2 implies Theorem 1 from [27].

Corollary 1.2. Let there exist a positive number γ, a γ-consistent pair (p, (ℓ_{i})_{i=1}^{n}) of continuous operators p : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) × C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → L_{α,β}(]a, b[ ; ℝ^{m}), (ℓ_{i})_{i=1}^{n} : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) × C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → ℝ^{mn} and functions η : ]a, b[ ×ℝ_{+} → ℝ_{+} and η_{0} : ℝ_{+} → ℝ_{+} such that the inequalities

∥f(x)(t) − p(x, x)(t)∥ ≤ η(t, ∥x∥_{C_{α,β}^{n−1}}),  (1.20)
∑_{i=1}^{n} ∥h_{i}(x) − ℓ_{i}(x, x)∥ ≤ η_{0}(∥x∥_{C_{α,β}^{n−1}})  (1.21)

are fulfilled for any x ∈ C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) and almost all t ∈]a, b[ . Moreover, η(·, ρ) ∈ L_{α,β}(]a, b[ ; ℝ_{+}) for ρ ∈ ℝ_{+} and

lim sup_{ρ→+∞} (\frac{η_{0}(ρ)}{ρ} + \frac{1}{ρ} ∫_{a}^{b}(s − a)^{α}(b − s)^{β}η(s, ρ) ds) < \frac{1}{γ} .  (1.22)

Then problem (1.1), (1.2) is solvable.

As an example, in C_{α,0}^{n−1}(]a, b[ ; ℝ^{m}) consider the boundary value problem

x^{(n)}(t) = g(t, x(τ_{1}(t)), …, x^{(n−1)}(τ_{n}(t))),  (1.23)
\begin{matrix}lim_{t→a} x^{(i−1)}(t) = c_{i}(x) (i = 1, …, k), \\ lim_{t→b} x^{(i−1)}(t) = c_{i}(x) (i = k + 1, …, n). \end{matrix}  (1.24)

Here k ∈ {1, …, n − 1}, α ∈ [0, n − k], τ_{i} : [a, b] → [a, b] (i = 1, …, n) are measurable functions, c_{i} : C̃_{α,0}^{n−1}(]a, b[ ; ℝ^{m}) → ℝ^{m} (i = 1, …, m) are continuous operators, and g : ]a, b[ ×ℝ^{mn} → ℝ^{m} is a vector function such that g(·, x_{1}, …, x_{n}) : ]a, b[ → ℝ^{m} is measurable for any x_{i} ∈ ℝ^{m} (i = 1, …, n) and g(t, ·, …, ·) : ℝ^{mn} → ℝ^{m} is continuous for almost all t ∈]a, b[ . We will also suppose that for i > n − α the inequality

τ_{i}(t) > a

holds almost everywhere on ]a, b[ .

The following statement is valid.

Corollary 1.3. Let there exist η_{0} : ℝ_{+} → ℝ_{+}, P_{i} ∈ L_{α,0}(]a, b[ ; ℝ_{+}^{mn}) (i = 1, …, n) and q : ]a, b[ ×ℝ_{+} → ℝ_{+}^{m} such that

∑_{i=1}^{n} ∥c_{i}(x)∥ ≤ η_{0}(∥x∥_{C_{α,0}^{n−1}}) for x ∈ C_{α,0}^{n−1}(]a, b[ ; ℝ^{m})  (1.25)

and on ]a, b[ ×ℝ^{mn} the inequality

|g(t, x_{1}, …, x_{n})|
≤ ∑_{i=1}^{n} (τ_{i}(t) − a)^{α_{i}}P_{i}(t)|x_{i}| + q(t, ∑_{i=1}^{n} (τ_{i}(t) − a)^{α_{i}}∥x_{i}∥)  (1.26)

holds. Let, moreover, q(·, ρ) ∈ L_{α,0}(]a, b[ ; ℝ_{+}^{m}) for every ρ ∈ ℝ_{+}, the components of q(t, ρ) are nondecreasing with respect to ρ,

lim_{ρ→+∞} (\frac{η_{0}(ρ)}{ρ} + \frac{1}{ρ} ∫_{a}^{b}(s − a)^{α}∥q(s, ρ)∥ ds) = 0  (1.27)

and

r(P) < 1,  (1.28)

where

P = ∑_{i=1}^{k} \frac{(b − a)^{n−k−1−α+α_{k+1}}}{(n − k − 1)!(k + 1 − i − α_{k+1})!} ∫_{a}^{b}(s − a)^{α}(τ_{i}(s) − a)^{k+1−i−α_{k+1}}P_{i}(s) ds

+ ∑_{i=k+1}^{n} \frac{(b − a)^{n−i−α+α_{i}}}{(n − i)!} ∫_{a}^{b}(s − a)^{α}P_{i}(s) ds.

Then problem (1.23), (1.24) is solvable.

Before passing to the formulation of the next corollary we introduce

Definition 1.3. An operator p : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → L_{α,β}(]a, b[ ; ℝ^{m}) (an operator ℓ : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → ℝ^{m}) is called positive homogeneous if the equality

p(λx)(t) = λp(x)(t) ( ℓ(λx) = λℓ(x) )

is fulfilled for all x ∈ C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}), λ ∈ ℝ_{+} and almost all t ∈]a, b[ .

Definition 1.4. A positive homogeneous operator p : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → L_{α,β}(]a, b[ ; ℝ^{m}) (a positive homogeneous operator ℓ : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → ℝ^{m}) is called strongly bounded (bounded) if there exists a function ζ ∈ L_{α,β}(]a, b[ ; ℝ_{+}) (a positive number ζ_{0}) such that the inequality

∥p(x)(t)∥ ≤ ζ(t)∥x∥_{C_{α,β}^{n−1}} ( ∥ℓ(x)∥ ≤ ζ_{0}∥x∥_{C_{α,β}^{n−1}})

holds for all x ∈ C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) and almost all t ∈]a, b[.

Corollary 1.4. Let there exist a linear, strongly bounded operator p : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → L_{α,β}(]a, b[ ; ℝ^{m}), a positive homogeneous, continuous, strongly bounded operator \overline{p} : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → L_{α,β}(]a, b[ ; ℝ^{m}), linear bounded operators ℓ_{i} : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → ℝ^{m} (i = 1, …, n), positive homogeneous, continuous, bounded operators \overline{ℓ}_{i} : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → ℝ^{m} (i = 1, …, m), and functions η :]a, b[ ×ℝ_{+} and η_{0} : ℝ_{+} → ℝ_{+} such that the inequalities

∥f(x)(t) − p(x)(t) − \overline{p}(x)(t)∥ ≤ η(t, ∥x∥_{C_{α,β}^{n−1}}),  (1.29)
∑_{i=1}^{n} ∥h_{i}(x) − ℓ_{i}(x) − \overline{ℓ}_{i}(x)∥ ≤ η_{0}(∥x∥_{C_{α,β}^{n−1}})  (1.30)

hold for any x ∈ C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) and for almost all t ∈]a, b[ . Moreover, η(·, ρ) ∈ L_{α,β}(]a, b[ ; ℝ_{+}) for any ρ ∈ ℝ_{+},

lim_{ρ→+∞} (\frac{η_{0}(ρ)}{ρ} + \frac{1}{ρ} ∫_{a}^{b}(s − a)^{α}(b − s)^{β}η(s, ρ) ds) = 0  (1.31)

and for any λ ∈ [0, 1] the problem

x^{(n)}(t) = p(x)(t) + λ\overline{p}(x)(t), ℓ_{i}(x) + λ\overline{ℓ}_{i}(x) = 0 (i = 1, …, n) (1.32)

has only a trivial solution. Then problem (1.1), (1.2) is solvable.

As an example, for the second order singular half-linear differential equation

u′′(t) = p_{1}(t)|u(t)|^{μ}|u′(t)|^{1−μ} sgn u(t) + p_{2}(t)u′(t) + p_{0}(t)  (1.33)

let us consider the two-point boundary value problems

lim_{t→a} u(t) = c_{1}, lim_{t→b} u(t) = c_{2}  (1.34_{1})

and

lim_{t→a} u(t) = c_{1}, lim_{t→b} u′(t) = c_{2}.  (1.34_{2})

We are interested in the case where μ ∈ [0, 1] and p_{i} : ]a, b[ → ℝ (i = 0, 1, 2) are measurable functions satisfying either the conditions

∫_{a}^{b}(t − a)(b − t)|p_{i}(t)| dt < +∞ (i = 0, 1), ∫_{a}^{b} |p_{2}(t)| dt < +∞, (1.35_{1})

p_{1}(t) ≥ −λ_{1}[σ(t)]^{1+μ}, [p_{2}(t) − \frac{σ′(t)}{σ(t)}] sgn(t_{0} − t) ≥ −λ_{2}σ(t) (1.36_{1})
for a < t < b,

or the conditions

∫_{a}^{b}(t − a)|p_{0}(t)| dt < +∞ (i = 0, 1), ∫_{a}^{b} |p_{2}(t)| dt < +∞,  (1.35_{2})

p_{1}(t) ≥ −λ_{1}[σ(t)]^{1+μ}, p_{2}(t) − \frac{σ′(t)}{σ(t)} ≥ −λ_{2}σ(t) for a < t < b. (1.36_{2})

Here t_{0} ∈]a, b[ , λ_{i} ∈ ℝ_{+} (i = 1, 2), and σ : ]a, b[ → ℝ_{+} is a locally absolutely continuous function such that either

σ′(t) sgn(t_{0} − t) ≤ 0 for a < t < b, ∫_{0}^{+∞}\frac{ds}{λ_{1} + λ_{2}s + s^{(1+μ)/μ}}

> \frac{μ}{2}[ ∫_{a}^{b} σ(s) ds + | ∫_{a}^{t_{0}} σ(s) ds − ∫_{t_{0}}^{b} σ(s) ds|],  (1.37_{1})

or

∫_{0}^{+∞}\frac{ds}{λ_{1} + λ_{2}s + s^{(1+μ)/μ}} > μ ∫_{a}^{b} σ(s) ds.  (1.37_{2})

By virtue of Theorems 3.1 and 3.2 from [9] Corollary 1.4 implies

Corollary 1.5. Let conditions (1.35_{i}), (1.36_{i}) and (1.37_{i}) be fulfilled for some i ∈ {1, 2}. Then problem (1.33), (1.34_{i}) has at least one solution.

This corollary is a generalization of the classical result of Ch. de la Vallée-Poussin [44] for equation (1.33).

2. Auxiliary Propositions

Lemma 2.1. Let ρ > 0, η ∈ L_{α,β}(]a, b[ ; ℝ_{+}), t_{0} ∈]a, b[ , and S be the set of (n − 1)-times continuously differentiable vector functions x : ]a, b[ → ℝ^{m} satisfying the conditions

∥x^{(i−1)}(t_{0})∥ ≤ ρ (i = 1, …, n),  (2.1)
∥x^{(n−1)}(t) − x^{(n−1)}(s)∥ ≤ ∫_{s}^{t} η(ξ) dξ for a < s ≤ t < b.  (2.2)

Then S ⊂ C̃_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) and S is a compact set of the space C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}).

Proof. Let x be an arbitrary element of the set S. Then by (2.2) the function x^{(n−1)} is locally absolutely continuous on ]a, b[ and

∥x^{(n)}(t)∥ ≤ η(t) for almost all t ∈]a, b[ .  (2.3)

Therefore

x^{(n)} ∈ L_{α,β}(]a, b[ ; ℝ^{m}),  (2.4)

x^{(i−1)}(t) = ∑_{j=i}^{n} \frac{(t − t_{0})^{j−i}}{(j − i)!} x^{(j−1)}(t_{0})

+\frac{1}{(n − i)!} ∫_{t_{0}}^{t}(t − s)^{n−i}x^{(n)}(s) ds for a < t < b (i = 1, …, n), (2.5)

800 I. KIGURADZE, B. PŮŽA, AND I. P. STAVROULAKIS

and

∥x^{(i−1)}(t)∥ ≤ ε_{i}(t) for a < t < b (i = 1, …, n),  (2.6)

where

ε_{i}(t) = ρ ∑_{j=i}^{n} \frac{(b − a)^{j−i}}{(j − i)!} + \frac{1}{(n − i)!}| ∫_{t_{0}}^{t}(t − s)^{n−i}η(s) ds| (i = 1, …, n). (2.7)

Let

i_{1} = max{i : α_{i} = 0}, i_{2} = max{i : β_{i} = 0}.

Then

n − i ≥ α, α_{i} = 0 for i ≤ i_{1}, α_{i} = α + i − n > 0 for i > i_{1}, (2.8_{1})
n − i ≥ β, β_{i} = 0 for i ≤ i_{2}, β_{i} = β + i − n > 0 for i > i_{2}. (2.8_{2})

Therefore

ε_{i}(t) ≤ ε_{i}(a+) < +∞ for i ≤ i_{1}, a < t ≤ t_{0},  (2.9)
∫_{a}^{t_{0}} ε_{1+i_{1}}(s) ds < +∞ if i_{1} < n − 1,  (2.10)
ε_{i}(t) ≤ ε_{i}(b−) < +∞ for i ≤ i_{2}, t_{0} ≤ t < b,  (2.11)
∫_{t_{0}}^{b} ε_{1+i_{2}}(s) ds < +∞ if i_{2} < n − 1.  (2.12)

If i > i_{1}, then, with (2.7) and (2.8_{1}) taken into account, for any δ ∈]0, t_{0} − a[ we find

lim sup_{t→a} [(t − a)^{α_{i}}ε_{i}(t)] = lim sup_{t→a} [\frac{(t − a)^{α+i−n}}{(n − i)!} ∫_{t}^{a+δ}(s − t)^{n−i}η(s) ds]

≤ \frac{1}{(n − i)!} ∫_{a}^{a+δ}(s − a)^{α}η(s) ds.

Hence, because of the arbitrariness of δ, it follows that

lim_{t→a} [(t − a)^{α_{i}}ε_{i}(t)] = 0 for i > i_{1}.  (2.13)

Analogously, it can be shown that

lim_{t→b} [(b − t)^{β_{i}}ε_{i}(t)] = 0 for i > i_{2}.  (2.14)

If i ≤ i_{1} (if i ≤ i_{2}), then by virtue of conditions (2.3) and (2.8_{1}) (conditions (2.3) and (2.8_{2})) we have

∫_{a}^{t_{0}}(s − a)^{n−i}∥x^{(n)}(s)∥ ds < +∞ ( ∫_{t_{0}}^{b}(b − s)^{n−i}∥x^{(n)}(s)∥ ds < +∞ ).

Hence (2.5) implies the existence of the limit

lim_{t→a} x^{(i−1)}(t) ( lim_{t→b} x^{(i−1)}(t) ).

If however i > i_{1} (i > i_{2}), then from (2.6) and (2.13) (from (2.6) and (2.14)) we have

lim_{t→a}(t − a)^{α_{i}}x^{(i−1)}(t) = 0 ( lim_{t→b}(b − t)^{β_{i}}x^{(i−1)}(t) = 0 ).

We have thereby proved the existence of limit (1.3). Therefore S ⊂ C̃_{α,β}^{n−1}(]a, b[ ; ℝ^{m}).

By the Arzela–Ascoli lemma, from estimates (2.3), (2.6) and conditions (2.9)– (2.14) it follows that S is a compact set of the space C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}).

Let (p, (ℓ_{i})_{i=1}^{n}) be a γ-consistent pair of continuous operators p : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) × C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → L_{α,β}(]a, b[ ; ℝ^{m}) and (ℓ_{i})_{i=1}^{n} : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) × C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → ℝ^{mn}, and q : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → L_{α,β}(]a, b[ ; ℝ^{m}), c_{0i} : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → ℝ^{m} (i = 1, …, n) be continuous operators. For any x ∈ C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}), consider the linear boundary value problem

y^{(n)}(t) = p(x, y)(t) + q(x)(t), ℓ_{i}(x, y) = c_{0i}(x) (i = 1, …, n). (2.15)

By condition (iii) of Definition 1.2, the homogeneous problem

y^{(n)}(t) = p(x, y)(t), ℓ_{i}(x, y) = 0 (i = 1, …, n)  (2.15_{0})

has only a trivial solution. By Theorem 1.1 this fact guarantees the existence of a unique solution y of problem (2.15). We write

u(x)(t) = y(t).

Lemma 2.2. u : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) is a continuous operator.

Proof. Let

x_{i} ∈ C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}), y_{i}(t) = u(x_{i})(t) (i = 1, 2)

and

y(t) = y_{2}(t) − y_{1}(t).

Then

y^{(n)}(t) = p_{2}(x_{2}, y)(t) + q_{0}(x_{1}, x_{2})(t),
ℓ_{i}(x_{2}, y) = c_{i}(x_{1}, x_{2}) (i = 1, …, n),

where

q_{0}(x_{1}, x_{2})(t) = p(x_{1}, y_{1})(t) − p(x_{2}, y_{1})(t) + q(x_{2})(t) − q(x_{1})(t),
c_{i}(x_{1}, x_{2}) = ℓ_{i}(x_{1}, x_{2}) − ℓ_{i}(x_{2}, y_{1}) + c_{0i}(x_{2}) − c_{0i}(x_{1}) (i = 1, …, n).

Hence, by condition (iii) of Definition 1.2 we have

∥u(x_{2}) − u(x_{1})∥_{C_{α,β}^{n−1}} ≤ γ( ∑_{i=1}^{n} ∥c_{i}(x_{1}, x_{2})∥ + ∥q_{0}(x_{1}, x_{2})∥_{L_{α,β}} ).

Since the operators p, q, ℓ_{i} and c_{0i} (i = 1, …, n) are continuous, this estimate implies the continuity of the operator u.

Lemma 2.3. Let k ∈ {1, …, n − 1}, α ∈ [0, n − k], and x ∈ C_{α,0}^{n−1}(]a, b[ ; ℝ^{m}) be a vector function satisfying conditions (1.24). Then on ]a, b[ the following inequalities are fulfilled:

|x^{(i−1)}(t)| ≤ ∑_{j=i}^{n}(b − a)^{j−i}|c_{j}(x)|

+\frac{1}{(n − i)!} (b − a)^{n−i−α+α_{i}}(t − a)^{−α_{i}}y(x) (i = k + 1, …, n), (2.16)

|x^{(i−1)}(t)| ≤ ∑_{j=i}^{n}(b − a)^{j−i}|c_{j}(x)|

+\frac{(b − a)^{n−k−1−α+α_{k+1}}}{(n − k − 1)!(k + 1 − i − α_{k+1})!} (t − a)^{k+1−i−α_{k+1}}y(x) (i = 1, …, k), (2.17)

where

y(x) = ∫_{a}^{b}(s − a)^{α}|x^{(n)}(s)| ds.  (2.18)

Proof. Let x_{0}(t) be a polynomial of degree not higher than n − 1 satisfying the conditions

x_{0}^{(i−1)}(a) = c_{i}(x) (i = 1, …, k), x_{0}^{(i−1)}(b) = c_{i}(x) (i = k + 1, …, n).

Then

|x_{0}^{(i−1)}(t)| ≤ ∑_{j=i}^{n}(b − a)^{j−i}|c_{j}(x)| for a ≤ t ≤ b (i = 1, …, n). (2.19)

On the other hand,

x^{(i−1)}(t) = x_{0}^{(i−1)}(t) − \frac{(−1)^{n−i}}{(n − i)!} ∫_{t}^{b}(s − t)^{n−i}x^{(n)}(s) ds  (2.20)
(i = k + 1, …, n),
x^{(i−1)}(t) = c_{i}(x) + ∫_{a}^{t} x^{(i)}(s) ds (i = 1, …, k).  (2.21)

By (1.4)

n − i − α − α_{i} ≥ 0 (i = 1, …, n).

Therefore

(s − t)^{n−i} ≤ (s − a)^{n−i−α+α_{i}}(s − a)^{−α_{i}}(s − a)^{α}
≤ (b − a)^{n−i−α+α_{i}}(t − a)^{−α_{i}}(s − a)^{α} for t ≤ s < b (i = 1, …, n).

If along with this we take into account inequality (2.19), then from (2.20) we obtain estimates (2.16).

It is clear that

α_{k+1} ≤ 1,

since α ≤ n − k. If α_{k+1} < 1, then by virtue of (2.16) and (2.19), from (2.21) follow estimates (2.17).

To complete the proof of the lemma it remains to consider the case where α_{k+1} = 1. Then α = n − k and thus from (2.19)–(2.21) we find

|x^{(k−1)}(t)| ≤ ∑_{j=k}^{n}(b − a)^{j−k}|c_{j}(x)|

+\frac{1}{(n − k − 1)!} ∫_{a}^{t} ( ∫_{τ}^{b}(s − τ)^{n−k−1}|x^{(n)}(s)| ds) dτ

= ∑_{j=k}^{n}(b − a)^{j−k}|c_{j}(x)|

+\frac{1}{(n − k − 1)!} [(t − a) ∫_{t}^{b}(s − a)^{n−k−1}|x^{(n)}(s)| ds + ∫_{a}^{t}(s − a)^{n−k}|x^{(n)}(s)| ds]

≤ ∑_{j=k}^{n}(b − a)^{j−k}|c_{j}(x)| + \frac{1}{(n − k − 1)!} y(x)

and

|x^{(i−1)}(t)| ≤ ∑_{j=i}^{n}(b − a)^{j−i}|c_{j}(x)| + \frac{1}{(n − k − 1)!(k − i)!} (t − a)^{k−i}y(x)

(i = 1, …, k).

Therefore estimates (2.17) are valid.

3. Proof of the Main Results

Proof of Theorem 1.1. Let B = C_{α,β}^{n−1}(]a, b[ ; ℝ^{m})×ℝ^{mn} be a Banach space with elements u = (x; c_{1}, …, c_{n}), where x ∈ C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}), c_{i} ∈ ℝ^{m} (i = 1, …, n), and the norm

∥u∥_{B} = ∥u∥_{C_{α,β}^{n−1}} + ∑_{i=1}^{n} ∥c_{i}∥.

Fix arbitrarily t_{0} ∈]a, b[ and, for any u = (x; c_{1}, …, c_{n}), set

p̃(u)(t) = ( ∑_{i=1}^{n} \frac{(t − t_{0})^{i−1}}{(i − 1)!} (c_{i} + x^{(i−1)}(t_{0}))

+\frac{1}{(n − 1)!} ∫_{t_{0}}^{t}(t − s)^{n−1}p(x)(s) ds; c_{1} − ℓ_{1}(x), …, c_{n} − ℓ_{n}(x)),

q̃(t) = (\frac{1}{(n − 1)!} ∫_{t_{0}}^{t}(t − s)^{n−1}q(s) ds; c_{01}, …, c_{0n}).

Problem (1.9), (1.10) is equivalent to the operator equation

u = p̃(u) + q̃  (3.1)

in the space B since u = (x; c_{1}, …, c_{n}) is a solution of equation (3.1) if and only if c_{i} = 0 (i = 1, …, n) and x is a solution of problem (1.9), (1.10). As for the homogeneous equation

u = p̃(u)  (3.1_{0})

it is equivalent to the homogeneous problem (1.9_{0}), (1.10_{0}).

From condition (1.8) and Lemma 2.1 it immediately follows that the linear operator p̃: B → B̃ is compact. By this fact and the Fredholm alternative for operator equations ([13], Ch. XIII, § 5, Theorem 1), equation (3.1) is uniquely solvable if and only if equation (3.1_{0}) has only a trivial solution. Moreover, if equation (3.1_{0}) has only a trivial solution, then the operator I − p̃ is invertible and (I − p̃)^{−1} : B → B is a linear bounded operator, where I : B → B is an identical operator. Therefore there exists γ_{0} > 0 such that for any q̃ ∈ B the solution u of equation (3.1) admits the estimate

∥u∥_{B} ≤ γ_{0}∥q̃∥_{B}.

However,

∥q̃∥_{B} ≤ ∑_{i=1}^{n} ∥c_{0i}∥ + γ_{1}∥q∥_{L_{α,β}},

where γ_{1} > 1 is a constant depending only on α, β, a, b, t_{0} and n. Hence

∥u∥_{B} ≤ γ( ∑_{i=1}^{n} ∥c_{0i}∥ + ∥q∥_{L_{α,β}} ),  (3.2)

where γ = γ_{0}γ_{1}.

Since problem (1.9), (1.10) is equivalent to equation (3.1), it is clear that problem (1.9), (1.10) is uniquely solvable if and only if problem (1.9_{0}), (1.10_{0}) has only a trivial solution. Moreover, if (1.9_{0}), (1.10_{0}) has only a trivial solution, then by virtue of (3.2) the solution x of problem (1.9), (1.10) admits estimate (1.11).

Proof of Corollary 1.1. We set

p(x)(t) = ∑_{i=1}^{n} P_{i}(t)x^{(i−1)}(τ_{i}(t))

for any x ∈ C_{α,β}^{n−1}(]a, b[; ℝ^{m}). Then equations (1.12) and (1.12_{0}) take respectively forms (1.9) and (1.9_{0}). On the other hand, in view of (1.13) and (1.14)

p : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → L_{α,β}(]a, b[ ; ℝ^{m})

is a strongly bounded linear operator. Therefore the conditions of Theorem 1.1 are fulfilled.

Proof of Theorem 1.2. Let δ, δ_{0} and γ be the functions and numbers appearing in Definitions 1.2 and 1.2′. We set

η(t) = 2ρ_{0}δ(t, 2ρ_{0}) + sup {∥f(x)(t)∥ : ∥x∥_{C_{α,β}^{n−1}} ≤ 2ρ_{0}},

η_{0} = 2ρ_{0}δ_{0}(2ρ_{0}) + ∑_{i=1}^{n} sup {∥h_{i}(x)∥ : ∥x∥_{C_{α,β}^{n−1}} ≤ 2ρ_{0}},

ρ_{1} = γ(η_{0} + ∥η∥_{L_{α,β}} ), η^{∗}(t) = δ(t, ρ_{1})ρ_{0} + η(t),  (3.3)
B_{0} = {x ∈ C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) : ∥x∥_{C_{α,β}^{n−1}} ≤ ρ_{1}},  (3.4)

χ(s) = \begin{cases}1 & for 0 ≤ s ≤ ρ_{0} \\ 2 − s/ρ_{0} & for ρ_{0} < s < 2ρ_{0} \\ 0 & for s ≥ 2ρ_{0} \end{cases} ,  (3.5)

q(x)(t) = χ(∥x∥_{C_{α,β}^{n−1}})[f(x)(t) − p(x, x)(t)],  (3.6)
c_{0i}(x) = χ(∥x∥_{C_{α,β}^{n−1}})[ℓ_{i}(x, x) − h_{i}(x)] (i = 1, …, n).  (3.7)

By (1.6) and (1.7)

η_{0} < +∞, η ∈ L_{α,β}(]a, b[ ; ℝ_{+}), η^{∗} ∈ L_{α,β}(]a, b[ ; ℝ_{+})

and for every x ∈ C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) and almost all t ∈]a, b[ we have the inequalities

∥q(x)(t)∥ ≤ η(t), ∑_{i=1}^{n} ∥c_{0i}(x)∥ ≤ η_{0}.  (3.8)

Let u : C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) → C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) be an operator which to every x ∈ C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) assigns the solution y of problem (2.15). By Lemma 2.1, u is a continuous operator. On the other hand, by conditions (ii) and (iii) of Definition 1.2, notations (3.3), (3.4) and inequalities (3.8), the vector function y = u(x) satisfies, for each x ∈ B_{0}, the conditions

∥y∥_{C_{α,β}^{n−1}} ≤ ρ_{1}, ∥y^{(n−1)}(t) − y^{(n−1)}(s)∥ ≤ ∫_{s}^{t} η^{∗}(ξ) dξ for a < s ≤ t < b.

By Lemma 2.2 this implies that the operator u maps the ball B_{0} into its own compact subset. Therefore, owing to Schauder’s principle, there exists x ∈ B_{0} such that

x(t) = u(x)(t) for a < t < b.

By notations (3.6), (3.7) the function x is a solution of problem (1.17), (1.18), where

λ = χ(∥x∥_{C_{α,β}^{n−1}}).  (3.9)

Let us show that x admits estimate (1.19). Assume the contrary. Then either

ρ_{0} < ∥x∥_{C_{α,β}^{n−1}} < 2ρ_{0},  (3.10)

or

∥x∥_{C_{α,β}^{n−1}} ≥ 2ρ_{0}.  (3.11)

If condition (3.10) is fulfilled, then by virtue of (3.5) and (3.9)

λ ∈]0, 1[ ,

which, by one of the conditions of the theorem, guarantees the validity of estimate (1.19). But this contradicts condition (3.10).

Assume now that inequality (3.11) is fulfilled. Then by virtue of (3.5) and (3.9)

λ = 0

and therefore x is a solution of problem (2.15_{0}). Thus x(t) ≡ 0 since problem (2.15_{0}) has only a trivial solution. But this contradicts inequality (3.11). The contradiction obtained proves the validity of estimate (1.19).

By (1.19), (3.5)–(3.7) and (3.9), it clearly follows from (1.17), (1.18) that λ = 1 and x is a solution of problem (1.1), (1.2).

Proof of Corollary 1.2. By (1.22) there is ρ_{0} > 0 such that

γ(η_{0}(ρ) + ∫_{a}^{b}(s − a)^{α}(b − s)^{β}η(s, ρ) ds) < ρ for ρ > ρ_{0}.  (3.12)

Let x be a solution of problem (1.17), (1.18) for some λ ∈]0, 1[ . Then y = x is also a solution of problem (1.15) where

q(t) = λ(f(x)(t) − p(x, x)(t)),
c_{0i}(x) = λ(ℓ_{i}(x, x) − h_{i}(x)) (i = 1, …, n).

Assume that

ρ = ∥x∥_{C_{α,β}^{n−1}}.

By the γ-consistency of the pair (p, (ℓ_{i})_{i=1}^{n}) and inequalities (1.20), (1.21) we have

ρ ≤ γ( ∑_{i=1}^{n} ∥c_{0i}(x)∥ + ∥q∥_{L_{α,β}} )

≤ γ(η_{0}(ρ) + ∫_{a}^{b}(s − a)^{α}(b − s)^{β}η(s, ρ) ds).

Hence by (3.12) it follows that ρ ≤ ρ_{0}. Therefore estimate (1.19) is valid, which due to Theorem 1.2 guarantees the solvability of problem (1.1), (1.2).

Proof of Corollary 1.3. Problem (1.23), (1.24) is obtained from problem (1.1), (1.2) when

f(x)(t) ≡ g(t, x(τ_{1}(t)), …, x^{(n−1)}(τ_{n}(t))),  (3.13)
\begin{aligned}h_{i}(x) & = lim_{t→a} x^{(i−1)}(t) − c_{i}(x) (i = 1, …, k), \\ h_{i}(x) & = lim_{t→b} x^{(i−1)}(t) − c_{i}(x) (i = k + 1, …, n). \end{aligned}  (3.14)

By virtue of the restrictions imposed on g, τ_{i}, c_{i} (i = 1, …, n) and the inequality α ≤ n − k it is obvious that f : C_{α,0}^{n−1}(]a, b[ ; ℝ^{m}) → L_{α,0}(]a, b[ ; ℝ^{m}) and h_{i} : C_{α,0}^{n−1}(]a, b[ ; ℝ^{m}) → ℝ^{m} (i = 1, …, m) are continuous operators satisfying conditions (1.6) and (1.7), where β = 0.

Assume for any x, y ∈ C_{α,0}^{n−1}(]a, b[ ; ℝ^{m}) and t ∈]a, b[ that

\begin{matrix}p(x, y)(t) = 0, ℓ_{i}(x, y) = lim_{t→a} y^{(i−1)}(t) (i = 1, …, k), \\ ℓ_{i}(x, y) = lim_{t→b} y^{(i−1)}(t) (i = k + 1, …, n). \end{matrix}  (3.15)

According to Definition 1.2′ and Theorem 1.2 the pair (p, (ℓ_{i})_{i=1}^{n}) of continuous operators p : C_{α,0}^{n−1}(]a, b[ ; ℝ^{m}) × C_{α,0}^{n−1}(]a, b[ ; ℝ^{m}) → L_{α,0}(]a, b[ ; ℝ^{m}), (ℓ_{i})_{i=1}^{n} : C_{α,0}^{n−1}(]a, b[ ; ℝ^{m}) × C_{α,0}^{n−1}(]a, b[ ; ℝ^{m}) → ℝ^{mn}, is consistent.

To prove Corollary 1.3, by Theorem 1.2 it is sufficient to show that for each λ ∈]0, 1[ an arbitrary solution x of problem (1.17), (1.18) admits the estimate

∥x∥_{C_{α,0}^{n−1}} ≤ ρ_{0},  (3.16)

where ρ_{0} is a non-negative constant not depending on λ and x.

By virtue of (3.13)–(3.15) problem (1.17), (1.18) takes the form

x^{(n)}(t) = λg(t, x(τ_{1}(t)), …, x^{(n−1)}(τ_{n}(t))),  (3.17)
\begin{aligned}lim_{t→a} x^{(i−1)}(t) & = λc_{i}(x) (i = 1, …, k), \\ lim_{t→b} x^{(i−1)}(t) & = λc_{i}(x) (i = k + 1, …, n). \end{aligned}  (3.18)

Let x be a solution of problem (3.17), (3.18) for some λ ∈]0, 1[ . Then by virtue of Lemma 2.3 we conclude that estimates (2.16), (2.17), where y(x) is the vector given by equality (2.18), are true. On the other hand, on account of (1.26) we have

y(x) ≤ ∑_{i=1}^{n} ∫_{a}^{b}(s − a)^{α}(τ_{i}(s) − a)^{α_{i}}P_{i}(s)|x^{(i−1)}(s)| ds

+ ∫_{a}^{b}(s − a)^{α}q(s, ∥x∥_{C_{α,0}^{n−1}}) ds.

If, along with (2.16) and (2.17), we take into account that α_{i} = 0 (i = 1, …, k), then from the latter inequality we obtain

y(x) ≤ Py(x) + y_{0}(x)

and therefore

(E − P)y(x) ≤ y_{0}(x),  (3.19)

where E is the unique m × m matrix and

y_{0}(x) = ∑_{i=1}^{n} ( ∫_{a}^{b} (τ_{i}(s) − a)^{α_{i}}P_{i}(s) ds) ∑_{j=i}^{n}(b − a)^{j−i}|c_{j}(x)|

+ ∫_{a}^{b}(s − a)^{α}q(s, ∥x∥_{C_{α,0}^{n−1}}) ds.  (3.20)

By the nonnegativeness of the matrix P and inequality (1.28), from (3.19) it follows that

y(x) ≤ (E − P)^{−1}y_{0}(x).

If along with this we take into account condition (1.25) and equality (3.20), then (2.16) and (2.17) imply that

∥x∥_{C_{α,0}^{n−1}} ≤ η_{1}(∥x∥_{C_{α,0}^{n−1}}),  (3.21)

where

η_{1}(ρ) = μ(η_{0}(ρ) + ∫_{a}^{b}(s − a)^{α}∥q(s, ρ)∥ ds),

and μ is a positive constant depending only on α_{i}, P_{i}, τ_{i} (i = 1, …, n), a and b. On the other hand, due to condition (1.27) we have

lim_{ρ→+∞} \frac{η_{1}(ρ)}{ρ} = 0

and therefore

η_{1}(ρ) < ρ for ρ > ρ_{0},

where

ρ_{0} = inf \{ρ > 0 : \frac{η(s)}{s} < 1 for s ∈ [ρ, +∞[ \}.

Therefore from (3.21) we obtain estimate (3.16). On the other hand, it is obvious that the constant ρ_{0} does not depend on λ and x.

Proof of Corollary 1.4. The strong boundedness of the operators p and \overline{p} and the boundedness of the operators (ℓ_{i})_{i=1}^{n} and (\overline{ℓ}_{i})_{i=1}^{n} guarantee the existence of ζ ∈ L_{α,β}(]a, b[ ; ℝ_{+}) and ζ_{0} ∈ ℝ_{+} such that the inequalities

\begin{aligned}∥p(x)(t)∥ + ∥\overline{p}(x)(t)∥ & ≤ ζ(t)∥x∥_{C_{α,β}^{n−1}}, \\ ∑_{i=1}^{n} (∥ℓ_{i}(x)∥ + ∥\overline{ℓ}_{i}(x)∥) & ≤ ζ_{0}∥x∥_{C_{α,β}^{n−1}} \end{aligned}  (3.22)

hold for each x ∈ C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}) and almost all t ∈]a, b[ .

By Theorem 1.1 and Definition 1.2′ the pair (p, (ℓ_{i})_{i=1}^{n}) is consistent since for λ = 0 problem (1.32) has only a trivial solution.

Let us consider for arbitrary λ ∈ [0, 1], q ∈ L_{α,β}(]a, b[ ; ℝ^{m}) and c_{0i} ∈ ℝ^{m} (i = 1, …, n) the boundary value problem

x^{(n)}(t) = p(x)(t) + λ\overline{p}(x)(t) + q(t),  (3.23)
ℓ_{i}(x) + λ\overline{ℓ}_{i}(x) = c_{0i} (i = 1, …, n)  (3.24)

and prove that every solution x of this problem admits the estimate

∥x∥_{C_{α,β}^{n−1}} ≤ γ( ∑_{i=1}^{n} ∥c_{0i}∥ + ∥q∥_{L_{α,β}} ),  (3.25)

where γ is a positive constant not depending on λ, q, c_{0i} (i = 1, …, n) and x. Assume the contrary that this is not so. Then for each natural k there are

λ_{k} ∈ [0, 1], q_{k} ∈ L_{α,β}(]a, b[ ; ℝ^{m}), c_{ki} ∈ ℝ^{m} (i = 1, …, n)

such that the problem

x^{(n)}(t) = p(x)(t) + λ_{k}\overline{p}(x)(t) + q_{k}(t),
ℓ_{i}(x) = λ_{k}\overline{ℓ}_{i}(x) + c_{ki} (i = 1, …, n)

has a solution x_{k} admitting the estimate

ρ_{k} =^{def} ∥x_{k}∥_{C_{α,β}^{n−1}} > k( ∑_{i=1}^{n} ∥c_{ki}∥ + ∥q_{k}∥_{L_{α,β}} ).

If we assume that

\overline{x}_{k}(t) = ρ_{k}^{−1}x_{k}(t), \overline{q}_{k}(t) = ρ_{k}^{−1}q_{k}(t), \overline{c}_{ki} = ρ_{k}^{−1}c_{ki} (i = 1, …, n),

then we have

∥\overline{x}_{k}∥_{C_{α,β}^{n−1}} = 1,  (3.26)
∥\overline{q}_{k}∥_{C_{α,β}^{n−1}} < \frac{1}{k} , ∑_{i=1}^{n} ∥\overline{c}_{ki}∥ < \frac{1}{k} ,  (3.27)
\overline{x}_{k}^{(n)}(t) = p(\overline{x}_{k})(t) + λ_{k}\overline{p}(\overline{x}_{k})(t) + \overline{q}_{k}(t),  (3.28)

ℓ_{i}(\overline{x}_{k}) = λ_{k}\overline{ℓ}_{i}(\overline{x}_{k}) + \overline{c}_{ki} (i = 1, …, n).  (3.29)

Let t_{0} = \frac{a+b}{2} . Then (3.28) implies

\overline{x}_{k}(t) = y_{k}(t) + z_{k}(t),  (3.30)

where

y_{k}(t) = ∑_{i=1}^{n} \frac{(t − t_{0})^{i−1}}{(i − 1)!} \overline{x}_{k}^{(i−1)}(t_{0})

+ \frac{1}{(n − 1)!} ∫_{t_{0}}^{t}(t − s)^{n−1}(p(\overline{x}_{k})(s) + λ_{k}\overline{p}(\overline{x}_{k})(s)) ds,  (3.31)

z_{k}(t) = \frac{1}{(n − 1)!} ∫_{t_{0}}^{t}(t − s)^{n−1}\overline{q}_{k}(s) ds.

By (3.27) we have

lim_{k→+∞} ∥z_{k}∥_{C_{α,β}^{n−1}} = 0, lim_{k→+∞} \overline{c}_{ki} = 0 (i = 1, …, n).  (3.32)

On the other hand, with (3.22) and (3.26) taken into account, from (3.31) we find

∥y_{k}^{(i−1)}(t_{0})∥ ≤ ρ^{∗} (i = 1, …, n),
∥y_{k}^{(n−1)}(t) − y_{k}^{(n−1)}(s)∥ ≤ ∫_{s}^{t} ζ(ξ) dξ for a < s < t < b,

where ρ^{∗} is a positive constant not depending on k. By virtue of these inequalities and Lemma 1.1, we can assume without loss of generality that the sequence (y_{k})_{k=1}^{+∞} is converging in the norm of the space C_{α,β}^{n−1}(]a, b[ ; ℝ^{m}). It can also be assumed without loss of generality that the sequence (λ_{k})_{k=1}^{+∞} is converging. Assume that

λ = lim_{k→+∞} λ_{k}, x(t) = lim_{k→+∞} y_{k}(t).

Then by (3.29)–(3.32) we have

lim_{k→+∞} ∥x_{k} − x∥_{C_{α,β}^{n−1}} = lim_{k→+∞} ∥y_{k} − x∥_{C_{α,β}^{n−1}} = 0  (3.33)

and

ℓ_{i}(x) = λ\overline{ℓ}_{i}(x) (i = 1, …, n),

x(t) = ∑_{i=1}^{n} \frac{(t − t_{0})^{i−1}}{(i − 1)!} x^{(i−1)}(t_{0}) + \frac{1}{(n − 1)!} ∫_{t_{0}}^{t}(t − s)^{n−1}(p(x)(s) + λ\overline{p}(x)(s)) ds.

Therefore x is a solution of problem (1.32). On the other hand, from (3.26) and (3.33) it clearly follows that

∥x∥_{C_{α,β}^{n−1}} = 1.

But this is impossible because for each λ ∈ [0, 1] problem (1.32) has only a trivial solution. The contradiction obtained proves the existence of a positive number γ that possesses the above-mentioned property.

By condition (1.31) there is ρ_{0} > 0 such that inequality (3.12) is fulfilled. To prove Corollary 1.4, it is sufficient due to Theorem 1.2 to establish that

for each λ ∈]0, 1[ an arbitrary solution x of the problem

x^{(n)}(t) = p(x)(t) + λ[f(x)(t) − p(x)(t)],  (3.34)
ℓ_{i}(x) = λ(ℓ_{i}(x) − h_{i}(x)) (i = 1, …, n)  (3.35)

admits estimate (1.19).

It is obvious that each solution x of problem (3.34), (3.35) is a solution of problem (3.23), (3.24), where

\begin{aligned}q(t) & = λ(f(x)(t) − p(x)(t) − \overline{p}(x)(t)), \\ c_{0i} & = λ(ℓ_{i}(x) + \overline{ℓ}_{i}(x) − h_{i}(x)) (i = 1, …, n). \end{aligned}  (3.36)

According to the above proof, x admits estimate (3.25) from which, with (1.29), (1.30) and (3.36) taken into account, we find

∥x∥_{C_{α,β}^{n−1}} ≤ γ(η_{0}(∥x∥_{C_{α,β}^{n−1}}) + ∫_{a}^{b}(s − a)^{α}(b − s)^{β}η(s, ∥x∥_{C_{α,β}^{n−1}}) ds).

Hence, by virtue of (3.12), we obtain estimate (1.19).

Acknowledgements

This work was supported by the Research Grant of the Greek Ministry of Development in the framework of Bilateral S&T Cooperation between the Hellenic Republic and the Republic of Georgia.

References

1. R. P. Agarwal and D. O’Regan, Nonlinear superlinear singular and nonsingular second order boundary value problems. J. Differential Equations 143(1998), No. 1, 60–95.

2. R. P. Agarwal and D. O’Regan, Second-order boundary value problems of singular type. J. Math. Anal. Appl. 226(1998), 414–430.

3. N. B. Azbelev, M. J. Alves, and E. I. Bravyi, On singular boundary value problems for linear functional differential equations of second order. (Russian) Izv. Vyssh. Uchebn. Zaved. Mat. 2(1996), 3–11.

4. N. V. Azbelev, V. P. Maximov, and L. F. Rakhmatullina, Introduction to the theory of functional differential equations. (Russian) Nauka, Moscow, 1991.

5. N. V. Azbelev, V. P. Maximov, and L. F. Rakhmatullina, Methods of the modern theory of linear functional differential equations. (Russian) R&C Dynamics, Moscow– Izhevsk, 2000.

6. E. Bravyi, A note on the Fredholm property of boundary value problems for linear functional differential equations. Mem. Differential Equations Math. Phys. 20(2000), 133– 135.

7. S. A. Brykalov, Problems for functional-differential equations with monotone boundary conditions. (Russian) Differentsial’nye Uravneniya 32(1996), No. 6, 731–738; English transl.: Differential Equations 32(1996), No. 6, 740–747.

8. S. A. Brykalov, A priori estimates and solvability of problems with nonlinear functional boundary conditions. (Russian) Differentsial’nye Uravneniya 35(1999), No. 7, 874–881; English transl.: Differential Equations 35(1999), No. 7, 880–887.

9. O. Došlý and A. Lomtatidze, Disconjugacy and disfocality criteria for second order singular half-linear differential equations. Ann. Polon. Math. 72(1999), No. 3, 273–284.

10. Sh. Gelashvili and I. Kiguradze, On multi-point boundary value problems for systems of functionaldifferential and difference equations. Mem. Differential Equations Math. Phys. 5(1995), 1–113.

11. R. Hakl, I. Kiguradze, and B. Půža, Upper and lower solutions of boundary value problems for functional differential equations and theorems of functional differential inequalities. Georgian Math. J. 7(2000), No. 3, 489–512.

12. R. Hakl, A. Lomtatidze, and B. Půža, On nonnegative solitions of first order scalar functional differential equations. Mem. Differential Equations Math. Phys. 23(2001), 51– 84.

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

14. I. Kiguradze, On some singular boundary value problems for nonlinear second order ordinary differential equations. (Russian) Differentsial’nye Uravneniya 4(1968), No. 10, 1753–1773; English transl.: Differential Equations 4(1968), 901–910.

15. I. Kiguradze, On a singular multi-point boundary value problem. Ann. Mat. Pura Appl. 86(1970), 367–399.

16. I. Kiguradze, On a singular boundary value problem. J. Math. Anal. Appl. 30(1970), No. 3, 475–489.

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

18. I. Kiguradze, On a singular problem of Cauchy–Nicoletti. Ann. Mat. Pura Appl. 104(1975), 151–175.

19. I. Kiguradze, On the modified problem of Cauchy–Nicoletti. Ann. Mat. Pura Appl. 104(1975), 177–186.

20. I. Kiguradze, Initial and boundary value problems for systems of ordinary differential equations, I. (Russian) Metsniereba, Tbilisi, 1997.

21. I. T. Kiguradze and A. G. Lomtatidze, On certain boundary-value problems for second-order linear ordinary differential equations with singularities. J. Math. Anal. Appl. 101(1984), No. 2, 325–347.

22. I. Kiguradze and N. Partsvania, On nonnegative solutions of nonlinear two-point boundary value problems for two-dimensional differential systems with advanced arguments. E. J. Qualitative Theory of Diff. Equ., 1999, No. 5, 1–22.

23. I. Kiguradze and B. Půža, On a certain singular boundary value problem for linear differential equations with deviating arguments. Czechoslovak Math. J. 47(1997), No. 2, 233–244.

24. I. Kiguradze and B. Půža, On boundary value problems for systems of linear functional differential equations. Czechoslovak Math. J. 47(1997), No. 2, 341–373.

25. I. Kiguradze and B. Půža, On the Vallée–Poussin problem for singular differential equations with deviating arguments. Arch. Math. 33(1997), No. 1–2, 127–138.

26. I. Kiguradze and B. Půža, Conti–Opial type theorems for systems of functional differential equations. (Russian) Differentsial’nye Uravneniya 33(1997), No. 2, 185–194; English transl.: Differential Equations 33(1997), No. 2, 184–193.

27. I. Kiguradze and B. Půža, On boundary value problems for functional differential equations. Mem. Differential Equations Math. Phys. 12(1997), 106–113.

28. I. Kiguradze and B. Půža, On the solvability of nonlinear boundary value problems for functional differential equations. Georgian Math. J. 5(1998), No. 3, 251–262.

29. I. T. Kiguradze and B. L. Shekhter, Singular boundary value problems for second order ordinary differential equations. (Russian) Itogi Nauki i Tekhniki 30(1987), 105–201; English transl.: J. Sov. Math. 43(1988), No. 2, 2340–2417.

30. I. Kiguradze and Z. Sokhadze, On the Cauchy problem for singular evolution functional differential equations. (Russian) Differentsial’nye Uravneniya 33(1997), No. 1, 48– 59; English transl.: Differential Equations 33(1997), No. 1, 47–58.

31. I. Kiguradze and Z. Sokhadze, On the structure of the set of solutions of the weighted Cauchy problem for evolution singular functional differential equations. Fasc. Math. (1998), No. 28, 71–92.

32. I. Kiguradze and G. Tskhovrebadze, On the two-point boundary value problems for systems of higher order ordinary differential equations with singularities. Georgian Math. J. 1(1994), No. 1, 31–45.

33. A. G. Lomtatidze, On one boundary value problem for linear ordinary differential equations of second order with singularities. (Russian) Differentsial’nye Uravneniya 22(1986), No. 3, 416–426.

34. A. G. Lomtatidze, On positive solutions of boundary value problems for second order ordinary differential equations with singularities. (Russian) Differentsial’nye Uravneniya 23(1987), No. 10, 1685–1692.

35. A. Lomtatidze and L. Malaguti, On a nonlocal boundary value problem for second order nonlinear singular equations. Georgian Math. J. 7(2000), No. 1, 133–154.

36. A. Lomtatidze and S. Mukhigulashvili, Some two-point boundary value problems for second order functional differential equations. Masaryk University, Brno, 2000.

37. V. E. Mayorov, On the existence of solutions of singular differential equations of higher order. (Russian) Mat. Zametki 51(1992), No. 3, 75–83.

38. S. Mukhigulashvili, Two-point boundary value problems for second order functional differential equations. Mem. Differential Equations Math. Phys. 20(2000), 1–112.

39. N. Partsvania, On a boundary value problem for the two-dimensional system of evolution functional differential equations. Mem. Differential Equations Math. Phys. 20(2000), 154–158.

40. B. Půža, On a singular two-point boundary value problem for the nonlinear m-th order differential equations with deviating argument. Georgian Math. J. 4(1997), No. 6, 557– 566.

41. B. Půža, On some boundary value problems for nonlinear functional differential equations. (Russian) Differentsial’nye Uravneniya 37(2001), No. 6, 797–806.

42. B. Půža and A. Rabbimov, On a weighted boundary value problem for a system of singular functional differential equations. Mem. Differential Equations Math. Phys. 21(2000), 125–130.

43. Š. Schwabik, M. Tvrdý, and O. Vejvoda, Differential and integral equations: boundary value problems and adjoints. Academia, Praha, 1979.

44. Ch. de la Vallée-Poussin, Sur l’équation différentielle linéaire du second ordre. Détermination d’une intégrale par deux valeurs assignés. Extension aux équations d’ordre n. J. Math. Pures Appl. 8(1929), 125–144.

45. P. J. Y. Wong and R. P. Agarwal, Singular differential equations with (n, p) boundary conditions. Math. Comput. Modelling 28(1998), No. 1, 37–44.

(Received 2.05.2001)

Authors’ addresses:

I. Kiguradze A. Razmadze Mathematical Institute Georgian Academy of Sciences 1, M. Aleksidze St., Tbilisi 380093 Georgia E-mail: kig@rmi.acnet.ge

B. Půža Masaryk University Faculty of Science Department of Mathematical Analysis Janáčkovo nám. 2a, 662 95 Brno Czech Republic E-mail: puza@math.muni.cz

I. P. Stavroulakis Department of Mathematics University of Ioannina 451 10 Ioannina Greece E-mail: ipstav@cc.uoi.gr