on the best constants in the solvability conditions for the periodic boundary value problem for...

ISSN 0012-2661, Differential Equations, 2012, Vol. 48, No. 6, pp. 779–786. c© Pleiades Publishing, Ltd., 2012.Original Russian Text c© E.I. Bravyi, 2012, published in Differentsial’nye Uravneniya, 2012, Vol. 48, No. 6, pp. 773–780.

ORDINARY DIFFERENTIAL EQUATIONS

On the Best Constants in the Solvability Conditionsfor the Periodic Boundary Value Problem

for Higher-Order Functional Differential Equations

E. I. BravyiPerm State Technical University, Perm, Russia

Received August 19, 2010

Abstract—We study the properties of the sequence of optimal constants in the conditionsof unique solvability of the periodic boundary value problem for nth-order linear functionaldifferential equations. These constants are expressed via the Euler–Bernoulli constants; simplerecursion relations between them and relations with other known mathematical constants arederived.

DOI: 10.1134/S001226611206002X

1. INTRODUCTION

In the last decade, numerous papers have been devoted to deriving solvability conditions for var-ious boundary value problems for linear functional differential equations. Efficient sharp solvabilityconditions have been obtained for many problems. Note the monographs [1, 2] and the papers [3–8]on the solvability of the periodic problem.

The derivation of sharp solvability conditions for the periodic boundary value problem for an nth-order functional differential equation [8] was an essential achievement. In this paper (see also [4]),sufficient conditions for the unique solvability are obtained for arbitrary n; the constants occurringin these conditions are defined recursively in a very complicated way. For n ≤ 7, it is shown that theobtained solvability conditions are in a sense sharp. For n ≥ 8, it was shown that if some auxiliaryassertion holds (which had not been proved in the general case), then the obtained conditions areoptimal as well, and optimal constants are defined by recursion relations. Necessary and sufficientconditions for the unique solvability of the periodic boundary value problem for families of nth-order functional differential equations with given norms of functional operators were obtained in [9]in a different way. It follows from these results that the conjectures [4, 8] on the optimality ofconstants and the solvability conditions obtained in [8] hold for an arbitrary n ≥ 2.

In the present paper, we continue the study of the properties of the sequence of optimal con-stants, for which we obtain a simple representation via the Euler–Bernoulli constants, analyze theirasymptotic behavior, and obtain a simple recursion formula. We describe the relationship betweenthe optimal constants and other known mathematical constants.

2. MAIN RESULTS

We use the following definitions and notation: R ≡ (−∞,∞); L is the Banach space of integrablefunctions z : [0, 1] → R with norm ‖z‖L =

∫ 1

0|z(t)| dt; C is the Banach space of continuous functions

x : [0, 1] → R with norm ‖x‖C = maxt∈[0,1] |x(t)|; ACn−1 is the Banach space of functionsx : [0, 1] → R with absolutely continuous (n − 1)st derivative, ‖x‖ACn−1 =

∑n−1

i=0 |x(i)(0)| +∫ 1

0|x(n)(t)| dt; a linear operator T : C → L is said to be positive if it takes each nonnegative

continuous function to a function that is nonnegative almost everywhere; the norm of such anoperator is given by the formula ‖T‖ =

∫ 1

0(T1)(t) dt, where 1 is the unit function.

779

780 BRAVYI

Consider the periodic boundary value problem

x(n)(t) = (T+x) − (T−x)(t) + f(t), t ∈ [0, 1], x(i−1)(0) = x(i−1)(1), i = 1, . . . , n. (1)

The solution x belongs to the space ACn−1, and the equality in the functional differential equationis understood as equality almost everywhere on [0, 1]. We assume that n ≥ 2, the linear operatorsT+, T− : C → L are positive, and f ∈ L.

The boundary value problem (1) is said to be uniquely solvable [10, pp. 32–36] if, for eachfunction f ∈ L, the problem has a unique solution x ∈ ACn−1. In this case, the solution admitsan integral representation, whose kernel is referred to as the Green function [10, pp. 113–122].

It is known that problem (1) is Fredholm [10, pp. 103, 113–122]; therefore, it is uniquely solvableif and only if the homogeneous problem

x(n)(t) = (T+x) − (T−x)(t), t ∈ [0, 1], x(i−1)(0) = x(i−1)(1), i = 1, . . . , n,

has only the trivial solution.Let us state the result in [9] (see also [4, 8]) on necessary and sufficient conditions for the unique

solvability of the periodic problem (1) for all functional differential equations with given norms ofthe operators T+ and T−. To this end, we define the constants Mn by the relation

Mn = maxt1,t2∈[0,1]

(Mn,t1,t2 − mn,t1,t2), (2)

where

Mn,t1,t2 = maxs∈[0,1]

(Gn(t1, s) − Gn(t2, s)), mn,t1,t2 = mins∈[0,1]

(Gn(t1, s) − Gn(t2, s)),

and Gn(t, s) is the Green function of the problem

x(n)(t) = f(t), t ∈ [0, 1], x(0) = 0, x(1) = 0, x(i)(0) = x(i)(1), i = 1, . . . , n − 2. (3)

The function Gn(t, s) is continuous for t, s ∈ [0, 1]. For each s ∈ (0, 1), its section is a polynomialof degree n − 1 on the intervals [0, s] and [s, 1], satisfies the boundary conditions of problem (3),and has the unit jump of the (n − 1)st derivative at the point t = s.

Theorem 1. Let T + = T − be given nonnegative numbers. Then, for the boundary value prob-lem (1) to be uniquely solvable for all positive operators T+, T− : C → L such that ‖T+‖ = T + and‖T−‖ = T −, it is necessary and sufficient that

Y

1 − Y≤ X ≤ 2(1 +

√1 − Y ),

where X = Mn max(T +,T −), Y = Mn min(T +,T −), and Mn is the constant given by relation (2).

Hence it follows that, for the problem

x(n)(t) = (Tx)(t) + f(t), t ∈ [0, 1], x(i−1)(0) = x(i−1)(1), i = 1, . . . , n,

with n ≥ 2 to be uniquely solvable for all operators T : C → L with norm ‖T‖ = T such thateither T or −T is a positive operator, it is necessary and sufficient that

0 < T ≤ 4Mn

.

Recursion relations were obtained in [4, 8] for the optimal constants Mn in the solvabilityconditions for n ≤ 7, and the conjecture on the validity of these relations for arbitrary n was putforward.

DIFFERENTIAL EQUATIONS Vol. 48 No. 6 2012

ON THE BEST CONSTANTS IN THE SOLVABILITY CONDITIONS 781

Let us find a closed-form expression for the optimal constants Mn via the Euler–Bernoulliconstants. To this end, we express the Green function Gn(t, s) via the Bernoulli polynomials.

Let Bn(t), n ≥ 2, be the unique solution of the boundary value problem

x(n)(t) = n!, t ∈ [0, 1], x(i)(0) = x(i)(1), i = 0, . . . , n − 2,

1∫

0

x(s) ds = 0, (4)

and let B1(t) be the solution of the problem

x(t) = 1, t ∈ [0, 1],

1∫

0

x(s) ds = 0.

Then Bn(t) is the sequence of Bernoulli polynomials [12, p. 608], which, in particular, satisfy therelations Bn(t) = (−1)nBn(1 − t). The periodic (with unit period) Bernoulli function is given bythe formula Bn(t) = Bn({t}), where {t} stands for the fractional part of the number t. One canreadily see that the function

Gn(t, s) =1n!

(Bn(t) − Bn(0) −Bn(t − s) + Bn(−s))

is the Green function of problem (3).For each continuous function v, we define the range R

sv(s) by the relation

Rsv = max

s∈[0,1]v(s) − min

s∈[0,1]v(s).

It follows from relation (2) that

n!Mn = maxt1,t2∈[0,1]

Rs(Bn(t2 − s) − B(t1 − s)) = max

t∈[0,1]Rs(Bn(t − s) − B(−s))

= maxt,s1,s2∈[0,1)

(Bn(t − s1) − Bn(−s1) − Bn(t − s2) + Bn(−s2)). (5)

Let us compute the constants Mn with the use of relation (5). If n = 1 and n = 2, then we haveB1(t) = t − 1/2 and B2(t) = t2 − t + 1/6, and the constants can readily be computed, M1 = 1,M2 = 1/4.

If n ≥ 3, then the function Bn(t) has the continuous derivative; therefore, a necessary conditionof maximum with respect to the variables t, s1, and s2 is given by the equality to zero of the partialderivatives of the expression occurring in (5) with respect to these variables.

Since, by (4), Bn(t) = nBn−1(t), it follows that the relations

Bn−1(−s2) = Bn−1(t − s2), Bn−1(−s1) = Bn−1(t − s1), Bn−1(t − s1) = Bn−1(t − s2)

should hold at the point of maximum in (5). Consequently,

Bn−1(−s1) = Bn−1(−s2) = Bn−1(t − s1) = Bn−1(t − s2); (6)

moreover, s1, s2 ∈ [0, 1) and s1 = s2, t ∈ (0, 1).Let n ≥ 3 be odd, n = 2m + 1. Then it follows from well-known properties of even-order

Bernoulli polynomials [11, p. 1092] that the function h(t) = (−1)mBn−1(t) with the propertiesh(0) = h(1) = h(−1) < 0 and h(−1/2) = h(1/2) > 0 is increasing on the interval [0, 1/2], isdecreasing on [1/2, 1], and satisfies the relation h(t) = h(1 − t) for all t ∈ [0, 1]. Let s2 ≥ s1.Then the relation Bn−1(−s2) = Bn−1(−s1) holds if and only if (s1 + s2)/2 = 1/2; i.e., s1 + s2 = 1.It follows from the definition of the function h that the relations

Bn−1(t − s2) = Bn−1(−s2) = Bn−1(−s1)


782 BRAVYI

hold only if t − s2 = s1; i.e., t = s2 − s1 = 1 − 2s1. Now the relations

Bn−1(t − s1) = Bn−1(−s2) = Bn−1(−s1)

hold only if t− s1 = s1. Hence we obtain t = 2s1 = 1− 2s1; therefore, t = 1/2 and either s1 = 1/4,s2 = 3/4 or s2 = 1/4, s1 = 3/4 (for s1 ≥ s2). Then we have

(−1)m+1n!Mn = Bn(−3/4) − 2Bn(−1/4) + Bn(1/4) = 4Bn(1/4).

By using the relation [12, p. 609]

4B2m+1(1/4) = −2m + 122m

E2m(1/2) = −2m + 142m

E2m,

where the E2m(t) are the Euler polynomials and the E2m = E2m(1/2) × 22m are the Euler con-stants [12, p. 608]; for odd n = 2m + 1, we obtain the representation

Mn =(−1)m+14Bn(1/4)

n!=

(−1)mnE2m

42mn!=

(−1)mEn−1

4n−1(n − 1)!. (7)

Let n ≥ 4 be even, n = 2m. Then it follows from the properties of odd-order Bernoullipolynomials [11, p. 1092] that the function g(t) = (−1)mBn−1(t) has the following properties:g(−1/2) = g(0) = g(1/2) = g(1) = 0; g(t) < 0 for t ∈ (0, 1/2); g(t) > 0 for t ∈ (1/2, 1);it is decreasing on the intervals [0, t∗1] and [t∗2, 1] and is increasing on the interval [t∗1, t∗2] for somet∗1 ∈ (0, 1/2) and t∗2 ∈ (1/2, 1); and g(t) = −g(1 − t) for all t ∈ [0, 1]. It follows from the form ofthe function g and relations (6) that all values occurring in relations (6) are zero. Therefore, themaximum in (5) is attained at t = 1/2 and either s1 = 0, s2 = 1/2 or s1 = 1/2, s2 = 0. Then foreven n = 2m, we have

Mn =(−1)m2(Bn(1/2) − Bn(0))

n!=

(−1)m4(1 − 2n)Bn

2nn!, (8)

where the Bn = Bn(0) are the Bernoulli constants and in the derivation, we have used the relationB2m(1/2) = (21−2m − 1)B2m(0) [12, p. 609].

From relations (7) and (8), we obtain a simple formula valid for any positive integer n. We havethereby proved the following assertion.

Theorem 2. The following representation holds :

Mn =

⎧⎪⎪⎨

⎪⎪⎩

(−1)m4(1 − 2n)Bn

2nn!if n = 2m is even

(−1)mEn−1

4n−1(n − 1)!if n = 2m + 1 is odd.

(9)

The Euler and Bernoulli constants can be found from the expansions

t

et − 1=

∞∑

n=0

Bn

tn

n!, |t| < 2π,

1cosh t

=∞∑

n=0

En

tn

n!, |t| <

π

4,

[11, p. 1092; 12, p. 607]. We put aside the computation of Mn until obtaining a simple recursionformula for the sequence Mn itself.

Euler obtained the representations of the Bernoulli and Euler constants in the form of theseries [11, p. 1091]

B2m =(−1)m+1(2m)!

22m−1π2m

∞∑

k=1

1k2m

, E2m =(−1)m22m+2(2m)!

π2m+1

∞∑

k=1

(−1)k−1

(2k − 1)2m+1.

This, together with relation (9), implies a new representation for Mn.



Theorem 3. One has

Mn(2π)n = 8(

1 − 12n

) ∞∑

k=1

1kn

= 8∞∑

k=1

1(2k − 1)n

(10)

for even n and

Mn(2π)n = 8∞∑

k=1

(−1)k−1

(2k − 1)n(11)

for odd n ≥ 3.

The representations (10) and (11) readily imply the following assertion.

Theorem 4. The sequence M2m(2π)2m is decreasing with increasing m, and the sequenceM2m+1(2π)2m+1 is increasing. In addition,

limn→∞

Mn(2π)n = 8.

By comparing the representation (9) for Mn and the well-known power-series expansions[11, p. 49]

1cos t

=∞∑

m=0

(−1)mE2m

(2m)!t2m, |t| <

π

2, tan t =

∞∑

m=1

(−1)m+1B2m(42m − 4m)(2m)!

t2m−1, |t| <π

2,

we obtain the following assertion.

Theorem 5. The sequence Mn is the generating sequence for the function

M(t) ≡ 1cos(t/4)

+ tan(t/4) =∞∑

n=0

Mn+1tn, |t| < 2π.

Since the function y(t) = 1/ cos t + tan t satisfies the differential equation 2y = 1 + y2 (e.g.,see [13, p. 16]), we see that the function M(t) satisfies the equation 8M = 1 + M2. Therefore,the coefficients of its expansion satisfy a simple recursion relation. We have thereby proved thefollowing assertion.

Theorem 6. The sequence Mn satisfies the relations M1 = 1, M2 = (M21 + 1)/8 = 1/4, and

Mn+1 =18n

n∑

k=1

MkMn+1−k, n ≥ 2.

Now one can readily compute the elements of the sequence Mn :

M3 =132

, M4 =1

192, M5 =

56144

, M6 =1

7680,

M7 =61

2949120, M8 =

175160960

, M9 =277

528482304, . . .

The recursion relations in the assertion stated below were proved in [4] for n ≤ 7, and it wasconjectured that they hold for arbitrary n (which was proved in [9]).


784 BRAVYI

Theorem 7. The sequence Mn satisfies the relations

M2m+1 = −m∑

k=1

(−1)kM2(m−k)+1

16k(2k)!, M2m = −8

m∑

k=1

(−1)kkM2(m−k)+1

16k(2k)!, m ≥ 1.

3. Mn AND WELL-KNOWN MATHEMATICAL CONSTANTS

3.1. The number An of snakes [13], i.e., of permutations of numbers {0, 1, . . . , n} satisfying theconditions x0 < x1 > x2 < x3 > . . . , is given by the relation

An = 4n+1(n + 1)!Mn+2.

3.2. The volume Vn of the polyhedron [14]

{(y1, . . . , yn) ∈ Rn : yi > 0, yi + yi+1 < 1, i = 1, . . . , n, yn+1 = y1}

can be determined as follows:Vn = 4nMn/8.

3.3. The solvability condition for the periodic boundary value problem for a functional differ-ential equation was obtained in [8] with the use of the constants Hn defined as the best constantsin the inequality

Rsx(s) ≤ Hn+1R

sx(n)(s),

which should hold for all functions x : [0, 1] → R with absolutely continuous nth derivativesatisfying the conditions

x(i)(0) = x(i)(1), i = 0, . . . , n.

Let us show thatHn = Mn for n ≥ 1. (12)

To this end, we obtain a representation of the constants Hn via the Bernoulli polynomials. LetACn−1

ω be the set of functions x ∈ ACn−1 such that

x(i)(0) = x(i)(1), i = 0, . . . , n − 1. (13)

The function Gn(t, s) = (Bn(t)−Bn(t, s))/n! is the Green function of the boundary value problem

x(n)(t) = f(t), t ∈ [0, 1], x(i)(0) = x(i)(1), i = 1, . . . , n − 2,

1∫

0

x(s) ds = 0.

Therefore, the relation

x(t) =

1∫

0

x(s) ds +

1∫

0

(Bn(t) − Bn(t − s))x(n)(s)n!

ds (14)

holds for each function x ∈ ACn−1ω .

Lemma 1. Let k : [0, 1] → R be a continuous function, and let y ∈ AC0ω. Then

1∫

0

k(t)y(t) dt ≤ Rsk(s)R

sy(s);

moreover , for a fixed function k, the constant Rsk(s) cannot be diminished while preserving the

inequality for all y ∈ AC0ω.



The representation (14) with x ∈ ACn−1ω and with arbitrary t1, t2 ∈ [0, 1], together with the

condition∫ 1

0x(n)(s) ds = 0, implies that

x(t1) − x(t2) =

1∫

0

(Bn(t1) − Bn(t1 − s) − Bn(t2) + Bn(t2 − s))x(n)(s)n!

ds

=

1∫

0

(Bn(t2 − s) − Bn(t1 − s))x(n)(s)n!

ds.

It follows from Lemma 1 and the properties of the function Bn that the inequality

Rsx ≤ HnR

sx(n−1)

holds for all x ∈ ACn−1ω , where

Hn = maxt1,t2∈[0,1]

Rs(Bn(t2 − s) −Bn(t1 − s))/n!; (15)

moreover, the constant Hn cannot be diminished. By comparing the representations (15) for Hn

and (5) for Mn, we find that relation (12) holds.

3.4. The formulaFn = Mn+1, n ≥ 1,

provides a relationship of Mn with the Favard constants Fn those are best constants in the Bohr–Favard inequality [15, p. 385 of the Russian translation, D. 27]

maxt∈[0,1]

|x(t)| ≤ Fn vrai supt∈[0,1]

|x(n)(t)|

for all functions x : [0, 1] → R with absolutely continuous (n − 1)st derivative and essentiallybounded nth derivative satisfying the conditions

x(i−1)(0) − x(i−1)(1) = 0, i = 0, . . . , n − 1,

1∫

0

x(t) dt = 0.

3.5. Levin [16] found the best constants Ln in the inequality

|x(t)| ≤ Ln maxt∈[0,1]

|x(n)(t)|, t ∈ [0, 1],

which holds for all functions x(t) n times continuously differentiable on the interval [0, 1] andsatisfying the conditions

x(t1) = x(t2) = · · · = x(n−1)(tn) = 0,

where t1, t2, . . . , tn ∈ [0, 1].The constants Mn are related to Ln by the formula

Ln =Mn+1

4n.

3.6. The best constants Sn [15, p. 385 of the Russian translation, D. 30] in the inequality

maxt∈[0,1]

|x(t)| ≤ Sn

1∫

0

|x(n)(t)| dt


786 BRAVYI

for all x ∈ ACn−1ω [see (13)] satisfying the condition

∫ 1

0x(t) dt = 0 are related to Mn for even n by

the formulaMn = 4Sn.

For odd n, one can use well-known estimates for Bernoulli polynomials and readily obtain theinequality

Mn < 4Sn <8

(2π)n.

ACKNOWLEDGMENTS

The research was supported by the Russian Foundation for Basic Research (project no. 10-01-96054-r-ural-a).

REFERENCES

1. Hakl, R., Lomtatidze, A., and Sremr, J., Some Boundary Value Problems for First Order Scalar Func-tional Differential Equations, Folia Fac. Sci. Natur. Univ. Masaryk. Brun. Math., 10, Brno, 2002.

2. Kiguradze, I. and Puza, B., Boundary Value Problems For Systems of Linear Functional DifferentialEquations, Folia Fac. Sci. Natur. Univ. Masaryk. Brun. Math., 12, Brno, 2003.

3. Lomtatidze, A.G., Puza, B., and Hakl, R., On the Periodic Boundary Value Problem for First-OrderFunctional Differential Equations, Differ. Uravn., 2003, vol. 39, no. 3, pp. 320–327.

4. Hakl, R. and Mukhigulashvili, S., On One Estimate for Periodic Functions, Georgian Math. J., 2005,vol. 12, no. 1, pp. 97–114.

5. Mukhigulashvili, S., On a Periodic Boundary Value Problem for Cyclic Feedback Type Linear FunctionalDifferential Systems, Arch. Math., 2006, vol. 87, no. 3, pp. 255–260.

6. Mukhigulashvili, S.V., On the Solvability of the Periodic Problem for Second-Order Nonlinear Functi-onal-Differential Equations, Differ. Uravn., 2006, vol. 42, no. 3, pp. 356–365.

7. Mukhigulashvili, S., On a Periodic Boundary Value Problem for Third Order Linear Functional Differ-ential Equations, Nonlinear Anal., 2007, vol. 66, no. 2 (A), pp. 527–535.

8. Hakl, R. and Mukhigulashvili, S., A Periodic Boundary Value Problem for Functional Differential Equa-tions of Higher Order, Georgian Math. J., 2009, vol. 16, no. 4, pp. 651–665.

9. Bravyi, E.I., On the Solvability of the Periodic Boundary Value Problem for a Linear Functional Differ-ential Equation, Vestn. Udmurt. Univ. Mat., 2009, vol. 3, pp. 12–24.

10. Azbelev, N.V., Maksimov, V.P., and Rakhmatullina, L.F., Vvedenie v teoriyu funktsional’no-differentsi-al’nykh uravnenii (Introduction to the Theory of Functional Differential Equations), Moscow: Nauka,1991.

11. Gradshtein, I.S. and Ryzhik, I.M., Tablitsy integralov, summ, ryadov i proizvedenii (Tables of Integrals,Sums, Series and Products), Moscow: Izdat. Fiz.-Mat. Lit., 1963.

12. Spravochnik po spetsial’nym funktsiyam (Reference Book on Special Functions), Abramovits, M. andStigan, I., Eds., Moscow, 1973.

13. Arnold, V.I., Snake Calculus and Combinatorics of the Bernoulli, Euler and Springer Numbers of CoxeterGroups, Uspekhi Mat. Nauk , 1992, vol. 47, no. 1 (283), pp. 3–45.

14. Beukers, F., Calabi, E., and Kolk, J.A.C., Sums of Generalized Harmonic Series and Volumes, NieuwArch. Wiskd., 1993, vol. 11, no. 3, pp. 217–224.

15. Hardy, G., Littlewood, J., and Polya, G., Inequalities , Cambridge, 1934. Translated under the titleNeravenstva, Moscow, 1948.

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