Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 36, 1-13; http://www.math.u-szeged.hu/ejqtde/

ON A NON-LOCAL BOUNDARY VALUE PROBLEM FOR LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS

Z. OPLUŠTIL AND J. ŠREMR

Abstract. We establish new efficient conditions for the unique solvability of a non-local boundary value problem for first-order linear functional differential equations. Differential equations with argument deviations are also considered in which case further results are obtained. The results obtained reduce to those well-known for the ordinary differential equations.

1. Introduction

On the interval [a, b], we consider the problem on the existence and uniqueness of a solution to the equation

u′(t) = ℓ(u)(t) + q(t)  (1.1)

satisfying the non-local boundary condition

h(u) = c,  (1.2)

where ℓ: C([a, b]; ℝ) → L([a, b]; ℝ) and h: C([a, b]; ℝ) → ℝ are linear bounded operators, q ∈ L([a, b]; ℝ), and c ∈ ℝ. By a solution to the problem (1.1), (1.2) we understand an absolutely continuous function u: [a, b] → ℝ satisfying the equation (1.1) almost everywhere on the interval [a, b] and verifying also the boundary condition (1.2).

The question on the solvability of various types of boundary value problems for functional differential equations and their systems is a classical topic in the theory of differential equations (see, e.g., [1,3–5,7–9,11–14] and references therein). Many particular cases of the boundary condition (1.2) are studied in detail (namely, periodic, anti-periodic and multi-point conditions), but only a few efficient conditions is known in the case, where a general non-local boundary condition is considered. In the present paper, new efficient conditions are found sufficient for the unique solvability of the problem (1.1), (1.2). It is clear that the ordinary differential equation

u′ = p(t)u + q(t),  (1.3)

2000 Mathematics Subject Classification. 34K10, 34K06.

Key words and phrases. linear functional differential equation, non-local boundary value problem, solvability.

For the first author, published results were acquired using the subsidization of the Ministry of Education, Youth and Sports of the Czech Republic, research plan 2E08017 ”Procedures and Methods to Increase Number of Researchers”. For the second author, the research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503.

where p, q ∈ L([a, b]; ℝ), is a particular case of the equation (1.1) and that the problem (1.3), (1.2) is uniquely solvable if and only if the condition

h (e^{∫_{a}^{·} p(s)ds}) ≠ 0  (1.4)

is satisfied. Below, we establish new solvability conditions for the problem (1.1), (1.2) in terms of norms of the operators appearing in (1.1) and (1.2) (see Theorems 2.1–2.4). Moreover, we apply these results to the differential equation with an argument deviation

u′(t) = p(t)u(τ(t)) + q(t)  (1.5)

in which p, q ∈ L([a, b]; ℝ) and τ: [a, b] → [a, b] is a measurable function (see Theorems 2.5 and 2.6), and we show that the assumptions of the statements obtained reduce to the condition (1.4) in the case, where the equation (1.5) is the ordinary one (see Remark 2.6). All the main results are formulated in Section 2, their proofs are given in Section 3.

The following notation is used throughout the paper:

(1) ℝ is the set of all real numbers, ℝ_{+} = [0, +∞[ .

(2) C([a, b]; ℝ) is the Banach space of continuous functions u: [a, b] → ℝ endowed with the norm ∥u∥_{C} = max{|u(t)| : t ∈ [a, b]}.

(3) L([a, b]; ℝ) is the Banach space of Lebesgue integrable functions p: [a, b] → ℝ endowed with the norm ∥p∥_{L} = ∫_{a}^{b} |p(s)|ds.

(4) P_{ab} is set of linear operators ℓ: C([a, b]; ℝ) → L([a, b]; ℝ) mapping the set C([a, b]; ℝ_{+}) into the set L([a, b]; ℝ_{+}).

(5) P F_{ab} is the set of linear functionals h : C([a, b]; ℝ) → ℝ mapping the set C([a, b]; ℝ_{+}) into the set ℝ_{+}.

2. Main Results

In theorems stated below, we assume that the operator ℓ admits the representation ℓ = ℓ_{0} − ℓ_{1} with ℓ_{0}, ℓ_{1} ∈ P_{ab}. This is equivalent to the fact that ℓ is not only bounded, but it is strongly bounded (see, e.g., [6, Ch.VII, §1.2]), i.e., that there exists a function η ∈ L([a, b]; ℝ_{+}) such that the condition

|ℓ(v)(t)| ≤ η(t)∥v∥_{C} for a. e. t ∈ [a, b] and all v ∈ C([a, b]; ℝ).

is satisfied.

We first consider the case, where the boundary condition (1.2) is understood as a non-local perturbation of a two-point condition of an anti-periodic type. More precisely, we consider the boundary condition

u(a) + λu(b) = h_{0}(u) − h_{1}(u) + c,  (2.1)

where λ ≥ 0, h_{0}, h_{1} ∈ P F_{ab}, and c ∈ ℝ. We should mention that there is no loss of generality in assuming this, because an arbitrary functional h can be represented in the form

h(v) =^{def} v(a) + λv(b) − h_{0}(v) + h_{1}(v) for v ∈ C([a, b]; ℝ).

Note also that we have studied the problem (1.1), (2.1) with λ < 0 in the paper [10].

Theorem 2.1. Let h_{0}(1) < 1 + λ + h_{1}(1) and ℓ = ℓ_{0} − ℓ_{1}, where ℓ_{0}, ℓ_{1} ∈ P_{ab}. Let, moreover,

λ(λ − h_{0}(1)) ≤ (1 + h_{1}(1))^{2}  (2.2)

and either the conditions

∥ℓ_{0}∥ < 1 − h_{0}(1) − (λ + h_{1}(1))^{2},  (2.3)
∥ℓ_{1}∥ < 1 − λ − h_{1}(1) + 2\sqrt{1 − h_{0}(1) − ∥ℓ_{0}∥} ,  (2.4)

be satisfied, or the conditions

∥ℓ_{0}∥ ≥ 1 − h_{0}(1) − (λ + h_{1}(1))^{2},  (2.5)
∥ℓ_{0}∥ + (λ + h_{1}(1))∥ℓ_{1}∥ < 1 + λ − h_{0}(1) + h_{1}(1),  (2.6)
(1 + h_{1}(1))∥ℓ_{0}∥ + λ∥ℓ_{1}∥ < 1 + λ − h_{0}(1) + h_{1}(1)  (2.7)

hold. Then the problem (1.1), (2.1) has a unique solution.

Remark 2.1. Geometrical meaning of the assumptions of Theorem 2.1 is illustrated on Fig. 2.1.

\begin{aligned}x_{1} & = 1 − h_{0}(1) − (λ + h_{1}(1))^{2} \\ x_{2} & = \frac{1 + λ − h_{0}(1) + h_{1}(1)}{1 + λ + h_{1}(1)} \\ y_{1} & = 1 + λ + h_{1}(1) \\ y_{2} & = 1 − λ − h_{1}(1) + 2\sqrt{1 − h_{0}(1)} \\ & Figure 2.1 \end{aligned}  \begin{aligned}x_{1} & = 1 + \frac{1 + h_{1}(1)}{λ} \\ x_{2} & = 1 − \frac{1 + h_{1}(1)}{λ} + 2\sqrt{1 − \frac{1}{λ} h_{0}(1)} \\ y_{1} & = 1 − \frac{1}{λ} h_{0}(1) − \frac{(1 + h_{1}(1)r}{λ^{2}} \\ y_{2} & = \frac{1 + λ − h_{0}(1) + h_{1}(1)}{λ + h_{1}(1)} \\ & Figure 2.2 \end{aligned}

Remark 2.2. Let ℓ = ℓ_{0} −ℓ_{1} with ℓ_{0}, ℓ_{1} ∈ P_{ab}. Define the operator φ: C([a, b]; ℝ) → C([a, b]; ℝ) by setting

φ(w)(t) =^{def} w(a + b − t) for t ∈ [a, b], w ∈ C([a, b]; ℝ).

For i = 0, 1, we put

ℓ̃_{i}(w)(t) =^{def} ℓ_{i}(φ(w))(a + b − t) for a. e. t ∈ [a, b] and all w ∈ C([a, b]; ℝ)

and

q̃(t) =^{def} −q(a + b − t) for a. e. t ∈ [a, b],

h̃(w) =^{def} h(φ(w)) for w ∈ C([a, b]; ℝ).

It is clear that if u is a solution to the problem (1.1), (1.2) then the function v =^{def} φ(u) is a solution to the problem

v′(t) = ℓ̃_{1}(v)(t) − ℓ̃_{0}(v)(t) + q̃(t),  h̃(v) = c,  (2.8)

and vice versa, if v is a solution to the problem (2.8) then the function u =^{def} φ(v) is a solution to the problem (1.1), (1.2).

Using the transformation described in the previous remark, we can immediately derive from Theorem 2.1 the following statement.

Theorem 2.2. Let λ > 0, h_{0}(1) < 1+λ+h_{1}(1), and ℓ = ℓ_{0}−ℓ_{1}, where ℓ_{0}, ℓ_{1} ∈ P_{ab}. Let, moreover,

1 − h_{0}(1) ≤ (λ + h_{1}(1))^{2}  (2.9)

and either the conditions

∥ℓ_{1}∥ < 1 − \frac{1}{λ} h_{0}(1) − \frac{(1 + h_{1}(1))^{2}}{λ^{2}} ,  (2.10)
∥ℓ_{0}∥ < 1 − \frac{1}{λ}(1 + h_{1}(1)) + 2\sqrt{1 − \frac{1}{λ} h_{0}(1) − ∥ℓ_{1}∥}  (2.11)

be satisfied, or

∥ℓ_{1}∥ ≥ 1 − \frac{1}{λ} h_{0}(1) − \frac{(1 + h_{1}(1))^{2}}{λ^{2}}  (2.12)

and the conditions (2.6) and (2.7) hold. Then the problem (1.1), (2.1) has a unique solution.

Remark 2.3. Geometrical meaning of the assumptions of Theorem 2.2 is illustrated on Fig. 2.2.

Remark 2.4. It is easy to verify that, for any λ ≥ 0 and h_{0}, h_{1} ∈ P F_{ab}, at least one of the conditions (2.2) and (2.9) is fulfilled and thus Theorems 2.1 and 2.2 cover all cases.

Theorems 2.1 and 2.2 yield

Corollary 2.1. Let λ > 0, h_{0}(1) < 1+λ+h_{1}(1) and ℓ = ℓ_{0}−ℓ_{1}, where ℓ_{0}, ℓ_{1} ∈ P_{ab}. If, moreover, the conditions (2.2), (2.6), (2.7), and (2.9) are fulfilled, then the problem (1.1), (2.1) has a unique solution.

In the case, where λ = 0 in (2.1), we consider the problem

u′(t) = ℓ(u)(t) + q(t),  u(a) = h_{0}(u) − h_{1}(u) + c  (2.13)

and from Theorem 2.1 we get

Corollary 2.2. Let h_{0}(1) < 1 + h_{1}(1) and ℓ = ℓ_{0} − ℓ_{1}, where ℓ_{0}, ℓ_{1} ∈ P_{ab}. Let, moreover, either the conditions

∥ℓ_{0}∥ < 1 − h_{0}(1) − h_{1}(1)^{2},  (2.14)
∥ℓ_{1}∥ < 1 − h_{1}(1) + 2\sqrt{1 − h_{0}(1) − ∥ℓ_{0}∥} ,  (2.15)

be satisfied, or the conditions

1 − h_{0}(1) − h_{1}(1)^{2} ≤ ∥ℓ_{0}∥ < 1 − \frac{h_{0}(1)}{1 + h_{1}(1)} ,  (2.16)
∥ℓ_{0}∥ + h_{1}(1)∥ℓ_{1}∥ < 1 − h_{0}(1) + h_{1}(1)  (2.17)

hold. Then the problem (2.13) has a unique solution.

Now we give two statements dealing with the unique solvability of the problem (1.1), (1.2). We assume in Theorems 2.3 and 2.4 that h = h^{+} − h^{−} with h^{+}, h^{−} ∈ P F_{ab}. There is no loss of generality in assuming this, because every linear bounded functional h: C([a, b]) → ℝ can be expressed in such a form.

Theorem 2.3. Let h(1) > 0, h = h^{+} − h^{−} with h^{+}, h^{−} ∈ P F_{ab}, and ℓ = ℓ_{0} − ℓ_{1}, where ℓ_{0}, ℓ_{1} ∈ P_{ab}. Let, moreover, the conditions

∥ℓ_{0}∥ + h^{+}(1)∥ℓ_{1}∥ < h(1)

and

h^{+}(1)∥ℓ_{0}∥ + ∥ℓ_{1}∥ < h(1)

be fulfilled. Then the problem (1.1), (1.2) has a unique solution.

Theorem 2.4. Let h(1) < 0, h = h^{+} − h^{−} with h^{+}, h^{−} ∈ P F_{ab}, and ℓ = ℓ_{0} − ℓ_{1}, where ℓ_{0}, ℓ_{1} ∈ P_{ab}. Let, moreover, the conditions

∥ℓ_{0}∥ + h^{−}(1)∥ℓ_{1}∥ < |h(1)|

and

h^{−}(1)∥ℓ_{0}∥ + ∥ℓ_{1}∥ < |h(1)|

be fulfilled. Then the problem (1.1), (1.2) has a unique solution.

Remark 2.5. Geometrical meaning of the assumptions of Theorems 2.3 and 2.4 is illustrated, respectively, on Fig. 2.3 and Fig. 2.4.

It is clear that, from Theorems 2.1–2.4, we can immediately obtain conditions guaranteeing the unique solvability of the problem (1.5), (1.2), whenever we replace the terms ∥ℓ_{0}∥ and ∥ℓ_{1}∥ appearing therein, respectively, by the terms ∫_{a}^{b}[p(s)]_{+}ds and ∫_{a}^{b}[p(s)]_{−}ds. In what follows, we establish two theorems, which can be also derived from Theorems 2.3 and 2.4, and which require that the deviation τ(t) − t is “small” enough. In order to simplify formulation of statements, we put

p_{0}(t) = σ(t)[p(t)]_{+} ∫_{t}^{τ(t)}[p(s)]_{+}e^{∫_{t}^{τ(s)} p(ξ)dξ}ds+

Figure 2.3  Figure 2.4

+ σ(t)[p(t)]_{−} ∫_{t}^{τ(t)}[p(s)]_{−}e^{∫_{t}^{τ(s)} p(ξ)dξ}ds+
+ (1 − σ(t))[p(t)]_{+} ∫_{τ(t)}^{t}[p(s)]_{−}e^{∫_{t}^{τ(s)} p(ξ)dξ}ds+

+ (1 − σ(t))[p(t)]_{−} ∫_{τ(t)}^{t}[p(s)]_{+}e^{∫_{t}^{τ(s)} p(ξ)dξ}ds for a. e. t ∈ [a, b] (2.18)

and

p_{1}(t) = σ(t)[p(t)]_{+} ∫_{t}^{τ(t)}[p(s)]_{−}e^{∫_{t}^{τ(s)} p(ξ)dξ}ds+
+ σ(t)[p(t)]_{−} ∫_{t}^{τ(t)}[p(s)]_{+}e^{∫_{t}^{τ(s)} p(ξ)dξ}ds+
+ (1 − σ(t))[p(t)]_{+} ∫_{τ(t)}^{t}[p(s)]_{+}e^{∫_{t}^{τ(s)} p(ξ)dξ}ds+

+ (1 − σ(t))[p(t)]_{−} ∫_{τ(t)}^{t}[p(s)]_{−}e^{∫_{t}^{τ(s)} p(ξ)dξ}ds for a. e. t ∈ [a, b], (2.19)

where

σ(t) = \frac{1}{2}(1 + sgn(τ(t) − t)) for a. e. t ∈ [a, b].

Moreover, having h^{+}, h^{−} ∈ P F_{ab}, we denote

μ_{0} = h^{+} (e^{∫_{a}^{·} p(s)ds})  and  μ_{1} = h^{−} (e^{∫_{a}^{·} p(s)ds}) .  (2.20)

Theorem 2.5. Let h = h^{+} − h^{−} with h^{+}, h^{−} ∈ P F_{ab}. Let, moreover, μ_{0} > μ_{1} and the conditions

∫_{a}^{b} p_{0}(s)ds + μ_{0} ∫_{a}^{b} p_{1}(s)ds < μ_{0} − μ_{1}

and

μ_{0} ∫_{a}^{b} p_{0}(s)ds + ∫_{a}^{b} p_{1}(s)ds < μ_{0} − μ_{1}

be fulfilled, where the functions p_{0}, p_{1} and the numbers μ_{0}, μ_{1} are defined, respectively, by the relations (2.18), (2.19) and (2.20). Then the problem (1.5), (1.2) has a unique solution.

Theorem 2.6. Let h = h^{+} − h^{−} with h^{+}, h^{−} ∈ P F_{ab}. Let, moreover, μ_{0} < μ_{1} and the conditions

∫_{a}^{b} p_{0}(s)ds + μ_{1} ∫_{a}^{b} p_{1}(s)ds < μ_{1} − μ_{0}

and

μ_{1} ∫_{a}^{b} p_{0}(s)ds + ∫_{a}^{b} p_{1}(s)ds < μ_{1} − μ_{0}

be fulfilled, where the functions p_{0}, p_{1} and the numbers μ_{0}, μ_{1} are defined, respectively, by the relations (2.18), (2.19) and (2.20). Then the problem (1.5), (1.2) has a unique solution.

Remark 2.6. Theorems 2.5 and 2.6 yield, in particular, that the problem (1.3), (1.2) is uniquely solvable if μ_{0} ≠ μ_{1}, i. e., if the condition (1.4) holds. However, it is well-known that, in the framework of the ordinary differential equations, the condition (1.4) is not only sufficient, but also necessary for the unique solvability of the problem (1.3), (1.2).

3. Proofs

It is well-known that the linear problem has the Fredholm property, i. e., the following assertion holds (see, e. g., [2,4]; in the case, where the operator ℓ is strongly bounded, see also [1, 14]).

Lemma 3.1. The problem (1.1), (1.2) has a unique solution for an arbitrary q ∈ L([a, b]; ℝ) and every c ∈ ℝ if and only if the corresponding homogeneous problem

u′(t) = ℓ(u)(t),  h(u) = 0  (3.1)

has only the trivial solution.

Proof of Theorem 2.1. According to Lemma 3.1, to prove the theorem it is sufficient to show that the homogeneous problem

u′(t) = ℓ_{0}(u)(t) − ℓ_{1}(u)(t),  (3.2)
u(a) + λu(b) = h_{0}(u) − h_{1}(u)  (3.3)

has only the trivial solution. Assume that, on the contrary, u is a nontrivial solution to the problem (3.2), (3.3).

First suppose that u changes its sign. Put

M = max{u(t) : t ∈ [a, b]},  m = − min{u(t) : t ∈ [a, b]},  (3.4)

and choose t_{M}, t_{m} ∈ [a, b] such that

u(t_{M}) = M,  u(t_{m}) = −m.  (3.5)

It is clear that

M > 0, m > 0. (3.6)

We can assume without loss of generality that t_{M} < t_{m}. The integration of the equality (3.2) from t_{M} to t_{m}, from a to t_{M}, and from t_{m} to b, in view of (3.4), (3.5), and the assumption ℓ_{0}, ℓ_{1} ∈ P_{ab}, yields

M + m = ∫_{t_{M}}^{t_{m}} ℓ_{1}(u)(s) ds − ∫_{t_{M}}^{t_{m}} ℓ_{0}(u)(s) ds ≤ MB_{1} + mA_{1}, (3.7)

\begin{aligned}M − u(a) + u(b) + m = & ∫_{a}^{t_{M}} ℓ_{0}(u)(s) ds − ∫_{a}^{t_{M}} ℓ_{1}(u)(s) ds+ \\ & + ∫_{t_{m}}^{b} ℓ_{0}(u)(s) ds − ∫_{t_{m}}^{b} ℓ_{1}(u)(s) ds ≤ MA_{2} + mB_{2}, \end{aligned} (3.8)

where

A_{1} = ∫_{t_{M}}^{t_{m}} ℓ_{0}(1)(s) ds, A_{2} = ∫_{a}^{t_{M}} ℓ_{0}(1)(s) ds + ∫_{t_{m}}^{b} ℓ_{0}(1)(s) ds,

B_{1} = ∫_{t_{M}}^{t_{m}} ℓ_{1}(1)(s) ds, B_{2} = ∫_{a}^{t_{M}} ℓ_{1}(1)(s) ds + ∫_{t_{m}}^{b} ℓ_{1}(1)(s) ds.

On the other hand, from the boundary condition (3.3), in view of the relations (3.5), (3.6) and the assumption h_{0}, h_{1} ∈ P F_{ab}, we get

u(a) − u(b) = −(1 + λ)u(b) + h_{0}(u) − h_{1}(u) ≤ (1 + λ)m + Mh_{0}(1) + mh_{1}(1)

and

u(a) − u(b) = (1 + \frac{1}{λ})u(a) − \frac{1}{λ}h_{0}(u) + \frac{1}{λ}h_{1}(u) ≤

≤ (1 + \frac{1}{λ})M + m\frac{1}{λ} h_{0}(1) + M \frac{1}{λ} h_{1}(1).

Hence, it follows from the relation (3.8) that

M − λm ≤ MA_{2} + mB_{2} + Mh_{0}(1) + mh_{1}(1)  (3.9)

and

m − \frac{1}{λ} M ≤ MA_{2} + mB_{2} + m\frac{1}{λ} h_{0}(1) + M \frac{1}{λ} h_{1}(1).  (3.10)

We first assume that ∥ℓ_{0}∥ ≥ 1. Then the conditions (2.6) and (2.7) are supposed to be satisfied. It is clear that the inequality (2.7) implies λ > 0 and ∥ℓ_{1}∥ < 1 − \frac{1}{λ}h_{0}(1) and thus

B_{1} < 1, B_{2} < 1 − \frac{1}{λ} h_{0}(1).

Using these inequalities and the relations (3.6), from (3.7) and (3.10) we obtain

0 < M(1 − B_{1}) ≤ m(A_{1} − 1),

0 < m(1 − \frac{1}{λ} h_{0}(1) − B_{2}) ≤ M(A_{2} + \frac{1}{λ}(1 + h_{1}(1))),

which yields that

(1 − B_{1})(1 − \frac{1}{λ} h_{0}(1) − B_{2}) ≤ (A_{1} − 1)(A_{2} + \frac{1}{λ}(1 + h_{1}(1))).  (3.11)

Obviously,

(1 − B_{1})(1 − \frac{1}{λ}h_{0}(1) − B_{2}) ≥ 1 − \frac{1}{λ}h_{0}(1) − ∥ℓ_{1}∥.  (3.12)

On the other hand, by virtue of (2.2), it follows from the inequality (2.7) that

∥ℓ_{0}∥ < 1 + \frac{λ − h_{0}(1)}{1 + h_{1}(1)} ≤ 1 + \frac{1}{λ}(1 + h_{1}(1)),

and thus we obtain

\begin{matrix}(A_{1} − 1)(A_{2} + \frac{1}{λ}(1 + h_{1}(1))) ≤ (∥ℓ_{0}∥ − 1)A_{2} + (A_{1} − 1)\frac{1}{λ}(1 + h_{1}(1)) ≤ \\ ≤ \frac{1}{λ}(1 + h_{1}(1))(A_{1} + A_{2} − 1) ≤ \frac{1}{λ}(1 + h_{1}(1))(∥ℓ_{0}∥ − 1). \end{matrix} (3.13)

Now, from (3.11), (3.12), and (3.13) we get

1 + λ − h_{0}(1) + h_{1}(1) ≤ (1 + h_{1}(1))∥ℓ_{0}∥ + λ∥ℓ_{1}∥.

which contradicts the inequality (2.7).

Now assume that ∥ℓ_{0}∥ < 1. Then, in view of the relations (3.6), the inequalities (3.7) and (3.9) yield

0 < m(1 − A_{1}) ≤ M(B_{1} − 1),
M(1 − h_{0}(1) − A_{2}) ≤ m(B_{2} + λ + h_{1}(1))

and thus we get ∥ℓ_{1}∥ ≥ B_{1} > 1 and

(1 − A_{1})(1 − h_{0}(1) − A_{2}) ≤ (B_{1} − 1)(B_{2} + λ + h_{1}(1)).  (3.14)

Obviously,

(1 − A_{1})(1 − h_{0}(1) − A_{2}) ≥ 1 − h_{0}(1) − ∥ℓ_{0}∥.  (3.15)

If ∥ℓ_{0}∥ ≥ 1−h_{0}(1)−(λ+h_{1}(1))^{2} then the conditions (2.6) and (2.7) are supposed to be satisfied. Therefore, we obtain from the inequality (2.6) that ∥ℓ_{1}∥ ≤ 1 + λ + h_{1}(1) and thus it is easy to verify that

\begin{matrix}(B_{1} − 1)(B_{2} + λ + h_{1}(1)) ≤ (∥ℓ_{1}∥ − 1)B_{2} + (B_{1} − 1)(λ + h_{1}(1)) ≤ \\ ≤ (λ + h_{1}(1))(B_{1} + B_{2} − 1) ≤ (λ + h_{1}(1))(∥ℓ_{1}∥ − 1). \end{matrix}  (3.16)

Now, it follows from (3.14), (3.15), and(3.16) that

1 + λ − h_{0}(1) + h_{1}(1) ≤ ∥ℓ_{0}∥ + (λ + h_{1}(1))∥ℓ_{1}∥,

which contradicts the inequality (2.6).

If ∥ℓ_{0}∥ < 1 − h_{0}(1) − (λ + h_{1}(1))^{2} then, taking the above-mentioned condition ∥ℓ_{1}∥ > 1 and the obvious inequality

(B_{1} − 1)(B_{2} + λ + h_{1}(1)) ≤ \frac{1}{4}(∥ℓ_{1}∥ − 1 + λ + h_{1}(1))^{2}

into account, from the relations (3.14) and (3.15) we get

1 − λ − h_{1}(1) + 2\sqrt{1 − h_{0}(1) − ∥ℓ_{0}∥} ≤ ∥ℓ_{1}∥,

which contradicts the inequality (2.4).

Now suppose that u does not change its sign. Then, without loss of generality, we can assume that

u(t) ≥ 0 for t ∈ [a, b].  (3.17)

Put

M_{0} = max{u(t) : t ∈ [a, b]},  m_{0} = min{u(t) : t ∈ [a, b]},  (3.18)

and choose t_{M_{0}}, t_{m_{0}} ∈ [a, b] such that

u(t_{M_{0}}) = M_{0},  u(t_{m_{0}}) = m_{0}.  (3.19)

It is clear that

M_{0} > 0, m_{0} ≥ 0, (3.20)

and either

t_{M_{0}} ≥ t_{m_{0}},  (3.21)

or

t_{M_{0}} < t_{m_{0}}.  (3.22)

Notice that if the assumptions of the theorem are fulfilled, then both inequalities

A + (λ + h_{1}(1))B < 1 + λ − h_{0}(1) + h_{1}(1)  (3.23)

and

(1 + h_{1}(1))A + λB < 1 + λ − h_{0}(1) + h_{1}(1)  (3.24)

hold, where A = ∥ℓ_{0}∥ and B = ∥ℓ_{1}∥.

The integration of the equality (3.2) from a to t_{M_{0}} and from t_{M_{0}} to b, in view of the relations (3.17), (3.18), and (3.19) and the assumption ℓ_{0}, ℓ_{1} ∈ P_{ab}, yields

M_{0} − u(a) = ∫_{a}^{t_{M_{0}}} ℓ_{0}(u)(s) ds − ∫_{a}^{t_{M_{0}}} ℓ_{1}(u)(s) ds ≤ M_{0}A

and

M_{0} − u(b) = ∫_{t_{M_{0}}}^{b} ℓ_{1}(u)(s) ds − ∫_{t_{M_{0}}}^{b} ℓ_{0}(u)(s) ds ≤ M_{0}B.

The last two inequalities yield

M_{0}(1 + λ) − u(a) − λu(b) ≤ M_{0}(A + λB)

and thus, using (3.3), (3.18), and the assumption h_{0}, h_{1} ∈ P F_{ab}, we get

m_{0}h_{1}(1) ≤ M_{0}(A + λB + h_{0}(1) − 1 − λ).  (3.25)

First suppose that (3.21) holds. The integration of the equality (3.2) from t_{m_{0}} to t_{M_{0}}, in view of (3.17), (3.18), and (3.19) and the assumption ℓ_{0}, ℓ_{1} ∈ P_{ab}, results in

M_{0} − m_{0} = ∫_{t_{m_{0}}}^{t_{M_{0}}} ℓ_{0}(u)(s) ds − ∫_{t_{m_{0}}}^{t_{M_{0}}} ℓ_{1}(u)(s) ds ≤ M_{0}A,

i. e.,

M_{0}(1 − A) ≤ m_{0}.

From this inequality and (3.25) we obtain

(1 + h_{1}(1))A + λB ≥ 1 + λ − h_{0}(1) + h_{1}(1),

which contradicts the inequality (3.24).

Now assume that (3.22) holds. The integration of the equality (3.2) from t_{M_{0}} to t_{m_{0}}, in view of (3.17), (3.18), and (3.19) and the assumption ℓ_{0}, ℓ_{1} ∈ P_{ab}, yields

M_{0} − m_{0} = ∫_{t_{M_{0}}}^{t_{m_{0}}} ℓ_{1}(u)(s) ds − ∫_{t_{M_{0}}}^{t_{m_{0}}} ℓ_{0}(u)(s) ds ≤ M_{0}B,

i. e.,

M_{0}(1 − B) ≤ m_{0}.

The last inequality, together with (3.25), results in

A + (λ + h_{1}(1))B ≥ 1 + λ − h_{0}(1) + h_{1}(1),

which contradicts the inequality (3.23).

The contradictions obtained prove that the homogeneous problem (3.2), (3.3) has only the trivial solution. □

Proof of Theorem 2.2. The assertion of the theorem can be derived from Theorem 2.1 using the transformation described in Remark 2.2. □

Proof of Corollary 2.1. The validity of the corollary follows immediately from Theorems 2.1 and 2.2. □

Proof of Corollary 2.2. It is clear that the assumptions of Theorem 2.1 with λ = 0 are satisfied. □

Proof of Theorem 2.3. Let the functionals h_{0} and h_{1} be defined by the formulae

h_{0}(v) =^{def} v(a) + h^{−}(v), h_{1}(v) = h^{+}(v) for v ∈ C([a, b]; ℝ).

By virtue of Corollary 2.2, the problem (1.1), (1.2) is uniquely solvable under the assumptions

∥ℓ_{0}∥ < 1 − \frac{1 + h^{−}(1)}{1 + h^{+}(1)} , ∥ℓ_{0}∥ + h^{+}(1)∥ℓ_{1}∥ < h^{+}(1) − h^{−}(1).

Moreover, using the transformation described in Remark 2.2, it is not difficult to verify that the problem (1.1), (1.2) is uniquely solvable also under the assumptions

∥ℓ_{1}∥ < 1 − \frac{1 + h^{−}(1)}{1 + h^{+}(1)} , ∥ℓ_{1}∥ + h^{+}(1)∥ℓ_{0}∥ < h^{+}(1) − h^{−}(1).

Combining these two cases we obtain the required assertion. □

Proof of Theorem 2.4. The validity of the theorem follows from Theorem 2.3 and fact that the problem

u′(t) = ℓ(u)(t) + q(t), h(u) = c

has a unique solution for every q ∈ L([a, b]; ℝ) and c ∈ ℝ if and only if the problem

v′(t) = ℓ(v)(t) + q(t), −h(v) = c

has a unique solution for every q ∈ L([a, b]; ℝ) and c ∈ ℝ. □

Proof of Theorem 2.5. According to Lemma 3.1, to prove the theorem it is sufficient to show that the homogeneous problem

u′(t) = p(t)u(τ(t)),  h(u) = 0  (3.26)

has only the trivial solution.

Let u be an arbitrary solution to the problem (3.26). Then it is easy to verify by direct calculation that the function

v(t) = u(t)e^{− ∫_{a}^{t} p(s)ds} for t ∈ [a, b]

is a solution to the problem

v′(t) = ℓ(v)(t),  h̃(v) = 0,  (3.27)

where the operators ℓ and h̃ are defined by the relations

ℓ(w)(t) =^{def} p(t) ∫_{t}^{τ(t)} p(s)e^{∫_{t}^{τ(s)} p(ξ)dξ}w(τ(s))ds

for a. e. t ∈ [a, b] and all w ∈ C([a, b]; ℝ)

and

h̃(w) =^{def} h (w(·)e^{∫_{a}^{·} p(s)ds}) for w ∈ C([a, b]; ℝ).

The operator ℓ can be expressed in the form ℓ = ℓ_{0} − ℓ_{1}, where ℓ_{0}, ℓ_{1} ∈ P_{ab} are such that ℓ_{0}(1) ≡ p_{0} and ℓ_{1}(1) ≡ p_{1} and, moreover, the functional h̃ admits the representation h̃ = h̃^{+} − h̃^{−} in which h̃^{+}, h̃^{−} ∈ P F_{ab} are such that h̃^{+}(1) = μ_{0} and h̃^{−}(1) = μ_{1}.

Consequently, by virtue of Theorem 2.3, the problem (3.27) has only the trivial solution and thus u ≡ 0. This means that the problem (3.26) has only the trivial solution. □

Proof of Theorem 2.6. The proof is analogous to those of Theorem 2.5, only Theorem 2.4 must be used instead of Theorem 2.3. □

References

[1] N. V. Azbelev, V. P. Maksimov, L. F. Rakhmatullina, Introduction to the theory of functional differential equations (in Russian), Nauka, Moscow, 1991.

[2] 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.

[3] R. Hakl, A. Lomtatidze, J. Šremr, Some boundary value problems for first order scalar functional differential equations, Folia Facult. Scien. Natur. Univ. Masar. Brunensis, Brno, 2002.

[4] R. Hakl, A. Lomtatidze, I. P. Stavroulakis, On a boundary value problem for scalar linear functional differential equations, Abstr. Appl. Anal. 2004 (2004), No. 1, 45–67.

[5] J. Hale, Theory of functional differential equations, Springer–Verlag, New York–Heidelberg– Berlin, 1977.

[6] L. V. Kantorovich, B. Z. Vulikh, A. G. Pinsker, Functional Analysis in Semi-Ordered Spaces (in Russian), Gostekhizdat, Moscow, 1950.

[7] I. Kiguradze, B. Půža, Boundary value problems for systems of linear functional differential equations, Folia Facult. Scien. Natur. Univ. Masar. Brunensis, Brno, 2003.

[8] V. Kolmanovskii, A. Myshkis, Introduction to the theory and applications of functional differential equations, Kluwer Acad. Publ., Dordrecht–Boston–London, 1999.

[9] V. Kolmanovskii, A. Myshkis, Applied theory of functional differential equations, Kluwer Acad. Publ., Dordrecht–Boston–London, 1992.

[10] A. Lomtatidze, Z. Opluštil, J. Šremr, On a nonlocal boundary value problem for first order linear functional differential equations, Mem. Differential Equations Math. Phys. 41 (2007), 69–85.

[11] A. D. Myshkis, Linear differential equations with retarded argument, Nauka, Moscow, 1972, in Russian.

[12] A. Rontó, V. Pylypenko, D. Dilna, On the unique solvability of a non-local boundary value problem for linear functional differential equations, Math. Model. Anal. 13 (2008), No. 2, 241–250.

[13] Z. Sokhadze, The Cauchy problem for singular functional-differential equations, Kutaisskij Gosudarstvennyj Universitet, Kutaisi, 2005, in Russian.

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

(Received April 24, 2009)

Institute of Mathematics, Faculty of Mechanical Engineering, Brno University of Technology, Technická 2, 616 69 Brno, Czech Republic

E-mail address: oplustil@fme.vutbr.cz

Institute of Mathematics, Academy of Sciences of the Czech Republic, Žižkova 22, 616 62 Brno, Czech Republic

E-mail address: sremr@ipm.cz