C om mun.Fac.Sci.U niv.A nk.Series A 1 Volum e 66, N umb er 2, Pages 91–99 (2017) D O I: 10.1501/C om mua1_ 0000000804 ISSN 1303–5991
http://com munications.science.ankara.edu.tr/index.php?series= A 1
DELAY DIFFERENTIAL OPERATORS AND SOME SOLVABLE MODELS IN LIFE SCIENCES
P. IPEK, B. YILMAZ, AND Z. I. ISMAILOV
Abstract. Using the methods of the spectral theory of di¤erential operators in Hilbert spaces L2-solvability of some models arising in mathematical biology are investigated. Particularly, concrete solvable models are given.
1. Introduction
It is known that the general theory of extension of densely de…ned linear opera-tors in Hilbert spaces was initiated by J. von Neumann in his seminal work [17] in 1929 (for more detail analysis see [18]). Later in 1949 and 1952 M.I. Vishik [22,23] has studied the boundedly (compact, regular and normal) invertible extensions of any unbounded linear densely de…ned operator in Hilbert spaces. The generaliza-tion of these results to the nonlinear and completely additive Hausdor¤ topological spaces have been done by B.K. Kokebaev, M. Otelbaev and A.N. Synybekov [for example, see 16].
Another approach to the description of regular extensions for some classes of linear di¤erential operators in Hilbert spaces of vector-functions on …nite intervals has been o¤ered by A.A. Dezin [3].
It is known that many problems arising in life sciences for example in elec-trodynamics, control theory, ecology, economy, chemistry, medicine, epidemiology, tumor growth, neutral networks, biology and etc. can be expressed as boundary or initial value problems the linear functional (time proportional or time delay) di¤erential equations in the corresponding functional spaces (for detailed analysis see [1,2,4,5,6,19,20]).
The main goal in all these theories is to obtain clear expressions of exact solutions of the considered problems. Note that all these problems are strongly connected to the with solvability of initial or boundary value problems for considered di¤er-ential equations in corresponding functional spaces (for special cases see [7-14,24]).
Received by the editors: September 19, 2016, Accepted: November 10, 2016. 2010 Mathematics Subject Classi…cation. 47A10, 92B05.
Key words and phrases. Hilbert space and direct sum of Hilbert spaces; di¤erential, bounded and boundedly solvable operators; Hutchinson, House‡ies, Drug-free and medical models.
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Problems in this area have not been investigated successfully. Of course, there are approximative and numerical results in these investigations.
Let us remember that an operator A : D(A) H ! H in a Hilbert space H is called boundedly solvable, if A is one-to-one, AD(A) = H and A 1 2 L(H). The
theory of boundedly solvable extensions of a linear densely de…ned closed operator in Hilbert or Banach spaces has been investigated by M. I. Vishik [22,23], M. O. Otelbayev (with his scienti…c group) [16], A. A. Dezin [3] and etc.
From the mathematical literature it is known that the general solution of di¤eren-tial–operator equation
u0(t) + A(t)u(t ) = 0; > 0
with continuous on uniformly operator topology coe¢ cient at …nite interval can be represented via evolution operators. Unfortunately, in the in…nite interval case since the structure of spectrum (A S ) of the operator A S 2 L H L2(J )
(H is a Hilbert space, J R in…nite interval S u(t) = u(t ); (A S ) u(t) = A(t)u(t )]) is not clear, then general solution of above retarded type delay dif-ferential equation can not be written via evolution operators in corresponding L2
spaces. This creates many theoretical di¢ culties. Note that investigation of these problems in special cases by di¤erence scheme method is reduced to the study of spectral properties of in…nite upper or lower triangular double-banded matrices over corresponding sequence spaces ( for example, see [15] and references therein). In this work to overcome this obstacle so-called many-Hilbert space-method is applied. In this paper before all of these, in section 2, using the many-Hilbert space-method all L2-boundedly solvable extensions of the minimal operator generated
by multipoint di¤erential-operator expression for …rst order in the Hilbert space of vector-functions are described. Lastly, in section 3, the obtained results to concrete mathematical models arising in life sciences are applied.
2. Boundedly Solvable Extensions
Throughout this work (an) is a real number sequence with properties a0< a1
a2< a3 ::: an 1< an < :::; lim
n!+1an= +1; Hn is a separable Hilbert space,
Jn = (an 1; an); Hn = L2(Hn; J n); n 1 and H = 1
L
n=1H n.
We consider the following linear multipoint di¤erential-operator expression of …rst order in H in form
l(u) = (ln(un)); u = (un);
where
(1): l1(u1) = u01(t)+A1(t)u1(t) and ln(un) = u0n(t)+An(t)un(t)+Bn 1(t)un 1(t);
(2): operator-function An(: ) : [an 1; an] ! L(Hn); Bn(: ) : [an 1; an] !
L(Hn); n 1 is continuous on the uniformly operator topology and
sup
n 1
sup
t2Jn
jJjkAn(t)k < 1:
Actually, the di¤erential expression l(: ) in H can be written in following form
l(u) = u0(t) + A(t)u(t) + B(t)u(t); (2.1) where: u = (un) and for t > a0
A(t) = 0 B B B B B B @ A1(t) A2(t) 0 . .. 0 An(t) . .. 1 C C C C C C A ; B(t) = 0 B B B B B B B B @ 0 B1(t) 0 0 B2(t) 0 . .. . .. 0 Bn 1(t) 0 . .. ... 1 C C C C C C C C A
The operators L0(M0) and L(M ) are minimal and maximal operators
corre-sponding to l(: )(and m(: ) =dtd + A(:)) in H, respectively. It is clear that L0= M0+ B(:); L = M + B(:):
On the other hand for each n 1 by standard way the minimal operator Mn0
and maximal operator Mn corresponding to the di¤erential expression mn( : ) = d
dt + An( : ) in Hn can be de…ned. It is clear that D(Mn0) =
o
W1
2(Hn; Jn) and
D(Mn) = W21(Hn; Jn); n 1:
Firstly, the main purpose in this section is to describe all boundedly solvable extensions of the minimal operator M0 in H in terms of boundary values. We …rst
give some results from work [10].
Theorem 2.1. If fM is any extension of M0 in H, then fM = 1 L n=1 f Mn; where fMn is a extension of Mn0 in Hn; n 1:
Theorem 2.2. For the boundedly solvability of any extension fM = L1
n=1
f Mn of
the minimal operator M0 in H, the necessary and su¢ cient conditions are the
boundedly solvability of the coordinate extensions fMn of the minimal operators
Mn0 in Hn; n 1 and sup n 1k f
M 1
Theorem 2.3. Let n 1: Each boundedly solvable extension fMn of the minimal
operator Mn0 in Hn; n 1 is generated by the di¤erential-operator expression
mn(:) and the boundary condition
(Kn+ En)un(an 1) = Knexp 0 @ Z Jn An(s)ds 1 A un(an); (2.2)
where Kn 2 L(Hn) and En : Hn ! Hn is an identity operator. The operator Kn
is determined by the extension fMn uniquely, i.e. fMn= MKn.
On the contrary, the restriction of the maximal operator Mn in Hn to the
lin-ear manifold of vector-functions satisfying the condition (2.2) for some bounded operator Kn 2 L(Hn) is a boundedly solvable extension of the minimal operator
Mn0.
On the other hand for each n 1 k fMK1
nk 2jJnj (1 + kKnk) exp 2jJnj sup
t2Jn
kAn(t)k
Theorem 2.4. Let us assumed that sup
n 1jJ
njkKnk < 1: Each boundedly solvable
extension fM of the minimal operator M0in H is generated by di¤erential-operator
expression m(:) and the boundary conditions
(Kn+ En)un(an 1) = Knexp 0 @ Z Jn An(s)ds 1 A un(an); n 1 where Kn 2 L(Hn); K = 1 L n=1 Kn 2 L 1 L n=1 Hn and En : Hn! Hn is an identity
operator, n 1. The operator K is determined by the extension fM uniquely, i.e. f
M = MK and vice versa.
Theorem 2.5. If K = L1 n=1 Kn2 L 1 L n=1 Hn and MK = 1 L n=1 MKn is a boundedly
solvable extension of the minimal operator M0and satis…es the following condition
kMK1B(:)k = sup n 1kM
1 Kn+1Bn(
:
)k < 1; then the operator LK = MK+ B(: ) is
boundedly solvable in H:
3. Application in Life Sciences
Example 3.1 [19]. Consider the following linearized logistic delay di¤erential equation (or Hutchinson’s model)
:
x(t) = rx(t ); > 0; t > 0
This problem can be written in form
:
x1(t) = r'(t ); 0 < t < ; :
xn(t) = rxn 1(t); (n 1) < t < n ; n 2: (3.1)
By Theorem 2.3. all boundedly solvable extension MKn of the minimal operator
Mn0 in Hn= L2((n 1) ; n ) is generated by the di¤erential expression mn(x) =
x0
n(t) and boundary condition
(kn+ 1)xn((n 1) ) = knxn(n ); kn2 C; n 1
and vice versa. On the other hand in this case for sup
n 1jknj < 1 we have sup n 1jJ nj (1 + jknj) = 1 + sup n 1jk nj < 1:
Then by Theorem 2.4. the extension L1
n=1
MKn in L
2(0; 1) is boundedly solvable.
Moreover, the following inequality sup n 1kM 1 kn+1Bn( : )k sup n 1kM 1 kn+1kkBn(t)k = sup n 1 p 2 (1 + jknj) jrjp = p 2 3=2jrj 1 + sup n 1jknj
implies that if p2 3=2jrj 1 + sup
n 1jknj < 1; then by Theorem 2.5. the following
equation (3.1) has a solution in the form
x1(t) = rk1 Z 0 '(t s)ds + r t Z 0 '( s)ds; 0 < t < ; xn(t) = rkn n Z (n 1) xn 1(s)ds + r t Z (n 1) xn 1(s)ds; (n 1) < t < n ; n 2:
Example 3.2 [19]. Consider the following linearized House‡ies Model (Musca domestica) in form
:
x(t) = dx(t) + bx(t ); t > 0;
where: x( :) is the number of adults, d > 0 denotes the death rate of adults, the
time delay > 0 is the length of the developmental period between oviposition and eslosion of adults, b > 0 is the number of eggs laid by pair adults, with history function
This problem ( that is, description of oscillations of the adults’numbers in lab-oratory populations) can be written in following form
:
x1(t) = dx1(t) + b (t ); 0 < t < ; :
xn(t) = dxn(t) + bxn 1(t); (n 1) < t < n ; n 2
in the direct sum of Hilbert spaces H = L1
n=1
L2((n 1) ; n ). Again by
The-orem 2.3. all solvable extensions MKn of the minimal operator Mn0 in Hn =
L2((n 1) ; n ) is generated by the di¤erential expression m
n(xn) = :
xn(t)+dxn(t)
and boundary condition (kn+1)xn((n 1) ) = knexp( d )xn(n ); kn2 C; n 1
and vice versa. If sup
n 1jk
nj < 1; then by Theorem 2.4. the extension 1
L
n=1
Mkn in L
2(0; 1) is
boundedly solvable. Moreover, from the following computations sup n 1kM 1 kn+1( b)k sup n 1[2 (1 + jknj) exp(2 d)b] implies that if b 1 + sup n 1jk nj exp(2 d) < 1 2; then by Theorem 2.5. the considered problem has a L2-solution.
Example 3.3 [21]. Consider the following delay case Drug-free model in the absence of immune response
TI0(t) = 2a4TM(t) d2TI(t) a1TI(t );
TM0 (t) = a1TI(t ) dTM(t); > 0; t > 0
with history functions
TI(t) = 0(t); t 2 [ ; 0]; TM(t) = 1(t); t 2 [ ; 0]:
Here TI(t) is a population of tumor cells during interphase at time t > 0; TM(t)
is a tumor population during mitosis at time t > 0; T > 0 is a resident time of cells in interphase, a1 and a4 represent the di¤erent rates at which cells cycle or
reproduce, d2TI and d3TM represent propositions of natural cell death or apoptosis,
d = d3+ a4:
To solve this system of delay di¤erential equations with mentioned history func-tions consider the following matrix form
T0(t) = AT (t) + BT (t ); > 0; t > 0 where: T (t) = (TI(t); TI(t))T; A = 0 @ 0d2 2ad4 1 A ; R2 ! R2 and B = 0 @ aa11 00 1 A ; R2 ! R2
in the direct sum of Hilbert spaces L2 R2; (0; 1) = L1
n=1
L2 R2; ((n 1) ; n ) : By Theorem 2.3 all solvable extensions Mkn of the minimal operator Mn0 in
Hn= L2 R2; ((n 1) ; n ) are generated by the di¤erential expression
mn(Tn) = :
Tn(t) + ( A)Tn(t); n 1
and boundary condition (kn+ 1)Tn((n 1) ) = knexp(A )Tn(n ); kn2 R2; n 1
and vice versa. In case when sup
n 1jknj < 1, we have (1 + jknj) exp(2 kAk) < 1: Consequently,
by Theorem 2.4. the extension L1
n=1
Mkn is boundedly solvable in
L2 R2; (0; 1) .
On the other hand it is clear that if sup
n 1kM 1
knBk sup
n 12 (1 + jknj) exp(2 kAk)kBk < 1;
then by Theorem 2.5. above considered problem has a unique L2-solution.
Example 3.4 [19]. Consider the following special multiple delay logistic model in the linear case
:
x(t) = ax(t ) + bx(t 2 ); > 0; t > 0; a; b 2 R
in L2(0; 1) with history functionx(t) = (t);: 2 < t < 0: These problems with
two delays appear in neurological models, physiological models, medical models, epidemiological models and etc.
This problem can be written in L2(0; 1) = L1
n=1 L2((n 1) ; n )) in following form l(x) = x01(t) + a (t ) + b (t 2 ); 0 < t < ; x0 n(t) + axn 1(t) + bxn 2(t); (n 1) < t < n ; n 2
Again by Theorem 2.3 all solvable extensions Mknof the minimal operator Mn0in
Hn= L2((n 1) ; n ) are generated by the di¤erential expression mn(xn) = :
xn(t)
and boundary condition (kn+ 1)xn((n 1) ) = knxn(n ); kn2 C; n 1 and vice
versa. If sup
n 1jknj < 1; then by Theorem 2.4 the extension M = 1 L n=1 MKnis boundedly solvable in L2(0; 1): In addition, if kMk1Ck p 2 1 + sup n 1kk nk 2jbj2+ 3jaj2 < 1;
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Current address, P. Ipek: Karadeniz Technical University, Institute of Natural Sciences, 61080, Trabzon, Turkey
E-mail address, P. Ipek: ipekpembe@gmail.com
Current address, B. Yilmaz: Marmara University, Department of Mathematics, Kad¬köy, 34722, Istanbul, Turkey
E-mail address, B. Yilmaz: bulentyilmaz@marmara.edu.tr
Current address, Z. I. Ismailov: Karadeniz Technical University, Department of Mathematics, 61080, Trabzon, Turkey
