Abstract
Inequalities for periodic solutions of firstorder functional differential equations are obtained. These inequalities are best possible in a certain sense.
MSC: 34K06, 34K10, 34K13.
Keywords:
functional differential equations; periodic solutions; periodic boundary value problem; estimates of solutions1 Introduction
Periodic solutions of functional differential equations are important in different applications (see, for example, [14] and the references therein, and also works on the general theory of boundary value problems for functional differential equations [511]). Conditions for the solvability of firstorder periodic problems are found in [1223]. In [15,16] the linear case is considered, and unimprovable sufficient conditions for the solvability of the periodic problem
are found in terms of the norms , of linear positive functional operators :
or
If both of these conditions are not satisfied for some norms , , there exist linear positive operators , with these norms such that problem (1)(2) has no solution. As to our knowledge, similar unimprovable estimates for solutions of (1)(2) in terms of norms , are yet unknown. Here we will fill this gap. Moreover, the estimates obtained here (in Theorems 1, 2, 3) can be expanded to some nonlinear functional differential equations (see Remark 1). Theorem 1 gives the best possible estimates of the norm of the Green operator for the periodic boundary value problem. In Theorem 2, we obtain unimprovable estimates of the solutions of (1)(2) for nonnegative f. In Theorem 3, unimprovable bounds of the difference between the maximum and the minimum of a solution are established.
We use the following notation: ℝ is the space of real numbers, C is the space of continuous functions with the norm ; L is the space of integrable functions with the norm ; a linear bounded operator is called positive if it maps nonnegative functions from C into almost everywhere nonnegative functions from L.
Consider the periodic boundary value problem (1)(2), where , are linear positive operators with norms , , 1 is the unit function. An absolutely continuous function is called a solution of the problem if it satisfies the periodic boundary condition (2) and equation (1) for almost all . We have to solve problem (1)(2) if, for example, we search for periodic solutions of the equation with delay
where are periodic locally integrable functions, is a measurable periodic nonnegative delay. Indeed, suppose that linear operators and are defined by the equalities
where and the integer numbers are such that for almost all . It is easy to show that problem (1)(2) has a solution if and only if equation (5) has a periodic solution with the period .
The conditions (3), (4) for the norms of the operators are well known [15]. They guarantee the existence and uniqueness of solutions of problem (1)(2). Note that these conditions are unimprovable in the following sense: if nonnegative numbers , satisfy neither (3) nor (4), then problem (1)(2) has no solution for some linear positive operators with norms , and for some .
2 The main results
In what follows, we suppose that one of conditions (3), (4) is fulfilled. First, we formulate the results only for the simplest problem (1)(2) with the null operator :
where is a linear positive operator with norm , . The assertions of the following Theorems 1, 2, 3 for problem (6) are as follows.
The solution x of (6) satisfies the estimates
If a function f is nonnegative, the solution x of (6) satisfies the estimates
All estimates (7), (8) and (9), which are proved in Theorems 1, 2, 3 in the general case, are best possible (see Remarks 3, 5, 6).
Remark 1 Consider also the nonlinear periodic problem
provided there exist nonnegative functions with norms
such that the operators satisfy the inequalities
It follows from Lemma 3 and the proofs of Theorems 1, 2, 3 that all statements of these theorems are also valid for solutions of periodic problem (10)(11) (if the solutions exist).
Theorem 1If the normsof the linear positive operatorssatisfy the conditions
andxis a solution of (1)(2), then the inequality
holds.
If the normsof the operatorssatisfy
andxis a solution of problem (1)(2), then the inequality
holds.
Remark 2 ([15])
If and both of the conditions (15), (17) are not fulfilled, then there exist linear positive operators , with norms , and a function such that problem (1)(2) has no solution.
Remark 3 From the proof of Theorem 1 it follows that estimates (16), (18) are best possible: if nonnegative numbers , satisfy (15) (or (17)), then equality holds in condition (16) (or (18)) for a unique solution x of problem (1)(2) for some linear positive operators , with norms , and for some function , .
The estimates of solutions (1)(2) for can be obtained in the same way.
Theorem 1^{∗}If the normsof the linear positive operatorssatisfy the conditions
andxis a solution of (1)(2), then the inequality
holds.
If the normsof the operatorssatisfy
andxis a solution of problem (1)(2), then the inequality
holds.
Remark 2^{∗} ([15])
If and both of conditions (19), (21) are not fulfilled, then there exist linear positive operators and with norms , and a function such that problem (1)(2) has no solution.
Remark 3^{∗} From the proof of Theorem 1 it follows that estimates (20), (22) are best possible: if nonnegative numbers , satisfy (19) (or (21)), then equality holds in condition (20) (or (22)) for a unique solution x of problem (1)(2) for some linear positive operators , with norms , and for some function , .
In the next statement we get the best possible lower bounds for solutions of problem (1)(2) for nonnegative f.
Theorem 2Letxbe a solution of problem (1)(2) for some nonnegativef.
If the norms, of the operatorssatisfy the conditions
then
if the norms, of the operatorssatisfy the conditions
then
if the norms, of the operatorssatisfy the conditions
then
Remark 4 ([15])
If and all of conditions (23), (25), (27) are not fulfilled, then there exist linear positive operators and with norms , and a function such that problem (1)(2) has no solution.
Remark 5 From the proof of Theorem 2 it follows that estimates (24), (26), (28) are best possible: if nonnegative numbers , satisfy (23) ((25) or (27)), then equality holds in condition (24) ((26) or (28)) for a unique solution x of problem (1)(2) for some linear positive operators , with norms , and for some function , .
Now we estimate the difference between the maximum and the minimum of solutions.
Theorem 3Let the solvability conditions (4) be fulfilled andxbe a unique solution of (1)(2). If
then
otherwise
Remark 6 From the proof of Theorem 3 it follows that inequalities (29) and (30) are unimprovable. It means that for every number , satisfying the conditions of the theorem, equality holds in conditions (29) or (30) for the solution x of problem (1)(2) for some positive operators with norms , , and for some nonnegative function , .
Remark 7 Theorems 2, 3, as Theorem 1, can be easily reformulated for the case when the solvability condition (3) holds.
3 Proofs
We need three lemmas to prove the main theorems.
Lemma 1Letbe linear positive operators, , , . Then there exist pointsand a functionsatisfying
such that the equality
holds.
Proof Let , . Since and the linear operators are positive, we have
Therefore, for some function satisfying (31), equality (32) holds. □
Lemma 2If, functionsare nonnegative, andsatisfies (31), then there exist linear positive operatorswith the norms
such that equality (32) holds.
Proof Let , , . Then the operators , defined by the equalities
satisfy the conditions of the lemma. □
Lemma 3Letsatisfy (13)(14), . Then there exist a functionsatisfying (31) and pointssuch that the equality
holds.
Proof Let , . Since and using (13), (14), we get
Therefore, for some function satisfying (31), equality (34) holds. □
Remark 8 It is obvious that one can choose the points and in Lemmas 1 and 3 in such a way that the solution y takes its maximum and minimum at these points.
Proofs of Theorems 1, 2, 3 If x is a solution of problem (1)(2) ((10)(11)), then by Lemma 1 (3) this solution satisfies the boundary value problem
where and nonnegative satisfy (31), (33). If condition (3) or (4) holds, then problem (35)(36) has a unique solution, which can be easily found explicitly. Since we are only interested in the maximal and minimal values of the solutions, by Remark 8, we have to obtain only representations for values and .
and
Suppose here that and condition (4) is fulfilled.
Define by P the set of all solutions of problem (35)(36) for all , for all functions and nonnegative such that conditions (12), (31) hold, and for all with .
Let S be the subset of P corresponding to nonnegative functions f.
From Lemmas 1 and 2, it follows that the set P coincides with the set of all solutions of problem (1)(2) for all linear positive operators with norms , and for all with . The subset S consists of all solutions of corresponding problems (1)(2) with nonnegative f.
Define the constants
From representations (37), (38), (39), it easily follows that all the constants are defined correctly and
Moreover, for every solution x of (1)(2), the following inequalities hold:
where the constants , , , are best possible.
The numerator and denominator of fractions in (37), (38), (39) are linear with respect to variables and . Therefore , , and take their minimal and maximal values at the bounds of restriction (31) with respect to variables on each of the sets E and I. Hence we have to consider only the following four different cases:
In case (i) we have
In case (ii) we have
In case (iii) we have
In case (iv) we have
Let , , , be the subsets of S for corresponding to cases (i), (ii), (iii), (iv).
We can easily calculate the minimal and maximal values in every case.
In case (iv) we have
In case (iii) we have
Therefore, in cases (iii) and (iv) we have
In case (i) we have
In case (ii) we have
Considering extremal values in all cases (i), (ii), (iii) and (vi), by elementary calculation, we obtain
This proves all Theorems 1, 2, 3. □
Competing interests
The author declares that he has no competing interests.
Author’s contributions
The author read and approved the final manuscript.
Acknowledgements
Research was supported by the Russian Foundation for Basic Research (14010033814). The author would like to thank both reviewers for their careful reading of the manuscript and valuable remarks.
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