On the constructive investigation of a class of linear boundary value problems for n th order differential equations with deviating arguments

R E S E A R C H

Open Access

On the constructive investigation of a class of

linear boundary value problems for

n

th order

differential equations with deviating

arguments

Aleksandr Rumyantsev

*_{Correspondence:}

[email protected] Department of Applied

Mathematics and Informatics, Perm State University, Bukireva street, 15, Perm, 614990, Russia

Abstract

A constructive technique for the study of boundary value problems fornth order differential equations with deviating arguments is described. A computer-assisted proof of the correct solvability of the considered problem is given.

MSC: Primary 34B05; 34K06; 65L10; secondary 34K10; 65L70

Keywords: differential equations with deviating arguments; boundary value problems; constructive methods; unique solvability; approximate solution

Introduction

Let us cite some facts from the theory of functional differential equations [] and the gen-eral description of the constructive approach to the investigation of boundary value prob-lems for such equations [, ].

Consider the linear boundary value problem

Ly=f, y=β, ()

whereL:DSn

p[,T](m)→Lnp[,T] is a bounded linear operator,:DSnp[,T](m)→Rnis a

bounded linear vector functional, andf ∈Ln_p[,T],β∈Rmn+n.Rndenotes the linear space of real columnsα, whereα=col{α, . . . ,αn}with the normαn=max≤i≤n|αi|;α

def =

col{|α|, . . . ,|αn|};Lnp[,T], ≤p<∞, denotes the Banach space of measurable functions

z: [,T]→Rn,z(·) =col{z(·), . . . ,zn(·)}, such that

zLn_p[,T]=max ≤i≤n

T

 zi(s)

p

ds



p < +∞;

let us fix a collection of points  =t<t< . . . <tm<tm+=T. PutBq= [tq–,tq),q= , . . . ,m;

Bm+= [tm,T];

χq(t) =

, ift∈Bq,

, ift∈/Bq;

DSn

p[,T](m) denotes the Banach space of all functionsy: [,T]→Rnwithy˙∈Lnp[,T]

and the representation:

y(t) =y() + t



˙

y(s)ds+

m

q=

χ[tq,T](t)y(tq),

wherey(tq) =y(tq) –y(tq– );χ[tq,T](t) =

, ift∈[tq,T], , ift∈/[tq,T];y

def

= col{y(),y(t), . . . ,y(tm)},

with the norm

yDSn_p[,T](m) =˙yLn_p[,T]+ymn_+n.

We assume that the principal boundary value problem

Ly=f, y=α, f ∈Ln_p[,T],α∈Rmn+n, ()

is correctly solvable. Then under these assumptions there exists an (mn+n)×(mn+n)

fundamental matrixYfor the homogeneous equation:

Ly= . ()

The following statement holds: the problem () is uniquely solvable for anyf,αif and only if the (mn+n)×(mn+n) matrix,

def=Y ()

(each column of the (mn+n)×(mn+n) matrixis the result of applying of the functional to the corresponding column of the matrixY) is invertible. The problem () is correctly solvable for a broad class of equations, including

• the ordinary differential equation

Lydef=y˙(t) +P(t)y(t) =f(t), t∈[,T];

P(·) =p(·)ij n

i,j=, pij∈L 

p[,T];

• and a differential equation with concentrated delays

Lydef=y˙i(t) + n

j=

pij(t)yj

hij(t)

=fi(t), t∈[,T];

hij(ξ) = , ξ< ,

pij∈Lp[,T], hij≤t, i,j= , . . .n.

It should be noted that in the more general case of a differential equation with deviating arguments, the problem () does not have this property and some further investigation is required. For example, define the spaceDS() on the partition [,, ] and consider the principal boundary value problem

˙

The key idea of the constructive study of the solvability of the problem () is as follows.

• Two(mn+n)×(mn+n)matrices,a_andv_{, with rational elements are constructed} according to a specially developed procedure based on a computer-assisted proof, such that

–a≤v;

letRn×n_{denotes the linear space of realn×n-matricesA={aij}}n

i,j=with the norm

ARn×n=max_≤i≤nn_j=|aij|;Adef= {|aij|}n i,j=.

• The invertibility of the matrixa_{is verified using exact arithmetic.} • If there exists an inverse matrixa–_{, it should be checked whether}

v Rmn+n<



a–_Rmn+n ()

holds, from which, by the theorem on the inverse operator [, p.], it follows that the matrixis invertible,i.e., the boundary value problem () is correctly solvable.

Further, the suggested general scheme of the constructive investigation will be applied to the boundary value problem for thenth order differential equation with deviating argu-ments.

A class of functions and operators

The constructive techniques for the study of equations with deviating arguments de-scribed below are based on a specific approximation of original problems within the class of computable functions and operators []. In what follows, we assume that the spaces

DSn

p[,T](m) andWSnp[,T](m) are constructed by means of the partition

 =t<t<· · ·<tm<tm+=T, () wheretq,q= , . . . ,m+ , are rational numbers. The setsBq= [tq–,tq),q= , . . . ,m;Bm+= [tm,T], and the corresponding characteristic functionsχq(·) are defined with respect to

the same partition.WSn

p[,T](m) denotes the Banach space of functionsy: [,T]→R,

withy(n)_∈L

p[,T],y(i)∈DSp[,T](m),i= , . . . ,n– , and the representation

y(t) = t



(t–s)(n–) (n– )! y

(n)_(s)ds+

n–

i=

ti i!y

(i)_()

+

n–

i=

m

q= (t–tq)i

i! χ[tq,T](t)y (i)_(t

q);

y(i)(tq) =y(i)(tq) –y(i)(tq– ),

ny=coly(),y()(), . . . ,y(n–)(),y(t),y()(t), . . . , y(n–)(t), . . . ,y()(tm),y()(tm), . . . ,y(n–)(tm) ,

with the norm

yWSn

Dn

p[,T] denotes the Banach space of absolutely continuous functionsx: [,T]→Rnsuch

thatx˙∈Ln

p[,T], with the norm xDn_p[,T]=x()

n+˙xLnp[,T].

Definition  A functiony∈DS_pn[,T](m) is said to possess the propertyC(is computable) if its components as well as the components of the functionsy˙(·) and_(·)y(s)dstake ratio-nal values at ratioratio-nal values of their arguments.

Lety∈DSn

p[,T](m). The propertyCis satisfied by functionsyof the form

y(t) =

m

q=

χq(t)pq(t), t∈[,T], ()

where the componentspq: [,T]→Rn,q= , . . . ,mare polynomials with rational

coeffi-cients. We denote byPn

mthe set of ally∈DSnp[,T](m) of the form (). Definition  A function y∈WSn

p[,T](m) is said to possess the property C (is

com-putable) if this function and the functionsy(i)_{(·),i= , . . . ,nand}(·)

 y(s)dstake rational values at rational values of their arguments.

Obviously, the functionsy∈WSn

p[,T](m) withy(n)∈Pmn possess the propertyC. Definition  A functionh: [,T]→Ris said to be computable over the partition () if

hpossesses the propertyCand for everyj= , . . . ,mthere exists an integerqj, ≤qj≤j,

such thath(t)∈Bqjast∈Bj.

An example of a function that is computable over the partition () ish: [,T]→R_such that

h(t) =

m+

q=

χq(t)hq, ≤hq≤tq,t∈[,T],

wherehq,q= , . . . ,m+ , are rational constants.

Definition  A functionh: [,T]→R_{is said to be computable over the partition () in} the generalized sense ifhpossesses the propertyC, and for everyj= , . . . ,m, there exists an integerqj, ≤qj≤m+ , such thath(t)∈Bqj, ast∈Bj.

An example of a function that is computable over the partition () in the generalized sense ish: [,T]→Rsuch that

h(t) =

m+

q=

χq(t)hq, t∈[,T],

Definition  A bounded linear operatorL:WSn

p[,T](m)→Lnp[,T] is said to possess

the propertyC(is computable) if it mapsP_mn into itself. An example of an operator that is computable is

Ln_y(t)≡y(n)_{(t) +}

n–

i=

ni

j=

pij(t)y(i)

hij(t)

=f(t),

y(i)(ξ) = , ξ∈/[,T],t∈[,T],

if the coefficientspijare the elements of the setPmn and the functionshijare computable

over the partition () in the generalized sense.

Problem setting

Consider the linear boundary value problem

Ln_y(t)≡y(n)_{(t) +}

n–

i=

ni

j=

pij(t)y(i)

hij(t)

=f(t), ()

y(i)(ξ) =

φ_i(ξ), ξ< , φT

i (ξ), ξ>T,

t∈[,T],

ky≡

T



ϕk(s)y(n)(s)ds+ n–

i=

ψ_ik_y(i)()

+

n–

i=

m

q=

ψ_iqky(i)(tq) =βk, k= , . . . ,mn+n,

where y∈WSn

p[,T](m), ≤p<∞, pij, andf ∈Lp[,T], thehij(·) are continuous and

strictly monotonic functions over everyBq,i= , . . . ,n– ,j= , . . . ,ni,

ϕk∈

CS_{[,T](m), p= ,}

L

p[,T], p> ,

βkandψiqk ∈R,k= , . . . ,mn+n,q= , . . . ,m.CSn[,T](m) denotes the Banach space of

functionsx: [,T]→Rn_{, defined by the equality}

x(t) =

m+

q=

χq(t)xq(t), t∈[,T],

xq(·) =col

x_q(·), . . . ,xn_q(·) , wherexi

q,q= , . . . ,m+ ,i= , . . . ,n, are continuous functions, and xCSn_[,T](m)=max

≤i≤n_t∈sup_[,T]

_xi_(t);

pdenotes the adjoint index top:

p=

p

p–, ifp> ,

Consider the principal boundary value problem corresponding to ():

Ln_y(t)≡y(n)_{(t) +}

n–

i=

ni

j=

pij(t)y(i)

hij(t)

=f(t), ()

y(i)(ξ) =

ϕ_i(ξ), ξ< , ϕT

i (ξ), ξ>T,

t∈[,T],

ny=α, α=col{α, . . . ,αmn+n},

under the same assumptions on the problem parameters.

The procedure for the constructive study of the problem () consists of the following steps:

• approximation of the problem () within the class of computable functions and operators,

• study of the principal boundary value problem (), • analysis of its solvability.

Approximation of the problem within the class of computable operators

Fixq= , . . . ,m+ . Approximatepijandf on the setBqby polynomialsapqijandafq with

rational coefficients and define the rational error bounds:

v_pq

ij≥pij–apqijLp[tq–,tq],

v_f

q≥f –afq_L_p[t

q–,tq].

Now define

a_p ij(t) =

m+

q=

χq(t)apijq(t), af(t) = m+

q=

χq(t)afq(t), t∈[,T],

fori= , . . . ,n–  andj= , . . . ,ni. Approximation of thehij

Find rational approximationsa_h¯q

ijofhij(tq),i= , . . . ,n– ,j= , . . . ,ni,q= , . . . ,m+ , and

define rationalv_{hsuch that} a_h¯q

ij–vh≤hij(tq)≤ah¯qij+vh,

and define a rational constanth=GCD{ah¯ij}q . Fixi= , . . . ,n– ,j= , . . . ,ni, and construct

the collection of points{ahν_ij}by the rule:

a_hν

ij=_≤min_q≤m+

_a¯

hq_ij +νmax≤q≤m+{

a_h¯q

ij}–min≤q≤m+{ah¯qij}

mij

,

ν= , . . . ,mij,mij=hν¯ij;

here the positive integer parameterν¯ijdefines the accuracy of the approximation tohij,

assuming thata_hν–

ofνsuch thata_hν

q ij

ij =ah¯ q

ij. Forq= , . . . ,m+ , we denotem q

ij+  for the number of values a_hν

ijthat belong reliably to the inverse image ofhijin the setBq. We define the elements of

the setI_ijq={a_h˜qν

ij} mqij

ν=as follows:

. ah˜q_ijν=ahν

q–

ij +ν

ij ,ν= , . . . ,m q

ij, ifhijis strictly increasing onBq; . a_h˜qν

ij =ah

νq_ij–ν

ij ,ν= , . . . ,m q

ij, ifhijis strictly decreasing onBq.

Next define the pairs of constants [a

t

qν

ij ,vt

qν

ij ], [at

qν

ij ,vt

qν

ij ],ν= , . . . ,m q

ij, such that

h–_ijah˜q_ijν∈a_t_ijqν,a_t_ijqν+v_t_ijqν;

h_ij–ah˜_ijqν+v_h∈a

t

qν

ij ,at

qν

ij +vt

qν

ij

.

We also assume that the following conditions hold:

a

t

qν

ij +vt

qν

ij <at

qν

ij ; tq–<at

q

ij; at

q(mq_ij–)

ij +vt

q(mq_ij–)

ij <T,

which can always be satisfied by means of the accuracy of the calculation of the function

h–

ij for the given points. Further, we construct the setJ q ij ={t

qν

ij },ν= , . . . ,m q ij,

t_ijqν=

a

t

qν

ij +vt

qν

ij +at

qν

ij

 .

From the order of the pointstq_ijν,ν= , . . . ,mq_ij, it follows that

tq–=tijq<t q

ij <· · ·<t q(mq_ij–)

ij <t qmq_ij

ij =tq; ()

a_h˜qν

ij –vh≤h

t_ijqν≤ah˜_ijqν+vh, ν= , . . . ,mq_ij.

Next put

Bq_ijν=tq_ij(ν–),t_ijqν, ν= , . . . ,mq_ij– ;

Bqm q ij ij = ⎧ ⎨ ⎩

[tq(m

q ij–)

ij ,tq), q<m+ ,

[tq(

q_mk ij–)

ij ,T], q=m+ ,

and denote by χ_ijqν(·) the characteristic function ofB_ijqν,ν= , . . . ,mq_ij. We construct the functiona_h

ij(·) by the following rule:

a_h ij(t) =

m+

q=

mq_ij

ν=

χ_ijqν(t)ah˜_ijq(ν–), t∈[,T]. ()

Define the set of pointsJ by

(excluding duplicate elements). By construction, thea_h

ij,i= , . . . ,n– ,j= , . . . ,ni, are

computable over the partition () in the generalized sense. Below suppose thatφ

i ∈Dp[h∗, ],φTi ∈Dp[T,h∗T],i= , . . . ,n– , and define the

con-stantsh∗_andh∗_Tby the equalities

h∗_= min

≤i≤n– ≤j≤ni

b_ij ,

b_ij=min

min t∈[,T]

hij(t) , min

≤ν≤mij _a

hν

ij–vh ,

h∗_T= max

≤i≤n– ≤j≤ni

bT_ij ,

bT_ij =max

min t∈[,T]

hij(t) , max

≤ν≤mij _a

hν_ij+vh .

We approximate the functionsφ_i,φT_i by polynomialsa_φ

i,aφiTwith rational coefficients

and with rational error bounds{v_φ_i,v_φ_i},{v_φT i ,vφiT}:

v

φi≥φi

h∗_–aφ_ih∗_, _vφ_i≥φ˙_i–aφ˙_i

Lp[h∗,], v

φiT≥φiT(T) –aφiT(T), vφiT≥φ˙Ti –aφ˙iTLp[T,h∗_T].

Approximation of the functional

The real numbers βi,i= , . . . ,mn+n, are approximated by rational numbersaβi with

rational error bounds vβi≥ |βi–aβi|. The constantsψiqk are approximated by rational

numbersaψ_iqk with rational error boundsvψ_iqk ≥ |ψ_iqk –aψ_iqk|,i= , . . . ,n– ;q= , . . . ,m;

k= , . . . ,mn+n. On eachBq,q= , . . . ,m+ , the functionsφj,j= , . . . ,mn+n, are

approx-imated by the polynomialsa_φq

j with rational coefficients and with rational error bounds v_φq

j ≥ aφ q j –φ

q

jL_p[tq–,tq]. Define the functions a_φ

j(·) by the equalities

a_φ j(t) =

m

q=

χq(t)aφjq(t), t∈[,T],j= , . . . ,mn+n.

Let us write the boundary value problem approximating the problem () as follows:

_a

Ln_y(t)≡y(n)_{(t) +}

n–

i=

ni

j=

a_p ij(t)y(i)

_a

hij(t)

=af(t), ()

y(i)(ξ) =

a_φ

i(ξ), ξ< , a_φT

i (ξ), ξ>T,

t∈[,T],

ak_y≡

T



a_ϕ

k(s)y(n)(s)ds+ n–

i=

a_ψk iy(i)()

+

n–

i=

m

q=

a_ψk

iqy(i)(tq) =aβk, k= , . . . ,mn+n.

Study of the principal boundary value problem

The aim of this study is to check whether the problem () is correctly solvable, having in mind the computer-assisted proof techniques. This issue is described in detail in []. Below suppose that the problem () is correctly solvable. Then there exists a fundamental systemyk,k= , . . . ,mn+nof the homogeneous equation

Ln_y(t)≡y(n)_{(t) +}

n–

i=

ni

j=

pij(t)y(i)

hij(t)

= , ()

y(i)(ξ) = , ξ∈/[,T];t∈[,T],

and a fundamental systemy˜k,k= , . . . ,mn+n, of the homogeneous equation

_a

Ln_y(t)≡y(n)_{(t) +}

n–

i=

ni

j=

a_p ij(t)y(i)

_a

hij(t)

= , ()

y(i)(ξ) = , ξ∈/[,T];t∈[,T].

Every functionykis defined as a solution of the principal boundary value problem

y(n)(t) +

n–

i=

ni

j=

pij(t)y(i)

hij(t)

= , ()

y(i)(ξ) = , ξ∈/[,T];t∈[,T],

ny=δk, δk={δkq}mnq=+n,δkq=

, k=q, , k=q,

and every functiony˜kis defined as a solution of the principal boundary value problem

y(n)(t) +

n–

i=

ni

j=

a_p ij(t)y(i)

_a

hij(t)

= , ()

y(i)(ξ) = , ξ∈/[,T];t∈[,T],

ny=δk, δk={δkq}mnq=+n,δkq=

, k=q, , k=q,

k= , . . . ,mn+n. Denote bya_y

k an approximation to the functionsy˜kand byvyk the

ap-proximation error bounds:

a_y k(t) =

m+

q=

χq(t)aqyk(t), vyk(t) = m+

q=

χq(t)vqyk, t∈[,T],

v

qyk≥y˜(kn)–aqy

(n)

k Lp[tq–,tq], k= , . . . ,mn+n.

A detailed description of the construction of the functionsa_y

kand the estimations ofvyk,

Analysis of solvability

Denote the elements of the matrices={γij}mn_i,j=+n,a_={a_γkj}mn+n

k,j= andv={vγkj}mnk,j=+nas follows:

γkjdef=kyj=

T



φk(s)y(jn)(s)ds+ψikjτj; ()

a_γ kj

def =

T



a_φ

k(s)ay(jn)(s)ds+aψikjτj;

v_γ kj

def

≥v_ψ ijτj+

m

q= a_ψq

kL_p[tq–,tq] v

qyj+vφq_k×vqyj+vφq_kaqy

(n)

j Lp[tq–,tq] ;

ij=

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

j, ≤j≤n;

j–n, n+ ≤j≤n; ..

.

j–mn, mn+ ≤j≤mn+n;

τj=

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

, ≤j≤n; , n+ ≤j≤n; ..

.

m, mn+ ≤j≤mn+n. From definition () it follows that

v_γ

kj≥γkj–aγkj, –a_R(mn+n)×(mn+n)≤v_R(mn+n)×(mn+n),

where the matrixis defined by (). Thus we arrived at the following.

Theorem  Let the matrixa_{defined by()be invertible,and let the inequality}

v

R(mn+n)×(mn+n)<



a–_R

(mn+n)×(mn+n)

()

hold.Then the boundary value problem()is correctly solvable.

Proof Under the condition of the theorem the inequality –a_R(mn+n)×(mn+n)<



a–

R(mn+n)×(mn+n)

()

holds. By the theorem on the inverse operator the matrixis defined by () to be

invert-ible,i.e., the problem () is correctly solvable.

Competing interests

The author declares that he has not competing interests.

Acknowledgements

The author expresses his sincere thanks to the referees for the careful and noteworthy reading of the manuscript and helpful suggestions that improved the manuscript.

References

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3. Rumyantsev, AN: The reliable computing experiment in the study of boundary value problems for functional differential equations. Funct. Differ. Equ.26(3-4), 499-519 (2002)

4. Akilov, GP, Kantorovich, LV: Functional Analysis. Nauka, Moscow (1978) (in Russian)

5. Rumyantsev, AN: The constructive study of differential equations with deviated argument. Mem. Differ. Equ. Math. Phys.26, 91-136 (2002)

10.1186/1687-2770-2014-53