On solutions of nonlinear boundary-value problems the components of which vanish at certain points
We show how an appropriate parametrization technique and successive approximations can help to investigate nonlinear boundary-value problems for systems of differential equations under the condition that the components of solutions vanish at some unknown points. The technique can be applied to nonli...
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2018
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507340354093056 |
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| author | Puza, B. Ronto, A. M. Ronto, M. I. Shchobak, N. Пуза, Б. Ронто, А. М. Ронто, М. Й. Щобак, Н. |
| author_facet | Puza, B. Ronto, A. M. Ronto, M. I. Shchobak, N. Пуза, Б. Ронто, А. М. Ронто, М. Й. Щобак, Н. |
| author_sort | Puza, B. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:17:34Z |
| description | We show how an appropriate parametrization technique and successive approximations can help to investigate nonlinear
boundary-value problems for systems of differential equations under the condition that the components of solutions vanish
at some unknown points. The technique can be applied to nonlinearities involving the signs of absolute value and positive
or negative parts of functions under various types of boundary conditions. |
| first_indexed | 2026-03-24T02:07:45Z |
| format | Article |
| fulltext |
UDC 517.9
B. Půža (Brno Univ. Technology, Czech Republic),
A. Rontó (Inst. Math. Czech Acad. Sci., Brno, Czech Republic),
M. Rontó (Inst. Math., Univ. Miskolc, Hungary),
N. Shchobak (Brno Univ. Technology, Czech Republic)
ON SOLUTIONS OF NONLINEAR BOUNDARY-VALUE PROBLEMS
THE COMPONENTS OF WHICH VANISH AT CERTAIN POINTS
ПРО РОЗВ’ЯЗКИ НЕЛIНIЙНИХ КРАЙОВИХ ЗАДАЧ,
КОМПОНЕНТИ ЯКИХ В ДЕЯКИХ ТОЧКАХ ОБЕРТАЮТЬСЯ В НУЛЬ
We show how an appropriate parametrization technique and successive approximations can help to investigate nonlinear
boundary-value problems for systems of differential equations under the condition that the components of solutions vanish
at some unknown points. The technique can be applied to nonlinearities involving the signs of absolute value and positive
or negative parts of functions under various types of boundary conditions.
Показано, як вiдповiдна процедура параметризацiї та послiдовнi наближення допомагають дослiджувати нелiнiйнi
крайовi задачi для систем диференцiальних рiвнянь за умови, що компоненти розв’язкiв обертаються в нуль у
деяких невiдомих точках. Ця процедура може бути застосована до нелiнiйностей, що включають знаки абсолютних
величин та додатнi або вiд’ємнi частини функцiй для рiзних типiв граничних умов.
1. Introduction and problem setting. The question on finding solutions of nonlinear differential
equations possessing a prescribed number of zeroes inside the given interval is interesting from many
points of view (see, e. g., [1, 3 – 5, 11], and the references therein). This is a rather complicated
problem and its investigation is generally based on considerations of purely qualitative character
which usually do not provide a way to obtain approximations to the solution in question. Further
difficulties arise when the equation is studied under nonlinear boundary conditions.
The aim of this paper is to show that this question can be efficiently treated by further exte-
sions of numerical-analytic techniques based upon successive approximations suggested at first by
A. M. Samoilenko [20, 21] for the periodic problem. Based on the schemes with interval divisions
developed in [12 – 15, 17], we construct here a suitable version of this approach for finding solutions
with a given number of zeroes.
We focus on the system of n nonlinear ordinary differential equations
u\prime (t) = f
\bigl(
t, [u(t)]+, [u(t)] -
\bigr)
, t \in [a, b], (1.1)
where [u]\pm for any u = \mathrm{c}\mathrm{o}\mathrm{l}(u1, . . . , un) stands for the vector \mathrm{c}\mathrm{o}\mathrm{l}([u1]\pm , . . . , [un]\pm ), and [s]+ :=
:= \mathrm{m}\mathrm{a}\mathrm{x}\{ s, 0\} , [s] - := \mathrm{m}\mathrm{a}\mathrm{x}\{ - s, 0\} for any real s. System (1.1) will be studied under the nonlinear
two-point boundary conditions of the general form
g(u(a), u(b)) = d. (1.2)
The functions f : [a, b] \times \Omega \times \Omega \rightarrow \BbbR n, g : \Omega \times \Omega \rightarrow \BbbR n are assumed to be continuous in their
domain of definition, the choice of \Omega \subset \BbbR n will be concretized later. Since [u]+ - [u] - = u and
[u]+ + [u] - = | u| , system (1.1) includes, e. g., Fučı́k type equations
c\bigcirc B. PŮŽA,A. RONTÓ, M. RONTÓ, N. SHCHOBAK, 2018
94 ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 1
ON SOLUTIONS OF NONLINEAR BOUNDARY-VALUE PROBLEMS . . . 95
u\prime \prime (t) = \alpha (t)[u(t)]+ + \beta (t)[u(t)] - + q(t, u(t)), t \in [a, b], (1.3)
equations of Emden – Fowler type
u\prime \prime (t) = p(t) | u(t)| \lambda u(t) + r(t), t \in [a, b], (1.4)
and various other systems of the form
u\prime (t) = h(t, u(t), | u(t)| ), t \in [a, b] .
In what follows we are looking for continuously differentable solutions u = \mathrm{c}\mathrm{o}\mathrm{l}(u1, u2, . . . , un)
of (1.1), (1.2) each component of which vanishes at some point from (a, b) and has prescribed signs
around it (see Section 2). The numerical-analytic approach [6, 8 – 11, 18] allows one to approximate
such solutions of problem (1.1), (1.2) and, moreover, rigorously prove their existence using the results
of computation [7, 16].
The form of system (1.1) is motivated, in particular, by equations of type (1.3), (1.4), where the
terms of type [u]\pm , | u| bring about additional difficulties for the practical realization of our scheme
due to the need of analytic integration of expressions depending on multiple parameters. We shall
see that, in the case of solutions of the kind mentioned above, the construction of approximations is
simplified and any additional approximation of integrands may not be needed.
2. Solutions with fixed signs on subintervals. For the convenience of notation, we introduce
two definitions (cf. [11]).
Definition 2.1. Let \{ \sigma 0, \sigma 1\} \subset \{ - 1, 1\} and t1 be a point from (a, b). We say that a function u :
[a, b] \rightarrow \BbbR is of type (\sigma 0, \sigma 1; t1) if u(t1) = 0 and
\sigma k - 1u(t) > 0 for t \in (tk - 1, tk), k = 1, 2,
where t0 := a, t2 := b.
Let us suppose that \{ \sigma i0, \sigma i1 : i = 1, 2, . . . , n\} \subset \{ - 1, 1\} and t1, t2, . . . , tn are such that
a =: t0 < t1 < t2 < . . . < tn < tn+1 := b. (2.1)
Definition 2.2. We say that a vector-function u = \mathrm{c}\mathrm{o}\mathrm{l}(u1, u2, . . . , un) : [a, b] \rightarrow \BbbR n is of type\bigl[
(\sigma 10, \sigma 11; t1), (\sigma 20, \sigma 21; t2), . . . , (\sigma n0, \sigma n1; tn)
\bigr]
if every uk, k = 1, 2, . . . , n, is of type (\sigma k0, \sigma k1; tk).
In what follows, we will look for solutions of (1.1) possessing the last mentioned property with
certain t1, t2, . . . , tn. Assumption (2.1) on the ordering of zeroes is no loss of generality because the
equations in (1.1) can always be renumbered accordingly.
Before applying the iteration techniques for finding this kind of solutions, it is convenient to
simplify the terms involving the positive and negative parts of a function using the information
known for its sign. For this purpose, put
j\sigma :=
1
2
(\sigma + 1) (2.2)
for any \sigma \in \{ - 1, 1\} and define the function \~f : [a, b]\times D \rightarrow \BbbR n by setting
\~f(t, u1, . . . , un) := f
\bigl(
t, j\sigma 11u1, . . . , j\sigma k - 1,1
uk - 1, j\sigma k0
uk, j\sigma k+1,0
uk+1, . . . , j\sigma n0un,
- j - \sigma 11u1, . . . , - j - \sigma k - 1,1
uk - 1, - j - \sigma k0
uk, - j - \sigma k+1,0
uk+1, . . . , - j - \sigma n0un
\bigr)
(2.3)
for u = (ui)
n
i from \Omega , t \in [tk - 1, tk], k = 1, 2, . . . , n+ 1.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 1
96 B. PŮŽA, A. RONTÓ, M. RONTÓ, N. SHCHOBAK
Lemma 2.1. Let \{ \sigma i0, \sigma i1 : i = 1, 2, . . . , n\} \subset \{ - 1, 1\} be fixed. Any
\bigl[
(\sigma 10, \sigma 11; t1), (\sigma 20, \sigma 21;
t2), . . . , (\sigma n0, \sigma n1; tn)
\bigr]
solution of (1.1) is a solution of the system
u\prime (t) = \~f(t, u(t)), t \in [a, b], (2.4)
where \~f is given by (2.3). Conversely, any
\bigl[
(\sigma 10, \sigma 11; t1), (\sigma 20, \sigma 21; t2), . . . , (\sigma n0, \sigma n1; tn)
\bigr]
solution
of (2.4) satisfies (1.1).
Proof. Let u = (ui)
n
i=1 be a solution of (1.1) having type
\bigl[
(\sigma 10, \sigma 11; t1), (\sigma 20, \sigma 21; t2), . . .
. . . , (\sigma n0, \sigma n1; tn)
\bigr]
. Since (2.1) is assumed on t1, t2, . . . , tn, it follows from the definition that
\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n} ui(t) = sik, t \in (tk - 1, tk), k = 1, 2, . . . , n+ 1, (2.5)
where S = (sik), i = 1, 2, . . . , n, k = 1, 2, . . . , n+ 1,
S :=
\left(
\sigma 10 \sigma 11 \sigma 11 . . . \sigma 11 \sigma 11
\sigma 20 \sigma 20 \sigma 21 . . . \sigma 21 \sigma 21
\sigma 10 \sigma 30 \sigma 30 . . . \sigma 31 \sigma 31
. . . . . . . . . . . . . . . . . .
\sigma n - 1,0 \sigma n - 1,0 \sigma n - 1,0 . . . \sigma n - 1,1 \sigma n - 1,1
\sigma n0 \sigma n0 \sigma n0 . . . \sigma n0 \sigma n1
\right)
.
By (2.2), we have j\sigma ij = (\sigma ij +1)/2, - j - \sigma ij = (\sigma ij - 1)/2, which, together with (2.5), implies that
u satisfies (2.4). The converse implication is obvious.
Lemma 2.1 is proved.
Note that, in contrast to (1.1), the expression on the right-hand side of the new system (2.4) does
not contain positive or negative parts of a function: instead of [ui]+ and [ui] - , one finds there either
ui, - ui or 0, depending on the subinterval considered.
The construction of \~f is rather easy and proceeds by changing the relevant terms in (1.1) according
to their sign. Namely, all the occurrences of [ui(t)]+ in (1.1) are replaced by ui(t) if t \in [a, ti],
\sigma i0 = 1 or t \in (ti, b], \sigma i1 = 1, and by 0 in all remaining cases. Similarly, the term [ui(t)] - is
replaced by - ui(t) if t \in [a, ti], \sigma i0 = - 1 or t \in (ti, b], \sigma i1 = - 1, and by 0 otherwise. For
example, if system (1.1) has the form
u\prime 1(t) = p11(t)[u1(t)]+ + p12(t)[u1(t)] - + q1(u1(t), | u2(t)| ),
u\prime 2(t) = p21(t)[u2(t)]+ + p22(t)[u2(t)] - + q2(u1(t), u2(t)), t \in [a, b],
(2.6)
and we take, e. g., \sigma 10 = 1, \sigma 11 = - 1, \sigma 20 = - 1, \sigma 21 = 1, then the corresponding system (2.4) is
written as
u\prime 1(t) = p11(t)u1(t) + q1(u1(t), - u2(t)),
u\prime 2(t) = - p22(t)u2(t) + q2(u1(t), u2(t))
(2.7)
for t \in [a, t1],
u\prime 1(t) = - p12(t)u1(t) + q1(u1(t), - u2(t)),
u\prime 2(t) = - p22(t)u2(t) + q2(u1(t), u2(t))
(2.8)
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 1
ON SOLUTIONS OF NONLINEAR BOUNDARY-VALUE PROBLEMS . . . 97
for t \in [t1, t2], and
u\prime 1(t) = - p12(t)u1(t) + q1(u1(t), u2(t)),
u\prime 2(t) = p21(t)u2(t) - + q2(u1(t), u2(t))
(2.9)
for t \in [t2, b] (recall that a < t1 < t2 < b). The assertion of Lemma 2.1 in this case means that,
if we restrict our consideration to solutions u = \mathrm{c}\mathrm{o}\mathrm{l}(u1, u2) of type [(1, - 1; t1), ( - 1, 1; t2)] in the
sense of Definition 2.2, then the original system (2.6), on the relevant subintervals, can be rewritten
equivalently as (2.7) – (2.9).
Remark 2.1. It is not difficult to verify that formula (2.3) for \~f can be represented alternatively
as
\~f(t, u) = f
\biggl(
t,
1
2
(Mk + I)u(t),
1
2
(Mk - I)u(t)
\biggr)
, (2.10)
where u = (ui)
n
i is from \Omega , t \in [tk - 1, tk], k = 1, 2, . . . , n+ 1, I is the unit matrix, and
Mk := \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}
\bigl(
\sigma 11, \sigma 21, . . . , \sigma k - 1,1, \sigma k0, \sigma k+1,0, . . . , \sigma n0
\bigr)
. (2.11)
Equality (2.10) implies, in particular, that possible occurrences of | ui| in the original system are
replaced by the ith component of Mku in \~f on [tk - 1, tk].
Using Remark 2.1 in the example above, it is easy to write down system (2.7) – (2.9) on the
three intervals directly because, in this case, in view of (2.11), the matrices M1 = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\sigma 10, \sigma 20),
M2 = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\sigma 11, \sigma 20), M3 = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\sigma 11, \sigma 21) have the form
M1 = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(1, - 1), M2 = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}( - 1, - 1), M3 = ( - 1, 1).
Therefore, on [tk - 1, tk], 1 \leq k \leq 3, the occurrences of [ui]+ (resp., [ui] - ) in (2.6) are replaced by
(1/2)[(Mku)i + ui] (resp., (1/2)[(Mku)i - ui]), and | u2| by (Mku)2.
3. Parametrization and auxiliary problems. We fix certain
\bigl\{
\sigma i0, \sigma i1 : i = 1, 2, . . . , n
\bigr\}
\subset
\subset \{ - 1, 1\} and focus on finding solutions of (1.1) which are of type
\bigl[
(\sigma 10, \sigma 11; t1), (\sigma 20, \sigma 21; t2), . . .
. . . , (\sigma n0, \sigma n1; tn)
\bigr]
for some t1, t2, . . . , tn from (a, b). The values of t1, t2, . . . , tn are a priori
unknown and have to be determined along with u. Without loss of generality we assume that these
points are ordered as indicated in (2.1).
The idea that we will use suggests to replace the boundary-value problem (2.4), (1.2) by a
suitable family of “model-type” problems with separated boundary conditions. The construction of
these problems is very simple. We “freeze” the values of u = \mathrm{c}\mathrm{o}\mathrm{l}(u1, u2, . . . , un) at points (2.1) by
formally putting
u(tk) = z(k), k = 0, 1, . . . , n+ 1, (3.1)
where z(k) = \mathrm{c}\mathrm{o}\mathrm{l}
\bigl(
z
(k)
1 , z
(k)
1 , . . . , z
(k)
n
\bigr)
, and consider the restrictions of system (2.4) to each of the
intervals [t0, t1], [t1, t2], . . . , [tn, tn+1]. This leads us to the n+1 two-point boundary-value problems
on the respective subintervals
u\prime (t) = \~f(t, u(t)), t \in [tk - 1, tk], (3.2)
u(tk - 1) = z(k - 1), u(tk) = z(k), (3.3)
where k = 1, 2, . . . , n+ 1, and \~f is given by (2.3). We fix certain nonempty bounded sets
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 1
98 B. PŮŽA, A. RONTÓ, M. RONTÓ, N. SHCHOBAK
\Omega k \subset \BbbR n, k = 0, 1, . . . , n+ 1, (3.4)
and treat the vectors z(j) appearing in (3.1), (3.3), and (3.4) as parameters with values in \Omega j , j =
= 0, 1, . . . , n+ 1. Using the family of problems (3.2), (3.3), we will study
\bigl[
(\sigma 10, \sigma 11; t1), (\sigma 20, \sigma 21;
t2), . . . , (\sigma n0, \sigma n1; tn)
\bigr]
solutions u = \mathrm{c}\mathrm{o}\mathrm{l}(u1, u2, . . . , un) of problem (1.1), (1.2) whose values at the
unknown points (2.1) lie in the corresponding sets (3.4), i. e., such that
u(tk) \in \Omega k, k = 0, 1, 2, . . . , n+ 1. (3.5)
After the simplification of the original system (1.1) using the sign properties of solutions, we will
impose restrictions needed to apply our method directly to the transformed system (2.4). Conditions
on \~f will be assumed over certain sets which are a somewhat wider than sets (3.4) fixed above.
Given sets (3.4), for any k = 1, 2, . . . , n+ 1, introduce the sets
\Omega k - 1,k :=
\bigl\{
(1 - \theta )\xi + \theta \eta : \xi \in \Omega k - 1, \eta \in \Omega k, \theta \in [0, 1]
\bigr\}
. (3.6)
It is clear that \Omega k - 1,k is constituted by all possible straight line segments joining points of \Omega k - 1with
the points of \Omega k. Further on, we shall need the componentwise \varrho (k)-neighbourhoods of \Omega k - 1,k,
k = 1, . . . , n+ 1:
\scrO \varrho (k)(\Omega k - 1,k), k = 1, 2, . . . , n+ 1, (3.7)
where
\scrO \varrho (\Omega ) :=
\bigcup
\xi \in \Omega
\scrO \varrho (\xi ) (3.8)
and \scrO \varrho (\xi ) :=
\bigl\{
\nu \in \BbbR n : | \nu - \xi | \leq \varrho
\bigr\}
for any \Omega \subset \BbbR n, \varrho \in \BbbR n
+, \xi \in \Omega . The values of \varrho (k),
k = 1, 2, . . . , n+ 1, to be used in (3.7) will be chosen later. The conditions to be formulated in the
sequel (see Section 4) are assumed over sets (3.7) with the respect to the space variables.
4. Assumptions. To study solutions of the auxiliary problems (3.2), (3.3) with z(j) \in \Omega j ,
j = 0, 1, . . . , n+1, we use suitable parametrised successive approximations constructed analytically
on the subintervals t \in [tk - 1, tk] , k = 1, 2, . . . , n + 1. Since we are looking for solutions of type\bigl[
(\sigma 10, \sigma 11; t1), (\sigma 20, \sigma 21; t2), . . . , (\sigma n0, \sigma n1; tn)
\bigr]
with the given \{ \sigma i0, \sigma i1 : i = 1, 2, . . . , n\} \subset \{ - 1,
1\} and some unknown t1, . . . , tn, we assume that the set \Omega 0 is chosen so that
\Pi i\Omega 0 \subset \sigma i0\BbbR +, i = 1, 2, . . . , n, (4.1)
where \Pi i\Omega := \{ si : (s1, . . . , si, . . . , sn) \in \Omega for some s1, . . . , sn\} .
Remark 4.1. Due to the nature of the problem under consideration, in addition to (4.1), it is
natural to suppose that the sets \Omega 0, \Omega 1, . . . ,\Omega n+1 have the properties
\Pi i\Omega j \subset vji\BbbR +, i = 1, 2, . . . , n, j = 0, 1, . . . , n+ 1, (4.2)
where vk = (vki)
n
i=1 are defined as
v0 := (\sigma 10, \sigma 20, . . . , \sigma n0),
vk = (\sigma 11, \sigma 21, . . . , \sigma k - 1,1, 0, \sigma k+1,0, \sigma n0), k = 1, 2, . . . , n,
vn+1 := (\sigma 11, \sigma 21, . . . , \sigma n1).
Although relations (4.2) are useful because they exclude from consideration sets which cannot contain
the values of solutions with type
\bigl[
(\sigma 10, \sigma 11; t1), (\sigma 20, \sigma 21; t2), . . . , (\sigma n0, \sigma n1; tn)
\bigr]
, it is enough to
assume in the sequel condition (4.1) fixing the signs of the solution at the initial subinterval.
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 1
ON SOLUTIONS OF NONLINEAR BOUNDARY-VALUE PROBLEMS . . . 99
Two assumptions on the function \~f appearing in (2.4), (3.2) will be needed.
Assume that there exist nonnegative vectors \varrho (1), \varrho (2), . . . , \varrho (n+1) such that
\varrho (k) \geq tk - tk - 1
4
\delta [tk - 1,tk],\scrO \varrho (k)
(\Omega k - 1,k)(
\~f) (4.3)
for all k = 1, 2, . . . , n+ 1, where
\delta [\alpha ,\beta ],\Omega ( \~f) := \mathrm{m}\mathrm{a}\mathrm{x}
(t,u)\in [\alpha ,\beta ]\times \Omega
\~f(t, u) - \mathrm{m}\mathrm{i}\mathrm{n}
(t,u)\in [\alpha ,\beta ]\times \Omega
\~f(t, u) (4.4)
for a < \alpha < \beta < b and a closed \Omega \subset \BbbR n.
Fix certain \varrho (1), \varrho (2), . . . , \varrho (n+1) for which (4.3) holds, consider the sets \scrO \varrho (k)(\Omega k - 1,k), k =
1, 2, . . . , n + 1, and suppose that, with some nonnegative matrices Kk, k = 1, 2, . . . , n + 1, the
function \~f satisfies the Lipschitz condition\bigm| \bigm| \~f(t, y1) - \~f(t, y2)
\bigm| \bigm| \leq Kk | y1 - y2| (4.5)
for t \in [tk - 1, tk], \{ y1, y2\} \subset \scrO \varrho (k)(\Omega k - 1,k), k = 1, 2, . . . , n+ 1. Finally, assume that
r(Kk) <
10
3(tk - tk - 1)
(4.6)
for all k = 1, 2, . . . , n+ 1.
Remark 4.2. When looking for solutions vanishing at certain points (which is the case, in
particular, for the class of solutions defined in Section 2), the direct verification of condition (4.6) is
impossible because the values of t1, t2, . . . , tn are unknown. Obviously, the fulfilment of (4.6) is
guaranteed if
\mathrm{m}\mathrm{a}\mathrm{x}
1\leq k\leq n+1
r(Kk) <
10
3(b - a)
. (4.7)
It does make sense, however, to keep inequalities (4.6) because they may lead one to conditions
considerably weaker than (4.7) if some estimates for t1, t2, . . . , tn are available (see Section 6).
Remark 4.3. In order to verify condition (4.3) on \varrho (0), . . . , \varrho (n+1), it is needed to compute
maximal and minimal values of the function \~f over \varrho (k)-neighbourhoods of sets \Omega k - 1,k, k =
1, 2, . . . , n+ 1, constructed according to (3.6). One may use computer software for this purpose. It
is convenient to specify suitable sets
\Omega (k) \supset \Omega k - 1,k, k = 1, 2, . . . , n+ 1, (4.8)
of simpler structure (e. g., parallelepipeds: if \Omega (k) is a parallelepiped, then, by (3.8), the set
\scrO \varrho (k)(\Omega
(k)) is a parallelepiped as well) and use the inequality \delta [\alpha ,\beta ],\~\Omega (
\~f) \geq \delta [\alpha ,\beta ],\Omega ( \~f) for any
\~\Omega \supset \Omega , which is an immediate consequence of (4.4). Then the fulfilment of (4.3) is guaranteed if
\varrho (k) \geq tk - tk - 1
4
\delta [tk - 1,tk],\scrO \varrho (k)
(\Omega (k))(
\~f) (4.9)
for k = 1, 2, . . . , n+ 1. The same observation concerns the Lipschitz condition (4.5), which may be
easier to check on the set \scrO \varrho (k)(\Omega
(k)) instead of \scrO \varrho (k)(\Omega k - 1,k), k = 1, 2, . . . , n+ 1.
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100 B. PŮŽA, A. RONTÓ, M. RONTÓ, N. SHCHOBAK
5. Successive approximations and determining equations. As we have seen above, the
question on
\bigl[
(\sigma 10, \sigma 11; t1), (\sigma 20, \sigma 21; t2), . . . , (\sigma n0, \sigma n1; tn)
\bigr]
solutions of the boundary-value problem
(1.1), (1.2) reduces to the same problem (1.2) for equation (2.4), where the function \~f is constructed
according to (2.3). To treat problem (2.4), (1.2), we can use the approach of [15, 17] using properties
of the auxiliary problems (3.2), (3.3). From now on till the end of the paper we assume that conditions
(4.3), (4.5), and (4.6) are satisfied.
Let us define the parametrized recurrence sequences of functions u
(k)
m
\bigl(
\cdot , z(k - 1), z(k), tk - 1, tk
\bigr)
,
m = 0, 1, . . . , by putting
u
(k)
0 (t, z(k - 1), z(k), tk - 1, tk) :=
:=
\biggl(
1 - t - tk - 1
tk - tk - 1
\biggr)
z(k - 1) +
t - tk - 1
tk - tk - 1
z(k), (5.1)
u(k)m (t, z(k - 1), z(k), tk - 1, tk) :=
:= u
(k)
0 (t, z(k - 1), z(k)) +
t\int
tk - 1
\~f
\bigl(
s, u
(k)
m - 1(s, z
(k - 1), z(k), tk - 1, tk)
\bigr)
ds -
- t - tk - 1
tk - tk - 1
tk\int
tk - 1
\~f
\bigl(
s, u
(k)
m - 1(s, z
(k - 1), z(k), tk - 1, tk)
\bigr)
ds (5.2)
for all m = 1, 2, . . . , z(0) \in \Omega 0, z
(k) \in \Omega k, t \in [tk - 1, tk], k = 1, 2, . . . , n+1. We recall that t0 = a,
tn+1 = b, while the intermediate time instants t1, . . . , tn are treated as unknown parameters.
It is clear that every function u
(k)
m (\cdot , z(k - 1), z(k), tk - 1, tk), m = 0, 1, . . . , satisfies conditions (3.3)
independently of the choice of z(k - 1) and z(k) :
u(k)m
\bigl(
tk - 1, z
(k - 1), z(k), tk - 1, tk
\bigr)
= z(k - 1), u(k)m
\bigl(
tk, z
(k - 1), z(k), tk - 1, tk
\bigr)
= z(k). (5.3)
The sequences given by (5.1), (5.2) are helpful for the investigation of the auxiliary problems (3.2),
(3.3) and, ultimately, of the given problem (1.1), (1.2).
Theorem 5.1. Assume (4.3), (4.5), and (4.6). Then, for any fixed z(k) \in \Omega k, k = 0, 1, . . . , n+1:
1. Functions (5.2) are continuously differentiable on t \in [tk - 1, tk], k = 1, . . . , n + 1, and the
inclusion \Bigl\{
u(k)m (t, z(k - 1), z(k), tk - 1, tk) : t \in [tk - 1, tk]
\Bigr\}
\subset \scrO \varrho (k)(\Omega k - 1,k) (5.4)
holds.
2. The limit
\mathrm{l}\mathrm{i}\mathrm{m}
m\rightarrow \infty
u(k)m
\bigl(
t, z(k - 1), z(k), tk - 1, tk
\bigr)
=: u(k)\infty
\bigl(
t, z(k - 1), z(k), tk - 1, tk
\bigr)
exists uniformly in t \in [tk - 1, tk], k = 1, 2, . . . , n+ 1.
3. The limit functions satisfy the separated two-point boundary conditions
u(k)\infty
\bigl(
tk - 1, z
(k - 1), z(k), tk - 1, tk
\bigr)
= z(k - 1),
u(k)\infty
\bigl(
tk, z
(k - 1), z(k), tk - 1, tk
\bigr)
= z(k).
(5.5)
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ON SOLUTIONS OF NONLINEAR BOUNDARY-VALUE PROBLEMS . . . 101
4. The function u
(k)
\infty
\bigl(
\cdot , z(k - 1), z(k), tk - 1, tk
\bigr)
is the unique continuously differentiable solution of
the integral equation
u(t) = u
(k)
0 (t, z(k - 1), z(k), tk - 1, tk)+
+
t\int
tk - 1
\~f(s, u(s))ds - t - tk - 1
tk - tk - 1
tk\int
tk - 1
\~f(s, u(s))ds, t \in [tk - 1, tk], (5.6)
with values in \scrO \varrho (k)(\Omega k - 1,k).
5. For any m \geq 0, the following estimate holds:\bigm| \bigm| u(k)\infty
\bigl(
t, z(k - 1), z(k), tk - 1, tk
\bigr)
- u(k)m
\bigl(
t, z(k - 1), z(k), tk - 1, tk
\bigr) \bigm| \bigm| \leq
\leq 5
9
\alpha 1(t, tk - 1, tk) Q
m
k (I - Qk)
- 1 \delta [tk - 1,tk],\scrO \varrho (k)
(\Omega k - 1,k)(f),
where Qk := (3/10)(tk - tk - 1)Kk,
\alpha 1(t, tk - 1, tk) := 2 (t - tk - 1)
\biggl(
1 - t - tk - 1
tk - tk - 1
\biggr)
for t \in [tk - 1, tk].
It follows from (5.6) that the function u
(k)
\infty
\bigl(
\cdot , z(k - 1), z(k), tk - 1, tk
\bigr)
, k = 1, 2, . . . , n + 1, is the
unique solution of the Cauchy problem for the system
u\prime (t) = \~f(t, u(t)) +
1
tk - tk - 1
\Delta (k)(z(k - 1), z(k), tk - 1, tk), (5.7)
u(tk - 1) = z(k - 1), (5.8)
where \Delta (k) : \Omega k - 1 \times \Omega k \times (a, b)2 \rightarrow \BbbR n, k = 1, . . . , n+ 1, is defined by the formula
\Delta (k)(\xi , \eta , s0, s1) := \eta - \xi -
tk\int
tk - 1
\~f
\bigl(
s, u(k)\infty (s, \xi , \eta , s0, s1)
\bigr)
ds (5.9)
for all \xi \in \Omega k - 1, \eta \in \Omega k, and \{ s0, s1\} \subset (a, b).
The proof proceeds by analogy to [17] (Theorem 1) and [15] (Theorem 5.1). The starting point
is to establish inclusion (5.4).
It is natural to expect that the limit functions u(k)\infty
\bigl(
\cdot , z(k - 1), z(k), tk - 1, tk
\bigr)
, k = 1, 2, . . . , n+1, of
the iterations (5.2) on the subintervals t \in [tk - 1, tk] will help one to formulate criteria of solvability
of the original problem (1.1), (1.2). It turns out that it is the functions
\Delta (k) : \Omega k - 1 \times \Omega k \times (a, b)2 \rightarrow \BbbR n, k = 1, 2, . . . , n+ 1, (5.10)
defined according to equalities (5.9) that provide such conclusions. Indeed, Theorem 5.1 guarantees
that under the conditions assumed, the functions u
(k)
\infty
\bigl(
\cdot , z(k - 1), z(k), tk - 1, tk
\bigr)
: [tk - 1, tk] \rightarrow \BbbR n,
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102 B. PŮŽA, A. RONTÓ, M. RONTÓ, N. SHCHOBAK
k = 1, 2, . . . , n + 1, are well defined for all (z(k - 1), z(k)) \in \Omega k - 1 \times \Omega k, (tk - 1, tk) \in (a, b)2.
Therefore, by putting
u\infty
\bigl(
t, z(0), z(1), z(2), . . . , z(n+1), t1, t2, . . . , tn
\bigr)
:= u(k)\infty
\bigl(
t, z(k - 1), z(k), tk - 1, tk
\bigr)
(5.11)
for t \in [tk - 1, tk], k = 1, 2, . . . , n + 1, we obtain a function u\infty (t, z(0), z(1), z(2), . . . , z(n+1), t1, . . .
. . . , tn) : [a, b] \rightarrow \BbbR n. This function is obviously continuous at the points tk, k = 1, 2, . . . , n,
because, by (5.5),
u(k)\infty
\bigl(
tk, z
(k - 1), z(k), tk - 1, tk
\bigr)
= u(k+1)
\infty
\bigl(
tk, z
(k), z(k+1), tk, tk+1
\bigr)
.
Along with (3.2), consider the equations with constant forcing terms
u\prime (t) = \~f(t, u(t)) +
1
tk - tk - 1
\mu (k), t \in [tk - 1, tk], (5.12)
under the initial conditions
u(tk - 1) = z(k - 1), (5.13)
where \mu (k) = \mathrm{c}\mathrm{o}\mathrm{l}(\mu
(k)
1 , \mu
(k)
2 , . . . , \mu
(k)
n ), k = 1, 2, . . . , n+ 1, are control parameters. Then, similarly
to [19] (Theorem 2), one obtains the following theorem.
Theorem 5.2. Assume (4.3), (4.5), and (4.6). Let z(j) \in \Omega j , j = 0, 1, . . . , n+1, be fixed. Then,
for the solutions of the Cauchy problems (5.12), (5.13) to have the properties
u(tk) = z(k), k = 1, 2, . . . , n+ 1, (5.14)
it is neccessary and sufficient that \mu (k) have the form
\mu (k) = \Delta (k)
\bigl(
z(k - 1), z(k), tk - 1, tk
\bigr)
, k = 1, 2, . . . , n+ 1, (5.15)
in which case the solution of (5.12), (5.13) coincides with u
(k)
\infty
\bigl(
\cdot , z(k - 1), z(k), tk - 1, tk
\bigr)
for any
k = 1, 2, . . . , n+ 1.
The next statement establishes the relation of function (5.11) to solutions of the original prob-
lem (1.1), (1.2) in the terms of zeroes of functions (5.10). Recall that \Omega 0 is chosen so that (4.1)
holds.
Theorem 5.3. Let (4.3), (4.5), and (4.6) hold. Then the function
u\infty (\cdot , z(0), z(1), z(2), . . . , z(n+1), t1, . . . , tn) : [a, b] \rightarrow \BbbR n
is a continuously differentiable solution of the boundary-value problem (1.1), (1.2) if and only if the
vectors z(k), k = 0, 1, 2, . . . , n+1, and the points t1, . . . , tn satisfy the system of n(n+2) numerical
determining equations
\Delta (k)
\bigl(
z(k - 1), z(k), tk - 1, tk
\bigr)
= 0, k = 1, 2, . . . , n+ 1, (5.16)
g
\bigl(
u(1)\infty (a, z(0), z(1), a, t1), u
(n+1)
\infty (b, z(n), z(n+1), tn, b)
\bigr)
= d. (5.17)
Furthermore, this solution has type
\bigl[
(\sigma 10, \sigma 11; t1), (\sigma 20, \sigma 21; t2), . . . , (\sigma n0, \sigma n1; tn)
\bigr]
.
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ON SOLUTIONS OF NONLINEAR BOUNDARY-VALUE PROBLEMS . . . 103
Proof. The statement is proved similarly to [19] (Theorem 3). We use Lemma 2.1 and take into
account the choice of the domain \Omega 0 according to (4.1), which, by virtue of the unique solvability of
the Cauchy problems (5.7), (5.8), excludes the existence of solutions not possessing the prescribed
property
\bigl[
(\sigma 10, \sigma 11; t1), (\sigma 20, \sigma 21; t2), . . . , (\sigma n0, \sigma n1; tn)
\bigr]
.
Theorem 5.3 is proved.
Finally, the following analogue of [19] (Theorem 4) shows that the determining equations (5.16),
(5.17) detect all possible solutions of the problem (1.1), (1.2) with graphs lying in the domains
specified.
Theorem 5.4. Let (4.3), (4.5), and (4.6) hold. If there exist some t1, . . . , tn from (a, b) and
z(j) \in \Omega j , j = 0, 1, . . . , n+1, that satisfy the determining equations (5.16), (5.17), then the function
u\ast (t) = u\infty
\bigl(
t, z(0), z(1), z(2), . . . , z(n+1), t1, . . . , tn
\bigr)
, t \in [a, b], (5.18)
is a
\bigl[
(\sigma 10, \sigma 11; t1), (\sigma 20, \sigma 21; t2), . . . , (\sigma n0, \sigma n1; tn)
\bigr]
solution of the boundary-value problem (1.1),
(1.2). Conversely, if problem (1.1), (1.2) has a solution u\ast (\cdot ) of type
\bigl[
(\sigma 10, \sigma 11; t1), (\sigma 20, \sigma 21; t2), . . .
. . . , (\sigma n0, \sigma n1; tn)
\bigr]
, which, in addition, satisfies the conditions
u\ast (tj) \in \Omega j , j = 0, 1, . . . , n+ 1,\bigl\{
u\ast (t) : t \in [tk - 1, tk]
\bigr\}
\subset \scrO \varrho (k)(\Omega k - 1,k), k = 1, 2, . . . , n+ 1,
then the system of determining equations (5.16), (5.17) is satisfied with the same t1, . . . , tn, and
z(j) := u\ast (tj), j = 0, 1, . . . , n+ 1.
Moreover, the solution u\ast (\cdot ) necessarily has form (5.18) with these values of parameters.
Remark 5.1. In the case of
\bigl[
(\sigma 10, \sigma 11; t1), (\sigma 20, \sigma 21; t2), . . . , (\sigma n0, \sigma n1; tn)
\bigr]
solutions, the pa-
rameters z(1), z(2), . . . , z(n) in the auxiliary two-point problems (3.2), (3.3) have the form
z(k) = \mathrm{c}\mathrm{o}\mathrm{l}
\bigl(
z
(k)
1 , . . . , z
(k)
k - 1, 0, z
(k)
k+1, . . . , z
(k)
n
\bigr)
, k = 1, 2, . . . , n, (5.19)
and, therefore, system (5.16), (5.17) involves n(n+ 1) variables.
6. Computation of approximate solutions. Although Theorem 5.4 describes theoretically all
the solutions of problem (2.4), (1.2) with graphs contained in the given region, its direct appli-
cation is difficult because the form of the limit functions of sequences and (5.1), (5.2) is usually
unknown and, as a consequence, the determining equations (5.16), (5.17) can rarely be written down
explicitly. The complication can be overcome in a customary way (see, e. g., [6, 12] and refe-
rences therein) if we replace in (5.11) the unknown limit u(k)\infty
\bigl(
\cdot , z(k - 1), z(k), tk - 1, tk
\bigr)
by an iteration
u
(k)
m (\cdot , z(k - 1), z(k), tk - 1, tk), k = 1, 2, . . . , n+1, of form (5.2) for a fixed m. In this way, we obtain
the function
um
\bigl(
t, z(0), z(1), z(2), . . . , z(n+1), t1, . . . , tn
\bigr)
:= u(k)m
\bigl(
t, z(k - 1), z(k), tk - 1, tk
\bigr)
(6.1)
for t \in [tk - 1, tk], k = 1, 2, . . . , n + 1. We see that (6.1) is an approximate version of the unknown
function (5.11). Its values can be found explicitly for all values of the parameters involved. Con-
sidering function (6.1), we arrive in a natural way to the mth approximate system of determining
equations
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104 B. PŮŽA, A. RONTÓ, M. RONTÓ, N. SHCHOBAK
\Delta (k)
m
\bigl(
z(k - 1), z(k), tk - 1, tk
\bigr)
= 0, k = 1, 2, . . . , n+ 1, (6.2)
g
\bigl(
u(1)m
\bigl(
a, z(0), z(1), a, t1
\bigr)
, u(n+1)
m (b, z(n), z(n+1), tn, b)
\bigr)
= d, (6.3)
where, by a direct analogy to (5.9), the functions \Delta (k)
m : \Omega k - 1\times \Omega k\times (a, b)2 \rightarrow \BbbR n, k = 1, . . . , n+1,
are defined as
\Delta (k)
m (\xi , \eta , s0, s1) := \eta - \xi -
tk\int
tk - 1
\~f
\bigl(
s, u(k)m (s, \xi , \eta , s0, s1)
\bigr)
ds (6.4)
for \xi \in \Omega k - 1, \eta \in \Omega k, and \{ s0, s1\} \subset (a, b). Note that, unlike system (5.16), (5.17), the mth appro-
ximate system (6.2), (6.3) contains only terms involving the functions u
(k)
m
\bigl(
\cdot , z(k - 1), z(k), tk - 1, tk
\bigr)
,
k = 1, 2, . . . , n+ 1, which are computable explicitly.
The approximate solutions of the original problem are obtained as usual (see, e.g., [6, 12]) by
substituting into (6.1) roots of the corresponding approximate determining system (6.2), (6.3). The
approximations, according to the approach described here, are constructed by “gluing” together the
curves obtained on every single interval [tk - 1, tk], k = 1, 2, . . . , n+ 1. This gluing is smooth.
Lemma 6.1. If z(k) \in \Omega k, k = 0, 1, 2, . . . , n + 1, satisfy equations (6.2) for a certain m, then
the corresponding function (6.1) is continuosly differentiable on [a, b].
Proof. Fix z(j), j = 0, 1, . . . , n+ 1, put v := um
\bigl(
\cdot , z(0), z(1), z(2), . . . , z(n+1), t1, . . . , tn
\bigr)
, and
consider the values of v around tk for a fixed k = 1, 2, . . . , n. By (6.1), it is enough to check only
u
(k)
m
\bigl(
\cdot , z(k - 1), z(k), tk - 1, tk
\bigr)
and u
(k+1)
m
\bigl(
\cdot , z(k), z(k+1), tk, tk+1
\bigr)
.
Indeed, it follows immediately from (5.2) that
v\prime (tk - ) = \~f
\bigl(
tk, u
(k)
m - 1(tk, z
(k - 1), z(k), tk - 1, tk)
\bigr)
+
+
1
tk - tk - 1
\Delta (k)
m
\bigl(
z(k - 1), z(k), tk - 1, tk
\bigr)
(6.5)
and
v\prime (tk+) = \~f
\bigl(
tk, u
(k+1)
m - 1 (tk, z
(k), z(k+1), tk, tk+1)
\bigr)
+
+
1
tk+1 - tk
\Delta (k+1)
m
\bigl(
z(k), z(k+1), tk, tk+1
\bigr)
. (6.6)
Since z(j), j = 0, 1, . . . , n+ 1, are supposed to satisfy (6.2), equalities (6.5), (6.6) imply that
v\prime (tk - ) = \~f
\bigl(
tk, x
(k)
m - 1(tk, z
(k - 1), z(k), tk - 1, tk)
\bigr)
,
v\prime (tk+) = \~f
\bigl(
tk, x
(k+1)
m - 1 (tk, z
(k), z(k+1), tk, tk+1)
\bigr)
.
(6.7)
However, in view of (5.3), we have
u
(k)
m - 1
\bigl(
tk, z
(k - 1), z(k), tk - 1, tk
\bigr)
= u
(k+1)
m - 1
\bigl(
tk, z
(k), z(k+1), tk, tk+1
\bigr)
= z(k),
which, together with (6.7), yields v\prime (tk - ) = v\prime (tk+).
Lemma 6.1 is proved.
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ON SOLUTIONS OF NONLINEAR BOUNDARY-VALUE PROBLEMS . . . 105
The solvability of the determining system (5.16), (5.17) can be analyzed similarly to [7, 16] using
topological degree methods [2] by studying some its approximate versions (6.2), (6.3) (this subject is
not treated here).
A special note should be made on the verification of the assumptions of Section 4. Namely,
both relations (4.3), which should be satisfied by \varrho (1), \varrho (2), . . . , \varrho (n+1), and inequalities (4.6) for
r(Kj), j = 1, . . . , n + 1, depend upon the unknown t1, t2, . . . , tn. Although one can replace the
subintervals by the entire [a, b] (cf. (4.7)), this would lead to more restrictive conditions. Another,
better opportunity is to use preliminary results of computation according to the scheme described
above.
Indeed, it is always expedient to start computations directly before checking conditions (4.3),
(4.6) because, by doing so, we may obtain a preliminary information on the space localization of
solutions and, as a consequence, a useful hint how to choose the regions where the conditions should
be verified. This concerns both the choice of the sets \Omega k, k = 0, 1, . . . , n + 1, with respect to the
space variables and intervals containing zeroes of solutions.
Suppose that we start computation directly and try to solve approximate determining equations.
If the computation shows reasonable, in some sense, results and we get certain approximate values
\^t1, \^t2, . . . , \^tn of t1, t2, . . . , tn, these values are natural to be used to set restrictions of the form
T -
k \leq tk \leq T+
k , k = 1, 2, . . . , n, (6.8)
by choosing appropriately the bounds T -
k , T+
k , k = 1, 2, . . . , n. Perhaps, the simplest choice is to
put
T -
k := \mathrm{m}\mathrm{a}\mathrm{x}
\biggl\{
a, \^tk -
b - a
n+ 1
\biggr\}
,
T+
k := \mathrm{m}\mathrm{i}\mathrm{n}
\biggl\{
\^tk +
b - a
n+ 1
, b
\biggr\}
for k = 1, 2, . . . , n, however, finer estimates may be available in concrete situations. Knowing
estimates of form (6.8), instead of (4.3), we can verify the relations
\varrho (k) \geq
T+
k - T -
k - 1
4
\delta [T -
k - 1,T
+
k ],\scrO
\varrho (k)
(\Omega (k))(
\~f), (6.9)
where T -
0 = T+
0 = a, T -
n+1 = T+
n+1 = b and \Omega (k), k = 1, . . . , n + 1, are suitably chosen sets
satisfying (4.8). Similarly, instead of (4.6), we will check the condition
r(Kk) <
10
3
\bigl(
T+
k - T -
k - 1
\bigr) , (6.10)
where Kk is the Lipschitz matrix for the restriction of \~f to [tk - 1, tk]\times \scrO \varrho (k)(\Omega
(k)), k = 1, 2, . . . , n+
+ 1. Condition (6.10) is, of course, preferable to (4.7).
Assuming (6.8), we formally make the problem more difficult since t1, t2, . . . , tn should satisfy
additional inequalities and cannot be arbitrary any more. However, with a reasonable choice of bounds
based on the results of computation, inequalities (6.8), in fact, say only that we restrict ourselves to
looking for the unknown time instants in the regions where we have reasons to believe they are.
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106 B. PŮŽA, A. RONTÓ, M. RONTÓ, N. SHCHOBAK
7. An illustrative example. We demonstrate the approach described above on a model example
of the three-dimensional system
u\prime 1(t) = u2(t)u3(t) - t2 +
67
10
t - 387
100
,
u\prime 2(t) = | u3(t)| u2(t) + q2(t),
u\prime 3(t) = | u1(t)| + q3(t)
(7.1)
for t \in [0, 1] with
q2(t) :=
\left\{
t2 - 6
5
t+
33
25
if t \in [0, t3],
- t2 +
6
5
t+
17
25
if t \in [t3, 1],
(7.2)
and
q3(t) :=
\left\{
- 11
4
t2 +
71
20
t+
2
5
if t \in [0, t1],
11
4
t2 - 71
20
t+
8
5
if t \in [t1, 1],
(7.3)
where t1 and t3, t1 < t3, are unknown points from the interval (0, 1). System (7.1) will be considered
under the two-point nonlinear boundary conditions
u21(0) - u22(1) = 0, u2(0)u3(1) = - 2
25
, u1(0) - u3(1) =
2
5
. (7.4)
Let us set the problem on finding
\bigl[
(1, - 1; t1), ( - 1, 1; t2), ( - 1, 1; t3)
\bigr]
solutions of (7.1), (7.4),
where t2 is a point lying between t1 and t3. The values of time instants t1, t2, and t3, where the
sign changes of the respective components of u occur, are to be determined.
It can be verified directly by computation that, for t1 = 1/5, t2 = 2/5, t3 = 4/5, the function
u\ast = (u\ast i )
3
i=1 with the components
u\ast 1(t) =
11
4
t2 - 71
20
t+
3
5
, u\ast 2(t) = t - 2
5
, u\ast 3(t) = t - 4
5
(7.5)
is a solution of the boundary-value problem (7.1), (7.4). This solution, as is easy to see, has type\bigl[
(1, - 1; 1/5), ( - 1, 1; 2/5), ( - 1, 1; 4/5)
\bigr]
in the sense of Definition 2.2.
Let us use the approach described above. It is clear that (7.1) is a particular case of (1.1) with
a = 0, b = 1, n = 3, and f of the form
f(t, x1, x2, x3, y1, y2, y3) :=
\left(
(x2 - y2)(x3 - y3) - t2 +
67
10
t - 387
100
(x3 + y3)(x2 - y2) + q2(t)
x1 + y2 + q3(t)
\right) , (7.6)
and, hence, the preceding argument is applicable. This explicit form (7.6) of f is, however, not needed
for writing down the corresponding system (2.4) because the function \~f = ( \~fi)
3
i=1 determining (2.4)
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ON SOLUTIONS OF NONLINEAR BOUNDARY-VALUE PROBLEMS . . . 107
can be constructed as in Remark 2.1 by using matrices (2.11):
M1 = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\sigma 10, \sigma 20, \sigma 30), M2 = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\sigma 11, \sigma 20, \sigma 30),
M3 = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\sigma 11, \sigma 21, \sigma 30), M4 = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(\sigma 11, \sigma 21, \sigma 31).
(7.7)
Since, in our case, \sigma 10 = 1, \sigma 11 = - 1, \sigma 20 = - 1, \sigma 21 = 1, \sigma 30 = - 1, \sigma 31 = 1, equalities (7.7)
yield
M1 = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}(1, - 1, - 1), M2 = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}( - 1, - 1, - 1),
M3 = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}( - 1, 1, - 1), M4 = \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g}( - 1, 1, 1).
Then | ui| , i = 1, 3, on the kth interval [tk - 1, tk], 1 \leq k \leq 4, where t0 = 0 and t4 = 1, should be
replaced by the ith component of Mku. By doing so, we obtain
\~f1(t, u1, u2, u3) = u2u3 - t2 +
67
10
t - 387
100
(7.8)
for all t \in [0, 1], while \~f2, \~f3 on the relevant subintervals are defined as follows:
\~f2(t, u1, u2, u3) = - u2u3 + t2 - 6
5
t+
33
25
,
\~f3(t, u1, u2, u3) = u1 -
11
4
t2 +
71
20
t+
2
5
(7.9)
for t \in [0, t1],
\~f2(t, u1, u2, u3) = - u2u3 + t2 - 6
5
t+
33
25
,
\~f3(t, u1, u2, u3) = - u1 +
11
4
t2 - 71
20
t+
8
5
(7.10)
for t \in [t1, t3] (the equations have the same form on [t1, t2] and [t2, t3]), and
\~f2(t, u1, u2, u3) = u2u3 - t2 +
6
5
t+
17
25
,
\~f3(t, u1, u2, u3) = - u1 +
11
4
t2 - 71
20
t+
8
5
(7.11)
for t \in [t3, 1]. The system (2.4) corresponding to (7.1) thus has the form
u\prime i(t) =
\~fi(t, u1(t), u2(t), u3(t)), i = 1, 2, 3, t \in [tk - 1, tk], \leq 1 \leq k \leq 4, (7.12)
with ( \~fi)
3
i=1 given by the respective equalities (7.8) – (7.11), and we pass from (7.1), (7.4) to prob-
lem (7.12), (7.4).
In order to apply the techniques described above, we need to choose suitable domains and verify
the conditions. Let us choose the sets \Omega 0, \Omega 1, . . . ,\Omega 4 in (3.4) as follows:
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108 B. PŮŽA, A. RONTÓ, M. RONTÓ, N. SHCHOBAK
(a) (b)
(c)
Fig. 1. Zeroth approximation: first component (a), second component (b), and third component (c).
\Omega 0 =
\bigl\{
(u1, u2, u3) : 0.5 \leq u1 \leq 0.7, - 0.6 \leq u2 \leq - 0.3, - 0.95 \leq u3 \leq - 0.6
\bigr\}
,
\Omega 1 =
\bigl\{
(u1, u2, u3) : - 0.1 \leq u1 \leq 0.1, - 0.3 \leq u2 \leq - 0.1, - 0.75 \leq u3 \leq - 0.5
\bigr\}
,
\Omega 2 =
\bigl\{
(u1, u2, u3) : - 0.5 \leq u1 \leq - 0.25, - 0.1 \leq u2 \leq 0.1, - 0.5 \leq u3 \leq - 0.3
\bigr\}
,
\Omega 3 =
\bigl\{
(u1, u2, u3) : - 0.55 \leq u1 \leq - 0.3, 0.3 \leq u2 \leq 0.5, - 0.1 \leq u3 \leq 0.1
\bigr\}
,
\Omega 4 =
\bigl\{
(u1, u2, u3) : - 0.3 \leq u1 \leq - 0.1, 0.5 \leq u2 \leq 0.7, 0.1 \leq u3 \leq 0.3
\bigr\}
.
(7.13)
This choice is motivated by the fact that the zeroth approximate determining system (i.e., (6.2), (6.3)
with m = 0) has roots lying in these sets, see the second column in Table 1. Figures 1 (a) – (c)
present the graph of the zeroth approximation U0 = (U0i)
3
i=1. Recall that, in order to obtain it, only
functions (5.1) are used, and no iteration is yet carried out. We see that this piecewise linear function
provides quite reasonable approximate values of the parameters (in particular, of the time instants
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 1
ON SOLUTIONS OF NONLINEAR BOUNDARY-VALUE PROBLEMS . . . 109
Table 1. Exact values of parameters for solution (7.5) and their computed approximations
u\ast m = 0 m = 1 m = 2 m = 3
z
(0)
1 0.6 0.5987479750 0.5999603161 0.6000012161 0.5999999745
z
(0)
2 –0.4 –0.4025198245 –0.4000793836 –0.3999975678 –0.4000000510
z
(0)
3 –0.8 –0.7704273551 –0.8000737079 –0.7999810861 –0.8000000219
z
(1)
2 –0.2 –0.2020709807 –0.2001071216 –0.199996485 –0.2000000620
z
(1)
3 –0.6 –0.5684050181 –0.6000878738 –0.5999800574 –0.6000000464
z
(2)
1 –0.38 –0.3838828957 –0.3801569730 –0.3799948547 –0.3800000942
z
(2)
3 –0.4 –0.3700284637 –0.3999682608 –0.3999842474 –0.3999999656
z
(3)
1 –0.48 –0.4842219409 –0.4800381441 –0.4799998221 –0.4800000226
z
(3)
2 0.4 0.3967271760 0.4000262560 0.3999978194 0.4000000159
z
(4)
1 –0.2 –0.2025151122 –0.2000954038 –0.1999968834 –0.2000000586
z
(4)
2 0.6 0.5987479750 0.5999603161 0.6000012161 0.5999999745
z
(4)
3 0.2 0.1987479750 0.1999603161 0.2000012161 0.1999999745
t1 0.2 0.1988295851 0.1999917615 0.2000000919 0.1999999900
t2 0.4 0.4003180698 0.4001083934 0.3999957228 0.4000000506
t3 0.8 0.7979815572 0.8000579299 0.7999969454 0.8000000377
t1, t2, and t3). In general, the quality of approximation by U0 grows with the number of equations
(which is equal to the number of intermediate nodes).
Given sets (7.13), we need to verify conditions of Section 4 on the corresponding sets \Omega 0,1, . . .
. . . ,\Omega 3,4 defined according to (3.6). For this purpose, we use Remark 4.3 and choose suitable
parallelepipeds \Omega (k) \supset \Omega k - 1,k, k = 1, . . . , 4:
\Omega (1) :=
\bigl\{
(u1, u2, u3) : - 0.1 \leq u1 \leq 0.7, - 0.6 \leq u2 \leq - 0.1, - 0.95 \leq u3 \leq - 0.5
\bigr\}
,
\Omega (2) :=
\bigl\{
(u1, u2, u3) : - 0.5 \leq u1 \leq 0.1, - 0.3 \leq u2 \leq 0.1, - 0.75 \leq u3 \leq - 0.3
\bigr\}
,
\Omega (3) :=
\bigl\{
(u1, u2, u3) : - 0.55 \leq u1 \leq - 0.25, - 0.1 \leq u2 \leq 0.5, - 0.5 \leq u3 \leq 0.1
\bigr\}
,
\Omega (4) :=
\bigl\{
(u1, u2, u3) : - 0.55 \leq u1 \leq - 0.1, 0.3 \leq u2 \leq 0.7, - 0.1 \leq u3 \leq 0.3
\bigr\}
.
(7.14)
We are going to verify conditions (4.9) on the sets \scrO \varrho (k)(\Omega
(k)), k = 1, . . . , 4, for which purpose
the vectors \varrho (1), . . . , \varrho (4) should be chosen. Let us put, for example
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110 B. PŮŽA, A. RONTÓ, M. RONTÓ, N. SHCHOBAK
\varrho (1) = \mathrm{c}\mathrm{o}\mathrm{l}(0.2, 0.2, 0.2), \varrho (2) = \varrho (1),
\varrho (3) = \mathrm{c}\mathrm{o}\mathrm{l}(0.6, 0.2, 0.3), \varrho (4) = \mathrm{c}\mathrm{o}\mathrm{l}(0.3, 0.2, 0.2).
(7.15)
Then, according to (3.8), we obtain from (7.14)
\scrO \varrho (1)(\Omega
(1)) =
\bigl\{
(u1, u2, u3) : - 0.3 \leq u1 \leq 0.9, - 0.8 \leq u2 \leq 0.1, - 1.15 \leq u3 \leq - 0.3
\bigr\}
,
\scrO \varrho (2)(\Omega
(2)) =
\bigl\{
(u1, u2, u3) : - 0.7 \leq u1 \leq 0.3, - 0.5 \leq u2 \leq 0.3, - 0.95 \leq u3 \leq - 0.1
\bigr\}
,
(7.16)
\scrO \varrho (3)(\Omega
(3)) =
\bigl\{
(u1, u2, u3) : - 1.15 \leq u1 \leq 0.35, - 0.3 \leq u2 \leq 0.7, - 0.8 \leq u3 \leq 0.4\} ,
\scrO \varrho (4)(\Omega
(4)) =
\bigl\{
(u1, u2, u3) : - 0.85 \leq u1 \leq 0.2, 0.1 \leq u2 \leq 0.9, - 0.3 \leq u3 \leq 0.5
\bigr\}
.
A direct computation using (7.8) – (7.11) shows that the Lipschitz condition (4.5) for \~f holds on
\scrO \varrho (1)(\Omega
(1)), . . . ,\scrO \varrho (4)(\Omega
(4)), respectively, with the matrices
K1 =
\left(
0 1.15 0.8
0 1.15 0.8
1 0 0
\right) , K2 =
\left(
0 0.95 0.5
0 0.95 0.5
1 0 0
\right) ,
K3 =
\left(
0 0.8 0.7
0 0.8 0.7
1 0 0
\right) , K4 =
\left(
0 0.5 0.9
0 0.5 0.9
1 0 0
\right) .
(7.17)
Then, taking into account the rough approximations of t1, t2, and t3 obtained at the zeroth step
(the second column of Table 1), we can assume, e. g., the following bounds for the regions (6.8)
where more precise values of these variables should be looked for:
T -
1 \leq t1 \leq T+
1 , T -
2 \leq t2 \leq T+
2 , T -
3 \leq t3 \leq T+
3 , (7.18)
where
T -
1 := 0.15, T+
1 := 0.25, T -
2 := 0.35,
T+
2 := 0.45, T -
3 := 0.75, T+
3 := 0.85.
(7.19)
Assuming (7.18), we obtain from (7.17)
r(K1) \approx 1.6383 <
40
3
=
10
3T+
1
,
r(K2) \approx 1.3268 <
100
9
=
10
3(T+
2 - T -
1 )
,
r(K3) \approx 1.3274 <
20
3
=
10
3(T+
3 - T -
2 )
,
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ON SOLUTIONS OF NONLINEAR BOUNDARY-VALUE PROBLEMS . . . 111
Table 2. Meaning of parameters in the example
z
(0)
1 z
(0)
2 z
(0)
3 z
(1)
2 z
(1)
3 z
(2)
1 z
(2)
3 z
(3)
1 z
(3)
2 z
(4)
1 z
(4)
2 z
(4)
3
u1(0) u2(0) u3(0) u2(t1) u3(t1) u1(t2) u3(t2) u1(t3) u2(t3) u1(1) u2(1) u3(1)
r(K4) \approx 1.2311 <
40
3
=
10
3(1 - T -
3 )
,
which means that conditions (6.10) hold. Furthermore, in view of (4.4), we have
\varrho (1) =
\left(
0.2
0.2
0.2
\right) >
T+
1
4
\delta [0,T+
1 ],\scrO
\varrho (1)
(\Omega (1))(
\~f) \approx 0.25
4
\left(
2.6475
1.2725
1.9156
\right) \approx
\left(
0.1655
0.0795
0.1197
\right) ,
\varrho (2) =
\left(
0.2
0.2
0.2
\right) >
T+
2 - T -
1
4
\delta [T -
1 ,T+
2 ],\scrO
\varrho (2)
(\Omega (2))(
\~f) =
0.3
4
\left(
2.59
0.94
1.57
\right) =
\left(
0.19425
0.0705
0.11775
\right) ,
\varrho (3) =
\left(
0.6
0.2
0.3
\right) >
T+
3 - T -
2
4
\delta [T -
2 ,T+
3 ],\scrO
\varrho (3)
(\Omega (3))(
\~f) \approx 0.5
4
\left(
3.59
0.9025
1.7401
\right) \approx
\left(
0.4488
0.1128
0.2175
\right) ,
\varrho (4) =
\left(
0.3
0.2
0.2
\right) >
1 - T -
3
4
\delta [T -
3 ,1],\scrO
\varrho (4)
(\Omega (4))(
\~f) \approx 0.25
4
\left(
1.9575
0.8575
1.3656
\right) \approx
\left(
0.1223
0.0536
0.0854
\right) .
This means that conditions (6.9) are satisfied.
Thus, taking account the observation of Section 6, we conclude that the scheme based on The-
orems 5.1 – 5.4 is applicable provided that bounds (7.18) for t1, t2, and t3 are assumed. Note that,
as the computation shows, the true values of these variables indeed satisfy estimates (7.18). This
situation is generic: when using this kind of computational schemes, it is always natural to choose
the sets in the conditions after we get some notion of where we are going to find the values of the
unknowns in the course of computation.
The scheme is now implemented as follows. We use equalities (5.1), (5.2) to construct the
corresponding functions
u(k)m
\bigl(
\cdot , z(k - 1), z(k), tk - 1, tk
\bigr)
: [tk - 1, tk] \rightarrow \BbbR 3, 1 \leq k \leq 4, m \geq 0. (7.20)
These functions depend on the 12 scalar parameters listed in Table 2 and on the unknown time
instants t1, t2, and t3. By Theorem 5.1, functions (7.20) form convergent sequences as m \rightarrow \infty .
Note that, according to Section 5, the function u
(k)
m
\bigl(
\cdot , z(k - 1), z(k), tk - 1, tk
\bigr)
is an approximation
to the solution of the kth auxiliary two-point problem (3.2), (3.3) on the respective subintervals
[tk - 1, tk], 1 \leq k \leq 4. For this example, system (3.2), (3.3) means the following four problems:
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 1
112 B. PŮŽA, A. RONTÓ, M. RONTÓ, N. SHCHOBAK
Eq. (7.12) on [0, t1] with \~f1 from (7.8) and \~f2, \~f3 from (7.9) under the conditions
u1(0) = z
(0)
1 , u2(0) = z
(0)
2 , u3(0) = z
(0)
3 ,
u1(t1) = 0, u2(t1) = z
(1)
2 , u3(t1) = z
(1)
3 ;
(7.21)
Eq. (7.12) on [t1, t2] with \~f1 from (7.8) and \~f2, \~f3 from (7.10) under the conditions
u1(t1) = 0, u2(t1) = z
(1)
2 , u3(t1) = z
(1)
3 ,
u1(t2) = z
(2)
1 , u2(t2) = 0, u3(t2) = z
(2)
3 ;
(7.22)
Eq. (7.12) on [t2, t3] with \~f1 from (7.8) and \~f2, \~f3 from (7.10) under the conditions
u1(t2) = z
(2)
1 , u2(t2) = 0, u3(t2) = z
(2)
3 ,
u1(t3) = z
(3)
1 , u2(t3) = z
(3)
2 , u3(t3) = 0;
(7.23)
Eq. (7.12) on [t3, 1] with \~f1 from (7.8) and \~f2, \~f3 from (7.11) under the conditions
u1(t3) = z
(3)
1 , u2(t3) = z
(3)
2 , u3(t3) = 0,
u1(1) = z
(4)
1 , u2(1) = z
(4)
2 , u3(1) = z
(4)
3 .
(7.24)
The auxiliary problems (7.21) – (7.24) are however not treated directly in the course of computa-
tion, which involves functions (7.20) only. Approximate solutions of the given problem (7.1), (7.4)
are constructed, on the respective subintervals, in the form
Um(t) := u(k)m
\bigl(
t, z(k - 1), z(k), tk - 1, tk
\bigr)
, t \in [tk - 1, tk], k = 1, . . . , 4,
where m is fixed and z(j), j = 0, . . . , 4, are vectors of form (5.19) satisfying the mth approximate
determining system (6.2), (6.3):
z(k) - z(k - 1) -
tk\int
tk - 1
\~f
\bigl(
s, u(k)m (s, z(k - 1), z(k), tk - 1, tk
\bigr)
ds = 0, k = 1, 2, . . . , 4,
\bigl(
u
(1)
1 (0, z(0), z(1), 0, t0)
\bigr) 2 - \bigl(
u
(4)
2 (1, z(3), z(4), t3, 1)
\bigr) 2
= 0,
u
(1)
2 (0, z(0), z(1), 0, t0)u
(4)
3 (1, z(3), z(4), t3, 1) = - 2
25
,
u
(1)
1 (0, z(0), z(1), 0, t0) - u
(4)
3 (1, z(3), z(4), t3, 1) =
2
5
.
(7.25)
In order to determine the values of parameters on step m, equations (7.25) are solved numerically for
z(j) \in \Omega j , j = 0, . . . , 4, and ti \in [T -
i , T+
i ], i = 1, 2, 3. An initial hint for the region where the roots
should be looked for is obtained by using the zeroth approximation (m = 0), the graphs of which
are shown on Fig. 1 (a) – (c). We have used Maple 14 to carry out all the computations.
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ON SOLUTIONS OF NONLINEAR BOUNDARY-VALUE PROBLEMS . . . 113
(a) (b)
(c)
Fig. 2. Exact solution (solid line) and third approximation: first component (a), second component (b), and third compo-
nent (c).
The numerical values of the 15 unknown parameters obtained from (7.25) for the first three steps
of iteration are shown in Table 1. We see that the approximate values from the third iteration are
very close to the exact ones.
The graphs of the respective components of the exact solution (7.5) and the approximate\bigl[
(1, - 1; t1), ( - 1, 1; t2), ( - 1, 1; t3)
\bigr]
solution U3 = (U3i)
3
i=1 of problem (7.1), (7.4) corresponding
to the numerical values from Table 1 are shown on Fig. 2 (a) – (c). The curves corresponding to the
subintervals [tk - 1, tk], k = 1, . . . , 4, with the values of t1, t2, and t3 computed on the third step are
drawn with different symbols.
References
1. Capietto A., Mawhin J., Zanolin F. On the existence of two solutions with a prescribed number of zeros for a
superlinear two-point boundary value problem // Topol. Methods Nonlinear Anal. – 1995. – 6, № 1. – P. 175 – 188.
2. Mawhin J. Topological degree methods in nonlinear boundary value problems // CBMS Region. Conf. Ser. Math. –
Providence, R.I.: Amer. Math. Soc., 1979. – 40.
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Received 09.10.17
ISSN 1027-3190. Укр. мат. журн., 2018, т. 70, № 1
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| id | umjimathkievua-article-1544 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:07:45Z |
| publishDate | 2018 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/07/b6e81a1fed22dd4f964d1623942b7407.pdf |
| spelling | umjimathkievua-article-15442019-12-05T09:17:34Z On solutions of nonlinear boundary-value problems the components of which vanish at certain points Про розв’язки нелiнiйних крайових задач, компоненти яких в деяких точках обертаються в нуль Puza, B. Ronto, A. M. Ronto, M. I. Shchobak, N. Пуза, Б. Ронто, А. М. Ронто, М. Й. Щобак, Н. We show how an appropriate parametrization technique and successive approximations can help to investigate nonlinear boundary-value problems for systems of differential equations under the condition that the components of solutions vanish at some unknown points. The technique can be applied to nonlinearities involving the signs of absolute value and positive or negative parts of functions under various types of boundary conditions. Показано, як вiдповiдна процедура параметризацiї та послiдовнi наближення допомагають дослiджувати нелiнiйнi крайовi задачi для систем диференцiальних рiвнянь за умови, що компоненти розв’язкiв обертаються в нуль у деяких невiдомих точках. Ця процедура може бути застосована до нелiнiйностей, що включають знаки абсолютних величин та додатнi або вiд’ємнi частини функцiй для рiзних типiв граничних умов. Institute of Mathematics, NAS of Ukraine 2018-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1544 Ukrains’kyi Matematychnyi Zhurnal; Vol. 70 No. 1 (2018); 94-114 Український математичний журнал; Том 70 № 1 (2018); 94-114 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1544/526 Copyright (c) 2018 Puza B.; Ronto A. M.; Ronto M. I.; Shchobak N. |
| spellingShingle | Puza, B. Ronto, A. M. Ronto, M. I. Shchobak, N. Пуза, Б. Ронто, А. М. Ронто, М. Й. Щобак, Н. On solutions of nonlinear boundary-value problems the components of which vanish at certain points |
| title | On solutions of nonlinear boundary-value problems the
components of which vanish at certain points |
| title_alt | Про розв’язки нелiнiйних крайових задач,
компоненти яких в деяких точках обертаються в нуль |
| title_full | On solutions of nonlinear boundary-value problems the
components of which vanish at certain points |
| title_fullStr | On solutions of nonlinear boundary-value problems the
components of which vanish at certain points |
| title_full_unstemmed | On solutions of nonlinear boundary-value problems the
components of which vanish at certain points |
| title_short | On solutions of nonlinear boundary-value problems the
components of which vanish at certain points |
| title_sort | on solutions of nonlinear boundary-value problems the
components of which vanish at certain points |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1544 |
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