ON THE SOLVABILITY OF THE PERIODIC PROBLEM FOR SYSTEMS OF LINEAR FUNCTIONAL DIFFERENTIAL EQUATIONS WITH REGULAR OPERATORS

Systems of two linear functional differential equations of the first order with regular operators are considered. General necessary and sufficient conditions for the unique solvability of the periodic problem are obtained. For one system with monotone operators we get effective necessary and sufficient conditions for the unique solvability of the periodic problem. We consider some classes of two-dimensional systems of first order linear func- tional differential equations with regular operators. General necessary and sufficient conditions for the solvability of the periodic problem for such classes are obtained. These conditions mean that some function on a set in a finite-dimensional space is positive (this functions is quadratic with respect to all variables). Moreover, in terms of norms of the operators appearing in the functional differential system, we get the necessary and sufficient conditions for the unique solvability of the periodic problem for one case of two-dimensional system with monotonic operators. It is found there exist two domains of parameters corresponding to the unique solvability. These result do not have analogues for systems. Non-improvable results for periodic problem are known only for cyclic first order functional differential systems (32). Necessary and sufficient conditions for the unique solvability of two-dimensional


Introduction
We consider some classes of two-dimensional systems of first order linear functional differential equations with regular operators.General necessary and sufficient conditions for the solvability of the periodic problem for such classes are obtained.These conditions mean that some function on a set in a finite-dimensional space is positive (this functions is quadratic with respect to all variables).Moreover, in terms of norms of the operators appearing in the functional differential system, we get the necessary and sufficient conditions for the unique solvability of the periodic problem for one case of two-dimensional system with monotonic operators.
It is found there exist two domains of parameters corresponding to the unique solvability.These result do not have analogues for systems.Non-improvable results for periodic problem are known only for cyclic first order functional differential systems [32].
Necessary and sufficient conditions for the unique solvability of two-dimensional functional differential systems with monotonic operators were achieved only for the Cauchy problems in [37,38,39].Here the similar problem is solved for periodic boundary conditions.Some criteria for the solvability of the periodic problem for ordinary differential equations can be found, for example, in [2,10,14,15,16,26].The works [11,12,13,17,18,20,25,36] are devoted to the investigation of the solvability conditions of the periodic problem for systems of ordinary differential equations.Conditions for the solvability of periodic problem for scalar functional differential equations are obtained in [8,9,19,24,27,28,29,30,31,35].Conditions for the solvability of the periodic problem for systems of functional differential equations are obtained in [7,21,22,23,32,33,34] (see also lists of literature in these articles).
All known conditions for the unique solvability were obtained with the help of some a priori estimates of solutions and fixed point theorems.In this paper it is proving that the unique solvability of periodic problem for all functional differential systems with regular operators from some class (where norms of positive and negative parts of operators are given) are equivalent to the existence only the trivial solutions for all systems from a corresponding class of systems with operators of simple structure.Every such an operator has the form where τ 1 and τ 2 are points from [a, b], functions p 1 and p 2 are integrable.We can often get all solutions of functional differential systems with such operators in the explicit form.So, we have necessary and sufficient conditions for the solvability of the whole class of the original problems.In [3,4,5,6] this approach is applied to other boundary value problems for functional differential equations and systems of such equations.
The main results are necessary and sufficient conditions for the unique solvability of the periodic problem (Theorem 7) for systems of two functional differential equations with regular operators and effective necessary and sufficient conditions of the unique solvability of the periodic problem for a system of functional differential equations with monotonic operators with given norms (Theorem 9, Corollaries 13,15,17).
Throughout the paper we use the following notation: where 1 is the unit function; an operator T is called monotonic if T or −T is a non-negative operator; if an operator can be represented by the difference of nonnegative operators, it is called regular; using the notation with a double index, for example T +/− , means two propositions: one for T + , another for T − .

The periodic problem for systems of functional differential equations
Consider the periodic problem for a two-dimensional system of functional differential equations: (1) where , are linear non-negative operators; the components x and y of the solution belong to the space of absolutely continuous functions AC.
Boundary value problem (1) is called uniquely solvable if it has a unique solution for all f 1 , f 2 ∈ L. It is well known that problem (1) has the Fredholm property (see, for example, [1,40]).Therefore (1) is uniquely solvable if and only if the homogeneous problem has only the trivial solution.
The following assertion is a basic for finding of solvability conditions.
Lemma 1.If problem (2) has a non-trivial solution, then the system has also a non-trivial solution for some points τ 1 , τ 2 , θ 1 , θ 2 ∈ [0, ω] and for some functions Proof.Suppose the homogeneous problem (2) has a non-trivial solution (x, y).Let min Using the inequalities and the non-negativeness of the operators T + ij , T − ij , from (2) we get the inequalities and for some function ξ : [0, ω] → [0, 1] the following equalities hold: , where It is clear that the functions ζ and ξ are measurable and conditions (4) hold.
The conditions for the solvability of problem (3) can be written in the explicit form.If every problem (3) under conditions (4) has only the trivial solution, then problem (2) has only the trivial solution, therefore, problem (1) is uniquely solvable.
Using Lemma 1, we get sufficient conditions for the unique solvability of (1).The inverse statement yields necessary conditions for the unique solvability of all systems with given T then problem (1) is not uniquely solvable for some operators Proof.Define the linear operators T )/2 are the positive and negative parts of the function p ij * , s k = τ k for j = 1, s k = θ k for j = 2, k = 1, 2. These operators are non-negative and T 3) is a solution of the homogeneous problem (2).Since problem (1) has the Fredholm property, we see that (1) is not uniquely solvable.
From Lemmas 1 and 2, we get necessary and sufficient condition for the unique solvability of all functional differential systems from a given class.

be given. Then boundary value problem (1) is uniquely solvable for all linear non-negative operators T
Let non-negative numbers T +/− ij , i, j = 1, 2, be given.Then problem (1) is uniquely solvable for all linear non-negative operators T Remark 4. In Lemma 3, it is sufficient to consider only the cases τ 1 < τ 2 and θ 1 < θ 2 .
Remark 5. Obviously, Lemma 3 is valid not only for the periodic problem but for any boundary value problem.
In the following lemma we get a condition for the existence of a unique solution to the Fredholm problem (3).This condition gives a possibility to obtain criteria of the unique solvability of problem (1).
Proof.The periodic problem for the simplest system ẋ = f 1 , ẏ = f 2 , x(0) = x(ω), y(0) = y(ω), has a solution if and only if In this case the solution is defined by the equalities for arbitrary constants x 0 , y 0 .Therefore, problem (3) has a solution (x, y) if and only if x(τ 2 ) = x(0) +  7), ( 8) hold and these two equations have a non-trivial solution with respect to the variables x(τ 1 ), x(τ 2 ), y(θ 1 ), y(θ 2 ), that is, if and only if equality ( 6) holds.Now we can get a necessary and sufficient condition for the solvability of the periodic problem for all systems with the operators of given norms T Theorem 7. Let non-negative numbers A +/− , B +/− , C +/− , D +/− be given.Periodic problem (1) is uniquely solvable for all non-negative operators T

if and only if
Remark 8.The problem on the necessary and sufficient conditions for the solvability of a class of functional differential equations is reduced to the problem on zeros of some algebraic function given on a finite dimensional set.This function is linear or quadratic with respect to every variable.Using the linearity of △ with respect to x A , y A , x B , y B , x C , y C , x D , y D , we get that to check the conditions of Theorem 7 it is sufficient to prove that the determinants (13) conserve their sign for all 14) and for all other variables at the ends of segments in ( 15)- (17).
Proof.Add the second column of the determinant in (6) to the first column, and the forth column to the third.Using conditions (4), we get EJQTDE, 2011 No. 59, p. 7 where If τ 1 < τ 2 and θ 1 < θ 2 (it follows from Remark 4 that it is sufficient to consider only this case), the function △, defined by equality (6), coincides with the function defined by equality (13).Using Lemmas 3 and 6 completes the proof.

Systems with monotonic operators
Let all operators T ij , i, j = 1, 2, in problem (1) be monotonic.By various substitutes of dependent and independent variables, we can reduce problem (1) to one of two cases: where every linear operator T ij , i, j = 1, 2, is non-negative.Consider here problem (18) only.The following statement will be proved in § 4 with the help of Theorem 7. To prove Theorem 9 we will find extrema of △ with respect to all variables.Two domains of the unique solvability have appeared.One of them corresponds to negative values of △, the other to positive ones.Theorem 9. Let non-negative numbers A, B, C, D be given.The periodic problem (18) is uniquely solvable for all linear non-negative operators T ij : C → L such that or Remark 10.Let inequalities (20) and (21) be fulfilled.Then the following condition is equivalent to inequality (22) from Theorem 9: Proof.Inequality (22) holds if the inequality holds for all t ∈ [0, 1].The left side of the latter inequality takes its maximum at t = 0 or t = 1 or t = 1/2.For t = 0 and t = 1 this inequality is equivalent the inequality which is fulfilled if inequality (21) holds.For t = 1/2 inequality (25) holds if and only if inequality (23) and the inequality hold.Inequality ( 26) is fulfilled if (21) holds.
From Theorem 9 and Remark 11, we obtain a simple sufficient condition for the solvability.
Corollary 12. Let non-negative numbers A, B, C, D be given.Periodic problem (18) is uniquely solvable for all linear non-negative operators T ij : C → L such that if the following inequalities are fulfilled: (19) or (20), ( 21), (23), or (20), ( 21), Necessary and sufficient conditions for the unique solvability has the simplest form when T 11 = T 22 .
Corollary 13.Let non-negative numbers A, C, D be given.Periodic problem (18) is uniquely solvable for all linear non-negative operators T ij : C → L satisfying the conditions Proof.Apply Theorem 9 for B = A. The left side of inequality (22) takes its maximum at t = 0 or t = 1 or t = 1/2.For t = 0 or t = 1 inequality ( 22) is equivalent to inequality which is valid if inequality (21) holds for A = B.
For t = 1/2 inequality ( 22) is equivalent to the inequality The latter inequality holds because for all A ∈ [0, 1) and inequality ( 21) is fulfilled for B = A. So, the corollary is proved.
Now with the help of Theorem 9 we write out the conditions for the unique solvability of (18) for the zero operator T 22 .

Lemma 6 .
Problem (3) has a non-trivial solution if and only if