Electronic Journal of Qualitative Theory of Differential Equations 2009, No. 59, 1-12; http://www.math.u-szeged.hu/ejqtde/

AN OPTIMAL CONDITION FOR THE UNIQUENESS OF A PERIODIC SOLUTION FOR LINEAR FUNCTIONAL DIFFERENTIAL SYSTEMS

S. MUKHIGULASHVILI, I. GRYTSAY

Abstract. Unimprovable effective efficient conditions are established for the unique solvability of the periodic problem

\begin{aligned}u′_{i}(t) & = ∑_{j=2}^{i+1} ℓ_{i,j}(u_{j})(t) + q_{i}(t)  for 1 ≤ i ≤ n − 1, \\ u′_{n}(t) & = ∑_{j=1}^{n} ℓ_{n,j}(u_{j})(t) + q_{n}(t), \\ u_{j}(0) & = u_{j}(ω)  for 1 ≤ j ≤ n, \end{aligned}

where ω > 0, ℓ_{ij} : C([0, ω]) → L([0, ω]) are linear bounded operators, and q_{i} ∈ L([0, ω]).

2000 Mathematics Subject Classification: 34K06, 34K13

Key words and phrases: Linear functional differential system, periodic boundary value problem, uniqueness.

1. Statement of Problem and Formulation of Main Results

Consider on [0, ω] the system

\begin{aligned}u′_{i}(t) & = ∑_{j=2}^{i+1} ℓ_{i,j}(u_{j})(t) + q_{i}(t) for 1 ≤ i ≤ n − 1, \\ u′_{n}(t) & = ∑_{j=1}^{n} ℓ_{n,j}(u_{j})(t) + q_{n}(t), \end{aligned}  (1.1)

with the periodic boundary conditions

u_{j}(0) = u_{j}(ω) for 1 ≤ j ≤ n,  (1.2)

where n ≥ 2, ω > 0, ℓ_{i,j} : C([0, ω]) → L([0, ω]) are linear bounded operators and q_{i} ∈ L([0, ω]).

By a solution of the problem (1.1), (1.2) we understand a vector function u = (u_{i})_{i=1}^{n} with u_{i} ∈ C̃([0, ω]) (i = \overline{1, n}) which satisfies system (1.1) almost everywhere on [0, ω] and satisfies conditions (1.2).

EJQTDE, 2009 No. 59, p. 1

Much work had been carried out on the investigation of the existence and uniqueness of the solution for a periodic boundary value problem for systems of ordinary differential equations and many interesting results have been obtained (see, for instance, [1–3,7–9,11,12,17] and the references therein). However, an analogous problem for functional differential equations, remains investigated in less detail even for linear equations. In the present paper, we study problem (1.1) (1.2) under the assumption that ℓ_{n,1}, ℓ_{i,i+1} (i = \overline{1, n − 1}) are monotone linear operators. We establish new unimprovable integral conditions sufficient for unique solvability of the problem (1.1),(1.2) which generalize the wellknown results of A. Lasota and Z. Opial (see Remark 1.1) obtained for ordinary differential equations in [13], and on the other hand, extend results obtained for linear functional differential equations in [5,14–16]. These results are new not only for the systems of functional differential equations (for reference see [2,4,6,10] ), but also for the system of ordinary differential equations of the form

\begin{aligned}u′_{i}(t) & = ∑_{j=2}^{i+1} p_{i,j}(t)u_{j}(t) + q_{i}(t) for 1 ≤ i ≤ n − 1, \\ u′_{n}(t) & = ∑_{j=1}^{n} p_{n,j}(t)u_{j}(t) + q_{n}(t), \end{aligned}  (1.3)

where q_{i}, p_{i,j} ∈ L([0, ω]) (see, for instance, [2, 7–9] and the references therein). The method used for the investigation of the problem considered is based on that developed in our previous papers [14–16] for functional differential equations.

The following notation is used throughout the paper: N(R) is the set of all the natural (real) numbers; R^{n} is the space of n-dimensional column vectors x = (x_{i})_{i=1}^{n} with elements x_{i} ∈ R (i = \overline{1, n}); R_{+} = [0, +∞[; C([0, ω]) is the Banach space of continuous functions u : [0, ω] → R with the norm ||u||_{C} = max{|u(t)| : 0 ≤ t ≤ ω}; C([0, ω]; R^{n}) is the space of continuous functions u : [0, ω] → R^{n}; C̃([0, ω]) is the set of absolutely continuous functions u : [0, ω] → R; L([0, ω]) is the Banach space of Lebesgue integrable functions p : [0, ω] → R with the norm ∥p∥_{L} = ∫_{0}^{ω} |p(s)|ds; if ℓ : C([0, ω]) → L([0, ω]) is a linear operator, then ∥ℓ∥ = sup_{0<||x||_{C}≤1} ∥ℓ(x)∥_{L}.

Definition 1.1. We will say that a linear operator ℓ : C([0, ω]) → L([0, ω]) is nonnegative (nonpositive), if for any nonnegative x ∈ C([0, ω]) the inequality ℓ(x)(t) ≥ 0 (ℓ(x)(t) ≤ 0) for 0 ≤ t ≤ ω is satisfied. We will say that an operator ℓ is monotone if it is either nonnegative or nonpositive.

Definition 1.2. With system (1.1) we associate the matrix A_{1} = (a_{i,j}^{(1)})_{i,j=1}^{n} defined by the equalities

a_{1,1}^{(1)} = −1, a_{n,1}^{(1)} = \frac{1}{4}||ℓ_{n,1}||, a_{i,1}^{(1)} = 0 for 2 ≤ i ≤ n − 1,

a_{i+1,i+1}^{(1)} = ||ℓ_{i+1,i+1}||−1, a_{i,i+1}^{(1)} = \frac{1}{4}||ℓ_{i,i+1}|| for 1 ≤ i ≤ n−1, (1.4)

a_{i,j}^{(1)} = 0 for i + 2 ≤ j ≤ n, a_{i,j}^{(1)} = ||ℓ_{i,j}|| for 3 ≤ j + 1 ≤ i ≤ n.

and the matrices A_{k} = (a_{i,j}^{(k)})_{i,j=1}^{n}, (k = \overline{2, n}) given by the recurrence relations

A_{2} = A_{1},  (1.5)
a_{i,j}^{(k+1)} = a_{i,j}^{(k)} for i ≤ k or j ∉ {k, k + 1},  (1.6)

a_{i,j}^{(k+1)} = a_{i,j}^{(k)} + \frac{a_{k,j}^{(k)}}{|a_{k,k}^{(k)}|}a_{i,k}^{(k)} for k + 1 ≤ i ≤ n, k ≤ j ≤ k + 1. (1.7)

Theorem 1.1. Let ℓ_{n,1}, ℓ_{i,i+1} : C([0, ω]) → L([0, ω]) (i = \overline{1, n − 1}) be linear monotone operators,

∫_{0}^{ω} ℓ_{n,1}(1)(s)ds ≠ 0, ∫_{0}^{ω} ℓ_{i,i+1}(1)(s)ds ≠ 0 for 1 ≤ i ≤ n − 1, (1.8)

and

a_{k,k}^{(k)} < 0 for 2 ≤ k ≤ n.  (1.9)

where the matrices A_{k} are defined by relations (1.4)-(1.7). Let, moreover,

∫_{0}^{ω} |ℓ_{n,1}(1)(s)|ds ∏_{j=1}^{n−1} ∫_{0}^{ω} |ℓ_{j,j+1}(1)(s)|ds < 4^{n} ∏_{j=2}^{n} |a_{j,j}^{(j)}|.  (1.10)

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

Definition 1.3. For the system (1.3) we define the matrix A_{1} = (a_{i,j}^{(1)})_{i,j=1}^{n} by the equalities (1.4)-(1.7) with

ℓ_{i,j}(x)(t) = p_{i,j}(t)x(t) for i, j ∈ \overline{1, n}, x ∈ C([0, ω]).  (1.11)

Corollary 1.1. Let

0 ≤ σ_{n}p_{n,1}(t) ≢ 0, 0 ≤ σ_{i}p_{i,i+1}(t) ≢ 0  for 1 ≤ i ≤ n − 1 (1.12)

where σ_{i} ∈ \{−1, 1\} (i = \overline{1, n}), the matrices A_{k} are defined by the relations (1.5)-(1.7), (1.11) and

a_{k,k}^{(k)} < 0  for 2 ≤ k ≤ n.  (1.13)

EJQTDE, 2009 No. 59, p. 3 Let, moreover,

∫_{0}^{ω} |p_{n,1}(s)|ds ∏_{j=1}^{n−1} ∫_{0}^{ω} |p_{j,j+1}(s)|ds < 4^{n} ∏_{j=2}^{n} |a_{j,j}^{(j)}|.  (1.14)

Then problem (1.3), (1.2) has a unique solution.

Now, assume that

ℓ_{1,j} ≡ 0 for j ≠ 2, ℓ_{i,j} ≡ 0 for j ∉ \{i, i + 1\}, i = \overline{2, n − 1},

ℓ_{n,j} = 0 for j = \overline{2, n − 1}. (1.15)

Then system (1.1) is of the following type

\begin{aligned}u′_{1}(t) & = ℓ_{1,2}(u_{2})(t) + q_{1}(t), \\ u′_{i}(t) & = ℓ_{i,i}(u_{i})(t) + ℓ_{i,i+1}(u_{i+1})(t) + q_{i}(t) for 2 ≤ i ≤ n − 1, \\ u′_{n}(t) & = ℓ_{n,1}(u_{1})(t) + ℓ_{n,n}(u_{n})(t) + q_{n}(t), \end{aligned} (1.16)

and from Theorem 1.1 we obtain Corollary 1.2. Let ℓ_{n,1}, ℓ_{i,i+1} (i = \overline{1, n − 1}) be linear monotone operators, the conditions (1.8) hold and

∫_{0}^{ω} |ℓ_{k,k}(1)(s)|ds < 1 for 2 ≤ k ≤ n.  (1.17)

Let, moreover,

\begin{matrix}∫_{0}^{ω} |ℓ_{n,1}(1)(s)|ds ∏_{j=1}^{n−1} ∫_{0}^{ω} |ℓ_{j,j+1}(1)(s)|ds < \\ < 4^{n} ∏_{j=2}^{n} (1 − ∫_{0}^{ω} |ℓ_{j,j}(1)(s)|ds). \end{matrix}  (1.18)

Then problem (1.16), (1.2) has a unique solution.

For the cyclic feedback system

\begin{aligned}u′_{i}(t) & = ℓ_{i}(u_{i+1})(t) + q_{i}(t) for 1 ≤ i ≤ n − 1, \\ u′_{n}(t) & = ℓ_{n}(u_{1})(t) + q_{n}(t), \end{aligned}  (1.19)

Corollary 1.2 yields Corollary 1.3. Let ℓ_{i} : C([0, ω]) → L([0, ω]) (i = \overline{1, n}) be linear monotone operators,

||ℓ_{i}|| ≠ 0 for i = \overline{1, n},  (1.20)

and

∏_{i=1}^{n} ||ℓ_{i}|| < 4^{n}.  (1.21)

Then problem (1.19), (1.2) has a unique solution.

EJQTDE, 2009 No. 59, p. 4 Remark 1.1. The problem

u′′(t) = p(t)u(t) + q(t), u(0) = u(ω), u′(0) = u′(ω),  (1.22)

is equivalent to the problem (1.19), (1.2) with n = 2, ℓ_{1}(x)(t) = x(t), ℓ_{2}(x)(t) = p(t)x(t), q_{1} ≡ 0 and q_{2} ≡ q.

Then if p, q ∈ L([0, ω]), p(t) ≤ 0 and ∫_{0}^{ω} p(s)ds ≠ 0 from the corollary 1.3 it follows that problem (1.19), (1.2) and therefore problem (1.22), has a unique solution if the condition ∫_{0}^{ω} |p(s)|ds < \frac{16}{ω} is fulfilled. This follows from the well-known result of A. Lasota and Z. Opial (see [13]).

Example 1.1. The example below shows that condition (1.21) in Corollary 1.3 is optimal and cannot be replaced by the condition

∏_{i=1}^{n} ||ℓ_{i}|| ≤ 4^{n}.  (1.21_{1})

Define the function u_{0} ∈ C̃([0, 1]) on [0, 1/2], and extend it to [1/2, 1] by the equalities

u_{0}(t) = \begin{cases}1 & for 0 ≤ t ≤ 1/8 \\ sin π(1 − 4t) & for 1/8 < t ≤ 3/8 \\ −1 & for 3/8 < t ≤ 1/2 \end{cases} ,

u_{0}(t) = u_{0}(1 − t) for 1/2 < t ≤ 1.

Now let measurable functions τ_{i} : [0, 1] → [0, 1] and the linear nonnegative operators ℓ_{i} : C([0, 1]) → L([0, 1])(i = \overline{1, n}) be given by the equalities

τ_{i}(t) = \begin{cases}1/8i & for 0 ≤ u′_{0}(t) \\ 1/2 − 1/8i & for 0 > u′_{0}(t) \end{cases} , ℓ_{i}(x)(t) = |u′_{0}(t)|x(τ_{i}(t)).

Then it is clear that u_{0}(0) = u_{0}(1), ℓ_{i} ≠ ℓ_{j} if i ≠ j, and ||ℓ_{i}|| = ∫_{0}^{1} |ℓ_{i}(1)(s)|ds = 16π ∫_{1/8}^{1/4} cos π(1 − 4s)ds = 4 for i = \overline{1, n}. Thus, all the assumptions of Corollary 1.3 are satisfied except (1.21), instead of which condition (1.21_{1}) is fulfilled with ω = 1. On the other hand, from the relations u′_{0}(t) = |u′_{0}(t)|u_{0}(τ_{i}(t)) = ℓ_{i}(u_{0})(t) (i = \overline{1, n}), it follows that the vector function (u_{i}(t))_{i=1}^{n} if u_{i}(t) ≡ u_{0}(t) (i = \overline{1, n}) is a nontrivial solution of problem (1.1), (1.2) with ω = 1, q(t) ≡ 0, which contradicts the conclusion of Corollary 1.3.

EJQTDE, 2009 No. 59, p. 5

2. Auxiliary Propositions

Lemma 2.1. Let the matrices A_{k} (k = \overline{1, n}) be defined by equalities (1.4)-(1.7). Then the following relations hold:

a_{i,j}^{(m)} ≥ 0  for i ≠ j, m = \overline{1, n},  (2.1_{m})
a_{n,1}^{(1)} = a_{n,1}^{(n)}  (2.2_{0})
a_{i,j}^{(λ)} ≤ a_{i,j}^{(m)}  for i ≥ m ≥ 2, j ≥ m, λ ≤ m.  (2.2_{m})

Proof. It immediately follows from the definition of A_{1}, A_{2} that inequalities (2.1_{1}) and (2.2_{2}) are true. Now, we assume that (2.1_{m}) holds for m = 3, .., m_{0} (m_{0} < n) and prove (2.1_{m_{0}+1}). If i ≤ m_{0} or j ∉ {m_{0}, m_{0} + 1}, relation (1.6) implies inequality (2.1_{m_{0}+1}), and if i ≥ m_{0} + 1, j ∈ {m_{0}, m_{0} + 1}, then (2.1_{m_{0}+1}) follows from (1.7).

Now we prove inequality (2.2_{m}). First assume that j ≥ m + 1. Then from (1.6) it is clear that

a_{i,j}^{(λ)} = a_{i,j}^{(λ+1)} = ... = a_{i,j}^{(m)} for j ≥ m + 1, i ≥ m, λ ≤ m. (2.3)

Now, let j = m. Then from (1.6) we get a_{i,m}^{(λ)} = a_{i,m}^{(λ+1)} = ... = a_{i,m}^{(m−1)} for i ≥ m, λ ≤ m. By the last equalities and (2.1_{m}), from (1.7) it follows

a_{i,m}^{(m)} = a_{i,m}^{(m−1)}+\frac{a_{m−1,m}^{(m−1)}}{|a_{m−1,m−1}^{(m−1)}|}a_{i,m−1}^{(m−1)} ≥ a_{i,m}^{(m−1)} = a_{i,m}^{(λ)} for i ≥ m, λ ≤ m,

From this inequality and (2.3) we conclude that (2.2_{m}) is fulfilled for all j ≥ m and i ≥ m. Equality (2.2_{0}) follows immediately from (1.5) and (1.6). □

Also we need the following simple lemma proved in the paper [17].

Lemma 2.2. Let σ ∈ {−1, 1} and σℓ : C([0, ω]) → L([0, ω]) be a nonnegative linear operator. Then

−m|ℓ(1)(t)| ≤ σℓ(x)(t) ≤ M|ℓ(1)(t)| for 0 ≤ t ≤ ω, x ∈ C([0, ω]),

where m = − min_{0≤t≤ω}{x(t)}, M = max_{0≤t≤ω}{x(t)}.

Now, consider on [0, ω] the homogeneous problem

v′_{i}(t) = ∑_{j=2}^{i+1} ℓ_{i,j}(v_{j})(t) for 1 ≤ i ≤ n, (2.4_{i})
v_{j}(0) = v_{j}(ω) for 1 ≤ j ≤ n,  (2.5)

where the operator ℓ_{n,n+1} and function v_{n+1} are defined by the equalities ℓ_{n,n+1} ≡ ℓ_{n,1} and v_{n+1} ≡ v_{1}. Also define the functional Δ_{i} :

EJQTDE, 2009 No. 59, p. 6 C([0, ω]; R^{n}) → R_{+} by the equality Δ_{i}(v) = max_{0≤t≤ω}{v_{i}(t)} − min_{0≤t≤ω}{v_{i}(t)}

(i = \overline{1, n}) for any vector function v = (v_{i})_{i=1}^{n} and put Δ_{n+1} ≡ Δ_{1}.

Lemma 2.3. Let ℓ_{i,i+1} : C([0, ω]) → L([0, ω]) (i = \overline{1, n}) be linear monotone operators,

∫_{0}^{ω} ℓ_{i,i+1}(1)(s)ds ≠ 0  for 1 ≤ i ≤ n,  (2.6)

the matrices A_{k} be defined by the equalities (1.4)-(1.7) and

a_{k,k}^{(k)} < 0  for 2 ≤ k ≤ n.  (2.7)

Let, moreover v = (v_{i})_{i=1}^{n} be a nontrivial solution of the problem ((2.4_{i}))_{i=1}^{n}, (2.5) for which there exists a k_{1} ∈ {2, ..., n} such that v_{k_{1}} ≢ 0. Then if

k_{0} = min{k ∈ {2, ..., n} : v_{k} ≢ 0},  (2.8)

the inequalities

0 < ||v_{k}||_{C} ≤ Δ_{k}(v)  for k = 1, k_{0} ≤ k ≤ n,  (2.9_{k})
0 ≤ a_{k,k}^{(k)}Δ_{k}(v) + a_{k,k+1}^{(k)}Δ_{k+1}(v)  for k_{0} ≤ k ≤ n,  (2.10_{k})

hold, where a_{n,n+1}^{(1)} = a_{n,1}^{(1)}.

Proof. Define the numbers M_{k}, m_{k} ∈ R, t′_{k}, t′′_{k} ∈ [0, ω] by the relations

M^{k} = v^{k}(t′k) = max_{0≤t≤ω}\{v^{k}(t)\}, −m^{k} = v^{k}(t{′′ \atop k}) = min_{0≤t≤ω}\{v^{k}(t)\}, (2.11_{k})

and introduce the sets I_{k}^{(1)} = [t′_{k}, t′′_{k}], I_{k}^{(2)} = I \ I_{k}^{(1)} for t′_{k} < t′′_{k}. It is clear from (2.8) that

v_{k_{0}} ≢ 0.  (2.12)

On the other hand, from (2.4_{k_{0}−1}) by (2.8) we obtain

∫_{0}^{ω} ℓ_{k_{0}−1,k_{0}}(v_{k_{0}})(s)ds = 0.  (2.13)

Equality (2.13), in view of (2.6) and Lemma 2.2 guarantees the existence of a t_{0} ∈ [0, ω] such that v_{k_{0}}(t_{0}) = 0. Then from (2.12) we get (2.9_{k_{0}}).

Let the numbers M_{k_{0}}, m_{k_{0}} ∈ R, t′_{k_{0}}, t′′_{k_{0}} ∈ [0, ω] be defined by the relations (2.11_{k_{0}}) and t′_{k_{0}} < t′′_{k_{0}} (the case t′′_{k_{0}} < t′_{k_{0}} can be considered analogously). The integration of (2.4_{k_{0}}) on I_{k_{0}}^{(r)}, by virtue of (2.5) and (2.8) results in

Δ_{k_{0}}(v) = (−1)^{r}[ ∫_{I_{k_{0}}^{(r)}} ℓ_{k_{0},k_{0}}(v_{k_{0}})(s)ds + ∫_{I_{k_{0}}^{(r)}} ℓ_{k_{0},k_{0}+1}(v_{k_{0}+1})(s)ds] (2.14)

for r = 1, 2. From the last equality, by virtue of (1.4), (2.7), (2.9_{k_{0}}) and (2.2_{k_{0}}) with λ = 1, i = j = k_{0} we get

0 < −a_{k_{0},k_{0}}^{(k_{0})}Δ_{k_{0}}(v) ≤ (−1)^{r} ∫_{I_{k_{0}}^{(r)}} ℓ_{k_{0},k_{0}+1}(v_{k_{0}+1})(s)ds  (2.15_{r})

for r = 1, 2. Assume that v_{k_{0}+1} is a constant sign function. Then in view of the fact that the operator ℓ_{k_{0},k_{0}+1} is monotone we get the contradiction with (2.15_{1}) or (2.15_{2}), i.e., v_{k_{0}+1} changes its sign. Then

M_{k_{0}+1} > 0, m_{k_{0}+1} > 0,  (2.16)

and the inequality (2.9_{k_{0}+1}) holds ((2.9_{1}) if k_{0} = n). If ℓ_{k_{0},k_{0}+1} is a nonnegative operator, from (2.15_{r}) (r = 1, 2) in view of (2.16) by Lemma 2.2 we get 0 < −a_{k_{0},k_{0}}^{(k_{0})}Δ_{k_{0}}(v) ≤ m_{k_{0}+1} ∫_{I_{k_{0}}^{(1)}} |ℓ_{k_{0},k_{0}+1}(1)(s)|ds, 0 <

−a_{k_{0},k_{0}}^{(k_{0})}Δ_{k_{0}}(v) ≤ M_{k_{0}+1} ∫_{I_{k_{0}}^{(2)}} |ℓ_{k_{0},k_{0}+1}(1)(s)|ds. By multiplying these es-

timates and applying the numerical inequality 4AB ≤ (A + B)^{2}, in view of the notations (1.4) we obtain 0 ≤ a_{k_{0},k_{0}}^{(k_{0})}Δ_{k_{0}}(v) + \frac{1}{4}(M_{k_{0}+1} +

m_{k_{0}+1})( ∫_{I_{k_{0}}^{(1)}} |ℓ_{k_{0},k_{0}+1}(1)(s)|ds+∫_{I_{k_{0}}^{(2)}} |ℓ_{k_{0},k_{0}+1}(1)(s)|ds) = a_{k_{0},k_{0}}^{(k_{0})}Δ_{k_{0}}(v)+

a_{k_{0},k_{0}+1}^{(1)}Δ_{k_{0}+1}(v), (0 ≤ a_{n,n}^{(n)}Δ_{n}(v) + a_{n,1}^{(1)}Δ_{1}(v) if k_{0} = n), from which by (2.2_{0}) if k_{0} = n and (2.2_{k_{0}}) with λ = 1, i = k_{0}, j = k_{0} + 1 if k_{0} < n, follows (2.10_{k_{0}}). Analogously, from (2.15_{r}) we get (2.10_{k_{0}}) in the case when the operator ℓ_{k_{0},k_{0}+1} is nonpositive.

Consequently, we have proved the proposition:

i. Let 2 ≤ k_{0} ≤ n, then the inequalities (2.9_{k_{0}}), (2.9_{k_{0}+1}) ((2.9_{1}) if k_{0} = n) and (2.10_{k_{0}}) hold.

Now, we shall prove the following proposition:

ii. Let k_{1} ∈ {k_{0}, ..., n − 1} be such that the inequalities (2.9_{k}),(2.10_{k}) for (k = \overline{k_{0}, k_{1}}), and (2.9_{k_{1}+1}) hold. Then the inequalities (2.9_{k_{1}+2}) if k_{1} ≤ n − 2, (2.9_{1}) if k_{1} = n − 1 and (2.10_{k_{1}+1}) hold too.

Define the numbers M_{k_{1}+1}, m_{k_{1}+1} ∈ R, t′_{k_{1}+1}, t′′_{k_{1}+1} ∈ [0, ω] by the relations (2.11_{k_{1}+1}) and let t′_{k_{1}+1} < t′′_{k_{1}+1} (the case t′′_{k_{1}+1} < t′_{k_{1}+1} can be proved analogously). The integration of (2.4_{k_{1}+1}) on I_{k_{1}+1}^{(r)}, by virtue of (2.5) and (2.8) results in

Δ_{k_{1}+1}(v) = (−1)^{r} ∑_{j=k_{0}}^{k_{1}+2} ∫_{I_{k_{1}+1}^{(r)}} ℓ_{k_{1}+1,j}(v_{j})(s)ds  (2.17)

for r = 1, 2. From this equality, by the conditions (1.4),(2.7),(2.9_{k}) with k = k_{0}, ..., k_{1} + 1, and (2.2_{k_{0}}) with λ = 1, i = k_{1} + 1, j = k_{0}, ..., k_{1} + 1

EJQTDE, 2009 No. 59, p. 8 we get

0 ≤ ∑_{j=k_{0}}^{k_{1}+1} a_{k_{1}+1,j}^{(k_{0})}Δ_{j}(v) + (−1)^{r} ∫_{I_{k_{1}+1}^{(r)}} ℓ_{k_{1}+1,k_{1}+2}(v_{k_{1}+2})(s)ds (2.18)

for r = 1, 2. By multiplying (2.10_{k}) with a_{k_{1}+1,k}^{(k)}/|a_{k,k}^{(k)}| for k ∈ {k_{0}, ..., k_{1}} in view of the inequalities (2.7) we obtain

0 ≤ −a_{k_{1}+1,k}^{(k)}Δ_{k}(v) + \frac{a_{k,k+1}^{(k)}}{|a_{k,k}^{(k)}|} a_{k_{1}+1,k}^{(k)}Δ_{k+1}(v).  (2.19_{k})

Now, summing (2.18) and (2.19_{k_{0}}) by virtue of (1.7) with k = k_{0}, i = k_{1} + 1, j = k_{0} + 1, we get

0 ≤ a_{k_{1}+1,k_{0}+1}^{(k_{0}+1)}Δ_{k_{0}+1}(v) + ∑_{j=k_{0}+2}^{k_{1}+1} a_{k_{1}+1,j}^{(k_{0})}Δ_{j}(v)+

+(−1)^{r} ∫_{I_{k_{1}+1}^{(r)}} ℓ_{k_{1}+1,k_{1}+2}(v_{k_{1}+2})(s)ds,

from which by (2.2_{k_{0}+1}) with i = k_{1} + 1, j ≥ k_{0} + 2, λ = k_{0}, we obtain

0 ≤ ∑_{j=k_{0}+1}^{k_{1}+1} a_{k_{1}+1,j}^{(k_{0}+1)}Δ_{j}(v) + (−1)^{r} ∫_{I_{k_{1}+1}^{(r)}} ℓ_{k_{1}+1,k_{1}+2}(v_{k_{1}+2})(s)ds (2.20)

for r = 1, 2. Analogously, by summing (2.20) and the inequalities (2.19_{k}) for all k = k_{0} + 1, ..., k_{1} we get

0 < −a_{k_{1}+1,k_{1}+1}^{(k_{1}+1)}Δ_{k_{1}+1}(v) ≤ (−1)^{r} ∫_{I_{k_{1}+1}^{(r)}} ℓ_{k_{1}+1,k_{1}+2}(v_{k_{1}+2})(s)ds (2.21)

for r = 1, 2. In the same way as the inequality (2.9_{k_{0}+1}) and (2.10_{k_{0}}) follow from (2.15_{r}), the inequalities (2.9_{k_{1}+2}) ((2.9_{1}) if k_{0} = n − 1) and (2.10_{k_{1}+1}) follow from (2.21).

From the propositions i. and ii. by the the method of mathematical induction we obtain that the inequalities (2.9_{1}), (2.9_{k}) and (2.10_{k}) (k = \overline{k_{0}, n}) hold. □

3. Proofs

Proof of Theorem 1.1. It is known from the general theory of boundary value problems for functional differential equations that if ℓ_{i,j} (i, j = \overline{1, n}) are strongly bounded linear operators, then problem (1.1), (1.2) has the Fredholm property (see [6]). Thus, problem (1.1), (1.2) is uniquely solvable iff the homogeneous problem (2.4_{i})_{i=1}^{n}, (2.5) has only the trivial solution.

EJQTDE, 2009 No. 59, p. 9

Assume that, on the contrary, the problem (2.4_{i})_{i=1}^{n},(2.5) has a nontrivial solution v = (v_{i})_{i=1}^{n}. Let

v_{1} ≢ 0, v_{i} ≡ 0 for 2 ≤ i ≤ n.  (3.1)

Thus from (2.4_{1}) and (2.4_{n}) it follows that v′_{1}(t) ≡ 0 and ℓ_{n,1}(v_{1})(t) ≡ 0, i.e., in view of the fact that the operator ℓ_{n,1} satisfies (1.8) we obtain that v_{1} ≡ 0, which contradicts (3.1). Consequently there exists k_{0} ∈ {2, ..., n} such that v_{k_{0}} ≢ 0. Then all the conditions of Lemma 2.3 are satisfied, from which it follows that 0 < ||v_{1}||_{C} ≤ Δ_{1}(v), i.e., v_{1} ≢ Const and in view of the condition (2.5) the function v′_{1} changes its sign. Thus from (2.4_{1}) by the monotonicity of the operator ℓ_{1,2}, we get that v_{2} changes its sign too. Consequently if M_{2}, m_{2} are the numbers defined by the equalities (2.11_{2}) then

M_{2} > 0, m_{2} > 0,  (3.2)

and if k_{0} is the number defined by the equality (2.8), then k_{0} = 2. Thus from Lemma 2.3 it follows that the inequalities (2.9_{1}), (2.9_{k}) and (2.10_{k}) (k = \overline{2, n}) hold.

Now, assume that the numbers M_{1}, m_{1}, and t′_{1}, t′′_{1} ∈ [0, ω[ are defined by the equalities (2.11_{1}) and t′_{1} < t′′_{1} (the case t′′_{1} < t′_{1} can be proved analogously). By integration of (2.4_{1}) on the set I_{1}^{(r)} we obtain

Δ_{1}(v) = (−1)^{r} ∫_{I_{1}^{(r)}} ℓ_{1,2}(v_{2})(s)ds  (3.3)

for r = 1, 2. First assume that the operator ℓ_{1,2} is nonnegative (the case of nonpositive ℓ_{1,2} can be proved analogously), then from (3.3) by (2.9_{1}), (3.2) and the Lemma 2.2 we obtain

0 < Δ_{1}(v) ≤ m_{2} ∫_{I_{1}^{(1)}} |ℓ_{1,2}(1)(s)|ds, 0 < Δ_{1}(v) ≤ M_{2} ∫_{I_{1}^{(2)}} |ℓ_{1,2}(1)(s)|ds.

By multiplying these estimates and applying the numerical equality 4AB ≤ (A+B)^{2} and the equalities (1.4) we get 0 ≤ a_{1,1}^{(1)}Δ_{1}(v)+ \frac{1}{4}(m_{2} +

M_{2})( ∫_{I_{1}^{(1)}} |ℓ_{1,2}(1)(s)|ds + ∫_{I_{1}^{(2)}} |ℓ_{1,2}(1)(s)|ds) = a_{1,1}^{(1)}Δ_{1}(v) + a_{1,2}^{(1)}Δ_{2}(v),

i.e., all the inequalities (2.10_{k}) (k = \overline{1, n}) are satisfied.

On the other hand from (1.4)–(1.6) and Lemma 2.1 it is clear that

a_{1,1}^{(1)} = −1, a_{n,1}^{(n)} = a_{n,1}^{(1)}, a_{k,k+1}^{(k)} = a_{k,k+1}^{(1)} = \frac{1}{4}||ℓ_{k,k+1}||  (3.4)

for 1 ≤ k ≤ n − 1. By multiplying all the estimates (2.10_{k}) (k = \overline{1, n}) and applying (3.4) we get the contradiction with condition (1.10). Thus our assumption fails, and hence v_{i} ≡ 0 (i = \overline{1, n}). □

EJQTDE, 2009 No. 59, p. 10 Proof of Corollary 1.1. From (1.11) and (1.12) it is clear that ℓ_{n,1} and ℓ_{i,i+1} are monotone operators and (1.8) holds. Also, from (1.13) and (1.14), the conditions (1.9) and (1.10) follow. Consequently all the conditions of Theorem 1.1 are fulfilled for system (1.3). □

Proof of Corollary 1.2. From (1.4), (1.6), and (1.15) it is clear that

a_{k,k}^{(k−1)} = a_{k,k}^{(k−2)} = ... = a_{k,k}^{(1)} = ||ℓ_{k,k}|| − 1 for 2 ≤ k ≤ n,  (3.5)

and

\begin{matrix}a_{k,k−i}^{(k−i−1)} = a_{k,k−i}^{(k−i−2)} = ... = a_{k,k−i}^{(1)} = 0 for 3 ≤ k − i ≤ n, \\ a_{2,1}^{(1)} = 0. \end{matrix} (3.6)

From (1.7), (1.15) and the first equality of (3.6) we get

\begin{matrix}a_{k,k−1}^{(k−1)} = a_{k,k−1}^{(k−2)} + \frac{a_{k−2,k−1}^{(k−2)}}{|a_{k−2,k−2}^{(k−2)}|}a_{k,k−2}^{(k−2)} = \frac{a_{k−2,k−1}^{(k−2)}}{|a_{k−2,k−2}^{(k−2)}|}a_{k,k−2}^{(k−2)} = \\ = \frac{a_{k−2,k−1}^{(k−2)}}{|a_{k−2,k−2}^{(k−2)}|} \frac{a_{k−3,k−2}^{(k−3)}}{|a_{k−3,k−3}^{(k−3)}|}a_{k,k−3}^{(k−3)} = ... = a_{k,2}^{(2)} ∏_{j=2}^{k−2} \frac{a_{j,j+1}^{(j)}}{|a_{j,j}^{(j)}|} = 0 \end{matrix}  (3.7)

for k ≥ 3. From (3.7) and the second equality of (3.6) it is clear that

a_{k,k−1}^{(k−1)} = 0 for 2 ≤ k ≤ n  (3.8)

Then from (1.7) by (3.5) and (3.8) we obtain

a_{k,k}^{(k)} = a_{k,k}^{(k−1)} + a_{k−1,k}^{(k−1)}a_{k,k−1}^{(k−1)}/|a_{k−1,k−1}^{(k−1)}| = ||ℓ_{k,k}|| − 1.

Thus from the conditions (1.17) and (1.18) it follows that (1.9) and (1.10) hold. Consequently all the conditions of Theorem 1.1 are fulfilled for the system (1.16). □

Acknowledgement

The research was supported by the Academy of Sciences of the Czech Republic, Institutional Research Plan No. AV0Z10190503, and the Grant of Georgian National Scientific Foundation # GNSF/ST06/3 − 002.

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Authors’ addresses: Sulkhan Mukhigulashvili

1. Mathematical Institute, Academy of Sciences of the Czech Republic, Žižkova 22, 616 62 Brno, Czech Republic.

2. I. Chavchavadze State University, Faculty of physics and mathematics, I. Chavchavadze Str. No.32, 0179 Tbilisi, Georgia.

E-mail: mukhig@ipm.cz

Iryna Grytsay

1. Taras Shevchenko National University of Kyiv, Faculty of Cybernetics, Department of Mathematical Analysis, Vladimirskaya Street, 64, 01033, Kyiv, Ukraine

E-mail: grytsay@mail.univ.kiev.ua

(Received July 30, 2009)