Volume 2009, Article ID 710386,20pages doi:10.1155/2009/710386
Research Article
On the Correct Solvability of
the Boundary-Value Problem for One Class
Operator-Differential Equations of the Fourth
Order with Complex Characteristics
Araz R. Aliev
1, 2and Aydin A. Gasymov
1 1Institute of Mathematics and Mechanics of NAS of Azerbaijan,9 F. Agaev street, Baku, AZ1141, Azerbaijan
2Baku State University, 23 Z. Khalilov street, Baku, AZ1148, Azerbaijan
Correspondence should be addressed to Araz R. Aliev,[email protected]
Received 19 February 2009; Revised 22 July 2009; Accepted 26 August 2009
Recommended by Ivan T. Kiguradze
Sufficient coefficient conditions for the correct and unique solvability of the boundary-value problem for one class of operator-differential equations of the fourth order with complex characteristics, which cover the equations arising in solving the problems of stability of plastic plates, are obtained in this paper. Exact values of the norms of operators of intermediate derivatives, which are involved in the perturbed part of the operator-differential equation under investigation, are found along with these in subspacesW4
2R;Hin relation to the norms of the
operator generated by the main part of this equation. It is noted that this problem has its own mathematical interest.
Copyrightq2009 A. R. Aliev and A. A. Gasymov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
It is well known that a number of problems in mechanics lead to studying the completeness of all or part of the eigenvectors and joint vectors of certain polynomial operator groups and the completeness of elementary solutions of the operator-differential equations corresponding to these groups see, e.g., 1, 2, and their references. In this case, it is first necessary to investigate the correct solvability of Cauchy or boundary-value problems for these equations, and only after this it will be possible to proceed to the abovementioned problems. The present paper is dedicated to the problem of correct solvability of the boundary-value problem for one class of operator-differential equations of the fourth order, considered on a semiaxis.
Let us consider the following operator-differential equation of the fourth order:
Q
d dt
ut≡
d dt−A
d dtA
3
ut
3
s 1
Asd
4−sut
dt4−s ft, t∈R 0;∞, 1.1
with the boundary conditions
dku0
dtk 0, k 0,1,2, 1.2
where ft ∈ L2R;H, ut ∈ W24R;H, As, s 1,2,3, are linear and generally unbounded operators inH. UnderL2R;HandW24R;H, the following Hilbert spaces
can be described:
L2R;H
ft:fL2R;H
∞
0
ft2
Hdt 1/2
<∞
,
W24R;H
⎧ ⎪ ⎨ ⎪
⎩ut:uW24R;H
⎛ ⎝∞
0
⎛ ⎝
d
4ut
dt4
2
H
A4ut2
H ⎞ ⎠dt
⎞ ⎠
1/2
<∞
⎫ ⎪ ⎬ ⎪ ⎭ 1.3
see3–5.
Definition 1.1. If the vector functionut∈W4
2R;Hsatisfies1.1almost everywhere inR, then
it is called a regular solution of 1.1.
Definition 1.2. If for anyft ∈ L2R;H, there exists a regular solution of 1.1which satisfies
boundary condition1.2in the sense that
lim t→0
A7/2−k d kut
dtk H
0, k 0,1,2, 1.4
and the inequality
uW4
2R;H≤const
f
L2R;H 1.5
Let us define the following subspaces of the spaceW24R;H:
o
W24R;H
ut:ut∈W24R;H, d
ju0
dtj 0, j 0,1,2,3
,
o
W24R;H;{k}
ut:ut∈W24R;H,
dku0
dtk 0, k 0,1,2
.
1.6
It should be noted that the solvability theory for the Cauchy problem and the boundary-value problems for first- and second-order operator-differential equations have been studied in more detail elsewhere. In addition to books6,7, these problems have been considered also by Agmon and Nirenberg 8, Gasymov and Mirzoev 9, Kostyuchenko and Shkalikov 10, and in works in their bibliographies. Other papers in which issues of the solvability of various problems for operator-differential equations of higher order have been studied have appeared alongside these works, and sufficiently interesting results have been obtained. Among these papers are those by Gasymov 11, 12, Dubinskii
13, Mirzoev 14, Shakhmurov 15, Shkalikov16, Aliev 17, 18, Agarwal et al. 19, Favini and Yakubov 20, the book by Yakubov 7, and other works listed in their bibliographies.
Sufficient coefficient conditions for regular solvability of the boundary-value problem stated in1.1 and 1.2are presented in this paper. To obtain these conditions, the main challenge is to find the exact values of the norms of operators of intermediate derivatives in
subspaces o
W4
2R;H,
o
W4
2R;H;{k}, the norms of which are expressed by the main part of 1.1. This problem has its own mathematical interestsee, e.g.,21,22, and works given in their bibliographies. Estimation of the norms of operators of intermediate derivatives, which are involved in the perturbed part of 1.1, is performed with the help of a factorization method for one class of polynomial operator groups of eighth order, depending on a real parameter. A similar approach has been presented in 9, 14, which makes it possible to formulate solvability theorems for the boundary-value problems, with conditions which can be easily checked.
It should be noted that if the main part of the equation has the operator in the form
−d2/dt2A22, then a biharmonic equation results, which is of mathematical interest not only theoretically, and also from a practical point of view. Many problems of elasticity theory e.g., the theory of bending of thin elastic slabs 23 can be reduced to studying the boundary-value problems for such equations. Much research has been performed to investigate the solvability of such problems, for example, that reported in 24. Operator-differential equations, which are studied in the present paper, include the fourth-order equations which arise when solving the stability problems of plates made of plastic material
see25, pages 185–196. It is very difficult to solve such problems because the differential equation must be solved in a more complete form, that is, when the main part of the equation has terms containingdut/dtandd3ut/dt3. As a result, the equation has more complex
characteristics, and1.1is of this type.
2. Auxiliary Results
First, let us study the main part of1.1:
Q0
d dt;A
ut≡
d dt−A
d dtA
3
ut ft, 2.1
whereft∈L2R;H.
The following theorem is true.
Theorem 2.1. OperatorQ0k, acting from the space o
W4
2R;H;{k}toL2R;Hin the following
way:
Q0kut≡Q0
d dt;A
ut, ut∈
o
W24R;H;{k}, 2.2
is an isomorphism between the spaces o
W4
2R;H;{k}andL2R;H.
Proof. It holds that Q0kut ft has a solution ut ∈
o
W24R;H;{k} for anyft ∈
L2R;H. In fact, the vector function
vt 1
2π
∞
−∞−iξEA
−3−iξE−A−1
∞
0
fse−iξsds
eitξdξ, t∈R, 2.3
satisfies the equation
d dt−A
d dtA
3
vt ft 2.4
inR almost everywhere. Let us prove thatvt ∈ W24R;H R −∞;∞. As is made
clear here, this means that
L2R;H
ft:fL
2R;H
∞
−∞ ft2
Hdt 1/2
<∞
,
W24R;H
⎧ ⎪ ⎨ ⎪
⎩ut:uW24R;H
⎛ ⎝∞
−∞ ⎛ ⎝
d
4ut
dt4
2
H
A4ut2
H ⎞ ⎠dt
⎞ ⎠
1/2
<∞
⎫ ⎪ ⎬ ⎪ ⎭.
From the Plancherel theorem, it follows that it is sufficient to show that A4vξ, ξ4vξ ∈
L2R;H, wherevξis the Fourier transform of the vector functionvt. From the spectral
theory of self-adjoint operators,
A4vξ L2R;H
A4−iξEA−3−iξE−A−1fξ L2R;H
≤A4−iξEA−3−iξE−A−1 H→H
fξ L2R;H
≤ sup
σ∈σA
σ4−iξσ−3−iξ−σ−1fξ
L2R;H
constfξ L2R;H
constfL
2R;H.
2.6
Herefξis the Fourier transform of the vector functionft. Analogously, it is possible to prove thatξ4vξ∈L2R;H. Consequently,vt∈W24R;H. Furthermore, let us denote by
u1tthe narrowing of the vector functionvton0;∞. It is clear thatu1t∈W24R;H.
Now,
ut u1t e−tAη0tAe−tAη1t2A2e−tAη2, t∈R, 2.7
where the vectors ηl ∈ DA7/2−l, l 0,2, and are defined by the condition ut ∈ o
W24R;H;{k}. This is why the following system of equations can be obtained relatively
toηl, l 0,2:
u10 η0 0,
du10
dt −Aη0Aη1 0, d2u10
dt2 A 2η
0−2A2η12A2η2 0.
2.8
From this, it is possible to obtain the operator equation,
MEη ζ, 2.9
where
ME
⎛ ⎜ ⎜ ⎝
E 0 0
−E E 0
E −2E 2E
⎞ ⎟ ⎟ ⎠, η
⎛ ⎜ ⎜ ⎝
η0
η1
η2
⎞ ⎟ ⎟ ⎠, ζ
⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
−u10
−A−1du10
dt
−A−2d2u10
dt2
⎞ ⎟ ⎟ ⎟ ⎟ ⎟
⎠. 2.10
Because u1t ∈ W24R;H, then from the theorem on trace 3–5, Chapter 1, it follows
that the operator matrix ME is boundedly invertible in H3 3
p 1H. Therefore, all
ηl ∈ DA7/2, l 0,2. Consequently, ut ∈ o
W24R;H;{k}. In the same way, it can be
established that the equationQ0kut 0 has only a trivial solution. OperatorQ0kis bounded, because
Q0ku2
L2R;H
d 4u dt4 2
L2R;H
4Ad
3u dt3 2
L2R;H
4A3du dt
2
L2R;H
A4u2
L2R;H
4 Re
d4u
dt4, A
d3u
dt3
L2R;H
−4 Re
d4u
dt4, A 3du
dt
L2R;H
−2 Re
d4u
dt4, A 4u
L2R;H −8 Re
Ad
3u
dt3, A 3du
dt
L2R;H
−4 Re
Ad
3u
dt3, A 4u
L2R;H
4 Re
A3du dt, A
4u
L2R;H
≤3d
4u dt4 2
L2R;H
6Ad
3u dt3 2
L2R;H
6A2d
2u dt2 2
L2R;H
4A3du dt
2
L2R;H
A4u2
L2R;H
≤constu2W4 2R;H,
2.11
because forut∈
o
W4
2R;H;{k}
2 Re
d4u
dt4, A 3du
dt
L2R;H
0, 2 Re
d4u
dt4, A 4u
L2R;H
2A2d2u
dt2
2
L2R;H
,
2 Re
Ad
3u
dt3, A 3du
dt
L2R;H
−2A2d
2u dt2 2
L2R;H
, 2 Re
Ad
3u
dt3, A 4u
L2R;H
0,
2 Re
A3du dt, A
4u
L2R;H 0.
The theorem on intermediate derivatives 3–5, Chapter 1 can be used to obtain the last inequality, with the inequality
Ajd
4−ju
dt4−j
L2R;H
≤cjuW4
2R;H, j 0,4, 2.13
assumed. Moreover, the Bunyakovsky-Schwartz and Young inequalities,
Re
d4u
dt4, A
d3u
dt3
L2R;H
≤d4u
dt4
L2R;H Ad
3u
dt3
L2R;H
≤ 1
2 d
4u
dt4
2
L2R;H
1
2 Ad
3u
dt3
2
L2R;H
,
2.14
are used in the expression Red4u/dt4, Ad3u/dt3
L2R;H.
As a result, Q0k is bounded and acts mutually and uniquely from the space o
W4
2R;H;{k} to the space L2R;H. Then, taking into account the Banach theorem on
the inverse operator, it can be established that the operatorQ0kcarries out the isomorphism from the space
o
W4
2R;H;{k}toL2R;H. Thus, the theorem is proved.
Denoting by Q1k the operator which acts from o
W4
2R;H;{k} to L2R;H in the
following way:
Q1kut≡
3
s 1
Asd
4−sut
dt4−s , ut∈ o
W24R;H;{k}, 2.15
the following statement results.
Lemma 2.2. LetAsA−s, s 1,2,3,be bounded operators inH. Then the operatorQ1kis a bounded
operator from o
W4
2R;H;{k}toL2R;H.
Proof. Because for any vector functionut∈
o
W4
2R;H;{k},
Q1ku
L2R;H≤
3
s 1
A4−sA−4s H→H
A4−sd
su
dts
then, from the theorem on intermediate derivatives3–5, Chapter 1, and from2.16, we get
Q1ku
L2R;H≤constuW24R;H. 2.17
Thus, the lemma is proved.
Now certain properties of polynomial operator groups will be investigated, which will have in the future a special role.
Let the following hold:
as 1 256s
s4−s4−s, s 1,2,3. 2.18
Consider the following polynomial operator groups which depend on the parameterα ∈
0;a−1
s , s 1,2,3:
Qsλ;α;A
λ2E−A24−αiλ2sA8−2s, s 1,2,3 2.19
The following can then be established.
Lemma 2.3. Letα ∈ 0;a−s1, s 1,2,3. Then the polynomial operator groupsQsλ;α;A, s 1,2,3, are invertible on the imaginary axis and can be represented as follows:
Qsλ;α;A Fsλ;α;AFs−λ;α;A, s 1,2,3; 2.20
moreover,
Fsλ;α;A
4
n 1
λE−ωs,nαA
≡λ4Ed1,sαλ3Ad2,sαλ2A2d3,sαλA3A4,
2.21
whereReωs,nα<0, n 1,2,3,4,and the numbersd1,sα, d2,sα, d3,sαsatisfy the following systems of equations:
1fork 1
−d21,1α 2d2,1α 4 0,
d2
2,1α−2d1,1αd3,1α−4 0,
−d23,1α 2d2,1α 4 α;
2fork 2
2d2,2α−d21,2α 4 0,
d22,2α−2d1,2αd3,2α−4 −α,
−d2
3,2α 2d2,2α 4 0;
2.23
3fork 3
−d21,3α 2d2,3α 4 α,
d22,3α−2d1,3αd3,3α−4 0,
−d23,3α 2d2,3α 4 0.
2.24
Proof. Characteristic polynomials of the operator groupsQsλ;α;A, s 1,2,3, are
Qsλ;α;σ
λ2−σ24−αiλ2sσ8−2s, s 1,2,3, 2.25
whereσ ∈ σA. Letλ iξ, ξ ∈ R −∞;∞. Then it is clear that for these characteristic polynomials, the following correlations are true:
Qsλ;α;σ Qsiξ;α;σ
σ8
ξ2
σ2 1
4
1−α ξ
2/σ2s ξ2/σ214
≥σ8
ξ2
σ2 1
4
1−α sup ξ2/σ2≥0
ξ2/σ2s ξ2/σ214
, s 1,2,3.
2.26
Because
sup ξ2/σ2≥0
ξ2/σ2s
ξ2/σ214 as, s 1,2,3, 2.27
then
Qsiξ;α;σ>0 2.28
forα ∈0;a−1
s , s 1,2,3. From2.28, it becomes clear that the polynomialsQsλ;α;σdo not have roots on the imaginary axis for α ∈ 0;a−1
these polynomials are homogeneous with respect to the argumentsλandσ, they can be stated in the following form:
Qsλ;α;σ Fsλ;α;σFs−λ;α;σ, s 1,2,3, 2.29
where
Fsλ;α;σ
4
n 1
λ−ωs,nασ
≡λ4d1,sαλ3σd2,sαλ2σ2d3,sαλσ3σ4,
2.30
and moreover Reωs,nα < 0, n 1,2,3,4, and the numbersd1,sα, d2,sα, d3,sαsatisfy the systems of equations shown inLemma 2.3, which are obtained from2.29in the process of comparing the coefficients for the same degrees. Then, from the spectral decomposition of operator A, the proof of the lemma can be obtained from 2.29. Thus, the lemma is proved.
The next step is to prove the theorem, which will play an important role in future investigations and will show the special importance of the spectral properties of the polynomial operator groupsQsλ;α;AandFsλ;α;A, s 1,2,3.
Theorem 2.4. Letα∈0;a−1
s . Then for anyut∈W24R;H, the following equality is true:
d dt−A
d dtA
3
u
2
L2R;H
−αA4−sd
su
dts 2
L2R;H
Fs
d dt;α;A
u
2
L2R;H Rsαϕ, ϕ
! H4,
2.31
where
H4 "4
p 1
H,
Rsα ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝
d3,sα−2 d2,sα d1,sα 2 2
d2,sα d2,sαd3,sα−d1,sα−2 d1,sαd3,sα 2 d3,sα 2
d1,sα 2 d1,sαd3,sα 2 d1,sαd2,sα−d3,sα−2 d2,sα
2 d3,sα 2 d2,sα d1,sα−2
⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠,
ϕ
ϕl A4−l−1/2d lu0
dtl 3
l 0
.
Proof. First define the space D4R
;Has the set of infinitely differentiable functions with
values in DA4, having compact support inR
. Because the spaceD4R;His dense in
W4
2R;H see3–5, Chapter 1, it is sufficient to prove the theorem for the vector functions
ut∈D4R
;H. Then
Fs
d dt;α;A
u
2
L2R;H
d 4u dt4 2
L2R;H
d12,sαAd
3u dt3 2
L2R;H
d22,sαA2d
2u dt2 2
L2R;H
d23,sαA3du dt
2
L2R;H
A4u2
L2R;H2d1,sαRe
d4u
dt4, A
d3u
dt3
L2R;H
2d2,sαRe
d4u
dt4, A 2d2u
dt2
L2R;H
2d3,sαRe
d4u
dt4, A 3du
dt
L2R;H
2 Re
d4u
dt4, A 4u
L2R;H
2d1,sαd2,sαRe
Ad
3u
dt3, A 2d2u
dt2
L2R;H
2d1,sαd3,sαRe
Ad
3u
dt3, A 3du
dt
L2R;H
2d1,sαRe
Ad
3u
dt3, A 4u
L2R;H
2d2,sαd3,sαRe
A2d
2u
dt2, A 3du
dt
L2R;H
2d2,sαRe
A2d
2u
dt2, A 4u
L2R;H
2d3,sαRe
A3du dt, A
4u
L2R;H.
2.33
After integration by parts,
Fs
d dt;α;A
u
2
L2R;H d 4u dt4 2
L2R;H
d2
1,sα−2d2,sα Ad
3u dt3 2
L2R;H
d23,sα−2d2,sα
A3du dt
2
L2R;H
A4u2 L2R;H
2−2d1,sαd3,sα d22,sαA2
d2u
dt2
2
−d1,sαϕ32−2d2,sαRe ϕ3, ϕ2
!
d3,sα−d1,sαd2,sαϕ22
−2d3,sαRe ϕ3, ϕ1
!
−2 Re ϕ3, ϕ0
!
2−2d1,sαd3,sαRe ϕ2, ϕ1
!
−2d1,sαRe ϕ2, ϕ0
!
d1,sα−d2,sαd3,sαϕ12
−2d2,sαRe ϕ1, ϕ0
!
−d3,sαϕ02.
2.34
Calculatingd/dt−Ad/dtA3u2L2R;Hanalogously toFsd/dt;α;Au 2
L2R;H,
d dt−A
d dt A
3
u
2
L2R;H
d
4u
dt4
2
L2R;H
4Ad
3u
dt3
2
L2R;H
6A2d
2u
dt2
2
L2R;H
4A3du dt
2
L2R;H
A4u2
L2R;H−2ϕ3
2
−2ϕ224 Re ϕ3, ϕ1
!
2 Re ϕ3, ϕ0
!
6 Re ϕ2, ϕ1
!
4 Re ϕ2, ϕ0
!
−2ϕ12−2ϕ02. 2.35
Substituting2.35into2.34, fromLemma 2.3,2.31can be obtained. Thus, the theorem is proved.
FromTheorem 2.4, it follows that:
Corollary 2.5. Ifut∈
o
W24R;Handα∈0;a−s1, then
d dt−A
d dtA
3
u
2
L2R;H
−αA4−sd
su
dts 2
L2R;H
Fs
d dt;α;A
u
2
L2R;H
. 2.36
Note that fromTheorem 2.1,Q0kuL
2R;His the norm in the space
o
W24R;H;{k},
which is equivalent to the initial normuW4
derivatives
A4−sd
s
dts : o
W24R;H;{k}−→L2R;H, s 1,2,3, 2.37
are continuous 3–5, Chapter 1, then the norms of these operators can be estimated using Q0kuL
2R;H. It is also easy to demonstrate that the norms uW4
2R;H and
d/dt−Ad/dtA3uL2R;Hare equivalent in the space o
W24R;H.
3. Norms of the Operators of Intermediate Derivatives
The rest of this paper will be related to the calculation of the following numbers:
m0,s sup
0/u∈Wo4 2R;H
A4−sd
su
dts
L2R;H
d dt−A
d dt A
3
u
−1
L2R;H
,
mk,s sup
0/u∈Wo4
2R;H;{k}
A4−sd
su
dts
L2R;H
d dt −A
d dtA
3
u
−1
L2R;H
.
3.1
First, let us calculatem0,s.
Lemma 3.1. It holds thatm0,s as1/2, s 1,2,3.
Proof. As 2.36 goes to the limit as α → a−s1, it is apparent that for any vector function
ut∈
o
W24R;H, the following inequality:
d dt−A
d dtA
3
u
2
L2R;H ≥a−s1
A4−sd
su
dts 2
L2R;H 3.2
is true. Thus, m0,s ≤ a1s/2, s 1,2,3. Furthermore, it is necessary to show that here the equalitiesm0,s a1s/2, s 1,2,3 also hold. This can be done by taking an arbitrary number
δ > 0 and showing that there exists a vector function uδt ∈ o
W4
2R;H such that the
following holds functional:
Λuδ≡
d dt−A
d dtA
3
uδ
2
L2R;H
−a−s1δ
A4−sd
su δ
dts 2
Let the vectorς∈DA8andς 1,htbe the numeral function; moreover,htς∈
W4
2R;H. Then using the Parseval equality, it is possible to obtain
Λhtς
d dt−A
d dtA
3
htς
2
L2R;H
−a−s1δ
A4−sd
sht
dts ς 2
L2R;H
∞
−∞ #
Q0−iξ;Aς, Q0−iξ;Aςhξ 2
−a−s1δ
ξ2sA4−sς, A4−sςhξ2 $
dξ
∞
−∞
Q0−iξ;AQ0−iξ;Aς−
a−s1δ
ξ2sA8−2sς, ςhξ2dξ
∞
−∞φξ;ς
hξ2dξ,
3.4
whereφξ;ς Qsiξ;a−s1δ;Aς, ς.
It will next be shown that φξ;ςfor a given vector ς has negative values in some intervalε0;ε1. Ifμ0 is an eigenvalue of the operatorAμ0 > 0, and ifςis its eigenvector,
then it is obvious that
φξ;ς Qs
iξ;a−s1δ;A
ς, ς Qs
iξ;a−s1δ;μ0
ς, ς, 3.5
and, as can be seen from the properties of the polynomial Qsiξ;α;μ0, is negative for α
a−s1δfor sufficiently smallδ > 0. Ifμ0 ∈σAis not the eigenvalue, thenμ0 is close to an
eigenvalue, that is, there existsςδsuch thatςδ 1 and
φξ;ςδ Qs
iξ;a−s1δ;μ0
ςδoδ asδ−→0, 3.6
because in this case, for sufficiently smallδ, the smallest value is negative for someςδ. Then there exists an intervalε0;ε1such thatφξ;ς< δforξ∈ε0;ε1.
Now consider the four times differentiable functionhξ, support of which comes from the intervalε0;ε1. Then from3.4and from the negativity ofφξ;ςδin the intervalε0;ε1,
it can be determined that
Λhtςδ ε1
ε0
φξ;ςδhξ
2
dξ <0. 3.7
Because o
W4
2R;H ⊂
o
W4
2R;H;{k}, then mk,s ≥ m0,s a1s/2, s 1,2,3. It is necessary to note that, for any vector functionut ∈
o
W4
2R;H;{k}and α ∈ 0;a−s1, the equality
d dt−A
d dtA
3
u
2
L2R;H
−αA4−sd
su
dts 2
L2R;H
Fs
d dt;α;A
u
2
L2R;H Rsα;kϕ, ϕ
! H
3.8
is true, whereRsα;k d1,sα−2 is obtained fromRsαby removing the first three rows and columns,ϕ A1/2d3u0/dt3. The correctness of3.8follows directly fromTheorem 2.4.
The following statement indicates when the numbersmk,s, s 1,2,3, can be equal to
a1s/2, s 1,2,3.
Lemma 3.2. To establish the conditionmk,s a1s/2, it is necessary and sufficient thatRsα;kbe positive for anyα∈0;a−1
s .
Proof. Necessity will be shown first. Letmk,s a1s/2. Then, from3.8, for any vector function
ut∈
o
W4
2R;H;{k}andα∈0;a−s1,
Fs
d dt;α;A
u
2
L2R;H Rsα;kϕ, ϕ
! H≥
d dt−A
d dtA
3
u
2
L2R;H
1−αm2k,s>0.
3.9
Because the polynomial operator groupFsλ;α;Aforα∈0;a−s1has the form
Fsλ;α;A
4
n 1
λE−ωs,nαA 3.10
seeLemma 2.3, where Reωs,nα<0, n 1,2,3,4, then the Cauchy problem,
Fs
d dt;α;A
ut 0, 3.11
dku0
dtk 0, k 0,1,2, 3.12
d3u0
dt3 A−
1/2ϕ, ϕ∈H, 3.13
has a unique solutionuαt∈W24R;H, which can be presented in the form
where ψ0, ψ1, ψ2, ψ3 ∈ DA7/2 are uniquely determined from the conditions at zero in 3.12 and 3.13. As a result, writing inequality 3.9 for the vector function uαt, for
α∈0;a−1
s Rsα;kϕ, ϕH>0. Necessity is thereby proved. Now sufficiency must be proved. If for anyα∈0;a−1
s ,Rsα;kis positive, then from
3.8, it follows that for allut∈
o
W4
2R;H;{k}andα∈0;a−s1,
d dt−A
d dtA
3
u
2
L2R;H
≥αA4−sd
su
dts 2
L2R;H. 3.15
As this expression goes to the limit asα → a−1
s , it can be observed thatmk,s≤a1s/2, and from this,mk,s a1s/2. Sufficiency is thereby proved, and thus the lemma is completely proved.
It is interesting that for somes, it may occur thatmk,s> a1s/2.
Lemma 3.3. It holds that mk,s > a1s/2 if and only if Rsα;k 0 has a solution in the interval
0;a−1
s ; moreover, this root is equal tom−k,s2.
Proof. Letmk,s> a1s/2, thenm−k,s2 ∈0;as−1. From3.8, forα∈0;m−k,s2,
Fs
d dt;α;A
u
2
L2R;H
Rsα;kϕ, ϕ !
H≥
d dt−A
d dtA
3
u
2
L2R;H
1−αm2
k,s
>0.
3.16
Substituting the solution of 3.11–3.13 into the last inequality, the result is that
Rsα;kis positive for α ∈ 0;m−k,s2. From the definition ofmk,s, for α ∈ mk,s−2;a−s1, there
exists a vector functionvαt∈ o
W4
2R;H;{k}such that
d dt −A
d dtA
3
vα
2
L2R;H
< αA4−sd
sv α
dts 2
L2R;H. 3.17
From the last inequality in3.8, it is possible to obtain
Fs
d dt;α;A
vα 2
L2R;H Rsα;kϕα,ϕα
!
H<0, 3.18
where
ϕα A1/2d
3v
α0
dt3 . 3.19
Thus, there exists a vectorϕα∈Hsuch that forα∈m−k,s2;a−s1,Rsα;kϕα,ϕαH<0. Because
Inversely, ifRsα;k 0 has a root in the interval0;a−s1, then this means that for any
α∈0;a−1
s ,the numberRsα;kcannot be positive. This is why, fromLemma 3.2,mk,s> a1s/2. Denoting the root ofRsα;k 0 byμk,s, it can be seen thatm−k,s2 ≤μk,s, because from the proof of the lemma, it was obtained that forα ∈ 0;m−2
k,s,Rsα;kis positive. Moreover, because
Rsm−k,s2;k 0, it can be determined thatm−k,s2 μk,s. The lemma is thereby proved.
By generalizing the last two lemmas, the following theorem can be derived.
Theorem 3.4. The following equality is true:
mk,s ⎧ ⎨ ⎩
a1s/2 forRs γ;k !
/0, γ∈ 0;a−1
s !
,
μ−k,s1/2 otherwise. 3.20
Remark 3.5. In the same way, it is possible to determine the results for boundary-value problems of the form1.1,1.2forkhaving any three values from the collection{0; 1; 2; 3}.
By considering concretely the casess, the following statement results.
Theorem 3.6. mk,1 1/
√
3; mk,2 1/2
√
3; mk,3 a13/2.
Proof. Taking into account the abovementioned procedure for finding the numbersmk,s, it is necessary to solve the systems from the proof ofLemma 2.3together with the equation
Rsα;k 0.
In the cases 1, it can be determined thatd1,1α 2 ⇒d2,1α 0⇒d3,1α −1⇒
α 3 ∈ 0;a−11. This is whymk,1 1/
√
3. To find the numbermk,2, it is necessary to solve
the system fromLemma 2.3fors 2 together with the equationd1,2α−2 0. In this case,
d1,2α 2, d2,2α 0, d3,2α ±2, and consequentlyα 12∈0;a−21andα −4/∈0;a−21.
As a result,mk,2 1/2
√
3. In the cases 3, it is found that d1,3α 2. Then, from the
corresponding system, it can be obtained that 2d2,3α αandd24,3α−8d22,3α−32d2,3α−
48 0 ord2,3α2d32,3α−2d22,3α−4d2,3α−24 0. It is clear thatd2,3α −2⇒d3,3α
0⇒α −4/∈0;a3−1. From the other side, if in the equationd23,3α−2d22,3α−4d2,3α−24 0,
it is assumed that 2d2,3α α, then the result is thatα3−4α2−16α−192 0, which has
only one real root,α 1/3%3 −29442√2113423−%329442√2113423∈/0;a−31. Therefore,
mk,3 a13/2, and the theorem is proved.
Theorem 4.1. Let the operatorsAsA−s, s 1,2,3, be bounded inHso that the inequality
mk,1A1A−1
H→Hmk,2
A2A−2
H→Hmk,3
A3A−3
H→H<1 4.1
is satisfied, where the numbersmk,s, s 1,2,3, are as defined inTheorem 3.6. Then the boundary-value problem1.1,1.2is regularly solvable.
Proof. The boundary-value problem1.1,1.2can be presented in the form of the operator
equationQ0kut Q1kut ft, whereft ∈ L2R;H,ut ∈
o
W24R;H;{k}. From
Theorem 2.1, it follows that the operator Q0khas a bounded inverse operator Q0k−1 which acts from the space L2R;H into the space
o
W4
2R;H;{k}. Then, after substitution of
ut Q0k−1vt, wherevt ∈ L2R;H, the equationEQ1kQ
k−1
0 vt ftresults.
Now it must be shown that whenever the conditions of the theorem are met, the norm of the operatorQ1kQ0k−1is less than one. AssumingTheorem 3.6, the following can be obtained:
Q1kQ0k−1v L2R;H
Q1ku
L2R;H
≤A1d 3u
dt3
L2R;H
A2d 2u
dt2
L2R;H
A3du
dt
L2R;H
≤A1A−1
H→H Ad
3u
dt3
L2R;H
A2A−2
H→H A2d
2u
dt2
L2R;H
A3A−3
H→H A3du
dt
L2R;H
≤mk,1A1A−1
H→Hmk,2
A2A−2
H→Hmk,3
A3A−3
H→H Qk
0 uL
2R;H
mk,1A1A−1
H→Hmk,2
A2A−2
H→Hmk,3
A3A−3
H→H
υL2R;H.
4.2
As a result:
Q1kQ0k−1
L2R;H→L2R;H≤mk,1
A1A−1
H→Hmk,2
A2A−2
H→H
mk,3A3A−3
H→H<1.
4.3
Then, in this case, the operatorEQ1kQ0k−1 has an inverse in the spaceL2R;H, and it is
possible to determineutfrom the following formula:
Moreover,
uW4 2R;H≤
Q0k−1
L2R;H→W4 2R;H
EQ1kQ0k−1−1
L2R;H→L2R;H
f L2R;H
≤constfL2R;H.
4.5
Thus, the theorem is proved.
Remark 4.2. The conditions of regular solvability obtained here for the boundary-value problem1.1,1.2are not improvable in terms of the operator coefficients of1.1.
Following is an example in which the conditions ofTheorem 4.1are verified. Consider the following problem on the semi-axisR×0;π:
∂ ∂t
∂2
∂x2
∂ ∂t−
∂2
∂x2
3
ut, x p1x∂ 5ut, x
∂x2∂t3 p2x
∂6ut, x
∂x4∂t2 p3x
∂7ut, x
∂x6∂t ft, x
∂ku0, x
∂tk 0, k 0,1,2,
∂2iut,0
∂x2i
∂2iut, π
∂x2i 0, i 0,1,2,3,
a b
4.6
wherepsx, s 1,2,3,are bounded on segment0;πfunctions,ft, x ∈L2R;L20;π,
which is a partial case of problem1.1,1.2. On the condition that
mk,1sup 0≤x≤π
p1xmk,2sup
0≤x≤π
p2xmk,3sup
0≤x≤π
p3x<1, 4.7
the given problem has a unique solution in the spaceWt,x,4,82R;L20;π.
Acknowledgment
The authors have dedicated this paper, in gratitude for useful consultation in their work, as a sign of deep respect to the memory of the Academician of the Azerbaijan National Academy of Sciences, Professor M. G. Gasymov.
References
1 I. I. Vorovich and V. E. Kovalchuk, “On the basis properties of one system of homogeneous solutions,” Prikladnaya Matematika i Mekhanika [Journal of Applied Mathematics and Mechanics], vol. 31, no. 5, pp. 861–869, 1967.
2 Y. A. Ustinov and V. I. Yudovich, “On the completeness of the system of elementary solutions of biharmonic equation on a semi-strip,” Prikladnaya Matematika i Mekhanika [Journal of Applied Mathematics and Mechanics], vol. 37, no. 4, pp. 706–714, 1973.
3 J. L. Lions and E. Magenes,Non-Homogeneous Boundary Value Problems and Applications, Dunod, Paris, France, 1968.
5 J. L. Lions and E. Magenes, Non-HomogeneousBoundary Value Problems and Applications, Springer, Berlin, Germany, 1972.
6 S. G. Krein,Linear Differential Equations in a Banach Space, Nauka, Moscow, Russia, 1967.
7 S. Y. Yakubov,Linear Differential-Operator Equations and Their Applications, Elm, Baku, Azerbaijan, 1985.
8 S. Agmon and L. Nirenberg, “Properties of solutions of ordinary differential equations in Banach space,”Communications on Pure and Applied Mathematics, vol. 16, pp. 121–239, 1963.
9 M. G. Gasymov and S. S. Mirzoev, “Solvability of boundary value problems for second-order operator-differential equations of elliptic type,”Differentsial’nye Uravneniya [Differential Equations], vol. 28, no. 4, pp. 651–661, 1992.
10 A. G. Kostyuchenko and A. A. Shkalikov, “Self-adjoint quadratic operator pencils and elliptic problems,”Funktsional’nyi Analiz i Ego Prilozheniya [Functional Analysis and Its Applications], vol. 17, no. 2, pp. 38–61, 1983.
11 M. G. Gasymov, “The multiple completeness of part of the eigen- and associated vectors of polynomial operator bundles,”Izvestija Akademii Nauk Armjansko˘ıSSR. Serija Matematika, vol. 6, no. 2-3, pp. 131–147, 1971.
12 M. G. Gasymov, “On the theory of polynomial operator pencils,”Doklady Akademii Nauk SSSR [Soviet Mathematics. Doklady], vol. 199, no. 4, pp. 747–750, 1971.
13 Y. A. Dubinskii, “On some differential-operator equations of arbitrary order,”Matematicheskii Sbornik [Mathematics of the USSR-Sbornik], vol. 90132, no. 1, pp. 3–22, 1973.
14 S. S. Mirzoev, “Conditions for the well-defined solvability of boundary-value problems for operator differential equations,”Doklady Akademii Nauk SSSR [Soviet Mathematics. Doklady], vol. 273, no. 2, pp. 292–295, 1983.
15 V. B. Shakhmurov, “Coercive boundary value problems for strongly degenerate operator-differential equations,”Doklady Akademii Nauk SSSR, vol. 290, no. 3, pp. 553–556, 1986.
16 A. A. Shkalikov, “Elliptic equations in a Hilbert space and related spectral problems,”Trudy Seminara Imeni I. G. Petrovskogo, no. 14, pp. 140–224, 1989.
17 A. R. Aliev, “Boundary-value problems for a class of operator differential equations of high order with variable coefficients,”Matematicheskie Zametki [Mathematical Notes], vol. 74, no. 6, pp. 803–814, 2003.
18 A. R. Aliev, “On the boundary value problem for a class of operator-differential equations of odd order with variable coefficients,”Doklady Akademii Nauk [Doklady Mathematics], vol. 421, no. 2, pp. 151–153, 2008.
19 R. P. Agarwal, M. Bohner, and V. B. Shakhmurov, “Maximal regular boundary value problems in Banach-valued weighted space,”Boundary Value Problems, vol. 2005, no. 1, pp. 9–42, 2005.
20 A. Favini and Y. Yakubov, “Higher order ordinary differential-operator equations on the whole axis in UMD Banach spaces,”Differential and Integral Equations, vol. 21, no. 5-6, pp. 497–512, 2008.
21 N. P. Kuptsov, “Exact constants in inequalities between norms of functions and of their derivatives,” Matematicheskie Zametki [Mathematical Notes], vol. 41, no. 3, pp. 313–319, 1987.
22 G. A. Kalyabin, “Some problems for Sobolev spaces on the half-line,”Trudy Matematicheskogo Instituta Imeni V. A. Steklova [Proceedings of the Steklov Institute of Mathematics], vol. 255, pp. 161–169, 2006.
23 G. A. Grinberg, “On the method proposed by P.F. Papkovich for solving the plane problem of elasticity theory for the rectangular domain and the problems of bending of a rectangular thin slab with two fixed selvages, and on some of its generalizations,”Prikladnaya Matematika i Mekhanika [Applied Mathematics and Mechanics], vol. 37, no. 2, pp. 221–228, 1973.
24 R. Z. Gumbataliyev, “On generalized solutions of one class of operator-differential equations of the fourth order,”Izvestiya Akademii Nauk Azerbajdzhana. Seriya Fiziko-Tekhnicheskikh i Matematicheskikh Nauk, vol. 18, no. 2, pp. 18–21, 1998.
