Green’s functional for higher-order ordinary differential equations with general nonlocal conditions and variable principal coefficient
The method of Green’s functional is a little-known technique for the construction of fundamental solutions to linear ordinary differential equations (ODE) with nonlocal conditions. We apply this technique to a higher order linear ODE involving general nonlocal conditions. A novel kind of adjoint pro...
Saved in:
| Date: | 2019 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2019
|
| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/1422 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507170525675520 |
|---|---|
| author | Özen, K. Озен, К. |
| author_facet | Özen, K. Озен, К. |
| author_sort | Özen, K. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T08:54:16Z |
| description | The method of Green’s functional is a little-known technique for the construction of fundamental solutions to linear ordinary
differential equations (ODE) with nonlocal conditions. We apply this technique to a higher order linear ODE involving
general nonlocal conditions. A novel kind of adjoint problem and Green’s functional are constructed for the completely
inhomogeneous problem. Several illustrative applications of the theoretical results are provided. |
| first_indexed | 2026-03-24T02:05:03Z |
| format | Article |
| fulltext |
UDC 517.9
K. Özen (Namık Kemal Univ., Turkey)
GREEN’S FUNCTIONAL FOR HIGHER-ORDER ORDINARY DIFFERENTIAL
EQUATIONS WITH GENERAL NONLOCAL CONDITIONS
AND VARIABLE PRINCIPAL COEFFICIENT
ФУНКЦIОНАЛ ГРIНА ДЛЯ ЗВИЧАЙНИХ ДИФЕРЕНЦIАЛЬНИХ РIВНЯНЬ
ВИЩОГО ПОРЯДКУ З НЕЛОКАЛЬНИМИ УМОВАМИ
ЗАГАЛЬНОГО ВИГЛЯДУ ТА ЗМIННИМ ГОЛОВНИМ КОЕФIЦIЄНТОМ
The method of Green’s functional is a little-known technique for the construction of fundamental solutions to linear ordinary
differential equations (ODE) with nonlocal conditions. We apply this technique to a higher order linear ODE involving
general nonlocal conditions. A novel kind of adjoint problem and Green’s functional are constructed for the completely
inhomogeneous problem. Several illustrative applications of the theoretical results are provided.
Метод функцiонала Грiна є маловiдомою технiкою побудови фундаментальних розв’язкiв лiнiйних звичайних ди-
ференцiальних рiвнянь (ЗДР) з нелокальними умовами. В роботi цю технiку застосовано до лiнiйних ЗДР вищого
порядку з нелокальними умовами загального вигляду. Спряжену проблему нового типу та функцiонал Грiна побудо-
вано для повнiстю неоднорiдної задачi. Наведено також кiлька iлюстративних застосувань теоретичних результатiв.
1. Introduction. As is known, a fundamental function is a distributional formulation and plays a cru-
cial role in mathematical analysis of differential equations. Once the fundamental function is found,
the desired solution of the original equation can be easily obtained by superposition principle. Green’s
function is a kind of fundamental function which is principally based on the theory of linear operators
and the theory of generalized functions in mathematical analysis [7 – 9, 11, 12, 18, 20, 32, 34 – 36],
and its construction by classical methods [35] such as Green’s function method and the method of
variation of parameters for a class of nonclassical problems such as differential problems with nonlo-
cal condition(s) may not be possible or may be troublesome. In order to overcome, in the literature, a
small number of studies exist on the investigations of this class of nonclassical problems by Green’s
functional method, the origin of which dates back to S. S. Akhiev [1 – 5, 21 – 25, 27 – 29, 33]. The
governing equation of the considered problems in most of these studies is in the standard form with
principal coefficient equal to one. The case with variable principal coefficient has been considered
only in [3]. Moreover, the case of a third order ODE with variable principal coefficient has been stud-
ied in [26]. To the best of our knowledge, any generalization on the case of the mth order ODE with
variable principal coefficient does not exist for an integer m greater than or equal to four. In order to
contribute to the enrichment of the literature in this context, we aim to extend the method for the sec-
ond and third order linear ODEs in [3, 26] to the higher order linear ODEs with nonlocal conditions.
The structure of our work is organized as follows. In Section 2, the problem considered through-
out the work is stated in detail. In Section 3, the solution space and its adjoint space are introduced.
In Section 4, the adjoint operator, adjoint system and solvability conditions for the completely nonho-
mogeneous problem are presented. In Section 5, Green’s functional and the special adjoint system are
defined. In Section 6, several applications are provided. In Section 7, the conclusions are emphasized.
2. Statement of the problem. Let R be the set of real numbers, X = (x0, x1) be a bounded
open interval in R, and Lp with 1 \leq p < \infty be the space of the p-integrable functions on X, let
L\infty be the space of the measurable and essentially bounded functions on X. The aim in this work is
to investigate the solvability conditions and Green’s function of the mth order ordinary differential
equation
c\bigcirc K. ÖZEN, 2019
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1 99
100 K. ÖZEN
(Vmu)(x) \equiv
\bigl(
gm - 1(x)(gm - 2(x) . . . (g2(x)(g1(x)u
\prime (x))\prime )\prime . . .)\prime
\bigr) \prime
+A0(x)u(x) = zm(x), x \in X,
(1)
subject to the nonlocal conditions
Viu \equiv a0iu(x0) + a1iu1(x0) + a2iu2(x0) + . . .+ am - 1
i um - 1(x0)+
+
x1\int
x0
Bi(\xi )u
\prime
m - 1(\xi ) d\xi = zi, i = 0, 1, 2, . . . ,m - 1, (2)
where m \geq 3 is an integer, zm \in Lp, zi \in R for i = 0, 1, 2, . . . ,m - 1, u1(x) = g1(x)u
\prime (x),
u2(x) = g2(x)u
\prime
1(x), u3(x) = g3(x)u
\prime
2(x), . . . , um - 1(x) = gm - 1(x)u
\prime
m - 2(x). Here, the following
assumptions are hold: g1, g2, . . . , gm - 1 \in L\infty are the given functions such that
1
g1
,
1
g2
, . . . ,
1
gm - 1
\in
\in Lp and the following integrals belong to the space Lp as functions of sm - 1 :
sm - 1\int
x0
dsm - 2
g2(sm - 2)g1(sm - 1)
\in Lp,
sm - 1\int
x0
sm - 2\int
x0
dsm - 3 dsm - 2
g3(sm - 3)g2(sm - 2)g1(sm - 1)
\in Lp,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
sm - 1\int
x0
sm - 2\int
x0
. . .
s2\int
x0
ds1 . . . dsm - 3 dsm - 2
gm - 1(s1) . . . g2(sm - 2)g1(sm - 1)
\in Lp
for 1 \leq p \leq \infty , and A0 \in Lp, Bi \in Lq for i = 0, 1, 2, . . . ,m - 1, where
1
p
+
1
q
= 1,
a0i , a
1
i , . . . , a
m - 1
i \in R for i = 0, 1, 2, . . . ,m - 1, and integration variables si are considered for
i \geq 1 throught the work.
Equation (1) is one of the generalized forms of the mth order linear ODEs in Sturm – Liouville
theory [19]. This equation is assumed to have generally nonsmooth coefficient becoming some
general properties such as p-integrability and boundedness, and its principal part\bigl(
gm - 1(gm - 2 . . . (g2(g1u
\prime )\prime )\prime . . .)\prime
\bigr) \prime
may have weak singularities at finite number points (in the clo-
sure X of X ) where some or all of functions gm - 1, gm - 2, . . . , g1 are continuous and zero.
Form (2) for the conditions can be considered as a generalization of the linearly local and non-
local conditions for such mth order ODEs. Many conditions such as the initial and classical type
boundary conditions and also multipoint [13, 14] and integral type conditions arising in modelling
of many physical phenomena by such an equation are specific forms of (2) for a suitable choice of
a0i , a
1
i , . . . , a
m - 1
i and Bi(\xi ).
Problem (1), (2) may not have a classical adjoint problem. In such a case, some serious difficul-
ties arise in applying the classical methods for this problem. In order to overcome these difficulties,
a novel method based on [1 – 3, 5] is presented. By this method, the isomorphic decompositions
of a weighted space of solutions and its adjoint space are used. A novel adjoint problem called as
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
GREEN’S FUNCTIONAL FOR HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS . . . 101
adjoint system for problem (1), (2) is introduced by these decompositions. This adjoint system is
constructed by more different and easier calculations than constructing the traditional adjoint prob-
lem. This system consists of m + 1 integro-algebraic equations for an unknown (m + 1)-tuple
(Fm(\xi ), Fm - 1, Fm - 2, . . . , F0) of a function Fm(\xi ) and m real numbers Fm - 1, Fm - 2, . . . , F0. One
of these equations is an integral equation and the others are algebraic equations. This system has a
similar role to that of the adjoint operator in general theory of the linear operators in Banach spaces
[10, 15 – 17]. The solvability conditions of the completely nonhomogeneous problem and corre-
sponding adjoint system are derived. Green’s functional concept for the problem is introduced as a
solution
\bigl(
Fm(\xi , x), Fm - 1(x), Fm - 2(x), . . . , F0(x)
\bigr)
of the adjoint system with a free term depending
on x \in X as a parameter. This concept is more natural than the classical Green’s function concept.
Fm(\xi , x) corresponds to Green’s function for the problem.
3. The solution space and its adjoint space. Equation (1) can be considered as a system
of m first order ODEs with unknowns u, u1, u2, . . . , um - 1 \in W
(1)
p , where W
(1)
p is the space of
all functions u \in Lp having derivative u\prime \in Lp. Hence, the solvability of problem (1), (2) can be
studied in space Wp with the weights gm - 1,gm - 2, . . . , g1 consecutively of all u \in Lp with u\prime \in Lp,
(g1u
\prime )\prime \in Lp, (g2(g1u
\prime )\prime )\prime \in Lp, . . . ,
\bigl(
gm - 1(gm - 2 . . . (g2(g1u
\prime )\prime )\prime . . .)\prime
\bigr) \prime \in Lp and also
\| u\| Wp = \| u\| Lp + \| u\prime \| Lp + \| u\prime 1\| Lp + \| u\prime 2\| Lp + . . .+ \| u\prime m - 1\| Lp ,
u1 = g1u
\prime , u2 = g2u
\prime
1, . . . , um - 1 = gm - 1u
\prime
m - 2.
This problem, which is linear completely nonhomogeneous, can be considered as an equation in the
form
V u = z (3)
with operator V = (Vm, Vm - 1, . . . , V0) and (m + 1)-tuples z = (zm(x), zm - 1, . . . , z0). By the
assumptions considered, V is a linear bounded operator from Wp into the Banach space
Ep \equiv Lp \times R\times R\times . . .\times R\underbrace{} \underbrace{}
m times
consisting of the (m + 1)-tuples z = (zm(x), zm - 1, . . . , z0) with \| z\| Ep = \| zm\| Lp + | zm - 1| + . . .
. . .+ | z0| .
Some of the principal features concerning with solution space Wp can be given as follows: The
trace or value operators
D0u = u(\gamma ), D1u = u1(\gamma ), . . . , Dm - 1u = um - 1(\gamma )
are surjective and bounded from Wp onto R for a given point \gamma of X. The operator
Nu =
\bigl(
u\prime m - 1(x), um - 2(\gamma ), . . . , u1(\gamma ), u(\gamma )
\bigr)
is a linear homeomorphism from Wp onto Ep and has a bounded inverse. On the other hand, any
function u \in Wp can be represented as
u(x) = u(x0) + u1(x0)
x\int
x0
dsm - 1
g1(sm - 1)
+ u2(x0)
x\int
x0
sm - 1\int
x0
dsm - 2 dsm - 1
g2(sm - 2)g1(sm - 1)
+
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
102 K. ÖZEN
+u3(x0)
x\int
x0
sm - 1\int
x0
sm - 2\int
x0
dsm - 3 dsm - 2 dsm - 1
g3(sm - 3)g2(sm - 2)g1(sm - 1)
+ . . .
. . .+ um - 1(x0)
x\int
x0
sm - 1\int
x0
. . .
s2\int
x0
ds1 . . . dsm - 2 dsm - 1
gm - 1(s1) . . . g2(sm - 2)g1(sm - 1)
+
+
x\int
x0
u\prime m - 1(\xi )
x\int
\xi
sm - 1\int
\xi
. . .
s2\int
\xi
ds1 . . . dsm - 2 dsm - 1
gm - 1(s1) . . . g2(sm - 2)g1(sm - 1)
d\xi . (4)
The structure of adjoint space W \ast
p is determined by the following theorem for the general case of
gm - 1, gm - 2, . . . , g1 [1 – 3, 5, 26].
Theorem 1. If 1 \leq p < \infty , then any linear bounded functional F \in W \ast
p can be represented by
F (u) =
x1\int
x0
u\prime m - 1(x)\varphi m(x) dx+ um - 1(x0)\varphi m - 1+
+um - 2(x0)\varphi m - 2 + . . .+ u1(x0)\varphi 1 + u(x0)\varphi 0, u \in Wp, (5)
by means of a unique element \varphi =
\bigl(
\varphi m(x), \varphi m - 1, \varphi m - 2, . . . , \varphi 1, \varphi 0
\bigr)
\in Eq, where
1
p
+
1
q
= 1. Any
linear bounded functional F \in W \ast
\infty can be represented by
F (u) =
x1\int
x0
u\prime m - 1(x) d\varphi m + um - 1(x0)\varphi m - 1+
+um - 2(x0)\varphi m - 2 + . . .+ u1(x0)\varphi 1 + u(x0)\varphi 0, u \in W\infty , (6)
by means of a unique element
\varphi =
\bigl(
\varphi m(e), \varphi m - 1, \varphi m - 2, . . . , \varphi 1, \varphi 0
\bigr)
\in \widehat E1 =
\Bigl(
BA
\Bigl( \sum
, \mu
\Bigr) \Bigr)
\times R\times R\times . . .\times R\underbrace{} \underbrace{}
m times
,
where \mu is Lebesgue measure on R,
\sum
is \sigma -algebra of the \mu -measurable subsets e \subset X and
BA
\bigl( \sum
, \mu
\bigr)
is the space of all bounded additive functions \varphi m(e) defined on
\sum
such that \varphi m(e) = 0
when \mu (e) = 0 [15]. The inverse is also true: That is, if \varphi \in Eq, then (5) is bounded on Wp for
1 \leq p < \infty and
1
p
+
1
q
= 1. If \varphi \in \widehat E1, then (6) is bounded on W\infty [3, 26].
Proof. Theorem 1 can be proved as in [3, 26]. The adjoint operator N\ast of N with \gamma = x0
is a linear homeomorphism from E\ast
p onto W \ast
p . Thus, for any linear bounded functional F \in W \ast
p
there exists one and only one \varphi \in E\ast
p such that F = N\ast \varphi or F (u) = \varphi (Nu) for all u \in Wp. Any
\varphi \in E\ast
p is an (m+ 1)-tuple
\varphi = (\varphi m, \varphi m - 1, \varphi m - 2, . . . , \varphi 1, \varphi 0) \in L\ast
p \times R\ast \times R\ast \times . . .\times R\ast \underbrace{} \underbrace{}
m times
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
GREEN’S FUNCTIONAL FOR HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS . . . 103
of the linear bounded functionals \varphi m, \varphi m - 1, \varphi m - 2, . . . , \varphi 1, \varphi 0 defined on Lp, R, R, . . . , R, R
respectively. That is
F (u) = \varphi m(u\prime m - 1) + \varphi m - 1
\bigl(
um - 1(x0))+
+\varphi m - 2(um - 2(x0)
\bigr)
+ . . .+ \varphi 1
\bigl(
u1(x0)) + \varphi 0(u(x0)
\bigr)
, u \in Wp.
Furthermore, R\ast = R, L\ast
p = Lq for 1 \leq p < \infty and L\ast
\infty = BA
\bigl( \sum
, \mu
\bigr)
in the sense of an
isomorphism [15]. Therefore, any F \in W \ast
p can be represented by (5) for 1 \leq p < \infty and by (6) for
p = \infty . The inverse is obtained from (5) and (6).
As can be seen from Theorem 1, undoubtedly, conditions (2) are the most general ones as linear
conditions for the continuous differential operator Vm : Wp \rightarrow Lp corresponding to equation (1).
Namely, each condition Viu = zi where Vi, i = 0, 1, 2, . . . ,m - 1, is a continuous linear functional
on Wp can be written in form (2) provided that p < \infty .
4. Adjoint operator, adjoint system and solvability conditions. An explicit expression for the
adjoint operator V \ast of V is investigated in this section. To this end, any F = (Fm(x), Fm - 1, . . .
. . . , F0) \in Eq is taken as a linear bounded functional on Ep, and
F (V u) \equiv
x1\int
x0
Fm(x)(Vmu)(x) dx+
m - 1\sum
i=0
Fi(Viu), u \in Wp, (7)
can be supposed. By substituting expressions (1), (2) and (4) into (7), we have
F (V u) \equiv
x1\int
x0
Fm(x)
\left[ u\prime m - 1(x) +A0(x)
\left\{ u(x0) + u1(x0)
x\int
x0
dsm - 1
g1(sm - 1)
+
+u2(x0)
x\int
x0
sm - 1\int
x0
dsm - 2 dsm - 1
g2(sm - 2)g1(sm - 1)
+
+u3(x0)
x\int
x0
sm - 1\int
x0
sm - 2\int
x0
dsm - 3 dsm - 2 dsm - 1
g3(sm - 3)g2(sm - 2)g1(sm - 1)
+ . . .
. . .+ um - 1(x0)
x\int
x0
sm - 1\int
x0
. . .
s2\int
x0
ds1 . . . dsm - 2 dsm - 1
gm - 1(s1) . . . g2(sm - 2)g1(sm - 1)
+
+
x\int
x0
u\prime m - 1(\xi )
x\int
\xi
sm - 1\int
\xi
. . .
s2\int
\xi
ds1 . . . dsm - 2 dsm - 1
gm - 1(s1) . . . g2(sm - 2)g1(sm - 1)
d\xi
\right\}
\right] dx+
+
m - 1\sum
i=0
Fi
\left[ a0iu(x0) + a1iu1(x0) + a2iu2(x0) + . . .+ am - 1
i um - 1(x0) +
x1\int
x0
Bi(\xi )u
\prime
m - 1(\xi ) d\xi
\right] .
Hence, we can obtain the following result by rearrangement:
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
104 K. ÖZEN
F (V u) \equiv
x1\int
x0
Fm(x)(Vmu)(x) dx+
m - 1\sum
i=0
Fi(Viu) =
=
x1\int
x0
(wmF )(\xi )u\prime m - 1(\xi ) d\xi +
+
m - 1\sum
i=1
(wiF )ui(x0) + (w0F )u(x0) \equiv (wF )(u) \forall F \in Eq \forall u \in Wp, (8)
where
(wmF )(\xi ) =
x1\int
\xi
Fm(x)A0(x)
x\int
\xi
sm - 1\int
\xi
. . .
s2\int
\xi
ds1 . . . dsm - 2 dsm - 1
gm - 1(s1) . . . g2(sm - 2)g1(sm - 1)
dx+
+Fm(\xi ) +
m - 1\sum
i=0
FiBi(\xi ), \xi \in X,
wm - 1F =
x1\int
x0
Fm(x)A0(x)
x\int
x0
sm - 1\int
x0
. . .
s2\int
x0
ds1 . . . dsm - 2 dsm - 1
gm - 1(s1) . . . g2(sm - 2)g1(sm - 1)
dx+
m - 1\sum
i=0
Fia
m - 1
i ,
wm - 2F =
x1\int
x0
Fm(x)A0(x)
x\int
x0
sm - 1\int
x0
. . .
s3\int
x0
ds2 . . . dsm - 2 dsm - 1
gm - 2(s2) . . . g2(sm - 2)g1(sm - 1)
dx+
m - 1\sum
i=0
Fia
m - 2
i ,
wm - 3F =
x1\int
x0
Fm(x)A0(x)
x\int
x0
sm - 1\int
x0
. . .
s4\int
x0
ds3 . . . dsm - 2 dsm - 1
gm - 3(s3) . . . g2(sm - 2)g1(sm - 1)
dx+
m - 1\sum
i=0
Fia
m - 3
i ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)
w3F =
x1\int
x0
Fm(x)A0(x)
x\int
x0
sm - 1\int
x0
sm - 2\int
x0
dsm - 3 dsm - 2 dsm - 1
g3(sm - 3)g2(sm - 2)g1(sm - 1)
dx+
m - 1\sum
i=0
Fia
3
i ,
w2F =
x1\int
x0
Fm(x)A0(x)
x\int
x0
sm - 1\int
x0
dsm - 2 dsm - 1
g2(sm - 2)g1(sm - 1)
dx+
m - 1\sum
i=0
Fia
2
i ,
w1F =
x1\int
x0
Fm(x)A0(x)
x\int
x0
dsm - 1
g1(sm - 1)
dx+
m - 1\sum
i=0
Fia
1
i ,
w0F =
x1\int
x0
Fm(x)A0(x) dx+
m - 1\sum
i=0
Fia
0
i .
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
GREEN’S FUNCTIONAL FOR HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS . . . 105
The operators wm, wm - 1, . . . , w1 and w0 are linear and bounded from the space Eq of the (m+
+ 1)-tuples F =
\bigl(
Fm(\xi ), Fm - 1, Fm - 2, . . . , F0
\bigr)
into the spaces Lq(X), R, R, . . . , R, respectively.
So, the operator w = (wm, wm - 1, . . . , w0) : Eq \rightarrow Eq represented by
wF = (wmF,wm - 1F, . . . , w0F )
is linear and bounded. By (8) and Theorem 1, V \ast becomes w for 1 \leq p < \infty and w\ast becomes
V S for 1 < p \leq \infty where S is the inverse of N with \gamma = x0. The operators V S and w can be
considered as adjoint operators to each other. Consequently, the equation
wF = \varphi , (10)
with an unknown F =
\bigl(
Fm(\xi ), Fm - 1, Fm - 2, . . . , F0
\bigr)
\in Eq and a given
\varphi =
\bigl(
\varphi m(\xi ), \varphi m - 1, \varphi m - 2, . . . , \varphi 0
\bigr)
\in Eq
can be considered as an adjoint equation of (3) for all 1 \leq p \leq \infty . The restrictions imposed on
the coefficients A0, B0, B1, . . . , Bm - 1 and the aforementioned assumptions assure that the operator
Q \equiv w - Iq : Eq \rightarrow Eq is completely continuous, where Iq is the identity operator on Eq and
1 < p < \infty . Thus, (10) is a canonical Fredholm equation and S is a right regularizer of V [3, 5, 6,
16, 17, 26].
(10) can be written in explicit form as the following system of the integro-algebraic equations:
(wmF )(\xi ) = \varphi m(\xi ), \xi \in X,
wm - 1F = \varphi m - 1,
wm - 2F = \varphi m - 2, (11)
. . . . . . . . . . . . . . .
w0F = \varphi 0.
As can be seen from (9), the first equation in (11) is generally an integral equation for Fm(\xi ) and
it may include F0, F1, . . . , Fm - 1 as parameters; on the other hand, the other m equations in (11)
are algebraic equations for the unknowns F0, F1, . . . , Fm - 1 and they may include some integral
functionals on Fm(\xi ). In other words, (11) is a system of m + 1 integro-algebraic equations. This
system, called the adjoint system for (3) is constructed by using (8), which is actually a formula of
integration by parts in a nonclassical form, and the traditional (classical) form of an adjoint problem is
defined by the classical Green’s formula of integration by parts [3, 4, 26, 35], therefore, the traditional
form only works for some restricted class of problems.
From this point, Fredholm alternative can be stated by the following theorem in the context of
solvability of the problem:
Theorem 2. If 1 < p < \infty , then V u = 0 has either only the trivial solution or a finite number
of linearly independent solutions in Wp :
(1) If V u = 0 has only the trivial solution in Wp, then also wF = 0 has only the trivial solution
in Eq. Then, the operators V : Wp \rightarrow Ep and w : Eq \rightarrow Eq become linear homeomorphisms.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
106 K. ÖZEN
(2) If V u = 0 has n linearly independent solutions u(1), . . . , u(n) in Wp, then wF = 0 has also
n linearly independent solutions
F \star 1 \star =
\bigl(
F \star 1 \star
m (x), F \star 1 \star
m - 1, . . . , F
\star 1 \star
0
\bigr)
, . . . , F \star n \star =
\bigl(
F \star n \star
m (x), F \star n \star
m - 1, . . . , F
\star n \star
0
\bigr)
in Eq. In this case, (3) and (10) have solutions u \in Wp and F \in Eq for given z \in Ep and \varphi \in Eq
if and only if the conditions
x1\int
x0
F \star i \star
m (\xi )zm(\xi ) d\xi + F \star i \star
m - 1zm - 1 + . . .+ F \star i \star
0 z0 = 0, i = 1, . . . , n, (12)
and
x1\int
x0
\varphi m(\xi )u\prime (i),m - 1(\xi ) d\xi + \varphi m - 1u(i),m - 1(x0) + . . .
. . .+ \varphi 1u(i),1(x0) + \varphi 0u(i)(x0) = 0, i = 1, . . . , n, (13)
where
u(i),1(x) = g1(x)u
\prime
(i)(x), u(i),2(x) = g2(x)u
\prime
(i),1(x), . . . , u(i),m - 1(x) = gm - 1(x)u
\prime
(i),m - 2(x)
are satisfied, respectively [3, 4, 26].
5. Green’s functional and a specific adjoint system. Consider the following equation given in
the form of a functional identity
(wF )(u) = u(x) \forall u \in Wp, (14)
where F = (Fm(\xi ), Fm - 1, . . . , F0) \in Eq is an unknown (m+ 1)-tuple and x \in X is a parameter.
Identity (14) is equivalent to the following system:
(wmF )(\xi ) = H(x - \xi )
x\int
\xi
sm - 1\int
\xi
. . .
s2\int
\xi
ds1 . . . dsm - 2 dsm - 1
gm - 1(s1) . . . g2(sm - 2)g1(sm - 1)
,
wm - 1F =
x\int
x0
sm - 1\int
x0
. . .
s2\int
x0
ds1 . . . dsm - 2 dsm - 1
gm - 1(s1) . . . g2(sm - 2)g1(sm - 1)
,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
w3F =
x\int
x0
sm - 1\int
x0
sm - 2\int
x0
dsm - 3 dsm - 2 dsm - 1
g3(sm - 3)g2(sm - 2)g1(sm - 1)
, (15)
w2F =
x\int
x0
sm - 1\int
x0
dsm - 2 dsm - 1
g2(sm - 2)g1(sm - 1)
,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
GREEN’S FUNCTIONAL FOR HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS . . . 107
w1F =
x\int
x0
dsm - 1
g1(sm - 1)
,
w0F = 1,
where H(x - \xi ) is a Heaviside function on R, and \xi \in X.
Definition 1. Let F (x) =
\bigl(
Fm(\xi , x), Fm - 1(x), Fm - 2(x), . . . , F0(x)
\bigr)
\in Eq be an unknown
(m+ 1)-tuple with a parameter x \in X. If F (x) is a solution of (15) for a given x \in X, then F (x)
is called a Green’s functional of V [3, 4, 26, 27, 29].
Theorem 3. If a Green’s functional
F (x) =
\bigl(
Fm(\xi , x), Fm - 1(x), Fm - 2(x), . . . , F0(x)
\bigr)
of V exists, then any solution u \in Wp of (3) can be represented by
u(x) =
x1\int
x0
Fm(\xi , x)zm(\xi ) d\xi +
m - 1\sum
i=0
Fi(x)zi. (16)
Additionally, \mathrm{K}\mathrm{e}\mathrm{r}V = \{ 0\} where \mathrm{K}\mathrm{e}\mathrm{r}V denotes the kernel of V.
Proof. By (8), we can obtain the following identity:
x1\int
x0
Fm(\xi , x)zm(\xi ) d\xi +
m - 1\sum
i=0
Fi(x)zi =
=
x1\int
x0
H(x - \xi )
x\int
\xi
sm - 1\int
\xi
. . .
s2\int
\xi
ds1 . . . dsm - 2 dsm - 1
gm - 1(s1) . . . g2(sm - 2)g1(sm - 1)
u\prime m - 1(\xi ) d\xi +
+
x\int
x0
sm - 1\int
x0
. . .
s2\int
x0
ds1 . . . dsm - 2 dsm - 1
gm - 1(s1) . . . g2(sm - 2)g1(sm - 1)
um - 1(x0) + . . .
. . .+
x\int
x0
sm - 1\int
x0
dsm - 2 dsm - 1
g2(sm - 2)g1(sm - 1)
u2(x0) +
x\int
x0
dsm - 1
g1(sm - 1)
u1(x0) + u(x0).
By (4), the right-hand side of the above identity equals to u(x). Thus, (16) is valid. The verity of
Ker V = \{ 0\} is obtained from (16).
Theorem 3 is related to the necessary condition for the existence a Green’s functional of V, and
the following theorem is related to the sufficient condition for the existence of a Green’s functional.
Theorem 4 [3, 26]. If V has a priori estimate as follows:
\| u\| Wp \leq c0\| V u\| Ep , u \in Wp, (17)
where c0 is a positive constant, then a Green’s functional of V exists, where 1 \leq p \leq \infty .
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
108 K. ÖZEN
Proof. V has a bounded left inverse by (17). Then the image Im w of w equals to Eq [3, 15].
Hence (15) has a solution
F (x) =
\bigl(
Fm(\xi , x), Fm - 1(x), Fm - 2(x), . . . , F0(x)
\bigr)
\in Eq
for all x \in X.
Remark 1. If w has a priori estimate as follows:
\| F\| Eq \leq c1\| wF\| Eq , F \in Eq, (18)
where c1 is a positive constant, then (1), (2) has always a solution u \in Wp. If both (17) and (18)
are valid, then V and w become homeomorphisms and a unique Green’s functional of V exists.
Estimates (17) and (18) are valid if
\sum m - 1
i=0
\| Bi\| Lq is sufficiently small and\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
a00 a10 \cdot \cdot \cdot am - 1
0
a01 a11 \cdot \cdot \cdot am - 1
1
...
...
. . .
...
a0m - 1 a1m - 1 \cdot \cdot \cdot am - 1
m - 1
\bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm| \bigm|
\not = 0.
Theorem 5 [3, 4, 26, 27, 29]. Assume that 1 < p < \infty . If there exists a Green’s functional,
then it is unique. Additionally, a Green’s functional exists if and only if V u = 0 has only the trivial
solution.
6. Several applications. In this section, we consider mth order problems involving generally
nonlocal condition(s) in order to support the theoretical presentation and to demonstrate the validity,
utility and advantages of the proposed approach.
Example 1. Firstly, in order to demonstrate the applicability for a problem with principal coeffi-
cient equal to one, we consider the following problem for which Green’s function has been presented
in [30, 31],
(Vmu)(x) \equiv u(m)(x) = f(x), x \in X = (0, 1),
Vm - 1u \equiv u(1) - \gamma u(\eta ) = 0,
Vm - 2u \equiv u(m - 2)(0) = 0,
Vm - 3u \equiv u(m - 3)(0) = 0,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
V1u \equiv u\prime (0) = 0,
V0u \equiv u(0) - \beta u(\alpha ) = 0,
where f(x) \in Lp, \alpha , \eta \in X and \beta , \gamma \in R. Here gi(x) = 1 for i = 1, 2, . . . ,m - 1, A0(x) = 0,
zm(x) = f(x),
a0m - 1 = 1 - \gamma , zm - 1 = 0,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
GREEN’S FUNCTIONAL FOR HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS . . . 109
a1m - 1 =
1\int
0
dsm - 1 - \gamma
\eta \int
0
dsm - 1,
a2m - 1 =
1\int
0
sm - 1\int
0
dsm - 2 dsm - 1 - \gamma
\eta \int
0
sm - 1\int
0
dsm - 2 dsm - 1,
a3m - 1 =
1\int
0
sm - 1\int
0
sm - 2\int
0
dsm - 3 dsm - 2 dsm - 1 -
- \gamma
\eta \int
0
sm - 1\int
0
sm - 2\int
0
dsm - 3 dsm - 2 dsm - 1,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
am - 1
m - 1 =
1\int
0
sm - 1\int
0
. . .
s2\int
0
ds1 . . . dsm - 2 dsm - 1 -
- \gamma
\eta \int
0
sm - 1\int
0
. . .
s2\int
0
ds1 . . . dsm - 2 dsm - 1,
Bm - 1(\xi ) =
1\int
\xi
sm - 1\int
\xi
. . .
s2\int
\xi
ds1 . . . dsm - 2 dsm - 1 -
- \gamma H(\eta - \xi )
\eta \int
\xi
sm - 1\int
\xi
. . .
s2\int
\xi
ds1 . . . dsm - 2 dsm - 1,
aim - 2 =
\left\{ 1 for i = m - 2,
0 for i \not = m - 2,
zm - 2 = 0, Bm - 2(\xi ) = 0,
aim - 3 =
\left\{ 1 for i = m - 3,
0 for i \not = m - 3,
zm - 3 = 0, Bm - 3(\xi ) = 0,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ai1 =
\left\{ 1 for i = 1,
0 for i \not = 1,
z1 = 0, B1(\xi ) = 0,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
110 K. ÖZEN
a00 = 1 - \beta , z0 = 0,
a10 = - \beta
\alpha \int
0
dsm - 1,
a20 = - \beta
\alpha \int
0
sm - 1\int
0
dsm - 2 dsm - 1,
a30 = - \beta
\alpha \int
0
sm - 1\int
0
sm - 2\int
0
dsm - 3 dsm - 2 dsm - 1,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
am - 1
0 = - \beta
\alpha \int
0
sm - 1\int
0
. . .
s2\int
0
ds1 . . . dsm - 2 dsm - 1,
B0(\xi ) = - \beta H(\alpha - \xi )
\alpha \int
\xi
sm - 1\int
\xi
. . .
s2\int
\xi
ds1 . . . dsm - 2 dsm - 1,
where H(\alpha - \xi ) and H(\eta - \xi ) are Heaviside functions on R. System (15) corresponding to the
problem can be written in the following form:
Fm(\xi ) + F0
\left[ - \beta H(\alpha - \xi )
\alpha \int
\xi
sm - 1\int
\xi
. . .
s2\int
\xi
ds1 . . . dsm - 2 dsm - 1
\right] +
+Fm - 1
\left[ 1\int
\xi
sm - 1\int
\xi
. . .
s2\int
\xi
ds1 . . . dsm - 2 dsm - 1 -
- \gamma H(\eta - \xi )
\eta \int
\xi
sm - 1\int
\xi
. . .
s2\int
\xi
ds1 . . . dsm - 2 dsm - 1
\right] =
= H(x - \xi )
x\int
\xi
sm - 1\int
\xi
. . .
s2\int
\xi
ds1 . . . dsm - 2 dsm - 1, \xi \in (0, 1), (19)
F0
\left[ - \beta
\alpha \int
0
sm - 1\int
0
. . .
s2\int
0
ds1 . . . dsm - 2 dsm - 1
\right] +
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
GREEN’S FUNCTIONAL FOR HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS . . . 111
+Fm - 1
\left[ 1\int
0
sm - 1\int
0
. . .
s2\int
0
ds1 . . . dsm - 2 dsm - 1 -
- \gamma
\eta \int
0
sm - 1\int
0
. . .
s2\int
0
ds1 . . . dsm - 2 dsm - 1
\right] =
=
x\int
0
sm - 1\int
0
. . .
s2\int
0
ds1 . . . dsm - 2 dsm - 1, (20)
F0
\left[ - \beta
\alpha \int
0
sm - 1\int
0
. . .
s3\int
0
ds2 . . . dsm - 2 dsm - 1
\right] + Fm - 2+
+Fm - 1
\left[ 1\int
0
sm - 1\int
0
. . .
s3\int
0
ds2 . . . dsm - 2 dsm - 1 -
- \gamma
\eta \int
0
sm - 1\int
0
. . .
s3\int
0
ds2 . . . dsm - 2 dsm - 1
\right] =
=
x\int
0
sm - 1\int
0
. . .
s3\int
0
ds2 . . . dsm - 2 dsm - 1, (21)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
F0
\left[ - \beta
\alpha \int
0
sm - 1\int
0
dsm - 2 dsm - 1
\right] + Fm - 1
\left[ 1\int
0
sm - 1\int
0
dsm - 2 dsm - 1 -
- \gamma
\eta \int
0
sm - 1\int
0
dsm - 2 dsm - 1
\right] + F2 =
x\int
0
sm - 1\int
0
dsm - 2 dsm - 1, (22)
- F0\beta
\alpha \int
0
dsm - 1 + F1 + Fm - 1
\left[ 1\int
0
dsm - 1 - \gamma
\eta \int
0
dsm - 1
\right] =
x\int
0
dsm - 1, (23)
F0(1 - \beta ) + Fm - 1(1 - \gamma ) = 1. (24)
In order to solve (19) – (24), firstly, F0 and Fm - 1 are uniquely obtained from (20) and (24) under
the condition
\Delta \equiv (1 - \beta )(1 - \gamma \eta m - 1) + (1 - \gamma )\beta \alpha m - 1 \not = 0
and then substituting the obtained values of F0 and Fm - 1 into the other equations in (19) – (24), we
have
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
112 K. ÖZEN
F0(x) =
1 - \gamma \eta m - 1 - (1 - \gamma )xm - 1
\Delta
,
Fi(x) =
xi
i!
+
\biggl[
1 - \gamma \eta m - 1 - (1 - \gamma )xm - 1
\Delta
\biggr]
\beta
\alpha m - 2
(m - 2)!
-
-
\biggl[
\beta \alpha m - 1 + (1 - \beta )xm - 1
\Delta
\biggr] \biggl\{
1 - \gamma \eta i
i!
\biggr\}
, i = 1, 2, . . . ,m - 2,
Fm - 1(x) =
\beta \alpha m - 1 + (1 - \beta )xm - 1
\Delta
,
Fm(\xi , x) = H(x - \xi )
(x - \xi )m - 1
(m - 1)!
-
-
\biggl[
1 - \gamma \eta m - 1 - (1 - \gamma )xm - 1
\Delta
\biggr]
H(\alpha - \xi )( - \beta )
(\alpha - \xi )m - 1
(m - 1)!
-
-
\biggl[
\beta \alpha m - 1 + (1 - \beta )xm - 1
\Delta
\biggr]
\times
\times
\biggl[
(1 - \xi )m - 1
(m - 1)!
- \gamma H(\eta - \xi )
(\eta - \xi )m - 1
(m - 1)!
\biggr]
.
Hence, under the condition (1 - \beta )(1 - \gamma \eta m - 1) + (1 - \gamma )\beta \alpha m - 1 \not = 0 we have Green’s functional
F (x) =
\bigl(
Fm(\xi , x), Fm - 1(x), Fm - 2(x), . . . , F0(x)
\bigr)
. The first component Fm(\xi , x) of this functional
corresponds to Green’s function for the problem. After substitution \xi = s into Fm(\xi , x) for notational
compatibility, one can see easily that the first component equals to G(x, s) in Example 3 of [31].
Consequently, by Theorem 3, Green’s solution can be represented by
u(x) =
1\int
0
\biggl[
H(x - \xi )
(x - \xi )m - 1
(m - 1)!
-
-
\biggl[
1 - \gamma \eta m - 1 - (1 - \gamma )xm - 1
\Delta
\biggr]
H(\alpha - \xi )( - \beta )
(\alpha - \xi )m - 1
(m - 1)!
-
-
\biggl[
\beta \alpha m - 1 + (1 - \beta )xm - 1
\Delta
\biggr]
\times
\times
\biggl[
(1 - \xi )m - 1
(m - 1)!
- \gamma H(\eta - \xi )
(\eta - \xi )m - 1
(m - 1)!
\biggr] \biggr]
f(\xi ) d\xi
since zm - 1 = zm - 2 = . . . = z0 = 0 for the problem.
Example 2. Now, we consider the following nonlocal problem:
(Vmu)(x) \equiv (e - xu(m - 1)(x))\prime = 0, x \in X = (0, 1),
Viu \equiv u(i)(0) = 1, for i = m - 1,m - 2, . . . , 1,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
GREEN’S FUNCTIONAL FOR HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS . . . 113
V0u \equiv
1\int
0
u(\tau ) d\tau = e - 1.
Here gi(x) = 1 for i = 1, 2, . . . ,m - 2, gm - 1(x) = e - x, A0(x) = 0, zm(x) = 0,
aim - 1 =
\left\{ 1 for i = m - 1,
0 for i \not = m - 1,
zm - 1 = 1, Bm - 1(\xi ) = 0,
aim - 2 =
\left\{ 1 for i = m - 2,
0 for i \not = m - 2,
zm - 2 = 1, Bm - 2(\xi ) = 0,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ai2 =
\left\{ 1 for i = 2,
0 for i \not = 2,
z2 = 1, B2(\xi ) = 0,
ai1 =
\left\{ 1 for i = 1,
0 for i \not = 1,
z1 = 1, B1(\xi ) = 0,
a00 =
1\int
0
d\tau , z0 = e - 1,
a10 =
1\int
0
\tau \int
0
dsm - 1 d\tau ,
a20 =
1\int
0
\tau \int
0
sm - 1\int
0
dsm - 2 dsm - 1 d\tau ,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
am - 2
0 =
1\int
0
\tau \int
0
sm - 1\int
0
. . .
s3\int
0
ds2 . . . dsm - 2 dsm - 1 d\tau ,
am - 1
0 =
1\int
0
\tau \int
0
sm - 1\int
0
. . .
s2\int
0
es1 ds1 . . . dsm - 2 dsm - 1 d\tau ,
B0(\xi ) =
1\int
\xi
\tau \int
\xi
sm - 1\int
\xi
. . .
s2\int
\xi
es1 ds1 . . . dsm - 2 dsm - 1 d\tau .
System (15) corresponding to the problem can be written in the form
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
114 K. ÖZEN
Fm(\xi ) + F0
\left[ 1\int
\xi
\tau \int
\xi
sm - 1\int
\xi
. . .
s2\int
\xi
es1 ds1 . . . dsm - 2 dsm - 1 d\tau
\right] =
= H(x - \xi )
x\int
\xi
sm - 1\int
\xi
. . .
s2\int
\xi
es1 ds1 . . . dsm - 2 dsm - 1, \xi \in (0, 1),
F0
\biggl[
e - 1
(m - 1)!
- 1
(m - 2)!
- . . . - 1 - 1
\biggr]
+ Fm - 1 =
= ex - xm - 2
(m - 2)!
- xm - 3
(m - 3)!
- . . . - x - 1,
F0
(m - 1)!
+ Fm - 2 =
xm - 2
(m - 2)!
, (25)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
F0
3!
+ F2 =
x2
2!
,
F0
2!
+ F1 = x,
F0 = 1.
Substituting the obtained value of F0 in (25) into the other equations, we have
F0(x) = 1,
Fi(x) =
xi
i!
- 1
(i+ 1)!
, i = 1, 2, . . . ,m - 2,
Fm - 1(x) = ex - e -
m - 2\sum
i=0
xi
i!
+
m\sum
i=1
1
(i - 1)!
,
Fm(\xi , x) = H(x - \xi )
x\int
\xi
sm - 1\int
\xi
. . .
s2\int
\xi
es1 ds1 . . . dsm - 2 dsm - 1 -
-
1\int
\xi
\tau \int
\xi
sm - 1\int
\xi
. . .
s2\int
\xi
es1 ds1 . . . dsm - 2 dsm - 1 d\tau .
Thus, Green’s functional F (x) =
\bigl(
Fm(\xi , x), Fm - 1(x), Fm - 2(x), . . . , F0(x)
\bigr)
has been determined.
The first component Fm(\xi , x) of this functional corresponds to Green’s function for the problem.
Consequently, by Theorem 3, we have
u(x) = ex
since zm(\xi ) = 0, zm - 1 = zm - 2 = . . . = z1 = 1, z0 = e - 1 for the problem.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
GREEN’S FUNCTIONAL FOR HIGHER-ORDER ORDINARY DIFFERENTIAL EQUATIONS . . . 115
7. Conclusion. The introduced method for the second and third order linear ODEs with variable
principal coefficient involving nonlocal conditions in [3, 26] is extended to the mth order linear ODEs
with variable principal coefficient involving nonlocal conditions. As can be seen from the theory
and illustrations, the proposed method principally is different from the known classical construction
methods of Green’s function. The structural properties of the space of solutions instead of the classical
Green’s formula of integration by parts are used. The proposed method can successfully be employed
also for the problems resulting from the addition of some delayed, loaded (forced) or neutral terms
to the operator Vm, provided that the linearity for the operator is conserved. The applicability easily
to a very wide class of linear and some linear boundary-value problems involving linear nonlocal
conditions is a natural property of the method. Obviously, this method has a significant advantage
especially for cases when the adjoint problem can not be constructed in the classical sense. Because
of these properties, we believe that it is one of the scarce methods which aim at the derivation of
a solution to such problems by reducing to an integral equation in general manner, the results may
be useful for the studies which aim at searching for the existence and uniqueness of the solutions
to the problems of interest, and the work is significant for its contribution to the feasibility of such
studies also with the weak assumptions such as Lp-integrability and boundedness instead of the
strong assumptions such as continuities for the coefficient and right-hand side functions of equation.
References
1. Akhiev S. S. Representations of the solutions of some linear operator equations // Sov. Mat. Dokl. – 1980. – 21, № 2. –
P. 555 – 558.
2. Akhiev S. S. Fundamental solutions of functional differential equations and their representations // Sov. Mat. Dokl. –
1984. – 29, № 2. – P. 180 – 184.
3. Akhiev S. S. Solvability conditions and Green functional concept for local and nonlocal linear problems for a second
order ordinary differential equation // Math. Comput. Appl. – 2004. – 9, № 3. – P. 349 – 358.
4. Akhiev S. S. Green and generalized Green’s functionals of linear local and nonlocal problems for ordinary integro-
differential equations // Acta Appl. Math. – 2007. – 95. – P. 73 – 93.
5. Akhiev S. S., Oruçoğlu K. Fundamental solutions of some linear operator equations and applications // Acta Appl.
Math. – 2002. – 71. – P. 1 – 30.
6. Azbelev N., Maksimov V., Rakhmatullina L. Introduction to the theory of linear functional differential equations. –
Atlanta, Georgia: World Federation Publ. Co., 1995.
7. Berezanskii Ju. M. Expansions in eigenfunctions of nonselfadjoint operators. – Providence, RI: Amer. Math. Soc.,
1968.
8. Berezansky Yu. M., Tesko V. A. Spaces of fundamental and generalized functions associated with generalized translati-
on // Ukr. Math. J. – 2003. – 55, № 12. – P. 1907 – 1979.
9. Boichuk A. A., Samoilenko A. M. Generalized inverse operators and Fredholm boundary-value problems. – Utrecht,
Boston: De Gruyter, 2004.
10. Brown A. L., Page A. Elements of functional analysis. – New York: Van Nostrand, 1970.
11. Duffy D. G. Green’s functions with applications. – Boca Raton, FL: Chapman and Hall/CRC, 2015.
12. Gel’fand I. M., Shilov G. E. Generalized functions: spaces of fundamental and generalized functions. – New York:
Acad. Press, 1968.
13. Il’in V. A., Moiseev E. I. Nonlocal boundary value problem of the first kind for a Sturm – Liouville operator in its
differential and finite difference aspects // Different. Equat. – 1987. – 23, № 7. – P. 803 – 810.
14. Il’in V. A., Moiseev E. I. Nonlocal boundary value problem of the second kind for a Sturm – Liouville operator //
Different. Equat. – 1987. – 23, № 8. – P. 979 – 987.
15. Kantorovich L. V., Akilov G. P. Functional analysis. – New York: Pergamon Press, 1982.
16. Krein S. G. Linear equations in Banach space. – Moscow: Nauka, 1971 (in Russian).
17. Krein S. G. Linear equations in Banach spaces. – Boston: Birkhäuser, 1982.
18. Lanczos C. Linear differential operators. – Great Britain: Van D. Nostrand Co. Ltd., 1964.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
116 K. ÖZEN
19. Liouville J. Premier memoire sur la theorie des equations differentielles lineaires et sur le developpement des fonctions
en series // Math J. Pures et Appl. – 1838. – 3. – P. 561 – 614.
20. Naimark M. A. Linear differential operators. – Great Britain: George G. Harrap and Ltd Co., 1968.
21. Oruçoğlu K. A new Green function concept for fourth-order differential equations // Electron. J. Different. Equat. –
2005. – 28. – P. 1 – 12.
22. Oruçoğlu K., Özen K. Investigation of a fourth-order ordinary differential equation with a four-point boundary
conditions by a new Green’s functional concept // AIP Conf. Proc. – 2011. – 1389. – P. 1160.
23. Oruçoğlu K., Özen K. Green’s functional for second-order linear differential equation with nonlocal conditions //
Electron. J. Different. Equat. – 2012. – 2012, № 121. – P. 1 – 12.
24. Özen K. Green’s function to the forced Duffing equation involving nonlocal integral conditions by Green’s functional
concept // AIP Conf. Proc. – 2012. – 1479. – P. 1982.
25. Özen K. Construction of green or generalized Green’s functional for some nonlocal boundary value problems: PhD
Thesis. – Istanbul, 2013 (in Turkish).
26. Özen K. Construction of Green’s functional for a third order ordinary differential equation with general nonlocal
conditions and variable principal coefficient // Georg. Math. J. (to appear).
27. Özen K., Oruçoğlu K. A representative solution to m-order linear ordinary differential equation with non local
conditions by Green’s functional concept // Math. Model. and Anal. – 2012. – 17, № 4. – P. 571 – 588.
28. Özen K., Oruçoğlu K. Green’s functional concept for a nonlocal problem // Hacet. Math J. Statist. – 2013. – 42,
№ 4. – P. 437 – 446.
29. Özen K., Oruçoğlu K. A novel approach to construct the adjoint problem for a first-order functional integro-differential
equation with general nonlocal condition // Lith. Math. J. – 2014. – 54, № 4. – P. 482 – 502.
30. Roman S. Green’s functions for boundary-value problems with nonlocal boundary conditions: Doct. Dis. – Vilnius
Univ., 2011.
31. Roman S. Linear differential equation with additional conditions and formulae for Green’s function // Math. Model.
and Anal. – 2011. – 16, № 3. – P. 401 – 417.
32. Schwabik S., Tvrdy M., Vejvoda O. Differential and integral equations: boundary value problems and adjoints. –
Prague: Academia Praha, 1979.
33. Sırma A. Adjoint systems and Green functionals for second-order linear integro-differential equations with nonlocal
conditions // Electron. J. Different. Equat. – 2015. – 2015, № 183. – P. 1 – 10.
34. Shilov G. E. Generalized functions and partial differential equations. – New York: Gordon and Breach, 1968.
35. Stakgold I. Green’s functions and boundary value problems. – New York: Wiley-Intersci. Publ., 1998.
36. Vladimirov V. S. Equations of mathematical physics. – New York: Marcel Dekker, 1971.
37. Zettl A. Sturm – Liouville theory. – Providence, RI: Amer. Math. Soc., 2005.
Received 18.03.16
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 1
|
| id | umjimathkievua-article-1422 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:05:03Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/1e/cee31024d9c403d73acb51e5251aa81e.pdf |
| spelling | umjimathkievua-article-14222019-12-05T08:54:16Z Green’s functional for higher-order ordinary differential equations with general nonlocal conditions and variable principal coefficient Функцiонал Грiна для звичайних диференцiальних рiвнянь вищого порядку з нелокальними умовами загального вигляду та змiнним головним коефiцiєнтом Özen, K. Озен, К. The method of Green’s functional is a little-known technique for the construction of fundamental solutions to linear ordinary differential equations (ODE) with nonlocal conditions. We apply this technique to a higher order linear ODE involving general nonlocal conditions. A novel kind of adjoint problem and Green’s functional are constructed for the completely inhomogeneous problem. Several illustrative applications of the theoretical results are provided. Метод функцiонала Грiна є маловiдомою технiкою побудови фундаментальних розв’язкiв лiнiйних звичайних диференцiальних рiвнянь (ЗДР) з нелокальними умовами. В роботi цю технiку застосовано до лiнiйних ЗДР вищого порядку з нелокальними умовами загального вигляду. Спряжену проблему нового типу та функцiонал Грiна побудовано для повнiстю неоднорiдної задачi. Наведено також кiлька iлюстративних застосувань теоретичних результатiв. Institute of Mathematics, NAS of Ukraine 2019-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1422 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 1 (2019); 99-116 Український математичний журнал; Том 71 № 1 (2019); 99-116 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1422/406 Copyright (c) 2019 Özen K. |
| spellingShingle | Özen, K. Озен, К. Green’s functional for higher-order ordinary differential equations with general nonlocal conditions and variable principal coefficient |
| title | Green’s functional for higher-order ordinary differential equations with general nonlocal
conditions and variable principal coefficient |
| title_alt | Функцiонал Грiна для звичайних диференцiальних рiвнянь
вищого порядку з нелокальними умовами
загального вигляду та змiнним головним коефiцiєнтом |
| title_full | Green’s functional for higher-order ordinary differential equations with general nonlocal
conditions and variable principal coefficient |
| title_fullStr | Green’s functional for higher-order ordinary differential equations with general nonlocal
conditions and variable principal coefficient |
| title_full_unstemmed | Green’s functional for higher-order ordinary differential equations with general nonlocal
conditions and variable principal coefficient |
| title_short | Green’s functional for higher-order ordinary differential equations with general nonlocal
conditions and variable principal coefficient |
| title_sort | green’s functional for higher-order ordinary differential equations with general nonlocal
conditions and variable principal coefficient |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1422 |
| work_keys_str_mv | AT ozenk greensfunctionalforhigherorderordinarydifferentialequationswithgeneralnonlocalconditionsandvariableprincipalcoefficient AT ozenk greensfunctionalforhigherorderordinarydifferentialequationswithgeneralnonlocalconditionsandvariableprincipalcoefficient AT ozenk funkcionalgrinadlâzvičajnihdiferencialʹnihrivnânʹviŝogoporâdkuznelokalʹnimiumovamizagalʹnogoviglâdutazminnimgolovnimkoeficiêntom AT ozenk funkcionalgrinadlâzvičajnihdiferencialʹnihrivnânʹviŝogoporâdkuznelokalʹnimiumovamizagalʹnogoviglâdutazminnimgolovnimkoeficiêntom |