Component-wise positivity of solutions to periodic boundary problem for linear functional differential system

The classical Ważewski theorem claims that the condition p_ij ≤ 0, j ≠ i, i, j =1,...,n, is necessary and sufficient for non-negativity of all the components of solution vector to a system of the inequalities , x_i (0) ≥ 0, i =1, ..., n. Although this result was extent on various boundary value problems and on delay differential systems, analogs of these heavy restrictions on non-diagonal coefficients p_ij preserve in all assertions of this sort. It is clear from formulas of the integral representation of the general solution that these theorems claim actually the positivity of all elements of Green's matrix. The method to compare only one component of the solution vector, which does not require such heavy restrictions, is proposed in this article. Note that comparison of only one component of the solution vector means the positivity of elements in a corresponding row of Green's matrix. Necessary and sufficient conditions of this fact are obtained in the form of theorems about differential inequalities. It is demonstrated that the sufficient conditions of positivity of the elements in the nth row of Green's matrix, proposed in this article, cannot be improved in corresponding cases. The main idea of our approach is to construct a first order functional differential equation for the nth component of the solution vector and then to use assertions, obtained recently for first order scalar functional differential equations. This demonstrates the importance to study scalar equations written in a general operator form, where only properties of the operators and not their forms are assumed. Note that in some cases the sufficient conditions, obtained in the article, does not require any smallness of the interval [0, ω], where the system is considered.

Mathematics Subject Classification 2000: 34K06; 34K10.

Consider the following system of functional differential equations

subjected to the periodic conditions

where x = col(x₁,...,x_n), B_ij : C_{[0, ω]} → L_{[0, ω]}, i, j =1, ..., n, are linear bounded operators, C_{[0, ω]} and L_{[0, ω]} are the spaces of continuous and summable functions y : [0,ω] → R¹, respectively, f_i ∈ L_{[0, ω]}, c_i ∈ R¹ , i = 1, ..., n.

The operators B_ij can be, for example, of the following forms:

and also their linear combinations or superpositions are allowed, where , h_k : [0, ω] → [0, ω], and K_ij : [0, ω] × [0, ω] → R¹ are suitable functions.

If the homogeneous periodic problem

has only the trivial solution, then the boundary value problem (1.1), (1.2) has for each f = col(f₁, ..., f_n) with f_i ∈ L_[0,ω] and c = col(c₁, ..., c_n) with c_i ∈ R¹ , i =1, ..., n, a unique solution x, which has the following representation [1] (see also [2,3])

where the n × n matrix G(t, s) is called Green's matrix of periodic problem (1.1), (1.4) and the n × n matrix X(t) is the fundamental matrix of the system (1.3) such that X(0)−X(ω)= E, where E is the unit n × n matrix. It is clear from the solution representation (1.5) that the matrices G(t, s) and X(t) determine all properties of solutions.

The following property is the basis of the approximate integration method by Tchaplygin [4]: from the conditions

where is a linear bounded functional, it follows that

Series of articles, started with the known article by Luzin [5], were devoted to the various aspects of Tchaplygin's approximate method. The monograph by Lakshmikantham and Leela [6] was one of important in this area. In the book by Krasnosel'skii et al. [7], which was devoted to approximate methods for operator equations, property (1.6) ⇒ (1.7) was also essentially used. These ideas have been developing in scores of books on the monotone technique for approximate solution of boundary value problems for systems of differential equations. Note in this connection the works by Kiguradze and Půža [3,8,9]. From formula of solution representation (1.5) it is clear that property (1.6) ⇒ (1.7) with lx ≡ x(0)−x(ω) is true if and only if all elements of the matrices G(t, s) and X(t) are non-negative.

As a particular case of system (1.1) let us consider the following system

where p_ij ∈ L_{[0, ω]} and h_ij : [0, ω] → [0, ω] are measurable functions. The classical Ważewski's [10] theorem claims that the condition

is necessary and sufficient for the property (1.6) ⇒ (1.7) with lx ≡ x(0) for system of ordinary differential equations

Extensions of this Ważewski's theorem on various boundary value problems with functional differential equations and boundary conditions were obtained in [3] which defines the modern level in this topic. But all these results are based on analogs of condition (1.9), which, of course, are close to necessary conditions for the property (1.6) ⇒ (1.7).

In many problems in practice one needs to compare only one component of solution vector. Let us change the formulation and focus our attention upon the problem of comparison for only one of the components of solution vector. Let k_i be either 1 or 2. In this article we consider the following property: when from the conditions

it does follow that, for a fixed r ∈ {1,...,n}, the components x_r and y_r of vectors x and y satisfy the inequality

This property is a weakening of the property (1.6) ⇒ (1.7) and, as we will obtain below, leads to essentially less hard limitations on the given system. From formula of solution's representation (1.5) it follows that this property is reduced to sign-constancy of all elements standing only in the rth row of Green's matrix.

The article is built as follows. Auxiliary results on positivity (negativity) of Green's matrices in the case when all nondiagonal operators B_ij (i ≠ j, i, j =1, ..., n) are negative (positive) are obtained in Section 2. In Theorem 2.1 the results of this sort are presented as three equivalent facts: the first of them is an assertion about differential inequality, the second--about positivity (negativity) of elements of Green's matrix, and the third--about the spectral radius of a corresponding auxiliary operator. In Theorem 2.2 we choose a constant vector as a test vector function, that simplifies the assertion about differential inequalities and allows us to obtain then estimates of elements of Green's matrix.

The main results of this article about positivity or negativity of elements in the nth row of Green's matrix are formulated in Section 3. General assertions are presented in Theorems 3.1 and 3.2. In Corollary 3.1 we avoid assumptions about positivity (negativity) of Green's functions of auxiliary equations and formulate all assumptions in the form of simple algebraic inequalities.

The proofs of formulated assertions of Sections 2 and 3 are presented in Section 4. The main idea of our approach is to construct a corresponding scalar functional differential equation of the first order

for nth component of a solution vector, where B : C_[0,ω] → L_[0,ω] is a linear continuous operator, f^∗ ∈ L_[0,ω]. This equation is built in Section 4 [11,12]. Then the technique of analysis of the first order scalar functional differential equations, developed, for example, in the works [13,14], is used. On this basis we prove assertions of Section 3.

The results about positivity and negativity of elements of Green's matrices of second order systems with deviating arguments are presented in Section 5.

Note that results of this sort for the Cauchy problem (i.e., if lx ≡ x(0)) and Volterra (according to Tikhonov's definition) operators B_ij : C_[0,ω] → L_[0,ω] were proposed in the recent article [11], where the obtained operator B : C_[0,ω] → L_[0,ω] became a Volterra operator. In this article we consider the periodic problem that implies that the operator B : C_[0,ω] → L_[0,ω] is not a Volterra one even in the case when all B_ij : C_[0,ω] → L_[0,ω], i, j =1, ..., n, are Volterra operators.

For our purpose we recall that the Tikhonov's definition can be generalised in the following manner: An operator B : C_[0,ω] → L_[0,ω] is called 0-Volterra, respectively ω-Volterra, if (Bx)(t) = 0 for t ∈ [0, c], respectively for t ∈ [c, ω], whenever c ∈ [0, ω] and x(t) = 0 for t ∈ [0, c], respectively for t ∈ [c, ω].

In this section we consider the inhomogeneous system (1.1) with the homogeneous boundary conditions (1.4), where as above the operators B_ij : C_[0,ω] → L_[0,ω] are supposed to be linear and bounded.

Assuming that, for any i = 1, ..., n, the scalar periodic problem

is uniquely solvable for every f_i ∈ L_{[0, ω]}, we define the operator acting in the space of n-dimensional vector functions with continuous elements x_i : [0, ω] → R¹ endowed with the norm as follows:

where g_i(t, s) denotes Green's function of the problem (2.1), (2.2).

Note that properties of Green's functions are studied in the book [1], where foundations of the general theory of functional differential equations are established. It was obtained that g_i(·, s) is absolutely continuous for almost every (a.e.) s ∈ [0, ω] on each of the intervals [0, s) and (s, ω]. Below, talking, for example, about positivity of Green's function in the rectangle t, s ∈ [0, ω], we actually mean g_i(t, s) > 0 for t ∈ [0, ω] and a.e. s ∈ [0, ω]. Analogously, if we say that Green's functions g_i(t, s) for 1 = 1, ..., n preserve their signs, we mean that for every i either g_i(t, s) ≥ 0 or g_i(t, s) ≤ 0 for t ∈ [0, ω] and a.e. s ∈ [0, ω].

The following assertion follows from Theorem 3.1 of the article [11], similar results can be found also in the monograph [3].

Theorem 2.1. Let the following conditions be fulfilled:

(1) Green's functions g_i(t, s), i =1, ..., n, of n scalar periodic problems (2.1), (2.2) exist, preserve their signs and are such that

for each positive measurable essentially bounded function φ,

(2) the non-diagonal operators B_ij, i, j =1, ..., n, i ≠ j, are positive or negative such that the operator determined by the formula (2.3) is positive.

Then the following assertions are equivalent:

(a) there exists a vector function with positive absolutely continuous components v_i : [0, ω] → (0, +∞) such that

(b) boundary value problem (1.1) and (1.4) is uniquely solvable for each right hand side f = col(f₁, ..., f_n) such that f_i ∈ L_{[0, ω]}, i =1, ..., n, and the elements G_ij(t, s), i, j = 1, ..., n, of its Green's matrix G(t, s) preserve their signs and satisfy the inequality

while

for i, j =1, ..., n,

(c) the spectral radius of the operatoris less than one.

The following assertion follows from Theorem 2.1 if we set v(t) ≡ col(z₁, ..., z_n) as a constant vector with positive components.

Theorem 2.2. Let the following conditions be fulfilled:

(1) Green's functions g_i(t, s), i =1, ..., n, of n scalar periodic problems (2.1), (2.2) exist, are non-negative (respectively, non-positive), and satisfy inequality (2.4) for each positive measurable essentially bounded function φ,

(2) all non-diagonal operators B_ij, i, j =1, ..., n, i ≠ j, are negative (respectively, positive),

(3) there exists a vector z = col(z₁, ..., z_n) with all positive components such that

respectively,

Then boundary value problem (1.1), (1.4) is uniquely solvable for each right hand side f = col(f₁, ..., f_n), all elements G_ij(t, s), i, j = 1, ..., n, of its Green's matrix G(t, s) are non-negative (respectively, non-positive) and the following inequalities

hold.

Remark 2.1. It is clear that assumption (3) in Theorem 2.2 is fulfilled if and only if there exists a vector with all positive components and ε > 0 such that

respectively,

For system with argument deviations (1.8), the condition (2.10) looks as the following one

that coincides with the fact that is M-matrix (see [15]).

If we set for j = 1, ..., n, then we get

which can be interpreted as the main diagonal dominance and condition (2.11) as its generalization.

Remark 2.2. If instead of assumption (3) in Theorem 2.2 we assume that there exists a constant matrix with non-negative components possessing the property

such that

respectively,

where

then the elements G_ij(t, s), i, j =1, ..., n, of the Green's matrix G(t, s) satisfy the inequalities

Notation 2.1. Let B : C_[0,ω] → L_[0,ω] be a linear operator. For any t ∈ [0, ω] define the sets

and

Note that it can be shown that the sets H*(t) and H*(t) are nonempty for almost every t ∈ [0, ω] and thus we can define

and

Corollary 2.1. Let the following conditions be fulfilled:

(1) the operators B_ii, i = 1, ..., n, admit the representation , where are linear positive operators, and at least one of the following assumptions (1a) or (1b) is satisfied:

(a), are 0-Volterra operators,

and

(b) the inequalities

and

hold for i = 1, ..., n,

(2) all non-diagonal operators B_ij, i, j = 1, ..., n, i ≠ j, are negarive,

(3) there exists a vector z = col(z₁, ..., z_n) with all positive components such that (2.7) holds.

Then boundary value problem (1.1), (1.4) is uniquely solvable for each right hand side f = col(f₁, ..., f_n), all elements G_ij(t, s), i, j = 1, ..., n, of its Green's matrix G(t, s) are non-negative and

The proof is based on Theorem 2.2 and Lemmas 4.2 and 4.4 from the next section.

Remark 2.3. If instead of assumption (3) in Corollary 2.1 we assume that there exists a constant matrix with non-negative components possessing the property (2.12) such that relations (2.13) are fulfilled with δ_ij given by (2.15), then the elements G_ij(t, s), i, j =1, ..., n, of the Green's matrix G(t, s) satisfy the inequalities

Corollary 2.2. Let the following conditions be fulfilled:

(1) the operators B_ii, i = 1, ..., n, admit the representation , where , are linear positive operators, and at least one of the following assumptions (1a) or (1b) is satisfied:

(a) , are ω-Volterra operators,

and

(b) the inequalities (2.18) and

hold for i =1, ..., n,

(2) all non-diagonal operators B_ij, i, j =1, ..., n, i ≠ j, are positive,

(3) there exists a vector z = col(z₁, ..., z_n) with all positive components such that (2.8) holds.

Then boundary value problem (1.1), (1.4) is uniquely solvable for each right hand side f = col(f₁, ..., f_n), all elements G_ij(t, s), i, j =1, ..., n, of its Green's matrix G(t, s) are non-positive and

The proof is based on Theorem 2.2 and Lemmas 4.3 and 4.5 from the next section.

Remark 2.4. If instead of assumption (3) in Corollary 2.2 we assume that there exists a constant matrix with non-negative components possessing the property (2.12) such that relations (2.14) are fulfilled with δ_ij given by (2.15), then the elements G_ij(t, s), i, j =1,..., n, of the Green's matrix G(t, s) satisfy the inequalities

Consider now the particular case of system (1.1)

where are summable functions and are measurable functions, i, j =1, ..., n, k =1, ..., m.

Corollary 2.3. Let the following conditions be fulfilled:

(1) at least one of the following assumptions (1a) or (1b) is satisfied:

(a) , t ∈ [0,ω], i =1, ..., n, k =1, ..., m, and the inequalities

hold, where , t ∈ [0, ω], i =1, ..., n,

(b) the inequalities

and

hold for i =1, ..., n,

where and denote the positive and negative parts, respectively, of the function ,

(2), i, j =1, ..., n, i ≠ j, k =1, ..., m,

(3) there exists a vector z = col(z₁, ..., z_n) with all positive components such that

Then boundary value problem (2.26), (1.4) is uniquely solvable for each right hand side f = col(f₁, ..., f_n) and all elements G_ij(t, s), i, j =1, ..., n, of its Green's matrix G(t, s) are non-negative and satisfy relation (2.20).

Remark 2.5. If instead of assumption (3) in Corollary 2.3 we assume that there exists a constant matrix with non-negative components possessing the property (2.12) such that relations

are fulfilled with δ_ij given by (2.15), then the elements G_ij(t, s), i, j =1, ..., n, of the Green's matrix G(t, s) satisfy the inequalities (2.21).

Remark 2.6. In the case of constant coefficients , the vector z = col(z₁, ..., z_n) appearing in assumption (3) of Corollary 2.3 can be found as a positive solution of the algebraic system

which then guarantees that

Analogously we can get the equality

in Corollary 2.3 if is a non-negative solution of the matrix equation

with δ_ij given by relation (2.15) possessing the property (2.12).

In this section, along with problem (1.1), (1.4) we consider the following auxiliary problem consisting of the n − 1-dimensional system

and the boundary conditions

Assuming that problem (3.1), (3.2) is uniquely solvable for each col(f₁, ..., f_n-1), we denote by its Green's matrix. As above we denote (if exists) by Green's matrix of periodic problem (1.1), (1.4).

Theorem 3.1. Let the following conditions be fulfilled:

(1) Green's functions g_i(t, s), i =1, ..., n − 1, of n − 1 scalar periodic problems (2.1), (2.2) exist, are non-negative, and satisfy inequality (2.4) for each positive measurable essentially bounded function φ,

(2) all non-diagonal operators B_ij, i, j =1, ..., n − 1, i ≠ j, are negative,

(3) there exists a vector z = col(z₁, ..., z_n-1) with all positive components such that

Then periodic problem (3.1), (3.2) is uniquely solvable for each col(f₁, ..., f_n-1), all elements K_ij(t, s), i, j =1, ..., n - 1, of its Green's matrix K(t, s) are non-negative,

and the following assertions are true:

(a) if Green's function g(t, s) of the scalar periodic problem

where the operator B : C_[0,ω] → L_[0,ω] is defined by the equality

and f ^* ∈ L_[0,ω], exists and is positive, then periodic problem (1.1), (1.4) is uniquely solvable for each right hand side f = col(f₁, ..., f_n) and the elements G_nj(t, s), j = 1, ..., n, of its Green's matrix G(t, s) satisfy the relations

in the case of negative operators B_ni, i =1, ..., n - 1, and relations

in the case of positive operators B_ni, i =1, ..., n - 1,

(b) if Green's function g(t, s) of the scalar periodic problem (3.5), (3.6), where the operator B : C_[0,ω] → L_[0,ω] is defined by equality (3.7) and f ^* ∈ L_[0,ω], exists and is negative, then periodic problem (1.1), (1.4) is uniquely solvable for each right hand side f = col(f₁, ..., f_n) and the elements G_nj(t, s), j =1, ..., n, of its Green's matrix G(t, s) satisfy the relations

in the case of negative operators B_ni, i =1, ..., n − 1, and relations

in the case of positive operators B_ni, i =1, ..., n − 1.

Remark 3.1. If instead of assumption (3) in Theorem 3.1 we assume that there exists a constant matrix with non-negative components possessing the property

such that relations

are fulfilled with δ_ij given by (2.15), then the elements K_ij(t, s), i, j =1, ..., n − 1, of the Green's matrix K(t, s) satisfy the inequalities

Theorem 3.2. Let the following conditions be fulfilled:

(1) Green's functions g_i(t, s), i =1, ..., n − 1, of n − 1 scalar periodic problems (2.1), (2.2) exist, are non-positive, and satisfy inequality (2.4) for each positive measurable essentially bounded function φ,

(2) all non-diagonal operators B_ij, i, j =1, ..., n − 1, i ≠ j, are positive,

(3) there exists a vector z = col(z₁, ..., z_n−1) with all positive components such that

Then periodic problem (3.1), (3.2) is uniquely solvable for each col(f₁, ..., f_n−1), all elements K_ij(t, s), i, j =1, ..., n − 1, of its Green's matrix K(t, s) are non-positive,

and the following assertions are true:

(a) if Green's function g(t, s) of the scalar periodic problem (3.5), (3.6), where the operator B : C_[0,ω] → L_[0,ω] is defined by equality (3.7) and f* ∈ L_[0,ω], exists and is positive, then periodic problem (1.1), (1.4) is uniquely solvable for each right hand side f = col(f₁, ..., f_n) and the elements G_nj(t, s), j =1, ..., n, of its Green's matrix G(t, s) satisfy the relations (3.8) in the case of positive operators B_ni, i =1, ..., n−1, and relations (3.9) in the case of negative operators B_ni, i =1, ..., n − 1,

(b) if Green's function g(t, s) of the scalar periodic problem (3.5), (3.6), where the operator B : C_[0,ω] → L_[0,ω] is defined by equality (3.7) and f* ∈ L_[0,ω], exists and is negative, then periodic problem (1.1), (1.4) is uniquely solvable for each right hand side f = col(f₁, ..., f_n) and the elements G_nj(t, s), j = 1, ..., n, of its Green's matrix G(t, s) satisfy the relations (3.10) in the case of positive operators B_ni, i = 1, ..., n -1, and relations (3.11) in the case of negative operators B_ni, i = 1, ..., n -1.

Remark 3.2. If instead of assumption (3) in Theorem 3.2 we assume that there exists a constant matrix with non-negative components possessing the property (3.12) such that relations

are fulfilled with δ_ij given by (2.15), then the elements K_ij(t, s), i, j = 1, ..., n - 1, of the

Green's matrix K(t, s) satisfy the inequalities

Corollary 3.1. Let the following conditions be fulfilled:

(1) the operators B_ii, i = 1, ..., n - 1, admit the representation , whereare linear positive operators, and at least one of the following assumptions (1a) or (1b) is satisfied:

(a) , are 0-Volterra operators,

and

(b) the inequalities (2.18) and (2.19) hold for i = 1, ..., n - 1,

(2) all non-diagonal operators B_ij , i, j = 1, ..., n - 1, i ≠ j, are negative,

(3) there exists a constant matrix with non-negative components possessing the property (3.12) such that inequalities (3.13) are fulfilled, where δ_ij is given by relation (2.15),

(4) the operator B_in, i = 1, ..., n, admit the representation , where are linear positive operators, and moreover .^a

Then the following assertions are true:

(i) if B_ni, i = 1, ..., n - 1, are negative operators and the inequalities

are fulfilled, where

and

then periodic problem (1.1), (1.4) is uniquely solvable for each right hand side f = col(f₁ , ..., f_n) and the elements G_nj(t, s), j = 1, ..., n, of its Green's matrix G(t, s) satisfy the inequalities (3.8),

(ii) if B_ni, i = 1, ..., n - 1, are negative operators and the inequalities

are fulfilled, where the numbers A⁺ and A^- are given by the relations (3.19) and (3.20), respectively, then periodic problem (1.1), (1.4) is uniquely solvable for each right hand side f = col(f₁, ..., f_n) and the elements G_nj(t, s), j = 1, ..., n, of its Green's matrix G(t, s) satisfy the inequalities (3.10),

(iii) if B_ni, i = 1, ..., n - 1, are positive operators and the inequalities (3.18) are fulfilled, where

and

then periodic problem (1.1), (1.4) is uniquely solvable for each right hand side f = col(f₁, ..., f_n) and the elements G_nj(t, s), j = 1, ..., n, of its Green's matrix G(t, s) satisfy the inequalities (3.9),

(iv) if B_ni, i = 1, ..., n - 1, are positive operators and the inequalities (3.21) are fulfilled, where the numbers A+ and A^- are given by the relations (3.22) and (3.23), respectively, then periodic problem (1.1), (1.4) is uniquely solvable for each right hand side f = col(f₁, ..., f_n) and the elements G_nj(t, s), j = 1, ..., n, of its Green's matrix G(t, s) satisfy the inequalities (3.11).

The main results of the article are based on the following assertion, which explains how the scalar functional differential equation for the component x_n of the solution vector can be constructed.

Lemma 4.1. Let there exist Green's matrix of periodic problem(3.1), (3.2). Then col(x₁, ..., x_n) is a solution of periodic problem (1.1), (1.4) if and only if x_n is a solution of scalar periodic problem (3.5), (3.6) and

where the operator B is defined by the equality (3.7) and

Proof. If (x₁, ..., x_n) is a solution of periodic problem (1.1), (1.4) then, using Green's matrix of problem (3.1), (3.2), we obtain that the components x_i, i = 1, ..., n - 1, satisfy the relations (4.1). Substitution of these representations in the nth equation of the system (1.1) leads to the fact that x_n is a solution of scalar periodic problem (3.5), (3.6) in which the operator B and the function f ^* are defined by formulas (3.7) and (4.2), respectively.

The converse implication can be verified easily by using Green's matrix K(t, s) of the periodic problem (3.1), (3.2).

Lemma 4.2. [16]Let the operator B admit the representation B = B⁺ - B^-, where B⁺, B^- : C_[0,ω] : L_[0,ω] are linear positive operators. If

and

then problem (3.5), (3.6) is uniquely solvable for each f ^* ∈ L_[0,ω] and its Green's function g(t, s) is positive.

Lemma 4.3. [16]Let the operator B admit the representation B = B⁺ - B^-, where B⁺, B^- : C_[0,ω] → L_[0,ω] are linear positive operators. If the inequalities (4.3) hold and

then problem (3.5), (3.6) is uniquely solvable for each f ^* ∈ L_[0,ω] and its Green's function g(t, s) is negative.

Lemma 4.4. [13]Let B be an operator admitting the representation B = B+ -B^-, where B⁺, B^- : C_[0,ω] → L_[0,ω] are linear, positive, and 0-Volterra operators. If

and

then problem (3.5), (3.6) is uniquely solvable for each f ^* ∈ L_[0,ω] and its Green's function g(t, s) is positive.

Analogously to Lemma 4.5 one can prove the following assertion:

Lemma 4.5. Let B be an operator admitting the representation B = B⁺ - B^-, where B⁺, B^- : C_[0,ω] → L_[0,ω] are linear, positive, and ω-Volterra operators. If

and

then problem (3.5), (3.6) is uniquely solvable for each f ^* ∈ L_[0,ω] and its Green's function g(t, s) is negative.

Now we are in a position to prove the statements formulated in Sections 2 and 3.

Proof of Corollary 2.1. Lemma 4.4 (respectively, Lemma 4.2) implies that condition (1a) (respectively, condition (1b)) is sufficient for the existence and positivity of Green's functions g_i(t, s), i = 1, ..., n, of periodic problems (2.1), (2.2) and for the validity of relation (2.4).

Consequently, the assertion of the corollary follows from Theorem 2.2.

Proof of Corollary 2.2. It can be proven analogously as Corollary 2.1 on the basis of Lemmas 4.3 and 4.5.

Proof of Corollary 2.3. It follows from Corollary 2.1 as a particular case of the operators B_ij.

Proof of Theorem 3.1. In view of the assumptions (1)-(3), Theorem 2.2 yields that Green's matrix K(t, s) of the periodic problem for auxiliary system (3.1) exists and its elements K_i,j(t, s), i, j = 1, ..., n-1, are non-negative and satisfy the inequalities (3.4). The assertions (a) and (b) follow from analysis of signs in formula (4.2).

Indeed, consider the case when Green's function g(t, s) of the scalar periodic problem (3.5), (3.6) with the operator B : C_[0,ω] → L_[0,ω] defined by equality (3.7) and f ^* ∈ L[0,ω] exists and is positive. Then, according to Lemma 4.1, the periodic problem (1.1), (1.4) has a unique solution col(x₁, ..., x_n) for an arbitrary right hand side f = col(f₁, ..., f_n) whereas

and x_i, i = 1, ..., n - 1, satisfy equalities (4.1), where the function f ^* is given by formula (4.2). It is clear that non-negativity of K_ij(t, s), i, j = 1, ..., n - 1, in the case of negativity of the operators B_ni, i = 1, ..., n - 1, implies that non-negative f_j, j = 1, ..., n, give non-negative f *. From the equalities (4.8) and

it follows desired relation (3.8). In the case of positivity of the operators B_ni, i = 1, ..., n - 1, the inequalities f_i ≤ 0, i = 1, ..., n - 1, f_n ≥ 0 give non-negative f *. Consequently, equalities (4.8) and (4.9) lead to desired relation (3.9).

Consider now the case when Green's function g(t, s) of the scalar periodic problem (3.5), (3.6) with the operator B : C_[0,ω] → L_[0,ω] defined by equality (3.7) and f ^* ∈ L_[0,ω] exists and is negative. One can show analogously as above that non-negativity of K_ij(t, s), i, j = 1, ..., n - 1, in the case of negativity of the operators B_ni, i = 1, ..., n - 1, implies relation (3.10) and in the case of positivity of the operators B_ni, i = 1, ..., n - 1, yields relation (3.11).

Proof of Theorem 3.2. It can be proven analogously as Theorem 3.1.

Proof of Corollary 3.1. According to Lemmas 4.2 and 4.4, each of the condition (1a) and (1b) implies that Green's functions g_i(t, s), i = 1, ..., n - 1, of periodic problems (2.1), (2.2) exist, are positive, and satisfy relation (2.4). Therefore, in view of assumptions (2) and (3), Theorem 3.1 (see also Remark 3.1) guarantees that the elements K_ij, i, j = 1, ..., n - 1, of Green's matrix K(t, s) of the periodic problem (3.1), (3.2) are positive and satisfy inequalities (3.14).

(i) Let us define the operator B : C_[0,ω] → L_[0,ω] by equality (3.7). Then, in view of assumption (4) and relations (3.14), the operator B admits the representation B = B⁺ - B^-, where B⁺, B^- : C_[0,ω] → L_[0,ω] are linear positive operators such that

Therefore, using Lemma 4.2, we get the existence and positivity of Green's function g(t, s) of the periodic problem for scalar equation (3.5), and thus the assertion follows from Theorem 3.1(a).

(ii) According to assumption (4) and relations (3.14), it is clear that the operator B :C_[0,ω] → L_[0,ω] defined by equality (3.7) admits the representation B = B⁺ - B^-, where B⁺, B^- : C_[0,ω] → L_[0,ω] are linear positive operators such that

Therefore, Lemma 4.3 yields that Green's matrix of periodic problem (3.5), (3.6) exists and is negative, and thus the assertion follows from Theorem 3.1(b).

Assertions (iii) and (iv) can be proven analogously.

Consider the periodic problem for the system

where p_ij, f_i : [0,ω] → R¹ are summable functions and the functions h_ij : [0,ω] → [0,ω] are measurable, i, j = 1, 2.

Let us introduce the notation:

means that p₁₁(t) ≥ 0, p₁₂(t) ≤ 0, p₂₁(t) ≤ 0, p₂₂(t) ≥ 0, t ∈ [0,ω]. Moreover, having a measurable essentially bounded function p: [0,ω] → [0,ω], we set

Theorem 5.1. Let the coefficients p_ij, i, j = 1, 2, be essentially bounded and the following conditions be fulfilled:

(1) ,

(2) Δ : = |p₁₁|_* |p₂₂|_* - |p₁₂|^* |p₂₁|^* > 0.

Then Green's matrix G(t, s) of the periodic problem for system (5.1) exists and satisfies:

(a)

(b)

(c)

(d)

Proof. Let the operators B_ij : C_[0,ω] → L_[0,ω] be defined by the relations

Then system (5.1) is a particular case of system (1.1) with n = 2.

According to assumption (2), the algebraic system

has a solution z₁ = (|p₂₂|_* + |p₁₂|*) / Δ > 0 and z₂ = (|p₁₁|_* + |p₂₁|*) / Δ > 0. This solution

satisfies also the inequalities

t ∈ [0, ω], and thus the assertions of the theorem follows from Theorem 2.1 with v(t) ≡ col(z₁, z₂) and Lemmas 4.2 and 4.3.

Theorem 5.2. Let the coefficients p_ij, i, j = 1, 2, be essentially bounded and the following conditions hold:

(1) ,

(2) ,

(3)

Then Green's matrix G(t, s) of the periodic problem for system (5.1) exists and satisfies:

(a)

(b)

(c)

(d)

To prove Theorem 5.2 we need the following lemma.

Lemma 5.1. Let the operator B admit the representation B = B⁺ - B^-, where B⁺, B^-: C_[0,ω] → L_[0,ω] are linear positive operators, and let the Green's function of the problem

exist and be negative. Let, moreover,

Then the Green's function of the problem (3.5), (3.6) exists and is negative.

Proof. It is sufficient to show that any nontrivial absolutely continuous function y : [0, ω] → R¹ satisfying

is negative. Obviously, if y is non-positive, then from (5.5) it follows that

whence, according to the non-negativity of the Green's function of the problem (5.3), we get y(t) < 0 for t [0, ω]. Assume therefore that

and put

Then w(t) ≥ 0 for t ∈ [0, ω] and there exist t₀ ∈ [0, ω] such that

Furthermore, in view of (5.4) and (5.5) we have

Consequently, since the Green's function of the problem (5.3) is negative, on account of (5.6) we obtain w ≡ 0, i.e., y(t) is a constant function. Now from (5.5) we obtain (B1)(t) ≥ 0 for t ∈ [0, ω], which contradicts (5.4).

Proof of Theorem 5.2. Let the operators B_ij : C_[0,ω] → L_[0,ω] be defined by relations (5.2). Then system (5.1) is a particular case of system (1.1) with n = 2.

We will show only the assertion (a). Let the operator B : C_[0,ω] → L_[0,ω] be defined by formula (3.7) with n = 2 in which K₁₁(t, s) denotes Green's function of the scalar periodic problem

the existence and positivity of which follows from Lemma 4.2 and assumption (1). Then we have

and B = B₂₂ - A, where the operator A : C_[0,ω] → L_[0,ω] is defined by the equality

The operator A is positive and non-zero and thus, according to assumption (2) and relation (5.8), Lemma 4.3 guarantees that Green's function of the problem

exists and is negative. Therefore, it follows from Lemma 5.1 that Green's function g(t, s) of the problem (3.5), (3.6) exists and is negative if there exists a positive constant z₂ and ε > 0 such that

Let us prove that such constants z₂ and ε exist. According to assumption (3), the algebraic system

has a solution z₁ = (-|p₂₂|^* -|p₁₂|_*) / Δ > 0 and z₂ = (-|p₁₁|^* - |p₂₁|_*) / Δ > 0. This solution satisfies also the inequalities

t ∈ [0.ω], Therefore, col(z₁, z₂) is a solution of the periodic problem for system (5.1) with f₁(t) ≡ |p₁₁(t)|z₁ - |p₁₂(t)|z₂ and f₂(t) ≡ -|p₂₁(t)|z₁ + |p₂₂(t)|z₂, and Lemma 4.1 with n = 2 then yields that

Consequently, relation (5.9) is satisfied with ε = 1 and thus Green's function g(t, s) of the problem (3.5), (3.6) exists and is negative.

Now it follows from Lemma 4.1 with n = 2 that the periodic problem for system (5.1) has a unique solution col(x₁, x₂) for an arbitrary right hand side col(f₁, f₂) whereas

It is clear that positivity of K₁₁ and negativity of g imply that non-negative f₁, f₂ give non-positive x₂ and thus the representation

guarantees the desired relations G₂₁(t, s) ≤ 0 and G₂₂(t, s) = 0, t, s∈ [0, ω].

Example 5.1. Let us demonstrate that in very natural cases only elements in corresponding rows of Green's matrix and not all its elements can preserve their signs. Consider the periodic problem for a second order scalar differential equation

and the corresponding differential system

with the periodic conditions

For problem (5.12), (5.13) the following assertion follows from Theorem 5.2.

Corollary 5.1. Let p₁₁ and p₁₂ be essentially bounded and let the following conditions hold:

(1) ,

(2) |p₁₂|^* ω < p_11*,

(3)

Then elements G₂₁(t, s) and G₂₂(t, s) of Green's matrix of the problem (5.12), (5.13) are non-positive for t, s ∈ [0, ω].

The element G₂₁(t, s) of Green's matrix of the problem (5.12), (5.13) coincides with Green's function W(t, s) of the periodic problem for the scalar second order equation (5.10), and G₁₁(t, s) corresponds to . It is clear that for any f₁, the derivative x' of a non-constant solution to the problem (5.10), (5.11) changes its sign, and thus the element G₁₁(t, s) also changes its sign.

Consider the system with constant coefficients

where p_ij ∈ R¹, f_i : [0, ω] → R¹ are summable functions and h_ij : [0, ω] → [0, ω] are measurable, i, j = 1, 2.

Corollary 5.2. Let p_ij, i, j = 1, 2, be such that

(1)

(2) 0 < p_ii ω < 1, i = 1, 2,

(3) p₁₂p₂₁ω < min{p₁₁, p₂₂}.

If p₁₁p₂₂ -p₂₁p₁₂ > 0, then Green's matrix G(t, s) of the periodic problem for system (5.14) exists and satisfies

If p₁₁p₂₂ - p₂₁p₁₂ < 0, then Green's matrix G(t, s) of the periodic problem for system (5.14) exists and satisfies

Proof. It follows from Theorems 5.1 and 5.2.

Theorem 5.3. Let the coefficients p₁₁ and p₁₂ be essentially bounded and the following conditions hold:

(1) ,

(2) ,

(3)

Then Green's matrix G(t, s) of the periodic problem for system (5.1) exists and satisfies:

(a)

(b)

(c)

(d)

Proof. Let the operators B_ij : C_[0,ω] → L_[0,ω] be defined by relations (5.2). Then system (5.1) is a particular case of system (1.1) with n = 2.

We will show only the assertion (b). It is clear that condition (1) yields the validity of the assumption (1b) of Corollary 3.1. Moreover, we have

and thus the assumption (3) of Corollary 3.1 is satisfied with Y = y₁₁ = 1/ |p₁₁|_*. If we define numbers A⁺ and A^- by formulas (3.22) and (3.23), respectively, we obtain

Therefore, inequalities (3.21) are fulfilled, because in this case. Consequently, the assertion follows from Corollary 3.1(iv).

The authors declare that they have no competing interests.

AD, RH, and JS obtained the results in a joint research. All authors read and approved the final manuscript.

^aAs usual, denotes the space of measurable and essentially bounded functions y : [0, ω] → R¹ endowed with the norm .

Robert Hakl and Jiří Šremr were supported by RVO: 67985840.

1. Azbelev, NV, Maksimov, VP, Rakhmatullina, LF: Introduction to the theory of functional differential equations. Adv Ser Math Sci Eng, World Federation Publisher Company, Atlanta (1995)

2. Hakl, R, Mukhigulashvili, S: On a boundary value problem for n-th order linear functional differential systems. Georgian Math J. 12(2), 229–236 (2005)

3. Kiguradze, I, Půža, B: Boundary value problems for systems of linear functional differential equations. Folia Facul Sci Natur Univ Masar Brun, Mathematica, Masaryk University, Brno (2003)

4. Tchaplygin, SA: New Method of Approximate Integration of Differential Equations. GTTI, Moscow-Leningrad (1932)

5. Luzin: About method of approximate integration of acad. S A Tchaplygin Uspekhi Mat Nauk. 6(6), 3–27 (in Russian) (1951)

6. Lakshmikantham, V, Leela, S: Differential and integral inequalities. Academic Press, New York/London (1969)

7. Krasnosel'skii, MA, Vainikko, GM, Zabreiko, PP, Rutitskii, JaB, Stezenko, VJa: Approximate methods for solving operator equations.

8. Kiguradze, I, Půža, B: On boundary value problems for systems of linear functional differential equations. Czechoslovak Math J. 47(2), 341–373 (1997). Publisher Full Text

9. Kiguradze, I: Boundary value problems for systems of ordinary differential equations. J Sov Math. 43, 2 (1988)

10. Ważewski, T: Systemes des equations et des inegalites differentielled aux deuxieme membres et leurs applications. Ann Soc Polon Math. 23, 112–166 (1950)

11. Agarwal, R, Domoshnitsky, A: On positivity of several components of solution vector for systems of linear functional differential equations. Glasgow Math J. 52, 115–136 (2010). Publisher Full Text

12. Domoshnitsky, A: Differential inequalities for one component of solution vector for systems of linear functional differential equations. Adv Diff Equ. 2010, 14 (Article ID 478020), doi:10.1155/2010/478020 (2010)

13. Domoshnitsky, A: Maximum principles and nonoscillation intervals for first order Volterra functional differential equations. Dyn Contin Discr Impuls Syst Ser A Math Anal. 15, 769–814 (2008)

14. Hakl, R, Lomtatidze, A, Šremr, J: Some Boundary Value Problems for First Order Scalar Functional Differential Equations. Facul Sci Natur Univ Masar Brun, Mathematica, Masaryk University, Brno (2002)

15. Berman, A, Plemmons, JR: Nonnegative Matrices in the Mathematical Sciences. Academic Press, New York/London (1979)

16. Hakl, R, Lomtatidze, A, Šremr, J: On a boundary-value problem of periodic type for first-order linear functional differential equations. Nonlinear Oscil. 5(3), 408–425 (2002). Publisher Full Text