Certain remarks to integral equations generated by the numerical-analytical method for solution of the boundary problems
In the paper the comparison method and the chosing of the appropriate norm of the comparison operator is used to establish the solvability of the integral-functional equation resulted by the application of the numeric-analytic method of solving of boundary value problems for ordinary differential-de...
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| Дата: | 1992 |
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| Мова: | Англійська |
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Institute of Mathematics, NAS of Ukraine
1992
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512765361258496 |
|---|---|
| author | Kwapisz, M. Kwapisz, M. |
| author_facet | Kwapisz, M. Kwapisz, M. |
| author_sort | Kwapisz, M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2023-09-14T12:03:56Z |
| description | In the paper the comparison method and the chosing of the appropriate norm of the comparison operator is used to establish the solvability of the integral-functional equation resulted by the application of the numeric-analytic method of solving of boundary value problems for ordinary differential-delay equations of the neutral type. |
| first_indexed | 2026-03-24T03:33:59Z |
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| fulltext |
I{O POTKI IIOBI)];OMJIEHHH
U DC 519.624. I
M. Kwapisz, prof. (Th e Gdanst Univ., P o l and)
Some remarks on an integral equation arising in applications of numerical
a na lytic method of solving of boundary value problems
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In t he paper the comparison methob anb the chosing of the appropriate norm of the compari•
son oper ator is used to est ablish the solvability of the integral-functional equation resulted by
the appl icati on of the numeric-ana lytic method of solving of boundary value problems fm Ot•
<lin an· differential-delay equations of the neutra l type.
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TpaJibHOro THny.
The app lica tion of the numerical-analytic method (see
and N. I. Ron to I 1, 2]) to the solution of the problem:
x' (t) = f (t, x (a (t)), x' ((:) (t))), t E J = [0, T],
by the su bstitution
Ax (0) + Bx (1) = d
I
x (t) = x 0 + ) z (s) ds
0
leaas us (see 131) to the following system of equations
A. M. Samoilenko
T>O, (I)
(2)
(3)
T a (t) a. <n
z tt) = f (t, x, + a (t) cp (x 0) + - 7 ) z (s) ds -
with
- a-,;.t) S z (s) ds, z (13 (t))),
et( I)
I T
cp (x0) - T ) z (s) ds = 0
0
0
cp (x0 ) = T- 1 B-1 (d - (A + B) x 0).
(4)
(5)
It is assumed that fE C(J x R" x R", R"), a , f3E C(J, J), dER". A and B
are n x n matrices.
It is quite clear that the solution x* of (1), (2) is found when a solution
(x~, z*) of the system (4), (5) is gi ven . The formu la (3) serves for the relation
between the solutions. In the method we are dea ling with the sy tern 14), (5)
is solved successively . As the fir st the equation (4) is solved with respect
to z at the same time x11 is considered as a parameter. As a result
of this we fin d a function z* ( .• x0 ) which together with eq uation (5) is used
to find x0• • Finally the solution of (1), (2) is given by the equation x* (t) =-=
= z* (t, x~).
C M. KWAPISZ, 1992
128 1SSN 0041-6053. Ytcp. MaT. tJ1Cyp11., 1992, T. 44, M 1
The a im of the presen t paper is to di scuss the conditions under which the
solution z* ( •, x0) of equa tion (4) can be obtained by the method of success ive
approx imations
Z1,-1- 1 (t, x 0) = (Fz,, ) (t , x 0) , k = 0, l, ... (6)
with z0 choosen arb itr a ril!y . Here the operator F is defin ed by
the ri ght han J s ide of equation (4). The convergence of t he seq uence {z1.} will
be obt ained by the com para tive method (see e. g. [4-6]).
We introduce the fo llowing assumptions
Ass u m p t i on H1 . There are n x n matrices K and L with positive
entries such that
If (t, X , y) - f (t, X, !/) I :s;;; KI X - XI+ LI y - y I (7)
for all t E J, x , y, x, y E Rn. Here I • I denotes the absolute value of the vec
T
tor, so Jx l =(lx1 l,-- -,l xn l) -
Ass um p ti on H 2 . For any positiv e function h EC (J x R", R+) there
exists the unique solution u0 EC (J X R", R.'.j.. ) of the comparison equation
u (t, x0) = (Qu) (t, x0) + h (t, x0), (8)
wi th the operator Q defined by the equation
T - a (t) a(t) a (t) T
(Qu) (t , x0 ) = T S Ku (s, x0 ) ds + ---;y- S Ku (s, x0 ) ds+ Lu (~ (t), x0).
o a(t)
ow for a fixed initial approximation 20 we define
h (t, X 0) = I (Fz0) (t, X 0) - z0 (t, X0 ) I (9)
and we construct the sequence {uh} by the formula
Uk+l (t, X0) = {Qu") (t, X0) , k = l, 2, ... , t E J, X 0 E Rn, (10)
for u0 defined by equat ion (8) with h given by equation (9). From Assumption
H 2 it fo llows easily {see for the compara tive theory) tha t the sequence {uh}
is nonincreas ing and converges a lmost uniformly {uniformly on any compact
subset of J x Rn) to the zero fu ncti on. Moreover by the mathematical
induct ion rule we get eas ily the following esti mations
lz1<(t,x.)-z0 (t, x~)l :s;;; u0 (t,x0 ), k= O, 1,.... (11)
I Zk+P (t, x0)- zk (t, X 0 ) :s;;; u ,, (t, X 0), k, P = 0, l, .... (12)
From these estimations it follows the followin g
Theorem. //' the assumptions H1 and H2 ·are satisfied then for every
x0 E R_n there exists a unique solution z* ( ·, x0) of equation (4), moreover
z* (t, Xe) = lim zh (t , x0), k-+ + oo, (13)
(14)
It follows from the given genera l discussion that now the crutial point
is to find practica l sufficient condi tions wh ich quarantee that the Assumption
H 2 holds true. There is an immedia te answer to our problem. It is enough to
assume that some norm of the operator ~2 is less th an one . However there are
many different norms of the same operator Q (they depend on the norm taken
in the space of continuous functions where the opera tor is defined). ret us
consider some specia l cases .
I. If we take in the space C (J, R11 ) the standard supremum norm (the
Tchebysheff norm) then we see tha t the Assumption H 2 holds if the condition
holds.
m\\kll+IIL!l<l, m = 21-1 max[{T-a(t))a(t):tEJJ (15)
ISSN 0041-6053. YKp. Mar. "'YPH.., 1992, T. 44, M I 9-1-812 129
2. If we take in C (J, Rn) the following weighted norm
II u l[1, = max [I u (t) [1, : t E J], I u (t) lb= max [b1 1 I ui (t) I: i = 1, 2, . .. , nl
with b the positive eigenvector of the matrix mk + L corresponding to the
eigenva lue which is equal to the spectral radius of that matrix (the existen
ce of such eigenvector is quaranteed by the well known Perron theorem) then
the Assumpt ion H 2 holds true when the spectral radius of mk + L is less
than one (see 13]), i. e.
p = p (mK + L) < I.
In fact in this case we have
I u (t) I~ Ii u l[ 1, • b, I (Qu) (t) I ~ II u li t (mK + L) b = p Ii u II,,· b
what means that II Q lib~ p < I.
(16)
3 . In the case a (t) = t, B (t) = t the condition (16) can be replaced by
the following one
(17)
To get the result we need to employ the fo llowing norm
II u !l ,,,c = max [(t (T - t) + c)- 1 I u (t) lb: t E JJ
for the corresponding eigenvector b and suffic iently small positive number
c. From this definit ion we have
I u (i) I~ II u llb,c • (t (T- t) + c) b
and consequently
l T
I (Qu) (t) I~ II u 11 ,, .c [r- 1 (T- t) 5 (s (T- s) + c) Kbds + r-'t 5 (s (T - s) +
0
+ c) Kbds + (t (T - t) + c) Lb l ~ II u llb,c rr-'t (T - t ) (b- 1 (T2 + 4Tt - 4f2)+
+ 2c) Kb+ (t (T - t) + c) LbJ.
This implies the followin g inequality
(t (T - t) + c)- 1 I (Qu) (t) I~ II u llb,c ((3- 'T + 2cT-1) Kb + Lb]=
= II U ll h,c (p'b + 2cT-1/(b) ~ II U llb,c (p' + 2cT- 1 IJ Kb lib) b_
because Kb ~ II Kb ll bb. From the last inequali ty we find
II Qu llb,c ~ (p ' + 2cT-' I! I< b IJ,,) JI U llb,c
which means that II Q II"·' < I if p' < I and c is sufficiently small.
4. Observe that in the case a (t) = t, f:, (t) = t one can reason in a dif
ferent way . lnder the condition p \l)< 1 the eq uation can be rewr itten in
the equ ivalent iorm
t T
u (t) = r-' ( (T- t) S 1( 1u (s) ds + t I K1u (s) ds) + hi(t), (18)
0 t
where
Ki=(/ - L)-1K , h1 (t) = (/ - L)- 11i (t , Xg) -
Now the Assumption H; is satisfied if the fo llowing conditions hold
p (L) < I, p (3- 1T (/- L)- 1/() < 1.
In this case it is worthy to observe that the substitution u (t) - hi(t) = v (t)
in the equation (1 8) reduces this eq uation to the form which can be considered
in the subsp ace of function of C (J, R") having the property
max [(t (T- t))-1 1 v (t) I: t E J] < + =·
130 ISSN 0041-6053. Y1'p. Mar. ~ ypH., 1992, r. 44, M l
Now the norm II u ll1;,c can be replaced by that which correspon ds to c = 0.
5 . Fina lly we will seek the solution of equation (18) by the Neumann
series. For the conveni ence we will write K 1 and h1 as s imply K and h. Let
l [
(Gu) (t) = r- 1 [iT - t) .\' K.u (s) ds + t S K.u (s) ds] , ( 19)
0 t
+=
U0 (t) = ) (Gih) (t).
"-'
i = O
(20)
It is clear that the series (20) converges uniformly when the norm of the opera tor
G is less than one . So one can use the results of our di scuss ion. Now we will
presen t the aproach ba sed on some po intwise estimations. This will be close
to the considera t ions given in the books l l, 2J. Let the sequence {a ;} be defi
ned as follows
' T
a 1+1 (t) = r-1 ( (T - t) .\' a; (s) ds + t Sa; (s) ds), a 0 (t) = l, i = 0, I, .. .
0 I
(2 1)
Take H E R", H>0 such that jh(t)l ~ H for tEJ. Now by the mathema
tical induction rule we infer
(22)
By the dirrect calculations (see fl l) one can find that
a 1 (f) ~ 2- 1Ta0 (f) , CG 2 (t) ~ 3-1Ta1 (t), a 3 (i) ~ 10-1T2a 1 (f),
_a2; (t) ~ ( Jio r-~a2 (t), a 2;+!(t) ~ ( Jio r ai(t), i = l, 2, ....
From th ese inequalities we get the estirnation
i = l, 2, ... ,
- VTii
ai(t) = - 3- a 1 (t) . (23)
From (22) a nd (23) we have
o ~ (G' (hJ (t) ~ ( .Ji0 xy-1a1 (t) KH, t E J, i = 1, 2, .....
Now we see that the series (20) converges uniformly when
. T
p ( V10 I< )< l. (24)
Clearly the function u0 defined by (20) is a solution of ( 18) and
u, (t) ~ h (t) + a,l (t) (' _ :ll) 1, rl KH .
We observe a lso that in the case considered the corresponding functions u,.
(see fo rmula ( 10'\ ) satisfy the re lation
u,._(f) = r(G;h)(t)~a(t)(VT )''- 1 (1 - ·vT 1<·)- 1KI-J . (25)
<'=k JU I d '
Now the estimation (14) takes the fo r,m
for fl = I, 2, •.•.
ISSN 0041-6058. Y,cp. MaT. (JfCypH., 1992, T. 44. NJ I 131
We observe that in the book [2, p. 13, 59] in t he correspondi ng es timations
the the number 10 does not appear, it is replaced by the number n. In t he
mer it there are no special reasons for the presence of this specific number.
The onl y justificat ion is t ha t these two numbers are very close. Indeed , we
haYe n = 3,141 59265 ... < -V IO = 3,1 6227766 .. ..
As it was ment ioned at the beginning of the paper we are not in tend to di s
cuss ia rther dete ils which are concerned with the solv ing the corresponding
eq uation wh ich determines x~ . The readers are advised to consult t he publi
cations l 1, 2, 7] for the necessary details.
I. Samoilenko A . M., Ronto N. I. Numerica l-analytic methobs of investigating of peri odi c
so lut ions (Russian).- Kiev: Vyssha Sk., 1976 .- 180 p.
2. Samoiienlw A. M., Ranlo N. 1. N um er ica I- ana lytic methods of invest igati ng bound ary
value problems (Russian). - Kiev: Na uk. Dumka, 1986 .- 224 p.
3. Augustynowicz A . , Ku..•cpi,z M . On a nu merical -analitic methob of solv ing of boundary va ·
Jue prob lem for functional hifirenli al equat ion of neutral type// Math. Nachr.- 1990.-
145 .- P. 255- 269.
4. Kwapisz NI. A comparison theorem on the existence anb uniqueness of the solut ion ot an
inlegra l-f unctiona I equati on / / Y 11 In t. Konf. uder N ichtlineare Schwingun!!en, Abhand
lum'.en i der Adw.- Ber li n: Aka b.-Verl., 1977.- P. 481-487.
5. Kwap isz M. On the comparison m_e lhob in the ex istence anb uniqueness t heory fo r eq ua tions
in spaces of funct ions (Russian) / / Differentia I equations with deviated ar('.ument.- Kiev :
Na uk. Dumka, 1977.- P. 148- 158.
6. Kwa pisz M . Some remarks on abstract form of iterative methods in functional equati on the
ory // Comment. Math.- 1984. - 24.- P . 281-294.
7, Samoilenko A. M ., Ranlo N. I. A modifi cati on of numerical-analytic method of successive
approx imations for boun dary va lue prob lems for ordinary dif ferenti al equa ti ons (Russian)/;
Ukr. Math. J .- 1990,- 42 , N 8.- P, 1107-I I l6,
Received 30.01.91
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| id | umjimathkievua-article-7803 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:33:59Z |
| publishDate | 1992 |
| publisher | Institute of Mathematics, NAS of Ukraine |
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| resource_txt_mv | umjimathkievua/87/a9042bac54a28b8bc667b59d743e6987.pdf |
| spelling | umjimathkievua-article-78032023-09-14T12:03:56Z Certain remarks to integral equations generated by the numerical-analytical method for solution of the boundary problems Некоторые замечания к интегральным уравнениям , порожденным численно-аналитическим методом решения граничных задач Kwapisz, M. Kwapisz, M. - In the paper the comparison method and the chosing of the appropriate norm of the comparison operator is used to establish the solvability of the integral-functional equation resulted by the application of the numeric-analytic method of solving of boundary value problems for ordinary differential-delay equations of the neutral type. С помощью метода сравнения и надлежащего подбора нормы оператора сравнения устанавливается разрешимость интегро-функционального уравнения, порожденного численно-аналитическим методом решения граничных задач для обыкновенных дифференциальных уравнений нейтрального типа. За допомогою методу порівняння і належного добору норми оператора порівняння встановлюється розв’язність інтегро-функціонального рівняння, породженого чисельно-аналітичним методом розв’язування граничних задач для звичайних диференціальних рівнянь нейтрального типу. Institute of Mathematics, NAS of Ukraine 1992-02-04 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/7803 Ukrains’kyi Matematychnyi Zhurnal; Vol. 44 No. 1 (1992); 128-132 Український математичний журнал; Том 44 № 1 (1992); 128-132 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7803/9471 Copyright (c) 1992 M. Kwapisz |
| spellingShingle | Kwapisz, M. Kwapisz, M. Certain remarks to integral equations generated by the numerical-analytical method for solution of the boundary problems |
| title | Certain remarks to integral equations generated by the numerical-analytical method for solution of the boundary problems |
| title_alt | Некоторые замечания к интегральным уравнениям , порожденным численно-аналитическим методом решения граничных задач |
| title_full | Certain remarks to integral equations generated by the numerical-analytical method for solution of the boundary problems |
| title_fullStr | Certain remarks to integral equations generated by the numerical-analytical method for solution of the boundary problems |
| title_full_unstemmed | Certain remarks to integral equations generated by the numerical-analytical method for solution of the boundary problems |
| title_short | Certain remarks to integral equations generated by the numerical-analytical method for solution of the boundary problems |
| title_sort | certain remarks to integral equations generated by the numerical-analytical method for solution of the boundary problems |
| topic_facet | - |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7803 |
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